Viscoelastic Effective Rheologies for Modelling Wave Propagation in Porous Media

Abstract

Biot's poroelastic differential equations are modified for including matrix–fluid interaction mechanisms. The description is phenomenological and assumes a solid–fluid relaxation function coupling coefficient. The model satisfies basic physical properties such as, for instance, that P-wave velocities at low frequencies are lower than those predicted by Biot's theory. In many cases, the results obtained with the Biot (two-phase) modelling are equal to those obtained with single-phase elastic modelling, mainly at seismic frequencies. However, a correct equivalence is obtained with a viscoelastic rheology, which requires one relaxation peak for each Biot (P and S) mechanism. The standard viscoelastic model, which generalizes compressibility and shear modulus to relaxation functions, is not appropriate for modelling the Biot complex moduli, since Biot's attenuation is of a kinetic nature (i.e. it is not related to bulk deformations). The problem is solved by associating relaxation functions with each wave modulus. The equivalence between the two modelling approaches is investigated for a homogeneous water-filled sandstone and a periodically layered poroelastic medium, alternately filled with gas and water. The simulations indicate that, in the homogeneous case, particle velocities in the solid skeleton, caused by a source applied to the matrix, are equivalent to viscoelastic particle velocities. In a finely layered medium, viscoelastic modelling is not, in principle, equivalent to porous modelling, due to substantial mode conversion from fast wave to slow static mode. However, this effect, caused by local fluid-flow motion, can be simulated by including an additional relaxation mechanism similar to the squirt-flow.