PETROLEUM RESERVOIR SIMULATION - BASIC CONCEPTS Khalid Aziz February 24, 2005 Stanford University, Stanford, CA 94305-2220
Contents 1 Introduction and Overview
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1.1
Why Simulate Reservoirs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
How Simulators are Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.4
Major Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5
Major Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Modern Tools for Forecasting Reservoir Performance
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2.1
Introduction to Reservoir Simulation Technology . . . . . . . . . . . . . . . . . .
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2.2
Fundamental Concepts
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2.1
Fluid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2.2
Reservoir Grid and Rock Properties . . . . . . . . . . . . . . . . . . . . .
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2.2.3
Flow Equations and Recovery Mechanism . . . . . . . . . . . . . . . . . .
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2.2.4
Solution of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2.5
History Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Types of Reservoir Simulators in Use 3.1
3.2
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Type Based on Fluid Description . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1.1
Black Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1.2
Extended Black Oil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1.3
Compositional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1.4
Limited Compositional . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Type Based of Reservoir System . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2.1
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Single Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
3.3
3.4
3.5
3.2.2
Dual Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2.3
Dual Porosity, Dual Permeability . . . . . . . . . . . . . . . . . . . . . . .
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Type Based on Recovery Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.1
Steam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.2
In Situ Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.3
Thermal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.4
Chemical Flood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3.5
Gas Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Type Based on Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.1
IMPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.2
Implicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.3
Adaptive Implicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4.4
Sequential Implicit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Type Based on Gridding Techniques . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.1
Block-Centered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.2
Point-Distributed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.3
Curvilinear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.4
Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.5
Voronoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5.6
Flexible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Data Requirements and Sources
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4.1
Data Needed and Data Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2
Sources of Reservoir Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.1
Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.2
Log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.3
Well Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.4
Seismic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2.5
Performance History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Sources of Fluid Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3.1
PVT Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3.2
Correlations and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Impact of Data of Various Types . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.3
4.4
5 Limitations of Current Technology and Current Research
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6 Computer-Requirements
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2
7 Concluding Remarks
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8 Nomenclature
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1
Introduction and Overview
In every endeavor that involves costs and possible benefits, the risk can be reduced with the use of tools that can mimic future behavior. Because in complex systems like petroleum reservoirs it is impossible to make forecasts that are precise, or even accurate under all conditions, both the forecasts and forecasting tools are constantly refined to make more and more reliable predictions. Because of the limited information that is normally available about the system being modeled – the petroleum reservoir and the fluids within it – there is always considerable degree of uncertainty associated with predictions of performance. Furthermore, incomplete understanding of the mechanisms involved in fluid displacements in porous media may also cause the results to deviate from reality. The job of a simulation team is to minimize this uncertainty in a way that is consistent with the objectives of the organization that pays the cost of the study. The degree of sophistication of the model and its use has to be carefully balanced with the benefits to the organization. For example even if it was possible to do so, a pore scale simulation of an entire reservoir is not desirable. This would be like tracking the motion of each molecule in a pipeline to calculate the pressure drop in that system. Such a model would be inconsistent with the objectives of the study, which is to predict the pressure drop in the pipeline due to the average behavior of the molecules in a control volume that is large with respect to the size of the molecules but small with respect to the size of the pipe. Similarly for modeling flow in a petroleum reservoir it reasonable to average the flow over several pores, but in most cases the system would be over simplified if an entire reservoir is treated as a single block as in tradition material balance calculations. All experts agree that great benefits in improved accuracy result from improvements in reservoir description. Identification of major flow units, faults and fractures can probably improve the predictions more than any other single factor. Further improvements result when reservoir heterogeneities can be defined in greater and greater detail. The intelligent use of information about the reservoir obtained at different scales with new and traditional measurement tools provides significant challenges to the reservoir engineer. In this review the state-of-the-art of reservoir simulation will be discussed from the point of view those who have scientific training but are not experts in this field. These notes do not provide a comprehensive literature review, but they do provide suitable references for further study. The interested reader should consult books like that of Aziz and Settari (1979) [3].
1.1
Why Simulate Reservoirs?
Petroleum engineers use reservoir simulators to enhance the efficiency of oil and gas production or storage operations. Simulators are used – or can be used – at every stage of the operation in answer important questions: • After the discovery of a field and during the development of an economic feasibility study, simulators can be used to obtain valuable information about the expected performance of the reservoir. This kind of information can be used to study the economic feasibility
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of the project and for risk assessment. Because of the limited information that would be normally available about the reservoir at this stage, only rough calculations can be made. In many cases standard reservoir engineering calculations, rather than reservoir simulation, are adequate during early stages of reservoir development. One of the advantages of using reservoir simulation early in the life of a reservoir is that analyses conducted at this stage would be completely consistent with more sophisticated calculations at future stages. Furthermore, early initiation of the reservoir management team to the simulator helps (i) in the identification of data that would be important at future stages, and (ii) the organization of the data in a way that would make it most useful. • Once it has been decided to develop a field and additional information has been obtained from field delineation wells, the simulator can be used to design field development (full field or pilot). This would include number and location of wells, type of wells (conventional or horizontal) and design of wells (casing program, tubing size, completion type) to be drilled initially. Information obtained from the simulator about production rates, breakthrough times for gas and water, gas and water rates, and wellhead pressures can be used for the design of production facilities. • After initial field development and as production data are obtained, the simulator can be used to refine initial design and optimize field operations. Matching of field production history with predictions from the simulator provides confidence about future predictions and helps in the understanding of mechanisms that are active in the reservoir to help or hinder the recovery of oil. This process continues through the entire production phase of the field. • Starting with the initial development phase and ending with field abandonment, the simulator can be used to evaluate different schemes for improving oil recovery and their economic impact. Expected performance of waterflooding, gas injection or some sophisticated enhanced oil recovery scheme can all be investigated with an appropriate simulator. As economic conditions change, the simulator can provide valuable information on possible options for enhancing profitability. • Finally the simulator can be used to study the optimum time for field abandonment and the manner in which the field should be abandoned. Possible advantages of new technologies like horizontal wells can be investigated to see if productive life can be extended. When used intelligently, the simulator is a powerful tool that can assist the petroleum engineer and the management with important decisions. As we will discuss later; improper use of this tool can also lead lo disastrous results.
