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About this book

The first Symposium on Recent Advances in Problems of Flow and Transport in Porous Media was held in Marrakech in June '96 and has provided a focus for the utilization of computer methods for solving the many complex problems encountered in the field of solute transport in porous media. This symposium has been successful in bringing together scientists, physicists, hydrogeologists, researchers in soil and fluid mechanics and engineers involved in this multidisciplinary subject. It is clear that the utilization of computer-based models in this domain is still rapidly expanding and that new and novel solutions are being developed. The contributed papers which form this book reflect the recent advances, in particular with respect to new methods, inverse problems, reactive transport, unsaturated media and upscaling. These have been subdivided into the following sections: I. Numerical methods II. Mass transport and heat transfer III. Comparison with experimentation and simulation of real cases This book contains reviewed articles of the top presentations held during the International Symposium on Computer Methods in Porous Media Engineering which took place in Giens (France) in October 1998. All of the presentations and the optimism shown during the meeting provided further evidence that computer modeling is making remarkable progress and is indeed becoming an essential toolkit in the field of porous media and solute transport. I believe that the content of this book provides evidence of this and furthermore gives a comprehensive review of the theoretical developments and applications.

Table of Contents


Numerical methods


The simulation of the transport of contaminants in groundwater flow: error estimates for a finite volume scheme

We present some error estimates for a finite volume scheme applied to a nonlinear advection-reaction-diffusion equation, where the velocity is deduced from Darcy’s law together with a uniformly parabolic equation for the liquid pressure. This system describes the transport of a contaminant in groundwater flow.
R. Eymard, R. Herbin, D. Hilhorst, N. Ramarosy

Inertial-Flow Anisotropy in Oblique Flow through Porous Media

The extension of the Darcean momentum equation to the inertial-flow is considered using the results of direct numerical simulation of flow through two-dimensional ordered porous media. Using oblique flows, the inertial-flow regime is examined for Reynolds numbers (based on the unit-cell length) up to 300. The results show that the inertial-flow regime is marked at the very beginning by a low-Reynolds number subregime (0 ≤ R e ≤ 10), where the deviation from the Darcean-flow pressure drop is quadratic in R e (cubic in the Darcean velocity). After a relatively extended intermediate-Reynolds number subregime (10 ≤ R e ≤ 50), a high-Reynolds subregime is observed (for R e > 50) which seems to be linear in R e (quadratic in the velocity). It is shown that for ordered arrangements the Darcean-isotropic structures become inertially anisotropic, i.e., the pressure gradient and the Darcean velocity vectors are not parallel in the inertial-flow regime, even though they are in the Darcean regime. The Darcean-anisotropic structures remain anisotropic in the inertial-flow regime. For strongly inertial flows, we study the transition to unsteady periodic solutions. The values of the critical Reynolds number depend on the angle θ that the incoming flow makes with the x-axis. The critical reynolds number increases with the angle of the flow. The R e dependence of the angle a that the drag force makes with the x-axis don’t confirm the new results of [5] The authors claim that for high Reynolds numbers, for low angle oblique flows (< 12°), the angle α reaches a maximum and the drag force becomes horizontal, and that for high angle oblique flows (> 12°, < 45°), the drag force makes a 45° angle with the x-axis. Concerning the Darcean-anisotropic structures, we present results which contradict their conclusions.
M. Firdaouss, P. Tran

An Adaptive Method for Characteristics-Finite Element Method for Solute Transport Equation in Unsaturated Porous Media

An adaptive method for the solution of equation modeling the transport of solute by dispersion and advection in unsaturated porous media is presented. In many applications, when the peclet number is quite large, advection dominates diffusion and the concentration often develops sharp fronts. So finite elements are combined with the method of characteristics to treat this problem. Because a good approximation of velocities is necessary to calculate the advective term of the equation, the flow equation is approximated by parabolic mixed finite element method.
An a posteriori error estimator is presented for adaptivity. This estimator yield upper and lower bounds on the error measured in the energy norm with constants which do not depend neither on meshsize nor on time step.
Numerical examples presented here indicate that this method gives nearly exact approximations of sharp fronts.
M. Gabbouhy, Z. Mghazli

