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About this book

Porous media, and especially phenomena of transport in such materials, are an impor1ant field of interest for geologists, hydrogeologists, researchers in soil and fluid mechanics, petroleum and chemical engineers, physicists and scientists in many other disciplines. The development of better numerical simulation techniques in combination with the enormous expansion of computer tools, have enabled numerical simulation of transport phenomena (mass of phases and components, energy etc. ) in porous domains of interest. Before any practical application of the results of such simulations can be used, it is essential that the simulation models have been proven to be valid. In order to establish the greatest possible coherence between the models and the physical reality, frequent interaction between numericians, mathematicians and the previously quoted researchers, is necessary. Once this coherence is established, the numerical simulations could be used to predict various phenomena such as water management, propagation of pollutants etc. These simulations could be, in many cases, the only financially acceptable tool to carry out an investigation. Current studies within various fields of applications include not only physical comprehension aspects of flow and energy or solute transport in saturated or unsaturated media but also numerical aspects in deriving strong complex equations. Among the various fields of applications generally two types of problems can be observed. Those associated with the pollution of the environment and those linked to water management. The former are essentially a problem in industrialized countries, the latter are a major source of concern in North-Africa.

Table of Contents


Mathematical methods


Convergence of a Finite Volume Scheme for a Parabolic Degenerate Equation

In this note we prove the convergence of explicit and implicit finite volume schemes for the numerical solution of the Stefan-type problem u t — Δφ(u) = v, together with the homogeneous Neumann boundary condition.
R. Eymard, T. Gallouët, D. Hilhorst, Y. Naït Slimane

Error Estimate for the Finite Volume Approximate of the Solution to a Nonlinear Convective Equation

This paper is mainly concerned with the study of an error estimate of the finite volume approximation to the solution u L (ℝ N X ℝ) of the equation u t +div(v f(u)) = 0, where v is a vector function depending on time and space. A “h 1/4” error estimate for an initial value in BV(ℝ N ) is shown for a large variety of finite volume monotoneous flux schemes, with an explicit or implicit time discretization. For this purpose, the error estimate is given for the general setting of solutions of approximate continuous entropy solutions, where the error is expressed in terms of measures in ℝ N X ℝ. All the proofs of this paper can be found in [7].
R. Eymard, T. Gallouët, M. Ghilani, R. Herbin

An Error Estimate for the Approximate Solution of a Porous Media Diphasic Flow Equation

In this paper we present an error estimate for the approximate solution of the nonlinear hyperbolic equation u t + div (f(u(x, t))v(x)) = 0 by an implicit finite volume scheme, using an Engquist-Osher numerical flux. We show that the error is of order \(\sqrt {k + \sqrt h } \) , where h and k are respectively the space and the time steps size parameters. The error estimate shows that the convergence of this scheme is possible with an unbounded CFL condition. This result is extended to other numerical fluxes and explicit scheme in [3].
M. Ghilani

An Analogue of Schwartz’s Method for the Construction of the Green Function of the Zaremba Problem and Its Application in Underground Hydrodynamics

In the present paper the classical problem of underground hydrodynamics, modeling the flow of incompressible fluid in porous medium, (see e.g. [1,2]) is studied.
A. I. Ibragimov, A. A. Nekrasov

New approaches


Size and Double-Layer Effects on the Macroscopic Behavior of Clays

The modeling of clays has been developed in two directions: macroscopic constitutive modeling and microscopic phenomenological studies. Starting with the linear poroelastic models of Biot and Terzaghi, several constitutive models have been developed at the macroscopic level. Creep and stress relaxation and rate effects, have been studied using viscoelastic or elasto-viscoplastic constitutive models (see Liang and Ma [9], Perzyna [15], Cristescu [4]) and yield, irreversible deformation and hysteretic soil behavior has been modeled using plasticity models (Roscoe and Burlan [16]). At the microscopic level recent experimental work (Low [11], Newman [14], Velde [18]) has contributed to the understanding of the behavior of clays. The influence of the salt concentration in the water phase, the double-layer forces and the structural changes in the water under the influence of the particle surface has also been studied, to explain for the nonlinear behavior of clays.
Bogdan Vernescu

Parameter’s Sensitivity in Water Storage Modelling for Unsaturated Soils

In many branchs of soil science, hydraulic engineering and agriculture, modelling of water infiltration in unsaturated soil is a major problem. The study of two-phases flow in unsaturated zones has started since the announcement of the famous law of Darcy for infiltration flow (1856). This law was extended by Buckingham (1907) to some infiltration flows in unsaturated porous media. The resulting equation, combined with the mass conservation law, involves a strong non linearity in some terms which generates some numerical disruptions that affect the simulation results.
K. Benhalima, A. Benhammou, E. K. Lakhal, B. Dahhou

