In previous work (Berryman and Wang, 1995) chose
the external confining pressure, *p*_{c},
and the fluid pressures in the
matrix, *p ^{(1)}*, and in the fracture,

(

) =
(

)
(

)
The six coefficients occur in three classes, which
correspond to the three original Biot coefficients.
The coefficient *a _{11}* = 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,

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 *S*_{ij}'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 *p*_{c,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* = *a _{11}a_{22}a_{33}* -

4/20/1999