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]
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 I N T RO D U C T I O N
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 = − ∇
p,
(1)
μ
p, and μ denote the specific discharge, the pore pressure gradient and the dynamic fluid viscosity, respectively. In poroelasticity
where q , ∇
(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’s law (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.
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2006 The Authors
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Journal compilation 685
GJI Volcanology, geothermics, fluids and rocks
Accepted 2005 October 4. Received 2005 September 28; in original form 2005 July 4
686
I. Song and J. Renner
The emphasis of previous experimental and theoretical studies has clearly been on determining permeability (e.g. Bernabé 1987; 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
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 T H E O R E T I C A L A N A LY S I S
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.
−
∂x2
κ ∂t
(5)
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)
= ṗu
dt
for t > 0,
(7)
and by the sensitivity of the downstream reservoir (x = 0):
μSd ∂ p(0, t) ∂ p(0, t)
−
=0
kA
∂t
∂x
for t > 0,
(8)
where
p(x, t):
x:
t:
A:
L:
k:
μ:
βs:
S d, S u:
ṗu :
pore fluid pressure along a sample as a function of x and t (Pa)
distance along the sample from the downstream boundary (m)
time from the start of the experiment (s)
cross-sectional area of the sample (m2 )
length of the sample (m)
permeability of the sample (m2 )
dynamic fluid viscosity (Pa s)
specific storage of the sample (Pa−1 )
storage capacity of the down- and upstream reservoir (m3 Pa−1 )
constant pressurization rate of the upstream fluid (Pa s−1 ).
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2006 The Authors, GJI, 164, 685–696
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Journal compilation Linear pressurization method for determining hydraulic permeability
pressure
upstream
reservoir
pressure
downstream
reservoir
687
sample
time
time
x
x=0
(a)
x=L
downstream reservoir
upstream reservoir
displacement
transducers
bypass valve
spacers
main intensifier
sample
auxiliary intensifier
(b)
pressure
transducer
confining pressure cell
pressure
transducer
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 ṗu is found as
∞
cos ϕm Lx − ϕm δ sin ϕm Lx
κϕm2
μSd
2L 2 x2 − L2
p(x, t)
exp − 2 t ,
+
(x − L) +
=t+
(9)
ṗu
2κ
kA
κ m=1 ϕm2 ϕm2 δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
L
where we introduced the dimensionless ratio of the storage capacities of the downstream reservoir and the sample
δ=
Sd
.
βs AL
(10)
The eigenvalues ϕ m are the roots of tan ϕ = (δϕ)−1 . 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, ṗu , 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 = 2,
(11)
L
and dimensionless position
x
ξ= ,
(12)
L
yielding a dimensionless pore pressure
P(ξ, τ ) ≡
κ p(ξ, τ )
.
L 2 ṗu
The dimensionless version of the solution (eq. 9) reads
∞
cos(ϕm ξ ) − ϕm δ sin(ϕm ξ )
1
exp −ϕm2 τ .
P(ξ, τ ) = τ + [ξ 2 + 2δξ − (1 + 2δ)] + 2
2
2
2
ϕm δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
m=1 ϕm
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2006 The Authors, GJI, 164, 685–696
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688
I. Song and J. Renner
Dimensionless differential pressure
Dimensionless pressure, P
10
upstream
δ =0.0
δ =0.2
δ =0.4
δ =0.6
δ =0.8
δ =1.0
8
6
4
2
dashed lines for downstream pressure
0
(a)
0
2
4
6
8
10
Dimensionless time,τ
1.6
1.2
0.8
δ =0.0
δ =0.2
δ =0.4
δ =0.6
δ =0.8
δ =1.0
0.4
0.0
(b)
0
2
4
6
8
Dimensionless time,τ
10
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
Pd (τ ) ≡ P(0, τ ) = τ −
∞
exp −ϕm2 τ
1
,
−δ+2
2
2
ϕm2 δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
m=1 ϕm
(16)
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:
Pu∞-d ≡ Pu (∞) − Pd (∞) =
∞
κ pu−p
1
= + δ.