Mathematical Investigation for Groundwater Pollution

In this study we give a mathematical formulation of the phenomenon of groundwater pollution, wich is due to a non—Darcy flow in an unsaturated zone and in the presence of solute transport and of heat transfer. The obtainde model is a system of coupled partial nonlinear equations of evolution type. The principal aim of this investigation is to treat theoretically the system obtained (existance and unicity of solution), within an appropriate functional framework. Thus we give the necessary mathematical conditions to be verified by the system’s data.
M. El Hatri

Solution of Degenerate Parabolic Problems by Relaxation Schemes

We present a new numerical approximations to nonlinear and degenerate parabolic systems of the form
$$ {\partial _t}b\left( u \right) - \nabla a\left( {t,x,u,\nabla \left( u \right)} \right) = f\left( {t,x,\beta \left( u \right)} \right) $$
J. Kačur

Mass transport and heat transfer


Transport in Saturated Porous Media

Adequacy between Theoretical, Numerical and Physical Modellings
A physical 3D model, making it possible to study the movement of a non-reactive pollutant in saturated and homogenous porous media, taking into account the contrast of density and viscosity was perfected on a laboratory scale. A sodium chloride solution was chosen as a pollutant because it enables concentration measurements using electric conductivity to be taken. Salt water was introduced in the section upstream from the porous media strata (one or two dimensional configuration).
Numerical modelling was used in combination with this approach. The computer code was based on the mixed hybrid finite elements method. At first it was tested on examples found in literature (Henry’s problem and Elder’s problem) and subsequently used for the simulation of experiments carried out on the physical model. The numerical model links the equations that describe flow in saturated porous media as well as solute transport equation, taking into account density and/or viscosity. The variation of density in the mixture is assumed to be linear as a function of the concentration.
C. Oltean, M. A. Buès

Perfection of the Simulation of Freshwater / Saltwater Interface Motion

We recall briefly the mathematical model that describes the abrupt freshwater/saltwater interface motion in a coastal aquifer. Then we give its coupled Galerkin formulation and the numerical methods solving both the steady and unsteady state problems. An algorithm decoupling the resolution in the unsteady state case is proposed and an efficient physical generation of boundary conditions at the coast is detailed. Several tests and comparisons of solutions show good results.
D. Esselaoui, Y. Loukili, A. Bourgeat

Numerical Dispersivity in Modelling of Saltwater Intrusion into a Coastal Aquifer

A numerical dispersion is often observed in the simulation of flow and mass transport in porous media. This phenomenon, which is well known by numericians, is characterized by the occurrence of negative values for the solute concentration and it is important to master this parasite effect in a simulation software. During the validation of such a software, this disturbance was studied in the case of the Henry’s problem. Several technical possibilities have been reviewed and a better solution can be obtained.
J. M. Crolet, F. Jacob

Theoretical Study and Experimental Validation of Transport Coefficients for Hydrocarbon Pollutants in Aquifers

Recent work has been done in order to better understand the transport of hydrocarbons in the saturated zone. Phenomenological correlarions were proposed for the mass transfer coefficient. Unfortunately there is little agreement between those correlations as the mass transfer coefficient varies over several orders of magnitude. In this paper, a new method is proposed to estimate this transport coefficient in a binary system. Dissolution and transport experiments of trapped residual non aqueous phase liquids (NAPLs) in water saturated homogeneous porous media were carried in a laboratory column. Saturation fields were measured by the gamma ray attenuation technique and pollutant concentration was measured by gas chromatography. A macroscopic model based on the averaged pore-scale mass balance equations was used to simulate experiments. The model takes into account convection, dispersion and interfacial mass transfer. The mass transfer coefficient was calculated in pore-scale periodic unit cells containing the solid phase, an immobile phase (NAPL), and a flowing phase (water) by the means of averaging theory results. Confrontation of experimental and numerical results shows that the model predicts well the total dissolution time and the shape of the dissolution front. An estimate of the mass transfer coefficient, and the construction of a representative (of the geometry of trapped NAPL blobs) unit cell are therefore possible.
G. Radilla, A. Aigueperse, M. Quintard, H. Bertin

Heat and Mass Transfer in Cylindrical Porous Medium of Activated Carbon and Ammonia

Theoretical and experimental approach
In the framework of research in the thermal cooling machines of adsorption, we have developed a model of heat and mass transfer in porous medium of activated carbon reacting by adsorption with ammonia.
For the initial conditions and for a given incident heat flux, we compute in every point and at each time the temperature of the porous medium and the adsorbed mass.
The validity of the model has been tested with experimental installation simulating an adsorption machine working within a large temperature range (-20 to 250°C) and pressure range (0 to 2.5 106 Pa).
A. Mimet, J. Bougard

Transient Natural Convection in a Square Porous Cavity Submitted to Different Time-Dependent Heating Modes

