Linear Pressurization Method for Determining Hydraulic Permeability and Speciﬁc Storage of a Rock Sample

Geophys. J. Int. (2006) 164, 685–696 doi: 10.1111/j.1365-246X.2005.02827.x

Linear pressurization method for determining hydraulic permeability and speciﬁc storage of a rock sample

I. Song and J. Renner Institute for Geology, Mineralogy and Geophysics, Ruhr-University Bochum, D-44780 Bochum, Germany. E-mail: [email protected]

Accepted 2005 October 4. Received 2005 September 28; in original form 2005 July 4 Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 SUMMARY We studied the methodology and applicability of linear pressurization for determining simultaneously hydraulic permeability and specific storage of a rock sample. We analytically solved the governing equation for 1-D pressure diffusion through a homogenous, isotropic cylindrical rock specimen located between a downstream and an upstream reservoir considering linear pressurization of the upstream fluid as the boundary condition. The solution consists of a transient and a steady state for which both differential pressure between the two reservoirs and injection rate at the upstream reservoir assume constant values. Since the steady-state differential pressure is linearly proportional to the compressive storage of the downstream reservoir, tests conducted with a systematic variation of the size of the downstream reservoir permit determining permeability and specific storage from the intercept and slope of the linear relationship. Alternatively, simultaneous measurement of differential pressure and fluid injection rate at steady-state conditions provides a basis for calculation of the hydraulic properties as previously presented for linear injection. Experiments were performed on Fontainebleau sandstone samples to document the feasibility of the method. The determined storage capacity value is used to calculate various poroelastic parameters, such as bulk and pore-space compressibility, Skempton and the effective pressure coefficient. Key words: fluids in rocks, linear pressurization, permeability, poroelasticity, specific storage capacity.

1 INTRODUCTION Rocks are characterized by two hydraulic properties, the capacity for storing the fluid and the ability to transmit it. If fluid is injected into a saturated porous medium, one part of the fluid is stored in the pore space due to the deformation of the fluid and voids, while the other part is transported through interconnected conduits due to the pore pressure gradient. In a homogenous isotropic porous rock, laminar fluid flow is governed by permeability k defined through Darcy’s empirical relationship: =−k ∇ , q μ p (1)

where q, ∇ p, and μ denote the specific discharge, the pore pressure gradient and the dynamic fluid viscosity, respectively. In poroelasticity GJI Volcanology, geothermics, fluids and rocks

(e.g. Biot 1941; Detournay & Cheng 1988, 1993; Rice & Cleary 1976), the specific storage capacity, β s, denotes the proportionality constant between the volume of fluid stored per unit volume of rock, ζ , and the pore fluid pressure, p, in a sample under constant applied stress (e.g. Wang 2000):

ζ = βs p. (2) Consideration of mass conservation during the fluid flow yields a fluid continuity constraint expressed as a relationship between ζ and the specific discharge, q (e.g. Detournay & Cheng 1988, 1993; Rice & Cleary 1976; Wang 2000): ∂ζ =−∇· q. (3) ∂t Combining Darcy’slaw (eq. 1), the constitutive relation (eq. 2), and the continuity eq. (3) with the assumption that the hydraulic properties, β s, k and μ, are constant yields the partial differential equation describing temporal and spatial variations of pore fluid pressure in homogeneous, isotropic porous media (e.g. Rice & Cleary 1976): 1 ∂p ∇2 p − = 0, (4) κ ∂t where κ = k/(μβ s) denotes the hydraulic diffusivity.

C 2006 The Authors 685 Journal compilation C 2006 RAS 686 I. Song and J. Renner

The emphasis of previous experimental and theoretical studies has clearly been on determining permeability (e.g. Bernabé1987; Brace et al. 1968; Kwon et al. 2001; Lin 1982; Lin et al. 1986; Zeynaly-Andabily & Rahman 1995; Zoback & Byerlee 1975), though the hydraulic diffusivity, that is, the ratio between transport and storage efficiency, controls the pore pressure variations for time-dependent pressure gradients. The neglect of the storage parameter is partly related to the simple realization of steady-state methods applying Darcy conditions (eq. 1) to highly permeable rocks. For low-permeability rocks, the evaluation of transient pressure records circumvents long experimental durations (Hsieh et al. 1981; Song et al. 2004a; Zeynaly-Andabily & Rahman 1995) and the explicit measurement of flow rate. However, the procedure of calculating both hydraulic parameters, k and β s, using the transient pressure curve often involves cumbersome curve matching routines (Hsieh et al. 1981; Neuzil et al. 1981; Zeynaly-Andabily & Rahman 1995). Experimental design for the commonly used pressure pulse method (Brace et al. 1968) aims at negligible storage capacity of the sample compared to storage capacities of the reservoirs of the pore pressure system greatly simplifying analysis but leaving sample storage capacity undetermined. From a practical point of view, it is clearly desirable to measure storage capacity routinely together with permeability rather than performing separate tests (e.g. Green & Wang 1986; Hart & Wang 1995; Tokunaga & Kameya 2003). From a scientific perspective, focus should be on determination of the central parameter, hydraulic diffusivity, and its sensitivity to changes in conditions to which a sample Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 is subjected rather than just the transport contribution. Furthermore, storage capacity is an important physical property on its own. Firstly, it determines the yield of reservoirs. The common practice to approximate the storage capacity by the product of fluid compressibility and porosity may fail for compliant pore space, fractured and jointed reservoirs. Secondly, the storage capacity controls the change in pore pressure as a result of a change in external pressure. Thus, if routinely performed in a systematic way, measurements of the specific storage capacity may contribute significantly to our understanding of stress transfer occurring during earthquakes (e.g. Lockner & Stanchits 2002; Pride et al. 2004; Stein 1999). The oscillatory pore pressure method (Fischer 1992) permits determination of both hydraulic parameters. However, it appears that in many cases the determination of the storage capacity remains rather uncertain, that is, not better constrained than to an order of magnitude (e.g. Rutter & Faulkner 1996). Recently, a linear flow injection technique was introduced permitting straightforward graphical determination of both, permeability and specific storage (Song et al. 2004b). The experimental arrangement consisted of a cylindrical specimen between two fluid reservoirs one of which is connected to a pump. By minimizing the storage capacity of the reservoirs, this method worked for tight micritic limestone with very low permeability down to 4 × 10−21 m2 (Song et al. 2002). Here, we suggest a technique in which instead of a constant flow-rate injection a linear pressurization is conducted at the upstream reservoir. This technique can be easily applied with a pressure-controlled hydraulic actuator. We present the analytic solution of the governing equation with the boundary condition of linear pressurization of the upstream reservoir inducing injection of fluid into a specimen located between a downstream and an upstream reservoir (Fig. 1a). The theoretical analysis of the solution provides two different approaches for determination of permeability and specific storage for which we present design considerations and experimental examples.

