Stress-dependence of the permeability and porosity of sandstone and shale from TCDP Hole-A

International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

Contents lists available at ScienceDirect

International Journal of
Rock Mechanics & Mining Sciences
journal homepage: www.elsevier.com/locate/ijrmms

Stress-dependence of the permeability and porosity of sandstone and shale
from TCDP Hole-A
Jia-Jyun Dong a,n, Jui-Yu Hsu a, Wen-Jie Wu a, Toshi Shimamoto b, Jih-Hao Hung c, En-Chao Yeh d,
Yun-Hao Wu c, Hiroki Sone e
a

Graduate Institute of Applied Geology, National Central University, No. 300, Jungda Road, Jungli, Taoyuan 32001, Taiwan
Department of Earth and Planetary Systems Science, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan
c
Institute of Geophysics, National Central University, Jungli, Taoyuan, Taiwan
d
Department of Geosciences, National Taiwan University, Taipei, Taiwan
e
Department of Geophysics, Stanford University, California, USA
b

a r t i c l e in fo

abstract

Article history:
Received 7 October 2009
Received in revised form

13 April 2010
Accepted 28 June 2010
Available online 10 July 2010

We utilize an integrated permeability and porosity measurement system to measure the stress
dependent permeability and porosity of Pliocene to Pleistocene sedimentary rocks from a 2000 m
borehole. Experiments were conducted by first gradually increasing the confining pressure from 3 to
120 MPa and then subsequently reducing it back to 3 MPa. The permeability of the sandstone remained
within a narrow range (10 À 14–10 À 13 m2). The permeability of the shale was more sensitive to the
effective confining pressure (varying by two to three orders of magnitude) than the sandstone, possibly
due to the existence of microcracks in the shale. Meanwhile, the sandstone and shale showed a similar
sensitivity of porosity to effective pressure, whereby porosity was reduced by about 10–20% when the
confining pressure was increased from 3 to 120 MPa. The experimental results indicate that the fit of
the models to the data points can be improved by using a power law instead of an exponential
relationship. To extrapolate the permeability or porosity under larger confining pressure (e.g. 300 MPa)
using a straight line in a log–log plot might induce unreasonable error, but might be adequate to predict
the stress dependent permeability or porosity within the experimental stress range. Part of the
permeability and porosity decrease observed during loading is irreversible during unloading.
& 2010 Elsevier Ltd. All rights reserved.

Keywords:
Permeability
Porosity
Specific storage
Effective confining pressure
Stress history

1. Introduction
Rock permeability, porosity and storage capacity are key fluid
flow properties. Precise knowledge of these parameters is crucial

for modeling fluid percolation in the crust [1–14]. Based on
laboratory work, the stress dependent permeability and porosity
of rocks and fault gouge are well documented [10,15–27], and are
postulated to be described by an exponential relationship
[10,15,19,21,28–31]. However, Shi and Wang [4] suggested that
the relationship between effective stress and permeability of fault
gouge should follow a power law, based on the laboratory
permeability measurements of Morrow et al. [18]. Therefore, the
stress dependent model of fluid flow properties for rock is still a
controversial issue [10]. Furthermore, it is well recognized that
the permeability and porosity is dependent not only on the
current loading condition, but also on the stress history within a
sedimentary basin [32]. The influence of the stress history for
deriving stress dependent models of permeability and porosity

n

Corresponding author at: Tel./fax: +886 3 4224114.
E-mail address: (J.-J. Dong).

1365-1609/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijrmms.2010.06.019

requires further study. In addition, surface rock samples are
frequently used when determining the fluid flow properties of
rocks in the laboratory. However, surface rock samples may be
altered by weathering processes and thus the experimental
results from surface rocks may differ from the values obtained
from drill holes [33]. As a result, fresh samples free from the
effects of weathering are preferable for the derivation of fluid flow

properties, although stress relief induced fractures are occasionally observed in both surface and borehole samples.
A deep drilling project (Taiwan Chelungpu fault Drilling
Project, TCDP) was conducted in the Western Foothills of Taiwan,
which is known to be a classic fold-and-thrust belt. The aim in
this study is to measure the fluid flow properties in sedimentary
rock samples from cores from TCDP Hole-A (2 km in depth). An
integrated permeability and porosity laboratory measurement
system was utilized to determine the permeability and porosity of
fresh core samples under different effective confining pressures.
Representative rock samples from depths of 900–1235 m were
selected. The samples included Pliocene to Pleistocene sandstone
and silty-shale. The maximum applied effective confining pressure was about 120 MPa, which roughly equals the effective
overburden of Cenozoic sediments in the Taiwan region


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J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

(a thickness of about 8 km [34], assuming a hydrostatic pore
pressure). From the experimental results, exponential and power
law relationships for describing the effective pressure dependency of permeability and porosity were compared, and their
corresponding parameters were determined. The specific storage
of the tested rocks, which is an important input for fluid flow
analysis [4,5], was computed based on the measured stress
dependent porosity. Based on the laboratory measurements, the
influence of stress history on the permeability and porosity of
rocks is discussed. In addition, the relationship between permeability and porosity of sandstone and shale, induced by mechanical compaction, is elucidated. Finally, a simplified form of the
power law model is suggested for describing the stress dependent
specific storage of the tested sedimentary rocks.

2. Description of the rock samples
The Taiwan Chelungpu fault Drilling Project (TCDP) was
conducted in order to further understand the faulting mechanics
of a large thrust earthquake, the 1999 Taiwan Chi-Chi earthquake.
Two deep holes (Hole-A and Hole-B; ground surface elevation
247 m) were drilled in Dakeng, Taichung City, western Taiwan.
The location of the Dakeng well (Hole-A) is shown in the general
geological map, along with the interpreted structural profile
across the Dakeng well (Fig. 1). The holes were drilled through the
Chelungpu fault which was ruptured during the Chi-Chi
earthquake. Hole-A of the TCDP penetrates the Chelungpu and
Sanyi faults at 1111 and 1710 m depth, respectively [35]. The
hanging wall of the Chelungpu fault is comprised of the late
Pliocene to early Pleistocene Chinshui Shale and Cholan
Formations. The boundary of the Cholan Formation and
Chinshui Shale occurs at a depth of 1013 m. Below 1111 m, a
thrust fault displacing the Chinshui Shale and early Pliocene
Kueichulin Formation over the Cholan Formation was observed at
a depth of 1710 m. Furthermore, the boundary of the Chinshui
Shale and the Kueichulin Formation was determined to be at a
depth of 1300 m. The Cholan Formation reappears as the footwall
of the Sanyi fault, as observed in cores taken from below 1710 m
depth to the end of Hole-A (2000 m deep). Although the pore
pressure during drilling was not measured, the mud pressure
profile was calculated based on the mud log. The profile showed a
hydrostatic distribution, indicating that during the drilling period
(5–6 years after the Chi-Chi earthquake), there was no
overpressure around the drill site. The detailed geological
setting of the Chelungpu thrust system in Central Taiwan has

been described in [36].
Our rock samples were taken from depths (below the ground
surface of Hole-A) between 800 and 1300 m. The mean effective
stress at 1112 m is estimated via the anelastic strain recovery
method to be about 13 MPa [37]. The rock samples are identified
as being from the lower Cholan Formation and upper Chinshui
Shale at depths of about 3500 m. The maximum vertical effective
stress of the tested rocks is about 49 MPa, assuming a hydrostatic
pore pressure.
The Chinshui Shale is dominated by claystone with minor
amounts of siltstones and muddy sandstones [38]. The sedimentary structures indicate that the Chinshui Shale was deposited in
shallow marine and intercalated tidal environments [39]. The
shale is mainly comprised of silts with a clay fraction of about
25%. According to the classification method for fine-grained
clastic sediments proposed in [40], the tested shale samples can
be categorized as a silt-shale. Clay minerals are composed of a
mixture of illite (25%), chlorite (25%), kaolonite (4%), and
montmorillonite (17%) [41].

