Double porosity modeling in elastic wave propagation for reservoir characterization

Transcription

1 Stanord Exploration Project, Report 98, September 2, 1998, pages Double porosity modeling in elastic wave propagation or reservoir characterization James G. Berryman 1 and Herbert F. Wang 2 keywords: poroelasticity, wave propagation, double porosity ABSTRACT To account or large-volume low-permeability storage porosity and low-volume high-permeability racture/crack porosity in oil and gas reservoirs, phenomenological equations or elastic wave propagation in a double porosity medium have been ormulated and the coeicients in these linear equations identiied. The generalization rom single porosity to double porosity modeling increases the number o independent inertial coeicients rom three to six, the number o independent drag coeicients rom three to six, and the number o independent stress-strain coeicients rom three to six or an isotropic applied stress and assumed isotropy o the medium. The analysis leading to physical interpretations o the inertial and drag coeicients is relatively straightorward, whereas that or the stress-strain coeicients is more tedious. In a quasistatic analysis o stress-strain, the physical interpretations are based upon considerations o extremes in both spatial and temporal scales. The limit o very short times is the one most relevant or wave propagation, and in this case both matrix porosity and ractures are expected to behave in an undrained ashion, although our analysis makes no assumptions in this regard. For the very long times more relevant or reservoir drawdown, the double porosity medium behaves as an equivalent single porosity medium. At the macroscopic spatial level, the pertinent parameters (such as the total compressibility) may be determined by appropriate ield tests. At the mesoscopic scale pertinent parameters o the rock matrix can be determined directly through laboratory measurements on core, and the compressiblity can be measured or a single racture. We show explicitly how to generalize the quasistatic results to incorporate wave propagation eects and how eects that are usually attributed to squirt low under partially saturated conditions can be explained alternatively in terms o the double-porosity model. The result is thereore a theory that generalizes, but is completely consistent with, Biot s theory o poroelasticity and is valid or analysis o elastic wave data rom highly ractured reservoirs. 1 berryman@sep.stanord.edu 2 Department o Geology and Geophysics, University o Wisconsin, Madison, WI wang@geology.wisc.edu 223

2 224 Berryman & Wang SEP 98 INTRODUCTION It is well-known in the phenomenology o earth materials that rocks are generally heterogeneous, porous, and oten ractured or cracked. In situ, rock pores and cracks/ractures can contain oil, gas, or water. These luids are all o great practical interest to us. Distinguishing these luids by their seismic signatures is a key issue in seismic exploration and reservoir monitoring. Understanding their low characteristics is typically the responsibility o the reservoir engineer. Traditional approaches to seismic exploration have oten made use o Biot s theory o poroelasticity [Biot, 1941; 1956; 1962; Gassmann, 1951]. This theory has always been limited by an explicit assumption that the porosity itsel is homogeneous. Although this assumption is known to be adequate or acoustic studies o many rock core samples in a laboratory setting, it is probably not a very good assumption or applications to realistic heterogeneous reservoirs. One approach to dealing with the heterogeneity is to construct a model that is locally homogeneous, i.e., a sort o inite element approach in which each block o the model satisies Biot-Gassmann equations. This approach may be adequate in some applications, and is certainly amenable to study with large computers. However, such models avoid the question o how we are to deal with heterogeneity on the local scale, i.e., much smaller than the size o blocks typically used in the codes. Although it is clear that porosity in the earth can and does come in virtually all shapes and sizes, it is also clear that two types o porosity are most important: (1) Matrix porosity that occupies a inite and substantial raction o the volume o the reservoir. This porosity is oten called the storage porosity, because this is the volume that stores the luids o interest to us. (2) Fracture or crack porosity that may occupy very little volume, but nevertheless has two very important eects on the reservoir properties. The irst eect is that ractures/cracks drastically weaken the rock elastically, and at very low eective stress levels introduce nonlinear behavior since very small changes in stress can lead to large changes in the racture/crack apertures (and at the same time change the racture strength or uture changes). The second eect is that the ractures/cracks oten introduce a high permeability pathway or the luid to escape rom the reservoir. This eect is obviously key to reservoir analysis and the economics o luid withdrawal. It is thereore not surprising that there have been many attempts to incorporate ractures into rock models, and especially models that try to account or partial saturation eects and the possibility that luid moves (or squirts) during the passage o seismic waves [Budiansky and O Connell, 1975; O Connell and Budiansky, 1977; Mavko and Nur, 1979; Mavko and Jizba, 1991; Dvorkin and Nur, 1993]. Previous attempts to incorporate have generally been rather ad hoc in their approach to the introduction o the ractures into Biot s theory, i Biot s theory is used at all. The present authors have recently started an eort to make a rigorous extension o Biot s poroelasticity to include ractures/cracks by making a generalization to double-porosity/dual-permeability modeling [Berryman and Wang, 1995]. The previ-

