Filtration in Porous Media and Industrial Application: Lectures given at the 4th Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy August 24–29, 1998 PDF
Preview Filtration in Porous Media and Industrial Application: Lectures given at the 4th Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy August 24–29, 1998
Lecture Notes in Mathematics 1734 Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris Subseries: Fondazione C. I. M. E., Firenze Adviser: Arrigo Cellina regnirpS Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore oykoT M. S. Espedal A. Fasano A. Mikeli6 Filtration in Porous Media and Industrial Application Lectures given at the 4th Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy August 24-29, 1998 Editor: A. Fasano enoizadnoF C.I.M.E. r e g ~ n i r p S Authors and Editor Magne Espedal Antonio Fasano Department of Mathematics Department of Mathematics "U. Dini" University of Bergen University of Florence Johs. Brunsgt. 12 Viale Morgagni 67a 5008 Bergen, Norway 50134 Florence, Italy magne.espedal @mi.uib.no [email protected] Andro Mikeli6 Laboratoire d' Analyse Num6rique, B~t. 101 Universit6 Lyon I 43 Bd. du 11 novembre 1918 69622 Villeurbanne Cedex, Frace [email protected] Cataloging-in-Publication Data applied for CIP-Einheitsaufnahme - Die Deutsche Bibliothek Filtration in media porous and held application : industrial in August Cetraro, Italy, 24 CIME. / - Fondazione 29, 1998 . M.S Especial, ... Heidelberg New York ; Barcelona ; Berlin ; - ; A. Fasano. Ed.: London ; Hong Kong ; Milan Tokyo Paris Singapore : ; ; ; ,,regnirpS 2000 (Lectures given at the ... session of Centro the Intemazionale Estivo (CIME) Matematico ... ; 1998,4) notes (Lecture in mathematics Fondazione Vol. Subseries: ; : 1734 CIME) 3-540-67868-9 ISBN Mathematics Subject Classification (2000): 76S05, 76M50, 74A, 35R35, 76M10 ISSN 0075-8434 ISBN 3-540-67868-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10725018 41/3142-543210 - Printed on acid-free paper preface This volume contains the notes of some of the lectures given at the CIME Course held in Cetraro during the week 24-29 August, 1998. The course was organized by myself and by H. Van Duijn (CWI, Amsterdam) and was addressed to an audience of 33 people from various countries. The subjects covered a large spectrum in the area of filtration theory, emphasizing its importance for applications and its specific interest for mathematicians. The course included the following series of Lectures: .1 Magne Espedai (University of Bergen, Norwey): Reservoirs flow models (3 lectures) 2. Antonio Fasano (University of Florence, Italy): Filtration problems in various industrial processes (6 lectures) 3. Peter Knabner (University of Erlangen, Germany): Reactive transport in porous media (4 lectures) 4. Andro Mikeli6 (University of Lyon, France): Homogenization theory in porous media (6 lectures) .5 Hans Van Duijn (CWI, Amsterdam, The Netherlands): Nonlinear model in subsurface transport (6 lectures). The idea was to present a substantial overview of this very diversified re- search subject, giving at the same time an idea of its impact on industry and of its rich mathematical content. Therefore the stress was not only on flue mathematical problems, but also on modeling and on numerical analysis. Here we give a short account of each series of lectures. 1) M. Espedai (lecture notes included). This is a remarkable overview of nu- mericai methods widely applied in the engineering of oil recovferroym reservoirs, with particular reference to the extraction technique consisting in the displace- ment of oil by another liquid. A self-contained exposition having a great value as a reference point for the study of this class of problems, including the math- ematical background. 2) A. Fasano (lecture notes included). Three main themes were addressed: the so-called espresso-coffee problem (a familiar experience but a formidably compli- cated problem), various questions in the manufacturing of composite materials, and a very recent subject concerning injections of liquid in diapers. 3) P. Knabner. A series of lectures devoted to groundwater accompanied by chemical reactions with two main (opposite) applications: soil contamination (organic, inorganic, radioactive), soil remediation (flushing, venting, microbial degradation). Typical situations have been sketched and the main theoretical results summarized, together with the illustration of numerical schemes. Iv 4) A. Mikelid (lecture notes included). A fundamental question in filtration theory is the following: how to justify the empirical laws (e.g. Darcy's law relating the fluid velocity to the pressure gradient) on the basis of the general laws of fluid dynamics. The answer to this question lies in homogenenization theory: solve the fluid dynamical problem in a cell which, repeated with peri- odicity provides an idealized model of the porous medium and then pass to the limit when the size of cell tends to zero. Here we have an extended explanation of this theory of remarkable clarity. 