The Cholan Formation consists of a series of upward coarsening successions. Each succession is characterized by claystones at
its base, graded upwards into siltstone and very thick sandstone
beds at its top [38]. The sandstone in the Cholan Formation is
predominantly composed of monocrystalline and polycrystalline
quartz (50%), feldspars (1%), and sedimentary (42%) and metasedimentary (7%) lithic fragments [42]. The mean and effective grain
sizes (50% and 10% of the particles finer than the sizes on a grain
size diagram) are 0.06–0.09 mm (very fine sand) and 0.005–
0.03 mm, respectively. The grain shape of the sandstone in
the Cholan Formation is subangular to angular and the clay
content is generally less than 10% [43]. Based on the sedimentary
structures and fragment composition, the Cholan Formation

can be interpreted as having been deposited in a delta
environment [39].
The lithology between 800 and 1300 m, determined from the
core [44] and the Gamma-ray log [35], is shown in Fig. 1. Based on
the correlation between gamma-ray radiation and core-derived
lithology, we can set 75 and 105 API as the boundaries separating
clean sand, silt and pure clay [35]. For the gamma-ray-derived
lithology, the colors green, brown, and yellow represent shale,
siltstone, and sandstone, respectively. The locations of selected
rock cores are also marked in Fig. 1. In summary, all rock cores are
either fine- to very fine-grained sandstone (with grain diameters
of 0.06–0.2 mm) or silty-shale. These are the representative rock
types in the cores from TCDP Hole-A.
Samples selected for measurements were cored by a laboratory coring machine using 20 and 25 mm diamond cores (cooling
with water) and were shaped into cylinders with smooth ends by
polishing machine. Cylindrical axes of all samples were parallel to
the axes of rock cores from TCDP Hole-A. That is, the axis of the
cylindrical samples were inclined at about 301 with respect to the
normal to the bedding planes. However, only the relatively
homogeneous cores were selected, and cores with interbedded
layers were discarded. Prior to permeability and porosity
measurements, the samples were oven dried at 105 1C for more
than three days. Samples were prepared with care to minimize
the occurrence of microcracks during the experimental procedures. The rock type, sample size, and dry density of the tested
rock samples, along with their corresponding drilling depth in
TCDP Hole-A, are listed in Table 1. Meanwhile, the sandstone and
shale samples are shown in Fig. 2. Two samples (R351_sec2 and
R390_sec3) with sample lengths less than 3 mm were selected for
SEM observation after permeability experiments. A carbon coater
was used for coating the surface of the samples under vacuum

conditions.

3. Laboratory measurement system
In this study we utilized an integrated permeability/porosity
measurement system – YOYK2 – for measuring the fluid flow
properties of rock samples from TCDP Hole-A. The tests were
performed using an intra-vessel oil pressure apparatus at room
temperature. A pressure generator was used with the oil
apparatus to increase the confining pressure to 200 MPa. Fig. 3a
and b shows the permeability and porosity measurement
systems, respectively. Fig. 3c shows the sample assembly. For
permeability measurement, two porous spacers with grooves
were used to ensure even pore pressure distribution across the
sample width. The sample was jacketed in two heat shrinkable
polyolefin tubes of 1 mm in thickness.
The steady state flow method was employed to assess the
permeability of the rock samples. The intrinsic permeability under
a constant hydraulic gradient for a body of compressible gas
flowing at constant flow rate Q (steady state) can be calculated as


J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

1143

900
R255
R261

Cholan Formation

Depth (m)
800

R287
1000
R307

Ele. 247m

R316

1100
R351

Chi-Chi
Rupture

Epicenter of
R390

Chi-Chi earthquake
1200

Sandstone
Intensive bioturbation
Major sand/minor silt

Chinshui Shale

TCDP Hole-A

R437

Major silt/minor sand
Siltstone or shale
1300
Fig. 1. Location of the Dakeng well (Hole-A, elevation at 247 m); generalized geological map and interpreted structural profile across the Dakeng well. The lithology
between 800 and 1300 m below the ground surface of Hole-A, determined from the core and the Gamma-ray log, is summarized, and locations of selected rock samples are
marked. In the gamma ray-derived lithology, green, brown, and yellow represent shale, siltstone, and sandstone, respectively (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this article.)

follows [45]:
2Q mg L Pd
,
K¼
A ðPu2 ÀPd2 Þ

ð1Þ

where K denotes the permeability, mg represents the viscosity
coefficient of the gas, L and A are the length and cross-sectional
area of the core sample, and Pu and Pd denote the pore pressure in
the upper and lower ends of the sample (Fig. 3a). The pore
pressure in the upper end, Pu, was controlled by the gas regulator,
and was kept constant at a value between 0.2 and 2 MPa during

testing. The pore pressure at the lower end, Pd, was at atmospheric
pressure, which is assumed to be 0.1 MPa. The viscosity of the
nitrogen gas, mg, is 16.6 Â 10 À 6 Pa s. The flow flux of nitrogen gas

was measured using a digital gas flowmeter (ADM) which ranged
from 1.0 to 1000.0 ml/min. The precision of the flowmeter was
0.5 ml/min. To increase the precision for flow rate measurement,
the ADM flowmeter was calibrated using a high resolution VP-1
gas flowmeter (this was done at Kyoto University, Japan). Note
that the intrinsic permeability is not dependent on the pore fluid.
Therefore, the permeabilities measured by gas and by water


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J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

Table 1
Descriptions of rock samples for permeability and porosity measurement.
Sample number

Corrected depth (m)

Rock type

Dry density (g/cm3)

R255_sec2

902.68

Silty-shale

2.59

R261_sec2

915.24

Fine-grained sandstone

2.21

R287_sec1
R307_sec1
R316_sec1
R351_sec2
R390_sec3
R437_sec1

972.42
1009.62
1028.43
1114.33
1174.24
1232.46

Silty-shale
Fine-grained sandstone
Silty-shale
Silty-shale
Silty-shale
Silty-shale

2.58
2.18
2.60
2.59
2.66
2.58

a

Sample length/diameter (mm)
Permeability

Porosity

11.73/20.39
4.58/19.97
13.40/20.36
4.36/24.88
4.28/25.76
13.65/19.65
–
2.93/25.80
2.68/24.65
4.99/25.54

30.10/19.76
–
18.96/25.12
14.64/19.60
25.24/25.63

27.64/25.32
33.96/25.28
18.59/25.05
19.69/25.11
30.34/25.57

Formationa

CL
CL
CL
CL
CL
CL
CS
CS
CS
CS

CL: Cholan formation; CS: Chinshui Shale.

Fig. 2. Photographs of the sandstone and silty-shale of the late Pliocene to early Pleistocene Chinshui Shale and Cholan Formation. Fine-grained sandstone with mean grain
diameters of 0.06–0.09 mm from (a) R261_sec2 and (b) R307_sec1; and silty-shale with clay content less than 25% from (c) R255_sec2 and (d) 437_sec1.

should be identical. Some laboratory results do show that
the intrinsic permeability to gas is generally higher than that to
water [26,46]. The influence of this bias on the permeability
estimated following the stress dependent model will be discussed
later.
Rock sample porosities are calculated based on the balanced

pressure Pf attained when two airtight spaces with known initial
pressure (Pi1, Pi2) are connected (Fig. 3b). One of the airtight
spaces comprises a sample with an attached tube. The volume of
this space therefore includes : (1) the single tube volume (Vl),
which is linked to the sample (the volume of the tube between
valve #2 and the rock sample); and (2) the pore volume (Vp) of the
rock sample. The other space includes a tube system only and has
volume Vs (the volume of the tube between valve #1 and valve
#2). Since the two airtight spaces are isolated and the gas is
assumed to be ideal, the pressure multiplied by the volume

should remain constant after opening the valve between the two
spaces, and can be expressed as
Pi1 Vs þ Pi2 ðVl þ Vp Þ ¼ Pf ðVs þVl þVp Þ:

ð2Þ

If the volumes Vs and Vl can be determined in advance, the pore
volume of the sample can be calculated as follows:


Pi1 ÀPf
ð3Þ
Vs ÀVl :
Vp ¼
Pf ÀPi2
Consequently, the sample effective porosity f can be calculated from f ¼ Vp =Vt , where Vt is the sample volume. The volume
of the two isolated systems (volumes Vs and Vl) was minimized to
enhance the accuracy of the pore volume measurement. It is not
easy to ‘‘directly measure’’ the volumes Vs and Vl. Therefore,

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

1145

Pressure guage

Pressure gauge Pu

Valve #1

Valve #2

Pi1 , Vs

Steady state
gas flow (N2)

Pi2 ,Vl +Vp

Sample
Flowmeter
Sample
Q, Pd

Permeability

Porosity

1.

Lower piston (permeability)

2.

Spacer with a pore pressure
hole

8

7

11

3.

Porous spacers (permeability)

4.

