Estimation of water saturation from nuclear magnetic resonance

Journal of Petroleum Science and Engineering 108 (2013) 40–51
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Journal of Petroleum Science and Engineering
journal homepage: www.elsevier.com/locate/petrol
Estimation of water saturation from nuclear magnetic resonance
(NMR) and conventional logs in low permeability sandstone reservoirs
Xiao Lianga,b,n, Zou Chang-chuna,b, Mao Zhi-qiangc, Shi Yu-jiangd, Liu xiao-penge, Jin Yanf,
Guo Hao-pengd, Hu Xiao-xine
a
Key Laboratory of Geo-detection, China University of Geosciences, Ministry of Education, Beijing, PR China
School of Geophysics and Information Technology, China University of Geosciences, Beijing, PR China
c
State Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum, Beijing, PR China
d
Research Institute of Exploration & Development, Changqing Oilfield Comapny, PetroChina, Shaanxi, PR China
e
Geological Exploration and Development Research Institute in Sichuan-Changqing Drilling and Exploration Engineering Corporation,
CNPC, Sichuan, PR China
f
Southwest Oil and Gas Field Branch Company, PetroChina, Sichuan, PR China
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 22 March 2012
Accepted 19 May 2013
Available online 15 June 2013
It is difficult to obtain rock resistivity parameters by using the cross plots of porosity vs. formation factor
and water saturation vs. resistivity index to calculate reservoir water saturation in low permeability
sandstones. The cementation and saturation exponents (m and n separately) are divergent, and no fixed
values can be obtained due to the complicated pore structure. This leads to a problem in water saturation
calculation. To investigate the main factors that heavily affect the cementation and saturation exponents,
36 core samples, which were drilled from low permeability sands of Xujiahe Formation, Sichuan basin,
southwest China, are chosen for laboratory resistivity and nuclear magnetic resonance (NMR) measurements, 20 of them for mercury injection capillary pressure (MICP) measurements and 10 of them for
casting thin-section analysis. The results show that these two parameters are associated with rock pore
structure. For rocks with good pore structure, the proportion of macropore components is dominant,
high cementation exponents and low saturation exponents can be obtained, and on the contrary, rocks
with poor pore structure will be dominated by the proportion of small pore components, and they will
contain low cementation exponents and high saturation exponents. To quantitatively acquire reliable
cementation and saturation exponents for water saturation estimation, a logarithmic function is
established to calculate cementation exponent from porosity. Irreducible water saturation (Swi), which
is estimated from NMR logs by using the optimal T2cutoff, is presented to characterize the proportion of
small pore components. A technique of calculating saturation exponent by combining with Swi, (1−Swi)
and the logarithmic mean of NMR T2 spectrum (T2 lm) is proposed, and the corresponding model is
established. The credibility of these techniques is confirmed by comparing the predicted cementation
and saturation exponents with the core analyzed results. The absolute errors between the predicted
cementation exponents and the experimental results are lower than 0.08, and the absolute errors
between the predicted saturation exponents and the experimental results are lower than 0.2. These
techniques proposed in this study are extended to several low permeability sands for field applications;
the field examples illustrate that cementation and saturation exponents can be accurately estimated in
the intervals with which NMR logs were acquired. By using the variable rock resistivity parameters,
precisely water saturation can be calculated for low permeability sandstones evaluation.
& 2013 Elsevier B.V. All rights reserved.
Keywords:
low permeability sandstone reservoirs
cementation exponent
saturation exponent
nuclear magnetic resonance (NMR) logs
irreducible water saturation
T2 lm
pore structure
1. Introduction
Hydrocarbon saturation is an important input parameter in
formation evaluation, fluid identification and reserves estimation,
and it also plays a very important role in reservoir simulation and
n
Corresponding author at: School of Geophysics and Information Technology,
China University of Geosciences, No. 29, Xueyuan Road, Haidian, Beijing 100083,
PR China. Tel.: +86 10 8232 0692 (office); +86 152 1087 9138 (mobile).
E-mail address: [email protected] (L. Xiao).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.petrol.2013.05.009
development program formulation. Hydrocarbon saturation is
always predicted after water saturation is first estimated. Hence,
obtaining water saturation as accurate as possible is of great
importance in reservoir evaluation, especially in low permeability
sandstones with complicated pore structure. Generally, water
saturation can be accurately calculated from conventional logs
once the reliable equation is used. Archie's equation was first
proposed by Archie (1942) to calculate water saturation from
conventional logs, and it had been widely used in conventional
reservoir for a long time. With the discovery of more and more
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
sandstones; n is the saturation exponent, and its value is affected
by many factors, such as pore structure and/or wettability; a, m
and n are referred to as the rock resistivity parameters.
Transforming Eq. (1) and substituting it into Eq. (2), a derivative
formula could be written as
complicated reservoirs (such as low resistivity contrast reservoir
and low permeability sandstones), Archie's equation had been
found not to be always effective. Since the 1960s, many derived
models had been proposed to estimate water saturation from
conventional logs in different types of reservoirs. Waxman and
Smits (1968) and Waxman (1974) proposed Waxman–Smits' equation, Clavier et al. (1977) proposed the dual-water model, and
Givens (1986, 1987) and Givens and Schmidt (1988) proposed the
conductive rock matrix model (CRMM) to predict water saturation
in low resistivity contrast reservoirs. Rasmus and Kenyon (1985)
and Rasmus (1987) proposed the appropriate model (known as
Rasmus's model) to calculate water saturation in reservoirs with
porous media of fracture and cavern. However, in low permeability
sandstones with complicated pore structure, no relevant model
had been proposed to estimate water saturation at present, and
Archie's equation is still being used. Archie's equation can be
expressed as
F¼
R0
a
¼ m
φ
Rw
ð1Þ
Ir ¼
Rt
1
¼ n
R0
Sw
ð2Þ
41
sffiffiffiffiffiffiffiffiffiffiffi
n aRw
Sw ¼
φm R t
ð3Þ
This formula illustrates that the values of a, m, n, Rw, φ and
Rt must be first acquired for water saturation calculation.
