from utexas.edu - The University of Texas at Austin

PETROPHYSICS, VOL. 46, NO. 1 (FEBRUARY 2005); P. 14–32; 30 FIGURES, 3 TABLES
The Influence of Water-Base Mud Properties and Petrophysical
Parameters on Mudcake Growth, Filtrate
Invasion, and Formation Pressure
Jianghui Wu1, Carlos Torres-Verdín2, Kamy Sepehrnoori2, and Mark A. Proett3
ABSTRACT
The work described in this paper models the complete
invasion process quantitatively with a finite-difference
invasion simulator that includes the dynamically coupled
effects of mudcake growth and multiphase, immiscible
filtrate invasion. A fully coupled mudcake growth model
is assumed and the flow rate of filtrate invasion is determined from both mud parameters and rock formation
properties.
Specific parametric representations of the assumed
invasion model are based on previously published laboratory experiments on mudcake buildup. As part of the
numerical validation of the simulator, we reproduced
available experimental data and obtained very good
agreements.
INTRODUCTION
Mud filtrate invasion takes place in permeable rock formations penetrated by a well that is hydraulically overbalanced by mud circulation. The invasion of mud filtrate into
permeable rock formations is responsible for the development of a mudcake on the borehole wall (solids deposition),
as well as for the lateral displacement of existing in-situ fluids from the borehole. Drilling variables such as mud density and chemistry, mud circulation pressure, and time of filtration may significantly affect the spatial extent of mud-fil-
The influence of several mud and petrophysical parameters on both mudcake growth and filtrate invasion is
quantified with a sensitivity analysis. These parameters
include mudcake permeability, mudcake porosity, mud
solid content, relative permeability, capillary pressure,
formation permeability, cross flow between adjacent layers, and gravity segregation. Our simulations reveal the
physical character of invasion profiles taking place under
realistic petrophysical conditions. Results also characterize formation pressure changes and pressure supercharging observed during wireline formation testing.
Keywords: mud filtrate, mudcake, invasion, supercharge
trate invasion. In-situ rock formation properties such as
porosity, absolute permeability, relative permeability, pore
pressure, shale chemistry, capillary pressure, and residual
fluid saturations, also play important roles in controlling
both the dynamic formation of mudcake and the time
evolution of the invasion process.
One of the technical problems often considered in
mud-filtrate invasion studies is the description of mudcake
buildup and invasion rates. Over the years, many laboratory
investigations have undertaken the phenomenological
description and quantification of this problem (e.g., Fergu-
Manuscript received by the Editor July 30, 2004; revised manuscript received December 17, 2004.
Baker Atlas, 2001 Rankin Road, Houston, Texas 77073; e-mail: [email protected]
2
Department of Petroleum and Geosystems Engineering, 1 University Station C0300, The University of Texas at Austin, Austin, Texas
78712; e-mail: [email protected], [email protected]
3
Halliburton Energy Services, 3000 N. Sam Houston Parkway E., P. O. Box 60070 (77204), Houston, Texas 77032; e-mail:
[email protected]
©2005 Society of Petrophysicists and Well Log Analysts. All rights reserved.
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PETROPHYSICS
February 2005
The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
son and Klotz, 1954; Bezemer and Havenaar, 1966;
Fordham et al., 1988 and 1991; Dewan and Chenevert,
2001). Based on laboratory experiments of mud circulation,
Dewan and Chenevert (2001) reported a methodology to
predict the time evolution of mudcake buildup, as well as
the effective petrophysical properties of mudcake. Dewan
et al.’s description is entirely based on six mud filtrate
parameters, all of which can be determined from a standard
on-site mud filtrate test.
A theoretical basis for laboratory and field observations
was first presented by Outmans (1963). He described a
method applicable to high permeability formations assuming that, even at the onset of the process of mud-filtrate
invasion, the full overbalance pressure was absorbed across
the mudcake. In his single-phase fluid flow study, Outmans
derived the well-known t law. However, the latter description is not accurate when the net flow resistance offered by
the formation is comparable to that of the mudcake.
Semmelbeck et al. (1995) introduced a two-phase fluid
flow simulator that assumed several of the mud properties
described in Dewan and Chenevert (2001). Gravity force
was ignored in Semmelbeck et al.’s work because only low
vertical permeability cases were considered in the analysis.
Dewan and Chenevert (2001) presented a single-phase flow
mathematical model to reproduce laboratory measurements. The simulation results matched laboratory measurements using a 0.25-inch core to represent the rock formation. However, this model cannot be applied to low permeability formations, where the net flow resistance offered by
the formation is comparable to that of the mudcake. Proett
et al. (2001) developed an immiscible invasion simulator
using a fully coupled mudcake growth model. Wu et al.
(2004) proposed a methodology to simulate mud-filtrate
invasion in deviated and horizontal wells.
In this paper, a new invasion simulator, termed INVADE,
is developed that accurately reproduces the process of
mud-filtrate invasion in multi-layer formations, including
the effect of salt mixing.
conditions. Because of these features, INVADE can be used
to numerically simulate the transport of salt due to invasion
of mud filtrate when there is a difference in salinity between
mud filtrate and connate water.
It is generally acknowledged that static filtration governs
the initial growth of mudcake and that the fundamental role
of dynamic filtration is to limit this growth (e.g., Chin,
1995; Dewan and Chenevert, 2001). In this paper, we
assume that mudcake stops growing after reaching its limiting thickness. By using a static filtration model other than
the dynamic filtration model, mudcake growth is accelerated because the dynamic filtration tends to decrease the
rate of mudcake growth. Well conditioning can also affect
the mudcake and influence the process of invasion. Frequently, the well is reconditioned prior to a logging run or in
the process of drilling. To account for reconditioning, we
can assume that the mudcake is either partially or completely removed, and that it is henceforth allowed to reform.
In the worst-case scenario, the mudcake is completely
removed from the borehole wall. Therefore, the limiting
mudcake thickness is a parameter input to INVADE, and so
is the option to remove the mudcake in order to simulate
reconditioning of the well.
The INVADE software has one extra input file in addition to the input files used by UTCHEM. This file, named
MUD contains parameters such as mud properties, mud
pressure, and “rub-off ” time. The flow rate of mud filtrate
calculated with equation (A.29) will be treated as a standard
rate of injection for the well.
In the Appendix, the mudcake growth model is coupled
with the two-phase immiscible Darcy flow boundary value
problem developed for radial invasion. It is noted that equation (A.29) used for the calculation of filtrate flow rate is
derived from a one-dimensional model. For multi-layer
cases, cross-flow between layers is simulated with
UTCHEM, while equation (A.29) is used for the calculation
of the rate of flow of mud filtrate.
Relative permeability and capillary pressure curves
FINITE-DIFFERENCE SIMULATOR-“INVADE”
The finite difference simulator INVADE was developed
based on the solution of fluid-flow differential equations
and boundary conditions for immiscible radial flow and
coupled mudcake growth. INVADE was built upon the
existing multi-phase, multi-component, and multi-chemical
species fluid-flow simulator UTCHEM, developed by The
University of Texas at Austin (Center for Petroleum and
Geosystems Engineering, 2000). UTCHEM can simulate
the advection, dispersion, diffusion, and transformation of
different species (oil, surfactant, water, salt, polymer, etc.)
in porous media under various production and injection
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In the development of the immiscible flow model, no
assumptions were made concerning the behavior of relative
permeability and capillary pressure curves. Therefore,
these curves are completely arbitrary and can be a power
function adjusted to match core measurements. The curves
shown in Figure 1 correspond to a typical behavior of relative permeability for a water-wet sandstone (f = 0.25, K =
300 md). In this case, the relative permeability curves can
be characterized by the saturation-dependent BrooksCorey-type equations (Brooks and Corey, 1966), given by
PETROPHYSICS
k rw = k rw0 × ( S wt* ) ew ,
(1)
15
Wu et al.
k ro = k ro0 × [1- ( S wt* )]eo ,
(2)
and
S wt* =
S w - S wi
,
1- S wi - S or
(3)
where S wt* is normalized water saturation, Sw is water saturation, Swi is irreducible water saturation, Sor is residual oil saturation, krw is water relative permeability, kro is oil relative
permeability, k rw0 is water relative permeability endpoint,
k ro0 is oil relative permeability endpoint, ew is the exponent
for krw, and eo is the exponent for kro.
