Geometrical models for poroelastic behaviour

Geophys. J . Int. (1997) 128, 522-532

Geometrical models for poroelastic behaviour

Anthony L. Endres* Waterloo Centrefor Groundwater Research, Department of Earth Sciences, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail: [email protected]

Accepted 1996 October 16. Received 1996 September 29; in original form 1995 August 7

SUMMARY Poroelasticity implicitly incorporates pore structure information through the use of empirically determined macroscopic parameters; hence, quantitative analysis of pore geometry effects on poroelastic behaviour cannot be performed. Analogues for poro-. elastic parameters with explicit dependence on pore structure are derived here by using an inclusion-based model where inclusions represent individual pores. The inclusion- based formulation used in this paper permits uniform pore fluid pressure throughout the pore space, a requirement of poroelasticity. When specific inclusion-based approximations resulting from different descriptions of inclusion interactions were considered, it was found that the analogue quantities obtained from the dilute volumetric average, Kuster-Toksoz and equivalent inclusion-average stress approximations replicated the relationships between poroelastic parameters predicted by poroelasticity. This equivalence implies that pore geometry information can be consistently incorporated into poroelastic parameters with an inclusion-based model when one of these approximations is used.

This connection is used to examine pore geometry effects on poroelastic behaviour by considering a simple model with identically shaped pores. The results of this modelling study show that variations in pore shape significantly affect poroelastic parameters and that the nature of these effects is controlled by the specified applied stress-strain and fluid pressure conditions. The following observations were made about specific poroelastic quantities. The Skempton ratio is relatively insensitive to porosities below 0.1. Its minimum value for a given solid matrix, pore fluid and total porosity level is obtained when all pores are spherical; this value can be significantly greater than the theoretical lower limit of zero. Specific storage terms increase by several orders of magnitude as pores become more crack-like. The difference between the traditional definition of specific storage and that proposed by Green & Wang ( 1990) for ‘normal’ aquifer conditions grows as the pore aspect ratio decreases.

Key words: effective medium approximations, inclusion-based models, poroelasticity, reservoir properties.

INTRODUCTION

The deformation of porous rocks and soils is an important factor in many problems encountered in the earth sciences, such as estimating the productivity of hydrocarbon and groundwater reservoirs or predicting surface subsidence due to compaction. Commonly, deformation is described in the context of poroelasticity as presented by Biot (1941) and subsequent works [see the reviews by Kumpel (1991) and Wang ( 1993)]. This approach characterizes the behaviour of

*Now at: Boston College, Institute for Scientific Research, PL/GPE, 29 Randolph Road, Hanscom AFB, MA 01731-3010, USA.

a porous medium in terms of a relatively small number of empirically determined macroscopic parameters. While this approach provides a relatively compact form for describing experimental observations, it is not well suited for a priori predictions of porous medium behaviour. In particular, these macroscopic parameters implicitly incorporate pore structure information into poroelasticity. Hence, this approach cannot be used to analyse quantitatively the relationship between pore space geometry and poroelastic behaviour. Laboratory results (Fatt 1959; van der Knaap 1959; Mesri, Adachi & Ullrich 1976; Coyner 1984; Green & Wang 1986; Hart & Wang 1995) show that changes in pore structure, such as those due to variations in confining and pore fluid pressures, can

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Models for poroelasticity 523

cause significant alterations in macroscopic poroelastic quantities.

To investigate the effects of pore structure, it is necessary to use a formulation that explicitly incorporates this information. This can be achieved by using an inclusion-based model that views a porous medium as a solid matrix with embedded inclusions representing individual pores. Mathematical formulations for elastic behaviour using an inclusion-based model commonly assume that the individual pores are isolated in terms of fluid pressure communication between pores. This is in contrast to the complete fluid pressure communication (i.e. uniform fluid pressure throughout the pore space) required by poroelasticity. The inclusion-based formulation used in this paper incorporates this pore fluid pressure condition in a manner that permits the description of the response of a porous medium to arbitrary changes in overall stress, strain and pore fluid pressure.

This expanded inclusion-based formulation allows the derivation of inclusion-based analogues for various macroscopic poroelastic parameters. However, it is necessary to establish the connection between these inclusion-based analogues and their poroelastic counterparts. This was done by considering specific inclusion-based approximations that result from particular descriptions of inclusion interactions; for three approximations, it was found that the relationships between the inclusion-based analogues are identical to those predicted by poroelasticity. This implies an equivalence between these approximations and poroelasticity which permits the explicit and consistent incorporation of pore geometry information into the definition of the macroscopic poroelastic quantities through the use of an inclusion-based model.

This correspondence is used to examine the effects of pore space geometry on poroelastic behaviour by varying the pore aspect ratio and porosity in an inclusion-based model containing identically shaped pores. The results of this modelling study demonstrate that pore shape significantly affects the poroelastic response of a porous medium and that the nature of these effects is determined by the specified applied stress- strain and fluid pressure conditions. Furthermore, a number of interesting observations can be made about the nature of specific poroelastic parameters, such as the lower limit of the Skempton ratio and the variance between differently defined specific storage terms.

POROELASTIC PARAMETERS

The response of a porous medium to incremental changes in overall stress, strain and pore fluid pressure is summarized by poroelastic constitutive relationships, such as those given by Rice & Cleary (1976). These relationships contain poroelastic parameters that characterize the properties of a porous medium. For this paper, the porous medium is assumed to be homogeneous and isotropic on a macroscopic scale, with a solid framework that is homogeneous and isotropic on a microscopic level. Its pore structure is interconnected such that complete fluid pressure communication occurs (i.e. the pore fluid pressure is uniform throughout the pore space). We will consider the behaviour of a representative volume element (RVE) of the porous medium with volume I/ which is large in comparison to the individual pores, but small relative to the overall system.