Generally, φ can be calculated from conventional or NMR logs
(Wyllie et al., 1956; Coates et al., 2000), and the deep lateral
resistivity (RLLD) or deep induction resistivity (RILD) can be
directly used as Rt; Rw can be checked from the formation water
salinity by using Schlumberger's log interpretation charts
(Schlumberger Well Services, 1986).
The determination of a, m and n is of great importance in
estimating water saturation by using Eq. (3). Generally, the values
of a, m and n are obtained from laboratory resistivity measurements of the target core samples by using the statistical regression
method of power function. To obtain the values of a, m and n,
several needed procedures should be covered as follows: (1) drilling the representative core samples from the interested intervals,
porosities and permeabilities of the selective plug samples should
cover all the target formations; (2) saturating core samples with
the same salinity as actual formation water, and all the saturated
core samples are taken for laboratory resistivity measurements by
using the porous plate method. In our studied Xujiahe Formation,
the salinity of water that was used to saturate core samples
where R0 is the rock resistivity with fully saturated water, Rw is the
formation water resistivity, Rt is the rock resistivity with saturated
hydrocarbon; their units are Ω m; F is the formation factor, Ir is the
resistivity index; they are all zero dimension; φ is the rock porosity
in fraction, Sw is the water saturation in fraction, a is the tortuosity
factor; its value always ranges from 0.6 to 1.6; m is the cementation exponent and its value is always ranges from 1.0 to 2.0 for
1000
Formation factor
Formation factor
100
10
y = 1.8389x-1.2473
R2 = 0.9312
1
0.01
0.1
1
100
10
y = 1.7819x-1.6083
R2 = 0.9253
1
0.01
0.1
1
Porosity, fraction
Porosity, fraction
Formation factor
1000
100
y = 2.8892x-1.1602
R2 = 0.8986
10
1
0.01
0.1
1
Porosity, fraction
Fig. 1. (a) The cross plot of porosity vs. formation factor for core samples drilled from low permeability sandstones of Xujiahe Formation in Sichuan basin, southwest China.
(b) The cross plot of porosity vs. formation factor for core samples drilled from low permeability sandstones of Chang 8 Formation in Ordos basin, northwest China. (c) The
cross plot of porosity vs. formation factor for core samples drilled from low permeability sandstones of the third basin.
42
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
10
10
no.2
no.4
no.6
no.8
no.10
no.12
no.14
no.16
no.18
no.20
no.22
no.24
no.26
no.28
no.30
no.32
no.34
no.36
no.x1
no.x3
no.x5
no.x7
no.x9
no.x11
no.x13
no.x15
no.x17
no.x19
no.x21
no.x23
no.x25
no.x27
no.x29
Resistivity index
Resistivity index
no.1
no.3
no.5
no.7
no.9
no.11
no.13
no.15
no.17
no.19
no.21
no.23
no.25
no.27
no.29
no.31
no.33
no.35
1
0.1
1
no.x2
no.x4
no.x6
no.x8
no.x10
no.x12
no.x14
no.x16
no.x18
no.x20
no.x22
no.x24
no.x26
no.x28
no.x30
1
0.1
1
Water saturation, fraction
Water saturation, fraction
Resistivity index
10
no.s1
no.s2
no.s3
no.s4
no.s5
no.s6
no.s7
no.s8
no.s9
no.s10
no.s11
1
0.1
1
Water saturation, fraction
Fig. 2. (a) The cross plot of water saturation vs. resistivity index for core samples drilled from low permeability sandstones of Xujiahe Formation in Sichuan basin, southwest
China. (b) The cross plot of water saturation vs. resistivity index for core samples drilled from low permeability sandstones of Chang 8 Formation in Ordos basin, northwest
China. (c) The cross plot of water saturation vs. resistivity index for core samples drilled from low permeability sandstones of the third basin.
Table 1
The laboratory resistivity experimental data sets for 36 core samples drilled from low permeability sands of Xujiahe Formation in Sichuan basin, southwest China.
Wells
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
A
A
A
A
A
A
A
A
A
B
B
C
C
C
C
C
C
C
C
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
Sample
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Depth
Temperature
Rw
Porosity
Permeability
R0
(m)
Brine
concentration
(g/l)
(1C)
(Ω m)
(%)
(mD)
(Ω m)
xx12.56
xx13.47
xx24.58
xx92.22
xx14.70
xx10.50
xx09.20
xx07.20
xx00.00
xx22.80
xx07.70
xx31.00
xx19.30
xx02.00
xx95.80
xx87.85
xx85.90
xx81.50
xx71.30
xx61.50
xx46.20
xx40.50
xx35.15
xx90.60
xx83.80
xx81.00
xx78.45
xx66.20
xx54.35
xx40.40
xx22.30
xx95.50
xx86.65
xx45.80
xx98.40
xx90.30
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
130.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
25.00
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
15.73
17.36
16.17
13.88
4.60
5.86
6.70
7.28
5.57
14.68
12.91
5.92
11.71
17.44
12.31
21.27
15.89
17.14
15.62
16.43
13.13
10.13
9.62
10.58
8.54
12.20
11.38
13.34
9.04
8.20
7.80
10.59
14.00
9.33
9.41
8.36
5.87
5.24
1.08
0.58
0.08
0.16
0.20
0.28
0.15
3.06
12.30
0.13
0.23
0.23
0.20
90.60
1.05
0.58
0.17
0.27
0.21
0.21
0.16
0.62
0.33
0.89
0.91
0.57
0.62
0.27
0.50
1.03
0.49
0.24
0.86
0.50
1.31
1.10
1.34
1.63
4.63
3.52
3.19
2.82
3.16
1.31
1.46
2.85
1.92
1.39
1.96
0.77
1.31
1.24
1.45
1.41
1.82
2.31
2.40
1.75
2.71
1.66
1.74
1.59
2.34
2.84
2.54
2.00
1.65
2.73
2.15
2.64
Formation
factor (F)
20.03
16.76
20.44
25.00
70.82
53.89
48.84
43.19
48.33
20.02
22.34
43.53
29.37
21.21
29.96
11.72
20.00
18.99
22.20
21.54
27.85
35.27
36.71
26.75
41.49
25.35
26.61
24.35
35.84
43.41
38.88
30.62
25.20
41.67
32.91
40.42
Saturation
exponent (n)
1.93
2.34
2.57
2.55
2.79
2.59
2.45
2.34
2.62
2.04
1.85
2.87
1.69
2.23
2.17
1.66
1.89
2.14
1.94
1.80
1.78
2.01
2.39
3.48
3.02
2.56
3.00
2.45
3.31
2.23
2.72
2.21
1.63
2.86
2.21
1.63
T2cutoff
T2 lm
(ms)
(ms)
22.85
19.59
13.34
10.86
7.19
22.77
12.46
11.62
4.74
14.41
13.73
9.58
41.81
28.83
22.54
55.76
34.58
25.52
27.14
30.60
21.81
21.19
21.39
19.13
19.03
20.49
23.48
13.36
18.59
14.70
21.10
25.84
17.90
14.67
22.55
22.01
29.09
39.74
28.56
17.56
13.13
17.90
18.89
17.92
8.26
39.50
63.73
8.64
26.93
26.73
44.70
123.96
75.18
49.93
25.01
43.67
35.60
35.33
33.36
20.41
10.12
29.11
30.53
19.98
29.09
28.18
34.05
47.14
30.62
25.21
43.19
35.02
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
43
Table 2
The pore type information obtained from 10 core samples with casting thin-section.