Estimation of parameter geometry

It is well known that to use a groundwater model as a predictive tool, model parameters have to be calibrated against measurements (heads, concentrations or other state variables). For this reason, groundwater literature is plenty of results and theory on inverse problems. Usually, the physical parameters are discretized using a parameterization defined through some variables (the so called model parameters) that can be estimated (calibrated). Most of the effort on inverse modeling has been done in estimating the values of the model parameters, but not their spatial variability, that customarily is considered as fixed except in some few works.
Among different alternatives, we have chosen zonation as the way of discretizing spatial variability of parameters, that is one of the most employed ways of parameterization. In this paper a methodology for the estimation of both parameter values at the zones and their shape (geometry) is presented. A model structure identification criterion (developed on the framework of the bayesian theory) has been defined to account for the consistency between model and real system. This criterion leads to a optimization problem with a objective function that is minimized using several integer algorithms. The differences and similarities between our proposed methodology and other approaches are highlighted, as well as their respective limitations. Examples are included to show the applicability and restrictions of the methodology.
J. Heredia, A. Medina Sierra, J. Carrera

Fast Real Space Renormalization for Two-Phase Porous Media Flow

Upscaling of hydraulic conductivity data in space is one of the important problems in hydrological modelling.
In this article the basics of Real Space Renormalization (RSR) for up-scaling hydraulic conductivity are introduced. The RSR procedure is described on a 2 — D hydraulic conductivity grid. RSR is chosen because it can cope with correlated and anisotropic media. An up to now unanswered problem is the non-unique relationship between water fluxes and water contents or heads.
In special cases analytical solutions for two phase flow systems allow RSR to be as fast as single phase RSR. In the general case, numerical solution of the small scale flow equations are required. A new solution procedure which reduces the amount of computations is described.
Marc R. Hoffmann

Solution of Convection-Diffusion Problems with the Memory Terms

In this paper 1 an approximation solution of the following convection diffusion problem is discussed
$${\partial _t}b\left( u \right) + div\left( {\bar F\left( {t,x,u} \right) - k\left( {t,x,u} \right)\nabla u} \right) = f\left( {t,x,u,s} \right),s\left( {t,x} \right) = \int\limits_0^t {K\left( {t,z} \right)\psi \left( {u\left( {z,x} \right)} \right)dzin\left( {0,T} \right)} \times \Omega ,$$
where Ω ⊂ ℝ N is a bounded domain with a Lipschitz continuous boundary Ω, T < ∞.
J. Kačur

Optimal Control Approach for a Flow in Unsaturated Porous Media

The aim of this paper, dealing with the management of fresh water, is to present an optimal control approach for the steady flow in a rectangular aquifer there are two wells. The classical problem is a free boundary problem. After a change of variable transformation, we obtain an optimal control problem in a fixed domain, where the control appears in a Dirichlet boundary condition and in the coefficients of the state equation. After a finite element discretization, we obtain an optimization problem where the cost function is differentiable and the gradient could be computed analytically.
C. M. Murea, J.-M. Crolet

Splitting The Saturation and Heterogeneity for Time Dependent Effective Phase Permeabilities

The method of splitting the saturation and heterogeneity is developed as a new fast numerical tool to compute the effective relative permeabilities (e.r.p), including in case of time-dependent, dynamic, permeabilities. Description of dynamic model, explanation of dynamic capillary non-nequilibrium effects, tensor properties of e.r.p., solution of cell problems, calculation of e.r.p. and qualitative analysis are presented.
Mikhail Panfilov, Alexei Tchijov

