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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113, B09211, doi:10.1029/2007JB005055, 2008

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Hydromechanical properties of Fontainebleau sandstone: Experimental determination and micromechanical modeling Insun Song1,2 and Jo¨rg Renner1 Received 19 March 2007; revised 29 April 2008; accepted 17 June 2008; published 26 September 2008.

[1] We measured ultrasonic velocity, hydraulic permeability, and specific storage capacity

of eight blocks of Fontainebleau sandstone, a well-sorted, medium-grained, almost pure quartz-sandstone of Oligocene age, covering a range in connected porosity from 3% to 10% and varying significantly in pore geometry for a given porosity. Ultrasonic P-wave velocity measured on water-saturated samples covers the full range predicted by variational Hashin-Shtrikman bounds. Permeability and specific storage were determined by the linear pressurization method at different effective pressures up to 180 MPa and room temperature. The permeability of tested samples varies from 1013 m2 down to 1020 m2, depending primarily on porosity and pore geometry, and subordinately on effective pressure. Specific storage capacity always exceeds the contribution of pore fluid compression. The excess corresponds to the contribution of pore deformability controlled by pore geometry and matrix material properties. Since the sandstone samples are composed of a single solid component, quartz, we were able to calculate various poroelastic parameters (Biot-Willis and Skempton coefficients, drained and undrained bulk moduli) crucial for estimating hydromechanical coupling in fluid-saturated aggregates. The micromechanical modeling of the effective pressure sensitivity of the hydraulic transport and poroelastic parameters by the contiguity model of Takei (1998) yields good agreement between static and dynamic elastic parameters for saturated samples, partial agreement between the pressure dependence of effective elastic parameters and Hertzian contact mechanics, and an explanation for pressure-insensitive permeability of samples with the highest porosity. Citation: Song, I., and J. Renner (2008), Hydromechanical properties of Fontainebleau sandstone: Experimental determination and micromechanical modeling, J. Geophys. Res., 113, B09211, doi:10.1029/2007JB005055.

1. Introduction [2] Subsurface fluid flow controls the efficiency of freshwater, oil and gas production, the transport of environmental contaminants, the recovery rates of geothermal energy, and a large range of geological processes. The shape and connectivity of hydraulic conduits, such as pores and (micro-) cracks on the grain scale but also joints and faults, are affected by the state of stress and its variations due to long-term tectonic loading or sudden stress redistribution associated with earthquakes and anthropogenic activities (e.g., reservoir impoundment, pumping operations). Two fundamental equations describe the hydromechanics in macroscopically homogeneous (i.e., on length scales much larger than individual grains or pores), isotropic porous media saturated with a single fluid phase. The linear poroelastic constitutive equations link components eij of the strain tensor 1 Institute for Geology, Mineralogy, and Geophysics, Ruhr-University Bochum, Bochum, Germany. 2 Now at Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania, USA.

Copyright 2008 by the American Geophysical Union. 0148-0227/08/2007JB005055$09.00

and pore fluid increment per unit volume of the composite z with components sij of the stress tensor and pore fluid pressure pf [e.g., Detournay and Cheng, 1993; Pride, 2005; Wang, 2000]:

eij ¼

1 1 1 a d ij pf d ij skk þ sij 2G 6G 9Kd 3Kd a skk 1 z¼ þ pf : Kd 3 B

ð1Þ

ð2Þ

[3] The tensors sij and eij correspond to bulk stresses and strains of the solid matrix, respectively, with compressive stresses and strains negative. Four independent parameters characterize the fluid-saturated aggregate: shear modulus G and drained bulk modulus Kd are elastic properties of the solid skeleton, and the Biot-Willis coefficient a [Biot and Willis, 1957] and Skempton coefficient B [Skempton, 1954] are poroelastic coupling parameters. Combining equation (2) with Darcy’s law for fluid flow and mass conservation yields the hydromechanical diffusion equation for pore fluid

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SONG AND RENNER: FONTAINEBLEAU SANDSTONE HYDROMECHANICS

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flow when no explicit fluid source is present [e.g., Pride, 2005; Rice and Cleary, 1976]: r2

B 1 @ B skk þ pf ¼ skk þ pf : 3 k @t 3

ð3Þ

[4] The central parameter, hydraulic diffusivity k = k/(sh), specifies the ratio between transport and storage, represented by permeability k divided by fluid viscosity h and specific storage capacity s, respectively. The above equations constitute continuum mechanical descriptions, i.e., all parameters represent effective properties describing relations between bulk phenomena in fluid flow and poroelasticity, neglecting the structure of solid constituents and pores on the microscale. [5] Constraining effective properties by micromechanical modeling has received great attention in the past given their central importance for quantitative understanding of hydromechanical processes. Clearly, such modeling has to go beyond accounting for the volume proportions of solid and liquid components by including geometric details of the phase arrangement. Models for pore geometry traditionally differ when addressing fluid transport or elastic deformation parameters. Bernabe´ [1991] proposed to distinguish three categories, spherical pores (or nodal pores), tube-like throats at three-grain edges, and sheet-like conduits at two-grain faces. Tubes and sheets dominate transport properties. The nodes contain the largest amount of fluid and thus dominate the fluid’s contribution to storage capacity. Yet, the compliant sheet-like conduits are also quite sensitive to changing pressures in contrast to the spherical and tube-like pores. [6] Besides the many more approaches to effective elastic properties employing inclusions of different shapes in (isotropic) solid media [e.g., Budiansky and O’Connell, 1976; Mavko, 1980], grain contiguity 8 was suggested as a microstructural parameter capable of combining the effects of porosity and pore topology [Takei, 1998]. [7] Experimental studies are crucial for benchmarking of derived model equations. Fontainebleau sandstone has frequently been used to study the relation between pore geometry and effective properties [e.g., Bourbie and Zinszner, 1985; David and Darot, 1989; Doyen, 1988; Fredrich et al., 1993; Song and Renner, 2006a, 2007]; its simple composition facilitates the comparison of experimental results with predictions of models. Here, we report measurements of hydraulic permeability and specific storage capacity of eight varieties of Fontainebleau sandstone that vary in porosity and pore structure. The employed linear pressurization method provides accurate simultaneous measurements of permeability and storage capacity [see Song and Renner, 2006b]. The latter can be used to compute effective poroelastic parameters. We test the ability of the grain contiguity concept [Takei, 1998] to model observed pressure-induced variations of both, transport and storage properties.