Well
Well
Well
Well
Well
Well
Well
Well
Well
Well
A
A
A
B
C
C
D
D
D
D
Sample number
2
3
7
10
16
18
20
21
22
32
Depth
The relative content
Residual
intergranular
pore
Intragranular
dissolved pore
(m)
Primary
intergranular
pore
(%)
xx13.47
xx24.58
xx09.20
xx22.80
xx87.85
xx81.50
xx61.50
xx46.20
xx40.50
xx95.50
40.00
20.00
5.00
25.00
70.00
65.00
65.00
65.00
15.00
0.00
55.00
75.00
95.00
70.00
30.00
35.00
35.00
35.00
85.00
15.00
5.00
5.00
0.00
5.00
0.00
0.00
0.00
0.00
0.00
85.00
is 130.0 g/l. The data sets of core porosity, formation factor, water
saturation and the corresponding resistivity index under different
water saturation are collected; (3) mapping the cross plots of
porosity vs. formation factor, and water saturation vs. resistivity
index for all experimental core samples in log–log coordinates;
(4) using the statistical regression method of the power function
to acquire the fixed values of a and m from the cross plot of
porosity vs. formation factor, and the constant value of n from the
cross plot of water saturation vs. resistivity index.
For conventional reservoirs, with the above mentioned procedures, the fixed a, m and n can be regressed, separately. However,
for low permeability sands, the relationships of porosity and
formation factor, and water saturation and resistivity index cannot
be rigidly expressed by the power function due to the complicated
pore structure and strong heterogeneity (Mao et al., 1995; Shi
et al., 2008). Fig. 1a–c shows the cross plots of porosity and
formation factor for three different types of low permeability
reservoirs in China, and Fig. 2a–c displays the corresponding cross
plots of water saturation and resistivity index for the same rocks.
From Figs. 1 and 2, it can be observed that for low permeability
sandstones, the relationships between porosity and formation
factors are not rigid power function; the tendency of core samples
with porosity lower than 8.0% is not coincident with those with
porosity higher than 8.0%; if the statistical regression method of
power function is still used to obtain a and m for all core samples,
the deviated values should be obtained. Moreover, the cross plot of
water saturation and resistivity index is divergent and the power
function cannot be used to describe the characterization of all core
samples; a fixed saturation exponent is difficult to acquire.
In this study, 36 core samples drilled from low permeability sands
of the Xujiahe Formation in Sichuan basin, southwest China, are
chosen for laboratory resistivity experiments based on the above
mentioned procedures; the experimental data sets are listed in Table 1.
To investigate the correlation of pore structure and rock resistivity
parameters of a, m and n, all 36 core samples are tested for laboratory
NMR measurements; 20 of them are chosen for mercury injection
capillary pressure (MICP) measurements and 10 of them for the
experiments of core casting thin-section. From these experimental
measurements, the NMR spectra, MICP data and the pore type
information for the core samples are obtained. The experimental
parameters of laboratory NMR measurement are listed as follows:
polarization time (TW): 6.0 s; inter-echo spacing (TE): 0.2 ms; the
number of echoes per echo train (NE): 4096; and scanning number:
128. The laboratory NMR measurements for all 36 core samples and
pore type information for 10 core samples are listed in Tables 1 and 2,
respectively.
The average
The average
pore-throat ratio coordination
number
17.05
13.13
0.00
16.69
17.30
13.80
13.64
19.10
18.63
0.00
0.40
0.57
0.00
0.88
1.22
0.73
0.85
0.80
0.46
0.00
100
y = 2.5177x-1.129
R2 = 0.91
Formation factor
Wells
y = 2.0255x-1.1412
R2 = 0.9934
10
no.36: por.=8.4%; perm.=0.5 mD
no.31: por.=7.8%; perm.=0.5 mD
1
0.01
0.1
1
Porosity, fraction
Fig. 3. Comparison of relationship between porosity and formation factor for two
core samples with similar physical parameters.
2. Effects of pore structure on rock resistivity parameters
of a, m and n in low permeability sands
2.1. Effects of pore structure on the parameters of a and m
To illustrate the effects of pore structure on a and m, the data
sets listed in Table 1 are reused, and the cross plot of porosity and
formation factor is shown in Fig. 3. Two core samples, which are
numbered as no. 31 and no. 36 and have the same permeability
and similar porosity, are highlighted. Fig. 3 shows that the
regularity of no. 31 and no. 36 is completely different. Core sample
no. 31 follows the regularity established by core samples with
relative low porosity and low permeability, while the relationship
between porosity and formation factor for core sample no. 36 is
similar with those core samples with high porosity and high
permeability.
Fig. 4(a) shows the NMR spectra of core sample no. 31 and no.
