Exact results for generalized Gassmann’s equations in composite porous media with two constituents

Abstract

Wave propagation in fluid‐filled porous media is goverened by Bilot's equations of poroelasticity. Gassmann's relation gives an exact formula for the poroelastic parameters when the porous contains only one type of solid constituent. The present paper generalizes Gassmann's relation and derives exact formulas for two elastic parameters needed to describe wave propagation in a conglomerate of two porous phases. The parameters were first introduced by Brown and Korringa when they derived a generalized form of Gassmann's equation for conglomerates. These elastic parameters are the bulk modulus [Formula: see text] associated with changes in the overall volume of the conglomerate and the bulk modulus [Formula: see text] associated with the pore volume when the fluid pressure [Formula: see text] and confining pressure (p) are increased, keeping the differential pressure [Formula: see text] fixed. These moduli are properties of the composite solid frame (drained of fluid) and are shown here to be completely determined in terms of the bulk moduli associated with the two solid constituents, the bulk moduli of the drained conglomerate and the drained phases, and the porosities in each phase. The pore structure of each phase is assumed uniform and smaller than the grain size in the conglomerate. The relations found are completely independent of the pore microstructure and provide a means of analyzing experimental data. The key idea leading to the exact results is this: Whenever two scalar fields (in our problem [Formula: see text] and [Formula: see text]) can be independently varied in a linear composite containing only two constituents, there exists a special value γ of the increment ratio for these two fields corresponding to an overall expansion or contraction of the medium with no change of relative shape. This fact guarantees that a set of consistency relations exists among the constituent moduli and the effective moduli, which then determines all but one of the effective constants. Thus, [Formula: see text] and [Formula: see text] are determined in terms of the drained frame modulus K and the constituents’ moduli. Because the composite is linear, the coefficients found for the special value of the increment ratio are also the exact coefficients for an arbitrary ratio. Since modulus K is commonly measured while the other two are not, these exact relations provide a significant advance in our ability to predict the response of porous materials to pressure changes. It is also shown that additional results (such as rigorous bounds on the parameters) may be easily obtained by exploiting an analogy between the equations of thermoelasticity and those of poroelasticity. The method used to derive these results may also be used to find exact expressions for three component composite porous materials when thermoelastic constants of the components and the composite are known.