2. Material and Methods 2.1. Sample Material [8] Experiments were performed on cylindrical samples cored and ground to 30 mm diameter and 60 mm length

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from eight blocks of Fontainebleau sandstone, a quartz arenite of Oligocene age collected in the Ile de France region around Paris [e.g., Haddad et al., 2006]. Three distinct cementation characteristics are found (Figure 1). (1) Grains of two blocks (FS1 and FS6) can be rubbed off by hand. Scanning electron microscopy reveals that the strikingly serrated grain boundaries are essentially open in these blocks (Figure 1a). The serration in two-dimensional sections results from quartz-cement with micron-sized crystal faces (Figure 1b). (2) In some blocks, boundaries between neighboring grains cannot be discerned by electron microscopy; in these the pore shape suggests that entire grain edges are constituted by rational faces mimicking the hexagonal habitus of quartz (Figure 1c). (3) In the last group of blocks, the pores between such idiomorphic grains are partially filled by a further, coating-like quartz-cement (Figure 1d). [9] Preparation of the blocks with little internal cohesion was difficult and corresponding samples deviate from perfect cylindrical geometry. Porosity and velocity values were measured on at least 4 samples of each block at atmospheric pressure while hydraulic tests were performed only on a single sample of each block that we simply identify by the block number in the following. Total porosity ftot calculated from the ratio between geometrically determined sample density and the crystal density of quartz (rqtz = 2630 kg/m3) ranges from 3 to 14% with moderate variation in a given block. Connected porosity fcon was determined by the weight gain upon saturating samples with water under vacuum. Nonconnected voids could not be resolved within the average accuracy of ±0.5% characteristic for the two methods of porosity determination (Figure 2a). Determining travel times of axially transmitted ultrasonic waves revealed that P-wave velocity of water-saturated samples shows little variation for a given block (Figure 2b) but the entirety of the data covers the full range between the upper and lower HashinShtrikman bounds [Hashin and Shtrikman, 1963] calculated for a two-phase material composed of quartz and water. Throughout the paper we use the Hashin-Shtrikman bounds derived for bulk and shear moduli of two-phase aggregates to calculate bounds for related properties, such as velocity or poroelastic parameters, as was previously done for the various compressibilities of a two-phase aggregate [see Zimmerman et al., 1986]. 2.2. Experimental Procedure [10] Our test arrangement consists of the pore fluid system, a confining pressure vessel, and a personal computer-aided control device. In the pore fluid system, a saturated sample is located between two reservoirs, downstream and upstream, connected to two hydraulic pressure intensifiers (Figure 3). The operation of the pore pressure intensifiers was conducted servohydraulically. Two displacement and three pressure transducers measured the position of each intensifier piston and the fluid pressures of the reservoirs and the vessel, respectively. These signals were digitally recorded and were available as feedback signal for the control unit. The intensifier piston velocity vip determined from the position measurement is converted to the upstream flow rate V_ u = vipAip using the known intensifier piston area Aip.

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Figure 1. Scanning electron microscope images of polished surfaces of samples (a) FS6 and (b) secondary electrons, (c) FS7 (backscattered electrons), and (d) FS8 (backscattered electrons). Samples FS6 and FS7 have a similar porosity of 9% but significantly different ultrasonic P-wave velocities (Figure 2), while FS8 has a lower porosity of 5.6% but an ultrasonic velocity similar to FS7. Quartz and pores (filled with epoxy) appear light and dark gray, respectively. Grains are poorly attached to their neighbors in variety FS6 (Figure 1a). The image of actual grain surfaces of sample FS6 (Figure 1b) demonstrates that the serrated grain boundaries are related to authigenic quartz cement exhibiting low index crystal faces. In contrast, grain boundaries can hardly be discerned in variety FS7 (Figure 1c), in which pores are often formed by crystal faces characteristic of the hexagonal habitus of quartz. The pores between such idiomorphic grains are partially filled by a further, coating-like cementation in sample FS8 (Figure 1d).

[11] Before tests, samples were saturated with degasseddistilled water under vacuum at room temperature for at least 24 hours. A saturated sample with end plugs at both ends held by a rubber jacket was inserted into the pressure vessel. After subjecting the sample to a hydrostatic oil confining pressure of up to 200 MPa, upstream and downstream reservoirs were simultaneously pressurized using the auxiliary intensifier while holding the upstream volume constant by keeping the position of the main intensifier piston fixed using position control. After equilibration of pore pressure throughout the sample, we disconnected the auxiliary intensifier from the pore fluid system and disconnected the upstream from the downstream reservoir by closing the corresponding valves. Then, the upstream pres-

sure was increased with a constant rate using the main intensifier in pressure control mode. For comparison, we also performed some classical steady state measurements of permeability by imposing a constant pressure difference between the two sample ends (Darcy’s method). Experiments were performed at successively increasing pore fluid pressures corresponding to decreasing effective pressures. For a few samples, we also determined hydraulic properties at successively decreasing effective pressure, applying a constant depressurization rate at each step. [12] Determination of hydraulic properties from transient methods requires precise calibrations of reservoir storage capacities, defined as the proportionality constant between fluid flow into or out of a reservoir and the resulting change

