Robust Control Volume Methods for Reservoir Simulation on Challenging Grids PhD Thesis Eirik Keilegavlen Department of Mathematics University of Bergen November 2009 Preface and Acknowledgements Preface This dissertation is submitted as a partial fulﬁlment of the requirements for the degree Doctor of Philosophy (PhD) at the University of Bergen. The research has been conducted at the Centre for Integrated Petroleum Research at the University of Bergen. Outline The dissertation consists of three parts. The ﬁrst part is devoted to background theory, and is structured as follows. Chapter 1 gives a high-level overview over important concepts and challenges in reservoir simulation. In Chapter 2, mathematical models for ﬂow in porous media are presented. The next three chapters are devoted to numerical techniques for solving the equations. In Chapter 3, we consider various aspects of reservoir simulation. Discretisation techniques for elliptic and hyperbolic equations are discussed in Chapter 4 and 5, respectively. Summary of papers produced are given in Chapter 6, while in the last chapter, conclusions are drawn, and future research directions are pointed out. The second part consists of in total ﬁve papers and manuscripts that have been produced during the work with the thesis. Three of these are published in journals or accepted for journal publication, one is submitted, and one is still in a draft stage. The third part contains a conference proceedings paper that is not considered part of the thesis, however, some of the results therein is useful for the understanding of the rest of the work. Acknowledgements The completion of this dissertation marks the end of three years of happiness and enthusiasm, as well as frustrations. First and foremost, it has been a period of acquiring experience and knowledge. My principle supervisor has been Ivar Aavatsmark. He has been a source of inspiration for me, and he has always been willing to share his knowledge. For this I am deeply grateful. Magne Espedal and Edel Reiso have been my co-supervisors. Their help and guidance have been of great importance. I will especially thank Magne for always taking time to listen and give advice whenever needed. Funding provided by the Centre for Integrated Petroleum Research at the University of Bergen is gratefully appreciated. During the work with this dissertation, I have greatly beneﬁted from interaction with many colleagues. Above all, I will thank Sissel Mundal for professional cooperation, as well as for being a great friend. Randi Holm has read the dissertation, and given valuable feedback. I will further apologise to both Sissel and Randi for all the convenient (perhaps less convenient for them) invasions of their ofﬁce whenever I did not feel like working. To the rest of the PhD students and employees at CIPR and the Department of Mathematics, thanks for providing a good research and social environment. While working with this thesis, I had the pleasure to visit Stanford University for a couple of months. This was an instructive and exciting time for me, both professional and socially, and I will thank Hamdi Tchelepi for giving me the opportunity to visit Palo Alto. My main collaborators there were Jeremy Kozdon at Stanford, and Brad Mallison at Chevron. I highly appreciate their hospitality and their willingness to discuss whatever I was wondering about. I am lucky to have great friends to remind me there is a world outside the concrete walls of the science building. Their support and encouragement has been of great importance to me. Furthermore, I will thank my family for always being there whenever I need them. And ﬁnally, I will thank Lena for making the ﬁnal year of the dissertation a joy. Your love, patience, and understanding have been an invaluable support to me. Eirik Keilegavlen Bergen, November 2009 Contents I Background 1 1 Introduction 1.1 Reservoir Characteristics . . . 1.1.1 Reservoir Geology . . 1.1.2 Fluid Characterisation 1.2 Production Processes . . . . . 1.3 Numerical Simulations . . . . 1.4 Scope of the Thesis . . . . . . 2 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 5 6 7 8 Mathematical Modelling 2.1 Physical Parameters . . . . . . . . 2.1.1 Rock Properties . . . . . . 2.1.2 Fluid Properties . . . . . . 2.1.3 Petrophysics . . . . . . . 2.2 Single Phase Flow . . . . . . . . . 2.3 Immiscible Two-Phase Flow . . . 2.4 Three-Phase Flow . . . . . . . . . 2.4.1 The Black Oil Formulation 2.5 Further Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 9 10 12 15 16 17 19 20 23 23 24 25 28 . . . . . . Reservoir Simulation 3.1 Solution of non-Linear Systems 3.1.1 Linear Solvers . . . . . 3.2 Grids . . . . . . . . . . . . . . 3.3 Time Stepping Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elliptic Discretisation Principles 4.1 Preliminaries . . . . . . . . . . 4.2 Control Volume Methods . . . . 4.3 Two Point Flux Approximations 4.3.1 Non-Linear Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . 31 . 32 . 33 . 34 4.4 4.5 4.6 4.7 5 Multi-Point Flux Approximations . . . . . . . . . . . . . . . . . 4.4.1 The O-Method . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The L-Method . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Other Multi-Point Methods . . . . . . . . . . . . . . . . . Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Maximum Principles . . . . . . . . . . . . . . . . . . . . 4.5.3 Sufﬁcient Condition for Monotonicity of Control Volume Methods . . . . . . . . . . . . . . . . . . . . . . . . . . Near-Well Discretisations . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Characteristics of Near-Well Regions . . . . . . . . . . . 4.6.2 Well Index Approaches . . . . . . . . . . . . . . . . . . . 4.6.3 Applications of MPFA Methods to Near-Well Modelling . Other Locally Conservative Methods . . . . . . . . . . . . . . . . 4.7.1 Mixed Finite Elements . . . . . . . . . . . . . . . . . . . 4.7.2 Mimetic Finite Differences . . . . . . . . . . . . . . . . . 4.7.3 Multiscale Methods . . . . . . . . . . . . . . . . . . . . . Hyperbolic Discretisation Principles 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Riemann Problems . . . . . . . . . . . . . . . . . . . . 5.1.2 Hyperbolic Formulations . . . . . . . . . . . . . . . . . 5.2 Control Volume Methods . . . . . . . . . . . . . . . . . . . . . 5.2.1 Upstream Weighting . . . . . . . . . . . . . . . . . . . 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting 5.3.1 Multi-Dimensional Transport . . . . . . . . . . . . . . 5.3.2 Discretisation of Two-Phase Flow . . . . . . . . . . . . 5.4 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Riemann Solvers and Front Tracking . . . . . . . . . . 5.4.2 Control Volume Methods . . . . . . . . . . . . . . . . . 5.4.3 Lagrangian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 38 39 40 40 41 41 42 44 46 47 48 50 53 54 54 56 57 57 58 58 60 61 62 63 66 71 71 72 74 6 Summary of the Papers 7 Conclusions and Future Directions 85 7.1 Control Volume Methods for Elliptic Problems . . . . . . . . . . 85 7.2 Simulation in Near-Well Regions . . . . . . . . . . . . . . . . . . 87 7.3 Multi-Dimensional Upstream Weighting . . . . . . . . . . . . . . 88 Bibliography 79 91 II Included Papers 111 A Sufﬁcient Criteria are also Necessary for Monotone Control Volume Methods B Monotonicity of MPFA-methods on triangular grids C Non-hydrostatic pressure in σ-coordinate ocean models D Discretisation Schemes for Anisotropic Heterogeneous Media on Near-well Grids E Multi-dimensional upstream weighting on general grids for ﬂow in porous media III Related publications F Monotonicity of Control volume methods on triangular grids Part I Background Chapter 1 Introduction Understanding of ﬂow in subsurface porous media is of importance for several reasons. Recovery of petroleum stored in the underground has been a vital source of energy for the last 100 years or so, and it likely to be so for several decades to come. Also, the subsurface stores vast amounts of geothermal energy that can be exploited. Further, ground water is the main water supply in large parts of the world, and thus contamination of water resources by industrial pollution or nuclear waste is a major concern. Geological storage of CO2 is one of the proposed solutions to reduce global warming, which is related to an increase in the concentration of CO2 in the atmosphere. These are some of the motivating factors for getting better knowledge of processes that takes place in porous media. The main focus in this work is recovery of hydrocarbons, however, many of the results can be relevant for other applications as well. In this introductory chapter, we provide a brief overview of important concepts in reservoir technology that serve as a motivating background for the thesis. Some of the concepts will be studied in more depth in the following chapters, for further information, see, e.g., [65, 97, 98, 180, 208, 235]. At the end of the chapter, the main directions of the thesis are outlined. 1.1 1.1.1 Reservoir Characteristics Reservoir Geology A petroleum reservoir consists of rock perforated by small channels, or pores, that are ﬁlled with hydrocarbons. To act as a reservoir, the pores must be connected, so that the ﬂuids can ﬂow. The rock in a reservoir has originated from sedimentary deposition processes that took place millions of years ago. It therefore has a layered structure, where each layer consists of a different type of rock, as seen in 4 Introduction Figure 1.1: An picture of a meter-scale outcrop. Note the layering, and the irregularities in the geology. Figure 1.1. Geological activity can later have modiﬁed the structure. The reservoir may contain fractures, that is cracks in the rock. Further, the layers may have been displaced along a fracture, creating a fault. These are some examples of the highly heterogeneous nature of a porous medium; it contains characteristic features on a continuum of length scales. To create a viable model of the reservoir is a difﬁcult task. Hard data is only available from core samples obtained from wells. Due to rock heterogeneities, information from the core samples is only locally representative. To obtain more knowledge of the reservoir, we must resort to indirect methods. A much used source of information is seismic data. In seismic surveys, sound waves are sent from the surface towards the rock. When the waves propagate through the rock, the signal is altered. Indeed, different types of rock have different impact on the waves. Hence, a careful interpretation of the reﬂected signal can expose features of the rock. Seismic surveys are thus based on the elastic properties of the rock. In electromagnetic surveys, one instead utilise the rock’s electric conductivity, by sending electromagnetic waves through the rock, and analyse the reﬂected signal. Further, dynamic data from wells, such as pressure and ﬂow rate can also inform on the structure of the reservoir. We emphasis that all these data sources can inform on the distribution of ﬂuids in the reservoir as well as on the geology. Based on these various sources of information, a geological model of the reservoir is build. Due to the lack of hard data, the geo-model relies heavily upon statistical methods. The sparsity of information on the rock means it is impossible to model each individual pore. Moreover, if such a description of the pore system was available, numerical simulations of ﬂow based on this model would require computational resources far beyond what is available. Therefore, the rock parameters in the geological model describe average properties. Despite the introduction 1.1 Reservoir Characteristics 5 Seal GAS OIL Impermeable rock WATER Figure 1.2: Schematic overview of a petroleum reservoir. The hydrocarbons are trapped by a sealing rock. The ﬂuids are in hydrostatic equilibrium. of average parameters, the geo-model can consist of up to 107 − 108 grid cells. This is at least one to two orders of magnitude more than what is possible for a traditional reservoir simulator to handle. Therefore, a coarser grid is constructed, and the rock properties are represented on the new cells. The coarsening process is known as upscaling. Ideally, the upscaling should not be detrimental to the quality of the reservoir model. It is therefore crucial to construct the coarser grid so that the rock parameters can be represented accurately in each cell. 1.1.2 Fluid Characterisation Together with the sediments that later became the reservoir rock, organic material was also deposited. This material has since evolved into oils and gasses by chemical reactions. It is commonly believed that the hydrocarbons are not formed in the reservoir, but in some source rock. Later, the hydrocarbons migrate through the reservoir. The main driving forces of the migration are buoyancy effects, but also capillary forces can affect the movement. Eventually, the hydrocarbons may reach a low conductivity rock formation, a seal, which prevents them from propagating any further. When trapped, the ﬂuids are in hydrostatic balance, as illustrated in Figure 1.2. The petroleum in a reservoir can consist of hundreds of different chemical components. In general, they are either liquids or vapour, commonly referred to as oil and gas, respectively. The vapour phase has lower density than the liquid phase. Moreover, all the hydrocarbons are in general lighter than water. The gas components can be highly mobile due to low viscosities, whereas the heavy oils can have much larger viscosity than water. During a recovery process, the hy- 6 Introduction drocarbons experience a wide range of pressure and temperature regimes. Both viscosity and density can be highly dependent on pressure and temperature. Further, the hydrocarbons can undergo phase transitions during production. A proper description of ﬂow in the reservoir requires knowledge of these PVT-properties of the ﬂuids. The properties can be quantiﬁed by laboratory analysis of the produced hydrocarbons. However, due to the complex chemical composition of the ﬂuids, this is a highly challenging task. When more than one phase is present, there are forces acting on the interface between the phases. The ﬂuids feel the action of these capillary forces in the form of reduced rock conductivity. To accurately measure these petrophysical properties is difﬁcult at best. 1.2 Production Processes The reservoir is initially at rest. This balance is disturbed when a well is opened, and the ﬂuids start to ﬂow. The initial reservoir pressure is often high enough to bring the ﬂuids all the way to the surface without external supply of energy. This period is called primary production. There are a few physical processes that can provide the energy necessary to recover the hydrocarbons. They are all initiated by the pressure drop, and tend to counteract the decline in pressure. The reservoir rock can experience a deformation and a decrease in pore volume, causing a compaction drive. Water located beneath the hydrocarbons can ﬂow upwards and maintain the pressure. This is called (natural) water drive. A decline in pressure also cause the highly compressible gas to expand, and cause a gas expansion drive. Moreover, at high pressures, gas can be dissolved into the liquid phase. As the pressure decreases, the gas is released again, sustaining the pressure. The primary production is rather ineffective, and can at best lead to a recovery rate of about 20%. Sometimes the rate is much lower. To produce more hydrocarbons, energy must be supplied to the reservoir. An easy and cheap way to do this is to maintain the pressure by injecting water. This water ﬂooding stage is called secondary production. The injection is most often done in the water layer beneath the hydrocarbons. The injection of water can be effective, and yield recovery rates of up to 40-50 %. However, the reservoir is heterogeneous, and water often ﬁnd a highly conductive path between the injection well and a producer. After water breakthrough, water ﬂooding have limited effect, and large amounts of hydrocarbons are left in less conductive parts of the reservoir. The effect of water ﬂooding is also limited by the high viscosity of heavy oil relative to water, and by capillary trapping of oil. To recover even more petroleum enhanced oil recovery (EOR) techniques are employed. This is called the tertiary production period. There are many EOR- 1.3 Numerical Simulations 7 methods, we will mention some of those most commonly applied. The adverse viscosity ratio between heavy oil and water can essentially be improved in two ways. The water viscosity can be increased, for instance by adding polymers to the injected water. The oil viscosity is reduced with an increase in temperature, which can be achieved by injection of hot water or steam. Also, if the supply of oxygen is carefully controlled, one can start in situ combustion of some of the viscous oil, and thereby heat the surrounding hydrocarbons. Other EOR-techniques aim at reduced interfacial tensions between the phases. If a mixture of gasses, notably containing CO2 , is injected, these can mix with the oil, and initiate a miscible displacement, where the gas and oil components act as a single phase with higher mobility. Gas injection is often employed in combination with injection of water. The interface tension can also be reduced by injection of surfactants, or by a careful accommodation for growth of micro-organisms, that can help change the ﬂow pattern. 