Macroscopic Governing Equations

In the double-porosity formulation, two distinct phases are assumed to exist at the macroscopic level: (1) a porous matrix phase with the effective properties K⁽¹⁾, K_m⁽¹⁾, $\phi^{(1)}$ occupying volume fraction V⁽¹⁾/V = v⁽¹⁾ of the total volume and (2) a macroscopic crack or joint phase occupying the remaining fraction of the volume V⁽²⁾/V = v⁽²⁾ = 1 - v⁽¹⁾. In earlier work (Berryman and Pride 2002), methods were developed to determine the coefficients of this system within a set of specific modeling assumptions. But the general laws presented in this section are independent of all such modeling assumptions, and the analysis to be presented in later sections is also independent of them as well.

The main difference between the single-porosity and double-porosity formulations is that we allow the average fluid pressure in the matrix phase to differ from that in the joint phase (thus the term ``double porosity'') over relatively long time scales. Altogether we have three distinct pressures: confining (external) pressure $\delta p_c$,pore-fluid pressure $\delta p_f^{(1)}$, and joint-fluid pressure $\delta p_f^{(2)}$.(See Figure dblporpic.) Treating $\delta p_c, \delta p_f^{(1)},$ and $\delta p_f^{(2)}$ as the independent variables in the double-porosity theory, we define the dependent variables to be $\delta e \equiv \delta V/V$,$\delta\zeta^{(1)} = (\delta V_\phi^{(1)} - \delta V_f^{(1)})/V$,and $\delta\zeta^{(2)} = (\delta V_\phi^{(2)} - \delta V_f^{(2)})/V$,which are respectively the total volume dilatation, the increment of fluid content in the matrix phase, and the increment of fluid content in the joints. Finally, we assume that the fluid in the matrix is the same kind of fluid as that in the joints.

Linear relations among strain, fluid content, and pressure then take the general form  

$$\left(\begin{array} {c} \delta e - \delta\zeta^{(1)} - \del... ..._c - \delta p_f^{(1)} - \delta p_f^{(2)} \end{array}\right).$$ (15)

By analogy with the single-porosity result (all), it is easy to see that a₁₂ = a₂₁ and a₁₃ = a₃₁. The symmetry of the new off-diagonal coefficients may be demonstrated using Betti's reciprocal theorem in the form  

$$\left(\begin{array} {ccc} \delta e & - \delta\zeta^{(1)} & ... ...\begin{array} {c} 0 0 -\delta p_f^{(2)} \end{array}\right),$$ (16)

where nonoverlined quantities refer to one experiment and overlined to another experiment to show that  

$$\delta\zeta^{(1)}\delta\overline{p}_f^{(1)} = a_{23}\delta p... ...lta p_f^{(2)} = \delta\overline{\zeta}^{(2)}\delta p_f^{(2)}.$$ (17)

Hence, a₂₃ = a₃₂. Thus, we have established that the matrix in (generalstrainstress) is completely symmetric, so we need to determine only six independent coefficients.