Recent work by Pride et al. (1992) using volume averaging to derive the form of the equations of motion for sound traveling through a fluid-saturated porous medium (Biot, 1962) has been restricted by the assumption that the solid part of such a solid/fluid composite was microhomogeneous, i.e., composed of only a single solid constituent. Motivated by the results of Berryman (1992b) showing that at least two solid constituents are needed to explain a variety of laboratory data on fluid-saturated rocks, we show in the present work how to generalize the results of Pride et al. (1992) to multicomponent solid frames. Some of the earlier approaches to volume averaging include those of Slattery (1967), Whittaker (1969), and Burridge and Keller (1981). The approach of Pride et al. (1992) and the closely related approach taken here have much in common with the methods of Slattery and Whittaker. However, those authors were studying fluid flow through a rigid solid matrix, whereas the present approach necessarily includes the effects of solid deformation.
It will also prove important to make connection with the definitions of Brown and Korringa (1975), which are themselves based on the well-known jacketed and unjacketed thought experiments of Biot and Willis (1957). While generally similar ideas have also been presented by Rice (1975) and Rice and Cleary (1976), we will stress the more detailed discussion presented by Brown and Korringa. If the total volume of the porous sample is V and the pore volume contained in that sample is (where the porosity is given by ), then Brown and Korringa (1975) define constants so that
-VV = p_dK^* + p_fK_s and
-V_V_ = p_dK_p + p_fK_. The independent variables in these formulas are the changes in differential pressure and pore-fluid pressure .The differential pressure is the difference between the external (confining) pressure and the fluid pressure, so .The coefficients are written in terms of the jacketed (or frame) bulk modulus K*, the unjacketed (or solid grain) bulk modulus Ks, and the unjacketed pore bulk modulus .The remaining modulus Kp (which is called the jacketed pore modulus) can be shown to be related to K*, Ks, and the porosity by the formula
K_p = 1K^* - 1K_s, assuming only that an energy density for the bulk deformations exists. Measurements of Kp have been made by Zimmerman et al. (1986) for some rocks. One other important fact is that, if the porous solid frame is composed of a single constituent (microhomogeneity), then -- and only then -- , where Km is the bulk modulus of the single type of mineral grain present.
In the second section of the paper, we briefly review the averaging theorem and the single solid component results. In the third section, we present the new results for volume averaging of multicomponent solid frames and derive the macroscopic equations of motion, which are just Biot's equations of poroelasticity with coefficients that must be found through an homogenization procedure.