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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.


next up previous print clean
Next: STRESS-STRAIN FOR SINGLE POROSITY Up: INERTIAL AND DRAG COEFFICIENTS Previous: Inertial coefficients
Stanford Exploration Project
8/21/1998