2 THEORETICAL ANALYSIS For 1-D pressure diffusion through a homogeneous, isotropic porous medium with pressure-independent hydraulic properties, the diffusion eq. (4) is expressed as: ∂2 p(x, t) 1 ∂p(x, t) − = 0. (5) ∂x 2 κ ∂t A constant pore pressure along the rock specimen equilibrated with up and downstream pressure constitutes the initial condition: p(x, 0) = 0 for 0 ≤ x ≤ L. (6)

During the test (t > 0), boundary conditions are determined by the pressurization of the upstream reservoir (x = L) with a constant rate: dp(L, t) = p˙ for t > 0, (7) dt u and by the sensitivity of the downstream reservoir (x = 0): μS ∂p(0, t) ∂p(0, t) d − = 0 for t > 0, (8) kA ∂t ∂x where

p(x, t): pore fluid pressure along a sample as a function of x and t (Pa) x: distance along the sample from the downstream boundary (m) t: time from the start of the experiment (s) A: cross-sectional area of the sample (m2) L: length of the sample (m) k: permeability of the sample (m2) μ: dynamic fluid viscosity (Pa s) −1 β s : specific storage of the sample (Pa ) 3 −1 Sd, Su: storage capacity of the down- and upstream reservoir (m Pa ) −1 p˙ u: constant pressurization rate of the upstream fluid (Pa s ).

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downstream upstream reservoir reservoir

sample pressure pressure time time x x = 0 x = L (a)

downstream reservoir upstream reservoir

displacement transducers Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 bypass valve

spacers main intensifier

sample

auxiliary intensifier confining pressure cell pressure pressure transducer transducer (b)

Figure 1. Schematic diagrams showing (a) the boundary conditions for linear pressurization and (b) the experimental test system composed of a cored rock specimen located between two reservoirs one of which is connected to two pressure intensiﬁers. The auxiliary intensiﬁer was used for maintaining the initial volume of upstream reservoir constant at any initial pressure level. The volume of downstream reservoir was controlled by placing up to three steel spacers (white blocks) into the reservoir.

The boundary conditions and their experimental realization are illustrated in Figs 1(a) and (b), respectively. The partial differential eq. (5) with the initial and boundary conditions given by eqs (6)–(8) was solved using Laplace transforms (see Appendix A). The pore ﬂuid

pressure divided by the upstream pressurization rate p˙ u is found as 2 2 2 ∞ x x 2 p(x, t) x − L μS 2L cos ϕm − ϕm δ sin ϕm κϕ = t + + d (x − L) + L L exp − m t , (9) κ κ ϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ 2 p˙ u 2 kA m=1 m m 4 cos m m (1 5 ) sin m L where we introduced the dimensionless ratio of the storage capacities of the downstream reservoir and the sample S δ = d . (10) βs AL −1 The eigenvalues ϕ m are the roots of tan ϕ = (δϕ) . The analytic solution (eq. 9) consists of two parts: a transient exponentially decaying with time and a steady state in the form of a linear function of time t and a parabolic function of position x. At steady state, the pressure at a given

position increases linearly with the rate of the upstream pressurization, p˙ u, and the pore pressure distribution along the specimen is described

by a parabolic curve characterized by the hydraulic properties of the specimen, β s and k, the compressive storage of the downstream reservoir, S d and the dynamic viscosity of the ﬂuid, μ. For a systematic discussion, we introduce dimensionless time κt τ = , (11) L2 and dimensionless position x ξ = , (12) L yielding a dimensionless pore pressure κ p(ξ,τ) P(ξ,τ) ≡ . (13) 2 L p˙ u The dimensionless version of the solution (eq. 9) reads ∞ 1 cos(ϕ ξ) − ϕ δ sin(ϕ ξ) P(ξ,τ) = τ + [ξ 2 + 2δξ − (1 + 2δ)] + 2 m m m exp −ϕ2 τ . (14) ϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ m 2 m=1 m m 4 cos m m (1 5 ) sin m

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS 688 I. Song and J. Renner

10 1.6 upstream δ P =0.0 8 δ =0.2 1.2 δ =0.4 δ =0.6 6 δ =0.8 δ =1.0 0.8 4 δ =0.0 δ =0.2 δ 0.4 =0.4 2 δ =0.6

Dimensionless pressure, δ =0.8 dashed lines for downstream pressure δ =1.0 Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 0 Dimensionless differential pressure 0.0 02468100246810 (a) Dimensionless time, τ (b) Dimensionless time, τ

Figure 2. Theoretical curves of (a) ﬂuid pressures at the upstream and downstream reservoirs and (b) the differential pressures between them as a function of time in a dimensionless domain for different values of δ, the ratio of the compressive storage of the downstream to that of the specimen. The solid line represents the linear variation of the upstream pressure P u(τ), as given by the boundary condition (eq. 7). The dashed lines clearly show a transient stage in the downstream pressure during which P d(τ) increases with increasing rate until its slope becomes equal to that of P u(τ). The reaction of the downstream pressure is slower and thus the pressure difference becomes larger for the larger δ. In a real test, the pore ﬂuid pressure is measured only at the upstream and downstream reservoirs corresponding to ξ = 1 and ξ = 0, respectively, thus the dimensionless up and downstream pressures become