1.2
How Simulators are Used
While in a broad sense the simulator is used to predict performance of petroleum reservoirs, its use in any give case can be for a variety of specific reasons: • Improving reservoir description through history matching. After the best possible of description of a reservoir has been obtained from the analysis of cores, logs, well tests and seismic data, there is usually still considerable uncertainty about reservoir properties. It is often possible to identify major flow units, and barriers near wells, through the matching of good production data with model predictions. Because the history matching process is non-unique (several different adjustments of reservoir properties can provide equally
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“good” matches of data), it has to be repeated as more and more information about the reservoir becomes available. • Analysis of laboratory experiments. Most information is extracted from laboratory experiments if they can also be modeled. Traditionally simple analytical models – based on assumptions that may not be always valid – have been used. More recently the reservoir simulator itself has been used for the analysis of experiments that are conducted for determining parameters like relative permeability and capillary pressure, and experiments that are conducted for understanding of the displacement of oil by some injected fluid (core displacements, slim-tube displacements, etc.) Once an appropriate laboratory experiment has been matched, the simulator provides a natural tool for scale-up to the field or pilot scale. • Understanding of flow mechanisms. In some cases the mechanisms of oil displacement at a small scale (order of few millimeters) is quite different from what must be modeled at the field scale. Consider for example the displacement of oil by a gas (nitrogen, carbon dioxide or some hydrocarbon gas). Because the viscosity of the injected fluid is much less than that of the oil in the reservoir, the injected fluid tends to “finger” through the oil. These fingers are caused both by the hydrodynamic instability of the displacement process and spatial variations in reservoir permeability (heterogeneities). But these fingers that are only a few millimeter wide are too small to be modeled at the field scale where the block dimensions can be many meters or even kilometers in each direction. Hence it is important to incorporate their averaged behavior in field scale simulators. Fine grid simulation of these fingers can provide important information for the averaging rules. Other examples of the use of models to understand flow mechanisms include in situ combustion and steam displacement (with or without foam). • Development of simple models and correlations. There are situations where it is desirable to develop a simple model or correlation for quick forecasts or for imbedding them inside a full field model. Such correlations have been traditionally developed with data collected through laboratory or field experimentation. In recent years such physical experiments have been augmented with “numerical” experiments with simulators to yield “data” for the development of simple models or correlations. Examples of such applications are conning correlation, correlations of block effective permeabilities in heterogeneous rocks, steam injection (cyclic and displacement) models. • Reservoir performance forecasting. All of the above applications of simulators are simply preliminary steps towards getting to the final goal of making reliable forecasts of reservoir performance. Such forecasts can be made for the initial design of production systems, expansion of production facilities, reservoir optimization, troubleshooting, resolution disputes among parties with interest in the same field, assessment of risks at each stage of field development, management decisions, monitoring reservoir performance, training of engineers and geologists, timing and implementation of enhanced oil recovery schemes, establishing need for additional data collection, location of new wells, and to assess value of new technologies like horizontal wells. The availability of data and the sophistication of the simulation at each stage of reservoir development is different. • Education and training. Simulators are powerful learning tools for engineers.
1.3
Historical Perspective
Reservoir simulation is nothing but a natural evolution – aided by developments in computer software and hardware technology – of classical reservoir engineering and the use of physical
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and analog models. Since the early 1930’s reservoirs have been mathematically modeled as single homogeneous blocks for material balance calculations. Such a model was simple enough to perform calculations even by hand (initially by slide rules, then by mechanical calculators, now by electronic calculators and personal computers). Such calculations ignore reservoir heterogeneities and treat the reservoir as a “tank” with uniform rock and fluid properties, pressure and temperature. Enhancements in such model calculations were obtained by incorporating simple analytical models for pressure gradients around a well (solution of the diffusion equation for one dimensional radial flow) and for one-dimensional two-phase displacements (Buckley-Leverett theory). Starting with the development and availability of digital computers in early 1950’s, engineers started solving nonlinear equations for simple flows (one-dimensional, single-phase) by using finite-difference methods. This meant that the reservoir could be replaced in the model by a series of blocks, rather than a single block as in conventional material balance calculations. As the power of computers has increased, the simulators have become more and more sophisticated. The period from about 1970 to the present can be considered the time of the birth of modern reservoir simulators. During this period tremendous advances have been made in several areas: • Understanding reservoir flow mechanisms and reservoir geology. • Improved understanding and mathematical modeling through equations of state of the phase behavior of petroleum fluids (how oil and gas phases and their composition changes with pressure and temperature). • Numerical techniques for the solution of flow equations, which form a large system of highly nonlinear coupled set of equations. • Numerical techniques for the solution of large sparse matrix equation. • Development of super computers, workstations and parallel computers. • Software tools with Graphical User Interface (GUI) for data entry and visualization of results. • Improvements in compilers (FORTRAN, C and C++) and development of new programming concepts (object oriented design). Whereas a typical simulator in the 1960’s was the result of one persons efforts, the modern simulator is developed by a team of experts. As a result of the feature demanded by the user, modern simulator can contain well over a million lines of code. Emerging object oriented approaches to the design of simulators are likely to have a major impact in the design and use of future simulators (Verma and Aziz, 1996) [11]. While developments of the past have been significant, much work remains. For example, our understanding of simultaneous three phase (oil, gas and water) flows in porous media is still primitive. This and other limitations of current technology will be discussed later in this paper.