36, and (b) is the comparison of MICP curves for the same two core
samples. These comparisons illustrate that the difference of the
relationship between porosity and formation factor for core
samples no. 31 and no. 36 is caused by the completely disparate
pore structure. Although they have the same permeability and
similar porosity, their pore structure is quite different. The NMR
T2 distribution illustrates that the proportion of macropore components of core sample no. 36 is dominant while core sample no. 31 is
44
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
Mercury injection pressure, MPa
100
0.24
no.31
Amplitude, v/v
0.2
no.36
0.16
0.12
0.08
0.04
0
0.1
1
10
100
1000
no.36
no.31
10
1
0.1
0.01
0.001
100
10000
80
T2, ms
60
40
20
0
Mercury injection saturation, %
Fig. 4. (a) The NMR T2 spectra of core sample no. 31 and no. 36. (b) Comparison of MICP curves for core samples no. 31 and no. 36.
10
0.6
Resistivity index
R2 = 0.9968
no.3
y = 0.9991x-2.5703
R2 = 0.9953
no.10
no.16
-2.0358
0.5
no.10
y = 0.9974x
no.3
R2 = 0.9975
Amplitude, v/v
no.16
y = 0.9772x-1.6628
no.12
no.12
y = 0.9437x-2.8656
R2 = 0.9704
no.16: n=1.66; T2lm=123.96ms
no.10: n=2.04; T2lm=39.50ms
no.3: n=2.57; T2lm=28.56ms
no.12: n=2.87; T2lm=8.64ms
0.4
0.3
0.2
0.1
0
0.1
1
0.1
1
10
1
100
1000
10000
T2, ms
Water saturation, fraction
Mercury injection pressure, MPa
100
10
1
0.1
0.01
0.001
100
no.16
no.10
no.3
no.12
80
60
40
20
0
Mercury injection saturation, %
Fig. 5. (a) The cross plot of water saturation vs. resistivity index for four representative core samples in low permeability sands. (b) The NMR T2 distribution for the same four
core samples. (c) The MICP curves for the same four core samples.
dominated by the proportion of small pore components, the
threshold pressure of core sample no. 36 is lower. These two
comparisons illustrate that in low permeability sands, the values
of a and m (especially m) should be variable in rocks with
complicated pore structure; if fixed values of a and m are defined
in the whole intervals, inaccurate water saturation will be
calculated.
2.2. Effects of pore structure on saturation exponent
Generally, rock saturation exponent is affected by two main
factors: wettability and pore structure (Sweeney and Jennings,
1960; Suman and Knight, 1997; Xiao et al., 2013). Based on the
laboratory measurements of sandstones, Sweeney and Jennings
(1960) and Suman and Knight (1997) found that the saturation
exponent can reach to 8.0 for rock with wetting oil phase. Xiao
et al. (2013) pointed out that rocks with poor pore structure will
have a high saturation exponent. Rock wettability determination
experiments in the target Xujiahe Formation showed that formations are water wetted. Hence, the effect of wettability on the
electrical resistivity can be ignored.
To illustrate the relationship between rock pore structure and
saturation exponent in low permeability sands, laboratory resistivity and NMR experimental results for 36 core samples, mercury
injection measurements for 20 core samples and casting thinsection experiments for 10 core samples have been studied.
Four representative core samples (which were numbered as no.
16, no. 10, no. 3 and no. 12 separately) with saturation exponent
increasing from 1.6628 to 2.827 are compared and displayed in
Fig. 5a. To illustrate the difference of pore structure among them,
the corresponding NMR T2 distributions for these four core
samples are displayed in Fig. 5b, the MICP curves are shown in
Fig. 5c, and the casting thin-sections for core samples no. 16, no. 10
and no. 3 are displayed in Fig. 6. From these casting thin-sections,
the information of pore type can be obtained, and the average
coordination number, which was defined as the average number
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
150 μm
45
75 μm
50 μm
Fig. 6. The casting thin-section for 3 representative core samples of (a) no. 16, (b) no. 10 and (c) no. 3.
80
no.16
no.10
Relative content, %
60
no.3
40
20
0
Primary intergranular pore Residual intergranular pore
Intragranular dissolved
pore
Pore type
Fig. 7. The pore type of three representative core samples of no. 16, no. 10 and no. 3.
of pore throats connected to a pore body and used to characterize
the connectivity (Wang and Sharma, 1988), was also acquired.
The average coordination number and pore type for core samples
no. 16, no. 10 and no. 3 are showed in Figs. 6 and 7, respectively.
Based on the experimental results shown in Figs. 5–7, the
relationship of saturation exponent and rock pore structure is
analyzed as follows: (1) core sample no. 16 contains the lowest
saturation exponent of 1.6628; the corresponding laboratory NMR
measurement shows that the T2 distribution is wide; the longest
T2 transverse relaxation time reaches to 2000.0 ms, T2 lm is
123.96 ms; the T2 spectrum is bimodal, and the proportion of
macropore components is dominating; the MICP curve shows that
the pore structure of core sample no. 16 is the best; the threshold
pressure is lower than 0.05 MPa; the average coordination number
is 1.22, and the relative content of primary intergranular pore and
residual intergraular pore is 75.0% and 25.0%, separately. (2) The
saturation exponent of core sample no. 10 is higher than no. 16;
the laboratory NMR T2 spectrum illustrates that the longest T2
transverse relaxation time of core sample no. 10 is 570.0 ms; T2 lm
is 39.50 ms; it is bimodal; and the proportion of macropore
components is dominating; the threshold pressure is close to
0.1 MPa; the average coordination number is 0.88, and the pore
type is dominant with residual intergraular pore; the relative
content of primary intergranular pore is lower than that of the
core sample no. 16. These show that the pore structure of core
sample no. 10 is poorer than no. 16. (3) The saturation exponent of
core sample no. 3 is lower than core samples no. 16 and no. 10; the
corresponding NMR T2 distribution illustrates that the T2 spectrum
is narrower than those of core samples no. 16 and no. 10; T2 lm is
28.56 ms. The T2 distribution is bimodal, but the proportion of
small pore components is dominating. The threshold pressure is
higher than 1.5 MPa, and the average coordination number is 0.57;
the relative content of primary intergranular pore is further
reduced and the relative content of residual intergraular pore is
incremental. (4) The saturation exponent of core sample no. 12 is
the highest; the corresponding laboratory NMR measurement and
MICP curve show that the pore structure is the poorest; the NMR
spectrum is unimodal, and the main T2 transverse relaxation time
is lower than 100 ms; T2 lm is 8.64 ms, and the MICP curve lies on
the top.
