Propagation of acoustic waves through saturated porous media.

The homogenization approach to wave propagation through saturated porous media is extended in order to include the compressibility of the interstitial fluid and the existence of several connected pore components which may or not percolate. The necessary theoretical developments are summarized and the Christoffel equation whose solutions provide the wave velocities is presented. Some analytical developments are proposed for isotropic media. Finally, a systematic application to a synthetic porous medium illustrates the methodology and its results.

Thermal convection in three-dimensional fractured porous media.

Thermal convection is numerically computed in three-dimensional (3D) fluid saturated isotropically fractured porous media. Fractures are randomly inserted as two-dimensional (2D) convex polygons. Flow is governed by Darcy's 2D and 3D laws in the fractures and in the porous medium, respectively; exchanges take place between these two structures. Results for unfractured porous media are in agreement with known theoretical predictions. The influence of parameters such as the fracture aperture (or fracture transmissivity) and the fracture density on the heat released by the whole system is studied for Rayleigh numbers up to 150 in cubic boxes with closed-top conditions. Then, fractured media are compared to homogeneous porous media with the same macroscopic properties. Three major results could be derived from this study. The behavior of the system, in terms of heat release, is determined as a function of fracture density and fracture transmissivity. First, the increase in the output flux with fracture density is linear over the range of fracture density tested. Second, the increase in output flux as a function of fracture transmissivity shows the importance of percolation. Third, results show that the effective approach is not always valid, and that the mismatch between the full calculations and the effective medium approach depends on the fracture density in a crucial way.

Percolation in three-dimensional fracture networks for arbitrary size and shape distributions.

The percolation threshold of fracture networks is investigated by extensive direct numerical simulations. The fractures are randomly located and oriented in three-dimensional space. A very wide range of regular, irregular, and random fracture shapes is considered, in monodisperse or polydisperse networks containing fractures with different shapes and/or sizes. The results are rationalized in terms of a dimensionless density. A simple model involving a new shape factor is proposed, which accounts very efficiently for the influence of the fracture shape. It applies with very good accuracy in monodisperse or moderately polydisperse networks, and provides a good first estimation in other situations. A polydispersity index is shown to control the need for a correction, and the corrective term is modelled for the investigated size distributions.

Comparative assessment of continuum-scale models of bimolecular reactive transport in porous media under pre-asymptotic conditions.

We compare the ability of various continuum-scale models to reproduce the key features of a transport setting associated with a bimolecular reaction taking place in the fluid phase and numerically simulated at the pore-scale level in a disordered porous medium. We start by considering a continuum-scale formulation which results from formal upscaling of this reactive transport process by means of volume averaging. The resulting (upscaled) continuum-scale system of equations includes nonlocal integro-differential terms and the effective parameters embedded in the model are quantified directly through computed pore-scale fluid velocity and pore space geometry attributes. The results obtained through this predictive model formulation are then compared against those provided by available effective continuum models which require calibration through parameter estimation. Our analysis considers two models recently proposed in the literature which are designed to embed incomplete mixing arising from the presence of fast reactions under advection-dominated transport conditions. We show that best estimates of the parameters of these two models heavily depend on the type of data employed for model calibration. Our upscaled nonlocal formulation enables us to reproduce most of the critical features observed through pore-scale simulation without any model calibration. As such, our results clearly show that embedding into a continuum-scale model the information content associated with pore-scale geometrical features and fluid velocity yields improved interpretation of typically available continuum-scale transport observations.

Transport properties of a Bentheim sandstone under deformation.

The mechanical and transport properties of a Bentheim sandstone are studied both experimentally and numerically. Three classical classes of loads are applied to a sample whose permeability is measured. The elasticity and the Stokes equations are discretized on unstructured tetrahedral meshes which precisely follow the deformations of the sample. Numerical results are presented, discussed, and compared to the available experimental data.

Solid matrix partition by fracture networks.

The geometrical properties of the matrix blocks formed by a random fracture network are investigated numerically, for a wide range of fracture shapes and for fracture densities ranging from the dilute limit to well above the threshold where the material is entirely partitioned into finite blocks. The main block characteristics are the density and volume fraction, the mean volume and surface area, and their number of faces. In the dilute limit, general expressions for these characteristics are obtained, which provide a good approximation of the numerical data for any fracture shape. In the dense regime, most properties are governed by power laws, which involve two fitted exponents independent of the fracture shape. The shape factors identified in the dilute limit remain relevant for dense networks and can be used to formulate a general model for the block characteristics, valid up to the total matrix fracturation. The transition density when this occurs is determined. It can also be used to account for the fracture shape effects in a very simple and fairly accurate general model. Beyond the transition density, the block characteristics converge as expected toward those in the space tesselation by infinite planes.

