Different theoretical and laboratory studies on the propagation of
elastic waves in real rocks have shown that the presence of heterogeneities
larger than the pore size but smaller than the predominant wavelengths
(mesoscopic-scale heterogeneities) may produce significant attenuation
and velocity dispersion effects on seismic waves. Such phenomena
are known as "mesoscopic effects" and are caused by equilibration
of wave-induced fluid pressure gradients. We propose a numerical
upscaling procedure to obtain equivalent viscoelastic solids for
heterogeneous fluid-saturated rocks. It consists in simulating oscillatory
compressibility and shear tests in the space-frequency domain, which
enable us to obtain the equivalent complex undrained plane wave and
shear moduli of the rock sample. We assume that the behavior of the
porous media obeys Biot's equations and use a finite-element procedure
to approximate the solutions of the associated boundary value problems.
Also, because at mesoscopic scales rock parameter distributions are
generally uncertain and of stochastic nature, we propose applying
the compressibility and shear tests in a Monte Carlo fashion. This
facilitates the definition of average equivalent viscoelastic media
by computing the moments of the equivalent phase velocities and inverse
quality factors over a set of realizations of stochastic rock parameters
described by a given spectral density distribution. We analyzed the
sensitivity of the mesoscopic effects to different kinds of heterogeneities
in the rock and fluid properties using numerical examples. Also,
the application of the Monte Carlo procedure allowed us to determine
the statistical properties of phase velocities and inverse quality
factors for the particular case of quasi-fractal heterogeneities.