2
ṗu
L
2
(17)
1.0
1.0
τ = 0.02
τ = 0.08
τ = 0.16
τ = 0.32
τ = 0.64
τ = 1.28
τ = 2.56
asymptotic
0.6
0.4
0.8
Normalized ΔP(ξ,τ)
Normalized ΔP(ξ,τ)
0.8
δ=0
0.2
0.0
(a)
τ = 0.32
τ = 0.64
τ = 1.28
τ = 2.56
τ = 5.12
τ = 10.24
τ = 20.48
Asymptotic
0.6
0.4
δ = 10
0.2
0
0.2
0.4
0.6
0.8
Dimensionless position,ξ
1
0.0
(b)
0
0.2
0.4
0.6
0.8
Dimensionless position,ξ
1
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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Normalized pressure
1.0
0.8
p /p
u
p
δ = 1000
0.6
δ = 10 δ = 100
δ=1
0.4
0.2
S =S
u
0.0
0.001
p /p
d
0.01
d
0.1
p
1
10
100
1000
4
10
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, S u = S d . Up and downstream pressures were normalized by the height of the pulse, p p , for the pulse technique. For the linear pressurization method,
the pressure difference between up and downstream was normalized to its asymptotic value, P u−d (τ )/
P ∞
u−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 ):
P(ξ, τ ) ≡ P(ξ, τ ) − Pd (τ ) =
∞
1 2
cos(ϕm ξ ) − ϕm δ sin(ϕm ξ ) − 1
exp − ϕm2 τ .
(ξ + 2δξ ) + 2
2
2
2
ϕm δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
m=1 ϕm
(18)
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 D E T E R M I N AT I O N O F P E R M E A B I L I T Y A N D S P E C I F I C S T O R A G E
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, p ∞
u−d , and the upstream pressurization rate, ṗu , is a linear function of the storage
capacity of the downstream reservoir (from eq. 9)
∞
pu−d
pu∞
pd∞
μβs L 2
μL
=
−
=
(19)
+
Sd ,
ṗu
ṗu
ṗd
2k
kA
where ṗu = ṗd 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
/ ṗu 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 v p of the piston in the actuator (Fig. 1b). The
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2006 The Authors, GJI, 164, 685–696
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Journal compilation I. Song and J. Renner
690
0
-1
10
-2
10
-3
10
-4
10
(a)
0
20
40
60
80
100
Dimensionless time,τ
Normalized differential pressure (s)
8
δ=0
δ=2
δ=4
δ=6
δ=8
δ = 10
10
7
10
10000
5
6
10
1000
10
100
3
10
1
0.1
10
4
10
1
10
2
10
0.01
2
μβ L /k
-1
10
0
10
Duration (s) / -log(e)
Normalized transient part, e
10
s
-2
10
-3
10
0.001
(b)
0.01
0.1
1
10
100
1000
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, p ∞
u−d , as an approximate bound for experimental resolution. (b) Normalized differential pressure (
pu−d / ṗ 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
for two samples differing by three orders of magnitude in normalized hydraulic properties. If for the sample with ‘low permeability’ (μβ s L 2 /k = 10)
downstream storage capacity is chosen such that δ = 0.01 and for the ‘high-permeability’ sample (μβ s L 2 /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 = A p v p , associated with the advancing piston of cross-section A p is given as
Q u = Q p − Su ṗu .
(21)
The steady-state pressure gradient at x = L can be obtained from the derivative of eq. (9) with respect to x:
μ ṗu
∂ p(L , t)
(βs AL + Sd ) .
=
∂x
kA
Substituting eqs (21) and (22) into the 1-D Darcy law (20) yields
Q∞
Ap vp∞ − Stot ṗu
p − (Su + Sd ) ṗu
=
,
βs =
AL ṗu
AL ṗu
which upon insertion into eq. (19) gives
k=
μL
2A
Q∞
p − (Su − Sd ) ṗu
∞
pu−d
=
μL
2A
Ap vp∞ − Su−d ṗu
∞
pu−d
(22)
(23)
,
(24)
where v ∞
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
∞
reservoir (Fig. 2b). For a given rock sample characterized by μβ s L 2 /k, the normalized differential pressure, pu−d
/ ṗu , and the experimental
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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 Q u = Q p − Su ṗu and Q d ≡ Q(0, t) = Sd ṗd , and ṗu = ṗd at steady-state
conditions, eq. (24) can be rewritten as
∞
k pu−d
Qu + Qd
=
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.
