Petrophysical Properties

Transcription

Petrophysical Properties from
small rock samples using
Image Analysis techniques
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus, prof. ir. K.F. Wakker,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 14 november 2000 om 10.30 uur
door
Theodoor Wouter FENS
electrotechnisch ingenieur
geboren te Den Haag
III
Dit proefschrift is goedgekeurd door de promotoren prof. dr. I.T. Young en prof. ir. M. Peeters.
Dr. P.W. Verbeek heeft als toegevoegd promotor in hoge mate bijgedragen aan het tot stand
komen van dit proefschrift.
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. I.T. Young (TU Delft, promotor)
Prof. ir. M. Peeters (Colorado School of Mines, USA, promotor)
Dr. P.W. Verbeek (TU Delft, toegevoegd promotor)
Prof. dr. N.F. Hurley (Colorado School of Mines, USA)
Prof. dr. J.L. Urai (RWTH Aachen, Deutschland)
Prof. dr. ir. L.J. van Vliet (TU Delft)
Dr. C.K. Harris (Shell Research, adviseur)
ISBN 90-9014-338-6
Copyright © 2000 by T.W. Fens
All rights reserved
No part of this material protected by its copyright notice may be reproduced or utilized in any
form or by any means, electronic or mechanical, including photocopying, recording or by any
information storage and retrieval system, without permission from the publisher:
Delft University Press, Stevinweg 1, 2628 CN, Delft, the Netherlands.
Printed in the Netherlands.
IV
Aan Yvonne, Niki & Taco.
V
Het in dit proefschrift behandelde onderzoek is uitgevoerd bij Research & Technology
Services van Shell Research B.V. te Rijswijk. Ik wil het management van Shell Research B.V.
bedanken voor de medewerking die is verleend bij het tot stand komen van dit proefschrift.
VI
Table of contents
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Introduction………………………………………………………………………………. 1
Reservoir, clastics………………………………………………………………………. 1
Field development plan, reservoir model…………………………………………….. 2
Data required to populate the models………………………………………………… 4
Range of scales…………………………………………………………………………. 5
Representativity of data, up-scaling…………………………………………………… 7
Problem statement………………………………………………………………………. 8
Approach and outline of the thesis…………………………………………………..… 9
2
2.1
2.2
2.2.1
2.2.2
2.2.3
2.2.4
2.3
2.3.1
2.3.2
2.4
2.4.1
2.4.2
2.4.3
2.4.4
2.5
2.5.1
2.5.2
The main petrophysical properties…………………………………………………..… 12
Introduction…………………………………………………………………………….… 12
Porosity, the available space in reservoir rock……………………………………..… 13
Simple cubic packing………………………………………………………………….... 14
Rhombohedral packing…………………………………………….…………………… 15
Porosity alteration……………………………………………………………………….. 16
Porosity in the context of images……………………………………………..……….. 17
Permeability, the hydraulic conductance of reservoir rock……………………….…. 17
Pore structure models for permeability……………………………………………….. 20
Permeability in the context of images…………………………………………………. 24
Saturation, the fraction of porosity occupied by fluids………………..……………… 25
Archie’s models………………………………………………………….………………. 26
The Waxman-Smits model……………………………………………………………… 27
Effective medium models……………………………………………………………….. 28
Saturation, the relation to images……………………………………………………… 29
Capillarity, the interplay between pore fluids and reservoir rock…………………… 29
Capillary pressure curves………………………………………………………………. 29
Capillary pressure curves from images…………………………….…………………. 30
3
3.1
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.3
3.3.1
3.3.2
3.3.3
3.3.4
3.3.5
3.3.6
Wireline logging and core analysis……………………………………………….….… 31
Introduction………………………………………………………………………….…… 31
Wireline logs……………………………………………….…………………………..… 31
The gamma ray log, a shale indicator…………………………………………….…… 31
The density log, a porosity indicator…………………………………………….…….. 32
The neutron log, porosity, shale and gas indicator………………..……………….… 32
The resistivity log, hydrocarbon saturation indicator ………………..………………. 33
The NMR log, porosity and permeability indicator………………..………………….. 33
Core analysis………………………………………………………….…………………. 34
Porosity………………………………………………………….……………………..…. 34
Permeability………………………………………………………….…………………… 34
Formation resistivity factor……………………………………………………………... 35
Resistivity index curve……………………………………………………..……………. 36
Capillary pressure curve………………………………………………………………... 37
Qv, the shale indicator…………………………………………………………………... 37
4
4.1
4.2
4.3
4.4
The sample sets for rock images and plug measurements………………...……….. 38
Extreme values for porosity and permeability………………..……………………..… 39
Reference sample set for thin section analysis……………..……………………...… 40
Reference sample set for SEM/BSE analysis……………..………………………..… 40
Test sample sets for SEM/BSE analysis……………..……………………………..… 42
5.
5.1
5.2
5.2.1
5.2.2
5.3
5.4
5.4.1
5.4.2
5.4.3
5.5
5.5.1
5.5.2
5.5.3
5.5.4
Image analysis with optical microscopy on thin sections……………….…..……….. 43
Introduction………………………………………………………….……………….…… 43
Equipment and image collection………………………………….……………….…… 44
Image collection…………………………………………………….………….………… 44
Selection of appropriate magnification………………………………………………… 44
Extraction and measurement of pores, clays and matrix……………………….…… 46
Petrophysical parameters……………………………………………………………..… 49
Porosity……………………………………………………………….…………………… 49
Permeability………………………………………………………….…………………… 52
Porosity-permeability relationship………………………….………………………...… 54
Discussion………………………………………………………….…………………..… 54
Porosity, the Holmes effect……………………………………….………………….… 54
Permeability prediction………………………………………….…………………….… 56
Colour segmentation, definition of pores, clays and rock matrix…………………… 57
Conclusions………………………………………………………….…………………… 57
6
6.1
Scanning electron microscopy applied to reservoir rock samples………………..… 59
Introduction………………………………………………………….………………….… 59
VII
6.2
6.2.1
6.2.2
6.2.3
6.3
6.3.1
6.3.2
6.4
6.4.1
6.4.2
6.4.3
6.4.4
6.4.5
6.4.6
Scanning electron microscopy (SEM) ………………………….…………………...… 59
Working principle of the SEM…………………………………………………………… 59
Interaction of the electron beam and the sample…………………………..………… 61
Image formation in the SEM………………………………….………………………… 64
Secondary electron (SE) imaging………………………………….……………………64
Sample preparation for SE imaging………………………………………….………… 65
Image formation in SE imaging………………………………………………………… 65
Backscatter electron (BSE) imaging…………………………….……………...……… 67
Sample preparation for BSE imaging………………………………………..………… 69
Atomic number discrimination……………………………………………………..…… 69
Pores and minerals in BSE images…………………………….……………………… 71
Resolution in BSE images…………………………………………………………….… 72
Calibration of grey-values in BSE images……………………………..……………… 75
Optimal magnification of reservoir rock BSE images……..…….…………………… 77
7
7.1
7.2
7.2.1
7.2.2
7.2.3
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Processing and analysis of BSE images of reservoir rock samples…...…………… 80
Introduction………………………………………………………….……….…………… 80
Extraction of pores, clays and matrix…………………..………….…………………… 80
Two-level thresholding……………………………………………………………………80
Grey-level noise filtering…………………………………………….……………………83
Morphological filtering…………………………………………………………….………84
Semi-automatic segmentation……………………………………………………..…… 86
Micro-porosity in the clays…………………………………………………….………… 88
Conversion from BSE images to DEF images, overview……………….…………… 93
Feature extraction from DEF images………………………………………...………… 95
Pore size distributions, from DEF images to POR images……………………..…… 96
Pore networks, from POR images to NET images…………………………………… 100
Representativity, selection of the number of images………………………………… 102
8
8.1
8.2
8.2.1
8.2.2
8.3
8.3.1
8.3.2
8.3.3
8.3.4
8.3.5
8.4
8.5
8.6
8.7
8.7.1
8.7.2
Petrophysical parameters estimated from BSE images……………...……………… 106
Introduction………………………………………………………….…………….……… 106
Porosity………………………………………………………….………………………… 107
Porosity estimation for the calibration samples, test of the method…………………108
Porosity prediction from reservoir rock samples………………………………...…… 110
Permeability………………………………………………………….…………………… 113
Relationship porosity and permeability, the k-Phi relation……………………………114
Permeability from a Hagen-Poiseuille type model……………………………….…… 115
Permeability from Kozeny-Carman type models………………………………………117
Permeability from Network models…………………………………………………..… 121
Discussion on the various models for permeability prediction…………….…………126
Formation resistivity factor………………………………………………………….……127
Clay content, Qv…………………….……………………………….……………………129
Capillary pressure curves from BSE images……………….…….…………………… 130
Discussion………………………………………………………….…………………..… 135
Porosity under-estimation in SEM/IA……………………………………………...…… 135
Porosity over-estimation in SEM/IA………………………………………….………… 139
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
SEM/BSE analysis applied to field cases………………..……….…………………… 143
Introduction………………………………………………………….………….………… 143
Depositional environment and core description well A………………………….…… 143
Petrophysical evaluation………………………………………...….……………………145
Thomas-Stieber analysis of the neutron/density log…………………………….…… 147
The role of DEF images in core description……………………………………………150
Petrophysical parameters from BSE images, well A………………………….……… 155
Petrophysical parameters from BSE images, well B…………………………….…… 159
Appendices
A
Monte Carlo simulations for BSE resolution…………………………………...……… 164
B
Morphological filtering, openings and closings…………………………..…………… 169
C
Directional permeability from network images…………………………………………174
D
SEM/BSE analysis; the equipment and the use in an operational environment….. 178
Conclusions & Recommendations…………………………………………………….………… 183
References………………………………………………………….……………………………… 185
Bibliography………………………………………………………….…………………..………… 189
List of symbols………………………………………………………….……………………..…… 190
Summary………………………………………………………….…………………………………194
Samenvatting………………………………………………………….…………………………… 196
Dankwoord………………………………………………………….……………………………… 198
VIII
Chapter 1
1.
Introduction
Introduction
Petrophysical parameters are very important for the oil industry because they determine the
economic viability of hydrocarbon-bearing reservoirs. A reservoir is a subsurface layer or a
sequence of layers of porous rock that contain hydrocarbons. Depending on their geological
origin, these layers are usually sandstone rock or carbonate rock.
Figure 1.1
Microscopic image from a sandstone,
reservoir rock sample
The hydrocarbons reside in the open spaces in the rock matrix called pores. Figure 1.1
shows a light microscopic image of a sample from a sandstone reservoir, in which the pores
are the blue areas enclosed by the white/yellow sand grains. These pores constitute a pore
system that stores the hydrocarbons and acts as a pathway for hydrocarbons to flow out of
the rock into the borehole. The parameters that determine the behaviour of the pore system
are known as petrophysical properties and are porosity, permeability, saturation and
capillarity. Porosity determines the storage capacity, while the permeability indicates the flow
capacity of the rock for fluids. Saturation is used when more than one fluid is present and is
defined as the fraction of the porosity that is occupied by a certain fluid such as oil, gas or
water. Finally, capillarity determines the affinity between the reservoir fluids and the rock
matrix and has therefore a strong influence on how much of the hydrocarbons that are stored
in the pores can be produced.
1.1
Reservoir, clastics
A hydrocarbon-bearing reservoir can only exist if a number of conditions are fulfilled. First a
mature source rock should generate oil or gas. Second a hydraulic path should exist along
which the hydrocarbons can migrate to the porous rock that makes up the reservoir. Third a
non-permeable layer, the cap rock, should confine the hydrocarbons to the reservoir layers,
and finally, the rock should remain undisturbed over geological times to avoid spillage of the
1
Chapter 1
Introduction
hydrocarbons by faulting and erosion. Sandstone reservoirs, often termed clastic reservoirs,
are made up of sand grains. Sands are classified according to the Wenthworth scale, which
runs from 63 micron in diameter (very fine sand) to 2 mm in diameter (very coarse sand).
Mineral sediments that consist of particles with diameters below 4 micron are clays. By burial
to depths more than 3 to 4 km these clays are transformed to shales. Sandstones with clays
present in the pores are known as shaly sandstones. The range between 4 micron and 63
micron is a transition from clay to sand and is termed silt. Silt and shale are considered nonreservoir because they usually have a permeability that is too low to produce large volumes
of oil or gas in a reasonable timespan of say 20 years.
1.2
Field development plan, reservoir model
Prior to the production of hydrocarbons from a reservoir, petroleum engineers make a field
development plan. This plan determines where the wells are drilled, from which depth the
hydrocarbons are produced and what production mechanism will be used. Possible
production mechanisms are primary depletion, water flood, chemical flood, gas injection etc.
An essential part of the field development programme is a model of the reservoir
[Schlumberger Publications, 1992]. This subsurface model is used for reservoir management
throughout the lifetime of a field, to control and plan the production, and to predict the
changes in pressure and fluid saturation in the reservoir layers. There are two types of
models in use: the static and the dynamic model.
The static model entails the reservoir structure, geometry and storage capacity (porosity and
fluid saturation). The static model is mainly used to calculate oil and gas volumes in place
(GIIP, gas initially in place, and OIIP, oil initially in place). The static model should contain all
relevant geological and petrophysical details and can nowadays contain as many as tens of
millions of grid blocks, which represent the reservoir volume. Each individual grid block with
typical dimensions of tens of meters has assigned properties for porosity and fluid saturation.
The dynamic model, also known as the simulation model, is used to predict the flow of the
reservoir fluids from one grid block to another [Schlumberger Publications, 1996, 1994]. In
addition to the static parameters each block in the dynamic model contains information about
permeability and capillarity. This model enables estimation of the recoverable reserves under
the various scenarios defined by well density, completion pattern and production mechanism.
The economic viability of a field is based on the calculations that are carried out with the
dynamic model. The flow of hydrocarbons and water through the reservoir and into the
borehole requires many time-steps and interactions, hence dynamic models are
computationally much more intensive than static models. Dynamic models are therefore
currently restricted to a few hundred thousand grid blocks. Figure 1.2 shows a flow diagram
of the way a reservoir model is constructed and configured. Data obtained from production
tests and core calibrated wireline logs are important input parameters for the dynamic model.
Chapter 2 provides more details on the petrophysical parameters while in chapter 3 we
discuss logging and the core measurements.
2
Chapter 1
Introduction
Core plugs
Whole cores
Borehole geophysics
Outcrop studies
Large-scale structure
Well tests
3D Seismic data
Geological expertise
1st generation model
Calibration
Small-scale structure
Dynamic reservoir model
Wireline logs
Calibration
Static reservoir model
Execution reservoir model
Figure 1.2
Flow chart presenting the steps required to build a reservoir
model (Courtesy Schlumberger, Oil Field Review)
3
Chapter 1
Introduction
The workflow of a reservoir model in a field development programme has undergone
substantial changes over the past 20 years. The greatest impact has been caused by the
availability of high-speed computer power at an acceptable price that allows rapid and early
development of reservoir models. In the past the development of a reservoir model was a
sequential process in which the data and the interpretations from one discipline were
transferred to the next in a conveyer belt style. Presently a reservoir model is developed very
early in the life of a field, sometimes even before the first well is drilled.
The difference between the traditional sequential development and the modern concurrent
development is depicted in Figure 1.3 [Schlumberger publications, 1992b]. The concurrent
approach means that as much detail as possible has to be incorporated in the reservoir
models as early as possible. This is vital because the quality of reservoirs that are currently
discovered is diminishing as the majority of the large and high permeable reservoirs has
been developed, leaving mainly marginal reservoirs with low permeabilities.
Discipline/time, sequential development
Data 1
Data 2
Concurrent development
Petrophysics
Data 3
Geophysics
Geology
Data 1
Geophysics
Geology
Petrophysics
Reservoir
engineering
Reservoir
model
Reservoir
model
Data 2
Data 3
Timeline
Reservoir
engineering
Figure 1.3
Development of a reservoir model, left the traditional sequential
development; right the modern concurrent development
Specifically for low permeability reservoirs it is vital that at a very early stage the amount of
producible hydrocarbons is determined, preferably before large investments in the form of
production facilities like platforms and pipelines are committed.
The concurrent development track can benefit greatly from the method that has been
developed in this study, because this method can provide essential petrophysical parameters
in an early stage of the reservoir model construction at a fraction of the price of a full core.
The capability of deriving petrophysical parameters from small rock samples enables the use
of drill cuttings and sidewall samples, which are available as soon as the first well is drilled.
Moreover, detailed information about mineralogy and pore structure can also be obtained in
this early stage because pore geometry and mineral dispersion can be derived from images
that are shown in Figure 1.1.
1.3
Data required to populate the models
The dynamic model can only simulate the flow of fluids through the reservoir layers in a
proper manner if each of the grid blocks depicted in Figure 1.2 contains appropriate values
for the petrophysical rock properties porosity, permeability, saturation and capillarity. The grid
block properties are derived from many sources as shown in the top rows of Figure 1.2 that
are further explained below:
4
Chapter 1
Introduction
1.
Seismic stratigraphy.
Seismic surveys use the reflections of acoustic waves to determine the position of the layer
boundaries [Robertson, 1989]. An image of these reflections is called a seismic section. The
seismic sections give, under favourable conditions, the rock type and even indications of the
presence of hydrocarbons. The maximum resolution of seismic images is approximately 10
m at a depth of some 1 - 3 km, where most reservoirs are found. On a seismic scale the
entire reservoir can usually be imaged and delineated.
2.
Wireline logs.
Geophysical instruments can be moved through the borehole at the end of a wireline cable or
can be incorporated in the drill-string. In both cases physical properties such as density,
resistivity, magnetic resonance, natural gamma radiation and others are recorded as a
function of the depth. These physical properties can be used to estimate porosity, saturation,
permeability and mineralogy. Wireline logs are sometimes termed open-hole logs because
many logs are obtained when the borehole is still open, prior to setting the casing. The
casing is a large steel pipe needed to prevent the borehole from collapsing. Conventional
wireline logs have a resolution of some 30 cm and measurements are restricted to the
immediate volume around the borehole.
3.
Cores.
Cylindrical rock samples can be cut from the reservoir and brought to the surface by using a
hollow drill bit. This process, known as coring, resembles the removal of the core from an
apple with an "apple bore". Cores are usually the only intact rock samples on which physical
properties can be measured in the laboratory. If cores and logs are obtained in the borehole,
the rock properties derived from core samples can then be used to calibrate the
measurements made with the wireline logging tools. Rock properties like porosity and
permeability are determined on core plugs with dimensions of some 2.5 cm diameter and 7.5
cm length. Core plug measurements give high resolution rock properties in depth, but cover
only a tiny part of the reservoir, and are therefore often less representative.
4.
Sidewall samples.
A sidewall sampler is an instrument that is run at the end of a wireline cable and can extract
a small piece of reservoir rock from the borehole wall. Hollow bullets are fired individually
from the instrument by electrically ignited powder charges that are placed behind the bullet.
The bullets penetrate the rock and cut a more or less cylindrical plug from the formation. The
bullets remain attached to the instrument by two heavy steel wires. When the sidewall
sampler is pulled upward, the bullet is detached from the borehole wall and retains the rock
sample. Sidewall samples measure about 2.5 cm in diameter and 5.0 cm in length. Because
of the impact, sidewall samples can be severely damaged and grains may be shattered. This
can hamper proper determination of porosity and permeability. In the late 1980s a wireline
tool was developed that did not have the disadvantage of damaging the samples. This rotary
sidewall tool uses a small hollow drill bit that can take small plugs from the borehole wall. An
advantage of this tool is that it works best in hard formations where the conventional sidewall
sampler often fails. Plugs from a rotary sidewall tool are mostly undamaged which results in
reliable porosity and permeability measurements.
5.
Cuttings.
Cuttings are rock debris that result from the drilling process and are carried to the surface by
the mud when the drilling fluids are circulated through the borehole. Depending on the drill bit
5
Chapter 1
Introduction
that was used, cuttings can have sizes from sub-micron up to a few mm in diameter. Cuttings
are routinely inspected during mud-logging, which is the analysis of the mud during the
drilling of a well. Contrary to the above data-collection methods such as cores and logs,
which are expensive to acquire and are usually only taken over the reservoir interval, cuttings
are always obtained over the total length of the well with a sampling rate from one to a few
meters.
1.4
Range of scales
Apart from the sources of data that are used for the configuration of the reservoir model, the
resolution of the information that is obtained plays an important role. In geo-sciences we
distinguish several scales, each with their own resolution. Figure 1.4 presents the range of
scales that encompasses reservoir heterogeneity.
Reservoir heterogeneity
Scale
non-sealing fault
sealing fault
Giga
> 300 m
(>1000 ft)
Closed fracture
Open fracture
Boundaries
Genetic units
Mega
3 m - 300 m
(10 -1000 ft)
Permeability zonations
within genetic units
Baffles within
genetic units
Macro
3cm - 3 m
(inch - ft)
Laminations
Cross-bedding
Texture & fabric
Pore geometry and structure
Micro
microns - cm’s
Figure 1.4
Range of scales encompassing the information that is used
in building a reservoir model (Courtesy Schlumberger, Oil Field review)
6
Chapter 1
Introduction
The giga-scale in units of 300 m or larger reflects the total extent of the reservoir and seismic
is the appropriate tool for gaining information at this scale. The next scale is the mega-scale
with units of 3 - 300 m, which defines reservoir layers and permeability zones. The
boundaries of the permeable zones are of paramount importance in the configuration of
subsurface models; to arrive at realistic models grid block boundaries in subsurface models
should coincide as much as possible with the zones of different permeabilities. The prime
measurement tools at the mega-scale are borehole seismic, well logging, well tests and in
some cases high-resolution seismic. The macro-scale ranges from 3 - 300 cm and the
appropriate tools are the core measurements, core imaging and borehole imaging.
Finally, the micro-scale covers features with sizes from centimeters to microns. At the microscale the grains and the rock's texture can be observed. As earlier indicated, the pores are
the spaces enclosed by the grains, and can be observed as well. Tools for observation at
these scales are microscopes because in general the pores cannot be observed with the
human eye, which can only resolve details of some 100 microns. Evaluation of reservoir rock
at this scale provides information about pore structure and pore geometry, rock texture,
fabric, and mineralogy. All these features are important for the accurate interpretation and
calibration of higher scale measurements that are used in subsurface modeling.
1.5
Representativity of data, Up-scaling
The parameters needed to adequately configure the subsurface model are often obtained on
a much smaller scale than the size of the grid blocks used in the models. Figure 1.5 shows a
reservoir that clearly depicts the scarceness of the available data. For instance, the data
about the reservoir rock and the reservoir fluids that are collected in wells usually represent
less than 0.0001% (1 PPM) of the total reservoir volume [Thakur, 1989].
Figure 1.5
Reservoir model visualised to show the sparse, paucy well data that is
used to assign properties to the rest of the model (Courtesy Schlumberger, Dr. I.
Bryant)
7
Chapter 1
Introduction
The large variation in resolution and volume of investigation of exploration methods leads to
large uncertainties in the extrapolation of borehole data with high resolution into the inter-well
reservoir areas, where only low-resolution data is available. Only when the up-scaling
process is properly performed, can we obtain a reservoir model that adequately simulates the
flow process on a reservoir unit scale [Peaceman, 1989]. Many reservoir models failed to
correctly predict production and remaining hydrocarbon volumes because capillarity and
permeability were not properly accounted for in both the configuration and the up-scaling
stages.
This demonstrates the need to calibrate the borehole and seismic data that are used for the
configuration of reservoir models [Harvey et all, 1998; Dake, 1978]. Normally this calibration
is carried out with core data for which cores are required. However, the acquisition of core is
expensive and therefore often limited, which causes a scarcity of data points for calibration.
This induced us to develop methods that can provide the same information as cores, but are
cheaper to obtain, such as cuttings and sidewall samples (SWS). SWS's are not as small as
cuttings but are often damaged, which hampers the analysis of porosity and permeability with
conventional methods. SWS's can however be used in the same way as cuttings by taking
small volumes that can be studied in detail with microscopic techniques. This study therefore
concentrates on the determination of petrophysical parameters from microscopic images. We
will demonstrate that it is essential to analyse image data with the help of pore models to
derive rock properties on a micron scale, which can be up-scaled and used to calculate
petrophysical parameters at the reservoir unit mega-scale as indicated in Figure 1.4.
1.6
Problem statement
Considerable effort was devoted in the past to establish a theoretical framework to
understand and explain transport properties of fluids through porous media. It is not
surprising that many researchers demonstrated that the transport properties could be
explained much better by incorporating microscopic features of porous media: the pore
geometry and pore connectivity. The macroscopic flow phenomena, observed on samples
which have dimensions that are 2 to 3 orders of magnitude larger than the pore sizes, could
be better understood when investigated and analysed on the pore scale.
The flow through individual pores can as a rule not be observed directly, and microscopic
techniques were therefore introduced to assess and investigate pore geometry and pore
connectivity. Many researchers developed flow models at the pore level that were based on
microscopic observations. As these observations were descriptive and qualitative in nature,
the pore level theories were largely based on simplified and idealised porous media
consisting of either bundles of capillaries, or spherical grains with uniform properties. It was
only in the mid-80s that quantitative microscopic techniques became available with the
introduction of computerised image analysis techniques. This in turn enabled further
development of pore level theories and associated models, which were now more based on
quantitative information.
The models that were developed from theory alone are idealised porous media that are not
representative of natural porous media such as reservoir rock, because rocks are
geometrically and mineralogically too complex to fully describe with a simple model.
Occasionally, the models that were developed have been tested against real rock samples.
8
Chapter 1
Introduction
These samples were in general clean sandstones that essentially consist of two parts: the
rock matrix and the pores. However, real reservoir rock generally contains a third phase that
is residing in the pores: clay. Specifically in marginal, low permeability reservoirs, clays
dominate the petrophysical properties. Therefore, the scope of the available models, which
are based on only two components to predict reservoir rock behaviour, is limited. Moreover, a
rigorous test of the validity of the models that were developed over the past decades with a
sample set encompassing a large variety of rock types, has not been undertaken to date. In
this study, we carried out such a rigorous test on samples that represent many different
sandstone reservoir rock types. The objective was to test the general applicability of the
existing models, and to enhance these models to account for the presence of clay in the
pores. In addition, we developed a new approach that is based on the connectivity of the
pores. In this approach we used topological information for the prediction of one of the most
important reservoir parameters, the permeability.
Better geophysical exploration techniques, especially 3D seismic, have demonstrated the
presence of many small reservoirs in the vicinity of the larger reservoirs that have been in
production for quite some time. In the past, wells were drilled through these marginal
reservoirs, without taking sufficient log and core data. However, cuttings are always collected
when a well is drilled and the cuttings are normally stored for possible later analysis. It is this
source of information that can be analysed with the method that was developed in this study,
and can help to assess these marginal reservoirs at acceptable cost. With this new
technique, it is possible to return to these old, overlooked reservoirs and investigate possible
development.
The technique developed in this thesis derives 3D pore system parameters from 2D images.
The question however, is whether the available models can convert in a reliable way the 2D
image features into 3D rock properties, and how close the results derived from images
resemble the results of physical measurements? In addition, it is not clear whether the
models need amending to improve the accuracy of the predictions for shaly samples. This
study indicates the limitations of the current models and recommends adaptations of these
models to incorporate shaly sandstones.
1.7
Approach and outline of the thesis
This thesis started with a description of a hydrocarbon bearing reservoir and the most
important petrophysical properties that govern the behaviour of the reservoir during
production. Emphasis has been placed on clastic reservoirs because the techniques that
were developed in this study focus on sandstone and shale lithologies. The subsurface
model was introduced as the heart of the field development program, which in turn is the
road map that is followed during the lifetime of a reservoir. The subsurface model can only be
used meaningfully when it is configured such that it adequately simulates reservoir
behaviour. This configuration requires proper calibration of the larger scale logging and
seismic data with core analysis data. The core analysis data can be partly replaced by the
image-derived data that will be described in this thesis. One of the key advantages of this
new technique is that it is less expensive than conventional core analysis and can still be
applied when only cuttings and/or sidewall samples are available. In the remainder of this
thesis the research carried out for this study and the results that were obtained are presented
as follows:
9
Chapter 1
Introduction
•
Chapter 2 provides an introduction to the main petrophysical parameters porosity,
permeability, saturation and capillarity. We will discuss why the current work extends the
calibration/verification process to the pore size level by bringing in microscopic images
from optical and scanning electron microscopy. Grains are the main constituents of
sandstones, therefore we focus on granular media for the definitions of the petrophysical
properties. The models to determine petrophysical properties will be discussed form the
perspective of planar images, which of course can only provide 2D information.
•
Wireline logging and core analyses, which are the cornerstones of petrophysical
evaluation, will be discussed in chapter 3. Emphasis is put on those wireline logs that are
important for this study. Coring and core-analysis will be discussed also because the
results of core measurements will be used to develop the models that can predict
petrophysical parameters from images.
•
The three sample sets from which we collected images that were used in this study will
be discussed in chapter 4. The first sample set enabled prediction of petrophysical
parameters from thin sections using optical microscopy and image analysis. Electron
microscopy and image analysis was used on the second and third sample set. The
second sample set served as reference for the definition of the predictive models. The
third sample set was part of a case study and was used to test the predictive capabilities
of the models and to demonstrate how the mages can support petrophysical and
geological evaluations.
•
In chapter 5 we will discuss the processing and analysis of colour images. The images of
thin sections of reservoir rock samples were obtained with optical microscopy. The
features extracted from these images were correlated with porosity and permeability
values that were measured on the same core plugs from which the thin sections were
prepared. The correlations led to the definition of predictive models for porosity and
permeability. The validity and utilisation of these models will be discussed in detail. We
also show the shortcomings that are introduced by the analysis of images from thin
sections, particularly for shaly samples.
•
Electron microscopy has distinct advantages over optical microscopy and we will discuss
these advantages together with the principles on which electron microscopy is based in
chapter 6. An imaging method that differs from conventional electron microscopy
imaging, the backscatter electron mode (BSE) will be introduced for use with reservoir
rock samples. We could not use the thin section samples for BSE image generation
because the thin sections were not suited for electron microscopy imaging. Therefore, a
special sample preparation method was developed that was specifically tuned to BSE
imaging. The properties of BSE imaging and the engineering involved in this type of
imaging will be discussed and followed by optimal equipment settings for reservoir rock
samples.
•
In chapter 7, we will develop algorithms to extract the pore system from the BSE images
using image-processing techniques. In this chapter we also develop a new method for
quantification of micro-porosity based on a statistical model of electron scattering in
clays. The micro-porosity will be defined as the volume fraction of small pores that are
residing in clays. These small pores cannot be resolved by optical microscopy, but can
10
Chapter 1
Introduction
be detected with electron microscopy. We also present an automatic feature extraction
method that is based on grey-level thresholding. Hitherto, automatic feature extraction
from BSE images of reservoir rock was not possible and had to be carried out manually.
Finally, we will present a new procedure that extracts a topographic representation of the
pore system that will be used for permeability prediction.
•
The prediction of porosity and permeability from BSE images is the topic of chapter 8.
We will show that the accuracy of the prediction of porosity can be improved substantially
by incorporating the micro-porosity according to the model we developed in chapter 7.
We also will discuss the various models that are available for permeability prediction from
image data based on the capillary bundle models that were introduced in chapter 2. In
addition, a new approach, to predict permeability form network representations derived
from BSE images, will be introduced. We finally will explain why the prediction of porosity
fails in some cases.
•
In chapter 9 we will apply the techniques that were developed in this study to two field
cases. In the first case we will show that porosity and permeability predictions from
images can be reliably integrated in the total petrophysical evaluation of a well, amending
the results of the interpretation of wireline logs, core analyses and core photos. We will
also show that important mineralogical information can be obtained from the BSE images
and that this information can be used to better understand and explain wireline log
responses, and thereby improve the quality of the evaluation.
•
Appendix A deals with the Monte-Carlo experiments we carried out with three objectives
in mind. First, to assess the optimal settings of the electron microscope to obtained BSE
images that are suitable for the type of analysis we developed in this study. Second, to
develop a calibration method of the grey-levels in the BSE images. Third, the
experiments were used to evaluate and calculate the resolution of the BSE images in
order to use this for the development of the model for prediction of micro-porosity.
•
The morphological techniques we used in the processing of the images are the subject of
Appendix B. Mathematical morphology was used on binary representations of the pore
system and the rock matrix that were extracted from the BSE images.
•
Appendix C describes the ideas and the scouting experiments we carried out to
investigate whether directional permeability could be obtained from BSE images. We
used the network approach that was developed and explained in chapter 7 and propose
to combine this with an electric analogy of the pore network.
•
In Appendix D we will discuss the equipment, the software package, and the engineering
aspects that are involved in the implementation of this technique on the well site. The
procedures that were used in this study are summarised and practical considerations will
be discussed. Finally some economic issues will be touched upon for actual
implementation in an operational environment.
11
Chapter 2
The main petrophysical parameters
2.
The main petrophysical properties
2.1
Introduction
Petrophysical measurements are made in the borehole and on cores in the laboratory to
determine the major reservoir properties: porosity, permeability, fluid saturations and
capillarity. The term petrophysics was coined by J.H.M. Thomeer, the first Professor of
Petroleum Engineering at Delft University of Technology in the Netherlands. Petrophysics
is sometimes called an art because the laws that relate petrophysical parameters to the
physical properties that are measured in the borehole such as resistivity, density, and
neutron absorption are not universal.
The petrophysical parameters that are obtained from cores are used to calibrate borehole
measurements. The borehole measurements are subsequently used to calculate the
volume of hydrocarbons present in the reservoir, at what rate they can be produced, and
how much of the hydrocarbons are left behind in the reservoir after the production is
terminated. The calibration of wireline log data and seismic data with results from core
analyses is necessary to find the specific relations that are valid for one particular
reservoir. The current work extends this calibration / verification process to the pore size
level by bringing in microscopic images from optical and scanning electron microscopy
(SEM). After all, the flow processes that sustain the oil industry take place in small pore
systems that can only be observed with microscopic techniques. In this chapter, we review
the main petrophysical parameters in the context of their derivation from images. After the
introduction of porosity, we will discuss permeability in more detail because this is the most
important reservoir parameter. Permeability governs the flow of hydrocarbons through
reservoir rock. We will briefly review the established models for permeability that are
reported in the literature from a pore system perspective. Finally, we will discuss saturation
and capillarity.
For a rock to contain hydrocarbons, it has to be porous; the rock should contain cavities
where the hydrocarbons can reside. In this study, we will use the definition of a porous
medium as given by Dullien [1979], which constitutes two requirements:
1
2
The medium must contain spaces, so-called pores or
voids, embedded in a solid matrix.
The medium must be permeable to liquid or gas, which
requires that the pores be connected into a system.
The total porosity encompasses all the void spaces in the rock; including isolated pores
that are not a part of the interconnected pore system. The effective porosity is defined as
all the connected pores in the pore system that can contribute to flow.
In sedimentary rock, there are two main porous classes, siliciclastics and carbonates.
Siliciclastic rock consists of a granular framework that is known as the detrital assemblage.
Siliciclastic rocks have a purely physical origin; their sediments result from the weathering
of non-porous rock. The main constituents of siliciclastic rock are quartz grains ( SiO2 )
and feldspar grains ( ( K , Na , Ca )[ AlSi 3 O8 ] ). Siliciclastic rocks are more commonly
known as sandstones, silts and gravel; and about 50% of all hydrocarbon-bearing
reservoirs are composed of these rocks. Siliciclastics is the name for the rock type that is
12
Chapter 2
investigated in this study. Carbonate rocks generally have a biogenic origin. They are
formed by the remnants of microscopic planctonic organisms, corals and shellfish. After
death, the calcareous skeletons of these organisms accumulate on the sea floor to form
sediments. Carbonate sediments represent roughly the other 50% of hydrocarbon bearing
reservoirs.
After deposition, the mineralogy and porosity of a sediment can alter substantially over
geological times. This alteration is of physio-chemical origin and is known as diagenesis.
In this process, during progressive burial and corresponding increase of the effective
stress and temperature, the sediment compacts and its mineralogy is altered by chemical
reactions of the rock matrix with minerals that are dissolved in the pore waters. The most
common mineralogical alteration is the precipitation of pore lining clays and cements of
silica, calcite and dolomite. These diagenetic processes have a profound influence on the
pore geometry and the physical properties of the rock, and it is therefore important to
recognise what type of diagenesis has occurred. This in turn will enable a more precise
assessment of pore geometry and pore wall mineralogy and leads to a better
understanding of the larger scale reservoir properties that control the flow of fluids through
the reservoir. The technique that was developed in this study uses images as a basis for
the prediction of petrophysical properties. These images also enable the detailed
observation of the pore geometry and the pore wall mineralogy to identify diagenetic
processes.
2.2
Porosity, the available space in reservoir rock
The porosity is the fraction of the total rock volume that can be filled with oil, gas, water or
a mixture of these fluids. For a petrophysicist, the porosity is the first parameter to evaluate
because it determines the amount of hydrocarbons that can be present in the reservoir.
The total porosity Φ is defined as the fraction of bulk volume of the reservoir rock that is
occupied by pore space:
Φ=
V pore
Vbulk
=
Vbulk − Vrock Vbulk − ( w / ρ rock )
=
Vbulk
Vbulk
2.1
Where Vpores and Vrock are the volumes of the pores and the rock respectively, Vbulk is the
bulk volume, w is the weight and ρrock is the specific density of the rock.
The reservoir rock type determines the porosity, which can vary from zero to 70%.
Diatomites, a very outlandish sediment, can have a porosity up to 70%. For sandstones
the porosity has a limited range, from near zero for ill-assorted and cemented grains, to a
theoretical maximum of 47.64% for a perfect cubic stacking of monosized, spherical
grains. The porosity in reservoir rock is a function of the grain size distribution, the grain
shape distribution and of the packing. The latter is the special arrangement of the grains.
A macroscopic parameter can be defined by assuming a Representative Elementary
Volume (REV). The REV is a conceptual space unit that has the same properties as a
physical point in continuum.
13
Chapter 2
2.2.1 Simple cubic packing
In the case of the simple-cubic centred packing, the REV is a unit cube that can be
obtained by connecting the centres of the spheres, which surround one single sphere, and
is shown in Figure 2.1.
dsphere
LREV
Figure 2.1
Representative elementary volume in cubic centred packing
The REV of which a cross-section through the centre sphere is shown in the right part of
Figure 2.1, is built up of 27 spheres, the unit cube then contains a volume of 8 spheres.
These spheres occupy a volume:
Vbulk = LREV 3
2.2
The total number of spheres occupying that unit cube is:
(
n = LREV d sphere
)
3
2.3
Using the equation for the volume of a sphere, we can calculate the volume occupied by
the spheres:
(
)
Vsphere = n πd sphere 3 / 6
2.4
Combining equations 2.1 and 2.4 the porosity can be derived:
Φ=
(
LREV 3 − LREV d sphere
LREV
) (πd
3
3
sphere
3
) = 1 − (π 6) = 0.4764
/6
2.5
Note that the porosity is independent of the size of the spheres in case of mono-sized
spheres.
2.2.2 Rhombohedral packing
Granular sediments are normally deposited under the influence of gravity and forces that
are exerted by water and/or wind, and that have random components. The interplay
between water, wind and gravity forces causes sediments to devolve into a denser type of
14
Chapter 2
packing, which approaches the rhombohedral packing. This packing is depicted in Figure
2.2. The rhombohedral packing is the densest packing possible with mono-sized spheres
[Mayer, 1965]. If we derive the porosity for this packing, our representative elementary
volume is a tetrahedron that is again obtained by connecting the centres of the spheres
that surround one sphere.
dsphere
LREV
Figure 2.2
Representative elementary
volume in rhombohedral packing
The smallest unit that we can obtain encompasses 10 spheres. Six spheres are placed on
the legs of the tetrahedron of which 16 of the volume of each sphere is present inside the
tetrahedron. Four spheres are placed at the corners of the tetrahedron, of which 112 of the
volume of each sphere is present inside the tetrahedron. The total volume of the spheres
inside the tetrahedron is then 4 3 of one sphere volume. The volume of the unit tetrahedron
is:
Vbulk = LREV 3 2 / 12 = 8d sphere 3 2 / 12
2.6
With a volume occupied by spheres inside the tetrahedron:
(
)
Vrock = 4 3 πd sphere 3 / 6
2.7
Combining equations 2.6 and 2.7 yields the porosity:
(
)
3
3
4
Vbulk − Vrock 8d sphere 2 / 12 − 3 πd sphere / 6
Φ=
=
= 0.2595
Vbulk
8d sphere 3 2 / 12
2.8
In a medium with the densest packing consisting of mono-sized spheres, the porosity is
almost 26% and, as was the case for the cubic packing, is independent of the size of the
spheres. In reality, sediments are of course not composed of mono-sized spheres. A
distribution of sizes produces a lower porosity than in the idealised cases. In
sedimentological terms, a wide grainsize-distribution is called poorly sorted; a narrower
distribution is considered well sorted. Well-sorted systems approach the ideal
rhombohedral packing, which in general gives a higher porosity than poorly sorted
systems.
15
Chapter 2
2.2.3 Porosity alteration
When deposited, a sediment has an initial porosity known as the primary porosity. Over
geological times this primary porosity can be altered by the following phenomena:
1) Diagenesis is the physio-chemical precipitation of minerals from the pore fluids, as
mentioned in section 2.1. Cementation is the most important diagenetic effect; it binds
the grains together and thereby reduces the porosity.
2) Compaction is caused by the load of the overlying sediments. Compaction leads to a
more compact arrangement of the grains. This is caused by sliding of grains along
their contacts, by indentation of grains through pressure solution, and by fracturing of
grains into smaller particles, which fill the pore space between larger fragments.
3) Leaching occurs when some minerals dissolve in the pore fluids. This type of porosity
is normally referred to as secondary porosity because it is produced after deposition of
the sediment.
4) Bioturbation is the mixing of the deposited sediment. The mixing is caused by
bioactivity, small specimens such as worms that move through the sediment and leave
a trace of fine particles behind. Bioturbation can only occur in freshly deposited
sediments.
5) Clay coating involves phyllosilicate crystals that grow on the surface of the grains over
geological times and can even completely fill pores. This process has a profound
influence on the pore geometry and pore wall mineralogy and thereby on reservoir
properties.
Diagenesis, compaction, bioturbation and clay coating decrease the porosity while
leaching can increase the porosity. In general, all phenomena with the exception of
bioturbation can take place in a reservoir over geological times. If the reservoir remains as
it was at the time of deposition, the grains will have only point-contacts. In that case, the
reservoir is considered unconsolidated. If diagenesis occurs, the grains are bound together
by grain coating clays and by mineral cements. The reservoir is then considered
consolidated. In addition, the pressure solution of grain material caused by compaction
also leads to a more consolidated reservoir.
Leaching processes mostly attack specific, easy to dissolve minerals like feldspars,
dolomites and calcites in their detrital form. Quartz grains are very rarely dissolved fast
enough by pore waters at the depths at which hydrocarbon bearing reservoirs occur.
Moreover, temperatures are usually too low to establish a chemical reaction with quartz.
Finally, quartz can only be dissolved by very strong acidic or basic fluids and pore fluids at
those depths seldom contain high concentrations of such ionic components.
Diagenesis and leaching can occur as long as the reservoir is water bearing. Once
hydrocarbons have entered the reservoir, they force most of the water out of the pores by
gravity segregation. Hydrocarbons seldom contain the chemicals that can dissolve or
precipitate minerals, and diagenesis and leaching will therefore diminish in the charging
phase. The very thin layer of water remaining around the grains, which is termed connate
water, is usually not continuous and fully saturated with dissolved minerals. Connate water
is in a state of equilibrium with the rock minerals and this prevents further physio-chemical
reactions.
16
Chapter 2
Compaction is a purely mechanical phenomenon and dependent on the rock type, the
pore fluid type and the weight of the overburden. [Schutjens, 1991]. Compaction can occur
during the entire lifetime of a reservoir. In many cases, the production of hydrocarbons
contributes to compaction. When fluids are extracted from the reservoir, the pore pressure
is reduced. This reduction increases the fraction of the overburden pressure that the grains
bear, and leads to higher pressures on the grain to grain contacts. Grains may break, shift
or indent and these effects together decrease the porosity and can lead to a reduction in
permeability.
2.2.4 Porosity in the context of images
The major impediment to derive porosity from images is that an image is evidently a twodimensional (2D) representation of a three-dimensional (3D) medium. As porosity has
been defined as the fraction of empty space in reservoir rock in 3D, a conversion is
required to make the step from 2D to 3D. Stereological concepts [Underwood, 1970] show
that in a homogeneous and isotropic medium the 2D area fraction of a constituent is
identical to the 3D volume fraction of that constituent. Hence, to make the link from planar
image porosity (2D) to volumetric porosity in real rock (3D), the porous medium has to fulfil
the requirements of homogeneity and isotropy. The homogeneity requirement of the
porous medium is related to the magnification at which images from that medium are
taken. The image should contain enough pores and grains in order to capture a
representative part of the pore system. The isotropy requirement is related to the
orientation. The rock can be considered isotropic when image features are independent of
orientation. For instance, the measured area percentage of pores determined from the
images should not alter significantly when images with different orientations are taken from
the same rock. If these requirements are not fulfilled the prediction of the petrophysical
properties like porosity and permeability will be less precise.
In order to derive porosity from images we have to define the reservoir rock as a twophase medium, in which the pore system constitutes one phase and the rock minerals,
also know as the matrix, the other phase. This process will be presented and discussed in
more detail in chapter 7.
2.3
Permeability, the hydraulic conductance of reservoir rock
In analogy to the electric conductance, which is defined by Ohm's law as the ratio of the
electric current and the electric potential, we can define the hydraulic conductance as the
ratio of the fluid flow and the pressure difference. The permeability or hydraulic
conductivity of a porous medium for fluid flow depends on both the properties of the fluid,
and on the pore system properties. The latter properties comprise pore geometry and pore
connectivity. Permeability determines the velocity of the flow a fluid through a porous
medium. The concept of permeability originates from the basic work that was carried out
by Henri Darcy in the 19th century [Darcy, 1856].
Darcy investigated the hydrology of sand-beds in sanitary systems. His experimental
studies on steady state uni-directional flow revealed the proportionality that exists between
the fluid flow rate and the applied pressure drop. This linear relationship is commonly
known as Darcy’s law:
17
Chapter 2
Q=k
At ∆p
µL
2.9
3
In which Q is the volumetric flow rate in m /s through the porous medium with a total
cross-sectional area At perpendicular to the flow direction, and µ the dynamic viscosity of
the fluid. ∆p is the pressure drop across the porous medium with length L, and k is the
permeability [Darcy, 1856]. Figure 2.3 depicts the definition of Darcy's law.
∆p
µ,Q
At
L
Figure 2.3
Definition of Darcy's law
In equation 2.9 the force exerted by gravity is neglected. Equation 2.9 can be rewritten to
determine flow velocity:
v=
Q k∆p
=
At
µL
2.10
In some textbooks like Scheidegger’s [1972] and Liu & Masliyah [1994], v is termed the
superficial flow velocity. In this study, we will use the term flow velocity. Darcy's law
describes the flow through a porous medium for low velocities. In engineering sciences
three flow velocity regimes with smooth transitions from one to another are generally
distinguished [Liu & Masliyah, 1994]:
1
2
3
Pre-Darcy flow that is dominated in porous media by surface effects and therefore
strongly dependent on the fluid/solid interaction like the adhesive forces between
the fluid and the pore wall.
In the Darcy flow regime, the inertial effect of the fluid is negligible and the flow
velocity is a linear function of the pressure gradient. Viscous forces prevail in the
Darcy regime.
In the Forchheimer flow regime inertial forces become significant compared with
viscous forces. The pressure gradient becomes a quadratic function of the flow
velocity and turbulence occurs. Turbulent flow is characterised by unstable and
chaotic movements of fluid particles. Random flow vortexes can be formed in this
regime.
Darcy's law was derived for uni-directional, non-turbulent flow. If we generalise Darcy's law
for multidimensional flow, we have to replace the scalar permeability k by a second-order
tensor permeability k that is dependent on the directional properties of the pore system.
The macroscopic flow velocity v is replaced by a velocity vector v and we arrive at the
differential form of Darcy's law that can be expressed in the pressure gradient ∇p :
18
Chapter 2
∇p = − µk −1 ⋅ v
2.11
Brinkman [1949] included a diffusion term into Darcy's equation to account for viscous
diffusion effects that occur in the Pre-Darcy regime:
∇p = − µ k −1 ⋅ v + µ∇ 2 v
2.12
Brinkman’s equation is, similar to Darcy’s law free of inertial effects and hence only valid
for very low flow velocities. The pressure drop is however still a linear function of the flow
velocity. Forchheimer's [1901] work on higher flow rates through porous media led to the
hypothesis that the non-linear behaviour is caused by inertial forces. In the turbulent flow
regime, the driving force ∇p not only causes movement of the fluid in the porous medium
but also increases the kinetic energy of the fluid resulting in the formation of local vortexes
or eddies. To incorporate these inertial forces, Forchheimer modified Darcy's law by
including a term for kinetic energy of the fluid ρv . Incorporating this in the Darcy2
Brinkman equation yields:
∇p = − µ k −1 ⋅ v + µ∇ 2 v − ρβv 2
2.13
In which ρ is the density of the fluid and β a tensorial term that represents the drag effect
of the porous material and is known as the turbulence factor. For a one-dimensional
situation Ward [1964] proposed:
β=
cF
k
1
2.14
2
Where k is the permeability and c F the form drag coefficient of the porous medium. The
Forchheimer hypothesis has been generally accepted as an extension to Darcy for high
flow rates. In the literature, the flow in the Darcy regime is often referred to as laminar flow
[Liu & Masliyah, 1994].
Macroscopic observations in producing wells show flow velocities in reservoir rock that are
typically one ft per day (1 cm per hour) for oil and some 4 ft per day for gas. In gas
reservoirs, specifically near wells, flow velocities can be much higher and are sometimes
in the turbulent Forchheimer regime. However, the flow velocities throughout the reservoir
rock away from the wells are in general so low that they fit in the Darcy regime. Therefore,
the permeability of reservoir rock samples in core analysis is determined with flow in the
Darcy regime. Consequently, the development of models to derive permeability from other
means than conventional physical measurements, such as from images of reservoir rock,
in this study are limited to flow in the Darcy regime.
2.3.1 Pore structure models for permeability
Based on the assumption that the permeability is a function of the pore geometry and the
pore connectivity, one intuitively can argue that if these properties are quantified, the
19
Chapter 2
permeability can be determined. However, this requires a complete description of the pore
size and shape distributions and of the pore system connectivity. The pore structure in a
natural sandstone is very complex and the effort to obtain such a complete description
would be a formidable task, impractical for routine use. Moreover, the models that are
presently available from literature do not have the sophistication that is required to
accommodate such a complete and complex description. We therefore start with less
complicated models that can accommodate the parameters that we can derive from
images.
A pore size distribution f (δ) is generally defined as the probability density function of the
pore volumes having a pore size δ . [Liu & Masliyah, 1994] The pore size density function
at a value δ is the fraction of the total pore volume with a pore size between δ and dδ ,
and is mathematically expressed as:
δ
F ( δ ) = ∫ f ( δ )dδ with
0
∞
∫ f ( δ)dδ = 1
2.15
0
Where F (δ) is the probability distribution function. Similar reasoning can be followed for
the pore shape distribution. The pore size is always calculated from a physical
measurement and interpreted using a pore structure model. As there are many pore
structure models, the pore size is not a well-defined property and therefore depends very
much on the method that is used for its determination. Some researchers defined a pore
size by the pore entry diameter derived from capillary pressure curves, other used the pore
diameter as observed on 2D rock sections. Many researchers, notably Dullien [1979],
reviewed numerous methods and definitions, and showed that the pore sizes obtained for
the same sandstone using different methods can vary significantly. We therefore argue
that any measurement that leads to a correct prediction of permeability is useful.
The pore system connectivity is characterised by the co-ordination number. This
parameter is defined as the average number of pores to which a single pore is connected.
If we define a porous medium as spaces that are interconnected by constrictions, the pore
space can then be assigned to the pore bodies and the constrictions can be considered as
the pore necks [Dullien, 1979]. An office building that contains rooms can be considered
as an analogy to this. The rooms are connected by doors and to move from one room to
another one has to pass a door. The rooms in the building are analogous to the pore
bodies while the open doors have the role of the pore necks. From the concept of pore
bodies connected by pore necks, we can define the co-ordination number Zpores as:
Z pores =
number _ of _ porenecks
number − of _ porebodies
2.16
From the above definition of a porous medium, one can argue that the permeability is
essentially governed by the pore neck properties. We therefore start with very simple
models in which the pore space consists of pore necks solely. Based on these simple
models we investigated how image features can be related to permeability. The simplest
pore structure model one can imagine is that of a bundle of straight capillaries in which all
capillaries have the same diameter. In this model, the capillaries connect one side of the
20
Chapter 2
porous medium to the other and the capillaries are not interconnected as shown in Figure
2.4.
Figure 2.4
Capillary bundle model
When a pressure difference ∆p is applied over this porous medium with length L , the
flow through each individual capillary is given by Poiseuille's law [Dullien, 1979]:
Q=
πr 4 ∆p
8µL
2.17
In which r is the radius of each capillary. Similarly, Darcy expressed the flow through a
medium of area A containing just one tube by:
Q=
kA∆p
µL
2.18
Combining equations 2.17 and 2.18 leads to:
πr 4
k=
8A
2.19
The medium with area A and tube area At can be expressed as the porosity:
Atube πr 2
πr 2
Φ=
=
⇒ A=
Φ
A
A
2.20
Inclusion in equation 2.19 then gives:
k=
Φr 2
8
2.21
Hence, the permeability k for a porous medium with n parallel and separate capillaries
is:
21
Chapter 2
k=
Φ 2
nr
8
2.22
When the capillaries have a discrete range of radii, the permeability is given by:
k=
Φ
ri 2
∑
8 i
2.23
Alternatively, in case of a continuous distribution of pore radii that may be obtained from
images:
Φ
k=
8
rmax
∫ rdr
2.24
0
In the above simple model, the permeability is a function of the radii of the capillaries and
the porosity. Kozeny [1927] and Carman [1937] extended this model using the hydraulic
radius theory. In analogy with the established practice in hydraulics, the diameter of the
capillaries can be represented by the hydraulic diameter d h , which can be related to the
equivalent pore diameter:
dh =
pore _ volume
pore _ area
2.25
The quantity dh is also known as the volume-to-surface ratio of the porous medium. If we
define A as the area of the cross-section of the capillaries perpendicular to the flow
direction and P as the so-called wetted perimeter of the cross-section of the capillaries,
we can transform 2.22 to image parameters that are inherently 2D:
dh =
π A pores A pores
≈
Ppores
4 Ppores
2.26
In equation 2.23 we neglect the term π/4 which is approaching unity. The equation can be
used to derive pore structure models for permeability that belong to the family of KozenyCarman models.
A cylindrical sample of reservoir rock, known as a core plug, usually contains thousands of
pores. When a permeability measurement is carried out one can distinguish two different
fluid flow velocities, the microscopic pore flow-velocity vF and the macroscopic velocity v
which gives the total flow-velocity through the plug. This is depicted in Figure 2.5 where
the ratio Le/L indicates the longer trajectory a hypothetical fluid particle has to travel
through the tortuous pore system. In the macroscopic situation this particle travels a path
L with velocity v while in the same time an actual fluid particle flowing with velocity vF
travels an effective path Le.
The Dupuit-Forchheimer assumption accounts for the difference between the flow velocity
in the pores and the flow velocity through the plug. The generally accepted DupuitForchheimer velocity vDF is defined as v Φ .
22
Chapter 2
Le
L
Figure 2.5
The tortuous path through the
pores in a core plug
v F = v DF
Le
1 L
= v e
L Φ L
2.27
A Hagen-Poiseuille type model gives the microscopic flow velocity in the pores:
d h 2 ∆p
vF =
32µk 0 Le
2.28
In which dh is the hydraulic diameter, Le the average tortuous path length and k0 a
dimensionless shape factor that describes the shape of the perimeter. The macroscopic
flow-velocity through the plug is given by Darcy's law:
v=
k∆p
µL
2.29
Equations 2.28 and 2.29 can be substituted into equation 2.27:
d h 2 ∆p
1 k∆p Le
=
32µk 0 Le Φ µL L
2.30
This can be reduced into:
Φd h 2  L 
 