3 SEP 98 Elastic waves or double porosity 225 ously published work concentrated on the luid low aspects o this problem in order to deal with the interactions between luid withdrawal and the elastic response (closure) o ractures during reservoir drawdown. It is the purpose o the present work to point out that a similar analysis applies to the wave propagation problem. We expect it will be possible to incorporate all o the important physical eects in a very natural way into this double-porosity extension o poroelasticity or seismic wave propagation. The price we pay or this rigor is that we must solve coupled equations o motion locally. Within traditional poroelasticity, there are two types o equations that are coupled. These are the equations or the elastic behavior o the solid rock and the equations or elastic and luid low behavior o the pore luid. In the double-porosity extension o poroelasticity, we will have not two types o equations but three. The equations or the elastic behavior o the solid rock will be unchanged except or a new coupling term, while there will be two types o pore-luid equations (even i there is only one luid present) depending on the environment o the luid. Pore luid in the matrix (storage) porosity will have one set o equations with coupling to racture luid and solid; while luid in the ractures/cracks will have another set o equations with coupling to pore luid and solid. Although solution o these equations is no doubt more diicult than or simple acoustics/elasticity, it is probably not signiicantly more diicult than traditional single-porosity poroelasticity. We are not going to solve these equations in the present paper. We will instead derive them and then show that the various coeicients in these equations can be readily identiied with measurable quantities. EQUATIONS OF MOTION The seismic equations o motion or a double-porosity medium have been derived recently by Tuncay and Corapcioglu [1996] using a volume averaging approach. (These authors also provide a thorough review o the prior literature on this topic.) We will present instead a quick derivation based on ideas similar to those o Biot s original papers [Biot, 1956; 1962], wherein a Lagrangian ormulation is presented and the phenomenological equations derived. Physically what we need is quite simple just equations embodying the concepts o orce = mass acceleration and dissipation due to viscous loss mechanisms. The orces are determined by taking a derivative o an energy storage unctional. The appropriate energies are discussed at length later in this paper, so or our purposes in this section it will suice to assume that the constitutive laws relating stress and strain are known, and so the pertinent orces are the divergence o the solid stress ield τ ij,j and the gradients o the two luid pressures p (1),i and p (2),i or the matrix and racture luids, respectively. (In this notation, i, j index the three Cartesian coordinates x 1, x 2, x 3 and a comma preceding a subscript indicates a derivative with respect to the speciied coordinate direction.) Then, the only work we need to do to establish the equations o motion or dynamical double-porosity systems concerns the inertial terms arising rom the kinetic energy o the system, and drag terms arising

4 226 Berryman & Wang SEP 98 rom dissipation due to relative motion o the constituents. Generalizing Biot s approach [Biot, 1956] to the ormulation o the kinetic energy terms, we ind that, or a system with two luids, the kinetic energy T is determined by 2T = ρ 11 u u + ρ 22 U (1) U (1) + ρ 33 U (2) U (2) +2ρ 12 u U (1) + 2ρ 13 u U (2) + 2ρ 23 U (1) U (2), (1) where u is the displacement o the solid, U (k) is the displacement o the kth luid which occupies volume raction φ (k), and the various coeicients ρ 11, ρ 12, etc., are inertial mass coeicients that take into account the act that the relative low o luid through the pores is not uniorm, and that oscillations o solid mass in the presence o luid leads to induced mass eects. Clariying the precise meaning o these displacements is beyond our current scope, but recent publications help with these interpretations [Pride and Berryman, 1998]. Dissipation plays a crucial role in the motion o the luids and so cannot be neglected in this context. The appropriate dissipation unctional will take the orm 2D = b 12 ( u U (1) ) ( u U (1) ) + b 13 ( u U (2) ) ( u U (2) ) +b 23 ( U (1) U (2) ) ( U (1) U (2) ). (2) This ormula assumes that all dissipation is caused by motion o the luids either relative to the solid, or relative to each other. (Other potential sources o attenuation, especially or partially saturated porous media [Miksis, 1988], should also be considered, but will not be discussed here.) We expect the coeicient b 23 will generally be small and probably be negligible, whenever the double-porosity model is appropriate or the system under study. Lagrange s equations then show easily that and that ( T t ( ) T + D = τ ij,j, or i = 1, 2, 3, (3) t u i u i U (k) i ) + D U (k) i = p (k) i, or i = 1, 2, 3; k = 1, 2. (4) These equations now account properly or inertia and elastic energy, strain, and stress, as well as or the speciied types o dissipation mechanisms, and are in complete agreement with those developed by Tuncay and Corapcioglu [1996] using a dierent approach. In (4), the parts o the equation not involving the kinetic energy can be shown to be equivalent to a two-luid Darcy s law in this context, so b 12 and b 13 are related to Darcy s constants or two single phase low and b 23 is the small coupling coeicient. Explicit relations between the b s and the appropriate permeabilities (see Eqs. (53) and (54) o Berryman and Wang [1995]) are not diicult to establish (see

