Patchy-Saturation a_ij Coefficients

To obtain the a_ij for the patchy-saturation model, we note that each patch has the same $\alpha$ and K. The poroelastic differences between patches is entirely due to B₁ being different than B₂. Upon volume averaging equation (3) and using $\nabla \cdot {\bf v} = \nabla \cdot \left(v_1 \dot{\overline{\bf u}}_1\right) + \nabla \cdot \left(v_2 \dot{\overline{\bf u}}_2\right)$, where an overline again denotes a volume average over the appropriate phase, and using the fact that the a_ij are defined in the extreme high-frequency limit where the fluids have no time to traverse the internal interface $\partial \Omega_{12}$ (i.e., the a_ij are defined under the condition that $\dot{\zeta}_{\rm int}=0$), one has

$$\begin{eqnarray} \nabla \cdot {\bf v}\, &=& - \frac{v_1}{K} \dot{\overline p}_{c... ...erline p}_{c2} - \frac{v_2 \alpha}{K B_2} \dot{\overline p}_{f2}.\end{eqnarray}$$ (37) (38) (39)

The average confining pressures $\overline{p}_{ci}$ in each phase are not a priori known; however, they are necessarily linear functions of the three independent applied pressures of the theory $P_c (= v_1 \overline{p}_{c1} + v_2 \overline{p}_{c2})$, $\overline{p}_{f1}$, and $\overline{p}_{f2}$. It is straightforward to demonstrate that if and only if the average confining pressures take the form

$$\begin{eqnarray} v_1 \dot{\overline p}_{c1} &=& v_1 \dot{P}_{c} + \beta \dot{\ov... ...{c} - \beta \dot{\overline p}_{f1} + \beta \dot{\overline p}_{f2},\end{eqnarray}$$ (40) (41)

will equations (37)-(39) produce a_ij that satisfy the thermodynamic symmetry requirement of a_ij = a_ji [i.e., these a_ij constants are all second derivatives of a strain-energy function as demonstrated by Pride and Berryman (2003a). Upon placing equations (40) and (41) into equations (37)-(39), we then have

$$\begin{eqnarray} a_{11} &=& {1}/{K} \\ a_{22} &=& \left (-\beta + {v_1}/{B_1}\ri... ... \\ a_{13} &=& - v_2 {\alpha}/{K} \\ a_{23} &=& \beta {\alpha}/{K}\end{eqnarray}$$ (42) (43) (44) (45) (46) (47)

where $\beta$ is a constant to be determined.

To obtain $\beta$, we note that in the high-frequency limit, each local patch of phase i is undrained and thus characterized by an undrained bulk modulus $K^u_{i} = K/(1-\alpha B_i)$ and a shear modulus G that is the same for all patches. In this limit, the usual laws of elasticity govern the response of this heterogeneous composite. Under these precise conditions (elasticity of an isotropic composite having uniform G and all heterogeneity confined to the bulk modulus which in the present case corresponds to K_i^u), the theorem of Hill (1963) applies, which states that the overall undrained-unrelaxed modulus of the composite K_H is given exactly by  

$$\frac{1}{K_{ H} + 4G/3} = \frac{v_1}{K^u_{1}+ 4G/3} + \frac{v_2}{K^u_{2} + 4G/3}.$$ (48)

In terms of the a_ij, this same undrained-unrelaxed Hill modulus is given by  

$$\frac{1}{K_{H}} = a_{11} + a_{12} \left(\frac{\delta {p}_{f1... ...a_{13} \left(\frac{\delta {p}_{f2}}{\delta {P}_c}\right)_{\! U}$$ (49)

where, upon using $\nabla \cdot {\bf q}_i =0$ and $\dot{\zeta}_{\rm int}=0$ in equation (8) and then using (42)-(47), the undrained-unrelaxed pressure ratios are

$$\begin{eqnarray} \left(\frac{\delta {p}_{f1}}{\delta {P}_c}\right)_{\! U} &=& \f... ...beta - v_1 v_2/B_1}{\beta(v_1/B_1 + v_2/B_2) - v_1 v_2/(B_1 B_2)}.\end{eqnarray}$$ (50) (51)

Thus, after some algebra, equation (49) yields the exact result

$$\beta =v_1 v_2 \!\!\left(\frac{v_1 }{ B_2}+\frac{v_2}{B_1}\r... ...v_2 B_2)} {\alpha - (1 - K/K_{H})(v_1/ B_1 + v_2/ B_2)} \right]$$ (52)

with K_H given by equation (48). All the a_ij are now expressed in terms of known information.