1.3 Numerical Simulations Numerical simulations of ﬂow in the reservoir have been employed in the industry for several decades. To optimise the recovery rate, a long term ﬁeld development plan is made, that involves drilling of wells, application of EOR-techniques, etc. The planning is supported by numerical predictions of the impact of different production scenarios. Also, numerical simulations can help day to day decisions on closing and opening wells etc. Reservoir models have an inherently high level of uncertainty, due to for instance the sparsity of geological information. To quantify the uncertainty, one can apply Monte Carlo simulations, where several realisations of possible geological models are employed in simulations, and statistics are computed from the results. Uncertainty estimation and prediction of worst case scenarios will also play a major role for large scale geological storage of greenhouse gasses. We emphasis that modelling and simulation of porous media ﬂow are challenging mathematical and numerical problems. The geology is highly heterogeneous, and the ﬂuid behaviour can be complex with frequent phase transitions. The time scale of the processes of interest spans from day-to-day production management, to the fate of CO2 hundreds of years after it is injected in the porous medium. A typical well has a diameter on the scale of centimetres, whereas the size of the reservoir can be several kilometres, yielding a vast difference in spatial scale. To give a proper mathematical description of the ﬂow is therefore nontrivial. Further, the model is discretised using advanced numerical techniques. The discretisation often leads to a large system of ill-conditioned equations. The choice of a mathematical model and numerical schemes is a trade-off between ac- 8 Introduction curacy and computational cost. Due to the large span in processes and simulations of interest, there is a need for diverse numerical methods with different forte. 1.4 Scope of the Thesis Even though numerical simulations successfully have been applied in the industry for decades, there is a constant need for further improvements of the methods. The work can be aimed at developing new techniques, or to extend already existing methods to new areas of applications. Also, it is important to achieve a better understanding of properties and limitations of already existing approaches. The focus in this work has been development and analysis of a class of numerical schemes known as control volume methods, which are prevailing in commercial reservoir simulators. In the present work, we have investigated a broad range of subjects related to control volume methods. The main computational overhead in reservoir simulations come from solution of linear systems of equations. So far, most simulation grids have been structured, partly due to a lack of efﬁcient linear solvers that allows for more general grid structures. However, usage of such general solvers seems to become more widespread, and thus ﬂexible grids can be used to a larger extent. This will render possible a better representation of complex geological structures. In this work, we analyse the robustness of control volume methods on realistic and ﬂexible grids. Moreover, we study applications of transport schemes designed for simulating EOR-scenarios with adverse mobility ratios on unstructured grids. All the recovered hydrocarbons are transported through wells. A proper description of ﬂow in near-well regions is therefore a key issue. In the vicinity of a well, the ﬂow rates are high, and the ﬂow pattern radial like. Moreover, to optimise the recovery rate, skew and horisontal wells are drilled, yielding highly heterogeneous near-well regions. The numerical schemes must be adapted to these challenges. In this work, we perform systematic tests of different grids and discretisation techniques in the vicinity of wells. Chapter 2 Mathematical Modelling The goal of this chapter is to provide insight in some physical effects that are important for ﬂow in porous media. Since this work mainly has been devoted to development and analysis of numerical methods, we will put emphasis on mathematical modeling. We start by describing some important physical parameters. Further, we discuss several sets of governing equations that model increasingly complex physical processes. During the presentation, we will highlight properties that are important for a numerical solution procedure. 2.1 Physical Parameters The physical properties used to describe the ﬂow in a reservoir can be divided into three groups, describing the rock, the ﬂuid, and the interaction between rock and ﬂuid. To characterise any of these properties is a research area in itself, thus we will only give a high-level presentation. 2.1.1 Rock Properties A reservoir consists of rock perforated by channels, or pores, ﬁlled with ﬂuids. It is impossible to obtain an accurate description of the channel system for a fullscale reservoir, and the computational cost of simulations on such a model will be far too high. Therefore, we must base the modelling on a macroscopic description of the reservoir. To bridge the gap between the microscopic pores and the spatial resolution feasible for simulations, we introduce the concept of representative elementary volumes (REV). The size of an REV must be much larger than the size of an individual pore. On the other hand, we deﬁne parameters describing the properties of the REV based on averaging. Hence, the size of the volume must be smaller than 10 Mathematical Modelling the length scale on which the rock properties vary considerably. It is important to keep in mind that all parameters, equations and so on are deﬁned for REVs, and are not valid for smaller scales. See e.g. [32] for a further discussion. For simplicity, we treat the rock properties as constant in time, i.e. we ignore compaction. See however Section 2.5. Porosity To describe the ﬂow through the pores, we need a measure of pore volume. Denote by Vpore the pore volume in an REV. The porosity, φ, is deﬁned as the volume fraction occupied by pores to the total volume, φ= Vpore , Vtot (2.1) where Vtot is the total volume of the rock. Note that Vpore only consists of the volume of interconnected pores, that is, we have neglected solitary caves. Absolute Permeability The ﬂow in the reservoir is driven by differences in the pressure. The ﬂow rate generated by a pressure gradient depends on the distribution of pores. In reservoirscale modelling, the ﬂow conductivity of the rock is described by the permeability, denoted by K. The permeability is a tensor, and according to Onsager’s principle, it is symmetric and positive deﬁnite. A realistic permeability ﬁeld is highly heterogeneous. Further, the permeability might be anisotropic, that is, the conductivity is different in different directions. 2.1.2 Fluid Properties The ﬂuid in the reservoir can exist in several phases. In this work, we allow for three different phases; aqueous, liquid, and vapour, denoted by a, l, and v, respectively. For our modelling, we need expressions for viscosity and density for each of the phases. The process of quantifying these properties is known as PVT-analysis. The aqueous phase most often consist of water only, although some models, notably those related to CO2 -sequestration, can allow for other species in the aqueous phase. Since the chemical composition of the aqueous phase is fairly homogeneous, the PVT-characteristics are not very complex. On the contrary, the hydrocarbons in a reservoir can consist of hundreds of different chemical components. Since the PVT-properties are dependent on the chemical composition of the phase, as well as on temperature and pressure, state of the art experimental 2.1 Physical Parameters pb (a) Vapour viscosity 11 pb (b) Density Figure 2.1: Qualitative behaviour of viscosity and density for a black oil model (Section 2.4.1). For high pressures, all of the hydrocarbons are in the vapour phase. When pressure drops below the bubble point pressure pb , the components with small molecules evaporate. The bubble point pressure is dependent on temperature and chemical composition of the hydrocarbons. The y-axes are not to a speciﬁc scale. analysis is needed for quantiﬁcation. As a rule of thumb, the ﬂuid properties can be characterised by the molecular weight of the components. Components that consist of large molecules, usually exhibit fairly stable behaviour under changes in pressure and temperature. If the molecular weight of a component is small, its PVT-characteristics is often highly sensitive to pressure and temperature changes. In general, a hydrocarbon component is found in both the liquid and vapour phase simultaneously. The larger the molecular weight, the larger portion of the component can be expected to be found in the liquid phase. We remark that in the literature, the phases are called water, oil, and gas. Further, the hydrocarbon components are often called to as oil and gas components referring to their state under some standard conditions. The present terminology is chosen to avoid confusion between phases and components. Throughout the thesis, α denotes a generic phase. Viscosity The viscosity, μα , represents the internal friction in the ﬂuids. For the aqueous phase, μa is usually only weakly pressure dependent, it is often considered constant. The viscosity of the liquid phase depends upon both the pressure, and the molar composition of the phase. A possible viscosity behaviour is illustrated in Figure 2.1(a). For the vapour phase, the viscosity is generally increasing with pressure. We remark that viscosity is highly dependent on temperature, this must be taken into account for non-isothermal ﬂow. 12 Mathematical Modelling Compressibility and Density The compressibility c, of a ﬂuid is deﬁned by c= 1 ∂ρ , ρ ∂p (2.2) where ρ is the density of the ﬂuid. For ideal ﬂuids the compressibility is constant, and Equation (2.2) can be integrated to yield ρ = ρ0 e(c(p−p0 )) , (2.3) where ρ0 is a reference density, measured at a reference pressure p0 . Water can usually be considered an ideal ﬂuid in reservoir simulation, moreover, the water compressibility is small. So can the liquid phase, if it is composed by large hydrocarbons. However, hydrocarbons with small molecular weight, which are likely to be found in the vapour phase are generally not ideal ﬂuids. In that case, the relation is given by more advanced equations of state, see e.g. [191]. See also Figure 2.1(b) for density behaviour of black oil ﬂuids. For simplicity, we often make an assumption of an incompressible ﬂuid by setting c = 0. Although not physically valid, this assumption eases both implementation and analysis of the problems. The impact of compressibility varies from one problem to another. Sometimes it can be seen as the limiting case when compressibility goes to zero, see for instance the remark at the end of Section 4.5. Under other circumstances, incompressibility is not a limiting case, but a singularity, and the properties of the model change signiﬁcantly when incompressibility is assumed. For instance, we will see that the pressure variable no longer needs initial values if the ﬂuids are assumed incompressible. Throughout the thesis, we often make this assumption. It is important to keep in mind that extensions to physically valid conditions may add signiﬁcant challenges. 2.1.3 Petrophysics When more than one phase is present in the pore system, there is a surface tension on the interface between the ﬂuids. This generates a capillary pressure on the interface. The interactions are determined by the geometry and the topology of the pore system, as well as on the ﬂuids. For a two-phase system, the wetting phase tends to stick to the pore walls, whereas the non-wetting phase is found in the middle of the pores. We can also have mixed-wet conditions, where which phase is wetting depends on the pore size. The wettability depends on the rock, as well as on the ﬂuids, see e.g. [113, 214]. For three-phase systems, the wettability of a phase is deﬁned relative to each of the other phases. 2.1 Physical Parameters 13 To ﬁnd functions for the relative permeabilities and the capillary pressure (deﬁned below) is a non-trivial task. Since they both are caused by interface tension, they must be deﬁned consistently. For two-phase ﬂow, there are several models,see [110, 113] and the references therein. These models contain tuning parameters that must be determined from laboratory experiments. To measure the ﬂow functions for three-phase ﬂow is nearly impossible. Instead, one must resort to interpolation of data from two-phase experiments. Saturations The forces between the phases depend on how large portion of the pore space is occupied by each phase. The saturation of a phase α, Sα , is deﬁned by Sα = Volume occupied by phase α . Total pore volume (2.4) By deﬁnition, 0 ≤ Sα ≤ 1. In general, some ﬂuid will be immobile due to capillary forces. The saturation of this immobile ﬂuid is called residual saturation, Sα,r . We deﬁne the effective saturation as Sα,e = Sα − Sα,r . 1 − α Sα,r (2.5) An assumption that the ﬂuids together ﬁll the entire pore space yields the constitutive relation Sα = 1 . (2.6) α Capillary Pressure On the interface between two phases, the pressure is discontinuous due to the surface tension. We deﬁne the capillary pressure as pcla = pl − pa , pcvl = pv − pl , (2.7) where pα , α = a, v, l is the phase pressures. The capillary pressures are considered functions of phase saturations. Capillary forces are of great importance for small scale processes, but the inﬂuence diminishes for larger length scales. Therefore, the capillary pressure is often neglected in simulations. Nevertheless, capillary forces have been shown to be important for simulation of certain EOR-processes [66]. 14 Mathematical Modelling Pc kr,nw l j j l m m k Sw,r (a) So,r 1 k Sw Sw,r Sw (b) So,r 1 Figure 2.2: Curves for capillary pressure (left) and relative permeability (right) with hysteresis in a water-wet system. The ﬁgure shows the bounding curves for drainage (j) and imbibition (k), and two examples of scanning curves for drainage (l) and imbibition (m). Note that both the capillary pressure and the relative permeability is considered functions of water saturation. Relative Permeability The absolute permeability represents the conductivity when the pores are occupied by a single phase. When more than one phase is present, the phases will interact, and thereby reduce the ﬂow. To describe this reduced conductivity, we introduce the relative permeability, kr,α , for each phase. The relative permeability for a phase is zero for saturation values below Sα,r . If S > Sα,r , kr,α increases, but it never exceeds unity. The effective permeability, Kα , is then deﬁned by Kα = kr,α · K . (2.8) The relative permeabilities are strongly non-linear functions of the phase saturations. Moreover, the ﬂuid interactions also reduce the total ﬂow, that is, α kα < 1. We now deﬁne the mobility of a phase, λα as relative permeability divided by viscosity, λα = kr,α . μα (2.9) Hysteresis The ﬂow functions depend not only on the saturations, but also on the saturation history. This is known as hysteresis. Figure 2.2(a) shows how the capillary pressure is different during drainage (Sw decreases) and imbibition (Sw increases). If 2.2 Single Phase Flow 15 the saturation change is reversed before it reaches the endpoint (Sw,r or 1 − So,r ), the capillary pressure will follow a scanning curve instead of the bounding curve. The relative permeabilities can also experience hysteresis, mainly for the nonwetting phase, see Figure 2.2(b), albeit the effect is not that important. 2.2 Single Phase Flow A model for single phase ﬂow is of limited importance in practical simulations, although it can be representative for the dynamics in the primary stages of production. However, the equations governing this ﬂow will serve as model problems in later chapters. When only one phase is present, the ﬂux, q, is computed from the gradient of the pressure by Darcy’s law, which reads 1 q = − K∇(p − ρgz) , μ (2.10) where g is the acceleration of gravity, and z is the vertical coordinate. Darcy’s law was originally found experimentally, however, it can be derived as a ﬁrst-order approximation of a volume averaged Navier-Stokes equation, see e.g. [227]. The Darcy model is only accurate for laminar ﬂow; in high ﬂow-regimes, this assumption breaks down. Such non-Darcy effects can be modelled by Forchheimer’s law, see e.g. [26]. Consider an arbitrary volume Ω. For conservation of mass within the volume, the total ﬂux out of Ω must equal the accumulation in the volume. Thus we get the conservation equation ∂ρ φ dV + n · (qρ) dS = q dV , (2.11) Ω ∂t ∂Ω Ω where n is the outer normal vector of the boundary ∂Ω, and we have taken possible source or sink terms, q, into account. By using the divergence theorem and Darcy’s law, we get the equation for single phase ﬂow φ ρ ∂ρ −∇· K∇(p − ρgz) = q . ∂t μ (2.12) Since density is a function of pressure, Equation (2.12) is non-linear. For weakly compressible ﬂuids, i.e. water or hydrocarbons with large molecular weight, Equation (2.11) can be approximated by a linear parabolic equation 1 q ∂p K∇(p − ρgz) = , (2.13) cφ − ∇ · ∂t μ ρ 16 Mathematical Modelling where we have neglected terms of order (∇p)2 . In the limit of incompressible ﬂuids, we get the elliptic equation q 1 K∇(p − ρgz) = . (2.14) −∇ · μ ρ 2.3 Immiscible Two-Phase Flow In some cases, the hydrocarbons are solely in the liquid phase, and the liquid and aqueous phase make up a two-phase system. Especially during secondary production, the ﬂuids are separated into two phases, provided the pressure is above the bubble point pressure. Moreover, many of the mathematical properties of the equations governing ﬂow in porous media can be illustrated by the ﬂow of two immiscible ﬂuids. It is therefore of interest for us to study two-phase ﬂow before we continue to more complex models. Let the two phases present be aqueous (a) and liquid (l). For each phase α, the ﬂux is modeled by a generalised Darcy law qα = −λα K∇(pα − ρα gz) . (2.15) For a discussion of the validity of this generalisation, we refer to [128] and the references therein. The conservation of mass for each phase can be stated on integral form as ∂(ρα Sα ) dV + φ n · (qα ρα ) dS = qα dV . (2.16) ∂t Ω ∂Ω Ω Now, the divergence theorem gives us ∂(ρα Sα ) + ∇ · (qα ρα ) = qα . (2.17) ∂t The Equations (2.15) and (2.17) for both phases, together with two-phase versions of (2.6) and (2.7) constitute a closed set of equations for the phase pressures and saturations. To the end of this section, we assume incompressibility of the ﬂuids. After dividing by ρα , summing Equation (2.17) over the phases, and using Equation (2.6), we get the pressure equation qα ∇ · qT = , (2.18) ρ α α=a,l φ where the total velocity, qT = qa + ql , is given by qT = −(K(λT ∇pl − λw ∇pcla − (λa ρa + λl ρl )g∇z)) . (2.19) 2.4 Three-Phase Flow 17 Here λT = λa + λl is the total mobility. This is an elliptic equation, which can be solved for the pressure. To get an equation for the saturation, we combine (2.15) with (2.19), and insert the result in Equation (2.17) for water to get φ λ λ λw qa ∂Sw a l +∇· K ∇(pcla + (ρl − ρa )g∇z) + qT . = ∂t λT λT ρa (2.20) This is a parabolic equation describing conservation of the aqueous phase. However, the capillary term is often dominated by the advection term, thus the saturation is governed by an almost hyperbolic equation. We point out that the governing equations can be modiﬁed to apply other primary variables, notably by introducing a global pressure [52]. The change of variables can both ease analysis of the equations, and make the problem better posed for numerical methods [57]. We will, however, not pursue this any further. 