4 E X P E R I M E N TA L E X A M P L E S
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 ±
90 m s−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 (V f ) in the test system is composed of
the upstream fluid (V uf ), the downstream fluid (V df ), and the pore fluid in the rock sample (V pf ). The movement of the piston induces a change
of total fluid volume
δVf = δVuf + δVdf + δVpf ,
(26)
where δV uf and δV df 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
Q∞
Q∞
Q∞
δVpf
δVf
δVuf
δVdf
p
p − (Su + Sd ) ṗ
p − Stot ṗ
=
≡
,
(27)
=
+
+
= Su + Sd + Vβs ⇒ βs =
ṗ
ṗV
ṗV
δp
δp
δp
δp
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 × 10−16 m2 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
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80
2.5
Experiment
Calibration
Vf (mm3)
δV f
40
δp
-14
= 2.65 x 10
3
m /Pa
δ Vuf
δ Vdf
+
δp
δp
-14
= S + S =2.54 x 10
u
20
d
-11
3
m /Pa
-1
β =2.6 x 10
Pa
s
2.0
Fontainebleau sandstone
(Pc = 170 MPa) Δp
p
u
u-d
60
p
d
55
1.5
1.0
50
(a)
52
53
54
-15
m /Pa
u
d
0.5
3
3
-8
45
3
Q = 2.65 x 10 m /s
p
0.0
0
(b)
pu (MPa)
m /Pa
S = 7.12 x 10
-16
0
51
-14
S = 1.83 x 10
Pore pressure (MPa)
60
Differential pressure (MPa)
Fontainebleau sandstone
(Pc = 170 MPa)
1.98 MPa
692
k = 3.5 x 10
5
10
15
20
m
2
25
40
30
Time (s)
12
Fontainebleau sandstone
Peff = 118 MPa
Normalized ΔP (s)
10
8
6
4
-14
-11
β = 3.5 x 10
2
s
-16
k = 3.5 x 10
0
3
slope = 2.42 x 10 Pa s/m
intercept = 0.171 s
0
1
(c)
2
3
-14
S (x 10
d
-1
Pa
m
2
4
5
3
m /Pa)
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 s AL. (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, S u and S d , and the
∞ / ṗ and the downstream storage capacity, S ,
flow rate, Qp , are also available. (c) The linear relationship between the normalized differential pressure, pu−d
u
d
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
∞
of the pressure difference (δ
p ∞
u−d /
p u−d < 2 per cent) and the storage capacity values of the downstream reservoirs (δS 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
pressure of 118 MPa, we gain k = (3.5 ± 0.1) × 10−16 m2 and β s = (3.5 + 1.9/−1.2) × 10−11 Pa−1 , thus κ = (1.0 + 0.5/−0.3) × 10−2 m2 s−1 .