Pu(τ) ≡ P(1,τ) = τ, (15) and ∞ 1 exp −ϕ2 τ P (τ) ≡ P(0,τ) = τ − − δ + 2 m , (16) d ϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ 2 m=1 m m 4 cos m m (1 5 ) sin m respectively. The dimensionless analytic solution (eq. 14) depends on only one parameter, the dimensionless ratio of storage capacities δ. Thus, it is of fundamental importance to understand how δ affects fluid flow along the porous medium. First, we examine the role of δ for the response of downstream pressure to the variation of upstream pressure. Theoretical curves of dimensionless fluid pressure at upstream and downstream reservoirs as a function of the dimensionless time τ for different values of δ demonstrate that the response of the downstream pressure is slower for larger δ, that is, the duration of the transient stage increases with the magnitude of δ (Fig. 2a). After the transient stage (for large τ), the differential pressure between the two reservoirs stabilizes at a constant value (Fig. 2b) that is linearly proportional to the value of δ:

∞ κ p − 1 P∞ ≡ P (∞) − P (∞) = u p = + δ. (17) u-d u d 2 L p˙ u 2

1.0 1.0 τ = 0.02 τ = 0.32 τ = 0.08 τ = 0.64 0.8 τ = 0.16 0.8 τ ) = 1.28 ) τ τ , τ = 0.32 , τ = 2.56 ξ ξ ( τ = 0.64 ( τ = 5.12 P 0.6 P Δ 0.6 τ = 1.28 Δ τ = 10.24 τ = 2.56 τ = 20.48 asymptotic Asymptotic 0.4 0.4 δ = 0 δ = 10 Normalized Normalized Normalized Normalized 0.2 0.2

0.0 0.0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 (a) Dimensionless position,ξ (b) Dimensionless position,ξ

Figure 3. Normalized dimensionless differential pore pressure, P(ξ, τ)/ P(1, ∞), along the specimen at different dimensionless times for (a) δ = 0 and (b) δ = 10. As δ is raised, the duration of the transient stage becomes longer, and the pore pressure variation at a given time becomes more linear along the specimen.

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1.0

0.8 p /p u p δ = 1000 0.6 δ δ = 10 = 100

δ = 1 0.4

Normalized pressure Normalized 0.2 S = S p /p u d d p Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021

0.0 0.001 0.01 0.1 1 10 100 1000 104 Dimensionless time, τ

Figure 4. Comparison between the evolution of pressure with dimensionless time, τ, for the linear pressurization method (solid lines) and the pressure pulse technique (dashed lines) at various values of the dimensionless ratio of sample and downstream storage capacity, δ, and identical size of up and downstream reservoirs, Su = Sd. Up and downstream pressures were normalized by the height of the pulse, pp, for the pulse technique. For the linear pressurization method, τ / ∞ the pressure difference between up and downstream was normalized to its asymptotic value, P u−d( ) Pu−d.

For the analysis of the pore pressure distribution along the specimen, we introduce the difference between the dimensionless pore pressure

along the sample P(ξ, τ) and the dimensionless downstream pore pressure P d(τ):

∞ 1 cos(ϕ ξ) − ϕ δ sin(ϕ ξ) − 1 P(ξ,τ) ≡ P(ξ,τ) − P (τ) = (ξ 2 + 2δξ) + 2 m m m exp − ϕ2 τ . (18) d ϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ m 2 m=1 m m 4 cos m m (1 5 ) sin m

As δ is raised, the duration of the transient stage becomes longer (Fig. 2). At steady state, the pressure distribution along the sample is a parabolic function of position characterized only by δ. With increasing δ, the pore pressure variation along the specimen becomes more linear (solid lines in Fig. 3). In comparison to the widely applied pressure pulse technique (e.g. Hsieh et al. 1981), the linear pressurization method

requires slightly longer experimental duration for the same δ and S u = S d (Fig. 4).

3 DETERMINATION OF PERMEABILITY AND SPECIFIC STORAGE Linear pressurization of the upstream reservoir permits determining hydraulic properties in two ways.

3.1 Varying the storage capacity of the downstream reservoir ∞ The ratio between the steady-state differential pressure, pu−d, and the upstream pressurization rate, p˙ u, is a linear function of the storage capacity of the downstream reservoir (from eq. 9) ∞ ∞ ∞ μβ 2 μ pu−d pu pd s L L = − = + Sd, (19) p˙ u p˙ u p˙ d 2k kA

where p˙ u = p˙ d at steady-state condition. If linear pressurization experiments are performed with several different sizes of the downstream ∞ / reservoir, evaluation of the relation between pu−d p˙ u and S d permits the determination of permeability and speciﬁc storage. On a ﬁrst glance, the necessity to perform several experiments appears as a disadvantage. However, the advantage of this approach lies in the determination of storage capacity based on pressure readings alone after the characteristics of the downstream reservoir are thoroughly determined once. The insertion of geometrically simple objects into the downstream reservoir constrains the relative sizes of the reservoir realizations (Fig. 1b).

3.2 Measuring ﬂow rate and the storage capacity of upstream reservoir For our choice of coordinate system, the 1-D Darcy law is expressed as: k ∂p(x, t) Q(x, t) =−q(x, t)A = A, (20) μ ∂x where Q(x, t) is the volume of ﬂuid crossing an area A per unit time in the direction of the x-axis. For increasing upstream pressure, the

actual ﬂow rate in the specimen is neither constant nor equal to the rate derived from the speed vp of the piston in the actuator (Fig. 1b). The

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS 690 I. Song and J. Renner

100 108 7 δ = 0 10 e δ = 2 10000 6 -1 δ = 4 10 -log( / (s) Duration 10 5 δ = 6 10 1000 δ 100 = 8 4 δ 10 = 10 3 10 -2 10 10 1 102 1 0.1

10 0.01 e -3 0 10 10 ) -1 μβ L2/k 10 s Normalized transient part, 10-2 -4 -3 Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 10 Normalized differential pressure (s) 10 0204060801000.001 0.01 0.1 1 10 100 1000 (a) Dimensionless time, τ (b) Storage capacity ratio, δ

