Two-phase fluid flow through fractured porous media with deformable matrix

Authors


Abstract

[1] The governing equations for a fully coupled analysis of two-phase fluid flow through fractured porous media with deformable matrix are presented. Two porosities, three constitutes (one solid and two fluids), and five phases are identified. The porosities are referred to as pores and fractures, and the two fluid constitutes are taken as water and air. The governing equations are derived using a systematic macroscopic approach based on the theory of poroelasticity, the effective stress principle, and the balance equations of mass and momentum. Matrix displacement vector, pore air pressure, pore water pressure, fracture air pressure and fracture water pressure are introduced as primary variables. Special attention is given to cross-coupling effects between the phases within the system. A general formulation is derived that reduces to all previously presented models in the field. When the pore air volume is reduced to zero, the fully coupled equations of flow and deformation for saturated double-porosity media are recovered. Similarly, when the matrix deformation is neglected and full saturation is assumed, Barenblatt et al.'s (1960) classical theory of double porosity is obtained.

1. Introduction

[2] Fluid flow and deformation in fractured porous media has been the subject of great interest in many engineering disciplines particularly in water resources, petroleum and mining engineering as well as the nuclear waste industry. Typical examples include: primary and tertiary enhancement of oil production, utilization of geothermal energy, remediation of contaminated sites, and isolation of hazardous wastes. The phenomenon has also bearing on process, mechanical, geotechnical and agricultural engineering. Concerted efforts have been made over the past few decades to arrive at predictive capabilities for fractured porous media associated with diverse topics such as protection of fresh water supplies, reservoir production and recovery, earthquake engineering, storage of nuclear waste, movement of pollutant plumes, etc. Several mathematical models have been proposed.

[3] The early models were based on the single porosity or continuum concept. In this approach, a fractured porous medium was grossly treated as an equivalent continuum with a single fluid constituent. The physical quantities such as potential, porosity and fluid pressure were averaged over representative blocks of the fractured porous medium containing sufficiently large number of fractures. The single porosity approach was associated with a number of drawbacks [e.g., see Long et al., 1982; Wilson et al., 1983] including, identification of the representative blocks and determination of the equivalent model parameters. Moreover, because of the averaging process involved, the model results, i.e., the fluid pressure and the fluid flux distribution, represented neither the situation in the pores nor in the fractures. In fact, the application of the single porosity models was only justified if one was dealing with large-scale flow fields, in which a detailed description of the field variables within the fractured porous medium was not of particular concern.

[4] A major departure from the single porosity approach was first made by Barenblatt et al. [1960] and Warren and Root [1963]. They modeled flow through rigid fractured porous media as a complex of two interacting flow regions: one representing the fracture network and the other the porous blocks. The fracture network was characterized by high permeability and low storage, and the porous blocks were characterized by low permeability and high storage. The two flow regions were in turn coupled through a leakage term controlling the transfer of fluid mass between the pores and fractures. An extension of Barenblatt's model to deformable fractured porous media was later given by Duguid and Lee [1977]. Also motivated by Barenblatt's work, Aifantis [1977, 1979, 1980] used the theory of mixtures and proposed a coupled double-porosity model for deformable fractured porous media. Alternatives to Aifantis' formulation were given by Wilson and Aifantis [1982], Khaled et al. [1984], Valliappan and Khalili-Naghadeh [1990], Khalili-Naghadeh and Valliappan [1991], Auriault and Boutin [1993], Bai et al. [1993], and Chen and Teufel [1997], among others. However, Aifantis' model was incomplete, in the sense that it related pore and fracture volume changes only to the overall volume change of the fractured porous medium. More specifically, it ignored the cross-coupling effects between the volume change of the pores and fractures within the system. This deficiency was eliminated in the formulations proposed by Khalili and Valliappan [1996], Tuncay and Corapcioglu [1996], Wang and Berryman [1996], Khalili et al. [1999], and Loret and Rizzi [1999]. The significance of the cross-coupling effects on the pore and fracture fluid pressure response of double-porosity media was highlighted by Khalili [2003].