k=
16k 0  Le 
2
2.31
Equation 2.31 is the basic form of all capillary models, which only differ from each other in
the method to determine the mean squared diameter d h and the term (1 k 0 )( Le L ) .
2
This term is supposed to be a function of the pore geometry and of the pore connectivity.
According to Carman [1937] the hydraulic diameter d h for a set of capillaries can be
expressed as:
dh = 4
Φ
S 0 (1 − Φ )
2.32
23
Chapter 2
In which S0 is the specific surface area that is based on the volume of the rock matrix.
Combining equations 2.31 and 2.32 we arrive at the usual form of the Kozeny-Carman
equation:
Φ3
1  Le 
k= 2
 
2
S 0 (1 − Φ ) k 0  L 
2
2.33
2
Where the term (Le/L) is generally called the tortuosity factor. The two theories that have
been discussed above are those by Hagen-Poiseuille and Kozeny-Carman. Models
derived from these theories will be tested with several data sets to assess their power for
prediction of permeability from rock thin section colour images and scanning electron
microscopy BSE images. In chapter 5 and chapter 7 respectively, we will discuss how the
parameters that are required to populate these models are extracted from the images.
2.3.2 Permeability in the context of images
The models for permeability prediction that were discussed above are based on geometric
properties of the individual pores. We used simple capillary bundle models because these
can be configured from 2D observations, in our case images. However, as indicated
earlier, permeability is a tensor property that is dependent on direction. Since the images
that we used were taken from reservoir rock samples in a fixed orientation, the information
we extracted from the images may be used to assess directional permeability.
The capillary bundle models will be configured from images that have been obtained from
reservoir rock samples oriented perpendicular to the flow direction in which the
permeability has been measured. However, it should be possible to devise a procedure to
assess conductivity in the plane of the flow direction, hence parallel with the tubes. For
this, we require information about the connectivity of the pores, which can be achieved
with the so-called network approach. In the network approach, we represent the pore
system as a fully connected network. The resulting network is a 2D reflection of the 3D
pore system. Figure 2.6 depicts the above reasoning showing the permeability
components provided by the different approaches.
DP
Direction of flow
Vertical
Network
Perpendicular
Poiseuille
Kozeny-carman
L
Horizontal
Network
Figure 2.6
Directional components for permeability
determination from images
24
Chapter 2
In Figure 2.7 the pore structure model is presented from which we derived the capillary
bundle approach and the network approach. The models for permeability prediction, that
are discussed above: Poiseuille, Kozeny, Kozeny-Carman and the Network approach, will
all be tested on image data. Optical microscopy, colour image analysis and permeability
prediction with Kozeny-Carman are discussed in chapter 5. Furthermore, in chapter 8, we
will use all above models in combination with image data obtained from Scanning Electron
Microscopy.
Network approach
Poiseuille &
Kozeny-Carman
approach
Figure 2.7
Combining the capillary bundle approach and the network approach to
predict directional permeability from images
2.4
Saturation, the fraction of porosity occupied by fluids
Another important reservoir parameter is the fluid saturation, which is the fraction of pore
space occupied by a certain fluid. More formally, the saturation of fluid i is defined as
[Dullien, 1979]:
Si =
V phasei
2.34
V pore
In which V phasei is the volume of fluid i and V pore the volume constituting the total porosity.
Reservoir rock often contains two (oil and water), or even three (oil, water and gas) fluids.
From an industry perspective, one is especially interested in oil and gas saturations and the
distributions of these fluids in the pore space.
The most established way to determine the hydrocarbon saturation in a reservoir is with
resistivity measurements. This is an indirect method based on the differences in conductivity
of water and hydrocarbons. Water in a formation usually contains dissolved salts that facilitate
ionic conductivity while hydrocarbons do not conduct. In addition, most of the minerals
constituting the rock matrix, with the exception of shales and pyrite, have a very high
resistance. Therefore, a relationship between electric conductivity and saturation can be
25
Chapter 2
established. Conducting clays form a key element in the assessment of hydrocarbon
saturation in shaly sand reservoirs, because their presence leads to over-estimation of the
amount of water if they are not taken into account.
2.4.1 Archie's models
The conductivity of a hydrocarbon-bearing reservoir can be measured with resistivity logging
tools. These measurements have to be interpreted in order to obtain an estimate of the water
saturation, which in turn is required to estimate the total amount of hydrocarbons in place.
There are a large number of saturation interpretation models using resistivity logs as input.
Most of these interpretation models are variations of the Archie model that was formulated in
1942 [Archie, 1942]. Archie published the results of an investigation concerning the electric
conductivity of clean, sandstone reservoir rocks. This publication had the most impact on the
birth of petrophysics. Archie found that the resistivity R0 of a brine saturated sample
increased linearly with the brine resistivity Rw . The ratio R0 Rw was found to be a rock
specific property and was called, the rock formation factor F :
F=
R0
Rw
2.35
The formation factor was supposed to be determined by the pore geometry and pore
connectivity. Hence, by the same properties that determine permeability. This is however only
valid for clean sandstones consisting of quartz and pore space, thus in the absence of clays.
Archie related the formation factor F to porosity Φ for a single lithology (rock type) and
demonstrated that the relationship was a straight line on a log-log plot, thus leading to the
now well known first Archie equation:
ln( F ) = − mln( Φ) or F =
1
Φm
2.36
In which m is called the cementation exponent that reflects both the pore geometry and pore
connectivity. The latter two are affected by cementation, a diagenetic effect that was
discussed in the section on porosity.
After his research on brine saturated samples, Archie carried out experiments with partially
saturated, oil bearing samples. He defined the resistivity index I, which is the ratio of the
resistivity of a partially water saturated rock Rt, and the resistivity R0 of the same sample fully
brine saturated.
I=
Rt
R0
2.37
He then applied a similar reasoning as in the case of the formation factor leading to the
second Archie equation:
ln( I ) = − nln( S w ) or I =
1
Swn
2.38
26
Chapter 2
In which Sw is the water saturation and n the saturation exponent. The saturation exponent
reflects the spatial distribution of water in the pore space, which in turn is controlled by the
pore geometry and the affinity between the pore fluids and the pore wall. The two Archie
equations can be combined into the general Archie’s law:
Rt =
Rw
Φ m S wn
2.39
Equation 2.39 can be re-written such that Sw is a function of a set of laboratory determined
parameters m, n and a set of parameters that can be measured in the borehole by wireline
logs. The latter are the total resistivity of the rock, Rt the resistivity of the formation water Rw,
and the porosity Φ . Note that Rw, can only be determined from a resistivity log or an SP log
when the formation has a 100% water-bearing interval. Another important requirement is that
the formation water, the mineralogical properties and the textural rock properties should be
the same in the water bearing and the hydrocarbon-bearing intervals. Once Sw is known, the
hydrocarbon saturation SHC can be calculated by:
S HC = 1 − S w
2.40
2.4.2 The Waxman-Smits model
In the early fifties, Patnode and Wyllie [1950] found that Archie’s relationships do not apply to
shaly sandstones. They argued that the clays cause a conductivity in addition to the value
predicted by Archie's first law based on porosity and formation water resistivity alone:
C0 = Cclay +
Cw
F
2.41
In which C0 is the total sample conductivity, Cclay the clay conductivity, Cw the water
conductivity and F Archie’s formation factor. Many researchers investigated the influence of
clays on the total conductivity and tried to understand and explain the physics associated with
clay conductivity. This culminated in more than 30 different saturation interpretation models of
which the Waxman-Smits model is one of the more commonly used. Waxman and Smits
[1967] argued that the electric pathways for clay conductivity through a sample are similar to
those of the ionic conductivity through the pore waters. Hence, they assumed that a single
formation factor F* can be used to describe the total tortuosity of the pores:
C0 =
1
( BQv + Cw )
F*
2.42
Where B is the cation equivalent conductance (the ion mobility) and Qv the cation exchange
capacity (the number of ions per volume unit). B is dependent on water salinity and
temperature, and is usually derived from laboratory charts. Qv Is measured on reservoir rock
samples in the core laboratory. The contribution of the clay to the total conductivity is covered
by the term BQv . The term F* includes the clay conductivity and differs from Archie's
formation factor F which reflects the conductivity of clean sands. Based on the assumptions
27
Chapter 2
that oil only displaces water in the centre of the pores and the that the sample is water wet,
Waxman and Smits argued that the conduction mechanisms in the clays are unaffected by
the presence of hydrocarbons. They then extended their model from brine saturated samples
to oil bearing shaly sands leading to the well known Waxman-Smits saturation model:
*

S wn  BQv
Ct = * 
+ Cw 
F  Sw

2.43
The Waxman-Smits model allows estimation of the in-situ hydrocarbon saturation in shaly
sand formations based on the resistivity measured in the borehole and on core measurement
*
of F, Qv and the saturation exponent n
2.4.3 Effective medium models
Following the work of Waxman and Smits other researchers hypothesised that the use of one
single formation factor for both the clay conductivity and for the ionic fluid conductivity is not
always valid. The use of one formation factor is only permitted when a clay layer of uniform
thickness covers the entire pore-wall throughout the pore space, and when the rock is entirely
water-wet. Pape and Worthington [1983] researched this subject and proposed two different
formation factors: one for the ionic conductivity and one for the clay conductivity. The latter is
depended on the clay type and on the clay morphology in the pore space. Indeed,
observations in this study corroborates this hypothesis, and inspection with scanning electron
microscopy shows that the clay morphology does not uniquely follow the pore wall with
uniform thickness. Moreover, it was found that different clay types can occur in adjacent pores
at distances of less than 100 micron. This confirms that clays are not uniformly distributed.
These observations put question marks on the use of a unique formation factor for both ionic
conductivity and clay conductivity and favour the approach of Pape and Worthington.
In the mid-80s, various researchers tried to arrive at physically more realistic saturation
interpretation models based on an effective medium theory. Bussian [1984] applied the
Hanai-Bruggeman equation which was originally used to describe dielectric properties of
random mixtures to determine the conductivity of shaly sands. More recently, Schwartz and
Sen [1988] provided practical expressions for evaluating resistivity logs obtained over shaly
sands. The most recent effective medium based model is that of De Kuijper et al [1995]. In
this model, the bulk electrical properties can be calculated from the volume fraction, shape
factors of the grains, conductivity and connectivity of the fluids and the minerals. The effective
medium models include shale properties intrinsically and experiments have shown that they
can describe the relationship between conductivity and saturation with more accuracy than
the conventional ionic models.
2.4.4 Saturation, the relation to images
A choice has to be made which model is the most appropriate one for a certain reservoir rock,
because there is not a universally valid interpretation model available for shaly sands. Electric
properties of clays play a very important role in this choice. The image analysis technique that
was developed in this study can help in the selection of the most appropriate saturation model
because the pore structure and the clay dispersion can be observed in detail in the images.
Presently it is not possible to derive quantitative parameters such as the cementation and
saturation exponents directly from images. It is however possible to obtain a qualitative idea
28
Chapter 2
of these parameters based on observations of pore structure and the pore connectivity. For
instance, the more tortuous the pore system the higher the cementation factor will be.
Moreover, grain shape factors can be estimated from images and thereby may support the
use of effective medium models.
2.5
Capillarity, the interplay between pore fluids and reservoir rock
All reservoir rock can be considered as a capillary system where, in the absence of other
forces, the distribution of the fluids is determined by the interaction of gravity, interfacial
tension and wettability. The fluid/rock interaction is dominated by adhesive forces and
referred to as wettability. The wettability describes the affinity between a liquid and a solid.
The fluid/fluid interaction is dominated by intermolecular van der Waals forces and is
known as interfacial tension. Gravity plays a role in both fluid/fluid interaction and in
fluid/solid interaction. In order to understand the distribution of fluids in a reservoir and the
position of oil/water and oil/gas contacts, knowledge of interfacial tension and wettability is
essential.
In the majority of oil reservoirs, oil and water are both present in the pore system. When
the adhesive affinity between water and the rock matrix is larger than the force between oil
and rock, the reservoir is considered water-wet. Conversely, in oil-wet rock the affinity
between oil and the rock matrix is larger than the affinity between water and rock. Most
sandstone reservoirs are assumed to be water-wet. Water wet rocks have a very thin layer
of water remaining at the pore wall after the oil migrated into the pores. The charging of a
reservoir comprises the invasion of (non-wetting) oil under the influence of gravity. The
pores are assumed to be initially filled with water that was forced out by oil because it
'floats' on the water. We will address capillarity using the same capillary bundle models as
were used for permeability prediction. For further details, we like to point the reader to
textbooks that are available on this topic. [Dullien, 1979; Scheidegger, 1972].
2.5.1 Capillary pressure curves
The wettability of a rock for a fluid can be expressed by the contact angle σ . In case of
contact angles smaller than 90° between the water surface and the solid, the solid is
considered water wet. For contact angles larger than 90° between the same substances
the rock is considered oil wet. This is depicted in Figure 2.7 where a drop of water is
placed on an oil-wet surface and a water-wet surface respectively:
Oil-wet, σ > 90°
Water-wet, σ < 90°
σ
Figure 2.8
σ
Contact angles for oil-wet water-wet surfaces
For a real reservoir rock, the charging process can be simulated by a so-called capillary
pressure curve. In a capillary pressure measurement, a non-wetting fluid is invading the
pore system that was originally occupied by a wetting fluid, whereby the latter is displaced
by the former. The capillary pressure of a non-wetting fluid invading into the pores is
described by Laplace's equation [Dullien, 1979]:
29
Chapter 2
Pc =
2γ
rpore
2.44
Or in case of a non-zero contact angle:
Pc =
2 γ cos σ fluid − solid
2.45
rpore
In which Pc is the capillary pressure, γ the interfacial tension and r the average radius of
the pores. The Laplace equation describes the relationship between the radius of the pore
neck and the pressure that is required for the fluid to enter a pore. In a reservoir rock, we
distinguish pore bodies and pore necks as discussed in section 2.3.1. According to the
Laplace equation, the diameter of the pore necks controls the invasion of a non-wetting
fluid.
2.5.2 Capillary pressure curves from images
The capillary bundle models that we use for permeability prediction consist of tubes. The
Laplace equation allows us to relate the capillary pressure to tube diameters. The tube
diameters in turn can be obtained from images, which will be discussed in chapter 7. A
method to derive a capillary pressure curve from image information will be presented and
discussed in chapter 8.
30
Chapter 3
Wireline logging and core analysis
3.
3.1
Introduction
In this chapter, we will briefly describe well logging and core analysis. Well logging is usually
carried out with instruments that are either suspended from a steel cable (wireline) or
embedded in the drill-string (logging while drilling, LWD). Only those logs that are of interest
for this study are discussed, for a more detailed treatment of well logging, the reader may
consult petrophysics textbooks. [Rider, 1996; Oil Field Review Publications, 1996]. We will
also discuss core analysis because the results of core analysis will serve as a reference for the
image analysis methods developed in this study: to predict petrophysical properties from small
rock samples.
3.2
Wireline logs
The most established way to gather information about hydrocarbon-bearing reservoirs over the
total length of the well is by means of wireline logging. The objective of logging is to obtain
information about the location, the amount and the producibility of hydrocarbons. Physical
properties such as density, resistivity, magnetic resonance, natural gamma radiation are recorded
as a function of depth. These physical properties have to be converted into petrophysical
properties, which require interpretation models. These models can vary from one reservoir to the
next because there are no known universal relationships between the physical parameters that
are measured in the well and petrophysical rock properties. In addition, the variation of the models
requires careful calibration with core data.
Wireline logging tools can be divided into passive and active tools. The passive tools measure
a natural occurring phenomenon such as the gamma radiation that is emitted by elements in
the rock or the electric potential caused by the difference in salinity of the mud in the well and
the formation waters. The former is known as the gamma ray (GR) log, the latter as the
Spontaneous Potential (SP) log. Both passive logs are used to distinguish reservoir layers
from non-reservoir layers.
Active tools are for example the density tool, the neutron tool, the resistivity tools and the more
modern NMR (Nuclear Magnetic Resonance) tools. In the active tools the response of the
formation to some form of excitation is measured. A separate family of active tools are the
borehole imaging tools that bridge the gap between core images and conventional low
resolution wireline logs. Core images have a resolution of 1 mm and conventional wireline logs
a resolution that varies from 0.15 m for the high-resolution density logs up to 0.6 m for
induction tools. Borehole imaging logs have a resolution of approximately 2 to 5 mm.
3.2.1 The gamma ray log, a shale indicator
The GR log detects gamma ray photons that are emitted by the radioactive decay of isotopes
in reservoir rock. These elements; Potassium, Thorium and Uranium, are present in feldspars
and micas that occur in many sandstone reservoirs. Feldspars decompose into clay minerals
and the radioactive elements are contained in the newly formed clay structures. Because clay
minerals like illite, smectite and chlorite are principle constituents of shales, they are detected
31
Chapter 3
by the GR log. The GR measurements are used to calculate the amount of shale as a function
of depth. The vertical resolution of GR tools is approximately 0.6 m with a depth of
investigation of 0.15 - 0.3 m depending on the density of the rock.
Sandstone reservoirs can contain significant amounts of feldspar grains. This type of rock is
known as arkosic sandstone and can have excellent reservoir quality, high porosity and high
permeability concurrent with high readings on the GR log. This leads to erroneously high shale
indication in genuine reservoir rock. Micaceous sandstones suffer from the same effect and
can show the same high GR value as the surrounding shales. Moreover, in some sandstones
the quartz grains can contain small amounts of Uranium that are detected by the GR log and
may also lead to classifying these sandstone layers as shale. Hence, in these cases the GR
log can give false information on the mineralogy and this may lead to of reservoir layers.
The techniques developed in this study can be invaluable to solve these log interpretation
dilemmas. Our technique uses images of small rock samples, and derives quantitatively the
fractions of minerals such as quartz, feldspar, mica and various clays using image analysis
techniques. The abundance of various minerals can then contribute to the interpretation of the
GR log, and the porosity logs that will be discussed below. Moreover, image analysis also
provides information on compaction and on grainsize distribution, which can be used to verify
and extend the determination of the environment of deposition, information usually derived
from the GR log.
3.2.2 The density log, a porosity indicator
The density tool emits medium energy gamma ray photons into the formation and counts the
photons returning to a detector after a number of scattering events. The number of photons
detected in the 100-300 keV range is dominated by Compton scattering and therefore
indicative for the electron density of the rock. The electron density is closely related to the bulk
density from which the porosity can be derived. The porosity determination is the main
application of the density log. The count-rate of low energy photons is dominated by the
photoelectric effect. The combination of the photoelectric effect and electron density is used to
determine the porosity and the lithology (sand, limestone, anhydrite, dolomite). The image
analysis technique that is developed in this study gives quantitative information about the
mineralogy. The results can therefore be used to confirm and even calibrate the interpretation
of the density log. The depth of investigation of the density tool is some 0.15 m while its
resolution is approximately 0.3 m.
3.2.3 The neutron log, porosity, shale and gas indicator.
The neutron tool emits high-energy neutrons into the formation from a radioactive source or an
electrically powered accelerator source called a minitron. The neutrons react primarily with
hydrogen atoms in the formation. The neutron log is a good indicator for porosity in clean
reservoirs when oil or water is filling the pores because water and oil have almost the same
hydrogen density. In shaly reservoirs or gas-bearing reservoirs, the neutron log is normally
combined with the density log to provide conclusive answers for porosity. Gas contains much
less hydrogen per unit mass leading to a low neutron response. Shales contain bounded and
interstitial water that contains hydrogen but do not represent effective porosity in which
hydrocarbons can be stored. The high reading in shales may be erroneously attributed to
32
Chapter 3
porosity. The depth of investigation of the neutron log is approximately 0.25 m with some
0.4 m depth resolution. Neutron tool readings are very sensitive to clays and vary from one
clay type to the other. Our image analysis techniques identify and quantify clays and can
therefore be used to enhance the neutron log interpretation.
3.2.4 The resistivity log, hydrocarbon saturation indicator
Resistivity logs send electric currents into the formation and measure the specific resistance of
the rock. Depending of the conductance of the drilling mud, the coupling with the formation is
galvanic or inductive. The resolution of resistivity tools depends on the depth of investigation.
Shallow reading tools can have resolutions down to 2.5 centimetres while the deeper reading
tools have resolutions of approximately one metre. The resolution becomes particularly important
in the case of thin bedded sand/shale sequences where separate determination of the sand
resistivity and the shale resistivity can often be difficult.
As was explained in chapter 2 the resistivity of the formation can be converted into hydrocarbon
saturation with a saturation model. However, a whole suite of saturation models is available for
different lithologies and different environments of deposition. A problem with all these models is
that they require local calibration as the relationship of saturation and resistivity is complicated
and non-linear. Again, this study can provide details about clay type, clay dispersion and pore
geometry and this information can narrow the selection of the appropriate saturation model.
3.2.5 The NMR log, porosity and permeability indicator
NMR (Nuclear Magnetic Resonance) logging is a newer family of logs that became popular in
the 90s. In NMR logging the nuclei of hydrogen atoms are excited by a magnetic pulse upon
which the relaxation time is measured. Because of this, NMR logs primarily "see" reservoir
fluids like oil and water, which can be distinguished by the differences in polarisation and
diffusion times of the hydrogen atoms that are present in these fluids.
The porosity can be derived by evaluating the amount of nuclei that is excited. Nuclei that
interact with the pore-wall have a faster relaxation than those residing in the centre of a pore,
because of this the signal that is measured can be related to surface area. The link to
permeability is then made via the Kozeny-Carman theory that relates specific surface area and
porosity to permeability. The NMR log and the Stonely wave acoustic log are the only logs that
can provide a continuous estimation of permeability as a function of depth. The depth of
investigation of the NMR logs varies on the tool type; for the CMR mandrel tool this is 0.15 m,
for the MRIL this is some 0.25 m.
Calibration of the NMR log is essential because, similarly to the neutron log, the hydrogen
atoms residing in the clays contribute to the measurements. The relaxation time of the
hydrogen atoms in clays is very short. The relaxation of clay bound water is incorporated in the
total NMR spectrum of the latest tools which can measure relaxation times down to 0.1
milliseconds. The NMR spectrum is divided into three regions representing free water (FFI,
free fluid index) in the centre of the pores, bound water (BVI, bound volume index) near the
pore wall and near the clay surface, and interstitial (CBW, clay bound water) water. Specifically
the definition of the regions for FFI and BVI is important because these two parameters
determine the prediction of permeability using an empirical relation. The image analysis
33
Chapter 3
techniques developed in this study provide next to clay typing and clay dispersion, also an
estimation of permeability. The permeability prediction is based same Kozeny-Carman theory
as the permeability prediction with NMR using Cotes equation and can therefore be used to
calibrate the permeability obtained with a NMR log.
3.3
Core analysis
The wireline logs discussed above are usually calibrated with analysis results from formation
samples, especially for appraisal wells when the reservoir rock is not yet properly characterised.
Reservoirs that have been discovered in the last decade tend to have much lower porosities and
permeabilities than the large accumulations that were discovered and developed earlier. As
discussed in the introduction, lower quality reservoirs require earlier calibration of the logs to
assess the economic viability of the discoveries.
The calibration takes place by direct measurements that are carried out on reservoir material such
as core samples. The core analysis is usually carried out on so-called core plugs, samples that
are taken from the bulk core. Both the cutting of core and the core analysis are expensive. In the
core laboratory, core plugs are drilled from the whole core that typically have a length of about 5
cm and a diameter of 2.5 cm. The petrophysical properties porosity, permeability, formation factor,
resistivity index curves and capillary pressure curves are then measured on these core plugs.
These measurements will be discussed in more detail in the remainder of this section because
they form the basis for the image analysis techniques that have been developed in this study.
3.3.1 Porosity
Porosity can be measured with a variety of methods. The buoyancy or imbibition method and the
Boyle’s law porosimetry are the most frequently used in the oil industry [Dullien, 1979; Dake,
1978; Ruth, 1991]. The buoyancy method is considered the most reliable approach and is also
known as the wet-and-dry-weight method. Porosity determined with the buoyancy method has an
overall accuracy of 0.1 to 0.2 porosity units (P.U.) for a 20% porosity sample. The accuracy is
fairly constant over the range for sandstone reservoirs. At very low porosities, say below 8%, the
accuracy decreases rapidly. Both methods measure the accessible, hence the effective porosity.
3.3.2 Permeability
Permeability is measured with a so-called Ruska permeameter and a modified version of Darcy's
law is used to relate fluid flow and pressure drop over a core sample. Because the Ruska
permeameter uses gas as a flowing medium through the sample, a conversion to liquid
permeability is required.
Darcy's law is different for gases, because the gas that passes through the sample is not only
dependent on the pressure drop over the sample but also on the absolute pressure itself. [Dullien,
1979; Juhasz, 1986; Ostermeier, 1993]. When this so-called mass flow is taken into account we
arrive at the modified Darcy equation:
k=
QLµ p
At ∆p p
3.1
34
Chapter 3
In which Q is the flow rate, L the length of the plug, At the total area perpendicular to the flow
direction, µ the viscosity and ∆p the pressure difference. The term p p corrects for the mass
flow in which p is the ambient pressure and p is the average pressure over the sample. The
conversion to liquid permeability requires another modification known as the Klinkenberg
correction. Contrary to liquids, gases flowing through a porous medium are barely affected by the
affinity between the gas and the pore wall. In the laminar flow regime the adhesive forces
between liquids and the pore wall give a fluid velocity near the pore wall of zero and a maximum
velocity in the centre of the pore. Klinkenberg [Dullien, 1979; Darcy, 1856] researched this and
proposed a correction that is now generally accepted:


k a = k 1 + b 
p

3.2
Where k a is the "apparent" measured gas permeability, k the fluid permeability and b the
Klinkenberg factor. Permeability measurements are normally carried out at various pressure
differences. If k a is plotted versus 1 p and the relation between k a and 1 p is extrapolated to
the point 1 p = 0 , we find k. At this point, there is no difference between the gas and liquid
permeability.
In the SI system permeability is expressed in m2, in the oil industry a more old fashioned unit, the
Darcy in cm2cP/atm = 0.987·10-12 m2 is used. To prevent very small numbers, the permeability is
normally expressed in milliDarcy (mD). The accuracy of the Ruska permeameter depends on the
actual permeability:
Permeability range [mD]
0.01 - 10
1 - 50
50 - 2000
2000 - 10000
Table 3.1
Accuracy
+/- 20%
+/- 10%
+/- 5%
> 20%
Accuracy of the Ruska Permeameter
3.3.3 Formation Resistivity Factor
The formation resistivity factor is measured on core plugs, which are saturated with water of a
known conductivity. According to Archie the formation resistivity factor (FRF) is the ratio of the
resistivity of the sample saturated with water, and the resistivity of the water itself. In the core
laboratory the FRF is measured on a sample set with varying porosity on plugs from the same
reservoir. The different values for FRF reflect the change in pore geometry caused by
cementation effects. The porosities and measured FRFs are plotted on a double logarithmic
35
Chapter 3
scale. For clean sands, this provides a straight line of which the slope m is known as the
lithological exponent or the cementation factor, according to equation 2.32.
This is depicted in Figure 3.1. Once the cementation factor for a specific lithology is known, the
resistivity of the formation water can be calculated from the resistivity and porosity logs taken in a
water-bearing leg of the reservoir. The resistivity of the formation water is needed for the
estimation of the hydrocarbon saturation.
First Archie equation
Formation Resistivity factor
100
10
1
0.1
1
Porosity
Figure 3.1
Cross-plot of FRF and porosity
reflecting the first Archie equation
3.3.4 Resistivity Index curve
A resistivity index measurement is carried out on partly saturated samples. The resistivity index is
the ratio between the resistivity Rt of the same partly water saturated sample and the resistivity R0
of the fully water saturated sample as given by equation 2.37. During this measurement, the
conductive water is replaced by the non-conductive oil in small volume steps. The resistivity index
is measured as a function of decreasing water saturation.
Similar reasoning as for the formation resistivity factor can be followed for partly oil-saturated
samples and is given by equation 2.38. Figure 3.2 depicts the relation of water saturation and
resistivity index.
Resistivity Index
Rt/Ro
100
10
1
0.1
1.0
Sw
36
Chapter 3
The saturation exponent n is required in the relation between the resistivity measured in the well
and the water saturation in the formation.
3.3.5 Capillary Pressure curve
The standard technique to obtain a capillary pressure curve is the air-mercury method. Other
techniques are the porous plate method and centrifuge method. In the air-mercury technique, a
capillary pressure curve is generated by forcing mercury into an evacuated sample. The vacuum
is considered the wetting phase and the mercury the non-wetting phase. During this invasion
process, the amount of mercury injected is measured as a function of the pressure. The
differences in contact angles and interfacial tensions between the air-mercury system and the oilwater system are corrected for by conversions. The conversion transforms the measurements
Figure 3.2
Watersaturation in the pores
versus the resistivity index reflecting the second
Archie equation
made under laboratory conditions to reservoir conditions such that a saturation-height function is
obtained from which the oil/water contact in the reservoir can be derived. Another application of
the capillary pressure curve is the estimation of the irreducible water saturation, which is used to
Capcurve
10000
Pressure
1000
100
10
1
0.0
0.2
0.4
0.6
0.8
1.0
Saturation
determine the maximum volume of hydrocarbons in the reservoir. The image analysis techniques
developed in this study can provide quantitative information on the pore dimensions. From the
parameters a "pseudo" capillary pressure curve can be determined. Figure 3.3 shows a measured
capillary pressure curve from a sandstone sample.
3.3.6 Qv, the shale indicator
The clay content of reservoir rock samples is measured with an indirect method that uses the
additional electric conductivity attributed to the clay particles. This excess conductivity was
discussed in detail in chapter 2 when we reviewed the Waxman-Smits model. The negative
charge on the surface of the clay particles attracts cations. The additional conductivity is caused
by the exchange of cations between the fluid and clay surface. The exchange capacity is
expressed as the CEC, the Cation Exchange Capacity, in milli-equivalents per unit weight of dry
rock. The Qv that is used in the Waxman-Smits model has the dimension of milli-equivalent per
unit pore volume. The CEC is commonly determined by using the titration method [Yuan, 1995].
The method is destructive; a small piece of a core plug (mostly a plug end of approximately 5 mm
thickness) is crushed and weighed. Then, a solution is used to exchange the cations from the clay
Figure 3.3
37 pressure curve
Capillary
Chapter 3
particles. The solution is titrated while the conductivity is monitored. From this, the CEC is
measured. Qv can also be measured with the membrane potential technique, which is nondestructive [Worthington, 1975]. This technique is more complicated to carry out but has the
advantage that the clays structures stay intact leading to a better determination of conductivity
contribution that is caused by the clays. The membrane potential technique is therefore more
reliable than the titration method.
38
Chapter 4
4.
The sample sets for rock images and plug measurements
The prediction of petrophysical parameters from images in this study was based on two
approaches. In the first approach we used colour images from thin sections. In the second
approach we used grey-level images that were obtained with electron microscopy in backscattered electron mode, (SEM and BSE images respectively). SEM/BSE images have distinct
advantages over images that are obtained with optical microscopy. These advantages will be
reviewed in chapter 6. Petrophysical parameters were measured on all rock samples prior to
the sample preparation that is required for image collection. The measured petrophysical
properties will serve as reference for the definition of the predictive models.
The first set of rock samples that was used for the prediction of petrophysical parameters
consisted of 195 samples. Thin sections were prepared from these samples and the procedure
to predict petrophysical parameters will be discussed in chapter 5 and is depicted in Figure
4.1.
Thin section
Reference
sample set
Thin
Sections
Optical
Microscopy
Image
Analysis
Predictive
model
Porosity : N = 195
Permeability : N = 149
Figure 4.1
Model definition from thin sections using optical microscopy
The second sample set consisted of 230 rock samples from which polished blocks were
prepared. The procedures required for prediction of petrophysical parameters is shown in
Figure 4.2 and will be discussed in chapters 6, 7 and 8 respectively.
SEM/BSE
Reference
sample set
Polished
blocks
Electron
Microscopy
Image
Analysis
Predictive
model
Porosity : N = 230
Permeability : N = 217
FRF : N = 164
Qv : N = 174
Figure 4.2
Model definition from polished blocks using electron
i
The third sample set encompassed 44 samples and was used to test the predictive capability
of the method developed with electron microscopy. Figure 4.3 shows the test procedure.
SEM/BSE
sample sets
Polished
blocks
Electron
Microscopy
Image
Analysis
Prediction
Comparison
Porosity : N = 36 + 8
Permeability : N = 34 + 8
Figure 4.3
Test of the predictive capability
38 of the SEM/BSE method
Chapter 4
4.1
Extreme values for porosity and permeability
Figures 4.4 and 4.5 show images of rock samples with the extreme values in porosity and
permeability that occur in the SEM/BSE reference data set. These are shown to indicate how
pore geometry controls these petrophysical properties. The images are coded as follows:
Colour
Constituent
Pore space
Clays
Quartz
Feldspars/Calcite
Heavy Minerals
Table 4.1
Figure 4.4
t
Colour code in the images
Processed images showing lowest and highest porosity, 2nd sample
Figure 4.5
Processed image of samples with the lowest and highest
permeability, 2nd sample set
39
Chapter 4
The left part of Figure 4.4 shows an image from a sandstone with a low porosity of 2.7% BV,
while the right part shows an image of a sample with a porosity of 33.9 % BV. In the left image
calcite cement almost entirely fills the pore space. The cementation in this sandstone leads to
the classification of a consolidated sandstone. Some pores of the fine-grained sandstone in
the right image are filled with clays. The high porosity and the point-to-point contacts of the
grains this sample is the reason that this sandstone is classified as unconsolidated.
The left image in Figure 4.5 is obtained from a sandstone sample of low permeability, 0.004
mD, and the right image from a sample with a permeability of 3000 mD. The left image in
Figure 4.5 shows a cemented, coarse grained sandstone, in which a large part of the primary
porosity is filled with quartz cement. It is also observed that clays fill a substantial number of
pores, thereby reducing both the porosity and the permeability. The coarse grained clean
sandstone in the right image of Figure 4.5 shows large pores that causes the high
permeability. In the following sections, we present and discuss the distributions of the
petrophysical properties of the three sample sets we used.
4.2
Reference sample set for thin section analysis
The sandstone sample set that was used for the prediction of petrophysical parameters from
thin sections originates from a fluvial environment of deposition, the fields are located in the
Middle East. Porosity was measured on 195 samples, governing a porosity range of 1 to 26 %
BV. Permeability was measured on 149 samples, which provided a range from 0.01 to 1770
mD.
Permeability could not be measured on all samples because some samples were too tight to
measure permeability. Other samples were so unconsolidated that they did not fit in the Ruska
permeameter. From all samples thin sections were prepared that were subjected to the image
processing and analysis that is discussed in chapter 5. Figure 4.6 shows the distributions of
porosity and permeability respectively.
Thin sections
Thin sections
60
40
50
Frequency
Frequency
30
40
30
20
20
10
10
0
0
0
5
10
15
20
25
Porosity [%BV]
Figure 4.6
t
4.3
30
35
-3
-2
-1
10
0
1
2
3
4
Log Permeability [mD]
Distributions of porosity and permeability of the thin section sample
Reference sample set for SEM/BSE analysis
The reference sample set for SEM/BSE (Scanning Electron Microscopy/BackScattered
Electron) analyses contains sandstone samples from a wide variety of environments of
deposition. Geographically the samples originate from fields in Western Africa, Northern
40
Chapter 4
Europe, the Middle East, the Far East and Australia. Lithologies range from coarse clean
sandstones to very fine-grained shaly sands and tightly cemented sands. The depositional
environments represented in this study are shown in Table 4.2.
Environments of deposition
Continental
Continental shelf
Continental margin
Turbiditic
Braided river
Desert environment
Glacial environment
Table 4.2
Sedimentological
systems from which the samples
in the reference sample set
The variety in depositional environments and sandstone lithologies of the reference sample set
is expected to make the models that were developed for prediction of petrophysical properties
more generally applicable. It is a great advantage when the range in petrophysical parameters
in the data set spans global variability and the main geological settings. This implies that the
predictions will be based on interpolation, which is much more reliable than extrapolation.
Consultation of sedimentologists and petrophysicists confirmed that the range in porosity and
permeability in the reference samples indeed encompasses the ranges that are normally
encountered in reservoirs. From this, we conclude that our reference sample set is considered
representative of the majority of sandstone reservoirs.
The porosity in the reference sample set ranges from 3 to 34 % BV. Permeability
encompasses the range from 0.01 to 3000 mD, almost 5 orders of magnitude. Figure 4.7
presents the distributions of porosity and permeability respectively.
Reference sample set
70
70
60
60
50
50
Frequency
Frequency
Reference sample set
40
30
20
10
40
30
20
10
0
0
5
10
15
20
25
30
0
35
-2
Porosity [%BV]
Figure 4.7
-1
10
0
1
2
3
4
Distributions of porosity and permeability in the reference sample set
The range of the Formation Resistivity Factor in the reference sample set is 5 to 263. Qv is the
parameter that indicates the presence of clay in a sample. In the reference sample set the Qv
varies from 0.005 meq/l for a very clean sandstone, to 1.23 meq/l for a very shaly sample that
contains a substantial amount of clay in the pores. Figure 4.8 shows the distributions of the
Formation Resistivity factor and the Qv.
41
Chapter 4
Validation sample sets
12
16
Frequency
Frequency
10
Alluvial
Deltaic
12
8
4
Alluvial
Deltaic
8
6
4
2
0
0
0
5
10
15
20
25
30
-1
35
0
1
2
3
4
5
10
Porosity [%BV]
Figure 4.8
Distributions of the formation resistivity factor and the Qv in the
reference sample set
4.4
Test sample sets for SEM/BSE analysis
The test sample sets that we used to validate the models for porosity and permeability
originate from an alluvial deposit in the Middle East and a deltaic deposit in West Africa. The
well in the Middle East penetrates two sand intervals and two shale intervals. The separation
between reservoir and non-reservoir was based on the geological core description. Core
analysis data of 36 samples was available only from the sand intervals because samples are
normally taken only from the reservoir layers. The porosity of the samples from the reservoir
layers varies between 13 and 35 % BV, the permeability range has values from 0.3 to 13000
mD. For this study some samples were taken from the two shale intervals and the
interpretation of the images obtained from the shale samples will be discussed in chapter 9.
The reservoir interval in the well in West Africa consists of homogeneous sand. Eight samples
were taken from this interval and the porosity ranges from 22 to 27 % BV. The permeability
12
10
Alluvial
Deltaic
12
Frequency
Frequency
16
8
4
Alluvial
Deltaic
8
6
4
2
0
0
0
5
10
15
20
25
30
-1
35
Porosity [%BV]
Figure 4.9
0
1
2
3
4
5
10
Distributions of porosity and permeability in the test sample sets
has values from 45 to 9000 mD. Figure 4.9 shows respectively the distributions of the porosity
and the permeability in both wells.
42
Chapter 5
Image analysis with optical microscopy on thin sections
5.
5.1
Introduction
A thin section is a very thin slice of rock that is mounted on a glass plate called a microscope
glass. The thin layer of rock is obtained by grinding and polishing the rock sample after mounting
it on the microscope glass. The rock layer is polished to a thickness of approximately 30 microns,
because at this thickness most minerals are transparent and can therefore be observed through
an optical microscope in transmission mode. Thin section analysis is a form of qualitative image
analysis and has been used by geologists and sedimentologists for more than 60 years [Durrel,
1949; Larsen, 1934]. Thin section analysis is carried out on all types of rock, reservoir and nonreservoir rock and its objective is, among others, the assessment of mineralogy.To produce a thin
section from reservoir rock, the rock sample has to be cleaned to remove the remnants of the
reservoir fluids and of the drilling mud. In addition, the grains have to be fixed to withstand the
forces exerted during the grinding and polishing process. Grain fixation is achieved by
impregnating the sample with a blue dyed epoxy. The epoxy fills the pore space in the rock, and
blue is a colour that does not normally occur in reservoir rock, therefore the blue areas can be
recognised as pore space.
Hitherto, the interpretations of thin section images from reservoir rock were mainly qualitative
because of their descriptive nature. However, quantification of the minerals and the pore space
from thin sections is possible by means of a technique called point counting. In point counting, a
square grid of points is placed over the microscopic image, upon which the number of grid points
falling in the pore space or in some mineral of the rock matrix are separately counted. This
process is repeated for some 300 points. Quantification is achieved by calculating the fraction of
points representing either pore space or a certain mineral. It is clear that point counting is a
tedious, labour intensive method to quantify porosity or mineral content from the thin sections.
Although labour intensive and thereby costly, point counting until recently was the only way for
extracting quantitative information from thin sections. Presently, the availability of colour image
processing with PC based software enables automation of the point counting process. In this
chapter, we shall describe such a method.
We will first discuss the equipment used for automated point counting of thin sections. Special
attention will be paid to the image collection process because automated quantification of image
properties requires calibration. The image processing will be discussed, followed by quantification
of the image features, which will be correlated with petrophysical parameters. Finally, the results
will be discussed and we will give recommendations for possible improvement of the predictions
of petrophysical parameters. The process that we followed to arrive at a predictive model is
depicted in Figure 5.1.
Thin section
Colour images
Extraction:
Pore space
Clays
Matrix
Quantification:
image features
Correlation:
Porosity
Permeability
Predictive
model
Plug data:
Porosity
Permeability
Figure 5.1
Procedure to define a model that enables to predict petrophysical parameters
from thin section images
43
Chapter 5
5.2
Equipment and image collection
Automation of point counting with PC based image analysis involves digitisation of microscopic
images. Therefore, a system was designed and built that consists of a microscope equipped with
a high-resolution digital camera and a PC with image processing and analysis software. The
microscope used was a Leitz petrographic microscope that was specifically designed for thin
section observation. The digital colour camera was a single CCD-chip Kontron Progres 3012 that
enabled the collection of full colour images up to 2300 x 3000 pixels with a 36 bit colour depth
resolution. It should be noted that a full colour image of 2300 x 3000 pixels in 36 bit depth would
give a file size of some 30 Mb for only one image. Because of the high storage requirements and
processing times involved, we limited the image size to 1024 x 1024 pixels with a 24 bit colour
depth resulting in an image size of 3 Mb. The reduction of the image size was achieved by subsampling. The image processing and analysis package we used was the KS400 package from
Zeiss, which was equipped with the full colour option.
5.2.1 Image collection
The quantification of the features that were extracted from images requires that these images
have to be calibrated with respect to colour and intensity. We need to collect images under
constant illumination conditions to enable consistent and possibly automatic extraction of the
pore, clay and matrix areas. Moreover, the spectral properties of the blue dye; the brightness,
colour (hue) and colour intensity, should also be constant. This demand requires that the
thickness of rock samples on the microscope glass does deviate not more than a few microns
from the standard of 30 microns. Moreover, the concentration of the blue dye in the epoxy prior to
fabricating the thin sections should also be constant.
The requirement of constant illumination conditions was easily met because the light source in the
microscope proved to be very stable. Intensity measurements showed variations of less than 1%
over several hours of observation. Inspection of the thin sections that were available for this work
showed, however, that the blue colour intensity of the epoxy in the pores varied significantly.
Furthermore, the thickness of the rock samples that constituted the thin sections also varied
substantially. The variation in both properties was too large to enable fully automatic extraction of
pore, clay and matrix areas. Therefore, we concluded that the segmentation of pores, clays and
matrix for this sample set unfortunately had to be carried out manually on all thin sections.
5.2.2 Selection of the appropriate magnification
In conventional point counting, thin sections of sandstone samples are normally inspected under a
petrographic microscope using varying magnifications. The magnification is generally adapted to
the dominant grain size to obtain a representative image for point counting. Representative in this
context means that an image should contain a sufficient number of pores and grains. Depending
on the grain size, the magnification can vary between 20 and 100 times. The value of 20 times is
normally used for coarse-grained sand and 100 times for very fine-grained sand. We followed a
different track in this study and selected a single magnification of 20 times for all the samples in
the data set. The low magnification ensures that a representative image is obtained, even for the
coarsest sandstone. Another justification of this single magnification is that the scale for all
images is the same, which greatly simplifies the quantification of the image features, and a single
software routine could therefore be used for all the measurements.
44
Chapter 5
The magnification of 20 X was achieved with an objective lens of 2 times and an ocular of 10
times. The numerical aperture (NA) of the objective lens was 0.04. The Nyquist criterion demands
that:
fs > 2 fc =
4 NA
λ
5.1
With fc the Nyquist frequency, fs the sampling frequency and λ the wavelength of light.
Using a wavelength of 0.5 micron, the sampling frequency to meet the Nyquist criterion should be:
f s > 8 NA ≅
1
3
samples / micron
5.2
This leads to a minimum sampling distance of approximately 3 micron. At a magnification of 20 X,
the field of view is 4 x 4 mm. An image of 1024 x 1024 pixels then corresponds to a sampling
distance of 4 micron. From this, we concluded that the images were sampled with approximately
the Nyquist frequency.
Figure 5.2
Thin section images from a coarse-grained and a fine-grained sandstone.
Note the differences in colour of the rock matrix
From each thin section 4 to 8 images were collected and stored for later processing and analysis.
Four images were collected from the fine-grained samples and eight images from the coarser
grained samples because these images contained less grains and pores. The left image in Figure
5.2 was obtained from a coarse-grained sandstone and the right image presents a fine-grained
sample, both images were collected at a magnification of 20 times.
5.3
Extraction and quantification of pore, clay and matrix fractions
The processing of the thin section images encompassed colour segmentation and noise filtering.
The images were processed in the so called RGB (red, green and blue) domain in which a full
colour image is composed of three grey-value images representing the primary colours red, green
and blue. A single grey-value image can be defined as a two dimensional function of grey values:
45
Chapter 5
I = f ( x, y ) : 0 ≤ f ( x, y ) ≤ 255
5.3
In which x , y is the location in the image. Consequently, a full colour image is then represented
by three grey-value images:
I c = f c ( x, y ) : 0 ≤ f c ( x, y ) ≤ 255 with c = r , g , b
5.4
Figure 5.3
Thin section image in which the dark
brown clay rims can be clearly observed
The pores, the clays and the rock matrix can be identified in the images by their colour. In Figure
5.3 the blue dyed epoxy in the pores can easily be distinguished from the dark brown clay rims
and the pale yellow and whitish quartz grains.
Red
Matrix
Green
Clays
Pore space
Blue
Figure 5.4
Locations of pore space, clays and
rock matrix in RGB space
Multivariate segmentation [Ehlers, 1987] was used to extract the pore-, clay- and rock matrix
areas respectively. In segmentation, a range of grey-values is extracted by thresholding. This
46
Chapter 5
operation is a transformation from a grey-value image to a binary image and is know as the
threshold operator T . The operator T is defined as follows: all pixels whose grey-values fall in
the range between low and high are assigned the value 1 and all pixels outside that range are
assigned the value 0. This is mathematically expressed as:
(
1 → if t c − low ≤ f c ( x , y ) ≤ t c − high
T t low , t high f c ( x , y ) = 
0 → otherwise
(
)
)

with c = r , g , b
5.5
In the case of a colour image, using the RGB model, the segmentation takes place in the three
images representing red, green and blue. It is therefore called multivariate segmentation. Figure
5.4 schematically depicts the locations of the pores, the clays and the rock matrix in RGB space.
Each axis in RGB space has a range from 0 to 255.
After the segmentation, the pores, clays and rock matrix are each represented by a single binary
image having pixel values of 0 or 255. These single binary images were subsequently used to
create a new grey-value image that we termed a DEF image, abbreviated from Display of
Elementary Fractions. A DEF image contains three grey-values only, representing the pores
(value 3), the clays (value 6) and the rock matrix (value 10) respectively. Prior to the creation of
the DEF image, morphological filtering of the individual binary images was applied to reduce noise
that inevitably occurs in the individual segmentation steps. A combination of opening and closing
operations with a 5 x 5 hexagon operator on the binary image was applied to remove noise. The
morphological filtering for the pores is shown in Figure 5.5; the opening removes the noise outside
the pores (white pixels) while the closing removes the noise inside the pores (dark pixels). Details
on the applied morphological filtering with a hexagon operator can be found in Appendix B.
Multivariate
segmentation
Figure 5.5
Opening
with
hexagon
Closing
with
hexagon
Morphological filtering on the binary representation of the pore space
In the creation of the DEF image, overlap can occur because the pores, clays and rock matrix are
segmented independently from the RGB image. This overlap was overcome by using modified
addition. First the pores were extracted, morphologically filtered and the resulting binary image
was added into the DEF image. All pixels that are 0 in the binary image will be 0 in the DEF
image, the pixels with value 255 are converted into value 3 in the DEF image:
0 + 0 = 0
DEF ( x , y ) = DEF ( x , y ) + f pores ( x , y ) with pixel values 

0 + 3 = 3
5.6
Following, the matrix was extracted and added into the DEF image. The possible overlap between
rock matrix and pores was removed by modified addition:
47
Chapter 5
DEF ( x , y ) = DEF ( x , y ) + f pores ( x , y )
0 + 0 = 0 


with pixel values 3 + 10 = 10 
0 + 10 = 10


5.7
Finally, the clays were segmented and added into the DEF image:
DEF ( x , y ) = DEF ( x , y ) + f pores ( x , y )
0 + 0 = 0 
3 + 6 = 6 