5 SEP 98 Elastic waves or double porosity 227 the next section). The harder part o the analysis concerns the constitutive equations required or the right hand side o (3). Ater the section on inertia and drag, the remainder o the paper will necessarily be devoted to addressing some o these issues concerning stress-strain relations. In summary, equations (3) and (4) can be combined into ρ 11 ρ 12 ρ 13 ρ 12 ρ 22 ρ 23 ρ 13 ρ 23 ρ 33 ü i Ü (1) i Ü (2) i + b 12 + b 13 b 12 b 13 b 12 b 12 + b 23 b 23 b 13 b 23 b 13 + b 23 u i U (1) i U (2) = i τ ij,j p (1),i p (2),i showing the coupling between the solid and both types o luid components., (5) In the next section we show how to identiy the inertial and drag coeicients with physically measureable quantities. INERTIAL AND DRAG COEFFICIENTS Inertial coeicients It is easy to understand that the inertial coeicients appearing in the kinetic energy T must depend on the densities o solid and luid constituents ρ s and ρ, and also on the volume ractions φ (1) and φ (2) o the matrix and racture porosities. The total porosity is given by φ = φ (1) + φ (2) and the volume raction occupied by the solid material is thereore 1 φ. For a single porosity material, there are only three inertial coeicients and the kinetic energy can be written as ( ) ( ) ρ11 ρ 2T = ( u U ) 12, (6) ρ 12 ρ 22 u U where U is the velocity o the only luid present. Then, it is easy to see that, i u = U, the total inertia ρ ρ 12 + ρ 22 must equal the total inertia present in the system (1 φ)ρ s + φρ. Furthermore, Biot [1956] has shown that ρ 11 + ρ 12 = (1 φ)ρ s, and that ρ 22 + ρ 12 = φρ. These three equations are not linearly independent and thereore do not determine the three coeicients. So we make the additional assumption that ρ 22 = αφρ, where α was termed the structure actor by Biot [1956] but has more recently been termed the tortuosity [Brown, 1980; Johnson et al., 1982]. Berryman [1980] has shown that α = 1 + r ( ) 1 φ 1, (7)

6 228 Berryman & Wang SEP 98 ollows rom interpreting the coeicient ρ 11 as resulting rom the solid density plus the induced mass due to the oscillation o the solid in the surrounding luid. Then, ρ 11 = (1 φ)(ρ s + rρ ), where r is a actor dependent on microgeometry that is expected to lie in the range 0 r 1, with r = 1 or spherical grains. For example, 2 i φ = 0.2 and r = 0.5, equation (7) implies α = 3.0, which is a typical value or tortuosity o sandstones. For double porosity, the kinetic energy may be written as ρ 11 ρ 12 ρ 13 2T = ( u U (1) U (2) ) ρ 12 ρ 22 ρ 23 ρ 13 ρ 23 ρ 33 u U (1). (8) U (2) We now consider some limiting cases. First, suppose that all the solid and luid material moves in unison. Then, we again have the result ρ 11 + ρ 22 + ρ ρ ρ ρ 23 must equal the total inertia o the system (1 φ)ρ s + φρ. Next, i we suppose that the two luids can be made to move in unison, but independently o the solid, then we can take U = U (1) = U (2), and telescope the expression or the kinetic energy to ( 2T = ( u U ) ρ 11 (ρ 12 + ρ 13 ) (ρ 12 + ρ 13 ) (ρ ρ 23 + ρ 33 ) ) ( u U ). (9) We can now relate the matrix elements in (9) directly to the barred matrix elements appearing in (6), which then gives us three equations or our six unknowns. Again these equations are not linearly independent, so we still need our more equations. Next we consider the possibility that the racture luid can oscillate independently o the solid and the matrix luid, and urthermore that the matrix luid velocity is locked to that o the solid so that u = U (1). For this case, the kinetic energy telescopes in a dierent way to ( ) ( ) 2T = ( u U (2) (ρ11 + 2ρ ) 12 + ρ 22 ) (ρ 13 + ρ 23 ) u (ρ 13 + ρ 23 ) ρ 33 U (2). (10) This equation is also o the orm (6), but we must be careul to account properly or the parts o the system included in the matrix elements. Now we treat the solid and matrix luid as a single unit, so ρ ρ 12 + ρ 22 = (1 φ)ρ s + φ (1) ρ + (α (2) 1)φ (2) ρ, (11) ρ 13 + ρ 23 = (α (2) 1)φ (2) ρ, (12) and ρ 33 = α (2) φ (2) ρ, (13) where α (2) is the tortuosity o racture porosity alone.