Natural convection induced in a square porous enclosure with its horizontal sides insulated and its vertical sides submitted to various periodic heating modes is studied numerically using the Darcy model. Parameters of the problem are the Rayleigh number (100 ≤ R ≤ 500), the amplitude of the variable temperatures) (0 ≤ a ≤ 1) and its (their) period (0.001 ≤ τ ≤ 1.2). The effect of the heating modes on the flow field and heat transfer is studied. It is found that the flow becomes multicellular and the heat transfer can be significantly improved for all the variable heating modes considered. The best improvement. is obtained when the active wall temperatures vary in opposition of phase. The resonance phenomenon is also observed when the period of the imposed temperature(s) is varied.
T. Daoui, M. Hasnaoui, A. Amahmid

Multiphase studies


Adaptive Mesh for Two-Phase Flow in Porous Media

When one fluid displaces others, the predominating physical process occurs along the moving interface between the fluids. This paper is devoted to the description of a computational algorithm for two-phase immiscible, incompressible flows in an adaptive composite grid to track the moving front. This algorithm is based on an IMPES method applied to a convergent and conservative finite volume scheme for the pressure equation, together with the finite difference schemes, upwind and MUSCL, for the transport equation on grids with local refinement.
M. Saad, H. Zhang

Upscaling Two-Phase Flow in Double Porosity Media: Nonuniform Homogenization

Problem of upscaling the transport through highly heterogeneous media characterized by a period of heterogeneity ε, a ratio of global permeabilities ωK, and a ratio of the order of capillary forces ω c is studied. Upscaling corresponds to the limit transition when ε tends to zero at small ωK and large ω c . The limit of upscaling transition is shown to be not equivalent to simple homogenization, in case of the high heterogeneity. In this case more general notion as nonuniform homogenization becomes more constructive. The outline of general nonuniform homogenization method is developed, which leads to decomposition of initial heterogeneous medium on two continuous homogeneous media, in limit. The method is applied to the problem of the two-phase transport through dual-porosity media.
A macroscopic model is constructed for one ratio between determining parameters. First, it allows to effective computing of macroscopic relative permeability tensors. Then, two macroscopic capillary pressure corresponding to each part of the medium result from upscaling procedure. One of them shows the non-equilibrium properties and is related to another one by kinetic equation, where relaxation time depends on saturation.
M. Panfilov

Flow of Multicomponent Gas Condensate Mixtures in Fractured Porous Media

Modeling a gases and liquids flow in fractured porous reservoirs was significant developed in last decades. To the present time there are already various mathematical models of fractured porous medium. The most known of them is double-porosity model offered by Barenblatt, Zheltov and Kochina [1], Warren and Root. P.1. [2]. The main assumptions of this model are the follows. Global flow occurs primarily through the high-permeability, low-effective-porosity fracture system surrounding matrix rock blocks. The matrix blocks. contain the majority of the reservoir storage volume and act as local source to the fracture system. There is pseudo-stationary exchange between fractures and porous blocks (a flow rate is proportional to a difference of average pressure in medium). Later, Kazemi [3], Duguid and Lee [4] and Evans [5] incorparated the double-porosity concept into numerical model. Improvement of a double-porosity approach and also double-permeability concept then was made by many authors. As a generalization of the double-porosity concept, Pruess et all [6,7] developed the MINC method, which treats the multiphase and multidimensional transient flow in both fractures and matrix blocks by a numerical approach. The MINC method involves discretization of matrix blocks into a sequence of nested volume elements. In this way, it is possible to resolve in detail the gradients (pressure and etc.) that drive iuterporosity flow.
A. Gritsenko, R. Ter-Sarkisov, A. Shandrygin

Two-Scale Percolation-Difference Method for Simulation of Transport with Trapping in Porous Media

We examine the phenomenon of isolated traps formation during two-phase displacement in porous media. Two main problems are connected with its description. First, the problem of non-locality is an obstacle, which results from the origin of traps. However, some algorithms to solve it are known. In this paper we study the second problem, dealing with the minimum scale of simulation and consisting in the following. A disordered medium generates traps of various sizes. For any finite scale of simulations h, traps, whose length is smaller than h, can not be caught in such a model. This requires to minimize the simulation step to zero.
We propose a solution of this problem in form of two-scale network method, applied to saturation transport equations, which allows to take into account of desired trap volume. It consists of three essential parts: a) involving two steps of different scale for the pressure field and for the saturation field; b) approximation of saturation transport equation by its analytical solution in scale of one step; c) checking for the non-locality condition.
In result, the pressure field is computed by finite difference method and determination of the saturation field is reduced to an invasion percolation algorithm of lgcal fronts tracing.
The method is realized to compute Darcy’s flow in heterogeneous porous medium, as well as Poiseuille motion in capillary network.
I. Panfilova, J. Muller


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