2.4 Three-Phase Flow In general, there will be hundreds of different chemical species present in the reservoir. To describe the motion of each individual component is far beyond the power of the computer resources available. It is therefore common to deﬁne pseudo-components, each representing several chemical species. Denote by nc the number of (pseudo-) components present. For notational simplicity, we assume the water to be separated from the hydrocarbons, i.e., the water component is only found in the aqueous phase, and the aqueous phase contains only water. Throughout this section, the subscript c denotes a generic component. Denote by ξc,α the molar density of component c in phase α. The molar density of phase α is found by ξα = nc ξi,α , α = l, v . (2.21) i=1 Then the mole fraction of component c in phase α is deﬁned as Cc,α = ξc,α . ξα (2.22) We can now write the conservation equation for component c as ∂ φ Cc,α ξα Sα + ∇ · Cc,α ξα qα = ρc,α qα , ∂t α=l,v α=l,v α=l,v (2.23) 18 Mathematical Modelling where qα are the phase velocities found by the generalised Darcy law. The liquid and vapour phase are assumed to be in equilibrium, fc,l = fc,v c = 1, . . . , nc , (2.24) where fc,α are fugacity coefﬁcients. Additional constraints that must be fulﬁlled are the capillary pressure relations (2.7), and further (2.6). Moreover, the sum of the mole fractions for each phase must be 1, that is nc Ci,α = 1 , α = a, l, v . (2.25) i=1 In total, this gives us 2 · nc + 6 equations. The variables are nc mole fractions for both the liquid and the vapour phase, three saturations, and three phase pressures, in total 2 · nc + 6. However, the ﬂow is assumed to be iso-thermal, thus there is one less degree of freedom. If we further apply Gibb’s phase rule, we ﬁnd that the state of the system is determined by nc variables, see e.g. [48, 107]. We will refer to these variables as primary variables, denoted by xp . The remaining, secondary variables, xs , can be found from the primary variables. We indicate this by writing xs = xs (xp ), perhaps unintentionally implying a simple functional relation. In reality to compute the secondary variables most often requires solving the equilibrium relation (2.24), which can take a considerable part of the total computation time. However, our main concern is not equilibrium calculations, thus we will not go into details. Regarding choice of primary equations and variables, there are several options available. These issues should also be seen in connection with the selection of a time stepping scheme (see next chapter). To give a presentation of the options available is beyond the scope of this work, see instead [48] and the references therein. A large effort has been put into investigations of the mathematical structure of the equations for compositional ﬂow. It is shown in [217] that the pressure is governed by a parabolic equation. If the compressibility is small, the pressure ﬁeld will rapidly reach a steady state, thus the equation is almost elliptic. The component transport equations are hyperbolic or close to being hyperbolic, depending on the relative permeability functions, we refer to [33, 100, 101, 219]. Miscible Displacement and Dispersion In the compositional model presented above, the hydrocarbons can exist in two phases. Under some circumstances, the liquid and vapour phase can mix, and act as a single phase. This is called miscible displacement. Since there is no surface tension between the hydrocarbons under miscible displacement, the ﬂow 2.4 Three-Phase Flow 19 increases, and so can the recovery rate do. Therefore, initiation of miscible displacement is a popular EOR technique. The mathematical model for miscible displacement is similar to the above compositional model. However, a process called mechanical dispersion that can make a considerable contribution to the overall ﬂow pattern, is not included in Equation (2.23). The irregular pore network will naturally disperse mass at the micro scale. Frictional forces along the walls in the pores also contribute to the dispersion. To include dispersion effects, we augment the transport term ∇ · ( α Cc,α ξα qα ) in Equation (2.23) to read ∇· Cc,α ξα qα − ξα Sα Dc ∇Cc,α , (2.26) α=l,v where Dc is a dispersion tensor, see [108, 201]. For more information on dispersion, see e.g. [32]. When dispersion effects are included, the transport equations become advection-diffusion equations, where the advection dominates. We will consider numerical methods for transport problems in Chapter 5. The schemes primarily studied in this work add an artiﬁcial numerical diffusion to the problem which often is of much larger magnitude than the mechanical dispersion. Therefore, we will hardly consider transport equations with second order terms, see however Section 5.4.3. 2.4.1 The Black Oil Formulation The computational burden of solving the compositional model can be quite heavy. By making some assumptions, a model known as the black oil formulation can be derived from the compositional model. The black oil formulation is computationally cheaper to solve than a compositional model. Especially during secondary production, if the liquid pressure drops below the bubble point, black oil models are appropriate. The black oil model has two hydrocarbon pseudo-components, referred to as heavy and light, in addition to water. To go from a compositional to a black oil model, we assume the following: • Water can only exist in the aqueous phase, and the aqueous phase contains only water. • The heavy component can only be in the liquid phase. • The light component can be found both in the liquid and in the vapour phase. With these assumptions, the conservation equations read 20 Mathematical Modelling ∂ bα Sα + ∇ · (ρα qα ) = qα , α = a, l , ∂t ∂ bv Sv + Rs bl Sl + ∇ · (bv qv + Rs bl ql ) = qv . φ ∂t φ (2.27) (2.28) Here, we have replaced the densities by the inverse volume factors bα = VVREF , RC where VRC and VREF are the volumes occupied by a unit mass under reservoir and reference conditions, respectively. The amount of the liquid phase consisting of light component is described by the dissolved gas-oil rate Rs . The water and vapour ﬂux are given by a three-phase version of the generalised Darcy’s law (2.15), with the density replaced by the inverse volume factor bα . For the liquid phase, Darcy’s law reads, in terms of the variables used in the black oil formulation, ql = −λl K(∇pl + bl (ρl + Rs ρv )g∇z) . (2.29) The conservation equations for each of the components, together with the constraints (2.6) and (2.7) form a closed set of equations. The primary variables are liquid pressure, and two phase saturations, commonly aqueous and vapour. If the pressure is sufﬁciently high, all of the light component will be in the liquid phase. In such cases, the vapour saturation is replaced by Rs , or equivalently by the bubble point pressure pb , as primary variable. As for the above models, the pressure variable exhibits elliptic behaviour, whereas the saturations are governed by hyperbolic equations in the limit of zero capillary pressure [185, 218]. The black oil model can be extended to allow for vaporised heavy hydrocarbons. This is known as the extended, or volatile, black oil model [185]. We will not pursue this further. 2.5 Further Extensions The models considered in the previous section cannot capture all of physical processes taking place in a porous medium during production. We here brieﬂy mention two possible extensions. Thermal Flow So far, all processes have been considered iso-thermal. In many cases this assumption can be justiﬁed by the large heat capacity of the rock. However, if EORtechniques such as steam injection or in situ combustion are applied, thermal effects must be accounted for by introducing a conservation equation for energy. For 2.5 Further Extensions 21 an introduction to modelling of thermal processes, and associated computational challenges, see for instance [59, 107, 108]. Rock Compaction All of the above models consider the pore volume to be constant. In reality, the pressure drop during production will change the geometry and topology of the pore system. Thus in general, the rock properties described in Section 2.1.1 will change. If these changes have signiﬁcant impact on the ﬂow pattern, both the porosity and the permeability should be considered functions of time. Moreover, rock compaction contributes signiﬁcantly to sustain pressure during production. Proper handling of these issues requires a coupling between geo-mechanical and reservoir models, which will further complicate the simulations. Confer [193] and the references therein for more information. Chapter 3 Reservoir Simulation So far, we have given governing equations for ﬂow in a porous medium. Even though the physical properties included in the modelling are simpliﬁed, we have to solve a set of highly non-linear equations for each time step. The number of grid blocks in the reservoir model can be of order 106 . As we saw in the previous chapter, there may be several unknowns in each grid block, depending on which equations are employed to model the ﬂow. In this chapter, we will describe some aspects of reservoir simulations that will serve as background for the next two chapters. We focus on solution of equations (both non-linear and linear), gridding, and on time stepping methods. The presentation will be rather brief, and the interested reader should consult the references for more information. Treatment of two topics that naturally belong to this chapter is postponed to Chapter 4: Gridding in near-well regions will be considered in Section 4.6. The multiscale methods introduced during the last decade are closely related to linear solvers. These methods are treated in Section 4.7. 3.1 Solution of non-Linear Systems Except from some very simpliﬁed cases, all the models presented in Chapter 2 contain equations that are non-linear. Thus we need to solve equations of the form F(xn+1 xs (xn+1 )) = 0 , p p (3.1) where F is a discretisation of the governing equations written in residual form (F = 0). The primary and secondary variables are represented by the vectors xp and xs , respectively. Superscripts refer to time steps. 24 Reservoir Simulation In reservoir simulation, non-linear equations are usually solved by the iterative Newton-Raphson method, or Newton’s method for short. Let xn,l p be a member of n,0 the sequence of approximate solutions at time n + 1, with xp = xnp . A Taylor expansion of F around xn,l p,i with respect to the primary variables gives us n,l F(xn,l+1 , xs (xn,l+1 )) ≈ F(xn,l p p p , xs (xp )) + n,l ∂F(xn,l p , xs (xp )) Δxp,i , (3.2) ∂xp where Δxp = xn,l+1 − xn,l p p . The differentiation is carried out with respect to the primary variables at iteration l. If the solution obtained from the next iteration is exact, the residual is zero, and thus n,l ∂F(xn,l p , xs (xp )) n,l Δxp = −F(xn,l p , xs (xp )) . ∂xp (3.3) This is a linear system to be solved for the update Δxp . If F (xn,l+1 , xs (xn,l+1 )) p p n+1 n,l+1 is smaller than some tolerance, we set xp = xp . If not, we recompute the Jacobian matrix, and obtain a new linear system. We also remark that columns corresponding to explicit variables are non-zero on the diagonal only, and thus these variables can be eliminated, yielding a smaller linear system. The algorithm presented above is a straightforward application of the Newton scheme. Provided with a good initial guess, the iterations will eventually experience second order convergence. However, in practical simulations, the equations are badly conditioned, and the iterations may not converge. The classical way to overcome convergence problems is to disregard the non-convergent iterations, reduce the time step size and start computing a new sequence of approximations. By taking into account the physical properties described by the equations, it is possible to design algorithms that are more robust and more efﬁcient. Since the transport equations have a hyperbolic nature, the approximation obtained by the Newton iteration might be improved by an update using local information. This approach is investigated in [20, 145], where the pressure ﬁeld obtained from an iteration is used to reorder the unknowns. This allows for a local update of variables governed by hyperbolic equations. Another possibility is to rewrite the Newton-Raphson scheme to a form where it is not necessary to disregard nonconvergent iterations, see [233]. 3.1.1 Linear Solvers In the Newton algorithm, a linear system of equations must be solved in each iteration. The size of the system can be several times the number of cells in the simulation grid. Thus, to solve linear systems is often a bottleneck, and the 3.2 Grids 25 availability of efﬁcient linear solvers plays an important role when the simulation strategy is decided. The large number of unknowns makes direct methods like LU-decomposition inappropriate. Instead, iterative schemes are applied, mainly methods that seek solutions in Krylov subspaces. An overview of such methods can be found in e.g. [31, 220]. For Krylov methods to be effective, they should be accompanied by a suitable preconditioner. A survey of preconditioning techniques can be found in, for example [35]. For the elliptic part of the problem (the pressure unknowns), the global domain of dependence makes multigrid methods with domain decomposition appropriate. The multigrid method could be geometric or algebraic, depending on the whether the grid is structured or unstructured (see next section). More information on multigrid methods can be found in e.g. [221], applications to reservoir simulation are considered in [49, 210]. We point out that the classical algebraic multigrid approaches require the system matrix (for the restricted pressure problem) to be an M-matrix. For transport equations, local methods like incomplete LU-factorisation can be expected to perform well. Different strategies for decoupling the system into pressure and transport parts are considered in [49, 148, 210, 225] 3.2 Grids Traditionally, reservoir simulations have employed quadrilateral grids in 2D, and hexahedrals in 3D, see Figure 3.1(a) for an example. The logical ordering of the cells eases the implementation. Moreover, the natural ordering of the unknowns in the linear system makes the solution procedure faster. If higher resolution is needed in some areas, this can be achieved by using local grid reﬁnement, see Figure 3.1(b). Note that this will introduce hanging nodes, which can cause difﬁculties for some numerical methods. To ﬁt a hexahedral grid to a complex reservoir with multiple geological layers, faults, and fractures is a non-trivial task. This is illustrated by Figure 3.2, which shows part of a simulation grid from a real North Sea ﬁeld. It may be less complicated to grid the reservoir with other polyhedra. For a three dimensional domain, tetrahedra are very well suited to grid a complex domain. Another option is to construct a two-dimensional triangular grid, and extend it prismatically to three dimensions. Such a grid will have a large degree of ﬂexibility. In a reservoir, the geological layering deﬁne a natural division into horisontal layers, which can be used for prismatic extensions of 2D-grids. However, geometrical constraints such as faults and skew and horisontal wells pose a challenge for construction of such 2-1/2-D grids. Another class of grids is the Voronoi-grids, which are duals of a primary grid. 26 Reservoir Simulation (a) (b) Figure 3.1: To the left, a quadrilateral grid. To the right, an example of local grid reﬁnement. Figure 3.2: Part of a hexahedral simulation grid from a real North Sea ﬁeld. Note the complex geometry. 3.2 Grids 27 (a) A triangular grid (b) The dual Figure 3.3: A triangular grid, and its dual. Note how cell centres and vertices change roles. The vertices and cell centres switch roles when going from a primary to a dual grid, see Figure 3.3(a) and 3.3(b). A Voronoi-tessellation can be constructed for any grid, however, they are mostly used for unstructured (triangular or tetrahedral) grids. The Voronoi-grids are well suited for discretising the ﬂow equations, see Chapter 4. However, to adjust a Voronoi-grid to discontinuities in the medium can be non-trivial. Unstructured grids are appealing because of their ﬂexibility, and there is a literature on generation of such grids in difﬁcult domains, see e.g. [96, 146, 147]. However, the number of unknowns in the linear system is high, and the system matrix will in general not be banded. In three dimensions, tetrahedral grids suffer more from a high number of unknowns than a prismatic extension of 2D triangular grids. The dual of a tetrahedral grid in 3D can have a large number of neighbours, which can lead to computationally very demanding simulations. The structure and bandwidth of the system matrix are determined by grid topology and geometry. Hence, properties of the numerical methods should ideally be used as constraints in the gridding process. We end this section by pointing to Figure 3.4, which shows an example taken from [114] on how a domain with constraints (fractures) can be gridded using triangles. Note that away from the fractures, the triangles are close to uniform. In these areas, other cells, such as quadrilaterals or Voronoi cells, could equally well have been ﬁtted to the geometry. Further considerations can be found in [25, 183, 223]. We also remark that this approach can be taken one step further by introducing adaptive grid reﬁnement based on a posteriori error estimates. 28 Reservoir Simulation Figure 3.4: A triangular grid ﬁtted to multiple fractures (bold lines). Note that the fractures resemble the outcrop shown in Figure 1.1. 