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In the absence of driving forces that can maintain pressure differentials over long time periods, double porosity models must reduce to single porosity models in the long time limit when the matrix pore pressure and crack pore pressure become equal. It is therefore necessary to remind ourselves of the basic results for single porosity models in poroelasticity. One important role these results play is to provide constraints for the long time behavior in the problems of interest. A second significant use of these results (see Berryman and Wang [1995]) arises when we make laboratory measurements on core samples having properties characteristic of the matrix material. Then the results presented in this section apply specifically to the matrix stiffnesses, porosity, etc.

For isotropic materials and hydrostatic pressure variations, the two independent variables in linear mechanics of porous media are the confining (external) pressure pc and the fluid (pore) pressure pf. The differential pressure $p_d \equiv p_c - p_f$ is often used to eliminate the confining pressure. The equations of the fundamental dilatations are then

-VV = p_dK + p_fK_s   for the total volume V,

- V_V_ = p_dK_p + p_fK_   for the pore volume $V_\phi= \phi V$, and

-V_fV_f = p_fK_f   for the fluid volume Vf. Equation (totalV) serves to define the various constants of the porous solid, such as the drained frame bulk modulus K and the unjacketed bulk modulus Ks for the composite frame. Equation (poreV) defines the jacketed pore modulus Kp and the unjacketed pore modulus $K_\phi$.Similarly, (fluidV) defines the bulk modulus Kf of the pore fluid.

Treating $\delta p_c$ and $\delta p_f$ as the independent variables in our poroelastic theory, we define the dependent variables $\delta e \equiv \delta V/V$and $\delta\zeta\equiv (\delta V_\phi- \delta V_f)/V$,both of which are positive on expansion, and which are respectively the total volume dilatation and the increment of fluid content. Then, it follows directly from the definitions and from (totalV), (poreV), and (fluidV) that

e - = 1/K & 1/K_s - 1/K - /K_p & (1/K_p + 1/K_f - 1/K_) - p_c - p_f .  

Now we consider two well-known thought experiments: the drained test and the undrained test [Gassmann, 1951; Biot and Willis, 1957; Geertsma, 1957]. (For a single porosity system, these two experiments are sometimes considered equivalent to the ``slow loading'' and ``fast loading'' limits respectively. However, these terms are relative since, for example, the fast loading -- equivalent to undrained -- limit is still assumed to be slow enough that the average fluid and confining pressures are assumed to have reached equilibrium.) The drained test assumes that the porous material is surrounded by an impermeable jacket and the fluid is allowed to escape through a tube that penetrates the jacket. Then, in a long duration experiment, the fluid pressure remains in equilibrium with the external fluid pressure (e.g., atmospheric) and so $\delta p_f = 0$and hence $\delta p_c = \delta p_d$;so the changes of total volume and pore volume are given exactly by the drained constants 1/K and 1/Kp as defined in (totalV) and (poreV). In contrast, the undrained test assumes that the jacketed sample has no passages to the outside world, so pore pressure responds only to in confining pressure changes. With no means of escape, the increment of fluid content cannot change, so $\delta\zeta= 0$.Then, the second equation in (defs) shows that

0 = -/K_p(p_c - p_f/B),   where Skempton's pore-pressure buildup coefficient B [Skempton, 1954] is defined by

B . p_fp_c|_ = 0   and is therefore given by

B = 11+K_p(1/K_f-1/K_).   It follows immediately from this definition that the undrained modulus Ku is determined by (also see Carroll [1980])

K_u = K1-B,   where we introduced the combination of moduli known as the Biot-Willis parameter $\alpha= 1 - K/K_s$. This result was apparently first obtained by Gassmann [1951] for the case of microhomogeneous porous media (i.e., $K_s = K_\phi= K_m$, the bulk modulus of the single mineral present) and by Brown and Korringa [1975] and Rice [1975] for general porous media with multiple minerals as constituents.

Finally, we condense the general relations from (defs) together with the reciprocity relations [Brown and Korringa, 1975] into symmetric form as

e - = 1K 1 & - -& /B - p_c - p_f .  

The storage compressibility, which is a central concept in describing poroelastic aquifer behavior in hydrogeology, is related inversely to one defined in Biot's original 1941 paper by

S . p_f|_p_c = 0 = BK.   This storage compressibility is the change in increment of fluid content per unit change in the fluid pressure, defined for a condition of no change in external pressure. It has also been called the three-dimensional storage compressibility by Kümpel [1991].

We may equivalently eliminate the Biot-Willis parameter $\alpha$ and write (all) in terms of the undrained modulus so that

e - = 1K 1 & - (1-K/K_u)/B -(1-K/K_u)/B & (1-K/K_u)/B^2 - p_c - p_f .   Equation (allundrained) has the advantage that all the parameters have very well defined physical interpretations, and are also easily generalized for a double porosity model. Finally, note that (all) shows that $K_p = \phi K/\alpha$, which we generally refer to as the reciprocity relation.

The total strain energy functional (including shear) for this problem may be written in the form

2E = _ije_ij + p_f ,   where $\delta e_{ij}$ is the change in the average strain with $\delta e_{ii}
\equiv \delta e$ being the dilatation, $\delta\tau_{ij}$ being the change in the average stress tensor for the saturated porous medium with ${1\over3}\delta\tau_{ii} = -\delta p_c$. It follows that

p_c = - E(e)   and

p_f = E(),   both of which are also consistent with Betti's reciprocal theorem [Love, 1927] since the matrices in (all) and (allundrained) are symmetric. The shear modulus $\mu$ is related to the bulk modulus and Poisson's ratio by $\mu= 3(1-2\nu)K/2(1+\nu)$. Then, it follows that the stress equilibrium equation is

_ij,j = (K_u+13)e_,i + u_i,jj - B K_u _,i = 0   and Darcy's law takes the form

kp_,ii = .,   where $\eta$ is the single-fluid shear viscosity.

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