https://www.southampton.ac.uk/~nwb/lectures/GoodPracticeCFD/Articles/Validation_SAND2002-0529.pdf

Verification and Validation in Computational Fluid Dynamics

One major cause of elastic wave attenuation in heterogeneous porous media is wave-induced flow of the pore fluid between heterogeneities of various scales. It is believed that for frequencies below 1 kHz, the most important cause is the wave-induced flow between mesoscopic inhomogeneities, which are large compared with the typical individual pore size but small compared to the wavelength. Various laboratory experiments in some natural porous materials provide evidence for the presence of centimeter-scale mesoscopic heterogeneities. Laboratory and field measurements of seismic attenuation in fluid-saturated rocks provide indications of the role of the wave-induced flow. Signatures of wave-induced flow include the frequency and saturation dependence of P-wave attenuation and its associated velocity dispersion, frequency-dependent shear-wave splitting, and attenuation anisotropy. During the last four decades, numerous models for attenuation and velocity dispersion from wave-induced flow have been developed with varying degrees of rigor and complexity. These models can be categorized roughly into three groups according to their underlying theoretical framework. The first group of models is based on Biot’s theory of poroelasticity. The second group is based on elastodynamic theory where local fluid flow is incorporated through an additional hydrodynamic equation. Another group of models is derived using the theory of viscoelasticity. Though all models predict attenuation and velocity dispersion typical for a relaxation process, there exist differences that can be related to the type of disorder periodic, random, space dimension and to the way the local flow is incorporated. The differences manifest themselves in different asymptotic scaling laws for attenuation and in different expressions for characteristic frequencies. In recent years, some theoretical models of wave-induced fluid flow have been validated numerically, using finite-difference, finite-element, and reflectivity algorithms applied to Biot’s equations of poroelasticity. Application of theoretical models to real seismic data requires further studies using broadband laboratory and field measurements of attenuation and dispersion for different rocks as well as development of more robust methods for estimating dissipation attributes from field data.

/pdf/seismic-wave-attenuation-and-dispersion-resulting-from-wave-5duht6vzak.pdf

Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks — A review

We study a composite medium consisting of insulating magnetodielectric spherical particles embedded in a background matrix. Using results from the literature going back as far as Lewin (1947), we show that the effective permeability and permittivity of the mixture can be simultaneously negative for wavelengths where the spherical inclusions are resonant and that the medium results in an effective "double negative (DNG) media". Materials of this type are also called negative-index materials, backward media (BW), and left-handed materials. This type of material belongs to a more general class of metamaterials. The theoretical results presented here show that composite media having much simpler structure than those reported in the literature can exhibit negative permeability and permittivity over significant bandwidths.

A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix

Based on a modified Darcy's law, Stokes’ first problem was investigated for a second grade fluid in a porous half-space with a heated flat plate. Exact solutions of the velocity and temperature fields were obtained using Fourier sine transforms. In contrast to the classical Stokes’ first problem, there is a steady-state solution for the second grade fluid in the porous half-space, which is a damping exponential function with respect to the distance from the flat plate. The well-known solutions for Newtonian fluids in non-porous or porous half-space appear in limiting cases of our solutions.

Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary

The equations governing the linear acoustics of composites with two isotropic porous constituents are derived from first principles using volume-averaging arguments. The theory is designed for modeling acoustic propagation through heterogeneous porous structures. The only restriction placed on the geometry of the two porous phases is that the overall composite remains isotropic. The theory determines the macroscopic fluid response in each porous phase in addition to the combined bulk response of the grains and fluid in the composite. The complex frequency-dependent macroscopic compressibility laws that are obtained allow for fluid transfer between the porous constituents. Such mesoscopic fluid transport between constituents within each averaging volume provides a distinct attenuation mechanism from the losses associated with the net Darcy flux within individual constituents as is quantified in the examples.

Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation

Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media

As in our previous paper, we investigate the macroscopic quasi-static description of a porous medium with a double porosity constituted by pores and fractures. For this purpose, we use a homogenization technique. As expected, the macroscopic description is sensitive to the ratios between the different scales,l/l′ andl/l′ wherel, l′, l″ are characteristic lengths of the pores, the fractures, and the macroscopic medium, respectively. In the first paper, we investigated the casel′/l″=O(l
2/l′2)≪1 (case 1) which exhibits a coupling between the flows through the pores and the fractures. In the present paper, we deal with the other homogenizable cases. The case 3 wherel/l′=O(l′2/l″2)≪1 gives a macroscopic description similar to that of a single porosity medium. The main result, however, is the case 2, wherel/l′=O(l′/l″)≪1, which exhibits memory effects. These are due to the seepage through the micropores.

/pdf/deformable-porous-media-with-double-porosity-quasi-statics-27yjravvay.pdf

Deformable porous media with double porosity. Quasi-statics. II: Memory effects

Double conductivity media: a comparison between phenomenological and homogenization approaches

A model for pollutant transport in a heterogeneous medium, including diffusion accompanied by adsorption, has been developed. The macroscopic equivalent diffusion tensor and the macroscopic governing equations for the average concentration field can be deduced by homogenization. The necessary condition for this theory to be applied is the existence of a small parameter E defined as the ratio of the scale of the pores and the scale of a sample, which can be distinguished in the heterogeneous medium. Transport processes with respect to different dominating phenomena, related to E, have been examined. Using the analysis performed the condition has been specified when the homogenization is not possible, i.e. when it is impossible to find the rigorous macroscopic equivalent description. Un modele de transport de polluants, prenant en compte diffusion et adsorption, a ete developpe. Le tenseur de diffusion macroscopique equivalant ainsi que les equations d'etat macroscopiques du champ de concentration moyen peu...

Homogenization analysis of diffusion and adsorption macrotransport in porous media: macrotransport in the absence of advection

This work is aimed towards deriving macroscopic models that describe pollutant migration through fractured porous media. A homogenisation method is used, that is, macroscopic models are deduced from the physical description over a representative elementary volume (REV), which consists of an open fracture surrounded by a porous matrix block. No specific geometry is at issue. The fractured porous medium is saturated by an incompressible fluid. At the REV's scale, the transport is assumed to be advective-diffusive in the porous matrix and due to convection and molecular diffusion in the fracture's domain. It is also assumed that there is no diffusion in the solid. We demonstrate that the macroscopic behaviour is described by a single-continuum model. Fluid flow is described by Darcy's law. Four macroscopic single-continuum models are obtained for the contaminant transport: a diffusive model, an advective-diffusive model and two advective-dispersive models. One of the two advective-dispersive models accounts for the advection process in the porous matrix. The domains of validity of these models are defined by means of the orders of magnitude of the local Peclet numbers in the porous matrix block and in the fracture's domain.

/pdf/continuum-modelling-of-contaminant-transport-in-fractured-3jio5fdt4u.pdf

J.-L. Auriault

Papers

Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media

Deformable porous media with double porosity. Quasi-statics. II: Memory effects

Double conductivity media: a comparison between phenomenological and homogenization approaches

Homogenization analysis of diffusion and adsorption macrotransport in porous media: macrotransport in the absence of advection

Continuum Modelling of Contaminant Transport in Fractured Porous Media