Table Of ContentMULTIPHASE FLUID
FLOW IN POROUS
AND FRACTURED
RESERVOIRS
YU-SHU WU
Department of Petroleum Engineering
Colorado School of Mines
Golden, CO, USA
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DEDICATION
This book is dedicated to my family, teachers, colleagues
and students.
PREFACE
Thisbookfocusesonthephysicsofmultiphasefluidflowanddisplacement
in porous and fractured media as well as quantitative approaches and
analyses for describing such physical processes in reservoirs. The book is
intendedtocomplementtheexistingliteraturebypresentingnewadvances
and updated developments in multiphase fluid flow in porous media. The
material of this book is based primarily on (1) a series of peer-reviewed
papers, published by me or with co-authors and (2) the course notes that I
haveusedtoteachundergraduateandgraduatecoursesonpetroleumreservoir
engineeringandmultiphasefluidflowinporousmediaattheColoradoSchoolof
Mines. The publications that this book is based on are related to the
research on the subject of multiphase fluid flows in porous and fractured
media, whichI have carriedout orbeen involved with since thelate1980s
attheUniversityofCalifornia,Berkeley,California(CA);HydroGeoLogic,
Inc., Reston, Virginia; the Lawrence Berkeley National Laboratory,
Berkeley, CA; and the Colorado School of Mines, Golden Colorado.
The book can be used as a textbook or reference for senior under-
graduate and graduate students in petroleum engineering, hydrogeology or
groundwater hydrology, soil sciences, and other related engineering fields,
such as civil and environmental engineering. It can also serve as a reference
book for hydrogeologists, petroleum reservoir engineers, and other engi-
neers and scientists working in the area of flow and transport in porous
media.
The content of the book is organized to cover fundamentals of
multiphase fluid flow in porous media. It discusses the physical processes
and principles governing multiphase porous-medium flow using Darcy’s
law,relative permeability,andcapillary-pressureconcepts.Thisbookuses
the black-oil model as an example of immiscible multiphase fluid flow to
discuss flow-governing equations and approaches for their solution to
quantity flow and displacement processes in reservoirs. Specifically, this
book presents the extensions of the classical Buckley–Leverett fractional
flow theory to one-dimensional linear and radial composite systems, to
analysis of immiscible displacement of non-Newtonian fluids in porous
media, and to non-Darcy displacement using Forchheimer and Barree and
Conway non-Darcy flow models. In addition, the book reviews the
concept, approach, and development for modeling multiphase flow in
xiii
xiv Preface
fractured porous media and multiphase fluid flow and heat transfer in
reservoirs. In an effort to include the new developments, the book also
presents mathematical formulations and numerical modeling approaches
for multiphase flow coupled with geomechanics and for flow in
unconventional petroleum reservoirs.
Yu-Shu Wu
Professor of Petroleum Reservoir Engineering
Department of Petroleum Engineering
Colorado School of Mines
Golden, Colorado, USA
Foundation CMG Chair in Reservoir Modeling
ACKNOWLEDGMENTS
I would like to thank Dr Zhaoqin Huang at the China University of
Petroleum (Huadong) for his tremendous help in preparing the manuscript
during his postdoctoral research at the Colorado School of Mines (CSM).
Inaddition,hehascontributedtheMATLABcodesforcalculationofmost
oftheanalyticalsolutionsformultiphaseflowinporousmediaofthebook,
which are provided in the appendixes. His MATLAB programs have
reproduced the plots from previous publications, and these programs will
findmoreapplicationsintheanalysisofmultiphaseflowinporousmediaas
well as the examination of the numerical simulation results. I could not
have completed this book without his support and help.
IwouldalsoliketothankmycolleagueatCSM,Dr.PhilipH.Winterfeld,
forhisthoroughtechnicalreviewofthemanuscriptandsharinghisexpertisein
reservoir simulation. I am highly indebted to Prof. Jun Yao at the China
UniversityofPetroleum(Huadong)forofferingandhostingmeforasabbatical
leaveinthefallof2014,whileItaughtseveralcoursesrelatedtomultiphasefluid
flowinporousmediaandhadthetimetowritethemajorityofthemanuscript.
I would like to take this opportunity to thank many of my current and
formercolleaguesandfriendsintheUSandChinawhohavemadethisbook
possible. Specifically, I would like to thank the faculty and students of the
Petroleum Engineering Department at CSM for their criticisms and sug-
gestions to improve the materials of the book. I am grateful to many of my
former colleagues in the Earth Sciences Division of the Lawrence Berkeley
National Laboratory, Berkeley, California, and in HydroGeoLogic Inc.,
Reston, Virginia, for the opportunities of working together on studies of
flow and transport in porous media over past decades.
I would like to thank the editorial and production staff of Elsevier for
their work and professionalism.
