UCRL-JC-119722PREPRINT

The Elastic Coefficients of Double-Porosity Modelsfor Fluid Transport in Jointed Rock

J. G. BerrymanH. F. Wang

This paper was prepared for subnfittal toGeophysical Research

January 1995

This is a preprint of a paper intended for publication in a journal or proceedings.Since ehangss may be made before publication, this preprint is made available withthe understandin$ that it will not be.. cited or re p..:ro~]gc-d_. ,....,.,without, the.~ermispiono f..~. ̄. :.~-x ~ ~’~’~.,.~.

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The Elastic Coefficients of Double-Porosity

Models for Fluid Transport in Jointed Rock

James G. BerrymanLawrence Livermore National Laboratory

P. O. Box 808 L-202Livermore, CA 94551-9900

and

Herbert F. Wang*Department of Geology and Geophysics

University of ~Visconsin1215 West Dayton Street

Madison. WI 53706

"Address for correspondence.

ABSTRACT

Phenomenological equations for the poroelastic behavior of a duaJ porosity medium are for°mulated and the coefficients in these linear equations are identified. The generalization fromthe single porosity c~.se increases the number of independent coe~cients from three to six for anisotropic applied stress. The physical interpretations are based upon considerations of differenttemporal and spatial scales. For very short times, both matrix and fractures behave in anundrained fashion. For very long times, the double porosity medium behaves as an equivalentsingle porosity medium. At the macroscopic spatial level, the pertinent parameters {such asthe total compressibility) may be determined by appropriate field tests. At the mesoscopicscale pertinent parameters of the rock matrix can be determined directly through laboratorymeasurements on core, and the compressibility can be measured for a single fracture. Allsix coefficients are determined from the three poroelastic matrix coefficients and the fracturecompressibility from the single assumption that the solid grain modulus of the composite isapproximately the same as that of the matrix for a small fracture porosity. Under this assump-tion, the total compressibility and three-dimensional storage compressibility of the compositeare the volume averages of the matrix and fracture contributions.

[. INTRODUCTION

Analysis of the quasistatic behavior of porous fluid-saturated ,nechanical systems is generallybased on "poroelastic theory." The first detailed studies of the coupling between the pore-fluidpressure and solid stress fields were described using a linear elastic theory by Blot [1941]. Thequasistatic constitutive equations relate the strain tensor linearly to both the stress tensor andthe fluid pressure. Time depe,~dcnt, quasistatic fluid flow is incorporated by combining thecontinuity requirement with Darcy’s law. As originally formulated. Blot’s theory applies to abomogeneotLs, porous medium. However. porosity and permeability often occur in rock masseson several distinct spatial scales. Thus. the need arises for more general models it~corporatingqu~tl]tativelv different types of porosity (e.g.. matrix versus fracture) ~ well as different typesof rock mass for code calculatmns at the re~er~olr or aquifer scale. Biot’s theory neverthelesscontinues to play an important role in these more ,:on~plcx models: the mechanicaJ behaviorof the matrix materials from which they a.re usually constructed is often described by Biot’sequations supple,nented with fluid-tlow coupliug terms between the bloCks.

I,~ problems of fluid flow in hydrocarbon reservoirs and aquifers, the si,nplest and mostfrequent idealization is the dual porosity mediu,n in which po,’ous matrix blocks are dissectedby a fracture network [Barenblatt et al.. 1960: Warre,L and Root. 1963]. It is generally assumedthat the fract~)re permeability is much greater than the matrix permeability, while the fractureporosity is much smaller than the ,natrix porosity. Therefore. fluid flow occurs primarily throughthe fracture network, but fluid storage occurs ,nostly in the porous matrix.

The typical pictorial representation of the double porosity medium shows disaggregated~natrix blocks surrounded entirely by fluid in fractures. In this scenario, the blocks must besupported entirely by fluid p,’essure. The poroelastic ~nodel for a double porosity medium tobe presented here accounts more realistically for so,ne of the external stress to be supportedpartially by asperities bridging fracture surfaces.

Since the pore-pressure field is not in general decoupled from the stress field, it is neces-sary to incorporate Blot’s concepts of poroelasticity into the dual porosity model. First thestatic constitutive equations ,nust be formulated. Then tim equations of fluid transport canbe included via the continuity requirement. The independent variables are chosen to be the

external confi,~ing pressure, p~. a,~d the fluid pressures i,1 the matrix p~) and in the fracture

p~2). respectively. The dependent variables are chosen to be the volumetric strain, and Blot’sfluid mass content (fluid volume accumulation per unit bulk volume) in the matrix ((t) fracture ((2). The phenomenological approach the,~ relates each dependent variable ]inearh-to the indepe,~de,~t variables. This choice of variables leads to a symmetric coefficient matri~xbecause the scalar product of the dependent and iudepende,~t variables is an energy de,~sity.

The governing equations ]br fluid tra,~sport i,~ a dual porosit.v medium are written as ap,’essure diffusion equa.tio,~ in the fractured medium with a tn~trix-to-fractt, re source ter,n that.in its simplest fo,’m, is proportional to the difference in p,’essure between fluid i,~ the matrixand fluid in the adjacent fracture [Wilson a,td Aifantis. 1982: Khaled ct al.. 1984; Beskos andAifantis. 1986: ].".lsworth and Bai. 1990: Elsworth and Bai. 1992: Bai et al.. 1993].

The focus of this paper is on the rigorous identification of the coefficients in a linear for-mulation of the phe,mmenological equations. Although some of the parameters identified are,mr readily measurable, the analysis nevertheless clarifies the significat~ce of all the coetficients."[’he procedure is analogous to tha~ of Blot and Willis [1957] in which they establish physical

interpretations of the coefficients in Blot’s equations as well as relations between macroscopicparameters and properties of the more microscopic constituents. The analysis is performedhy considering both different temporal and different spatial scales. Three temporal scales areconsidered: very short times (both matrix and fracture phases are undrained}, intermediate(only one phase is drained), and long times (matrix and fracture pressures are equilibrated}.Similarly, three levels of spatial scale are considered: macroscopic (reservoir scale), mesoscopic(core sample scale), and microscopic (individual grain and fracture scale). At the macroscopiclevel, the pertinent parameters (such as the fracture compressibility) may in principle be de-ter~nined by appropriate field tests or large block tests. At the scale of core samples, pertinentparameters of the rock matrix may be determined directly through laboratory measurements.Two special cases are identified: (1) When the fracture tluid pressure is set equal to that of theconfining pressure, the matrix material is effectively isolated from the fracture behavior and theanalysis may be conveniently simplified. (2} When a special choice of the matrix fluid pressureis made, the matrix is constrained to behave rigidly on the average, and so the fracture behavioris effectively isohtted from that of the matrix. These two cases allow a natural decoupling ofthe double-porosity system so the coefficients in the equations may be determined in easilydescribed laboratory experiments.