To describe the universality of the analyzed relationship of
saturation exponent and rock pore structure above, the data sets
listed in Tables 1 and 2 are reused, and the cross plots of T2 lm vs.
saturation exponent for 36 core samples, the average coordination
number vs. saturation exponent for 10 core samples are drawn,
and they are shown in Fig. 8a and b, separately. These two figures
illustrate that the negative correlations between T2 lm and saturation exponent, the average coordination number and saturation
exponent are ubiquitous. This means that the pore structure is
really the main factor that affects the saturation exponent in the
Xujiahe Formation.
From the displayed experimental results in Figs. 5–8, some
conclusions can be observed: (1) saturation exponent is mainly
related with the rock pore structure. For core samples with low
saturation exponents, they will contain good pore structure, wide
T2 distribution and high T2 lm. The proportion of macropore
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
4
4
3
3
Saturation exponent
Saturation exponent
46
2
1
0
2
1
0
1
10
100
0
1000
0.5
1
1.5
The average coordination number
T2lm, ms
Fig.8. (a) The cross plot of T2 lm vs. saturation exponent for 36 core samples. (b) The cross plot of the average coordination number vs. saturation exponent for 10 core
samples.
components is dominated, and the corresponding MICP curve lies
in the bottom; the threshold pressure is the lowest; the average
coordination number is high, and thus depicts good connectivity;
the main pore space is the primary intergranular pore. (2) For rock
dominated by microporosity, the proportion of small pore components is large; the value of T2 lm is low; the threshold pressure of
MICP curve is high; the average coordination number is low and
the connectivity is poor; the corresponding relative content of
primary intergranular pore is reduced and the relative content of
residual intergraular pore is incremental; the saturation exponent
will increase.
With the above conclusions, it can be observed that the rock
pore structure must be first characterized to estimate accurately
the saturation exponent for water saturation calculation in low
permeability sandstones.
Saturation exponent is related to the proportion of small pore
components, MICP curve and parameters extracted from core
casting thin-section. However, the data sets cannot be acquired
consecutively in the whole target intervals except for NMR logs in
field applications. Hence, the best method is to estimate variable
saturation exponent from NMR logs. Xiao et al. (2013) proposed an
empirical statistical formula to estimate saturation exponent from
NMR logs after the values of Swirr_30, Swirr_40 and Swirr_50 (irreducible water saturations calculated from NMR logs by defining 30.0,
40.0 and 50.0 ms as T2 cutoffs). However, field applications
illustrated that this empirical method was not always effective.
For example, in low permeability sandstones, the T2 cutoff is
always lower than 30.0 ms (Mao et al. 2013). If all formations are
classified by using fixed 30.0, 40.0 and 50.0 ms as T2 cutoffs, many
effective pore spaces are considered as useless. In the next section,
an alternative technique and a general equation of estimating
saturation exponent from NMR logs is proposed.
3. Estimation of rock resistivity parameters from NMR logs
3.1. Estimation of variable cementation exponent in low
permeability sands
Taking logarithm on both sides in Eq. (1), a derived formula can
be expressed as follows:
logðFÞ ¼ logðaÞ−m logðφÞ
ð4Þ
Eq. (4) illustrates that formation factor and porosity are linear
in log–log coordinate once the value of cementation exponent is
fixed. For low permeability sandstones, the cementation exponent
is variable. Hence, the relationship between formation factor and
porosity in log–log coordinate is nonlinear.
Plenty of experimental results have illustrated that the relationship between formation factor and porosity in log–log coordinate can be expressed by using quadratic function (Mao et al.,
1997; Liu et al., 2012); this means that the formation factor can be
estimated from porosity by using
2
logðFÞ ¼ x log ðφÞ þ y logðφÞ þ z
ð5Þ
where x, y and z are the undetermined coefficients.
Comparing Eq. (4) with (5), the coefficient z can be defined as
z ¼ logðaÞ
ð6Þ
Based on the analysis of rock resistivity experimental data sets
acquired from different basins, Mao et al. (1997) and Liu et al.
(2012) pointed out that the constant mentioned in Eq. (5) was
close to 0.0 by using the quadratic function to express the
relationship of formation factor and porosity. Hence, the value of
a approximates to 1.0. This viewpoint was also verified by our
listed data sets in Table 1. Thereby, Eq. (5) can be rewritten as
2
logðFÞ ¼ x log ðφÞ þ y logðφÞ ¼ ðx logðφÞ þ yÞlogðφÞ
ð7Þ
Comparing Eqs. (4) and (7), we can observe that the expression
of cementation exponent can be written as
m ¼ x logðφÞ þ y
ð8Þ
Submitting Eq. (8) into (1), the relationship between porosity
and formation factor can be expressed as
F¼
1
φx logðφÞþy
ð9Þ
Formation porosity φ can be directly obtained from field NMR
logs in oil bearing reservoir or water saturated layers, and it can be
accurately calculated from NMR and conventional logs in gas
bearing formation (Xiao et al., 2012). In the meanwhile, φ can also
be calculated from conventional logs while enough core samples
were drilled for routine analysis and the core scale logging method
was applied. Once the values of x and y are calibrated, the variable
m can be precisely estimated from porosity.
Using the data sets listed in Table 1, the values of x and y are
calibrated, and Eq. (9) is expressed as
F¼
1
;
φ0:48 logðφÞþ2:00
Correlationcoefficient : 0:99
ð10Þ
Fig. 9 illustrates the principle of acquiring the optimal cementation exponent from porosity. It can be observed that the trend
line passes by the vast majority of core samples; this means the
regressed equation can be used to express the relationship of
porosity and formation factor, and this also ensures the responsibility
of the obtained cementation exponent. The variable cementation
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
100
47
18
Frequency
Formation factor
15
10
y =1/φ 0.48×log(φ)+2.00
Correlation coefficient: 0.99
12
9
6
3
0
1
0.01
0.1
3~10
10~17
17~24
24~31
31~38
>38
T2 cutoff, ms
1
Porosity, fraction
Fig. 9. The principle of acquiring the optimal cementation exponent from porosity
by using the proposed technique.
exponent can be obtained consecutively in the intervals with which
precise porosity was acquired from field NMR or conventional logs
once the proposed technique is extended to field applications.