The capillary pressure curve shown in Figure 2 is characterized by a relationship of the form
Pc = Pc0
f
× [1- ( S wt* )]e p ,
K
(4)
where Pc is capillary pressure between oil and water, f is
formation porosity, K is formation permeability, Pc0 is the
coefficient for capillary pressure, and ep is the exponent for
capillary pressure.
The exponents in equations (1) through (4) (i.e., ew, eo,
ep) control the shape of the curves, whereas the coefficients
in the same equations (i.e., k rw0 , k ro0 , Pc0 ) control the location
of the endpoints. The coefficient for capillary pressure, Pc0 ,
is set to 2 for the case of a 300 md formation, while for the
case of low-permeability formations, Pc0 is set to 0.2. These
FIG. 1 Water-oil relative permeability curves assumed in the
numerical simulations described in this paper. The solid and
dashed curves describe relative permeabilities as a function of
water saturation for water and oil fractions, respectively.
16
saturation-dependent functions can approximate most practical cases of relative permeability and capillary pressure
curves and are used for the INVADE examples described in
this paper.
COMPARISON OF INVADE PREDICTIONS WITH
LABORATORY MEASUREMENTS
To validate the numerical simulator, we attempted to
reproduce the experimental data acquired during a static filtration test performed with field Mud 97074 (Dewan and
Chenevert, 2001). In this particular case, mud properties are
as follows: mudcake reference permeability K mc0 = 0.003
md, mudcake reference porosity fmc0 = 0.59, solid fraction
fs = 0.231, mudcake thickness = 0.25 cm, compressibility
exponent for mudcake permeability v = 0.63, and exponent
multiplier for mudcake porosity d = 0.1. In step 1, a pressure
of 300 psi is applied during 30 minutes and the filtrate volume and slowness (inverse of flow rate) are recorded after
the onset of invasion. In step 2, the pressure is raised to 1000
psi and the recording continues for another 30 minutes.
A radial invasion model is constructed such that the filtration area is the same as that of filter paper (45.8 cm2). We
assume a wellbore radius equal to 10 cm. The distance
between the wellbore and the formation’s outer-boundary is
0.635 cm (0.25 inch) to match the thickness of the filtration
medium used in the experiment. INVADE enforces a constant-pressure condition at the outer boundary, whereas fluids can move freely out of the outer boundary. The influ-
FIG. 2 Capillary pressure curve used in the sensitivity analysis
of mud-filtrate invasion. This curve represents two formations: a
300 md formation with Pc0 = 2, and a 3 md formation with Pc0 =
0.2.
PETROPHYSICS
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The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
ence of the radial geometry of the simulation is negligible in
this case. Only single-phase fluid-flow (water) was considered in the experiment, whereupon the initial water saturation for the model was set to 1.0.
In Figure 3, S1 represents the slowness at the end point of
step 1, while S2 represents the slowness at the start point of
step 2. Figure 3 indicates that INVADE results exhibit a
good agreement with slowness measurements of step 1 and
S1, but S2 calculated with INVADE is about 30% higher
than the measured S2. This discrepancy causes a transient
pressure kick during the measurement sequence when pressure increases from 300 psi to 1000 psi.
The compressibility index v is calculated from the measured slowness, S1 to S2, when pressure is increased from P1
to P2 in the two-step recording shown in Figure 3, i.e.,
v =1-
log(S 1 / S 2 )
.
log(P2 / P1 )
(5)
Conversely, S2 could be predicted if other parameters
were known. In the example shown in Figure 3, we find S1 =
37500 sec/cm, v = 0.63, P1 = 300 psi and P2 = 1000 psi,
resulting in S2 = 24000 sec/cm. This value agrees well with
the calculation performed with INVADE.
To assess mudcake growth in low-permeability formations, we make use of the following constants applicable to
UT mud: kmc0 = 0.03 md, fmc0 = 0.8, fs = 0.06, v = 0.9 and d =
0. The wellbore radius of the radial model is 10 cm and the
distance between the wellbore and the formation’s
FIG. 3 Comparison of the time evolution of the measured and
simulated slowness of static filtration for Mud 97074. Slowness
is the inverse of invasion flow rate (the filtration area of filter
paper is 45.8 cm2).
February 2005
outer-boundary is set to 0.635 cm (0.25 in.) in order to
match the thickness of the filtration medium used in the
experiment. Again, only single-phase fluid-flow (water)
was assumed in the experiment, whereupon the initial water
saturation for the model was set to 1.0.
Figure 4 (solid lines) compares the calculated volume of
filtrate as a function of t against the experimental measurements performed at UT Austin. Aside from a 1.2 cc offset at t = 0 for the experimental data, (commonly referred to
as “spurt loss”) the agreement is good. By shifting the time
axis, simulation results will properly match experimental
measurements. To match this experiment in the presence of
“spurt loss,” it is expected to take 4.6 minutes less than the
calculated time (400 minutes) for the mudcake to reach the
thickness of 0.25 cm.
Also shown in Figure 4 (dashed line) is the calculated
mudcake thickness, which grows linearly with t, reaching
a value of 0.25 cm in approximately 6.7 hours. Figure 5
shows the pressure buildup across the mudcake as a function of time after enforcing a 50 psi overbalance pressure.
The 3 md curve in Figure 5 shows that even though it takes
hours to fully build, sufficient mudcake deposits within 2
seconds to absorb 90% of the full overbalance pressure.
FIG. 4 Comparison between measurements and numerical
simulations. Time evolution of volume of filtrate and mudcake
thickness during a static filtration test performed through a 3 md
rock core sample. For convenience, time-dependence is
described with the square root of the actual time of invasion
(adapted from Dewan et al., 1993). The square-dashed and
solid lines describe measured and simulated volumes of filtrate,
respectively, whereas the dashed line describes the simulated
mudcake thickness.
PETROPHYSICS
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Wu et al.
TABLE 1 Summary of mudcake, petrophysical, fluid, and
invasion parameters used in the numerical simulations of
mud-filtrate invasion considered in this paper.