2. EQUATIONS FOR DEFORMATION DEPENDENT FLOW

The equations for deformation-coupled flow in a single porosity medium are easily derivedfrom Biot’s equations of poroelasticity by taking the low frequency limit, assuming only that in-ertial effects and second derivatives with respect to time are negligible. If the solid displacementis ui and the fluid pressure is p (positive in compression}, the resulting equations are

3K 3(1 - 2v}K2(l+v) eJ+ 2(l+v) Ui’jj’-aP’i’ (1)

where the solid dilatation is e = ~tj.j, K and v are the drained bulk modulus and Poisson’sratio, respectively, and

¢tP.ii - Bh.,,[~ + ~.(2)

where K~, is the undrained bulk modulus and B is Skempton’s coefficient [Skempton, 1954].The dots indicate time derivatives. The remaining constants are the Biot-Willis parameterc~ = 1 - K/K~ = (1 - K/K,,)/B. the permeability k. and the fluid viscosity #. [Anotherconstant we will need, but that does not appear explicitly here. is the undrained Poisson’sratio u~,.] A detailed discussion of all these constants will be given in the next section. Theseequations are well known and may be found in Rice and Clcary [1976], Cleary [1977]. etc.

The generalization of (1) and (2} to double porosity media is straightforward. First, assume that there are two types of porosity and corresponding pressures p{D and p{~) withinthe fluids contained in each pore type. (See Figure 1.) Then, to generalize the equation for thesolid displacement, we merely change the forcing term on the right hand side to allow for morecomponents, so (1) becomes

3K 3(1 - 2v)Ka{~}.(~} (3)

where we have introduced phenomenological constants ~(l) and (~) whose precise physicalsignificance must be determined. An important characteristic of this equation is that eachpoiat in space now has two fluid pressures associated with it, and therefore these pressures are{not the true microscopic pressures in the fluid, but actually) averages over sorae representativevolume element. Similarly, Khaled et al. [1984] write two equations in place of (2) having theforms

and

IZ ’ Bi~) h’(.,1)

B(.~_~(~i~t~) + c~(-~)~ + ~,.(p(.a) _ ply)). (,5

and we will show how to interpret their coefficients in terms of a more complete derivation.The coefficients k(~) and k(~) are permeabilities associated with matrix and fracture poros-

ity, respectively. The remaining factors are straightforward generalizations of the constantsappearing in (2). Long time analysis to be presented later shows that k(1) + k(~) = also provides other relations among the various constants. The terms proportional to pressuredifferences have been introduced to drive the pressures in the two types of porosity towards asingle equilibrium pressure that will be approached at long times. More detailed derivationsof these equations may be found in the papers by Wilson and Aifantis [1983], Khaled et al.[1984], and Beskos and Aifantis [1986]. In particular, note that setting K = a(~) = o~(~) results in the classical double porosity model of Barenblatt et al. [1960]. The main point to beemphasized here is that these equations are just linear relations between stresses and strains.Except for questions about the neglect of possible cross-coupling terms (.which we show later isa real issue but fortunately a small one), the form of the equations is not in doubt: however,the meanings and values of various coefficients may be nontrivial to deduce from the physicsand mechanics of the underlying microscopic problem.

The remainder of the paper is devoted to a careful analysis of the precise physical significanceof the various parameters appearing in equations (3)-(5}.

3. SINGLE POROSITY MODELS AND LONG TIME BEHAVIOR

In the absence of driving forces that can maintain pressure differentials over long time peri-ods, double porosky models ~nust reduce to single porosity models in the long time limit whenthe matrix pore pressure and crack pore pressure beco~ne equal. It is therefore necessary toremind ourselves of the basic results for single porosity models in poroelasticity. So, one impor-tant role these results play is to provide constraints for the long time behavior in the problemsof interest. A second significant use of these results (which we address later in this paper) ariseswhen we make laboratory measurements on core samples having properties characteristic of thematrix material. Then the results presented in this section apply specifically to the matrixstiffnesses, porosity, etc.

For isotropic materials and hydrostatic pressure variations, the two independent variables inlinear mechanics of porous media are the contining (external) pressure Pc and the fluid (pore)

4

pressure pl. The differentiaJ pressure P,l = Pc - Pl is often used to eliminate the confiningpressure. The equations of the flmdamental dilatations are then

6I," ~p~ 6p! (6)-P-’:-= If + h’~

for the total volume V,

-E~"= A’~ + A’,~ (7)

for the pore volume V,~ = ¢I,’. and

for the fluid volume VI. Equation {6) serves to define the various constants of the poroussolid, such as the drained frame bulk modulus g and the unjacketed bulk modulus K~ for thecomposite frame. Equation (7) de~nes the jacketed pore modulus h’p and the unjacketed poremodulus g~. Similarly, (8) de~nes the bulk modulus KI of the pore fluid.

Treating ~p~ and ~p/ as the independent v~riabhs in our poroelastic theory, we deiinethe dependent variables #e e ~V/V and ~ ~ (6¢g - ~~)/g, both of which ~re positive onexpansion, and which ~re respectively the total volume dilatation and the increment of ~uidcontent. Then. it bllows directly from the de~nitions and from (6), (7), and (8)

k-¢/K~ ¢(1/K~+I/K/-1/K¢)J k-~Pl]" (9)

Now we consider two well-known gedanken experiments: the drained test and the undrainedtest [Gassm~nn, 1951~ Blot and Wi~s, 1957; Geertsma, 1957]. (For a single porosity system,these two experiments are sometimes considered equivalent to the "’slow loading" and "fastloading" limits respectively. However, these terms are relative since, for example, the fast load-ing -- equivalent to undrained ~ ~mit is still assumed to be slow enough that the average fluid~nd confining pressures are assumed to have reached equilibrium.} The drained test ~ssumesthat the porous ~naterial is surrounded by an impermeable jacket and the fluid is a~owed toescape through a ~ube that penetrates the jacket. Then. in a long duration experiment, thefluid pressure remains in equilibrium with the external fluid pressure (e.g., atmospheric} andso 6p] = 0 and hence 6Pc = b’pu; so the changes of total volume and pore vohlme are givenexactly by the drained constants 1/K and 1/K, as defined in (6) ~nd {7). On the other hand.the undrained test assumes that the jacketed sample has no tubes to the outside world, sopore pressure responds only to the confining pressure changes. With no means of escape, theincrement of fluid content cannot change, so b~ = 0. Then, the second equation in (9} showst h~t

0 = -¢/Kt,(6p~ - 6p.f/B),

where Skempton’s pore-pressure buildup coefficient B [Skempton, 1954] is defined by

(I0)

6Pc 6(=0( 11

and is therefore given by

LB = 1 + g’p(l/Ks - tlK, )’ (12)

It follows immediately from this definition that the undrained modulus Ku is determined by(also see Carroll [19S0])

h"K,, -- l - c~B" (13)

where we introduced the combination of moduli known as the Blot-Willis parameter a ---- 1 -h’/h’~. This result was apparently first obtained by Gassmann [1951] (t hough not in this form~for the case of microhomogeneous porous media (i.e.. h’~ = h’d = K,,, the bulk modulus of thesingle mineral present.) anti by Brown and Korringa [1975] and Rice [1975] for general porousmedia with multiple minerals as constituents.