Sample

5.

Pore-pressure-line connector

6.

Heat shrinkable flexible

polyolefin jacket

3
4
5

3
2

10

7.

Upper piston (permeability)

8.

Galvanized wire

9.

Lower piston (porosity)

10. Porous spacer (porosity)
1

6

9

11. Upper piston (porosility)

1
2

10

3

Fig. 3. (a) Permeability and (b) porosity measurement systems and equations for determining flow properties.

standard samples of hollow metal cylinders with known inner
diameters (the pore volume Vp for standard samples is a known)
were used to calibrate the volumes Vs and Vl indirectly. Two sets
of standard metal samples were used, with outer diameters of 20
and 25 mm. From the calibrated results, we see that Vl ¼0.625 ml
and Vs ¼3.135 ml for the 20 mm diameter sample, while Vl ¼0.604
ml and Vs ¼3.126 ml for the 25 mm diameter sample.
The effective confining pressure Pe is defined as the difference
between the confining pressure Pc and the pore pressure Pp. That

is, a Terzaghi effective pressure law (Pe ¼ Pc À Pp) is adopted where
the effective stress coefficient n in the general form of effective
stress law (Pe ¼ Pc À nPp; [47–49]) is simply assumed as unity. A
sample-average pore pressure Pav ¼ 2LðPu2 þ Pu Pd þ Pd2 Þ=3ðPu þ Pd Þ
was used to calculate the effective pressure. The pore pressure for
measuring the porosity is the balance pore pressure Pf in Eq. (2).
The average pore pressures for permeability measurement were
0.13–1.40 MPa and the balance pressures for porosity measurement were 0.30–1.41 MPa.

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J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

Notably, the porosity measurements made using the above
method are sensitive to the pore space volume of the samples. If
we consider Pi1 ¼2.0 MPa and use the given values of Pi2 ¼0.1
MPa, for the range of measured porosity (0.05–0.2) with a sample
volume of 12.55 ml (average sample volume in Table 1 with
$25 mm in diameter), the balanced pressure Pf varies in the
range 1.05–1.46 MPa. With a precision of 0.01 MPa for the
pressure measurement system, a porosity of 12.43% with an error
of 0.179% will be obtained when the balance pressure is 1.25 MPa.

atmospheric pressure is Ko ¼ 2:17 Â 10À12 m2 for Boise sandstone
(porosity 34.9%) whereas Ko ¼ 1:48À6:48 Â 10À14 m2 for the other
four sandstones (porosity 13.8–20.7%).
On the other hand, Shi and Wang [4] suggested that the
relationship between effective pressure and rock permeability
should follow a power law, based on the laboratory permeability
measurements made by Morrow et al. [18] for fault gouges. A
power law for describing the stress dependency of permeability
can be expressed as follows:
K ¼ Ko ðPe =Po ÞÀp ,

4. Experimental results
Fig. 4 shows the permeability and porosity measurement
results. The experiments were conducted first while gradually
increasing (loading) the confining pressure Pc from 3 to 5 MPa,

then to 10, and finally (in 10 MPa increments) to 120 MPa. Pc was
then gradually reduced (unloading) back to 3 MPa in the reverse
order. The horizontal axis of Fig. 4 is the effective confining
pressure Pe( ¼Pc ÀPp). Fig. 4 indicates that the unloading paths for
both permeability and porosity are consistently lower than the
loading path, because the compaction of the geomaterials is not
fully reversible [18]. Notably, the permeability and porosity of the
sandstones (10À14 À10À13 m2 and 15–19%) significantly exceeds
that of the silty-shale (10À20 À10À15 m2 and 8–14%). In other
words, aside from the influence of the fracture network, it is the
rock type (sandstone or shale) that dominates the permeability
and porosity of the wall rocks around the fault.
The permeability of silty-shale is more sensitive to changes in
the effective confining pressure than the sandstone, particularly
at low confining pressure (Fig. 4a). The permeability of the
sandstone was reduced to less than 50% when the confining
pressure Pc was increased from 3 to 10 MPa (the effective
confining pressure Pe is slightly smaller than the indicated value).
On the other hand, the permeability of silty-shale at Pc ¼10 MPa
was one to two orders of magnitude smaller than that at
Pc ¼3 MPa. In contrast, the porosities of different rock types
(sandstone and shale) were almost identical in terms of the stress
sensitivity (Fig. 4b). Generally, the porosity of tested sandstone
and silty-shale samples was reduced by 10–20% when the
confining pressure was increased from 3 to 120 MPa. A quantitative evaluation of the stress dependency of permeability and
porosity is discussed in detail below.
4.1. Models for describing the effective confining pressure
dependency of permeability
Fluid flow simulation in the crust requires models that reflect
the relationship between permeability and depth (effective

stress). David et al. [10] suggested that an exponential relationship would be suitable for describing the stress dependent
permeability. Their results were based on laboratory experiments
(with pressures up to 400 MPa) for five different sandstones and
are consistent with those of a previous study [19]. Evans et al. [21]
also noted that the stress dependent permeability (for effective
pressures up to 50 MPa) for granitic rocks near a fault zone
exhibited an exponential relationship. The exponential relationship for the stress dependent permeability can be expressed as
follows:
K ¼ Ko exp½ÀgðPe ÀPo ފ,

ð4Þ

where K denotes the permeability under the effective confining
pressure Pe, Ko represents the permeability under atmospheric
pressure Po which is assumed to be 0.1 MPa, and g is a material
constant. David et al. [10] reported that g ¼ 9:81À18:1 Â
10À3 MPaÀ1 for five different sandstones. The permeability under

ð5Þ

where p is a material constant. For pure clay, clay-rich and clayfree fault gouges, the material constant p is found to range from
1.2 to 1.8 as the effective pressure increases (during loading) from
5 to 200 MPa, and from 0.4 to 0.9 as the effective pressure
decreases (during unloading) from 200 to 5 MPa [4]. The
permeability under atmospheric pressure for the tested fault
gouge is Ko ¼ 10À18 À10À14 m2 [4]. Ghabezloo et al. [27] also
reported that the permeability of a limestone under different
confining pressures closely fits a power law.
Based on the permeability measurement results (Fig. 4a), we
can easily determine the parameters in Eqs. (4) and (5) using

curve fitting (Fig. 5a and b). The measured parameters (Ko,g and
Ko,p) are listed in Table 2. The determined parameters in the
exponential relationship for the sandstones under loading are
Ko ¼ 5:85À7:08 Â 10À14 m2 and g ¼ 2:84À7:68 Â 10À3 MPaÀ1 . The
measured Ko for the sandstone is almost identical to that
previously reported for sandstone by David et al. [10] (with the
exception of Boise sandstone), while the measured g for the
sandstone approaches the lower bound of g obtained by David
et al. [10]. In other words, the permeability of the sandstone
exhibits less stress dependency than that shown in the results
reported by David et al. [10]. Compared with the measurements
for sandstone, significantly lower Ko ð ¼ 2:80 Â 10À19 À1:45 Â
10À16 m2 Þ and much higher values of gð ¼ 16:78À43:47 Â
10À3 MPaÀ1 Þ are obtained for the silty-shale under loading.
For the power law, the determined parameter values p for the
tested silty-shale are 0.588–1.744 (loading) and 0.196–0.855
(unloading), similar to those reported by Morrow et al. [18], where
p ranged from 1.2 to 1.8 (loading) and 0.4 to 0.9 (unloading) for
fault gouges. Under atmospheric pressure Ko the permeability of
the silty-shale during loading was found to range from 3:34 Â
10À18 m2 to 4:42 Â 10À13 m2 . The measured Ko is also of the same
order as that reported by Morrow et al. [18]. A much lower p was
determined for the sandstone, 0.120–0.303 under loading conditions and 0.057–0.114 under unloading conditions, which indicates
lower stress sensitivity compared with the silty-shale.
It is notable that the measured permeability in gas flow
experiments generally leads to an overestimate of water permeability. Faulkner and Rutter [46] suggested that water permeability in the fault gouge is typically one or more orders of
magnitude less than that of gas permeability. The influence of
using gas as a fluid for measuring the permeability will be
evaluated and discussed in Section 5.2.
4.2. Models for describing the effective confining pressure

dependency of porosity
The model describing the relationship between effective
confining pressure and porosity (effective porosity) includes the
following exponential relationship developed for shale [28,29],
sandstone [31], and carbonate [30]:

f ¼ fo exp½ÀbðPe ÀPo ފ,

ð6Þ

where f denotes the porosity under the effective confining
pressure, fo represents the porosity under atmospheric pressure,