Variable
Units
Mudcake reference permeability
Mudcake reference porosity
Mud solid fraction
Mudcake maximum thickness
Mudcake compressibility exponent v
Mudcake exponent multiplier d
Water viscosity (filtrate)
Oil viscosity
Rock compressibility
Water compressibility
Initial formation pressure
Mud hydrostatic pressure
Formation permeability
Formation porosity
Permeability anisotropy
Coefficient for capillary pressure Pc0
Exponent for capillary pressure ep
Total invasion time
Mudcake rub-off time
Wellbore radius
Formation outer-boundary
Mud filtrate salinity
Formation water salinity
Archie’s tortuosity/cementation factor a
Archie’s cementation exponent m
Archie’s saturation exponent n
Temperature
Value
md
0.03
fraction
0.30
fraction
0.06
cm
1.00
fraction
0.40
fraction
0.10
cp
1.00
cp
3.00
1/psi
0.0E-6
1/psi
0.0E-6
psi
5000.00
psi
5500.00
md
300.00
fraction
0.25
fraction
1.00
psi
2.00
n/a
6.00
hours
48.00
hours
N/A
cm
10.00
cm
610.00
ppm
43,900.00
ppm
102,500.00
n/a
1.00
n/a
2.00
n/a
2.00
°C
24.00
mize the numerical dispersion effects, a third-order spatial
discretization option is selected in the simulation input file
for INVADE.
Radial profiles of formation water salinity are shown in
Figure 7. Salt concentrations are converted into equivalent
values of connate water resistivity, Rw, using the formula
(Dresser Atlas Inc., 1982)
r é
3647.5 ù 82
R w ( r ) =ê 0.0123 +
,
r
ú
C w ( r ) 0.955 û1.8T + 39
ë
(6)
where T is temperature measured in degreesr centigrade, Cw
is salt concentration measured in ppm, and r is the location
of the observation point. In turn, electrical resistivities are
calculated via Archie’s law from the corresponding spatial
distribution of water saturation shown in Figure 6. The calculated radial profiles of electrical resistivity are plotted in
Figure 8.
Mud filtrate invasion flow rate, volume of filtrate,
buildup of pressure across the mudcake, and mudcake
thickness are shown as solid lines (Kmc0 = 0.03 md) in Figures 9, 10, 11 and 12, respectively. The coupled mudcake
model allows the thickness to increase to a maximum of 1
cm, where it is assumed that dynamic filtration limits its
growth. It takes about eight seconds (0.0001 day) before a
noticeable increase of mudcake properties is observed. At
this time, the invasion flow rate is reduced by nearly 95%
from its initial rate. The pressure drop across the mudcake
has also increased to 450 psi of the 500 psi overbalance
INVADE SIMULATION RESULTS
Base Case
A “Base Case” was chosen as a reference example for the
simulation of mud filtrate invasion. Input data for this base
case include the relative permeability and capillary pressure
curves shown in Figures 1 and 2, respectively, with the
remaining variables described in Table 1. Results from the
INVADE simulation are shown in Figure 6 in the form of
profiles of water saturation as a function of radial distance
away from the borehole wall. The first curve shows the
most significant invasion advance, with the front advancing
to 0.2 m within six hours. This advance is due to the fact
that initially there is no mudcake and it takes about 19 hours
for it to reach a thickness of one centimeter (see also Figure
12). The water saturation curves do not exhibit the piston-like behavior that would be expected with immiscible
invasion that does not consider capillary pressure. To mini18
FIG. 5 Pressure at the sandface during static filtration. Time
evolution of pressure at the sandface for six different values of
rock-core permeability. The simulation of static filtration was
performed through a 0.25-in. core sample maintained at a confining pressure of 50 psi.
PETROPHYSICS
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The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
FIG. 6 Time-lapse simulation: radial profiles of water saturation. The curves describe water saturation in the radial direction
from the wellbore and across the center of the permeable layer
at 6-hour increments after the onset of mud-filtrate invasion.
FIG. 7 Time-lapse simulation: radial profiles of salt concentration. The curves describe salt concentration in the radial direction from the wellbore and across the center of the permeable
layer at 6-hour increments after the onset of mud-filtrate invasion. The corresponding radial distributions of water saturation
are shown in Figure 6.
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FIG. 8 Time-lapse simulation: radial profiles of electrical resistivity. The curves describe electrical resistivity in the radial direction from the wellbore and across the center of the permeable
layer at 6-hour increments after the onset of mud-filtrate invasion. The corresponding radial distributions of water saturation
and salt concentration are shown in Figures 6 and 7, respectively. Electrical resistivity was calculated assuming Archie’s
equations with tortuosity/cementation factor a = 1, cementation
exponent m = 2, and saturation exponent n = 2.
FIG. 9 Sensitivity analysis of mudcake reference permeability
and flow rate across the mudcake. Time evolution of the flow
rate of mud filtrate for three different values of mudcake reference permeability.
PETROPHYSICS
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Wu et al.
pressure. The mudcake continues to thicken until a 1 cm
thickness is reached at about 19 hours after the onset of
invasion. At this point, the invasion flow rate has been
reduced to a small fraction of its initial value and the
sandface pressure has reached its steady state.
Mudcake permeability decreases with increasing pres-
sure across the mudcake. The solid line (Kmc0 = 0.03 md) in
Figure 13 shows that mudcake permeability is stabilized at
2.5´10–3 md after eight seconds of invasion.
Case of mudcake removal
In this case, mudcake is removed after one day of inva-
FIG. 10 Sensitivity analysis of mudcake reference permeability
and accumulated volume of mud filtrate. Time evolution of the
accumulated volume of mud-filtrate for three different values of
mudcake reference permeability.
FIG. 12 Sensitivity analysis of mudcake reference permeability
and mudcake thickness. Time evolution of mudcake thickness
for three different values of mudcake reference permeability.
FIG. 11 Sensitivity analysis of mudcake reference permeability
and pressure across the mudcake. Time evolution of the pressure across the mudcake for three different values of mudcake
reference permeability.
FIG. 13 Sensitivity analysis of mudcake reference permeability
and mudcake permeability. Time evolution of mudcake permeability for three different values of mudcake reference permeability.
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The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
sion. As shown in Figure 14(a), during the next day of invasion the thickness increases. Mudcake thickness increases
at nearly the same rate as before the removal of mudcake,
taking about 19 hours to reach the 1 cm maximum thickness. Figure 14(b) shows that the invasion flow rate
increases nearly instantaneously to about 30% of the initial
rate and then is slowed down to a fraction of its original rate as
the mudcake reforms. Comparison of the water saturation profiles described in Figure 15 against those shown in Figure 6
indicates that there is a sudden increase of invasion length
after mudcake is removed at the end of one day, and the final
invasion front is 8 cm deeper than in the base case (46 cm).
SENSITIVITY ANALYSIS
A sensitivity analysis was performed using the base case
example as reference by varying the selected parameters
shown in Table 2. In the base case, mudcake reference permeability (0.03 md) is very small compared to that of the
formation permeability (300 md). Therefore, the mudcake
will control the flow rate of filtrate invasion. Three parameters related to mud properties are selected for the sensitivity
analysis, i.e., mudcake reference permeability, porosity,
and filtrate solid fraction.
Mudcake Permeability
TABLE 2 Summary of the test cases considered in the sensitivity analysis of mud-filtrate invasion reported in this
paper. The table shows input variables and results obtained
from the simulations.
Mudcake properties
Case
no.
Kmc0
(md)
fmc0
Base
1
2
3
4
5
6
0.030
0.010
0.003
0.030
0.030
0.030
0.030
0.3
0.3
0.3
0.5
0.8
0.3
0.3
fs
0.06
0.06
0.06
0.06
0.06
0.20
0.40
Invasion Results
Filtrate
Mudcake
volume
buildup
after 2 days
time
(m3/m)
(hours)
0.124
0.066
0.037
0.117
0.106
0.097
0.091
19.0
56.9
189.5
15.1
9.3
4.8
1.8
high permeability zones since the mudcake is the primary
regulator of the flow rate of filtrate invasion.