1.4
Major Benefits
• Design and performance analysis of a complex system like an oil field requires that one consider the interaction of many different parts of the system. For example, we have to
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consider fluid/fluid interaction (mass transfer between phases, changes in the amount and composition of each phase), fluid/rock interaction (multiphase, multicomponent flow in irregular pore spaces, surface properties of the rock system), reservoir/well interaction (inflow of fluids from the reservoir to production wells and outflow from injection wells), reservoir/reservoir interaction in a field with several reservoirs, and the interaction of gathering or injection system with the reservoir (control of reservoirs by constraints placed at some point in the production or injection system). Traditionally, each component of such systems has been analyzed separately, and simplifying assumptions have been made to make the problem tractable. While reservoir simulation allows a more rigorous analysis of the problem than is possible by any other means, significant limitations still exist. With the expanding power of modern computers, it is just becoming possible to not only model each component accurately but also consider the interaction of different components of the system with each other. • Another important benefit of simulation is that it allows the integration of data from different sources and their use. In addition it becomes possible to look at the value of information of different types through sensitivity studies with simulators. • Modern simulation tools coupled with stochastic simulation provide an opportunity to get an assessment of uncertainty for each development or production scenario that may be under consideration. This can be accomplished through simulations using multiple reservoir descriptions that all honor available data.
1.5
Major Problems
Notwithstanding many major technological breakthroughs over the past twenty years or so, there are still many areas where the demands of the users and our need for understanding far exceed the capabilities of modern reservoir simulators: • Three-phase flow. The relative resistance to the flow of different phase (oil, gas or water) in the porous media is handled in the simulator through the introduction of relative permeability as a multiplier to absolute permeability in Darcy’s law. While reasonably reliable techniques are available for measuring two-phase relative permeabilities, the same is not true for three-phase relative permeabilities. As a matter of fact, three phase measurements are so difficult that they are rarely made even in research laboratories. Normally two sets of two-phase data (oil-gas and oil-water) are to predict three-phase permeabilities in the simulator. Several such models are available, but there is no way to tell which, if any, of the models will be reliable in a given situation. • Thermodynamic equilibrium. All commercial simulators assume that as fluids flow in to a grid block they instantaneously reach thermodynamic equilibrium. This assumption becomes reasonable if the grid blocks are sufficiently small. However, in some situations (e.g. simulation of gas injection with large grid blocks) this assumption can lead to improper handling mass transfer between phases during the simulation process. • Simulation of large, complex fields. Typically reservoir engineers handle the simulation of large fields either by using very large grid blocks or by dividing the reservoir into smaller units. In the case where large grid blocks are used all fluid and rock properties are averaged over the entire grid block. Furthermore, only one set of saturations and pressures are associated with each grid block. This means that any variations in these parameters within the grid block are ignored. This can lead to unreliable (often optimistic) predictions. Physical parameters are replaced by “pseudo functions” that are designed to mimic true
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behavior. Unfortunately there are no rigorous methods for calculating pseudo functions and their use can lead to incorrect results. Furthermore, pseudo functions change with changing conditions and hence they are not robust. When the reservoir is divided in into small units that can be handled with blocks of reasonable size, the interaction among various units is difficult to handle and often ignored. New domain decomposition techniques and parallel computations offer significant opportunities to overcome these limitations. Many of the major problems are related to the modeling of large reservoirs with gridblocks that are too large to adequately represent either the reservoir or the recovery process. Even with the largest and most powerful computers of today, difficulties arise in the simulation of large reservoirs. Gridding a large reservoir can also be a very difficult and time consuming task, requiring many man months of time for just a single possible reservoir description (Pettersen, 1992). • Modeling of wells. Field scale reservoir simulation is done with gridblocks that are orders of magnitude larger than the magnitude the size of the well. This means that the pressure of a large gridblock cannot be assumed to be the pressure of one or more wells that might be in that grid block. Usually single phase flow assumptions are made to relate the well pressure to the block pleasure through a well geometric factor. Well models have only been derived for two-dimensional flow in homogeneous reservoirs for steady-state or pseudo steady-state cases. Only limited information exists for three-dimensional flow (partial well completion, wells that are not oriented along grid lines) and transient flow conditions. Another complication in modeling wells arises due to the variation of saturations within a gridblock (gas or water coning, edge water encroachment). Since the flow of various phases into the well is controlled by the relative permeabilities of those phases – and hence saturations – near the well, using average gridblock saturations can lead to large errors. Both of the above problems are reduced by grid refinement around wells. But current techniques for local grid refinement lead to the substantial increases in computational time. • Modeling of surface facilities. Petroleum engineers typically model reservoirs and surface facilities separately. This leads to a miss-match at the interface between these two systems. The reason for doing this is again to reduce computational time. New techniques utilizing domain decomposition and parallel processing are emerging to handle this problem. • Integration of data from different sources. Reservoir description should be based on reservoir data at different scales and from all sources. Integration of all available data into one or more usable reservoir descriptions has been a problem. New research on stochastic techniques is starting to provide some powerful tools to handle this problem. • Fluid description. Reservoir fluids are extremely complex mixtures of hundreds of different components. It is difficult to obtain and use a complete description (analysis) of reservoir oil and gas. Analytical techniques for making the analysis are difficult to use and expensive. Traditionally petroleum engineers have lumped the hydrocarbon mixture in the reservoir into two pseudo components. They are oil and gas at standard (stock tank) conditions. This highly simplified description is inadequate for many purposes, especially when EOR techniques are being considered. We have the problem: how to find an optimum set of pseudo components and how to characterize them? All necessary fluid properties of pseudo components must be calculated from the characterizing parameters. Unfortunately in many reservoir simulations inadequate description of fluids is used.
