The drag coefficients may be determined by first noting that the equations presented here reduce to those of Berryman and Wang (1995) in the low frequency limit by merely neglecting the inertial terms. What is required to make the direct identification of the coefficients is a pair of coupled equations for the two increments of fluid content and . These quantities are related to the displacements by and .

The pertinent equations from Berryman and Wang (1995) are

.^(1).^(2) =
k^(11) & k^(12)
k^(21) & k^(22)
p_,ii^(1) p_,ii^(2) ,
where is the shear viscosity of the fluid, the *k*^{(ij)} are
permeabilities including possible cross-coupling terms. The
pressures appearing here are the actual pore pressures in the storage
and fracture porosity.
We can extract the terms we need from (finaleom), and then take
the divergence to obtain

1/(1-v^(2))^(1) & 0
0 & 1/v^(2)^(2)
b_12+b_23 & - b_23
- b_23 & b_13+b_23
(B<>U^(1) - B<>u)
(B<>U^(2) - B<>u) =
- p_,ii^(1) p_,ii^(2) .
Comparing these two sets of equations and solving for the
*b* coefficients, we find

b_12 = (1-v^(2))^(1)[(1-v^(2))^(1)k^(22) -v^(2)^(2)k^(21)] k^(11)k^(22)-k^(12)k^(21),

b_13 = v^(2)^(2)[v^(2)^(2)k^(11) -(1-v^(2))^(1)k^(12)] k^(11)k^(22)-k^(12)k^(21), and

b_23 = v^(2)(1-v^(2))^(1)^(2)k^(21)
k^(11)k^(22)-k^(12)k^(21)
= v^(2)(1-v^(2))^(1)^(2)k^(12)
k^(11)k^(22)-k^(12)k^(21).
For wave propagation, it will often be adequate to assume that the
cross-coupling vanishes, as this effect is presumably more important
for long term drainage of fluids than it is for short term
propagation of waves. When this approximation is satisfactory,
we have *b _{23}* = 0, and

b_12 = (1-v^(2))^2(^(1))^2k^(11) and

b_13 = (v^(2)^(2))^2k^(22), which also provides a simple interpretation of these coefficients in terms of the porosities and diagonal permeabilities.

This completes the identification of the inertial and drag coefficients introduced in the previous section.

4/20/1999