3.3 Time Stepping Methods There are several commonly used time stepping methods in reservoir simulations. They differ in the choice of implicit in time variables. Due to its elliptic behaviour, the pressure is always treated implicitly. For the advection dominated mass variables, there are several options. In the following we brieﬂy describe some common time stepping strategies. For more a more detailed description and analysis of the methods, we refer to [26, 59]. As in the previous chapter, we let nc be the number of components, which will be equal to the number of primary variables, also for two-phase and black oil models. Further, let ne be the number of cells in the grid. Fully Implicit In the fully implicit method (FIM), or simultaneous simulation, all the primary variables are treated implicitly. FIM is unconditionally stable with respect to time step size, however, in practice the time steps must be small enough for the Newton iterations to converge. The robustness of the fully implicit method makes it popular for industrial simulations with relatively few variables, i.e. black oil 3.3 Time Stepping Methods 29 simulations. For problems with more unknowns, the computational cost of the simulations becomes severely high. IMPES As previously mentioned, it is only the pressure variable that deﬁnitely needs to be discretised implicitly in time. If we treat all other variables explicitly, we arrive at the implicit pressure explicit saturation (IMPES) or implicit pressure explicit molar mass (impem) formulation. The two models differ in which variables are used to represent masses. Common choices of primary variables lead to the use of IMPES for two-phase or black-oil simulations, whereas impem is used in compositional simulations. In each Newton iteration with IMPES/impem, there will only be ne linear equations to solve. Thus one time step with the method is computationally cheap, and the method is especially attractive for compositional simulations. However, the time step sizes will be limited due to a stability criterion (CFL criterion), which is typically determined by cells in high-ﬂow regions. The restrictions on the time step size can in many situations be severe. One possible remedy is to apply Asynchronous time stepping [167], where a local time step is applied for each cell. IMPSAT To choose the number of implicit variables is a trade off between robustness and computational cost. FIM and IMPES can be seen as the two extrema in that sense. An alternative is to treat pressures and saturations implicitly, and the rest of the primary variables explicitly. This method is known as Implicit pressure and saturations (IMPSAT). For IMPSAT to be different from FIM, we of course need nc to be larger than 3. The IMPSAT method can be considerably cheaper than FIM, and allows for much larger time steps than IMPES, see [48]. For more information on IMPSAT and some models employing this strategy, we refer to [107]. The Adaptive Implicit Method So far, we have considered three time stepping methods that treat a various number of variables implicitly. However, the degree of implicitness has been static, in the sense that the choice of implicit variables is the same for every grid cell. Moreover, it is customary to maintain the same degree of implicitness during the entire simulation. With an explicit method, the time step size is dictated by the cell with the highest CFL-number, which is typically found in high ﬂow regions, e.g. near fractures or wells. In such regions, an implicit time discretisation is 30 Reservoir Simulation appropriate. In the Adaptive implicit method, different time stepping strategies are employed in different cells. Thus the discretisation scheme can be tuned to increase robustness and time step size. The adaptive implicit method was introduced to the reservoir community in [213]. Criteria for switching between implicit and explicit treatment are analysed in i.e. [199]. Compositional ﬂow was studied in [48], where FIM, IMPSAT, and IMPES all were employed in the same simulations. In [68, 69], AIM is proven to be inconsistent on the boundary between explicit and implicit cells. A remedy is also proposed, together with higher order schemes in time and space. Chapter 4 Elliptic Discretisation Principles As shown in Chapter 2, the pressure variable exhibits parabolic behaviour. After discretising the time derivatives, we are left with solving an elliptic equation for the pressure. This chapter is devoted to numerical methods for elliptic equations, with emphasis on control volume methods. These methods make it feasible both to solve the entire system of governing equations fully implicit (FIM), and to use some sort of operator splitting (IMPES,IMPSAT, etc.). Other methods for elliptic equations that are applied in the reservoir simulation community are mentioned in the last section. 4.1 Preliminaries Consider the equation for an incompressible single phase, (2.14). We introduce a ﬂow potential u = p − ρgz, set the viscosity to 1, and get −∇ · (K∇u) = q , (4.1) where q represents sink/source terms. The solution u behaves similarly to the pressure variable for the more complex physical processes presented in Chapter 2, and we therefore employ Equation (4.1) as a model equation for elliptic problems. Before introducing discretisation schemes, we will brieﬂy state some results from the theory of elliptic equations: • Regularity: For Equation (4.1) to be well posed, q must be continuous, and K must be continuously differentiable. However, in porous media applications q can be a point sink/source, and the permeability is in general only piecewise constant. Therefore, the equation is often casted into in a weak form, and we seek a week solution of the problem. The weak, or variational, formulations are starting points for many numerical methods, 32 Elliptic Discretisation Principles see Section 4.7. The control volume methods investigated in this work are based on an integral formulation, however, they are closely related to approximations derived from the variational formulations [137]. • Symmetry: The differential operator −∇ · (K∇·) is self adjoint in the L2 inner product. The analogy for a discrete method is a symmetric system matrix. • Positive deﬁniteness: The eigenvalues of the differential operator are positive. For a numerical method to mimic this property, it should render a positive deﬁnite system matrix for the potential. This can be considered as a stability condition for the discretisation. • Maximum principles: The solution obeys a maximum principle. The analogy for numerical methods is covered by the consideration of monotonicity properties. These issues are thoroughly investigated in Section 4.5. • Global domain of dependence: A point sink/source will inﬂuence the solution everywhere in the computational domain. Discretisations of elliptic problems therefore render systems of algebraic equation where all cells are connected. Hence, it can be computationally demanding to solve the equations. The literature on elliptic equations is extensive, we refer to [6, 90] for more information. 4.2 Control Volume Methods Let Ωi be a cell in our computational domain. We integrate Equation (4.1) over Ωi and use the divergence theorem to get − ∇·(K∇u)dV = − n·(K∇u)dS = n·qdS = qdV .(4.2) Ωi ∂Ωi ∂Ωi Ωi Here we have introduced the ﬂux variable q = −K∇u. Equation (4.2) is a conservation law, the ﬂux out of Ωi is equal to the sinks/sources inside the cell. The control volume methods approximate the gradient of u by the pressure values in nearby cells. Let ∂Ωi,j be an edge of Ωi . The numerical ﬂux fj through ∂Ωi,j , is given by n · (K∇u) dS ≈ fj = tj,k uk , (4.3) ∂Ωi,j k 4.3 Two Point Flux Approximations 33 where uk is the potential in cell k, and the sum is taken over all cells contributing to the ﬂux expression. The quantities tj,k are called transmissibilities. They depend on both permeability and grid geometry. The ﬂuxes are approximated for all edges ∂Ωi,j , and conservation of mass is obtained by requiring fj = tj,k uk = q dV . (4.4) j j k Ωi There is a large number of control volume methods, see [93] for a recent overview. In the sequel, we will discuss some methods that are relevant for reservoir simulations. 4.3 Two Point Flux Approximations The two-point ﬂux approximation (TPFA) is one of the simplest control volume methods. As the name indicates, the ﬂux over ∂Ωi,j is approximated by using the pressure in the two cells sharing the edge. Consider Figure 4.1(a). The ﬂux through the edge seen from cell number 1, f1 , can be expressed as v1 f1 = − (ue − u1 ) dS , n · K1 · (4.5) v1 ∂Ωi,j where n is the normal vector out of cell 1, see Figure 4.1(a). The gradient is approximated by a normalised vector v1 , pointing from the cell centre x1 to the midpoint of the edge xe , scaled by the potential difference between ue = u(xe ) and u1 = u(x1 ). Similarly, the ﬂux can be seen from cell number 2 as v2 (u2 − ue ) dS . n · K2 · (4.6) f2 = − v2 ∂Ωi,j Here v2 represents a vector from xe to x2 , cell centre of cell 2, and u2 = u(x2 ). By requiring continuity of the ﬂux we obtain f1 = f2 = t(u1 − u2 ) . (4.7) The transmissibility can be found by combining Equations (4.5) and (4.6). Note that we have implicitly required continuity of the potential in xe . TPFA does not render a consistent ﬂux approximation, e.g. [5, 7, 82]. It is only convergent in the special case where the grid is aligned with the principle axes of the permeability tensor. Such grids are said to be K-orthogonal. Figure 4.1(b) shows two Voronoi cells with a primary triangular grid. The vector ν is normal to the straight line that connects the cell centres, and n is normal to the edge between the cells. A sufﬁcient condition for K-orthogonality 34 Elliptic Discretisation Principles ν .v x1 xe 1 v2 .x n n 2 (a) TPFA (b) K-orthogonality of Voronoi grids Figure 4.1: To the left, the ﬂux stencil for TPFA. To the right, a primal triangular (stippled) and a dual grid. The vector ν is orthogonal to the line connecting the cell centres. reads ν T Kn = 0. We see that dual grids are K-orthogonal if the permeability is isotropic, and the triangles have the Delaunay property. However, Voronoi grids can yield good results also for anisotropic media. Dual grids for reservoir simulations are studied in i.e. [109, 133, 184]. Even though TPFA is not convergent, and in many cases not capable of resolving the ﬂow ﬁeld accurately, it is widely used in industrial simulations [204]. There are several reasons for this, apart from historical ones. The method is easy to formulate and apply. The resulting system matrix is symmetric with small bandwidth, and it is an M-matrix provided all transmissibilities are positive. This renders possible design of efﬁcient linear solvers. Moreover, since the system matrix is an M-matrix, TPFA does not suffer from artiﬁcial oscillations related to monotonicity issues. This makes it more robust than the multi-point methods studied in Section 4.4. 4.3.1 Non-Linear Methods Despite the shortcomings of the TPFA method, its small cell stencil is appealing. Before we introduce methods with larger stencil, we will therefore mention some schemes that keep a two-point approximation of the ﬂuxes, but consider the transmissibilities as functions of the solution u, as well as of permeability and grid geometry. This yields a signiﬁcant improvement of the traditional TPFA method, to the price of increased computational complexity. We denote the resulting schemes non-linear two point ﬂux approximation (NTPFA) methods. 4.3 Two Point Flux Approximations 35 An Upscaling-Related Approach For the two-point ﬂux approximation to be good, the grid must be aligned with the permeability ﬁeld. The ﬁne scale geological models often have isotropic permeability, but anisotropies and heterogeneities are introduced by grid effects and upscaling. Still, upscaling procedures based on TPFA can provide accurate results. This motivated Chen et. al. to develop a NTPFA method applicable both to upscaling and as a single level method [55]. The main idea is to compute a reference solution by multi-point ﬂux approximations; either globally or locally in combination with a global TPFA. Then, transmissibilities are computed based on a local two-point ﬂux approximation. The aim is to tune the transmissibilities so that the reference ﬂow ﬁeld is reproduced as accurately as possible. To accomplish this, iterations might be necessary, hence the transmissibilities depend on u. If more than one phase is present, the ﬂow pattern and thereby the reference potential solution will gradually change as components are transported. The transmissibilities, which are tuned to mimic the reference ﬂow pattern, must thus be updated regularly. Compared to the computational cost of a multiphase simulation, the update is not a signiﬁcant burden. The results obtained with this method are promising, and indicate that in many cases, a two-point method might be sufﬁcient, provided the transmissibilities are carefully computed. An Unconditionally Monotone NTPFA Scheme In [152], LePotier introduced an unconditional monotone NTPFA method. The scheme has later been extended to a class of methods, see [159, 161, 234]. The methods approximate the potential in collocation points that need not coincide with the cell barycentres. In most of the methods, auxiliary unknowns are introduced in the vertices of the grid. The ﬂux through an edge from one cell can then be approximated by the potential values in the collocation point of the cell and the vertices of the edge. A similar expression is found for the other cell sharing the edge. Requiring continuity of the ﬂux gives a two point ﬂux expression, i.e. the auxiliary unknowns are eliminated. However, the transmissibility becomes a function of the potential in the vertex nodes, thus the method is non-linear. Further, the vertex values are computed by interpolation from values in the collocation points, and the choice of interpolation scheme affects the accuracy of the scheme. An interpolation-free method is presented in [161]. The framework described here provides methods that are provably monotone, also in cases where linear methods with increased cell stencils cannot be monotone (see Section 4.5). The methods render algebraic systems of non-linear equations. To guarantee positivity of the potentials in each iteration step, the system is solved 36 Elliptic Discretisation Principles x2 x3 x1 (a) An interaction region (b) A variational triangle Figure 4.2: To the left, an interaction region (stippled) for a cluster of cells. To the right, a variational triangle used to construct basis functions. by Piccard iterations instead of by Newton’s method. The number of iterations needed is quite high, partly because not much effort has been put into solving the system efﬁciently yet. It has been suggested that the schemes are best suited for non-linear problems [205]. We are not aware of any applications of the method to multiphase ﬂow. Therefore, it is an open question how the non-linear transmissibilities will behave in difﬁcult scenarios such as counter-current ﬂow and potentials close to the bubble point. We also remark that if rock compressibility is signiﬁcant, the permeability is no longer constant in time. Hence the transmissibilities must be recomputed anyhow, and the overburden due to the non-linear ﬂux approximation is easier to justify. 4.4 Multi-Point Flux Approximations So far, we have stated that the ﬂux over an edge cannot be approximated consistently by a linear combination of the potentials in the two adjacent cells. One possible remedy is to introduce a non-linear approximation. Another way to amend to the TPFA, is to modify the ﬂux expression (4.7) to include more points in the ﬂux stencil. The result is the so called multi-point ﬂux approximations (MPFA), which were developed independently by Aavatsmark et. al. [8], and Edwards and Rogers [80], and further studied in i.e. [9, 10, 11, 12, 81, 82, 153]. To obtain a local ﬂux expression, MPFA methods introduce interaction regions. For most methods we are aware of, these are cells or subcells in the dual grid, see Figure 4.2(a). The only exception is the rarely used Z-method. The shape of the interaction region differs from method to method. However, the construction of the schemes have many similarities. We will therefore describe the general procedure before we go on to discuss the two most commonly applied methods. All considerations in this thesis are in two dimensions. From Figure 4.2(a), we see that the interaction regions divide each cell into 4.4 Multi-Point Flux Approximations 37 several subcells. In each of these subcells, the potential is approximated by a linear function. The function is determined by the values in the cell centre, and in one point on each edge, see Figure 4.2(b). These three points span a variational triangle. Thus, on subcell i, the potential approximation can be expressed as ui = 3 ui,j φi,j . (4.8) j=1 Here ui,j is the potential values in the cell centre and the points at the edge, see Figure 4.2(b). The basis functions φi,j are linear, with value 1 in one vertex of the variational triangle, and 0 in the others. The gradient of the potential can be used to ﬁnd an approximation of the ﬂux through a half-edge seen from one cell. The ﬂux through the half-edge can also be approximated by the potential in the adjacent cell. This can be done for all half-edges in the interaction region. By requiring continuity of all the ﬂuxes, we get a linear system of equations Aū = Bu , (4.9) where ū and u are potential values at the edges and at the cell centres, respectively. We invert A to get ū = A−1 Bu. The ﬂux over the edges can be expressed as f = Cū + Du , (4.10) and by substituting the expression for ū, we get f = (CA−1 B + D)u = Tu . (4.11) The elements in T are the transmissibilities for the half-edges in the interaction region. The ﬂuxes over the edges are found by summing the half-edge ﬂuxes. The set of all cells contributing to the ﬂux over an edge form a f lux stencil, see Figure 4.3(a), whereas the cell stencil of a cell is the union of the ﬂux stencils for all the cell’s edges, confer Figure 4.3(b). Requiring conservation of mass yields a global system of linear equations with the potentials in the cell centres as unknowns. The bandwidth of the resulting system matrix equals the size of the cell stencils. Boundary conditions can be handled by modifying the Equations (4.9)-(4.10). In reservoir simulation, homogeneous Neumann conditions are prevailing. They can easily be implemented by applying ghost cells with zero permeability. Ghost cells can also be used to implement interpolated Dirichlet conditions. For handling of non-homogeneous Neumann conditions and non-interpolated Dirichlet conditions, we refer to [50, 87, 171]. We also remark that the use of interaction regions based on the dual grid allows for assigning different permeability tensors to each subcell, and thus incorporate more information from the geological model. However, this approach is rarely used in practice. 