Earth materials composing either aquifers or oil and gas reservoirs are generally heterogeneous, porous, and often fractured or cracked. Distinguishing water, oil, and gas using seismic signatures is a key issue in seismic exploration and reservoir monitoring. Traditional approaches to seismic monitoring have often used Biot's theory of poroelasticity (Biot, 1941, 1956a,b, 1962; Gassmann, 1951). Many of the predictions of this theory, including the existence of the slow compressional wave, have been confirmed by both laboratory and field experiments (Plona, 1980; Berryman 1980a; Johnson et al., 1982; Chin et al., 1985; Winkler, 1985; Pride and Morgan, 1991; Thompson and Gist, 1993; Pride, 1994). Nevertheless, this theory always has been limited by an explicit assumption that the porosity itself is homogeneous. Although this assumption is often applied to acoustic or ultrasonic studies of many core samples in the laboratory setting, heterogeneity of porosity still exists in the form of both pores and cracks. One approach to dealing with this source of heterogeneity is to construct a model that is locally homogeneous (i.e., a finite element). This approach may be adequate for some applications, and is certainly amenable to study with large computers. However, such methods necessarily avoid the question of how we are to deal with heterogeneity on the local scale (i.e., much smaller than the block size or wavelength in the cases being studied). Double porosity models have been introduced as a means of dealing with these problems. Rather than trying to deal with all the heterogeneity at once, we choose to consider a model intended to capture two main features of importance. Just two types of porosity are often key at the reservoir scale: (1) Matrix porosity occupies a finite and substantial fraction of the volume of the reservoir. This porosity is often called the storage porosity since it stores the fluids of interest. (2) Fracture or crack porosity may occupy very little volume overall, but nevertheless has two very big effects on reservoir behavior. First the fractures/cracks drastically weaken the rock mechanically, so that a change in a very low effective stress level may introduce nonlinear geomechanical responses. The second effect is that fractures/cracks introduce a fast pathway for the fluid to escape from the reservoir. This effect is obviously key to reservoir analysis and the economics of fluid withdrawal.
Many attempts have been made to incorporate fractures into rock models, and especially models that try to account for compressional wave attenuation in rocks containing fluids. But these models have often been viscoelastic rather poroelastic (Budiansky and O'Connell, 1976; O'Connell and Budiansky, 1977). Berryman and Wang (1995) showed how to make a rigorous extension of Biot's poroelasticity to include fractures/cracks by making a generalization to double-porosity/dual-permeability media modeling. That work concentrated on geomechanics and fluid flow aspects of the problem in order to deal with the interactions between fluid withdrawal and elastic closure of fractures during reservoir drawdown. The resulting equations were later applied to the reservoir consolidation problem by Lewallen and Wang (1998). Berryman and Wang (2000) then showed how the double-porosity approach could be applied to wave propagation problems, thereby generalizing Biot's work on waves to allow for heterogeneous porosities and permeabilities.
The present paper addresses the question of scale-up in heterogeneous reservoirs. If Biot's equations of poroelasticity are the correct equations at the mesoscale, then what are the correct equations at the macroscale? We show that Biot's equations are not the correct equations at the macroscale when there is significant heterogeneity in fluid permeability. However, the double-porosity dual-permeability approach appears to permit consistent modeling of such reservoirs and also shows that no further up-scaling is required beyond the double-porosity stage.