Finally’, we condense the general relations from (9) together with the reciprocity" relations[Brown and Korringa. 1975] into symmetric form as

-~i = ~-( -r,~ o~/B -~Pl ] ’ (1-1)

A storage compressibility, which is a central concept in describing poroelastic aquifer be-havior in hydrogeology, related inversely to one defined in Blot’s original 1941 paper is

S -- = (15)

"[’his storage compressibility is the change in increment of fluid content per unit change in thefluid pressure, defined for a condition of no change in ezternal pre, ssure. It has also been calledthe three-dimensional storage compressibility by" Kfimpel [1991].

We may equivalently eliminate the Blot-Willis parameter ~. and write ( 14 } in terms of theundrained modulus so that

.-~.t," = ~ -(1 - h’/h’~)/B (l - K/K,,)/B "~ -6p] " (16)

Equ~ttion ( 16} has the advantage that all the parameters trove very well defined ph.vsical inter-pretations, and are also easily’ generalized for ~ double porosity model. Finally, note t.hat (14)shows that Kp = tK/a. which we generally refer to as the reciprocity relation.

The total strain energy functional (including shear) for tiffs problem may be written in theform

2E = br,:be 0 + (17)

where b’eij is the change in the average strain with 6eli =~ be being the dilatation. (rii beingthe change in the average stress tensor for the saturated porous medium with ~brii = -~Pc. Itfollows that

OE

and

OE~ps = 0(~¢)’ (19)

both of which are also consistent with Betfi’s reciprocal theorem [Love, 1927] since the matricesin (14) and (16) are symmetric. The shear modulus G is related to the bulk modulus Poisson’s ratio by G - 3(I - 2v)K/2(l + v). Then, it follows that the stress equi]]briumequation is

rij,j = (K,, + ½G)e,i + G,q.jj - BI(~C,.i =

and Darcy’s law takes the form

k-~p.,, = ~’.

From these equations, (1) and (2) may be easily derived using the identity

which follows easily from (16).from the equations.

(20)

(21)

( p eK,, -K - ( BK.)2 + BK----~’ (22)

Equation (22) is used to eliminate the confining pressure

4. DOUBLE POILOSITY I~,[ODELS: IDENTIFYING PHENOMENOLOGICAL COEFFICIENTS

~Ve ,low assume two distinct phases at the macroscopic level: a porous matrix phase with

tile effective properties K{U, G(I), h’~1, ¢{1} occupying volume fraction V(1)/V = v{U of thetotal volume and ̄ macroscopic crack or joint phase occupying the remaining fraction of thevolume V(2}/I" = v{2) = 1 - o(x). The key feature distinguishing the two phases .-- and thereforerequiring this analysis -- is the very high fluid permeability k~221 of the crack or joint phaseand the relatively lower permeabi~ty k(tO of the matrix phase. We could also introduce a thirdindependent permeabifity k02) = k{2~) for fluid flow at the interface between the in~trix andcrack phases, but for simplicity we assume here that this third permeability is essentia~y ~hesame as that of the matrix phase, so k{~2) = k{~x}.

We lmve three distinct pressures: confining pressure 8pc. pore-fluid pressure ~p~}. and

joint-fluki pressure ~p~}. Treating ~pc,~p}~}, and *p~2’ as the independent variables in ourdouble porosity theory, we define the dependent variables ~e ~ $V/V (as betbre), 6~(1}

(~)- ~v~lJ)/V, and ~({2) = (~,.~2) _ ~1..~21)/i,,, which are respectively the total volumedilatation, the increment of fluid content in the mntrix phase, and the increment of fluid contentin the joints. We assume that the fluid in the matrix is the sanle kind of fluid as that in thecracks or joints, but that the two fluid regions may be in different states of ~verage stress andtherefore need to be distinguished by their respective superscripts.

Linear relations among strain, fluid content, and pressure then take the general form

_~¢-(1) = a2, a22 a23 _~p~U . (23)_~((2) a31 a3 2 a33

By analogy with (1,1) and (16), it is easy to see that al~ = a~l and a~3 = a31. The symmetryof the new off-diagonal coefficients may be demonstrated by using Betti’s reciprocal theorem inthe form

where unbazred quantities refer to one experhnent and barred to ~mothcr experiment to showt hat

Hence. a~-3 = o3~. Similar arguments have often been used to establish the symmetry of theother off-diagoual components. Thus, we have established that the matrix in (23) is completelysymmetric, so we need to determine only six independent coefficients. To do so, we considera series of gedanken experiments, including tests in both the short time and long time limits.The key idea here is that at long times the two pore pressures must come to equilibrium

(P~) = P~") = as t - ~ .~)as longas the crosspermeability k ivy) is fin ite . However, at ver yshort times, we may assume that the process of pressure equilibration has not yet begun, orequivalently that k(12} = 0 at t = 0. We nevertheless assume that the pressure in each of thetwo components have individually equilibrated on the average, even at short times.

We should emphasize that lhese are thought experiraents, and as such may not necessarilybe realizable in the laboratory in all cases.

Also, note that the e~stence of a second pore pressure and increment of fluid contentleads to the definitions of several Skempton-like or Biot-Willis-like coefficients. :[’lie somewhatcomplicated notation we introduce will attempt to emphasize the defining boundary conditionsfor the various cases. We will clarify these differences in the Discussion Section.

4.1 Undrained joints, undrained matriz, short time

There are several different, hut. equally valid, choices of time scale on which to defineSkempton-like coefficients for the matrix/fracture system under consideration. Elsworth andBN [1992] use a definition based on the idea that for very short time both tluid systems wi~independently act undrained after the addition of a sudden change of confining pressure. Thisidea imp~es that ~((~} = 0 = ~((’~1 which, when substituted into (23),

-6e = all’Pc + a126p~11 + al3bP~~

0 = an~p~ + a.n6p~~ + a~abp~~)

0 = al3¢~pc + ~/23~p~1) ~ a33~p "~).

Defining

a,n d EB ---- ~Pc (27)

8

we can solve (26) for the two Skempton’s coefficients and find the results

and

](I} a23a13 -- a12a33EB --

a22a33 _ a]3 (2S)

/j(2) a23a12 -- a13a22EB "-- a22a33 _ t~223

The effective undrained modulus is found to be given by

1 (~e _ r~(l) a

These definitions will be compared to others as our analysis progresses.

(29)

(30)

Drained joints, undrained matrix, intermediate time

Again consider a sudden change of confining pressure on a jacketed sample, but this time

with tubes inserted in the joint {fracture} porosity so 6p~2} = 0, while ~C"0} = 0. We will callthis the drained joint, undrained matrix limit. The resulting equations are

~e = -ai16pc - a]~p~~)

0 = -a2~pc - a225P~])

-~(’~) = -a3t3pc - aa2~p~~),

showing that the pore-pressure buildup in the matrix is

B[u(l)]___ 6P~1) ---- _a21(~Pc 6~0)=6p(1~)=0 a22’

(31)

(32)

Similarly, the effective undrained modulus for the matrix phase is found from (31) to be deter-mined by

K[u(’)] ~ (~p~_ _ = art + a~2B[u(t)]. {33}

Notice that if a23 = 0 then (28) and (32) are the same.