3.2. Estimation of saturation exponent from NMR logs
Through Figs. 5–8, we have concluded that saturation exponent
is proportional to the proportion of small pore components.
Hence, the proportion of small pore components must be first
characterized to precisely estimate saturation exponent. In this
aspect, NMR logs have unique advantages. It is known that
irreducible water saturation (Swi) is an effective parameter in
characterizing the proportion of small pore components (Coates
et al., 2000), and rocks with poor pore structure will contain high
proportion of small pore components, and thus high irreducible
water saturations; on the contrary, for rocks with low proportion
of small pore components, the corresponding irreducible water
saturations are low. In this study, irreducible water saturation
under a defined T2 cutoff is introduced to characterize rock pore
structure and the proportion of small pore components.
To compare pore structure of all core samples, the unified
optimal T2 cutoff is used to calculate irreducible water saturation
from NMR logs to characterize the proportion of small pore
components. To determine the optimal T2 cutoff, T2 cutoffs
acquired from laboratory NMR measurements for all core samples
are showed in a histogram, and the T2 cutoff with the maximum
frequency is defined as the optimal T2 cutoff. By using the optimal
T2 cutoff, Swi can be estimated by using (Straley et al., 1994)
R T 2cutoff
Swi ¼
T 2 min
R T 2max
T 2 min
SðTÞdt
SðTÞdt
Fig. 10. The statistical histogram of T2 cutoff for all 36 core samples drilled from the
Xujiahe Formation in Sichuan basin.
maximum frequency is 20.75 ms. Hence, 20.75 ms is chosen as the
optimal T2 cutoff in the Xujiahe Formation.
T2 lm is the overall signature of NMR T2 distribution, and it is
associated with the rock pore structure. For rock with good pore
structure, the value of T2 lm is high, and on the contrary, low T2 lm
corresponds to poor pore structure. Hence, T2 lm is inversely
proportional to saturation exponent; in the proposed saturation
exponent estimation model, T2 lm is chosen as another input
parameter besides irreducible water saturation.
Considering Swi can be used to characterize the proportion of
small pore components, then, (1−Swi) can be used to characterize
the proportion of macropore components; it is inversely proportional to saturation exponent. Hence, in the novel saturation
exponent estimation model, the parameters of Swi, (1−Swi) and
T2 lm are used, and the corresponding model is established as
1−Swi p q
n¼C
T 2 lm
ð12Þ
Swi
where C, p and q are the statistical model parameters, and their
values can be calibrated by using the laboratory NMR and
resistivity experimental results. Considering the relationship
between saturation exponent and rock pore structure, the values
of p and q should be negative.
In this study, the data sets of laboratory resistivity experiments
and NMR measurements listed in Table 1 are reused; the parameters of C, p and q involved in Eq. (12) are calibrated, and the
formula of estimating saturation exponent from NMR logs in the
Xujiahe low permeability sands of Sichuan basin is expressed as
1−Swi −0:156 −0:0215
n ¼ 2:548
T 2 lm ;
Swi
Correlation coefficient : 0:79
ð11Þ
where Swi is the estimated irreducible water saturation from NMR
logs by using the optimal T2 cutoff in fraction, T2min is the
minimum horizontal relaxation time, T2max is the maximum
horizontal relaxation time, T2cutoff is the defined optimal T2 cutoff,
and their unit is microsecond. S(T) is the porosity distribution
function, which is associated with the T2 relaxation time.
Fig. 10 shows the statistical histogram of T2cutoff for all 36 core
samples listed in Table 1; it can be observed that the main
distribution of T2 cutoffs is 17.0–24.0 ms, and T2 cutoff with the
ð13Þ
Once this technique is extended to field applications, precise
saturation exponent can be consecutively calculated by using
Eq. (13) in the intervals with which field NMR logs were acquired.
4. Reliability verification
To verify the credibility of the proposed model in calculating
cementation and saturation exponents in low permeability sands,
comparisons of cementation exponents obtained from core samples and calculated by using Eq. (9), and of saturation exponents
48
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
14
15
12
12
Frequency
Frequency
10
9
6
8
6
4
3
2
0
0
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
Absolute for calculated cementation exponent
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
Absolute error for calculated saturation exponent
Fig. 11. (a) Comparison of cementation exponents obtained from the core samples and estimated from NMR logs by using the proposed technique. (b) Comparison of
saturation exponents obtained from core samples and estimated from NMR logs by using the proposed technique.
obtained from core samples and calculated by using Eq. (13) are
plotted in Fig. 11a and b, separately. From these two comparisons,
it can be observed that the predicted cementation and saturation
exponents from NMR logs by using the proposed technique are
close to the laboratory measured results; the absolute error of the
estimated cementation exponents is low than 0.08, and the
absolute error of the calculated saturation exponents is lower
than 0.2. From Fig. 11a and b, it can be concluded that the
proposed techniques of estimating cementation and saturation
exponents are reliable.
5. Case studies
The technique for estimating cementation and saturation
exponents proposed in this study is based on the laboratory
NMR measurements, and if it is extended to field low permeability
sandstone reservoirs, consecutive cementation and saturation
exponents can be obtained in the intervals with which field
NMR logs were acquired, and precise water saturation can be
estimated. Some field examples of estimating water saturation are
showed in the following section.
5.1. Estimation of water saturation in low permeability sandstone
reservoirs of Sichuan basin
With the calibrated Eqs. (9) and (13) by using 36 core samples
drilled from the Xujiahe Formation of Sichuan basin, southwest
China, several wells in Sichuan basin are processed and water
saturations are estimated. Fig. 12 shows a field example of well A
from Sichuan basin. In the first track, the displayed curves are
gamma ray (GR), spontaneous potential (SP) and borehole diameter (CAL), and their contribution is effective formation indication. The second track is depth and its unit is meter. RLLD
displayed in the third track is deep lateral resistivity, and RLLS is
shallow lateral resistivity. In the fourth track, we show density log
(RHOB), compensated neutron log (NPHI) and interval transit time
log (DT). They are used for porosity estimation. T2_distribution
displayed in the fifth track is field NMR spectrum which was
acquired from Halliburton's MRIL-C tool, and T2cutoff is the
optimal T2 cutoff that is obtained from the statistical histogram
of core derived results for 36 core samples, and its value is
20.75 ms; T2 lm is the logarithmic mean of the NMR T2 spectrum.