[5] Recently, Lewis and Ghafouri [1997], Bai et al. [1998], Pao and Lewis [2002], and Nair et al. [2005] extended the double-porosity concept to include multiphase flow, albeit subject to a number of restrictions. Lewis and Ghafouri [1997] introduced the assumption that the applied loads were taken only by the porous blocks, and hence neglected the fracture deformation. Bai et al. [1998] and Nair et al. [2005] considered the fracture deformation, but treated the porous blocks and fracture network separately. They introduced two effective stress equations for the quantification of the deformation field, in contradiction of the definition of the effective stress principle. Moreover, none of the models took into account the cross-coupling effects between the fluid phases completely and/or consistently.

[6] The main objective in this paper is to provide a rigorous treatment of the theory of flow and deformation in fractured porous media saturated with two immiscible fluids. The governing differential equations are formulated using a systematic macroscopic approach based on the theory of poroelasticity satisfying the balance equations of mass and momentum. The effective stress equation of the system is derived, and the cross-coupling terms between the phases within the system are identified. Matrix displacement vector, pore air pressure, pore water pressure, fracture air pressure and fracture water pressure are introduced as seven primary variables for a three-dimensional boundary value problem. The governing equations are closed using the static equilibrium or balance of momentum of the fractured porous medium (three equations) and the balance of momentum of the fluid phases (four equations). All model parameters are identified in terms of measurable physical entities.

2. Notation and Sign Convention

[7] Compact matrix-vector notation is used throughout. Bold letters indicate matrices and vectors. ∇() = ∂()/∂x is the spatial gradient, and div() = ∇ · () is the divergence operator. Sign convention of continuum mechanics is adopted with compressive stresses and strains taken as positive. The mean normal stress, p, and the volumetric strain, ɛv, are defined as p = −equation imagetrσ and ɛv = −trequation image so that they are positive in compression following the reservoir engineering convention. Fluid pressure is taken as positive in compression.

3. Basic Concepts

[8] The theory of fluid flow and deformation for saturated double-porosity media, proposed by Khalili and Valliappan [1996], will be the starting point of the present derivations. Two porosities, three constitutes (one solid and two fluids), and five phases are identified. Each phase is viewed as an independent entity, endowed with its own kinematics, mass and momentum, occupying the entire volume of the double porous medium, V. The two porosities are referred to as pores (p) and fractures (f), and the two fluid constitutes as water (w) and air (a). The phases identified are: solid, water in pores, water in fractures, air in pores and air in fractures. The solid constituent is considered slightly compressible, but it can be made incompressible, if necessary. The fractured network is assumed to be preexisting and of intensity suitable for representation as a continuum. The macroscopic response of the solid skeleton to mechanical loading is assumed to be linearly elastic for the stress range of interest. Formation of new fractures due to mechanical and fluid pressure loading is ignored. The pressure difference between the air and water constituents in pores and fractures is referred to as the matric suction of pores and fractures, respectively. In keeping with the terminology used in the double-porosity literature and starting from the smallest material scale, three primary media are identified in a fractured porous medium. These are solid medium, porous medium (also referred to as the porous blocks), and fractured porous medium, Figure 1. The pores and fractures are identified using the script α = p, f and the constituents are identified using the script β = s, w, a. Each constituent has a mass Mβ and a volume Vβ, which make up the total mass M = Ms + Mw + Ma and the total volume V = Vs + Vw + Va. Intrinsic quantities are defined using subscripts, and apparent quantities using superscripts. For example, intrinsic mass density of β phase in α pore is denoted ραβ = Mαβ/Vαβ, whereas the apparent mass density is written as ραβ = Mαβ/V; hence ραβ = nαβραβ, where nαβ = Vαβ/V is the apparent volume fraction of constituent αβ. The apparent volume fractions satisfy the constraint ns + nw + na = 1. All measurements are made from configuration at time t = 0. The initial configuration is assumed to be free of strain and stress. dαβ()/dt = ∂()/∂t + ∇() · vαβ is used to denote the total derivative, with vαβ representing the velocity field of constituent β in pore space α.

