Modeling Phenomena of Flow and Transport in Porous Media

761 Pages·2018·13.406 MB·English

by  Jacob Bear (auth.)

Theory and Applications of Transport in Porous Media
Jacob Bear
Modeling
Phenomena
of Flow and
Transport in
Porous Media
Theory and Applications of Transport in Porous
Media
Volume 31
Series editor
S.MajidHassanizadeh,DepartmentofEarthSciences,UtrechtUniversity,Utrecht,
The Netherlands
Founding series editor
Jacob Bear
More information about this series at http://www.springer.com/series/6612
Jacob Bear
Modeling Phenomena
of Flow and Transport
in Porous Media
123
JacobBear
Department ofCivil andEnvironmental
Engineering
Technion—Israel Institute of Technology
Haifa
Israel
ISSN 0924-6118 ISSN 2213-6940 (electronic)
Theory andApplications of Transportin PorousMedia
ISBN978-3-319-72825-4 ISBN978-3-319-72826-1 (eBook)
https://doi.org/10.1007/978-3-319-72826-1
LibraryofCongressControlNumber:2017961494
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To Siona,
Eitan and Jennifer,
Alon, Lior, Yoav, and Ido,
Iris and Moshe,
Sharon, Efrat, and Gilad,
with love.
Preface
The objective of this book is to present and discuss the construction of mathe-
matical models that describe phenomena offlow and transport in porous media as
encountered in many disciplines, e.g., civil and environmental engineering, soil
mechanics,groundwaterhydrology,petroleumengineering,drainageandirrigation
in agricultural engineering, reactors in chemical engineering, and geothermal
engineering. The book can also serve as a text for courses on modeling in these
disciplines.
Soil, sand, fissured and fractured rock, cemented sandstone, Karstic limestone,
ceramics, filters, foam rubber, bread, wood, concrete, and kidneys, are just a few
examples of the large variety of natural and man-made porous materials encoun-
tered in these disciplines.
Phenomena of transport of extensive quantities, like mass offluid phases, mass
of chemical species dissolved in fluid phases, momentum and energy of the solid
matrix and offluid phases, as well as electric charge, in porous medium domains,
are encountered in the disciplines mentioned above and in others not mentioned.
For example, civil engineers deal with the movement of moisture through con-
crete walls and with water flow through and under hydraulic structures. They also
deal with the movement of heat in structures, and with stresses under building
foundations.Hydrogeologistsdealwiththeflowofwaterinaquifers.Environmental
engineers deal with the transport of contaminants in the subsurface. Agricultural
engineers deal with the movement of water and solutes in the root zone during
irrigation and drainage. Heat and mass transport in packed bed reactors are
encountered in chemical engineering. Petroleum engineers deal with the flow of
(liquidandgaseous)hydrocarbonsandwaterinpetroleumreservoirs.Themovement
of fluids and chemical species in lungs and kidneys is studied in biomedical
engineering.
Partofthevoidspaceinthesubsurfacemaybeoccupiedbyanonaqueoustoxic
liquid phase. Components of such a liquid may dissolve in and move with per-
colating water, thus constituting a source for groundwater contamination. Often,
dissolved chemical species interact with each other and with the soil, especially
withtheclayandorganicfractionsofthelatter.Phenomena,suchasadsorption,ion
vii
viii Preface
exchange, dissolution, volatilization, and biological decay, continuously affect the
concentration of chemical constituents in the water. The solid matrix itself may be
porous, with tiny pores. Obviously, modeling all these phenomena is essential in
any effort to understand and combat subsurface contamination, or to produce
hydrocarbons from petroleum reservoirs.
Landsubsidence,asaconsequenceofpumping,orupheavalasaconsequenceof
fluid injection, are examples involving solid matrix deformation as a result of
changes inpressure distributioninthesubsurface. Non-isothermal flowoccurs and
the solid matrix may undergo deformation when hot or cold water is injected into
thesubsurfaceforenergystoragepurposes,whenCO ,capturedfrompowerplants’
2
emissions, is injected for disposal in depleted petroleum reservoirs, or in deep
brine-containing formations, and when compressed air or gas is injected into
anticlinal geological formations for short or long term storage purposes.