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Figure 2. (a) Cross plot of open and total porosity; error bars indicate standard deviation found for four samples; accuracy is approximately ±0.5%. The discrepancy between total and connected porosity of block FS1 is probably related to problems with preparation of cylindrical samples. (b) Ultrasonic P-wave velocities of the eight tested varieties of Fontainebleau sandstone as varying with total porosity. For comparison, upper and lower Hashin-Shtrikman bounds (HS+, HS) are presented. Measured velocities are within isotropic bounds, except for variety FS3. We identify three groups with 4%, 5.5%, and 9% porosity. in fluid pressure. Replacing the rock specimen with an impermeable steel cylinder, we could separately calibrate the storage capacity of the total fluid system (Stot = Su + Sd) and the upstream reservoir (Su) by pressurizing the pore fluid system with open and closed bypass valve, respectively. The dominant contribution to the storage capacity of the reservoirs comes from the compressibility of the fluid (Cf), e.g., Si ’ CfVi. Using the auxiliary pressure intensifier

ensures identical reservoir storage capacities for calibrations and actual tests. 2.3. Determination of Hydraulic Properties Using the Linear Pressurization Technique 2.3.1. Principle [13] Linear pressurization testing was recently proposed for simultaneously measuring the hydraulic permeability

Figure 3. Sketch of the permeameter. The components of the upstream and downstream reservoirs are distinguished by dark and light gray shading, respectively. 4 of 16


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Figure 4. Examples of test records for pressurization and depressurization tests on FS4 at a confining pressure pc = 170 MPa. (a) For the pressurization (infusion) test, the upstream fluid pressure was linearly increased with a pressurization rate of 1 MPa/s starting from 50 MPa until 70 MPa was reached. The retarded response of the downstream pressure pd is characterized by an increasing rate during a transient phase before the rate imposed at the upstream reservoir is reached. The differential pressure (Dpud = pu pd) increases rapidly during the transient stage and eventually reaches a nearly constant value. After pressurization of the upstream reservoir is terminated, the downstream pressure approaches asymptotically the upstream pressure fixed by the control system. (b) During the subsequent depressurization (withdrawal) test, the upstream pressure decreased at a constant rate until the initial pore pressure of 50 MPa was reached.

and specific storage capacity of rock samples [Song et al., 2004a, 2004b; Song and Renner, 2006b]. The evaluation employs solutions of the one-dimensional diffusion equation for appropriate boundary and initial conditions. Equation (3) simplifies to a diffusion equation for pore fluid pressure alone when hydromechanical coupling is linearly approximated and inertia effects in the fluid are neglected [e.g., Pride, 2005]. Whether fluid injection is conducted with constant flow [Song et al., 2004a] or pressurization rate [Song and Renner, 2006b], the same asymptotic solution applies giving specific storage capacity s and fluid permeability k as s¼

1 V_ u ðSu þ Sd Þp_ u ALp_ u

! 1 hL V_ u ðSu Sd Þp_ u ; k¼ Dp1 2A ud

ð4Þ

ð5Þ

respectively. Here, A, L, p_ u, V_ 1 u , and Dpud denote the cross-sectional area and length of the sample, the upstream pressurization rate, and the asymptotic values of the upstream flow rate and of the difference between upstream and downstream pressure, respectively. The latter three _ u) quantities are obtained from linear regressions (V_ 1 u and p and simple averaging (Dpud) over the appropriate segment of the test record. For the majority of tests, initial and final upstream pressure of a single (de)pressurization differed by 10 MPa, but in some by 20 MPa. The initial transient stage during a (de)pressurization was excluded for the calculation

of hydraulic properties, thus, the pore pressure range of the _ u, and Dpud did not segment actually used to obtain V_ 1 u , p exceed 3 MPa in most cases. In the following, hydraulic parameters are given for the average pore pressure value in the segment where the asymptote is reached. Furthermore, we represent our results for the specific storage capacity normalized by the fluid compressibility, i.e., s/Cf. For a macroscopic isotropic sample, this ratio is related to porosity according to Cpp s ; ¼ fcon 1 þ Cf Cf

ð6Þ

where Cpp denotes the compressibility of pore space as a result of a change in pore pressure [Zimmerman et al., 1986]. Thus, connected porosity should constitute a lower bound for the ratio because Cpp 0 where the equality is reached for a perfectly rigid pore space. 2.3.2. Pressure Dependence and Hysteresis [14] The analysis of hydraulic tests usually relies on the diffusion equation (3) that is exactly valid only for pressureindependent material properties. Clearly, when cracks constitute a significant portion of the hydraulic conduits, permeability and storage capacity of rocks are sensitive to pressure variations. To stay within the validity of the approximation, the pressure perturbation necessary to induce flow is kept small in a single test and the pressure dependence of hydraulic properties is constrained from a sequence of experiments for which the effective pressure (peff pc pf) is successively changed in steps that exceed the perturbation. In steady state flow tests (according to Darcy’s method), the problem of properties evolving with

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Figure 5. (a) Permeability of sample FS4 as dependent on effective pressure. The linear pressurization and Darcy tests were performed at pc = 170 and 157 MPa, respectively. The symbol shape indicates flow direction, i.e., a pressurization test corresponds to a Darcy test with flow from the upstream to the downstream reservoir. (b) Normalized specific storage capacity of FS4 as dependent on effective pressure. The average and standard deviations in porosity are indicated by the dashed lines; accuracy is approximately ±0.005.

