Drag coefficients

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 $\zeta^{(1)}$ and $\zeta^{(2)}$. These quantities are related to the displacements by $\zeta^{(1)} = -(1-v^{(2)})\phi^{(1)}\nabla\cdot({\bf U}^{(1)} - {\bf u})$and $\zeta^{(2)} = -v^{(2)}\phi^{(2)}\nabla\cdot({\bf U}^{(2)} - {\bf u})$.

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 $\eta$ 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 b₂₃ = 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.