5) H. Van Duijn. These lectures were also dealing with underground transport of chemical substance, but with special emphasis on solutions. The ultimate aim is to study coastal aquifers and interfaces between fresh water and salt water. This problem has become classical, but there are new developments too (e.g. brine transport) that have been illustrated. Of particular relevance is the problem of cusps formation at fresh/salt water interface which may cause the breakthrough of salt water with obvious negative consequences when pumping drinkable water from the aquifer. The lecture notes here collected are preceded by a short introduction to some basic facts concerning the flow of fluids through porous media, providing a general mathematical framework to this class of phenomena. In conclusion I can say that this CIME Course was a unique opportunity of presenting classical and modern problems in a field that is rich of real applica- tions and is also a great source of nice and profound mathematical problems. I wish to thank the Scientific Council of CIME for inviting me to organize this Course and to express my gratitude to the other lecturers for their enthusiastic response and their extremely valuable contribution. A. ~'hsano (Director of the Course) Table of Contents A. Fasano Some General Facts about Filtration Through Porous Media M. Espedal, K.H. Karlsen Numerical Solution of Reservoir Flow Models Based on Large Time Step Operator Splitting Algorithms A. Fasano Filtration Problems in Various Industrial Processes 79 A. Mikeld~ Homogenization Theory and Applications to Filtration ThroughP orous Media 721 Antonio Fasano Some general facts about filtration through porous media Filtration of fluids through porous media is a phenomenon occurring in sev- eral natural processes, as well as in an incredible variety of technological appli- cations. A porous medium is typically composed by solid particles (grains) and interstitial spaces (pores) that are connected, permitting a fluid to flow. Such a definition is quite generic, since it includes coarse materials, like granules, and soils with extremely fine texture, like clays, with a great range of intermediate possibilities. However, the solid component is not necessarily a loose pack of grains, but it can also be connected, forming a skeleton. Examples can be found among artificial media (e.g. ceramics) and material products. Moreover the solid com- ponent may be rigid or it can be deformed to some extent (like in the extreme case of sponges). In normal cases a porous medium can be modelled as a continuum by consid- ering a representative volume element on which we define averaged quantities, next extended to the whole domain occupied by the system as regular functions of the space coordinates. The volume element must be large enough to contain a sufficiently high number of grains and pores, but much smaller than the typical size of the system. In this way we can define a basic geometric property, the porosity ¢, i.e. the volume fraction occupied by the pores. When the flowing fluid is incompressible we may likewise introduce the saturation S, i.e. the frac- tion of the volume available to the flow (i.e. the pore volume) which is occupied by the fluid. These are both numbers between 0 and .1 We say that the medium is saturated when S -- 1, and unsaturated if 0 < S < 1. When S = 0 the medium is dry. We may have multicomponent flows (with a partial saturation defined for each component) and also quite complex porous structures (highly heteroge- neous, with fissures, etc.). Even the case of grains which are in turn porous have been considered [19] [18]. Making an exhaustive list of filtration problems is a hopeless task, but we may roughly distinguish two large classes: filtration of gases and filtration of liquids, although this distinction is rather artificial, because flowing liquids quite often displace air and, when the media is not saturated, the presence of vapor may be important (like e.g. in drying processes). The most normal situation ni the former class is the flow of a gas in a porous medium previously occupied by another gas (typically air) under small concentration gradients. This process is