Interaction between a fracture network and a cubic cavity.

The intersection between a network of polygonal fractures and a cubic cavity is numerically studied. Several probabilities are defined and particular attention is paid to the probabilities of intersection or not of the percolating cluster with the cavity; they depend on the size of the domain, on the fracture density, and on the relative size of the fractures and of the cavity. These probabilities are extrapolated to infinite domains. Analytical approximations are proposed which are in good agreement with the numerical data for sufficiently large densities. Some extensions of practical and theoretical interests are given in the concluding remarks.

Microscale simulation and numerical upscaling of a reactive flow in a plane channel.

A bimolecular homogeneous irreversible reaction of the kind A+B→C is simulated in a plane channel as a base example of reactive transport processes taking place at the microscale within porous and/or fractured media. The numerical study explores the way microscale processes embedded in dimensionless quantities such as Péclet (Pe) and Damköhler (Da) numbers propagate to upscaled coefficients describing effective system dynamics. The microscale evolution of the reactant concentrations is obtained through a particle-based numerical method which has been specifically tailored to the considered problem. Key results include a complete documentation of the process evolution for a wide range of Pe and Da, in terms of the global reaction rate, space-time distribution of reactants, and local mixing features leading to characterization of effective reaction and dispersion coefficients governing a section-averaged upscaled model of the system. The robustness of previously presented theoretical analyses concerning closures of volume-averaged (upscaled) formulations is assessed. The work elucidates the dependence of the effective dispersion and reactive parameters on the microscale mixing and reactive species evolution. Our results identify the role played by Da and Pe on the occurrence of incomplete mixing of reactants, which affects the features of the reactive transport scenario.

Percolation and permeability of fracture networks in excavated damaged zones.

Generally, the excavation process of a gallery generates fractures in its immediate vicinity. The corresponding zone which is called the excavated damaged zone (EDZ), has a larger permeability than the intact surrounding medium. Therefore, some of its properties are of crucial importance for applications such as the storage of nuclear wastes. Field observations suggest that the fracture density is an exponentially decreasing function of the distance to the wall and that the fracture orientation is anisotropic and well approximated by a Fisher law whose pole is orthogonal to the wall. Numerical samples are generated according to these prescriptions and their percolation status and hydraulic transmissivity are systematically determined for a wide range of decay lengths and anisotropy parameters. All the numerical data are presented and discussed. A heuristic analytical expression for the percolation threshold is proposed which unifies and accurately represents all the numerical data. A simple parallel flow model yields an explicit analytical expression for the transmissivity as a function of the density, heterogeneity, and anisotropy parameters; the model also successfully accounts for all the numerical data.

Permeability of isotropic and anisotropic fracture networks, from the percolation threshold to very large densities.

The asymptotic behaviors of the permeability of isotropic fracture networks at small and large densities are characterized, and a general heuristic formula is obtained which complies with the limiting behaviors and accurately predicts the permeability of these networks over the whole density range. Theses developments are based on extensive numerical calculations and on theoretical arguments inspired by the examination of the flow distribution in the fractures at large densities. Then, the results are extended to anisotropic networks with a Fisher distribution of the fracture orientations, to polydisperse networks, and to fractured porous media. Finally, guidelines are provided for the practical evaluation of the required parameters from typical field data. A summary of the results is given in Table III.

Grain reconstruction of porous media: application to a Bentheim sandstone.

The two-point correlation measured on a thin section can be used to derive the probability density of the radii of a population of penetrable spheres. The geometrical, transport, and deformation properties of samples derived by this method compare well with the properties of the digitized real sample and of the samples generated by the standard grain reconstruction method.

Trace analysis for fracture networks with anisotropic orientations and heterogeneous distributions.

Since only intersections with lines or planes are usually available to quantify the properties of real fracture networks, a stereological analysis of these intersections is a crucial issue. This article-the second of a series-is devoted to the derivation of the direct relations between the properties and the observable quantities. First, this derivation is achieved for anisotropic networks whose orientations obey a Fisher probability distribution function; second, it is extended to networks which are heterogeneous in space, i.e., whose density decays according to an exponential law. Five major quantities are determined: the excluded volume, the average number of intersections with a line and with a plane, the average trace length and the surface density of trace intersections. Some of these relations are valid for any convex fracture shape and some only for circular disks; however, numerical simulations show that excellent approximations are obtained by considering disks with the same area as the noncircular fractures.

