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Double-Porosity Transport

Pride and Berryman (2003b) obtain the internal transport coefficient of equation (9) as
 (21)
where the parameter that holds in the final stages of internal fluid-pressure equilibration is given by
 (22)
Since the more compressible embedded phase 2 typically has a permeability much greater than the host phase 1, the O(k1/k2) correction can be neglected. The transition frequency corresponds to the onset of a high-frequency regime in which the fluid-pressure-diffusion penetration distance between the phases becomes small relative to the scale of the mesoscopic heterogeneity. It is given by
 (23)
The length L1 characterizes the average distance in phase 1 over which the fluid pressure gradient still exists in the final approach to equilibration and has the formal mathematical definition
 (24)
where is the region of an averaging volume occupied by phase 1 and having a volume measure V1. The potential has units of length squared and is a solution of an elliptic boundary-value problem that under conditions where the harmonic mean is a good approximation for the overall drained modulus and where the permeability ratio k1/k2 can be considered small, reduces to
 (25) (26) (27)
where is the external surface of the averaging volume coincident with phase 1, and where is the internal interface separating phases 1 and 2. Multiplying equation (25) by and integrating over , establishes that second integral of equation (24).

For complicated geometry, L1 can only be determined numerically. For idealized geometries it can be analytically estimated. For example, if phase 2 is taken to be small spheres of radius a embedded within each sphere R of the composite, Pride and Berryman (2003b) obtain
 (28)
The volume fraction v2 of small spheres is then v2 = (a/R)1/3 which can be used to eliminate R since R=a v2-1/3. The other length parameter is the volume-to-surface ratio V/S where S is the area of in each volume V of composite. For the simple spherical-inclusion model, it is given by V/S=R3/(3a2)= a v2/3.

The coefficient governing shear generally has a non-zero viscosity'' associated with the mesoscopic fluid transport between the compressional lobes surrounding a sheared phase 2 inclusion. Both of the frequency functions and are real and are Hilbert transforms of each other. The frequency dependence of was not modeled by Pride and Berryman (2003b). However, if the inclusions of phase 2 are taken to be spheres, then exactly and is a constant that can be approximately modeled using a simple harmonic average 1/G = v1/G1 + v2/G2 of the underlying shear moduli of each phase.

Finally, the dynamic permeability to be used in the effective Biot theory can be modeled in several ways. Perhaps the simplest modeling choice when phase 2 is modeled as small inclusions embedded in phase 1 is to again take a harmonic average .

Next: Phase Velocity and Attenuation Up: REVIEW OF THE DOUBLE-POROSITY Previous: Double-Porosity aij Coefficients
Stanford Exploration Project
10/14/2003