Figure 10 shows that the total filtrate volume after two
days of invasion increases with increasing mudcake permeability. As expected, pressure across the mudcake decreases
with increasing values of mudcake permeability. The latter
Mudcake properties, particularly permeability, have a
significant influence on the invasion process. The assertion
that mudcake permeability is significant is clearly true for
FIG. 14 Time evolution of (a) mudcake thickness, and (b) flow
rate of mud filtrate during the process of invasion. Mudcake is
removed after one day of invasion and is allowed to re-grow during the subsequent one-day interval.
February 2005
FIG. 15 Time-lapse simulation: radial profiles of water saturation. The curves describe water saturation in the radial direction
from the wellbore and across the center of the permeable layer
at 6-hour increments after the onset of mud-filtrate invasion.
Mudcake is removed after one day of invasion and is allowed to
re-grow during the subsequent one-day interval.
PETROPHYSICS
21
Wu et al.
observation is confirmed by the family of curves shown in
Figure 11.
The mudcake buildup time is reduced with increasing
mudcake permeability because the deposition of solids
depends on the flow rate of filtrate, which is higher for
high-permeability mudcake. It takes 190 hours for a
0.003-md mudcake to thicken to 1 cm, while it takes 19
hours for a 0.03 md mudcake to reach the same thickness.
Mudcake porosity
Mudcake porosity has an influence on mudcake growth.
The mudcake buildup time decreases with increasing
mudcake porosity because the total amount of solid deposition decreases with an increase of porosity.
It takes 9.3 hours for a 0.8 porosity mudcake to build
while it takes 19 hours for a 0.3 porosity mudcake to reach
the same thickness (Table 2).
In Figure 16, the flow rate for a high-porosity mudcake
decreases because of relatively thicker mudcake. The total
filtrate volume decreases with increasing mudcake porosity
(Table 2). In Figure 17, the pressure across mudcake
increases with increasing mudcake porosity because of a
thicker mudcake.
Solid fraction
FIG. 16 Sensitivity analysis of mudcake porosity and flow rate
across mudcake. Time evolution of the flow rate of mud-filtrate
for three different values of mudcake reference porosity.
FIG. 17 Sensitivity analysis of mudcake porosity and pressure
across mudcake. Time evolution of the pressure across
mudcake for three different values of mudcake reference porosity.
22
Filtrate solid fraction also has a considerable influence
on mudcake growth. The time of mudcake buildup
decreases with increasing solid fraction because the speed
of solid deposition increases.
Figure 18 shows that the flow rate for a high solid-content mud decreases because of relatively thicker mudcake.
The total filtrate volume decreases with an increase of mud
solid fraction (Table 2). In addition, Figure 19 shows that
the pressure across the mudcake increases with increasing
mud solid fraction because of thicker mudcake.
FIG. 18 Sensitivity analysis of filtrate solid fraction and flow
rate across mudcake. Time evolution of the flow rate across
mudcake for three different values of filtrate solid fraction.
PETROPHYSICS
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The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
As shown in Table 2, it takes 1.8 hours for a mud with a
solid fraction of 0.4 to build while it takes 19 hours for a
0.06 solid fraction mud to reach the same thickness.
Invasion in low-permeability formations
When formation permeability exceeds a few millidarcies, sufficient mudcake forms in a matter of seconds,
and virtually the entire overbalance pressure driving the
invasion is absorbed across the mudcake. Therefore, the
rate of invasion is entirely controlled by mudcake properties rather than by formation properties. However, Dewan
and Chenevert (1993) showed that conditions are different
when permeability is lower than a few millidarcies. The initial invasion rate, limited by formation permeability, is sufficiently low so that the pressure drop across the mudcake
increases very slowly.
To study how formation properties will affect invasion,
we make use of the set of formation and mudcake variables
given in Table 3. The mud properties are the same as those
of UT Austin Mud and the distance between the wellbore
and the formation’s outer-boundary is set to 610 cm (20 ft).
Two-phase immiscible flow is simulated with the
water-oil relative permeability curves shown in Figure 1.
The coefficient for the capillary pressure equation, Pc0 , is
set to 0.2 and the exponent for the capillary pressure equation, ep, is set to 6.
Two formation properties, i.e., formation permeability
and oil relative permeability endpoint, are selected for this
sensitivity analysis. Capillary pressure influences the rate
of change of water saturation in the invaded zone but the
changes are small. Simulation tests indicate that other formation properties will have only a slight influence on the
process of mudcake buildup.
Figure 20 shows the pressure across the mudcake as a
function of time after the enforcement of a 50 psi overbalance pressure. It takes the 3 md formation 6.7 hours to build
a 45 psi pressure across the mudcake rather than the several
seconds indicated in Figure 5. The maximum pressure
across the mudcake takes place when the mudcake reaches
its maximum thickness. When the mudcake stops growing,
mud filtrate continues to invade into even deeper regions in
the formation, and therefore the resistance resulting from
the formation will increase. As a result, the pressure across
the mudcake will decrease and the pressure at the sandface
will increase. Figure 21(a) shows that the flow rate is higher
for a high permeability formation and will converge after
3.8 hours of invasion. After that, the flow resistance from
the 0.3 md formation continues to increase, resulting in a
gradually decreasing flow rate. As expected, Figure 21(b)
shows that the volume of filtrate is higher for a high permeability formation.
February 2005
TABLE 3 Case of a low-permeability formation. Summary
of mudcake, petrophysical, fluid, and invasion parameters
used in the numerical simulations of mud-filtrate invasion
considered in this paper.
Variable
Mudcake reference permeability
Mudcake reference porosity
Mud solid fraction
Mudcake maximum thickness
Mudcake compressibility exponent v
Mudcake exponent multiplier d
Water viscosity (filtrate)
Oil viscosity
Rock compressibility
Water compressibility
Initial formation pressure
Mud hydrostatic pressure
Formation permeability
Formation porosity
Permeability anisotropy
Coefficient for capillary pressure Pc0
Exponent for capillary pressure ep
Total invasion time
Mudcake rub-off time
Wellbore radius
Formation outer-boundary
Units
Value
md
fraction
fraction
cm
fraction
fraction
cp
cp
1/psi
1/psi
psi
psi
md
fraction
fraction
psi
n/a
hours
hours
cm
cm
0.03
0.80
0.06
0.25
0.90
0.00
1.00
1.00
0.0E-6
0.0E-6
5000.00
5050.00
3.0/1.0/0.3
0.25
1.00
0.20
6.00
48.00
N/A
10.00
610.00
FIG. 19 Sensitivity analysis of filtrate solid fraction and pressure across mudcake. Time evolution of the pressure across
mudcake for three different values of filtrate solid fraction.
PETROPHYSICS
23
Wu et al.
The oil relative permeability endpoint affects invasion
through the resistance offered by the formation to the flow
of filtrate. As shown in Figure 22, increasing the value of
the endpoint translates to a decreasing resistance and, consequently, the overbalance pressure lost in the formation
decreases, and pressure across the mudcake increases.
Figure 23(a) shows that higher values of k ro0 cause higher
flow rates of invasion. Therefore, as shown in Figure 23(b),
mudcake buildup is faster and, consequently, the volume of
filtrate is slightly higher.
FIG. 20 Time evolution of (a) pressure across mudcake, and
(b) pressure at the sandface during a static filtration test for six
different values of formation permeability. The static filtration
test was performed assuming a 20-ft formation maintained at a
confining pressure of 50 psi.
FIG. 22 Time evolution of pressure at the sandface during a
static filtration test performed assuming a 20-ft formation at a
confining pressure of 50 psi for three different values of formation k ro0 .
