Deriving the equations of motion for porous isotropic media

Abstract

The equations of motion and stress/strain relations for the linear dynamics of a two-phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

Two-Dimensional Fractional Order Generalized Thermoelastic Porous Material

“…A short list of papers pertinent to the present study includes Biot(1941Biot( , 1956, Gassmann (1951), Biot and Willis (1957), Biot (1962), Deresiewicz and Skalak (1963), Mandl (1964), Nur and Byerlee (1971), Brown and Korringa (1975), Rice and Cleary (1976), Burridge and Keller (1981), Zimmerman et al (1986Zimmerman et al ( ,1994, Berryman and Milton (1991), Thompson and Willis (1991)], Pride et al (1992), Berryman and Wang (1995), Tuncay and Corapcioglu (1995), Alexander and Cheng (1991), Charlez, P. A., and Heugas, O. (1992), Abousleiman et al (1998), Ghassemi and Diek (2002), Tod (2003).…”

Wave propagation in fractured porous media

“…Fatt (1959) calculated Biot's constants for sandstone. The elastic coefficients were also studied by Geertsma and Smith (1961) and Pride et al (1992). We refer to Corapcioglu (1991) for an extensive review of Biot's theory.…”

“…In a subsequent paper, de la Cruz and Spanos (1989) extended their theory to include the thermodynamic considerations. Pride et al (1992) obtained Biot's (1941Biot's ( , 1956a equations for saturated porous media by employing the volume averaging technique. The resulting constitutive relations of Pride et al (1992) contained the same parameters as of Biot and Willis (1957).…”

Consolidation of Elastic Porous Media Saturated by Two Immiscible Fluids

“…In a subsequent paper, de la Cruz and Spanos (1989) extended their theory to include the thermodynamic considerations. Pride et al (1992) obtained Biot's (1941Biot's ( , 1956) equations for satur~ted porous media by employing the volume averaging techmque. The resulting constitutive relations of Pride et al (1992) contained the same parameters as those of Biot and Willis (1957).…”

“…Pride et al (1992) obtained Biot's (1941Biot's ( , 1956) equations for satur~ted porous media by employing the volume averaging techmque. The resulting constitutive relations of Pride et al (1992) contained the same parameters as those of Biot and Willis (1957). As seen in this brief review, there is no theory available to obtain constitutive relations as well as mass and…”