In previous work (Berryman and Wang, 1995) chose the external confining pressure, pc, and the fluid pressures in the matrix, p(1), and in the fracture, p(2), to be the independent variables. The dependent variables were chosen to be the volumetric strain, e, and Biot's increment of fluid content (fluid volume accumulation per unit bulk volume) in the matrix, , and fracture, ,separately. The phenomenological approach then relates each dependent variable linearly to the independent variables. This choice of variables leads to a symmetric coefficient matrix because the scalar product of the dependent and independent variables is an energy density. The double-porosity theory with six independent coefficients, aij, for hydrostatic loading is a straightforward generalization of Biot's original equations.
) = (
) The six coefficients occur in three classes, which correspond to the three original Biot coefficients. The coefficient a11 = 1/K is an effective compressibility of the combined fracture-matrix system. The coefficients a12 and a13 are generalized poroelastic expansion coefficients, i.e., the ratio of bulk strain to matrix pressure and fracture pressure, respectively. The terms a22, a23 , a32=a23, and a33 are generalized storage coefficients, i.e., aij is the volume of fluid that flows into a control volume (normalized by the control volume) of phase i-1 due to a unit increase in fluid pressure in phase j-1.
Formulas relating these parameters to properties of the constituents are summarized in TABLE 1, in which values are used for Berea sandstone from the results tabulated in TABLE 2. The definitions of the input parameters are: K and K(1) are the (jacketed) frame bulk moduli of the whole and the matrix respectively, Ks and Ks(1) are the unjacketed bulk moduli for the whole and the matrix, and are the corresponding Biot-Willis parameters, Kf is the pore fluid bulk modulus, v(2) = 1-v(1) is the total volume fraction of the fractures in the whole, and B(1) is Skempton's pore-pressure buildup coefficient for the matrix. Poisson's ratio and the porosity of the matrix are and , respectively. It was observed by Berryman and Wang (1995) that the fluid-fluid coupling term a23 was small or negligible for the examples considered, and that it is expected to be small or negligible in most situations in which it makes sense to use the double-porosity model at all. Since our main goal for this paper is to extract and evaluate a somewhat simplified version of these formulas, from the more general analysis presented so far, we will therefore make the approximation in the remainder of this paper that . This physically reasonable choice will also make the subsequent analysis somewhat less tedious.
We need to express the vector on the right hand side of (fouriertime) in terms of the macroscopic variables, using the constitutive relations in (compliances), plus the usual relations of linear elasticity. The basic set of equations for assumed isotropic media [analogous expressions for single-porosity with and without elastic anisotropy are given in Berryman (1998)] has the form
S_11 & S_12 & S_12 & -^(1) & -^(2) & & & S_12 & S_11 & S_12 & -^(1) & -^(2) & & & S_12 & S_12 & S_11 & -^(1) & -^(2) & & & -^(1) & -^(1) & -^(1) & a_22 & a_23 & & & -^(2) & -^(2) & -^(2) & a_23 & a_33 & & & & & & & & 12 & & & & & & & & 12 & & & & & & & & 12 _11 _22 _33 -p^(1) -p^(2) _23 _31 _12 = e_11 e_22 e_33 -^(1) -^(2) e_23 e_31 e_12 , where the Sij's are the usual drained elastic compliances, and the 's are poroelastic expansion coefficients approximately of the form , where is the Biot-Willis parameter (Biot-Willis, 1957) for single-porosity and K is the drained bulk modulus. We will not show our work here, but it is not hard to derive the following three relations:
_ij,j = (+ )e_,i + u_i,jj -3K[^(1)p_,i^(1) + ^(2)p_,i^(2)],
-3K[^(1)p^(1)_,i+^(2)p^(2)_,i] = -p_c,i - Ke_,i, and
-3[^(1)p_,i^(1) + ^(2)p_,i^(2)] = B^(1)[-^(1)_,i - 3^(1)p_c,i] + B^(2)[-^(2)_,i - 3^(2)p_c,i]. Appearing in (taustress) are and , which are the Lamé parameters for the drained medium. A linear combination of the last two equations can be found to eliminate the appearance of pc,i, and then this result can be substituted into (tau) to show that
_ij,j = (K_u + 13)e_,i + u_i,jj +K_u[B^(1)^(1)_,i + B^(2)^(2)_,i], where
K_u = K1 - 3K[^(1)B^(1)+^(2)B^(2)] is the undrained bulk modulus for the double porosity medium -- specifically, it is the undrained bulk modulus for intermediate time scales, i.e., undrained at the representative elementary volume (REV) scale but equilibrated between pore and fracture porosity locally. (This statement is consistent with our assumption that ,but needs some qualification if .)
Combining (stressijj) with (compliances) and taking the divergence, we finally obtain the expression we need:
_ij,ji - p_,ii^(1) - p_,ii^(2) = K_u + 43& B^(1)K_u & B^(2)K_u -a_12a_33/D & (a_11a_33-a_13^2)/D & a_12a_13/D
-a_13a_22/D & a_12a_13/D & (a_11a_22-a_12^2)/D e_,ii -^(1)_,ii -^(2)_,ii , where D = a11a22a33 - a122a33 - a132a22.