Mavko G., Mukerji T., Dvorkin J. The Rock Physics Handbook
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Uses
These results can be used to estimate the geometric relations of the packing of
granular materials, which is useful for understanding porosity, transport properties,
and effective elastic constants.
Assumptions and limitations
The preceding results assume idealized spheres.
5.2 Thomas–Stieber model for sand–shale systems
Synopsis
Thomas and Stieber (1975, 1977) developed a model for the porosity of thinly bedded
sands and shales, under the assumption that all rocks in the interval, including dirty
sands, can be constructed by mixing clean, high-porosity sand and low-porosity shale
(see also Pedersen and Nordal, 1999). Figure 5.2.1 illustrates some of the mixtures
1
2
3
4
5
6
Shale
Clean sand
Sand with some
dispersed shale
Sand with pore
space completely
filled with shale
Shale with
dispersed quartz
grains
Sand with
structural shale
clasts
Figure 5.2.1 Laminations of shale, clean sand, sand with dispersed shale, and sand with
structural shale clasts.
237 5.2 Thomas–Stieber model for sand–shale systems
considered, and Figure 5.2.2 shows the associated porosities. The approach to ana-
lyzing porosity can be illustrated as a ternary diagram with the following limiting
points and boundaries (Figure 5.2.2):
1. Shale (layer 1 in Figure 5.2.1; point C in Figure 5.2.2), with total porosity
f
total
¼f
shale
.
2. Clean sand (layer 2 in Figure 5.2.1; point A in Figure 5.2.2), with total porosity
total
¼
clean sand
.
3. Sand with some dispersed shale in the sand pore space (layer 3 in Figure 5.2.1;
line A–B in Figure 5.2.2), with total porosity
total
¼
clean sand
1
shale
ðÞV
disp shale
where V
disp shale
is the volume fraction of microporous dispersed shale (clay) in the
sand. It is assumed that the sand packing is undisturbed by the shale in the pore
space, as long as V
disp shale
clean sand
. The total porosity is reduced by the solid
fraction of the pore-filling shale, ð1
shale
ÞV
disp shale
. This is sometimes called an
ideal binary mixture model (Cumberland and Crawford, 1987; Marion et al., 1992)
and corresponds to the limiting case of very large particles (sand) mixed with very
small particles (clay).
4. Sand with pore space completely filled with shale (layer 4 in Figure 5.2.1; point
BinFigure 5.2.2), with total porosity
Porosity
A
Structural shale
Increasing shale laminations
N/G =0.8 N/G =0.6
N/G =0.4
N/G =0.2
C
B
D
E
F
V = 0.2f
clean
sand
0.4f
clean
sand
0.6f
clean
sand
0.8f
clean
sand
V
disp shale
= f
clean sand
Dispersed shale
0% 100%
f
shale
Shale volume
disp
shale
φ
clean
sand
f
clean
sand
f
disp
shale
Figure 5.2.2 Porosity versus shale volume in the Thomas–Stieber model. V
disp shale
is the volume of
dispersed shale in the sand pore space. The minimum porosity occurs when the sand porosity is
completely filled with shale, V
disp shale
¼ F
clean sand
. Net-to-gross, N/G, is the thickness fraction of
sand in the laminated composite.
238 Granular media
total
¼
clean sand
shale
In this case V
disp shale
¼
clean sand
.
5. Shale with dispersed quartz grains (layer 5 in Figure 5.2.1; line B–C in
Figure 5.2.2), with total porosity
total
¼ V
shale
shale
In this case regions of microporous shale are replaced by solid quartz grains,
reducing the total porosity.
6. An additional case falls outside the triangle A–B–C: sand with structural shale
clasts (layer 6 in Figure 5.2.1; line A–D in Figure 5.2.2), with porosity
total
¼
clean sand
þ
shale
V
shale clast
In this case, it is assumed that some of the solid quartz grains are replaced by
porous shale clasts, resulting in a net increase in total porosity (see also Florez,
2005).
