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The main results used here can be derived using uniform expansion, or self-similar, methods analogous to ideas used in thermoelasticity by Cribb (1968) and in single-porosity poroelasticity by Berryman and Milton (1991). Cribb's method provided a simpler and more intuitive derivation of Levin's earlier results on thermoelastic expansion coefficients (Levin, 1967). Our results also provide a simpler derivation of results obtained by Berryman and Pride (2002) for the double-porosity coefficients. Related methods in other applications to micromechanics are called ``the theory of uniform fields'' by some authors (Dvorak and Benveniste, 1997). First assume two distinct phases at the macroscopic level: a porous matrix phase with the effective properties Kd(1), Gd(1), Km(1), $\phi^{(1)}$ (which are drained bulk and shear moduli, grain or mineral bulk modulus, and porosity of phase 1 with analogous definitions for phase 2), occupying volume fraction V(1)/V = v(1) of the total volume and a macroscopic crack or joint phase occupying the remaining fraction of the volume V(2)/V = v(2) = 1 - v(1). The key feature distinguishing the two phases -- and therefore requiring this analysis -- is the very high fluid permeability of the crack or joint phase and the relatively lower permeability (but higher fluid volume content) of the matrix phase.

Figure 1
Schematic of the random polycrystals of laminates model.

Figure 2
Blowup showing a detail that illustrates how each one of the grains is composed of two very different types of porous materials: one being a storage material (high porosity and low permeability) and one a transport material (low porosity and high permeability).