The constitutive relationships of Rice & Cleary (1976) can

be stated in terms of the following three expressions. The first describes the response of the RVE to incremental changes in the fluid pressure apf and the hydrostatic pressure ap. These quantities are related to the cubical dilatation af3 by

- k o ( a o ) v = ( a ~ ) v - ( 1 -kolk,)aPf, (1) where k, is the solid matrix (i.e. constituent) bulk modulus and (...)vs denotes the volumetric average over a specified volume V', as defined by

The term ko is the effective bulk modulus of the RVE under open (i.e. drained) conditions. Pore fluid is allowed to move freely across the boundaries of the RVE in response to an incremental stress change under these conditions; there is no resulting change in pore fluid pressure (i.e. dpf= 0). Since k, is independent of the pore fluid, ko is an intrinsic property of a porous medium.

The second expression summarizes the incremental fluid mass variation in the RVE am due to incremental hydrostatic and pore fluid pressure changes. This relationship is given by

(3) where k, and pf are the pore fluid bulk modulus and density, respectively, and 4 is the porosity. The $/kf term is due to the fluid density change associated with apf; the other terms are the result of the net pore volume change in the RVE.

The behaviour of the porous medium described by these first two constitutive relationships is dependent on the incremental pressure conditions as specified by apf and ( 3 ~ ) ~ . The discussion of these relationships will be separated into two sets of incremental pressure conditions. In the first instance, it is assumed that = 0; this is equivalent to unconfined conditions where a porous medium is subjected to an incremental pore fluid pressure change without an accompanying confining pressure variation. The elastic behaviour of the RVE system in eq. (1 j is described by

k,(aoV = aPf9 (4)

where

( 5 )

is the effective bulk modulus of the RVE system under unconfined conditions. Since k, 2 k,, eq. ( 5 ) implies that k, 2 ko. The incremental fluid mass change from eq. (3) for unconfined conditions is

(7) is the specific storage defined by van der Kamp & Gale (1983).

The second set of incremental pressure conditions occurs when Cap)" # 0; the fluid pressure change is characterized by the ratio

B = aPf/<aP)V. (8) From eq. (11, it is seen that the RVE behaviour for this set of pressure conditions is characterized by

(9)

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where

k , = kmko/Ckm + P(k0 - k m ) l (10) is the effective bulk modulus of the RVE for the specified pressure conditions. Eq. (3) now gives the incremental fluid mass change as

am, = Pf vs, <dP>V 1 (11)

s , = ( & l ) / k o + C1-B(1+4)l/km+B4lkf (12)

where

is the equivalent specific storage term for these pressure conditions.

The third expression for the poroelastic constitutive relationships describes the response of the RVE to an incremental change in deviatoric stress components asij. It is assumed that asij is decoupled from the net pore volume change in the RVE; hence, the effects of as,, are independent of the fluid pressure conditions when complete fluid pressure communication occurs. The incremental deviatoric strain components in the RVE, aeij, are connected to asij under all pressure conditions by

2~ (aeij > v = (asij >Y > (13) where p is the effective shear modulus of the RVE system. Like ko, p is independent of the pore fluid and represents an intrinsic property of a porous medium.

In addition to the elastic moduli and the specific storage terms in the poroelastic constitutive relationships, a number of other parameters have been defined to characterize the poroelastic response of a medium. The coefficient of effective pressure 5 is used to define an effective stress law for the deformation of porous media (Nur & Byerlee 1971). This term is also referred to as the Biot-Willis parameter (Biot & Willis 1957). In eq. (l), (as), is proportional to an effective pressure change

aPeff = m), - u p , , (14)

{ = 1 -ko/km. (15)

where

Porous media under closed (i.e. undrained) conditions are frequently examined; no pore fluid is exchanged between the RVE and its surroundings during an incremental stress change under these conditions. Closed system behaviour can be described in terms of the ratio between the induced apf and the applied (ap>V (i.e. the value of /? for an undrained system), which is referred to as the Skempton ratio (Skempton 1954). In terms of k,,, the Skempton ratio B is (Green & Wang 1986)

B = kf(km - ko)/CkAkm - ko) + 4ko(km - kf)I. (16)

Both the Skempton ratio and the specific storage terms have associated quantities that are defined for different applied stress-strain conditions. Van der Kamp & Gale (1983) have proposed two induced pore fluid pressure ratios for closed conditions resulting from barometric/ocean tide and earth tide loading scenarios. For barometric/ocean tide loading, it is assumed that the horizontal displacement is negligible and the incremental vertical stress change is aabo. The barometric/ ocean tide loading efficiency (i.e. the ratio between apf and aab,) is defined as

BbO= -3Bko/[3ko+4~(1 -cB)]. (17)

For earth tide loading, it is assumed that the vertical stress is negligible and the incremental horizontal displacement change is ace = (a&ll)v + ( C ~ E ~ ~ ) ~ . The ratio between apf and a&e, referred to as the earth tide loading efficiency, is

Green & Wang (1990) suggested that a specific storage term should be defined for ‘normal’ aquifer conditions: no strain in two orthogonal directions and no stress in the third orthogonal direction. This leads to a specific storage quantity

Unlike the specific storage S, defined in eqs (6) and (7), the change a p f is associated with a non-zero ( a p ) ” under ‘normal’ aquifer conditions.