basically diffusive and we are not going to deal with it. Completely different is the problem of the expansion of a gas through a porous medium which is void. This is a celebrated problem that leads to the so-called porous media equation, a parabolic equation which degenerates at the propagation front. The degeneracy gives rise to typically non-parabolic features like waiting times and finite speed of the invasion front. Basic contributions in this field are the papers ,2[ 4], that have been the seed of a massive research work. Filtration of liquids is the real subject of this CIME Course. By liquids we mean Newtonian incompressible fluids although theories for non-Newtonian flows through porous media have been developed [8]. For the basic theory we refer to classical books like ]6 []7[ [5], [21], [26] and to the survey papers [24] [25] [23] [16] [17]. It may be useful to expose some fundamental fact, about incompressible flows through porous media. The fundamental experimental law, very well known and most frequently used to describe the flow of liquids through porous media is Darcy's law, which dates back to 1856 [15]. The experimental basis of Darcy's law is parallel to Fourier's law for heat conduction and Fick's law for diffusion: consider the flow through a homogeneous layer under a prescribed pressure gradient. The quantity to be measured is the volume of liquid passing through the unit section per unit time (volumetric velocity). For a vertical saturated flow in a horizontal layer, if p(0) -p(b) is the driving pressure difference between the top surface (z = 0) and the bottom surface (z = L), L being the layer thickness, and p the liquid density, Darcy's law states that the volumetric velocity q is q = k(P(O) LP(L) + pg), (1) where g is the gravity acceleration. We may have an upward flow (q < 0) if 5p -Z- < "gP-- The positive constant k is the hydraulic conductivity of the system. Experi- menting with different liquids one realizes that k can be expressed as K k = --, (2) # when K is a typical constant of the medium (permeability) and # is the liquid viscosity. Extending (1) to a general porous media, the volumetric velocity becomes a vector and (1) is replaced by q = - 1gV(p - pgz), (3) # where K is the permeability tensor and z the downward directed vertical coor- dinate. Of course we have K = KI if and only if the medium is isotropic. The quantity p - pgz plays the role of a potential and is called piezometric head. For an incompressible saturated flow on an indeformable porous medium, the conservation of mass is expressed by div q = 0. (4) Typically, for an isotropic medium (3) and (4) take the respective forms q = -kV(p - pg ), (5) zxp = o, (6) A denoting the Laplacian operator. When the flow is not saturated, then k is a decreasing function of saturation S (vanishing for S = 0) and mass conservation is no longer expressed by (4), but it becomes O(¢S) + div q = 0 (7) (in the absence of sources or sinks). In this case (5) and (7) are no longer sufficient to describe the flow, but we .ytirallipac need a relationship between S and p, which is physically provided by p > Ps (saturation pressure) The function S = S(p) is equal to 1 for and is Ps. monotonically decreasing for p < Once it has been wet a porous medium will contain at least a thin film of liquid surrounding the grains, no matter how Ps. far p is from For unsaturated flows, with ¢ =constant, (5) and (7) lead to a parabolic equation for pressure, possibly degenerating at the interface with the saturated region. One can also consider the limit case in which the effect ofc apillarity is totally disregarded (such an approximation makes sense in particular in the presence of wetting front). a sufficiently fast In this extreme situation the medium is either saturated or dry and in invasion problems the wetting front coincides with the saturation front. The prototype of a penetration problem with no capillarity is the Green- Ampt model, developed in 1911 for the injection of water into a dry soil [20]. This is a one-dimensional model of a vertical flow in which the pressure pene- s(t) tration front z = is the atmospheric pressure (p = 0), while at the inflow surface (z = 0) the injection pressure is specified as p = Po > 0. Since the advancing front is a material surface, its velocity is given by Darcy's law, hence pa (s) zi---bk- + kp9. = 02p Because of incompressibility equation (6) applies, i.e. zO 2 = 0, and therefore we findp(z,t)=po(1 - s-~) , and (8) reduces to a simple first order o.d.e. =o. (9) pgs, Setting a = we find Po k(pg) a - log(1 + a) - - - t. (10) Poe
The list of books you might like
