A numerical upscaling procedure to estimate effective plane wave and shear moduli in heterogeneous fluid-saturated poroelastic media

Abstract

An important loss effect in heterogeneous poroelastic Biot media is the dissipation mechanism due to wave-induced fluid flow caused by mesoscopic scale heterogeneities, which are larger than the pore size but much smaller than the predominant wavelengths of the fast compressional and shear waves. These heterogeneities can be due to local variations in lithological properties or to patches of immiscible fluids. For example, a fast compressional wave traveling across a porous rock saturated with water and patches of gas induces a smaller fluid-pressure in the gas patches than in the water-saturated parts of the material. This in turn generates fluid flow and slow Biot waves which diffuse away from the gas–water interfaces generating significant energy losses and velocity dispersion. To perform numerical simulations using Biot’s equations of motion, it would be necessary to employ extremely fine meshes to properly represent these mesoscopic heterogeneities and their attenuation effects on the fast waves. An alternative approach to model wave propagation in these type of Biot media is to employ a numerical upscaling procedure to determine effective complex P-wave and shear moduli defining locally a viscoelastic medium having in the average the same properties than the original Biot medium. In this work the complex P-wave and shear moduli in heterogeneous fluid-saturated porous media are obtained using numerical gedanken experiments in a Monte Carlo fashion. The experiments are defined as local boundary value problems on a reference representative volume of bulk material containing stochastic heterogeneities characterized by their statistical properties. These boundary value problems represent compressibility and shear tests needed to determine these moduli for a given realization. The average and variance of the phase velocities and quality factors associated with these moduli are obtained by averaging over realizations of the stochastic parameters. The Monte Carlo realizations were stopped when the variance of the computed quantities stabilized at an almost constant value. The approximate solution of the local boundary value problems was obtained using a Galerkin finite element procedure, and the method was validated by reproducing known solutions in the case of periodic layered media. For the spatial discretization, standard bilinear finite element spaces are employed for the solid phase, while for the fluid phase the vector part of the Raviart–Thomas–Nedelec mixed finite element space of order zero was used. Results on the uniqueness of the continuous and discrete problems as well as optimal a priori error estimates for the Galerkin finite element procedure are derived. Numerical experiments showing the implementation of the procedure to estimate the average and variance of the fast compressional and shear phase velocities and inverse quality factors in these kind of highly heterogeneous fluid-saturated porous media are presented.