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• Modeling physical dispersion. One of the outstanding problems of reservoir simulation is the modeling of physical dispersion. The difficulty results from the fact that numerical techniques (finite-difference methods) used in reservoir simulation with large gridblocks lead to large errors that behave just like physical dispersion. Fortunately in many case the effects of physical dispersion need not be considered explicitly, but in other situations – for example in the simulation of tracer flow – one must control numerical dispersion so that physical dispersion can be modeled. High order finite-difference and finite-element techniques have been successful only in limited situations to solve this problem. • Assessment of risk through reservoir modeling. Every step of the simulation process involves assumptions and approximations. The predictions made by a single simulation run have only a finite probability of being right. Assessment of this uncertainty and through it risk involved in making investments based on reservoir performance predictions is bound to be a major future challenge for simulation teams (Ballin et al. 1992) [4].
2
Modern Tools for Forecasting Reservoir Performance
2.1
Introduction to Reservoir Simulation Technology
Reservoir simulation is a blend of engineering, physics, chemistry, mathematics, numerical analysis, computer programming, and engineering experience and practice. Basically the reservoir is divided into grid blocks and the transient flow of fluids between various grid blocks and between grid blocks and wells is computed under certain specified conditions (rate or pressure) at the wells. But considerable experience is needed in determining how the reservoir is divided into blocks, how properties are assigned to these blocks and what assumptions are made in writing and solving the flow equations. Unless sound judgment based on experience is used at every step of this process, the entire exercise becomes completely unwieldy. This is the reason why successful development and application of this technology has remained primarily in the hands of those who understand (to varying degrees) all aspects of this technology. Some practical aspects are discussed by Aziz (1989) [1].
2.2 2.2.1
Fundamental Concepts Fluid System
After a reservoir has been divided into grid blocks, equations of mass balance are written over each grid block. Before writing the equations of mass balance we must understand that mass is conserved for each component (methane, ethane, propane, etc.) of a system. In system where mass transfer between phases (oil, gas and water) is allowed, the mass of a phase is not conserved. This immediately leads to a serious difficulty. First of all natural hydrocarbon systems contain hundreds of components. Secondly, determination of exact amount of each component in a specific oil is a highly complex task. Even if we could determine the complete analysis of a hydrocarbon (consisting of oil and gas phases) system, the use of this information in actual reservoir simulation would be virtually impossible with current technology, except perhaps for highly simple (consisting of only a few components) systems. There are two main reasons for this: 1. The number of equations to be solved for each grid block at each timestep depends on the number of components. The computational work increases sharply as the number of equations increases.
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2. Characterization of all of the hydrocarbon components is usually difficult and sometimes impossible. Fortunately there is no need to use a large number of components in doing reservoir simulation. Instead the actual components in the reservoir are lumped together into a few pseudo components, and the mass balance is written for these pseudo components. Most of the standard reservoir studies are done with what are called “black oil” models. These models are a natural evolution of traditional reservoir engineering where the reservoir fluids were assumed to consist of only three pseudo components with the composition of oil, gas and water at standard or stock tank conditions. For example, in such a model reservoir oil would be made up of oil and gas pseudo components (which by definition are oil and gas at standard conditions). 2.2.2
Reservoir Grid and Rock Properties
Grid selection is one the most difficult and time consuming tasks in the simulation of geologically complex reservoirs (Palagi and Aziz, 1991 and 1993; Aziz, 1993; Verma and Aziz, 1996) [7, 9, 2, 11]. A reservoir simulator predicts reservoir performance by solving flow equations on a discrete grid that is chosen by the simulation engineer to represent the reservoir. The grid is normally selected with one or more of the following considerations in mind: 1. Geology and size of the reservoir and the available data used for reservoir description. 2. Type of fluid displacement or depletion process to be modeled. 3. Past and anticipated field development (location and type of wells). 4. Numerical accuracy desired. 5. Available software options. 6. Objectives of the simulation study. 7. Competence of the simulation engineer or team. 8. Available computer resources, time constraints or project budget. In the early days of reservoir simulation it was often the last of the above considerations that determined the number of grid blocks, and then the available software limited the choice of grid types to usually block-centered Cartesian or cylindrical grid. Research in reservoir simulation and hardware developments, especially over the past ten years, have greatly extended the gridding options available to the user. Modern commercial simulators typically offer one or more of the following techniques: • Standard Cartesian or cylindrical block centered grid • Local Grid Refinement • Hybrid Grid • Curvilinear (Stream-Tube) Grid • Voronoi or PEBI Grid (Generalization of Point-Distributed Grid) • Corner Point Geometry
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• Dynamic Grid • Automatic Grid Generation • Elastic Grid Adjustment Methods • Control Volume Finite Element Methods • Mixed Flexible Grids While this abundance of options provides flexibility, it can also make the choice of the appropriate grid bewildering for the person using the simulator. 2.2.3
Flow Equations and Recovery Mechanism
Conservation of mass is the mathematical expression of (Aziz and Settari, 1979) [3]: [mass influx overtime] = [mass accumulation overtime] Before writing the mathematical equation we have to know the mechanisms that will contribute to the influx of mass. In most situations convective flow is all that is important and this is the basis of standard reservoir simulators discussed below. The conservation of mass for component c (for Black-Oil models c = oil, gas and water at standard conditions) combined with Darcy’s law yields the following set of flow equations: Np Nn X h i X n+1 n w ∆t Tc,pi,j (Φp,j − Φp,i ) − qc,i = (Mc,i ) − (Mc,i ) (1) j
p
where the transmissibility between nodes i and j is defined by: fc AKkrp ωc,p Tc,pi,j = dµp i,j and mass of component c in the grid block i at timestep n is given by: n Np X n Mc,i = Vb,i φSp ωc,p p
(2)
(3)
i
In the equations in this volume index Np stands for number of phases (oil, water and gas in standard simulators). All symbols are defined in the nomenclature section. The well flow rate of a component is related to the flow rate of phase into the well: Np X w = ωc,p qpw (4) qc,i p
i
The flow rate of a phase to a well is given by: w = W Ii qp,i
w Kiw kr,p Φp,i − Φw p,i µp
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(5)