In all the above examples, extensive quantities (mass, momentum, energy, or
electric charge) are transported through porous material domains. The term trans-
port is used here to describe the movement, storage, and transformation of a
considered extensive quantity.
To investigate all these phenomena, as encountered when solving problems in
practice, mathematical models have to be constructed. By solving the latter, fore-
casts are obtained of the response of the considered system to excitations in the
form of changes in controllable source terms, or in boundary conditions. Pressure,
stress,strain,velocity,soluteconcentration,temperature,etc.,foreachphaseinthe
system,andfortheporousmediumasawhole,serveasexamplesofstatevariables.
In this way, models serve as an essential step in the solution of real-life problems.
In this book, we present and discuss various phenomena offlow, transport, and
transformation that take place in porous medium domains, and the construction of
conceptual and well-posed mathematical models that describe them. The objective
istodevelopmodelsofphenomenaoftransportinporousmediumdomainsthatare
based on visualizing such domains as continua.
Analytical solutions of the mathematical models considered here are seldom
possibleforproblemsofpracticalinterest.Theusualwayofsolvingmodelsofsuch
cases is to transform the mathematical model into a numerical one, and then use
computerprogramstosolvethelatter.Thus,themathematicalmodelsdevelopedin
this book should serve as the background to computer codes such as TOUGH
(developed at LBNL), NUFT (developed at LLNL), PFLOTRAN (open source,
multi-institution code), and many other freely available or commercial codes.
Numerical models and computer codes are not discussed in this book.
The book is divided into nine chapters:
Chapter1startswiththequestion“Whatisaporousmedium?”.Wethenpresent
the continuum approach, define microscopic, macroscopic, and megascopic levels
ofdescriptionofporousmediaandphenomenaoftransportthattakeplaceinthem.
The continuum description is obtained by employing the phenomenological
approach,orbyimplementinganaveragingorhomogenizationapproach.Wealso
mention the molecular (or nano-), microscopic (or pore-), macroscopic (or labo-
ratory,field-),andmegascopic(orformation-)scalesofdescriptionandtheissueof
Preface ix
upscaling. Finally the role, content and construction of transport models are
presented.
Chapter 2 introduces some fundamentals of thermodynamics of phases and
chemical species that are required for the presentation in this book. Among the
concepts defined and discussed are the pressure, density, chemical and other
potentials, Gibbs function, internal energy, enthalpy and capillary pressure.
Equations of state and their role in modeling are discussed.
InChap.3,westartbypresentingthegeneralbalanceequationforanyextensive
quantity, such as mass momentum and energy, of phases and chemical species
encountered in porous medium domains. The equation is stated at the microscopic
level,i.e.,atapointwithinafluidphasepresentinthevoidspace,andatapointin
the solid matrix. We make use of the phenomenological approach to present this
balance equation at the macroscopic level. The macroscopic balance equation can
alsobeobtainedbyvolumeaveraging,orbyhomogenization.Theseapproachesare
briefly described. We then derive equations for mass, momentum and energy.
Chapter 4 is devoted to the macroscopic fluid motion equation, emphasizing its
origin as a momentum balance equation. We start with the momentum balance
equation of a phase. Then, under certain simplifying assumptions, this equation is
reduced to the well known (linear) Darcy’s law. Other forms of the motion equa-
tion, e.g., which take into account also inertial effects, are also presented.
Chapter5isdevotedtomodelingmasstransportofasinglefluidphase,liquidor
gas, that completely occupies the void space. The core of such model is the mass
balance equation of the phase. The specific storativity is introduced to account for
fluid and solid matrix compressibility. Boundary and initial conditions are pre-
sented, leading to a well-posed flow (= mass transport) model.
InChap.6,weconstructmasstransportmodelsforthecaseofmultiple(i.e.,two
or three) fluid phases that occupy the void space simultaneously. Again, the
objective is to lead to complete, well-posed mathematical models, taking into
account the coupling that takes place between the phases.