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
∞
contrast, the uncertainty in the steady-state piston velocity, δv ∞
p = δ Q p /A p , 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 Q∞
p ṗu δStot ṗu δSu ṗu δSd /2 ṗu δSu−d /3, and negligible uncertainty in differential pressure, that is, δ
p u−d /
p u−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 (Q ∞
p − Stot ṗ u ). Since both terms in the difference
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Journal compilation Linear pressurization method for determining hydraulic permeability
-15
-10
10
k (m2)
-1
β (Pa )
10
-11
s
10
Fontainebleau sandstone
(P = 118 MPa)
Fontainebleau sandstone
(P = 118 MPa)
-12
10
693
eff
10
(a)
-16
100
∞
∞
Qp /( Qp −
1000
Stot pu )
10
(b)
0.5
eff
1
1.5
∞
2
∞
Qp /( Qp − Su −d pu )
Figure 7. Specific storage capacity and permeability values determined from single pressurization phases relying on eqs (23) and (24) (data points) in
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
∞
chosen x-axes represent the weighting factors between the relative error of asymptotic flow, δ Q ∞
p /Q p , and relative errors in (a) storage capacity, δβ s /β 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
δ Q∞
δ Q∞
2Q ∞
2
δβs
p + ṗ u δStot
p
p
δ Q∞
and
∞
,
δβs p
AL ṗ u
AL ṗ u
βs
Q p − Stot ṗ u Q ∞
p
(28)
∞
respectively. In our experiments, the magnitude of δβ s /β s is around 50–1500 times larger than δ Q ∞
p /Q p (Fig. 7a). Only tests characterized
∞
∞
by a small weighting factor Q p /(Q p − Stot ṗ 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 Q∞
δ Q∞
4Q ∞
δ
pu−d
δk
p + ṗ u δSu−d
p
p
+
.
(29)
∞
∞
k
pu−d
Q∞
Q∞
p − Su−d ṗ u
p − Su−d ṗ u Q p
∞
In contrast to the weighting factor of relative uncertainty in storage capacity, the weighting factor Q ∞
p /(Q p − Su−d ṗ 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 compressibility of water, β f = β w = 4.25 × 10−10 Pa−1 , the gained values correspond to apparent porosities φ̂ = βs /βw = (0.083 + 0.045/−0.029)
consistent with the actual measured connected porosity φ = 0.040 ± 0.004, that is, φ̂ > φ. For Fontainebleau sandstone, the compressibility
of the matrix-forming mineral quartz is well constrained to c r = c qtz = 2.64 × 10−11 Pa−1 (Gebrande 1982). Following the notation of
Zimmerman et al. (1986), pore and bulk compressibility calculate to c pp = (4.5 + 3.5/−2.4) × 10−10 Pa−1 and c bc = (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
almost coincides with the arithmetic average of the theoretical bounds 2+φ
≤ α ≤ 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 C O N C LU S I O N
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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I. Song and J. Renner
Table 1. Poroelastic parameters resulting from our measurement on
Fontainebleau sandstone.
Property
Value
Upper bound
Lower bound
Unit
φ
βs
β ∗w
c∗qtz
c bc
c pp
α
B
cu
0.040
3.5 × 10−11
4.2 × 10−10
2.64 × 10−11
4.5 × 10−11
4.5 × 10−10
0.44
0.55
3.5 × 10−11
0.044
5.4 × 10−11
4.3 × 10−10
2.66 × 10−11
6.4 × 10−11
8.0 × 10−10
0.54
0.68
5.0 × 10−11
0.036
2.3 × 10−11
4.1 × 10−10
2.62 × 10−11
3.3 × 10−11
2.1 × 10−10
0.28
0.38
2.9 × 10−11
—
Pa−1
Pa−1
Pa−1
Pa−1
Pa−1
—
—
Pa−1
∗ 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
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.
AC K N OW L E D G M E N T S
This research project was generously funded by the German Science Foundation (SFB526 ‘Rheology of the Earth’).
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APPENDIX A:
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
∞
1 ∞ −st ∂ p
∂2 p
dt = 0.
(A1)
e−st 2 dt −
e
∂x
κ 0
∂t
0
Accounting for the initial condition (eq. 6), we can express eq. (A1) in the form of an ordinary differential equation
d 2 p̄
s
− q 2 p̄ = 0 with q 2 =
dx2
κ
that has the general solution
p̄(s) = C1 eq x + C2 e−q x .
(A2)
(A3)
Taking the Laplace transforms of the boundary conditions, eqs (7) and (8), we can determine C 1 and C 2 yielding
p̄(s) =
ṗ u q cosh(q x) + sλd sinh(q x)
,
s 2 q cosh(q L) + sλd sinh(q L)
where λ d = μS d /kA. The inversion of the Laplace transform (A4) is obtained by the usual inverse formula
γ +i∞
1
est p̄(s) ds,
p(x, t) =
2πi γ −i∞
(A4)
(A5)
where γ has to be sufficiently large that all the singularities of p̄(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
where sm are poles of the integrand, est p̄(s), and Res(sm ) are the associated residues. The residue at a simple pole of order m > 1 at z = a of
a function f (z) is given by
m−1
1
d
m
Res f (z) =
lim
[(z − a) f (z)] .
