It is well-known in the phenomenology of earth materials that rocks are generally heterogeneous, porous, and often fractured or cracked. In situ, rock pores and cracks/fractures can contain oil, gas, or water. These fluids are all of great practical interest to us. Distinguishing these fluids by their seismic signatures is a key issue in seismic exploration and reservoir monitoring. Understanding their flow characteristics is typically the responsibility of the reservoir engineer.
Traditional approaches to seismic exploration have often made use of Biot's theory of poroelasticity (Biot, 1941; 1956a,b; 1962; Gassmann, 1951). Many of the predictions of this theory, including the observation of the slow compressional wave, have been confirmed by both laboratory and field experiment (Berryman, 1980; Plona, 1980; Johnson et al., 1982; Chin et al., 1985; Winkler, 1985; Pride and Morgan, 1991; Thompson and Gist, 1993; Pride, 1994). Nevertheless, this theory has always been limited by an explicit assumption that the porosity itself is homogeneous. Although this assumption is often applied to acoustic studies of many core samples in a laboratory setting, heterogeneity of porosity nevertheless exists in the form of pores and cracks. Also, single homogeneous porosity is often not a good assumption for application to realistic heterogeneous reservoirs in which porosity exists in the form of matrix and fracture porosity. One approach to dealing with the heterogeneity is to construct a model that is locally homogeneous, i.e., a sort of finite element approach in which each block of the model satisfies Biot-Gassmann equations. This approach may be adequate in some applications, and is certainly amenable to study with large computers. However, such models necessarily avoid the question of how we are to deal with heterogeneity on the local scale, i.e., much smaller than the size of blocks typically used in such codes.
Although it is clear that porosity in the earth can and does come in virtually all shapes and sizes, it is also clear that just two types of porosity are often most important 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, because this is the volume that stores the fluids of interest to us. (2) Fracture or crack porosity may occupy very little volume, but nevertheless has two very important effects on the reservoir properties. The first effect is that fractures/cracks drastically weaken the rock elastically, and at very low effective stress levels introduce nonlinear behavior since very small changes in stress can lead to large changes in the fracture/crack apertures (and at the same time change the fracture strength for future changes). The second effect is that the fractures/cracks often introduce a high permeability pathway for the fluid to escape from the reservoir. This effect is obviously key to reservoir analysis and the economics of fluid withdrawal.
It is therefore not surprising that many attempts have been made to incorporate fractures into rock models, and especially models that try to account for partial saturation effects and the possibility that fluid moves (or squirts) during the passage of seismic waves (Budiansky and O'Connell, 1975; O'Connell and Budiansky, 1977; Mavko and Nur, 1979; Mavko and Jizba, 1991; Dvorkin and Nur, 1993). Previous attempts to incorporate fractures have generally been rather ad hoc in their approach to the introduction of the fractures into Biot's theory, if Biot's theory was used at all. The present authors have recently started an effort to make a rigorous extension of Biot's poroelasticity to include fractures/cracks by making a generalization to double-porosity/dual-permeability modeling (Berryman and Wang, 1995). The previously published work concentrated on the fluid flow aspects of this problem in order to deal with the interactions between fluid withdrawal and the elastic behavior (closure) of fractures during reservoir drawdown. The resulting equations have been applied recently to the reservoir consolidation problem by Lewallen and Wang (1998).
It is the purpose of the present work to point out that a similar analysis applies to the wave propagation problem. Just as Biot's early work on poroelasticity for consolidation (Biot, 1941) led to his later work on wave propagation (Biot, 1956; 1962), the present work follows our own work on consolidation (Berryman and Wang, 1995) with its extension to wave propagation. We expect it will be possible to incorporate all of the important physical effects in a very natural way into this double-porosity extension of poroelasticity for seismic wave propagation. The price we pay for this rigor is that we must solve a larger set of coupled equations of motion locally. Within traditional poroelasticity, there are two types of equations that are coupled. These are the equations for the elastic behavior of the solid rock and the equations for elastic and fluid flow behavior of the pore fluid. In the double-porosity extension of poroelasticity, we have not two types of equations but three. The equations for the elastic behavior of the solid rock will be unchanged except for the addition of a new coupling term, while there will be two types of pore-fluid equations (even if there is only one fluid present) depending on the environment of the fluid. Pore fluid in the matrix (storage) porosity will have one set of equations with coupling to fracture fluid and solid; while fluid in the fractures/cracks will have another set of equations with coupling to storage fluid and solid. Although solving these equations is surely more difficult than for simple acoustics/elasticity, finding solutions for the double-porosity equations is not significantly more difficult than for traditional single-porosity poroelasticity. We will solve these equations in the present paper. We will first derive them and then show that the various coefficients in these equations can be readily identified with measurable quantities. Then we develop and solve (numerically) the dispersion relation.