A laminated system of rocks has total porosity
total
¼
X
i
X
i
i
where f
i
is the total porosity of the ith layer and X
i
is the thickness fraction of the ith
layer. For a laminated sequence of shale and dirty sand (sand with dispersed shale),
the total porosity is
¼ N=G
clean sand
1
shale
ðÞV
disp shale
þ 1 N=GðÞ
shale
where net-to-gross, N/G, is the thickness fraction of sand layers. N/G is not identical to
the shale fraction, since some dispersed shale can be within the sand. The laminated
systems fall within the triangle A–B–C in Figure 5.2.2. As mentioned above, the line
A–C corresponds to laminations of clean sand and shale. Points inside the triangle
correspond to laminations of shale with dirty sand (sand with dispersed shale).
Figure 5.2.3 is similar to Figure 5.2.2 except that the horizontal axis is the gamma
ray response of the thinly laminated system. A practical workflow is to construct the
model curves shown in Figure 5.2.3 and then superimpose well-log-derived total
porosity and gamma ray data for interpretation to identify laminar, dispersed, and
structural shale. For example, point E in Figures 5.2.2 and 5.2.3 can be interpreted as
a laminar sequence where the net-to-gross is 60%. Dispersed clay fills 40% of the
pore space in the sand layers.
Figure 5.2.4 is similar to Figure 5.2.2, except that the vertical axis now represents
bulk density. Table 5.2.1 summarizes total porosity and density at several key
locations in the model.
239 5.2 Thomas–Stieber model for sand–shale systems
Density
A
Structural shale
Increasing shale laminations
N/G = 0.8
N/G = 0.6
N/G = 0.4
N/G = 0.2
C
B
D
E
F
V
disp
= 0.2f
clean
shale
sand
0.4f
clean
sand
0.6f
clean
sand
0.8f
clean
sand
Dispersed shale
0% 100%
V
disp shale
= f
clean sand
Shale volume
r
clean
sand
r
shale
Figure 5.2.4 Bulk density versus V
shale
in the Thomas–Stieber model, similar to Figure 5.2.2.
GR
clean
+ f
clean
sand
sand
f
clean
f
disp
sand shale
f
clean
sand
φ
shale
Porosity
A
F
Structural shale
Increasing shale laminations
N/G =0.8
N/G =0.6
N/G =0.4
N/G =0.2
C
B
D
E
=0.2f
clean
sand
V
disp
shale
0.4f
clean
sand
0.6f
clean
sand
0.8f
clean
sand
V
disp shale
= f
clean sand
Dispersed shale
GR
clean
sand
GR
shale
Gamma Ray
(GR
shale
−GR
clean
sand
)
Figure 5.2.3 Porosity versus gamma ray in the Thomas–Stieber model, similar to Figure 5.2.2.
240 Granular media
Table 5.2.1 Total porosity and bulk density at key locations in Figures 5.2.2 and 5.2.4, corresponding
to the Thomas–Stieber model.
Total porosity Bulk density
Point A
clean
sand
clean
sand
Point C
shale
shale
Line A–C ¼ N=G
clean
sand
þð1 N=GÞ
shale
N=G
clean
sand
þð1 N=GÞ
shale
Point B
clean
sand
shale
clean
sand
þ
clean
sand
ð
shale
water
Þ¼
clean
sand
shale
þð1
clean
sand
quartz
Line A–B
total
¼
clean
sand
ð1
shale
ÞV
disp
shale
clean
sand
þ V
shale
ð
shale
water
Þ
Line B–C
total
¼ V
shale
shale
V
shale
shale
þð1 V
shale
Þ
quartz
Notes:
clean sand
,
shale
¼ porosities of clean sand and shale; N/G ¼ fractional thickness of sand layers; V
disp shale
¼ volume fraction of dispersed shale in sand;
V
shale
¼ total volume fraction of shale in the laminated stack; and, r
clean sand
, r
shale
, r
water
, r
quartz
¼ densities of clean sand, water-saturated shale, water,
and quartz.
