next up previous print clean

Patchy-Saturation aij Coefficients

To obtain the aij 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 aij 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 aij are defined under the condition that $\dot{\zeta}_{\rm int}=0$), one has

\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)
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
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)
will equations (37)-(39) produce aij that satisfy the thermodynamic symmetry requirement of aij = aji [i.e., these aij 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
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)
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 Kiu), the theorem of Hill (1963) applies, which states that the overall undrained-unrelaxed modulus of the composite KH 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}.\end{displaymath} (48)
In terms of the aij, 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}\end{displaymath} (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
\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)
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]\end{displaymath} (52)
with KH given by equation (48). All the aij are now expressed in terms of known information.

next up previous print clean
Stanford Exploration Project