IalsoacknowledgefinancialsupportfromFoundationCMG(Computer
Modelling Group).
xv
CHAPTER 1
Introduction
1.1 BACKGROUND
Processes of flow and displacement of multiphase fluids though porous
mediaoccurinmanysubsurfacesystemsandhavefoundwideapplicationin
many scientific, technical, and engineering fields, such as petroleum engi-
neering,groundwaterandvadosezonehydrology,soilsciences,geothermal
energy development, subsurface contamination investigation and remedi-
ation, and subsurface resource storage or waste disposal. Because of the
needs in these areas, tremendous research and development efforts have
been devoted to investigating the physics of multiphase flow and transport
in porous and fractured media in the past century. As a result of multi-
disciplinaryresearchefforts,significantprogressandscientificadvanceshave
been made in the understanding of the dynamics of porous-medium flow
of multiple immiscible fluids.
The continual research and development in analyzing flow and trans-
port processes of multiphase fluids in porous media over the past half
centuryhavebeenhelpedandacceleratedbyconcertedeffortsoflaboratory
experiments on porous-medium samples at various scales, field studies,
theoretical analysis, and mathematical modeling. Many quantitative ap-
proaches and models have been developed and applied successfully to
describe, understand, and predict flow behavior of multiphase fluids in
reservoirs. During the same period, our knowledge in understanding
porous-medium flow phenomena and our ability to apply those un-
derstandingstosolvingpracticalproblemshavebeensignificantlyimproved
using the quantitative modeling approach. Mathematical modeling ap-
proaches have matured and become standard practices in subsurface natural
resource development, storage system design, contamination investigation,
and remediation scheme evaluation. In addition, numerical modeling
studies are routinely carried out to investigate physical phenomena for
insights as well as to optimize field project design and operation, produc-
tion,orcleanup,whichinmanycasesmaynotbepossiblewithoutthehelp
of a numerical modeling tool.
MultiphaseFluidFlowinPorousandFracturedReservoirs
ISBN978-0-12-803848-2 ©2016ElsevierInc.
1
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2 MultiphaseFluidFlowinPorousandFracturedReservoirs
1.2 LITERATURE REVIEW, DEVELOPMENT, AND ADVANCE
Humankindmayhavenoticedflowinporousmediafromthebeginningof
civilization, when they stood on beach sands facing tides, looked at rain
percolating into the ground, or began to grow plants in soils.
The development of the theory of flow through porous media, as a
branchofappliedscience,beganwithHenryDarcy(1856)whodetermined
experimentally the proportionality of pressure gradient and flux, now
knownasDarcy’slaw.Darcy’slawwasfirstdevelopedforwaterinsaturated
flow, but has since been extended to multiphase and unsaturated flow, and
to incorporate other phenomena, such as surface tension, gravity, fracture
flow,chemicalreaction,andchangingfluidproperties.Suchextensioninto
various coupled processes has been made possible by the use of numerical
computation, which in turn depends on powerful computer technology
and advances in computational mathematics. Darcy’s law and its extensions
have been applied to a wide range of activities, including agriculture,
environmental management, and most notably, petroleum reservoir
engineering.
Darcy’s law has been the foundation for studies of flow and transport
phenomena through porous media. The empirical Darcy’s law can also be
derived from the Navier–Stokes equations via a volume averaging method
or homogenization theory (e.g., Neuman, 1977; Whitaker, 1986). Even
though originally obtained only for describing flow of a single-phase fluid,
Darcy’s law has been extended and generalized to describe the flow of
multiple,immisciblefluids(e.g.,Scheidegger,1974;HassanizadehandGray,
1979a,b).ThemultiphaseextensionofDarcy’slawhasbeenusedexclusively
as the basis for quantitative studies of dynamics of multiphase fluid flow in
porous and fractured media.
Most significant contributions to understanding multiphase flow in
porous media have been made since the 1920s (Willhite, 1986), 1930s
(Richards, 1931), and 1940s (Buckley and Leverett, 1942) with the rapid
advances in the petroleum industry, groundwater hydrology, and soil sci-
ences. The fundamental understanding of immiscible flow and displace-
ment of Newtonian fluids in porous media was initially contributed by
Buckley and Leverett (1942) in their classical study of fractional flow the-
ory. The Buckley–Leverett solution provides insight into immiscible-fluid
displacement processes, describing a saturation profile advancing with a
sharp front along the flow direction, while capillary pressure and gravity
effectsareignored.Effectsofgravityandcapillarypressureonalinearwater
Introduction 3
flood have since been included by later studies (e.g., Fayers and Sheldon,
1959; Hovanessian and Fayers, 1961; Codreanu et al., 1966). Some special
analytical solutions of immiscible displacement including the effects of
capillarypressurewere obtainedintheliterature,suchasYortsos andFokas
(1983), Chen (1988), and McWhorter and Sunada (1990).