J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

100

1E-013

Fine-grained sandstone
R261_sec2_1
R261_sec2_2
R307_sec1

1E-014

10

Permeability (m2)

Silty-shale
R255_sec2_1
R255_sec2_2
R287_sec1
R351_sec2
R390_sec3
R437-sec1

1E-016

1E-017

0.1

0.01

1E-018

0.001

1E-019

0.0001

0

10

20

30 40 50
60 70 80 90
Effective Confining Pressure (MPa)

100 110

Permeability (md)

1

1E-015

1E-020

1147

1E-005
120

19.0
Fine-grained sandstone
R261_sec2_1
R261_sec2_2
R307_sec1

18.0
17.0
16.0

Porosity (%)

15.0
14.0
13.0

Silty-shale
R255_sec2_2

12.0

R287_sec1
R316_sec1

11.0

R351_sec2

10.0

R437_sec1

R390_sec3

9.0
8.0
7.0

0

10

20

30 40 50
60 70 80
90
Effective Confining Pressure (MPa)

100 110

120

Fig. 4. Stress dependent (a) permeability and (b) porosity of the sandstone and silty-shale, for sandstone (red dashed lines) and silty-shale (solid black lines).
(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

and b is a material constant. The exponential relationship for
stress dependent porosity has been used for analyzing
the compaction flow in sediment basins [4–6,8]. The pore

compressibility of rocks can be simply expressed as
@f
bf ¼ À
¼ bf
@Pe

ð7Þ


1148

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

Curve fitting results
Power law (loading)
Exponential relation (loading)
Power law (unloading)
Exponential relation (unloading)

8E-014

80

7E-014

70

6E-014

60

5E-014
4E-014

50

0

Permeability (m2)

Permeability (m2)

Experimental data points
Loading
Unloading

1E-014

10

1E-015

1

1E-016

0.1

1E-017

0.01

1E-018

0.001

1E-019

40
20

40
60
80 100 120
Effective Confining Pressure (MPa)

0

0.0001
20
40
60
80 100 120
Effective Confining Pressure (MPa)

9.5

18

9

Porosity(%)

Porosity(%)

17.5
17
16.5

8.5

16
15.5

0

8

20
80 100 120
40
60
Effective Confining Pressure (MPa)

0

20
40
60
80
100 120
Effective Confining Pressure (MPa)

Fig. 5. Loading and unloading curves of the stress dependent permeability for (a) sandstone (R261_sec2_1) and (b) silty-shale (R390_sec3); and stress dependent porosity
for (c) sandstone (R307_sec1) and (d) silty-shale (R351_sec2). Both an exponential relationship and a power law were utilized to fit the experimental data.

if the stress dependency of porosity follows an exponential
relationship. That is, the material constant b reflects the pore
compressibility of the sediments. David et al. [10] found that
fo ¼13.8–34.9% and b ¼ 0:44À3:30 Â 10À3 MPaÀ1 for sandstone
under loading conditions. Curve fitting for the porosity measurements can be used to obtain the material constants for the

exponential model of porosity, as illustrated in Fig. 5c and d. The
determined parameters (fo, b) of the exponential relationship for
the stress dependent porosity are listed in Table 2. The parameter
values determined for the sandstone under loading are
fo ¼17.06–17.67% and b ¼ 0:91À1:58 Â 10À3 MPa-1 . These measured parameters fall within the range of the measured
parameters reported by David et al. [10]. For unloading, the
determined parameters for the sandstone are fo ¼ 16.68–16.94%
and b ¼ 0:69À1:15 Â 10À3 MPaÀ1 .
A power law of the form

f ¼ fo ðPe =Po ÞÀq

For the silty-shale, the measured fo is again higher for a power
law (fo ¼10.23–14.76% for loading and fo ¼8.96–13.78% for
unloading) than the exponential relationship (fo ¼8.84–13.86%
for loading and fo ¼8.28–13.39% for unloading). In Table 2 it can
be seen that the stress sensitivity parameters (b, q) for the
sandstone and silty-shale are similar. These results suggest that
the stress sensitivity of porosity for the sandstone and silty-shale
will be similar, regardless of whether the exponential relationship
(Eq. (6)) or power law (Eq. (8)) is used. Generally, for the Pliocene
to Pleistocene sandstone and silty-shale the calculated values of b
range from 0.41 to 1:58 Â 10À3 MPaÀ1 (loading) and 0.14 to 1:15 Â
10À3 MPaÀ1 (unloading), while calculated values of q range from
0.014 to 0.056 (loading) and 0.006 to 0.040 (unloading).

4.3. Stress dependent specific storage

ð8Þ

appears to better describe the relationship between the effective
confining pressure and the porosity of the sandstone and siltyshale (based on curve fitting) than the exponential relationship
(Fig. 5), where q is a material constant. The determined
parameters (fo, q) for describing the stress dependent porosity
power law are listed in Table 2. The determined values of q for the
sandstone are 0.037–0.056 (loading) and 0.024–0.040 (unloading). The value of fo obtained for the power law is greater than
that for the exponential relationship. For the tested sandstone
we find fo ¼20.20–22.45% (loading) and fo ¼18.52–20.14%
(unloading).

The stress dependent specific storage of sediments should be
incorporated into fluid flow analysis in a sedimentary basin [5].
The specific storage Ss can be expressed as follows [4]:
Ss ¼

bj
ð1ÀfÞ

þ fbf ,

ð9Þ

where bf and bf are the compressibility of the porosity and pore
fluid, respectively. The compressibility of the solid grains ( $ 10 À 5
MPa À 1) is ignored in Eq. (9). The compressibility of the porosity
bf is equal to À@f=@Pe and the compressibility of water is about
4 Â 10À4 MPaÀ1 [4]. Using Eq. (9), the stress dependent specific


Sample number

Permeability

Porosity

Exponential relationship K ¼ Ko exp½ÀgðPe ÀPo ފ
Loading
Ko (m2)
Fine-grained sandstone
R261_sec2_1
6.55 Â 10 À 14
R2 ¼0.855
R261_sec2_2
5.85 Â 10 À 14
R2 ¼0.941
R307_sec1
7.08 Â 10 À 14
R2 ¼0.880
Silty-shale
R255_sec2_1
R255_sec2_2
R287_sec1
R351_sec2

1.01 Â 10 À 18
R2 ¼0.378
2.40 Â 10 À 18
R2 ¼0.423
6.06 Â 10 À 18
R2 ¼0.868

7.07 Â 10 À 19
R2 ¼0.647

Power law K ¼ Ko

Unloading

Loading

h iÀp
Pe
Po

Unloading

R437_sec1

1.45 Â 10 À 16
R2 ¼0.894
2.80 Â 10 À 19
R2 ¼0.521

Power law f ¼ fo

Loading

Loading

Unloading

g (MPa À 1)

Ko (m2)

g (MPa À 1)

Ko (m2)

p

Ko (m2)

p

fo (%)