7 SEP 98 Elastic waves or double porosity 229 Finally, we consider the possibility that the matrix luid can oscillate independently o the solid and the racture luid, and urthermore that the racture luid velocity is locked to that o the solid so that u = U (2). The kinetic energy telescopes in a very similar way to the previous case with the result ( ) ( ) 2T = ( u U (1) (ρ11 + 2ρ ) 13 + ρ 33 ) (ρ 12 + ρ 23 ) u (ρ 12 + ρ 23 ) ρ 22 U (1). (14) We imagine that this thought experiment amounts to analyzing the matrix material alone without ractures being present. The equations resulting rom this identiication are completely analogous to those in (11)-(13), so we will not show them explicitly here. We now have nine equations in the six unknowns and six o these are linearly independent, so the system can be solved. The result o this analysis is that the o-diagonal terms are given by 2ρ 12 /ρ = (α (2) 1)φ (2) (α (1) 1)φ (1) (α 1)φ, (15) 2ρ 13 /ρ = (α (1) 1)φ (1) (α (2) 1)φ (2) (α 1)φ, (16) and 2ρ 23 /ρ = αφ α (1) φ (1) α (2) φ (2). (17) The diagonal terms are given by ρ 11 = (1 φ)ρ s + (α 1)φρ, (18) and ρ 33 is given by (13). ρ 22 = α (1) φ (1) ρ, (19) Estimates o the three tortuosities α, α (1), and α (2) may be obtained using (7), or direct measurements may be made using electrical methods as advocated by Brown [1980] and Johnson et al. [1982]. Drag coeicients The drag coeicients may be determined by irst noting that the equations presented here reduce to those o Berryman and Wang [1995] in the low requency limit by merely neglecting the inertial terms. What is required to make the direct identiication o the coeicients is a pair o coupled equations or the two increments o luid content ζ (1) and ζ (2). These quantities are related to the displacements by ζ (1) = φ (1) (U (1) u) and ζ (2) = φ (2) (U (2) u).

8 230 Berryman & Wang SEP 98 The pertinent equations rom Berryman and Wang [1995] are ( η ζ(1) ) ( k (11) k ζ (2) = (12) ) ( p (1) ),ii k (21) k (22) p (2), (20),ii where η is the shear viscosity o the luid, and the ks are permeabilities including possible cross-coupling terms. We can extract the terms we need rom (5), and then take the divergence to obtain ( b12 + b 23 b 23 b 23 b 13 + b 23 ) ( ( U (1) ) ( (1) ) u) p ( U,ii (2) = u) p (2). (21),ii Comparing these two sets o equations and solving or the b coeicients, we ind b 12 = ηφ(1) (k (22) k (21) ), (22) k (11) k (22) k (12) k (21) and b 23 = b 12 = ηφ(2) (k (11) k (12) ), (23) k (11) k (22) k (12) k (21) ηφ (1) k (21) k (11) k (22) k (12) k = ηφ (2) k (12). (24) (21) k (11) k (22) k (12) k (21) For many applications it will be adequate to assume that the cross-coupling vanishes. In this situation, b 23 = 0, and b 12 = ηφ(1) k (11) (25) b 13 = ηφ(2), (26) k (22) which also provides a simple interpretation o these coeicients in terms o the porosities and diagonal permeabilities. This completes the identiication o the inertial and drag coeicients introduced in the previous section. STRESS-STRAIN FOR SINGLE POROSITY In the absence o driving orces that can maintain pressure dierentials over long time periods, double porosity models must reduce to single porosity models in the long time limit when the matrix pore pressure and crack pore pressure become equal. It is thereore necessary to remind ourselves o the basic results or single porosity models