4.3 Drained matrix, undrained joints, intermediate time

The next thought experiment might be difficult or impossible to realize in the laboratory.Nevertheless. consider another sudden change of confining pressure o,t a jacketed sample, but

this time the tubes are inserted in the matrix porosity so 6p~t) = O, while/iq’l~) = O. We willcall this the drained matrix, undrained joint limit. The equations are

~e = -allgPe - alSP~~)

-b~(1) ---- --a216pc -- a23(~p~2)

0 -- -a3! 6pc -- (t336P~2), (34)

showing that the pore-pressure buildup in tit(: cracks is

6P~2)I _ a31. (35)

Similarly, the effective undrained modulus for the joint phase is found from (34) to be deter-mined by

K[u(~)] =(36)~l=.;p’)"=o = at1 + a~3B[u(~)].

We m;~y properly view Eqs. (32), (33), (35), and (36) as "defining" relations among parameters.

Notice that if a23 = 0 then (29) and (35) are the s;~me.

Drained test. long time

The long duration drained (or "’jacketed") test for a double porosity system should reduceto the same results as in the single porosity limit. The conditions on the pore pressures are

6p~~} = dip~~) = 0, and the total volume obeys be = -a~6p~.. It follows therefore that

1all =

where K is the overall drained bulk modulus of the system including the fractures.

4.5 Undrained test, lot~g time

The long duration undrained test for a double porosity system should also produce thesame physical results as a single porosity system (assuming only that it makes sense at someappropriate larger scale to view the medium as homogeneous). The basic equations are

~; -- 6((~) + ,~(’(~)

assuming the total mass of Ihfid is confined. ’.[’hen, it follows that

(38)

be = -all’P,:- (a12 0 = -(a~ ~ az~)~p¢ - (a2~ + a~3 + aa: + a33)~p],

showing that the overall pore-pressure buildup coefficient is given by

-- OP/I = _ a21 + (L31

OPc’Jc’r..’=O fl’2"2 "1- a23 -’1- a32 q- a33’

(39)

(40)

I0

Similarly, the undrained bulk [nodulus is fouud to be given by

= all+Cat2+ (41)

Fluid injection test, long time

The conditions on the pore pressures for the long duration, single porosity limit for the

three-dimensional storage compressibility 5’ are $p~t} = ~p~2) = ~pl, while the confining pressureremains constant. It follows therefore that

2( = a~_2 + a23 + a32 + a33.S = 8p-~l~p,=°

4.7 Generalized Blot-Willis Parameters

Equation (37) has already determined the coefficient all. Then. (-11) shows

1~K-IlK.a12 + at3 = B = -a/K, (43)

while (33) and (37) show

1/K - 1/K[u0}]-_a12 = --

S[?./.(1)](44}

and similarly (36) and (37) show

IlK- l/K[u c~)] = _~C))IK.a13 =

B[u(2)](45)

In (.i,l) and (45}, we have introduced generalized Blot-Willis parameters ~(t) and ~(2) matrix and joint phases, defined by these equations. These two parameters are defined thisway for txotational convenie,xce, but for example a-(1) should not be confused with the trueBlot-Willis parameter nO} of the matrix material, which we anticipate will generally differ in

value from ~(t}.Combining (44) and (45) with (43} shows

a = ~(~) + ~2). (46)

Relations such as (46) showing dependencies a,nong the various constants are useful becausethey show that consta,lts potentially difficult to measure (such as the undrained joint modulusK[u(2)]} can actually be determined from other more easily accessible data.

These results show that "all the constants in the first row of the nlatrix in (23} have now beendetermined in tcrrns of quantities that could in principle be measured. Similar manipulationsgive the remaining coI~stants as we will show.

11

¯ t.8 Summary of reswlts

Combining these results, we obtain the following general relations

where the expressio,~ for the remaining off-di~,.gonal term

a~.a =- (~/13 - ~{’) i13[uf’)] - ~) l B[u(~)])12h"

(47)

(48)

follows from (40) after substituting for other coefficients using (32), (35). and (43)-(,15).

pari,tg (14)with the long time behavior (bp~’~) -. ’1) of ( -17)provides a si mple means ofchecking the validity of (47).

The comparative simplicity of (.17) may be sufficient reason to consider this set of definitionsas the most "’natural" one. ttowever, sine,, not all of the parameters are easily measured, wewill need to cortti,tue our analysis in the next two sections. For ,tow. we make use of theseresults to derive equations for fluid flow.

In analogy with (22), we need to write equations for ¢-(t) and (,’(~) elimi,mting the confiningpressure. Using (44) and (45}, the results of these calculations are:

and

~(1) ~(I )~(2))P~’~)

(-19)¢c,~ = zT(’~ + ~[,,(,~]A.[,,(~]~~I + (~,.~ A----:---

-~(’2) (’2) +( a..,:{ ~L)~(~) )Pc(") a~e + ~[~,c~l]A[,(~l]" ~ A--=- (50)

it is important to note that the presence of the nonzero coet’ticient [a.,.:~ - (a--(’)e-(~))/K)] cross-coupling terms is a departure front the approach of Khaled et (d. [1984] and Elsworth andBai [1992]. ’.Fhese authors have simply assumed that this coeflicient vanishes identically. Thisis an appro.,dmation that needs further justitication.

Since confining pressure has been eliminated, we can drop the subscripts (but lint thesuperscripts) on the two fluid pressures. Then. our final equations are:

-~ ~’,. + # ~,. = Bru(~)]A.[~t(~)]~)(~) + E(t)# +_a~3 h" ~(~) (51)

and

These ecLnations should be compared to (-l) and (5). A detailed comparison of differences nume,ical predictio,ts is beyond the scope of the present paper, but will be pursued in futurework.

L2

LABORATORY MEASUREMENTS ON CORZ SAMPLES

Although the preceding section gives rigorous definitions of the various constants neededto generalize from single-porosity to double-porosity poroelasticity, nevertheless some of theseconstants are not easily measured directly either in the laboratory or in the field. Therefore,in this section and the next, we show how to relate all the coefficients to more easily measuredand interpreted paratneters.

It follows immediately from (14) and the definitions of the constants that the pertinentequations for pure matrix material must be given by

(~e(1)-(~(~’{1)/W{1)) -" h’O)l_[ I -a(I, _~{1) a{~)/B(,}) ( (53)

These constants cart all be estimated [Rice and Cleary, 1976; Detournay and Cheng, 1993] orfound in the laboratory by ~nalysis of core samples of matrix material [Hart and Wang, 1995].This fact is important because it suggests another way of identifying certain combinations ofthe general double-porosity coefficients in (23}.

