SINGLE-POROSITY GEOMECHANICS

In the absence of external driving forces that can maintain fluid-pressure differentials over long time periods, double-porosity and multi-porosity models must all reduce to single-porosity models. This reduction occurs in the long-time limit when the matrix fluid pressure and joint fluid pressure become equal. It is therefore necessary to remind ourselves of the basic results for single-porosity models in poroelasticity (Biot 1941; Detournay and Cheng 1993; Wang 2000), as the long-time behavior may be viewed as providing limiting temporal boundary conditions (for $t \to \infty$)on the analysis of multi-porosity coefficients. Further, in the specific models we adopt for the geomechanical constants in the multi-porosity theory, extensive use of the single-porosity results will be made.

The volume changes of any isothermal, isotropic material can only be created by hydrostatic pressure changes. The two fundamental pressures of single-porosity poroelasticity are the confining (external) pressure p_c and the fluid (pore) pressure p_f. The differential pressure (or Terzaghi effective stress) $p_d \equiv p_c - p_f$ is often used instead of the confining pressure. The volumetric response of a sample due to small changes in p_d and p_f take the form [e.g., Brown and Korringa (1975)]  

$$-{{\delta V}\over{V}} = {{\delta p_d}\over{K^*}} + {{\delta p_f}\over{K_s}}$$ (2)

for the total volume V,  

$$- {{\delta V_\phi}\over{V_\phi}} = {{\delta p_d}\over{K_p}} + {{\delta p_f}\over{K_\phi}}$$ (3)

for the pore volume $V_\phi= \phi V$ (where $\phi$ is the porosity), and  

$$-{{\delta V_f}\over{V_f}} = {{\delta p_f}\over{K_f}}$$ (4)

for the fluid volume V_f. Equation (totalV) serves to define the drained (or ``jacketed'') frame bulk modulus K^* and the unjacketed bulk modulus K_s for the composite frame. Equation (poreV) defines the jacketed pore modulus K_p and the unjacketed pore modulus $K_\phi$.Similarly, (fluidV) defines the bulk modulus K_f of the pore fluid.

Treating $\delta p_c$ and $\delta p_f$ as the independent variables, we define the dependent variables to be $\delta e \equiv \delta V/V$ and $\delta\zeta\equiv (\delta V_\phi- \delta V_f)/V$, which are termed respectively the total volume dilatation (positive when a sample expands) and the increment of fluid content (positive when the net fluid mass flow is into the sample during deformation). Then, it follows directly from these definitions and from (totalV), (poreV), and (fluidV) that  

$$\left(\begin{array} {c} \delta e -\delta\zeta \end{array}\... ...in{array} {c} - \delta p_c - \delta p_f \end{array}\right).$$ (5)

Now we consider two well-known thought experiments: the drained test and the undrained test (Gassmann 1951; Biot and Willis 1957; Geertsma 1957; Wang, 2000). In the drained test,the porous material is surrounded by an impermeable jacket and the fluid is allowed to escape through a conduit penetrating the jacket. Then, in a long duration experiment, the fluid pressure remains in equilibrium with the external fluid pressure (e.g., atmospheric) and so $\delta p_f = 0$. Hence, $\delta p_c = \delta p_d$.So changes of total volume and pore volume are given by the drained constants 1/K^* and 1/K_p as defined in (totalV) and (poreV). In contrast, for the undrained test, the jacketed sample has no connection to the outside world, so pore pressure responds only to the confining pressure changes. With no way out, the total fluid content cannot change, so the increment $\delta\zeta= 0$.Then, the second equation in (defs) shows that  

$$0 = -\phi/K_p(\delta p_c - \delta p_f/B),$$ (6)

where Skempton's pore pressure buildup coefficient B (Skempton 1954) is defined by  

$$B \equiv \left. {{\delta p_f}\over{\delta p_c}}\right\vert _{ \delta\zeta= 0} = {{1}\over{1+K_p(1/K_f-1/K_\phi)}}.$$ (7)

It follows immediately from this definition that the undrained modulus K_u is determined by [also see Carroll (1980)]  

$$K_u = {{K^*}\over{1-\alpha B}},$$ (8)

where $\alpha$ is the combination of moduli known as the Biot-Willis parameter, or the total volume effective-stress coefficient. The precise definition of $\alpha$ follows immediately from the form of (totalV), by substituting $\delta p_d = \delta p_c - \delta p_f$and rearranging the equation into the form  

$$-{{\delta V}\over{V}} = {{\delta p_c - \alpha\,\delta p_f}\over{K^*}},$$ (9)

with $\alpha= 1 - K^*/K_s$.The result (Gassmann) was apparently first obtained by Gassmann (1951) (though not in this form) 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. We will sometimes use the term ``Gassmann material'' when making reference to a microhomogeneous porous medium.

Next, to clarify the structure of (defs) further, note that Betti's reciprocal theorem (Love 1927), shows that the drained and undrained pressures and strains satisfy a reciprocal relation, from which it follows that  

$${1\over{K_u}} = {1\over{K^*}} - {{\phi B}\over{K_p}}.$$ (10)

Comparing (Gassmann) with (reciprBetti), we obtain the general reciprocity relation (Brown and Korringa 1975)  

$${{\phi}\over{K_p}} = {{\alpha}\over{K^*}}.$$ (11)

This reciprocity relation and the form of the compressibility laws (defs) also follow directly from general thermodynamic arguments [e.g., Pride and Berryman (1998)]. Then, Skempton's pore-pressure buildup coefficient (Skempton 1954) may be written alternatively as  

$$B = {{1/K^*-1/K_s}\over{1/K^*-1/K_s + \phi(1/K_f-1/K_\phi)}}.$$ (12)

Finally, the condensed form of (defs) -- incorporating the reciprocity relations -- is  

$$\left(\begin{array} {c} \delta e - \delta\zeta \end{array}... ...gin{array} {c} -\delta p_c - \delta p_f \end{array}\right),$$ (13)

where the Biot-Willis (1957) parameter $\alpha$ can now be expressed as  

$$\alpha= (1-K^*/K_u)/B.$$ (14)

The parameter $\alpha$ is also known as the total volume effective-stress coefficient [see Berryman (1992) for elaboration]. This form of the compressibility laws is especially convenient because all the coefficients are simply related to the three moduli K^*, K_u, and B that have the clearest physical interpretations. This now completes our review of the standard results concerning the single-porosity compressibility laws.