9 SEP 98 Elastic waves or double porosity 231 in poroelasticity. One important role these results play is to provide constraints or the long time behavior in the problems o interest. A second signiicant use o these results (see Berryman and Wang [1995]) arises when we make laboratory measurements on core samples having properties characteristic o the matrix material. Then the results presented in this section apply speciically to the matrix stinesses, porosity, etc. For isotropic materials and hydrostatic pressure variations, the two independent variables in linear mechanics o porous media are the conining (external) pressure p c and the luid (pore) pressure p. The dierential pressure p d p c p is oten used to eliminate the conining pressure. The equations o the undamental dilatations are then or the total volume V, δv V δv φ V φ = δp d K + δp K s (27) = δp d K p + δp K φ (28) or the pore volume V φ = φv, and δv V = δp K (29) or the luid volume V. Equation (27) serves to deine the various constants o the porous solid, such as the drained rame bulk modulus K and the unjacketed bulk modulus K s or the composite rame. Equation (28) deines the jacketed pore modulus K p and the unjacketed pore modulus K φ. Similarly, (29) deines the bulk modulus K o the pore luid. Treating δp c and δp as the independent variables in our poroelastic theory, we deine the dependent variables δe δv/v and δζ (δv φ δv )/V, both o which are positive on expansion, and which are respectively the total volume dilatation and the increment o luid content. Then, it ollows directly rom the deinitions and rom (27), (28), and (29) that ( ) ( ) ( ) δe 1/K 1/Ks 1/K δpc =. (30) δζ φ/k p φ(1/k p + 1/K 1/K φ ) δp Now we consider two well-known thought experiments: the drained test and the undrained test [Gassmann, 1951; Biot and Willis, 1957; Geertsma, 1957]. (For a single porosity system, these two experiments are sometimes considered equivalent to the slow loading and ast loading limits respectively. However, these terms are relative since, or example, the ast loading equivalent to undrained limit is still assumed to be slow enough that the average luid and conining pressures are assumed to have reached equilibrium.) The drained test assumes that the porous material is surrounded by an impermeable jacket and the luid is allowed to escape through

10 232 Berryman & Wang SEP 98 a tube that penetrates the jacket. Then, in a long duration experiment, the luid pressure remains in equilibrium with the external luid pressure (e.g., atmospheric) and so δp = 0 and hence δp c = δp d ; so the changes o total volume and pore volume are given exactly by the drained constants 1/K and 1/K p as deined in (27) and (28). In contrast, the undrained test assumes that the jacketed sample has no passages to the outside world, so pore pressure responds only to in conining pressure changes. With no means o escape, the increment o luid content cannot change, so δζ = 0. Then, the second equation in (30) shows that 0 = φ/k p (δp c δp /B), (31) where Skempton s pore-pressure buildup coeicient B [Skempton, 1954] is deined by B δp (32) δp c δζ=0 and is thereore given by B = K p (1/K 1/K φ ). (33) It ollows immediately rom this deinition that the undrained modulus K u is determined by (also see Carroll [1980]) K u = K 1 αb, (34) where we introduced the combination o moduli known as the Biot-Willis parameter α = 1 K/K s. This result was apparently irst obtained by Gassmann [1951] or the case o microhomogeneous porous media (i.e., K s = K φ = K m, the bulk modulus o the single mineral present) and by Brown and Korringa [1975] and Rice [1975] or general porous media with multiple minerals as constituents. Finally, we condense the general relations rom (30) together with the reciprocity relations [Brown and Korringa, 1975] into symmetric orm as ( ) δe = 1 ( ) ( ) 1 α δpc. (35) δζ K α α/b δp The storage compressibility, which is a central concept in describing poroelastic aquier behavior in hydrogeology, is related inversely to one deined in Biot s original 1941 paper by S δζ δp δpc=0 = α BK. (36) This storage compressibility is the change in increment o luid content per unit change in the luid pressure, deined or a condition o no change in external pressure. It has also been called the three-dimensional storage compressibility by Kümpel [1991].

11 SEP 98 Elastic waves or double porosity 233 We may equivalently eliminate the Biot-Willis parameter α and write (35) in terms o the undrained modulus so that ( ) δe = 1 ( ) ( ) 1 (1 K/K u )/B δpc δζ K (1 K/K u )/B (1 K/K u )/B 2. (37) δp Equation (37) has the advantage that all the parameters have very well deined physical interpretations, and are also easily generalized or a double porosity model. Finally, note that (35) shows that K p = φk/α, which we generally reer to as the reciprocity relation. The total strain energy unctional (including shear) or this problem may be written in the orm 2E = δτ ij δe ij + δp δζ, (38) where δe ij is the change in the average strain with δe ii δe being the dilatation, δτ ij being the change in the average stress tensor or the saturated porous medium with 1 3 δτ ii = δp c. It ollows that δp c = E (δe) (39) and δp = E (δζ), (40) both o which are also consistent with Betti s reciprocal theorem [Love, 1927] since the matrices in (35) and (37) are symmetric. The shear modulus µ is related to the bulk modulus and Poisson s ratio by µ = 3(1 2ν)K/2(1 + ν). Then, it ollows that the stress equilibrium equation is and Darcy s law takes the orm where η is the single-luid shear viscosity. τ ij,j = (K u µ)e,i + µu i,jj BK u ζ,i = 0 (41) k η p,ii = ζ, (42) STRESS-STRAIN FOR DOUBLE POROSITY We now assume two distinct phases at the macroscopic level: a porous matrix phase with the eective properties K (1), µ (1), K m (1), φ (1) occupying volume raction V (1) /V = v (1) o the total volume and a macroscopic crack or joint phase occupying the remaining raction o the volume V (2) /V = v (2) = 1 v (1). The key eature

