The approach that we take instead
is to first reduce these double-porosity laws (6)-(10) to an effective
single-porosity Biot
theory having complex frequency-dependent coefficients. The easiest
way to do this is to assume that phase 2 is entirely embedded in phase 1 so that
the average flux into and out of the averaging volume across the external
surface of phase 2 is zero. By
placing into the compressibility laws (8),
the fluid pressure *p*_{f2} can be entirely eliminated from the theory.
In this case the double-porosity laws reduce to effective single-porosity
poroelasticity governed by laws of the form
(3) but with effective poroelastic
moduli given by

(11) | ||

(12) | ||

(13) |

The complex frequency dependent ``drained''
modulus *K*_{D} again defines the total volumetric response
when the
average fluid pressure throughout the entire composite is unchanged; however, the
local fluid pressure in each phase may be non-uniform even though the average is zero
resulting in mesoscopic
flow and in *K*_{D} being complex and frequency dependent. Similar interpretations
hold for the undrained moduli *K*_{U} and *B*.
An undrained response is when no fluid can escape or enter through the
external surface of an averaging volume;
however, there can be considerable internal exchange of fluid between the
two phases resulting in the complex frequency-dependent nature of both
*K*_{U} and *B*.

10/14/2003