Fuzzy Simulation of Waterflooding

A New Approach to Handling Uncertainties in Multiple Realizations
Evaluation of field performance and a long-term production forecast require considerable resources spent on [numerical] reservoir simulation. Uncertainty in reservoir characterization and future prospects concerning oil price, operating cost, etc. advocates for sound sensitivity analysis which requires even more resources.
Most of the problems associated with the uncertainty of evaluation can be handled either by running a sensitivity analysis or by probabilistic methods. The latter being extensively used in the past for resovling numerous engineering problems are often limited by lack of data with statistical properties. Moreover, in many engineering applications amount of accessible information is often not sufficient for its processing by statistical methods. In such cases fuzzy methods seem to be more appropriate technique to solve the problems.
From a mathematical perspective, the difference between probabilistic and fuzzy methods is based on the definition of membership function that does not necessarily rest on probability, but rather on relative preference among the members of the reference set. As a result, probability theory evaluates the likelihood of outcomes, while fuzzy mathematics models the possibility of occurence. Fuzzy methods can handle uncertainty directly, without running the sensitivity analysis. Another advantage of fuzzy technique is that it links uncertainty of input data to the reliability estimation of the final decision.
From a computational point of view, fuzzy methods, being based on rules resembling axioms of deterministic mathematics, are much faster as compared to stochastic methods. However, little effect can be gained when applying those methods to a volumetric reserve estimate, material balance equation, decline curve analysis, etc. Considerable effect can be foreseen in handling problems related to reservoir characterization. In areas of [numerical] reservoir simulation fuzzy technique outperforms probabiliastic methods in the most effective way and seems to have no rivals.
Examples of the application of fuzzy methods to petroleum engineering problems like resources and reserves estimate, reservoir description and characterization, reservoir simulation, optimization and decision making, have been discussed earlier in the literature[16, 11, 5, 19, 20]. However, little attention has been paid to numerical simulation of a multiphase flow in porous media. The emphasis in this paper is given to reservoir simulation problems illustrated by comparizon of classical deterministic and fuzzy solutions to a two-phase flow of incompressible fluids in porous media known as a fractional flow or a Buckley-Leverett problem.
A. B. Zolotukhin

Mass transport and heat transfer


A Two-Domain Model for Infiltration into Unsaturated Fine-Textured Soils

In this paper, a one-dimensional model handling the vertical infiltration process within a compacted clay column submitted on its top to a constant hydraulic head is presented. The model is based on the flow mechanism described by Philip (1968) and Jayawickrama (1990). The infiltration is supposed to take place simultaneously in two different porosity-domains. The first is the soil intact matrix and the second represents a net of interconnected channels presenting much higher hydraulic conductivities. The flow in the two domains is described by Richards’ pressure-based formulation and an exchange term reflects the interaction on the basis of Darcy’s law. A time-centered Finite Difference scheme is used to accommodate the transient nature of the problem. No attempt is made here to reproduce the geometry of the two domains, thus no assumption is needed concerning macropore-distribution, only an estimation of macropore hydraulic conductivity is required. Numerical simulation of laboratory infiltration tests yielded good agreement with measured data.
A. Abdallah, F. Masrouri

Numerical study of heat and mass transfer in a cubical porous medium heated by solar energy “Boubnov-Galerkin method”

This paper presents The modelization of heat and mass transfer in cubical reactor of solar adsorption cooling machine. The reactor is heated by solar energy and contains a porous medium constituted of activated carbon reacting by adsorption with ammonia. From real solar data, the model computes the performances of the machine and shows the existence of the optimal dimensioning of the reactor.
For the resolution of the equations describing the coupling between heat and mass transfer, we have adapted a “Boubnov-Galerkin” method combined to an iterative process, this method provides a continuous distribution of the temperature and adsorbed mass. The convergence of the method is discussed and the numerical results are compared with the results provided by finite-difference method.
Considering the rapidity of convergence and the order of Algebraic system ( That is generally inferior to 10), the proposed method appeared to be very effective in solving such problem.
A. Al Mers, A. Mimet, M. Boussouis

Cylindrical Reactor Performance Evaluation for a Solar Adsorption Cooling Machine

In this paper, a numerical simulation of solar adsorption cooling machine is presented for the region (Tetouan) real climate conditions. From the computed collected mass, we determine the produced cold quantity and the performance coefficient for typical clear — sky daily global radiation for each month. The numerical results are in good agreement with experimental ones and have been used to design a solar installation producing cold.
H. El Aroudam, A. Mimet