pore fluid pressure is eliminated because fluid pressure does not change during the test. Yet, a spatial variation (i.e., along the sample axis) may still affect the analysis. We suppose that the best possible constraint on the pressure dependence of hydraulic properties comes from our Darcy tests with pressure differences applied of only 2 MPa. [15] The change in fluid pressure during a linear pressurization test may exceed the typical perturbations applied during alternative transient tests, pulse tests [Brace et al., 1968] or oscillatory tests [Fischer, 1992; Kranz et al., 1990]. In fact, a constant pressure difference between upstream and downstream reservoir, Dp1 ud, was not observed for some samples but a small decrease with time prevailed, i.e., @Dp1 ud/@t < 0 (Figure 4), which was neglected for the analysis. Still, the pressure dependence of permeability determined from sequential linear pressurization tests compares closely to the results of sequential steady state tests (Figure 5a). [16] While permeability values can be checked against results of steady state tests, benchmarking is not possible for specific storage capacity [see also Tokunaga and Kameya, 2003; Wang and Hart, 1993]. The classical pulse method [Brace et al., 1968] totally neglects storage capacity and oscillatory tests encounter serious uncertainties at certain conditions or exhibit an as yet not fully understood period dependence [Bernabe´ et al., 2004; Song and Renner, 2007]. The relative uncertainty of storage capacity determined by the linear pressurization method strongly depends on the storage capacity of the reservoirs. In this study we ensured a _1 _ u) < 60 most likely keeping the ratio V_ 1 u /(V u Stot p relative uncertainty to ds/s < ±30% [see Song and Renner, 2006b, Figure 7a]. We note that the difference between

pressurization and depressurization tests remains within the scatter of specific storage capacity (Figure 5b). 2.4. Calculation of Poroelastic Parameters [17] For hydrostatic confining pressure pc, equations (2) and (3) are expressed in terms of volumetric components of stress pc = skk/3 and strain q = ekk: 1 ðpc apf Þ Kd

ð7Þ

a 1 pc pf : Kd B

ð8Þ

q¼

z¼

The associated effective parameters, drained bulk modulus Kd, Biot-Willis coefficient a (nb in Zimmerman et al. [1986]), and Skempton coefficient B, are related to the specific storage capacity and properties and proportions of the constituents according to [e.g., Ku¨mpel, 1991; Zimmerman et al., 1986]

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1 Cbc ¼ s fcon Cf þ ð1 þ fcon ÞCr ; Kd

ð9Þ

a Cbp ¼ s fcon ðCf Cr Þ; Kd

ð10Þ

a s: Kd B

ð11Þ


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Figure 6. Results of the quantitative microstructure analysis. (a) Distributions of Feret diameters of grains (bars) and pores (open circles) and coordination numbers of grains. (b) Correlations between average Feret diameter of grains and two-dimensional coordination number and connected porosity. The dashed line indicates the average coordination number expected for two-dimensional sections of aggregates composed of space-filling polyhedra (5 1/7) [Smith, 1952]. The undrained bulk modulus Kud can be calculated as [e.g., Wang, 2000] Kud

dpc Kd ¼ : dq z¼0 1 aB

ð12Þ

[18] For our calculations, measurements of specific storage capacity s and connected porosity fcon are combined with the well constrained compressibility of the rock matrix, Cr ’ Cqtz = 2.64 1011 Pa1 [Gebrande, 1982], and the pore fluid, Cf, obtained using the software package FLUIDCAL (http://www.ruhr-uni-bochum.de/thermo/ Software/Seiten/Fluidcal-eng.htm) [Wagner and Pruß, 2002].

Figure 7. (a) Normalized specific storage capacity and (b) pore space compressibility calculated according to equation (6) as dependent on effective pressure. 7 of 16

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Figure 8. (a) Permeability and (b) permeability normalized by the permeability observed for 100 MPa effective pressure as a function of effective pressure. 2.5. Quantitative Microstructure Analysis [19] Scanning electron microscope (SEM) images and polarized light microscope (transmission) microphotographs of thin sections were digitized to determine average phase proportions, grain and pore geometry as well as the coordination number of grains (Figure 6). Samples are devoid of any significant shape preferred orientation. Average grain and pore sizes as expressed by Feret diameters, i.e., the longest line fully inside the two-dimensional sections of the grains/pores, range between 180 and 270 mm and 20 and 50 mm, respectively. Grain size distributions are fairly symmetric Gaussian distributions. Pore size distributions exhibit long tails toward pore sizes comparable to the largest grain sizes. The average coordination numbers determined from 2D sections, Z2D, do not deviate significantly from the average coordination number for space-filling polyhedra of 5 1/7 [Smith, 1952] but tend to decrease slightly with increasing porosity (Figure 6b). The grain diameter exhibits a weak anticorrelation with porosity possibly related to cementation as the pore space-reducing mechanism.

3. Experimental Results [20] We conducted hydraulic tests using the linear pressurization method on seven samples at a range of effective pressures from 20 to 180 MPa. In the following, we first represent results for specific storage capacity, then for permeability, and finally for poroelastic parameters. 3.1. Specific Storage Capacity and Pore Compressibility [21] Values of normalized specific storage capacity s/Cf ranging between 0.05 and 0.5 exceed the corresponding sample porosity as expected from equation (6) (Figure 7a). Despite data scatter for some samples it is obvious that all samples but one (FS7) exhibit a decrease in storage capacity with increasing effective pressure by at least a factor of 2.

The sensitivity to changes in effective pressure decreases with increasing pressure for most samples. The magnitude of s/Cf depends on sample porosity, i.e., ratios are higher for samples with higher porosity. Yet, significant variations occur within a group of samples with similar porosity. [22] The pore compressibility, Cpp, quantifies changes in pore volume due to changes in pore fluid pressure. In comparison to storage capacity, the contribution of the compressible water in the pores is removed and Cpp characterizes the contribution of pore geometry alone (see equation (6)). Pore compressibility decreases notably with increasing effective pressure except for FS7 (Figure 7b). The decrease levels off with increasing effective pressure. Note that for the calculation we neglected the variation of porosity with effective pressure but used the connected porosity measured at ambient conditions. Owing to this procedure, Cpp values are increasingly underestimated with increasing effective pressure. However, the relative error remains well below 10% for the highest values of applied pressure and encountered pore compressibility. 3.2. Permeability [23] Our experiments reveal a large range of permeability values from 1020 to 1013 m2 for the varieties of Fontainebleau sandstone covering a range in connected porosity from >3 to 10%. The primary variable for permeability is porosity, i.e., high permeability values ranging from 1014 to 1013 m2 correlate with large porosity (8 10%), lower permeability values are determined for samples with smaller porosities (Figure 8a), but notable exceptions occur. For example, the permeability of FS8 with intermediate porosity (5.6%) is much lower than that of sample FS4 with smaller porosity (4.0%). [24] Only two samples (FS3 and FS5) exhibit a pronounced sensitivity of permeability to effective pressure (Figure 8b); for both, the decrease in permeability with increasing effective pressure prevails uniformly over the