2.84 Â 10 À 3

5.57 Â 10 À 14
R2 ¼0.882
3.15 Â 10 À 14
R2 ¼0.824
6.07 Â 10 À 14
R2 ¼0.919

1.37 Â 10 À 3

1.14 Â 10 À 13
R2 ¼ 0.997
2.33 Â 10 À 13

R2 ¼ 0.961
1.37 Â 10 À 13
R2 ¼ 0.983

0.120

7.24 Â 10 À 14
R2 ¼ 0.991
5.36 Â 10 À 14
R2 ¼ 0.995
9.07 Â 10 À 14
R2 ¼ 0.987

0.057

17.06
0.91 Â 10 À 3
R2 ¼ 0.923
17.67
1.58 Â 10 À 3
R2 ¼ 0.864
17.32
1.03 Â 10 À 3
R2 ¼ 0.900

4.66 Â 10 À 19
R2 ¼0.259
5.68 Â 10 À 19
R2 ¼0.300
2.18 Â 10 À 19

R2 ¼0.625
2.07 Â 10 À 19
R2 ¼0.628

7.91 Â 10 À 3

5.95 Â 10 À 17
R2 ¼ 0.730
1.80 Â 10 À 15
R2 ¼ 0.837
1.15 Â 10 À 14
R2 ¼ 0.987
3.99 Â 10 À 17
R2 ¼ 0.952

0.844

3.48 Â 10 À 18
R2 ¼ 0.549
2.83 Â 10 À 17
R2 ¼ 0.692
1.93 Â 10 À 18
R2 ¼ 0.928
1.93 Â 10 À 18
R2 ¼ 0.971

0.416

7.68 Â 10

À3

3.46 Â 10 À 3

16.78 Â 10 À 3
35.29 Â 10

À3

43.47 Â 10 À 3
25.93 Â 10 À 3

2.65 Â 10

À3

2.16 Â 10 À 3

18.88 Â 10

À3

10.58 Â 10 À 3
13.90 Â 10 À 3

0.303
0.143

1.478
1.677

0.937

0.114
0.087

0.855
0.466
0.514

R316_sec1
R390_sec3

Exponential relationship f ¼ fo exp½ÀbðPe ÀPo ފ

42.90 Â 10 À 3
22.78 Â 10 À 3

2.41 Â 10 À 18
R2 ¼0.871

4.84 Â 10 À 3

4.42 Â 10 À 13
R2 ¼ 0.993
3.34 Â 10 À 18
R2 ¼ 0.838

1.744
0.588

5.94 Â 10 À 18
R2 ¼ 0.977

0.196

9.78
R2 ¼ 0.994
10.65
R2 ¼ 0.951
9.00
R2 ¼ 0.911
8.84
R2 ¼ 0.984
10.83
R2 ¼ 0.977
13.86
R2 ¼ 0.988

b (MPa À 1)

0.95 Â 10 À 3
0.94 Â 10 À 3
1.04 Â 10 À 3
1.30 Â 10 À 3
1.01 Â 10 À 3
0.41 Â 10 À 3

fo (%)

b (MPa À 1)

fo (%)

h iÀq
Pe
Po

Unloading
q

fo (%)

q

16.68
0.69 Â 10 À 3
R2 ¼ 0.901
16.94
1.15 Â 10 À 3
R2 ¼ 0.969
16.87
0.75 Â 10 À 3
R2 ¼ 0.925

20.20
0.037
R2 ¼ 0.986
22.45
0.056
R2 ¼ 0.995

20.75
0.040
R2 ¼ 0.991

18.52
0.024
R2 ¼ 0.986
20.14
0.040
R2 ¼ 0.991
19.17
0.028
R2 ¼ 0.980

0.42 Â 10 À 3

11.27
0.033
R2 ¼ 0.860
12.51
0.036
R2 ¼ 0.968
10.23
0.032
R2 ¼ 0.966
10.79
0.046
R2 ¼ 0.911
12.72
0.036

R2 ¼ 0.936
14.76
0.014
R2 ¼ 0.904

9.91
0.017
R2 ¼ 0.981
10.70
0.016
R2 ¼ 0.988
9.86
0.030
R2 ¼ 0.977
8.96
0.023
R2 ¼ 0.968
11.70
0.028
R2 ¼ 0.975
13.78
0.006
R2 ¼ 0.950

9.16
R2 ¼ 0.800
9.95
R2 ¼ 0.778
8.68
R2 ¼ 0.661

8.28
R2 ¼ 0.685
10.25
R2 ¼ 0.713
13.39
R2 ¼ 0.644

0.37 Â 10 À 3
0.82 Â 10 À 3
0.54 Â 10 À 3
0.65 Â 10 À 3
0.14 Â 10 À 3

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

Table 2
Parameters determined using curve fitting techniques based on measured permeability and porosity of the tested sandstone and silty-shale.

1149


1150

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

0.01

storage of sediments can be estimated if the stress dependent
porosity can be obtained.
Combining Eq. (9) and (7) for the exponential relationship

describing the stress dependent porosity, the specific storage can
be expressed as
ð10Þ

On the other hand, if the stress dependent porosity is
described as a power law, Eq. (8), the specific storage can be
expressed as

fq
þ fbf :
Ss ¼
ð1ÀfÞPe

0.001

Silty-shale
R255_sec2
R287_sec1
R351_sec2
R316_sec1
R390_sec3
R437_sec1

R261_sec2_1
R261_sec2_2
R307_sec1

0.0001

ð11Þ

The specific storage as a function of effective confining
pressure calculated by Eqs. (10) and (11) is illustrated in Fig. 6a
and b, respectively. Clear differences exist between specific
storage estimated using different stress dependent models of
porosity. The specific storage calculated using an exponential
relationship (Fig. 6a) ranged from 0.06 to 0:4 Â 10À3 MPaÀ1 for the
tested sandstone and shale when the confining pressure was
increased from 3 to 120 MPa. Domenico and Mifflin [50] reported
the specific storage of dense sand and medium-hard clay to be
about 10 to 100 Â 10À3 MPaÀ1 . Consequently, estimates of specific
storage under low effective confining pressure for sediments at
shallow depths can be seriously underestimated if a power law is
utilized to describe the stress dependency of the porosity. The
calculated specific storage is more sensitive to the effective
confining pressure if a power law (Eq. (8)) is adopted than when
an exponential relationship (Eq. (6)) is adopted. Sharp and
Domenico [51] noted that the specific storage of sediments was
sharply reduced with increasing effective confining pressure. In
other words, the specific storage of sediments should be highly
dependent on the variation of effective confining pressure. It is
thus suggested that a power law should be used to describe the
stress dependent porosity when deriving the specific storage of
the tested Pliocene to Pleistocene sedimentary rocks. The specific
storage calculated using a power law (Fig. 6b) ranges from 2 Â
10À3 to 0:2 Â 10À3 MPaÀ1 for the sandstone, and from 0:7 Â 10À3
to 0:07 Â 10À3 MPaÀ1 for the silty-shale, when the confining
pressure is increased from 3 to 120 MPa. Generally, the estimated
specific storage of the tested sedimentary rocks is reduced by
about one order of magnitude when the confining pressure is

increased from 3 to 120 MPa. Wibberley [23] demonstrated that
the specific storage of fault gouges was reduced by approximately
two orders of magnitude (0:1À10 Â 10À3 MPaÀ1 ) when the
effective pressure increased from about 30 to 125 MPa. This
indicates that fluid flow analysis of sedimentary basins should
account for the stress dependency of the specific storage.
Notably, when a power law is adopted, the calculated specific
storage of the tested sandstone or silty-shale will be concentrated
within a narrow range (Fig. 6b). Rather than using a complex form
of Eq. (11), here we propose the following explicit power law
model to represent the stress dependent specific storage of
sediments:
Ss ¼ Ss,Po ðPe =Po ÞÀr ,

Specific Storage (MPa-1)

þ fbf :

ð12Þ

where Ss,Po denotes the specific storage under atmospheric
pressure Po and r represents a material constant. Based on the
laboratory work, the parameters in Eq. (12) are calculated as
Ss,Po ¼ 42:3 Â 10À3 ðMPaÀ1 Þ and r ¼0.823 for sandstones, and
Ss,Po ¼ 11:5 Â 10À3 ðMPaÀ1 Þ and r ¼0.734 for shales. Values of r
determined for sandstones and shales are similar (about 0.7–0.8),
and are represented by similarly shaped specific storage –

1E-005

0

10 20 30 40 50 60 70 80 90 100 110 120
Effective Confining Pressure (MPa)

0.01
Specific storage model (Eq. (11))
Fine-grained sandstone Silty-shale
Specific Storage (MPa-1)

fb
ð1ÀfÞ

R261_sec2_1
R261_sec2_2
R307_sec1

0.001

R255_sec2
R287_sec1
R351_sec2
R316_sec1
R390_sec3
R437_sec1

0.0001

1E-005

0

10 20 30 40 50 60 70 80 90 100 110 120
Effective Confining Pressure (MPa)

0.01
Experiment results:
Silty-shale
Fine-grained sandstone
Specific Storage (MPa-1)

Ss ¼

Specific storage model (Eq. (10))

Fine-grained sandstone

R261_sec2_1
R261_sec2_2
R307_sec1

0.001

R255_sec2
R287_sec1
R351_sec2
R316_sec1
R390_sec3
R437_sec1

0.0001
Specific storage model (Eq.(12))
Sandstone: Ss,Po=42.3x10-3 (MPa) and r=0.823
-3

Sandstone: Ss,Po=11.5x10

1E-005

0

(MPa) and r=0.743

10 20 30 40 50 60 70 80 90 100 110 120
Effective Confining Pressure (MPa)

Fig. 6. Stress dependent specific storage calculated based on (a) an exponential
relationship; and (b) a power law, for the sandstone (red dashed lines) and siltyshale (solid black lines). (c) The explicit form of the stress dependent specific
storage (Eq. (12)). The symbols represent the experimental data points. (For
interpretation of the references to color in this figure legend, the reader is referred
to the web version of this article.)