with pixel values 

10 + 6 = 6
0 + 6 = 6 
5.8
Visual inspection showed that this approach resulted in images that best approached the original
full colour images. Figure 5.6 presents the flow chart in which the whole process of segmentation,
morphological noise filtering and modified addition is depicted for the pores, the clays and the
rock matrix.
Add
Segmented pores
Opening and
closing
Add
Colour image
Segmented matrix
Opening and
closing
Add
Segmented clays
Figure 5.6
Opening and
closing
DEF image
The steps in the procedure to transform a thin section colour image into a DEF image
The quantification of the areas of the pores, the clays and the matrix was the next step. For each
of these, the image features area percentage and total perimeter were measured. The area
percentage pores is the ratio of pore pixels and the total number of pixels in one image:
48
Chapter 5
N
A% Pores =
∑ Pixel
1
Pores
5.9
N
∑ Pixel
1
The total perimeter of the pores is determined by counting the pixels at the edge of the pores in
one image and multiplying that with the scale of one pixel. In this process, the multiplication factor
f accounts for the position of one edge pixel to the next neighbouring edge pixels.
N
PPores = ∑ ( EdgePixel Pores × Scale × f )
5.10
1
Equivalent approaches were used for clays and rock matrix. For each pore, we also measured the
maximum diameter, which is defined as the maximum of the Feret diameters projected over a
range of 0 to 360° in steps of 5°:
[
Dmax = max Feret θ
]
5.11
θ = 5 ,...,360
Feret
q
Figure 5.7
Definition of the
Feret diameter
The results of the measurements were averaged over the number of images taken from each thin
section. The area percentage was needed to estimate porosity and the total pore perimeter was
used in the prediction of permeability with the models we defined in chapter 2.
5.4
Petrophysical parameters
5.4.1 Porosity
The area percentage of the pores that was measured in the thin section images should be related
to the measured plug porosity if the homogeneity and isotropy requirements discussed in chapter
2 are met. Since our images contain 1024 x 1042 pixels, more than 1 million points in total, the
representativity and the statistical validity of this method should be substantially better than that of
traditional point counting with some 300 points. We first calculated the porosities of the 195
samples in the thin section sample set, based on the area percentage of the pores. The results
are shown in Figure 5.8 in which we plot the area percentage of the pores against the measured
plug porosity. This graphical representation is often used in petrophysics and is known as a crossplot. Figure 5.8 shows that there is a significant scatter over the total range of porosities.
49
Chapter 5
Porosity
30
Plug Porosity
25
20
15
10
2
5
R = 0.66
0
0
5
10
15
20
25
30
Area% Pores
Figure 5.8
Area percentage pores from thin sections versus plug porosity
Statistical analysis of the relation between plug porosity and area percentage pores revealed that
the correlation coefficient is 0.81 leading to an explained variance (R2) of 66%. The explained
variance measures the reduction in the total variation of the plug porosity due to the variation of
the area percentage pores. The standard error of the estimate was 3.15 porosity unit (P.U. or
%BV, Bulk Volume) leading to an accuracy range (2s) of ± 6.3 P.U., which then comprises 95%
of all cases. In this analysis, it was explicitly assumed that the residuals are normally distributed.
Linear regression was applied to this data to obtain equation 5.12, which can be used to predict
porosity from the image feature area percentage pores:
Φ Area _ pores = 0.7 A% pores + 81
.
5.12
The statistical analysis also showed an underestimation of the porosity by 8.1 P.U., a zero offset
in the regression. Apparently, when we use area percentage pores, we 'miss' porosity that is
present in the samples according to the plug measurements. It is well known that a significant
portion of the porosity that is measured with conventional methods resides in the clays. The
porosity determination from thin sections does not distinguish the porosity in the clays, while many
samples of our data set do contain substantial amounts of clay. Given the limited resolution of the
thin section images, one can argue that this porosity in the clays is not included when we only use
the area fraction of pores. Consequently, we will be underestimating the porosity. Including the
area percentage clays in a regression analysis indeed leads to a reduction of the offset from 8.1
P.U. to 5.8 P.U. The standard error reduces slightly to 2.9 P.U. and the explained variance
increases to 0.69. Figure 5.9 presents the cross-plot of the porosity from thin sections based on
area% pores plus area% clays and the plug porosity. From comparing Figures 5.8 and 5.9, it
shows that specifically the predicted value of the very low porosities with relatively high amounts
of clay have been increased by including clay area in the regression equation.
Further statistical analysis showed that a relationship exists between the plug porosity and the
total perimeter in microns of the rock matrix. At this point, we do not have a physical explanation
for this relationship. We included this parameter in the regression analysis, which led to a
50
Chapter 5
Porosity
30
Plug Porosity
25
20
15
10
2
R = 0.69
5
0
0
5
10
15
20
25
30
Area% Pores + Area% Clays
Figure 5.9
Thin section porosity including microporosity in the clays versus plug porosity
Porosity
30
Plug Porosity
25
20
15
10
2
R = 0.79
5
0
0
5
10
15
20
25
30
Thin Section Porosity
Figure 5.10
Cross-plot of thin section porosity according to
equation 5.13 and the plug porosity
reduction of the standard error to 2.5 P.U., and an increase in explained variance to 0.79. The
cross-plot of the porosity from thin sections (using A% clay, A% pores and total pore perimeter)
and the plug porosity is shown in Figure 5.10. The predictive model is represented by equation
5.13.
Φ thin sec = 0.7 A% pores + 0.06 A%clays + 0.4 Pmatrix + 3.9
5.13
5.4.2 Permeability
For the prediction of permeability, we used a Kozeny-Carman type model as formulated in chapter
2 and given by equation 5.14:
51
Chapter 5
k Kozeny = C Kozeny
Φ3
Sv 2
5.14
These models use the porosity and the surface to volume ratio of the pores, the latter is also
known as the specific surface area Sv . The surface to volume ratio can be approximated with the
ratio of the pore area and the total pore perimeter, Sva ,as shown in equation 5.12.
S va =
Ppores
5.15
A pores
Figure 5.11 shows the cross-plot of the permeability predicted from thin sections using the
Kozeny-Carman model given by equation 5.14, and the permeability that was measured on core
plugs. The constant C was taken equal to 5 as suggested by Carman [1937]. The predicted
permeability deviates sometimes as much as a factor 100 from the measured plug permeability.
Figure 5.11 also shows a significant scatter around the trend line. Statistical analysis revealed an
explained variance (R2) value of 0.50, which means that only 50% of the variance in the data set
is explained by the relation between the two variables. The standard error was a factor 10.
Permeability
10000
Plug Permeability
1000
100
10
1
2
R = 0.50
0.1
0.01
0.01
0.1
1
10
100
1000
10000
Thin Section Permeability, eqn. 5.11
Figure 5.11
Cross-plot of permeability predicted from
thin section by the Kozeny-Carman models and the
permeability measured on core plugs
Further statistical analysis showed that the total matrix perimeter and the maximum pore diameter
also correlate with permeability. Therefore, we included these parameters in the regression
analysis, which led to a substantially higher R2 value of 0.69, and a reduction in standard error to
a factor 6. We applied stepwise regression with an F-test to investigate whether addition of the
two variables total matrix perimeter and the maximum pore diameter contributed significantly to
the results. The α value in the test was taken 0.05. The calculated Fcritical for inclusion of the two
variables was 22.71 and 13.45 respectively. The values for Fobserved in the regression analysis
were 92.20 and 78.49. The values for Fobserved were larger than the values for Fcritical, we therefore
52
Chapter 5
concluded that the contribution of the total matrix perimeter and the maximum pore diameter was
significant.
Figure 5.12 presents the cross-plot of the thin section permeability obtained by including the total
matrix perimeter and the maximum pore diameter, and the plug permeability.
Permeability
10000
Plug Permeability
1000
100
10
2
1
R = 0.69
0.1
0.01
0.01
0.1
1
10
100
1000
10000
Thin Section Permeability, eqn. 5.13
Figure 5.12
Thin section permeability according to
equation 5.16 versus plug permeability
Equation 5.16 gives the formula of the predictive model. The specific surface of the pores has a
fitted coefficient of -2.54, which is not far from -2 used in the Kozeny-Carman model. Similarly, the
coefficient of the porosity is 3.14, again close to the value of 3 confirming the Kozeny-Carman
model.
k thin sec tion = A% Pores 3.14 × S v −2 .54 × PMatrix 3.24 × D Max _ Pores 1.95 × 10 −9 .4
5.16
A Student's t-test was used to assess the significance for each individual variable and their values
are presented in Table 5.1. A Student's t distribution table provided a value of 1.65 for tcritical given
the 143 degrees of freedom and an α of 0.05. All t-Stat values in Table 5.1 are higher than the
value for tcritical and we concluded that all variables contributed significantly to the prediction of
permeability using equation 5.16.
Parameter
Intercept
Area percentage pores
Svratio pores
Perimeter matrix
Maximum pore diameter
Table 5.1
t-Stat
6.15
5.65
3.67
4.83
5.82
t-statistics of the parameters used in equation 5.13 for permeability
53
Chapter 5
5.4.3 Porosity-permeability relationship
A classic method of estimating permeability in the absence of permeability measurements is the
so-called k-Phi method. Specifically for unconsolidated sandstones it is often difficult to measure
permeability because of irregularly shaped core plugs, which do not fit in the Ruska permeameter.
However, porosity can reliably be measured on irregular samples using the buoyancy method and
the permeability is then estimated using the k-Phi method. The k-Phi method is based on an
assumed logarithmic dependence of permeability on porosity. Since both plug porosity and plug
permeability are available for the 147 samples in the thin section data set, we applied this method
to investigate the relationship between porosity and 10log permeability. By using the logarithm of
the permeability, we imply a very simple power relationship between porosity and permeability
that resembles the Kozeny-Carman approach. The cross-plot of this relationship is presented in
Figure 5.13. Statistical analysis showed that the explained variance (R2) is 0.8.
Porosity/Permeability relationship
10000
Plug Permeability
1000
100
10
1
2
R = 0.80
0.1
0.01
0
5
10
15
20
25
30
Plug Porosity
Figure 5.13
Relationship of plug porosity and plug
permeability of the thin section sample set
5.5
Discussion
5.5.1 Porosity, the Holmes effect
In the prediction of porosity, the best results were obtained using a thin section porosity that was
based on a regression analysis which included the pore area%, the clay area% and the total
matrix perimeter. This regression had an explained variance of 0.79 and a standard error of ± 5
P.U. but also with a zero offset of some 4 P.U, leading to an underestimation. The deficit in
porosity can be explained by investigating the process of image collection. Figure 5.14 presents a
thin section image of a clean sandstone. The edges of the grains have a blurred appearance,
which is not caused by out of focus setting of the microscope lenses. The range over which the
image remains sharp in microscopy is known as the depth of focus. The depth of focus is given
by:
Dof ≅
λ
2 NA 2
5.17
54
Chapter 5
With a wavelength of λ = 0.5 micron and the 2 X lens with a numerical aperture of 0.04, the Dof
is approximately 156 micron. This means that the full 30 micron thickness of the rock will be
always in focus when the microscope is properly adjusted and that the observed blurring was not
caused by the optics of the microscope.
Figure 5.14
Thin section image from a clean sandstone
The blurring is caused by the so-called Holmes effect by which a thin section image does not
present an infinitely thin section through the rock. The latter was explicitly assumed in the above
analysis for the prediction of porosity and permeability from images. The image that is observed
from a thin section is the cumulative projection through about 30 microns of impregnated rock
material.
The Holmes effect is explained by using Figure 5.15, in which a part of a thin section is depicted.
The quartz grains are presented in yellow, and the visible pore space in blue. The pore space that
cannot be clearly distinguished is presented in hashed blue and is termed hidden pore space.
When grains are totally opaque, the hidden pore space cannot be observed at all, and this leads
to a significant underestimation of porosity. In the case of fully transparent grains, all of the hidden
pore space can be observed in principle, which then would lead to an over-estimation of porosity.
In reality, grains are neither totally opaque nor fully transparent. Moreover, the colour intensity of
the blue that is observed decreases with the thickness of the epoxy. The blue colour becomes
increasingly faint and makes it more difficult to distinguish pores from grains. This effect is most
significant at grain edges, as shown in Figure 5.14. The Holmes effect is further amplified by the
55
Chapter 5
presence of clays at the grain edges. Clays generally appear as dark rims around grains making
the edges of grains essentially opaque.
Hidden pore space
Grain
Visible
Pore space
30 microns
Grain
Hidden pore space
Illumination
Figure 5.15
The Holmes effect, underestimation of pore space
The origin of the Holmes effect is further explained by Figure 5.16, in which a section with a
thickness of 30 microns and a length of 500 microns is shown. This section was taken from a
shaly sandstone and the image was obtained with an electron microscope operating at submicron resolution with a magnification of 200 times. According to the standard colour coding in
this study, quartz is presented in yellow, clays in green and pore space in blue.
500 microns
30 microns
Figure 5.16
Record of the rock slice on a thin section with a electron microscope
It is easy to see that an important part of the blue dyed epoxy can not be seen in transmission
observation if the grains are fully transparent and the clays are opaque. In addition, the small
pores that contain very little epoxy cannot be seen either due to the faint blue colour. Based on
the foregoing, we conclude that the Holmes effect is the cause that not all pores can be observed
in the analysis of thin section images, which results in an underestimation of porosity. As can be
observed from the cross-plot presented in Figure 5.10, this is an offset error, which can be
corrected. The regression equation 5.10 corrects for this underestimation by the intercept value. It
should be noted that this correction is not universal, because the samples originate from the same
environment of deposition . For another sample from a different environment of deposition, a new
factor has to for correction has to be determined.
5.5.2 Permeability prediction
The limited accuracy of the porosity prediction from thin sections also has a detrimental effect on
the prediction of permeability when we used Kozeny-Carman type models, because porosity is
rated to the power 3. In these models, the porosity plays a major role; hence poor porosity
prediction will lead to poor permeability prediction. Even if four image features were included
(area percentage pores, specific surface area of the pores, feret-diameter of pores and total
perimeter of the matrix), and using a statistical regression approach, the best regression has still a
56
Chapter 5
rather poor explained variance of 0.69. This value is considered inadequate for operational use
when permeability has to be predicted from thin sections in a field study. Furthermore, when the
classic k-Phi method was used for the same data set, a substantially better explained variance of
0.80 was obtained, indicating a good relation between porosity and permeability. This relationship
was explained from the fact that all sandstone samples used in this thin section study originated
from the same environment of deposition, essentially confirming the validity of the k-Phi method if
a good data set is available. However, the k-Phi method does not always works so well. In
chapter 8.3.1, we will discuss the prediction of permeability from images obtained with Scanning
Electron Microscopy for a data set in which the relationship between plug porosity and plug
permeability is rather poor. Statistical analysis of the relation between porosity and permeability
for that data set gave an explained variance of 0.46 as shown Figure 8.10. This lack of correlation
was caused by the fact that the samples originate from different environments of deposition.
5.5.3 Colour segmentation, definition of pores, clays and rock matrix
In the applied segmentation procedure, the pores, clays and rock matrix are recognised by their
colour. When the grains in the rock matrix can be clearly distinguished from clays and pores, the
segmentation is unique. However, not all grains in a sandstone sample have the same pale yellow
colour, nor do all clays appear dark brown. Overlap in colour can occur as can be observed in
Figure 5.17, where the kaolinite clays in the pores that can be recognised by their shape, have the
same yellow colour as the majority of the quartz grains. Conversely, some areas on the feldspar
grains have the same brown colour as the clay rims occurring at the grain edges. This effect
causes non-uniqueness in the segmentation and thereby influences the quantification of pores,
clays and rock matrix areas.
Figure 5.17
Overlap of clays and quartz grains hampering
accurate segmentation, note that the kaolinite clays in the pores
have the same yellow colour as the quartz grains
5.5.4 Conclusions
Table 5.2 presents the explained variance for the methods used to predict porosity and
permeability from thin sections. The prediction of porosity from thin sections using image analysis
did not provide very reliable results. Similarly, the use of a Kozeny-Carman type model for the
57
Chapter 5
prediction of permeability also gave a rather poor result. Both methods show explained variance
values of less than 70%, which are considered too poor to warrant accurate predictions in an
operational environment. In addition, relations based on regression analysis between image
features and plug measured porosities and permeabilities did not improve the explained variance
values significantly.
Explained
variance
Deterministic
Regression
k-Phi
Porosity
R2
Permeability
R2
0.66
0.79
-
0.50
0.69
0.80
Table 5.2
Explained variance for the
various methods to predict porosity and
permeability
We observed that in thin section images the projection through about 30 microns of impregnated
rock blurs the images and thereby hampered accurate assessment of the porosity that is residing
in small wedges or in clays. Moreover, the Holmes effect makes it difficult to observe all pores in a
thin section. This leads to a value for the area percentage of pores that is too low and
consequently to an underestimation of porosity. Both effects combined limit the use of thin
sections and optical microscopy for the accurate prediction of petrophysical parameters. In
addition, the limited spatial resolution of several microns makes optical microscopy unsuitable for
detailed clay studies because the typical dimensions of clay particles are in general smaller than a
few microns. This effect is amplified by the uncertainty in the segmentation of pores, clays and
matrix, where it may be difficult to distinguish clays from rock matrix based solely on colour.
It is remarked that our experiments were carried out on conventional thin section that we had
available at that time. Presently, ultrathin section can be obtained with substantially better
polishing quality. These ultrathin sections will enable a better definition of the pore space, clays
and matrix and may lead to a reduction of the Holmes effect. Consequently, the accuracy of the
predictions of porosity and permeability may be improved.
From the above analysis, we concluded that the representation of the pores, clays and rock matrix
in thin section images we used is not sufficiently accurate. We argue that the predictions of
porosity and permeability could be made more accurate if we would have a better representation
of pores clays, rock matrix in images. This better representation can be found in images obtained
with electron microscopy of which the resolution is up to 100 times better than that of optical
microscopy. In the following chapters we will discuss and develop a method for the prediction of
porosity and permeability from electron microscope images.
58
Chapter 6
Scanning Electron Microscopy on reservoir rock samples
6.
Scanning Electron Microscopy applied to reservoir rock
samples
6.1
Introduction
Electron microscopy originated in the late 1930s. There are two principle methods to obtain
images; transmission electron microscopy and scanning electron microscopy. In transmission
electron microscopy (TEM), a high-energy electron beam penetrates an ultra-thin sample and the
image is formed from the projection of the transmitted electrons. Scanning electron microscopy
(SEM) uses a beam of medium-energy electrons to scan the surface of a sample. Among other
effects, electrons are produced from the interaction between the beam electrons and the matter of
the sample, and these electrons are used to construct images. In this study, we use SEM in the
backscatter electron (BSE) mode to collect images.
We start with discussing the use of SEM for the observation of reservoir rock from the perspective
of resolution. Next, we will present the basics of scanning electron microscopy by applying the
analogy to optical microscopy. Then we will describe secondary electron (SE) imaging and
explain how this type of imaging is used on reservoir rock samples and what type of information
can be obtained. We will continue with a discussion on backscattered electron (BSE) imaging and
explain why this type of imaging is particularly suited for the derivation of petrophysical
parameters from reservoir rock samples. Special attention will be paid to the preparation of
samples intended for BSE imaging. We will also discuss the properties of BSE images in detail
and will define the conditions for image collection that are required for the image analysis
technique that will be developed in this study.
6.2
Scanning electron microscopy
The limited resolution of optical microscopy does not allow detailed inspection of reservoir rock
and particularly clay structures. Clay structures with dimensions below 0.5 micron cannot be
resolved because the physical resolution of optical microscopy is limited to the wavelength of
visible light, approximately 0.5 micron. Moreover, the morphology of clay structures cannot be
observed properly with optical microscopy as discussed in chapter 5. Specifically, marginal
reservoirs that were discovered in the past decades demand more detailed microscopy because
these reservoirs often contain clays. The resolution that is required for inspection of clay
structures can be provided by scanning electron microscopy, which is able to reveal details down
to several nano-metres.
Initially, SEM was used only for qualitative descriptions of reservoir rock samples and was mainly
applied by geologists and sedimentologists. SEM is often combined with the spectral analysis of
X-rays (Energy Dispersive X-ray analysis (EDX)) that are induced by the electron beam. Upon
electron excitation, elements emit X-rays with specific energies that are used to identify, classify
and quantify minerals that contain these elements. The combination of SEM/EDX is very powerful
because it not only enables one to view the pore structure of reservoir rock in detail, but also
provides information on the rock's mineralogy at a resolution of several microns.
59
Chapter 6
6.2.1 Working principle of the SEM
Although at first sight vastly different, the functionality and limits of both optical and electron
microscopy are essentially based on the same physical principles of optics and wave mechanics.
The main differences between the two types of microscopy are the media used for image
formation; they differ in the wavelength of the electromagnetic radiation and the type of lenses.
Table 6.1 presents these major differences between optical microscopy and electron microscopy.
Property
Optical Microscopy
Scanning Electron Microscopy
Medium
Light
Accelerated electrons
Wavelength
700 (Visible) – 300 (UV) nm
0.7 (20kV) – 0.3 (100 kV) nm
Environment
Atmospheric
Vacuum
Lens
Glass
Magnetic, Electrostatic
Aperture angle
70
35
Resolution
500 – 200 nm
0.2 – 0.3 nm
Magnification
10X – 2,000X
30X – 1,000,000X
Focusing
Mechanical
Electrical, magnetic, continuous
Contrast
Absorption, reflection, phase
Scattering, absorption, diffraction
Table 6.1
Comparison optical microscopy and electron microscopy
A scanning electron microscope consists of the same basic components as an optical
microscope: an illumination system, a set of lenses and an image projection device. The whole
assembly is kept under high vacuum to prevent dispersion of the electrons in the beam. Figure
Accelerating
voltage
Electron gun
Electron beam
Focusing lens
Scanning coils
Detector
Sample
Figure 6.1
Components of a Scanning Electron Microscope
60
Chapter 6
6.1 depicts the components of a scanning electron microscope, and below we will briefly discuss
these components.
The illumination system in a SEM consists of an electron gun that generates electrons either by
thermal excitation of by field emission. The SEM used in this study was equipped with a field
emission electron gun because this has the advantage of high electron intensity and high stability.
The high electron intensity brings about a very good signal-to-noise ratio in the images, which is
essential for estimation of petrophysical parameters from these images. In order to generate an
electron beam required for scanning the surface of a sample, the emitted electrons have to be
accelerated and focused. Acceleration takes place directly below the electron gun by a voltage
gradient that can range from 1 to 25 kV. The wavelength is determined by the kinetic energy of
the electrons according to the formula of De Broglie, the wavelength for an electron in free space
is:
λ=
h
me v e
6.1
In which h is Planks’ constant, me the electron mass and ve its speed. The analogy between
electron mechanics and optics indicates lens properties for a rotationally symmetric electric or
magnetic field. This analogy enables to construct instruments based on electron optics with far
better resolution than ‘standard’ optics. If an electron with charge e is accelerated by a voltage
U , then the kinetic energy is:
eU =
1
2
me v e
6.2
Combining equations 6.1 and 6.2 leads to an expression for the wavelength:
λ=
h
2 ⋅ me eU
6.3
Equation 6.3 shows that there is a relationship between the kinetic energy of accelerated
electrons and their wavelength. For example, an accelerated electron with a kinetic energy of 100
keV has a wavelength of 0.0037 nm (0.037 Å). This can be compared with the wavelength of
visible light of 0.5 micron that determines the physical limit for optical microscopy. Images with
very high-resolution can be generated, if we combine this wavelength/energy dependence with
the possibility of manipulating electrons and a system to detect electrons.
The column that contains the lenses, apertures, the electron gun and the sample is kept at high
vacuum to prevent dispersion of the electron beam. The magnetic fields, generated by toroidal
coils, are used to focus the electron beam, as depicted schematically in Figure 6.1. These coils
function similar to the lenses in optical microscopy. The required vacuum is maintained by a
cascade of vacuum pumps. The low vacuum pre-vac rotary pump is followed either by a turbomolecular pump or by an oil-diffusion pump.
61
Chapter 6
6.2.2 Interaction of the electron beam and the sample
When the electrons in the beam hit the sample surface, interactions between the atoms in the
sample and the beam electrons take place. This process can roughly be divided into elastic
scattering and inelastic scattering. In the case of elastic scattering, the beam electrons are
scattered by the atomic structure in the sample whereby the direction of the beam electrons is
altered over a wide range and only very small energy losses occur. In the case of inelastic
scattering, the beam electrons are scattered inside the sample at low angles, losing energy. In
essence, energy from the beam electrons is conveyed to the atoms in the sample. This energy
transformation results in phenomena with two principal types of responses. First, the generation of
other electrons: secondary electrons, backscattered electrons, auger electrons, transmitted
electrons and absorbed electrons. Second, the generation of photons: luminescence photons and
X-ray photons. Figure 6.2 depicts the different particles and photons that can emerge.
Electron beam
Backscattered electrons
X-ray photons
Secondary electrons
Auger electrons
Luminescence
sample
Absorbed electrons
Transmitted electrons
Figure 6.2
Interaction between the beam
and the sample, generation of different signals
In principle, all these phenomena can be used to construct images each with their own application
field. However, for this study the secondary electrons, backscattered electrons and the X-ray
photons are of major interest. The secondary and backscattered electrons were used for the
image formation. The X-rays were used for analytical purposes that enable assessment of
mineralogy in energy dispersive X-ray analysis (EDX).
The energy spectrum of the electrons emerging from the sample surface is schematically shown
in Figure 6.3. In this spectrum, the density of the emerged electrons is shown as a function of the
normalised log-scaled beam energy and two principal peaks are observed.
Backscattered
electrons
Electron density
Secondary
electrons
Relative energy E/E0 , log scale
Figure 6.3
1.0
Energy distribution of the electrons that emerge from the sample's surface
62
Chapter 6
The electrons with low energy are the secondary electrons. Secondary electrons originate from
the conduction band where they are weakly bound and absorbed sufficient energy from the beam
electrons to escape from the sample surface. Their energy ranges from near zero up to 50 eV.
The modal value of measured distributions is around 4 eV [Goldstein, 1992]. The electrons
emerging from the sample surface that have an energy only slightly less than the electrons in the
beam are known as backscattered electrons and are mainly the result of elastic scattering. All
electrons with energy above 50 eV are considered backscattered electrons. Both secondary and
backscattered electrons emerge from the sample surface with scattering angles that have a
cosine distribution at normal beam incidence. [Goldstein, 1992]
This cosine distribution of the backscattered electrons implies a geometrical maximum around the
electron beam. Therefore, the BSE detector is placed just above that sample in order to capture a
maximum number of electrons which leads to a maximum signal-to-noise ratio. Placement of the
SE detector is less restricted because of the electric measures that are taken for optimal S/N
ratio, this will be discussed in the sections below concerning SE and BSE images.
Secondary electrons can only emerge from the sample surface at very shallow depths because of
their low energy. In general, secondary electrons originate from depths up to a few nano-metres
and hence can only carry information about the sample material and morphology very near the
surface. The higher energy backscattered electrons can escape from locations deeper in the
sample, and therefore they can bear information from these locations. The X-ray photons also
escape from deeper in the sample because photons are less hampered by charged particles such
as electrons and protons.
The secondary and backscattered electrons as well as the X-rays that are the result of the
interaction of the beam electrons and the atoms in the sample, are generated in a volume that is
known as the excitation volume. The geometry of this excitation volume is shown schematically in
Electron Beam
Sample surface
Secondary electrons
Characteristic X-rays
Continuous X-rays
Fluorescent X-rays
Figure 6.4
Geometry of the excitation
volume and the origin of the various signals
63
Chapter 6
Figure 6.4 for beam electrons with energy around 25 keV and a sample with a relatively low
atomic mass. The shape and size of the excitation volume depends on the beam energy, the
beam electron density and the average atomic density of the sample material that is present in the
excitation volume.
6.2.3 Image formation in the SEM
In Figure 6.5 we depict the image formation in the SEM. The focused electron beam is deflected
in x and y direction by the scanning system. The deflection is controlled such that a small
rectangular frame is scanned on the sample’s surface. The scanning system synchronises the
electron beam in the microscope with the electron beam inside the monitor that is used to display
the image. The intensity of the beam inside the monitor is modulated by the signal that is obtained
from the electron detector in the SEM.
Electron gun
Lens
Scanning system
Aperture
Detection system
Electron beam
Sample
Monitor
Figure 6.5
The scanning and detection
system in a SEM
In this way, the image is built up point by point and line by line, with an intensity in each point that
is proportional to the signal from the electron detector. The signal from the electron detector in
turn is dependent on the amount of electrons collected. The magnification is defined as the ratio
of the size of the image on the monitor and the size of the frame that is scanned on the sample’s
surface. The magnification can be increased by reducing the size of the scanning frame, which is
controlled by the scanning coils.
64
Chapter 6
6.3
Secondary electron (SE) imaging
Secondary electron imaging is mainly used for morphological observation. As earlier indicated,
the low energy of the secondary electrons is the main reason that they can only carry information
from the first few nm below the surface of the sample. The information depth for conductors is
typically 1 nm, for insulators this can be up to 10 nm. The resolution for SE imaging depends
primarily on the diameter of the electron beam at the surface of the sample, and is typically better
than 1 nm (10 Å).
6.3.1 Sample preparation for SE imaging
The amount of secondary electrons that are generated depends on the ionisation energy of the
material at the sample surface and on the energy of the beam electrons [Robards & Wilson,
1993]. The number of the secondary electrons is roughly proportional to the atomic number of the
material at the spot where the beam hits the sample. In order to avoid contrast attributed to
elemental composition, a thin (0.2 to 10 nm) conductive layer is applied to the sample to be
investigated. This conductive layer is usually gold ( Au ) or a gold-palladium alloy ( Au − Pd ) and
is applied by a plasma deposition process under vacuum conditions. In this way, the ionisation
energy is kept constant for all locations on the sample which optimises the morphological contrast
in the images.
The thin conductive layer serves also as a drain path to ground for the surplus electrons that are
present at the sample surface. The image formation involves only a small fraction of the beam
electrons. Thus, a surplus of electrons builds up at the sample's surface when the sample is not
sufficiently conductive. This is the case for the vast majority of the mineral samples because most
minerals do not conduct. The surplus electrons cause a negative charge at the sample's surface,
which leads to the formation of an electric field. This electric field can be so strong that all new
incoming beam electrons are repelled. Hence, when beam electrons are not reaching the sample,
no interaction can take place and images cannot be constructed. It is therefore essential that a
dynamic balance is maintained between the arrival of beam electrons on the one hand, and the
electrons used for image formation on the other hand. The drainage of surplus electrons by the
thin conductive metal layer assures this dynamic balance.
6.3.2 Image formation in SE imaging
With constant ionisation energy and constant beam energy, the generation of secondary electrons
is now solely dependent on the sample/detector geometry. This property makes SE imaging a
very suitable method for morphological studies of reservoir rock samples. As discussed in the
introduction of this chapter, in geo-sciences SE imaging is mainly used to qualitatively inspect the
surface structure of the pores in reservoir rocks. This inspection is carried out on freshly broken
samples of reservoir rock material, to study the pore geometry, pore wall mineralogy, the rock
matrix mineralogy and the clay morphology. The large depth of focus of SE images provides very
sharp images over a substantial range of the optical axis, which is an important advantage over
optical microscopy.
Especially the combination of SE imaging and X-ray microanalysis is very powerful in the
identification of mineralogy of reservoir rock samples. Figure 6.6 shows a SE image at 50X
magnification of a shaly sandstone sample. A scale bar of 100 micron is inserted in the lower left
of the image as size reference. The sand grains are coated with a thin layer of clay and only very
65
Chapter 6
few open pores are observed. Figure 6.7 presents a close-up of the sandstone in Figure 6.6. The
magnification of Figure 6.7 is 3200X, roughly 3 times better than the maximum attainable with
optical microscopy.
Figure 6.6
Secondary electron image, magnification 50 X
Figure 6.7
Secondary electron image, magnification 3200 X
The large depth of focus and the high resolution of the SE image can clearly resolve the plately
type kaolinite clay structures in the centre of the image. The presence of this clay type was
66
Chapter 6
confirmed by X-ray analysis, which revealed that the elements Si , Al and O were present.
These are the constituent elements for kaolinite. The grain coating clay with a flakey appearance
was identified as smectite/illite, also by X-ray analysis. The presence of this type of clay in
significant amounts often indicates a low permeability because the flakes can narrow or even
close off pore necks that normally connect the pore bodies.
For morphological SEM analysis with SE imaging as discussed above, only a very small sample is
required, in the range of a few millimetres in diameter. Therefore, this type of analysis can also be
applied to drill cuttings. The use of cuttings may reduce the requirement of taking expensive core.
However, depth matching of cuttings can pose problems because of fall out of cuttings in the
drilling fluid. Depth matching is the process of aligning the depth origin of the cuttings with other
measurements such as wireline logs.
The above example illustrates the power of SE imaging in the characterisation of reservoir rock.
However, SE imaging has also disadvantages. Morphological analysis of the surface of freshly
broken reservoir rock samples is descriptive and qualitative. Quantitative assessment of reservoir
rock from this type of images is very difficult if not impossible because of the roughness of the
samples. Quantitative analysis becomes possible when the roughness is removed. Therefore,
polished blocks are used, which are the electron microscopy equivalents of thin sections. The
contrast mechanism for these samples is based on backscattered electrons (BSE), and differs
from the contrast mechanism that is used in morphological analysis, which is based on secondary
electrons (SE). BSE imaging enables accurate quantification without the disadvantages of thin
sections that were discussed in the last section of chapter 5. In the following section, we will
discuss BSE imaging in the context of the analysis of reservoir rock samples.
6.4
Backscattered Electron (BSE) imaging
SE primarily provides images in which the surface structure can be observed because the
contrast is dominated by the geometry of the sample. Therefore, SE imaging is very suitable for
morphological observation, as we have seen in the previous section. SE imaging at an acceptable
signal-to-noise ratio can only be carried out by taking special precautions that ensure that a
sufficient number of electrons reach the detector. To this end, the SE detector is supplied with a
bias voltage of 10 kV that attracts and accelerates the low energy electrons from the sample
surface [Lloyd, 1985]. However, backscattered electrons cannot easily be controlled by
conventional bias voltages without mixing backscattered electrons with secondary electrons
because backscattered electrons have high energies. Moreover, as BSE detectors are primarily
used in the so-called composite mode [Robards & Wilson, 1993] to obtain images with atomic
number contrast, the mixing will pollute the BSE signal with topographic contrast and thereby
obscure the atomic number contrast. This effect leads to a poor definition of the atomic number
contrast and yields BSE images in which details are more difficult to observe. The limited
definition of the atomic number contrast in turn hampers the accurate quantification of image
features. The mixing of atomic number contrast with topographic contrast can be largely avoided
by using the BSE detector in compositional mode and by optimal placement of the detector. Both
measures will be discussed below.
The BSE detector that was used in this study is a solid-state, semi-conductor detector that can
produce signals from which images with atomic number contrast can be constructed. Solid state
BSE detectors are based on the phenomenon that a high-energy electron passing through a P-N
67
Chapter 6
layer that is maintained under a reverse voltage, produces an avalanche of electrons. In BSE
imaging, the number avalanche electrons is detected and used for image formation. The spatial
distribution of the high-energy BSE electrons that emerge from a flat sample surface can be
described by a cosine function under the condition of normal beam incidence and uniform
composition of the excitation volume. This is schematically shown in Figure 6.8, in which both the
distribution of the backscattered electrons and the excitation volume are depicted.
Electron beam
BSE Detector
BSE distribution
Sample
Excitation
volume
Figure 6.8
Detection of
backscattered electrons
The BSE detector should physically subtend a solid angle that is large enough to collect a
sufficient amount of electrons for image formation at acceptable signal-to-noise ratios. Therefore,
solid-state BSE detectors are constructed such that they can be positioned just above the sample
to maximise the amount of captured backscattered electrons. A central aperture in the detector
assembly lets the electron beam through as shown in Figure 6.9.
1
2
3
4
Figure 6.9
Four
quadrant BSE detector
Each of the four quadrants of the detector is an individual detecting element and provides a
separate BSE signal. When these four signals are added, the influence of the topographic
contrast is minimised and the detector works in the compositional mode. When the signals of
quadrants 1 and 4 are subtracted from quadrants 2 and 3, the detector works in the so-called
68
Chapter 6
topographic mode. In topographic mode, the compositional contrast is minimised and images that
resemble SE imaging are generated.
6.4.1 Sample preparation for BSE imaging
The preparation of samples for BSE imaging differs substantially from that for SE imaging. In SE
imaging, a thin conductive layer is applied to the surface of a freshly broken and dried sample. In
this way, the morphology of the sample is preserved and emphasised by the conductive layer, this
observation mode is therefore known as SE topographic imaging. For BSE imaging in
compositional mode, we want to distinguish the mineral constituents of reservoir rock and the
pore space. Therefore, the influence of the sample’s topography should be excluded. This was
achieved by totally removing the topology leaving a flat surface. To this end the sample was
impregnated with epoxy resin and polished on one side once the epoxy has hardened.
The sample preparation procedure for BSE imaging was derived from the procedure that is
normally used for preparation of thin sections. Because of the higher resolution of the SEM, more
severe requirements were posed to the polishing procedure compared to thin section preparation.
For thin sections, the final polishing step was carried out with a polishing agent containing
abrasive particles of some 30 microns in diameter. For BSE imaging, this final polishing step
entailed a mechanical/chemical process with abrasive particles of some 50 nano-metres in
diameter. For this study, a special preparation procedure was developed of which more detailed
information can be found in work carried out by [Dijkshoorn, 1990]. This sample preparation
procedure was applied to all the samples that were used in this study. Minerals in reservoir rock
either have a very high resistance or are insulators Therefore, precautions have to be taken to
avoid charging, which can deteriorate BSE images beyond utility. The charging effects were
minimised by coating the sample with a thin conductive layer that is transparent to backscattered
electrons. This is based on similar reasoning as for SE images. However, a conductive layer of
metal applied by plasma deposition as is common for SE imaging, is not suitable for BSE imaging.
Experiments showed that gold ( Au ) or a gold-palladium alloy ( Au − Pd ) coating on polished
samples resulted in BSE images with a very poor signal-to-noise ratio, because only a limited
number of backscattered electrons can leave the sample surface and can be detected. This is
caused by the high absorption level of the heavy atoms in the conductive metal layer.
Experiments with non-coated samples were also carried out. The generation of BSE images was
impossible with non-coated samples due to charge effects. The solution to this problem was found
in applying a thin layer of carbon using a vacuum deposition process. Carbon is a light element
resulting in a low absorption for backscattered electrons but still has an acceptable conductivity
when applied to flat and polished surfaces. In the sample preparation procedure developed for
this study, the areas of interest on the samples were coated with a carbon layer with a thickness
of approximately 70 to 100 nm. All other sides of the samples were coated with gold to assure the
presence of a conductive path to ground. This sinked the surplus electrons and thus avoided
charge effects.
6.4.2 Atomic number discrimination
The contrast mechanism that we used for BSE imaging in this study is dominated by differences
in atomic number. This can be explained by electron scattering phenomena. When a beam
electron enters the sample at the surface, there are two principal scattering phenomena; the first
one is caused by electron-electron interaction and the second one by electron-proton interaction.
69
Chapter 6
In both these interactions, atoms with high atomic numbers will have a higher probability of
bouncing back electrons than atoms with a low atomic number. Atoms with high atomic numbers
contain a high amount of electrons and have a large nucleus. Therefore, their apparent diameter
for accelerated electrons is high and is roughly proportional to the atomic number. In both
electron-electron interaction and electron-proton interaction, we deal with elastic collisions. In
elastic collisions, the electrons will leave the sample surface with an energy that is just below
energy of the electron beam. This explains the peak at the high-energy side for backscattered
electrons in the energy spectrum in figure 6.3. Hence, the amount of backscattered electrons that
emerge from a sample depends on the average atomic density in the excitation volume that is
present on the spot where the beam hits the sample surface. Therefore, the backscattered
electron signal can provide information about the atomic density, which in turn can be used to
determine the composition of the reservoir rock sample.The backscatter coefficient η is in
essence the relative amount of high-energy electrons emerging from the sample. η depends on
the range of parameters that includes the incident angle ψ , the beam energy U , the energy
spread of the electron beam ∆U , the spot size of the electron beam at the samples surface d e ,
the atomic number Z , and the effect of charging q e :
η = f (U , ψ , Z , ∆U ,d e ,q e )
6.4
When a flat polished sample is used, the incident angle ψ is eliminated and the backscatter
coefficient η is a function of the atomic number Z , the beam parameters U , ∆U , d e and the
charge effect q e . The charge effect q e is eliminated by applying the thin C layer on the sample
surface and the conductive path to ground provided by the Au layer. Experiments showed that
the effects of the beam energy spread ∆U and the spot size d e of the electron beam are
negligible for the reservoir rock samples we use in this study. This is due to the use of the field
emission electron gun, which is very stable. The foregoing reduces the variables that determine
the backscatter coefficient η to the beam energy U and the atomic number Z of the sample.
Figure 6.10 shows the relation published by Lloyd [1985] of the backscatter coefficient η and
atomic number Z for different beam energies when all other variables are considered constant. It
can be observed that for higher energies in the range of 10 to 30 keV, the backscatter coefficient
increases with atomic number.
Normalised BSE coefficient
[%]
Atomic number discrimination
100
90
80
0.5 keV
70
1 keV
60
2 keV
50
5 keV
40
10 keV
30
20 keV
20
30 keV
10
0
0
10
20
30
40
50
60
Atomic number
Figure 6.10
beam energy
Normalised BSE coefficient versus electron
70
Chapter 6
For lower beam energies the reverse can even occur, at higher atomic numbers the backscatter
coefficient decreases. The strongest dependence is found around a beam energy of 5 keV. The
relationship of the backscatter coefficient and the atomic number shows an almost linear
dependence in the range where Z is between 6 and 20. This range is of particular interest in the
context of this study because the majority of bulk minerals in reservoir rock contain elements with
atomic numbers in this range. In 1985, it was shown by Lloyd [1985] that BSE imaging could be
used for delineating minerals in petrographic analysis of polished thin sections. In this study, we
used this effect for the delineation of pore space
From the above analysis, we concluded that beam energies between 2 keV and 5 keV are optimal
for BSE imaging of reservoir rock. At these energies, the discrimination for atomic number
contrast in BSE images reaches a maximum for the range of minerals that occur in reservoir rock,
the derivative of η = f (Z ) has a maximum in this range. However, experiments also showed
that the efficiency of semiconductor solid-state detectors decreases with decreasing beam
energy, which resulted in a poor signal-to-noise ratio. Therefore, from a detector efficiency point of
view, a beam energy of 5 keV was selected for BSE image formation in this study. However, other
parameters also influence the selection of the beam energy, and this will be discussed in the
following sections.
6.4.3 Pores and minerals in BSE images
In BSE images, the pore space and the minerals in the rock matrix can be distinguished because
they differ in atomic density. The grey-values in BSE images represent atomic density. However,
in order to assess the pore space and the rock minerals in BSE images quantitatively, the images
have to be calibrated against a set of known standards. This is discussed in the next sections.
First we first explain how pore space and rock minerals are recognised in BSE images, and
discuss the boundary conditions required for optimal BSE imaging of reservoir rock.
Figure 6.11 presents a typical BSE image from a clean sandstone. The higher the grey-value the
higher the atomic number of the constituent. The impregnating epoxy has a low atomic density
compared to reservoir rock minerals and it therefore appears dark in BSE images. The
impregnating epoxy is filling the pore space, hence the dark parts in the image represent the pore
system. The dark grey areas represent the clays and the medium grey areas the quartz or
dolomites. Brighter areas contain either feldspars or calcites and the brightest areas represent
heavy minerals like baryte, siderite, anhydrite etc [Lloyd, 1985]. In some cases, ambiguity can
arise, for instance when minerals have approximately the same atomic density, as is the case
when dolomite and quartz are present simultaneously. When this happens, X-ray elemental
analysis, available on most electron microscopes, can help distinguish the two.
In BSE images, the total grey-value range can be divided in sub-ranges representing the various
constituents of reservoir rock. The procedure to extract these sub-ranges is similar to that used in
the segmentation of colour images from thin sections, as will be discussed in chapter 7. The
segmentation is now applied to the grey-value range of BSE images instead of colour images.
71
Chapter 6
Figure 6.11
Backscattered electron image from a clean sandstone sample
6.4.4 Resolution in BSE images
To make meaningful use of BSE images in the prediction of petrophysical properties, the
boundary conditions that determine the attainable resolution should be known. The resolution in
BSE imaging depends primarily on the energy of the beam electrons and on the average atomic
density of the excitation volume below the spot where the beam hits the sample surface.
In BSE images, the resolution is defined by the area on the sample's surface from which
backscattered electrons can emerge after a number of collisions. This area is known as the point
spread function (PSF). Figure 6.12 depicts this PSF for backscattered electrons. The PSF has a
circular shape on the sample's surface when the excitation volume consists of one single element
or one single compound. We defined the size of the PSF as the diameter of the circular area from
which the electrons can escape the sample's surface. If we keep the beam energy constant, the
size of the PSF depends solely on the atomic density of the excitation volume.
72
Chapter 6
Electron Beam
Sample surface
Secondary electrons
BSE PSF
or
lateral
resolution
Characteristic X-rays
Continuous X-rays
Fluorescent X-rays
Figure 6.12
Point spread function of a BSE
signal on the sample's surface
When a BSE image is digitised and stored on a PC we have to assign a length scale to the side of
the square pixels in the image. This length scale depends on the set magnification of the SEM.
The length of a pixel in an image is calculated from the area that is scanned on the sample's
surface by the electron beam and the number of pixels in the image:
sampling _ dis tan ce =
scanned _ area _ in _ BSE _ image
number _ of _ pixels _ in _ BSE _ image
6.5
For example, if the scanned area is 2 x 2 mm and the image size is 2048 x 2048, then the
sampling distance in the image is approximately 1 micron.
The resolution of BSE images also depends on the beam energy. This was assessed by analysis
of simulations followed by physical experiments to test the validity of the simulations. The random
nature of electron scattering in solid matter can be described with statistical analysis that is known
as Monte Carlo simulation [Joy, 1995]. Simulations were carried out for the major minerals in
reservoir rock and for the pore filling epoxy at electron beam energies between 1 and 30 keV.
Appendix A presents the details of the simulations that were carried out for this study. Figure 6.13
shows the relation between beam electron energy and the diameter in microns of the PSF, which
is essentially the BSE lateral resolution. It can be seen that below a beam energy of 7 keV the
lateral resolution for the major minerals in this study and for epoxy is around 0.5 micron. The
Monte Carlo experiments showed that for a beam energy of 5 keV the lateral resolution for the
sample constituents in this study varies between 0.4 and 0.6 micron. For the densest mineral
dolomite, this is 0.4 micron and for the pore filling epoxy the PSF has a diameter of 0.6 micron.
We concluded that the lateral resolution of BSE imaging is of the same order of magnitude as the
resolution of visible light, about 0.5 micron. However, the contrast mechanism in BSE images
enables one to distinguish pore space from rock matrix and this is the most important property we
used in this study.
73
Chapter 6
Resolution/energy dependence
20
PSF [microns]
Epoxy
Quartz
15
Calcite
Dolomite
10
Feldspar
5
0
0
5
10
15
20
25
30
Beam energy [keV]
Figure 6.13
Electron beam energy versus the diameter of
the PSF at the sample's surface for the various constituents
It is also remarked that the required detail for clay studies can be obtained in SE morphological
imaging with resolution of a few nm's. The combination of quantitative BSE imaging and
qualitative SE imaging has been applied to the majority of the samples used in the current study.
Another justification for the selection of the electron beam energy of 5 keV is the optimum signalto-noise ratio of the BSE detector has as was discussed in one of the previous sections.
Moreover, at this beam energy the atomic number discrimination for the minerals occurring in
reservoir rock is optimal as was discussed in section 6.4.2. In the Monte Carlo simulations, the
BSE coefficient η was evaluated for various electron beam energies using different reservoir
rock minerals and epoxy. The BSE coefficient was defined as the fraction of the primary beam
electrons that emerges from the sample surface and is available for detection. Figure 6.14
presents the results of these simulations and shows that at 5 keV beam energy, the BSE
coefficient for all constituents lies between 0.075 and 0.108.
BSE coefficient/energy dependence
0.16
BSE coefficient
0.14
0.12
Epoxy
0.10
Quartz
0.08
Calcite
0.06
Dolomite
0.04
Feldspar
0.02
0.00
0
5
10
15
20
25
Beam energy [keV]
Figure 6.14
Electron beam energy versus the BSE
coefficient for the various reservoir rock constituents
74
30
Chapter 6
For a beam energy of 3 keV approximately equal amounts of backscattered electrons are
generated for all the main sample constituents. This essentially means that at 3 keV only minor
atomic density contrast exists for the minerals and the epoxy in the reservoir rock samples.
Experiments confirmed that at a beam energy of 3 keV it was impossible to generate a BSE
image with sufficient contrast between the rock matrix material and the pore filling epoxy. This
effect is amplified by the limited penetration power of the 3 keV beam electrons because their
energy is too low to twice traverse the carbon layer. Consequently, the image formation was
limited to the carbon layer only very limited atomic number contrast could be obtained. The above
conclusion has been confirmed by Monte Carlo simulations at 3 keV, of which the details can be
found in Appendix A. At a beam energy of 5 keV sufficient atomic number contrast occurs in BSE
images to enable quantitative analysis.
6.4.5 Calibration of the grey-values in BSE images
We have established that pore space and rock matrix minerals could be accurately distinguished
in BSE images that were collected at a beam energy of 5 keV. Next, the outcome of this analysis
had to be translated into a set of practical guidelines to generate BSE images that would be
suitable for image analysis. Quantification of image features requires some form of calibration of
the grey-values in the BSE images.
The first concern was the signal-to-noise (S/N) ratio in the images. In order to understand the
concept of S/N ratio in BSE imaging, the image collection process had to be assessed. As earlier
discussed, the BSE images are generated stepwise: the image is built up point by point and line
by line. The electron beam in the SEM is positioned at a location on the sample’s surface that is
associated with the first position in the image, the first pixel on the first image line. At that point,
backscattered electrons are emerging from the sample’s surface from a circular area of which the
diameter is described by the PSF. A fraction η of these backscattered electrons reaches the
detector where they cause a signal that is proportional to the BSE intensity. The signal from the
detector is then digitised during a fixed time interval and converted into a grey-value that is
assigned to the associated pixel. In fact, the grey-value represents the amount of electrons
detected within that collection time interval. The time interval should be long enough to distinguish
the signal from the background noise in the detector. Following, the electron beam is positioned to
the next location of the sample’s surface that is associated with the second pixel on the first image
line. This process is repeated until the whole image is generated.
The S/N ratio in BSE images can be improved by statistical averaging. We controlled the S/N ratio
by averaging several measurements taken at one point, hence one image pixel. Experiments
were carried out in which the number of measurements per pixel was varied. Hereby we had to
compromise between S/N ratio in the BSE image for each pixel and the obtained total collection
time for the image. In the experiments, we varied the number of measurements per pixel in a
logarithmic fashion: we started with 2 measurements and doubled this in each subsequent
experiment. We concluded that with 1024 measurements per pixel, an acceptable trade-off
between image collection time and S/N ratio in the image was reached.
The next concern is the actual grey-value calibration. A calibration method was developed based
on the two main constituents, the quartz and the pore filling epoxy, in order to enable semiautomatic routine processing of the BSE images. The settings of the detector electronics
controlling the grey-value scaling were adjusted such that these two main constituents could be
75
Chapter 6
imaged properly. The dynamic range of 255 grey-values in the image was optimised for these two
main phases in reservoir rock, but sufficient headroom was left to fit in the signals for heavier
minerals. Experiments with artificial reservoir rock samples were carried out to determine the
optimal detector settings.
Figure 6.15
space only
BSE image of one of the calibration samples that consist of quartz and pore
These artificial reservoir rock samples consisted of fused glass-beads, made from amorphic
quartz that resembled the detrital quartz grains found in real reservoir rock. The epoxy in the pore
space was the same material that we used for all the samples in this study.
After applying the sample preparation procedure that was developed for this study, we obtained
reservoir rock samples with only two phases: the rock matrix of quartz and the pore filling epoxy.
Figure 6.15 shows a BSE image taken from one of these calibration samples. The associated
grey-value histogram is presented in Figure 6.16.
76
Chapter 6
Grey-value histogram
0.10
0.09
Relative frequency
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0
50
100
150
200
250
Grey-value
Figure 6.16
Grey-level histogram obtained from the BSE
shown in Figure 6.15
As expected, the grey-value histogram of the calibration sample only shows two peaks, one from
the pore filling epoxy and another one from the quartz matrix. From our experiments, we
concluded that optimal values for pore space and for quartz were obtained when the detector
electronics were adjusted such that the pore space has a peak at a grey-value of 8 and the quartz
has a peak at grey-value 128. These settings make proper use of the dynamic range in the
images for the main constituents. Furthermore, it also provided ample dynamic room for the
heavier minerals to be properly imaged.
6.4.6 Optimal magnification for BSE images
Similar to the reasoning we applied in the case of thin section analysis, we investigated the
optimal magnification for the BSE images. Also in this case the magnification should be
sufficiently low to ensure representative images of the sandstones that we use in this study. In the
thin section study, we selected a magnification of 20X with an image resolution of 4 micron. The
main criterion for the selection of the appropriate magnification was that images from even the
coarsest sandstone should reveal a sufficient number of grains and pores to be considered
representative.
In the SEM study, the attainable resolution of the BSE images ranged from 0.4 micron for the rock
matrix minerals to 0.6 micron for the pore space. The lowest magnification limit of the SEM that
we could use was 35 X, resulting in a scanning frame on the sample surface of some 2 x 2
millimetre. Larger scanning frames corresponding with lower magnifications could not be achieved
due to physical constraints of the electron lenses and the scanning coils. Similar to the thin
section study we varied the magnification on a large number of samples and inspected the result.
It was found that even for very coarse sandstones acceptable images could be obtained. Figure
6.17 presents a BSE image from the coarsest sandstone we had available in the sample set. It
can be seen that this sandstone contains grains with diameters of 0.5 mm and larger. In terms of
the Wenthworth classification, this sandstone was considered medium coarse to very coarse. The
selection of a magnification of 35X enabled us to encompass the total range indicated by the
77
Chapter 6
Wenthworth classification, which starts at very fine sands with grain diameters of 64 micron to
very coarse sands with grain diameters of 0.5 mm.
Figure 6.17
BSE image of the coarsest samples in the reference sample set
The 2 x 2 mm area scanned by the SEM with an image size of 2048 x 2048 pixels, led to a
resolution of approximately 1 micron per pixel. The PSF (projected on the sample's surface) for
epoxy describes a circle of 0.6 micron in diameter. The sampling distance of 1.0 micron is
sufficient to prevent that one pixel contains information from adjacent spots on the sample.
Therefore, the image pixels can be considered mutually independent.
With a diameter of the PSF of 0.5 micron and a sampling distance of 1.0 micron the images are
essentially under-sampled. Maximum information in a signal is preserved when the standard
deviation σ of the Gaussian shaped PSF is approximately equal to sampling distance. With a
PSF diameter of 0.5 micron, σ is approximately 0.1 micron assuming the total width of the PSF to
be 6σ. This clearly leads to under-sampling by a factor 10. However, sandstone rock properties
and the methods we used for image processing minimised the influence of this under-sampling.
The grains in the sandstone reservoir rock are much larger (30 to 500 micron) than the sampling
78
Chapter 6
distance (1 micron). Therefore hardly any loss of information about the grain edges occurred. The
finer details in the clay structures with dimensions below 1 micron were accounted for in the
model for micro-porosity that we will develop in chapter 7. Hence, in the finer details no loss of
information occurred for the parameters we used in the prediction of petrophysical parameters.
Only small deviations in the predictions of petrophysical parameters were observed when images
were collected at higher magnifications. Important disadvantages were that the number of images
increased quadratically with higher magnification to cover the same scanned area. The required
logistical effort that was associated with a higher number of images disqualified the use of higher
magnifications.
In conclusion: because the grains in sandstone reservoir rock were significantly larger than the
sampling distance in the images, the under-sampling did not hinder accurate definition of the pore
system we used for analysis. Moreover, loss of information about fine clay details was accounted
for by a model that included the non-visible micro-pores in the estimation of porosity. This
approach was confirmed and justified by the excellent agreement between porosity derived from
images using the method that we will develop in chapter 8 and the porosity measured from core
plugs using conventional methods.
79
Chapter 7
Processing and analysis of BSE images from reservoir rock
7.
samples
7.1
Introduction
For the analysis of BSE images, we devised a similar process as was applied for the analysis of
thin sections. The process of extracting pores, clays and rock matrix from BSE images is simpler
compared to the process for thin sections. A colour thin section image contains three grey-level
images in RGB space, while in BSE analysis only one grey-level image is involved. In BSE
images, the rock matrix can be subdivided into ranges associated with different minerals using the
atomic density contrast as was discussed in chapter 6.
Two-level thresholding was used to extract the clays and rock matrix minerals in addition to the
pores. Filtering was applied prior to the thresholding in order to remove noise from the BSE
images. The filtering modified the grey-level histogram such that thresholding could be applied
more easily. Subsequently, morphological filtering was used to remove noise from the images that
contain the segmented pores, clays and rock matrix minerals. This noise arose from the
thresholding process. The parameters of the applied morphological filter were selected such that
images containing the pores, clays and rock matrix resembled the BSE images as best as
possible.
A segmentation procedure was developed that enabled automatic extraction of pore, clays and
rock matrix to cope with the large volume of BSE images (approximately 1200) to be processed
and measured. This procedure is based on fitting three Gaussian functions to the grey-level
histogram. The discussion on micro-porosity, which was started in chapter 5 to explain the underestimation of porosity, was extended to BSE images. Atomic density in clays can be related to the
micro-porosity when we assume that pores smaller than approximately 1 micron in diameter
primarily reside in the clays.
The capillary tube models we discussed in chapter 2 require input of parameters that define the
pore system. We therefore developed a process that enables the extraction of geometric
information on the pores from BSE images that can be used for the prediction of permeability.
Similarly, we developed a method to extract topographic parameters from BSE images. The
topographic parameters provide information about the connectivity of the pores in the pore
network and can be used to predict permeability.
Finally, we will discuss the experiments that were carried out to determine the optimal number of
images for each sample that is required to arrive at representative predictions of petrophysical
parameters.
7.2
Extraction of pores, clays and matrix
7.2.1 Two-level thresholding
A BSE image is a grey-value image that can be defined as a two dimensional function of discrete
grey-values:
I = f ( x , y ) ⇒ 0 ≤ f ( x , y ) ≤ 255
7.1
80
Chapter 7
In BSE images the grey-values represent atomic density. Prior to processing and analysis, these
grey-values have to be calibrated which takes place using a set of standards with known greyvalues. This calibration is essential to make quantitative use of the data provided by the analysis.
The calibration standards we used consisted of artificial reservoir rock samples that contain only
quartz and epoxy.
Figure 7.1 presents a typical BSE image of a rock sample from a shaly sand interval in a
sandstone reservoir. The grey-level histogram of this image is shown in Figure 7.2. Prior to
collection of this image, the settings of the BSE detector electronics were calibrated against the
standards that were introduced in chapter 6. In BSE images, the epoxy appears as dark areas
Figure 7.1
BSE image of a shaly sandstone
caused by its low atomic density. The epoxy fills the accessible pore space and thereby
represents the pore system and the porosity. The medium grey areas represent the quartz and/or
dolomites. The brightest areas in BSE images represent the heavy minerals because of their high
atomic density.
81
Chapter 7
As earlier indicated in chapter 6, the total grey-value range in BSE images can be divided in subranges. In Figure 7.2, the colour bar below the histogram shows the division in these sub-ranges
representing pores, clays, quartz/dolomite, feldspar/calcite and the heavy minerals. Two-level
thresholding is used to extract the pixels in each range of grey-values. Thresholding is an imageto-image transformation, in this case a transformation from a grey-value image to a binary image.
0.08
Quartz
Relative frequency
0.07
0.06
0.05
0.04
0.03
0.02
Pore space
0.01
Feldspar
Clay region
0.00
0
50
100
Heavy minerals
150
200
250
Grey-value
F
Clays + wedges
QZ
Feldspar
Heavy
Minerals
Ranges for thresholding
0
Grey-value
255
Figure 7.2
The grey-level histogram calculated from the BSE image
shown in Figure 7.1
Thresholding entails applying the Threshold operator T that is defined as follows: all pixels
whose grey-values fall in the range between low and high are assigned to 1 and all pixels outside
that range are assigned to 0. This is mathematically expressed as:
[T (t
t
low , high

(t ≤ f ( x , y) ≤ t )
)] f ( x , y) = 10 →→ ifotherwise

low
high

7.2

The binary images representing the reservoir rock constituents, pores, clays, quartz, feldspars
and heavy minerals are transferred into a new image using the modified addition that we
introduced in chapter 5 page 47. We term this new image a DEF image, an abbreviation of
Display of Elementary Fractions. In the DEF images the constituents are each represented by one
single grey-level and are colour coded according to Table 7.1. The colours were selected from an
internationally accepted legend used by sedimentologists in petroleum Geology.
82
Chapter 7
Colour
Constituent
Pore space
Clays
Quartz
Feldspars/Calcite
Heavy Minerals
Table 7.1
Colour code
used for the constituents in BSE
images
7.2.2 Grey-level noise filtering
During the image collection process noise was introduced which hampered accurate
segmentation. The solution to this problem was found in applying a filter that removes the noise.
Various filter types were tested in order to determine which filter provided the best result for the
BSE images. Initially we applied a low-pass filter and a median filter. However, these filters have
the disadvantage that they blur the grain edges and this makes the segmentation of the pore
space less accurate. Therefore, we applied a selective filter that only averages grey-values in a
specific range and leaves the sharp edges in the images with large differences in grey-values
unaffected. The so-called Sigma filter can effectively remove noise without blurring sharp edges
like transitions from pore space to minerals. The Sigma filter is well known in the field of image
enhancement and is available as a standard function in the image processing and analysis
package that we used in this study. The main property of the Sigma filter is that it preserves fine
structures in an image as long as these structures fall outside the range defined by the σ
parameter. The Sigma filtering averages only those grey-values in a square kernel that do not
exceed the specified limits of deviation of the grey-values with respect to the value of the centre
pixel in the kernel. The size of the kernel defines the strength of the smoothing. Sigma filtering is
an image to image transformation that is mathematically defined by the mask function:
((
) )
((
) )
1 → if f x , y − σ ≤ f ( x , y ) ≤ f x , y + σ
µ
µ
µ
µ
m( x , y ) = 
0 → otherwise

7.3
followed by:
f ( x , y ) out =
∑ m( x , y) f ( x , y )
window ( xc , yc )
7.4
∑ m( x , y)
window ( xc , yc )
Experiments demonstrated that for the BSE images used in this study a square kernel size of 15 x
15 pixels combined with a value for σ of 7 was optimal. Applying the Sigma filter to the BSE
images not only removes the noise without blurring edges, but it also has the effect of sharpening
the peaks in the histogram. It appeared to be easier to define the segmentation levels from the
more pronounced histogram. Figure 7.3 presents the histograms of the BSE image shown in
83
Chapter 7
Figure 7.1. The original histogram is presented in red while the sigma-filtered histogram is shown
in blue.
Relative frequency
0.12
Original
0.10
Sigma Filtered
0.08
0.06
0.04
0.02
0.00
0
50
100
150
200
250
Grey-value
Figure 7.3
The original and the sigma filtered histogram
from the BSE image in Figure 7.1
7.2.3 Morphological filtering
Despite the Sigma filtering that was used to reduce noise in the BSE image, the DEF image
resulting from the thresholding procedure did not resemble the original BSE image sufficiently.
Inspection showed that anomalies occurred in the thresholding procedure, which produced an
artificial "clay" coating on all quartz and feldspar grains. This can be clearly observed in Figure 7.4
where a section of the BSE image and the associated DEF image from Figure 7.1 are presented.
Figure 7.4
grain edges
BSE image and the uncorrected DEF image, note the green rims at the
84
Chapter 7
In Figure 7.4, the amount of clay is over-represented because of the BSE equivalent of the
Holmes effect that was discussed in chapter 5. The same effect is observed at the edges of the
olive-green feldspars at the bottom part of Figure 7.4. This edge effect can be minimised by
applying the same morphological filtering as was used in the processing of thin section images
that was discussed in chapter 5.
Grain damage produces artefacts in both the rock matrix and the pore space. The grain damage
is caused by inadequate sample preparation and appears in the BSE images in the form of small
holes in the grains. This can lead to erroneous assignment of pore space and thereby to a large
number of very small pores. Small rock fragments, removed from the rock matrix in the grinding
and polishing phase, can move into the pore space, again leading to erroneous assignments in
DEF images. It appeared impossible to eliminate these effects by more stringent control
procedures in the preparation phase.
After experiments on BSE images the solution to the BSE version of the Holmes effect and the
grain damage effect was found in a combination of morphological filtering and a fixed sequence
for constructing the DEF image. First the pore space was extracted and all pores with areas
smaller than 128 pixels were deleted to correct for artificial pores caused by grain damage. This
process is known as binary scrapping. The 128 pixels represent an area of 128 micron2 and this
cut-off value was based on measurements of the dominant size of the damage holes in the grains.
From the reasoning that the inverted pore space should be the rock matrix, the image was
inverted and again subjected to binary scrapping to delete the rock fragments that are present in
the pore space. In this case, it was also found that an area of 128 pixels represents the dominant
size of the rock fragments. The resulting binary image was transferred into the DEF image by
modified addition, in a similar fashion as described in chapter 5.3 for thin section images. The
pore space was coded in blue and the rock matrix that represented quartz in yellow.
After this, the clays, which have grey-values that fall between the peaks for pore space and
quartz, were extracted. To correct for edge effects, a binary opening with a hexagon structuring
element of size 5 x 5 was applied. The resulting binary image was transferred into the DEF image
and coded in green. Similarly, the feldspars were extracted and as was the case for quartz, small
grains were deleted to correct for grain damage, again with a binary scrap of 128 pixels.
To correct for edge effects, the feldspars were dilated with a hexagon structuring element of size
5 x 5. In order to prevent reduction of pore space caused by the dilation, the area of the dilated
feldspars were reduced with the pore space leaving only the feldspar dilation into the quartz area.
Finally, the heavy minerals were extracted and transferred into the DEF image and coded light
yellow. The heavy minerals in general do not suffer from edge effect as they are in the vast
majority of cases embedded in the detrital assemblage as grain inclusions. More details on the
applied morphological filtering can be found in Appendix B. Figure 7.5 shows the BSE image and
the associated corrected DEF image.
85
Chapter 7
Figure 7.5
BSE image and the corrected DEF image, note that the artificial clay rims
at the grain edges have vanished
7.3
Semi-automatic segmentation
Initially we manually carried out the selection of the grey-level segmentation boundaries tlow and
thigh for each phase. However, as some 1200 BSE Images had to be processed, we developed a
method that enabled automatic selection of the segmentation boundaries. Figure 7.2 illustrates
that in general three peaks occur in the grey-level histogram of a BSE image from a sandstone
sample. These peaks are associated with pore space, quartz/dolomite and feldspar/calcite. The
grey-value range for clays is bounded by the peaks for pore space and quartz. The range for
heavy minerals is bounded by the peak for feldspar/calcite on the one hand and the end of the
grey-value scale (255) on the other hand. We concluded that when it is possible to locate these
three peaks, we would be able to devise a method for automatic definition of all required
segmentation boundaries.
From our experiments with the calibration standards, we observed that the grey-values under the
peaks are distributed such that they can be fitted to a Gaussian function. Gaussian functions have
the advantage that they can be fully described by three parameters: the mean value µ , the
standard deviation σ , and the amplitude α. The three peaks for the pore space (Φ), the quartz
(QZ) and the feldspars (FS) had to be fitted simultaneously. Therefore, a fit procedure for 9
parameters was developed in which a least squares approach was used as minimisation criterion.
The squared difference between the sum of the three Gauss functions representing pore space,
quartz and feldspar and the grey-value histogram was minimised. For the minimisation procedure,
we used the solver in EXCEL that contains the Generalised Reduced Gradient (GRG2) non-linear
optimisation. Two options were available in this routine, the Newton-Raphson and the Conjugate
Gradient method. We applied the Newton-Raphson method because this required less iterations
to reach convergence. The minimisation criterion is mathematically defined as:
255
(
(
)
min ∑ f ( g ) − α Φ G( g ,µ Φ , σ Φ ) − α QZ G g ,µ QZ ,σ QZ − α FS G( g ,µ FS ,σ FS )
g =1
In which:
86
)
2
7.5
Chapter 7