Many extensions, generalizations, and improvements to Buckley–
Leverett theory have been made to obtain and enhance understandings of
complicated flow behavior of multiple phases in porous media. In partic-
ular, the Buckley–Leverett fractional flow theory has been generalized and
appliedby various authors to studyEnhanced OilRecovery (EOR) (Pope,
1980), surfactant flooding (Larson, 1978), polymer flooding (Patton et al.,
1971), mechanisms of chemical methods (Larson et al., 1982), and alkaline
flooding (DeZabala et al., 1982). An extension to more than two immis-
cible phases using “coherence theory” was described by Helfferich (1981).
More recently, studies have extended the Buckley–Leverett solution to
flow in a composite, one-dimensional heterogeneous, composite-reservoir
system (Wu et al., 1993), to non-Newtonian fluid flow (Wu, 1990; Wu
et al., 1991, 1992; Wu and Pruess, 1996, 1998), and to non-Darcy
displacement of immiscible fluids in porous media (Wu, 2001, 2002a,b;
Wu et al., 2011a,b).
Most fundamentalsof thephysics offlowof multiphasefluidsin porous
media have been understood by laboratory experiments, theoretical anal-
ysis, mathematical modeling, and field studies (Collins, 1961; Bear, 1972;
Scheidegger, 1974). Analysis of porous medium flow processes relies
traditionally on Darcy’s law-based approaches, and application of such
analysis has provided quantitative methodologies and modeling tools for
many related scientific and engineering disciplines. Among them, one of
thebestbeneficiariesofthetheoryandadvancesdevelopedinflowthrough
porous media is perhaps petroleum reservoir engineering, in which calcu-
lationofoil,gas,andwaterflowandproduction,andassessmentofprimary
and secondary oil recovery and reservoir dynamics, are all based on the
physics of multiphase flow in reservoirs (Craft et al., 1959; Dake, 1983;
Prats, 1985; Willhite, 1986; Honarpour et al., 1986; Ahmed and
McKinney, 2011). Recent development of EOR applications, multiphase
flow under chemical flooding, supplementary fluid injection, or thermal
recovery is also modeled using various modifications of Darcy’s law and
multiphase flow concepts (Prats, 1985; Lake, 1989; Sorbie, 1990; Green
and Willhite, 1998).
4 MultiphaseFluidFlowinPorousandFracturedReservoirs
Oneofthemainreasonsforthesignificantscientificadvanceandrelated
technical development has been development and application of mathe-
matical modeling methodology. Numerical simulation of multiphase sub-
surface flow and transport phenomena may be one of the most important
developments in the earth sciences during the second half of the twentieth
century.Duetoitsgeneralityandeffectivenessinhandlingmultiphaseflow
and transport problems in porous and fractured media, the numerical
simulation technique has evolved to become the major tool used by sci-
entists and engineers in studies of flow and transport processes in a porous
medium. The continual development of numerical approaches has been
motivated by a variety of needs in many industries and earth sciences from
developing subsurface natural resources to addressing environmental con-
cerns.Sincethelate1950s,significantprogresshasbeenmadeindeveloping
and applying numerical simulation techniques in petroleum engineering
(e.g.,Peaceman,1977; Azizand Settari, 1979;Thomas,1981; Coats,1987;
Mattax and Dalton, 1990; Ertekin et al., 2001; Fanchi, 2005), in ground-
water literature (e.g., Huyakorn and Pinder, 1983; Istok, 1989; Zheng and
Bennett, 2002), and more recently in computational science (e.g., Chen
et al., 2006; Chen, 2007). It should be mentioned that the advances in
numerical simulation have benefited significantly from rapidly advancing
modern computer technology, hardware, and computational algorithms.
Numerical modeling approaches currently used for simulating coupled
multiphase flow and transport processes are generally based on methodol-
ogies developed for petroleum and geothermal reservoir simulation and
groundwater modeling. They involve solving fully coupled formulations
describing these processes using finite-difference or finite-element schemes
with a volume-averaging approach. Earlier researches on modeling multi-
phase flow in porous media were primarily motivated during the devel-
opment of petroleum reservoirs (Douglas et al., 1959; Peaceman and
Rachford, 1962; Coats et al., 1967) and geothermal reservoirs (Mercer
et al., 1974; Thomas and Pierson, 1978; Pruess, 1990, 1991). During the
same decades, problems involving unsaturated and two-phase flow and
transport in aquifers and subsurface soils were increasingly recognized and
investigated in groundwater hydrology and soil science. Many numerical
approaches were developed in parallel and applied to modeling flow
and transport phenomena in the vadose zone (e.g., Narasimhan and
Witherspoon, 1976; Cooley, 1983; Huyakorn et al., 1984; Morel-Seytoux
and Billica, 1985; Celia et al., 1990; Wu and Pruess, 2000; Looney and
Falta, 2000).
