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

300, Jungda Road, Jungli, Taoyuan 32001, Taiwan b Department of Earth and Planetary Systems Science, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan c Institut

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

Jia-Jyun Donga,n

, Jui-Yu Hsua, Wen-Jie Wua, Toshi Shimamotob, Jih-Hao Hungc, En-Chao Yehd,

a

Graduate Institute of Applied Geology, National Central University, No 300, Jungda Road, Jungli, Taoyuan 32001, Taiwan

b

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

Article history:

Received 7 October 2009

Received in revised form

13 April 2010

Accepted 28 June 2010

Available online 10 July 2010

Keywords:

Permeability

Porosity

Specific storage

Effective confining pressure

Stress history

a b s t r a c t

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 (1014–1013m2) 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

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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

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 occasion-ally 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 pres-sure was about 120 MPa, which roughly equals the effective overburden of Cenozoic sediments in the Taiwan region

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n

Corresponding author at: Tel./fax: +886 3 4224114.

E-mail address: jjdong@geo.ncu.edu.tw (J.-J Dong).

(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

depen-dency 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

perme-ability and porosity of sandstone and shale, induced by

mechan-ical 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

sedimen-tary 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 coarsen-ing 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 metase-dimentary (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 labora-tory 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 proce-dures 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

K ¼2QmgL

A

Pd

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 Puand Pddenote 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  106Pa 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

R255 R261

R287

R316

R351

R390

R437 Ele 247m

Depth (m)

Chi-Chi Rupture

TCDP Hole-A Epicenter of Chi-Chi earthquake Sandstone Intensive bioturbation Major sand/minor silt Major silt/minor sand Siltstone or shale

1300 1200 1100 1000 900 800

R307

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.)

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 Pfattained 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

Pi1VsþPi2ðVlþVpÞ ¼PfðVsþVlþVpÞ: ð2Þ

If the volumes Vsand Vlcan be determined in advance, the pore volume of the sample can be calculated as follows:

Vp¼ Pi1Pf

PfPi2

Consequently, the sample effective porosity f can be calcu-lated fromf¼Vp=Vt, where Vtis the sample volume The volume

of the two isolated systems (volumes Vsand Vl) was minimized to enhance the accuracy of the pore volume measurement It is not easy to ‘‘directly measure’’ the volumes V and V Therefore,

Table 1

Descriptions of rock samples for permeability and porosity measurement.

Sample number Corrected depth (m) Rock type Dry density (g/cm 3

) Sample length/diameter (mm) Formation a Permeability Porosity

4.36/24.88 14.64/19.60 CL

a

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.

standard samples of hollow metal cylinders with known inner

diameters (the pore volume Vpfor standard samples is a known)

were used to calibrate the volumes Vsand Vlindirectly Two sets

of standard metal samples were used, with outer diameters of 20

and 25 mm From the calibrated results, we see that Vl¼0.625 ml

and Vs¼3.135 ml for the 20 mm diameter sample, while Vl¼0.604

ml and Vs¼3.126 ml for the 25 mm diameter sample

The effective confining pressure Peis defined as the difference

between the confining pressure P and the pore pressure P That

is, a Terzaghi effective pressure law (Pe¼PcPp) is adopted where the effective stress coefficient n in the general form of effective stress law (Pe¼PcnPp; [47–49]) is simply assumed as unity A sample-average pore pressure Pav¼2LðP2þPuPdþP2Þ=3ðPuþPdÞ was used to calculate the effective pressure The pore pressure for measuring the porosity is the balance pore pressure Pfin Eq (2) The average pore pressures for permeability measurement were 0.13–1.40 MPa and the balance pressures for porosity measure-ment were 0.30–1.41 MPa

hole

polyolefin jacket

10 Porous spacer (porosity)

11 Upper piston (porosility)

Pressure gauge P u

Steady state gas flow (N 2 )

P i2 ,V l +V p

Pressure guage Valve #1

Sample

P i1 , V s

Sample Flowmeter

Q, P d

1

2 3

3 4

5

6

9 10

11 Permeability

1 2

3 10

Porosity

Valve #2

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

Notably, the porosity measurements made using the above

method are sensitive to the pore space volume of the samples If

we consider Pi1¼2.0 MPa and use the given values of Pi2¼0.1

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

4 Experimental results

Fig 4 shows the permeability and porosity measurement

results The experiments were conducted first while gradually

increasing (loading) the confining pressure Pcfrom 3 to 5 MPa,

then to 10, and finally (in 10 MPa increments) to 120 MPa Pcwas

then gradually reduced (unloading) back to 3 MPa in the reverse

order The horizontal axis of Fig 4 is the effective confining

pressure Pe( ¼PcPp) 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 (10141013m2 and 15–19%) significantly exceeds

that of the silty-shale (10201015m2 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 Peis slightly smaller than the indicated value)

On the other hand, the permeability of silty-shale at Pc¼10 MPa

was one to two orders of magnitude smaller than that at

Pc¼3 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

quanti-tative 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

relation-ship 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

relation-ship for the stress dependent permeability can be expressed as

follows:

where K denotes the permeability under the effective confining

pressure Pe, Ko represents the permeability under atmospheric

pressure Powhich is assumed to be 0.1 MPa, andgis a material

constant David et al [10] reported that g¼9:8118:1 

103MPa1for five different sandstones The permeability under

atmospheric pressure is Ko¼2:17  1012m2for Boise sandstone (porosity 34.9%) whereas Ko¼1:486:48  1014m2for 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:

where p is a material constant For pure clay, rich and clay-free 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 Ko¼10181014m2 [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,gand

Ko,p) are listed in Table 2 The determined parameters in the exponential relationship for the sandstones under loading are