Chapter7isdevotedtothetransportofchemicalspeciesdissolvedinfluidphases
thatoccupythevoidspace,withorwithoutchemicalreactions.Thephenomenonof
dispersionisintroducedandsolutefluxesduetodiffusion,dispersion,andadvection,
are discussed. Like any other model of transport, the core of the solute transport
model is the balance of mass of the considered chemical species. Various sources
andsinks,aswellaschemicalreactionsandinterphasetransfersarepresented.The
discussion leads to awell-posed model of thesolute transport problem.
Chapter8dealswithnon-isothermalsituations.Thisrequirestheconstructionof
a model that also describes the transport of energy, or heat, together with appro-
priate initial and boundary conditions.
InChap.9,wedealwithcasesoftransportofmassandenergyinwhichthesolid
matrix, as the entire porous medium domain, undergoes deformation. Models of
consolidationasaconsequenceofconstruction,andmodelsoflandsubsidencedue
toheavypumping,aredevelopedandpresented.Also,thepropagationofwavesin
a porous medium domain, in which soil deformability plays an essential role, is
briefly discussed.
x Preface
Three appendices, written by appropriate experts, are added at the end of the
book.AppendixA,writtenmainlybyProf.RaphaelSemiat(ChemicalEngineering,
Technion) demonstrates how the material presented in the book is applied in
Chemical Engineering, especially in the design of reactors. Appendix B, by Dr.
JonathanAjo-FranklinandDr.MarcoVoltolini(LBNL)dealswithmodelingatthe
microscopic level, while Appendix C, by Dr. David Trebotich (LBNL), discusses
the advent of supercomputing.
Thebookiswrittenasatextsuitableforgraduateandupperlevelundergraduate
students,andfor practitioners.Althoughoneofthemainobjectivesofthisbookis
toconstructmathematicalmodels,theamountofmathematicalknowledgerequired
is kept minimal. The emphasis is on understanding the physical and chemical
phenomenathattakeplaceinporousmediumdomains.Mathematicsisusedonlyas
a compact language for expressing these phenomena.
Much of the material included in this book has been developed in the period
2009–2017, during which I have been an active researcher in three projects:
MUSTANG. PANACEA and TRUST, conducted and funded by the European
Commissionwithinthe7thprogramoftheEuropeanUnionFrameworkProgramme
for Research and Innovation. I wish to acknowledge the support I received from
these three projects which dealt with CO disposal in deep saline water containing
2
geological formations. Obviously, an important part of these research programs,
conductedbyateamofexpertsandresearchersfrommorethan20universitiesand
industries,hasbeenthedevelopmentandextensiveuseofmodelsthatdescribethe
transport of mass–energy and momentum, as well as chemical reactions, encoun-
tered in such projects. These models can be used for predicting the spreading and
trapping of CO injected into geological formations. Many of these developments
2
are incorporated in this book.
Finally, I wish to thank Prof. Dr. S. Majid Hassanizadeh (Utrecht University,
NL), Prof. George J. Moridis (LBNL), Prof. Brian Berkowitz (Weizmann Institute
of Science, Israel), and Prof. Henry Power (Nottingham University, UK) for their
important suggestions and contributions. Special thanks are due to Dr. Peter
Lichtner of OFM Research for his expert advice and for devoting many days to
review parts of the book. Thanks are also due to Dr-Ing. Patrick Kurzeja (TU
Dortmund), Dr. Hui-Hai Liu (ARAMCO), Dr. Ehsan Nikooee (Shiraz University),
Yin Xiaoguang (Utrecht University), Dr. Vahid Joekar-Niasar (University of
Manchester), Dr. Chaozong Qin (Eindhoven Technical University), Prof. Alex
Cheng (University of Mississippi), Prof. Alex Furman (IIT, Israel), and Prof. Dr.
Ing.JenniferNiessner(HeilbronnUniversity)fortheirimportantcomments.Thanks
are also due to Diana Swantek (LBNL) for her devoted and meticulous work on
some of the figures presented in this book.
Haifa, Israel Jacob Bear
2017