Table Of ContentFunctional Approach to Nonlinear Models of Water Flow in Soils
MATHEMATICAL MODELLING:
Theory and Applications
VOLUME 21
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Functional Approach to
Nonlinear Models of
Water Flow in Soils
by
Gabriela Marinoschi
Institute of Mathematical Statistics and Applied Mathematics,
Romanian Academy, Bucharest, Romania
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4879-3 (HB)
ISBN-13 978-1-4020-4879-1 (HB)
ISBN-10 1-4020-4880-7 (e-book)
ISBN-13 978-1-4020-4880-7 (e-book)
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Contents
Foreword ...................................................... ix
Introduction-motivation ....................................... xi
Part I Modelling water infiltration in soils
1 Brief overview of unsaturated flow concepts ............... 3
1.1 Some basic definitions in the unsaturated flow............... 3
1.2 Richards’ equation....................................... 6
1.3 Presentation of the empirical hydraulic models .............. 11
1.4 Comments.............................................. 15
2 Settlement of the mathematical models of nonhysteretic
infiltration ................................................ 17
2.1 Physical context and mathematical hypotheses .............. 18
2.2 Strongly nonlinear saturated-unsaturated diffusive model ..... 22
2.3 Weakly nonlinear saturated-unsaturated diffusive model ...... 28
2.4 Quasi-unsaturated model ................................. 29
2.5 Degenerate models....................................... 33
2.6 Extensions of the functions below the field capacity .......... 34
2.7 Dimensionless form of the diffusive models.................. 35
2.8 Comments.............................................. 36
Part II Analysis of infiltration models
3 Basic existence theorems for evolution equations
with monotone operators in Hilbertsp.a.c.e.s.................. 43
3.1 The semigroup approach ................................. 43
3.2 Nonlinear m-accretive operators in Hilbert spaces............ 46
3.3 The Cauchy problem within the semigroup approach......... 48
vi Contents
3.4 The Cauchy problem within the variational approach ........ 60
3.5 Comments.............................................. 65
4 Functional approach to the quasi-unsaturated infiltration
model ..................................................... 67
4.1 Basic hypotheses for the quasi-unsaturated model ........... 68
4.2 Preliminary results ...................................... 70
4.3 Weakly nonlinear conductivity. Homogeneous Dirichlet
boundary conditions ..................................... 76
4.4 Strongly nonlinear conductivity. Homogeneous Dirichlet
boundary conditions..................................... 101
4.5 Weakly nonlinear conductivity. Nonhomogeneous Dirichlet
boundary conditions..................................... 105
4.6 Comments..............................................128
5 Functional approach to the saturated-unsaturated
infiltration model ..........................................133
5.1 Basic hypotheses for the saturated-unsaturated model........134
5.2 The approximating problem ..............................141
5.3 The original problem.....................................166
5.4 The weak solution in the pressure form.....................177
5.5 Existence of the free boundary ............................185
5.6 Uniqueness of the weak solution ...........................198
5.7 Comments..............................................200
6 Specific problems in infiltration............................205
6.1 Analysis of the diffusivity-degenerate model.................205
6.2 Analysis of the porosity-degenerate model ..................209
6.3 Analysis of an infiltration hysteretic model..................224
6.4 Comments..............................................238
Part III Inverse problems in infiltration
7 Identification of the boundary conditions from recorded
observations ..............................................243
7.1 Basic concepts in the theory of optimal control..............243
7.2 The identification problem settlement ......................245
7.3 Identification using time average observations ...............248
7.4 Case of a plane soil surface ...............................271
7.5 Identification problem using final time observations ..........273
7.6 Comments..............................................276
Contents vii
Part IV Appendix
A Background tools ..........................................281
A.1 Some definitions and results in Banach spaces...............281
A.2 Lp spaces and Sobolev spaces .............................284
A.3 Vectorial distributions and Wk,p spaces ....................291
A.4 Operators in Banach spaces...............................295
A.5 Convex functions and subdifferential mappings ..............299
A.6 Various formulas ........................................303
References.....................................................305
Index..........................................................313
Foreword
... a pure mathematician does
what he can do as well as he should,
whilst an applied mathematician does
what he should do as well as he can...