FIG. 21 (a) Time evolution of filtrate flow rate for three different
values of formation permeability; (b) Time evolution of total filtrate volume per formation thickness for three different values of
formation permeability.
FIG. 23 (a) Time evolution of filtrate flow rate for three different
values of formation k ro0 ; (b) Time evolution of the total filtrate volume per formation thickness for three different values of formation k ro0 .
24
PETROPHYSICS
February 2005
The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
FLOW RESISTANCE RATIO OF MUDCAKE
Formation properties will affect invasion only when the
formation flow resistance is comparable to that of the
mudcake. We denote the ratio of mudcake flow resistance
over total flow resistance by the variable q. The relationship between q and the remaining parameters is summarized by the equation
q=
1
,
ln(R f / R well )
ln(R out / R f )
K mc0
mo × K mc0
1+
+
K × krw (q × DP ) v ln(R well / R mc ) mw × K × kro × (q × DP ) v ln(R well / R mc )
(7)
This last expression is derived from Darcy’s law by making
four assumptions: (1) there is no capillary pressure, (2) the
invasion profile is piston-like, (3) water is the only flowing
fluid in the invaded zone, and (4) oil is the only flowing
fluid in the virgin zone.
Supercharging
The flow resistance parameter q can also be used to
assess supercharging effects. Supercharging is defined as
the increased pressure observed at the wellbore sandface
caused by invasion. This is an important factor in wireline
formation testing since the pressures recorded are influenced by supercharging. The supercharge pressure can now
be estimated by subtracting the pressure across the
FIG. 24 Graphical description of the relationship between the
supercharge index (1 – q) and K/Kmc0 and mo /mw. The plot was
constructed assuming Rw = 10 cm, Rf = 18.3 cm, Rout = 610 cm,
Rmc = 9.75 cm, overbalance pressure = 50 psi, kro = 1.0, krw =
0.2, and compressibility exponent for mudcake permeability v =
0.4 to 0.9.
February 2005
mudcake from the overbalance pressure. Consequently, the
degree of supercharging is proportional to 1 – q.
Figures 24 and 25 show the relationship borne by the
supercharging index (1 – q) with K/Kmc0 and uo /uw. The
value of (1 – q) will decrease with increasing values of
K/Kmc0 and the value of (1 – q) will increase with increasing
values of uo /uw.
Now consider the time when the mudcake reaches its
maximum thickness. For the three cases (K = 3, 1, and 0.3
md) shown in Figure 20, the corresponding flow resistance
ratios are 0.92, 0.77, and 0.32, respectively. This ratio multiplied by the overbalance pressure will yield the pressure
across the mudcake when the mudcake reaches its maximum thickness. Pressures across the mudcake for these
cases are 46.0, 38.5, and 16.0 psi, respectively. The calculated pressures are in good agreement with the values
shown in Figure 20. As a rule of thumb, when q is smaller
than 0.9, the flow resistance offered by the formation
becomes comparable to that of the mudcake.
Figure 26 shows that (1 – q) will increase with increasing invasion front radii. This agrees well with the fact
observed in Figure 20 that after the mudcake reaches its
maximum thickness, the pressure at the sandface increases
while the radius of the invasion front increases.
Figure 27 shows that (1 – q) will decrease with increasing values of oil relative permeability. This agrees well with
the results shown in Figure 22.
In summary, the value of q calculated from equation (7)
FIG. 25 Family of curves that graphically describe the relationship between the supercharge index (1 – q) and K/Kmc0 and
mo /mw. The curves were constructed assuming Rw = 10 cm, Rf =
18.3 cm, Rout = 610 cm, Rmc = 9.75 cm, overbalance pressure =
50 psi, kro = 1.0, krw = 0.2, and compressibility exponent for
mudcake permeability v = 0.4 to 0.9.
PETROPHYSICS
25
Wu et al.
is a close approximation of the simulation results. A template of q can be constructed to guide the assessment of
whether the resistance to flow offered by the formation is
comparable to that of mudcake flow resistance.
The influence of supercharging on wireline formation
tester measurements was described by Stewart and
Whittman (1979) and Proett et al. (2001).
INVASION IN MULTIPLE-LAYER FORMATIONS
All the invasion processes discussed above take place in
a single layer formation. In practice, cross flow will exist
when mud filtrate invades non-isolated multiple-layer formations.
A three-layer formation model was constructed to illustrate the effect of cross flow between adjacent layers. The
mud properties are the same as those listed in Table 1. Layer
permeabilities are 30 md, 300 md, and 1000 md. Each layer
has a thickness of 61 cm (2 ft) and the formation’s
outer-boundary is set to 610 cm (20 ft). Densities for water
and oil are 1.0 and 0.85 g/cm3, respectively.
Figure 28 shows radial profiles of filtrate saturation after
two days of invasion. It can be observed that the low permeability formation lost some filtrate to the higher permeability formation due to cross flow between layers. The profile
of filtrate saturation in the 1000 md layer suggests the influence of gravity effects.
In addition to supercharging, gravity force can influence
FIG. 26 Graphical description of the relationship between the
supercharge index (1 – q) and K/Kmc0 for three different value of
Rf. The curves were constructed assuming RW = 10 cm, Rout =
610 cm, Rmc = 9.75 cm, overbalance pressure = 50 psi, kro = 1.0,
krw = 0.2, compressibility exponent for mudcake permeability v =
0.9, and mo /mw = 1.
26
the pressure gradient. In the three-layer case, the pressure
gradients for filtrate and oil were 0.433 and 0.368 psi/ft,
respectively. Figure 29 shows the time evolution of pressures at the sandface for the 1000 md layer. At the initial
condition, the oil-bearing layer exhibits a pressure gradient
of 0.368 psi/ft. After one hour of mud-filtrate invasion, the
pressure gradient decreases to 0.328 psi/ft because of the
influence of the upper layer (K = 30 md), which sustains a
higher pressure due to supercharging. The pressure gradient
gradually increases to 0.378 psi/ft at 10 hours after the onset
of invasion. At the end of two days, the gradient becomes
0.398 psi/ft, i.e. 8% higher than the initial condition. As
suggested by Figure 29, the oil-water-contact location calculated based on pressure tests after invasion may be raised
because of pressure-curve shifts, assuming that no changes
occur in the pressure gradient of the underlying water-bearing zone.
SUMMARY AND CONCLUSIONS
1. The simulator described in this paper, referred to as
“INVADE”, can be used to match laboratory measurements as well as to calculate invasion into multi-layer
formations. INVADE can also be used to simulate the
process of salt mixing between mud filtrate and connate
water.
2. For high permeability zones, both mudcake growth rate
and mud filtrate invasion rate are controlled primarily by
FIG. 27 Graphical description of the relationship between the
supercharge index (1 – q) and K/Kmc0 for three different value of
kro. The curves were constructed assuming Rw = 10 cm, Rf =
18.3 cm, Rout = 610 cm, Rmc = 9.75 cm, overbalance pressure =
50 psi, krw = 0.2, compressibility exponent for mudcake permeability v = 0.9, and mo /mw = 1.
PETROPHYSICS
February 2005
The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
mud properties (i.e., mudcake permeability, mudcake
porosity and mud solid fraction).
3. For low permeability zones, both mudcake growth rate
and mud filtrate invasion rate will be influenced by formation properties (i.e., formation permeability, oil relative permeability endpoint) in addition to mud properties.
4. A mudcake flow resistance parameter was introduced in
this paper to characterize the potential for invasion. This
parameter can be used to closely approximate the pressure differential supported by the mudcake and potential
supercharging sensed by wireline formation testers.