Information about the pore structure of the medium is implicitly incorporated into all of these poroelastic parameters. An explicit connection between pore structure and poroelastic behaviour requires the use of effective medium approximations that describe the porous medium in terms of geometrical components. This will now be done by using an inclusion- based formulation to derive quantities analogous to the poroelastic parameters.

INCLUSION-BASED FORMULATION

The inclusion-based formulation used in this paper will depict the porous medium as a solid matrix with embedded inclusions representing the individual pores; pore geometry information is contained in the range of inclusion shapes used. Using the volumetric averaging method (Hill 1963), the incremental changes in stress, pore fluid pressure and strain in the RVE can be related. Starting with the definition of the volumetric average given by eq. (2) for both stress and strain and using the constitutive relationships for the solid matrix material, the following are obtained:

and

(asij>v = 2 ~ m C ( a e i j > v - 4<aeij>vl , (21)

where P is the pore volume in the RVE and pm is the shear modulus of the solid material.

Evaluation of eqs (20) and (21) requires the determination of the incremental strain components in the pores; this can be a very difficult procedure for arbitrarily shaped inclusions. To simplify this step, the individual pores are represented as oblate spheroids, shapes for which solutions have been derived. The pore shape is characterized by its aspect ratio

a = a / b , (22) where a and b are the lengths of the minor and major axes of the oblate spheroid, respectively; the range of LX values is from 0 to 1, inclusively. The pore structure in the RVE is described in terms of the volumetric distribution of pore shapes V(a), where

V = [ @(a) da . (23)

An estimate of the incremental strain within an individual pore is obtained by using the equivalent inclusion method

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of Eshelby (1957). In this analysis, we consider an oblate spheroidal inclusion with aspect ratio a embedded in a homogeneous infinite background medium having elastic moduli kb and pb. This system is subjected to an applied incremental strain having uniform cubical dilatation doA and deviatoric components ae; at infinity, resulting in an incremental fluid pressure change apf in the inclusion. The macroscopic isotropy of the porous medium implies that an individual pore is randomly oriented. Endres & Knight (1997) found that the expected values of the incremental strain components inside such a randomly oriented inclusion are

ae'n"(a) = (doA + apf/kb)P(kb, 0, 0, - apf/kb (24)

i?ep(a) = Q(kb, pb, 0, 0, cc)de$. (25)

and

The terms P(kb, pb, ki, p i , a) and Q(kb, p b , ki, pi, a) are complicated functions that are defined by Berryman (1980) for an inclusion with elastic moduli ki and p i ; the expressions for P and Q are reproduced in Appendix A.

The background moduli and applied strain that appear in this analysis of the inclusion response are used to simulate the interaction between the inclusions in V when eqs (24) and (25) are used in an inclusion-based formulation. The explicit form of these quantities is determined by the type of approximation used and the incremental pressure conditions specified for the system. When necessary, the dependence of the background moduli and applied strain on pressure conditions will be denoted by an appropriate subscript.

With these estimates of the incremental strain components within the individual pores, the following expressions are obtained. It is assumed that all pores in the RVE experience the same form of inclusion interactions as specified by the apparent background material and applied incremental strain (i.e. the values of kb, pb, doA and de$ are the same for all inclusions). Hence eqs (20) and (21) become

( 3 ~ ) ~ = krnC4y(kb, pb, 0, - <aQ,l +4{1 +krn[y(kb, pb3 O , O ) - 1I/kb}dpf (26)

and

(asij>y=2pmC<aeij>v-4X(kb, pb> O,O)ae$l> (27)

y(kb, pb, 0,O) = v-l

where

O(a)P(kb, pb, 0, 0, a) da (28) sb sd

and

X(kb, pb, 0,O) = v-' O(a)Q(kb, pb, 0, 0, a) dcc. (29)

The incremental fluid mass change in the RVE due to pressure variations can be derived for an inclusion-based model by starting with

am = Vap, + pfaV, (30) where dp, and av are the resulting incremental fluid density and net pore volume variations, respectively. The term associated with the fluid density change can be expressed as (Coyner 1984)

pap, = P f 4 VdPflkf. (31) To obtain an expression for av, one uses the relationship

between the incremental volume and cubical dilatation changes within an individual pore,

&(a) = v(a)ae'n'(a), (32)

and integrates over all pore shapes. Using the estimate for d@"(a) given by eq. (24), one obtains

av= V4[(aoA + dpf/kb)7(kb, pb, 0,O) - apf/kb]. (33)

Incorporating the expressions for apf and d v into eq. (30), the dm value given by the inclusion-based formulation is

am = pfV4[(aQA + dpf/kb)y(kb, pb, 0,O) + ( l / k f - l/kb)apf].