In the double-porosity model, there are three distinct pressures: confining pressure $\delta p_c$,pore-fluid pressure $\delta p_f^{(1)}$ [for the storage porosity], and joint-fluid pressure $\delta p_f^{(2)}$ [for the transport porosity]. (See Figures 1 and 2.) Treating $\delta p_c, \delta p_f^{(1)},$ and $\delta p_f^{(2)}$ as the independent variables in our double porosity theory, I define the dependent variables $\delta e \equiv \delta V/V$,$\delta\zeta^{(1)} = (\delta V_\phi^{(1)} - \delta V_f^{(1)})/V$,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. The fluid in the matrix is the same as that in the cracks or joints, but the two fluid regions may be in different states of average stress and, therefore, need to be distinguished by their respective superscripts. Linear relations among strain, fluid content, and pressure take the symmetric form  
{c}\delta e \\  - \delta\zeta^{(1)} \\  ...
 ... - \delta p_f^{(1)} \\  - \delta p_f^{(2)}\end{array}\right),
 \end{displaymath} (1)
following Berryman and Wang (1995) and Lewallen and Wang (1998). It is easy to check that a11 = 1/Kd*, where Kd* is the overall drained bulk modulus of the system. I now find analytical expressions for the remaining five constants for a binary composite system. The components of the system are themselves porous materials 1 and 2, but each is assumed to be what I call a ``Gassmann material'' satisfying  
{c}\delta e^{(1)} \\  - \delta\zeta^{(1)...
 ...}- \delta p_c^{(1)} \\  - \delta p_f^{(1)} \end{array}\right)
 \end{displaymath} (2)
for material 1 and a similar expression for material 2. The new constants appearing on the right are the drained bulk modulus Kd(1) of material 1, the corresponding Biot-Willis (Biot and Willis, 1957) coefficient $\alpha^{(1)}$, and the Skempton (1954) coefficient B(1). The volume fraction v(1) appears here in order to correct for the difference between a global fluid content and the corresponding local variable for material 1. The main special characteristic of a Gassmann (1951) porous material is that it is composed of only one type of solid constituent, so it is ``microhomogeneous'' in its solid component, and in addition the porosity is randomly, but fairly uniformly, distributed so there is a well-defined constant porosity $\phi^{(1)}$ associated with material 1, etc. To proceed further, I ask this question: Is it possible to find combinations of $\delta p_c = \delta p_c^{(1)} = \delta p_c^{(2)}$,$\delta p_f^{(1)}$, and $\delta p_f^{(2)}$ so that the expansion or contraction of the system is spatially uniform or self-similar? Or equivalently, can I find uniform confining pressure $\delta p_c$, and pore-fluid pressures $\delta p_f^{(1)}$ and $\delta p_f^{(2)}$, so that all these scalar conditions can be met simultaneously? If so, then results for system constants can be obtained purely algebraically without ever having to solve equilibrium equations of the mechanics. I initially set $\delta p_c = \delta p_c^{(1)} = \delta p_c^{(2)}$, as this condition of uniform confining pressure is clearly a requirement for the self-similar thought experiment to be a valid solution of stress equilibrium equations. So, the first condition to be considered is the equality of the strains of the two constituents:  
\delta e^{(1)} = -\frac{1}{K_d^{(1)}}(\delta p_c - \alpha^{(...{1}{K_d^{(2)}}(\delta p_c - \alpha^{(2)} \delta p_f^{(2)}).
 \end{displaymath} (3)
If this condition is satisfied, then the two constituents are expanding or contracting at the same rate and it is clear that self-similarity prevails, since  
\delta e = v^{(1)}\delta e^{(1)} + v^{(2)}\delta e^{(2)} = 
\delta e^{(1)} = \delta e^{(2)}.
 \end{displaymath} (4)
If I imagine that $\delta p_c$ and $\delta p_f^{(1)}$ are fixed, then I need an appropriate value of $\delta p_f^{(2)}$, so that (3) is satisfied. This requires that  
\delta p_f^{(2)} =
\delta p_f^{(2)}(\delta p_c,\delta p_f^{(...
 ...alpha^{(1)}K_d^{(2)}}{\alpha^{(2)}K_d^{(1)}}\delta p_f^{(1)},
 \end{displaymath} (5)
showing that, for undrained conditions, $\delta p_f^{(2)}$ can almost always be chosen so the uniform expansion takes place. Using (5), I now eliminate $\delta p_f^{(2)}$ from the remaining equality so  
\delta e = -\left[a_{11}\delta p_c + a_{12}\delta p_f^{(1)} ...
 ...(1)}}\left[\delta p_c - \alpha^{(1)} \delta
 \end{displaymath} (6)
where $\delta p_f^{(2)}(\delta p_c,\delta p_f^{(1)})$ is given by (5). Making the substitution and then noting that $\delta p_c$ and $\delta p_f^{(1)}$ were chosen independently and arbitrarily, I find the resulting coefficients must each vanish. The two equations I obtain are  
a_{11} + a_{13}\left(1-K_d^{(2)}/K_d^{(1)}\right)/\alpha^{(2)} = 1/K_d^{(1)}
 \end{displaymath} (7)
a_{12} + a_{13}\left(\alpha^{(1)}K_d^{(2)}/\alpha^{(2)}K_d^{(1)}\right)
= -\alpha^{(1)}/K_d^{(1)}.
 \end{displaymath} (8)
Since a11 is assumed to be known, (7) can be solved directly, giving  
a_{13} = - \frac{\alpha^{(2)}}{K_d^{(2)}}
 \end{displaymath} (9)
Similarly, with a13 known, substituting into (8) gives  
a_{12} = -
 \end{displaymath} (10)
So, formulas for three of the six coefficients are now known. [Also, note the similarity of the formulas (9) and (10), i.e., interchanging indices 1 and 2 on the right hand sides takes us from one expression to the other.] To evaluate the remaining coefficients, I consider what happens to fluid increments during the self-similar expansion. I treat only material 1, but the equations for material 2 are completely analogous. From the preceding equations,  
\delta\zeta^{(1)} = a_{12}\delta p_c + a_{22} \delta p_f^{(1...
 ...)}\delta p_c +
(\alpha^{(1)}/B^{(1)})\delta p_f^{(1)}\right].
 \end{displaymath} (11)
Again substituting for $\delta p_f^{(2)}(\delta p_c,\delta p_f^{(1)})$ from (5) and noting that the resulting equation contains arbitrary values of $\delta p_c$ and $\delta p_f^{(1)}$, the coefficients of these terms must vanish separately. Resulting equations are  
a_{12} + a_{23}(1-K_d^{(2)}/K_d^{(1)})/\alpha^{(2)} = -
 \end{displaymath} (12)
a_{22} + a_{23}\left(\alpha^{(1)}K_d^{(2)}/\alpha^{(2)}K_d^{(1)}\right) 
= \alpha^{(1)}v^{(1)}/B^{(1)}K_d^{(1)}.
 \end{displaymath} (13)
Solving these equations, I obtain  
a_{23} = \frac{K_d^{(1)}K_d^{(2)}\alpha^{(1)}\alpha^{(2)}}
 ...)}} + \frac{v^{(2)}}{K_d^{(2)}}
- \frac{1}{K_d^*}\right],\,\,
 \end{displaymath} (14)
a_{22} = \frac{v^{(1)}\alpha^{(1)}}{B^{(1)}K_d^{(1)}}
- \lef...
 ...(1)}} + \frac{v^{(2)}}{K_d^{(2)}}
- \frac{1}{K_d^*}\right].\,
 \end{displaymath} (15)
Performing the corresponding calculation for $\delta\zeta^{(2)}$ produces formulas for a32 and a33. Since (14) is already symmetric in component indices, the formula for a32 provides nothing new. The formula for a33 is easily seen to be identical in form to a22, but indices 1 and 2 are interchanged. Formulas for all five of the nontrivial coefficients of double porosity have now been determined. These results also show how the constituent properties Kd, $\alpha$, B up-scale at the macrolevel for a two-constituent composite (Berryman and Wang, 1995; Berryman and Pride, 2002). I find  
\alpha = - \frac{a_{12}+a_{13}}{a_{11}}
= \frac{\alpha^{(1)}...
 \end{displaymath} (16)
\frac{1}{B} = - \frac{a_{22}+2a_{23}+a_{33}}{a_{12}+a_{13}}.
 \end{displaymath} (17)
Note that all the important formulas [(8),(9),(11)-(14)] depend on the overall drained bulk modulus Kd* of the system. So far this quantity is unknown and therefore must still be determined independently either by experiment or by another analytical method. It should also be clear that some parts (but not all) of the preceding analysis generalize to the multi-porosity problem (i.e., more than two porosity types). A discussion of the issues surrounding solvability of the multiporosity problem has been presented elsewhere (Berryman, 2002).

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