38 Elliptic Discretisation Principles (a) Flux stencil (b) Cell stencil Figure 4.3: A ﬂux stencil and a cell stencil for the MPFA O-method on a quadrilateral grid. xe η x0 x̄j η (a) The continuity point (b) The O(1/3)-method for triangles Figure 4.4: The continuity point for the O-method is determined by η, as shown to the right. To the left, the O(1/3)-method on triangles. Note that the variational triangles are part of parallelograms. 4.4.1 The O-Method The O(η)-method can be considered the basic MPFA scheme. It can be applied both in two and three dimensions, and for most grid topologies. However, it has problems with handling hanging nodes, see [17]. The interaction region for the O-method consists of all cells sharing a vertex, yielding an O-shaped region like the one showed in Figure 4.2(a). Thus there are 3 nc degrees of freedom, where nc is the number of cells in the interaction region. We use 2 nc degrees of freedom to impose continuity of the ﬂuxes, and to lock the values in the cell centres. With the remaining nc degrees of freedom, continuity of the potential is enforced in one point on each half-edge. The location of the continuity point is determined by the parameter η, by the relation x̄j = η x0 + (1 − η)xe , (4.12) where xe is the midpoint of the edge, and x0 is the vertex in the centre of the interaction region. Hence, η measures the relative distance from the midpoint of the edge to the continuity point for the potential, see Figure 4.4(a). On quadrilateral grids, the two common choices for η are 0 and 0.5, see e.g. [5] and [81], 4.4 Multi-Point Flux Approximations 39 x4 x3 x̄4 x0 x1 x̄1 x0 x2 (a) x1 x̄1 x̄2 x2 (b) Figure 4.5: The two possible variational triangles for the L-method. respectively. On triangular grids η = 1/3 has been investigated in [95, 135]. For quadrilateral grids in two dimensions, the O-method in general yields a 9-point cell stencil. In three dimensions, the cell stencil consists of 27 cells. Note that for K-orthogonal grids, the O(0)-method will reduce to TPFA. For quadrilateral grids, K-orthogonality is also the only case in which the system matrix yields an M-matrix for this choice of η. If the transmissibilities are computed in physical space, the O-method applied to a quadrilateral grid will be symmetric only if the cells are parallelograms. Symmetry can be obtained by a mapping to reference space, however, to this is achieved to the price of deteriorated the convergence properties [13]. On triangular grids, the O(1/3)-method with discretisation in physical space will always be symmetric. This is due to the variational triangles being parallelograms, see Figure 4.4(b). 4.4.2 The L-Method For high anisotropy ratios, the O-method is known to suffer from spurious internal oscillations. This will be further discussed in Section 4.5. The shortcomings of the O-method have motivated development of other MPFA methods. The Lmethod was introduced for quadrilateral grids in [17], although it was originally discovered in [177] as a method with optimal properties with respect to avoiding internal oscillations. Further, the L-scheme can handle hanging nodes, thus it is a natural choice for local grid reﬁnement. A method for three dimensional problems was introduced in [15]. In two dimensions, the interaction region for the L-method consists of three cells. This gives us 9 degrees of freedom, out of which 3 are used to lock the values in the cell centres. The remaining degrees of freedom are sufﬁcient to ensure continuity of both the potential and the ﬂux over the entire half-edges. Consider Figure 4.5(a). The ﬂux over half-edge x0 x̄1 can be computed by using the cells 1,2, and 4. However, as indicated by Figure 4.5(b), we can also use the cells 1,2, 40 Elliptic Discretisation Principles and 3. The two possible interaction regions give two sets of transmissibilities for the half-edge x0 x̄1 . Let t1 and t2 be the transmissibilities for the half-edge computed with cell 1 and 2 as the central cell in the interaction region, respectively. If |t1 | < |t2 |, we use the values computed with cell 1 as the central cell, if not use cell 2. For a uniform grid in a homogeneous medium, this selection criterion will chose cells aligned with directions of high conductivity. The selection criterion is deﬁned so that the system matrix will be an M-matrix for a large range of grid parameters, confer Section 4.5.3. A further motivation can be found in [17]. The cell stencil of the L-method consists of 7 cells if the cells are uniform parallelograms and the medium homogeneous, and increases to 9 for general quadrilaterals and permeabilities. In three dimensions, the stencil has 13 cells. We also remark that if the grid is K-orthogonal, the L-method reduces to a two-point ﬂux approximation. The L-method does not render a symmetric system matrix. 4.4.3 Convergence There is no independent theory of convergence for MPFA methods. However, the schemes can be considered as either mixed ﬁnite element methods or mimetic ﬁnite difference methods (see Section 4.7), with numerical quadrature. Analytical convergence results based on this relation can be found in [138, 135, 136, 139, 226] for the O-method, and [209] for the L-method. Numerical convergence of MPFA methods is studied in e.g. [17, 14, 16, 86, 182]. For smooth solutions and rough grids, these numerical investigations indicate second order convergence for the potential, and ﬁrst order convergence for the edge ﬂux. However, the half-edge ﬂuxes are known to exhibit a reduced convergence rate [139]. For solutions with low regularity, the convergence rates are in agreement with ﬁnite element theory. 4.4.4 Other Multi-Point Methods For completeness, we mention some MPFA methods and related schemes that are not discussed here. The MPFA Z-method, is constructed by using parts of two cells in the dual grid, see [178]. This improves the monotonicity properties of the O-method in highly anisotropic media. The U-method was introduced together with the O-method [8] to reduce the size of the cell stencils, and thereby the bandwidth of the system matrix. There is also an Enriched MPFA method [53], which is an extension of the O-method, with improved monotonicity properties. Similar work is also done in [85]. The notion of multi-point methods used in this thesis can somewhat loosely be deﬁned as methods that are formulated with an interaction-region framework, and use more than two cell values to approximate the ﬂux. A more straightforward interpretation of the name could consider all schemes that yield ﬂux stencils 4.5 Monotonicity 41 with more than two points, which would include a vast number of methods. The deﬁnition used here is, however, commonly used in the reservoir simulation community. 4.5 Monotonicity As mentioned in the beginning of this chapter, the solution of elliptic equations fulﬁls a maximum principle (for a precise deﬁnition, see below). This guarantees that the solution will be free of internal oscillations, and the maximum principle should be preserved when the equation is discretised. Questions related to preservation of maximum principles and to avoid spurious internal oscillations for elliptic equations have become known as monotonicity issues. A major part of the present work has been devoted to investigations related to monotonicity and control volume methods, with special emphasis on MPFA methods. The results are given in Papers A and B, and also the additional Paper F. In the following, we give background information on monotonicity for elliptic equations. 4.5.1 Motivation We say that a numerical method is monotone if it fulﬁls the discrete maximum principle stated below. The importance of applying monotone methods is illustrated in Figure 4.6. The ﬁgure shows two solutions of Equation (4.1), where the right hand side is a point source, and the boundary conditions are u = 0. The transition from the peak at the source to the minima on the boundary should be smooth, as seen in Figure 4.6(a). However, if the numerical solution scheme is non-monotone, we can get internal oscillations, and false global extrema, see Figure 4.6(b). Monotonicity issues also arise for the Multiscale Finite Volume method discussed in Section 4.7.3. False internal oscillations in the potential can also cause problems for streamline methods. Further, reordering methods for non-linear problems [145] and transport methods [172] also show sensitivity to monotonicity issues. It is important to notice that since the MPFA methods by construction reproduce uniform ﬂow exactly, they tend to robust even if applied to problems outside their region of monotonicity. One example of this can be seen to the end of Paper B. In other words, even if a discretisation violates necessary conditions for monotonicity, this is not sufﬁcient to yield oscillating solutions when the discretisation is applied to a speciﬁc problem. Non-monotone behaviour is most often seen in highly non-linear ﬂow regimes, e.g. near wells. 42 Elliptic Discretisation Principles (a) Monotone (b) Non-monotone Figure 4.6: The ﬁgure shows two numerical solutions of a problem with a point source in the middle of a domain, and homogeneous Dirichlet boundary conditions. The smooth solution to the right is obtained from a monotone method. To the left, a highly oscillatory solution produced by a non-monotone method. Note that the entire setup (including the grid) is identical for the two ﬁgures. 4.5.2 Maximum Principles We consider the elliptic model equation (4.1), stated for a domain Ω. This solution then fulﬁls a maximum principle, which can be stated as follows [115]: Hopf’s ﬁrst lemma Let the permeability tensor be continuously differentiable and let the boundary be sufﬁciently smooth. Further, let the source/sink term q be non-negative in a domain ω ⊆ Ω. The, the solution u of Equation (4.1) has no local minima in ω, i.e., there is no point x0 ∈ ω such that u(x0 ) < u(x) for all other x in a neighbourhood of x0 . The maximum principle, or equivalently Hopf’s ﬁrst lemma, should be preserved in the discretisation. Since the term internal oscillation is not deﬁned for a discrete solution, it is not straightforward to deﬁne a discrete analogue of Hopf’s lemma. To motivate the deﬁnition, we introduce a Green’s function G(ξ, x) to express the solution of the elliptic equation. To test if the solution is free of local minima on a domain ω, we impose homogeneous Dirichlet conditions on ∂ω. Then, the solution of Equation (4.1) can be written u(x) = ω G(ξ, x)q(ξ) dτξ . (4.13) 4.5 Monotonicity 43 If the solution has no internal oscillations, it should decrease from a maximum at the sources to 0 at the boundary, i.e. the solution is positive inside ω. Hence, we see that if a non-negative q is to imply a non-negative u, G must be everywhere non-negative; G(ξ, x) ≥ 0 . (4.14) A linear discretisation method will give a system of linear equations Au = q , (4.15) where A is the discretised operator, and u and q are the solution and the sink/source term in the grid blocks, respectively. The solution of the linear system reads u = A−1 q, thus A−1 can be considered a discrete Green’s function. Now, a positive right hand side should render a positive solution. This is fulﬁlled if the discrete equivalent of Eq. (4.14) A−1 ≥ 0 , (4.16) where the inequality is interpreted in the element-wise sense. When the inequality (4.16) holds, we say that the matrix A−1 is monotone, while A is inverse monotone. To impose homogeneous Dirichlet conditions on a subdomain gets a bit involved. The boundary conditions are implemented using ghost cells. Thus, the boundary is deﬁned as a set of cell centres, and the linear interpolation between them. This curve resembles a possible iso-contour for the solution. If there is a positive source/sink inside the curve, and the curve happens to be an iso-contour, we want our numerical solution to honour Hopf’s lemma; it should be positive inside the curve. This is achieved if the restricted system matrix is inverse monotone. However, for Hopf’s lemma to be valid on the domain bounded by this piecewise linear curve, the boundary must have certain properties speciﬁed below. Thus we get the following deﬁnition of the discrete maximum principle, formulated in Paper A: A discrete maximum principle For any subgrid bounded by a closed Jordan curve, with homogeneous Dirichlet conditions, the discretisation must yield a system matrix, whose inverse has no negative elements. The boundary conditions are implemented by using ghost cells. The boundary of the subgrid is deﬁned by the linear interpolation of the cell centres of the surrounding cells. 44 Elliptic Discretisation Principles We call numerical schemes which have this property monotone methods. It is important to note that violations of the discrete maximum principle does not rule out convergence of the numerical solution and vice versa. Indeed, a convergent solution can exhibit spurious oscillations on every reﬁnement level. The discrete maximum principle does not involve discretisation of boundary conditions on ∂Ω. That is partly covered by the following result [116] Hopf’s second lemma If the solution of Equation (4.1) with a smooth boundary and continuously differentiable K has a maximum on the boundary, there must be a non-zero ﬂow through the boundary. From a physical point of view, it is obvious that a stationary solution (no time dependency) cannot have a maximum on a no-ﬂow boundary. However, by Hopf’s second lemma, non-physical values on the boundary are related to the maximum principles. Moreover, in reservoir simulation, homogeneous Neumann conditions are prevailing. Investigations of whether discretisations fulﬁll discrete maximum principles should therefore involve tests with no-ﬂow boundaries. 4.5.3 Sufﬁcient Condition for Monotonicity of Control Volume Methods To test directly whether a discretisation of Equation (4.1) fulﬁls the discrete maximum principle we must invert system matrices for all subgrids fulﬁlling the requirements. The computational complexity makes this task undesirable. Therefore, there has in recent years been an effort to derive sufﬁcient conditions for a control volume method to yield a monotone discretisation of the elliptic model equation. One class of matrices that are inverse monotone is the M-matrices. An Mmatrix is an invertible matrix with positive diagonal elements and non-positive off-diagonal elements [102]. The TPFA method always yields an M-matrix on the main grid, as well as on subgrids, hence it is always monotone. For MPFA methods, the situation is more complex. For instance, the O(0)-method on quadrilaterals does not render an M-matrix unless the grid is K-orthogonal, when the discretisation reduces to a TPFA. Still, the system matrix for the O(0)-method is inverse monotone for more general grids. M-matrix analysis for other choices of η can be found in e.g. [81]. Thus, we need criteria that take into account matrices with positive off-diagonal elements, and still fulﬁll the discrete maximum principle. 4.5 Monotonicity 45 b |c| 1 5 4 3 6 1 2 7 8 0.6 0.4 a1 9 (a) Local numbering 0.8 a2 (b) Parameters O(0) O(0.5) L 0.2 0 0 0.2 0.4 0.6 0.8 1 a |c| (c) Monotonicity regions Figure 4.7: To the left, the local numbering for the monotonicity conditions is shown. The ﬁgure in the middle shows the parameters in Equation (4.18). To the right, monotonicity regions for three MPFA methods for a parallelogram grid in √ a homogeneous medium are shown. The green line shows the elliptic bound ab = |c|. In [175] Nordbotten and Aavatsmark derived sufﬁcient conditions for monotonicity on uniform parallelogram grids in homogeneous media. The derivation is based on constructing the discrete Green’s function (i.e. A−1 ) in a dimension by dimension manner. Their arguments are, however, quite complicated and extensions of the technique to more general media and grid topologies seem out of reach. By utilising a splitting of the system matrix, Nordbotten et. al. [177] extended the results from [175] to general quadrilateral grids. The splitting follows the grid topology, based on the two natural directions in a quadrilateral grid. The analysis holds for all linear control volume methods that lead to a 9-point cell stencil. Let mi be the contribution to the system matrix corresponding to cell i, the local numbering is shown in Figure 4.7(a). The results in [177] require m1 > 0 and max(m2 , m4 , m6 , m8 ) < 0. Further, the contribution from the corner cell 3 must obey an inequality of the form i,j−1 mi,j − mi,j−1 mi,j 2 m4 3 1 > 0. (4.17) Here, the superscripts (i, j) refer to grid row and column index. Similar constraints apply to m5 , m7 , and m9 . Thus, the system matrix can be inverse monotone even if the elements corresponding to corner cells are positive. The analysis in [177] also considered the special case of MPFA methods applied to uniform grids in homogeneous media. Let K denote the permeability in the medium. Further, let a1 and a2 be normal vectors to the cells, with length equal to their respective edges, see Figure 4.7(b). Then the monotonicity properties of the MPFA methods can be visualised by quantities a, b, and c, deﬁned 46 Elliptic Discretisation Principles by T 1 a c a1 a2 K a1 a2 = , V c b (4.18) where V is the area of the cell. Note that for K to be positive deﬁnite, we must have ab > c2 . The monotonicity regions are shown in Figure 4.7(c). The sufﬁcient conditions seem to indicate that no linear methods are monotone below the line a/|c| = b/|c|. The MPFA L-method is ideal in the sense that it is monotone whenever b ≥ a, whereas both the O-methods have reduced monotonicity regions. We remark that the L-method yields an M-matrix whenever it is monotone. The analysis and numerical experiments in [177] did not, however, include tests on convex subgrids. Such grids were considered in Paper A, with the result that most of the conditions derived in [177] are necessary as well as sufﬁcient. Monotonicity for MPFA methods on triangular grids are considered in Paper B. We also remark that for Voronoi grids, all cells that share a vertex will also share an edge. By imposing the discrete maximum principle on subgrids consisting of two cells sharing an edge, it can be shown that for the discretisation to be monotone, it must yield a system matrix that is an M-matrix. Monotonicity for Parabolic Equations The elliptic model Equation (4.1) was derived from the equation for single phase ﬂow (2.12) under an assumption of incompressibility. The ﬂow of a weakly compressible ﬂuid is governed by a linear parabolic equation, see Section 2.2. Monotonicity properties for discretisations of such equations were studied in [120]. The compressibility terms impose additional restrictions for the solution to be free of spurious oscillations. The corner-elements m3 , m5 , m7 , and m9 can still be positive, but their magnitudes must be smaller than what is sufﬁcient for incompressible ﬂow. Also worth noticing is that the region where the discretisation is monotone increases when the time step is increased. 