(34)

The RVE behaviour described by eqs (26) and (34) is now considered for the two sets of incremental pressure conditions specified above to obtain inclusion-based analogues for the bulk moduli and specific storage terms in the poroelastic constitutive relationships. An asterisk is used to denote the inclusion-based analogue of a poroelastic parameter. When ( 3 ~ ) ~ = 0, the inclusion-based equivalents of k , and S , are

(35)

and

For all incremental pressure conditions, the inclusion-based definition of p obtained from eq. (27) is

P* = pm[1- 42(kb> pb> 0, o)ae$/<aeij>,I. (39)

In addition to quantities from the poroelastic constitutive relationships, inclusion-based analogues can be obtained for the other poroelastic parameters. To derive the inclusion-based counterpart of the effective pressure coefficient, we use the equivalence between 5 and the ratio of the incremental changes under open conditions of the net pore volume 86 to the RVE volume avo (Kiimpel 1991):

< = a v , / a v , . (40)

From eq. (33), the net pore volume change of an inclusion-based model under open conditions is

d vo = V#y(k;, p; , 0, 0)ae: . (41)

With the relationship of the incremental RVE volume change to the cubical dilatation given by

av= v(ae),, (42) the following inclusion-based definition for the coefficient of effective pressure is obtained:

t* = ~ k ; , P;, 0, o)awao>,. (43)

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For the inclusion-based expression of the Skempton ratio, we start with the requirement that no pore fluid is exchanged between the RVE and its surroundings during closed conditions (i.e. am* = 0). This implies that S* = 0 for this system; hence, eqs (37) and (38) give

(44) For the poroelastic parameters requiring anisotropic applied

stress-strain conditions (i.e. Bb,, Be and $,), it is necessary to determine the relationships between the stress and strain components in terms of inclusion-based elastic moduli. For barometric/ocean tide and earth tide loadings, the relationships between the forcing quantities and (8~)~ are

= -(4p* + 3k,*)(ap)V/(3k,*) (45)

(46)

and

ace = -(4p* + 3k:)(aP),/(6P*k:), respectively. The bulk modulus for the RVE under closed conditions is

(47)

which is obtained from eqs (37) and (44). It follows that the inclusion-based definitions of the loading efficiencies are

B&= -[3k,*/(3k,* +4p*)]B* (48)

B,*= -[6p*k,*/(3k,* + 4 p * ) ] B * , (49)

and

where the terms multiplying B* are the coupling factors due to the specified applied conditions. The 'normal' aquifer conditions imply that

( ~ P ) v = 4~*(dQ)vP. (50)

This result, with eqs (26) and (34), gives the following inclusion-based equivalent of S,:

To this point, the analogues for the various poroelastic parameters have been derived for a general inclusion-based formulation using the volumetric averaging method. However, this study has found that only a limited class of effective- medium approximations using this approach gives results that are consistent with those of poroelasticity. We will now consider this class of inclusion-based approximations and discuss how the equivalence between these formulations and poroelasticity is established.

RELATIONSHIP BETWEEN INCLUSION- BASED FORMULATIONS A N D POROELASTICITY

Poroelasticity predicts a set of unique relationships between the various poroelastic parameters that characterize a porous medium. It is desirable that this set of relationships be repli-

cated by the inclusion-based approximation used explicitly to incorporate pore structure information into an analysis of poroelastic behaviour. To establish the consistency between a given inclusion-based approximation and poroelasticity, it is necessary to obtain the explicit form of the analogue quantities associated with the approximation. This requires defining the nature of the inclusion interactions in terms of the apparent background material (kb and pb) and applied incremental strain components (doA and 8e;) which appear in the expressions for the analogue quantities derived in the previous section. This will now be done for three specific approximations: the dilute volumetric average, Kuster-Toksoz, and equivalent inclusion-average stress approximations. In all three cases, the apparent background material is assumed to be the actual solid matrix of the porous medium (i.e. kb = k, and pb = A); the inclusion interactions are characterized by the apparent applied incremental strain.

The dilute volumetric average approximation assumes a sufficiently small inclusion concentration to permit one to neglect the alterations of the strain in the RVE due to the presence of the inclusions. This implies that the applied incremental strain is identical to the volumetric average of the incremental strain in the RVE (i.e. d o A = (aQ), and ae$ = (aeij)v) for all conditions. This leads to the cancellation of the 83 and deij terms in all of the analogue expressions. Unfortunately, there are significant limitations on inclusion concentration with this approach; results of the dilute volumetric average approximation can become inconsistent with the Hashin-Shtrikman bounds (Hashin & Shtrikman 1963) at very small inclusion densities.

Kuster & Toksoz (1974) used a method referred to as the average T matrix approximation (Berryman 1992) in an attempt to overcome dilute inclusion concentration limitations. This approach analyses a sphere composed of the porous rock system which is embedded into an infinite background having moduli k, and p,; this assemblage is subjected to an applied incremental strain that is uniform at infinity. The behaviour of this system is compared with that of an identically sized sphere composed of the homogeneous effective medium and subjected to the same strain conditions. If it is assumed that the elastic-wave scattering observed in the far field is the same for both systems, the volumetric averages of the incremental stress and strain in the porous medium sphere are equal to the uniform incremental stress and strain in the homogeneous sphere in the quasi-static case.

This requirement leads to the following expressions for the inclusion interactions. For all conditions, the deviatoric components are related by

In general, the cubical dilatation terms are connected by

this expression takes on various forms when one considers the different incremental pressure conditions used to derive the inclusion-based analogues of the poroelastic parameters. When (ap), = 0, one obtains

For the (8~)~ # 0 case, the inclusion interactions are described

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by

a@ = (3kJ + 4prn)(dQ),/(3krn + 4prn). (55)

This expression can be rewritten in the following manner for the closed-system conditions imposed during the derivations of induced fluid pressure ratios (B* , B&, and B,*) and for the 'normal' aquifer conditions used to specify S,*. Using eqs (37) and (55), it is found that

( 3 k m +4&) {km + k f F ( k , LL, 090) - 1 11 (86) v ae: = (3k,+4&)(k- kf)+ C k d 3 L + 4 k ) - 34km(k,- km)17((6, &, 0,O)

(56)

for a closed system. For the 'normal' aquifer conditions, eqs (50) and (53) give

80: = 4 ( / - ~ - ~ * ) ( a 0 ) , / ( 3 k r n + 4 ~ r n ) > (57)

where p* is obtained from eqs (39) and (52). However, the Kuster-Toksoz approximation also has concentration limitations; it violates the Hashin-Shtrikman bounds when non-spherical inclusions are present in sufficient quantities (Berryman 1980).