The sixth track of Fig. 12 indicates a reasonable match between
core analyzed porosity (CPOR) and NMR derived porosity (PHIT).
In the Xujiahe Formation, there are collapsed boreholes, and the
conventional logs (especially density log) are heavily affected,
whereas, the effects of collapsed boreholes to NMR porosity
acquired from MRIL-C tool can be ignored due to the centralized
measurement pattern. It suggests that PHIT is reliable and little
error will be introduced when it is applied in water saturation
estimation. Calc_m displayed in the seventh track and Calc_n
showed in the eighth track are estimated cementation and
saturation exponents from field NMR logs by using the proposed
technique, separately. Core_m is the cementation exponent, and
Core_n is the saturation exponent, and they are all obtained from
laboratory resistivity measurements. These comparisons illustrate
that cementation and saturation exponents estimated from field
NMR logs are close to the core analyzed results. In the meanwhile,
from the displayed NMR T2 spectrum in the interval of xx24–
xx60 m, it can be observed that the proportion of macropore
components decrease; this means reservoir pore structure is
poorer than that obtained in the above interval of xx20–xx80 m,
while their NMR total porosities are proximate. Hence, the
estimated cementation exponents from NMR total porosity
in these two layers are similar, while the calculated saturation
exponents in these two intervals are quite different. The estimated
saturation exponents are high in the intervals with poor pore
structure, and they all match with the core analyzed results
very well. The ninth track displays the comparisons of water
saturations obtained from three different methods. Sw is the
water saturation estimated from conventional logs by using
variable cementation and saturation exponents, Sw_cal is calculated by using fixed cementation and saturation exponents,
while Core_Sw is the analyzed water saturation by using the
sealed coring method. From these comparisons, it can be observed
that water saturation estimated by using variable cementation
and saturation exponents matches with the core analyzed results
very well in the whole interval, while water saturation estimated
by using fixed cementation and saturation exponents is underestimated. This means that the proposed technique of estimating
cementation and saturation exponents from field NMR logs in
this study is practicable in the low permeability reservoirs of
Xujiahe Formation in Sichuan basin. Additionally, if we observe
the value of Sw_cal, the whole interval is considered as gas bearing
formation besides a thin barrier bed of xx80–xx81 m, while
from Sw, we can observe that the upper 100 m interval of
xx22–xx24 m is gas bearing formation, and the lower interval of
xx24–xx56 m is not effective gas bearing formation, and considering the relatively high resistivity and poor pore structure, it is
identified as a dry layer. This identification is confirmed by drill
stem testing data.
5.2. Estimation of water saturation in low permeability sandstone
reservoirs of Ordos basin
To verify the wide applicability of the proposed technique in
this study, we applied it to Chang 8 Formation of Ordos basin,
northwest China, which is another typical low permeability
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
49
Fig. 12. Comparison of water saturations estimated by using variables m and n, fixed m and n, and obtained from the core samples in the Xujiahe Formation of Sichuan basin,
southwest China.
Fig. 13. Comparison of water saturation estimated by using the proposed technique and obtained from the core samples in the Chang 8 Formation of Ordos basin,
northwest China.
sandstone reservoir in China. To calibrate the used parameters in
the formulae of estimating cementation and saturation exponents
from NMR logs, 20 core samples, which were drilled from Chang
8 Formation, are applied for laboratory resistivity and NMR
measurements. From the laboratory NMR measurements, the
optimal T2 cutoff is determined as 18.05 ms, and the corresponding irreducible water saturation (Swi) is predicted. Combining the
experimental results and the predicted Swi, formulae of estimating
50
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
cementation and saturation exponents are calibrated, and they are
expressed respectively as
F¼
1
;
φ0:78 logðφÞþ2:69
n ¼ 2:361
1−Swi
Swi
Correlationcoefficient : 0:99
−0:0605
ð14Þ
T −0:111
2 lm ;
Correlation coefficient : 0:81
Field examples from two different low permeability sandstone
reservoirs in China show that the proposed technique of estimating cementation and saturation exponents from field NMR logs is
applicable widespread, and after they are used for water saturation calculation, precise results can be obtained. This will be of
great importance in improving the level of exploration and
reducing the risk of development in low permeability sandstones
reservoirs.
ð15Þ
A field example of well B from Ordos basin is processed by
using Eqs. (14) and (15), and water saturation is calculated, as
shown in Fig. 13. In well B, no core samples are drilled for
laboratory resistivity experiments. Comparisons of water saturation calculated by using the obtained variable cementation and
saturation exponents, fixed cementation and saturation exponents
and derived from core samples are showed in the last track. It can
be observed that the estimated water saturation by using variable
cementation and saturation exponents is consistent with the core
derived result very well, and this ensures that the estimated
cementation and saturation exponents from field NMR logs are
dependable, and the proposed technique and models are valuable,
while the calculated water saturation by using fixed cementation
and saturation exponents is overestimated.
These two field examples from different basins illustrate that
water saturation cannot be precisely calculated from conventional
logs by defining fixed cementation and saturation exponents.
However, once the proposed technique in this study is introduced,
credible cementation and saturation exponents can be effectively
estimated from field NMR logs, and accurate water saturation can
be calculated in low permeability sandstones with complicated
pore structure.
6. Conclusions
In low permeability sandstone reservoirs, the relationship
between porosity and formation factor, water saturation and
resistivity index cannot be expressed by power functions due to
the complicated pore structure. Fixed rock resistivity parameters
cannot be obtained from laboratory resistivity experiments for
precise water saturation estimation. To calculate water saturation
by using Archie's equation as accurate as possible, variable
cementation and saturation exponents should be first estimated.
Based on the analysis of the laboratory resistivity experiments,
the relationship of quadratic function between porosity and
formation factor is established, and a technique of estimating
variable cementation exponent from porosity is proposed.
Saturation exponent is proportional to the proportion of small
pore components and inversely proportional to T2 lm. Irreducible
water saturation (Swi) calculated by using the optimal T2 cutoff can
be used to characterize the proportion of small pore components.
