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 $\zeta^{(1)} = -\phi^{(1)}\nabla\cdot({\bf U}^{(1)} - {\bf u})$and $\zeta^{(2)} = -\phi^{(2)}\nabla\cdot({\bf U}^{(2)} - {\bf u})$.

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, and the ks are permeabilities including possible cross-coupling terms. We can extract the terms we need from (finaleom), and then take the divergence to obtain

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)(k^(22)-k^(21)) k^(11)k^(22)-k^(12)k^(21),  

b_12 = ^(2)(k^(11)-k^(12)) k^(11)k^(22)-k^(12)k^(21),   and

b_23 = ^(1)k^(21) k^(11)k^(22)-k^(12)k^(21) = ^(2)k^(12) k^(11)k^(22)-k^(12)k^(21).   For many applications it will be adequate to assume that the cross-coupling vanishes. In this situation, b₂₃ = 0,

b_12 = ^(1)k^(11)   and

b_13 = ^(2)k^(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.