Transport properties of real metallic foams.

Three dimensional samples of three different foams are obtained by microtomography. The macroscopic conductivity and permeability of these foams are calculated by three different numerical techniques based on either a finite volume discretization or Lattice Boltzmann algorithm. Permeability is also measured and an excellent agreement is obtained between the various estimations. Calculated conductivities are successfully compared to available data.

Random packings of spiky particles: geometry and transport properties.

Spiky particles are constructed by superposing spheres and oblate ellipsoids. The resulting star particles (but nonconvex) are randomly packed by a sequential algorithm. The geometry, the conductivity, and the permeability of the resulting packings are systematically studied. Overall correlations are proposed to approximate these properties when the geometry of the particle is known.

Percolation and permeability of networks of heterogeneous fractures.

Networks composed by heterogeneous fractures whose local permeability is a binary correlated random field are generated. The percolation and permeability properties of a single heterogeneous fracture are strongly influenced by finite size effects when the correlation length is of the order of the fracture size. For fracture networks, a mean-field approximation is derived which approximates well the macroscopic permeability while an empirical formula is proposed for the percolation properties.

Permeability and percolation of anisotropic three-dimensional fracture networks.

The percolation properties and permeability of a group of anisotropic three-dimensional fracture networks are studied numerically. Finite-size scaling is used to extrapolate the percolation thresholds of infinite networks in three spatial directions, i.e., X , Y , and Z directions. The influence of the angular dispersion parameter of fracture orientations on percolation thresholds is analyzed. In this analysis, we considered a family of fractures in a three-dimensional space that are oriented around the Z axis based on the Fisher distribution. We revealed that increased anisotropy leads to decreased percolation thresholds in both X and Y directions, and in these two directions percolation thresholds in anisotropic networks demonstrate a declining trend as anisotropy goes up. However, in the Z direction the trend is the opposite. The fracture networks are triangulated via an advancing front technique and the macroscopic permeability of the networks is determined by solving the two-dimensional Darcy equation in each fracture. We found that the macroscopic permeability in the X and Y directions is higher than the associated permeability of isotropic fracture networks, and this property for anisotropic networks in the Z direction is lower compared with that of the isotropic case. Furthermore, as the anisotropy of networks increases the differences become more remarkable.

Wave propagation through saturated porous media.

The homogenization procedure is applied to the problem of wave propagation in the biphasic mode in porous media saturated with a Newtonian fluid. The local problems corresponding to the solid and fluid phases have been solved separately for complex three-dimensional media. The effective rigidity tensor, some effective coefficients, the dynamic permeability, the celerities, and the attenuation of the three waves are systematically determined. The characteristic length Lambda was successfully used to gather results for the dynamic permeability as well as for the attenuation coefficients for all media.

Geometrical and transport properties of random packings of polydisperse spheres.

Loose packings of spheres with bidisperse or log-normal distributions are generated by random sequential deposition. Porosity, conductivity, and permeability are determined. The porosities correspond to loose packings, but they follow the usual trends for bidisperse packings. The conductivity and permeability follow power laws as functions of the porosity of the packings. Several other quantities such as the classical Kozeny constant are successfully represented as functions of porosity. Some dimensionless representations gather the numerical data on curves valid for all particle distributions. Finally, comparisons with experimental data are satisfactory.

Effective permeability of fractured porous media with power-law distribution of fracture sizes.

The permeability of geological formations which contain fractures with a power-law size distribution is addressed numerically by solving the coupled Darcy equations in the fractures and in the surrounding porous medium. Two reduced parameters are introduced which allow for a unified description over a very wide range of the fracture characteristics, including their shape, density, size distribution, and possibly size-dependent permeability. Two general models are proposed for loose and dense fracture networks, and they provide a good representation of the numerical data throughout the investigated parameter range.

Nuclear magnetic resonance diffusion with surface relaxation in porous media.

Nuclear magnetic resonance (NMR) diffusion simulations with surface relaxation were performed numerically in unconsolidated and consolidated porous media by a random walk technique. Two uniform and nonuniform models of surface relaxation were proposed and compared. The apparent diffusion coefficient and extinction function were determined and studied in the fast, slow and intermediate diffusion regimes of relaxation. According to theoretical predictions, it was observed that the extinction function does not depend on surface relaxivity parameter rho 2 in the slow diffusion regime. The apparent diffusion coefficients are independent of rho 2 in the fast diffusion regime and tend to be superposed onto a single curve in the slow one. The evolution of the apparent diffusion coefficients is gathered by a reduced representation in the fast diffusion regime.