A novel technique, which connects saturation exponent, Swi,
(1−Swi) and T2 lm, is proposed to estimate saturation exponent
from NMR logs, and the corresponding model is established.
Parameters mentioned in this model need to be calibrated by
using the laboratory resistivity experiments and NMR measurements that are obtained from the target core samples in different
reservoirs.
The predicted cementation and saturation exponents from field
NMR logs by using the proposed model in this study are credible,
and they are all close to the core derived results. The absolute
errors of these two kinds of cementation exponents are lower than
0.08, and the absolute errors of these two kinds of saturation
exponents are lower than 0.2. These ensure the proposed techniques and models are reliable.
Acknowledgments
We sincerely acknowledge the anonymous reviewers whose
correlations and comments have greatly improved the manuscript.
This research work was supported by the China Postdoctoral
Science Foundation funded project (No. 2012M520347,
2013T60147), National Science and Technology Major Project
(No. 2011ZX05044), the Fundamental Research Funds for the
Central Universities, China (2652013036) and Open Fund of Key
Laboratory of Geo-detection (China University of Geosciences,
Beijing), Ministry of Education (No. GDL1204).
References
Archie, G.E., 1942. The electrical resistivity log as an aid in determining
some reservoir characteristics. Trans. Am. Inst. Min. Metall. Pet. Eng. 146,
54–61.
Clavier, C., Coates, G., Dumanoir, J., 1977. Theoretical and experimental bases
for the dual-water model for interpretation of shaly sands. SPE J. 24 (2),
153–168.
Coates, G.R., Xiao, L.Z., Primmer, M.G., 2000. NMR Logging Principles and Applications. Gulf Publishing Company, Houston, USA, pp. 1–256.
Givens, W.W., 1986. Formation factor, resistivity index, and related equations based
upon a conductive rock matrix model (CRMM). In: Paper P Presented at the
27th SPWLA Annual Logging Symposium.
Givens, W.W., 1987. A conductive rock matrix model (CRMM) for the analysis of
low-contrast resistivity formations. Log Anal. 28 (2), 138–151.
Givens, W.W., Schmidt, E.J., 1988. A generic electrical conduction model for lowcontrast resistivity sandstones. In: Paper E Presented at the 29th SPWLA Annual
Logging Symposium.
Liu, X.P., Hu, X.X., Xiao, L., 2012. Effects of pore structure to electrical properties
in tight gas reservoirs: an experimental study, In: Paper SPE150926 presented
at the SPE/EAGE European Unconventional Resources Conference and
Exhibition.
Mao, Z.Q., Zhang, C.G., Lin, C.Z., Ouyang, J., Wang, Q., Yan, C.J., 1995. The effects of
pore structure on electrical properties of core samples from various sandstone
reservoirs in Tarim basin. In: Paper LL presented at the 36th SPWLA Annual
Logging Symposium.
Mao, Z.Q., Tan, T.D., Lin, C.Z., Wang, Q., 1997. The laboratory studies on pore
structure and electrical properties of core samples fully-saturated with brine
water. Acta Pet. Sin. 18 (3), 51–55 (in Chinese).
Mao, Z.Q., Xiao, L., Wang, Z.N., Jin, Y., Liu, X.G., Xie, B., 2013. Estimation of
permeability by integrating nuclear magnetic resonance (NMR) logs with
mercury injection capillary pressure (MICP) data in tight gas sands. Appl.
Magn. Reson. 44 (4), 449–468.
Rasmus, J.C., Kenyon, W.E., 1985. An improved petrophysical evaluation of Oomoldic
Lansing-kansas city formations utilizing conductivity and dielectric log
measurements. In: Paper V Presented at the 26th SPWLA Annual Logging
Symposium.
Rasmus, J.C., 1987. A summary of the effects of various pore geometries and their
wettabilities on measured and in-situ values of cementation and saturation
exponents. Log Anal. 38 (2), 152–164.
Schlumberger Well Services, 1986. Log Interpretation Charts. Schlumberger Well
Services. pp. 1–112.
Shi, Y.J., Li, G.R., Zhou, J.Y., 2008. Study on litho-electric character and saturation
model of argillaceous low-permeability sandstone reservoir. Well Logg. Technol.
32 (3), 203–206, In Chinese.
Straley, C., Rossini, D., Vinegar, H., Tutunjian P., Morriss, C., 1994. Core Analysis by
Low Field NMR. The Society of Core Analysts, SCA-9404.
Suman, R.J., Knight, R.J., 1997. Effects of pore structure and wettability on the
electrical resistivity of partially saturated rocks—a network study. Geophysics
62 (4), 1151–1162.
Sweeney, S.A., Jennings, H.Y., 1960. Effect of wettability on the electrical resistivity of carbonate rock from a petroleum reservoir. J. Phys. Chem. 64 (5),
551–553.
L. Xiao et al. / Journal of Petroleum Science and Engineering 108 (2013) 40–51
Wang, Y.M., Sharma, M.M., 1988. A network model for the resistivity behavior of
partially saturated rocks. In: Paper G presented at the 29th SPWLA Annual
Logging Symposium.
Waxman, M.H., Smits, L.J.M., 1968. Ionic double-layer conductivity in oil-bearing
shaly sands. SPE Form. Eval. 4 (1), 20–32.
Waxman, M.H., 1974. Electrical conductivities in shaly sands: (I) the relation
between hydrocarbon saturation and resistivity index; (Ⅱ) the temperature
coefficient of electrical conductivity. J. Pet. Technol. 26 (3), 213–225.
51
Wyllie, M.R.J., Gregory, A.R., Gardner, L.W., 1956. Elastic waves velocities in
heterogeneous and porous media. Geophysics 21 (1), 41–70.
Xiao, L., Mao, Z.Q., Li, G.R., Jin, Y., 2012. Calculation of porosity from nuclear
magnetic resonance and conventional logs in gas-bearing reservoirs. Acta
Geophys. 60 (5), 1030–1042.
Xiao, L., Mao, Z.Q., Li, G.R., Jin, Y., 2013. Estimation of saturation exponent from
nuclear magnetic resonance (NMR) logs in low permeability reservoirs. Appl.
Magn. Reson. 44 (3), 333–347.