− 12 
1
G( g , µ , σ ) =
e 
σ 2π
g −µ  2

σ 
7.6
Equation 7.6 represents the three Gaussians αG(g,µ,σ) in which α is the volume fraction of the
constituent while in eqn. 7.5 f(g) is the grey-value distribution function of the BSE image. The
peaks appeared more or less at the same grey-level values for all images, because the BSE
detector electronics were always adjusted within calibrated grey-value ranges during image
collection. It was possible to set a number of conditions in the solver that allowed automatic
processing of approximately 95% of the 1200 images that were treated. We considered this as
semi-automatic thresholding because not all BSE images could be subjected to the automatic
definition of segmentation levels in the thresholding procedure. For instance, if the three
mentioned peaks for pores space, quartz and feldspars were not all prominently present, the
segmentation levels had to be defined manually. The developed method failed especially in shaly
samples due to high clay volumes. The grey-values ranges were expressed in the parameters
that were fitted in the procedure to obtain the segmentation levels. This is mathematically defined
as follows:
f ( x , y ) ∈ Φ ⇒ 0 ≤ f ( x , y ) ≤ µ Φ + 4σ Φ
Pore space:
7.7
f ( x , y ) ∈ CL ⇒ µ Φ + 4σ Φ < f ( x , y ) ≤ µ QZ − 4σ QZ
Clays:
f ( x , y ) ∈ QZ ⇒ µ QZ − 4σ QZ < f ( x , y ) ≤ µ QZ + 4σ QZ
Quartz/dolomite
f ( x , y ) ∈ FS ⇒ µ QZ + 4σ QZ < f ( x , y ) ≤ µ FS + 4σ FS
Feldspar/calcite
f ( x , y ) ∈ HM ⇒ µ FS + 4σ FS < f ( x , y ) ≤ 255
Heavy minerals
7.8
7.9
7.10
7.11
Figure 7.6 shows the result of the fit to the histogram of the example BSE image in Figure 7.1.
Greylevel histogram
0.12
BSE Histogram
Relative Frequency
0.10
Pores Fit
Quartz Fit
0.08
Feldspar Fit
Pore space
Clays
0.06
Quartz
Feldpar
0.04
Heavy Minerals
0.02
0.00
0
50
100
150
200
Grey-value
Figure 7.6
Grey-level histogram with the fitted
Gaussians for pore space, quartz and feldspars
87
250
Chapter 7
The fit of individual Gaussians with the peaks for pore space, quartz and feldspar is not perfect.
However, we were primarily interested in the segmentation levels that were derived from the
parameters µ and σ . The non-perfect Gaussian fit did therefore not hamper the use of this
procedure. The segmentation boundaries are colour coded and presented in the horizontal line in
Figure 7.6.
The fit procedure was also tested on a carbonate sample consisting of a macro-sucrosic dolomite.
The BSE image and the associated histogram, and the Gaussian fits are presented in Figure 7.7.
The procedure provided in this case the appropriate segmentation levels, because the grey-level
histogram largely resembles that of sandstones.
Greylevel histogram
0.06
BSE Histogram
Relative Frequency
0.05
Pores Fit
Dolomite Fit
0.04
Calcite Fit
Pore space
Clays
0.03
Dolomite
Calcite
0.02
Heavy Minerals
0.01
0.00
0
50
100
150
200
250
Grey-value
Figure 7.7
Gaussian fits
7.4
BSE image from a carbonate sample and the histogram with the three
Micro-porosity in the clays
Pore space has an average grey-value of 8 and quartz has an average value of 128 in the BSE
images that were used in this study. These grey-values were obtained from an excitation volume
of approximately 0.5 micron in diameter as was explained in chapter 6. Consequently, the greyvalue that arises from an excitation volume that contains both pores and quartz will lie in the
range between 8 and 128. Special attention should be paid to the grey-levels that fall in this
range, because this is the range for pixels that represent clays. The porosity represented by the
small pores in the clays can only be obtained from this region by using the concept of total
porosity as defined in Chapter 2. In chapter 5 we attributed the observed deficit in porosity
estimation from thin sections to the small pores in the clays.
The total porosity concept encompasses all pore space that is present in the sample, including the
accessible as well as the non-accessible pores. Accessible porosity is defined as the pores that
are connected and can thereby conduct fluids. Consequently, the accessible pores can be
reached by the epoxy in the impregnation phase of the sample preparation procedure, and can
thus be viewed on BSE images. The non-accessible porosity consists of enclosed cavities in the
rock matrix that are the result of diagenetic events and that we discussed in chapter 2. The
diagenetic changes excluded these pores from the pore system over geological times. These
pores are currently not accessible because they are not part of the connected pore system and
thereby not of interest from a petroleum engineering point of view.
88
Chapter 7
Pores with dimensions smaller than 0.5 micron are below the resolution of the BSE images and
cannot be analysed. However, from visual inspection with the SEM in Secondary Electron (highresolution) mode, we found that pores with sizes down to several nano-metres in diameter can
still be filled with epoxy. In general, these small pores exist either as wedges at grain to grain
contacts or as pores in clay structures. In this study, we shall call these small pores micro-porosity
and propose an operational porosity model, which is defined as:
Φ total = Φ accessible + Φ non − accessible = Φ effective + Φ micro + Φ non − accessible
7.12
The conventional buoyancy porosity measurements carried out in the core-laboratory provide
accessible porosity. The wireline logs provide the total porosity including the non-accessible
porosity. The non-accessible porosity cannot be observed in BSE images because these pores
are not filled with epoxy. Therefore, the technique developed here provides a porosity that is
equal to the accessible porosity. We conclude that the porosity that can be derived from
SEM/BSE analysis reflects the accessible porosity and consists of the sum of the effective
porosity and the micro-porosity:
Φ SEM / IA = Φ effective + Φ micro
7.13
Mineralogically, clay consists of aluminium silicates with approximately the same atomic density
as quartz. The grey-value of clays falls in the range between the quartz peak and the pore space
peak because of the small pores that are present in the clays. Hence, the grey-value range
between the peak of the pore space and the peak of quartz is defined as the clay region:
f ( x , y ) = Clay → ( µ pores + 4σ pores ) ≤ f ( x , y ) ≤ ( µ quartz − 4σ quartz )
7.14
An exception to the rule of clay particles with sub-micron sizes is the clay type kaolinite, which is a
pure aluminium silicate. Kaolinite occurs in a plately habitus (see Chapter 5.5.3) in which the
plates can have dimensions from 0.5 to 4 micron. Consequently, kaolinite can be resolved on
BSE images with resolutions between 0.4 and 0.6 micron. However, as kaolinite contains both Al
and Si , it will have a slightly lower BSE coefficient than quartz, and therefore the grey-value of
solid kaolinite will fall in the clay-range that is defined above.
Figure 7.9 shows a BSE image of a sandstone in which a pore is filled with kaolinite. The
associated histogram is presented in Figure 7.10, which demonstrates that the grey-value is
slightly lower than that of quartz. The peak caused by the kaolinite booklets is situated at the low
side of the peak representing quartz, precisely in the region that we designated for clay.
In Chapter 5, we introduced the concept of micro-porosity, being defined as the porosity that
cannot be resolved by optical microscopy in thin section analysis. At this point, we extend that
definition to SEM/BSE images. The pores in BSE images with dimensions below 0.6 micron
cannot be resolved because of the limited resolution. Therefore, we assign pores with dimensions
below 0.6 micron to the micro-porosity. Conversely, the pores with dimensions above 0.6 micron
can be resolved in BSE imaging and are therefore assigned to effective porosity.
89
Figure 7.9
Example of pore filling kaolinite
0.07
0.06
Relative frequency
Chapter 7
Quartz
0.05
0.04
0.03
0.02
0.01
Kaolinite
Pore space
Clay region
0.00
0
50
Heavy
minerals
Feldspar
100
150
200
250
Grey-value
Figure 7.10
in Figure 7.9
Grey-level histogram from the BSE image
90
Chapter 7
When only a low amount of micro-porosity is present in the excitation volume, the average greyvalue of the associated pixels will be near the quartz value of 128. In contrast, when a high
amount of micro-porosity is present in the excitation volume, the average grey-value of the pixels
will be near the pore space value of 8. We assume that the amount of micro-porosity in the
excitation volume is a linear function of the grey-value in the clay region, i.e. between the peaks
for pore space and for quartz. Mathematically this is defined as:
µ QZ − 4 σ QZ


1
 µ QZ − 4σ QZ −

,
f
x
y
(
)
n µ∑


Φ +4σΦ
=
 Area%Clay
µ
4
σ
µ
4
σ
−
−
+
(
)
QZ
QZ
Φ
Φ






(
Φ micro
)
(
)
7.15
in which
1
µ QZ − 4 σ QZ
n
∑ f ( x , y)
7.16
µΦ +4σΦ
is the mean grey-value of the n pixels in the image that fall in the grey level range for clays. Figure
7.11 depicts the porosity model as defined above. The amount of micro-porosity in clays can vary
substantially, from a few percent in dense clays like smectite, up to 85% in non-compacted clays
such as fibrous Illite.
Porosity
1.0
Feffective
0.0
Fmicro
mQZ - 4sQZ
mΦ + 4sΦ
Grey-value
Figure 7.11
Effective porosity and micro-porosity
expressed as grey-value regions in BSE images
We assumed in the above porosity model that the micro-porosity is solely associated with clays.
However, the wedges, the small pore ends at the grain to grain contacts, also contribute to the
grey-level range that we assigned to clays. This is caused by the size of the wedges, which are
too small to be resolved in BSE images with a resolution of some 0.6 micron.
In our porosity model the wedges belong to the effective porosity and not to the micro-porosity.
The porosity contribution of the wedges should therefore be added to the effective porosity. The
solution to this problem was found in a correction function derived from our set of calibration
samples. The calibration samples are totally clay-free, because they were constructed from glass
bead mixtures. In order to investigate this, we selected a set of 6 glassbead samples with
increasing porosity and permeability. The permeability is controlled by the magnitude of the
91
Chapter 7
mixing fractions of glassbeads with different sizes. The calibration samples were impregnated and
subjected to the same procedures as was used for the reference and validation sample sets. BSE
images were collected and subjected to the developed processing and analysis. The glassbead
samples with lower permeability contain a higher amount of small glassbeads than the samples
with higher permeabilities. Consequently, the samples with lower permeability show a higher
number of pores and consequently a higher number of wedges.
Figure 7.12 shows this effect in the BSE images from two samples from the calibration sample
set. This effect was quantified by calculating the number of pores and the area percentage of the
regions with grey-values that do fall in the clay range. The cross-plot of these parameters for the
six glassbead samples is presented in Figure 7.13.
Two samples from the calibration sample set
Glassbead calibration samples
1.4
1.2
1
Areal% Clay
Figure 7.12
0.8
2
R = 0.73
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
# of Pores
Figure 7.13
Correction function for porosity
associated with wedges
92
800
900
Chapter 7
The trendline defines a linear relation between the number of pores and the area percentage clay
with an explained variance (R2) of 0.73. In this way, we established a relationship between the
amount of pores and the associated effective porosity in the wedges that otherwise erroneously
would have been assigned to micro-porosity.
For the glassbead sample with the highest amount of pores, the correction is approximately 1
P.U. (Porosity Unit). When this correction is applied to a shaly sample from the reference sample
set with 10% to 20% clay, this may be negligible, however, for a sample with 2 % clay it can be
substantial. We refer to this issue in chapter 8, where the results of the statistical models for
prediction of petrophysical parameters are discussed.
7.5
Conversion from BSE images to DEF images, overview
The total procedure starting with a Sigma filtered BSE image and ending with a corrected DEF
image is depicted in the flow chart that is shown in Figure 7.14. Note that the segmentation by
thresholding has to be carried out for each constituent, except for quartz, which was found by
using the mass balance principle, hence the quartz is the inverse of then pores. Feldspars, clays
and heavy minerals are inserted into the DEF image using the modified addition method that was
explained in chapter 5. In addition, different morphological filters are applied to the different
constituents.
We concluded that a robust and repeatable, semi-automated method was developed to extract
the pore space, clays and rock matrix minerals from BSE images, which was based on verified
statistical and image processing algorithms. The DEF images that were produced by this
processing string were suitable for the extraction of image features that in turn will be used for
prediction of porosity and permeability.
93
Chapter 7
Segment porespace, scrap 128, invert, scrap 128
BSE, s-filtered
Modified
addition
Opening
Segment clays, opening operation with hexagon
Scrap 128
Modified
addition
Segment feldspars, dilate & truncate with pore space
Modified
addition
Segment heavy minerals
Figure 7.14
Final DEF image
The procedure that converts a BSE image into a DEF image
94
Chapter 7
7.6
Feature extraction from DEF images
The transformation from a BSE image into a DEF image was required to evaluate the image
features of the constituents quantitatively. Various features related to the pores, clays,
quartz/dolomite, feldspar/calcite and heavy mineral fractions were extracted from the DEF image
and subsequently measured, which is schematically shown in Figure 7.15.
Measurement
global
parameters
Constituent
extraction
Repeat for all 5 constituents
DEF image
Figure 7.15
Data
base
Measurement of the entities in the DEF images
The quantitative evaluation comprised the measurement of global and local geometric and
intensity properties. Intensity parameters includes the moments of the grey-value distribution
functions such as the mean grey-value per object or, in case of global intensity parameters,
summed over all objects in the image. In this study, we defined global parameters as parameters
that provide one number per image. This includes the area and the perimeter summed over all the
objects in the image. It also includes the percentage area encompassed by all objects in the
image and the number of objects. Local geometric parameters include area, perimeter and
diameters of the individual objects in the image, e.g. the pores and grains.
The global properties were employed in the prediction of petrophysical parameters like porosity,
permeability, formation factor and shalyness. The local properties were used in the prediction of
permeability according to the Hagen-Poiseuille model that was discussed in chapter 2. In addition,
local properties were also used to estimate capillary pressure curves from images and this will be
presented and discussed in chapter 8. Table 7.2 shows the global geometric and global intensity
parameters that were measured.
Pore space
Clays
Quartz/dolomite
Feldspars/calcite
Heavy minerals
Table 7.2
Area%
Perimeter
Count
[
[
[
[
[
[
[
[
Greyval
low
Greyval
high
Greyval
Mean
[
[
[
Global parameters of the various entities that are measured
The definition of the parameters is depicted in Figure 7.16. Averaging was applied over the
images analysed from each sample to obtain one number for each of the global properties. In the
case of local properties, the data from the four images were accumulated into a population. From
the populations, statistical moments were extracted for further analysis. The image features that
were used in this study were defined as follows:
95
Chapter 7
Parameter
Depicted
Local
1 object of n pixels
Global
N objects per image
n
N
Area
∑ pixel
∑ Area
Perimeter
Perimeter algorithm
∑ Perimeter
i =1
i
i =1
i
N
i =1
i
N
∑ Area
Area%
i =1
i
image _ size _ x × image _ size _ y
min[ pixel _ clay ]
Low grey-value clays
max[ pixel _ clay ]
High grey-value clays
1
N
Mean grey-value clays
Figure 7.16
7.7
1 n

 ∑ pixel _ clayi 
∑
j =1  n i = 1
j
N
Definition of the parameter that are measured in the DEF images
Pore size distributions, from DEF to POR images
In chapter 2, we discussed pore structure models and the concept of pore size. We also stated
that the definition of pore size depends primarily on the model that is used to describe the pore
system. We argued that any measurable parameter reflecting pore size can be used in a model to
predict a reservoir property, provided that the model can accommodate this parameter. For
instance, if we use a Hagen-Poiseuille type model for prediction of permeability, we obviously
would use the pore diameter distribution because Hagen-Poiseuille theory is based on the radius
of circular tubes or capillaries. Hence, to extract a distribution of diameters of capillaries from the
pore space we have to convert the concave shaped pores into convex shaped tubes that
resemble capillaries, which can be measured. Following the Hagen-Poiseuille model of capillaries,
a single pore should then be represented as a bundle of capillaries. In Figure 7.17, this
conversion of a single pore into a set of capillary tubes is depicted stylistically.
96
Chapter 7
Pore
Figure 7.17
Pore filled with
convex shapes
Pore filled with
capillaries
The process of filling a pore with a bundle of capillaries
In chapter 2, we also announced that capillary bundle models will be used for the prediction of
permeability. Therefore, we developed a process in which the pore space in the DEF images was
converted into sets of capillary tubes. In addition, we merged these tubes into bundles in which
each bundle represents a single pore.
In this process, we applied mathematical morphological and Boolean operations on binary images
that represent the pore space. The resulting capillary tubes were grouped into bundles such that
they fit the pores from which they were extracted. We termed the resulting image with the bundles
a POR (Pores) image. Figure 7.18 presents the flow chart that shows how a DEF image is
transformed into a POR image.
N=N-1
No
DEF
image
Pore
space
N = 20
N opening
operations
Subtract from
pore space
Add in TUB
image
N = 0?
Yes
Merge
tubes
POR
image
Figure 7.18
Flow chart showing the steps in the procedure that converts a DEF image
into a POR image
The process to transform a DEF image into a POR image entails the use of successive opening
operators on the binary representation of the pore space. The opening operator has an important
property that makes it very suitable for the conversion of pores into sets of tubes. This property is
the smoothing effect and facilitates the transformation of the concave shaped pores into convex
shapes tubes. The binary pore space was subjected to N opening operators with a hexagon
structuring element of 5 x 5 pixels representing 5 x 5 micron. The hexagon operator was chosen
because it best resembles a circle and thereby enables the conversion of the concave shaped
pores into sets of convex shaped tubes. We started with N=20 opening operations that resulted in
the survival of the largest pores, now represented by convex shapes that largely resemble tubes.
These tubes fit entirely inside the pores and therefore fulfil the requirement that no extra pore
97
Chapter 7
space was generated in this process. The largest tubes were assigned to the largest class of
pores and were included into a new image, which we term a TUB image. Following, the largest
tubes were subtracted from the binary pore space. The remaining pore space was then subjected
to N-1 opening functions, which resulted in an image with the one but the largest tubes. These
one but largest tubes were also added into the TUB image by a Boolean OR operation and pixelwise subtracted from the binary pore space. This sequence was repeated until the smallest pores
had been processed.
Pores in DEF image
DEF image
Figure 7.19
Pores filled with capillaries
Capillaries merged into a pores
TUB image
POR image
The conversion of a pore into a bundle of capillaries depicted in images
Figure 7.19 shows a process step for a single large pore. The left image in Figure 7.19 is the DEF
image from which the pore space is extracted. The centre image is the TUB image in which the
tubes resulting from the 20 opening functions are coded in different shades of blue; the largest
tubes have the darkest shade of blue. The right-hand image in Figure 7.19 shows the resulting
POR image. All the tubes that belong to one pore are assigned to the class of the largest tube and
colour coded accordingly. This is based on the capillary behaviour of such a pore, when a fluid
interface enters this pore, it would fill the whole pore at once. Therefore, we consider this bundle
as one total pore. This approach has another important advantage; we exclude the pore wall
roughness that is coded in red in the TUB and POR images. The pore roughness contribute very
little to flow and hence very little to permeability. Literature shows that for Darcy flow, the flow
velocity near the pore wall is zero [Liu & Masliyah, 1996]. Therefore, we argue that the roughness
that we observe in the images does not contribute to Darcy flow. Consequently, the roughness
should be excluded from models for prediction of permeability. One can argue that just a single
hexagon opening operator would suffice to arrive at the POR (capillary bundle) image shown in
right-hand image in Figure 7.19. If this would have been the only objective, then this is indeed the
case. However, the intermediate stage, the TUB image, was required to produce a range of
capillaries with different diameters. The latter was used to simulate capillary pressure curves and
to populate Hagen-Poiseuille type models for prediction of permeability.
Experiments were carried out to determine the optimal number of opening operations. For these
experiments, we selected two samples, one a coarse, arkosic sandstone and the other a fine
grained, well-sorted sandstone. These samples were considered representative of both the
reference sample set and the validation sample set with respect to the total range of porosity and
permeability values. We found that with 20 classes, which resulted in maximum 20 successive
opening functions, the large pores were properly covered. In this way, the tubes encompassed the
full range of pore sizes in the reference sample set. Figure 7.20 shows the normalised area of
each pore class versus the number of applied openings for these two samples. Normalisation was
achieved by setting the largest pore class area to unity and scaling the rest accordingly. The pore
area class number was defined as the number of opening operations that led to the largest tube in
98
Chapter 7
the pore. The pore size then follows from the sum of the area of the tubes residing in a particular
pore as shown in figure 7.19 and is described as:
n
Area pore = ∑ Area tube _i
7.16
i =1
Pores fine sample
1
Pore class area (normalised)
Pore class area (normalised)
Pores coarse sample
0.9
0.8
0.7
Φ = 25.7 %BV
k = 880 mD
0.6
0.5
0.4
Pore classes
0.3
Roughness
0.2
0.1
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
1
0.9
0.8
Φ = 15.7 %BV
k = 75 mD
0.7
0.6
0.5
0.4
Pore classes
0.3
Roughness
0.2
0.1
0
1
2
3
Number of openings (Pore size)
Figure 7.20
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21
Number of openings (Pore size)
Pore area classes for two representative samples from the reference sample set
In Figure 7.20, the 20 classes are shown in blue and the remaining roughness class is shown in
red. The coarse sample has a wide distribution dominated by the largest pore class. The fine
sample has a narrow distribution, in which the smaller pores dominate and larger pores do not
occur.
POR image
DEF image
Roughness
1
Class
20
No. of openings
Figure 7.21
Transformation of a DEF image into a POR image, note that the pores
are colour coded according to the area class they belong
The roughness area in the fine sample with low permeability is substantially higher than that in the
coarser sample. Since these two samples were considered end members with respect to pore
99
Chapter 7
size distribution, we concluded that 20 opening functions are adequate for the conversion of pores
into sets of capillary bundles. Figure 7.21 shows the POR image next to the DEF image of the
coarse sample. The pores are coded with different shades of blue according to the pore area
class. In the subsequent quantitative analysis, the area of each pore class was measured and this
information was used in the prediction of permeability that will be discussed in chapter 8.
7.8
Pore networks, from POR images to NET images
The pore network reflects the connectivity of the pores. A pore network consists of pore bodies
that are interconnected by pore necks, which is based on the concepts we defined in chapter 2,.
Pore necks are the constrictions a fluid particle has to pass when travelling from one pore to the
next. In 3D, these pore necks can easily be observed. However, BSE images are 2D
representations of the 3D pore network, which makes it much more difficult to locate and assess
the pore necks. The probability that the narrowest part of a pore neck is intersected in 2D images
is extremely small [Dullien, 1979; Scheidegger, 1972; Koplik & Lassiter, 1982]
Many researchers have tried to configure 3D pore network models from 2D observations of
reservoir rock. These attempts all followed the same track; 2D properties were measured and
represented as distributions. Random sampling from these distributions was then used to
configure the 3D models. The 2D to 3D conversion was intrinsically entailed in the sampling
process, because areas were converted into volumes based on an assumed shape for the pore
bodies. The more-difficult-to-obtain pore neck properties were treated similarly. Lengths of lines in
constrictions in 2D were assigned to the diameters of the tubes.
It was assumed that a section through these 3D models is statistically equal to the images from
which the information was obtained. Note that no real 3D information was used and the accuracy
of the predictions for permeability with these models depended strongly on the requirements of
heterogeneity and isotropy of the samples. However, one can argue that when these
requirements for the geometry of the pore system are met, then the conditions should also be
valid for the connectivity of the pore network. Hence, the pore network that can be observed in 2D
BSE images may be representative of the actual pore network in 3D. Based on this reasoning we
will investigate in chapter 8 whether there is a relationship between the 2D pore network
parameters and the actual 3D permeability. First, we will discuss the development of a method to
convert the POR image into an image in which the pore network can be quantified. From this
network image, which we term a NET image, we can extract the network parameters that will be
used for permeability prediction in chapter 8. This network image will be based on the skeleton of
the pore system that can be observed in the images.
A traditional network consists of nodes that are interconnected by bonds. In such a general pore
network [Koplik & Lassiter, 1982], the nodes are represented by the pores while the bonds are
associated with the pore necks. It is not possible in 2D BSE images to accurately determine the
3D pore neck properties and therefore we have to amend the definition of a pore network.
In this study, we defined a network as the web of lines that run through the pores equidistant from
the pore/grain boundaries and follow the grain-to-grain boundaries in the rock matrix. We also
state that the pore network should percolate from one side of the image to the other side and that
all bonds should be connected to the nodes. We finally impose the condition that nodes can only
be present in the pores.
100
Chapter 7
Based on the above conditions, we developed a set of image processing steps that produced a
2D pore network, which can be used for permeability prediction. Figure 7.22 shows a flow chart
reflecting this procedure that converts a POR image into a NET (Network) image.
Pore space
Skeleton
POR
image
Merge
Rock matrix
Define
bonds
and
nodes
Raw
NET
image
Closing
Binary
thinning
Skeleton
Remove
matrix
nodes
Remove
obsolete
matrix
bonds
NET
Image
Figure 7.22
Flowchart of the process that facilitates transformation of a POR image
into a NET image
The total pore space, encompassing the 20 pore classes and the roughness, was extracted from
the POR image and converted into a binary pore image in which all pore pixels are 255 and all
matrix pixels are 0. Next, the pore space was subjected to binary thinning to obtain the pore
skeleton, which is the part of the network that is only present in the pore space. In order to meet
the condition of percolation, the pore skeleton had to be connected through the rock matrix
following the grain-to-grain boundaries. Thus, to obtain the full network, the bonds in the
complement of the pore space, the rock matrix including the clays, was also subjected to a
thinning process. This thinning process differed from the one that we used for the pore space and
was based on the watershed transformation, which is a combination of region growing and edge
detection. [Serra & Soille, 1994; Soille, 1998]. The two resulting skeletons were combined into a
new image by using a Boolean OR operation. It may happen that the pore skeleton and the matrix
skeleton do not connect precisely. Therefore, we applied a closing operation that was followed by
binary thinning in order to arrive at a network with loops that are totally closed. The remaining
loose parts of the network were removed in this process as well. The left-hand image in Figure
7.23 presents a POR image from which the NET image at the right was obtained. In the NET
image, the bonds falling in the rock matrix are colour coded blue, and the bonds falling in the pore
space are coded in yellow. The nodes are coded in red. Precautions were taken that only the
pores can contain nodes.
101
Chapter 7
POR image
Figure 7.23
Net image
A POR image and the associated NET image
The pore network can be used in two ways. First, we correlate the statistics of the bond-and nodeproperties with permeability. The bonds and nodes are also used to calculate the co-ordination
number, which quantitatively describes the pore connectivity, and was defined in chapter 2.
Second, conductivities can be assigned to the pore bonds and the matrix bonds, upon which the
conductivity of the total network can be calculated. This conductivity can then be related to the
measured permeability.
We applied the same averaging approach that was used in the measurements of the image
features from DEF images to merge the data from the four images that were available from each
sample. Table 7.3 shows the parameters that were measured from the pore network and will be
used in the prediction of permeability in chapter 8.
Length
Matrix bonds
Pore bonds
Pore nodes
Pores in network
Table 7.3
7.9
[
[
Count
[
[
[
[
Area
Coding NET image
[
Blue
Yellow
Red
Dark blue
Parameters that are extracted and measured from the NET images
Representativity, selection of the number of images
We investigated the number of BSE images that were required to obtain statistical validity in the
same way as for thin section analysis, in which the appropriate magnification and the number of
images were selected. The selection of the magnification of 35 X was already discussed in
Chapter 6. The main criterion was that images from even the coarsest sandstone should show a
sufficient number of grains and pores to be considered representative. At an image size of 2048 x
2048 pixels, and a set image resolution of 1 micron, the scanned field of view is approximately 2 x
102
Chapter 7
2 mm. The scanned field of 2 x 2 mm for a BSE image is the maximum because 35 X was the
minimum magnification the SEM could accommodate. This boundary condition implied that
samples that contain heterogeneities larger than 2 mm could not be imaged and evaluated
properly. In the sample sets we used in this study, only three samples were found with such
heterogeneity. These samples contained extremely large grains with diameters larger than 1 mm.
Therefore, these samples were excluded from the study. In discussion with sedimentologists, it
arose that these coarse sandstone reservoirs occur only very rarely. It should however be noted
that the current method is not applicable to extremely coarse sandstone samples when SEM
instruments with a minimum magnification of 35 X are used.
The total area that could be scanned at the cross-section from a standard 1-inch plug is
approximately 500 mm2. The area encompassed by one BSE image is only 4 mm2 and thereby
represents less than 1 % of the total plug area. In order to determine the number of images
required for sufficient representativity of the measured image features, a set of images were
collected from two typical samples. These samples were considered best representatives of end
members over the total range of porosity and permeability. These samples were the same
samples that we used for the determination of the number of opening functions to convert DEF
images into POR images. Figure 7.24 shows the BSE images from these two test samples.
Figure 7.24
BSE images of two samples from the reference sample set that are
considered representative of the total grainsize range
From the cross-section of a standard 1-inch plug, 64 square BSE images can be collected.
Because of manpower and disc-space restrictions, 24 BSE images, representing some 38% of
the total area available for imaging, were collected from each sample. Furthermore, it was
assumed that a sufficient number of images had been analysed when the addition of one image
more did not result in a significant change in the value of the image features. In this, the notion
"significant change" was defined as a certain cut-off value for the selected image feature. The
pore space and quartz are the dominant reservoir rock constituents, we therefore selected their
area percentage as the image features to test significant changes. Other global geometric
parameters, such as the perimeter of the pore space or the clays, depend very strongly on larger
scale heterogeneity, and were therefore considered not sufficiently descriptive for this type of
analysis. The mean value of the areal percentage was calculated for 2,3,4,…12 images. We
103
Chapter 7
calculated the average µ of k images over n rotations of the 24 available images. This is
mathematically expressed as:
1 i = n + k −1 mod 24
∑ x in which k = 2 ,3,....,12 and n = 1,2 ,3,....,24
k i = n mod 24i
µ k ,n =
7.17
From the resulting population of averages of k images, we calculated the average absolute
deviation of the mean δ as follows:
δk =
24
n
n =1
i =1
∑ ∑x
i
− µ k ,n
7.18
24 ⋅ n
The result of this analysis is presented in Figure 7.25 for both the coarse and the fine sample
respectively.
3.0
1.0
2.5
Quartz, areal%
Pore space, areal%
Average absolute deviation, coarse test sample
1.2
0.8
0.6
0.4
2.0
1.5
1.0
0.5
0.2
0.0
0.0
0
5
10
0
15
# of images
5
10
15
# of images
3.0
1.0
2.5
Quartz, areal%
Pore space, areal%
Average absolute deviation, fine test sample
1.2
0.8
0.6
0.4
2.0
1.5
1.0
0.5
0.2
0.0
0.0
0
5
10
0
15
# of images
Figure 7.25
5
10
# of images
Average absolute deviation versus the number of images
104
15
Chapter 7
For the fine test sample the average absolute deviation of areal percentage pore space
decreased from around 0.4 %BV for two images, to a value of around 0.1 % for 5 images.
Addition of more images did not change the average absolute deviation substantially. A similar
decreasing behaviour was observed for the areal percentage of quartz in the fine test sample. At
5 images the average absolute deviation decreased to a value around some 0.25%. The small
variation was caused by the fine-grained nature of this test sample, which contains an average
500 pores per image.
However, this level of stability was not reached for the coarse test sample. The area percentage
pore space in the coarse sample showed a steep descent until 4 images. For more images, a
gradual reduction of the absolute deviation occurs. In the coarse sample, an average of 200 pores
was counted per image. A similar behaviour was seen at 5 images for the area percentage quartz.
From this analysis, we concluded that the minimum number of images that was required is
probably some complicated function of the “coarseness” of the sample. As it was impractical to
collect a tuned amount of images for each sample, (some 260 samples) we selected 4 images per
sample as an operational standard. This number was a trade-off between statistical validity,
limited memory storage capacity (each image is 4 Mb in size amounting to 4.8 Gb for the two data
sets) and the required effort in manpower.
105
Chapter 8
Petrophysical parameters from BSE images
8
PETROPHYSICAL PARAMETERS FROM BSE IMAGES
8.1
Introduction
In chapter 5 we discussed the prediction of porosity and permeability from thin section images
that were obtained with optical microscopy. Following the discussion in chapter 5, we concluded
that the uncertainty in the predictions is caused by the poor definition of the pore system in the
images. We showed that this is an inherent property of images obtained with optical microscopy
and suggested that this uncertainty may be reduced substantially by using images obtained with
electron microscopy. In chapter 6 we discussed electron microscopy and explained why BSE
images are more suitable for the prediction of petrophysical parameters than optical microscopy
images. The process of extracting the pore space and mineral fractions, the pore size distribution
and the pore network from BSE images, and the subsequent analysis of these was discussed
chapter 7. This process of extracting quantitative information from BSE images is schematically
shown in Figure 8.1.
BSE image
Processing
Processing
Processing
Measurement
Global features:
Areal %, Perimeter
Count, Grey-value moments
Figure 8.1
NET image
POR image
DEF image
Measurement
Pore features:
Area
Measurement
Pore network features:
# Pore bonds, # Matrix bonds
# Pore nodes
Various stages in the image processing and the extracted features (# = number of)
This chapter deals with the models for the prediction of porosity, permeability, clay content and
formation resistivity factor from BSE images, and will demonstrate that the predictions based on
BSE images are more accurate than those obtained with optical microscopy. The predictive
models for porosity and permeability are based on the theoretical models that were discussed in
chapter 2. For porosity estimation, we suggested that inclusion of micro-porosity should
substantially improve the accuracy of the porosity prediction, particularly for shaly sandstones.
The relation between the plug porosity and the plug permeability will be investigated to assess
whether the permeability can be reliably predicted from plug porosity alone. The prediction of
106
Chapter 8
permeability from BSE image data will start with using the classical Hagen-Poiseuille model and
the pore size distribution that was obtained from the POR images. Next, the Kozeny model with
the extension as proposed by Carman will be discussed. Permeability can also be predicted from
the pore network that includes information about the connectivity of the pores.
We will also introduce a model for the prediction of the formation resistivity factor from BSE
images. This model is based on similar assumptions as the model that we will use for
permeability. Finally, the relation between image derived clay content and plug measured Qv is
discussed.
Figure 8.2 depicts the process that will be followed to define various models for the prediction of
petrophysical parameters from BSE images.
BSE imaging
+
analysis
Impregnated
sample
Image features
Physics based model
&
Statistical
correlation
Physical
measurements
Regression
model
Petrophysical
parameters
Cleaned
sample
Figure 8.2
8.2
Flowchart of the process for the definition of a regression model
Porosity
The concept of porosity that we will use for granular media such as sandstones was introduced in
chapter 2. From this, a porosity model was devised that we used to derive porosity from images.
In chapter 5, we showed that porosity can be extracted with only moderate accuracy from thin
section images obtained with optical microscopy. From the discussion in chapter 5 we concluded
that the underestimation of porosity from thin sections, when compared with porosity measured on
core plugs, is attributed to the poor definition of the pore system. We therefore concluded that the
inadequacy for porosity prediction from thin section images is in fact a (depth) resolution problem.
In order to arrive at a better definition of the pore system we introduced electron microscopy. This
instrument when used in BSE mode, has a spatial resolution of 0.5 micron that is comparable with
the limit of optical microscopy. The 0.5 micron resolution was confirmed by Monte-Carlo
experiments that were discussed in chapter 6 and in Appendix A.
In the BSE images that we used in this study, the spatial resolution is approximately 1 micron per
pixel, and hence a factor four better than that of thin section images. However, the definition of the
pore system in a BSE image is substantially better than that in thin section images because the
depth resolution in BSE images is approximately 0.5 micron compared to the 30 micron of thin
sections. Given the spatial resolution being four times better and the depth resolution 60 times
better than in thin section images, we conclude that the definition of the pore system from BSE
107
Chapter 8
images is superior to that of thin section images obtained with optical microscopy. We argue that
this should also lead to a more accurate estimation of porosity provided the homogeneity and
isotropy requirements are met.
In chapter 7, we defined an operational model for porosity derived from BSE images, the SEM/IA
porosity:
Φ SEM / IA = Φ effective + Φ micro + Φ wedges
8.1
in which:
Φ effective = Area% Porespace
8.2
The area percentage pore space encompasses the pores that can be directly observed on BSE
images. As stated in chapter 7, the micro-porosity can only be determined indirectly. The microporosity is a linear function of the grey-level range that entails the clays and the wedges. The
micro-porosity was derived in chapter 7 (eqn. 7.15):
Φ micro
(
(
)
) (
 µ