Figure 5. (a) The normalized contribution of transient flow, e, as a function of dimensionless time for different δ . The dimensionless time doubles for every order of magnitude reduction in the contribution of transient flow. The dashed horizontal line indicates a contribution of less than 1 per cent to ∞ ∞ / the differential pressure, pu−d, as an approximate bound for experimental resolution. (b) Normalized differential pressure ( pu−d p˙u) and experimental duration to reach a contribution e of the transient solution as a function of δ. The open and the closed circles indicate how the diagram can be used 2 for two samples differing by three orders of magnitude in normalized hydraulic properties. If for the sample with ‘low permeability’ (μβ s L /k = 10) 2 downstream storage capacity is chosen such that δ = 0.01 and for the ‘high-permeability’ sample (μβ s L /k = 0.01) downstream storage capacity is chosen such that δ 600, then for either sample, the normalized differential pressure of about 6 s (left y-axis) corresponds to an easily measurable difference of 1 MPa at a pressurization rate of 1/6 MPa/s and a contribution e = 10−2 of the transient is reached after a duration of −log (10−2) × 10 s = 20 s (right y-axis). relationship between the actual flow rate into the specimen at the upstream boundary (x = L), Q(L, t) ≡ Q u, and the rate of upstream volume change, Q p = Apvp, associated with the advancing piston of cross-section Ap is given as

Qu = Qp − Su p˙ u. (21) The steady-state pressure gradient at x = L can be obtained from the derivative of eq. (9) with respect to x: ∂p(L, t) μp˙ = u (β AL + S ) . (22) ∂x kA s d Substituting eqs (21) and (22) into the 1-D Darcy law (20) yields ∞ − + v∞ − Qp (Su Sd)p˙ u Ap p Stot p˙ u βs = = , (23) ALp˙ u ALp˙ u which upon insertion into eq. (19) gives

∞ ∞ μ Q − (S − S )p˙ μ A v − S − p˙ = L p u d u = L p p u d u , k ∞ ∞ (24) 2A pu−d 2A pu−d v∞ = + = − where p denotes the steady-state piston velocity; S tot S u S d; and S u−d S u S d. The last two equations are the same as those previously derived for the boundary condition of constant pumping rate (Song et al. 2004b), because pressurization and injection rate are simultaneously constant at steady-state conditions. These equations provide the basis for calculation of speciﬁc storage and permeability if piston displacement is monitored and calibration measurements of the up and downstream compressive storages, S u and S d, are performed.

3.3 Design considerations The finite downstream storage capacity distinguishes our technique from conventional steady-state experiments at Darcy conditions where the storage capacity of the downstream is kept apparently infinite. The parameter of foremost importance for design considerations is δ, the ratio between the storage capacity of the downstream reservoir and that of the sample, owing to its effect on the duration of the transient stage and the magnitude of differential pressure. Theoretically, the transient never terminates, but asymptotically approaches zero as time increases. The normalized contribution, e, of the transient part defined as the ratio between transient and steady-state differential pressure decreases exponentially with dimensionless time τ (Fig. 5a). A reduction in the uncertainty of the steady-state pressure by an order of magnitude requires waiting a specific dimensionless time τ that is inversely proportional to L2. The dimensionless time τ at a specific value of e increases linearly with increasing storage capacity ratio δ, that is, the experimental duration increases linearly with the size of the downstream reservoir for a given rock specimen. The storage capacity ratio δ influences both, the experimental duration and the differential pressure between the up and downstream μβ 2/ ∞ / reservoir (Fig. 2b). For a given rock sample characterized by s L k, the normalized differential pressure, pu−d p˙ u, and the experimental

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS Linear pressurization method for determining hydraulic permeability 691

duration can be determined from the value of δ (Fig. 5b). The less permeable the rock specimen, the smaller the downstream reservoir should be to reduce experimental duration; yet, for highly permeable rocks, δ should be large enough to yield a significant differential pressure between the two reservoirs. Note that reducing the downstream storage capacity below about 10 per cent of the sample storage capacity (δ = 0.1) does not further affect pressure difference or duration. Unlike Darcy condition, our test condition yields a linearly varied flow rate from the upstream (x = L) to the downstream (x = 0) since the steady-state pressure variation is a parabolic function (Fig. 3). The variation of flow rate along sample length may prove to be a valuable

tool for investigating the heterogeneity of samples. Because Qu = Qp − Su p˙ u and Qd ≡ Q(0, t) = Sd p˙ d, and p˙ u = p˙ d at steady-state conditions, eq. (24) can be rewritten as ∞ Q + Q k p − u d = u d A. (25) 2 μ L This equation resembles a 1-D Darcy law (eq. 20) for the arithmetic average of flow rate. The difference between the flow rate at up and downstream end of the sample constrains the amount of fluid stored in the sample owing to the increase in pore fluid pressure. Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 4 EXPERIMENTAL EXAMPLES

4.1 Description of experiment Our set-up is composed of a pore fluid (water) and a confining fluid (oil) system (Fig. 1b). A cored rock specimen is located between two pore fluid reservoirs, a downstream and an upstream. Porous spacers distribute the pore fluid at the sample ends. A rubber jacket encloses the sample separating pore and confining fluid and imposing conditions approximating 1-D flow. The upstream reservoir is connected to two servo-hydraulically controlled pressure intensifiers. The position of the intensifier pistons and the up, downstream and confining pressures are measured using displacement and pressure transducers, respectively, and digitally recorded by a computer system. The auxiliary intensifier is used to apply the initial pore pressure and maintain constant initial volume of the upstream reservoir. After equilibration of pore pressure in the reservoirs and the sample, the auxiliary intensifier is disconnected from the upstream reservoir and the two reservoirs are separated using the bypass valve. Then, a linear pressurization of the upstream reservoir is conducted using the main intensifier in pressure control mode until

a steady-state (asymptotic) flow condition is reached. This procedure is repeated for different values of S d realized by placing up to three steel spacers in the downstream reservoir (Fig. 1b). The calibration of the pore pressure system is conducted using an impermeable steel specimen instead of a rock sample. We measure the storage capacity of the total pore fluid system and the upstream reservoir by pressurizing the pore fluid system at open and closed bypass

valve, respectively. The storage capacity of the downstream is calculated as S d = S tot − S u. Tests were performed on a cored Fontainebleau sandstone sample (0.06 m long and 0.03 m in diameter) with a porosity of 0.040 ± 0.004 and an ultrasonic P-wave velocity of 4700 ± 90ms−1 when water-saturated.