Ko¼5:857:08  1014m2 andg¼2:847:68  103MPa1 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 ofg 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 Koð ¼2:80  10191:45 

1016m2Þ and much higher values of gð ¼16:7843:47 

103MPa1

Þare 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 Kothe permeability of the silty-shale during loading was found to range from 3:34 

1018m2to 4:42  1013m2 The measured Kois 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 condi-tions and 0.057–0.114 under unloading condicondi-tions, 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 perme-ability Faulkner and Rutter [46] suggested that water perme-ability 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]:

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

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

bf¼ @f

Effective Confining Pressure (MPa)

1E-020 1E-019 1E-018 1E-017 1E-016 1E-015 1E-014 1E-013

2 )

1E-005 0.0001 0.001 0.01 0.1 1 10 100

Silty-shale

R255_sec2_1 R255_sec2_2 R287_sec1 R351_sec2 R390_sec3 R437-sec1

Fine-grained sandstone

R261_sec2_1 R261_sec2_2 R307_sec1

0

Effective Confining Pressure (MPa)

7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0

Silty-shale

R255_sec2_2 R287_sec1 R316_sec1 R351_sec2 R390_sec3 R437_sec1

Fine-grained sandstone

R261_sec2_1 R261_sec2_2 R307_sec1

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.)

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

fo¼13.8–34.9% and b¼0:443:30  103MPa1 for sandstone

under loading conditions Curve fitting for the porosity

measure-ments 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

fo¼17.06–17.67% andb¼0:911:58  103 MPa-1 These

mea-sured parameters fall within the range of the meamea-sured

parameters reported by David et al [10] For unloading, the

determined parameters for the sandstone arefo¼16.68–16.94%

andb¼0:691:15  103MPa1

A power law of the form

appears to better describe the relationship between the effective

confining pressure and the porosity of the sandstone and

silty-shale (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

(unload-ing) The value offoobtained for the power law is greater than

that for the exponential relationship For the tested sandstone

we find fo¼20.20–22.45% (loading) and fo¼18.52–20.14%

(unloading)

For the silty-shale, the measuredfois again higher for a power law (fo¼10.23–14.76% for loading and fo¼8.96–13.78% for unloading) than the exponential relationship (fo¼8.84–13.86% for loading andfo¼8.28–13.39% 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 ofb range from 0.41 to 1:58  103MPa1(loading) and 0.14 to 1:15 

103MPa1 (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 The stress dependent specific storage of sediments should be incorporated into fluid flow analysis in a sedimentary basin[5] The specific storage Sscan be expressed as follows[4]:

Ss¼ bj

wherebfandbfare the compressibility of the porosity and pore fluid, respectively The compressibility of the solid grains (  10 5 MPa1) is ignored in Eq (9) The compressibility of the porosity

bfis equal to @f=@Peand the compressibility of water is about

4  104MPa1[4] Using Eq (9), the stress dependent specific

Effective Confining Pressure (MPa)

4E-014 5E-014 6E-014 7E-014 8E-014

40 50 60 70 80

Experimental data points

Loading Unloading

Effective Confining Pressure (MPa)

1E-019 1E-018 1E-017 1E-016 1E-015 1E-014

2 )

2 )

0.0001 0.001 0.01 0.1 1 10

Curve fitting results

Power law (loading) Exponential relation (loading) Power law (unloading) Exponential relation (unloading)

0

Effective Confining Pressure (MPa)

15.5 16 16.5 17 17.5 18

Effective Confining Pressure (MPa)

8 8.5 9 9.5

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.

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

P o

Fine-grained sandstone

¼0.986

¼0.991

¼0.980 Silty-shale

0.416

¼0.981

¼0.988

¼0.977

¼0.968

¼0.975

¼0.950

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

Ss¼ fb

On the other hand, if the stress dependent porosity is

described as a power law, Eq (8), the specific storage can be

expressed as

Ss¼ fq

ð1fÞPe

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  103MPa1for 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  103MPa1 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 

103to 0:2  103MPa1for the sandstone, and from 0:7  103

to 0:07  103MPa1 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 (0:110  103MPa1) 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:

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,P o¼42:3  103ðMPa1Þ and r ¼0.823 for sandstones, and

Ss,P o¼11:5  103ðMPa1Þ 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 –

Effective Confining Pressure (MPa)

1E-005 0.0001 0.001 0.01

-1 )

Silty-shale

R255_sec2 R287_sec1 R351_sec2 R316_sec1 R390_sec3 R437_sec1

Fine-grained sandstone

R261_sec2_1 R261_sec2_2 R307_sec1

Specific storage model (Eq (10))

0

Effective Confining Pressure (MPa)

1E-005 0.0001 0.001 0.01

-1 )

Silty-shale

R255_sec2 R287_sec1 R351_sec2 R316_sec1 R390_sec3 R437_sec1

Fine-grained sandstone

R261_sec2_1 R261_sec2_2 R307_sec1

Specific storage model (Eq (11))

10 20 30 40 50 60 70 80 90 100 110 120

0

Effective Confining Pressure (MPa)

1E-005 0.0001 0.001 0.01

-1 )

Specific storage model (Eq.(12))

Sandstone: S s,Po =42.3x10 -3

(MPa) and r=0.823 Sandstone: S s,Po =11.5x10 -3

(MPa) and r=0.743

Silty-shale

R255_sec2 R287_sec1 R351_sec2 R316_sec1 R390_sec3 R437_sec1

Experiment results:

Fine-grained sandstone

R261_sec2_1 R261_sec2_2 R307_sec1

10 20 30 40 50 60 70 80 90 100 110 120

0 10 20 30 40 50 60 70 80 90 100 110 120

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 silty-shale (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.)