Double-Porosity Transport

Pride and Berryman (2003b) obtain the internal transport coefficient $\gamma$ of equation (9) as

$$\gamma(\omega) = {\gamma_m}\sqrt{1- i \frac{\omega}{\omega_m}}$$ (21)

where the parameter $\gamma_m$ that holds in the final stages of internal fluid-pressure equilibration is given by

$$\gamma_m = \frac{v_1 k_1}{\eta L_1^2} \left[1 + O(k_{1}/k_{2})\right].$$ (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 $\omega_m$ 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  

$$\omega_m = \frac{ B_1 K^d_1}{\eta \alpha_1}\frac{k_1 (v_1 V/... ...frac{k_1 B_2 K^d_2 \alpha_1}{k_2 B_1 K^d_1 \alpha_2}}\right)^2.$$ (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  

$$L_1^2 = \frac{1}{V_1} \int_{\Omega_1} \Phi_1 \, dV =\frac{1}{V_1} \int_{\Omega_1} \nabla \Phi_1 \cdot \nabla \Phi_1 \, dV$$ (24)

where $\Omega_1$ is the region of an averaging volume occupied by phase 1 and having a volume measure V₁. The potential $\Phi_1$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

$$\begin{eqnarray} \nabla^2 \Phi_1 &=& -1 \mbox{ \hskip1mm in } \Omega_1, \\ {\bf ... ...tial E_1, \\ \Phi_1&=&0 \mbox{\hskip1mm on } \partial \Omega_{12},\end{eqnarray}$$ (25) (26) (27)

where $\partial E_1$ is the external surface of the averaging volume coincident with phase 1, and where $\partial \Omega_{12}$ is the internal interface separating phases 1 and 2. Multiplying equation (25) by $\Phi_1$ and integrating over $\Omega_1$, 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

$$L_1^2 = \frac{9}{14} R^2\left[1 - \frac{7}{6} \frac{a}{R} + O(a^3/R^3) \right].$$ (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 $\partial \Omega_{12}$ 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 $G(\omega) - i \omega g (\omega)$ governing shear generally has a non-zero ``viscosity'' $g(\omega)$ associated with the mesoscopic fluid transport between the compressional lobes surrounding a sheared phase 2 inclusion. Both of the frequency functions $G(\omega)$ and $-\omega g({\omega})$ are real and are Hilbert transforms of each other. The frequency dependence of $g(\omega)$ was not modeled by Pride and Berryman (2003b). However, if the inclusions of phase 2 are taken to be spheres, then $g(\omega)=0$ exactly and $G(\omega)=G$ 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 $k(\omega)$ 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 $1/k(\omega) = v_1/k_1(\omega) + v_2/k_2(\omega) \approx v_1/k_1(\omega) [1 + O(v_2 k_1/k_s)]$.