J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

1151

Fig. 7. SEM images of silty-shale samples (a) R351 and (b) R390. Microcracks with a width about 1–3 mm can be clearly identified.

effective pressure curves. Fig. 6c shows the specific storage

calculated using Eq. (12). The model closely fits data points
calculated using the measured porosity. Since the specific storage
of sediments under atmospheric pressure is well documented
(e.g., [50]), the parameters in Eq. (12) can be estimated with
reasonable accuracy. Consequently, the explicit form of the stress
dependent specific storage can be easily incorporated into nonlinear fluid flow analysis.

5. Discussion
The proposed models adopting a power law empirically
describe the stress dependency of permeability and porosity.
Aside from the well constrained lithology and geological age of
the samples, the influence of the deformation mechanism and
stress range of the tested rocks on their stress dependency is
further elucidated.
5.1. The influence of the deformation mechanism
David et al. [10] suggested that microcrack closure, particle
rearrangement and crushing are the dominant mechanisms
controlling the evolution of rock permeability with the effective
confining pressure. Accordingly, they propose three types of
permeability evolution induced by mechanical compaction: Type I
for low porosity crystalline rock; Type II for porous clastic rock;
and Type III, typically for unconsolidated materials. The permeability evolution of the tested sandstones shown in Fig. 4a can be
categorized as type II. Type II permeability evolution is typically
observed in porous clastic rocks with relatively low stress

sensitivity and where the compaction is related to the relative
movement of grains. A rapid decrease in permeability was
observed in the porous clastic rock when the effective pressure
exceeded a critical value, which meant that the permeability
evolution curve could also be classified as type III [10]. The

absence of a sudden decrease in porosity as shown in Fig. 4b for
tested sandstone indicates that there was no particle crushing
mechanism involved. Therefore, the particle crushing mechanism
was not involved in the evolution of the permeability of the
sandstone in the sedimentary basin at depths of 8–9 km. As a
result, the variation of permeability and porosity of the sandstone,
induced by mechanical compaction, did not exceed 50% or 20%,
respectively.
The permeability of the tested silty-shale is relatively more
sensitive to changes in the effective confining pressure than that
of the sandstone. The tested shale can be classified as having Type
I permeability evolution [10], which typically occurs in low
porosity crystalline rock where the closure of microcracks plays
an important role. Walsh [52] reported a much larger stress
sensitivity of permeability for crack-like pores than equivalent
pore channels. Therefore, the higher stress sensitivity of the
permeability of shale compared to that of sandstone might
originate from the different pore shapes or from the presence of
microcrack networks in the shale. Kwon et al. [32] suggested that
the significant reduction in permeability of the silty-shale under a
low effective confining pressure was possibly dominated by
microcrack closure. The above postulate is supported by scanning
electron microscope (SEM) images of samples R351_sec2 (Fig. 7a)
and R390_sec3 (Fig. 7b). Microcracks with a width of about
1–3 mm can be clearly identified in Fig. 7a and b. Nevertheless, the
origin of these microcracks is unclear. In summary, the proposed


1152

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

power law models could be used to predict the permeability and
porosity of porous sandstone at depth if no particle crushing
mechanism is involved. In addition, the proposed power law
models also closely fit the experimental data for shale with
microcrack networks.
5.2. The influence of stress range and maximum overburden on
parameter estimates
5.2.1. Experimental stress range
To illustrate the effect of stress range on the performance of
the model prediction, we plotted the exponential relationship and
power law, calculated with the measured parameters in Table 2,
between effective pressures of 0.1 and 300 MPa. Two samples,
R261_sec2_2 (sandstone) and R390_sec3 (shale), were selected to
demonstrate the results of the model prediction within the stress
range. Fig. 8a and b shows the permeability and porosity between
effective pressures of 0.1 and 300 MPa predicted by the
exponential relationship (dash lines) and power law (solid
lines). The experimental data points between effective pressures
of 3–120 MPa are also plotted in these figures. Although the
power law shows a better fit with the experimental results than
that of the exponential relationship, the difference between the
exponential relationship and power law is relatively minor in a
log–log plot within the experimental pressure range. However,
the discrepancy between these two models is considerable under
low and high effective pressures. In general, the predicted
permeability or porosity of the power law is always higher than
that of the exponential relationship when the effective pressure is
greater than 100 MPa or less than 10 MPa.

To further elucidate the importance of the experimental stress
range, the permeability results presented in this study and the
permeability of independent measurements [53] of samples
obtained from the same borehole (TCDP Hole-A) are compared.
The initial permeability and porosity of the samples tested by Chen
et al. [53] (Ko ¼ 3:14À5:21 Â 10À14 m2 , fo ¼15.8–18.5%) are very
close to those measured in the present study. However, the values
of the pressure sensitivity coefficient g are quite different. Chen
et al. reported g ¼ 45À155 Â 10À3 MPaÀ1 for sandstone here we
obtained a much lower g ¼ 2:84À7:68 Â 10À3 MPaÀ1 . The experimental stress range of [53] was 5–40 MPa, which is much smaller
than that of the present study. As shown in Fig. 5, the exponential
relationship is a straight line in a semi-log plot. According to Eq. (4),
the slope of the straight line is proportional to the pressure
sensitivity coefficient g. If the exponential relationship is used to fit
the data points for a lower stress range, g will be greater than that
obtained from the data points for a higher stress range. Consequently, the experimental stress range plays an important role in
determining the parameters of an empirical model.
5.2.2. Maximum overburden
Generally, a power law is superior to an exponential relationship for describing the stress dependent permeability and
porosity of the Pliocene and Pleistocene sandstones and siltyshales, as demonstrated in Fig. 5. However, there is still a
discrepancy between the fitted curves obtained using the power
law (Fig. 5, solid lines) and the experimental data. It is well known
that the stress history affects the mechanical behavior of
geomaterials. The question then arises, should the effect of stress
history on the permeability and porosity of sediments be taken
into account in stress dependent models of the fluid flow
properties? Detection of the inflexion point in the slope of the
permeability effective pressure curve could serve as a useful
method for inferring the maximum effective stress that sediments
have experienced [32]. Yang and Aplin [54] found the perme-

ability effective pressure curves (on logarithmic axes) of mudstones to have inflexion points from which they successfully
estimated the maximum effective stresses of the tested samples.
Fig. 8 shows curves of permeability or porosity versus effective
pressure (on log–log axes) of the tested sandstone and silty-shale
samples. The inflexion points in the slope of the porosity-effective
pressure curves can be observed for the tested rocks (Fig. 8b). The
corresponding effective confining pressures of the inflexion points
(40–60 MPa) are close to the estimated maximum overburden
($49 MPa) of these rocks, at a depth of about 3500 m before
thrusting occurred. Consequently, the stress dependency of sediment porosity when the effective stress is greater than the
maximum effective stress sustained by the sediments is different
to that when the effective stress is smaller than the maximum.
Further, while a straight line might be adequate to predict the
stress dependent permeability or porosity within the experimental
stress range, extrapolating the permeability or porosity to larger
confining pressures (e.g. 300 MPa) using a straight line might
induce unreasonable error. A bi-linear model (in log–log scale)
to predict the permeability or porosity would be feasible to
account for the maximum imposed overburden. The effects of
the stress history, including maximum burial, uplift and erosion, on
the fluid flow properties of sedimentary rocks require further
study.
5.3. Influence of sample anisotropy and sample length
It is well known that stratified shale and sandstone with thin
alternation beds is anisotropic. The hydraulic conductivity parallel
to the bedding plane is larger than that perpendicular to the
bedding plane [22]. Chen et al. [53] reported that the permeability
of a shaly siltstone sample from TCDP Hole-A is anisotropic,
where the measured permeability in the direction parallel to the

bedding plane (Y-direction) was one to two orders of magnitudes
larger than that in the direction inclined at an angle of 301 or 601
to the bedding plane (X and Z directions). To minimize the
influence of anisotropy, only the relatively homogeneous cores
were selected, and cores with interbedded layers were discarded.
Meanwhile, all samples were cored parallel to the axes of rock
cores from TCDP Hole-A (the Z-direction in [53]). Aside from the
influence of anisotropy, sample length could influence the
permeability measurements. For example, microcracks could
penetrate through the thin samples and create a permeable
pathway. Microcracks with similar widths (about 1–3 mm) were
identified in the SEM images of samples R351_sec2 and
R390_sec3 (Fig. 7). However, a much higher permeability was
observed for sample R390_sec3. It is postulated that the
microcracks in R390_sec3 penetrated through the sample, which
had a length of only 2.68 mm. Anisotropy will enhance the
influence of sample length on permeability measurements. For
example, a thin, inclined relatively impermeable layer may only
cross cut part of a thin sample. However, this impermeable layer
may completely cross cut a larger sample, creating an impermeable barrier. For heterogeneous or anisotropic samples, permeability measurements in different directions are suggested and
the effect of sample aspect ratios should be considered.
5.4. The influence of using gas as a fluid on the stress dependency of
permeability
As stated previously, the measured permeability in gas flow
experiments is generally different from the water permeability.
Using the Klinkenberg Eq. [55]
Kg ¼ Kl ½1 þðb=Pav ފ,

