1. introduction

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Chapter 1

Introduction

1.1 Petroleum Reservoir SimulationIn mathematical terminology, a porous medium is the closure of a subset of the Euclideanspace R

d (d = 1, 2, or 3). A petroleum reservoir is a porous medium that contains hy-drocarbons. The primary goal of reservoir simulation is to predict future performance of areservoir and find ways and means of optimizing the recovery of some of the hydrocarbons.

The two important characteristics of a petroleum reservoir are the natures of the rockand of the fluids filling it. A reservoir is usually heterogeneous; its properties heavily dependon the space location. A fractured reservoir is heterogeneous, for example. It consists of aset of blocks of porous media (the matrix) and a net of fractures. The rock properties in sucha reservoir dramatically change; its permeability may vary from one millidarcy (md) in thematrix to thousands md in the fractures. While the governing equations for the fracturedreservoir are similar to those for an ordinary reservoir, they have additional difficulties thatmust be overcome. The mathematical models presented in this book take into accountthe heterogeneity of a porous medium, and computational methods are presented for bothordinary and fractured media.

The nature of the fluids filling a petroleum reservoir strongly depends on the stage ofoil recovery. In the very early stage, the reservoir essentially contains a single fluid suchas gas or oil (the presence of water can be usually neglected). Often the pressure at thisstage is so high that the gas or oil is produced by simple natural decompression without anypumping effort at the wells. This stage is referred to as primary recovery, and it ends whena pressure equilibrium between the oil field and the atmosphere occurs. Primary recoveryusually leaves 70%–85% of hydrocarbons in the reservoir.

To recover part of the remaining oil, a fluid (usually water) is injected into some wells(injection wells) while oil is produced through other wells (production wells). This processserves to maintain high reservoir pressure and flow rates. It also displaces some of the oiland pushes it toward the production wells. This stage of oil recovery is called secondaryrecovery (or water flooding).

In the secondary recovery, if the reservoir pressure is above the bubble point pressureof the oil phase, there is two-phase immiscible flow, one phase being water and the other

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2 Chapter 1. Introduction

being oil, without mass transfer between the phases. If the reservoir pressure drops belowthe bubble point pressure, then the oil (more precisely, the hydrocarbon phase) is split intoa liquid phase and a gaseous phase in thermodynamic equilibrium. In this case, the flowis of black oil type; the water phase does not exchange mass with the other phases, but theliquid and gaseous phases exchange mass.

Water flooding is not very effective, and after this stage 50% or more of hydrocarbonsoften remain in the reservoir. Due to strong surface tension, a large amount of oil is trappedin small pores and cannot be washed out using this technique. Also, when the oil is heavyand viscous, the water is extremely mobile. If the flow rate is sufficiently high, instead ofproducing oil, the production wells primarily produce water.

To recover more of the hydrocarbons, several enhanced recovery techniques have beendeveloped. These techniques involve complex chemical and thermal effects and are termedtertiary recovery or enhanced recovery. Enhanced oil recovery is oil recovery by injectingmaterials that are not normally present in a petroleum reservoir. There are many differentversions of enhanced recovery techniques, but one of the main objectives of these techniquesis to achieve miscibility and thus eliminate the residual oil saturation. The miscibility isachieved by increasing temperature (e.g., in situ combustion) or by injecting other chemicalspecies like CO2. One typical flow in enhanced recovery is the compositional flow, whereonly the number of chemical species is given a priori, and the number of phases and thecomposition of each phase in terms of the given species depend on the thermodynamicconditions and the overall concentration of each species. Flows of other types involvethermal methods, particularly steam drive and soak, and chemical flooding, such as alkaline,surfactant, polymer, and foam (ASP+foam) flooding. All flows of these types in petroleumreservoir applications are considered in this book.

1.2 Numerical MethodsIn general, the equations governing a mathematical model of a reservoir cannot be solved byanalytical methods. Instead, a numerical model can be produced in a form that is amenableto solution by digital computers. Since the 1950s, when digital computers became widelyavailable, numerical models have been used to predict, understand, and optimize complexphysical fluid flow processes in petroleum reservoirs. Recent advances in computationalcapabilities (particularly with the advent of new parallel architectures) have greatly expandedthe potential for solving larger problems and hence permitting the incorporation of morephysics into the differential equations. While several books are available on finite differencemethods as applied to the area of porous media flow (Peaceman, 1977B; Aziz and Settari,1979), there does not appear to be available a book that examines the application of finiteelement methods in this area. The purpose of this book is to attempt to provide researchers inthis area, especially in petroleum reservoirs, with the current, state-of-the-art finite elementmethods.