quartz − 4 σ quartz − average _ greyvalueclay 

=
 Area%Clay corrected
µ
−
4
σ
−
µ
+
4
σ
quartz
porespace
porespace 
 quartz
)
8.3
In which the parameters µ and σ were derived from the semi-automatic fit procedure that uses
the three Gauss functions fitting the grey-level peaks for pore space, quartz/dolomite and
feldspar/calcite in the histogram. The average_greyvalueclay is the mean grey-value calculated
over all pixels that fall in the grey-value range between the peak for pores and the peak for
quartz/dolomite. The parameter Area%Claycorrected is the area percentage clay that is corrected
for the effect of the wedges we discussed in chapter 7, and was derived from the glassbead
samples:
Area%Clay corrected = Area%Clay( 0.001Porecount − 0.0714)
8.4
Which then enables us to estimate the porosity that is associated with the wedges:
Φ wedges = Area% Clay − Area%Clay corrected
8.5
8.2.1 Porosity estimation for the calibration samples, test of the method
Clean sandstones do not contain clays and consequently there is no contribution of micro-porosity
to the total porosity. The glassbead samples that we used for the calibration of the SEM settings
in the collection of BSE images, are adequate representatives of clean sandstones, because they
consist of only two phases, quartz and pore space. This was confirmed by the occurrence of only
two peaks in grey-value histogram from one of the glassbead samples shown in Figure 8.3.
Therefore, the SEM/IA porosity for the glassbead samples is the sum of the effective porosity and
108
Chapter 8
the porosity in the wedges. The effective porosity can be seen directly in BSE images, while the
porosity in the wedges cannot be observed directly due to the resolution limit of 0.5 micron.
0.10
0.09
Quartz
Relative frequency
0.08
0.07
0.06
Pores
0.05
0.04
Pore space
0.03
Φ
0.02
Clays
0.01
0.00
0
50
100
150
200
250
Grey-value
Figure 8.3
sample
Grey-value histogram from the BSE image of a glassbead
The absence of clay means that all the pixels with grey-values in the range of the clays, indicated
by the Clays arrow in Figure 8.3, fully contribute to the total porosity. Consequently, the Pore
space arrow in Figure 8.3 encompasses the full grey-value range for SEM/IA porosity.
Figure 8.4 presents the cross-plot of the calculated SEM/IA porosity and the porosity that has
been physically measured on glassbead core plugs. Statistical analysis reveals that the explained
variance (R2) is 0.88 with a standard error of 1.65 porosity units. This example clearly shows that
it is possible to predict porosity of glassbead samples from images with a high accuracy.
Porosity
30
25
Plug Porosity
2
R = 0.88
20
15
10
5
0
0
5
10
15
20
25
SEM/IA Porosity
Figure 8.4
Cross-plot of porosity from BSE images
and the plug porosity
109
30
Chapter 8
8.2.2 Porosity prediction from reservoir rock samples
To assess the influence of micro-porosity we first calculated the SEM/IA porosity solely based on
the effective porosity observed directly in BSE images. The SEM/IA effective porosity is
comparable with the point count porosity we calculated from thin section images in chapter 5 (see
Figure 5.8). Figure 8.5 presents the cross-plot of the SEM/IA effective porosity and the plug
porosity of the 230 samples in the reference sample set.
Porosity
35
30
Plug Porosity
25
20
15
2
R = 0.68
10
5
0
0
5
10
15
20
25
30
35
SEM/IA effective Porosity
Figure 8.5
Cross-plot of the porosity directly observed
in BSE images and the plug porosity
The SEM/IA porosity underestimates the real (plug) porosity substantially. From the Qv
measurements on the samples in the reference set, we know that the samples contain varying
amounts of clay. Therefore, we state that the SEM/IA porosity substantially underestimates the
true total porosity because the contribution of the micro-porosity (pores with dimensions smaller
than 1 micron) is not included. Statistical analysis shows that the explained variance is 0.68. This
value compares with the value of 0.66 for the point count porosity from thin section images, which
also excludes the clays.
In chapter 7, we postulated a linear relation between the grey-value in the clay range in BSE
images and the amount of micro-porosity. If that relation is valid, there must also be a linear
relationship between the amount of clay and the deficit in SEM/IA porosity. Figure 8.6 presents
the cross-plot of the differences between the SEM/IA effective porosity and the plug porosity
versus the area percentage of clay (corrected for wedges). The SEM/IA porosity deficit can
indeed be approximated by a linear function of the amount of clay.
In chapter 2, we discussed porosity reduction effects caused by clay coating of grains and by
cementation. Both phenomena are diagenetic effects since minerals precipitate from the pore
waters onto the grain surface. One can argue that cements, similar to clays, can also contain
small pores with dimensions smaller than 1 micron and thereby contribute to micro-porosity.
Cements are not explicitly included in the current operational porosity model. Cements have
densities similar to those of clays and thereby have approximately the same grey-value in BSE
images. Therefore, micro-porous cements can be treated in the same manner as we treat clays.
Moreover, studies on diagenesis of reservoir rock reveal that micro-porosity is only rarely present
110
Chapter 8
in cements. We conclude that the deficit in SEM/IA porosity is attributed to the micro-porosity in
the clays and only sometimes in cements.
Porosity
Difference Plug PorositySEM/IA effective Porosity
30
25
20
15
10
5
0
0
5
10
15
20
25
Wedge corrected Area% Clays
Figure 8.6
Area percentage clays versus deficit of the
SEM/IA effective porosity
Figure 8.7 shows the cross-plot of the SEM/IA porosity, which was calculated by adding the
SEM/IA effective porosity and the SEM/IA micro-porosity, versus the plug porosity of the samples
in the reference sample set. The R2 of 0.86 explains 86% of the variance entailed in the data
points and confirms our porosity model. The increase in correlation coefficient from 0.68 to 0.93
shows that a significant relationship exists between the clay corrected porosities and the
porosities that have been measured on core plugs.
Porosity
35
Plug Porosity
30
25
20
15
10
2
R = 0.86
5
0
0
5
10
15
20
25
30
35
SEM/IA Porosity
Figure 8.7
Cross-plot of effective porosity added with
micro-porosity and wedge porosity versus the plug porosity
Since the range of porosity values in the reference sample set encompasses nearly the total
range of possible porosities that normally occur in sedimentary rock, generalisation of the model
seems justified. Therefore, we devised a predictive model for porosity that is based on multiple
linear regression. In this model we investigated the correlation between the image derived
111
Chapter 8
parameters Φeffective , Φmicro and Φwedges on the one hand and plug porosity on the other hand.
One should bear in mind that generalisation of the model outside the porosity range of the
reference sample set is risky. Statistical models give excellent interpolation results, as long as the
data point falls in the range of the reference sample set, the accuracy of the prediction will be
indicated by the standard error of the linear regression. It is less certain how accurate points that
are lying outside the porosity range of the reference set are predicted because a linear
relationship is not guaranteed in that area. Physics based models are better in this respect
because they are based on proven theory derived from first principle physics. A disadvantage of
physics based models is that these are far more complicated when high accuracy is required.
The cross-plot of Figure 8.7 is based on a simple summation of the image parameters Φeffective ,
Φmicro and Φwedges..The weighting factors of this "model" are implicitly set to unity. If we allow
these weighting factors to be free parameters and fit them in a multiple linear regression, we
arrive at a statistical model with four degrees of freedom: the three weighting factors and the
offset. Figure 8.8 presents the results of such a multiple regression analysis that was applied to
the combination of the three image derived parameters and the plug porosity. The obtained
regression equation is:
Φ SEM / IA = 0.91Φ effective + 0.94Φ micro + 7.1Φ wedges + 4.27
8.6
Porosity
35
Plug Porosity
30
25
20
15
10
2
R = 0.89
5
0
0
5
10
15
20
25
30
35
SEM/IA Porosity
Figure 8.8
Cross-plot of the porosity predicted from BSE images and the
porosity measured on core plugs
The explained variance increased from 0.86 to 0.89. This is not surprising, the four coefficients
enable more reduction of the residual values caused by the minimisation procedure that is used in
the regression. The coefficients for Φeffective and Φmicro are approaching unity, which confirms
that the statistical relation has a physical meaning. The rather high weighting coefficient for
112
Chapter 8
Φwedges indicates an underestimation. This was due to using the linear model defined in chapter 7
that was derived from the glassbeads samples to determine the porosity associated with the
wedges. The intercept value of 4.27 also confirms a systematic underestimation of porosity from
BSE images. A number of samples with errors larger than two times the standard error are
present, i.e. outside the ±2σ range around the regression line. The outliers and the reason for
their occurrence will be discussed later in this chapter.
We evaluated the processing sequence to determine whether the cause of this under-estimation
could be attributed to the processing. A prime candidate for the under-estimation was the nonlinear scrapping function that was applied to the pore space and to the rock matrix, to reduce the
effect of damage caused by the sample preparation procedure. This was described in chapter 7
section 7.2.3. We selected a number of samples with very limited preparation damage, and
subjected these to the total processing and analysis sequence but without the scrapping function.
We found that the difference in the value of the effective porosity with and without scrapping
function was less than 0.3 P.U. (% BV or Porosity Units). We therefore concluded that the
scrapping function cannot be held responsible for the offset of 4.27 P.U. in the predicted porosity.
The non-linear morphological filtering used in the processing could also be ruled out because it
was not applied to the pore space. Finally, the phase segmentation was revisited, as this was the
only processing candidate left. The automatic phase segmentation was compared with manual
evaluations for a limited number of samples with varying porosity values encompassing the total
porosity range. The difference in effective porosities between the two methods was less than 0.5
P.U. We concluded that currently we have no physical explanation for the rather high offset of
4.27 P.U. and we therefore recommend further work to assess the reason for this offset.
8.3
Permeability
The concept of permeability was discussed in chapter two, where it was argued that permeability
is a function of the pore geometry and the pore connectivity, both of which can be partly observed
in BSE images. We state ‘partly’ because BSE images are 2D representatives of what is
fundamentally a 3D pore system. This means that prediction of permeability from images will
suffer from a lack of dimensional information and consequently errors are introduced in the
predictions. These errors are minimised when the images are sufficiently representative of the 3D
pore system. This can only be the case when the image features that are used in the predictions
are independent of direction and of scale of the observations. Therefore, we state that image
features can be used for prediction of permeability only when the two requirements of
homogeneity and isotropy have been met.
Permeability is directionally dependent and thereby a tensorial property. However, when we limit
ourselves to non-fractured sandstone reservoirs, and approach it from a geo-sciences point of
view, only two permeability components are of interest: the horizontal and the vertical
permeability. The horizontal permeability is defined perpendicular to the borehole wall and the
vertical permeability is defined parallel to the borehole wall. The tensorial permeability is then
reduced to two scalar permeabilities. This can be translated to the permeabilities that are
measured on conventional core plugs of which two types are available: horizontal and vertical
plugs. The permeability measured on core plugs is just a scalar property because it is measured
in one single direction as is depicted in Figure 8.9.
113
Chapter 8
DP
Direction of flow
L
Figure 8.9
SEM/IA permeability is determined perpendicular to the image plane
In this Figure, it is also shown that the BSE images that we use in this study are taken
perpendicular to the measurement flow direction. This is in the same direction for which we
defined the capillary bundle models in chapter 2. Therefore, we argue that the concept of capillary
bundles we introduced in chapter 2, is adequate to predict permeability from images. The capillary
bundle concept reduces the porous medium to a one-dimensional system, and enables us to
relate the scalar permeability to 2D geometric properties of the pore system.
8.3.1 Relationship porosity and permeability, the k-Phi relation
In chapter 5, we predicted permeability from thin section optical microscopy images using a
Kozeny-Carman type model. The prediction from images showed an explained variance of 0.69,
which compares unfavourably with the value of 0.80 when the permeability was predicted solely
from the plug porosity. The high value of 0.80 for the k-Phi relation was explained by the fact that
all samples originated from the same environment of deposition. The samples in the reference
sample set originated from a variety of environments of deposition as was discussed in chapter 4.
This should disqualify the use of a generalised, simple k-Phi model based on a power law for
permeability prediction. Therefore, prior to discussing the results of the predictions of permeability
using the various models defined in chapter 2, we will investigate the k-Phi relationship for the
reference sample set. Figure 8.10 shows the cross-plot of the plug porosity and the plug
permeability.
Statistical analysis showed that the explained variance has a value of 0.46 and this is considered
too low for operational use of the k-Phi model to predict permeability. In Figure 8.10 a significant
scatter is observed. For instance, a porosity of 25 %BV can lead to a permeability between 3 and
3000 mD, a variation of 3 orders of magnitude. Next to the diversity caused by the different
environments of deposition, we put forward a second hypothesis to explain the poor relation
between porosity and permeability. The presence of clay in the samples leads to very small pores
that we assigned to micro-porosity in chapter 7. From Darcy’s law it can be deduced that the fluid
flow velocity is zero at the pore wall. In small pores, the distance from the centre of the pore to the
pore wall is very small. We argue that in these small pores, the flow velocity is negligible, and that
these small pores do not contribute significantly to the flow. Hence, micro-porosity does not
contribute to permeability.
114
Chapter 8
Permeability
10000
Plug Permeability
1000
100
10
1
2
R = 0.46
0.1
0.01
0
5
10
15
20
25
30
35
Plug Porosity
Figure 8.10
permeability
Relationship plug porosity and plug
The samples in the reference sample set contain different types of clay and associated microporosity, which deteriorates the relation between porosity and permeability. The above analysis
shows that for the reference sample set a more sophisticated model with more independent
parameters than porosity is required to obtain reliable estimates for permeability. In the following,
we will show that porosity information amended with geometric and topographic information from
images substantially improves the prediction of permeability based on the models we defined in
chapter 2.
8.3.2 Permeability from a Hagen-Poiseuille type model
The simplest model one can imagine is a bundle of straight capillaries. In chapter 2, we showed
that the flow trough these capillaries can be described by Poiseuille’s law and that the
permeability for the model is given by the discrete distribution of tube radii:
k Poiseuille =
20
1
8
∑Φ r
i =1
2
8.7
i i
The integral equation 2.21 was replaced by a sum of squared radii over the 20 classes that were
defined in chapter 7. The tube radii were obtained from the POR images as was described in
chapter 7. For each pore area class, the accumulated area was determined and from this value
the radius of the area equivalent circle was calculated. It should be remembered that the
roughness we found in the images was not included in this model, and that this part of the
porosity will not contribute to the prediction of permeability.
The permeability was obtained by the summation of the contributions of the individual pore
classes, equation 8.7 represents this simple model. In Figure 8.11 it is shown that the predicted
SEM/IA permeability encompasses only three orders of magnitude while the plug permeability
ranges over 5 orders of magnitude.
115
Chapter 8
Permeability Poiseuille
Plug Permeability
10000
1000
100
10
2
R = 0.64
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.11
Cross-plot of the Poiseuille permeability
according equation 8.7 and the plug permeability
Permeability values above some 200 mD are predicted quite well, but from 0.1 to 200 mD the
Poiseuille model fails significantly. Although the model is extremely simple, the 64% explained
variance value is much better than the 46 % of the permeability versus plug porosity (k-Phi)
relationship. We also deduce that pore geometry information, albeit given in this model a
rudimentary form of 20 pore size classes, substantially improves the prediction of permeability
compared with the k-Phi model.
In the Poiseuille model described above, we used equal weighting for each class of tubes. One
can argue that larger tubes contribute even more to the total permeability than the smaller ones
2
than based on the term r . This was investigated using a statistical process in which weight
factors wi were determined for each pore class:
k Poiseuille =
20
1
8
∑Φ w r
i =1
i
2
8.8
i i
In which wi is the weighting factor for pore area class i. These factors were determined from
multiple linear regression of the 10Log plug permeabilities with 10Log SEM/IA permeabilities.
Figure 8.12 presents the cross-plot of the permeability predicted with the Poiseuille type model
that is given by equation 8.8, and the plug permeability. The explained variance increases from
68% to 76%. One should bear in mind that this model is now partly based on physics and partly
controlled by the information in the sample set. However, considering the large range in
permeabilities in the reference sample set, we expect that the model can be used for the
prediction of permeability in clastic sediments on a global scale.
116
Chapter 8
Permeability Poiseuille
Pore class weighted
Plug Permeability
10000
1000
100
10
2
R = 0.76
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.12
Class fitted Poiseuille permeability using
8.3.3 Permeability from Kozeny-Carman type models
In the Hagen-Poiseuille type models, the permeability is controlled by the radii of the capillaries,
expressed in the pore size distribution. Pore bodies and pore necks are the same in these models
and the distribution of radii describes also the geometry of the pores in the reservoir rock. The
Kozeny type models are another class of models that are not based on a geometric distribution
but rather on effective, macroscopic parameters such as porosity and specific surface area. The
Kozeny type models can be derived from Poiseuille’s law as was discussed in chapter 2. Similar
to the Hagen-Poiseuille type models, the micro-porosity is not included in the Kozeny models that
we used in this study. The very small pores residing in the micro-porosity hardly contribute to the
flow and have no place in a predictive model for permeability.
The hydraulic diameter can be related to the specific surface area or the surface-to-volume ratio
as was derived in chapter 2. The 2D version of the surface-to-volume ratio of the pore space can
be obtained from the DEF images as the ratio between the pore perimeter Ppores and the pore
area Apores. For a capillary of length L and of which the axis is perpendicular to the image
surface, this ratio is exact:
Surface L2 πr 2 πr Perimeter
=
=
=
Volume Lπr 2 πr 2
Surface
hence: S v =
Ppores
A pores
8.9
The surface-to-volume ratio is one of the parameters in the basic Kozeny equation for
permeability. This equation contains further the effective porosity Φeffective and the Kozeny
constant CKozeny that represents the tortuosity [Schlueter, 1995]:
117
Chapter 8
k Kozeny = C Kozeny
Φ effective 3
8.10
Sv 2
The porosity that was used in the Kozeny model is the effective porosity. The effective porosity
was obtained by measuring the area percentage pore space from the images. For the Kozeny
constant, we took a value of 5 as suggested in the literature [Liu & Masliyah, 1996]. We calculated
the permeability according to equation 8.10 for the samples in the reference sample set. The
results were compared with the permeability that was measured on core plugs. Figure 8.13 shows
the cross-plot between the SEM/IA permeability and the plug permeability.
Permeability Kozeny
10000
Plug Permeability
1000
100
10
2
R = 0.71
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.13
SEM/IA permeability according to
Statistical analysis revealed that the explained variance is 0.71. There is significant scatter around
the regression line leading to uncertainty in the prediction of permeability. Similar to the situation
for the Poiseuille model in the previous section, where the weighting coefficients for the
parameters were fitted, the value for the coefficients in the Kozeny equation can be determined
with linear regression. To this end, we linearise the Kozeny model:
log k = a log C Kozeny + b log Φ effective + c log S v
8.11
We fitted the 10Log of the plug permeability with the 10Log of the parameters effective porosity and
surface-to-volume ratio from images and the Kozeny constant. Figure 8.14 presents the cross-plot
of SEM/IA permeability and the predicted permeability.
The SEM/IA permeability according to the Kozeny model with fitted coefficients does not
significantly improve the prediction of the physics based, theoretical Kozeny model, both models
lead to the same value of 0.71 for the explained variance. The distribution of the residuals
between the predicted and the plug permeabilities can be approached by a normal distribution,
which enables the calculation of a standard error. The standard error for this model amounts to a
factor of four. The regression equation that enables permeability prediction according to the fitted
Kozeny model is given by:
118
Chapter 8
k Kozeny = 2.7
Φ effective 2 .3
8.12
S v 2 .1
Permeability Kozeny, fitted
10000
Plug Permeability
1000
100
10
2
R = 0.71
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.14
SEM/IA permeability according to equation
8.12 versus plug permeability
The coefficients of the fitted model and the theoretical model listed in Table 8.1 are very similar:
Theoretical
Fitted
3
-2
2.3
-2.1
Φeffective
Sv
Table 8.1
Coefficients Kozeny model
The similarity of the coefficients for Φeffective and Sv confirms that the Kozeny model is indeed
suitable for permeability prediction for this particular sample set. It also honours the insight of
Kozeny who devised this theory at a time (1920's) that means like image analysis were not
available to confirm their hypothesis! The reference sample set is considered representative for
the majority of sandstone reservoirs, we therefore expect that in general the Kozeny model can be
also used for the prediction of permeability on a global basis.
Carman [1937] proposed an extension to the Kozeny model by using the specific surface area S0
based on the volume of rock, instead of the pore volume. The Kozeny-Carman model is then
written as:
k Kozeny − Carman = C Kozeny − Carman
(
Φ effective 3
S 0 2 1 − Φ effective
)
2
in which S 0 =
119
Ppores
Amatrix
8.13
Chapter 8
The Kozeny-Carman model represented by equation 8.13 was used to calculate the permeability
of the samples in the reference sample set. Figure 8.15 presents the cross-plot of the predicted
SEM/IA permeability and the plug permeability.
Permeability Kozeny-Carman
10000
Plug Permeability
1000
100
10
2
R = 0.69
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.15
SEM/IA permeability according to the
Kozeny-Carman model versus plug permeability
The explained variance for the Kozeny-Carman model is slightly lower than that of the Kozeny
model. The predicted permeability values cover less than 3 orders of magnitude while the plug
permeabilities range over 5 orders of magnitude. For the reference sample set, the Carman
extension does not improve the permeability predictions.
In the Kozeny-Carman model we used the same approach as for the Kozeny model, and applied
multiple linear regression to find the coefficients in the linearised equation:
(
log k = a log C Koz − Car + b log Φ effective + c log S v + d log 1 − Φ effective
)
8.14
The results of the regression are shown in the following equation:
k Kozeny − Carman = 4.6
Φ effective 2 .5
(
S 0 1.4 1 − Φ effective
)
8.15
20
Figure 8.16 shows a cross-plot of the predicted SEM/IA permeabilities versus the plug
permeability. Using the statistical model increased the explained variance substantially, from 69 to
77%. The standard error reduced from a factor 4 to 3.5.
120
Chapter 8
Kozeny-Carman Permeability, fitted
Plug Permeability
10000
1000
100
10
2
1
R = 0.77
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.16
Cross-plot of the SEM/IA permeability
according to the fitted Kozeny-Carman model and the plug
permeability
8.3.4 Permeability from Network models
Hitherto, we have predicted the permeability from pore structure models that only contained
information about the geometry of the pore system. The pores were represented by their size but
not by their location with respect to other pores. Moreover, in the models we used until now, the
pore shape was assumed convex instead of the real shape that that tends to be concave.
In chapter 7, we developed a procedure that converts a POR image into a NET image that
contained a network representation of the pore system in 2D. In such a 2D pore network, the
geometry of the pores is not present; the network is solely dependent on the location of the pores.
The network describes the topography of the pore system, which shows how each pore is
connected to the other pores.
A pore network consists of nodes that are connected by bonds. The image processing to obtain
the pore network was designed such that nodes can only occur in the pores while the bonds can
be present in both the pores and the rock matrix to arrive at a fully connected network. This
means that there is a continuous path from top to bottom and from left to right through the image.
The bonds in samples with high permeability proved to be much longer than the bonds in low
permeability samples. Evidently, a fine-grained sample shows a more densely branched network
than a coarse-grained sample. This is shown in Figure 8.17, in which the sample on the left has a
permeability of 880 mD and the sample on the right a permeability of 75 mD. The pore bonds are
coded in yellow, the matrix bonds are coded in blue and the nodes in the pores are coded in red.
121
Chapter 8
Figure 8.17
Φ = 25.7 %BV
Φ = 15.7 %BV
k = 880 mD
k = 75 mD
NET images of two samples from the reference sample set
In general, coarse-grained samples have a higher permeability than fine-grained samples. This
can be clearly observed in Figure 8.18 where the NET images of two samples with the same
porosity but very different permeabilities are shown. On the left in Figure 8.18, the pore network is
more densely branched than the pore network on the right. The average pore bond length in the
right image is larger that that of the left image.
Figure 8.18
permeability
Φ = 19.1 %BV
Φ = 19.2 %BV
k = 9.3 mD
k = 240 mD
NET images of two samples with equal porosity but vastly different
122
Chapter 8
Based on these observations, one can argue that the permeability should be a function of the
pore bond length, the matrix bond length and the complexity of the pore network. The complexity
of the pore network is related to the total number of bonds and nodes and can be quantified by
the co-ordination number Z that was discussed in chapter 2.
To investigate the above hypothesis, we cross-plotted, for the samples in the reference sample
set, the average length of the pore bonds and the matrix bonds against the 10Log of the plug
permeability. Figure 8.19 shows these results.
Permeability
3
3
Log Pemeability
4
2
1
2
R = 0.40
2
1
0
10
0
10
Log Pemeability
Permeability
4
2
R = 0.36
-1
-1
-2
-2
0
5
10
15
20
0
25
10
20
30
40
50
60
70
Mean Matrix Bond length
Mean Pore Bond length
Figure 8.19
Mean pore-bond and matrix-bond lengths calculated from NET images of
the samples in the reference sample set
The explained variances of 36 and 40% for the pore bonds length and the matrix bonds length
respectively, cannot be considered as statistically significant. The next step was to combine these
two parameters and fit them to the 10Log of the plug permeability in a simple power law:
log k = a log µ pb + b log µ mb + c
8.16
In which µpb and µmb are the mean pore bonds length and the mean matrix bond length
respectively. The coefficients a, b and c were, as usual, determined with linear regression and
produced the following equation:
k Bonds
 µ pb 5.6 
= 7
4 .1 
 µ mb 
8.17
The very significant increase in explained variance from less than 40% to 80% implies that the
ratio of mean lengths of the pore bonds over the mean of the matrix bonds is very sensitive for
permeability variations, and supports our hypothesis. Figure 8.20 presents the cross-plot of the
SEM/IA permeability according the Bonds model and the plug permeability, which is given by
equation 8.17.
123
Chapter 8
Permeability, Bonds model
Plug Permeability
10000
1000
100
10
2
R = 0.80
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 8.20
Cross-plot of SEM/IA permeability
according to equation 8.17 and the plug permeability
The other part of our hypothesis concerns the complexity of the pore network, which can be
quantitatively described by the co-ordination number Zpores. In chapter 2, we defined the coordination number as:
Z pores =
number _ of _ porenecks
number _ of _ porebodies
8.18
This definition was based on a 3D pore network and indicates how many pores are on average
connected to one single pore. In this definition, the pore necks connect the pore bodies in 3D. In
our 2D pore network we assumed that the pore necks can be represented by the bonds and that
the pore bodies are represented by the pores. Then this definition can be ported from a 3D to a
2D pore network as follows:
Z pores _ 2 D =
number _ of _ bonds
number _ of _ pores
8.19
The bonds in equation 8.19 are defined as the length of the lines that connect the nodes in the
network, irrespective whether they fall in the matrix or in the pores. The left hand plot in Figure
8.21 shows the cross-plot of the 2D co-ordination number Zpores_2D and the plug permeability. It
appeared that no strong relationship exists between the 2D co-ordination number and the
permeability. At this point, we emphasise that the processing was designed such that only the
pores can contain nodes. This leads in fact to a disguised relation with porosity, because the
number of nodes is in that case proportional to the pore size and the number of pores. We
investigated the relationship between the number of nodes and the plug permeability. The right
hand plot in Figure 8.21, is a cross-plot of the number of nodes and the permeability. A power law
relationship between the permeability and the number of nodes best described the relation
between these two variables, but the explained variance is only 38%.
124
Chapter 8
Permeability
10000
1000
1000
Plug Permeability
Plug Permeability
Permeability
10000
100
10
100
10
1
2
1
R = 0.38
2
R = 0.24
0
0.1
1
2
3
4
0
100
Z2D
200
300
400
500
600
700
800
Number of nodes
Figure 8.21
Relationships between the plug permeability and respectively the 2D coordination number (left) and the number of pore nodes (right)
The number of nodes can be brought into the calculation of the 2D co-ordination number in the
form of a normalisation factor as shown in equation 8.20.
Z pores _ 2 D _ weighted =
number _ of _ bonds
× norm( pore _ nodes)
number _ of _ pores
8.20
The number of pore nodes is normalised to the maximum number of pore nodes that was found
among the samples in the reference set. In this way, we arrive at a weighted 2D co-ordination
number in which the influence of the number of pores and their shape is included. Correlating the
plug permeability with the weighted 2D co-ordination number according to a power law relation
provides the following equation:
(
k Z 2 D = a Z pore _ 2 D _ weighted
)
b
8.21
In which the coefficients a and b are as usual found by regression analysis. Figure 8.22 presents
the cross-plot of the weighted 2D co-ordination number and the plug permeability. The explained
variance of 61% is not great but implies a significant relationship.
Permeability, Z2D
Plug Permeability
10000
1000
100
10
2
R = 0.61
1
0.1
0.1
1
10
Weighted Z2D
Figure 8.22
Permeability from SEM/IA based on the
weighted co-ordination number versus the plug permeability
125
Chapter 8
8.3.5 Discussion on the various models for permeability prediction
In table 8.2 we present the numbers for the explained variances, the standard error and the
correlation coefficients for the various models we used to predict permeability from BSE images
obtained from the reference sample set. To facilitate the use of multiple linear regression, we took
the 10log of the plug permeability. By using the 10log of the permeability, we implied a power law
relationship between the permeability and the image features. We also present the correlation
coefficients that do not imply any model. The 10log of the permeability enabled the use of multiple
linear regression to establish quantitative relationships with the image features. The standard
error is presented as a factor and is an indicator for the accuracy of the permeability prediction
when the residuals of the regression are normally distributed. For instance, the predicted
permeability can have a value of 10 mD with a standard error of a factor of 4. This means that the
2σ range goes from 2.5 mD to 40 mD, and that 67% of the predictions are expected to fall in this
range.
Approach
k-Phi
Poiseuille
Poiseuille fitted
Kozeny
Kozeny fitted
Kozeny-Carman
Kozeny-Carman fitted
Bonds
Network Z2D
Network
Table 8.2
Explained
variance
Standard
Error [factor]
Correlation
coefficient
0.46
0.64
0.76
0.71
0.71
0.69
0.77
0.80
0.61
0.80
6.5
4.5
4.0
4.2
4.2
4.5
3.6
3.2
5.1
3.0
0.68
0.80
0.87
0.84
0.84
0.83
0.88
0.89
0.78
0.91
Accuracy and error of the various approaches for permeability prediction
Table 8.2 shows that all approaches that use image features predict the permeability substantially
more accurately than the k-Phi approach. We conclude that when we included geometric and
topographic information from images the prediction was improved significantly. We also
investigated whether the correlation coefficients for the various approaches occurred by chance.
There is a meaningful relationship among the variables if the F-observed statistic is greater than
the F-critical value.
Approach
F-observed
F-critical]
216.8
30.8
265.1
243.3
427.6
326.7
194.1
3.0
1.6
2.6
2.4
2.6
3.0
2.1
k-Phi
Poiseuille
Poiseuille fitted
Kozeny
Kozeny fitted
Kozeny-Carman
Bonds
Network Z2D
Network
Table 8.3
T-tests applied to the various approaches for permeability prediction
126
Chapter 8
The probability of erroneously concluding that there is a relationship was set to 0.05. An F-test
was applied to all approaches and the results are presented in Table 8.3. The results of the F-test
in Table 8.3 show that in all approaches the relationship between the permeability and the image
features are significant.
Only minor differences in predictive capability occurred between the fitted Kozeny-Carman model
and the fitted Poiseuille model. Comparing the Poiseuille model, the Kozeny-Carman model and
the Network model, the latter provides the best prediction of permeability.
The Network model is a clear winner with a standard error of a factor three and explained
variance of 80%. Finally, we present the equations that are associated with the single models to
predict permeability from image features in Table 8.4.
Single Models
k-Phi
Predictive equation
k = 01
. Φ 0.13 (Φ in %BV)
k Poiseuille =
Poiseuille
k Poiseuille =
Poiseuille fitted
i =1
1
8
Kozeny-Carman
k Kozeny − Carman = 5
i =1
k Bonds
Φ3
Sv 2
(
Φ 2 .3
S v 2 .1
Φ effective 3
S 0 1.4 (1 − Φ)
 µ pb 5.6 
= 7
4 .1 
 µ mb 
(
)
2
Φ 2 .5
k Z 2 D = 19
. Z 2 Dweighted
Network Z2D
2
i i
S 0 2 1 − Φ effective
k Kozeny − Carman = 4.6
Bonds
2
i
∑w r
k Kozeny = 2.7
Kozeny fitted
8.4
∑r
20
k Kozeny = 5
Kozeny
Table 8.4
20
1
8
)
20
2 .9
Equations for the combined models governing permeability prediction
Formation Resistivity Factor
The formation resistivity factor and the hydraulic permeability are comparable rock features and
can both be explained with potential difference theory. The permeability gives the hydraulic
conductance while the formation resistivity factor is related to electrical conductance. The main
difference between the two is caused by the presence of clays. Clays do not contribute to flow
and are therefore not present in predictive models for permeability. However, clays do contribute
127
Chapter 8
to electrical conductivity and are therefore included in models that predict formation resistivity
factors.
The first Archie equation relates the formation resistivity factor to porosity. Therefore, we devised
a model that predicts formation resistivity factor from BSE images, accounting for both porosity
and clay content. Since the derivation of a pure analytical model for the prediction of formation
resistivity factor (FRF) from images proved to be complex, we took the same statistical approach
that was successfully used in the prediction of permeability. We will again use linear regression in
a similar way as applied for the derivation of permeability relations. We first tried out a power law
model for FRF that includes area percentage clay and the effective porosity:
(
)
b
FRFSEM / IA = a Φ effective Vclay c
8.34
Which leads to an equation that can be used for linear regression:
log FRF = a + b log Φ effective + c log Vclay
8.35
The results of the regression are shown in Figure 8.23, where the cross-plot between the
formation resistivity factor and the measured formation resistivity factor is presented.
1000
Plug FRF
100
10
2
R = 0.68
1
1
10
100
1000
SEM/IA FRF
Figure 8.23
Formation resistivity factor from images
versus formation resistivity factor determined from core
plugs
The explained variance of 68% indicates a loose relationship between these image features and
the formation resistivity factor. The standard error amounts to a factor 1.5. Also for the FRF we
applied an F-test with the probability of rejection that that there is a relationship set to 0.05. The
F-statistic observed was 168.6 and the F-critical was 3.0. We conclude that this relationship is
genuine because the F-observed was much larger that the F-critical.
The good results we obtained in permeability prediction with the Network model led to a similar
approach to predict the FRF. We argued that mean length of the pore bonds and the number of
128
Chapter 8
bonds should be inversely related to the FRF. We extended the power law model in equation 8.34
by incorporating the mean pore bond length µpb. and the number of bonds Nbonds.
(
)
b
FRFSEM / IA = a Φ effective Vclay c µ pb d N bonds e
8.36
This provides an equation to be used in linear regression:
log FRF = a + b log Φ effective + c log Vclay + d log µ pb + e log N bonds
8.37
Figure 8.24 presents the results of the regression. The explained variance increased slightly from
0.68 to 0.71. The F-observed was 96.1 for an F-critical of 2.2. We conclude that the prediction is
not substantially improved by incorporating two more variables in the equation. Apparently, the
network approach does work better for permeability prediction than for FRF prediction.
SEM/IA FRF
1000
100
10
2
R = 0.71
1
1
10
100
1000
Plug FRF
Figure 8.24
Formation resistivity factor including
network data versus formation resistivity factor from core
plugs
8.5
Clay content and Qv
The clay content of reservoir rock is particularly important for the determination of the
hydrocarbon saturation. The hydrocarbon saturation is determined with saturation models that use
the electrical conductivity of reservoir rock as discussed in chapter 2. Clays can cause an excess
conductivity and it is therefore important to obtain knowledge about the amount, the type and the
dispersion of clay in the reservoir. In chapter 3, we discussed the determination of Qv by the
conventional titration method. The Qv is based on conductivity because saturation models such
as Archie [1942] and Waxman-Smits [1967] also use electrical conductance.
A disadvantage of the Qv method is that the samples have to be crushed prior to the
measurement. In the crushing, the morphology of the clays and their dispersion in the pore space
are destroyed, which inevitably influences the conductivity behaviour. The in-situ clay conductivity
129
Chapter 8
can only be approximated by the Qv due to the crushing and it is therefore considered a rather
crude method. Moreover, the degree of crushing greatly influences the CEC, the cation exchange
capacity, from which the Qv is calculated. Statistical analysis of the relationship between Qv and
the volume of clay obtained from SEM/BSE images revealed a rather poor explained variance of
0.31. This was attributed to the differences in the methods of determination of the parameters Qv
and Vclay.
The Qv is a conductivity-based parameter that was primarily developed to fit the Waxman-Smits
type models. At that time, in the late sixties, more sophisticated methods to determine the amount
of clay in reservoir rock, like the method developed in this study, were not available. The V-shale
is a volume fraction that is obtained from BSE images by areal measurements as discussed in
chapter 7. Only the newer effective medium based saturation models, discussed in chapter 2, can
accommodate Vclay as a parameter reflecting the conductivity behaviour. Literature [De Kuijper,
1995] shows that the saturation models based on effective medium theory have similar accuracy
as the empirical Waxman-Smits model. The advantage of the effective medium models in that the
Qv measurement can be replaced by the method developed in this study.
The determination of Vclay from BSE images is based on the same principles and assumptions as
the determination of porosity. Therefore, we argue that similar accuracy for Vclay as for porosity
can be reached, which is a standard error of approximately 1.5 V-shale units.
8.6
Capillary pressure curves from BSE images
The idea of extracting a capillary pressure curve is based on the observation that the entry
pressure and the plateau of a capillary pressure curve are very closely related the permeability
[Mishra, 1988; Dias, 1996]. The permeability in turn is related to the pore size distribution, or,
more correctly, to the pore neck size distribution. This is no surprise; at the entry pressure and the
start of the plateau in a capillary pressure curve, the invading non-wetting fluid is entering into the
sample and the flow is in the Darcy regime.
Since capillary bundle models were used with success to predict permeability from images in this
study, we will use the same approach to obtain an estimate for the capillary pressure curve from
images. In order to make the link between the pores we see in the POR image, and a capillary
pressure curve, we distinguish four regions:
1) The entry region, the part of the curve where the invading fluid enters the pore system
via the pore necks.
2) The plateau region, this is the part of the curve where the invading fluid fills the pore
bodies.
3) The asymptotic region, being the part of the curve where very small pores, the
wedges and the pore-wall roughness are filled by the invading fluid.
4) The irreducible region, the part of the pore space that is not accessible for a nonwetting fluid at the maximum pressures that are normally used to obtain capillary
pressure curves.
These regions are indicated by the arrows in Figure 8.25 where the capillary pressure curve of a
low permeability sandstones is presented. Based on theses regions, we devised a procedure in
130
Chapter 8
which we will use morphological operations on the images to mimic the invasion process in a
capillary pressure curve.
Capcurve
10000
Pressure
1000
100
10
1
0.0
0.2
0.4
0.6
0.8
1.0
Saturation
Irreducible
Asymptotic
Plateau
Entry
Figure 8.25
A capillary pressure curve can be divided
into 4 regions that can be modelled from image data
Figure 8.26 presents that procedure in which the steps that are associated with the regions in a
capillary pressure curve are indicated in red. The procedure starts with the conversion of a DEF
image into a TUB (tubes in the pore space) image to extract a form of a 2D pore size distribution.
This pore size distribution was required for the prediction of permeability using Poiseuille type
models as was discussed previously in this chapter. The pore size distribution now forms the
foundation to obtain a capillary pressure curve.
When a hypothetical, non-wetting fluid interface is pressing against the surface of the sample
represented by the image, the largest pores will be accessible at the lowest pressure according to
the Laplace equation that was discussed in chapter 2. The entry region is associated with the
largest tubes in the pores that have to be filled first and are represented by pore class 1. The
centres of gravity of these largest tubes were selected as the starting points of the capillary
pressure curve. The points at the centres of gravity were dilated with a hexagon 5 x 5 operator in
subsequent steps. These steps were representing the pressure steps in the capillary pressure
curve. The 5 x 5 operator represented an area of 5 x 5 microns. The area percentage pore space
that was filled in this process represents the non-wetting saturation. The cumulative area
percentage pore space that was filled by the dilation was measured after each step. This stepwise
process was continued until the tubes that belong to class 1 were filled.
131
Chapter 8
N=N-1
No
Pore
space
N opening
functions
N = 20
Subtract from
pore space
Add in TUB
image
N = 0?
Yes
Merge
tubes
Conversion from a DEF image into a TUB image
No
Class 1
CGRAV
Measure
%pores
filled
Dilation
Class 1
pores
filled?
Dilation
+
add class N
Dilation
N = 20?
Plateau
Entry
Measure
%pores
filled
Measure
%pores
filled
Roughness
filled?
Yes
Yes
Capillary
pressure
curve from
image
Areal% clay
is
irreducible
No
Irreducible
Asymptotic
Figure 8.26
A flowchart representing the procedure to derive a capillary pressure curve from a
DEF image. The regions depicted in Figure 10.1 are used for the different image processing steps
However, the pores in the remaining classes consist of sets of tubes as depicted in Figure 8.27. In
the above procedure, only the largest tubes were filled. The remaining tubes are supposed to be
filled at higher pressures, hence later dilation steps, in the plateau region.
Roughness
Remaining tubes
Largest tube
Pore
Figure 8.27
Pore filled with
convex shapes
Pore filled with
capillaries
The process of converting a pore into a bundle of capillaries
132
Chapter 8
We associated the plateau region with the remaining 19 tube classes and assume that these are
filled at higher pressures, again according to the Laplace equation that relates tube diameter to
pressure. The largest tube in each pore class is filled instantaneously. The remaining tubes of that
pore are considered to be filled at higher pressure steps, hence later dilation steps, again with a
5 x 5 hexagon operator. Similarly to the above entry region, the cumulative areal percentage
pores were measured at each dilation/filling step.
After filling all 20 pore classes in the POR image, the pore space remaining to be filled was
considered roughness and representing the asymptotic region. The roughness is also filled by
dilation steps, but now with a smaller operator, a 3 x 3 cross operator representing 3 x 3 micron.
The smaller dilation operator was taken to enable finer steps. After each dilation step, again the
area percentage pore space filled was measured. When all the pore space was filled, there
remained the micro-porosity residing in the clays. This was considered to represent the irreducible
region, which agrees with the general assumption that these micro-pores are not filled in a
conventional mercury-air capillary pressure curve experiment.
We have carried out the above procedures on the images of several samples from the reference
sample set and on images of some of the glassbead samples. Physically measured capillary
pressure curves were available for comparisons. Based on observations of physically measured
capillary pressure curves, we postulated an exponential relationship between the area percentage
pore space in 2D and the cumulative dilation steps, the former representing the water saturation
and the latter the capillary pressure respectively:
Area% pores _ filled = e
# Dilations
⇒ Sw = e
a
steps
in which : Area% pores _ filled =
∑
i =1
Pc
a
10.1
  ( Area%)

