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.

Fast Laplace solver approach to pore-scale permeability.

We introduce a powerful and easily implemented method to calculate the permeability of porous media at the pore scale using an approximation based on the Poiseulle equation to calculate permeability to fluid flow with a Laplace solver. The method consists of calculating the Euclidean distance map of the fluid phase to assign local conductivities and lends itself naturally to the treatment of multiscale problems. We compare with analytical solutions as well as experimental measurements and lattice Boltzmann calculations of permeability for Fontainebleau sandstone. The solver is significantly more stable than the lattice Boltzmann approach, uses less memory, and is significantly faster. Permeabilities are in excellent agreement over a wide range of porosities.

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.

Three-scale analysis of the permeability of a natural shale.

The macroscopic permeability of a natural shale is determined by using structural measurements on three different scales. Transmission electron microscopy yields two-dimensional (2D) images with pixels smaller than 1 nm; these images are used to reconstruct 3D nanostructures. Three-dimensional focused ion beam-scanning electron microscopy (5.95- to 8.48-nm voxel size) provides 3D mesoscale pores of limited relative volume (1.71-5.9%). Micro-computed tomography (700-nm voxel size) provides information on the mineralogy of the shale, including the pores on this scale which do not percolate; synthetic 3D media are derived on the macroscopic scale by a training image technique. Permeability of the nanoscale, of the mesoscale structures and of their superposition is determined by solving the Stokes equation and this enables us to estimate the permeabilities of the 700-nm voxels located within the clay matrix. Finally, the Darcy equation is solved on synthetic 3D macroscale media to obtain the macroscopic permeability which is found in good agreement with experimental results obtained on the centimetric scale.

Two-scale analysis of a tight gas sandstone.

Tight gas sandstones are low porosity media, with a very small permeability (i.e., below 1 mD). Their porosity is below 10%, and it is mainly composed of fine noncemented microcracks, which are present between neighboring quartz grains. While empirical models of permeability are available, their predictions, which do not compare well with macroscopic measurements, are not reliable to assess gas well productivity. The purpose of this work is to compare the permeability measured on centimetric plugs to predictions based on pore structure data. Two macroscopic measurements are performed, namely dry gas permeability and mercury intrusion porosimetry (MIP), together with a series of local measurements including focused ion beam and scanning electron microscopy (FIB-SEM), x-ray computed microtomography (CMT), and standard two-dimensional (2D) SEM. Numerical modeling is performed by combining analyses on two scales, namely the microcrack network scale (given by 2D SEM) and the individual 3D microcrack scale (given by either FIB-SEM or CMT). The network permeability is calculated by means of techniques developed for fracture networks. This permeability is proportional to the microcrack transmissivity, which is determined by solving the Stokes equation in the microcracks measured by FIB-SEM or CMT. Good correlation with experimental permeability values is only found when using transmissivity from 3D CMT data.

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.

Application of level-set method for deposition in three-dimensional reconstructed porous media.

Three-dimensional (3D) porous structures which are usually discretized by voxels can also be discretized by the level-set method (LSM) and flow, reactive transport, and structure evolution can be modeled. The determination of the solid-liquid interface is detailed as well as the discretization of the governing equations. Comparisons between a one-dimensional analytical solution and LSM are conducted for validation. Deposition in 3D reconstructed media is studied under various flow and reaction conditions. The evolution of the structure is explored locally by means of the pore geometry and globally by means of the permeability and the porosity. The difference between the voxel method and LSM is discussed during the investigations.

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.

Continuum percolation of isotropically oriented circular cylinders.

The continuum percolation of circular cylinders has been studied for various values of the aspect ratio b;{'} . The percolation threshold is shown to have a maximum for b;{'} approximately 2 when the cylinder length is equal to its diameter. Other quantities such as the average intersection volume and the porosity also possess a maximum for this value.

Resurgence flows in porous media.

Porous media with resurgences can be described by a double structure, namely, a continuous porous medium and capillaries with impermeable walls which relate distant points of the continuous medium. The resurgences can be either punctual or extended. The equations for flow in such media are derived; some general properties of the resulting system, which involves nonlocal aspects, are deduced. A "dilute" approximation is detailed for punctual resurgences in two-dimensional media and is illustrated by a few examples.

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.