Compared with finite difference methods, the introduction of finite element methods isrelatively recent. The advantages of the finite element methods over the finite differences arethat general boundary conditions, complex geometry, and variable material properties canbe relatively easily handled. Also, the clear structure and versatility of the finite elementsmakes it possible to develop general purpose software for applications. Furthermore, thereD

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1.3. Linear System Solvers 3

is a solid theoretical foundation that gives added confidence, and in many cases it is possibleto obtain concrete error estimates for the finite element solutions. Finite element methodswere first introduced by Courant (1943). From the 1950s to the 1970s, they were developedby engineers and mathematicians into a general method for the numerical solution of partialdifferential equations.

Driven by the needs for designing technologies for exploration, production, and re-covery of oil and gas, the petroleum industry has developed and implemented a variety ofnumerical reservoir simulators using finite element methods (e.g., see the biannual SPEnumerical simulation proceedings published by the society of petroleum engineers since1968). In addition to the advantages mentioned above, finite element methods have somepeculiar features when applied to reservoir simulation, such as in the reduction of grid ori-entation effects; in the treatment of local grid refinement, horizontal and slanted wells, andcorner point techniques; in the simulation of faults and fractures; in the design of stream-lines, and in the requirement of high-order accuracy of numerical solutions. These topicswill be studied in detail.

The standard finite element methods and two closely related methods, control vol-ume and discontinuous finite element methods, are covered here. Control volume finiteelement methods possess a local mass conservation property on each control volume, whilediscontinuous methods are closely related to the finite volume methods that have been uti-lized in reservoir simulation. Two nonstandard methods, the mixed and characteristic finiteelement methods, are also discussed. The reason for the development of mixed methodsis that in many applications a vector variable (e.g., a velocity field in petroleum reservoirsimulation) is the primary variable in which one is interested, and then the mixed methodsare designed to approximate both this variable and a scalar variable (e.g., pressure) simul-taneously and give a high-order approximation for both variables. The characteristic finiteelement methods are suitable for advection-dominated (or convection-dominated) problems.They take reasonably large time steps, capture sharp solution fronts, and conserve mass.Finally, adaptive finite element methods are described. These methods adjust themselvesto improve approximate solutions that have important local and transient features.

1.3 Linear System SolversFor a petroleum reservoir simulator with a number of gridblocks of order 100,000, about80%–90% of the total simulation time is spent on the solution of linear systems. Thusthe choice of a fast linear solver is crucial in reservoir simulation. In general, a systemmatrix arising in numerical reservoir simulation is sparse, highly nonsymmetric, and ill-conditioned. While sparse, its natural banded structure is usually spoiled by wells thatperforate into many gridblocks and/or by irregular gridblock structure. Furthermore, thematrix dimension M often ranges from hundreds to millions. For the solution of suchsystems, Krylov subspace algorithms are the sole option.

Over a dozen parameter-free Krylov subspace algorithms have been proposed forsolving nonsymmetric systems of linear equations. Three such leading iterative algorithmsare the CGN (the conjugate gradient iteration applied to the normal equations), GMRES(residual minimization in a Krylov space), and BiCGSTAB (a biorthogonalization methodadapted from the biconjugate gradient iteration). These three algorithms differ fundamen-D

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tally in their capabilities. Examples of matrices can be constructed to show that each typeof iteration can outperform the others by a factor on the order of

√M or M (Nachtigal

et al., 1992). Moreover, these algorithms are often useless without preconditioning. TheKrylov subspace algorithms and their preconditioned versions are discussed. The discussionof these algorithms and of their preconditioners is for algorithms of general applicability.Some guidelines are also provided about the choice of a suitable algorithm for a givenproblem.

1.4 Solution SchemesSince the fluid flow models in porous media involve large, coupled systems of nonlin-ear, time-dependent partial differential equations, an important problem in the numericalsimulation is to develop stable, efficient, robust, accurate, and self-adaptive time steppingtechniques. Explicit methods like forward Euler methods require that a Courant–Friedrichs–Lewy (CFL) time step constraint be satisfied, while implicit methods such as backward Eulerand Crank–Nicolson methods are reasonably stable. On the other hand, the explicit methodsare computationally efficient, and the implicit methods require the solution of large systemsof nonlinear equations at each time step. Explicit methods, together with linearization bysome Newton-like iteration, have been frequently used in reservoir simulation. Due to theCFL condition, enormously long computations are needed to simulate a long time period(e.g., over ten years) problem in a field-scale model, and thus fully explicit methods cannotbe efficiently exploited, especially for problems with strong nonlinearities.

A variation to achieve better stability without suffering too much in computation isthe IMPES (implicit in pressure and explicit in saturation) scheme. This scheme works wellfor problems of intermediate difficulty and nonlinearity (e.g., for two-phase incompressibleflow) and is still widely used in the petroleum industry. However, it is not efficient forproblems with strong nonlinearities, particularly for problems involving more than twofluid phases.