( Area%) i
i
1 − 
−
Area%clay  
  ∑ Area% ∑ Area%
  i
10.2
We determined the parameter a by fitting the capillary pressure curve from the image with the
capillary pressure curve that was obtained by a physical mercury-air experiment. Figure 8.28
shows the capillary pressure curves that were calculated from the images of two of the glassbead
samples next to the mercury-air capillary pressure curves.
Capcurve glassbead sample 5
Capcurve glassbead sample 2
10000
10000
1000
1000
Pc [Psi]
Pc [Psi]
Measured Hg-air
Derived from IA
100
10
Measured Hg-air
100
Derived from IA
10
1
1
0.0
0.2
0.4
0.6
Sw [% PV]
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Sw [% PV]
Figure 8.28
Comparison of capillary pressure curves derived from images and physically
measured on the glassbead samples. Sample 2: Φ = 18.7 %BV and k = 170 mD, sample 5: Φ =23
%BV and k = 1500 mD
133
Chapter 8
The capillary pressure curves from images agree remarkably well with the ones that were
obtained from a mercury-air experiment, despite that fact that they were obtained from 2D
information only. The irreducible region is not present because the glassbead samples are totally
free of clays. Encouraging results were also obtained for the remaining four glassbead samples.
Figures 8.29 and 8.30 present the results of capillary pressure curves that were obtained from
real reservoir rock samples in our reference sample set. For samples 12, 13 and 8 the entry
pressure of the IA based capillary pressure curves agreed well with the measured curves. For the
high permeability (880 mD) sample 7, the entry pressure of the IA based capillary pressure curve
was far too low. In all cases, the irreducible saturation was predicted accurately. The asymptotic
region showed substantial mismatch for samples 7 and 8 while the agreement was acceptable for
samples 12 and 13.
Capcurve reference sample 13
10000
1000
1000
Pc [Psi]
Pc [Psi]
10000
100
10
100
Measured Hg-vac
10
Measured Hg-vac
Derived from IA
Derived from IA
1
1
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
Sw [% PV]
0.4
0.6
0.8
1.0
Sw [% PV]
Figure 8.29
Capillary pressure curves from samples 12 and 13. Sample 12: Φ = 24.8
% BV, k = 14 mD, sample 13: Φ = 13.8 % BV and k = 6.8 mD
10000
10000
1000
1000
Measured Hg-vac
Pc [Psi]
Pc [Psi]
Measured Hg-vac
Derived from IA
100
10
Derived from IA
100
10
1
1
0.0
0.2
0.4
0.6
0.8
1.0
0.0
Sw [% PV]
0.2
0.4
0.6
0.8
1.0
Sw [% PV]
Figure 8.30
Capillary pressure curves from samples 7 and 8. Sample 7: Φ = 25.7
%BV and k = 880 mD, sample 8: Φ = 15.7 %BV and k = 75 mD
From the few experiments we carried out, we concluded that there certainly is potential for the
estimation of capillary pressure curves from BSE images. However, the processing of particularly
the asymptotic region has to be investigated more thoroughly. In addition, the scale parameter a
that greatly influences the entry pressure, has to be related to a feature that can be determined
independently from the pore system in BSE images. We also emphasise that the isotropy and
homogeneity of the samples are absolute essential requirements for the prediction of capillary
pressure curves from BSE images. In contrast to the prediction of porosity and permeability, all
pore size classes have to represented properly to find an agreement over the full saturation
range.
134
Chapter 8
8.7
Discussion
Porosity derived from SEM/IA and porosity measured on 230 core plugs were cross-plotted in
Figure 8.8, which we reproduce for convenience of the reader in Figure 8.31.
Porosity
35
Plug Porosity
30
25
20
15
10
2
R = 0.89
5
0
0
5
10
15
20
25
30
35
SEM/IA Porosity
Figure 8.31
Cross-plot of the porosity predicted from BSE
images and the porosity measured on core plugs
In Figure 8.31, the blue points show samples with errors in the predicted porosity that are
larger than 2 times the standard error around the fit line, which covers 95% of the width of the
distribution. In the following, we will discuss these samples to find the main causes for these
deviations. Prior to the discussion, we remind the reader of the colour coding we apply for the
DEF images:
Colour
Constituent
Pore space
Clays
Quartz
Feldspars/Calcite
Heavy Minerals
Table 8.5
Colour code of reservoir
rock constituents in DEF images
8.7.1 Porosity under-estimation in SEM/IA
Inspection of the 11 samples for which the SEM/IA porosity is underestimated were subdivided
as follows: 1) high amounts of clay (8 of 11), 2) cementation and/or the presence of micaceous
minerals (2 of 11) and 3) no specific observation (1 of 11). In samples with high amounts of
clay, the reason for the under-estimation may be that not all the micro-porosity could be
detected. The very small pores in the clays may not have been filled totally in the impregnation
step that is part of the sample preparation. We have investigated this effect and observed that
pores in kaolinite clay structures down to 50 nm could be filled with epoxy. However, because
of differences in wettability characteristics of the various types of clay, it may occur that not all
micro-pores in clay structures are filled with epoxy during impregnation. Consequently, these
micro-pores cannot be observed in BSE images and hence they cannot be detected.
Furthermore, the presence of abundant clays may have closed off larger pores for the
135
Chapter 8
impregnating epoxy. This effect can occur during the diagenetic processes that form clays in
reservoir rock as discussed in chapter 2.2.3. The media that were used in the conventional
porosity measurements: Helium gas for the Boyle method and chlorothene for the buoyancy
method, have a far lower viscosity than the impregnating epoxy, and can enter micro-pores
that are not accessible for epoxy. Consequently, the porosity measured with these
conventional methods may be higher than the porosity predicted from BSE images. Inspection
showed indeed that the samples with a SEM/IA porosity that is lower than the measured plug
porosity contain significant amounts of clay. Therefore, we investigated the fraction of the pore
volume (PV) that is filled with clay for the eight samples with high clay content. According to
the definition of primary porosity in chapter 2, we calculated pore volume as the sum of the
effective porosity and the clay fraction bulk volume (BV). Equation 8.36 shows the calculation
of the fraction of pore space filled with clay:
Clay% PV =
Clay% BV
Clay% BV + Φ effective
8.36
On average 75 % of the pore volume of the eight shaly samples is filled with clays. This is shown
in Figure 8.32, where the primary porosity consisting of effective porosity (blue) and the clay infill
(green) for each sample are presented.
SEM/IA Porosity
30
Effective porosity
Percentage BV
25
Clay
20
15
10
5
0
1
2
3
4
5
6
7
8
Sample
Figure 8.32
Samples for which the SEM/IA porosity is
under-estimated, note the high amount of clay in the pores
An example of high clay content is shown in Figure 8.33, where the DEF image from sample
number 8 is presented. The calculated SEM/IA porosity was 21.4 %BV while the porosity
measured on the plug was 26.4 %BV. The majority of the pores is filled with clay and very little
effective porosity is left in the centres of the larger pores.
136
Chapter 8
Figure 8.33
DEF image of sample 8, note the abundant
pore filling clay
Figure 8.34
DEF image of sample 1, pore filling clay and
high feldspar content
137
Chapter 8
Figure 8.35
DEF image sample 3, pore filling cement
Figure 8.36
DEF image sample 2, micaceous sandstone
138
Chapter 8
The image in Figure 8.34 shows the DEF image of sample number 1. Abundant clays are present
in the pores next to a significant amount of feldspar. Analysis revealed a clay content of 13.6 %BV
of which 5.6 %BV is micro-porosity. The high amount of feldspar (32 %BV), indicates that this is
an arkosic sandstone. The feldspars were identified using X-ray analysis. The shape of the
feldspars indicates that they occur in the form of grains as part of the detrital assemblage. As
discussed in chapter 2, feldspars easily decompose and then convert into clay, which is the most
likely explanation for the abundant presence of clays in this sample.
Figure 8.35 presents the DEF image from sample 3. The calculated SEM/IA porosity was 6.0
%BV while the measured plug porosity was 10.9 %BV. Apart from the presence of clay in the
pores, it is clear that some pores are totally filled with carbonate cement, coded in olive green in
the DEF image. The cements can be recognised by their convex shapes and their mineral
composition was confirmed by X-ray analysis.
Finally, Figure 8.36 shows the DEF image from a micaceous sandstone with a calculated SEM/IA
porosity of 14.0 %BV, and a measured plug porosity of 18.3 %BV. Detailed inspection of the
physical sample showed regions with abundant biotite and feldspars, next to regions that are
almost homogeneous quartz. In this case, the sample heterogeneity has not been captured in the
BSE image collection, only images from the feldspar rich regions were collected. Therefore, the
images were not sufficiently representative to allow reliable prediction of porosity. As earlier
discussed, when heterogeneity beyond the scale of the images occurs, this method will fail to
accurately predict porosity.
8.7.2 Porosity over-estimation in SEM/IA
So far, we have discussed and explained under-estimation of porosity, which applied to the
points falling above the trend line in figure 8.31. Over-estimation of SEM/IA porosity occurred
for 7 samples. The causes of over-estimation of the SEM/IA porosity is attributed to omissions
in the sample preparation procedure and to the presence of significant amounts of clay and
possibly lack of representativity of the BSE image for the total plug, hence heterogeneity.
In the sample preparation procedure an effect that is known as grain plucking can occur.
Specifically unconsolidated sandstones suffer from this effect. Prior to impregnation, a slice is
cut from a core plug using a diamond saw [Dijkshoorn, 1990]. Grains that are only loosely
attached within the detrital assemblage may be removed by the forces exerted by the sawing
process, leaving a space the size of that grain. In the impregnation step, this space is filled
with epoxy and will be assigned pore space in BSE images. This effect can be clearly
observed in Figure 8.31 where several grains were removed from the matrix and are now
erroneously assigned to the pore space.
Grain plucking can not only be recognised by the pore size that is too large to fit in the detrital
assemblage, but also by the convex shape. Pores normally have a dendritic shape and are
therefore concave. The SEM/IA porosity that was calculated for the sample in Figure 8.37 was
32.1 %BV while the porosity measured on the core plug amounted to 26.6 %BV.
139
Chapter 8
Removed
grain
Removed
grain
Removed
grain
Figure 8.37
plucking
Over-estimation of porosity caused by grain
Figure 8.38
Over-estimation caused by expanding clays
in the sample preparation procedures, note the crack top left
140
Chapter 8
Over-estimation of SEM/IA porosity can also occur when abundant clays are present. In the
sample preparation procedure, the remnants of hydrocarbons and drilling mud are removed by
the so-called Soxletth cleaning. [Yuan & Schipper, 1995]. The Soxletth process uses water and
alcohol to rinse hydrocarbons and drilling mud from the sample. However, some clay types can
increase in volume by taking up water in the clay structures, and are then crystal-bound. The
forces associated with the water absorption cause cracks to propagate through the clay areas.
These cracks are filled with epoxy in the sample preparation procedure and are mistakenly
classified as effective pore space, artificially increasing the porosity. This effect was encountered
several times and is shown in Figure 8.38 in which a crack occurs in the upper part of the image.
In the previous section, we briefly discussed sample heterogeneity. Two types of heterogeneity
could be identified, heterogeneity at the scale of the BSE images (2 x 2 mm) and heterogeneity at
the plug scale. The former could be identified by comparing the four images that were used in the
analysis. Figure 8.39 presents an example of this type of heterogeneity. The left image shows a
very shaly part in which a crack artificially increases the SEM/IA porosity. The right image shows
a clean, fine-grained sandstone. These images were collected at a distance of some 2 mm from
one another. The SEM/IA porosity of this sample was 34.5 %BV and the porosity that was
measured on the core plug was 29.5 %BV.
Another form of heterogeneity observed at the scale of BSE images is the occurrence of very
coarse grains in a sample. In chapter 7 we discussed this effect in the context of selecting the
appropriate magnification. Several samples were removed from the reference sample set
because they were considered too coarse. Figure 8.40 shows an example of a borderline case
that was deliberately kept in the reference sample set. The two images were collected at a
distance of 2 mm. The images clearly illustrate the heterogeneity that can occur at the scale of the
BSE images. Poor sorting is clear because the left image shows grains with diameters of several
hundreds of microns, while the right image shows grains of less than 100 microns. The SEM/IA
porosity of this sample averaged over 4 images was 22.7 %BV and the porosity measured on the
core plug was 17.5 %BV.
In conclusion: sample heterogeneity should be and can be assessed prior to the application of the
methods that were developed in this study. Estimation of petrophysical parameters is primarily
dependent on a representative definition of the pore space and the clays. Therefore, proper
screening of the samples for this type of analysis is absolutely essential and of paramount
importance.
141
Chapter 8
Figure 8.39
heterogeneity at the scale of BSE images, the distance between these two
images is 2 mm
Figure 8.40
Heterogeneity at the scale of BSE images, note the difference in grains size of
these images that were collected at 2 mm distance
142
Chapter 9
SEM/BSE analysis applied to field cases
9
SEM/BSE ANALYSIS APPLIED TO FIELD CASES
9.1
Introduction
In chapter 8 we defined the models for prediction of petrophysical parameters from BSE images,
the central theme in this thesis. In this chapter, we will show how the information from BSE
images can be incorporated in a field study and how it can support both the petrophysical and
geological evaluation of a well.
We will start with a general description concerning the environment of deposition of the reservoir.
The information about the reservoir that is provided by the core description and the wireline log
information will be complemented by the information that was obtained form BSE images:
lithology and the rock type. Furthermore, we will explain how porosity and shale content from BSE
images is related to the plug measurements and to the wireline logs. A comparison between core
plug measurements and petrophysical parameters derived from BSE images will be made to
indicate the usability of the methods we have developed in this study. Finally, the same procedure
will be carried out for a second well from a different environment of deposition in order to
demonstrate the general applicability of the methods.
9.2
Depositional environment and core description well A
The geological evaluation of a reservoir normally starts with the determination of the environment
of deposition. This is based on information obtained from the seismic interpretation, regional
knowledge and on the paleontological analysis of cuttings and cores. Figure 9.1 stylistically
depicts the environment of deposition.
Well A
ALLUVIAL FAN
FLUVIAL CHANNEL
ABANDONED CHANNEL
CREVASSE SPLAY
FLOOD PLAIN
LAGOON
ISOLATED FLUVIAL CHANNELS
UPPER ALLUVIAL PLAIN
MIDDLE ALLUVIAL PLAIN
COASTAL CARBONATES
LOWER COASTAL PLAIN
Figure 9.1
plain
OFFSHORE BARS
Depositional environment of well A which penetrates a middle alluvial
The deposits found in well A over the reservoir interval are representative of a middle alluvial
plain, in layman’s terms, river deposits in the meandering part of the river where the water flow
velocity is relatively low. Over the entire reservoir interval, a core was taken in order to determine
143
Chapter 9
the lithology and to support the petrophysical evaluation. The cored interval was divided into five
major lithological zones consisting predominantly of either sand, shale/silt or heterolithics. A more
detailed lithological description of each zone was provided by sedimentologists and is
summarised in Figure 9.2.
DEPTH LITHO(m)
ZONE
MAIN
LITHOLOGY
CORES
LITHOLOGICAL DESCRIPTION
2070
2075
1
CORED INTERVAL (1275.65-1307.0 m)
2080
2
A
2085
3
2090
2095
B
B
C
C
4
The upper part comprises well-laminated
fissile sst with silty/shaly partings (mm
- cm scale), overlain by structureless,
silty, v.f sst with haematitic mottles.
The middle part consists of wavy-laminated
fine sst with localised cross-bedding &
silty lenses.
The lower part comprises strongly contorted,
moderately cemented sst overlain by wavy
laminated vf-f sst & shaly/silty sst. The shaly
beds are slightly bioturbated & are moderately
to intensely contorted towards the base.
The upper part comprises brecciated, v.f
sst with vertical branching, silt-filled
cracks. This is underlain by purple-red,
slightly variagated mudstone &silty mudstone.
Medium/fine grained sst with feint to welldeveloped parallel lamination (e.g. 2093.5 m)
& cross-lamination (e.g. 2094.3 m. The sst
has patchy cement.
The upper part comprises purple-red mudstone
with local rooted horizons. This is underlain by
mottled sandy mudstone with possible rootlets
& silt with sandy streaks.
D
D
2100
5
E
2105
Medium to coarse-grained sst with high-angle
cross-lamination (15-30 degrees). Some beds
contain scattered granule to pebble-size green
clay clasts locally concentrated above erosion
surfaces. Scattered calcite nodules (mm scale)
are present at 2005.8 m.
2110
Mainly sand
Figure 9.2
Mainly silt/shale
Mainly heterolithics
Detailed core description of well A
These interpretations allow classification of the different facies leading to the zonation presented
in Table 9.1. The sandstone intervals (zones C and E) were interpreted as fluvial channel sands
and the silts/shales zones B and D represent the associated channel fines, i.e. channel
144
Chapter 9
abandonment sequences, flood-plain deposits. The heterolithic sequence of zone A is interpreted
to be a sequence of inclined heterolithic stratification deposited as part of a laterally accreted
channel fills.
Zone
A
B
C
D
E
Table 9.1
9.3
Depth [m]
2075 – 2089
2089 – 2092
2092 – 2096
2096 – 2100
2100 - 2107
Interval
Laminated heterolithics
Silt/shale
Fine/medium grained shaly sand
Silt/shale
Coarse/medium grained sand
Lithological zones of well A from the core description
Petrophysical evaluation
The petrophysical evaluation of well A is limited to those logs that are important for the
comparison with the petrophysical parameters derived from BSE images with log derived
equivalents. We used the gamma-ray log as a shale indicator and compared the results with the
values for shale content from BSE images. The shale content can also be obtained with the socalled Thomas-Stieber method, which uses the density en neutron logs. The latter is sometimes
preferred over the gamma ray because of its slightly higher resolution, specifically in sand/shale
sequences as encountered in alluvial deposits. We also incorporated the resistivity logs that were
used for the calculation of the hydrocarbon saturation. As discussed in chapters 2, 3 and 8, the
amount of clay in reservoir rock has a profound influence on the conductivity and thereby on the
resistivity. Specifically knowledge of the clay dispersion in the pore space can support the
interpretation of the resistivity logs. The wireline porosity calculated from the density and the
neutron logs is normally compared with the porosity measurements that were obtained from plugs
measurements. We also compared the plug porosity with the porosity derived from BSE images.
The zonation that was provided by the geological interpretation revealed three potential oil
bearing intervals. First, the laminated sand/shale (heterolithic) interval at 2075 - 2089 m, second
the fine/medium grained shaly sand interval at 2092 - 2096 m, and third the coarse/medium
grained sand at depths 2000 - 2007 m.
Figure 9.3 presents the logging suite that we used for this limited petrophysical interpretation. At
the right, the intervals obtained from the core description are shown according to the standardised
patterns and colour schemes. The high values on the gamma-ray log indicate high amounts of
clay and these intervals are classified as shale. In the laminated heterolithic interval from
2075 - 2089 m, a high gamma-ray value indicates the presence of shale. This high gamma-ray
value continues until 2092 m, which is the bottom of the first shale interval. In the interval 2092 –
2096 m, the gamma-ray value is relatively low, indicating a shaly sandstone. In the second shale
interval, which starts at 2096 m, the gamma-ray log again reaches high values. Finally, the
gamma-ray value returns to low values at 2100 m pointing towards cleaner sands. From the
resistivity logs, MSFL, LLS and LLD, a true resistivity of the invaded zone Rt has been calculated.
The Rt log, shows high resistivities in the laminated heterolithics and the fine/medium-grained
sands. Low resistivities around 1 ohmm are found in the coarse sand below 2100 m indicating a
water-bearing zone, the oil/water transition is observed from about 2100 to 2005 m. The higher
resistivities in the laminated heterolithic interval at 2075 - 2089 m, and the fine/medium grained
145
Chapter 9
shaly sand interval at 2092 –2096 m are caused by the oil in the pores. These resistivities would
normally be higher, and are probably due to the presence of clay in the pores. This conclusion
cannot be derived from the wireline logs. The image analysis the methods developed in this study
can clarify this point as presented later in this chapter
Resistivity [ohmm]
Porosity [%BV]
2075
2075
2080
2080
2080
2085
2085
2085
2090
2090
2090
2095
2095
2095
2100
2100
2100
2105
2105
2105
2110
0.1
2110
0.0 0.1 0.2 0.3 0.4 0.5
2110
0
50
100
1
10
Corase Sand
Shale
2075
Heterolithics
Depth
Neutron
Density
Fine Sand
Rt
MSFL
LLD
LLS
Shale
Gamma ray [API]
Figure 9.3
part of the logging suite that was run in well A. The LLD, LLS and MSFL are
resistivity logs with different depth of investigation and resolution.
146
Chapter 9
In chapter 3 we discussed the use of the density and the neutron log as porosity indicators.
Furthermore, the neutron is also a shale indicator due to the presence of clay-bound water in the
shales. The neutron porosity in Figure 9.3 is significantly higher than the density porosity in the
shaly sands and the shale intervals. In the coarse, clean sandstone interval from 2100 – 2107 m
the density and the neutron porosity are roughly equal caused by the absence of clays. This can
be explained by evaluating Figures 9.4 and 9.6 in the discussion below. Prior to this discussion,
we will explain the Thomas-Stieber analysis of the neutron and density logs.
9.4
Thomas-Stieber analysis of the neutron/density log
The density and the neutron logs separately are not particularly suited for lithology determination.
However, the combined use of these logs can provide an excellent lithology indicator. The
different measurement principles of the neutron and density logs enable determination of
lithology. For determination of porosity in clean, water bearing sandstones and dolomites,
transformations for both the density and neutron vales are required. This is illustrated on the
cross-plot of the density and neutron values for the three main lithologies as shown in figure 9.4.
3
Bulk density [g/cm ]
Density / neutron cross-plot
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
Clean Limestone
Clean Sandstone
Clean Dolomite
-5
0
5
10
15
20
25
30
35
40
45
Neutron Porosity index [% BV]
(apparent limestone porosity)
Figure 9.4
Density-neutron cross-plot for the various
lithologies encountered in reservoir rock
Unfortunately, reservoirs are seldom totally clean and shales can have a profound influence on
the value of the neutron log due to the water present in the clays. This clay-bound water is the
cause of the increase of the neutron values. Consequently, the neutron value in shales is
significantly higher than the density value. The amount of clay-bound water is dependent on clay
properties and thereby on the type of clay. We can infer the type of shale by proper interpretation
of the neutron and density logs in the presence of clay. It is however important to have knowledge
about the distribution and the type of clays that leads to various shale types.
The resolution of the neutron/density combination is approximately 0.3 to 0.6 m. Shale intervals
with a thickness larger than this resolution can easily be distinguished on a neutron/density log.
Shale is often contained in fine laminated sand/shale sequences, with interval thickness below
this resolution and cannot be resolved. The laminated heterolithic interval from 2075 – 2089 is a
clear example of this case as shown in Figure 9.7 on the images of the coreslabs. The shale
layers are much smaller than the resolution of the neutron-density logs. Imaging, either from
147
Chapter 9
coreslabs or from borehole images, will prove to be very valuable for laminated intervals. With
resolutions down to millimetres and sub millimetres, imaging can resolve even the finest layers.
However, calibration of the images with either thin sections or SEM/BSE images will again be
required to arrive at reservoir parameters with acceptable accuracy.
Clean sandstone
Dispersed clay
Laminated clay
Structural clay
Figure 9.5
The various occurrences of clays in reservoir rock
according to Thomas & Stieber
Two types of shale can be distinguished: 1) detrital shale (consisting of clay grains) that was
deposited when the sediment was laid down, and 2), authigenic or diagenetic shale that was
formed in-situ. Thomas & Stieber [1975] proposed an operational model based on the neutron
and density logs to assess the distribution of clays in shales and shaly sands. The different
appearances of clays in sandstones according Thomas & Stieber; dispersed, structural and
laminated, are depicted in Figure 9.5. The quartz grains are coded in yellow, clays in green and
pore space in blue, according to the colour coding we also used in the DEF images. We
distinguish clays present at the pore scale and at a somewhat larger scale, leading to the above
mentioned laminated shale. Upper left shows a clean sandstone with no clays present. At the
pore scale, shale can occur as dispersed shale, partly filling the pore space reducing the porosity.
The clays are attached to the pore wall where they 'grew' over geological times. This type of clay,
shown in Figure 9.5 upper right, is known as authigenic clay as it precipitates from the pore
waters that are rich in dissolved minerals. Another type of shale consists of so called clay clasts
that do not reduce the porosity. This type of shale forms an integral part of the matrix as it was
deposited at the time of sedimentation together with the quartz grains, the above mentioned
detrital shale. The terms detrital and authigenic are geological terms. In petrophysical terms, the
detrital shale is known as structural shale and is shown in Figure 9.5 lower left. Finally, lower right
the laminated shale is depicted. In laminated shale there is a continuous layer of clay that acts as
a surplus conductor for electric current in the case of resistivity logs. The increase in conductance
may lead to erroneously low hydrocarbon saturation estimates. The thickness of the layers in
laminated shales can range from several tens of microns up to many metres in thickness and can
occur at a range of scales. In addition, laminated shales reduce the vertical permeability to very
low values.
In shaly reservoirs, we can use the neutron and density logs in combination with the model that
was proposed by Thomas & Stieber to determine whether the shale is occurring as laminated,
dispersed or structural. In Figure 9.6 the cross-plot of the neutron porosity and density porosity
148
Chapter 9
from well A is presented. It can be seen that at many points the neutron porosity is substantially
higher than the density porosity. Detailed inspection of both the coreslab images and the
SEM/BSE images showed that the increase in neutron porosity is attributed to the presence of
laminated and dispersed clay. From the images that were taken from the coreslabs form well A
we observed that laminated shale layers have a thickness in the range from millimetres to
centimetres, so-called shale drapes, which are below the resolution of the neutron/density logs.
From the position of each measurement point on the neutron/density cross-plot, the volume
fraction and type of shale can be estimated.
Neutron-density cross-plot well A
Density Porosity [%BV]
50
45
Clean sand
point
40
35
Structural
shale point
30
25
20
15
Laminated
shale point
10
Dispersed
shale point
5
0
0
10
20
30
40
50
Neutron porosity [% BV]
Figure 9.6
Density-neutron cross-plot of the logs from well A
The clean sand point was selected from the interval where the neutron end density logs had
approximately the same value and was confirmed by the coreslab images; at those locations
clean sand was found. The neutron porosity and density porosity are related to the effective
porosity found in the SEM/BSE images:
Φ neutron − sand = Φ density − sand = Φ effective
9.1
The laminated shale point was found by searching for a location on the Gamma ray log that
contained a high amount of shale and reading the values for neutron and density porosity
respectively. Also in this case confirmation had to be obtained by inspecting the coreslab images
and the density porosity is related to the SEM/BSE micro-porosity:
Φ density − shale = Φ micro
9.2
149
Chapter 9
The dispersed shale point followed from the assumption that all pore space in the detrital
assemblage is fully filled with clay. Thus, in the dispersed shale point the amount of shale is equal
to the clean porosity. The neutron porosity and density porosities then follow from:
Φ neutron − disp = Φ neutron − sand × Φ neutron − shale
9.3
Φ density − disp = Φ density − clean × Φ density − shale
9.4
The structural shale point in Figure 9.6 is based on the assumption that all quartz grains are
replaced by clay clasts (clay grains) and that the pore space between the clay grains is still
present. By taking into account the micro-porosity inside the clay grains the neutron and density
porosities for the structural point follow from:
Φ neutron − struct = (1 − Φ neutron − sand ) × Φ neutron − shale
(
)
Φ density − struct = 1 − Φ density − sand × Φ density − shale
9.5
9.6
Although the Thomas/Stieber analysis gives a semi-quantitative estimation of the type and volume
of clay, it should be noted that different clay types generally yield different contributions to the log
responses, especially for the neutron log. Therefore, the methodology for clay typing and
quantification that has been described in this study can serve as a useful, if not essential, aid in
the interpretation of wireline log data.
9.5
The role of DEF images in core description
The core description can be amended with accurate interpretation of DEF images. Figure 9.7
shows the core description and the coreslab images associated with the different lithofacies. In
the laminated heterolithics from 2075 – 2089 m the images of two coreslabs are shown. The
laminated structure can be clearly observed in both images, coarse laminations on the left
coreslab and very fine laminations (with shale drapes) on the right coreslab. The different shades
of brown and grey indicate different clay types and different amounts of clay in the sands.
Figure 9.10 shows the DEF images of two samples taken from the heterolithic interval some 0.5 m
apart. The left image is a fine/medium-grained sandstone and of excellent reservoir quality, a plug
porosity of 28.5 %BV and a plug permeability of 597 mD was measured with conventional
methods. The right image represents a shale layer, in this case a mud supported shale, with very
low porosity and negligible permeability. These lithologies are considered two end members of
this interval, all other lithologies will be a mixture of these two. A sandy shale also occurs in this
heterolithic interval and the DEF image of this lithology is shown in Figure 9.11. This sample has a
plug porosity of 25.3 %BV and a plug permeability of 5 mD. The interval 2089 – 2092 m is a pure
shale. Figure 9.12 shows the DEF image, and the high clay content indicates that this is a mud
supported shale. The interval 2092 - 2069 m is a fine/medium-grained shaly sandstone. This was
confirmed by the DEF image of a sample from this interval, which is shown in Figure 9.13. The
plug porosity was 30.9 %BV, plug permeability was 1113 mD. From the presence of the clays in
the pores, it is clear that this sample is slightly shaly.
Figure 9.14 shows the DEF image of sample taken from the second shale interval at depths 2096
– 2100 m. From the presence of large quartz grains and smaller feldspar grains we conclude that
150
Chapter 9
this is a grain supported shale. This interval differs substantially from the one at 2089 – 2092 m,
the latter is a mud-supported shale with only very small, silt sized quartz grains.
Finally, Figure 9.15 shows a DEF image from a sample that was taken in the coarse-grained,
water bearing, interval starting at depth 2100 m and contains large quartz and feldspar grains with
diameters above 250 micron. The plug porosity of this sample was 28.5 %BV and the plug
permeability was 1072 mD. This very clean sandstone is of excellent reservoir quality.
Unfortunately this interval has been evaluated to be water bearing and was therefore not of
interest from a petroleum engineering point of view.
Depth
2075
Laminated
heterolithics
2089
Shale
2092
Fine grained
shaly sand
2096
Shale
2100
Coarse grained
clean sand
2107
Figure 9.7
Core description presented with 1 m sections of the core from well A
151
Chapter 9
Figure 9.10
DEF images from two samples taken from the heterolithic interval. Left: a clean,
fine-grained sandstone, right: a mud supported shale. The samples were taken at 0.5 m distance
Figure 9.11
Sandy shale, grain supported shale
152
Chapter 9
Figure 9.12
content
Mud supported shale, note the high clay
Figure 9.13
Slightly shaly sandstone from the
fine/medium-grained interval
153
Chapter 9
Figure 9.14
Mud supported shale with high quartz content
Figure 9.15
Clean coarse sandstone
154
Chapter 9
The above analysis shows that the interpretation of DEF images can substantially contribute to
the geological and petrophysical evaluation. It is emphasised that the interpretation of DEF
images can only be meaningfully carried out in combination with other information such as
wireline logs and core images. The latter may be partly replaced by acoustic or resistivity borehole
image logs combined with the image analysis methods that were developed in this study.
9.6
Petrophysical parameters from BSE images, well A
As discussed in chapter 4, we used the samples from well A to test the models that we developed
for prediction of porosity and permeability. For this reason, the samples from well A were
excluded in the determination of the models in chapter 8. Figure 9.16 shows a flow chart of this
test procedure, the top line shows the image analysis part while the bottom line reflects the
physical part. The impregnated samples were taken after the physical measurement had been
carried out.
BSE imaging
+
analysis
Impregnated
sample
Petrophysical
parameters
from Regression model
Image features
Comparison
Physical
measurements
Cleaned
sample
Figure 9.16
Petrophysical
parameters
Test procedure for the models developed in chapter 8
Figure 9.17 presents the cross-plot of the porosity calculated according to equation 8.6 and the
porosity that was measured on 36 core plugs form well A. The porosity that was predicted with our
porosity model (from 4 images of each sample) agrees well with the plug porosity.
Porosity Well A
40
Plug Porosity
30
20
10
R2 = 0.75
0
0
10
20
30
SEM/IA Porosity
Figure 9.17
Cross-plot of the porosity predicted form
BSE images and the porosity from core plugs
155
40
Chapter 9
Statistical analysis showed a correlation coefficient of 0.87 calculated form 36 paired
observations. To investigate whether the correlation is significant we applied a statistical test. The
critical value for the correlation coefficient was obtained from a two-tailed test table with a value
for α = 0.05, and was 0.325 [CRC Handbook of Probability and Statistics, pg 390]. The calculated
correlation coefficient of 0.87 is higher than the critical value of 0.325 and we therefore conclude
that the correlation is significant.
The five samples that deviate more than 5 % BV from the plug porosity will be further discussed
below in order to assess whether a physical cause for the deviation could be found. Figure 9.18
shows the DEF image of the sample for which the SEM/IA porosity was 24.6 %BV while the plug
porosity was 30.6 %BV. This sample was taken from the coarse-grained clean sandstone in the
water-bearing interval. In the lower left of Figure 9.18 the calcite cement, filling the pores, is
clearly visible. This is a typical example of heterogeneity at the scale of BSE images and is the
cause of the underestimation of the SEM/IA porosity.
Figure 9.18
SEM/IA porosity is under-estimated because
of heterogeneity caused by the patches of calcite cement
Similarly, heterogeneity at the plug scale is the cause for the over-estimation of porosity of a
sample from the fine-grained shaly sand in the interval 2092 - 2096 m. Figure 9.19 presents a
DEF image in which the pore filling cement can be clearly seen. The SEM/IA porosity for this
sample was 28.3 %BV while the plug porosity was 22 %BV. It is remarked that heterogeneity at
the scale of BSE images can lead to both under- and over estimation of porosity, but can be
detected as was discussed in chapter 8.
156
Chapter 9
Figure 9.19
Heterogeneity caused by pore filling cement,
over-estimation of porosity
Figure 9.20
DEF image from a sample with overestimated porosity. Cause can be either the cracks in the clays
or heterogeneity
157
Chapter 9
Figure 9.20 presents a DEF image from a sample with a calculated SEM/IA porosity of 19.6 %BV
and a measured plug porosity of 14.6 %BV. In this case, it could not be established whether the
over-estimation of porosity is attributed to heterogeneity at the plug scale or to the cracks that are
the result of clay swelling in the cleaning prior to the sample preparation procedure. The cracks
can be clearly observed but their occurrence in area percentage is not sufficient to fully explain
the over-estimation of SEM/IA porosity.
Figure 9.21 shows the cross-plot of the permeability predicted with the Network model and the
permeability that was measured on core plugs. Statistical analysis showed that the correlation
coefficient is 0.89. The critical correlation coefficient for this data set was taken from a two-tailed
test table with a value for α = 0.05, and was 0.325 [CRC Handbook of Probability and Statistics,
pg 390]. We consider the correlation significant because the value of 0.89 is substantially higher
than the critical value of 0.325.
Permeability Well A
10000
Plug Permeability
1000
100
10
1
0.1
0.1
1
10
100
1000
10000
SEM/IA Permeability
Figure 9.21
SEM/IA permeability versus plug permeability
Three outliers were observed: two under-estimates and one over-estimate, indicated in blue in
Figure 9.21. The sample with the overestimated SEM/IA permeability of 2290 mD is the same
samples that showed an overestimated SEM/IA porosity that was larger than 5 % BV and that
was discussed above. The DEF image of this sample is shown in Figure 9.19.
Figure 9.22 presents a DEF image from another outlier on the permeability cross-plot. The
calculated SEM/IA permeability is 0.48 mD and the plug permeability is 60 mD. The pores are
almost fully filled with calcite cement, which was confirmed with X-ray analysis. Evaluation of all
four DEF images revealed that only very little effective porosity was present. Inspection of the
core plug indeed showed heterogeneity at the plug scale. Streaks of thin cemented bands were
observed. We concluded that the four BSE images taken from the plug originated from the
cemented areas.
158
Chapter 9
Figure 9.22
DEF image from a sample for which the
SEM/IA permeability is under-estimated because of local
cementation
9.7
Petrophysical parameters from BSE images, well B
A second small set of eight samples from well B was available for imaging with electron
microscopy. In addition to testing the predictive capability for porosity and permeability, we will
show that these images can be used to assess reservoir sand quality. Conventionally this
assessment is carried out on thin sections with optical microscopy. All samples originated from a
sandstone reservoir with plug porosities between 20 and 25% BV, and permeabilities in the Darcy
range.
Figure 9.23 shows the cross-plot of the predicted SEM/IA porosity and the plug porosity. The
explained variance was 0.38. The reason for this low value is that the porosity values extend over
a rather small range only. Therefore, we did not insert a trendline in the cross-plot but show the
x=y line instead. From a statistical point of view the correlation cannot be considered significant
because the correlation coefficient of 0.62 is lower than critical value of 0.71 for a set of 8
samples. The critical value is based on a two-tailed test with a probability of 5 % to reject the nullhypothesis that the data is un-correlated. Clearly, the problem here is the low number of samples
that hampers a representative correlation. However, from Figure 9.23 it can be seen that the
SEM/IA porosity agrees well with the porosity measured on plugs. Therefore, the average error of
1.34 % BV in the prediction is a better indicator for the quality of the prediction.
159
Chapter 9
Porosity well B
30
Plug Porosity
25
20
15
10
5
0
0
5
10
15
20
25
30
SEM/IA Porosity
Figure 9.23
Cross-plot of porosity from BSE images and
porosity measured from plugs
Figure 9.24 presents the cross-plot of permeabilities that were obtained from BSE images and the
permeability measured on core plugs. Similarly, in this case a very acceptable agreement is
obtained. The value of 0.85 for the explained variance leads to a correlation coefficient of 0.92. In
this case, the correlation is considered significant because the value of 0.85 is higher than the
critical value of 0.71. Note that the permeability ranges over 4 orders of magnitude.
Permeability well B
Plug Permeability
10000
1000
100
10
1
1
10
100
1000
10000
SEM/IA Permeability
Figure 9.24
Permeability predicted from SEM/BSE
images versus the plug permeability
The lithological classification for the eight samples is presented in Table 9.2. The lithological
description reveals the cause for the lower permeabilities of three samples 1,7 and 8. Figure 9.25
shows the DEF image from sample 1, which shows that the low permeability is caused by patchy
pore filling clays. Figure 9.26 presents a DEF image from the coarse clean sandstone. A DEF
image from a fine-grained, arkosic sandstone is shown in Figure 9.27. The combination of small
160
Chapter 9
grains and clay causes the low permeability. Finally, in Figure 9.28 an arkosic, shaly sandstone
can be seen, the poor sorting combined with the clays present in the pores diminish the reservoir
quality.
Observation of the BSE and DEF images revealed a depth trend. Based on the eight samples, a
lithological sequence can be recognised. As a function of depth, one can distinguish the following
sequence:
Sample
Lithology
1
2
3
4
5
6
7
8
Fine grained, well sorted, shaly, arkosic sandstone
Coarse grained, well sorted, clean arenite
Coarse grained, well sorted, clean arenite
Medium grained, well sorted, clean arenite
Medium grained, well sorted, clean arenite
Very fine grained, well sorted, shaly, arkosic sandstone
Fine grained, poorly sorted, shaly, arkosic sandstone
Medium grained, poorly sorted, slightly shaly, sandstone
Reservoir
quality
Poor
Excellent
Excellent
Good
Good
Medium
Poor
Medium
Figure
9.25
9.26
9.27
9.28
-
Table 9.2
Lithology derived from BSE images and the estimated reservoir quality for 8
samples from well B.
Particle shapes for the detrital assemblage were in all cases classified as angular to sub-angular,
indicating a low energy environment of deposition. It should be noted that the lithological
sequence was derived solely from the SEM/BSE images, eight sampling points over 87 m of
reservoir is heavily under-sampled. When only a limited amount of samples is available for
analysis, careful attention has to be paid to reservoir heterogeneity. For instance, the
heterogeneity in this reservoir is indicated by the vast differences in properties between sample 3
(Figure 9.26) and 5 (Figure 9.27), which were taken only 1 m apart. Therefore, more work will be
required to substantiate and further refine the observed lithological sequence in order to make
meaningful use of it in reservoir evaluation. The evaluation should be amended by other
information such as wireline logs, core and perhaps borehole images. The latter especially are
extremely useful to delineate sections with different reservoir quality, which, in turn, enables
optimisation of reservoir management.
161
Chapter 9
Figure 9.25
Patchy pore filling clays, an example of
heterogeneity occurring at the scale of BSE images
Figure 9.26
Coarse clean sandstone
162
Chapter 9
Figure 9.27
Fine-grained, arkosic sandstone
Figure 9.28
Fine-grained, very shaly arkosic sandstone,
poor reservoir quality
163
Appendix A
Monte Carlo simulations for BSE resolution
Appendix A. Monte Carlo simulation for BSE resolution
A1.
Introduction
The correct interpretation of BSE images requires a detailed understanding of the physical
processes that provide the data for the image formation. Backscattered electron images are the
result of high-energy electrons that emerge from the sample surface after a number of interactions
of the electrons in the primary beam with the atoms constituting the matter in the sample. These
interactions can be modeled by the so-called Monte Carlo method. Monte Carlo modeling
provides a stepwise simulation of the trajectory of an electron that enters matter from vacuum.
A2.
The electron scattering process
When an electron enters a sample, two end-results are possible: either the electron is absorbed in
the matter matrix in the sample or the electron leaves the sample. Both results occur after many
scattering events that are a mix of elastic and in-elastic events. In the in-elastic case, the electron
looses its energy by generating other phenomena that can be roughly divided in electron events
and photon events: electron events encompass the generation of secondary, backscatter, Auger
and absorbed electrons. It is remarked that in the SEM world an arbitrary boundary between
secondary and backscattered electron exists, all electrons with energies below 50 eV are
considered secondary, all electrons with energies above 50 eV are assigned to be backscattered
electrons. Non-electron events comprise of the generation of X-rays, cathodoluminescense and
heat. In the elastic case, the electron looses only minor amounts of energy and the electron is
capable of leaving the sample surface after many events with nearly the incident energy.
With respect to the interaction between the electron beam and the sample, functional
relationships can be derived. These relate sample parameters and electron beam parameters on
the one hand and macroscopic properties like backscatter coefficients on the other hand.
However, the vast amount of events an electron can encounter in a single trajectory precludes the
use of a detailed analytical model. A statistical model such as the Monte Carlo technique uses
random numbers as a method for predicting the occurrence of possible events. Within the
framework of this study, the Monte Carlo technique was used to assess the dependence of the
physical resolution in BSE images on controllable beam parameters and material parameters. The
most important beam parameter was the energy of the electrons in the beam, the major material
parameters were the elemental constituents in the sample material. These constituted the pore
filling epoxy and the rock matrix minerals such as quartz, feldspar and calcite.
The Monte Carlo technique statistically describes the trajectory that an electron can take when
entering solid matter. Note that the simulation cannot produce real electron trajectories. However,
if the physics of the events can be properly modeled, the result of many trajectories will accurately
describe effects that can be experimentally confirmed. The required calculations are based on two
basic parameters: 1) the angle by which the electron is deflected as it travels through the sample,
and 2), the estimate how far an electron will travel given a certain incident energy.
A3.
Approximations
The Monte Carlo technique we used in this study employs two approximations in order to reduce
the required computing time:
164
Appendix A
1) It is assumed that only elastic scattering events are significant in the calculation of the
electron trajectory in the solid matter. Elastic scattering, described by the screened Rutherford
cross-section and produced by the Coulomb interaction between the travelling electron and the
nuclei in the solid matter, can result in angular deflection of a few degrees up to 180º. It is
remarked that the majority of the in-elastic scattering events produce angular deflections that
are typically 0.5º or less. Therefore, elastic scattering effects dominate the trajectory an
incident electron takes. Ignoring the effects of in-elastic scattering introduces a negligible error
while greatly reducing the number of calculations that are required to arrive at a statistically
valid result.
2) The electron is assumed to loose its energy continuously at the rate indicated by the Bethe
relationship rather than as results of discrete in-elastic events. This simplification allows the
net result of all possible in-elastic scattering processes to be accounted for without concern for
the exact details of the individual events.
A4.
Simulations for the determination of BSE resolution.
The Monte Carlo simulations required for this study were carried out with the software Electron
Flight Simulator version 2.0 from Small World. This software enabled to simulate electron
scattering experiments for a multitude of configurations. The specific configuration that was
required for our experiments is a bulk substrate with a thin layer on top. Because we were
investigating the size of the point-spread function of the high-energy electrons that emerge from
the sample surface, the electron model was used. In this model, the generation of X-rays is
excluded and solely the trajectory of electron/electron events and electron/nucleus events is used.
The reservoir rock samples that were used in this study contain next to the epoxy in the pore
space, a variety of rock matrix minerals. Analysis of the detrital assemblage revealed minerals like
quartz, dolomite, feldspars or in some cases calcite. In our data set the bulk percentage other
than the pre-mentioned minerals never exceeded five volume percent. Therefore, we limited the
Monte Carlo experiments to the five main constituents: epoxy, quartz, potassium feldspars, calcite
and dolomite. As we used flat polished samples at an incident angle of 90º, all simulations were
carried out with perpendicular entrance of the electron beam into the sample. To prevent charge
effects, all samples physically had a thin carbon layer applied to the surface at the area of
interest. Therefore, this carbon layer was included in the simulations. The thickness of the applied
carbon layer varied between 70 and 100 nm, therefore in the simulations a thickness of 100 nm
was selected for the carbon layer. Consequently, the total configuration we used in the
simulations consists of a bulk substrate of either epoxy, quartz, calcite, dolomite or potassium
feldspar that was topped with a carbon layer of 100 nm. Another free simulation parameter was
the number of electron trajectories and experiments have been carried out to determine the
optimal value. We selected approximately 5000 electron trajectories because the shape of the
excitation volume did not alter substantially when more trajectories are taken. Therefore, a total
number of 5000 electron trajectories was selected as the stopping criterion for the simulations.
The Monte Carlo scattering experiments were carried out over a range of beam electron energies:
from 3 to 30 keV. The results of these simulations for the five reservoir rock constituents are
presented in Figure A1. It can be observed that the mineral constituents all show roughly similar
165
Appendix A
resolutions, in this case the diameter of the assumed circular point-spread function (PSF). Below
a beam energy of 7 keV the diameter of the PSF is less than one micron.
Resolution
PSF [microns]
20
Epoxy
15
Quartz
Calcite
10
Dolomite
Feldspar
5
0
0
5
10
15
20
25
30
Beam energy [keV]
Figure A.1
Size of the circular PSF as a function of the
electron beam energy for the main reservoir rock constituents
Figures A.2 to A.4 show the actual graphical results of a set of Monte Carlo experiments on
epoxy, quartz, potassium feldspar, calcite and dolomite at a beam energy of 5 keV. The actual
diameter of the circular PSF was obtained by measurement of the line distance around the
electron beam from which electrons can emerge from the sample surface. As this is a statistical
phenomenon, the criterion we used is 95% of the pixels present on that line in both directions
starting from the electron beam. When uniform constituent properties are assumed, this is then
the diameter of the point-spread function.
Micron
Electron beam
C-layer
0.6 micron
Bulk epoxy
Micron
Figure A.2
Results of the Monte Carlo experiments for
epoxy with a carbon layer of 100 nm
166
Appendix A
For the minerals involved in the simulations, the diameter of the PSF was in the range of 0.4 to
0.5 micron. In Figures A.3 and A.4 this diameter is indicated in yellow. This outcome is not
surprising; the atomic density of these minerals extends over a small range as can be seen in
table A1.
Micron
Micron
Electron beam
Electron beam
C-layer
C-layer
0.46 micron
0.48 micron
Bulk dolomite
Bulk quartz
Micron
Micron
Figure A.3
PSF's for quartz and dolomite at a beam energy of 5 keV
Micron
Electron beam
Micron
Electron beam
C-layer
C-layer
0.46 micron
0.50 micron
Bulk calcite
Bulk feldspar
Micron
Figure A.4
Micron
PSF's for calcite and potassium feldspar at a beam energy of 5 keV
Constituent
Quartz ( SiO2 )
Calcite ( CaCO3 )
Dolomite ( CaCO3 MgCO3 )
K-Feldspar ( KALSi3 O8 )
Epoxy ( C5 H8 O2 )
Table A.1
constituents
Density
2.654
2.710
2.850
2.600
1.185
Diameter PSF
0.48
0.46
0.46
0.50
0.60
Diameter of the PSF for the various reservoir
The value for epoxy was the highest, the low density caused a larger PSF and thereby a lower
resolution of 0.6 micron. From these Monte Carlo experiments, we concluded that at 5 keV beam
energy, the resolution for the BSE images varied slightly around 0.5 micron.
A5. Simulations for the determination of the BSE coefficient
The Electron Flight simulation software also enabled the calculation of the BSE coefficient. The
BSE coefficient was defined as the decimal fraction of the primary beam electrons that can
emerge from the sample surface and thereby become available for detection. The BSE coefficient
167
Appendix A
was calculated for the five constituents over the same range of electron beam energies as used to
BSE coefficient
0.16
BSE coefficient
0.14
0.12
Epoxy
0.10
Quartz
0.08
Calcite
0.06
Dolomite
Feldspar
0.04
0.02
0.00
0
5
10
15
20
25
30
Beam energy [keV]
Figure A.5
BSE coefficient as a function of the beam
determine the resolution. Figure A.5 presents the results of these simulations. At 5 keV beam
energy the BSE coefficient for all constituents range between 0.075 and 0.105. From Figure At a
beam energy of 3 keV approximately equal amounts of backscattered electrons were generated
for the main sample constituents. This essentially means that at 3 keV only minor atomic density
exists for the minerals and the epoxy in the reservoir rock samples. Indeed, experiments showed
that at a beam energy of 3 keV it rendered impossible to generate a BSE image with sufficient
contrast between the rock matrix material and the pore filling epoxy. This effect was amplified by
the limited penetration power of the 3 keV electrons, their energy was too low to twice traverse the
C layer that was applied to prevent charge effects. Consequently, mainly the C layer was
imaged and only very limited atomic number contrast could be obtained. Figure A.6 shows the
graphical representation of the results of a simulation at 3 kev beam energy. It can be observed
that only a limited number of electron can penetrate the 100 nm carbon layer. The probability of
an electron that reaches the quartz area and retains sufficient energy to escape the sample
surface is considered negligible. We therefore concluded that in terms of BSE coefficient a beam
energy of 3 keV is lower limit.
Micron
Electron beam
C-layer
Bulk quartz
Micron
Figure A.6
Excitation volume at 3 keV beam energy, note that the
majority of interactions takes place in the carbon layer
168
Appendix B
Morphological filtering, opening and closings
Appendix B. Morphological filtering, openings and closings
B1.
Introduction
In Chapter 7 we explained that the combination of Sigma filtering and thresholding in itself did not
provide DEF images that resembled the original BSE image sufficiently. This was caused by the
BSE version of the Holmes effect that occurred in thin section analysis (see Chapter 5). This socalled BSE edge effect can be substantially minimised by lowering the energy of the beam
electrons. With less energy, the travel path of these electrons in matter reduces. However,
restrictions in BSE detector efficiency and signal to noise considerations limited the beam energy
to 5 keV. Thereby, the lowest attainable diameter of the excitation volume was limited to some 0.6
micron for the epoxy in the pore space. Specifically the clays in BSE images suffered from this
edge effect because of their location; at the edges of grains on one side and the pore space on
the other side. Clays are the result of diagenetic precipitation on the grain surface and are thereby
always found at the pore-grain interface. This hampered a unique distinction between the clay
layers with thickness’ below 0.6 micron and the edge effect occurring at clean grain surfaces.
However, clay layers with thickness’ below 0.6 micron do not influence petrophysical properties,
such as permeability, dramatically. Therefore, such thin clay layers can be neglected and in image
terms, may therefore be removed from the DEF images. This removal could be established most
effectively by using morphological operators in the binary domain, a field that is known as
mathematical morphology.
Mathematical morphology is basically a theory for the analysis of spatial structures. The term
morphology stems from the analysis subject; it concerns the shape and the spatial structure of
objects. Morphological operators belong to the class of non-linear neighborhood operators. The
neighborhood for a certain morphological operator is known as a structuring element. The
operators are defined by testing whether or not a structuring element fits or does not fit image
objects by means of Boolean operations. Mathematical morphology can be applied to
n-dimensional sets in the grey-level domain or in the binary domain. In this study, the application
of mathematical morphology was restricted to the binary domain and 2D data sets, hence images.
It is possible to carry out all the required non-linear filtering in the grey-value domain but for
simplicity, we limited ourselves to binary images as we process individual phases such as pores,
clays and rock matrix minerals.
B2.
Binary erosion and dilation
Erosion and dilation are two main fundamental morphological operators because all other
operators are based on combinations of these. Erosion and dilation operators can be controlled
by two parameters, the size/shape of the structuring element and the number of times the
operator is applied. For our application, we selected a structuring element shaped as a hexagon
with a size of 5 x 5 pixels.
This structuring element was applied two times for erosion on the pores in Figure B.1.
The left image in Figure B.1 shows the original pore system as extracted from a BSE image. The
right image presents the eroded pore system with the original pore system in yellow. The pores
that fit the structuring element were removed by this operation. The erosion operator on a set X
by a structuring element B is denoted by ε B ( X ) and is mathematically defined as:
169
Appendix B
ε B ( X ) = {x| B x ⊆ X }
B1
In which x is the pixel that is excluded in X when the structuring element B is placed at x . In
other words, for the pixel at x to ‘survive’, all other pixels in B should be 1. This process can also
be described as an intersection of translations where the translations are determined by the size
and shape of the structuring element:
Figure B.1
Hexagon erosion operator applied to a binary pore image
ε B ( X ) = Ι X −b
B2
b ∈B
This definition can be extended to images: the erosion of an image f by a structuring element B
( ) and defined as the minimum of the translations of
is denoted by ε B f
f by the vectors − b of
B:
ε B ( f ) = ∧ f −b
B3
b ∈B
( )
Therefore, the eroded value at a certain pixel in an image f x , y , is the minimum value of the
( )
image in de window defined by the structuring element when its central pixel is placed at f x , y :
[ε ( f )]( x , y) = min f ( x + b, y + b)
B
B4
b ∈B
The dilation is a dual operator of the erosion. The dilation of the pore system in Figure B.1 is
presented in Figure B.2. A hexagon structuring element with size 5 x 5 pixels was applied two
times.
The right image shows the original pore system in yellow. The dilation of a set X using
structuring element B is denoted by δ B ( X ) and is defined as:
δ B ( X ) = {x| Bx ∩ X ≠ 0}
B5
170
Appendix B
Figure B.2
Hexagon dilation operator applied to a binary pore image
On which x is the pixel that is included in X when the structuring element B is placed at x .
Hence, a pixel is generated when at least one of the pixels in the window defined by the
structuring element is present. Similarly, to the case of erosion, the dilation can be expressed in a
union set of translations in which the translation is defined by the structuring element:
δ B ( X ) = Υ X −b
B6
b ∈B
and can be extended to images: the dilation of an image f by a structuring element B is
denoted by δ B and is defined as the maximum of the translation of by the vectors − b of B :
δ
(f)=
∨ f −b
B7
b ∈B
( )
Consequently, the dilated value at certain pixel in an image f x , y , is the maximum value of the
( )
image in de window defined by the structuring element when its central pixel is placed at f x , y :
[ε ( f )]( x , y) = max f ( x + b, y + b)
B
B3.
B8
b ∈B
Binary opening and closing
The erosion and dilation are the most basic building block operators in mathematical morphology
applied to binary images. The erosion destroys all objects that can be encompassed by the
structuring element while objects that are larger than the structuring element are only eroded. This
behaviour is controlled by the size of the structuring element. The need for an operator that
removes morphological roughness from images leads to the definition of the morphological
opening operator. The small objects are removed and the larger ones only smoothed. An opening
operator consists of an erosion with a certain structuring element followed by a dilation with the
same structuring element:
γ B ( X ) = ε B (δ B ( X ) )
B9
171
Appendix B
In terms of Boolean operations the opening is defined as follows:
γ B ( X ) = Υ{ B| B ⊆ X }
B10
Figure B.3 shows the result of two openings with a hexagon structuring element applied to the
pore system in Figure B.1.
Figure B.3
Hexagon opening operator applied to a binary pore image
The left image shows the original pore system, the right image the opened one. It can be
observed that the pores equal or smaller to twice the size of the structuring element are removed.
Similarly, the roughness of the larger surviving pores that is covered by the structuring element is
also reduced resulting in a smoothing effect. The dual operator of the opening is the closing:
φ B ( X ) = δ B (ε B ( X ) )
Figure B.4
B11
Hexagon closing operator applied to a binary pore image
172
Appendix B
With its Boolean representation:
[
]
φ B ( X ) = C Υ{ B| B ⊆ X c } = Ι { B c | X ⊆ B c }
B12
In which the notation c indicates the complement. A closing operator applied to the pore system of
Figure B.1 is presented in Figure B.4. It can be observed that holes and crevasses of the size of
twice the hexagon structuring element are filled.
B4.
Morphological operation applied to DEF images
The four morphological operators defined above were used to modify the clay phase in DEF
images such that an optimal fit was obtained with the BSE images of origin. Specifically the
opening and dilation operators were used to remove noise resulting from the thresholding step.
Figure B.5
hexagon opening operator applied to the clays. Note that the surplus in clays
caused by the edge effect in the thresholding step is largely corrected
In addition, the edge effect was largely removed by applying a hexagonal opening operator.
Chapter 7 described the processing of the individual phases in the DEF image in more detail. This
appendix merely defines the applied morphological filter operators that were used in the
processing. Figure B.5 compares the results applying a hexagon opening operator to the clay
phase. The left image shows the BSE images on which the non-corrected clays are indicated in
yellow. The accompanying image on the right shows the corrected clays in the yellow overlay.
173
Appendix C
Directional permeability from network images
Appendix C. Directional permeability from network images
C1.
Introduction
In the past substantial effort was devoted to use electric analogues of pore network
representations to determine permeability [Dullien, 1979]. This approach was based on the
observation that electrical and hydraulic conductances are based on the same linear potential
difference theory; and that Darcy’s law is the hydraulic equivalent of the electrical Ohms law. In
this appendix, we present an approach to configure 2D network models in a deterministic manner
and to derive directional conductivity from these. The analogy between electric conductance and
hydraulic conductance may allow prediction of permeability.
C2.
Network approach
Hitherto, a statistical approach was mostly taken in these efforts as was discussed in chapter 7.
This statistical approach is depicted in the flow chart in Figure C.1. A disadvantage of the
statistical approach is that these 2D networks have a fixed, predefined structure in many
configurations; for example square, hexagonal or hexa-triangular [Dullien, 1979].
Statistical approach
∆P
Heterogeneous
system
Statistical
averaging
Replace each
property by its
effective value
Homogeneous
system
Same macroscopic flow properties
∆P
Conductance
Figure C.1
Statistical approach to calculate hydraulic conductance from network
information using the electrical analogon
This predefined structure was taken because it is mathematically easier to derive and solve the
equations that are required to calculate the conductivity. Inspection of the networks we extracted
from BSE images demonstrated that the actual structure is a mixture of these configurations.
The pore network from BSE images is solely determined by the location of the pores and raises
the question whether the known topography of the pore network, albeit in 2D only, would lead to
an estimate of directional permeability. This idea was emphasised by the observation that the
permeability predicted with a statistical network model was superior to any other model that is
available to date, as we demonstrated in chapter 8. This possible deterministic network approach
is depicted in Figure C.2.
Deterministic approach
∆P
Figure C.2
Heterogeneous
system
Network
Kirchhoff
Conductance
Deterministic approach that preserves the network topography in 2D
It is not necessarily a disadvantage that the pore networks obtained from images are ‘only’ 2D.
The deterministic approach with networks implies that the total conductance becomes sensitive
for the direction in which the voltage is applied to the network. One can envisage that the
conductance can be calculated in two directions as indicated in Figure C.4. Moreover, we have
174
Appendix C
shown that the permeability perpendicular to the plane in which the image was taken could be
predicted with acceptable accuracy, using capillary bundle models such as the Poiseuille model
and the Kozeny-Carman model. The network approach may provide indicators for the
conductance, and thereby permeability, in the plane of the images. This is stylistically shown in
(the reproduced) Figure C.3.
DP
Direction of flow
Vertical
Network
Perpendicular
Poiseuille
Kozeny-carman
L
Horizontal
Network
Figure C.3
to the images
Components of the permeability tensor related
The NET images that are the result of the image processing that was developed in chapter 7,
provided the basis for this deterministic approach. Figure C.4 shows the conversion from a pore
network into an electrical network
Horizontal
NET image
Electrical network
V
I
Vertical
I
V
Figure C.4
Transformation of a NET image into an electrical network that preserves
the topography. The associated equations can be solved for situation where the driving
potential is applied in perpendicular directions
Once the topography of the network was known, resistances could be assigned to the bonds in
the network. The network nodes were considered to have zero resistance. The resistances in the
175
Appendix C
network were determined according to their length, and whether they were part of the pores or
part of the rock matrix. Measures were taken to incorporate the edges of the images into the
networks. This procedure transformed the pore network into an electrical network with the same
topography and values assigned to the resistors according to their length. A voltage was then
applied over the network, upon which the currents flowing in the network could be determined by
solving Kirchhoff’s law for each current loop. The total current flowing through the network should
then be an indicator for the overall horizontal or vertical conductance and we can compare this
conductance with permeability.
The left image in Figure C.4 presents the pore network from which the nodes and bonds were
extracted as shown in the middle image. Each bond was assigned a resistance, for three random
current loops this is indicated in red. The resistance was based on geometric properties that were
obtained from the images by measurement, for instance the length of the bond, the fraction of the
bond that lies in the pore space and the area of the associated pore.
By applying the voltage in two directions, horizontal and vertical, conductance could be calculated
as shown in the right two images in Figure C.4. In this way, the anisotropy for conductivity may be
assessed which is impossible when the statistical approach is used. The conversion from a POR
image into a NET image was designed such that only closed loops could occur. This was required
for the use of Kirchhoff’s law to solve the conductivity equations in the resistor network. Figure C.5
shows a simplified version of the resistor network from which we derive the equations that are
required to solve the total current in the network.
i2
R1
i5
R4
R2
i1
i3
R3
i4
imain
V
Figure C.5
The resistor network from which the main
current is obtained by solving Kirchhoff's law for each
current loop
The schematic in Figure C.5 shows 9 loops for which we defined the equations. For the centre
loop, the currents and the associated resistors are indicated in blue. In the electrical network, we
distinguished current loops inside the network with no driving voltage and the main current loop,
indicated in green, that has a driving voltage V. The Kirchhoff equation for the blue loop is:
R1 (i1 − i2 ) + R2 (i1 − i3 ) + R3 (i1 − i4 ) + R4 (i1 − i5 ) = 0
176
C.1
Appendix C
This equation can be divided in a forward current term and a negative counter-current term:
N
N