4.2 Experimental results Determining the difference between the change in total volume due to the piston movement versus the resultant pressure increase for a rock

sample and an impermeable sample constrains the storage capacity (Fig. 6a). The total volume of fluid (Vf) in the test system is composed of the upstream fluid (Vuf), the downstream fluid (Vdf), and the pore fluid in the rock sample (Vpf). The movement of the piston induces a change of total fluid volume

δVf = δVuf + δVdf + δVpf, (26)

where δVuf and δVdf are the changes in up and downstream fluid volumes, respectively, due to the fluid compressibility, and δV pf the change in fluid volume stored in the rock specimen owing to the combined effect of fluid and pore compressibility. At steady-state conditions, the change of fluid pressure in sample and reservoirs is identical yielding ∞ ∞ − + ∞ − δV Q δV δV δV Q (Su Sd)p˙ Q Stot p˙ f = p = uf + df + pf = S + S + Vβ ⇒ β = p ≡ p , (27) δp p˙ δp δp δp u d s s pV˙ pV˙ consistent with the derivation from the analytic solution of the governing eq. (5) (see eq. 23), where V denotes the volume of specimen (V ≡ AL). The slope of the calibration curve for the impermeable sample during which the bypass valve between the two reservoirs was open (Fig. 1b) is slightly lower than that of the experiment on a Fontainebleau sandstone sample (Fig. 6a) because less total fluid volume requires less additional fluid for the same increment in pressure. The difference in slope determines the specific storage to 2.6 × 10−11 Pa−1 (eqs 23 and 27). It should be noted that the difference in slope is small and may actually remain within the uncertainty of the reservoir storage capacity, in particular when the fluid volume in the reservoirs largely exceeds the fluid volume in the sample as will be discussed in more detail in the following paragraph. The permeability amounts to 3.5 × 10−16m2 according to eq. (24) using the steady-state pressure difference (Fig. 6b) yielding a hydraulic diffusivity κ = 1.3 × 10−2 m2 s−1. The determined hydraulic properties represent elastic behaviour; the sample did not suffer any measurable permanent change in dimensions. For the successive measurements with different sizes of the downstream reservoir, steady-state pressure differences linearly increase with increasing size of the downstream reservoir (Fig. 6c). Rather than performing a linear regression analysis we employed an inversion

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS 692 I. Song and J. Renner

80 2.5 Fontainebleau sandstone Fontainebleau sandstone p (P = 170 MPa) u 60 c (P = 170 MPa) Δp c u-d p (MPa) pressure Pore 2.0 d 60 Experiment Calibration 55

) δ 1.5 3 V f -14 3 δ = 2.65 x 10 m /Pa 40 p δδVV uf+ df (mm

f δδ 50 pp 1.0 V S = 1.83 x 10-14 m3/Pa = S + S =2.54 x 10-14 m3/Pa u u d 20 1.98 MPa S = 7.12 x 10-15 m3/Pa d 0.5 45 β -11 -1 Q = 2.65 x 10-8 m3/s

=2.6 x 10 Pa Differential pressure (MPa) s p k = 3.5 x 10-16 m2 0 0.0 40 Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 51 52 53 54 0 5 10 15 20 25 30 (a) p (MPa) u (b) Time (s)

12 Fontainebleau sandstone P = 118 MPa 10 eff

P (s) 8 Δ

4 slope = 2.42 x 10-14 Pa s/m3 Normalized intercept = 0.171 s 2 β = 3.5 x 10-11 Pa-1 s k = 3.5 x 10-16 m2 0 012345 (c) S (x 10-14 m3/Pa) d

Figure 6. (a) Flow versus fluid pressure at steady-state condition for a calibration on an impermeable sample and an experiment on a Fontainebleau sandstone sample. The difference between the two slopes corresponds to the storage capacity of the sandstone sample, β sAL. (b) An example of test records showing upstream, downstream and differential pressures from which the permeability can be obtained if the storage capacities of the reservoirs, Su and Sd, and the ∞ / flow rate, Qp, are also available. (c) The linear relationship between the normalized differential pressure, pu−d p˙ u and the downstream storage capacity, Sd, constraining the specific storage and permeability of the sample according to eq. (19).

algorithm (e.g. Sotin & Poirier 1984) ensuring that all quantities involved remain positive and conservatively accounting for the uncertainties δ ∞ / ∞ < δ / of the pressure difference ( pu−d pu−d 2 per cent) and the storage capacity values of the downstream reservoirs ( S d S d 15 per cent). This procedure provides constraints on the absolute uncertainties for permeability and speciﬁc storage of the sample. For an effective −16 2 −11 −1 −2 2 −1 pressure of 118 MPa, we gain k = (3.5 ± 0.1) × 10 m and β s = (3.5 + 1.9/−1.2) × 10 Pa , thus κ = (1.0 + 0.5/−0.3) × 10 m s . Permeability is obviously much better constrained than speciﬁc storage.

4.3 Uncertainty analysis

In principle, two main sources of uncertainty, pressure difference and steady-state piston velocity, have to be considered for an error analysis of the determined permeability and storage capacity values using eqs (23) and (24), that is, evaluating single pressurizations. Yet, the pressure difference would become critical only for very permeable rocks; in our tests the relative uncertainty barely exceeds 1 per cent. In δv∞ = δ ∞/ contrast, the uncertainty in the steady-state piston velocity, p Qp Ap, might be substantial owing to undetected leaks and temperature variations in addition to transducer sensitivity. Storage capacity values for the reservoirs result from piston velocity measurements, too. Thus for the following estimation we assume similar uncertainties for measurements on rock specimen and impermeable sample, that is, δ ∞ δ δ δ / δ / δ ∞ / ∞ Qp p˙ u Stot p˙ u Su p˙ u Sd 2 p˙ u Su−d 3, and negligible uncertainty in differential pressure, that is, pu−d pu−d 0. On a first glance, eqs (23) and (24) appear formally very similar, thus the huge difference in variability of storage capacity and permeability is striking (Fig. 7). Yet, for our design for which samples contain much less fluid than the two reservoirs, the calculation of specific storage ∞ − according to eq. (23) corresponds to searching for a small difference of two similar numbers (Qp Stot p˙u). Since both terms in the difference

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS Linear pressurization method for determining hydraulic permeability 693

10-10 10-15 ) ) -1 2 10-11 (m (Pa

s k β

Fontainebleau sandstone Fontainebleau sandstone (P = 118 MPa) (P = 118 MPa) eff eff 10-12 10-16 10 100 1000 0.5 1 1.5 2 ∞∞ ∞∞− − (a) QQpptotu/( Sp ) (b) QQppudu/( Sp− )