4.6 Near-Well Discretisations In our presentation of control volume methods, we have barely touched upon how to handle sink/source terms. Obviously, all the recovered hydrocarbons must ﬂow through wells, thus a proper representation of the sink/source terms is crucial for the simulations. The importance of the well terms can be illustrated by considering the pressure equation; for a reservoir with no-ﬂow conditions on the outer boundary, the ﬂow pattern will to a large extent be determined by the wells. Data on the ﬂow rate through a well, and the pressure at the well bore are among the 4.6 Near-Well Discretisations 47 few sources to knowledge of what happens in the reservoir that are available in real time. This information is utilised in day-to-day reservoir management. Thus, there are many reasons why an accurate description of ﬂow in the vicinity of wells is a key issue. A part of this work has been devoted to investigations of control volume discretisations for ﬂow in near-well regions. In the following, we give background information for numerical modelling of processes near wells. Results from the investigations can be found in Paper D. 4.6.1 Characteristics of Near-Well Regions To illustrate the nature of near-well ﬂow, we consider Equation (4.1) in a homogeneous reservoir. The reservoir contains a single well with ﬂow rate q. If the medium is isotropic, the solution of Equation (4.1) reads r q ln . (4.19) u = uw + 2πk rw Here uw is the pressure at the well bore, k is the (scalar) permeability, r is a radial coordinate, and rw is the well radius. We note that the ﬂow is purely radial, there is no angular dependency in the solution. Further, the solution varies logarithmically in the radial direction. Equation (4.1) should be equipped with appropriate boundary conditions. On the well bore, either the pressure or the ﬂux must be speciﬁed. For an isotropic medium, it seems evident that the well boundary condition should be uniform. If the permeability is anisotropic, the situation becomes somewhat more complicated. In a work addressing modelling of horisontal wells, Babu and Odeh apply a point sink/source representation of the well [27]. They argue that in an anisotropic medium, the iso-potential curves are ellipsis, and therefore the pressure varies along the well bore. According to Babu and Odeh, a uniform ﬂux condition is the most correct boundary condition [28, 29]. Peaceman disagreed with the sink/source representation of the well, and argued in favour of assigning uniform pressure along the well bore. The anisotropic medium can be transformed to an isotropic medium. The relation between the physical coordinates (x, y) and (ρ, η) in the isotropic space reads k 1/4 x x = b sinh ρ sin η , (4.20) ky k 1/4 y y = b cosh ρ cos η . (4.21) kx Here, we have assumed kx > ky . The constant b is given by b2 = rw2 (kx − ky ) . kx ky (4.22) 48 Elliptic Discretisation Principles Figure 4.8: The ﬂow pattern near a well in an anisotropic medium. The potential is uniform along the well bore, but the iso-contours turn ellipses away from the well. The streamlines are hyperbolas which are orthogonal to the potential contours. Now, the potential is given by [187, 188] q u = uw + (ρ − ρw ) , 2π kx ky (4.23) where tanh2 (ρw ) = kx /ky . Further, Peaceman showed that on the well bore, uniform pressure and uniform ﬂux are in fact equivalent. The iso-contours for the potential are circular at the well bore, but turn into ellipsis away from the well bore, as shown in Figure 4.8. The streamlines are orthogonal to the potential iso-contours, and thus are hyperbolas. The difference between the sink/source approach of Babu and Odeh and Peaceman’s ﬁnite well bore model decreases rapidly as the distance from the well bore increases [190]. So far, we have tacitly assumed a homogeneous permeability ﬁeld all the way up to the well bore. In reality, the rock near the well can be damaged during drilling. The permeability near the well can also be altered by injection of acids and/or hydraulic fracturing, both aimed at increasing the conductivity near the well. Such effects will modify the potential distribution described by (4.19), and the equation becomes [65] q r u = uw + ln +S , (4.24) 2πk rw where S is called a skin factor. The modiﬁcation of (4.23) is similar. For simplicity, we will set S = 0. We consider the well bore as the boundary of our domain, that is, we assume inﬁnite conductivity inside the well. In reality this is not true, and it might be necessary to couple the porous media model to a hydraulic model for ﬂow inside the well. These issues will not be pursued further in this work. 4.6.2 Well Index Approaches The main challenge in modelling near-well ﬂow is the difference in scale. The radius of the well bore will normally some centimetres, while the size of the reservoir can be several kilometres. For a well block with a size of meters, it may be 4.6 Near-Well Discretisations 49 reasonable to consider the well as geometric point. The numerical scheme itself is not adapted to the ﬂow in the near-well region. We emphasis that representing the well as a point sink/source in the numerical scheme is not equivalent to the point sink/source solution to the analytical problem, presented by Babu and Odeh. Indeed, models for the interaction between the well and the surrounding grid can be constructed both from Babu and Odeh’s point-source method, and Peaceman’s ﬁnite well bore approach. When the well is represented as a point sink/source, the computed pressure in the grid block containing the well, ub , can differ signiﬁcantly from the well bore pressure. The relation between the pressure difference ub − uw and the well ﬂow rate is usually calculated according to q= WI (ub − uw ) . μ (4.25) Here, W I is the well index, which depend on cell geometry, permeability ﬁeld, well radius etc. For multiphase ﬂow, (4.25) becomes qα = W I kr,α (ub − uw ) . μα (4.26) The above expressions are commonly applied in commercial simulators, with an appropriate value of W I. Thus, the dynamics in the near-well region is represented by a well index, which is computed by upscaling. Several authors have addressed the question of how to calculate the well index for various well conﬁgurations. The ﬁrst thorough investigation of these matters was done by Peaceman in a series of papers. He deﬁned an equivalent radius, re , where the steady state ﬂow pressure is equal to the well block pressure. In an inﬁnite, isotropic reservoir, with a uniform Cartesian grid with cell size h, the equivalent radius for an isolated well is given by [186] re = 0.2h . (4.27) Now, let the medium be anisotropic with permeabilities kx and ky in the x and y direction, respectively. Further Δx and Δy represent the grid spacing. Then, re is deﬁned by [187] 1/2 (ky /kx )1/2 (Δx)2 + (kx /ky )1/2 (Δy)2 re = 0.28 . (4.28) (ky /kx )1/4 + (kx /ky )1/4 In an anisotropic medium, the pressure at the equivalent radius is given by r q e ue = uw + ln . (4.29) r 2π kx ky w 50 Elliptic Discretisation Principles Thus, (4.25), (4.28) and (4.29) together give us (with viscosity set to unity) 2π kx ky q . (4.30) = WI = ue − uw ln( rrwe ) Peaceman’s approach builds upon several simplifying assumptions. The grid blocks are assumed to be rectangular, and aligned with the principle permeability directions. The wells must be vertical and isolated, although an extension to the case of multiple wells within one grid block is considered in [189]. Despite these shortcomings, Peaceman’s equivalent radii approach is applied in commercial simulators [204]. To increase the recovery factor, wells are drilled with highly complex trajectories. In onshore production, vertical wells are still predominating, partly due to low drilling costs. However, for offshore production, skew and horisontal wells with multiple branches are commonly drilled. Such non-conventional wells can intersect with grid blocks in an arbitrary manner. To upscale the near-well dynamics for such wells, more involved models are required. For general media, we have no analytical expressions for the potential near non-conventional wells. Instead, it is common to either consider simpliﬁed physical conﬁgurations where analytical solutions can be found, or resort to semi-analytical techniques, or even numerically computed solutions. Babu and Odeh proposed an analytical model to compute the well index in [27, 30]. Their method applies to horisontal wells in anisotropic, but homogeneous media. For more complex scenarios, a semianalytical approach based on Green’s functions was proposed in [229, 230]. Ding studied upscaling for heterogeneous near-well regions [71]. He found the classical upscaling methods that assume a linear ﬂow regime inappropriate in the near-well region. Instead he proposed to adapt the upscaling to the non-linearity in the near-well dynamics. He applied a ﬁne-scale pressure solve to compute equivalent coarse-scale transmissibilities and well indices. This approach was later developed in e.g. [72, 79]. Similar considerations can also be found in for instance [228]. We emphasis that the above presentation of approaches to near-well modelling by no means is a complete survey of the available schemes. Our intention is rather to point out that this is an unsolved topic, and thus there is a need for a more general and robust framework. 4.6.3 Applications of MPFA Methods to Near-Well Modelling We here present a methodology for application of local grid reﬁnement and control volume methods in near-well regions. The common numerical methods, such as 4.6 Near-Well Discretisations 51 the control volume methods previously introduced, employ a linear reconstruction of the potential. In near-well regions, we expect the potential to be dominated by the well, and thus exhibit a logarithmic behaviour. Ding and Jeannin perform a truncation error analysis near singularities of control volume methods based on linear reconstruction [73]. They split the error into a ’regular’ term, er , and a ’singular’ term, es , stemming from the well, and the overall error, e, can be written e = er + es = O(h2 ) + O h2 rw2 , (4.31) where h is a measure of the grid block size. Near the well, we expect e to be dominated by es , and hence the truncation error will be small only if the grid blocks are smaller than the well radius. To improve the performance of the control volume methods, Ding and Jeannin proposed to adapt the discretisation to the logarithmic potential proﬁle. They introduce a mapping from the physical space P to a logarithmic reference space R, according to x = eρ cos θ , y = eρ sin θ . (4.32) (4.33) Here, (x, y) and (ρ, θ) are the coordinates in P and R, respectively. A logarithmic potential in physical space will vary linearly in reference space, thus numerical methods based on linear reconstruction are appropriate. Moreover, the transformation is of polar-type, honouring the radial nature of near-well ﬂow. The potential in R is the solution of an elliptic equation similar to (4.1), but with a transformed permeability K̂. Further, let eP be an edge in physical space, and let eR be its curvilinear image in reference space. Then the ﬂux over the edge is preserved, that is f= n · (K∇u) dS = n̂ · (K̂∇u) dS . (4.34) eP eR where n̂ is normal to eR . Since ﬂuxes are conserved in the transformation between P and R, and further the potential varies linearly in R, it is natural to consider discretisation in reference space instead of physical space. In [73], Ding and Jeannin investigate this approach for a Cartesian grid. They show that if the transmissibilities are computed in R by an MPFA method, the total error is O(h2 /r). The well is considered a point sink/source inside the well block in R. However, the numerically computed potential in the cell containing the block is a good approximation to the actual pressure at the well bore. That is, the transmissibilities computed in reference space can be considered a well index. 52 Elliptic Discretisation Principles (a) Quadrilaterals (b) Triangles Figure 4.9: Illustration of the near-well grids in R. The spacing in radial direction is equidistant. (a) Quadrilaterals (b) Triangles Figure 4.10: Illustration of near-well grids with logarithmic reﬁnement shown in physical space. The transmissibilities are computed in R, and the edges are mapped to curved lines in P. (a) Quadrilaterals (b) Triangles Figure 4.11: Illustration of near-well grids with logarithmic reﬁnement, and discretisation in physical space. The edges are straight lines in P. Note that the well bore is approximated by a polygon. 4.7 Other Locally Conservative Methods 53 Another approach to well modelling is to create a grid that follows the well bore itself, and describe the physical parameters at there. Thus, the boundary condition on the well bore is treated as a boundary condition also in the numerical scheme. This is in contrast to all previously discussed methods, where the well is discretised as a point sink/source. Ding and Jeannin suggested this approach using ﬂexible grids, i.e. triangles or general polyhedra, in the near-well region [74]. In Paper D, we follow up their work with a systematic test of different options for near-well modelling with radial-like grids. We adapt to the logarithmic potential proﬁle by creating grids with equidistant spacing in the radial direction in R, thus when transformed to P, the grids adapt to the logarithmic potential proﬁle. The grid cells can be either isosceles triangles or quadrilaterals, see Figure 4.9(a) and 4.9(b). On both grids, the elliptic equation is discretised by MPFA methods, with the transmissibility calculation performed either in reference space, or in physical space. Since the potential varies linearly in R, we expect discretisations in reference space to do well. A discretisation in physical space apply a linear reconstruction of the potential, which was shown insufﬁcient by Ding and Jeannin for uniform Cartesian grids. However, since the grid cells are distributed according to the logarithmic behaviour, the drop in the potential between cells aligned in the radial directions is approximately constant. Therefore, we hope that a linear reconstruction sufﬁce to capture the ﬂow pattern. The grid edges are straight lines in the discretisation space. If the the transmissibilities are computed in R, the grid has curved edges in physical space, as shown in Figure 4.10(a) and 4.10(b). If the elliptic equation is discretised in P, the edges will be non-curved, see Figure 4.11(a) and 4.11(b). Note that when discretising in P, we approximate the well bore by a polygon. When local grid reﬁnement is applied in the near-well region, the near-well grid must be coupled with the surrounding global grid. In Paper D, the global grid is triangular, and the near-well region is considered a constraint in the global triangulation. Note that if the global grid was quadrilateral, say, the transition zone between the near-well grid and the surrounding grid can still be covered by triangles. In this way, we can ensure a smooth transition from logarithmic to linear gridding. Another way to handle the coupling can be found in [192]. 4.7 Other Locally Conservative Methods In addition to the control volume methods, there are several other locally conservative methods that handle discontinuous permeability tensors. A recent overview and comparison of some methods can be found in [137]. We here brieﬂy discuss some choices that are commonly considered in the reservoir simulation community. 54 Elliptic Discretisation Principles 4.7.1 Mixed Finite Elements As mentioned in Section 4.1, the elliptic model equation (4.1) can be written in weak form that reads; ﬁnd u ∈ L2 and q ∈ H(Div, Ω) such that pT K−1 q + u(∇ · p) dV = 0 , Ω v(∇ · q − q) dV = 0 , ∀ p ∈ H(Div, Ω) , (4.35) ∀ v ∈ L2 . (4.36) Ω For a derivation of the weak form, and deﬁnition of the Sobolev space H(Div, Ω) confer e.g. [41, 194]. The mixed ﬁnite element (MFE) method seeks for approximations (uh , qh ) to u and q in ﬁnite dimensional subspaces Vh ⊂ L2 and Wh ⊂ H(Div, Ω), respectively. Here, uh is associated with pressure values in the cells, while qh represents ﬂuxes over the edges. For ﬁrst order methods, there is one pressure unknown for each cell, and one ﬂux unknown for each edge. Higher order methods have more degrees of freedom. For stability of the MFE method, the approximation spaces Vh and Wh must be chosen such that a coersivity and an inf-sup condition hold, see e.g. [37, 44]. In two dimensions, families of approximation spaces are constructed for triangles and quadrilaterals, for deﬁnitions and consideration see e.g. [24, 42, 195]. For three-dimensional problems the MFE method is deﬁned for tetrahedral and hexahedrals, see e.g. [43, 173], and also [179] and the references therein. Note, however, that if the hexahedral cells are more irregular than slightly perturbed parallelepipeds, no approximation spaces have been published so far. The MFE method has a good theoretical foundation, see e.g. [76]. Applications to reservoir simulation can be found in for instance [56, 78]. However, the method has drawbacks that so far have limited the usage in industry. The Equations (4.35)-(4.36) form a saddle-point problem, and the discretisation renders an indeﬁnite symmetric linear system with both pressure and velocity variables as unknowns. Local ﬂux expressions for MFE methods are only known in special cases [224]. Because of its high computational complexity, the MFE method is mostly applied in IMPES formulations. 