Figure 1: Gridblock i and its connections The superscript w is intended to indicate well terms. Note that the permeability and relative permeability of well in block i can be different from the corresponding values for that block. The well index (W I) includes the influence of all geometric factors (well radius, skin, location of well in the block, completion interval, effective well block radius). For each block one equation of the type of Eq. (1) is written for each component or pseudo component, c, in the system. The required geometric propertied of the gridblock are: block volume Vb , the area A of each block face and distance d between i and j for each connection. This method of writing flow equations is known as the Control Volume Finite Difference (CVFD) method and it reduces to the standard finite-difference method for Cartesian grids. An important characteristic of this method is that in Eq. (1) the flow across the gridblock face between i and j depends only on the component of the potential gradient in me direction i − j. For nonorthogonal coordinate systems, the flow calculation across a block face would depend on all components of the gradient of the potential on the surface. Hence an error in flow calculations results, if in a non-orthogonal grid only the component of the gradient in the i − j direction is considered. The terms on the left hand side of Eq. (1) represent net influx for block i over time step ∆t and the right hand side represents the net accumulation of mass in block i over the same time. The first term on the left hand side represents flow across block boundaries and the second term represents flow from wells (positive for production wells). The index j is for blocks that are connected to block i, the block for which the mass balance is written. Note that the connections of block i need not be just to the neighbors of this block. A block can be “connected” to other blocks through wells or faults. The shape of the gridblock and location of the grid point within the block will influence the evaluation of each of these terms. The flow between blocks is calculated by multiplying the interblock transmissibility with the difference in potential between the blocks (Fig. 1). This flow term depends on both the grid geometry and the location of the grid point in the block. The grid points should be selected so that the finite-difference approximation of the gradient of pressure is as accurate as possible. In other words the difference in potential between the two nodes on either side of a boundary divided by the distance between the nodes should be a good approximation at the boundary for the average potential gradient normal to the boundary. The accumulation term uses the gridblock volume and the pressure at the node
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to calculate the mass in the block at different times. For this purpose the grid point should be as close to the mass center of the block as possible. Finally, for the calculation of well flow a well model is required to relate the well block pressure to the well pressure. The well model depends on the grid type and for certain kinds of grid the well model has to be adjusted as the flow field in the reservoir changes (Palagi and Aziz, 1992) [8]. We will consider these factors as we discuss different types of grids in the next section 2.2.4
Solution of Equations
Once a grid has been defined and flow equations for each grid block and all the wells have been written, the next step is to solve these equations. The system of equations to be solved is a highly nonlinear, tightly coupled set. Typically Newton’s method or one of its variations is used to linearize the system of equations. This linear set of equations has to be solved repeatedly for each timestep to achieve convergence to the solution of nonlinear equations. In most simulation studies, most of the computer time is used in solving linear systems of equations. For large practical problems, these equations have to solved by iterative methods. For standard grids the coefficient matrix for the linear system has a banded structure that can be exploited, but this structure is lost for problems with unstructured or flexible grids, because of the large amount of computer time required to solve the linear system of equations, attempts are made to speedup this process by taking advantage of the machine architecture (parallel, vector). 2.2.5
History Matching
A reservoir simulator provides the engineer with a tool for making rapid comparisons of alternate operating policies or a detailed prediction of the future reservoir performance. The simulated performance may be compared with actual performance to obtain an improved set of reservoir parameters that are consistent with the performance history of the reservoir. This process is known as “history mulching”. It is important to recognize that history matching actually uses a simulator in a reverse mode. The response variables (field observations) over some period of time are assumed known. The unknowns are the input parameters (reservoir description, fluid description).This so-called “inverse problem” or “parameter identification problem” involves an attempt to find the best set of input data coupled with appropriate control variables that reproduce the past performance This inverse problem has no unique solution. History of the reservoir over a limited time can be reproduced by many quite different sets of input data. Even unphysical parameters like negative permeabilities can produce what might appear to be reasonable results over the production history period. Obviously, however, use of such unreal parameters will lead to performance predictions that are completely unreliable. Hence bounds placed must be placed on the input parameters to constrain them within reasonable ranges. Furthermore, the parameters adjusted to achieve a desired change in simulator output should be carefully selected. Again because multiple adjustments of input data can produce very similar results over a limited period of time. A good understanding of how the simulator works the kind of recovery mechanism that id operative in the reservoir can be very important at this stage. A sensitivity analysis can be performed to examine the behavior of the model as the input parameters are individually varied over their ranges. Frequently, such an exercise may indicate a given reservoir response is insensitive to a certain input parameter. It is impossible to identify such parameters by history matching. On the other hand, if the response is highly sensitive to one or more parameters, then a good fix on its (their) values is possible through history matching. Even when a reasonable history match has been obtained, it is a mistake to
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assume that the final adjusted input parameters, say permeabilities, for example, are the actual values in the reservoir. They are simply a set of numbers, constrained within reasonable ranges that reproduce the desired response variables. There may be other values of the same variable, that are constrained within the same ranges, that would yield an equally “good” match. The data available for history matching are usually pressures, GOR’s and WOR’s on a field and individual well basis. In some instances, the total field rate may be a historical variable that is matched. Obviously, because of the many distributed parameters throughout the system, there are many degrees of freedom available to the model user in undertaking a history match. Random selection of input parameters, without regard to the cause/effect relationships between the input parameters and response variables is usually not productive. Basic reservoir engineering considerations can help in developing a systematic approach. The simulation engineer should understand how input parameters affect: • pressure distribution in the reservoir, • average pressure level, • saturation distribution in the reservoir; and • bottomhole flowing pressure. It is instructive to consider and understand the flow equations that are being solved by the simulator (Eq. (1)). • Pressure Distribution The pressure distribution in a reservoir is established as a result of fluid flow. This in turn is governed by the transmissibility (Eq. (2)) which is a function of the product of permeability and relative permeability (Kkr ). Consequently, the modification of one of these will effectively change the product and alter the pressure distribution. Obviously permeability affects the flow of all phases while relative permeability affects the flow of individual phases. So adjustments in overall pressure distribution should be obtained initially by adjusting the absolute permeability. • Pressure Level Pressure level in a reservoir is determined largely by the total reservoir pore volume and the change in pore volume with pressure. The pore volume is determined from reservoir dimensions and the porosity, while the change in pore volume with pressure is reflected by rock compressibility. When multiple phases are present the total compressibility of the fluid/rock system controls the pressure level. While liquid and rock compressibilities can be similar in magnitude, gas compressibility is much higher. Hence when gas is present, its compressibility dominates. • Saturation Distribution The distribution of fluid saturations in a reservoir will affect well injection and production rates, and consequently, WOR’s and GOR’s. They are functions of relative permeability ratios. Since simulators permit different sets of relative permeabilities for each well, WOR and GOR can be matched by either local or global alterations of these parameters. Of course local (well) values are adjusted when changes in the well region are desired. It is worth noting that any alteration in relative permeability also changes the transmissibility and hence the pressure distribution.