’lb obtain the connecting equations, first note that

~e = ~ ~p" = vO}~V0}V0---~ + -~’~V(~) = v0)~e0) + v(’~)~e(~}" (54)

for ~ = 1,2, (55}

and

v v v

The final relation follows from the identity/iV {’~) = ~V~u}, since the fracture volume is void

space.Thus. to obtain the desired relations in a form analogous to {53). we must rewrite (23}

--~0 "(I) -- a21 a22 (123 --

o(2)6e(~) ]-- a31 ~32 E33

where the third diagonal element has been modified to e~minate the fluid contribution suchthat

Now we consider another thought experiment: Suppose the confining pressure is equal tothe fracture pressure so ~Pc = ~P~}. "[’his situation mimics that of the matrix core sample in alaboratory experiment by completely surrounding the matrix material with a uniform pressure

13

field. (See Figure 2.) Then, by combining the appropriate rows and columns, we can telescopethe 3 × 3 system down to a 2 × 2 system of the form

~,-~(t~)/v {1) = av~ + a2a a22 ) (59)

So, except for an overall factor of vO) = V(I}/V (which is generally very close to unity in thecases of interest), equations (53) and (59} are of the same form. We are therefore led to identifications

A.(~) - a~l + 2a~a + ff:~3, (60)

and

-- - a~2 + a23, (61)

[(L}a¢]) O(|)o(L}( l/h’] L})

it-(~----~ + - = B(~llt.(~ ) - a~2.

The stress-strain re]ations may now be expressed in the form

-/f((2) -c~/h" - a12 -c(l)c¢(1)//t "(t) -

where

-v(l)a(~)//((~) - )

ass _,Sp~)

and frown (32)

a33 "-- v(2)/h "] + v(1)/K(1) - ( 1 - 2~)/h" -F 2ar.~.

a~ =

Combining (60) and (61) then shows

1/h’~~) = art + a12 + 2al3 + (/23 "]" ~’33,

while combining (61} and (62.} sltows that

or equivalently that

6(~ )( ~ .] -- 1"o ) = (Z12 "4- a22 "i-" a23,

where, by analogy to ( "

--0(I)/I((I) = a12 + ~’22 + a2s.

~(~}~22 -- (122 -- ~

(62)

4,63)

(64)

(65)

(66)

(67)

(68)

(69)

1.1

6. FRACTURES IN A R, IG[DLY CONSTRAINED MATRIX

In the last section, setting p(~) = pc created a situation where the matrix was completely

by an effective confining pressure pc = pl2). This trick essentially isolated thesurroundedmatrix materi~l from the fracture behavior and permitted a simple identification of certaincombinations of the double-porosity parameters in terms of the m~trix constants.

Now we would fike to isol~te the the fractures just as we isolated the matrix in the last

section. Unfortunately, the same trick does not apply. Setting p~l) = Pc does not genera~yisolate the fracture (although there might be p~rticular geometries where it does), becauseneither the confining pressure nor the matrix fluid pressure arc applied directly to the surfaceof the fractures. So instead we must consider some alternative thought experiment to achievethe desired decoupfing, ff we consider an experiment on a double porosity medium so thatthe m~trix material remains rigid (i.e., ~e(~) = 0) while the fractures deform, this condition

specifies a p~rticular choice of the m~trix fluid pressure p~) = p* (but norm~Hy p" ~ pc) we will show that the desired separation is then accomplished.

Consider equation (57). To determine whether ~ particular set of stresses results in them~trix remaining rigid or not. we need an equation isolating the nmtrix strain 6e(1} from theoverall strain be. Since ~e = v(1)~e{~} + v{~}~e{~), it follows directly from {57} that we have zeromatrix strain when

-o(1)~e(1) = {,11 + g.31)~Pe + ("13 + ~33)~p~2) +(a,2 + a32)~p~1) ~0. (T0)

This final equatity (setting the right hand side equal to zero) provides the needed rigidity

condition. Solving the resulting equation for 8p~t) in terms of 8pc and ~p~2} provides the relation

required to determine the value of matrix fluid pressure p~l) = p. needed to guarantee overallmatrix rigidity (see Fig. 3). Then, we obtain the desired equation for t’{216e(2) as a function of6p~ and

= (-opc) + , el ..a~2 + a3~ ax2 + a32

The definitions of the constants for the fracture system in a rigid matrix (Bert) = 0) given by

( = ~(’)/B(’))( ))" [72)

It is important to recognize that, whereas B(~) and K(~) have their well defined (standard)interpretations for the fracture phase as experimental observables, ~(~) is now a parameter thatstrictly speaking does not have the usual Biot-Wil~s [1957] micromechanical interpretationin terms of frame and gr~in modulus since there is no grain modulus associated with thefracture phase. We can nevertheless define ~n effective grain modulus according toff(~)/(1 - ~(:)), but care should be taken not to overinterpret this parameteL With definitions, we are finally led to the identifications:

t ’(’~) a~a3~ - a~aar~(73)

15

and

v(2)aTM ~.33a12 - a13a23-- (74)1((2) el2 + a23

Note that. if a23 vanishes, the right hand sides of these equations reduce to -au~ and ~’3~,respectively.

The analogous calculation of the fracture fluid increment 6~(2) shows that

_gq-(2) = a~3at2 - a23al[(_~pc) + a33(a12 + a23) - a23(a13 "~- ~’33)(_t~p~2)). a12 "~ a23 a12 -~- a’~3

Since reciprocity shows that the oil-diagonal terms of (72) are equ,’fi, we obtain a condition the coefficients by equating the numerators of these expressions, showing that

a13a12 -- alla23 = (z13a23 - ~33t,/12.

Substituting from (63) and solving for a~2. we obtain the result

(76)

which should be compared to (65).

a(~) h’!~)a~- h’(~) K., (77)

7. DISCUSSION

7.1 Comparison u:ith single po~vsity theory

The preceding analysis of the constitutive equations for a fractured or jointed, porousmedium is the logical extension of Blot’s linear theory for a double porosity material. Thepurpose of this section is to provide additional interpretation of the basic coefficient matrix in(23). Blot’s original formulation for an isotropic, poroelastic material requires three indepen-dent moduli for an isotropic state of stress (1.1). Blot’s constant, a/K, is analogous to thethermal expansion coefficient in thermoelasticity, and hence will be referred to as the poroelas-tic expansion coefficient. This coeIficient gives the amount of bulk expansion of the stress-freematerial for a uniform pore pressure increase. The other constant, S = a/BK. is the so-calledthree-dimensional storage compressiblity as used in hydrogeology [Van der Kemp and Gale,1982; Kiimpel. 1991; Wang, 1993]. It gives the volume of fluid that must flow into a controlvolume due to an increase in pore pressure at constant confining pressure.

The double porosity theory (23} with six independent coefficients, aij, is a straightforwardgeneralization of Blot’s original equations. The six coefficients occur in three categories thatcorrespond to the three original Blot coefficients. The coefficient a~ = l/A" is an effectivecornpressibility of the combined fracture-matrix system. The coefficients, a~ = -5(~)/K andat3 = -~")/K. are generalized poroelastic expansion coefficients. The overbar notation isused to emphasize that the bulk dilatation due to a pore pressure increase in phase 1 while theconfining pressure, and p(2)] are kept constant is not the same a.s the poroelastic exI)ansion ofthe matrix itself, in general, although the analysis shows that it is a good approximation for afractured, porous medium.

