Mixture theory for a fluid-saturated isotropic elastic matrix

The theory for a fluid saturated linearly isotropic elastic matrix is still the basis for many geophysical applications, and commonly adopts Biot’s symmetric stress–strain laws for the matrix stress and fluid pressure. These involve a shear modulus and three elastic moduli governing the mixture and constituent compressions, in contrast to four compression moduli if Biot’s invalid potential energy argument is not applied. We now show that an energy argument applied to undrained loading also leads to three compression moduli, but distinct from those derived by Biot (Biot symmetry). However, there are two distinct solutions of this energy balance, corresponding to the Voigt and Reuss limits of the analogous theory of a linear two-phase elastic composite, whereas a unique undrained modulus not at either limit would be expected. It is proposed that an energy contribution is lost due to the idealised assumptions made for undrained loading, which therefore does not determine a further restriction, so that there are four independent compression moduli. The general and restricted combinations of the total pressure and fluid pressure (effective stress) governing the matrix compression are then presented, together with the alternative forms of the partial differential equations governing the deformation and flow.