The third form of inclusion interactions that we will consider is the equivalent inclusion-average stress (EIAS) approximation (Benveniste 1987). While the EIAS approximation does not, in general, produce results that are consistent with the Hashin-Shtrikman bounds, this approach will obey these bounds regardless of the inclusion shape distribution and inclusion concentration when applied to two-phase systems composed of isotropic components having randomly oriented inclusions (Benveniste 1987; Norris 1989). The EIAS approximation presumes that the apparent applied strain is equivalent to the volumetric average of the solid matrix strain:

(ae), = (1 - 4)ae* + d(ae>,

(ae,), = (1 - 4)ae;"j + 4(aeij), .

( 5 8 )

and

(59)

The quantities (aO), and (deij)y are evaluated by using eqs (24) and (25).

For the deviatoric strain components, this condition leads to the following expression:

ae$= (1 + 4CX(krn, prnj 0,O) - 1 1 1 - ' (aei j>v. (60)

For the cubical dilatation, the inclusion interactions are described by

(a%/y(krn, p m , 0,O) (61)

when (dp), = 0. If ( d p ) , # 0, inclusion interactions are expressed as

a$=(1 -LV)(ae>,/{l+ 4C(1- B)y(krn, prn, 0,O) - 1 1 I . (62)

The following descriptions of the inclusion interactions are obtained for the closed system and the 'normal' aquifer conditions. Using eqs (44) and (47), we obtain

for a closed system. With eqs (26) and (50), the 'normal'

aquifer conditions give

{3krn-4~*[7(krn,~rn, O > O ) - 1 I I

x (aQ>v/C3krn7(krn, prn, O , O ) l > (64)

where p* is procured from eqs (39) and (60). Using these descriptions of the inclusion interactions, explicit

expressions for the inclusion-based analogues of the poroelastic parameters were derived for these three approximations; these quantities are given in Appendix B. However, it still needs to be demonstrated that these inclusion-based analogues are consistent with their poroelastic theory counterparts; this is done in the following manner. The moduli k, and p are used to characterize the porous medium in the definitions of the various poroelastic parameters [i.e. eqs (5), (7), (lo), (12) and (15)-(19)] discussed in this paper. For each of the three inclusion-based approximations, these moduli are replaced by the expressions for their inclusion-based analogues k8 and p* in these definitions of the poroelastic parameters. The expressions for the inclusion-based analogues kg and p* are found in Appendix B, where kd is derived from k,* using /s' = 0.

With some algebraic manipulation, it can be shown that the resulting expression for each poroelastic parameter is identical in form to that derived for its inclusion-based analogue. For example, the dilute volumetric average approximation has an open-system bulk modulus analogue given by

k8 = krn[ 1 -4y(krn, Prn, 0,O)l. (65)

When this is substituted for k, in the poroelastic definition for k, given by eq. (5), one obtains

ku = k m [ 1 - 4 y ( k m , Prn,Q O)IIC4y(km, ~ r n ? O * O ) l , (66)

which is the same expression as one gets for the inclusion-based analogue k,* [i.e. eq. ( B . l ) ] .

Furthermore, it was seen that k,* and p* are independent of the pore fluid properties (i.e. k,) for all three approximations, as are their poroelastic counterparts. Hence, the set of relationships between the poroelastic parameters is replicated by the analogue quantities obtained from these three approximations. This result implies that geometrical information can be incorporated into poroelastic parameters with these inclusion- based approximations in a manner that is consistent with poroelasticit y.

This result represents a generalization of those presented in Endres & Knight (1997). In this aforementioned work, it was found that these three inclusion-based approximations pre- served the Gassmann (1951) relationships between the elastic moduli of the open and closed systems. It has now been determined that these approximations can be used describe the poroelastic behaviour of a medium over a range of applied stress-strain and fluid pressure conditions in a manner that is consistent with poroelastic theory. However, it has also been found that the dilute interaction energy approximation (Eshelby 1957), which Endres & Knight (1997) found to be consistent with the Gassmann relationships, violates poroelastic relationships when parameters pertaining to unconfined conditions (i.e. ( a p ) , = 0) are considered.

In addition to the above-mentioned inclusion-based approximations, the differential effective medium (DEM) approximation (Norris 1985) and formulations assuming the self-consistency condition (e.g. Berryman 1980) have been used to model the elastic properties of porous rocks. When Endres & Knight (1996) used a framework similar to that presented

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in this paper to incorporate pore fluid pressure communication into these formulations, it was found that the resulting inclusion-based analogues for the open- and closed-system elastic moduli do not replicate the Gassmann relationships. A similar finding was reported by Thomsen (1985) with respect to the self-consistent formulation of Budiansky & O'Connell (1980). Hence, these approximations do not give results that are compatible with poroelasticity when pore fluid pressure communication is simply incorporated.