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• Bottomhole Flowing Pressure For given flow rate bottomhole pressure is determined by productivity or injectivity indices (Eq. (3)). Their initial values are based on well test and reservoir data. Adjustments are often necessary to match history. • Overall Procedure It is necessary to set up a systematic step-wise approach to history matching. The procedure involves uncoupling, as far as possible, the search for parameters that affect pressure from those that affect saturation. Because this is impossible to do entirely, some iterations are required to refine the set of input parameters. Most history matching is done in practice by the following steps: 1. Check initial volumes in place by adjusting reservoir dimensions, porosity’s and initial saturations. 2. Match pressure distribution by adjusting permeabilities. 3. Match saturations by adjusting relative permeabilities. 4. Match well pressures by adjusting well indices. 5. Iterate as necessary. In recent years efforts have been made to develop techniques that guide the simulation engineer through the development of sensitivity matrices, which contain information on the effect of changes in reservoir parameters on reservoir performance. These matrices are developed by making multiple runs through a systematic change in parameters. Such approaches are bound to become more common in the future.
3
Types of Reservoir Simulators in Use
While the trend in recent years has been to wards the development of multipurpose simulators, there are still a large variety of simulators available in the market. A brief description of various classes of simulators follows.
3.1
Type Based on Fluid Description
The most common classification of simulators bused on the fluid description is given here. 3.1.1
Black Oil
A large majority of simulations done in the oil industry (and all reservoir engineering calculations) are based on the black oil assumption. Under this assumption the reservoir oil is a mixture of stock-tank oil and gas produced at standard conditions. The reservoir gas is assumed to be the same as the gas produced at standard conditions. This means that only two mass balance equations are needed per gridblock to represent the hydrocarbon system. These two equations along with the equation for the conservation of water complete the system of mass balance equations needed for each gridblock.
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3.1.2
Extended Black Oil
A standard black oil model can be extended by allowing the oil (or condensate) to vaporize into the gas phase. This means that the reservoir gas is now a mixture of stock tank oil and gas produced at the surface. This simple extension does not increase the number of equations to be solved and the amount of additional computational effort is relatively small. Such models improve the ability of standard black oil models to simulate light oil or condensate reservoirs. Fluids are normally described through densities at standard conditions, formation volume factors and solubility ratios. 3.1.3
Compositional
A compositional simulator allows one to describe the hydrocarbon system by arbitrary number of real and/or pseudo components. These fluid components are based on a limited or full analysis of the produced oil and gas. One pseudo component may be formed by lumping several real components. All components are characterized by their critical properties. The properties of real (pure) components are available in standard handbooks and they are built in the program. Critical properties of pseudo components are calculated from other data (density or API gravity, normal boiling point, molecular weight) through empirical correlations. Once the hydrocarbon system is fully described, all properties required for fluid flow calculations are predicted from equations of state or other correlations. 3.1.4
Limited Compositional
A limited compositional model usually restricts the number of hydrocarbon components to three. This greatly enhances the ability to model light oil or condensate reservoirs over what is possible with standard black oil models.
3.2
Type Based of Reservoir System
The most common classification of simulators based on the rock type is given here. 3.2.1
Single Porosity
In standard (single porosity) reservoir simulators the reservoir rock is considered to be single continuum with properties that can vary in space and time. 3.2.2
Dual Porosity
In highly fractured reservoirs is not appropriate to average the fracture properties with the matrix properties. In such systems the reservoir rock is considered to be composed of two overlapping continua, one the fracture system and other the matrix system. Flow is allowed between the matrix and the fracture system, but the reservoir bulk flow is considered to take place only in fracture system. This doubles the number of equations that must be solved for each gridblock, but the matrix conservation equations are simple, because no flow take place between gridblocks through the matrix. This assumption is justified when the fracture permeability is orders of magnitude higher then the matrix permeability.
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3.2.3
Dual Porosity, Dual Permeability
In highly fractured reservoirs is not appropriate to average the fracture properties with the matrix properties. In such systems the reservoir rock is considered to be composed of two overlapping continua, one the fracture system and other the matrix system. Flow is allowed between the matrix and the fracture system. This doubles the number of equations that must be solved for each gridblock, and because the transmissibility of the two systems are very different, the equations are also much harder to solve. This assumption is justified when the fracture permeability is considerably higher then the matrix permeability, but matrix permeability is also significant for flow between blocks.
3.3 3.3.1
Type Based on Recovery Method Steam
In standard reservoir simulation the reservoir is assumed to stay at its original temperature. Thus there is no need to solve the energy balance equation. This additional equation is required if thermal energy is added to the reservoir through steam. The black oil steam simulator is special kind of thermal simulator in which the number of equations solved per gridblock is maintained at three by assuming that there is no hydrocarbon gas in the system. In this case only the gas phase that consists entirely of water can be modeled. 3.3.2
In Situ Combustion
An in situ combustion simulator allows chemical reactions among real or pseudo components. Such system require a minimum of around eight components to describe the process. With eight components nine conservation equations (one energy equation) must be solved for ever gridblock. This makes greatly increase the amount of work per timestep (because number of equations has increased) and the number of timesteps (because the equations to be solved are highly nonlinear). 3.3.3
Thermal
A thermal model can handle both steam and in situ combustion. 3.3.4
Chemical Flood
Chemical flood simulators are compositional model that are designed to handle complex chemical reactions and phase behavior involved in chemical flooding. 3.3.5
Gas Injection
Typically requires more than two hydrocarbon components.