Assumptions and limitations
There are several important assumptions made in the Thomas–Stieber model:
clean sand is homogeneous with constant porosity;
shale is the only destroyer of porosity; i.e., reduction of porosity by cementation
and/or sorting is ignored; and
the shale content in sands is primarily detrital, having essentially the same pro-
perties as the shale laminae.
Extensions
The Thomas–Stieber model focuses on volumetrics. Yin (1992) and Marion et al.
(1992) developed a model for predicting seismic velocities in the same dispersed clay
systems. The Backus average allows an exact prediction of elastic properties of a
thinly laminated composite in terms of the individual layer properties.
5.3 Particle size and sorting
Synopsis
Particle size
Particle size is one of the most fundamental parameters of clastic rocks. The Udden–
Wentworth scale uses a geometric progression of grain diameters in millimeters,
d ¼1, 2, 4, 8, 16, . . ., and divides sediments into seven grain-size grades: clay, silt,
sand, granules, pebbles, cobbles, and boulders. The Udden–Wentworth scale is
summarized in Table 5.3.1. The third column in the table is a subclassification:
vc ¼very coarse, c ¼coarse, m ¼medium, f ¼fine, and vf ¼very fine. Krumbein
introduced a logarithmic transformation of the Udden–Wentworth scale, known as
the phi-scale, which is the most used measure of grain size:
¼log
2
d
A lesser-known description is the psi-scale, c ¼–f.
Sorting
Sorting is the term that geologists use to describe the size distributions that are a
fundamental property of all naturally occurring packings of particles or grains.
Sorting is no more than a measure of the spread of the grain size s, and there are
standard statistical methods to describe the spread of any population. Two of
the main parameters to measure the spread of a population are the standard
deviation and the coefficient of variation (the standard deviation normalized by
the mean).
242 Granular media
In the phi-scale, sorting is usually expressed as being equal to the standard
deviation (Boggs, 2001). Sorting is often estimated graphically by plotting a frequency
histogram of grain sizes (measured by weight) expressed in the phi-scale, and normal-
ized by the total weight of grains (see Figure 5.3.1). The cumulative distribution
function (CDF) of grain sizes, ranging from 0% to 100%, is derived by successively
adding and smoothing the frequencies in the histogram (from small to large sizes). The
CDF makes it easy to identify the percentiles of the distribution. For example, the 25th
percentile, f
25
, is the value of f such that 25% of the grains by weight are smaller.
Folk and Ward (1957) suggested various measures of the grain size distribution that
can be computed from the graphically determined percentiles (after Tucker, 2001):
Median MD ¼
50
Mean M ¼
16
þ
50
þ
84
3
Standard deviation ¼
84
16
4
þ
95
5
6:6
Table 5.3.1 Grain sizes and classes, according to
the Udden–Wentworth scale.
Diameter (mm) f Class
4096 12 Block
2048 11 vc Boulder
1024 10 c
512 9m
256 8f
128 7 c Cobble
64 6f
32 5 vc Pebble
16 4c
8 3m
4 2f
2 1 Granule
1 0 vc Sand
0.50 1 c
0.25 2 m
0.125 3 f
0.063 4 vf
0.031 5 c Silt
0.015 6 m
0.008 7 f
0.004 8 vf
Clay
243 5.3 Particle size and sorting
Skewness SK ¼
16
þ
84
2
50
2ð
84
16
Þ
þ
5
þ
95
2
50
2ð
95
5
Þ
Statistics such as these, measured in phi units, should have f attached to them. The
Folk and Ward formula for phi standard deviation, sf, follows from assuming a
log-normal distribution of grain diameters in millimeters, which corresponds to a
normal distribution of sizes in f. If grain sizes have a normal distribution, then 68%
of the grains fall between 1 standard deviation, or between f
16
and f
84
. Then we
can write
68
¼ð
84
16
Þ=2. Similarly, in a normal distribution, 90% of the
grains fall between 1.65 standard deviations, or between f
05
and f
95
. Hence,
90
¼ð
95
05
Þ=3:30. The Folk and Ward phi standard deviation is the average
of these two estimates: ¼½ð
84
16
Þ=2 þð
95
05
Þ=3:30=2. The sorting
constant S
0
is equal to the phi standard deviation:
S
0
¼
84
16
4
þ
95
5
6:6
84
16
2
It is interesting to convert this expression back to the millimeter scale:
S
0
84
16
2
) 2
S
0
¼
ffiffiffiffiffiffi
d
16
d
84
r
Another popular way to express the variation of the population in millimeters is using
the coefficient of variation, which is the standard deviation divided by the mean.