Figure 1.

Media at different scales of a fractured porous medium.

4. Effective Stress

[9] An essential preliminary in the coupled analysis of multiphase, multiporous media is the determination of the effective stress of the solid skeleton. Expressed as that function of the total stress and pore fluid pressures which controls the mechanical effects of a change in stress, the effective stress enters the elastic as well as the elastoplastic constitutive equations of the solid phase, linking a change in stress to straining or any other relevant quantity of the solid skeleton [Khalili et al., 2005; Nuth and Laloui, 2008]. Within this context, the effective stress satisfies a priori the constraint

equation image

in which equation image is the elastic strain tensor of the solid skeleton, C is the drained elastic compliance tensor of the medium, and σ′ is the effective stress tensor. For isotropic loading, equation (1) reduces to dɛv ≡ −d(trequation image) = cdp′, in which ɛv is the volumetric strain, and c is the underlying tangent drained compressibility coefficient of the medium, and p′ is the mean effective stress.

[10] In its most general form, the effective stress for a fractured porous medium, saturated with air and water, is expressed as [Khalili et al., 2005]

equation image

in which σ is the total stress, βαβ is the incremental effective stress parameter, pαβ is the fluid pressure, and I is the second-order identity tensor. The effective stress parameters, βαβ, are derived by conducting an elastic strain equivalency analysis satisfying equation (1), much in line with the work of Nur and Byerlee [1971], Khalili and Valliappan [1996], and Khalili et al. [2000].

[11] To this end and to identify βαβ, in terms of physically measurable entities, consider an elemental volume of an unsaturated fractured porous medium subjected to isotropic loading cases 1–4 as shown in Figure 2.

Figure 2.

Loading cases with known volumetric response of the fractured porous medium.

[12] Case 1 corresponds to the stress increments (dσ, dppw, dppa, dpfw, dpfa) = (−dpI, dp, dp, dp, dp), in which the unsaturated fractured porous medium is subjected to an all around increment of the external pressure, dp, equal to the increments of the internal pressures of the pore water, pore air, fracture water and fracture air. For this case, the increment of stress and strain in the solids is the same as that, which would occur, if the pores and fractures were hypothetically filled up with solids, and the stress increments applied to the outer surface were left unchanged. Invoking the uniqueness theorem for the deformation of an elastic boundary value problem, and noting the work of Nur and Byerlee [1971] and Zimmerman [1991], it then follows that the overall compressibility of the system for case 1 is equal to the compressibility of the solid constituent, cs, and the macroscopic volumetric response of the system is calculated as dɛv(I) = csdp.

[13] Case 2 corresponds to equal increments of fracture water, fracture air and external isotropic pressures and zero increments of pore water and pore air pressures; (dσ, dppw, dppa, dpfw, dpfa) = (−dpI, 0, 0, dp, dp). Here, the stress and strain response of the system is identical to the condition where the fractures are filled up with the material forming the porous blocks, and the conditions on the sample boundaries are left unchanged. Therefore, the overall compressibility of the system is equal to that of the porous blocks, cp [see, e.g., Khalili and Valliappan, 1996; Loret and Rizzi, 1999; Pao and Lewis, 2002], and dɛv(II) = cpdp.

[14] Cases 3 and 4 correspond to the stress increments (dσ, dppw, dppa, dpfw, dpfa) = (−dpI, 0, dp, dp, dp) and (dσ, dppw, dppa, dpfw, dpfa) = (−dpI, 0, dp, 0, dp), respectively. These represent the stress conditions of the porous blocks and the fractured porous medium during a water expulsion or drying test. Such tests can be conducted by subjecting a representative volume of the porous blocks and fractured porous medium to increasing values of the air pressure at the sample boundaries, while maintaining the water pressure zero within the medium. Drying tests are routinely performed in soil engineering, with the results typically presented in terms of water content versus matric suction or degree of saturation versus matric suction. These are referred to as the soil water characteristic curves or water retention-pressure deficiency characteristic curves. During a drying test, the volume change of the sample can be measured, which can then be expressed as the compressibility coefficient of the test medium with respect to a change in matric suction. Here, the compressibility coefficient of the porous blocks with respect to a change in matric suction is represented by, cmp, and the compressibility coefficient of the fractured porous medium with respect to a change in matric suction (in which both pores and fractures are desaturated) is denoted by, cmfp. Accordingly, the volumetric response of the loading cases 3 and 4 can be expressed as dɛv(III) = cmpdp and dɛxv(IV) = cmfpdp, respectively.