Figure 7. Speciﬁc storage capacity and permeability values determined from single pressurization phases relying on eqs (23) and (24) (data points) in Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 comparison to the results of an inversion of a set of measurements at various sizes of the downstream reservoir relying on eq. (19) (solid and dashed lines). The δ ∞/ ∞ δβ /β chosen x-axes represent the weighting factors between the relative error of asymptotic ﬂow, Qp Qp , and relative errors in (a) storage capacity, s s, and (b) permeability, δk/k (see eqs 28 and 29).

result from measurements of piston velocities, we can approximate the absolute and relative uncertainties in storage capacity as δQ∞ + p˙ δS δβ 2Q∞ δQ∞ δβ p u tot 2 δ ∞ s p p , s Qp and β ∞ − ∞ (28) ALp˙u ALp˙u s Qp Stot p˙u Qp δβ /β δ ∞/ ∞ respectively. In our experiments, the magnitude of s s is around 50–1500 times larger than Qp Qp (Fig. 7a). Only tests characterized ∞/ ∞ − by a small weighting factor Qp (Qp Stot p˙u) yield an overlap between storage capacity calculated from a single pressurization and the range of storage capacity values constrained by the inversion of tests with varying size of the downstream reservoir. From eq. (24) and the above stated assumptions, we can express the relative uncertainty of permeability as ∞ ∞ ∞ ∞ δ δ p δQ + p˙uδSu−d 4Q δQ k u−d + p p p . ∞ ∞ − ∞ − ∞ (29) k pu−d Qp Su−d p˙u Qp Su−d p˙u Qp ∞/ ∞ − In contrast to the weighting factor of relative uncertainty in storage capacity, the weighting factor Qp (Qp Su−d p˙u) remains of order 1 (Fig. 7b). Consequently, the permeability estimates from the individual pressurizations agree closely with the range of values constrained by the inversion.

4.4 Calculation of poroelastic parameters With the constraints on storage capacity further poroelastic parameters can be calculated (see Appendix B). When compared to the compress- −10 −1 ibility of water, β f = β w = 4.25 × 10 Pa , the gained values correspond to apparent porosities φˆ = βs/βw = (0.083 + 0.045/−0.029) consistent with the actual measured connected porosity φ = 0.040 ± 0.004, that is, φ>φˆ . For Fontainebleau sandstone, the compressibility −11 −1 of the matrix-forming mineral quartz is well constrained to cr = cqtz = 2.64 × 10 Pa (Gebrande 1982). Following the notation of −10 −1 Zimmerman et al. (1986), pore and bulk compressibility calculate to cpp = (4.5 + 3.5/−2.4) × 10 Pa and cbc = (4.5 + 1.9/−1.2) × 10−11 Pa−1, respectively (see Table 1). The effective pressure coefficient for elastic bulk volume changes and the Skempton coefficient amount to α = 0.44 + 0.10/−0.16 and B = 0.55 + 0.13/−0.17, respectively. The effective pressure coefficient differs considerably from 1 and 3φ ≤ α ≤ almost coincides with the arithmetic average of the theoretical bounds 2+φ 1 (Zimmerman et al. 1986). Finally, the low frequency or Gassmann limit of the bulk compressibility (e.g. Winkler & Murphy 1995) of our water-saturated sandstone sample amounts to c f →0 = 4.4 × 10−11 Pa−1 and suggests a difference between the high- and low- frequency limits of the velocity of compressional waves of about 5– 10 per cent.

5 CONCLUSION

We studied 1-D pressure diffusion induced by linear pressurization at one end of a rock sample located between two separate reservoirs to determine the permeability and specific storage of the sample. Our analytic solution of the diffusion equation reveals that the differential pressure between the two reservoirs asymptotically approaches a steady-state value that is linearly proportional to the size of the downstream reservoir. Hydraulic permeability and specific storage of rock samples can be obtained from this relationship. The two hydraulic parameters can also be determined from only one test if the flow rate is monitored and the upstream storage capacity is calibrated. The key parameter for design consideration is the dimensionless ratio of storage capacities of the sample and the downstream reservoir because it controls the magnitude of the differential pressure and the length of the initial transient stage. Of the two proposed methods, the combination of pressure and flow measurements is less time consuming while successive measurements at varying size of the downstream reservoir have the advantage of determining storage capacity based on pressure readings alone. Our error analysis highlights the differences in uncertainty of storage capacity

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Table 1. Poroelastic parameters resulting from our measurement on Fontainebleau sandstone. Property Value Upper bound Lower bound Unit φ 0.040 0.044 0.036 — −11 −11 −11 −1 β s 3.5 × 10 5.4 × 10 2.3 × 10 Pa β∗ × −10 × −10 × −10 −1 w 4.2 10 4.3 10 4.1 10 Pa ∗ × −11 × −11 × −11 −1 cqtz 2.64 10 2.66 10 2.62 10 Pa −11 −11 −11 −1 cbc 4.5 × 10 6.4 × 10 3.3 × 10 Pa −10 −10 −10 −1 cpp 4.5 × 10 8.0 × 10 2.1 × 10 Pa α 0.44 0.54 0.28 — B 0.55 0.68 0.38 — −11 −11 −11 −1 cu 3.5 × 10 5.0 × 10 2.9 × 10 Pa ∗For water, the range in compressibility accounts for a variation between 30 and 40 MPa and between 17◦C and 23◦C in pressure and temperature, respectively; for quartz, the given range represents the differences between Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 the Voigt and Reuss averages of elasticity data for single crystals (Gebrande 1982). Obviously, the relative variations in ﬂuid and mineral compressibility are more than an order of magnitude smaller than in measured storage capacity. and permeability values. Reliable estimates of the speciﬁc storage capacity can be used to calculate a number of poroelastic parameters that are important for characterization of the subsurface.

ACKNOWLEDGMENTS This research project was generously funded by the German Science Foundation (SFB526 ‘Rheology of the Earth’).