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Figure 9. (a) Biot-Willis and (b) Skempton coefficients calculated according to equations (9) – (11) as dependent on effective pressure.

entire range in pressure and amounts to more than an order of magnitude. For the other samples, permeability values do not vary by more than a few percent. 3.3. Poroelastic Parameters [25] As for pore compressibility we neglected the change in connected porosity with effective pressure when calculating the poroelastic parameters. The values of the Biot-Willis coefficient, a, and Skempton coefficient, B, calculated using equations (9)– (12) are fairly similar and depend on porosity (Figure 9). In contrast to ultrasonic velocity that assumes values covering the entire range between the Hashin-Shtrikman (HS) bounds (Figure 2b), most of the values of these coefficients plot in the middle between the HS bounds. Only for the lowest pressures, the observed coefficients approach their respective upper HS bound (corresponding to the lower HS bound for velocity). In particular, observations for samples that exhibit velocities near the upper HS bound stay well separated from the corresponding lower bounds for a and B (e.g., FS7). Similar relations to the variational bounds are also found for the drained and undrained moduli, Ku and Kd (Figure 10). Extrapolating the pressure dependence of drained moduli to ambient pressure yields moduli values that correlate with average ultrasonic P-wave velocities measured at ambient conditions. Yet, samples with distinctly different ultrasonic velocity at ambient pressure exhibit almost identical poroelastic parameters at high effective pressure due to differences in sensitivity to effective pressure.

4. Modeling Pore Geometry by Grain Contiguity [26] In the following, we perform a micromechanical analysis of the pressure dependence of storage capacity and permeability, focusing on two main experimental observations: (1) at a given porosity, the two hydraulic

parameters vary significantly, and (2) the relative sensitivity of storage capacity to changes in effective pressure is significantly larger than that found for permeability. Samples with similar porosity exhibit strikingly different behavior probably because of variations in pore geometry. For example, the insensitivity of poroelastic parameters of sample FS7 to changes in effective pressure (Figure 7 and 9) corresponds well to the high stiffness of pores inferred from its ultrasonic velocity close to the upper HashinShtrikman bound (Figure 2b) and to the absence of grain separations with large aspect ratio in SEM images (Figure 1c). The significant difference between the values of s/Cf for FS6 and FS7 observed at the lowest pressures diminishes with increasing pressure (Figure 7a) suggesting that the grain separation found at ambient pressure in FS6 (Figure 1a and 1b) can be quite effectively reduced with increasing effective pressure. Early 2D work by Walsh [1965] analyzed the relation between the compliance of voids and their aspect ratio and proposed a quantitative relation for critical pressures necessary to close voids of a given aspect ratio. Most of the decrease in pore compressibility with increasing effective pressure is observed for peff < 50 MPa (Figure 7b) corresponding to aspect ratios on the order of c Eqtz/pclosure < 2000 [Mavko and Nur, 1978; Walsh, 1965]. Approximating the long crack dimension by the average grain size of 200 mm (Figure 6) yields an initial crack width of only 0.1 mm that appears low in comparison to the grain separation observed on SEM images (Figure 1a and 1b). The use of Young’s modulus for quartz is a conservative upper bound for the bulk elastic behavior of the porous aggregate yielding a lower bound for the initial crack width. [ 27 ] The variations in pore geometry can also be expressed by variations in grain contact geometry. In the following, we employ the parameter grain contiguity introduced by Takei [1998] to model our observations. Grain contiguity 8 quantifies the ratio between a grain’s surface

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Figure 10. (a) Drained and (b) undrained bulk moduli calculated according to equations (9) and (12) as dependent on effective pressure. The extrapolations of drained moduli to ambient pressure correlate with average ultrasonic P-wave velocities given in the legend. Variational Hashin-Shtrikman bounds (+, upper; , lower) are given for comparison. Note that the lower bound is identical ‘‘0’’ for the drained modulus. area in contact with neighboring grains and its total surface area and thus combines effects of total porosity and contact geometry. We use the determined drained moduli (Figure 10a) to calculate contiguity relying on the numerical approximation presented in the appendix of Takei [1998]. In particular, we approximate the contiguity at ambient pressure, 80, by extrapolating the relation between drained bulk modulus and effective pressure to zero effective pressure. 4.1. Relation Between Static and Dynamic Elastic Parameters [28] The elastic parameters derived from the hydraulic experiments constitute static values in the sense that linear pressurization tests involve timescales and impose strains comparable to the ones employed in triaxial tests to deduce static elastic moduli but strikingly different from the dynamic ultrasonic measurements. We use the estimates for the contiguity at ambient pressure, 80, as the mediator for a comparison between storage capacity (static) and ultrasonic velocity (dynamic) by comparing respective pairs of measured ultrasonic velocity vP and extrapolated initial contiguity 80 with predictions by the modeling of elastic moduli by Takei [1998] transformed into velocity values in the Biot-Gassmann low-frequency limit:

vPð8;fÞ

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 u uKuð8;fÞ þ Gð8Þ t 3 : ¼ rðfÞ

ð13Þ

The undrained modulus depends on the fluid compressibility according to Ku(8,f) = Kd(8) + (Kr Kd(8))2/[(1 f)Kr Kd(8) + fK2r /Kf], where the density of the aggregate r(f) = (1 f)rr + frf. The Biot-Gassmann low-frequency limit is defined by wrf k/h 1 [e.g., Mavko et al., 1998; Pride,