[15] Now, for the effective stress to be valid, it must be applicable to all loading cases. Applying equation (2) to loading cases 1–4 and making use of equation (1) the following equivalencies are established

equation image

[16] Rearranging (3) then yields

equation image

subject to the constraint βpw + βpa + βfw + βfa = 1 − equation image. Relations in (4) define the incremental effective stress parameters in terms of measurable compressibility coefficients, c, cs, cp, cmp, and cmfp. Imposing the incompressibility of the solid constituent, cs = 0, (4) reduces to

equation image

in which equation image and equation image, with equation image representing the contribution of fractures to the overall compressibility of the double porous medium, i.e., in excess of that if the entire system was made of the porous blocks. Similarly, cmfcmfpcmp denotes the contribution of the fracture to the overall compressibility of the double porous medium due to a change in the matric suction.

[17] Using the definitions in (5), the effective stress equation (2) can be written as

equation image

or

equation image

in which βp and βf are the conventional incremental effective stress parameters for saturated double porous media [Khalili and Valliappan, 1996]; ψp and ψf, a nonlinear function of suction and wetting and drying paths, are the unsaturated incremental effective stress parameters of the pores and fractures, respectively; and spppappw and sfpfapfw are the matric suction of pores and fractures, respectively.

[18] The physical interpretation of ψp, ψf, βp and βf is that ψp and ψf scale/average air and water pressure increments in the pores and fractures to “equivalent” increments of pore fluid pressure, dppeq and fracture fluid pressure, dpfeq, as

equation image
equation image

[19] βp and βf in turn quantify the contribution of the “equivalent” increments of pore and fracture pressures to the effective stress of the double porous medium. Khalili et al. [2004] showed that for single porosity media ψp may be approximated as

equation image

in which χp is the unsaturated total effective stress parameter of the pore defined as [Khalili and Khabbaz, 1998]

equation image

[20] The exponent Ω is a material parameter, with a best fit value of 0.55; sp is the matric suction in the pores; and sp(e) is the suction value separating saturated from unsaturated conditions in the pores. It is equal to the air entry value, sp(ae), for the main drying path, and the air expulsion value, sp(ex), for the main wetting path [Khalili et al., 2004]. For suction values between the main drying and the main wetting paths an interpolation function in line with the work of Khalili et al. [2008] may be used.

[21] Substituting for χp in (8) from (9), we obtain

equation image

[22] Now, extending the observation of Khalili and Khabbaz [1998] for single porosity media to fractures, one can write

equation image

and

equation image

where sf(e) represents the matric suction separating saturated from unsaturated conditions in the fractures, and sf is the matric suction in the fractures.

[23] Notice that at the limiting case of zero fracture volume we have cp = c thus, βp = 1 and βf = 0, and equation (6) reduces to the effective stress for unsaturated single porosity media dσ′ = d(σ + ppa)Iψpd(ppappw)I, first proposed by Bishop [1959]. When the volume fraction associated with the air in both pores and fracture is reduced to zero then cmp = cp and cmfp = cψp = ψf = 1, giving the incremental form of effective stress equation for saturated double porous media dσ′ = dσ + βpdppwI + βfdpfwI [Khalili and Valliappan, 1996]. When both the fracture volume and the volume of air are zero, then cmfp = cmp = cp = cβpa = βfw = βfa = 0, the familiar form of Skempton's [1961] effective stress equation is recovered dσ′ = dσ + (1 − equation image)dppwI, for the case of compressible solid grains. This reduces to the Terzaghi's effective stress for the incompressible solid constituent (cs = 0), i.e., dσ′ = dσ + dppwI.