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3.4 3.4.1
Type Based on Solution Method IMPES
In a simulator that solves three conservation equations per gridblock (e.g. black oil simulator) the three primary unknowns are usually one pressure and two saturations. The IMPES model solves for the pressure field implicitly (by solving pressure equations for all grid blocks simultaneously) and then solves for saturations explicitly. While this approach greatly reduces the computational effort per timestep, it places a limit on the maximum timestep size. 3.4.2
Implicit
In this approach all mass balance equations are solved simultaneously and there is no theoretical limit on the size of timestep. This however increases the computational effort required per time step. 3.4.3
Adaptive Implicit
This technique tries to take advantage of the best features of IMPES and Implicit methods by adapting the degree of implicitness to the needs of different regions of the reservoir. 3.4.4
Sequential Implicit
This technique is tries to relax the timestep limitation of IMPES by solving for saturations implicitly after the determination of pressure implicitly.
3.5 3.5.1
Type Based on Gridding Techniques Block-Centered
This the traditional grid used in reservoir simulation. The reservoir is divided into gridblocks and grid nodes (where pressures, saturations and other variables are calculated) are centered in each gridblock. 3.5.2
Point-Distributed
In this approach the grid nodes are placed first and the block boundaries are located so that they are bisect the line joining the grid nodes on either side and are normal to this line. This approach is a bit harder to use but the error of discretization is lower than in the block-centered grid. 3.5.3
Curvilinear
This kind of grid attempts to reduce errors by following streamlines and equipotential line. Since these lines change with changes in flow directions, such grids have limited general applicability.
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3.5.4
Hybrid
In this approach more than one kinds of grids are combined. For example, in order to model coning around wells, cylindrical grid is imbedded in Cartesian gridblocks. 3.5.5
Voronoi
This is a generalization of point-distributed grid where that a block can have arbitrary number of connections with other blocks. Every point inside a Voranoi gridblock is closer to its own grid node than to any other grid node. This condition is sufficient to generate a Voronoi grid. This type of grid offers tremendous flexibility in modeling sharp changes in pressure and saturations near well, and in modeling complex reservoir structures. 3.5.6
Flexible
Very general flexible grids based on tetrahedra are being developed. They are discussed in a paper by Verma and Aziz (1996) [12]. These grids can be used to model complex reservoir features and well configurations.
4
Data Requirements and Sources
4.1
Data Needed and Data Use
In order to proceed with simulating a petroleum reservoir using a standard black oil model, we must • describe the reservoir rock system by providing its dimensions (areal and vertical extent, depth, faults, fractures), properties (permeability, porosity, rock compressibility) as a function of position, • provide initial distribution of pressure and fluids (saturations) in the reservoir, • provide conditions (pressure, flux) at the boundaries of the region to be simulated, and • provide properties (density, viscosity, capillary pressure, relative permeability) of the fluids in the reservoir as a function of pressure and /or saturations. Data are used in simulators at the scale of the gridblock. Effective properties of gridblocks can be quite different from the measured values in the laboratory (from cores) or field tests. Proper use of data requires integration and upscaling, which in most cases is a difficult task.
4.2 4.2.1
Sources of Reservoir Data Core
Most of the direct measurements are made on selected reservoir cores to obtain porosity, permeability, lithology, capillary pressure and relative permeability. Some of these properties depend on the state of the core at the time of recovery and its preservation. Since cores may not be
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recovered from the entire pay zone and measurements are only made on few selected cores, they will differ from effective block properties even near the well from which the core is obtained. 4.2.2
Log
Well logs provide important information over the entire zone logged. Results can be analyzed to obtain porosity, fluid contacts, fractures and their orientation, lithology, and fluid types. 4.2.3
Well Test
Well test provide information for a region of influence around a well or between wells. Results can be analyzed to obtain pore volume, permeability, reservoir type (single or dual porosity) and distance to boundaries. Often the analysis can yield multiple solutions, in such cases additional information is required to obtain a unique picture of the reservoir around the well. 4.2.4
Seismic
Seismic is the only tool that can yield information on the entire reservoir. Important information on porosity and reservoir architecture. Recent work indicates that information on fluid distribution may also be obtained. Seismic is also being used to monitor reservoir performance. 4.2.5
Performance History
As discussed earlier history matching of reservoir performance helps tune information obtained from other sources.
4.3 4.3.1
Sources of Fluid Data PVT Studies
A PVT (pressure-volume-temperature) analysis provides information on how the volume of reservoir oil, gas and water changes with pressure. This includes the effects fluid compressibility and the solubility of gas in oil and water. The amount of gas dissolved in oil and water at different pressures is also reported. This coupled with the densities of fluids at standard conditions provides a complete PVT description of the reservoir fluids based on the black oil assumption. Fluid systems that can not be adequately described by the black oil model (two pseudo component) are treated as compositional systems. These systems are analyzed to obtain the amounts of various real and pseudo components that can be used to describe the fluid. 4.3.2
Correlations and Models
In black oil systems the PVT data, if obtained to mimic reservoir depletion, can be used directly in reservoir simulation studies. In compositional systems analysis (along with the characterization of pseudo components) is used with EOS (equations of state) to predict the PVT behavior of hydrocarbon fluids. In some cases when adequate information is not available from PVT studies of black oil systems, correlations are used. Such correlations usually lead to poor accuracy.