5. In a multi-layer formation zone, low permeability formations can lose filtrate to higher permeability formations
due to cross flow between layers. The initial pressure
gradient can be affected by mud filtrate invasion, which
may cause errors in the estimation of the location of
oil-water contacts.
NOMENCLATURE
C
Cw
eo
ep
ew
fs
H
i
constant
salt concentration, ppm
exponent for kro equation
exponent for capillary pressure equation
exponent for krw equation
solid fraction
height of the zone
grid number index
FIG. 28 Spatial cross-section (radial and vertical directions) of
water saturation simulated in a three-layer formation. The invasion time is two days. Individual layer permeabilities and porosities are indicated on the figure.
February 2005
K
Kc
Kmc
Kmc0
knw
krw
k rw0
kro
k ro0
kw
Pc
Pc0
Pnw
Pw
q
Q
Rf
Rmc
Rout
Rw
Rwell
rr
r
S1
S2
Sor
Sw
Swi
S wt*
t
formation permeability
core permeability
mudcake permeability
mudcake reference permeability
non-wetting phase permeability
water relative permeability
water relative permeability endpoint
oil relative permeability
oil relative permeability endpoint
wetting phase permeability
capillary pressure
coefficient for capillary pressure equation
pressure for non-wetting phase fluid
pressure for wetting phase fluid
mud filtrate flow rate
total sandface flow rate
invasion front radius
mudcake radius
formation outer boundary radius
connate water resistivity, ohm-m
borehole radius
radius in cylindrical grid system
location of the observation point
slowness at pressure P1
slowness at pressure P2
residual oil saturation
wetting phase or water saturation
residual wetting phase saturation
normalized wetting phase saturation
invasion time
FIG. 29 Time evolution of the pressures at the sandface for the
lower layer (K = 1000 md) of the three-layer formation described
in Figure 28.
PETROPHYSICS
27
Wu et al.
T
v
v(t)
|vn|
vnw
vw
Vl
Vs
xmc
temperature measured in degrees centigrade
compressibility exponent of cake permeability
total production Darcy velocity
Darcy velocity of the filtrate through the cake
Darcy velocity for non-wetting phase fluid
Darcy velocity for wetting phase fluid
volume of liquids in the mud suspension
volume of solids in the mud suspension
mudcake thickness
Greek Symbols
DP
overbalance pressure
Dp
pressure drop across the mudcake
d
exponent multiplier for mudcake porosity
f
formation porosity
mudcake porosity
fmc
fmc0
mudcake reference porosity
q
ratio of mudcake flow resistance over total
flow resistance
mf
filtrate viscosity
non-wetting phase fluid viscosity
mnw
oil viscosity
mo
mw
wetting phase fluid or water viscosity
ACKNOWLEDGMENTS
We would like to express our gratitude to Anadarko
Petroleum Corporation, Baker Atlas, Conoco-Phillips,
ExxonMobil, Halliburton Energy Services, the Mexican
Institute for Petroleum, Schlumberger, Shell International
E&P, and TOTAL, for funding of this work through UT
Austin’s Joint Industry Research Consortium on Formation
Evaluation. A special note of gratitude goes to Dick Woodhouse and David Kennedy for their constructive technical
criticism and editorial comments on the first version of this
paper.
REFERENCES
Bezemer, C., and Havenaar, I., 1966, Filtration behavior of circulating drilling fluids: Society of Petroleum Engineers Journal,
vol. 6, no. 4, p. 292–298.
Brooks, R. H., and Corey A. T., 1966, Properties of porous media
affecting fluid flow, Journal of the Irrigation and Drainage
Division, Proceedings of the American Society of Civil Engineers 92, no. IR2, p. 61–88.
Center for Petroleum and Geosystems Engineering, 2000,
UTCHEM Technical Documentation, The University of Texas
at Austin.
Chin, W. C., 1995, Formation invasion with applications to measurement-while-drilling, time-lapse analysis, and formation
damage, Gulf Publishing Company, Houston, Texas.
Dewan, J. T., and Chenevert, M.E., 1993, Mudcake buildup and
28
invasion in low permeability formations; application to permeability determination by measurement while drilling, paper
NN, in SPWLA/CWLS 34th Annual Logging Symposium
Transactions: Society of Professional Well Log Analysts, Calgary, Alberta.
Dewan, J. T., and Chenevert, M. E., 2001, A model for filtration of
water-base mud during drilling: determination of mudcake
parameters: Petrophysics, vol. 42, no. 3, p. 237–250.
Dresser Atlas Inc., 1982, Well Logging and Interpretation Techniques, Dresser Industries, Houston, Texas.
Ferguson, C. K., and Klotz, J. A., 1954, Filtration of mud during
drilling: Petroleum Transactions of AIME, vol. 201, p. 29–42.
Fordham, E. J., Ladva, H. K. J., and Hall, C., 1988, Dynamic filtration of bentonite muds under different flow conditions, SPE
18038, in SPE Annual Conference Proceedings: Society of
Petroleum Engineers, Houston, TX.
Fordham, E. J., Allen, D. F., and Ladva, H. K. J., 1991, The principle of a critical invasion rate and its implications for log interpretation, SPE 22539, in SPE Annual Technical Conference
Proceedings: Society of Petroleum Engineers, Dallas, TX.
Holditch, S. A., and Dewan, J. T., 1991, The evaluation of formation permeability using time lapse logging measurements during and after drilling: Annual Report, Contract No.
5089-260-1861, Gas Research Institute, December.
Outmans, H. D., 1963, Mechanics of static and dynamic filtration
in the borehole: Society of Petroleum Engineering Journal,
Sept., p. 236–244.
Proett, M. A., Chin, W. C., Manohar, M., Sigal, R. and Wu, J.,
2001, Multiple factors that influence wireline formation tester
pressure measurements and fluid contacts estimates, SPE
71566, in SPE Annual Technical Conference Proceedings:
Society of Petroleum Engineers, New Orleans, LA.
Semmelbeck, M. E., Dewan, J. T., and Holditch, S. A., 1995, Invasion-based method for estimating permeability from logs, SPE
30581, in SPE Annual Technical Conference Proceedings:
Society of Petroleum Engineers, Dallas, TX.
Stewart, G. and Whittman M., 1979, Interpretation of the pressure
response of the repeat formation tester, SPE 8362, in SPE
Annual Conference Proceedings: Society of Petroleum Engineers, Dallas, TX.
Wu, J., Torres-Verdín, C., Sepehrnoori, K., and Delshad, M., 2004,
Numerical simulation of mud-filtrate invasion in deviated
wells: SPE Reservoir Evaluation and Engineering, vol. 7, no.
2, p. 143–154.
APPENDIX
Consider a simple constitutive model for incompressible
mudcake buildup. The filtration of a fluid suspension of
solid particles by a porous but rigid mudcake can be constructed from first principles. First let xmc(t) > 0 represent
cake thickness as a function of time, where xmc(0) = 0 indicates zero initial thickness. Also, let Vs and Vl denote the
volumes of solids and liquids in the mud suspension,
respectively, and let fs denote the solid fraction defined as
PETROPHYSICS
February 2005
The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
fs =
Vs
.
(Vs + Vl )
(A.1)
Direct integration of this last expression under the assumption of initial zero filtrate gives
If the solid particles do not enter the formation, Chin
(1995) shows that the time evolution of mudcake thickness,
xmc(t), satisfies the ordinary differential equation
fs
dx mc
vn ,
=
dt
(1- f s ) (1- f mc )
(A.2)
where fs is the solid fraction of mud, fmc is the mudcake
porosity, and |vn| is the Darcy velocity of the filtrate through
the mudcake and through the filter paper.