(A7)
(m − 1)! z→a dz m−1
z=a
Now est p̄(s) has a simple pole at s = 0. For s = q 2 κ → 0, the functions sinh q x, cosh q x and est can be approximated by the first two terms in
the Taylor series
q3x3
q2x2
, cosh q x = 1 +
, and est = 1 + st.
3!
2!
Higher-order terms do not contribute to the pole at s = 0. The series approximation yields
sinh q x = q x +
p̄(s) =
ṗ u 6κ + 3x 2 s + 6κλd xs + λd x 3 s 2
.
s 2 6κ + 3L 2 s + 6κλd Ls + λd L 3 s 2
The residue of est p̄(s) at s = 0 for m = 2 is
x2 − L2
d
Res(0) = lim s 2 (1 + st) p̄(s) = ṗ u t +
+ λd (x − L) .
s→0 ds
2κ
(A8)
(A9)
In addition, if q is imaginary the function p̄ has multiple poles when the denominator in eq. (A4) vanishes:
q cosh(q L) + sλd sinh(q L) = 0.
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I. Song and J. Renner
For imaginary q, we can write:
q L = iϕ and q 2 L 2 = −ϕ 2 ,
(A11)
where ϕ is a real number. Then from eq. (A2) it is seen that s is always negative:
s L 2 = −κϕ 2 .
(A12)
The transcendental eq. (A10) can be rewritten by substituting eqs (A11) and (A12):
1
,
tan ϕ =
δϕ
(A13)
d
where δ = βsSAL
. The multiple poles are then determined by the roots ϕ m of eq. (A13). Now est p̄ is of the form N (s)/D(s) and the residues
are given by
Res(sm ) =
N (sm )
q cosh(q x) + sλd sinh(q x)
2
2
= ṗ u e−κtϕm /L .
D (sm )
s 2 q cosh(q L) + s 3 λd sinh(q L)
(A14)
With qL = iϕ m , s L 2 = −κϕ 2m , s 2 L 4 = κ 2 ϕ 4m , and s 3 L 6 = −κ 3 ϕ 6m , eq. (A14) becomes
Res(sm ) =
2 ṗ u L 2 [cos(ϕm x/L) − ϕm δ sin(ϕm x/L)]
exp −κϕm2 t L 2 .
2
2
κϕm ϕm δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
(A15)
The complete analytical solution of the diffusion equation is
∞
p(x, t) = Res(0) +
Res(sm ),
(A16)
m=1
and thus
μSd
1 2
(x − L 2 ) +
(x − L)
p(x, t) = ṗ u t +
2κ
kA
+
.
∞
2 ṗ u L 2 cos(ϕm x/L) − ϕm δ sin(ϕm x/L)
exp − κϕm2 t L 2
2
2
κ m=1 ϕm ϕm δ − 4 cos ϕm + ϕm (1 + 5δ) sin ϕm
(A17)
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, c pp , if fluid compressibility β f and porosity φ are known. In fact, the common practice β s φβ f 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 c r 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 p eff = p c − α p f
defines a coefficient for elastic changes of bulk volume
βs − φ(βf − cr )
cr
=
,
(A20)
α =1−
cbc
βs − φβf + (1 + φ)cr
closely related to the Skempton coefficient (Kümpel 1991)
β f − cr
αcbc
=1−φ
,
(A21)
B=
αcbc + φ(βf − cr )
βs
and the undrained compressibility
βf − c r
β f − cr
cbc + 1 − φ
(A22)
cr .
cu = K u−1 = (1 − B)cbc + Bcr = φ
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
of the compressibility of the composite for a change in fluid pressure, respectively (Kümpel 1991), thus, β s = R −1 and H −1 = c bp = c bc −
c r = β s − φ(β f − c r ).
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