4.7.2 Mimetic Finite Differences The mimetic ﬁnite difference (MFD) methods provide a framework to obtain discretisations that preserve, or mimic, certain properties of the continuous operator. For elliptic conservation laws, the MFD methods deﬁne approximation spaces Ph for the scalar variables (potentials) and Fh for the edge ﬂuxes. In Ph we deﬁne an 4.7 Other Locally Conservative Methods inner product by [p, q]Ph = |Vi |pi qi , p, q ∈ Ph , 55 (4.37) i where the sum is taken over all cells, |Vi | represents the volume of cell i, and pi and qi contain the cell values. The construction of the inner product in Fh is somewhat involved. For each cell C, we deﬁne [f , g]C = f T MC g . (4.38) Here MC is a symmetric, positive deﬁnite matrix of dimension ne × ne , where ne is the number of edges for the cell C. Further, f and g store ﬂuxes over the edges of C. The construction of the matrix MC is non-trivial. Sufﬁcient conditions for existence are given in [45]. Details on how to construct the matrix and analysis of the resulting schemes can be found in [47, 158]. Note that the construction is not unique. The inner product in Fh is now deﬁned as [f , g]Fh = [f , g]C , f , g ∈ Fh , (4.39) C with summation over all cells. The divergence operator DIV is readily deﬁned as a sum over edge ﬂuxes; ne 1 (DIV G)C = Gi |ei | , (4.40) |VC | i=1 where the sum is taken over the edges of the cell, Gi is the edge ﬂux, and ei is the length of the edges. Finally, we can write a discrete Green’s formula as [f , GRADp]Fh = [p, DIV f ]Ph , ∀ p ∈ Vn , ∀ f ∈ Wn . (4.41) This gives an implicit deﬁnition of the discrete gradient GRAD. The MFD discretisation of Equation (4.1) now reads; ﬁnd uh ∈ Ph and qh ∈ Fh such that qh = GRADuh , DIV qh = b , (4.42) (4.43) where b represents the sink/source terms. The MFD method is appealing due to its applicability to general grids, and a solid theoretical foundation for single phase problems. Convergence proofs for the method are given in [45, 46]. However, like in the mixed ﬁnite element formulation, the ﬂux in the MFD method is a global function of the potential, and the resulting linear system is of saddle point type. One notable exception is the local method presented in [160], which is identical to the MPFA O(1/3)-method. Regarding extensions to multiphase ﬂow, the only applications we are aware of are the multiscale methods mentioned next. 56 4.7.3 Elliptic Discretisation Principles Multiscale Methods The ﬂow in porous media have characteristic features on a multiple of length scales, and information on geological properties such as permeability is available on a much ﬁner resolution than what can be handled by standard techniques. The computational cost is especially severe for the pressure equation, due to its global domain of dependence. We touched upon this problem in Chapter 1, where we mentioned upscaling techniques as a tool to represent the parameters on a scale that can be handled by the computer resources available. However, this approach leaves much of the ﬁne-scale information available unused after the upscaling. In multiscale (MS) methods, coarse and ﬁne scale descriptions of the medium are combined in ways that attempt to combine the computational efﬁciency of a coarse scale algorithm and the accuracy of the ﬁne scale. A vast number of multiscale methods have been proposed, and we describe only some of the methods studied in the reservoir simulation community. For simplicity, we will stick to a two-level method in our outline. In the methods we consider, both a ﬁne and a coarse grid are constructed, so that the coarse grid cells deﬁne subdomains for the ﬁne grid. By solving local ﬁne scale problems, one can obtain a coarse scale operator for the pressure that incorporates ﬁne scale features. The coarse scale problem has fewer degrees of freedom than the original ﬁne scale problem, and its solution can be found faster. So far, the procedure resembles upscaling of the pressure equation. However, contrary to an upscaling method, a MS method provides a way to obtain the pressure and, for the methods applied to reservoir simulation, mass conservative ﬂuxes on the ﬁne scale. The velocity ﬁeld can be used to solve transport equations on the ﬁne scale. Since MS methods for elliptic problems were introduced in [118], a vast number of different methods have been introduced, see e.g. [131]. There are MS methods that applies techniques from MFE [1, 58] and MFD [4, 157]. The control volume methods that have been the primal focus of this work have their multiscale extension in the Multiscale Finite Volume (MSFV) method [121, 122]. The MSFV method has been formulated to handle a large range of physical problems [123, 154, 162, 163]. Also, since it is based on control volume methods, it is a natural candidate for incorporation into industrial codes. However, the monotonicity issues discussed in Section 4.5 may limit the applicability to realistic problems. The idea of considering the problem on multiple levels, and dividing the ﬁne scale grid into subdomains is also found in the domain decomposition (DD) methods. Indeed, links between multiscale methods and DD are studied in [2, 3, 176]. Chapter 5 Hyperbolic Discretisation Principles In Chapter 2 we saw that the mass variables are governed by equations that have an hyperbolic nature. This chapter is devoted to techniques for discretising hyperbolic equations. Like in the previous chapter, we mainly consider schemes that are used in the mainstream commercial simulators of today, i.e. control volume methods. At the end of the chapter, we will brieﬂy review alternative approaches. 5.1 Preliminaries Hyperbolic equations have a quite different nature from the elliptic equations considered in the last chapter. They propagate information with ﬁnite speed, allowing for explicit time stepping methods, cf Chapter 3. The solution of hyperbolic equations can exhibit discontinuities, or shocks. Consider the prototype hyperbolic model equation for an unknown c in one dimension ∂c ∂f (c) + = 0. ∂t ∂x (5.1) Here, f (c) is a possibly non-linear ﬂux function. Now, assume that the solution experience a discontinuity, with value c = cR on one side, and c = cL on the other. The speed of the discontinuity, σ, can be found by the Rankine−Hugoniot shock condition σ= f (cR ) − f (cL ) . cR − cL (5.2) However, there are two types of solutions that honours the Rankine-Hugoniot condition. One is a shock, where the solution jumps from one state to another. 58 Hyperbolic Discretisation Principles The other is a continuous transition between cR and cL in rarefaction wave. For the solution to exhibit a proper shock, it must honour Oleinik’s entropy condition f (c) − f (cL ) f (c) − f (cR ) ≥σ≥ . c − cL c − cR (5.3) This holds for all c between cL and cR , no matter which of the initial values are largest. For a general ﬂux function f , the solution can consist of multiple shocks and rarefaction waves. 5.1.1 Riemann Problems A fundamental building block in many numerical methods for hyperbolic problems is the much studied Riemann problem. The Riemann problem consist of a non-linear hyperbolic equation like (5.1) with initial condition cL x < 0, c(x, 0) = (5.4) x > 0. cR A numerical method that approximates the ﬂux over an edge using only function values in the two adjacent cells, must in practice solve a Riemann problem for each edge. The analytical ﬂux for a Riemann problem with a general shaped ﬂux function is given by f (c) = ⎧ ⎨ min f (c) , if cL ≤ cR , ⎩ max f (c) , if cR ≤ cL . cL ≤c≤cR cR ≤c≤cL (5.5) We remark that for a monotone f , the ﬂux is simply either f (cL ) or f (uR ), depending on the initial data. 5.1.2 Hyperbolic Formulations We now develop two formulations of the hyperbolic transport equations introduced in Chapter 2. For simplicity, we focus on ﬂow of two incompressible, immiscible phases. The governing equations for this case were introduced in Section 2.3. If we deﬁne the concentration of phase α as cα = ρα Sα , and for simplicity set the porosity to 1 everywhere and ignore capillary forces, Equation (2.17) can be written ∂cα − ∇ · (ρα λα K∇(p − ρα gz)) = qα . ∂t (5.6) 5.1 Preliminaries 59 f (c) f (c) c (a) Monotonic c (b) Non-monotonic Figure 5.1: Two possible ﬂux functions. The function to the left is monotonic. The function to the right has a sonic point. This is a hyperbolic transport equation for the concentrations. We refer to Equation (5.6) as the phase-based approach. In the absence of capillary forces, the total velocity qT can be written qT = − λα K∇(p − ρα gz) . (5.7) α Now, we can rewrite Equation (5.6) to read ∂cα − ∇ · (ρα fα (qT − K∇(p − ρβ=α gz)λβ=α )) = qα . ∂t (5.8) Here we have introduced the fractional ﬂow function fα = λλTα . Hereafter, we will refer to Equation (5.8) as the fractional ﬂow or total velocity formulation. The total velocity ﬁeld will normally vary slower in time than the phase velocities. Therefore, the fractional ﬂow might be the more robust formulation, however, it might be more difﬁcult to apply. Consider the methods in one spatial dimension, written on the form of Equation (5.1). The monotonic ﬂux function (f > 0) shown in Figure 5.1(a) is representative for the phase based formulation both for horisontal and vertical ﬂow. The ﬂux function for the fractional ﬂow formulation will only be monotonic if the ﬂow is horisontal. For vertical ﬂow, the ﬂux function can have a shape as shown in Figure 5.1(b). Note that it has a point where f = 0, and thus the solution of the Riemann problem becomes non-trivial. In two dimensions, the phase formulation can be written in the form ∂c + ∇ · (ζ(c)(σx ex + σz ez ) = q , ∂t where ζ is some generally non-linear function, and the velocity vector σ = σx ex + σz ez , (5.9) (5.10) 60 Hyperbolic Discretisation Principles is independent of c. Due to the gravity term, this will not be true for the fractional ﬂow formulation. To ease the presentation, we have neglected capillary forces. The character of the phase formulation is not altered if capillary forces are introduced, although the phase velocities will change. On the contrary, the fractional ﬂow formulation turns into an advection-diffusion equation, which has a parabolic nature. In the control volume framework presented next, the parabolic terms will naturally be treated with elliptic discretisation principles as discussed in the previous chapter. To ensure stability the diffusive terms should be treated implicitly. Treatment of diffusion terms in Lagrangian methods will be discussed at the end of this chapter. 5.2 Control Volume Methods For a multi-dimensional problem, a generic hyperbolic equation can be written ∂c + ∇ · f (c) = q . ∂t (5.11) As in the previous chapter, letΩi be a cell in the grid, and write the edges of the cell as ∂Ωi,j , so that ∂Ωi = j ∂Ωi,j . Integration in space over Ωi and in time from tn to tn+1 yields tn+1 tn+1 1 (cn+1 − cn )dV + f (c) · ndSdt = qdV dt,(5.12) Vi j Ωi ∂Ωi,j Ωi tn tn where we have used the divergence theorem, Vi is the area of Ωi , and n is the outer normal vector of ∂Ωi,j . By replacing c and q by cell centred variables, we get a numerical scheme tn+1 n+1 1 t n+1 n c −c = f (c) · ndSdt + qi dt . (5.13) Vi j tn ∂Ωi,j tn Our main focus is to construct approximations, fnum to the ﬂux integral fnum ≈ (f (c) · n)dS . (5.14) ∂Ωi,j However, before we do so, we need to introduce a concept from the theory for scalar transport schemes. If the approximation (5.13) only depends on values at time tn , the scheme can be written in the generic form Cin+1 = Φni (C1n , . . . , Cnne ) , (5.15) 5.2 Control Volume Methods 61 where ne is the number of cells in the grid. The scheme is said to be monotone if [155] ∂Φni (C1n , . . . , Cnnc ) ∂Cin+1 = ≥0 ∂Cjn ∂Cjn ∀j . (5.16) A monotone scheme cannot generate spurious oscillations in the computed solution, and requiring monotonicity may seem natural. This will be sufﬁcient to guarantee convergence to the correct entropy solution [62, 105]. However, as shown in [99], monotone methods are at most ﬁrst order accurate. Thus, it is natural to require monotonicity for ﬁrst order schemes, whereas higher-order schemes must meet other criteria. We will brieﬂy come back to this in Section 5.4. 5.2.1 Upstream Weighting The perhaps simplest way to approximate the ﬂux integral (5.14) is to consider a Riemann problem with the discontinuity at the edge ∂Ωi,j , and the initial values equal to the cell values in the adjacent cells. Now, the upstream method, hereafter denoted SPU (single point upstream) evaluate the direction of the ﬂux over the edge based on the local velocity ﬁeld v, and deﬁne the ﬂux according to |∂Ωi,j |f (ui ) , v · n > 0, fnum (cu ) = (5.17) v · n < 0, |∂Ωi,j |f (uj ) , where the normal vector of the edge n points from cell i into cell j, and |∂Ωi,j | denotes the length of the edge. The prevailing solution technique in industrial reservoir simulators is to solve the phase based formulation with upstream weighting [26]. The upstream approach is monotone, and proof of convergence for this approach to one-dimensional problems and homogeneous media can be found in [40, 203]. The upstream method suffers from some problems near discontinuities in the ﬂux function (i.e. discontinuous permeability) [19]. The difﬁculty stems from how the upstream direction is found. SPU deﬁnes the upstream direction according to the local velocity ﬁeld. However, the analytical solution of the Riemann problem applies upstreaming with respect to waves deﬁned by the ﬂux function. For complex ﬂux functions, the two approaches are not necessarily equal. The standard upstream method as presented above is essentially a onedimensional method; it computes the ﬂux based on the values in the two adjacent cells only. This methodology can readily be extended to multi-dimensional problems by doing a dimensional splitting, and this is the common approach in industrial simulations. The convergence proof for monotone methods given in [62] also holds for dimensional splitting on Cartesian grids. 62 Hyperbolic Discretisation Principles 1 3 v 2 (a) (b) (c) Figure 5.2: Figure a) illustrates cells involved in multi-dimensional upstream weighting. Figure b) and c) show saturation contours for a quarter-ﬁve spot simulation. The direction of the grid orientation is indicated in the ﬁgures. Except from grid orientation, the parameters for the simulations are identical. However, in displacement processes with adverse mobility ratios, i.e. the intruding ﬂuid is more mobile than the displaced ﬂuid, dimensional splitting can lead to grid orientation effects (GOE), ﬁrst observed in [215]. Consider Figure 5.2(a). The upstream method approximates the ﬂow over the edge between cells 1 and 2 by using cell 2 as the upstream cell. Any contribution from cell 3 is ignored. A possible result of this can be seen in the Figures 5.2(b) and 5.2(c). These shows saturation proﬁles from a simulation on a Cartesian grid. There is an injection in the lower left corner, and a producer well in the upper right corner. The displacing ﬂuid has lower viscosity than the displaced ﬂuid, and the mobility functions are taken from [232]. The saturation front tends to follow the coordinate lines, rendering solutions that completely depend upon grid geometry. GOE in reservoir simulation is a much studied topic, see for instance [38] and the references therein. The fundamental difﬁculty is that the displacement process is unstable on the level it is discretised; that is, the mathematical model is not well posed. Therefore, the instability does not disappear with grid reﬁnement. Parameters such as grid size, mobility ratio, and truncation errors for the discretisation schemes play an important role in triggering the instabilities. Several remedies have been proposed for reducing GOE, including applying higher order methods for transport [54], and constructing pressure discretisations aimed at capturing the ﬂow over corners [207, 232]. 