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.