Now consider a one-dimensional, constant density, single liquid flow. The corresponding Darcy velocity is given
by
vn =
K mc Dp
,
m f x mc
(A.3)
where Kmc is mudcakepermeability, Dp is the pressure drop
across the mudcake, and mf is filtrate viscosity.
Substitution of equation (A.3) into equation (A.2) leads
to
fs
dx mc
K mc Dp
=
.
dt
(1- f s ) (1- fmc ) m f x mc
2tDpf s
K mc
.
(1- f s ) (1- f mc ) m f
(A.5)
This latter result demonstrates that mudcakethickness in a
linear flow grows with time in proportion to t. Equation
(A.5) is valid only when Kmc, Dp, and fmc are constant. If
Kmc, Dp, and fmc are functions of time, equation (A.4) can
also be integrated numerically. To obtain the filtrate production volume, we combine the relation dVl = |vn|dAdt and
equation (A.3) to obtain
dVl =
K mc Dp
dAdt ,
m f x mc
(A.6)
where dA is an elemental area of filter paper. Further substitution of xmc from equation (A.5) yields
dVl =
February 2005
DpK mc (1- f s ) (1- f mc )
dAdt .
2t m f
fs
(A.7)
K mc (1- f s ) (1- f mc )
dA .
mf
fs
(A.8)
From equation (A.8) it follows that filtrate production volume in a linear flow also grows with time in proportion to
t.
By taking the derivative of equation (A.8) with respect to
time we obtain the mud filtrate invasion flow rate, q(t),
given by
q ( t ) = Dp
K mc (1- f s ) (1- f mc )
dA .
mf
2tfs
(A.9)
In reality, mudcake may be compressible, that is, its
mechanical properties may vary with the applied pressure
differential. Compressibility effects can be determined by
performing the filtration experiment at increasing pressure
differentials. Experiments on mudcake properties by
Dewan and Chenevert (2001) indicate that the permeability
during the initial mudcake buildup can be estimated with
the expression
(A.4)
If the mudcake thickness is infinitesimally thin at t = 0, with
xmc(0) = 0, equation (A.3) can be integrated to yield
x mc ( t ) =
Vl = 2tDp
K mc ( t ) =
K mc 0
,
v
pmc
(t )
(A.10)
where pmc is the mudcake pressure differential (psi), Kmc0 is a
reference permeability defined at 1 psi differential pressure
and v is a “compressibility” exponent. Typically, v is in the
range of 0.4 to 0.9. A value of zero for v corresponds to completely incompressible mudcake. Dewan and Chenevert’s
(2001) work also indicates a similar relationship for mudcake
porosity during the initial mudcake buildup, given by
f mc ( t ) =
f mc 0
.
dv
pmc
(t )
(A.11)
In the above expression, the new exponent multiplier d was
found to vary from 0.1 to 0.2 based on a porosity-permeability cross plot. Further experiments indicated that mudcake is
not completely elastic and that it exhibits a hysteresis on the
first compression and decompression cycle. While additional relationships can be determined for hysteresis effects,
in most cases mudcake builds to a stable thickness and the
hydrostatic pressure does not change appreciably.
Up to this point, we have discussed static mudcake
growth wherein the borehole fluids are not moving. In a
dynamic condition, annular flow can limit the growth of
mudcake by continuously shearing the mudcake surface.
Also, the mechanical action of the rotating drill pipe can
PETROPHYSICS
29
Wu et al.
directly remove the mudcake and can increase the surface
shear of the annular fluids. In essence, dynamic filtration
limits the growth of mudcake. Static filtration left
unchecked would continue to grow until the mudcake
would literally plug off the wellbore. Normally, static filtration is sufficiently slow such that total plugging does not
pose operational problems.
The dynamic filtration process has been studied extensively in the industry. Fordham, et al. (1991), for example,
introduced the concept of a “critical invasion rate” beyond
which mudcake will not form, while Holditch and Dewan
(1991) have attributed it to an “adhesion fraction” which
controls the buildup process. Most authors agree on the
physical principle that erosion occurs when the hydraulic
shear stress on the surface of the mud closest to the center of
the borehole exceeds the mudcake shear strength (e.g.,
Chin, 1995; Dewan and Chenevert, 2001). Only Chin
(1995) has fully developed the theoretical basis for
dynamic mudcake erosion for Newtonian and non-Newtonian annular flow and has offered solutions for a variety of
wellbore conditions. Factors that influence dynamic filtration include: mud rheology, pumping rates, annular flows
(concentric and eccentric) and borehole shape (round versus elliptical).
It is generally acknowledged that static filtration governs
the initial mudcake growth and that the role of dynamic filtration is to limit this growth (e.g., Chin, 1995; Dewan and
Chenevert, 2001). In this paper, we will assume that
mudcake reaches such a limiting thickness but will not present the analysis for the corresponding prediction. Mudcake
thickness is normally estimated using caliper logs from
openhole wireline or logging-while-drilling logs shortly
before a formation tester is run. Mudcake thickness estimated using the caliper log can be assumed to be the limiting filtrate thickness.
Well conditioning can also affect mudcake and influence
invasion. Frequently, the well is reconditioned prior to a
logging run or in the process of drilling. To account for well
reconditioning, we can assume that the mudcake is either
partially or completely removed, and henceforth allowed to
reform. In the worst-case scenario, the mudcake is completely removed during well conditioning. In practice, this is
unlikely because inaccessible mudcake forms within the
near-surface rock grain structure, and it is difficult to remove.
The flow equations for one-dimension immiscible, constant density radial flow are obtained by combining Darcy’s
law and the equation of mass conservation. The Darcy
velocities are
v w =-
30
k w ¶Pw
,
m w dr
(A.12)
and
v nw =-
k nw ¶Pnw
,
m nw dr
(A.13)
where m w and m n w are viscosities, k w and k n w are
permeabilities, and Pw and Pnw are pressures. The subscripts
w and nw here are used to denote wetting and non-wetting
phases.
The mass continuity equations in cylindrical radial coordinates take the form
¶v w v w
¶S w
- =-f
,
r
dt
¶r
(A.14)
¶ vnw vnw
¶ S nw
=-f
,
¶r
r
dt
(A.15)
and
where the signs of vw and vnw are taken positive for invasion
(i.e., injection). By combining equations (A.12) through
(A.15) we obtain the production Darcy velocities, given by
¶Sw
¶ æ k w ¶Pw ö
vw
÷
- ç
÷= ,
ç
dt
¶rè m w dr ø r
(A.16)
¶ S nw
¶ æ k nw ¶Pnw ö
vnw
÷
.
- ç
÷=
ç
dt
¶rè mnw dr ø r
(A.17)
f
and
f
Furthermore, by adding equations (A.14) and (A.15),
together with substitution from
S w + S nw =1,
(A.18)
¶( v w + v nw )
+ ( v w + v nw ) =0 ,
¶r
(A.19)
one obtains
r
or, equivalently, {r(vw + vnw)}r = 0. It then follows that
( v w + v nw ) = v ( t ),
(A.20)
where v(t) is the total production Darcy velocity. For incompressible flow, v(t) is the mud filtrate velocity at the
wellbore.
At this point, it is convenient to introduce the capillary
pressure function Pc and write it as a function of the wetting
phase saturation Sw, namely,
Pc ( S w ) = Pnw - Pw .