2005], where w denotes the angular frequency of the wave. Except for the most permeable samples for which wrf k/h 1, the low-frequency condition fully holds for our tests, i.e., pore fluid pressure variations are equilibrated within the pore space but samples remain undrained with respect to the surrounding environment on the timescale of wave propagation. [29] The correspondence between measured ultrasonic velocities and predictions of the contiguity model [Takei, 1998, 2002] is fair for dry samples and quite good for saturated samples (Figure 11), except for sample FS8 with 5.6% porosity exhibiting the internal coating-like cement (Figure 1d) which has been suggested to be at least partly amorphous [Haddad et al., 2006; Page et al., 2004]. The agreement for saturated samples is remarkable considering that the velocity data cover the entire range between Hashin-Shtrikman bounds (Figure 2b) and that the contiguity model relies on an analysis of a representative grain. The latter assumption appears in conflict with the documented grain-scale variability of the sandstone samples (Figure 1). The success of the predictions is probably related to their documented insensitivity to coordination number [Takei, 1998, 2002]. The better agreement between experimentally determined and theoretically predicted velocity values for water-saturated than for dry samples may result from a weaker effect of flaws (fractures) that exceed the grain scale in saturated samples than in dry samples due to the lesser contrast in compressibility between grain matrix and water than between grain matrix and air. 4.2. Effect of Pressure on Contiguity [30] The contiguity approach appears to reconcile static and dynamic measurements of elastic parameters. Is contiguity indeed the appropriate (microstructural) state variable for addressing changes in physical properties with changing

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Figure 11. Comparison between measured ultrasonic P-wave velocities for (a) dry and (b) watersaturated samples and predictions of the contiguity model [Takei, 1998]. Contiguities at ambient pressure 80 were determined from an extrapolation of the effective pressure dependence of drained moduli (Figure 10a) to peff = 0. The model predictions were calculated according to equation (13) using the numerical approximations of the moduli-contiguity relations given in the appendix of Takei [1998]. thermodynamic conditions? Two typical relations between contiguity and effective pressure are found (Figure 12). Either contiguity increases approximately proportionally to p2/3 eff or it is pressure-independent, i.e., exhibits a plateau. The plateaus may be preceded and/or succeeded by an increase. [31] An increase in contiguity (area of contact) with increasing effective pressure as p2/3 eff is in accord with Hertzian contact mechanics. Pressure-insensitive behavior, plateaus, can be associated with truncated spheres. When a sphere of radius R with a flat truncation of radius afl is pushed on an elastic half-space of identical elastic properties, the radius a of the contact area changes with normal load Fn on the sphere according to Fn ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E* 2 2a þ a2fl a2 a2fl ; 2R

ð14Þ

with E* = 4E/(3 3n 2) [see Maugis and Barquins, 1983; Meyer, 1989] where E and n denote Young’s modulus and Poisson’s ratio, respectively. The classical Hertzian result of two identical spheres in contact is recovered for afl a. Equation (14) reformulated in terms of contiguity and confining pressure reads pc

Fn E*a31 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð28 þ 8fl Þ 8 8fl ; 2 pR pR3

ð15Þ

where a1 denotes the radius of the maximum contact area for contiguity 8 = 1, 8fl = a2fl/a21 is the initial contiguity corresponding to the truncated flat before loading, and 8 = a2/a21 is the current contiguity at pressure pc. Two regimes can be distinguished for equation (15) (see Figure 12):

Figure 12. Contiguity derived from observed drained bulk moduli as dependent on effective pressure. The solid line labeled ‘‘spherical Hertzian contacts’’ indicates the expected pressure dependence according to Hertzian contact theory of spheres. The straight dashed lines repeat the power law relation of spherical Hertzian contacts to indicate that its slope is obeyed over certain pressure intervals that may be succeeded or preceded by plateaus. The curved line labeled ‘‘truncated 80 = 0.26’’ represents micromechanical modeling of grain contacts by truncated elastic spheres, according to equation (15).

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behavior affected by porosity. An increase of contiguity with pressure before a plateau cannot be modeled by equation (15) but requires an additional compliant element such as roughness of contact areas. Contact areas of grains in some samples deviate from perfectly flat as documented by the SEM observations (Figure 1b); authigenic cement modulates the grain contacts and reduces the actual area of contact. While the modeling by Takei [1998], on which our estimates of contiguity are based, envisioned ‘‘smooth’’ grain surfaces such that a single ‘‘holeless’’ contact area exists between neighboring grains such restrictions are not actually employed in the determination of the relation between contiguity and elastic parameters. The observed p2/3 eff relation preceding a plateau suggests that ‘‘asperities’’ or roughness reduce the effective elastic properties of the contact while bulk behavior is still linear elastic and that their cumulative effect can still be represented by an integrated contiguity.

Figure 13. Permeability normalized by grain size squared of the tested sandstone varieties as dependent on connected porosity. For normalization, we used Feret diameters, as determined from two-dimensional sections without a stereological correction. The symbols represent permeability values gained from an extrapolation of measured values to zero effective pressure. The bold lines emerging from the symbols represent the relation between permeability and porosity derived from tests at various effective pressures. The variation of porosity with effective pressure was derived by integrating the effect of the observed pore compressibilities over the course of an experiment. The dashed line labeled k = f3d2/330 represents the wellestablished cubic relation between permeability and porosity for f 0.08 [Bourbie and Zinszner, 1985]. The dotted lines represent modeling, according to equation (18), for two different percolation thresholds, fc.