12 234 Berryman & Wang SEP 98 p c p c p c p c p (1) p (2) Figure 1: The double porosity model eatures a porous rock matrix intersected by ractures. Three types o macroscopic pressure are pertinent in such a model: external conining pressure p c, internal pressure o the matrix pore luid p (1), and internal pressure o the racture pore luid p (2). A single porosity medium is one in which either matrix or racture porosity are present, but not both. jim-pic [NR]

13 SEP 98 Elastic waves or double porosity 235 distinguishing the two phases and thereore requiring this analysis is the very high luid permeability k (22) o the crack or joint phase and the relatively lower permeability k (11) o the matrix phase. We could also introduce a third independent permeability k (12) = k (21) or luid low at the interace between the matrix and crack phases, but or simplicity we assume here that this third permeability is essentially the same as that o the matrix phase, so k (12) = k (11). We have three distinct pressures: conining pressure δp c, matrix-luid pressure δp (1), and joint-luid pressure δp(2). Treating δp c, δp (1), and δp(2) as the independent variables in our double porosity theory, we deine the dependent variables δe δv/v (as beore), δζ (1) = (δv (1) φ δv (1) )/V, and δζ (2) = (δv (2) φ δv (2) )/V, which are respectively the total volume dilatation, the increment o luid content in the matrix phase, and the increment o luid content in the joints. We assume that the luid in the matrix is the same kind o luid as that in the cracks or joints, but that the two luid regions may be in dierent states o average stress and thereore need to be distinguished by their respective superscripts. Linear relations among strain, luid content, and pressure then take the general orm δe a 11 a 12 a 13 δp c δζ (1) = a 21 a 22 a 23 δp (1). (43) δζ (2) a 31 a 32 a 33 δp (2) By analogy with (35) and (37), it is easy to see that a 12 = a 21 and a 13 = a 31. The symmetry o the new o-diagonal coeicients may be demonstrated by using Betti s reciprocal theorem in the orm ( δe δζ (1) δζ (2) ) 0 δp (1) 0 = ( δe δζ (1) δζ (2) cr ) 0 0 δp (2), (44) where unbarred quantities reer to one experiment and barred to another experiment to show that δζ (1) δp (1) = a 23 δp (2) δp(1) = a 32 δp (1) δp(2) = δζ (2) δp (2). (45) Hence, a 23 = a 32. Similar arguments have oten been used to establish the symmetry o the other o-diagonal components. Thus, we have established that the matrix in (43) is completely symmetric, so we need to determine only six independent coeicients. To do so, we consider a series o thought 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 (1) = p (2) = p as t ) as long as the cross permeability k (12) is inite. However, at very short times, we may assume that the process o pressure equilibration has not yet begun, or equivalently that k (12) = 0 at t = 0. We nevertheless assume that the pressure in each o the two components have individually equilibrated on the average, even at short times.

14 236 Berryman & Wang SEP 98 Undrained joints, undrained matrix, short time There are several dierent, but equally valid, choices o time scale on which to deine Skempton-like coeicients or the matrix/racture system under consideration. Elsworth and Bai [1992] use a deinition based on the idea that or very short time both luid systems will independently act undrained ater the addition o a sudden change o conining pressure. This idea implies that δζ (1) = 0 = δζ (2) which, when substituted into (43), gives δe = a 11 δp c + a 12 δp (1) + a 13 δp (2) 0 = a 12 δp c + a 22 δp (1) + a 23 δp (2) 0 = a 13 δp c + a 23 δp (1) + a 33 δp (2). (46) Deining B (1) EB δp(1) δp c δζ (1) =δζ (2) =0 and B (2) EB δp(2) δp c δζ (1) =δζ (2) =0 (47) we can solve (46) or the two Skempton s coeicients and ind the results B (1) EB = a 23a 13 a 12 a 33 a 22 a 33 a 2 23 (48) and B (2) The eective undrained modulus is ound to be given by EB = a 23a 12 a 13 a 22. (49) a 22 a 33 a K u EB δe δp c δζ (1) =δζ (2) =0 = a 11 + a 12 B (1) EB + a 13B (2) EB. (50) These deinitions will be compared to others. Drained joints, undrained matrix, intermediate time Now consider a sudden change o conining pressure on a jacketed sample, but this time with tubes inserted in the joint (racture) porosity so δp (2) = 0, while δζ (1) = 0. We will call this the drained joint, undrained matrix limit. The resulting equations are δe = a 11 δp c a 12 p (1) δζ (2) = a 31 δp c a 32 δp (1), (51)

