We now assume two distinct phases at the macroscopic level: a porous matrix phase with the effective properties K(1), ,Km(1), 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 k(22) of the crack or joint phase and the relatively lower permeability k(11) of the matrix phase. We could also introduce a third independent permeability k(12) = k(21) for fluid flow at the interface between the matrix and crack phases, but for simplicity we assume here that this third permeability is essentially the same as that of the matrix phase, so k(12) = k(11).
We have three distinct pressures: confining pressure ,matrix-fluid pressure ,and joint-fluid pressure .Treating and as the independent variables in our double porosity theory, we define the dependent variables (as before), ,and ,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. We assume that the fluid in the matrix is the same kind of fluid as that in the cracks or joints, but that 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 then take the general form
e - ^(1) - ^(2) = a_11 & a_12 & a_13 a_21 & a_22 & a_23 a_31 & a_32 & a_33 - p_c - p_f^(1) - p_f^(2) . By analogy with (all) and (allundrained), it is easy to see that a12 = a21 and a13 = a31. The symmetry of the new off-diagonal coefficients may be demonstrated by using Betti's reciprocal theorem in the form
e & - ^(1) & - ^(2) 0 -p_f^(1) 0 = e & -^(1) & -^(2) cr 0 0 -p_f^(2) , where unbarred quantities refer to one experiment and barred to another experiment to show that
^(1)p_f^(1) = a_23p_f^(2)p_f^(1) = a_32p_f^(1)p _f^(2) = ^(2)p_f^(2). Hence, a23 = a32. Similar arguments have often been used to establish the symmetry of the other off-diagonal components. Thus, we have established that the matrix in (generalstrainstress) is completely symmetric, so we need to determine only six independent coefficients. To do so, we consider a series of thought experiments, including tests in both the short time and long time limits. The key idea here is that at long times the two pore pressures must come to equilibrium (pf(1) = pf(2) = pf as ) as long as the cross permeability k(12) is finite. However, at very short times, we may assume that the process of pressure equilibration has not yet begun, or equivalently that k(12) = 0 at t = 0. We nevertheless assume that the pressure in each of the two components have individually equilibrated on the average, even at short times.