RETRASO, a parallel code to model REactive TRAnsport of SOlutes

Reactive transport of solutes in porous media has received an increasing attention due to a growing of the social sensibilitation on environmental and health problems caused by contamination of solutes. It is important, then, the characterization of reactive transport in order to predict accurately the behavior of solutes.
RETRASO is a code capable to simulate REactive TRAnsport of SOlutes. The code solves the reactive transport problem by substituting the chemical equations into a source/sink term of the transport equation leading to a system of non-linear partial differential equations. This system is discretized by applying the finite element method and the obtained discretized system is solved with the Newton-Raphson method (NR). For interesting cases, this size could be huge so, large computing time is required.
A parallel version of RETRASO has been developed to reduce the simulation time. The method used to parallelize is a SPMD (Single Program Multiple Data) with message passing communication for distributed memory architecture. As a result of the CPU profiling analysis; parallelization was focused on the following most consuming CPU time (more than 90%) parts: 1) the building of the Jacobian matrix of the NR linear system, and 2) solving the system itself. As communication between processors should be optimized for message passing models, a specific algorithm that minimizes the communication needs was designed for part (1). For solving the system, a linear solver module was developed at CEPBA (Centro Europeo de Paralelización de Barcelona, UPC).
The performance of the parallelized version of RETRASO was checked in a SGI ORIGIN 2000 machine with a PVM version based on sockets.
I. Benet Llobera, C. Ayora, J. Carrera

A Numerical Study of the Coupled Evolutions of Micro-Geometry and Transport Properties of Simple 3D Porous Media

Simulating, understanding and predicting the evolution during mineral diagenesis of porous rocks physical properties is a very complex problem. When properties like effective diffusivity, formation factor or permeability are considered interest can be mainly focussed on the coupled evolutions of micro-geometry and transport properties.
D. Bernard, G. Vignoles

Pore-scale modelling to minimize empirical uncertainties in transport equations

As computer methods and the underlying numerical procedures improve dramatically with time the demands on transport equations to reflect the real physical conditions are also on the increase. Empiricisms in transport equations cannot always satisfactorily describe the extremal physical conditions enforced onto them by numerical strains and the present work is aimed at minimizing the need for empirical expressions in favor of simple but physically plausible models of the various processes taking place in porous media. Pore-scale modeling is used for closure of volume averaged transport equations. The model addresses the interstitial geometry and the physics of the particular phenomena to provide closure for the general volume averaged equations.
J. P. Du Plessis

Modeling Contaminant Transport and Biodegradation in a Saturated Porous Media

The present paper describes a numerical model, which allows to compute solute transport and biodegradation in a saturated porous media. Mathematical formulation of such processes leads to a set of non- linear partial differential equations coupled to ordinary differential equations. The transport equation is approximated by a finite volume scheme whereas biodegradation equations are treated separately as a system of ordinary differential equations. Numerical results for biorestoration using Monod kinetics are presented.
S. A. Kammouri, M. El Hatri, J. M. Crolet

Water Phase Change and Vapour Transport in Low Permeability Unsaturated Soils with Capillary Effects

A discussion of water phase change in unsaturated soils that develop capillary effects is first carried out in the paper. A distinction between the GR (geothermal reservoir) and the NUS (nonisothermal unsaturated soil) approaches is performed. Several aspects concerning advective and non advective fluxes of vapour are described secondly and some relationships concerning the case of mass motion in a closed system subjected to temperature gradients derived. Since the structure of unsaturated clays changes with moisture content, in order to correctly simulate the coupled phenomena induced by temperature gradients a model for intrinsic permeability as a function of humidity is required. A preliminary version of the model is presented and applied to interpret a laboratory test by means of a numerical simulation using CODE_BRIGHT.
S. Olivella, A. Gens, J. Carrera