PETROPHYSICS
(A.21)
February 2005
The Influence of Water-Base Properties and Petrophysical Parameters on Mudcake Growth, Filtrate Invasion, and Formation Pressure
Then the non-wetting velocity in equation (A.13) can be
written as
vnw =-
k nw æ ¶ Pc ( S w ) ¶ Pw ö
ç
÷.
+
m nw è dr
dr ø
é k nw æ D Pc ( S w ) ö æ k nw
D Pw ù
kw ö
÷
ç
÷+ rç
+
êr
ú =C ,
ç
÷
m w ø Dr ûi
ë mnw è Dr ø è mnw
(A.26)
(A.22)
where C is a constant and 2 £ i £ n. The latter equation can
be expressed for each i-th grid block as
We now combine equations (A.17), (A.18), and (A.22) to
obtain an expression for the non-wetting production per unit
volume per unit time in terms of the wetting saturation and
pressure, given by
é k nw æ Pc ,i - Pc ,i+1 ö æ k nw
ùæ Pw ,i - Pw ,i+1 ö
kw ö
ç
÷
÷
÷
+
ê
ç
÷+ç
ç
÷úç
ç
÷= C .
m w øûè ln( ri ) - ln(ri+1 ) ø
ë mnw è Pw ,i - Pw ,i+1 ø è mnw
(A.27)
¶(1- S w ) ¶ é k nw æ ¶Pc ( S w ) ¶Pw öù vnw
ç
÷ú=
.
f
- ê
+
dt
¶rë m nw è dr
dr øû r
(A.23)
By making use of equation (A.20), the wetting and non-wetting production per unit volume per unit time equations
(A.16 and A.23, respectively) can be combined to yield an
alternate form of equation (A.23), i.e.,
¶ é k nw æ ¶Pc ( S w ) ö æ
k nw
kw ö
¶ Pw ù v ( t )
÷
ç
÷+ç
.
- ê
+
ú=
ç
÷
r
¶rë m nw è dr ø è m nw
m w ø dr û
(A.24)
This last equation can be used to solve for Pw and subsequently equation (A.16) can be used to solve for Sw. Such a
procedure is known as the “implicit pressure-explicit saturation” or IMPES method and is widely used in reservoir simulation.
Having the immiscible two-phase flow formulations and
the properties of mudcake growth, we can now approach
the problem where mudcake forms and grows, thereby creating a Darcy flow that appears at the inlet of our radial
geometry. This flow satisfies its own pressure differential
equation and is characterized by the moving mud-tomudcake boundary and a fixed mudcake-to-rock interface
shown in Figure A.1.
At the wellbore surface, the total sandface flow rate is
defined as
Q ( t ) =2pHR well v ( t ),
(A.25)
where Rwell is the radius of borehole, H is the height of the
zone and v(t) is the velocity through the mudcake obtained
from equation (A.24). Using finite differencing, let i be the
grid block index where i = 1 at the mud-to-mudcake boundary, i = 2 at the fixed mudcake-to-rock interface, and i = n at
the outer boundary. For incompressible flow, the mass flow
rate between each grid block should be equal and this condition leads to
February 2005
For a mudcake sustaining single-phase flow, the mass
balance condition for equation (A.27) dictates that C be
equal to the mud filtrate flow, i.e.,
K mc
¶ Pw
× r×
=C ,
mf
¶r
(A.28)
where mf is filtrate viscosity and Kmc is mudcake permeability.
Mass balance can also be enforced for the total flow rate
through the entire radial model (i.e., i = 2 to n). This can be
done by combining equations (A.25), (A.27), and (A.28).
Accordingly, it can be shown that the inlet filtrate flow rate
is coupled to the formation flow by the expression
Q (t ) =
2 pH (Pw ,1 (t ) - Pw , n (t ))
.
æ R
ö
mf
ln(rr +1 ) - ln(ri )
+
× lnç well ÷
åæ öæ P (t ) P (t ) ö æ
K mc (t ) è R mc (t ) ø
- c, i +1 ÷
kn w
k ö
c, i
i= 2
ç kn w ÷ç
+ w÷
ç
÷+ç
è mn w øiè Pw , i (t ) - Pw , i +1 (t ) ø è mn w
mw øi
n
(A.29)
As the mudcake grows, the total inlet flow rate can be calculated directly from the above equation. The immiscible
radial Darcy flow equations are solved using equations
(A.24) and (A.16) for each grid block, i, for each time step.
In summary, the linkage between the mudcake model and
the immiscible invasion model is governed by the flow rate
using the coupled Darcy flow described by equation (A.29).
From equation (A.29), we notice that at the start of invasion Rmc = Rwell and hence the flow rate reaches its maximum. As the mudcake builds and Rmc becomes smaller than
FIG. A-1 Diagram of a one-dimensional mudcake-rock model.
The mudcake is located on the left and the rock formation
extends to the right of the diagram.
PETROPHYSICS
31
Wu et al.
Rwell the mudcake flow rate decreases. The flow rate is also
influenced by the saturation of the fluid in the formation, Sw.
As the mudcake builds and reaches a maximum, the
value of Rmc can remain unchanged in the numerical simulation. In this case, the invasion is governed by the invasion
front represented by the fluid saturations. In the case of
mudcake removal or “rub-off,” mudcake thickness is
reduced to 0 and allowed to rebuild thereafter. Again, the
mudcake growth is dynamically coupled to the invasion
model and grows at a time rate that is linked to the invasion
front that continues to penetrate the formation.
32
ABOUT THE AUTHORS
Jianghui Wu is a research scientist with Baker Atlas. His
research areas include wireline formation testing, reservoir simulation, and integration of logging data. He holds BS and MS
degrees from the University of Petroleum in China and a PhD
degree from The University of Texas at Austin, all in petroleum
engineering. Formerly, he was a reservoir engineer with CNPC.
Carlos Torres-Verdín received a PhD in Engineering
Geoscience from the University of California, Berkeley, in 1991.
During 1991–1997 he held the position of Research Scientist with
Schlumberger-Doll Research. From 1997–1999, he was Reservoir
Specialist and Technology Champion with YPF (Buenos Aires,
Argentina). Since 1999, he has been with the Department of Petroleum and Geosystems Engineering of The University of Texas at
Austin, where he currently holds the position of Associate Professor. He conducts research on borehole geophysics, formation evaluation, and integrated reservoir characterization. Torres-Verdín
has served as Guest Editor for Radio Science, and is currently a
member of the Editorial Board of the Journal of Electromagnetic
Waves and Applications, and an associate editor for Petrophysics
(SPWLA) and the SPE Journal.
Kamy Sepehrnoori is the Bank of America Centennial Professor in the Department of Petroleum and Geosystems Engineering
of The University of Texas at Austin. His teaching and research
interests include computational methods, reservoir simulation,
parallel computations, applied mathematics, and enhanced oil
recovery. Sepehrnoori holds a PhD degree in petroleum engineering from The University of Texas at Austin.
Mark A. Proett received a BSME degree from the University
of Maryland and a MS degree from Johns Hopkins. He has been
involved with the development of formation testing systems since
the early 1980s, and has published extensively. Proett holds 16 patents, 14 of which deal with well testing and fluid flow analysis
methods. He has served on the SPWLA and SPE technical committees and served as the Chairman for the SPE Pressure Transient
Testing Committee. He is currently a Senior Scientific Advisor for
Halliburton Energy Services in the Strategic Research group.
PETROPHYSICS
February 2005