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4.4
Impact of Data of Various Types
In the section on history matching, the impact of data on various reservoir variables (pressure, saturations, production rate, bottomhole pressure, WOR, GOR) was discussed. Reservoir simulator provides an ideal tool to investigate the impact of various data on key predictions to be made with a simulator. Sensitivity studies should be performed to see how simulator results for the reservoir in question change with variations in input data. A carefully planned sensitivity study can help in the identification of data that are most important for the study. In most cases accurate estimation of fluids in place (pore volume and saturations), and the identification of major flow paths (high permeability regions that connect wells and different regions of the reservoir, fractures) and barriers to flow (faults) are a crucial starting point for making reliable predictions. The simulator itself can be an important aid in refining this information - through history matching, provided sufficient good quality production history data are available.
5
Limitations of Current Technology and Current Research
While great advances have been made in the simulation technology since its birth approximately 40 years ago, there are still many unresolved problems that are me subject of current research. These limitations have been discussed throughout these notes, particularly in the section on major problems. The greatest improvement in performance predictions is expected to result from improved reservoir description and our ability to use this information in the simulator. Improved logging, well testing and seismic techniques along with geostatistical tools for data integration are providing new insight about reservoir heterogeneities. The proper use of this information requires simulators with flexible grids that can be easily generated to satisfy various criteria. Such simulators are in various stages of development, and some are starting to appear on the market. Great deal of work is still needed in the development of efficient ways of generating and using flexible grids. In the area of modeling EOR processes we are largely limited to modeling only portions (for example one-eighth of a five-spot) of the reservoir. Main reasons for this are numerical dispersions and the high computational effort required to solve the equations. In some cases inadequate process description also limits our ability to model EOR processes like in situ combustion and foam flooding. Reservoir cannot be isolated from production facilities, yet most simulations are done by making this assumptions. Full field simulation that includes all coupled production facilities is the next challenge. Current techniques for this are inaccurate and crude. Multiphase flow in the production systems and lack of adequate techniques for modeling them are a serious limitations. Once adequate full field models have been developed, then we have the possibility to do full field optimizations and risk assessment using such models. Domain decomposition and parallel computation are likely to have a major impact on success in this area. Traditional reservoir simulators with field oriented options desired by the users can easily approach one million or more line of code. Development, upgrading and maintenance of such large programs developed with traditional procedural languages (like FORTRAN) has become a monumental task. As a result, most of the major oil companies have opted to use software developed by commercial vendors. There are indications that this trend may change as object based design using the features of object oriented languages like C++ (Verma and Aziz, 1996) [11]. Object based technology will allow the development different objects by different
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teams. Furthermore, this approach will make it easier to exploit emerging hardware and other software technologies.
6
Computer-Requirements
Reservoir simulations of large reservoir are often limited by the capacity and speed of computers. While many routine simulations can be conducted on workstations, larger problems require super computers and parallel processors. Software currently in use is rarely able to take full advantage of available hardware. Utilization new hardware like parallel machines is even harder. With good software tools, reservoir simulation engineers will require the power of the most powerful computers.
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Concluding Remarks 1. Reservoir simulation is a powerful and essential tool for reservoir management. 2. While there are still some serious limitations, intelligent use of this technology can provide solutions to many problems. 3. Reservoir description, process understanding, flexible gridding, full field simulation and optimization, and risk assessment through simulation are some of important areas of current research. 4. New object based software design based on object oriented languages like C++ is expected to make it easier to take advantage of emerging hardware and software technologies.
8 Φ µ φ ωc,p ∆t A c d f i j K kr M n
Nomenclature flow potential viscosity porosity concentration of component c in phase p timestep size during simulation cross-sectional area of a block boundary between two grid blocks index for component for which the material balance equation is written distance between grid nodes transmissibility correction factor index for block for which the material balance is written index for blocks that are connected to the block i for which the material is written absolute permeability relative permeability mass of material (component) in the block at a given time timestep number
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Nn Np p qw S T Vb
number of connected blocks to block i number of phases in the system phase index = oil, water or gas well flow rate saturation transmissibility block volume
References [1] Aziz, K. Ten golden rules for the simulation engineer. JPT (November 1989). [2] Aziz, K. Reservoir simulation grids: Opportunities and problems. JPT (July 1993). [3] Aziz, K., and Settari, A. Petroleum Reservoir Simulation. Applied Science Publishers, 1979. [4] Ballin, P., Journel, A., and Aziz, K. Prediction of uncertainty in reservoir performance forecasting. Journal of Canadian Petroleum Technology 31, 4 (April 1992). [5] Batycky, R. P., Thiele, M. R., and Blunt, M. J. A streamline simulator to model field scale three-dimensional flow. Tech. Rep. SCRF Report 9, Stanford, May 1996. [6] Bissell, R. Calculation of optimal parameters for history matching. In ECMOR 4 (Roros, Norway, 1994). [7] Palagi, C., and Aziz, K. Use of voronoi grid in reservoir simulation. In SPE 66th Annual Technical Conference and Exhibition (Dallas, TX, October 6-9 1991). [8] Palagi, C., and Aziz, K. Handling of wells in reservoir simulators. In Fourth International Forum on Reservoir Simulation (Salzburg, Austria, August 31 - September 4 1992). [9] Palagi, C., and Aziz, K. The modeling of flow in heterogeneous reservoirs with voronoi grid. In SPE Symposium on Reservoir Simulation (New Orleans, Feb. 28 - March 3 1993). [10] Pettersen, O. The Gullfaks field – A modelling challenge. In Fourth International Forum on Reservoir Simulation (Salzburg, Austria, August 31 - September 4 1992). [11] Verma, S., and Aziz, K. FLEX: An object-oriented reservoir simulator. In the Petroleum Computer Conference (Dallas, Texas, 1996). [12] Verma, S., and Aziz, K. Two- and three-dimensional flexible grids for reservoir simulation. In ECMOR 5 (Leobon, Austria, 1996).
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