16

The terms a22, a23 , a32 = a23, and a33 are generalized storage coefficients, i.e., a~j is thevolume of fluid that flows into a control volume (normalized by the control volume) of phasei - I due to a unit iucrease in fluid pressure in phase j - I. They can be thought of as forminga tensor storage coefficient, linking the vector composed of the increments of fluid content

and ~’(~) to the two pore pressures p~) and p~2). In Biot’s single porosity theory, the storagecompressibility S = o~/BK. The diagonal components a~ and s33 for the double porositytheory are ~(0/B[s(~)]I( for ~ = 1: 2, where again the notation emphasizes the similaxity to single porosity case, but also emphasizes th~.t they are not necessarily the values of ph~e i asa single porosity material. ’]’he double porosity material is a composite, and hence has effectivemoduli that depend on the interactions between the two phases.

7. 2 Stress formulation

The six coefficients that describe completely the double porosity material can be obtainedfrom the three values for the matrix and the fracture compressibility with just one additionalassumption about the coupling between the matrix and fracture phases at constant stress. Inour stress-based formulation, the coefficient, a~a is a cross-storage coefficient for conditions ofconstant confining pressure. The assumption a~a = 0 is equivalent to K~~} = v(1)K~, whichobtains for a matrix of a single constituent and v(~) -’, 1. The assumption a~a = 0 immediately

leads to equality between the various Skcmpton-like coefficients: B(~) ~(t)= ~’E~ = B[u(~)] -a~2/a2~ and B(~) B(~} = B[u(2)] = -a~a/a33. This assumption also leads from (43) to the: EBresult that the overall compressibility a~ is the volume average of the matrix compressibility~/K(D and the fracture compressibility o(~)/K(~) = 1/(k,~s}, where k,, is the fracture stiffness(GPa/m} and s (m) is the fracture spacing (cf. Elsworth and Bai [1992]).

Numerical values for the matrix and fracture properties are given for Berea sandstone and~Vesterly granite in Table 1. Expressions and numerical values for all the aljs are given inTable 2. Note that if v(1) "-’ 1, then the results for a22 and a33 are the same as the normalthree-dimensional storage coefficient for the ~natrix and fracture phases separately. Also, thelong-time three-dimensional storage coefficient is the sum of the storage coefficients of theindividual phases. These results are intuitively reasonable, and provide means of predicting thedeformation and fluid storage behavior of the fractured, porous medium from knowledge of theindividual phases. In particular, note that all the coefficients in Table 2 were computed from thevalues of K(1), K~~), KI, ¢5(~’}, v(~), and K(~} quoted in Table 1. Thus. these six measurements(together with Poisson’s ratio) are sufficient to determine completely the behavior of the double-porosity model.

In contrast to the preceding example, Table 3 presents data for Chelmsford granite and~Veber sandstone taken from l~boratory measurements by Coyner [198,i]. The data availablein these experiments differs somewhat from the preceding case, since Coyner’s experimentsincluded a series of tests on several types of laboratory scale rock samples at different confiningpressures. The values quoted for K and K~ are those for a moderate confining pressure of 10MPa (values at lower confining pressures were also measaured but we avoid using these valuesbecause the rocks generally exhibit nonlinear behavior in thai region of the parameter space).while the values quoted for h"(~) and K~~) are at 25 MPa. Thus, based on the idea that thepressure behavior is associated with two kinds of porosity in the laboratory samples -- a crackporosity, which is being closed between 10 and 25 MPa, and a residual matrix porosity above

17

25 MPa, we assume the a.vailable data ,~re K, h’.~, K(1), K~~), KI, O{1). and v(2). We find thatthese data are sufficient to compute all the coefficients, and therefore no assumption need bemade about the value of a~a. In "_[’able 4, we find for both types of rock that this coefficient ispositive and small -- about an order of magnitude smaller than the other matrix elements. Theonly other unusual feature of the results computed using this laboratory data is the occurrenceof values larger than unity for B[u(0] in Chelmsford granite and for B[,~(~)] and ~(~) in WeberOEBsandstone. Note also that o~{~) for both rocks is very close to unity. In this example, sevenmeasurements ( together with Poisson’s ratio) are sufficient to determine completely the behaviorof the double-porosity model. The main difference between this example and the preceding oneis that having a direct measurement of A" eliminates the necessity of assu~ning a~3 = 0.

7.3 Strain formulation

Flsworth and B~ [1992], Khaled et al. [1984], and Wilson and Aifantis [1982] formulatedthe constitutive equations for ~(O) and (~) inter ms of bulk str ain. 6e [cf . (49 ) and(50)respectively] in pl~ce of pc. We wi~ show that the assumption a~a = 0 leads to f~r morereasonable results than their anMogous assumption A~:~ ~ a~a-a~a~a/a~ = O, where the uppercase coe~cients Aq signify corresponding coefficients for a strain-based formulation. Solvingfor a~a in the case of Bere~ s~ndstone le~ds to ~ value of 0.085 GPa-~. which is comparable inm~gnitude with the poroelastic expansion coe~cient a~. Furthermore, using this wlue forleads to ~ value of n(~) = 0.11 for the short-term Skempton’s coefficient in the matrix phase.The problem stems from the f~ct that A~ is not likely to be neg~gible. The coe~cient m~, bedefined by

~((:~ ~’=~v~~= ~ (78)¯ -1~3 bp~Z~=0

The fractures will expand within the constraint of zero total strain because ~n increase in thefluid pressure in the fractures will cause compression of the matrix. Therefore, fluid must bewithdrawn from the matrix in order ~o maintain 6p~t} = 0. Thus, the constant A.z3 is expectedto be negative, while in fact A2a = -ai2aia/a~ < 0 follows front the assumption that a23 = 0.

The diagonal storage coefficient in the strain formulation A33 = a:~3 - a~3/a~. As in thediscussion of .4~3, this coefficient is for the case of constant total strain. ~nd the fractures are

to exp~nd in response to an increase in p~Z)’- because the matrix contracts. Therefore. thefracture storage is not negfigible, as it would be for ~ rigid matrix. Elsworth ~nd Bai [1992]calculated an unreasonably smaU value of n{~) of 2.3x l0 -4. because they used the rigid matrixvalue for A.~a.

The strain formulation is as valid a formulation as the stress formulation we have used.However. the ad hoc ~ssumption that the cross storage coefficient .4~ = 0 is not justified:significant coupling occurs between ~he fracture and matrix for conditions of constant totalstrain. The assumption that .-t.z3 = 0 or .4~ ~ 0 leads to significant underestimation of theearly pressure buildup in the m~trix at short times. On the other hand. the assumption that thecross-storage coefficient a~3 = 0 is justified on the grounds that the overaU solid grain modulusis ~kely to be close to that of the matrix grains (also compare Table 4).

