Juan E. Santos

A model for wave propagation in a porous medium saturated by a two-phase fluid

A theory to describe the propagation of elastic waves in a porous medium saturated by a mixture of two immiscible, viscous, compressible fluids is presented. First, using the principle of virtual complementary work, the stress–strain relations are obtained for both anisotropic and isotropic media. Then the forms of the kinetic and dissipative energy density functions are derived under the assumption that the relative flow within the porous medium is of laminar type and obeys Darcy’s law for two‐phase flow in porous media. The equations of motion are derived, and a discussion of the different kinds of body waves that propagate in this type of medium is given. A theorem on the existence, uniqueness, and regularity of the solution of the equations of motion under appropriate initial and boundary conditions is stated.

Equivalent viscoelastic solids for heterogeneous fluid-saturated porous rocks

Differenttheoreticalandlaboratorystudiesonthepropagation ofelasticwavesinrealrockshaveshownthatthepresenceofheterogeneities larger than the pore size but smaller than the predominant wavelengths mesoscopic-scale heterogeneities may producesignificantattenuationandvelocitydispersioneffectson 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 finiteelementproceduretoapproximatethesolutionsoftheassociated boundary value problems. Also, because at mesoscopic scales rock parameter distributions are generally uncertain and of stochastic nature, we propose applying the compressibility and sheartestsinaMonteCarlofashion.Thisfacilitatesthedefinition 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,theapplicationoftheMonteCarloprocedureallowed us to determine the statistical properties of phase velocities and inverse quality factors for the particular case of quasi-fractal heterogeneities.

Static and dynamic behavior of a porous solid saturated by a two‐phase fluid

A method is presented to determine the elastic constants for an isotropic, porous, elastic solid saturated by a two‐phase fluid. Assuming that the shear modulus of the empty matrix is known, it is shown that the six additional coefficients in the stress–strain relations can be uniquely determined by performing two ideal experiments referred to as ‘‘generalized jacketed and partially jacketed compressibility tests,’’ in analogy with the single‐phase theory of Biot. Under reasonable assumptions on the behavior of the material, the experiments yield expressions for the coefficients in terms of the material properties of the individual phases and the capillary pressure function relating the pressures in the two fluid phases. Finally, numerical results showing properties of the phase velocities and attenuations for the four different types of body waves are presented and analyzed.

Computational poroelasticity — A review

Computational physics has become an essential research and interpretation tool in many fields. Particularly in reservoir geophysics, ultrasonic and seismic modeling in porous media is used to study the properties of rocks and to characterize the seismic response of geologic formations. We provide a review of the most common numerical methods used to solve the partial differential equations describing wave propagation in fluid-saturated rocks, i.e., finite-difference, pseudospectral, and finite-element methods, including the spectral-element technique. The modeling is based on Biot-type theories of dynamic poroelasticity, which constitute a general framework to describe the physics of wave propagation. We explain the various techniques and discuss numerical implementation aspects for application to seismic modeling and rock physics, as, for instance, the role of the Biot diffusion wave as a loss mechanism and interface waves in porous media.

FREQUENCY DOMAIN TREATMENT OF ONE-DIMENSIONAL SCALAR WAVES

A naturally parallelizable numerical method for approximating scalar waves in a single space variable is developed by going to a frequency domain formulation. General forms of attenuation are permitted. Convergence is established and numerical results are presented.

Reflection and transmission coefficients in fluid‐saturated porous media

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 sho...

APPROXIMATION OF SCALAR WAVES IN THE SPACE-FREQUENCY DOMAIN

A numerical method for approximating a pseudodifferential system describing attenuated, scalar waves is introduced and analyzed. Analytic properties of the solutions of the pseudodifferential systems are determined and used to show convergence of the numerical method. Experiments using the method are reported.

A NONCONFORMING MIXED FINITE ELEMENT METHOD FOR MAXWELL'S EQUATIONS

We present a nonconforming mixed finite element scheme for the approximate solution of the time-harmonic Maxwells equations in a three-dimensional, bounded domain with absorbing boundary conditions on artificial boundaries. The numerical procedures are employed to solve the direct problem in magnetotellurics consisting in determining a scattered electromagnetic field in a model of the earth having bounded conductivity anomalies of arbitrary shapes. A domain-decomposition iterative algorithm which is naturally parallelizable and is based on a hybridization of the mixed method allows the solution of large three-dimensional models. Convergence of the approximation by the mixed method is proved, as well as the convergence of the iteration.

Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modelling

We present a new finite element (FE) method for magnetotelluric modelling of three-dimensional conductivity structures. Maxwells equations are treated as a system of first-order partial differential equations for the secondary fields. Absorbing boundary conditions are introduced, minimizing undesired boundary effects and allowing the use of small computational domains. The numerical algorithm presented here is an iterative, domain decomposition procedure employing a nonconforming FE space. It does not use global matrices, therefore allowing the modellization of large and complicated structures. The algorithm is naturally parallellizable, and we show results obtained in the IBM SP2 parallel supercomputer at Purdue University. The accuracy of the numerical method is verified by checking the computed solutions with the results of COMMEMI, the international project on the comparison of modelling methods for electromagnetic induction.

Angular and Frequency-Dependent Wave Velocity and Attenuation in Fractured Porous Media

Wave-induced fluid flow generates a dominant attenuation mechanism in porous media. It consists of energy loss due to P-wave conversion to Biot (diffusive) modes at mesoscopic-scale inhomogeneities. Fractured poroelastic media show significant attenuation and velocity dispersion due to this mechanism. The theory has first been developed for the symmetry axis of the equivalent transversely isotropic (TI) medium corresponding to a poroelastic medium containing planar fractures. In this work, we consider the theory for all propagation angles by obtaining the five complex and frequency-dependent stiffnesses of the equivalent TI medium as a function of frequency. We assume that the flow direction is perpendicular to the layering plane and is independent of the loading direction. As a consequence, the behaviour of the medium can be described by a single relaxation function. We first consider the limiting case of an open (highly permeable) fracture of negligible thickness. We then compute the associated wave velocities and quality factors as a function of the propagation direction (phase and ray angles) and frequency. The location of the relaxation peak depends on the distance between fractures (the mesoscopic distance), viscosity, permeability and fractures compliances. The flow induced by wave propagation affects the quasi-shear (qS) wave with levels of attenuation similar to those of the quasi-compressional (qP) wave. On the other hand, a general fracture can be modeled as a sequence of poroelastic layers, where one of the layers is very thin. Modeling fractures of different thickness filled with CO2 embedded in a background medium saturated with a stiffer fluid also shows considerable attenuation and velocity dispersion. If the fracture and background frames are the same, the equivalent medium is isotropic, but strong wave anisotropy occurs in the case of a frameless and highly permeable fracture material, for instance a suspension of solid particles in the fluid.