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 heterogeneities͒ 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 fluidsaturated 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 finiteelement 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.

This paper presents a theory to describe wave propagation in a porous medium composed of two solids saturated by a single-phase fluid for spatially variable porosity. This problem has been previously solved for constant porosity when one of the solids is ice or clay, but that model is not useful for most realistic situations. The equations for variable porosity are derived from the virtual work principle, where the generalized coordinates are identified as the displacements of the two solid phases and a new variable associated with the relative fluid flow, whose divergence is the change in fluid content. The generalized forces are the fluid pressure and combinations of the stress tensor of each solid phase and the fluid pressure. The Lagrangian equations of motion are derived for the isotropic case and a theorem on the existence and uniqueness of their solution is given. The plane wave analysis reveals the existence of three compressional and two shear waves. The theory is applied to wave propagation in shaley sandstones showing that phase velocities of the faster P and S waves agree very well with experimental data for varying porosity and clay content. A simulation through a plane interface separating two frozen sandstones of different ice contents is presented.

Using Biot’s theory to describe the propagation of elastic waves in a fluid-saturated porous elastic solid (a Biot medium), the reflection and transmission coefficients were computed at a plane interface between a fluid and a Biot medium and at interfaces inside a Biot medium defined by either a change in saturant fluids or in the intrinsic rock permeability. The reflection and transmission coefficients were computed with and without the inclusion of a frequency correction factor that according to Biot has to be introduced in the equations above a certain critical frequency (‘‘frequency-dependent’’ versus ‘‘classic model’’). For a fluid–Biot medium interface and in the range 5 kHz–10 MHz for the example analyzed the two models show differences of the order of 11% for the reflection coefficients and between 11% and 31% for the type I, type II, and shear transmission coefficients. For the interfaces within a Biot medium, and for type II incident waves, in the same range of frequencies the cases examined showed differences in the reflection and transmission coefficients in the range 5%–80%. Because of the asymptotic properties of the frequency correction factor, the reflection and transmission coefficients coincide at very low and high frequencies. The analysis shows the importance of the inclusion of the frequency correction factor in analyzing wave propagation in Biot media for frequencies lying between the seismic and ultrasonic ranges.

We present an iterative algorithm formulated in the space-frequency domain to simulate the propagation of waves in a bounded poro-viscoelastic rock saturated by a two-phase fluid. The Biot-type model takes into account capillary forces and viscous and mass coupling coefficients between the fluid phases under variable saturation and pore fluid pressure conditions. The model predicts the existence of three compressional waves or Type-I, Type-II and Type-III waves and one shear or S-wave. The Type-III mode is a new mode not present in the classical Biot theory for single-phase fluids. Our differential and numerical models are stated in the space-frequency domain instead of the classical integrodifferential formulation in the space-time domain. For each temporal frequency, this formulation leads to a Helmholtz-type boundary value problem which is then solved independently of the other frequency problems, and the time-domain solution is obtained by an approximate inverse Fourier transform. The numerical procedure, which is first-order correct in the spatial discretization, is an iterative nonoverlapping domain decomposition method that employs an absorbing boundary condition in order to minimize spurious reflections from the artificial boundaries. The numerical experiments showing the propagation of waves in a sample of Nivelsteiner sandstone indicate that under certain conditions the Type-III wave can be observed at ultrasonic frequencies.

This paper studies the reflection and transmission of plane elastic waves at interfaces in fluid-saturated poroviscoelastic media in which the solid matrix is composed of two weakly coupled solids. The analysis of this problem, not formally performed before, is based on a theory recently developed by some of the authors, which allows us to derive expressions for the reflection and transmission coefficients at a plane interface within this kind of media and their relationship with the energy flux (Umov-Poynting) vector. The results of the present derivation were applied to study the energy splitting that takes place when a plane fast compressional wave strikes obliquely an interface defined by a change in ice content within a sample of water saturated partially frozen sandstone. The numerical results show wave mode conversions from fast to slow compressional and shear waves, with maximum energy conversion on the order of 20% from fast to slow wave modes near the critical angle. This phenomenon was observed at frequencies lying from the seismic to the ultrasonic range, showing that the role of the slow waves must be taken into account when considering wave propagation in this type of media.

Melting/freezing behavior of a fluid confined in porous glasses and MCM-41: Dielectric spectroscopy and molecular simulation J. Chem. Phys. 114, 950 (2001); 10.1063/1.1329343 Freezing of simple fluids in microporous activated carbon fibers: Comparison of simulation and experiment J. Chem. Phys. 111, 9058 (1999); 10.1063/1.480261Freezing and melting of water in a single cylindrical pore: The pore-size dependence of freezing and melting behavior A recent article ͓J. M. Carcione and G. Seriani, J. Comput. Phys. 170, 676 ͑2001͔͒ proposes a modeling algorithm for wave simulation in a three-phase porous medium composed of sand grains, ice, and water. The differential equations hold for uniform water ͑ice͒ content. Here, we obtain the variable-porosity differential equations by using the analogy with the two-phase case and the complementary energy theorem. The displacements of the rock and ice frames and the variation of fluid content are the generalized coordinates, and the stress components and fluid pressure are the generalized forces. We simulate wave propagation in a frozen porous medium with fractal variations of porosity and, therefore, realistic freezing conditions.

We derive the time-domain stress-strain relation for a porous medium composed of n − 1 solid frames and a saturating fluid. The relation holds for nonuniform porosity and can be used for numerical simulation of wave propagation. The strain-energy density can be expressed in such a way that the two phases (solid and fluid) can be mathematically equivalent. From this simplified expression of strain energy, we analogize two-, three-, and n-phase porous media and obtain the corresponding coefficients (stiffnesses). Moreover, we obtain an approximation for the generalized Gassmann modulus. The Gassmann modulus is the bulk modulus of a saturated porous medium whose matrix (frame) is homogeneous. That is, the medium consists of two homogeneous constituents: a mineral composing the frame and a fluid. Gassmann's modulus is obtained at the low-frequency limit of Biot's theory of poroelasticity. Here, we assume that all constituents move in phase, a condition similar to the dynamic compatibility condition used by Biot, by which the P-wave velocity is equal to Gassmann's velocity at all frequencies. Our results are compared with those of the Berryman-Milton (BM) model, which provides an exact generalization of Gassmann's modulus to the three-phase case. The model is then compared to the wet-rock moduli obtained by static finite-element simulations on digitized images of microstructure and is used to fit experimental data for shaly sandstones. Finally, an example of a multimineral rock (n > 3) saturated with different fluids is given.

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