Thomsen (1985) and Berryman (1992) have proposed methods for making the self-consistent and DEM approximations compliant with the Gassmann relationships. Endres & Knight (1997) found that the Thomsen approach produced Gassmann-consistent results when only spherical pores were used. Berryman ( 1992) did obtain Gassmann-consistent results for the DEM and coherent potential (a self-consistent approach) approximations; however, due to mathematical com- plexity, Berryman's results are restricted to the case where only spherical pores are used. Given the special nature of spherical inclusions (Endres & Knight 1996), it is not readily apparent that the Berryman method would give a Gassmann- consistent inclusion-based approximation when non-spherical pores are present. Further, Berryman only considered the relationship between open and closed systems. As the dilute interaction energy approximation demonstrates, it is possible for an inclusion-based approximation to be Gassmann-consistent without being compatible with general poroelasticity.

MODELLING RESULTS

The connection that has been established between this inclusion-based formulation and poroelasticity permits us to examine quantitatively the dependence of poroelastic parameters on the pore space geometry. This will be done in this paper by considering various simple models for porous media. In a given model, all pores are identically shaped oblate spheroids; hence, the pore space of a model is completely characterized by the aspect ratio CI of the pores and the porosity 4. For these inclusion-based models, the following quantities in the inclusion-based approximations become

and

All models have water-saturated pores (i.e. k f = 2.32 GPa) and a rock matrix with elastic moduli k,= 30GPa and p, = 17 GPa. The EIAS approximation was used to determine the poroelastic parameters in this paper.

The effective bulk moduli k,* and k: are given in Fig. 1 as functions of CI for several 4 values. These moduli describe the response of a medium for the two limiting cases of the incremental pressure conditions (i.e. dpf=O for k,* and ( i ? p ) " = O for k t ) . Both k,* and k,* decrease monotonically with c1 as the pore space compressibility increases, due to the fact that the pores change from spheres to flattened discs. The same models were used to investigate the dependence of p* on a and 4; it was found to be similar to that of k,*. However, a comparison of k,* and k,* in Fig. 1 shows a significant difference in the functional dependence on a, demonstrating that the effects of pore geometry on poroelastic behaviour are affected by the incremental pressure conditions. It can be seen in Fig. 1

n

m n U

c .L

k * \ * ku\A

Figure 1. Effective bulk moduli k,* and kS as functions of pore aspect ratio ct for simple models. The porosity 4 for each series of models is denoted on the figure.

that k,* decreases slowly until c( z 4, after which a more rapid decline occurs. The value of k: changes significantly at CI z 0.4 for the range of 4 values examined. The magnitude of k,* is scaled by a factor of approximately 1/4; this means that k,*/k,* for a given CI value grows steadily as 4 approaches zero. As CI decreases, the values of k,* and k,* converge; however, this occurs at smaller c( values as 4 lessens.

The Skempton ratio B* was determined as a function of 4 for various a values; these results are shown in Fig. 2. It can be seen that B* monotonically decreases to some non-zero value as 4 varies from 1 to 0; the value of B* is weakly dependent on porosity for 4 < 0.1. For a given value of 4, B* increases to its theoretical upper limit of 1 as CI goes to 0; the most rapid variation occurs for CI between 0.1 and 0.01. This increase is due to the improved coupling between dpf and ( d p ) y , which occurs as the pore space compressibility increases. While the theoretical lower limit of B* is zero, the modelling results in Fig. 2 suggest that the actual limiting value for a given set of k,, pm, k f and 4 values can be significantly higher. The lower limiting value is encountered when all pores are spherical (i.e. CI = 1); this case corresponds

0.8

c 0.6- PP

a = 0.1 0.4-

0.2- a = 1.

1x10.~ 1 ~ 1 0 - ~ i ~ i o - ~ IXIO-' ixio0 Porosity @

Figure2. Skempton ratio B* as a function of the porosity 4 for simple water-saturated models. The pore aspect ratio ct for each series of models is denoted on the figure.

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to the minimum pore space compressibility, which results in the least coupling between 8pf and ( 3 ~ ) " . For the water- saturated models used in this study, the lower limit of B* is approximately 0.16 when 4 < 0.1; this lower bound becomes substantially greater as 4 approaches 1.

Not only does pore structure affect B*, it determines the coupling between forcing quantities and (8~)" used to define the related closed-system loading efficiencies. Figs 3(a) and (b) give BZo and B,*, respectively, as functions of a for various 4 values. These quantities are negative due to the sign convention adopted for a pressure component relative to its associated stress; however, it is the magnitudes of B:,, and B: that are of interest. Unlike for B*, it can be seen in these figures that porosity is also an important parameter in determining the behaviour of the loading efficiencies, due to the coupling factors in eqs (48) and (49). In addition, a comparison of Figs 3(a) and (b) shows that the changing pore structure has different effects on BZo and B:, due to variations in the relative magnitudes of p* and k: in the coupling factors. The magnitude of Bto in Fig. 3(a) increases monotonically to a limiting value of 1 as a decreases; the relationship develops a region of relatively little change at intermediate tl values as 4 declines. In contrast, the magnitude of B,* in Fig. 3(b) increases to a

.