5. Governing Equations

5.1. Conceptual Model

[24] The formulation presented consists conceptually of two separate, yet overlapping models: the deformation model, and the flow model. The deformation model is based on the theory of elasticity, the effective stress principle and the equations of equilibrium; and the flow model is based on the theory of double porosity. Two interacting, two-phase fluid flow regions are identified: one representing the flow of air and water in the porous blocks, and the other representing the flow of air and water in the fracture network. The two flow regions are coupled through two leakage terms controlling the transfer of water and air from the porous blocks into the fracture network, and vice versa. The rate of fluid transfer at any point is assumed to depend on the fluid pressure difference between the like constituents in the porous blocks and the fracture network, the phase permeability of the porous blocks, and the fracture spacing. The coupling between the air and water flow, in each of the pore spaces, is established through the water retention-pressure deficiency characteristic curve of the pore. The coupling between the fluid flow and deformation is established through the effective stress parameters.

5.2. Deformation Model

[25] To derive the deformation model, consider a representative elementary volume of the fractured porous medium subjected to total stresses, σ. For quasi-static conditions, the linear momentum balance equation for this elemental volume can be written as

equation image

where F is the body force per unit volume. The stress and strain relationship of the element is in turn expressed as

equation image

in which D is the tangent drained stiffness matrix of the fractured porous medium. Assuming infinitesimal deformations, equation image is related to displacement, u, by

equation image

[26] Combining equations (12) to (14) and using the effective stress equation (2) yields the differential equation describing the deformation field for an unsaturated fractured porous medium

equation image

5.3. Flow Model

[27] The flow model is developed by combining the equation of linear momentum balance for the fluid phases with the mass balance equation of the fluids. Neglecting the inertial and viscous effects, the equation of linear momentum balance for the fluid phases (Darcy's law) is written as

equation image

where vαrβ is the relative velocity vector, kαβ is the phase permeability defined over the representative volume of the fractured porous medium, μαβ is the dynamic viscosity, and g is the vector of gravitational acceleration. The relative velocity of the fluid phase with respect to the moving solid is given by

equation image

where nαβVαβ/V is the apparent volume fraction of constituent β in flow region α, vαβ is the absolute velocity of constituent β in the flow region α, and vs is the solid skeleton velocity. The absolute velocities vαβ and vs are defined as

equation image

where uαβ and u are the displacement vectors of the fluid constituent and solid, respectively.

[28] Excluding mass exchange between water and air constituents due to vaporization and condensation, the mass balance equation for the fluid phases is given by

equation image

with

equation image

where Γβ is the leakage term controlling the exchange rate of constituent (β) between the porous block and fracture network. Both quasi-steady [Barenblatt et al., 1960; Warren and Root, 1963] and unsteady models [Huyakorn et al., 1983] may be used to represent Γβ. The model proposed by Barenblatt et al. [1960] is preferred here because of its simplicity and the ease of use, i.e.,

equation image

in which γβ is the leakage parameter. Substituting (17) into (19) yields

equation image

[29] Introducing the Lagrangian total derivative with respect to a moving solid ds()/dt = ∂()/∂t + ∇() · vs, and a moving fluid dαβ()/dt = ∂()/∂t + ∇() · vαβ, with the vector identity div{()vβ} = ()divvβ + ∇() · vβ, equation (21) can be rearranged to

equation image

Assuming the fluid constituents are barotropic, then

equation image

in which cαβ is the coefficient of fluid compressibility. From the definition of the apparent constituent volume fraction, nαβ = Vαβ/V, we have

equation image

Substituting equations (16), (23), and (24) into equation (22) and noting (dsV/dt)/V = div vs yields

equation image

[30] In equation (25), dsVαβ/V represents the change in the volume of the fluid constituent β in pore space, α, over the volume of the fractured porous medium. Equations (15) and (25) form the governing differential equations describing flow of two-phase fluids through deformable fractured porous media. They involve eleven unknowns (u, pαβ and Vαβ) and seven equations in a three-dimensional boundary value setting. Thus, four additional equations are required to close the system.