(Gr. C. Moisil
Romanian mathematician, 1906-1973)
Flows in porous media were initially the starting point for the study which
has evolved into this book, because the acquirement and improving of know-
ledgeabouttheanalysisandcontrolofwaterinfiltrationandsolutespreading
arechallenginganddemandingpresentissuesinmanydomains,likesoilscien-
ces, hydrology, water management, water quality management, ecology. The
mathematical modelling required by these processes revealed from the begin-
ning interesting and difficult mathematical problems, so that the attention
was redirected to the theoretical mathematical aspects involved. Then, the
qualitative results found were used for the explanation of certain behaviours
of the physical processes which had made the object of the initial study and
for giving answers to the real problems that arise in the soil science practice.
In this way the work evidences a perfect topic for an applied mathematical
research.
Thisbookwaswrittenintheframeworkofmyresearchactivitywithinthe
Institute of Mathematical Statistics and Applied Mathematics of the Roma-
nianAcademy.SomeresultswereobtainedwithintheprojectCNCSIS33045/
2004-2006, financed by the Romanian Ministry of Research and Education.
In a preliminary form, part of the results included here were lecture
notes for master and Ph.D. students during the scientific stages (November-
December 2003 and May-June 2004) of the author at the Center for Optimal
Control and Discrete Mathematics belonging to the Central China Normal
University in Wuhan.
ix
x Foreword
The book addresses mathematicians, applied mathematicians and all re-
searchersinterestedinmathematicalproblemssusceptibletobesolvedwithin
the semigroup and variational approaches, in particular applied to ground-
water flows, and can be used as a basis for a graduate course in Applied
Mathematics.
This work is a result of the suggestion made by Professor Viorel Barbu,
to whom I also owe my initiation in this elegant domain of mathematics and
theunderstandingoftheperspectivewhichthefunctionalapproachconfersto
appliedproblems.Itakethisopportunitytoexpressmygratitudeforthefruit-
fuldiscussionsandobservations,aswellasforthepermanentencouragements
provided during the elaboration of this work.
I am indebted to Professor Mimmo Iannelli for the helpful mathematical
discussions I have had with him in the last years.
Also, I would like to thank Springer and in particular Prof. R. Lowen for
the publication of my book, and Marlies Vlot, Marieke Mol and Werner
Hermens for their kind assistance.
Gabriela Marinoschi January 2006, Bucharest
Introduction-motivation
This book is a work of applied mathematics focusing on the functional study
ofthenonlinearboundaryvalueproblemsrelatingtothewaterflowinporous
media and it was written with the belief that the abstract theory may be
sometimes easier and richer in consequences for applications than standard
classicalapproachesare.Thevolumedealswithdiffusiontypemodelsandem-
phasizes the mathematical treatment of their nonlinear aspects. An unifying
approachtodifferentboundaryvalueproblemsmodellingthewatermovement
in porous media is presented, and the high degree of generality and abstrac-
tion,kepthoweverwithinreasonablelimits,isrewardedbytherichnessofthe
results obtained in this way.
Water infiltration, and transport and diffusion of solutes in porous media,
are two underground flow processes whose study is of great importance due
tothestrongimpacttheyhaveonthehumanlife.Watersuppliedbyrainfalls,
irrigationorleakagesfromotherwaterbodiescrossesthesoilcarryingwithit
various soluble substances provided by surface or subsurface sources and can
reach thephreatic aquifer from where thedrinkablewater isextracted. These
processes evolve in time and in principle the problem is to detect the state of
the system at any given time when knowing its initial state and the laws that
governthesystemchangeswithtime.Themathematicalmodelforsuchasys-
tem is an evolution equation, in the most cases a partial differential equation
(PDE). Water infiltration and solute dispersion are not the only processes
that develop in the underground, but we restricted the study especially to
the first one because it is basic and the models describing it are fundamental
in the theory of parabolic PDEs, being valid, with slight modifications, for
solute dispersion, too.
The practical demands lie generally on the quantitative and numerical
study of the system evolution, namely on its solution, but the mathematical
point of view occurs before this and directs the interest to the proof of facts
that allow the approach to make sense. These are the existence, uniqueness
and the regularity properties of the solutions. The final intention is to apply
theseresultstorealphysicalsystems,soitmustbetakenintoaccountthatthe
xi