18

7.4 Teraporal and spatial scales

The local mechanical and fluid pressure response of a double porosity material is time andscale dependent, because fluid is exchanged between the two phases and because the two fluidpressures can be m~intained independently {at least in a gedanken experiment). As demon-strated in our preceding detailed analysis and that of Wilson and Aifantis [1983], considerationof short, intermediate, and long time scales leads to theoretical relationships between measure-ments and phenomenological coefficients. By making separate analyses at the mesoscopic scaleof a typical core sample and of a typical fracture, we showed that all six constants are deter-mined from the matrix poroelastic coefficients and the fracture compressibility ba~ed on thesingle assumption that a23 --" 0. Then all the constants defined for different time scales can bedetermined. At the long-time scale, the double-porosity medium behaves as an equivalent singleporosity medium with a single macroscopic compressibility: Skempton’s coefficient, and stor-age compressibility. In general, a23 does not vanish identically, but is likely to be significantlysmaller in magnitude than the other matrix elements.

8. CONCLUSIONS

’[’he six coefficients in the double porosity theory can be broken down into three categoriescorresponding to the three coefficients in the single porosity theory. The effective mediumhulk modulus replaces the ordinary bulk modulus of a single porosity medium. Twoporoelastic expansion coefficients, one for the pore pressure in each phase, replace thesingle poroelastic expansion coefficient in a single porosity medium. A symmetric two-by-two poroelastic storage tensor consisting of three coefficients, two diagonal and oneoff-diagonal, replaces the single storage coefficient in a single porosity medium.

The magnitude of the off-diagonal coefficient a2a in a stress-based formulation can be as-sumed to be zero (or at least quite small) for a fractured, porous medium. The assumptionthat the strain-based cross-coefficient A23 - 0 is not justitied.

Consideration of very short, intermediate, and long time scales yields definitions of anumber of poroelastic moduli, many of which are physically realizable in the laboratoryor field, and the interrelationships between these poroelastic moduli.

Finally, an important direction for future work is to deal with those situations where itmight be either difficult or impossible to make the required measurements of the parameters(but predictive capability is vital}. Then, we will want to introduce a new microscopic pointof view in order to relate the phenomenological coefficients to quantities such as grain moduhlsand porosity at the microscale. The standard model of the microscale (and the one used in theexamples in the present paper) is that used by Gassmann [1951], which carries the restrictiveassumption that the solid frame is composed of only one type of elastic constituent. A moregeneral point of view has been introduced recently by Berryman and Miltou [1991. 1992], Norris[1992], aud Berryman [1992], who show how to relate macroscopic coefficients in poroelasticityto quantities at the microscale when two types of solid constituents are present. Making useof this approach will permit us to make the microscopic identification of coefficients in morecomplex and therefore more realistic geologic media.

19

ACKNOWLEDGMENTS

JGB thanks N. G. W. Cook and S. Ita for helpful conversations regarding reciprocity rela-tions. The work of JGB was performed under the auspices of the U. S. Department of Energyby the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-.18 and sup-ported specifically by the Geosciences Research Program of the DOE Office of Energy Researchwithin the Office of Basic Energy Sciences. Division of Engineering and Geosciences. The workof HFW was also supported by OBES under grant no. DE-FG02-91ER14194.

Bai. M., D. Elsworth, and .].-C. Roegiers, Modeling of naturally fractured reservoirs usingdeformation dependent flow mechanism, Int. J. Rock Mech. Min. Sci, ~ Geomech. Abstr.30. 1185-1191, 1993.

Barenblatt, G. I.. and Yu. P. Zl,eltov, Fundamental equations of filtration of homogeneousliquids in fissured rocks. Soy. Phys. Doklady 5. 522.-525 [English translation of: Doklady.-lkademii .Vauk 5’SSR 132. 545-548], 1960.

Berryman, J. G.. Effective stress tbr transport properties of inhomogeneous porous rock, J.Geophys. Res. 97, 17.109-17,12,1. 1992.

Berryman. J. G.. and G. W. Milton. Exact results for generalized Gassmann’s equations incomposite porous media with two constituents, Geophysics 56. 1950-1960, 1991.

Berryman, J. G., and G. W. Milton, Exact results in linear thermomechanics of fluid-saturatedporous media, Appl. Phys. Left. 61, 2030-2032, 1992.

Beskos. D. E., and E. C. Aifantis, On the theory of consolidation with double porosity, -- II,Int. J. Engng. Sci. 24, 1697.-1716. 1986.

Blot, .kI. A., General theory of three di~nensional consolidation, J..4ppl. Phys. 12. 155-164,1941.

Blot, M. A., anti D. G. Willis, The elastic coefficients of the theory of consolidation, J. App.Mech. 24, 594-601, 1957.

Brown, R. J. S., and J. Korringa, On the dependence of the elastic properties of a porous rockon the compressibility of a pore fluid, Geophysics 40. 608-616. 1975.

C.arroll. M. M.. Mechanical response of fluid-saturated porous materials, in Theoretical andApplied Mechanics. F. P..l. Rimrott and B. Tabarrok (eds.). Proceedings of the 15th In-ternational Congress of Theoretical and AI)plied Mechanics, Toronto. August 17-23, 1980.North-lfolland. :kmsterdam, 1980. pp. 251-262.

C.leary, M. P.. Fundamental sohltions for a ttuid-saturated porous solid, l,~t. J. 5"olids Structures13, 785-806, 1977.

C.oyner, K. B.. Effects of Stress. Pore Pressure, and Pore Fluids on Bvlk Strain, Velocity, a~dPermeability of Rocks. Ph. D. Thesis. Massachusetts Institute of Technology, 198.I.

2O

Detournay, E., and A. Ii.-D. Cheng, Fundamentals of poroelasticity, in Comprehensive RockEngineering, edited by J. A. Hudson, Vol. 2, Chapter 5, Pergamen Press, Oxford, 1993.

Elsworth, D., and M. Bai, Continuum representation of coupled flow-deformation response ofdual porosity media, in Mechanics of Jointed and Faulted Rock, Roesmanith (ed.}, Balkema,Rotterdam, 1990, pp. 681-688.

Elsworth, D., and M. Bai, Flow-deformatioa response of dual-porosity media, ASCE J. Geotech.Engng., 118, 107--124, 1992.

Gassmann, F., ~lber die elastizit/i.t porSser medien, Veirteljahrsschrift der NaturforschendenGesellschafl in Ziirich 96, 1-23, 1951.

Geertsma, J., The effect of fluid pressure decline on volumetric changes of porous rocks. Tans.:tIME 210. 331-340, 1957.

Hart. D. J., and It. F. Wang, Laboratory measurements of a complete set of poroelastic rnodullfor Berea sandstone and Indiana limestone, J. Geophys. Res., submitted. 1995.

Khaled. M. Y., D. E. Beskos, and E. C. Aifantis, On the theory of consolidation with doubleporosity -- III A finite element formulation, Int. J. Num. Anal. Methods Geomech. 8,101-123, 1984.

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

Love, A. E. H.. A Treatise on the Mathematical Theory of Elasticity. Dover, New York, 1927,pp. 173-174.