5.4. Constitutive Relationships

[31] Constitutive equations complement governing equations by providing additional relationships between the deformation and stress variables. Two classes of constitutive relationships are presented: constitutive equations for volumetric deformations of water and air phases to supplement the flow model, and constitutive equations for the stress-strain relationship of the soil skeleton to supplement the deformation model. For both groups, invoking the existence of potential Ψ(σ, ppw, ppa, pfw, pfa), quadratic in σ but a priori not in suction, the strain components (equation image, dVpw/V, dVpa/V, dVfw/V, dVfa/V) are related to the stress components (σ, ppw, ppa, pfw, pfa) such that enjoy the major symmetry as

equation image
equation image

in which aαβ,ζξ are the incremental cross-coupling coefficients. Notice that because of the suction dependency of the effective stress parameters, the underlying symmetry in (26), imposed because of the existence of Ψ, may only be established in the incremental form, and not in the secant form [Loret and Khalili, 2000]. This is an important aspect, which is often neglected in the literature, leading to less than accurate constitutive models.

[32] To identify aαβ,ζξ in terms of measurable entities, we again make reference to the loading cases 1–4, Figure 2. For each case, using the homothetic strain field, the volume change of constituent β in pore space α is written as

equation image

[33] For the loading cases 3 and 4, however, in addition to the homothetic strain field, the volume change of the constituents is influenced by the shift in the air-water interface because of the change in matric suction, dsαdpαadpαw, in pores and fractures. This is captured through the coefficient cαwa in (27), cases 3 and 4.

[34] Using the volumetric changes in (27), the cross-coupling coefficients aαβ,ζξ can readily be obtained from (26) as

equation image

Substituting (28) into (26) and rearranging then gives

equation image

[35] Equation (29) shows that the volume change of constituent β in pore space α depends on (1) the overall volume change of the fractured porous medium, (2) the pressure of the fluid constituent and (3) the pressure difference between the target fluid constituent and the other fluid constituents within the system. Of particular interest is the pressure difference between the fluid constituents occupying the same pore space. The volume change due to this component is captured through the cross-coupling terms apw,pa = (βpanpa)βpwc + cpwa and afw,fa = (βfanfa)βfwc + cfwa, which are fully defined in terms of measurable entities, except for the air-water interface translation coefficients cpwa and cfwa. These two coefficients can be conveniently determined by conducting a drying test on a representative sample of the fractured porous medium. The water content-suction curve from such a test will exhibit two distinct characteristic regions as shown in Figure 3. Region I corresponds to desaturation of the fractures while the pores remain saturated, whereas region II is characterized by draining of both the fractures and pores. The slope of the response in region I can be used to quantify afw,fa and hence cfwa, and that of region II to obtain apw,pa and cpwa. Noting the stress states for regions I and II; (dσ, dppw, dppa, dpfw, dpfa)I = (−ds, 0, 0, 0, ds) and (dσ, dppw, dppa, dpfw, dpfa)II = (−ds, 0, ds, 0, ds), then from (29) we have

equation image
equation image

and

equation image
equation image

in which Gs is the specific gravity, n is the porosity, and w is the gravitational water content defined as the mass of water over the mass of solid.

Figure 3.

Schematic representation of water retention-pressure deficiency characteristic curve for a fractured porous medium.

[36] The cross-coupling terms aαβ,ζξ may also be determined using the equivalent pressure concept, as expressed in equation (7), and the definition of the volume fraction for each of the fluid constituents. Both approaches lead to identical cross-coupling terms (Appendix A).

5.5. Coupled Equations

[37] The fully coupled equations are obtained by combining the governing equations for the flow and deformation models, equations (15) and (25) with the constitutive equation (26).