Another basic scheme for solving multiphase flow equations is the simultaneous so-lution (SS) method, which solves all of the coupled nonlinear equations simultaneously andimplicitly. This technique is stable and can take very large time steps while stability is main-tained. For the black oil and thermal models (with a few components) considered in thisbook, the SS scheme is a good choice. However, for complex problems that involve manychemical components (e.g., the compositional and chemical compositional flow problems),the size of system matrices to be solved is too large, even with today’s computing power.

A variety of sequential methods for solving equations in an implicit fashion without afull coupling have been developed. They are less stable but more computationally efficientthan the SS scheme, and more stable but less efficient than the IMPES scheme. The se-quential schemes are very suitable for the compositional and chemical compositional flowproblems that involve many chemical components.

Finally, an adaptive implicit scheme can be employed in reservoir simulation. Theprincipal idea of this technique is to seek an efficient middle ground between the IMPES(or sequential) and SS schemes. That is, at a given time step, the expensive SS scheme isconfined to those gridblocks that require it, while on the remaining gridblocks the IMPESscheme is implemented. The majority of research in the solution schemes has concentratedD

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1.5. Numerical Examples 5

on the stability of time stepping methods, and the efficient linearization and iterative solutionof the resulting equations. The accuracy of these schemes must be also addressed. All thesolution schemes mentioned are covered and compared in this book.

1.5 Numerical ExamplesMany numerical examples are presented to test and compare different numerical methods,linear system solvers, and solution schemes. These examples are based on the bench-mark problems of the first nine comparative solution projects organized by the Society ofPetroleum Engineers. Typically, about ten organizations participated in each project. Thenumerical examples presented include three-dimensional black oil reservoir simulations, aconing problem study, gas cycling analysis of retrograde condensate reservoirs, steam in-jection simulations, dual porosity model simulations, gridding techniques, horizontal wellmodeling, and large-scale reservoir simulations. A couple of numerical examples are basedon real field data analysis.

1.6 Ground Water Flow ModelingThere are many modeling and simulation processes that use technologies and techniquessimilar to those in petroleum reservoir simulation; one example is ground water flow mod-eling. Ground water is one of the most widely distributed and important resources on theearth. Over half of the population in the USA depends on ground water for its water supply,for example. Also, ground water is an important source of irrigation and industrial processwater. In a large part of the USA, available sources of ground water are a fundamentalconstraint on development and economic activity. Ground water quality is endangered byorganic, inorganic, and radioactive contaminants introduced into the ground by improperdisposal or accidental spill. Protecting this quality is a problem of broad economic andsocietal importance.

Water movement in the subsurface has been studied for many decades by soil scientistsand agricultural engineers. This research dates back to the classical work of Richards (1931).The subsurface is a multiphase system. It consists of at least three phases: the solid phase ofthe soil matrix, the water phase, and a gaseous phase. Other phases like a separate organicliquid phase or an ice phase may exist. The traditional approach of studying a subsurfacesystem has concentrated exclusively on water. Over the past few decades, interest hasgrown in problems where other phases can be important. These include the evaluation ofremediation technologies such as soil venting where the gas phase plays an important role.Soil venting is a technology that attempts to remove contaminants from the soil before theycan seriously pollute ground water supplies. It works by pumping air through a part of thesubsurface contaminated by a volatile contaminant and inducing it to volatilize so that itcan be removed by the gas phase flow. Previous evaluation of this technology has indicatedthat it is economical and efficient in contaminant cleanup. For such an application, couplednonlinear equations for an air-water system must be solved. While ground water modelinghas become increasingly important, it is beyond the scope of this book to study it. However,we emphasize that technologies and techniques similar to those used in petroleum reservoirsapply also to ground water flow (Chen and Ewing, 1997A; Helmig, 1997).D

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1.7 Basin ModelingBasin modeling is a term often used to describe three factors: the burial history of sediments,the thermal history of these sediments, and the generation, migration, and preservationof hydrocarbons. The burial history of sedimentary units is driven by sediment supply,chemical and mechanical compaction, tectonic forces, erosional and intrusive events, andsea-level changes. An understanding of this dynamical evolution of sediments is critical tobasin modeling since paleostructures, porosity, sedimentary thermal conductivity, solubility,faulting, and fluid flow all depend on the sedimentary patterns of behavior. When the burialhistory of the sediments is known, one needs to determine their thermal history. There aretwo approaches to this. The first approach assumes a priori models for heat flux evolution,and the determination is carried out by fiat. The second approach uses present-day data thatcontain some cumulative measure of thermal history and attempts to utilize these data toreconstruct the thermal history of the sediments. After determining the sedimentary thermalhistory, one needs to determine the generation, migration, and preservation of hydrocarbons.In this step, one needs to figure out the ways and means of providing thermokinetic modelsof hydrocarbon generation from organic material and to assess their accuracy. All thesefactors constitute crucial parts in attempts at basin modeling. Basin modeling is a veryimportant and complex process (Allen and Allen, 1990; Lerche, 1990; Chen et al., 2002B).However, due to the scope of this book, this topic will not be discussed further.