Behaviour of Infiltration Plume in Porous Media

Adequacy between Numerical Results and a Simplified Theory
The proposed numerical code simulates the movement of a fluid as well as the transport of a non-reactive pollutant into a saturated porous media (2D configuration). The model uses a combination of the mixed hybrid finite element method and the discontinuous finite element method. Coupling between flow and transport is carried out by an equation of state. In the mixing zone, the density is assumed to vary as a function of concentration. In a saturated media, the transport of an incompressible fluid is described by a set of initial and boundary conditions and by a system of equations constituted by Darcy’s law, the continuity equation, Fick’s law and the advection-dispersion equation. Precision in estimating the velocity field, which determines pollutant propagation, is essential. Results obtained by classical numerical methods (conforming finite element method or classical finite difference methods) are often not very satisfactory due to the diffusive character of these methods. In order to compensate for these disadvantages, a combination between the mixed hybrid finite element technique and the discontinuous finite element technique has been implemented. When applied to the problem under consideration, this technique makes it possible to simultaneously estimate the pressure field and the velocity field (hydrodynamic module) as well as the dispersive flux and concentration field (mass transport module). Furthermore, application of these methods makes it possible to preserve the mass balance at the scale of each element and to ensure the continuity of the normal components of the velocity and dispersive flux from one element to another. In order to analyse the infiltration of a salt solute punctually injected into a porous medium, a comparison between a simplified theory and numerical simulations is presented. The density contrast between the two miscible fluids, as also the injection flow rate, play an important role. Studies carried out on 2D physical models have shown the existence of a steady-state regime located in the middle of the mixing zone. With such observations, the equations describing the transport phenomenon can be modified in order to lead to a simplified analytic solution. This result experimentally established is bounded by numerical verifications.
C. Oltean, M. A. Buès

A comparison of two alternatives to simulate reactive transport in groundwater

Numerical simulation of reactive transport in groundwater (that is, transport of species undergoing chemical reactions) requires the solution of a large number of mathematical equations, which can be highly non linear. This can cause many problems of numerical nature. Therefore, the choice of a method to solve these equations is important. Two types of methods exist: The Direct Substitution Approach (DSA), based on Newton-Raphson, and the Picard or Sequential Iteration Approach (SIA). The advantage of the DSA is that it converges faster and is more robust than the SIA. Its disadvantage is that one has to solve simultaneously a much larger set of equations than for the SIA. We applied both methods to several examples and compared computational behaviour. Results showed that, for chemically difficult (that is, highly non linear) cases, the SIA often requires very small time steps leading to excessive computation times, whereas the DSA does not show this inconvenience due to its robustness. On the other hand, for chemically simple cases but with grids of many nodes, the DSA tends to be less favourable because of the size of the set of equations to be solved.
Maarten W. Saaltink, Jesús Carrera, Carlos Ayora

Case studies


Modeling of Organic Liquid Entrapment and Surfactant Enhanced Recovery in Heterogeneous Media

This paper provides an overview of modeling investigations designed to ex­plore the influence of textural heterogeneities on the migration, persistence, and surfactant enhanced solubilization of organic liquid contaminants in the subsur­face. Multiphase flow and transport simulators are used to model the entrapment and surfactant enhanced recovery of perchloroethylene from saturated media. A laboratory-based linear driving force expression is used to represent solubilization. Simulation results illustrate the influence of local permeability variations and mass transfer limitations on remediation performance. Additional model simulations of a controlled laboratory sandbox experiment are used to highlight the strengths and weaknesses of current modeling approaches.
Linda M. Abriola

Application of the Back-Tracking Method to the Definition of Sanitary Zones of Catchwork Protection for Drinking Water Supply

The implementation of sanitary zones of catchwork protection for drinking water supply has become a requirement stated by the legislator for years. However, this is a complex and technical issue which has often led to inadequate or questionable-and questioned-recommendations or implementations of sanitary zones. Sometimes, their implementations were even delayed.
However, new systems of mathematical groundwater flow modeling allow to determine these sanitary zones, at reasonable cost and in any case objectively, while referring to widely recognized normative criteria. This is particularly true for travel times to the catchwork — from the boundaries of the sanitary zone — that most of European laws laid down at 50 days.
This paper presents a method to calculate time contours around a catchwork by backtracking pathways. This method, known as the back tracking method can easily adapt to all kinds of flow mathematical models. Thereafter, we analyze and discuss examples of results obtained in real situations.
Marc Bonnet, François Bertone