i1 ( R1 + R2 + R3 + R4 ) − (i2 R1 + i3 R2 + i4 R3 + i5 R4 ) ⇒ i1 ∑ Rk − ∑ i j Rk 
C.2
k =1
 k =1
 j = adjacent _ loop
A similar equation can be set up for all the loops and for the main current loop k with driving
(
)
voltage V. The result is a set of linear equations that builds up a matrix:
i1
1
∑R
forward
2
..
k −1
α
....
α
k
α
In which:
α j ,k
i2
....
i j −1
ij
α
....
α
α
=
0
....
α
....
....
....
α
....
∑ R forward
α
....
α
=
=
=
0
0
0
α
....
α
=
1
∑R
forward
∑R
forward
n

α
=
−
Rcounter → j = adjacent
∑

m =1
=
α = 0 → j ≠ adjacent

V
C.3
C.4
In this resistor matrix, the columns represent the currents and the rows represent the loops. The
forward current terms appear on the diagonal of the matrix while the counter current terms are
placed off-diagonal. Note that the counter current terms are either zero or negative. All the
elements of the voltage vector are zero with the exception of the last element which is the driving
voltage. The equations can be written more compactly in matrix form:
R ⋅i = V
C.5
Because the number of variables j is equal to the number of equations k, and the determinant is
non-zero, the system is non-singular and has a unique solution. In this way, the total conductivity
for the network can be calculated, which can then be compared with the measured permeability.
177
Appendix D
SEM/BSE analysis in an operational environment
Appendix D
SEM/BSE analysis; the equipment and the
use in an operational environment
In this chapter, we will discuss some of the engineering aspects of the methods that we
developed in this study. In addition to the equipment, an overview of the procedures that are
required for prediction of petrophysical parameters from BSE images will be summarised and
presented in flow charts.
D.1
The equipment
The equipment that was used for this study to predict petrophysical parameters from SEM/BSE
images, was especially configured for research and development purposes. The equipment can
be divided into three main components: the sample preparation equipment, the Scanning Electron
Microscope (SEM) and the image analysis (IA) tools. The IA tools comprised a PC-software
package and the SEM was a Phillips XL40-FEG equipped with an EDAX energy dispersive X-ray
analysis system (EDX). The addition FEG stands for Field Emission Gun, an electron source
based on field emission, which was discussed in chapter 6. , Figure D.1 shows the SEM
equipment that was used for this study.
Figure D.1
The equipment that is used in this study: the SEM with EDX system
coupled to a dedicated image processing system based on a PC
The image analysis procedures that were used in this study were developed with an image
analysis package, KS400 from Zeiss. This development platform provides image processing and
analysis functions that can be executed as command sequences, known as macros. The macro
language has a C programming structure and was used in interpreter mode for the development
of the procedures. KS 400 runs on a standard PC and has facilities for batch processing, which
were used to process and analysis the 1200 BSE images in the sample sets. All image
processing and analysis procedures used in this study were developed on a low cost PC
178
Appendix D
containing a Pentium 166 MHz with 64 Mb memory and 4 Mb graphic memory to display the
images.
D.2
The procedures
The flow chart in Figure D.2 shows the total sequence starting from plug ends, sidewall samples
or cuttings, and ending with the predicted parameters accompanied with the interpreted DEF
images.
Parameter
prediction
Sample
preparation
Image
collection
Image
processing
Image
feature
extraction
Image
interpretation
Figure D.2
The total sequence governing the steps from sample
preparation to petrophysical parameters and lithological description
The sequence in Figure D.2 can be divided into components that are related to the equipment.
The first part is the sample preparation that was discussed in chapter 6. Figure D.3 presents the
flow chart depicting the sample preparation process.
Cuttings
Sidewall
samples
Cleaning
&
drying
Impregnation
Polishing
Mounting
Coating
C and Au
Plug ends
Figure D.3
The sample preparation process required for SEM/BSE imaging
The outcome of the sample preparation process is a set of samples that can be loaded into the
SEM. Note that after the polishing process the samples should be manipulated only with
protective gloves to prevent pollution of the SEM sample chamber. In order to obtain good quality
BSE images it is essential that the sample chamber of the SEM be not contaminated as this is
kept at high vacuum conditions. The image collection procedure with the most important settings
of the SEM is depicted in Figure D.4.
SE image
focus at 500 X
Figure D.4
Beam energy 5 keV
Working distance 10 mm
Spotsize 100 nm
magnification 35 X
BSE detector
Slow scan
Imsize 2k x 2k
1024 integrations
per pixel
Collect and store
4 images per sample
TIF Format
Collection of BSE images with important SEM settings
The image processing and image feature extraction are depicted in the flow chart in Figure D.5,
where the steps and the intermediate images are shown. For each of the steps indicated in green,
separate macros have been written that were executed in batch processing. The nomenclature of
the different image types were discussed in chapter 7 and 8.
179
Appendix D
BSE
DEF
Sigma
filter
Segmentation
Measure
global
features
Morphological
filtering
Porosity
Permeability
FRF
V-shale
POR
Model
NET
Measure
local pore
features
DEF to POR
image
Measure
network
features
POR to NET
image
Model
Permeability
Model
Permeability
Figure D.5
Sequence for processing and analysis of BSE images showing the
intermediate images and the data required for the various models for prediction of
petrophysical parameters
BSE
DEF
Φeffective
Kozeny-Carman
model
Sv-ratio
Permeability
Kozeny-Carman approach
BSE
DEF
TUB
POR
Pore
spectrum
Poiseuille
model
Permeability
Poiseuille approach
BSE
DEF
TUB
POR
NET
Network
params
Network
model
Permeability
Network approach
Figure D.6
Three methods for permeability estimation form BSE images
The prediction of porosity was obtained from the global measurements. For the permeability three
approaches were available, the Kozeny-Carman model, the Poiseuille model and the Network
models. These different approaches are depicted in figure D.6. The fastest route was the use of
global information and the Kozeny-Carman models. The most accurate was the Network model,
which required more processing and hence more time.
180
Appendix D
D.3
Practical considerations
An mandatory condition for the use of the methods that were developed in this study, is that the
procedures for extracting parameters are robust and fast. A short processing/evaluation time is
considered essential in operations where this technique would be applied. Modern exploration
activities concentrate on reduction of costs and often cutting of core is omitted. The absence of
core makes it hard to obtain accurate calibration points for petrophysical and geological
evaluation. The calibration of the wireline logs for adequate assessment of porosity, permeability
and hydrocarbon saturation is usually cumbersome without core. For geological evaluation, the
determination of the different facies and associated diagenetic history, which controls reservoir
quality may also be difficult without core.
Instead of core, sidewall samples can usually be obtained at low to moderate costs. In an
exploration well, sidewall samples are acquired routinely to detect oil shows and for palynology
(age dating). Contrary to core, which is only taken over the reservoir interval, sidewall samples
are taken in both reservoir sections and in the shales that form the cap-rock or source-rock. The
shale samples are required to determine shale parameters for petrophysical evaluation, to
evaluate cap-rock properties such as sealing potential and to assess source rock properties.
Sidewall samples have the advantage that they are retrieved from accurately determined depths.
A possible disadvantage may be the damage that is caused by the impact of the bullets. Figure
D.7 shows a BSE image from a sidewall sample. The cracks in the grains caused by the bullet's
impact can be seen clearly.
Figure D.7
BSE images taken from a sidewall sample showing severe cracks
caused by the impact of the bullet
Samples may be so shattered that the original pore structure and mineral assemblage cannot be
observed with enough detail to apply the methods that were described in this thesis. The image
analysis software can cope with a certain degree of grain damage; it intrinsically corrects for
limited crack damage with the morphological filters that were applied in the transformation of BSE
images into DEF images. When the reservoir rock is susceptible to severe damage using
conventional sidewall samples, the use of the MSCT (mechanical sidewall coring tool) may be
181
Appendix D
considered. This tool drills a 1” core plug from the borehole wall. An additional advantage of the
MSCT samples is that they may be used in conventional core analysis if the plugs are sufficiently
intact. Note however that MSCT samples are substantially more expensive than the sidewall
bullet samples. Furthermore, the risk of tool sticking is larger because of the longer resident time
of the tool in the borehole compared to the conventional bullet sidewall tool. The MSCT is not
widely used to date because of these two reasons.
Cuttings come almost free of charge as they are routinely collected by the mud logger. Similar to
sidewall samples, geochemical and palynological information is normally extracted from cuttings
in mud logging. Qualitative information about mineralogy, grain density, grain-size and pore
morphology can be obtained on request, however this is not common practice. The methods that
were developed in this thesis are in essence a quantitative extension of this process. In contrast
to the advantage of low cost however, stands the disadvantage of inaccurate depth determination.
One has to rely on the rather crude calculations of the mud engineer to arrive at depths for the
collected cutting samples. When a sufficient number of samples is available for analysis and
depth trends can be recognised, it is possible to arrive at a depth matching which may be
acceptable. Also, a combination of cutting analysis and sidewall samples may facilitate accurate
depth matching with wireline logs. The presence of observable geological markers that can be
tied in with wireline logs is essential for depth matching, irrespective whether cuttings or sidewall
samples are used as data source.
A final problem is mud contamination. This is valid for both sidewall samples and for cuttings. It is
therefore important that detailed information about the applied mud system is available prior to
SEM/BSE analysis in order to determine the optimal cleaning procedure, which minimises
damage to delicate clay structures.
D.4
Implementation in operations
As stated, the equipment that was used to develop the methods, was considered state-of-the-art
in research and developments at the time this research was conducted. However, given the low
magnification of the images, there is no need to have such a sophisticated and therefore
expensive, Scanning Electron Microscope for day-to-day use in operations. Similarly, the
computer controlled polishing machine we used for the sample preparation is also not mandatory
in routine operations. Robust and simple PC controlled grinding and polishing equipment is readily
available at affordable prices. Table-top SEM equipment, specifically designed for use in an
operational environment is also commercially available at a fraction of the price of the system we
have used in this study. These table-top SEMs can be equipped with very robust semi-conductor
BSE detectors. Finally, the developments in the field of PCs have dramatically accelerated the
processing power over the past 10 years. Therefore, the return times for image processing and
analysis are negligible and no matter of concern, contrary to the moment when this study was
taken up by the author. Therefore, we conclude that it may be very well possible to implement the
methods developed in this study in day-to-day operations. For instance, the sample preparation
equipment, the table-top SEM and the PC may be installed in a container at the well site. In this
way, geo-scientists can have crucial information about the reservoir available in a very early stage
of a field development.
182
Conclusions and recommendations
Conclusions & Recommendations
Conclusions
•
Petrophysical parameters can be predicted from small rock samples, such as drill cuttings
and side wall samples, using image analysis techniques, provided that samples are
homogenous and isotropic, at the scale dictated by the magnification. An indication of the
homogeneity and isotropy can be obtained from the standard deviation of the image
features that are used to predict the petrophysical parameters.
•
Porosity can be estimated from thin section images with limited accuracy. The method
developed in this study can replace manual point counting, because it is more objective
and does not require extensive expertise. Moreover, the image analysis method is far less
labour intensive. Estimation of permeability from thin section images should only be
applied when it is impossible to measure the permeability with conventional methods. This
is the case when undamaged core plugs or appropriate cutting samples are not available.
If proper plugs are available conventional methods are preferred.
•
Predicting petrophysical parameters from images obtained with electron microscopy in
backscattered mode is substantially more accurate than predicting the parameters from
thin section images obtained with optical microscopy. The reason for this higher accuracy
is the better definition of the pore network in backscattered electron images, compared to
thin section images. The lateral resolution limit of both imaging techniques is approx. 0.5
micron, but the axial resolution in backscattered images is far less influenced by the
Holmes effect and is therefore a factor 60 better than that of thin section images.
•
Multiple linear regression techniques that relate image features to porosity and
permeability give more accurate results than theoretical models based on first principles.
A very wide range of petrophysical parameters was used to derive the statistical models
This will allow general use of these models and was confirmed by the validation tests on
two independent sample sets. The porosity model can predict with an accuracy of 1.5
porosity units over a range of 5 to 35 porosity units. The Network model predicts the
within a factor 3 over a permeability range of 4 orders of magnitude: from 0.1 mD to 3000
mD.
•
The 2D-Network approach, whereby only topographic information (e.g pore surface, and
pore perimeters) was used for the prediction of permeability, gave more accurate
predictions than the traditional Kozeny-Carman and Poiseuille approaches.
•
The systematic under-estimation of the porosity by 4 porosity units from Back-Scatter
Electron (BSE) images remains unexplained. No cause could be found in the detailed
analysis of the processing steps that transform a BSE image into a Display of Elementary
Fractions (DEF) image. A possible explanation is that the sub-micron heterogeneity
and/or anisotropy are not captured in the images at a magnification of 35 X.
•
The analysis of Display of Elementary Fractions (DEF) images method developed in this
study provides a lithological evaluation conventionally carried out on thin sections. DEF’s
can also be used for clay typing which is difficult on thin sections due to the limited
resolution, and the Holmes effect.
183
Conclusions and recommendations
•
It is remarkable that our methods work so well given the very limited rock area of 16 mm2
that is used from each sample. The core measurements were carried out on plugs 25 mm
in diameter and a length of 30 mm. The good agreement for porosity and permeability
between the image analysis and core plug measurements confirms the robustness of the
models developed in this study.
•
The minimum size of sidewall and cutting samples for the new analysis is 3 x 3 x 3
millimetres. For sidewall samples the grain damage caused by the impact of the sidewall
bullet should be minimal, and for cuttings a sufficient number of samples should be
analysed in order to have enough measuring points to construct an artificial log.
Recommendations
•
The Network model, developed in this study to predict permeability, contains four
parameters of which only one is loosely related to porosity. Our empirical observations
point toward a more fundamental principle that is underlying this network approach. More
work in the field of quantitative stereology should be carried out to provide a more rigid
theoretical fundament of the Network approach. In addition, the excellent results that we
obtained with the statistical network approach warrant efforts for further investigation of
the Kirchhoff network approach, in which the electric analogy may be used to predict
permeability.
•
Sample homogeneity should be carefully assessed prior to the application of the methods
developed in this study. The price that is paid for omitting this screening is low accuracy
and limited reliability.
•
The experiments to derive capillary pressure curves from images show potential. Further
work will be required to arrive at a robust method that can be implemented in an
operational environment. In addition, work should be carried out to obtain a good
theoretical foundation for this approach.
•
Once the capillary behaviour can be properly modelled , the way is open for resistivity
index curve prediction from images. For each pressure step in the capillary pressure
curve, a resistance can be assigned to the pore that has been filled with a non-wetting,
hence a non-conductive fluid. After each step, the total conductivity can be calculated.
This will result in a relation between non-wetting saturation and resistivity, from which a
resistivity index curve may be calculated.
•
The methods derived in this study should be tested on non-clastic samples to investigate
whether they are valid for more complex lithologies such as carbonates and silicilytes.
•
To investigate the added value and possible cost savings, the new methods should be
further tested on exploration wells. In addition, these new methods be integrated with
borehole image data and conventional wireline data.
•
The use of advanced cutting collection procedures should be further pursued because it
can lead to substantial cost savings when used in combination with the new methods.
184
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Montpellier France, September 1996
Fens, 1998
Fens, T.W., Kraaijveld, M.A., Riepe, L. and Visser, R., Archie’s dream: petrophysics from
sidewall samples and cuttings, new ways to extract information from sidewall samples and
cuttings, Published at SCA conference The Hague, September 1998
189
Samenvatting
Summary
The productivity of wells in a hydrocarbon bearing (oil/gas) reservoir depends on petrophysical
properties. Hydrocarbon bearing reservoir rock consists of two components: the rock matrix and
an interconnected pore network. The pores can have dimensions varying from sub-microns for
tight sandstones to cm’s for vuggy carbonate rock.
The main petrophysical properties are porosity, permeability, saturation and capillarity. Porosity
determines the storage capacity for hydrocarbons and permeability determines the fluid flow
capacity of the rock. Saturation is the fraction of the porosity that is occupied by hydrocarbons or
by water. Finally, capillarity determines how much of the available hydrocarbons can be produced.
Accurate determination of these petrophysical properties is essential to assess the economic
viability of the development of reservoirs.
Petrophysical properties can be obtained from logging instruments that are lowered in the wells
and by core analysis on reservoir rock material that is obtained from the well with a hollow drill bit.
However, the acquisition of reservoir rock material is expensive, therefore methods have been
sought to carry out core analysis on small rock samples such as cuttings and sidewall samples,
which are much cheaper to obtain. Core measurements are used to calibrate the log readings.
Small samples and pores with sizes in the micron range imply the use of microscopic techniques.
The application of microscopy to reservoir rock samples to determine petrophysical parameters
from images is the subject of this study.
Microscopic optical mages from rock samples have been routinely evaluated to assess reservoir
quality and mineralogy in a qualitative sense. However, the link from microscopic images to
petrophysical properties requires a quantitative evaluation. The advancements in image analysis
techniques in the mid-1980s enabled this quantitative evaluation of microscopic images.
Quantitative microscopy combined with statistical analysis form a powerful combination to
determine pore features from images. The pore features are subsequently related to petrophysical
properties.
In the past, a theoretical framework was set up to understand and explain hydrocarbon transport
properties on a pore level, and to develop interpretation models. These models were developed
from idealised porous media because real reservoir rock is geometrically complex. The models
have seldom been tested against real rock samples. In this study, we carried out a rigorous test
on a large amount of samples that represent many different sandstone reservoir rock types. The
objective was to investigate the general applicability of the models and to devise a method that
can be implemented in an operational environment.
In this study images were provided by optical and electron microscopy. Pore structure features
were extracted from the images, and used as input for the models available from literature to
predict petrophysical properties. The models were amended and extended to accommodate
additional parameters such the micro-porosity residing in the clays that can be extracted from the
images. In addition, we developed new models for prediction of permeability based on networktheories and network representations of the pore system. The predicted petrophysical properties
were compared with petrophysical properties that have been measured from core plugs using
conventional core analysis techniques.
190
Samenvatting
The methods that we used were based on 2D images, a representation of what is fundamentally a
3D network. This study demonstrated that the approach we developed can convert the 2D
measurements into 3D reservoir rock properties. Application of the developed techniques on
actual field cases showed that this method has a substantial influence on operations. The final
aim of this study, a well-defined and robust method that can provide petrophysical properties from
small rock samples (3 x 3 x 3 mm) with acceptable accuracy, has been achieved. Porosity over a
range between 4 % BV and 35 % BV can be predicted with a standard error of ±1.5 %.
Permeability can be predicted within a factor of 3 over a range of 0.01 mD to 5000 mD. Formation
resistivity factor can be estimated with an accuracy of a factor 1.5. The accuracy of the prediction
for porosity, permeability, formation resistivity factor and clay content are adequate for use in an
operational environment. The results to predict capillarity from images look very promising but
more work is required for further development in order to apply this in operations. Similarly, the
experiments and ideas to extract directional permeability from images require more work to
assess utilisation. In addition, further theoretical work is required in order to provide the methods
developed in this study with a solid theoretical foundation.
191
Samenvatting
Samenvatting
De productiviteit van putten in olie- (of gas-) houdende lagen in reservoirs hangt af van de
petrofysische eigenschappen van het poreuze gesteente. De verzamelnaam voor olie en gas is
koolwaterstoffen, het zijn verbindingen van koolstof en waterstof. Reservoirgesteente dat
koolwaterstoffen kan bevatten bestaat uit twee componenten: het gesteente zelf en het poriesysteem. De poriën hebben afmetingen die kunnen variëren van minder dan een micrometer in
fijne zandstenen tot centimeters in grove kalkstenen. Productie van koolwaterstoffen vanuit deze
poriën vindt plaats doordat deze met elkaar verbonden zijn, op deze manier stromen de
koolwaterstoffen vanuit de poreuze gesteentelagen in de boorput vanwaar de olie vervolgens
geproduceerd wordt.
Het productiviteitsgedrag van een reservoir wordt bepaald door petrofysische eigenschappen die
op zich weer worden bepaald door de geometrie van de poriën en de mineralogie van het
gesteente. De belangrijkste petrofysische eigenschappen zijn: de porositeit, de permeabiliteit, de
saturatie en de capillariteit. De porositeit geeft aan hoeveel ruimte er in het gesteente aanwezig is
om koolwaterstoffen of water op te slaan. De permeabiliteit bepaalt de doorstroombaarheid, hoe
snel de koolwaterstoffen of het water door het gesteente kunnen stromen. De saturatie is de
fractie van de ruimte in het porie-systeem die wordt ingenomen door de koolwaterstoffen. De
capillariteit geeft tenslotte aan hoe de interactie tussen de koolwaterstoffen en de mineralen in het
gesteente verloopt, en bepaalt voor een groot deel hoeveel van de aanwezige koolwaterstoffen
geproduceerd kan worden. Om de economische levensvatbaarheid van een reservoir vast te
kunnen stellen is het noodzakelijk om deze petrofysische eigenschappen nauwkeurig beschikbaar
te hebben.
Petrofysische eigenschappen worden normaliter verkregen uit boorgatmetingen die gemaakt
worden gemaakt met meetsondes die in een boorgat worden neergelaten en een aantal fysische
eigenschappen van het gesteente meten. Om deze gemeten fysische eigenschappen te vertalen
naar petrofysische eigenschappen is een conversie nodig: bij elke meetsonde hoort een
interpretatie model wat de metingen vertaalt naar petrofysische eigenschappen. Om deze
conversie op juistheid te kunnen controleren en toe te kunnen passen, is calibratie vereist die
plaatsvindt door de resultaten te vergelijken met laboratorium-metingen die uitgevoerd zijn op
zogenaamde pluggen. Een plug is een cylindervormig stukje steen van 3 cm lang en 2.5 cm in
diameter, wat uit een kern genomen wordt. De kern wordt op zijn beurt uit het boorgat genomen
met een groot model appelboor wat een lange cylinder gesteente oplevert van ongeveer 10 meter
lengte en 20 cm diameter. Het nemen van een kern is erg duur, daarom is er naar methoden
gezocht die dezelfde informatie kunnen verschaffen op basis van kleine monsters die goedkoper
te verkrijgen zijn zoals boorwand-monsters en boorgruis. Kleine monsters en poriën met
afmetingen in het micrometerbereik impliceren het gebruik van microscopische technieken. Het
afleiden van petrofysische eigenschappen uit microscopische beelden van reservoir gesteente is
het onderwerp van deze studie.
Microscopische beelden van gesteentemonsters werden, en worden nog steeds, geanalyseerd
om een kwalitatief idee te krijgen van de mineralogie en de eigenschappen van
reservoirgesteente. Echter, om de verbinding te kunnen maken tusssen beelden en petrofysische
eigenschappen zijn kwantitatieve metingen nodig. De opkomst van computergestuurde
beeldanalyse in het midden van de jaren tachtig maakte deze kwantitatieve metingen mogelijk.
Hierdoor kunnen de geometrische en topografische parameters bepaald worden van het porie-
192
Samenvatting
systeem dat in de microscopische beelden te zien is. Deze parameters worden dan vervolgens
getransformeerd naar petrofysische eigenschappen.
In het verleden is een theorie opgezet om het transport van koolwaterstoffen op porieschaal te
kunnen verklaren en te begrijpen. Het doel van dit werk was een theoretische onderbouwing te
verkrijgen voor de interpretatie-modellen waarmee petrofysische eigenschappen afgeleid kunnen
worden uit microscopische beelden. Deze interpretatie-modellen zijn gestoeld op geïdealiseerde
poreuze media omdat reservoirgesteente een te complexe geometrie heeft om één op één
modellen af te leiden. Incidenteel zijn deze eenvoudige modellen getest op reservoirgesteente.
Een rigoreuze test van deze modellen op een grote variëteit van reservoir zandstenen heeft, voor
zover bekend bij de auteur, echter nooit plaatsgevonden. In de studie die beschreven wordt in dit
proefschrift is zo'n rigoreuze test uitgevoerd op groot aantal gesteentemonsters die een wijd
bereik van petrofysische eigenschappen vertegenwoordigen. De doelstelling hierbij was om de
algemene toepasbaarheid van de bekende modellen te onderzoeken, om uit te vinden of deze
modellen aangepast dienden te worden, en om een methode te ontwerpen die in een
operationele omgeving gebruikt kan worden.
In deze studie zijn van een groot aantal zandsteen-monsters beelden opgenomen. De beelden
zijn afkomstig van zowel optische microscopie als van elektronenmicroscopie. Eigenschappen
van de porie-structuur werden afgeleid vanuit de beelden en werden gebruikt als invoer in de
modellen die bekend waren vanuit de literatuur. Hierbij zijn, waar nodig, de modellen aangepast
aan de soort van invoerparameters die uit beelden gehaald werden. Nieuwe modellen, gebaseerd
op netwerktheoriën en netwerkrepresentanten van porie-systemen, werden ontwikkeld om de
permeabiliteit te kunnen voorspellen. De voorspelde petrofysische eigenschappen werden
vervolgens vergeleken met de petrofysische eigenschappen die fysiek aan pluggen gemeten zijn
met conventionele technieken.
De modellen die wij gebruikt hebben zijn gebaseerd op beelden die slechts een tweedimensionale weergave zijn van het werkelijke drie-dimenionale porie-systeem. Deze studie laat
zien dat het mogelijk is om 2D metingen te gebruiken om 3D eigenschappen te schatten. De
toepassing van de ontwikkelde techniek op zandsteen-monsters in twee praktijkgevallen laat zien
dat deze methode een belangrijke invloed kan hebben op reservoir evaluatie in operaties. Het
uiteindelijke doel van deze studie, een goed gedefinieerde en robuuste methode om petrofysiche
eigenschappen af te leiden vanuit microscopische beelden van zeer kleine stukjes
reservoirgesteente (3 x 3 x 3 mm), is bereikt. Porositeit kan voorspeld worden met een
nauwkeurigheid van ±1.5 %, over een totaal bereik van 4 % tot 35% bulk volume. De
permeabiliteit kan geschat worden binnen een factor 3 over een totaal bereik van 0.01 tot 5000
mD. De Formation resistivity factor kan geschat worden binnen een factor 1.5. De nauwkeurigheid
waarmee de porositeit, permeabiliteit, fomation resistivity factor en de hoeveelheid klei in een
poriesysteem bepaald kan worden vanuit microscopische beelden is voldoende voor operationeel
gebruik. De resultaten om vanuit beelden capillair gedrag te voorspellen zijn bemoedigend maar
er is meer onderzoekswerk nodig om dit naar volwassenheid te dragen. De experimenten en
ideeen die in deze studie aangedragen worden om de permeabiliteit in verschillende richtingen te
schatten verdienen eveneens meer aandacht voor verdere ontwikkeling. Meer fundamenteel werk
zal nodig zijn om een verdere theoretische onderbouwing van de ontwikkelde modellen te
verkrijgen.
193

Petrophysical Properties

Transcription

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