(1) low pressures have an insignificant effect and the initial radius of the flat characterizes the contact area and (2) above a characteristic pressure, the radius of the flat increases pffiffiffi similar to a Hertzian contact. Approximating a1 ’ 2R/ Z and Z ’ 12 yields a prediction for the pressure sensitive regime of 8¼

pp 2=3 c

4E*

Z ’ 4 103 MPa2=3 p2=3 c

ð16Þ

modeling the elasticity of individual grains with VoigtReuss-Hill averaged elastic properties of quartz, Eqtz = 95.3 GPa and n qtz = 0.08 [Gebrande, 1982]. The predicted power relation is found between contiguity and effective pressure peff = pc pf (Figure 12). However, observations exceed the estimate (16) by up to an order of magnitude. The quantitative disagreement probably originates from using single crystal elastic parameters for the estimate while samples may exhibit enhanced compliance due to locally amorphous cements [Haddad et al., 2006] and bulk

4.3. Permeability [32] Great effort has been spent on predicting fluid permeability from a characterization of pore geometry [e.g., Andrade et al., 1997, 1999; Bernabe´ and Bruderer, 1998; Berryman and Blair, 1987; Koponen et al., 1996; Martys and Garboczi, 1992; Schwartz et al., 1993; Walsh and Brace, 1984]. From a dimensional analysis, it is generally concluded that permeability must be related to the square of a characteristic length of hydraulic conduits. Indeed, permeability obeys k / r2ct for laminar viscous flow in simple capillary tubes of radius rct. Of the large number of permeability models in the literature (see reviews by Bernabe´ and Bruderer [1998] and Bernabe´ et al. [2003]), the most commonly used one is the Kozeny-Carman relation, k/

f3 ; S2

ð17Þ

where S denotes the specific surface area defined as the ratio of the pore surface area to the total volume of the pore. The proportionality ‘‘constant’’ is determined by interconnectivity and tortuosity of the conduit network. Originally developed considering networks of tubes equation (17) has often been applied to other geometries. Spheres of diameter d yield Sspheres = 1.5(1 f)/d, i.e., a dependence of permeability on grain size squared, as verified for Fontainebleau sandstone [Bourbie et al., 1987]. [33] The relation between permeability and porosity for high porosity (f 0.08) samples of Fontainebleau sandstone is consistent with the Kozeny-Carman model [Bourbie and Zinszner, 1985; Song and Renner, 2006a]. For samples with low porosity (f < 0.08), however, the porosity exponent significantly exceeds 3 (Figure 13) [see also Mavko and Nur, 1997]. The contribution of occluded or deadend pores to total porosity increases rapidly with decreasing porosity as the pore network approaches the percolation threshold [Katz and Thompson, 1986]. To explain the rapidly decreasing permeability at low porosity, Koponen et al. [1997] introduced the effective porosity defined as the ratio of the volume of the conducting pores to the total volume. Percolation theory suggests a power law as the universal scaling relation between permeability and porosity [e.g.,

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Koponen et al., 1997; Martys and Garboczi, 1992; Saar and Manga, 1999]: k ¼ koðf fc Þe ;

ð18Þ

where k0 = 2(1 f)/S2 and fc denotes the percolation threshold porosity, e.g., the pores become disconnected at fc in the direction of flow. On the basis of numerical simulation results e ’ 4 [Torquato, 2002], but experimental results may be fit with an exponent of 3, too [e.g., Mavko and Nur, 1997]. Here, we do not find a universal threshold porosity, fc (Figure 13); its variability may be related to differences in cementation. In variety FS8, grain contacts are ‘‘glued’’ together by the cementation (Figure 1d) while grain contacts in FS3, FS4, and FS5 (comparable to the ones documented for FS6 in Figure 1a and 1b) probably maintain some hydraulic conductivity and thus shift the percolation threshold to lower values. [34] At first glance, the insensitivity of permeability to pressure observed for several of our samples (Figure 8) appears contradictory to their significant pore compressibility (Figure 7b) and the marked sensitivity of permeability to porosity constrained from measurements on blocks with different porosity (Figure 13). Integrating the determined pore compressibility over the course of an experiment we find the connected porosity as a function of effective pressure. For samples with 4% porosity, the decrease in permeability with pressure mimics the permeability-porosity relation modeled according to equation (18). However, permeability of samples with 9% porosity stays constant although a significant porosity decrease occurs with increasing effective pressure for which permeability should decrease according to the cubic relation (Figure 13). In terminology of percolation theory [e.g., Stauffer and Aharony, 1994], the insensitivity of permeability to pressure may be related to a stiff conducting backbone in comparison to compliant dead ends. Alternatively, this apparent discrepancy may be resolved by the contiguity concept. According to the classic Kozeny-Carman approach (equation (17)) permeability should increase with decreasing surface area of pores at a given porosity. Less surface area will make conduits more tube-like. Specific surface area and contiguity are related by S(8) / (1 8). We find some indication that permeability obeys k / S2 = (1 8)2 for samples with 9% porosity. The hydraulic properties of these samples are probably still well represented by a model that employs a representative grain, as does the contiguity model. All pores are part of the conduit network. Employing a contiguity dependence in the extended Kozeny-Carman model, two contributions to the pressure dependence of permeability compete: 1 @k 1 @k @f @k @8 ¼ þ k @peff k @f @peff @8 @peff 3 @f 2 @8 ¼ þ ; ðf fc Þ @peff ð1 8Þ @peff

ð19Þ

where @f/@peff < 0 and @8/@peff > 0. The pressure insensitivity of permeability of samples FS2 and FS6 which is in stark contrast to their pronounced pressure dependence

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in storage capacity may result from an annihilation of the two contributions in equation (19). A quantitative evaluation of equation (19) for such samples with porosities ‘‘far’’ from the percolation threshold (f fc ’ f) yields 1 @k 1=3 ’ 3Cpp þ ð1 to 4Þ 102 MPa2=3 peff k @peff

’ 3 104 to 103 MPa1 þ 2 104 to 103 MPa1 ð20Þ

for the investigated range in effective pressure. The variation of the two terms with pressure is similar, such that they essentially cancel at all pressures. In contrast, the prefactor of the porosity change – related derivative increases without limits close to the percolation threshold (f ! fc) and permeability becomes very sensitive to (pressure-related) reductions in porosity. Sample FS5 constitutes an example of such behavior. Then, the shape of the conduits is subordinate compared to their interconnectivity.