[38] Substituting (26) into (25) yields

equation image

Noting −tr equation image = −div u, introducing the approximation ds(·)/dt ≈ ∂(·)/∂t by assuming ∇() · vβ ≪ ∂(·)/∂t, and assigning (α and ζ) = p, f and (β and ξ) = w, a the differential equation governing the flow of water and air through an unsaturated fractured porous medium can be written from (32) as

equation image

with

equation image

[39] Equations (15) and (33) form the set of differential equations governing flow and deformation phenomena in unsaturated fractured porous media. These equations are general in nature and reduce to all previously presented formulations in the field. For example, when pore air volume is reduced to zero the theory of double-porosity consolidation is recovered [Khalili and Valliappan, 1996]. Similarly, when the matrix deformation is neglected and full saturation is assumed within the two-pore system, Barenblatt et al.'s [1960] classical theory of double porosity is obtained. On the other hand, the equations reduce to Biot's theory of consolidation [Biot, 1941] when the fracture volume and pore air volume become zero.

6. Model Parameters

[40] The proposed model involves the following parameters.

[41] 1. D [unit: Pa] is the drained stiffness of the soil skeleton. For an isotropic elastic material, D is completely defined in terms of the drained modulus of elasticity, E, and the drained Poisson's ratio, ν.

[42] 2. βpw, βaw, βfw and βfa are nondimensional “incremental” effective stress parameters, relating changes in pore water, pore air, fracture water and fracture air pressures to a change in matrix deformation.

[43] 3. equation imagepw,pw, equation imagepa,pa, equation imagefw,fw and equation imagefa,fa [unit: Pa−1] are the apparent compressibilities of pore water, pore air, fracture water and fracture air, respectively.

[44] 4. apw,pa, apw,fw, apw,fa, apa,fw, apa,fa, afw,fa [unit: Pa−1] are the incremental cross-coupling terms relating the volume change of a fluid constituent to the changes in the pressure of other fluid constituents within the system,

[45] 5. γw and γa [unit: Pa−1/s] are the proportionality coefficients controlling the rate of water and transfer from pores to fractures and vice versa.

[46] 6. kpw, kpa, kfw, and kfa [unit: m2] are the permeability tensors of pore water, pore air, fracture water and fracture air, respectively.

[47] 7. μpw, μpa, μfw and μfa [unit: Pa × s] are the dynamic viscosities of pore water, pore air, fracture water and fracture air, respectively.

[48] The coefficients βpw, βaw, βfw, βfa, equation imagepw,pw, equation imagepa,pa, equation imagefw,fw, equation imagefa,fa, apw,pa, apw,fa, apa,fw, apa,fa are all related to the basic measurable parameters npw, npa, nfw, npa, cpw, cpa, cfw, cfa, c, cs, cp, cmp, cmfp, cpwa, cfwa, ρpw, ρpa, ρfw, ρfa through relations given in (34). Fluid phase volume fractions (npw, npa, nfw, npa) can be obtained from a knowledge of the volume fraction of pores, np, and the volume fraction of fractures, nf, the degree of saturation of pores, Sp, and the degree of saturation of fractures, Sf. nf and np can be obtained using a number of direct and indirect methods available in the literature [see, e.g., Bear [1972]. Sp and Sf are linked to matric suction through water retention-pressure deficiency characteristic curves. The compressibility coefficients of water cpw = cfw and solid, cs, as well as the densities of solid, ρs, and water, ρpwρfw, are available from the literature. The compressibility and density of the air constituents, assuming isothermal conditions, can be obtained from

equation image

in which ω is the molecular weight of the air mass; R is the universal gas constant; Pαa = (pαa + patm) is that absolute air pressure in the pore space α; c and cp are the compressibility coefficients of the fractured porous medium and the porous block which can be determined in the laboratory by isolating representative samples of each medium and subjecting them to isotropic loading; and cmp and cmfp are the incremental compressibility coefficients of the porous blocks and the fractured porous medium due to a change in matric suction. These are determined directly in the laboratory by subjecting the elemental volumes of the porous blocks and the fractured porous medium to changes in matric suction (in the range expected in the field) and measuring the volume change of the samples. In this case, cmp and cmfp are defined as

equation image

[49] Alternatively, cmp and cmfp can be obtained from the definition of the effective stress parameters as

equation image

in which ψp and ψf are determined from equations (10) and (11), respectively. The air-water interface translation coefficients cpwa and cfwa are obtained from the slope of the water retention-pressure deficiency characteristic curve of the fracture porous medium using equations in (31).