XlcTigue. D. F.. Thermoelastic response of fluid-saturated porous rock. J. Geophys. Res. ~1.9533-9542. 1986.

Norris, A. N.. On the correspondence between poroelasticity and thermoelasticity, J. Appl.Phys. 71. 1138-1141, 1992.

Rice. J. I~., On the stability of dilatant hardening for saturated rock masses. J. Geophys. Res.80. 1531-1536. 1975.

Rice, J. R.. and M. P. Cleary, Some basic stress diffusion solutions for fluid-saturated elasticporous media with compressible constituents, Rev. Geophys. Space Phys. 14.227-241. 1976.

Rosen, B. W.. and Z. IIashin, Effective thermal expansion coefficients and specific heats ofcomposite materials, Int. J. Engng. Sci. 8, 157-173, 1970.

Skcmpton. A. W., The pore-pressure coefficients A and B, Geotechnique 4, 143-b17, 1954.

Touloukian. Y. S., W. R. Judd, R. F. Roy: Physical Properties of Rocks and Minerals, McGraw-llill. New York. 1989, Chapter 11(2}.

Vart der Kamp. G.. and J. E. Gale. Theory of earth tide and barometric effects in porousfortnations with compressible grains. Water Resources Res. 19, 538-544. 1983.

21

Wang, It. F., Quasi-static poroeIastic parameters in rock and their geophysical applications.PAGEOPH 141,269-286, 1993.

Warren, J. E., and P. J. R,oot, The behavior of nal;urally fractured reservoirs, Soc. Pet. Eng. J.3, 2,15-255, 1963.

~Vilson, It. K., and E. C. Aifantis. On the theory of consolidation with double porosity, Int. J.Engng. Sci. 20, 1009-1035, 198,.

.2;.)

List of Symbolsoverall fluid permeabilitymatrix and fracture permeabilitiesconfining pressurePc - Pl. the differential pressurefluid pressure

matrix and fracture fluid pressures

drained and undrained fluid pressuresvolume fractions occupied by matrix and fractureswith V(1) "J- i)(2) 1Skempton’s coefficientshear modulusbulk modulus of drained porous frame (jacketed)fluid bulk modulusmaterial (or grain) bulk modulusCK/o~. an effective pore bulk modulus (jacketed)an effective solid bulk modulus (unjacketed)bulk modulus of undrained (confined) porous framean effective pore bulk modulus (unjacketed)a/BK, the storage compressibilitytotal volumetotal matrix and fracture volumes(1 - o)V, the solid volumeCV, the pore volumeBlot-Willis parametervolume thermal expansion coefficientpressure difference coefficientincrement of total fluid contentincrements of matrix and fracture fluid contenttemperaturefluid viscosityPoisson’s ratiototal porosity = v(1)¢(t) + v(2

matrix porosity (fracture porosity is unity)

’FABLE I. M,~terial Pro:)erties

BereaParameter SandstoneK(1) (GPa,) 8.0~

u(’) 0.204A’~~) (GPa} 36.0"a(~) 0.78KI (GPa) 3,3~¢)(1) 0.064b

B(1) 0,8475’(1) (GPa-~) 0,115c(2) 0.0064c

K(’~)(GPa) 0.00775c

WesterlyGranite25.0"0.25"

45.4~

0.453.3~

0.00106b

0.98’!0.01830.0106:0.0876c

~From Rice and Cleary [t976]bFrom Touloukian et al. [1989]CFrom Elsworth and Bai [1992]

TABLe. 2. Double porosity parameters coznputed from material properties in Table i.

Parameter

K (GPa)

I(~ (GPa)

a~t (GPa-~)

ar~ (GPa-’)ala (Gea-1)a.~ (GPa-~)a23 (GPa-t)

a33 (GPa-~)

B

B(~1, B[u(’)],B(’~-), B[u(~)], n(~)K~ (GPa)K[u(1)] (GPa)K[u(~)] (GPa)8 (GPa-~)S(~) (GPa-’)

Formula[v(’)/K0) + ~(~)/K(~)]-~

1 - K/K,

-~(Da(D/K(~)

_~(~)/K(~)

~O)aO)/B(~)K(~)

0.0v(~)Cl/Kl + L/K(:))

1.0-(al~ + a~:])/(a~ -a12/a22

-a13/a33

Jail -- (a12 + ala)2/(a22 a33)]-1

[~. - .h/a~]-’a/BKa(~)/B(~)K(~)

BereaSandstone

1.0520

36.20.9710.951

-0.0969-0.8260.11,140.00.8281.00.9790.$47

0.99821.51.1517.920.943

129.5

WesterlyGranite

6.25

45.90.86,10.1601

-0.0178-0.1210.018170.00.12-t1.00.977

0.984

0.97439.67.01

23.40.1415

11.67

TABLF; 3..’vla.t(:rial Pro)erties

Chehnsford WeberParameter Granite SandstoneK (GPa)K, (GPa)

I(I~) (GPa)

h’.~~1 (GPa.)c~(1)

It’] (GPa)o(~)B(~)S(x)(GPa-~) ,v(~)

8.0a 4.0a

5’1.5~ 37.0~

0.85~ 0.89a

17.0~ 10.0a

0.25 O. 15

, 55.5a 38.0~

0.69" 0.7.V3.3 3,30.0011 0.095"0.992 0.355(}.0409 0.2080.011~ 0.0095

"I:’rom Coyner [198.1]

TABLE 4. Double porosity parameters computed from material properties in Table 3.

Parametera~ (GPa-~)

a~ (GPa-~)

a~3 (GPa-~)

a~ (GPa-|)

a~ (GPa-~)

a33 (GPa-1)

~.’~3 (Gea-1)

BB(1)

EBB(~)

B(2}EB

K~ (GPa)K(2) (GPa)S (GPa-1)S(2) {GPa-~)

Formu|aChelmsfordGranite

0.125

-0.0413-0.06490.0,1050.001190.06640.06300.9970.9730.9921.022

0.9930.9500.978

0.96146.30.1790.10925.87

WeberSandstone

0.250

-0.076-0.1,170.2060.002700.1450.1420.9940.6240.3550.368

0.3550.9801.011

1.0048.990.06660.357

15.24

Figure 1,: The elements of a double porosity model are: porous rock matrix intersected byfractures. Three types of m~croscopic pressure arc pertinent in such a. modeh external confining

pressure pc. ]nterna~l pressure of the matrix pore fluid p~}, and internal pressure of the fracture

pore fluid p~’}.

oo ,o

o

Figure 2: Setting p~t.)1~ = Pc establishes a constant pressure around the matrix material which isthen equivalent to the situation experienced by a core sample of matrix material in a laboratorytest.

p~l)

¯ Pc

Pc

Figure 3: For any given value of confining pressure Pc and fracture fluid pressure p~2), there is

a special choice of the matrix fluid pressure p~} = p" that guarantees no macroscopic strain{$e(~) = 0} ia-.the matrix. This situation allows the behavior of the fracture to be isolated fromthat of the matrix essentially as if the matrix material were macroscopica]ly rigid.