15 SEP 98 Elastic waves or double porosity 237 showing that the pore-pressure buildup in the matrix is B[u (1) ] δp(1) δp c δζ (1) =δp (2) a 21 =. (52) a 22 =0 Similarly, the eective undrained modulus or the matrix phase is ound rom (51) to be determined by 1 K[u (1) ] δe = a δp c 11 + a 12 B[u (1) ]. (53) δζ (1) =δp (2) =0 Notice that i a 23 = 0 then (48) and (52) are the same. Drained matrix, undrained joints, intermediate time Next consider another sudden change o conining pressure on a jacketed sample, but this time the tubes are inserted in the matrix porosity so δp (1) = 0, while δζ (2) = 0. We will call this the drained matrix, undrained joint limit. The equations are showing that the pore-pressure buildup in the cracks is B[u (2) ] δp(2) δp c δe = a 11 δp c a 13 p (2) δζ (1) = a 21 δp c a 23 δp (2) 0 = a 31 δp c a 33 δp (2), (54) δζ (2) =δp (1) a 31 =. (55) a 33 =0 Similarly, the eective undrained modulus or the joint phase is ound 1 K[u (2) ] δe = a δp c 11 + a 13 B[u (2) ]. (56) δζ (2) =δp (1) =0 We may properly view Eqs. (52), (53), (55), and (56) as deining relations among these parameters. Notice that i a 23 = 0 then (49) and (55) are the same. Drained test, long time The long duration drained (or jacketed ) test or a double porosity system should reduce to the same results as in the single porosity limit. The conditions on the pore pressures are δp (1) = δp (2) = 0, and the total volume obeys δe = a 11 δp c. It ollows thereore that a 11 1 K, (57) where K is the overall drained bulk modulus o the system including the ractures.

16 238 Berryman & Wang SEP 98 Undrained test, long time The long duration undrained test or a double porosity system should also produce the same physical results as a single porosity system (assuming only that it makes sense at some appropriate larger scale to view the medium as homogeneous). The basic equations are δp (1) = δp (2) = δp, δζ δζ (1) + δζ (2) = 0, (58) assuming the total mass o luid is conined. Then, it ollows that δe = a 11 δp c (a 12 + a 13 )δp, 0 = (a 21 + a 31 )δp c (a 22 + a 23 + a 32 + a 33 )δp, (59) showing that the overall pore-pressure buildup coeicient is given by B p p c Similarly, the undrained bulk modulus is ound to be given by (60) δζ=0 1 δe = a K u δp c 11 + (a 12 + a 13 )B. (61) δζ=0 Fluid injection test, long time The conditions on the pore pressures or the long duration, single porosity limit or the three-dimensional storage compressibility S are δp (1) = δp (2) = δp, while the conining pressure remains constant. It ollows thereore that S ζ p = a 22 + a 23 + a 32 + a 33. (62) δpc=0 This completes the main analysis o the elastic coeicients or double porosity. Further details may be ound in Berryman and Wang [1995].