REFERENCES Kwon, O., Kronenberg, A.K. & Gangi, A.F., 2001. Permeability of Wilcox shale and its effective pressure law, J. geophys. Res., 106, 19 339– Bernabé,Y., 1987. A wide range permeameter for use in rock physics, Int. 19 353. J. Rock Mech. Min. Sci. & Geomech. Abstr., 24, 309–315. Lin, W., 1982. Parametric analysis of the transient method of measuring Biot, M.A., 1941. General theory of three-dimensional consolidation, J. permeability, J. geophys. Res., 87, 1055–1060. Appl. Phys., 12, 155–164. Lin, C., Pirie, G. & Trimmer, D.A., 1986. Low permeability rocks; labo- Brace, W.F.,Walsh, J.B. & Frangos, W.T.,1968. Permeability of granite under ratory measurements and three-dimensional microstructural analysis, J. high pressure, J. geophys. Res., 73, 2225–2236. geophys. Res., 91, 2173–2181. Carslaw, H.S. & Jaeger, J.C., 1959. Conduction of Heat in Solids. Clarendon Lockner, D.A. & Stanchits, S.A., 2002. Undrained poroelastic response Press, Oxford, UK. of sandstones to deviatoric stress change, J. geophys. Res., 107, Detournay, E. & Cheng, A.H.-D., 1988. Poroelastic response of a borehole 2353. in a non-hydrostatic stress field, Int. J. Rock Mech. Min. Sci. & Geomech. Neuzil, C.E., Cooley,C., Silliman, S.E., Bredehoeft, J.D. & Hsieh, P.A.,1981. Abstr., 25, 171–182. A transient laboratory method for determining the hydraulic properties of Detournay, E. & Cheng, A.H.-D., 1993. Fundamentals of poroelasticity, in tight rocks-II. Application, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Comprehensive rock engineering, pp. 113–71, ed. Hudson, J.A., Pergamon 18, 253–258. Press, Oxford, UK. Pride, S.R., Moreau, F. & Gavrilenko, P., 2004. Mechanical and electrical Fischer, G.J., 1992. The determination of permeability and storage capac- response due to fluid-pressure equilibration following an earthquake, J. ity: Pore pressure oscillation method, in Fault Mechanics and Transport geophys. Res., 109(B3), B03302. Properties of Rocks, pp. 187–211, eds Evans, B. & Wong, T.-F.,Academic Rice, J.R. & Cleary, M.P., 1976. Some basic stress-diffusion solutions for Press, London, UK. fluid saturated elastic porous media with compressible constituents, Rev. Gebrande, H., 1982. Elastic wave velocities and constants of rocks and rock Geophys. Space Phys., 14, 227–241. forming minerals, in Landolt-Börnstein:Zahlenwerte und Funktionen aus Rutter, E.H. & Faulkner, D.R., 1996. A critical assessment of the pore pres- Naturwissenschaft und Technik, Neue Serie, Guppe 5, Bd. 1b, pp. 1–34, sure oscillation technique to measure permeability and storage capacity, ed. Angenheister, G., Springer Verlag, Berlin, Germany. EOS, Trans. Am. geophys. Un., 77(46), 747. Green, D.H. & Wang, H.F., 1986. Fluid pressure response to undrained Song, I., Elphick, S.C., Main, I.G. & Ngwenya, B.T., 2002. A new method of compression in saturated sedimentary rock, Geophysics, 51(4), 948– determining the specific storage and the hydraulic conductivity of a rock 956. sample, EOS, Trans. Am. geophys. Un., 83, F609. Hart, D.J. & Wang, H.F., 1995. Laboratory measurements of complete set of Song, I., Elphick, S.C., Main, I.G. & Ngwenya, B.T., 2004a. Hydro- poroelastic moduli for Berea sandstone and Indiana limestone, J. geophys. mechanical behaviour of fine-grained calcilutite versus fault gouge Res. 100(B9), 17 741–17 751. from the Aigion Fault Zone, Greece. C. R. Geoscience, 336, 445– Hsieh, P.A.,Tracy, J.V.,Neuzil, C.E., Bredehoeft, J.D. & Silliman, S.E., 1981. 454. A transient laboratory method for determining the hydraulic properties of Song, I., Elphick, S.C., Main, I.G., Ngwenya, B.T., Odling, N.W. & Smyth, tight rocks-I. Theory, Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 18, N.F., 2004b. One-dimensional fluid diffusion induced by constant-rate 245–252. flow injection: theoretical analysis and application to the determination of Kümpel, H.-J., 1991. Poroelasticity: parameters reviewed, Geophys. J. Int., fluid permeability and specific storage of a cored rock sample, J. geophys. 105, 783–799. Res., 109, B05207.

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Sotin, C. & Poirier, J.P., 1984. Analysis of high-temperature creep experi- Ahrens, T.J., AGU, Washington, DC, USA. ments by generalized nonlinear inversion, Mech. Mat., 3, 311–317. Zeynaly-Andabily, E.M. & Rahman, S.S., 1995. Measurement of perme- Stein, R.S., 1999. The role of stress transfer in earthquake occurrence, Na- ability of tight rocks, Meas. Sci. Technol., 6, 1519–1527. ture, 402, 605–609. Zimmerman, R.W., Somerton, W.H. & King, M.S., 1986. Com- Tokunaga, T. & Kameya, H., 2003. Determination of speciﬁc storage of a pressibility of porous rocks, J. geophys. Res., 91(B12), 12 765– porous material from ﬂow pump experiments: theoretical analysis and 12 777. experimental evaluation, Int. J. Rock Mech. Min. Sci., 40, 739–745. Zoback, M.D. & Byerlee, J.D., 1975. The effect of microcrack dilatancy Wang, H.F., 2000. Theory of Linear Poroelasticity with Applications to Ge- on the permeability of Westerly granite, J. geophys. Res., 80, 752– omechanics and Hydrogeology, Princeton University Press, Oxford, UK. 755. Winkler, K.W. & Murphy, W.F., 1995. Acoustic velocity and attenuation in porous rocks, in Rock Physics and Phase Relations, Vol. 3, pp. 20–34, ed.

