Table Of ContentLecture Notes in Mathematics 1734
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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]
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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
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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