17 SEP 98 Elastic waves or double porosity 239 Table 1. Material Properties Barre Bedord Parameter Granite Limestone K (GPa) 13.5 a 23.0 a K s (GPa) 54.5 a 66.0 a α 0.75 a 0.65 a K (1) (GPa) 21.5 a 27.0 a ν (1) K s (1) (GPa) 55.5 a 66.0 a α (1) 0.61 a 0.59 a K (GPa) φ (1) a B (1) S (1) (GPa 1 ) v (2) a a From Coyner [1984] EXAMPLES Table 1 presents data or Barre granite and Bedord limestone taken rom laboratory measurements by Coyner [1984]. Coyner s experiments included a series o tests on several types o laboratory scale rock samples at dierent conining pressures. The values quoted or K and K s are those or a moderate conining pressure o 10 MPa (values at lower conining pressures were also measured but we avoid using these values because the rocks generally exhibit nonlinear behavior in that region o the parameter space), while the values quoted or K (1) and K s (1) are at 25 MPa, which is close to the value beyond which the constants cease depending on pressure and thereore or which we assume all the cracks were closed. Thus, based on the idea that the pressure behavior is associated with two kinds o porosity in the laboratory samples a crack porosity, which is being closed between 10 and 25 MPa, and a residual matrix porosity above 25 MPa, we assume the available data are K, K s, K (1), K s (1), K, φ (1), and v (2). We ind that these data are suicient to compute all the coeicients. In Table 2, we ind in both types o rock that the coeicient a 23 is positive and small about an order o magnitude smaller than the other matrix elements. Another unusual eature o the results computed using these laboratory data is the occurrence o values larger than unity or B and B[u (1) ] in Barre granite; similar results were observed by Berryman and Wang [1995] or Chelmsord granite, and in subsequent work or Westerly granite. Also note that α (2) or both rocks is close to unity or these calculations. In this example, seven measurements (together with Poisson s ratio) are suicient to determine completely the mechanical behavior o the double-porosity model. Having a direct measurement o K eliminates the necessity o assuming a 23 = 0, which we have ound is sometimes necessary when dealing with ield data [Berryman and Wang, 1995].

18 240 Berryman & Wang SEP 98 Table 2. Double porosity parameters computed rom material properties in Table 1. Barre Bedord Parameter Formula Granite Limestone a 11 (GPa 1 ) 1/K a 12 (GPa 1 ) α (1) K s (1) /K (1) K s a 13 (GPa 1 ) α/k a a 22 (GPa 1 ) v (1) α (1) /B (1) K (1) a 23 (GPa 1 ) v (1) α (1) /K (1) a a 33 (GPa 1 ) v (2) /K + v (1) /K (1) (1 2α)/K + 2a a 33 (GPa 1 ) a 33 v (2) /K α (2) (a 33 a 12 a 13 a 23 )/(a 11 a 23 a 13 a 12 ) B (a 12 + a 13 )/(a a 23 + a 33 ) B (1) (a 12 + a 23 )/a B[u (1) ] a 12 /a B (1) EB (a 23 a 13 a 12 a 33 )/(a 22 a 33 a 2 23 ) B (2) (a 33 a 12 a 13 a 23 )/[a 33 (a 12 + a 23 ) a 23 (a 13 a 33 )] B[u (2) ] a 13 /a B (2) EB (a 23 a 12 a 13 a 22 )/(a 22 a 33 a 2 23 ) K u (GPa) [a 11 (a 12 + a 13 ) 2 /(a a 23 + a 33 )] K (2) (GPa) v (2) (a 12 + a 13 )/(a 11 a 23 a 13 a 12 ) S (GPa 1 ) α/bk S (2) (GPa 1 ) α (2) /B (2) K (2) CONCLUSIONS We now have a complete set o equations or the double-porosity/dual-permeability model. The next step in the analysis will be to solve these equations and ind their wave propagation properties. This work is in progress and will be reported at a later date. REFERENCES Berryman, J. G., 1980, Conirmation o Biot s theory: Appl. Phys. Lett., 37, Berryman, J. G., and Wang, H. F., 1995, The elastic coeicients o double-porosity models or luid transport in jointed rock: J. Geophys. Res., 100, Biot, M. A., 1941, General theory o three dimensional consolidation: J. Appl. Phys., 12, Biot, M. A., 1956, Theory o propagation o elastic waves in a luid-saturated porous solid. I. Low-requency range: J. Acoust. Soc. Am., 28,

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20 242 Berryman & Wang SEP 98 Miksis, M. J., 1988, Eects o contact line movement on the dissipation o waves in partially saturated rocks: J. Geophys. Res., 93, O Connell, R. J., and Budiansky, B., 1974, Seismic velocities in dry and saturated cracked solids: J. Geophys. Res., 79, O Connell, R. J., and Budiansky, B., 1977, Viscoelastic porperties o luid-saturated cracked solids: J. Geophys. Res., 82, Pride, S. R., and Berryman, J. G., 1998, Connecting theory to experiment in poroelasticity: J. Mech. Phys. Solids, 46, Rice, J. R., 1975, On the stability o dilatant hardening or saturated rock masses: J. Geophys. Res., 80, Rice, J. R., and Cleary, M. P., 1976, Some basic stress diusion solutions or luidsaturated elastic porous media with compressible constituents: Rev. Geophys. Space Phys., 14, Skempton, A. W., 1954, The pore-pressure coeicients A and B: Geotechnique, 4, Tuncay, K., and Corapcioglu, M. Y., 1996, Wave propagation inractured porous media: Transp. Porous Media, 23,