1.8 UnitsBritish units are used almost exclusively in reservoir engineering in the USA. However,the use of metric systems, particularly the SI (Sisteme International) unit system, has beenincreasing. Hence we state the SI base units and some common derived units adapted fromCampbell and Campbell (1985) and Lake (1989). The SI base quantities and units are givenin Table 1.1. When the mole is used, the elementary entities must be specified; they can beatoms, molecules, ions, electrons, other particles, or specified groups of such particles inpetroleum engineering.

Some SI derived units are shown in Table 1.2, and a list of useful conversions arestated in Table 1.3. Two troublesome conversions are between pressure (1 MPa ≈ 147 psia)and temperature (1 K = 1.8 R, Rankine). Neither the Fahrenheit nor the Celsius scale isabsolute, so an additional conversion is required:

◦F = R − 459.67, ◦C = K − 273.16.

Table 1.1. SI base quantities and units.

Base quantity SI unit SI unit symbol SPE symbolTime Second s tLength Meter m LMass Kilogram kg MThermodynamic

temperature Kelvin K TAmount of substance Mole mol

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1.8. Units 7

Table 1.2. Some common SI derived units.

Quantity Unit SI unit symbol Formula

Pressure Pascal Pa N/m2

Velocity Meter per second m/sAcceleration Meter per

second squared m/s2

Area Square meter m2

Volume Cubic meter m3

Density Kilogram percubic meter kg/m3

Energy (work) Joule J N·mForce Newton N kg·m/s2

Viscosity (dynamic) Pascal second Pa·sViscosity Square meter(kinematic) per second m2/s

Table 1.3. Selected conversion factors.

To convert from To Multiply by

Day (mean solar) Second (s) 8.640000E + 04Darcy Meter2 (m2) 9.869232E − 13Mile (U.S. survey) Meter (m) 1.609347E + 03Acre (U.S. survey) Meter2 (m2) 4.046872E + 03Acres Feet2 (ft2) 4.356000E + 04Atmosphere (standard) Pascal (Pa) 1.013250E + 05Bar Pascal (Pa) 1.000000E + 05Barrel Feet3 (ft3) 5.615000E + 00Barrel (petroleum 42 gal) Meter3 (m3) 1.589873E − 01British thermal unit Joule (J) 1.055232E + 03Dyne Newton (N) 1.000000E − 05Gallon (U.S. liquid) Meter3 (m3) 3.785412E − 03Hectare Meter2 (m2) 1.000000E + 04Gram Kilogram (kg) 1.000000E − 03Pound (lbm avoirdupois) Kilogram (kg) 4.535924E − 01Ton (short, 2000 lbm) Kilogram (kg) 9.071847E + 02

The superscript ◦ is not used for the absolute temperature scales K and R. The volumeconversions are also troublesome due to the interchangeable use of mass and standardvolumes:

1 reservoir barrel (or bbl) = 0.159 m3,

1 standard barrel (or STB) = 0.159 SCM.

The symbol SCM (standard cubic meter) is not a standard SI unit; it indicates the amountof mass contained in one cubic meter calculated at standard pressure and temperature.

The use of unit prefixes is sometimes convenient (cf. Table 1.4), but it does requirecare. If a prefixed unit is exponentiated, the exponent applies to the prefix as well as theunit. For example, 1 km2 = 1 (km)2 = 1 (103 m)2 = 1 × 106 m2.D

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Table 1.4. SI unit prefixes.

Factor SI prefix Symbol Meaning (U.S.)

10−9 nano n One billionth of10−6 micro µ One millionth of10−3 milli m One thousandth of10−2 centi c One hundredth of10−1 deci d One tenth of10 deka da Ten times102 hecto h One hundred times103 kilo k One thousand times106 mega M One million times109 giga G One billion times1012 tera T One trillion times

There are several quantities that have the exact same or approximate numerical valuebetween the SI and practical units:

1 cp = 1 mPa·s, 1 dyne/cm = 1 mN/m,1 Btu ≈ 1 kJ, 1 darcy ≈ 1µm2, 1 ppm ≈ 1 g/m3.

There are several more useful unit conversions:

1 atm = 14.7 psia, 1 day = 24 hrs, 1 ft = 30.48 cm,1 bbl = 5.615 ft3, 1 darcy = 1,000 md, 1 hr = 3,600 sec.

More unit conversions will be stated in Chapter 16.

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1. introduction

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