ð13Þ

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

10000

1E-011
Permeability model of R261_sec2_2
Power law
Exponential relation

1E-012

1000
100

1E-013
Permeability model of R390_sec3
Power law
Exponential relation

Permeability (m2)

1E-015
1E-016

10
Fine-grained sandstone
R261_sec2_1
R261_sec2_2
R307_sec1

1
0.1

1E-017

0.01

1E-018

0.001

Permeability (md)

1E-014

0.0001

1E-019
Silty-shale
R255_sec2_1
R255_sec2_2
R287_sec1
R351_sec2
R390_sec3
R437-sec1

1E-020
1E-021
1E-022

1E-023

1153

0.1

1
10
100
Effective Confining Pressure (MPa)

1E-005
1E-006
1E-007
1E-008
1000

30
Porosity model of R261_sec2_2
Power law
Exponential relation
Fine-grained sandstone
R261_sec2_1
R261_sec2_2
R307_sec1

20

Porosity (%)

Porosity model of R390_sec3
Power law
Exponential relation

10.0
9
8
7
6
5
0.1

Silty-shale
R255_sec2_2
R287_sec1
R316_sec1
R351_sec2
R390_sec3
R437_sec1
1
10
100
Effective Confining Pressure (MPa)

1000

Fig. 8. Comparison between the models adopting a power law and exponential relationship for (a) permeability and (b) porosity.

where Pav is the average pore pressure, and b is the Klinkenberg
slip factor, the water permeability Kl could be estimated from

measurements of gas permeability Kg. In general, differences in
gas and water permeability due to the Klinkenberg effect would
be less than one order of magnitude in the surface samples of

sandstone and siltstone from the Taiwan Western foothills [26].
The Klinkenberg slip factor for these rocks was determined as
b¼0.15 Â KlÀ 0.37 (the units of b and Kl are Pa and m2, respectively)
[26]. The average pore pressure Pav ¼ 2LðPu2 þ Pu Pd þ Pd2 Þ=3ðPu þ Pd Þ
of each measurement was calculated with Pu and Pd ¼0.1 MPa.


J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

8E-014

Permeability of Sandstone: R261_sec2_1
Measured Kg using gas as fluid (Loading)
(Unloading)
7
Klinkenberg correction Kl (Loading)
(Unloading)
Power law (Loading)
(Unloading)

Permeability(m2)

7E-014

6E-014

6

5E-014

4E-014

5

0

10

20

30

40 50 60 70 80 90
Effective confining pressure (MPa)

100 110 120

Permeability of Shale: R390_sec3
Measured Kg using gas as fluid (Loading)

1E-015
Permeability(m2)

4

10

1E-014

(Unloading)
Klinkenberg correction Kl (Loading)

1E-016

(Unloading)

1
0.1

Power law (Loading)
(Unloading)

1E-017

0.01

1E-018

0.001

1E-019

Permeability(md)

8

0

10

20

30 40 50 60 70 80 90
Effective confining pressure (MPa)

Permeability(md)

1154

0.0001
100 110 120

Fig. 9. Permeability with the Klinkenberg correction for (a) sandstone (R261_sec2_1) and (b) silty-shale (R390_sec3).

The measured Kg of sandstone (R261_sec2_1) and silty-shale
(R390_sec3) and estimated Kl obtained using the Klinkenberg
correction are plotted in Fig. 9. In general, the estimated Kl is
smaller than the measured Kg. The discrepancy between Kg and Kl
is less than one order of magnitude.
The permeability model parameters with the Klinkenberg
correction are listed in Table 3. Generally, the level of fit
(R-squared) of the permeability models increased slightly after the
Klinkenberg correction. In addition, the stress sensitivity of
permeability (g or p) was consistently increased, so that the
difference between gas and water permeability will increase with
decreasing permeability. Although the calculated Ko of shale after

the Klinkenberg correction increased slightly when a power law was
adopted (i.e., a smaller intercept of the straight lines in Fig. 8a), the
overall permeability still decreased because the sensitivity of
permeability (p) is larger (the straight lines in Fig. 8a will be steeper).

5.5. Porosity sensitivity exponent
The permeability of sedimentary rocks is a function of porosity
and pore structure. The mechanical and chemical processes may
also play important roles in the permeability–porosity relationship.
Therefore, there is no simple direct relationship between porosity
and permeability [45]. However, the evolution of permeability and
porosity in rocks can be constrained provided that the processes
changing the pore space are known [56]. David et al. [10] proposed

the following power law to describe the permeability–porosity
relationship induced by mechanical compaction:
K ¼ Ko ðf=fo Þa ,

ð14Þ

where a is a material constant named the porosity sensitivity
exponent. If both the permeability and porosity of the rocks can be
described using an exponential relationship (Eqs. (4) and (6)), the
porosity sensitivity exponent can be easily determined as a ¼ g/b.
On the other hand, if power laws (Eqs. (5) and (8)) are used, the
porosity sensitivity exponent can be determined as a ¼p/q.
Table 4 lists the porosity sensitivity exponents of the tested
sandstone and silty-shale. Values of a for sandstone, obtained
from an exponential relationship, range from 3.12 to 4.86
(loading) and 1.99 to 2.88 (unloading); those obtained for a

power law range from 3.24 to 5.45 (loading) and 2.38 to 3.14
(unloading). There is therefore little difference in measured values
of a for sandstone obtained with the exponential relationship and
the power law. The estimated a of the sandstone is close to that
recommended by Brace et al. [15] where Kpf3, implying that
a ¼3, and by Rieke and Chilingarian [57] where Kpf5, implying
that a ¼5. David et al. [10] demonstrated that the porosity
sensitivity exponent was equal to 4.6 for Boise sandstone with
high porosity (fo ¼34.9%), before the onset of particle crushing
when the effective confining pressure exceeded a critical pressure.
Larger porosity sensitivity exponents a ¼ 14.7–25.4 were
determined for the remaining four porous sandstone samples,


J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

1155

Table 3
Klinkenberg correction of the measured parameters in the power law model of permeability.
Sample number

Permeability
Exponential relationship K ¼ Ko exp½ÀgðPe ÀPo ފ

Power law K ¼ Ko

Loading

Loading

Unloading

Ko (m2)
Fine-grained sandstone
R261_sec2_1
6.04 Â 10 À 14
R2 ¼ 0.854
R261_sec2_2
5.26 Â 10 À 14
R2 ¼ 0.941
R307_sec1
6.55 Â 10 À 14
R2 ¼ 0.881
Silty-shale
R255_sec2_1
R255_sec2_2
R287_sec1
R351_sec2
R390_sec3
R437_sec1

5.80 Â 10 À 19
R2 ¼ 0.393
1.45 Â 10 À 18
R2 ¼ 0.444
4.78 Â 10 À 18
R2 ¼ 0.887
4.06 Â 10 À 19
R2 ¼ 0.682

1.18 Â 10 À 16
R2 ¼ 0.916
1.30 Â 10 À 19
R2 ¼ 0.532

h iÀp
Pe
Po

Unloading

g (MPa À 1)

Ko (m2)

g (MPa À 1)