* 0

PP"

1 ~ 1 0 - ~ 1 ~ 1 0 - ~ I X I O - ~ IXIO-' 1 ~ 1 0 ~

Pore Aspect Ratio a

m n Y

1 x i o - ~ 1 x i 0" 1 XI o 2 1 XI 0-l 1 XI oo Pore Aspect Ratio a

Figure 3. (a) Barometric/ocean tide and (b) earth tide loading efficiencies (B& and BT, respectively) as a function of the pore aspect ratio c! for simple water-saturated models. The porosity Q for each series of models is denoted on the figures. Negative values are due to the definition of the pressure component relative to the associated stress.

relative maximum as a decreases from 1; the location of the maximum shifts to lower a values as 4 approaches 0. Hence, for a given 4 value, there is a pore structure for which the coupling between earth tide strain and induced pore fluid pressure is optimal. Furthermore, it can be seen that the magnitude of this optimal coupling increases as 4 goes to zero.

The specific storage terms S: and S,* are presented as a function of a for two 4 values in Fig. 4. It can be seen that both storage terms increase monotonically with decreasing a; the more rapid changes occur for GI < 0.1. The effects of pore shape on the magnitude of both S: and S,* are significant; the values of each term vary over several orders of magnitude due to the increasing pore space compressibility as the pores change from spheres to very thin discs. Porosity affects the size of the storage terms by a scaling factor of approximately 4. However, it can be seen in Fig.4 that the effects of different applied conditions used in defining the specific storage terms lead to notable differences between S,* and S,*. In particular, it is observed that S,* is less than S: for each model. Furthermore, the disparity between these two parameters increases with decreasing a and increasing 4; S,* is almost three times greater than S,* in the case where 4 = 0.1 and a = 0.0001. This result implies that the use of an inappropriate specific storage term can lead to significant errors in predicting reservoir response in highly fractured media.

CONCLUSIONS

While poroelasticity provides a convenient framework to describe the deformation of porous rocks and soils, pore structure information is implicitly incorporated into this formulation through the use of empirically determined macroscopic parameters. This makes it difficult to predict quantitatively the effects of pore geometry on poroelastic behaviour. To incorporate structural information, an inclusion-based model has been used to describe the pore space when defining analogues for the macroscopic poroelastic parameters. The pore space information is expressed in terms of the inclusion shapes used and their concentration in the medium. To overcome the assumption of isolated individual pores commonly used in inclusion-based formulations, the results of Endres & Knight (1997) were used to incorporate complete fluid

1 ~ 1 0 - ~ 1 ~ 1 0 - ~ i ~ i o - ~ I X I O - ~ 1 ~ 1 0 ~

Pore Aspect Ratio a Figure 4. Specific storage terms S: and S,* as a function of the pore aspect ratio CI for simple water-saturated models. The porosity 4 for each series of models is denoted on the figure.

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530 A. L. Endres

pressure communication throughout the pore space. Hence, the requirement of uniform pore fluid pressure within the medium used in poroelasticity can be satisfied. The resulting inclusion-based formulation can be used to describe the response of a porous medium to arbitrary incremental changes in applied stress-strain and pore fluid pressure.

To obtain the various inclusion-based approximations, it is necessary to specify the manner in which interactions between inclusions in the system are simulated. When this was done for the dilute volumetric averaging, Kuster-Toksoz and EIAS approximations, it was found that the relationships between these analogue quantities replicated the relationships predicted by poroelasticity. This consistency between inclusion-based approximations and poroelasticity is a desirable property that is not possessed by all inclusion-based approximations. Furthermore, this connection allows poroelastic behaviour to be related to other physical properties that can be described in terms of inclusion-based formulations, such as dielectric permittivity (Endres & Knight 1991) or thermal conductivity (Zamora, Cattin & Bienfait 1994).

Simple models containing identically shaped pores were used to examine the relationship between pore structure and poroelastic behaviour; a number of interesting observations can be made from these results. It was found that all poroelastic parameters change significantly as pores evolve from spheres to flattened discs. The effects of pore geometry on the poroelastic response of a medium vary with the incremental stress- strain and fluid pressure conditions specified in the definition of a given poroelastic parameter. In addition, a number of observations for specific parameters can be made. The Skempton ratio B* is relatively insensitive to porosity when 4 <0.1. Its minimum value for a given solid matrix, pore fluid and total porosity level is obtained when all pores are spherical; this lower limit may be significantly greater than the theoretical limit of zero. The size of specific storage terms increases by several orders of magnitude as pores become more crack-like. Furthermore, the difference between the specific storage terms S,* and S: grows as the pore aspect ratio decreases.

ACKNOWLEDGMENTS

This work was supported by an Individual Research Grant to the author from the Natural Sciences and Engineering Council of Canada. The author is an Associate of the Waterloo Centre for Groundwater Research.

REFERENCES

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Berryman, J.G., 1980. Long-wavelength propagation in composite elastic media 11. Ellipsoidal inclusions, J. acoust. Soc. Am., 68,

Berryman, J.G., 1992. Single-scattering approximations for coefficients in Biot’s equations of poroelasticity, J . acoust. Soc. Am., 91, 551-571.

Biot, M.A., 1941. General theory of three-dimensional consolidation,

Biot, M.A. & Willis, D.G., 1957. The elastic coefficients of the theory

Budiansky, B. & OConnell, R.J., 1980. Bulk dissipation in hetero-

Coyner, K.B., 1984. Effects of stress, pore pressure, and pore fluids on

1820-1831.

J. appl. Phys., 12, 155-164.

of consolidation, J . appl. Mech., 24, 594-601.

geneous media, Solid Earth Geophys. and Geotech., 42, 1-10,

bulk strain, velocity, and permeability in rocks, PhD thesis, Massachusetts Institute of Technology, MA.