APPENDIX A: Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 A1 Particular solution of the 1-D diffusion equation A procedure of solving the general diffusion equation is brieﬂy described here (Carslaw & Jaeger 1959; Hsieh et al. 1981). Applying the Laplace transforms to eq. (5) yields ∞ ∂2 p 1 ∞ ∂p e−st dt − e−st dt = 0. (A1) ∂ 2 κ ∂ 0 x 0 t Accounting for the initial condition (eq. 6), we can express eq. (A1) in the form of an ordinary differential equation d2 p¯ s − q2 p¯ = 0 with q2 = (A2) dx2 κ that has the general solution qx −qx p¯ (s) = C1e + C2e . (A3)

Taking the Laplace transforms of the boundary conditions, eqs (7) and (8), we can determine C1 and C2 yielding

p˙ q cosh(qx) + sλ sinh(qx) p¯ (s) = u d , (A4) 2 s q cosh(qL) + sλd sinh(qL)

where λd = μS d/kA. The inversion of the Laplace transform (A4) is obtained by the usual inverse formula γ +i∞ , = 1 st , p(x t) π e p¯ (s) ds (A5) 2 i γ −i∞ where γ has to be sufﬁciently large that all the singularities of p¯ (s) lie to the left of the line (γ − i∞, γ + i∞). The contour integral (A5) can then be evaluated by computing the contour to the left and summing residues. By the Residue theorem, eq. (A5) is rewritten as p(x, t) = Res(sm ), (A6) m st where sm are poles of the integrand, e p¯ (s), and Res(sm) are the associated residues. The residue at a simple pole of order m > 1atz = a of a function f (z)isgivenby 1 dm−1 Res f (z) = lim [(z − a)m f (z)] . (A7) → m−1 z=a (m − 1)! z a dz Now est p¯ (s) has a simple pole at s =0.Fors = q2κ → 0, the functions sinh qx, cosh qx and est can be approximated by the ﬁrst two terms in the Taylor series q3x 3 q2x 2 sinh qx = qx + , cosh qx = 1 + , and est = 1 + st. 3! 2! Higher-order terms do not contribute to the pole at s = 0. The series approximation yields

p˙ 6κ + 3x 2s + 6κλ xs + λ x 3s2 p¯ (s) = u d d . (A8) 2 2 3 2 s 6κ + 3L s + 6κλd Ls + λd L s

The residue of est p¯ (s)ats = 0 for m = 2is 2 − 2 d 2 x L Res(0) = lim s (1 + st)p¯ (s) = p˙u t + + λd(x − L) . (A9) s→0 ds 2κ In addition, if q is imaginary the function p¯ has multiple poles when the denominator in eq. (A4) vanishes: q cosh(qL) + sλd sinh(qL) = 0. (A10)

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS 696 I. Song and J. Renner

For imaginary q, we can write: qL = iϕ and q2 L2 =−ϕ2, (A11) where ϕ is a real number. Then from eq. (A2) it is seen that s is always negative: sL2 =−κϕ2. (A12) The transcendental eq. (A10) can be rewritten by substituting eqs (A11) and (A12): ϕ = 1 , tan δϕ (A13) where δ = Sd . The multiple poles are then determined by the roots ϕ of eq. (A13). Now est p¯ is of the form N(s)/D(s) and the residues βs AL m are given by

N(sm ) −κ ϕ2 / 2 q cosh(qx) + sλd sinh(qx) = = t m L . Res(sm ) p˙ue (A14) 2 3 Downloaded from https://academic.oup.com/gji/article/164/3/685/2133401 by guest on 23 September 2021 D (sm ) s q cosh(qL) + s λd sinh(qL)

= ϕ 2 =−κϕ2 2 4 = κ2ϕ4 3 6 =−κ3ϕ6 With qL i m , sL m , s L m , and s L m , eq. (A14) becomes

2p˙ L2[cos(ϕ x/L) − ϕ δ sin(ϕ x/L)] Res(s ) = u m m m exp −κϕ2 t L2 . (A15) m κϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ m m m 4 cos m m (1 5 ) sin m

The complete analytical solution of the diffusion equation is ∞ p(x, t) = Res(0) + Res(sm ), (A16) m=1 and thus 1 μS p(x, t) = p˙ t + (x 2 − L2) + d (x − L) u 2κ kA ∞ 2p˙ L2 cos(ϕ x/L) − ϕ δ sin(ϕ x/L) + u m m m exp − κϕ2 t L2 κ ϕ2 ϕ2 δ − ϕ + ϕ + δ ϕ m (A17) m=1 m m 4 cos m m (1 5 ) sin m .

A2 Relation between storage capacity and some common poroelastic parameters The speciﬁc storage capacity of a macroscopic isotropic sample reads

βs = φ(cpp + βf). (A18) (notation according to Zimmerman et al. 1986) and thus provides direct access to the compressibility of the pore space as a result of a change in pore pressure, cpp, if fluid compressibility β f and porosity φ are known. In fact, the common practice β s φβf may fail for compliant pore space, such as in fractured and jointed rock mass. The bulk compressibility as a result of a change in confining pressure cbc = βs − φβf + (1 + φ)cr, (A19) where cr denotes the compressibility of the material composing the solid matrix, assumed microscopically isotropic. This assumption is obviously not always justified but variational averaging will often provide close bounds. Introducing the effective pressure peff = pc − α pf defines a coefficient for elastic changes of bulk volume c β − φ(β − c ) α = 1 − r = s f r , (A20) cbc βs − φβf + (1 + φ)cr closely related to the Skempton coefficient (Kümpel1991) αc β − c B = bc = 1 − φ f r , (A21) αcbc + φ(βf − cr) βs and the undrained compressibility β − β − = −1 = − + = φ f cr + − φ f cr . cu Ku (1 B)cbc Bcr cbc 1 cr (A22) βs βs The latter two parameters constitute measures of the maximum pore pressure change resulting from a change in external pressure conditions. The principal unknown in these equations is storage capacity, which may vary by orders of magnitude; all other parameters can be constrained to within a factor of two at least when the type of fluid, the composition of the rock and the thermodynamic conditions are known. Note, the Biot coefficients, R−1 and H −1, were defined as a measure of the change in fluid content for a given change in fluid pressure and as a measure −1 −1 of the compressibility of the composite for a change in fluid pressure, respectively (Kümpel1991), thus, β s = R and H = cbp = cbc − cr = β s − φ(β f − cr).

C 2006 The Authors, GJI, 164, 685–696 Journal compilation C 2006 RAS