Double-Porosity a_ij Coefficients

The constants a_ij are all real and correspond to the high-frequency response for which no internal fluid-pressure relaxation can take place. They are given exactly as (Pride and Berryman, 2003a)

$$\begin{eqnarray} a_{11}& =& {1}/{K} \\ a_{22}&=& \frac{v_1 \alpha_1}{K^d_1} \lef... ...} \left(\frac{1}{K} - \frac{v_1}{K^d_1} - \frac{v_2}{K^d_2}\right)\end{eqnarray}$$ (14) (15) (16) (17) (18) (19)

where the Q_i are auxiliary constants given by

$$v_1 Q_1 = \frac{1-K^d_2/K}{1-K^d_2/K^d_1} \mbox{\hskip3mm and \hskip2mm} v_2 Q_2 = \frac{1-K^d_1/K}{1-K^d_1/K^d_2}.$$ (20)

Here, v₁ and v₂ are the volume fractions of each phase within an averaging volume of the composite. The one constant that has not yet been defined is the overall drained modulus K=1/a₁₁ of the two-phase composite (the modulus defined in the quasi-static limit where the local fluid pressure throughout the composite is everywhere unchanged). It is through K that the a_ij potentially depend on the mesoscopic geometry of the two porous phases. However, a reasonable modeling choice when phase 2 is embedded within phase 1 is to simply take the geometry-independent harmonic mean 1/K = v₁/K^d₁ + v₂/K^d₂. Although this choice actually violates the Hashin-Shitrikman bounds (Hashin and Shtrikman, 1961) for truly isotropic media, it is nevertheless a reasonable choice for earth systems where the assumed isotropy is itself an approximation. This choice is also a particularly convenient one because it results in Q₁=Q₂=1 as well as a₂₃=0. All dependence on the fluid's bulk modulus is contained within the two Skempton's coefficients B₁ and B₂ and is thus restricted to a₂₂ and a₃₃. In the quasi-static limit $\omega\rightarrow 0$ (fluid pressure everywhere uniform throughout the composite), equations (12) and (13) reduce to the known exact results of Berryman and Milton (1991) once equations (14)-(19) are employed.