Ko (m2)

p

Ko (m2)

p

2.93 Â 10 À 3

5.12 Â 10 À 14
R2 ¼ 0.883
2.77 Â 10 À 14

R2 ¼ 0.825
5.59 Â 10 À 14
R2 ¼ 0.919

1.41 Â 10 À 3

1.07 Â 10 À 13
R2 ¼ 0.997
2.19 Â 10 À 13
R2 ¼ 0.959
1.29 Â 10 À 13
R2 ¼ 0.982

0.123

6.70 Â 10 À 14
R2 ¼0.991
4.84 Â 10 À 14
R2 ¼0.996
8.46 Â 10 À 14
R2 ¼0.987

0.058

2.36 Â 10 À 19
R2 ¼ 0.265
2.83 Â 10 À 19
R2 ¼ 0.317
9.31 Â 10 À 20
R2 ¼ 0.638

9.23 Â 10 À 20
R2 ¼ 0.645
8.49 Â 10 À 19
R2 ¼ 0.874

9.52 Â 10 À 3

6.83 Â 10 À 17
R2 ¼ 0.746
2.85 Â 10 À 15
R2 ¼ 0.853
3.39 Â 10 À 14
R2 ¼ 0.983
5.24 Â 10 À 17
R2 ¼ 0.963
1.46 Â 10 À 12
R2 ¼ 0.987
2.93 Â 10 À 18
R2 ¼ 0.846

0.988

2.62 Â 10 À 18
R2 ¼0.555
2.86 Â 10 À 17
R2 ¼0.714
1.52 Â 10 À 18
R2 ¼0.936
1.57 Â 10 À 18
R2 ¼0.977

1.54 Â 10 À 18
R2 ¼0.976

0.499

7.96 Â 10

À3

3.57 Â 10 À 3

19.83 Â 10 À 3
41.16 Â 10

À3

51.98 Â 10 À 3
32.12 Â 10 À 3
51.40 Â 10

À3

2.78 Â 10

À3

2.23 Â 10 À 3

22.67 Â 10

À3

13.66 Â 10 À 3
17.88 Â 10 À 3
6.45 Â 10

À3

28.77 Â 10 À 3

0.313
0.147

1.699
1.979
1.136
2.059

0.119
0.090

1.013
0.598
0.655
0.261

0.738

Table 4
Porosity sensitivity exponents of the tested sandstone and silty-shale, for an exponential relationship and power law.

Sample number

a ¼ p/q (power law)

a ¼ g/b (exponential relationship)
Loading G/W (P)a

Unloading G/W (P)

Loading G/W (P)

Unloading G/W (P)

Fine-grained sandstone
R261_sec2_1
R261_sec2_2
R307_sec1

3.12/3.22 (3%)
4.86/5.04 (4%)
3.36/3.47 (3%)

1.99/2.04 (3%)
2.30/2.42 (5%)
2.88/2.97 (3%)

3.24/3.32 (2%)
5.45/5.59 (3%)
3.58/3.68 (3%)

2.38/2.42 (2%)
2.85/2.98 (5%)
3.14/3.21 (2%)

Silty-shale
R255_sec2_1
R255_sec2_2
R287_sec1
R351_sec2
R316_sec1
R390_sec3
R437_sec1

–
37.15/43.79
46.24/55.30
24.93/30.88
–
42.48/50.89
55.56/70.17

–
44.95/53.98 (20%)
28.59/36.92 (29%)
16.95/21.80 (29%)
–
7.45/9.92 (33%)
–

–

44.79/51.48
46.58/54.97
29.28/35.50
–
48.44/57.19
42.00/52.71

–
50.29/59.59 (18%)
29.13/37.38 (28%)
17.13/21.88 (28%)
–
7.00/9.32 (33%)
–

a

(18%)
(20%)
(24%)
(20%)
(26%)

(15%)
(18%)
(21%)
(18%)
(26%)

G: Gas permeability; W: water permeability with Klinkenberg correction; P: percentage increased of porosity sensitivity exponent after Klinkenberg correction.

which had lower porosities (fo ¼13.8–20.7%) than the Boise
sandstone (fo ¼34.9%).
The values of a determined using an exponential relationship
ranged from 24.93 to 55.56 (loading) and 7.45 to 44.95
(unloading) for the tested silty-shale (Table 4). The values of a
obtained using a power law ranged from 29.28 to 48.44 (loading)
and 7.00 to 50.29 (unloading). Here, the porosity sensitivity
exponent for the silty-shale is considerably higher than that of the
sandstone, such that a slightly decreased porosity induced by
compaction of silty-shale causes permeability to decrease dramatically. For example, in sample R255_sec2_2 (silty-shale), an
increase in the confining pressure from 3 to 120 MPa caused a
decrease in permeability by two orders of magnitude (2 Â 10 À 17–
10 À 19 m2) with only a small decrease in porosity from 10% to 9%
(a ¼44.79). It is postulated that the effect of microcracks closure
contributed to the high porosity sensitivity exponent.
Table 4 also lists the porosity sensitivity exponents when the
stress sensitivity of permeability (g or p) is modified after the

Klinkenberg correction. It is interesting to find that the porosity
sensitivity exponent increased by 2–5% for sandstone and
increased by 15–33% for shale when the Klinkenberg correction
was applied to estimate the water permeability. Consequently,
the porosity sensitivity exponent will be underestimated if the
gas permeability is used. David et al. [10] noted that the
accumulation of excess pore pressure in a crustal layer is easier
with a larger porosity sensitivity exponent. Consequently, the
efficiency of accumulation of excess pore pressure is also underestimated if the gas permeability is used.

6. Conclusions

A permeability and porosity measurement system was used to
measure and evaluate the stress dependent fluid flow properties
of sedimentary rock cores (Pliocene to Pleistocene) from TCDP
Hole-A. The permeabilities and porosities measured under


1156

J.-J. Dong et al. / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157

effective confining pressures of up to 120 MPa were
10À14 À10À13 m2 and 15–19% for the fine-grained sandstones,
respectively, and 10À20 À10À15 m2 and 8–14% for the silty-shales,
respectively. Part of the decrease in permeability and porosity with
increasing effective confining pressure is irreversible, which
indicates permanent deformation in the form of voids. The
permeability of shale was more sensitive to changes in effective
confining pressure than the sandstone, particularly at low effective
confining pressures. Meanwhile, the stress sensitivity of porosity of
different rock types (sandstone versus shale) was almost identical.
Based on the experimental results, it can be inferred that the
evolution of compaction and permeability of the tested sandstone
is related to the relative movement of the grains; the particle
crushing mechanism is not involved. It is also postulated that a
microcrack network in the tested shales contributes to the stress
sensitivity of permeability under low effective confining pressure.
This hypothesis is supported by the SEM images.
The laboratory work indicates that a power law is superior to
an exponential relationship for describing the stress dependency
of permeability and porosity of the tested sedimentary rocks.

Notably, the determined permeabilities will be underestimated
under either low (say, Pe o10 MPa) and high (say, Pe 4100 MPa)
effective confining pressures when an exponential relationship is
used with the measured parameters. Although the match between
the models and the experimental data can be largely improved by
using a power law, there is still a discrepancy between the models
and the data for both for the permeability and the porosity. The
models could be further improved in future if the influence of the
maximum overburden on the stress dependency of permeability
and porosity is considered.
After applying a Klinkenberg correction, the estimated water
permeability was less than the gas permeability. The difference is
less than one order of magnitude. Meanwhile, the stress
sensitivity of permeability is consistently increased. In permeability–porosity relationships, the porosity sensitivity exponent
determined for the sandstones is close to the theoretical values
calculated for porous mediums (a ¼3–5). In contrast, the porosity
sensitivity exponent of the silty-shale is much higher than that of
the sandstone.
Specific storage is another important fluid flow property.
Calculations showed that specific storage exhibits greater stress
sensitivity in a power law than in an exponential relationship. If a
power law is adopted, the specific storage calculated for both the
sandstone and the silty-shale is concentrated in a narrow range. It
is thus suggested that an explicit form of the power law model
should be incorporated into fluid flow analysis describing the stress
dependent specific storage of Pliocene and Pleistocene rocks.

Acknowledgements
The authors thank the National Science Council of the Republic
of China, Taiwan for financially supporting this research under

Contract nos. NSC-94-2119-M-008-018 and 95-2119-M-008-033.
The building of our testing machine at National Central University
was supported by the COE program of the 21st Century Active
Geosphere Program of Kyoto University. We would also like to
extend our thanks to Tom Mitchell, and the two anonymous
reviewers, for their very constructive comments that greatly
improved the manuscript.
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