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 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 to 120 MPa and then subsequently reducing it back to 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 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 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 4224114 E-mail address: jjdong@geo.ncu.edu.tw (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 1142 J.-J Dong et al / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157 (a thickness of about 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 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 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 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 Meanwhile, the sandstone and shale samples are shown in Fig Two samples (R351_sec2 and R390_sec3) with sample lengths less than mm were selected for SEM observation after permeability experiments A carbon coater was used for coating the surface of the samples under vacuum conditions 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 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 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 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  10 À 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 1144 J.-J Dong et al / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157 Table 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 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 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 Lower piston (permeability) Spacer with a pore pressure hole 11 Porous spacers (permeability) Sample Pore-pressure-line connector Heat shrinkable flexible polyolefin jacket 10 Upper piston (permeability) Galvanized wire Lower piston (porosity) 10 Porous spacer (porosity) 11 Upper piston (porosility) 10 Fig (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 1146 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 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  10À12 m2 for Boise sandstone (porosity 34.9%) whereas Ko ¼ 1:48À6:48  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 , Experimental results Fig shows the permeability and porosity measurement results The experiments were conducted first while gradually increasing (loading) the confining pressure Pc from to MPa, then to 10, and finally (in 10 MPa increments) to 120 MPa Pc was then gradually reduced (unloading) back to MPa in the reverse order The horizontal axis of Fig is the effective confining pressure Pe( ¼Pc ÀPp) Fig 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 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 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  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 to 200 MPa, and from 0.4 to 0.9 as the effective pressure decreases (during unloading) from 200 to 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 The determined parameters in the exponential relationship for the sandstones under loading are Ko ¼ 5:85À7:08  10À14 m2 and g ¼ 2:84À7:68  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  10À19 À1:45  10À16 m2 Þ and much higher values of gð ¼ 16:78À43:47  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  10À18 m2 to 4:42  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 10 20 30 40 50 60 70 80 90 Effective Confining Pressure (MPa) 100 110 Permeability (md) 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 10 20 30 40 50 60 70 80 90 Effective Confining Pressure (MPa) 100 110 120 Fig 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 Permeability (m2) Permeability (m2) Experimental data points Loading Unloading 1E-014 10 1E-015 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) 0.0001 20 40 60 80 100 120 Effective Confining Pressure (MPa) 9.5 18 Porosity(%) Porosity(%) 17.5 17 16.5 8.5 16 15.5 20 80 100 120 40 60 Effective Confining Pressure (MPa) 20 40 60 80 100 120 Effective Confining Pressure (MPa) Fig 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  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 The parameter values determined for the sandstone under loading are fo ¼17.06–17.67% and b ¼ 0:91À1:58  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  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 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  10À3 MPaÀ1 (loading) and 0.14 to 1:15  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 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 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  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  10 À 14 R2 ¼0.855 R261_sec2_2 5.85  10 À 14 R2 ¼0.941 R307_sec1 7.08  10 À 14 R2 ¼0.880 Silty-shale R255_sec2_1 R255_sec2_2 R287_sec1 R351_sec2 1.01  10 À 18 R2 ¼0.378 2.40  10 À 18 R2 ¼0.423 6.06  10 À 18 R2 ¼0.868 7.07  10 À 19 R2 ¼0.647 Power law K ¼ Ko Unloading Loading h iÀp Pe Po Unloading R437_sec1 1.45  10 À 16 R2 ¼0.894 2.80  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  10 À 5.57  10 À 14 R2 ¼0.882 3.15  10 À 14 R2 ¼0.824 6.07  10 À 14 R2 ¼0.919 1.37  10 À 1.14  10 À 13 R2 ¼ 0.997 2.33  10 À 13 R2 ¼ 0.961 1.37  10 À 13 R2 ¼ 0.983 0.120 7.24  10 À 14 R2 ¼ 0.991 5.36  10 À 14 R2 ¼ 0.995 9.07  10 À 14 R2 ¼ 0.987 0.057 17.06 0.91  10 À R2 ¼ 0.923 17.67 1.58  