[1] [1] () [1] [1] [1] [1] [1]   [1]   [1]   [1]   [1]   et al. [1] Proof:   Remark:  

$$\begin{eqnarray} { \enq{\end{eqnarray}$$ (1)

x x a b c d e E g h j J m n p ^ q r ^ s t u v w x . y z ^ ^ ^ ^ ^ 1_n 1_m A B C D F G H I J k K L M N O 0 P Q R S T T U V W X Y Z R 0&^T&0 &00& A C D E F G K L P R

S T V W min max LS [1]LS[#1] [1]LS[#1] Int Bdy k_- k_+ k_s i(r - t) i(r - t) i(z - t)

Equations of motion for isotropic porous media with multiple solid constituents

James G. Berryman and Steven R. Pride

ABSTRACT

The volume averaging technique for obtaining macroscopic equations of motion for materials that are microscopically inhomogeneous is extended to the situation in which multiple solid constituents form a porous matrix while a uniform fluid fills the pores. Previous volume averaging efforts of Pride et al. (1992) and others have concentrated on single solid constituent porous media. The analysis for multiple solid constituents is complicated by the presence of internal interfaces between the solid constituents within the averaging volume. These interfaces are characterized by constants that measure the fraction of the interface on which solid touches solid, fluid touches one solid, fluid touches the other solid, or fluid lies on both sides of the interface. These fractions are easily computed if the interface fractions are assumed to be uncorrelated, but real materials may be expected to exhibit some correlation. On the other hand, these interface fractions do not appear in the volume average equations at the macroscopic level. To complete the analysis, it is found that the jacketed and unjacketed tests of Biot and Willis (1957) together with the thought experiments of Berryman and Milton (1991) for solid matrix composed of two constituents are required in order to obtain definite results. Results are found to be in complete agreement with earlier work of Brown and Korringa (1975) concerning the most general possible form of the quasistatic equations for volume deformation and therefore of the equations of motion for wave propagation through such media.