It is worth pointing out that many serious sedimentologists have studied distribu-
tions of grain sizes using standard statistical methods, without attempting to define or
redefine the term sorting.
Frequency (%)
φ
φ
CDF
25th
percentile
−10123456
−10123456
100
90
80
70
60
50
40
30
20
10
0
50
40
30
20
10
0
Figure 5.3.1 Frequency histogram of grain sizes (left) and the corresponding cumulative
distribution function (after Tucker, 2001).
244 Granular media
5.4 Random spherical grain packings: contact models
and effective moduli
Synopsis
Contact stiffnesses and effective moduli
The effective elastic properties of packings of spherical particles depend on normal
and tangential contact stiffnesses of a two-particle combination, shown schematically
in Figure 5.4.1. The normal stiffness of two identical spheres is defined as the ratio of
a confining force increment to the shortening of a sphere radius. The tangential
stiffness of two identical spheres is the ratio of a tangential force increment to the
increment of the tangential displacement of the center relative to the contact region:
S
n
¼
@F
@d
S
t
¼
@T
@
where S
n
and S
t
are the normal and tangential stiffnesses, respectively, F is the
normal force, T is the tangential force, d is the normal displacement, and t is the
tangential displacement.
For a random sphere packing, effective bulk and shear moduli can be expressed
through porosity f, coordination number C (the average number of contacts per
sphere), sphere radius R, and the normal and tangential stiffnesses, S
n
and S
t
,
respectively, of a two-sphere combination:
K
eff
¼
Cð1 Þ
12pR
S
n
m
eff
¼
Cð1 Þ
20pR
ðS
n
þ 1:5S
t
Þ
The effective P- and S-wave velocities are (Winkler, 1983)
V
2
P
¼
3C
20p R
S
n
þ
2
3
S
t
t
F
F
T
T
R − d
R
Figure 5.4.1 Normal and tangential displacement in a two-particle system.
245 5.4 Random spherical grain packings
V
2
S
¼
C
20p R
S
n
þ
3
2
S
t
V
2
P
V
2
S
¼ 2
3S
n
=S
t
þ 2
2S
n
=S
t
þ 3
where r is the grain-material density. Wang and Nur (1992) summarize some of the
existing granular media models.
Observed values of the coordination number relative to the porosity are discussed
in Section 5.1.
The Hertz–Mindlin Model
In the Hertz model of normal compression of two identical spheres, the radius of the
contact area, a, and the normal displacement, d, are
a ¼
3FR
8m
ð1 nÞ
1=3
; d ¼
a
2
R
where m and v are the shear modulus and the Poisson ratio of the grain material,
respectively.
If a hydrostatic confining pressure P is applied to a random, identical-sphere
packing, the confining force acting between two particles is
F ¼
4p R
2
P
Cð1 Þ
Then
a ¼ R
3pð1 nÞ
2Cð1 Þm
P
1=3
The normal stiffness is
S
n
¼
4ma
1 n
The effective bulk modulus of a dry, random, identical-sphere packing is then
K
eff
¼
C
2
ð1 Þ
2
m
2
18p
2
ð1 nÞ
2
P
"#
1=3
Mindlin (1949) shows that if the spheres are first pressed together, and a tangential
force is applied afterward, slip may occur at the edges of the contact (Figure 5.4.1).
The extent of the slip depends on the coefficient of friction between the contacting
246 Granular media