[50] Here kpw, kpa, kfw, and kfa can be related to the intrinsic permeabilities of pores, Kp, and fractures, Kf, and the relative permeabilities kprw, kpra, kfrw, kfra as

equation image
equation image

The intrinsic permeabilities Kp and Kf can be obtained from saturated permeability tests conducted on representative samples of the porous blocks and the fractured porous medium. For fractured rock formations they can also be obtained from the pressure response of the fractured porous medium to a pumping test in the field. The slope of the early time response can be used to determine Kf and the late time response to capture Kp + Kf [Khalili-Naghadeh and Valliappan, 1991]. The relative permeabilities can be obtained directly in the laboratory or indirectly by semiempirical relationships between kαrβ and pore/fracture degree of saturation [Brooks and Corey, 1964]

equation image
equation image

where Sαe is the effective degree of saturation, defined as

equation image

in which Sαru is the residual degree of saturation and λp is the pore size distribution index. Both of these parameters are identified from the water retention-pressure deficiency characteristic curve presented in terms of Sα. The leakage parameters γw and γa for quasi steady state condition can be obtained as [Warren and Root, 1963]

equation image

with

equation image

n = 1, 2, 3 is the number of normal sets of fractures and d1, d2 and d3 are the fracture intervals in each direction.

7. Conclusions

[51] A fully coupled formulation is presented for two-phase fluid flow through deformable fractured porous media. The effective stress of the solid skeleton is derived and is used as a platform for coupling the deformation of the solid skeleton with the volume change of the water and air constituents in pores and fractures. Particular attention is given to the derivation of cross-coupling coefficients between various phases within the system. Inclusion of such coupling terms is crucial for satisfying pore-scale deformation compatibility between the phases in the system and accurate modeling of fluid pressure response of the constituents to external excitation. The governing equations presented enjoy the major symmetry of the constitutive coefficients, a requirement of the existence of elastic potential. All model parameters are identified in terms of physically measurable entities.

Appendix A

[52] Alternatively, the cross-coupling terms introduced in equation (26) may be determined using the equivalent pressure concept, expressed in equation (7), and the definition of the volume fraction for each of the fluid constituents. To demonstrate this, we first satisfy the kinematic constraint

equation image

in which Sαβ is the degree of saturation of the fluid constituent β in pore space α. On the other hand

equation image

with

equation image

in which equation imagedVα represents the change in the pore degree of saturation due to a change in the volume of the pore at constant suction, and is related to the surface area of the air-water interface per unit volume of the porous medium. It has a value of the same order of magnitude as the degree of saturation.

[53] Substituting (A2) and (A3) into (A1) yields

equation image
equation image

with

equation image

[54] In (A4), dVα/V represents the change in the pore space, α, over the current volume of the fractured porous medium, which from the analysis of Khalili and Valliappan [1996] can be related to the change in the equivalent fluid pressure of pores and fracture, pαeq, as

equation image
equation image

[55] Now, substituting for dppeq and dpfeq in (A5) from (7) yields

equation image
equation image
equation image
equation image

By symmetry, we then obtain equation imagep = ψp. In a more compact form, (A6) may be written as

equation image

in which

equation image

In (A8), equation image represents the change in the degree of saturation of the pore space with suction, which can be obtained from a drying test performed on an elemental volume of the system.

[56] To show this for the fracture network, we first write

equation image

Then, using the drying test condition (dσ, dppw, dppa, dpfw, dpfa) = (−ds, 0, 0, 0, −ds), and equations (26) and (A5) we obtain

equation image
equation image

Substituting (A10) and (A11) into (A9), and noting Sαwβαβαw, yields

equation image

Similarly, for equation image we obtain

equation image

Using (A12) and (A13) in (A8) renders equation imageαβ,ζξ identical to aαβ,ζξ in (28).

Acknowledgments

[57] Financial support of the Australian Research Council (ARC) in the form of an ARC Discovery Project is gratefully acknowledged.

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