5. Discussion [35] Since the early work of Terzaghi [1923], the concept of effective pressure has been recognized as helpful in systematizing variations of physical properties with pore fluid and confining pressure by using a weighted difference between the two pressures, i.e., g(pf, pc) = h(peff = pc agpf), where ag is the effective pressure coefficient of g [e.g., Wang, 2000]. The effective pressure laws for hydromechanical properties are of outmost importance for the understanding and modeling of geological processes involving redistributions of stresses or changes in pore fluid pressure due to fluid sources. Above, we specifically analyzed our experimental observations regarding the pressure dependence of several hydromechanical properties on effective pressure in the light of the contiguity concept. [36] Our experiments lack the systematic changes in confining and fluid pressure necessary to evaluate the effective pressure coefficient of permeability, ak. An extensive study by Bernabe´ [1987] demonstrated the experimental difficulties associated with precisely determining this parameter. An expression for the effective pressure coefficient of permeability was theoretically derived by Berryman [1992]. For the suite of tested samples, we find this theoretical effective pressure coefficient of permeability to deviate from 1 by a maximum of a few percent. [37] The effective pressure coefficient for specific storage capacity, as, has received less attention. The observed agreement between the pressure-related variations in contiguity constrained from bulk moduli with predictions of Hertzian contact mechanics, i.e., the p 2/3 eff relation (Figure 12), when peff = pc pf, i.e., an effective pressure coefficient of 1, is used suggests that contiguity has no simple relation to volumetric strain. Volumetric strain should obey an effective pressure law with the fluid pressure weighted by the Biot-Willis coefficient a (see equation (7)) that deviates significantly from 1 for our samples (Figure 9a). In fact, Biot-Willis and Skempton coefficients (Figure 9b) exhibit variations of as much as a

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model has to be tested on rocks composed of more than one mineral. [41] Finally, geological processes will most often be associated with transient changes in pore fluid pressure. Then, hydraulic diffusivity, i.e., the ratio between permeability and storage capacity, rather than permeability alone controls the spatiotemporal characteristics of the fluid pressure field. Our experiments reveal that diffusivity may exhibit contrasting variations with changing effective pressure. In some cases, diffusivity actually increases with increasing pressure because the associated decrease in storage capacity dominates over the decrease in permeability (Figure 14). An increase in diffusivity enlarges the characteristic length scale over which a fluid pressure perturbation is transmitted.

6. Conclusions

Figure 14. Hydraulic diffusivity of the tested Fontainebleau sandstone varieties as dependent on effective pressure. factor of 2 over a range in effective pressure between 0 and 50 MPa for a given sandstone variety and similar variations at a given porosity. [38] The pressure sensitivity of the poroelastic parameters is crucial in determining changes in regional stress associated with earthquakes. For example, Coulomb stress analysis considers the changes in shear and normal stress with respect to the orientation of the fault plane of a major earthquake and changes in fluid pressure [e.g., Cocco and Rice, 2002]: DCFF ¼ Dt þ mðDsn þ Dpf Þ:

ð21Þ

[39] The change in pore fluid pressure due to a change in mean stress Dpf = BDskk/3 (equation (3)) for an undrained fluid system, i.e., z = 0, with a sensitivity of changes in pore fluid pressure to changes in mean stress of @Dpf 1 @B ¼ þB : @Dskk 3 @ ln Dskk

ð22Þ

[40] When using our experimental observation of @B/@ ln peff 1.5 as an estimate for @B/@ ln Dskk, a dominance of the pressure sensitivity of the Skempton coefficient over the absolute value of the Skempton coefficient is indicated in equation (22). A full analysis has to account for the tensorial character of the Skempton coefficient. Experiments investigating the relation between Skempton coefficient and deviatoric loading have already been presented [Lockner and Stanchits, 2002] and contiguity can be expanded to a tensor, too. Thus, the contiguity model may provide a helpful tool for sophisticated numerical modeling in which changes in all associated physical properties is consistently tracked by changes in contiguity with changing thermodynamic conditions. Clearly, the usefulness of this grain-scale

[42] We measured hydraulic permeability and specific storage capacity of samples from blocks of Fontainebleau sandstone using the linear pressurization method at room temperature. The blocks exhibit ranges in open porosity from about 3 to 10% and in ultrasonic velocity of saturated samples from 2670 to 5470 m/sec. Scanning electron microscope analysis suggests that the type and degree of cementation controls ultrasonic velocity. Permeability varies from 1013 m2 down to 1020 m2, and normalized specific storage capacity exceeds the measured porosity by up to a factor of 10. The two hydraulic parameters depend on porosity and pore geometry and differ significantly in their sensitivity to changes in effective pressure. The generally large pressure sensitivity of storage capacity is controlled by elastic deformation of grain contacts that are partly modulated by roughness or individual ‘‘asperities’’ associated with authigenic cement. Sample permeability exhibits two regimes of pressure sensitivity, a pore geometry – dominated and a pore interconnectivity – dominated regime. For samples with porosity far away from the percolation threshold permeability is insensitive to pressure probably because the reduction in pore surface area with increasing pressure outweighs the reduction in total porosity. The permeability of samples with porosity close to the interconnectivity limit strongly decreases with increasing pressure due to changes in network characteristics. Modeling our results using the grain contiguity approach yields consistency between sample compressibility hydraulically determined at elevated pressure and ultrasonic velocity measurements on saturated samples at ambient pressure. Pronounced pressure sensitivity of poroelastic parameters has consequences for transient pressure fields associated with rapid stress redistribution. [43] Acknowledgments. We gratefully acknowledge generous funding by the German Science Foundation (Collaborative Research Center, ‘‘Rheology of the Earth,’’ SFB 526). Sincere thanks to A. Grafchikov for unselfishly sharing data, R. Neuser for his support during SEM work, F. Bettenstedt for excellently preparing samples and maintaining test equipment, and E. Meier-Katschmann for her dedicated contributions to the quantitative microstructure analysis. The stimulating and constructive comments of two anonymous reviewers are appreciated.

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J. Renner, Institute for Geology, Mineralogy, and Geophysics, RuhrUniversity Bochum, D-44780 Bochum, Germany. (renner@geophysik. ruhr-uni-bochum.de) I. Song, Department of Geosciences, Pennsylvania State University, University Park, PA 10682, USA.

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