Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Filtration in a Porous Granular Medium: 1. Simulation of Pore-Scale Particle Deposition and Clogging

This paper presents a numerical model for simulating the pore-scale transport and infiltration of dilute suspensions of particles in a granular porous medium under the action of hydrodynamic and gravitational forces. The formulation solves the Stokes’ flow equations for an incompressible fluid using a fixed grid, multigrid finite difference method and an embedded boundary technique for modeling particle–fluid coupling. The analyses simulate a constant flux of the fluid suspension through a cylindrical model pore. Randomly generated particles are collected within the model pore, initially through contact and attachment at the grain surface (pore wall) and later through mounding close to the pore inlet. Simple correlations have been derived from extensive numerical simulations in order to estimate the volume of filtered particles that accumulate in the pore and the differential pressure needed to maintain a constant flux through the pore. The results show that particle collection efficiency is correlated with the Stokes’ settling velocity and indirectly through the attachment probability with the particle–grain surface roughness. The differential pressure is correlated directly with the maximum mound height and indirectly with particle size and settling velocity that affect mound packing density. Simple modification factors are introduced to account for pore length and dip angle. These parameters are used to characterize pore-scale infiltration processes within larger scale network models of particle transport in granular porous media in a companion paper. Articlenote: Currently at GZA GeoEnvironmental Inc., 1 Edgewater Drive, Norwood, MA 02062, U.S.A.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

Analytical Determination of Flow Patterns in a Double Porosity Medium Containing Ellipsoidal Occluded Macro-Voids

This article presents the analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, occluded macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for the pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its size is characterized by the macro-porosity. Confrontations are made with numerical solutions in order to investigate the applicability of the analytical pressure and velocity solutions to microstructures of finite size.     

The Effect of Double Dispersion on Natural Convection Heat and Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

The effect of power law index parameter of the non-Newtonian fluid on free convection heat and mass transfer from a vertical wall is analyzed by considering double dispersion in a non-Darcy porous medium with constant wall temperature and concentration conditions. The Ostwald–de Waele power law model is used to characterize the non-Newtonian fluid behavior. In this case a similarity solution is possible. The variation of heat and mass transfer coefficients with the governing parameters such as power law index, thermal and solutal dispersion parameters, inertia parameter, buoyancy ratio, and the Lewis number is discussed for a wide range of values of these parameters.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.     

Two-Phase Flow in Heterogeneous Porous Media with Non-Wetting Phase Trapping

This article presents a mathematical model describing flow of two fluid phases in a heterogeneous porous medium. The medium contains disconnected inclusions embedded in the background material. The background material is characterized by higher value of the non-wetting-phase entry pressure than the inclusions, which causes non-standard behavior of the medium at the macroscopic scale. During the displacement of the non-wetting fluid by the wetting one, some portions of the non-wetting fluid become trapped in the inclusions. On the other hand, if the medium is initially saturated with the wetting phase, it starts to drain only after the capillary pressure exceeds the entry pressure of the background material. These effects cannot be represented by standard upscaling approaches based on the assumption of local equilibrium of the capillary pressure. We propose a relevant modification of the upscaled model obtained by asymptotic homogenization. The modification concerns the form of flow equations and the calculation of the effective hydraulic functions. This approach is illustrated with two numerical examples concerning oil–water and CO₂–brine flow, respectively.     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

Natural convection due to heat and mass transfer in a composite system

A numerical study is performed to analyse heat and mass transfer phenomena due to natural convection in a composite cavity containing a fluid layer overlying a porous layer saturated with the same fluid. The flow in the porous region is modelled using Brinkman–Forchheimer-extended Darcy model that includes both the effect of macroscopic shear (Brinkman effect) and flow inertia (Forchheimer effect). The vertical walls of the two-dimensional enclosure are isothermal whilst the horizontal walls are adiabatic. The two regions are coupled by equating the velocity and stress components at the interface. The resulting coupled equations in non-dimensional form are solved by an alternating direction implicit method by transforming them into parabolic form by the addition of false transient terms. The numerical results show that the amount of fluid penetration into the porous layer depends strongly upon the Darcy, thermal and solutal Rayleigh numbers. Average Nusselt number decreases while average Sherwood number increases with an increase of the Lewis number. The transfer of heat and mass on the heated wall near the interface depends strongly on the Darcy number. Received on 11 May 1998     

Generalized Macroscopic Models for Fluid Flow in Deformable Porous Media II: Numerical Results and Applications

In Part I of this study, generalized mathematical models were developed to describe the motion of fluids in porous media. The second part of this study solved the problem of fluid flow in small channels of a periodic elastic solid matrix at the pore scale numerically, and applied the volume-averaging technique to predict the macroscopic behavior of reservoirs. The numerical results demonstrated different macroscopic behavior of a porous medium due to cyclic excitation at various frequencies corresponding to the five separate characteristic macroscopic models identified in Part I. The results emphasize the need to use an appropriate model to interpret the corresponding responses of a saturated porous medium.     

Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow

We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Law of flow of a viscoplastic fluid through a porous medium with allowance for inertial losses

A macroscopic law of flow of a viscoplastic Schwedoff-Bingham fluid through a porous medium is obtained on the basis of percolation theory with allowance for viscous and inertial losses. The asymptotics of the flow law are estimated and expressions for determining the limiting pressure gradient as a function of the microinhomogeneity parameters are given. Satisfactory qualitative agreement between the theoretical and known experimental data is observed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 68–73, January–February, 1999.     

Automaton Simulations of Dispersion in Porous Media

A thermodynamic lattice gas (automaton) model is used to simulate dispersion in porous media. Simulations are constructed at two distinctly different scales, the pore scale at which capillary models are constructed and large scale or Darcy scale at which probabilistic collision rules are introduced. Both models allow for macroscopic (pore scale) phase separation. The pore scale models clearly show the effect of pore structure on dispersion. The large scale (mega scale) simulations indicate that when the pressure difference between the displacing phase and displaced phase is properly chosen (representing the average pressure gradient between the phases). The simulation results are consistent with both theoretical predictions and experimental observations.     

Modeling Large-deformation-induced Microflow in Soft Biological Tissues

The homogenization approach to multiscale modeling of soft biological tissues is presented. The homogenized model describes the relationship between the macroscopic hereditary creep behavior and the microflow in a fluid-saturated dual-porous medium at the microscopic level. The micromodel is based on Biot’s system for quasistatic deformation processes, modified for the updated Lagrangian formulation to account for coupling the fluid diffusion through a porous solid undergoing large deformation. Its microstructure is constituted by fluid-filled inclusions embedded in the porous matrix. The tangential stiffness coefficients and the retardation stress for the macromodel are derived for a time-stepping algorithm. Numerical examples are discussed, showing the strong potential of the model for simulations of deformation-driven physiological processes at the microscopic scale.     

Flux in Porous Media with Memory: Models and Experiments

The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient–flux and the pressure–density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux.     

Upscaling the flow of generalised Newtonian fluids through anisotropic porous media

The homogenisation method with multiple scale expansions is used to investigate the slow and isothermal flow of generalised Newtonian fluids through anisotropic porous media. From this upscaling it is shown that the first-order macroscopic pressure gradient can be defined as the gradient of a macroscopic viscous dissipation potential, with respect to the first-order volume averaged fluid velocity. The macroscopic dissipation potential is the volume-averaged of local dissipation potential. Using this property, guidelines are proposed to build macroscopic tensorial permeation laws within the framework defined by the theory of anisotropic tensor functions and by using macroscopic isodissipation surfaces. A quantitative numerical study is then performed on a 3D fibrous medium and with a Carreau–Yasuda fluid in order to illustrate the theoretical results deduced from the upscaling.     

Dependency of Tortuosity and Permeability of Porous Media on Directional Distribution of Pore Voids

Single-phase fluid flow in porous media is usually direction dependent owing to the tortuosity associated with the internal structures of materials that exhibit inherent anisotropy. This article presents an approach to determine the tortuosity and permeability of porous materials using a structural measure quantifying the anisotropic distribution of pore voids. The approach uses a volume averaging method through which the macroscopic tortuosity tensor is related to both the average porosity and the directional distribution of pore spaces. The permeability tensor is derived from the macroscopic momentum balance equation of fluid in a porous medium and expressed as a function of the tortuosity tensor and the internal structure of the material. The analytical results generally agree with experimental data in the literature.     

Calculation of the permeability and apparent permeability of three-dimensional porous media

In this study, creeping and inertial incompressible fluid flows through three-dimensional porous media are considered, and an analytical–numerical approach is employed to calculate the associated permeability and apparent permeability. The multiscale homogenization method for periodic structures is applied to the Stokes and Navier–Stokes equations (aided by a control-volume type argument in the latter case), to derive the appropriate cell problems and effective properties. Numerical solutions are then obtained through Galerkin finite-element formulations. The implementations are validated, and results are presented for flows through cubic lattices of cylinders, and through the dendritic zone found at the solid–liquid interface during solidification of metals. For the interdendritic flow problem, a geometric configuration for the periodic cell is built by the approximate matching of experimental and numerical results for the creeping-flow problem; inertial effects are then quantified upon solution of the inertial-flow problem. Finally, the functional behavior of the apparent permeability results is analyzed in the light of existing macroscopic seepage laws. The findings contribute to the (numerical) verification of the validity of such laws.