INTRODUCTION

The subject of ``geomechanics'' includes such topics as the study of rock mechanics, soil mechanics, and engineering geology, and has overlapping interests in some cases with ``hydrogeology'' when the mechanical behavior of the earth system of interest is strongly affected by the presence of water. In general ``geomechanics'' means the study mechanics of earth systems, and therefore ``microgeomechanics'' means the study of the effects of micromechanics on earth systems. Our main interest here will be in the interaction of fluid pressure changes (usually induced by reservoir depletion) with the mechanical properties of the reservoir.

Perhaps the most typical applications of geomechanics arise in the engineering disciplines of mining and oil reservoir assessment and production, and even earlier in soil mechanics. The history of the main features contained in the theory of geomechanics dates back at least to the work of Terzaghi (1925) on ``effective stress,'' which is the observation that, when external confining stress and internal pore pressure act simultaneously on a porous material, the pore pressure tends to counteract the confining pressure. Terzaghi's effective stress law for changes in volume was the simple statement that the effective stress was the confining stress minus the pore pressure, i.e. the differential stress. For soils, this approximation is often a very good one. For porous materials in general, theory and experiment have shown both that the concept of an effective stress is valid, and that the actual effective stress is not just the differential stress of Terzaghi, but rather the confining stress minus some fraction of the pore pressure. This fraction has often been taken to be some overall average number -- for example, in the range 0.85-0.88 (Brandt, 1955; Schopper, 1982) -- for the earth's crust. But it has been shown theoretically by Biot and Willis (1957) and experimentally by Fatt (1958; 1959) and Nur and Byerlee (1971) that the volume effective stress coefficient is actually related to the elastic properties of the porous system. If $\alpha$ is this effective stress coefficient, then -- in a microhomogeneous porous material (composed of voids and a single type of granular material) -- it is related to the bulk modulus of the grains K_g and the overall bulk modulus of the drained porous system K^* by $\alpha = 1 - K^*/K_g$. This rule reduces to Terzaghi's effective stress rule for soils when the soil is very poorly consolidated so that K^* << K_g, for then $\alpha \simeq 1$.

Terzaghi's early work was expanded into a theory of consolidation, both by himself and through the work of Biot (1941), Gassmann (1951), Skempton (1957), Geerstma (1966; 1973), and many others. Biot (1941) is usually given credit for the first comprehensive theory of consolidation, at least in the case of simple, single porosity systems. Gassmann (1951) was the first to obtain one of the fundamental results of the theory - sometimes called the fluid-substitution formula, relating the dry or drained bulk modulus K^* to the undrained (or saturated) modulus K_u by $K_u = K^*/(1-\alpha B)$, where the pore-pressure buildup coefficient B is Skempton's second coefficient (Skempton, 1954; Carroll 1980). Geertsma (1966; 1973) was one of the earliest users of this theory for studies of compaction of oil and gas reservoirs during and after drawdown of the reservoir. Wang (2000) also gives Geerstma (1966) credit for coining the term ``poroelasticity'' that is normally used today to describe Biot's theory in full. Early laboratory measurements (Biot and Willis 1957; Fatt 1958; 1959) of the constants in Biot's equations helped to establish the theory. Early engineering solutions of the equations of poroelasticity were given by Rice and Cleary (1976) and Cleary (1977) which helped to make it a standard tool in civil engineering. Some fundamental extensions of the theory to systems having multiple solid constituents have been given by Brown and Korringa (1975), Rice (1975), Berryman and Milton (1991), Berryman (1992), and Norris (1992). Poroelasticity is now a well-established subject having recent technical reviews by Detournay and Cheng (1993), Wang (1993), Pride and Berryman (1998), Berryman (1999), and books by Bourbié, Coussy and Zinzner (1987) and Wang (2000) describing the current state of our understanding.

Another fascinating use of the theory of poroelasticity in reservoirs is its relatively recent application to the studies of earthquakes induced by oil and gas reservoir pumping (Kovach 1974; Pennington et al. 1986; Segall 1985; 1989; 1992; Segall and Fitzgerald 1998; Gomberg and Wolf 1999; Pennington 2001). A related issue arising in the opposite physical extreme is the subject of CO₂ sequestration in the earth (Wawersik et al. 2001), where it is clear that pumping pressurized fluids into the ground must have a strong tendency to decrease the effective stress in the earth system used for sequestration. Decreasing effective stress implies weakening of the system, resulting in undesirable (for this application) increases in fluid permeability. Studies of partially saturated systems are also of continuing interest (Li, Zhong, and Pyrak-Nolte 2001) both for oil and gas exploitation and for environmental cleanup applications.

Biot's original single-porosity, microhomogeneous theory of poroelasticity has significant limitations when the porous medium of interest is very heterogeneous. One important generalization of poroelasticity that has been studied extensively started with the work on double-porosity dual-permeability systems by Barenblatt and Zheltov (1960) and Warren and Root (1963). These papers take explicit note of the fact that real reservoirs tend to be very heterogeneous in both their porosity and permeability characteristics. In particular, the two types of porosity normally treated are storage and transport porosities. Storage porosity holds most of the volume of the fluid underground but may have rather low permeability, while the transport porosity is low volume but high permeability. The transport porosity is usually treated as being in the form of fractures in the reservoir, or joints in the rock mass. The theory of double-porosity dual-permeability media has been expanding in both volume and scope during the last 20 years, and now includes work by Wilson and Aifantis (1983), Elsworth and Bai (1992), Bai, Elsworth, and Roegiers (1993), Berryman and Wang (1995), Tuncay and Corapciaglu (1995), Bai (1999), and Berryman and Pride (2002). Computations of transport and subsidence in double-porosity dual-permeability media include work by Khaled, Beskos, and Aifantis (1984), Nilson and Lie (1990), Cho, Plesha, and Haimson (1991), Lewallen and Wang (1998), and Bai, Meng, Elsworth, Abousleiman, and Roegiers (1999).

Some technical details follow on the single-porosity poroelasticity needed in the main arguments of the paper. Then equations are formulated for double-porosity systems, and finally multi-porosity systems are discussed. The focus will be on determining how the coefficients of the resulting equations depend on the physical properties of the microstructural constituents' of these complex geomechanical systems. The main results are obtained using new techniques in micromechanics that permit a rather elementary analysis of these complex systems to be carried through exactly. For systems containing two porosities and two types of solid constituents, exact results for all but one (which may be taken as the overall drained bulk modulus of the system) of the macroscopic geomechanical constants are derived.