10 À R2 ¼ 0.864 17.32 1.03  10 À R2 ¼ 0.900 4.66  10 À 19 R2 ¼0.259 5.68  10 À 19 R2 ¼0.300 2.18  10 À 19 R2 ¼0.625 2.07  10 À 19 R2 ¼0.628 7.91  10 À 5.95  10 À 17 R2 ¼ 0.730 1.80  10 À 15 R2 ¼ 0.837 1.15  10 À 14 R2 ¼ 0.987 3.99  10 À 17 R2 ¼ 0.952 0.844 3.48  10 À 18 R2 ¼ 0.549 2.83  10 À 17 R2 ¼ 0.692 1.93  10 À 18 R2 ¼ 0.928 1.93  10 À 18 R2 ¼ 0.971 0.416 7.68  10 À3 3.46  10 À 16.78  10 À 35.29  10 À3 43.47  10 À 25.93  10 À 2.65  10 À3 2.16  10 À 18.88  10 À3 10.58  10 À 13.90  10 À 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  10 À 22.78  10 À 2.41  10 À 18 R2 ¼0.871 4.84  10 À 4.42  10 À 13 R2 ¼ 0.993 3.34  10 À 18 R2 ¼ 0.838 1.744 0.588 5.94  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  10 À 0.94  10 À 1.04  10 À 1.30  10 À 1.01  10 À 0.41  10 À fo (%) b (MPa À 1) fo (%) h iÀq Pe Po Unloading q fo (%) q 16.68 0.69  10 À R2 ¼ 0.901 16.94 1.15  10 À R2 ¼ 0.969 16.87 0.75  10 À 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  10 À 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  10 À 0.82  10 À 0.54  10 À 0.65  10 À 0.14  10 À J.-J Dong et al / International Journal of Rock Mechanics & Mining Sciences 47 (2010) 1141–1157 Table 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  10À3 MPaÀ1 for the tested sandstone and shale when the confining pressure was increased from to 120 MPa Domenico and Mifflin [50] reported the specific storage of dense sand and medium-hard clay to be about 10 to 100  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  10À3 to 0:2  10À3 MPaÀ1 for the sandstone, and from 0:7  10À3 to 0:07  10À3 MPaÀ1 for the silty-shale, when the confining pressure is increased from 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 to 120 MPa Wibberley [23] demonstrated that the specific storage of fault gouges was reduced by approximately two orders of magnitude (0:1À10  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  10À3 ðMPaÀ1 Þ and r ¼0.823 for sandstones, and Ss,Po ¼ 11:5  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 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 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 (MPa) and r=0.743 10 20 30 40 50 60 70 80 90 100 110 120 Effective Confining Pressure (MPa) Fig 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 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 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  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  10À3 MPaÀ1 for sandstone here we obtained a much lower g ¼ 2:84À7:68  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 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 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 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 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 0.1 Silty-shale R255_sec2_2 R287_sec1 R316_sec1 R351_sec2 R390_sec3 R437_sec1 10 100 Effective Confining Pressure (MPa) 1000 Fig 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  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) Klinkenberg correction Kl (Loading) (Unloading) Power law (Loading) (Unloading) Permeability(m2) 7E-014 6E-014 5E-014 4E-014 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) 10 1E-014 (Unloading) Klinkenberg correction Kl (Loading) 1E-016 (Unloading) 0.1 Power law (Loading) (Unloading) 1E-017 0.01 1E-018 0.001 1E-019 Permeability(md) 10 20 30 40 50 60 70 80 90 Effective confining pressure (MPa) Permeability(md) 1154 0.0001 100 110 120 Fig 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 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 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 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 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  10 À 14 R2 ¼ 0.854 R261_sec2_2 5.26  10 À 14 R2 ¼ 0.941 R307_sec1 6.55  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  10 À 19 R2 ¼ 0.393 1.45  10 À 18 R2 ¼ 0.444 4.78  10 À 18 R2 ¼ 0.887 4.06  10 À 19 R2 ¼ 0.682 1.18  10 À 16 R2 ¼ 0.916 1.30  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  10 À 5.12  10 À 14 R2 ¼ 0.883 2.77  10 À 14 R2 ¼ 0.825 5.59  10 À 14 R2 ¼ 0.919 1.41  10 À 1.07  10 À 13 R2 ¼ 0.997 2.19  10 À 13 R2 ¼ 0.959 1.29  10 À 13 R2 ¼ 0.982 0.123 6.70  10 À 14 R2 ¼0.991 4.84  10 À 14 R2 ¼0.996 8.46  10 À 14 R2 ¼0.987 0.058 2.36  10 À 19 R2 ¼ 0.265 2.83  10 À 19 R2 ¼ 0.317 9.31  10 À 20 R2 ¼ 0.638 9.23  10 À 20 R2 ¼ 0.645 8.49  10 À 19 R2 ¼ 0.874 9.52  10 À 6.83  10 À 17 R2 ¼ 0.746 2.85  10 À 15 R2 ¼ 0.853 3.39  10 À 14 R2 ¼ 0.983 5.24  10 À 17 R2 ¼ 0.963 1.46  10 À 12 R2 ¼ 0.987 2.93  10 À 18 R2 ¼ 0.846 0.988 2.62  10 À 18 R2 ¼0.555 2.86  10 À 17 R2 ¼0.714 1.52  10 À 18 R2 ¼0.936 1.57  10 À 18 R2 ¼0.977 1.54  10 À 18 R2 ¼0.976 0.499 7.96  10 À3 3.57  10 À 19.83  10 À 41.16  10 À3 51.98  10 À 32.12  10 À 51.40  10 À3 2.78  10 À3 2.23  10 À 22.67  10 À3 13.66  10 À 17.88  10 À 6.45  10 À3 28.77  10 À 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 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 to 120 MPa caused a decrease in permeability by two orders of magnitude (2  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 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 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 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