Endres, A.L. & Knight, R., 1991. The effects of pore scale fluid distribution on the physical properties of partially saturated tight sandstones, J . appl. Phys., 69, 1091-1098.

Endres, A.L. & Knight, R.J., 1997. Incorporating pore geometry and fluid pressure communication into modeling the elastic behavior of porous rocks, Geophysics, in press.

Eshelby, J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond., A, 241, 376-396.

Fatt, I., 1959. The Biot-Willis elastic coefficients for a sandstone, J . appl. Mech., 26, 296-297.

Gassrnann, F., 1951. Uber die elastizitat poroser medien, Vierteljahresschr. Naturforsch. Ges. Zurich, 96, 1-21.

Green, D.H. & Wang, H.F., 1986. Fluid pressure response to undrained compression in saturated sedimentary rock, Geophysics, 51,948-956.

Green, D.H. & Wang, H.F., 1990. Specific storage as a poroelastic coefficient, Water Resour. Res.. 26, 1631-1637.

Hart, D.J. & Wang, H.F., 1995. Laboratory measurement of a complete set of poroelastic moduli for Berea sandstone and Indiana limestone, J . geophys. Rex, 100, 17 741-17 751.

Hashin, Z. & Shtrikman, S., 1963. A variational approach to the theory of the elastic behavior of multiphase materials, J . Mech. Phys. Solids, 11, 127-140.

Hill, R., 1963. Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids., 11, 357-372.

Kumpel, H.-J., 1991. Poroelasticity: parameters reviewed, Geophys. J. Int., 105, 783-799.

Kuster, G.T. & Toksoz, M.N., 1974. Velocity and attenuation of seismic waves in two-phase media: Part I. theoretical formulations, Geophysics, 39, 587-606.

Mesri, G., Adachi, K. & Ullrich, C.R., 1976. Pore-pressure response in rock to undrained change in all-round stress, Geotechnique,

Norris, A.N., 1985. A differential scheme for the effective moduli of composites, Mech. Mater., 4, 1-16.

Norris, A.N., 1989. An examination of the Mori-Tanaka effective medium approximation for multiphase composites, J. appl. Mech.,

Nur, A. & Byerlee, J.D., 1971. An exact effective stress law for the elastic deformation of rock with fluids, J . geophys. Res., 76, 64 14-64 19.

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26, 317-330.

56, 83-88.

APPENDIX A: EXPRESSIONS FOR P AND Q

Let us consider a randomly oriented oblate spheroidal inclusion composed of a material with elastic moduli ki and pi embedded in an infinite background medium having elastic

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moduli kb and pb; the aspect ratio of the inclusion is M. This system is subjected to an applied incremental strain field that has uniform cubical dilatation and deviatoric components BA and set, respectively, at infinity. Using the approach of Eshelby (1957), it can be shown that the expected value of the cubical dilatation and deviatoric components of the incremental strain inside this inclusion are

ae"'"(a) = P(kb, pb, ki, pi, a)dQA

dep(a) = Q(kb, pb, ki, pi, a)ae;,

(A.1)

64.2)

and

respectively, where the terms P and Q are complicated functions defined in Berryman (1980) and are reproduced below.

For a non-spherical inclusion (i.e. M # l ) ,

R k b , pb3 ki, pi, a) = FI/Fz, ('4.3) and

For an oblate spheroid (i.e. E < l) ,

2 ( 3 0 - 2) f = 1 - a 2

and

(A.17)

(A.18)

For a spherical inclusion (i.e. M = l),

(A.19)

and

(A.20)

where

(A.21) b(9kb + 8pb) 6kb + 12pb '

Fb=

where APPENDIX B: DEFINITION OF POROELASTIC ANALOGUES FOR INCLUSION-BASED APPROXIMATIONS

The following expressions are obtained for the inclusion-based analogues for the poroelastic quantities discussed in this paper when the form of the inclusion interactions specified by the dilute volumetric average, Kuster-Toksoz and EIAS approximations are used. The corresponding expressions for the loading efficiencies B:,, and B: are obtained from eqs (48) and (49) by using the expressions for k,*, p* and B* given below. The terms 7 and ,j represent 7(km, pm, 0,O) and z(km, pm, 0, 0),

A.5)

A.6)

(f+ 0) - - (3f + 50) + B( 3 - 4R) " 1 2

A 2

+ - ( A + 3B)(3 - 4R)[ f + 0 - R ( f - 0 + 2e')],

f', = 1 + A [I - (f + 28) + R(f + 0)] ,

A 4 F4 = 1 + - [ f + 30 - R ( f - 0)] ,

F , = A [ -f + R (1. + e - :)I + m ( 3 - 4R),

F,= 1 + A [ 1 +f - R ( f + 0)] + B( 1 - 0)(3 -4R),

.4 4 F, = 2 + -[3f + 90 - R(3f + so)] + BQ(3 - 4R),

('4.9)

(A.lO)

(A . l l )

+ B( 1 - 0)(3 -4R)

(A.12)

(A.13)

and

F, = A [ ( R - 1)f- Re] + B0(3 - 4R).

The terms A, B and R are given by

(A.14)

(A.15)

and

(A.16)

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532 A. L. Endres

15km + 20pm - 4(9km + Sp,)j 1% + 20pm + 64(km + 2pm)j ’ p*=pm (B.14)

(B.18)

(B.19)

(B.20)

(B.21)

(B.22)

(B.23)

(B.24)

(B.25)

(B.26)

$11 I (B.27)



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Geometrical models for poroelastic behaviour

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