Experimental and Numerical Investigation of Porosity Variations in Saline Media Induced by Temperature Gradients

Preliminary experimental results and a numerical interpretation of porosity variations in saline media induced by temperature gradients are presented in this paper. The experimental results show a pattern of porosity variation with the same features as obtained theoretically in Olivella et al (1996a). An interpretation of the experimental results using CODE_BRIGHT (Olivella et al, 1996b) has been carried out. Since it is possible to reproduce the experimental results with the numerical approach it can be concluded that the mechanism for porosity changes induced by temperature gradients proposed in a preceding work effectively takes place.
S. Castagna, S. Olivella, A. Lloret, E. E. Alonso

CLOG: A Code to Address the Clogging of Artificial Recharge Systems

Clogging of groundwater Artificial Recharge systems is a very ubiquitous problem that affects numerous recharge facilities and can have dramatic technological and economic impacts. A quantitative approach to clogging is presented in this paper by describing a new comprehensive numerical model.
The current version of this code, termed CLOG, has been obtained by using two existing codes, one for multiphase flow and the other for reactive transport, and by adding specific clogging subroutines. As a result the model is capable of treating the basic clogging processes: transport of particles, bacterial growth (attached to the medium), chemical reactions (homogeneous and heterogeneous kinetics, with biocatalysed paths), gas flow, and compaction. Therefore, the code is integrating the elementary processes, as determined by numerous experiences.
This paper summarises the numerical structure of CLOG, focusing on the implementation of the clogging-related issues. A real example is enclosed with the aim of remarking its capabilities and, also, discussing which are the main limitations. Finally, a rapid discussion of future trends and modifications is done.
A. Pérez-Paricio, I. Benet, M. W. Saaltink, C. Ayora, J. Carrera

Groundwater Flow Modelling of a Landslide

The development of pore water pressures is an important issue in the behaviour of landslides under rainfall infiltration. Thus, the understanding of groundwater flow is an important achievement. The modelling of these rainfall events has been attempted so as to develop a system to predict large movements in landslides triggered by rainfall.
The aim of this paper is the presentation of the results obtained in the groundwater flow modelling of Vallcebre landslide (Eastern Pyrenees) under both saturated and unsaturated conditions. This is a traslational landslide with two clay layers sliding over a limestone bed. The movements are concentrated in the deepest clay layer. There are some scarps and extension zones along the landslide. Some high intensity rainfall events cause a fast raising of water levels and a reactivation of movements. This behaviour has been verified in the monitoring devices: inclinometers, wire extensometers and piezometers in the landslide zone and a rain gauge in the basin of Vallcebre. A GPS survey device is also monitoring surface movements.
The landslide has been modelled by the finite element method and some rainfall events have been tested to calibrate some flow parameters like permeability or storativity. Also, an inverse analysis has been performed in order to estimate in a systematic manner those parameters. The model has been attempted in both two-dimensional and three-dimensional analyses. The effect of preferential flow paths produced by tension cracks has been included in the model as well. The results of the analyses are consistent with the measurements, and show the importance of preferential flow paths and boundary conditions on the simulation of the landslide behaviour.
J. Rius, J. Mora, A. Ledesma

Trace Gas Absorption by Soil

Simulation Study on Diffusion Processes of Trace Gases, CO, H2, and CH4 in Soil
A two-layered diffusion model was applied to the uptake process of trace gases as CO, H2, and CH4 which are utilized by soil microorganisms or enzymes assuming that its uptake obey first-order kinetics about its concentration. Analytical solutions for mono-layered model exhibit that the physical property as gas diffusivity in soil is more important for uptake process than emission process. The numerical simulation shows that the deposition of CO, H2, and CH4 are limited by the combination of transport process and the localization of the soil uptake zone, within 0.06cm s−1 for CO and 0.1cm s−1 for H2, respectively, which are in reasonable consistence with field measurements.
S. Yonemura, M. Yokozawa, S. Kawashima, H. Tsuruta


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