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting Here we present a framework for upstream methods that preserve the multidimensionality of the problem better than dimensional splitting approaches. The 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting 63 6 2 1 3 4 5 Figure 5.3: The ﬁgure shows local numbering of cells that are possible candidates for a transport scheme for the central cell. intention is to amend to SPU, and thereby reduce biasing with respect to the grid. The methods apply to all problems that can be written on the form (5.9) - (5.10). The framework was originally developed and studied by Kozdon and coworkers in [141, 142, 143]. To ease the presentation, we restrict ourselves to Cartesian grid. Extensions to other grids are considered in Paper E. 5.3.1 Multi-Dimensional Transport The reason for the poor performance of SPU shown in Figure 5.2(b) and 5.2(c), is that the scheme neglects ﬂow over corners, that is, from cell 3 to cell 1, referring to Figure 5.2(a). To formulate better schemes, we thus need to deﬁne methods with larger stencils. Consider the cells shown in Figure 5.3, and assume for the time being that both the velocity components are positive, i.e. the ﬂow is more or less from the lower left to the upper right corner. Then, the four cells that are most signiﬁcant for the state in cell 1 are number 1, 2, 3, and 4. We will formulate a framework which utilise schemes consisting of these four most important cells. We remark that also cell number 5 and 6 are in the upstream direction to cell 1, and thus it is possible to deﬁne stencils with ﬁve and six cells. This will not be pursued here, more information can be found in [222]. One of the simplest hyperbolic problems is that of linear transport. Referring to Equation (5.11), this corresponds to f = (cv1 , cv2 ), where v1 and v2 are the velocity components in the x- and y-direction, respectively. In [197], Roe and Sidilkover found that the linear four-point schemes on Cartesian grids can be described by a single parameter. Further, due to the simple structure of the problem, it is possible to gain insight in the properties of the methods by a modiﬁed equation analysis. Details can be found in [142, 197], and also in Paper E, where the analysis is extended to parallelogram grids. Even though our transport problem of interest is non-linear, we would like to base our framework upon the linear theory. As we saw in Chapter 2, the variables that are governed by hyperbolic equations typically represents mass or concentrations. Obviously, negative values of 64 Hyperbolic Discretisation Principles ζ̄3 ζ4 . U3 ζ̄4 ζ1 x0 xω U2 ζ̄2 U4 . U1 ζ̄1 U2 . ζ3 . ζ2 (a) An interaction region x̄2 U1 (b) Forward trace Figure 5.4: To the right, an interaction region for the multi-dimensional transport framework. The arrows show the positive directions of the velocities. The forward trace is shown to the left, ω ∗ = xω − x0 /x̄2 − x0 . these variables make no sense from a physical point of view, and the numerical schemes should be designed to avoid this situation. For two-phase systems, we can avoid such numerical artifacts by requiring the schemes to be monotone. That is, Deﬁnition (5.16) will be imposed for each cell in the grid. In general, all cells are included in several cell stencils, and hence the positivity requirement might yield a global coupling of the coefﬁcients in the transport scheme, which must be solved for each time step. In addition to being a cumbersome solution procedure, this can impose strong restrictions on the schemes. Moreover, the global coupling is in conﬂict with the local nature of hyperbolic problems. To avoid these issues, the numerical schemes are formulated with the help of interaction regions, which were presented in Section 4.4. As we will see, this framework allows for the coefﬁcients in the transport scheme to be deﬁned locally. The desire to use interaction regions is also the reason why we only seek inspiration from methods with four-point cell stencils. We now ﬁrst describe how the schemes constructed for linear transport can be formulated by interaction regions. Applications to two-phase ﬂow will be considered next. Assume that a velocity ﬁeld is known, and let the ﬂuxes over a half-edge i be represented by Ui . Moreover, denote by ζi cell centred values of a quantity to be treated with upstreaming. For the time being we do not assign any physical meaning to ζ, but instead emphasis the generality of the framework presented here. Later, we will let the upstreamed quantities be phase mobilities. Consider the interaction region shown in Figure 5.4(a). The transport over half-edge i is given by Fi = Ui ζ̄i , (5.18) where the function value at the half-edge, ζ̄i , is to be determined. Consider the 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting 65 case shown in Figure 5.4(b), when both U1 and U2 are positive. Then, a forward trace of the characteristic from the midpoint of edge 1 suggests that ζ̄2 can be expressed as a combination of the value of ζ in the centre of cell 2, ζ2 = ζ(C2 ), and ζ̄1 . This can be achieved if we deﬁne ζ̄2 by ζ̄2 = (1 − ω2 )ζ2 + ω2 ζ̄1 , (5.19) where ω2 will be deﬁned below. For a general velocity ﬁeld, the edge concentrations can be written as (1 − ωi )ζi + ωi ζ̄i−1 , if Ui ≥ 0 , ζ̄i = (5.20) if Ui < 0 , (1 − ωi )ζi+1 + ωi ζ̄i+1 , where i±1 is deﬁned cyclically on i = 1 . . . 4. This is of course not the only way to deﬁne edge quantities ζ̄i . It is however convenient, since Deﬁnition (5.20) allows the parameter ωi to describe the family found by Roe and Sidilkover of all possible linear, monotone four-point schemes for a constant velocity ﬁeld. Moreover, we can apply these schemes to more general cases as well. Equation (5.20 can be set up for all half-edges in the interaction region. Let ζ = (ζ1 , . . . , ζ4 ) and ζ̄ = (ζ¯1 , . . . , ζ¯4 ). Then ζ̄ can be found from the linear system ζ̄ = Sζ + Tζ̄ . (5.21) In this way, the coefﬁcients in the transport scheme are computed locally within each interaction region, thus, we avoid the issues with a global coupling. If the concentrations are positive, ζ (C) > 0, and ωi < 1 for at least one i, the scheme will be monotone [141, 142]. What remains is to deﬁne the parameter ω so that ﬂow not aligned with the grid is properly represented. Again, consider Figure 5.4(b). We do a local reconstruction of the velocity ﬁeld inside cell 2, based on U1 and U2 . A vector aligned with the velocity ﬁeld, starting in the midpoint of edge 1, x̄1 , will intersect with half-edge 2 in a point xω . Then deﬁne ω ∗ as the relative distance between xω and the centre in the interaction region x0 , compared to the length of the half-edge, ω∗ = xω − x0 . x̄2 − x0 (5.22) The functional relationship between ω ∗ and ω = ω(ω ∗ ) is inspired by methods designed for linear adevection. To guarantee monotonicity of the schemes, the weights of both ζ and ζ̄ should be positive. Therefore, we need 0 ≤ ω ≤ 1. Moreover, the schemes are not monotone unless ω ≤ ω ∗ [141]. For Cartesian grids, 66 Hyperbolic Discretisation Principles we consider three choices, single point upstream (SPU), tight multi-dimensional upstream (TMU), and smooth multi-dimensional upstream (SMU), given by ωSP U (ω ∗ ) = 0 , min(ω ∗ , 1) , ωT M U (ω ∗ ) = 0, ∗ ω , 1+ω ∗ ωSM U (ω ∗ ) = 0, ω∗ > 0 ω∗ < 0 , ω∗ > 0 ω∗ < 0 . (5.23) The schemes are motivated by the standard SPU scheme, the Narrow scheme [197] and Koren’s scheme [140], respectively. The reason for deﬁning the limiter functions in this way can most easily be seen from a modiﬁed equation analysis, confer [142] and Paper E. We point out that in [142], the method was formulated as a traceback instead of a forward trace. For Cartesian grids, the two approaches are equivalent. However, on an unstructured grid, forward tracing is conceptually somewhat easier to formulate than tracing backwards. In practice, the two different formulations yield approximately the same results. 5.3.2 Discretisation of Two-Phase Flow So far, we have only considered transport schemes for a single time step. We will now outline how the scheme presented above can be applied to two-phase ﬂow. As we saw in Section 2.3, the mobilities render a hyperbolic behaviour for the mass or saturation variables, thus they are treated with upstream weighting. We remark that we could have treated the mass variables with upstreaming instead of the mobilities. However, it has been found difﬁcult to obtain schemes that are provably monotone within such a framework [141]. Flow of two immiscible phases was studied in Section 2.3, where we derived the governing equation ∂(ρα Sα ) dV + φ ∂t Ω ∂Ω n · (ρα λα K∇(pα − ρα gz)) dS = qα dV . (5.24) Ω Throughout this section, we for simplicity consider the density to be constant. Now, let the control volume be a cell Ωi , and split the surface integral into one for each edge. We then recognise n · (ρα λα K∇(pα − ρα gz))dS , (5.25) ∂Ωi,j 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting 67 as a two-phase version of Equation (4.3). For each edge j, we therefore split the ﬂux term into a ﬂuid term λ̄α,j and a geometric term σα,j = n · (K∇(pα − ρα gz))dS , (5.26) ∂Ωi,j where σα,j is a phase ﬂux over edge j. This phase ﬂux should not be confused with the mass ﬂux of phase α, which is fα = ρα λ̄α σα . For simplicity, the densities are considered constant in this section. The geometric term can thus be treated by the control volume techniques from the previous chapter, and a discretisation of Equation (5.26) for all edges gives us σα = T(p − ρα gH)) . (5.27) Here σα is a vector of phase velocities for each edge, T is a transmissibility matrix, and the vectors p and H contain cell centre pressures and heights. The phase velocity ﬁeld can be used to deﬁne mobilities on the edges by the method previously presented. The vector of edge mobilities, λ̄α , can be written as λ̄α = Cα λα , (5.28) where Cα is a matrix that takes cell values to edge values, based on Equation (5.21), and λα is a vector of cell mobilities. Note that Cα depend on σα , hence the edge mobilities are functions of the pressure. The ﬂux over all edges can now be written on the form fα = λ̄α · σα , (5.29) where · denotes component wise multiplication. Each component in fα can be considered a ﬂux approximation fnum to its respective edge, confer Section 5.2. As long as the mobilities are non-decreasing functions of saturation, the phase formulation yield a ﬂux function with no sonic points. For more complex physical processes, this will no longer be the case. A strategy for how to handle sonic points is presented in [141]. The basic idea is to apply upstreaming to ﬂuxes instead of mobilities. We remark that this approach requires knowledge of the sonic points of the ﬂux functions. Explicit Time Stepping The schemes introduced in the previous section are provably monotone, and thereby free of spurious oscillations. For a consistent scheme, monotonicity is sufﬁcient to guarantee non-negative values of the transported variable cα [141]. 68 Hyperbolic Discretisation Principles However, we have no means to impose an upper bound on the variable. The common approach for a two-phase simulation would be to let the mass be represented by one saturation variable, and deﬁne the saturation of the other phase according to (2.6); α Sα = 1. However, the pressure equation for the multi-dimensional framework derived below has non-linearities which require an iterative solution approach, i.e. Newton’s method. In practice, the pressure solution will therefore not be exact, and transport of saturations according to the corresponding velocity ﬁeld may yield non-physical values, i.e. they do not honour the constraints 0 ≤ Sα ≤ 1, and α Sα = 1. One remedy is to use phase masses as primary variables. Let Mα,i be the average mass in cell i, Mα,i = ρα Sα dV , (5.30) Ωi where the phase can be aqueous or liquid. The discrete saturation of phase α in cell i is then deﬁned by Sα,i = Mα,i /ρα . Ma,i /ρa + Ml,i /ρl (5.31) This deﬁnition guarantees positive saturations if the masses are so, moreover, the physical constraints are satisﬁed. However, the discrete saturation concept, means we have in fact introduced a framework which allows for a volume residual. Another way to think of this is that we have added a small, artiﬁcial compressibility to the system. Again considering Equation (5.24), we now have a strategy for discretising all the surface integrals for all the edges. Moreover, we recognise the ﬁrst term as the time derivative of the discrete mass in cell i. We integrate in time from tn to tn+1 , and apply the fundamental theorem of calculus to get the discrete conservation equation for phase α as Mn+1 = Mnα (5.32) α −1 n n+1 n+1 n n+1 −ΔtV (Div((ρα λ̄α (Sα , p ) · σα (p ))) − ρα Qα (Sα , p )) . Here, Mα is a vector of cell averaged masses, V is a diagonal matrix with cell volumes on the diagonal, and Div is a discrete divergence operator. Further Sα is a vector of phase saturations, and (volumetric) source terms are represented by Qα . Superscripts indicate time steps and Δt is the time step size. Note that we have employed an IMPES strategy; pressure is discretised implicitly, while masses and saturations are treated explicitly. 5.3 A Framework for Truly Multi-Dimensional Upstream Weighting 69 If we divide by the densities, and sum the conservation equations for the two phases, we get a pressure equation which reads Mn+1 Mn α α = (5.33) ρ ρ α α α=a,l α=a,l −1 n n+1 n+1 n n+1 −ΔtV Div((λ̄α (Sα , p ) · σα (p ))) − Qα (Sα , p ) . α=a,l If the volume balance is exact, we have Mn Mn+1 α α = . ρα ρα α=a,l α=a,l (5.34) However, within the volume relaxed framework, this is not necessarily true. We aim for volume balance at time step n + 1 by setting n+1 Mi,α = 1. ρα α=a,l (5.35) Now, the pressure equation can be written in residual form, F = 0, where F(pn+1 ) = e − +ΔtV−1 Mn α ρ α α=a,l (5.36) Div((λ̄α (Snα , pn+1 ) · σα (pn+1 ))) − Qα (Snα , pn+1 ) , α=a,l where e is a vector of ones. This is a non-linear equation for pn+1 , which can be solved by Newton’s method. The Jacobian matrix, Jp,p , is also needed in the fully implicit method, and is presented below. The above equations represent a truly multi-dimensional framework for twophase transport. The computational cost of the scheme is somewhat increased compared to a standard two phase method with pressure and saturation as primary variables. First, more than one iteration might be needed to solve the pressure equation, although, in practice a residual tolerance of 10−3 has been sufﬁcient for the tests done so far. This is normally achieved after 1-3 iterations on the pressure equation, depending on how important role gravitational forces play. Second, we need to update two mass equations variables instead of one saturation equation. However, solving an explicit transport equation is cheap, and this does not add a signiﬁcant computational burden to the simulation. 70 Hyperbolic Discretisation Principles Implicit Time Stepping The multi-dimensional framework can also be used for fully implicit methods. Since the transport equation now is solved implicitly, it is tempting to use water saturation as our mass variable to limit the number of equations. We therefore abandon the volume relaxed framework. Treatment of non-physical saturation values is discussed in Paper E. Our governing equations are now one equation for pressure, and one for transport of the saturation of water. The discretised pressure equation can be written as −1 Fp (Sw , p) = ΔtV Div(λ̄α (Sα , p) · σα (p)) − Qα (Sα , p) . (5.37) α=a,l Similarly, the saturation equation is discretised by FS (Sw , p) = Sw −Snw +ΔtV−1 Div(λ̄w (Sw , p)·σw (p))−Qα (Sw , p)).(5.38) We solve the equations by casting them into residual form, (Fp , FS ) = 0, and apply Newton’s method. The Jacobian matrix reads JS,S JS,p J(S, p) = , (5.39) Jp,S Jp,p where Ja,b = ∂Fa . ∂b JS,S = JS,P = JP,S = JP,P = The elements in the Jacobian are deﬁned as follows ∂ λ̄w ∂Qw I + ΔtV , (5.40) Div diag(σw (p)) − ∂Sw ∂S ∂ λ̄w ∂Qw −1 ΔtV − ,(5.41) Div diag(λ̄w )T + diag(σw (p)) ∂p ∂p ∂ λ̄α ∂Qα −1 , (5.42) ΔtV Div diag(σα (p)) − ∂Sw ∂S α=a,l ∂ λ̄α (5.43) ΔtV−1 Div diag(λ̄α )T + diag(σα (p)) ∂p α=a,l ∂Qα − . ∂p −1 λ̄α to the JaThe multi-dimensional upstream formulation adds terms involving ∂∂p cobian matrix. Differentiation of Equation (5.21) with respect to the pressure, λ̄α gives a linear system that can be solved for ∂∂p . 5.4 Other Approaches 71 The methods presented in this section have been applied to Cartesian grids in [141, 143], and they have been shown to reduce grid orientation dependency. Development of the method presented above is in a starting phase, and only very simpliﬁed cases have been considered so far. Some thoughts on the road ahead are given in Chapter 7. 5.4 Other Approaches There are many methods developed for solving transport equations apart from the simple ﬁrst order upstream strategy. The aim of this section is to give an overview of some of the alternatives to the upstreaming presented above. There are two distinct groups of schemes for hyperbolic problems: control volume methods and Lagrangian methods. However, we ﬁrst discuss Riemann solvers in general, and present front tracking methods, which can be applied both in control volume and Lagrangian frameworks. 5.4.1 Riemann Solvers and Front Tracking In Section 5.2, we formulated control volume methods in terms of a set of Riemann problems that must be solved for each edge. Godunov proposed a method based on solving the Riemann problem exactly for each edge. For scalar equations, he also showed that this strategy gives a method with less artiﬁcial diffusion than any other monotone method. If this strategy is applied to problems with more than one hyperbolic conservation law, we need to solve systems of equations of the form ∂c + ∇ · (f (c)) = q , ∂t (5.44) with initial data cL c(x, 0) = cR x < 0, x > 0. (5.45) The solution will in general contain multiple shocks, rarefaction waves and contact discontinuities. One key difﬁculty for Riemann solvers is that the theory for hyperbolic systems is not fully developed, and we only know the solution in certain special cases, see for instance [127, 130]. If an analytical solution is not available, approximate Riemann solvers must be applied to ﬁnd the wavestructure, see for instance [181, 196, 216]. These procedure can be cumbersome, and this partly explains why reservoir simulators often use upstream evaluation based on the phase velocity ﬁeld instead of the Riemann waves. 72 Hyperbolic Discretisation Principles The Riemann solvers can be applied to reservoir simulations in (at least) two different ways. The edge ﬂuxes in a control volume method can be considered as solutions of Riemann problems. Another possibility is to represent the solution as a set of discontinuities, and propagate them in time. This is the philosophy of the front tracking methods. An important point here is that the ﬂux function is approximated by a piecewise linear function, while the initial data is piecewise constant [63, 112]. Thus, the speed of the shocks can be computed by the RankineHugoniot condition. Further, rarefaction waves are approximated by a sequence of small shocks. As the name indicates, the methods are especially well suited to follow fronts in the solution, e.g. the water front during secondary production. Now, the fronts can either be tracked on a Cartesian grid, with some dimensional splitting, e.g. [106], or a streamline method (explained below) can be applied [129, 156]. 5.4.2 Control Volume Methods Higher Order Upstream Methods So far, all methods presented have solved problems with constant initial data in each cell. As a result, the methods are only ﬁrst order accurate. Higher order methods can be constructed in a Reconstruct - Solve - Average fashion. Based on the solution from one time step, a higher order representation of the data is computed, often in the form of polynomials. These polynomials serve as initial data for the next time step. The solution is propagated, and an average value in each cell is computed for use in the next time step. As previously mentioned, a monotone method cannot be more than ﬁrst order accurate. A naive approach to the reconstruction of the solution may lead to self-sharpening, and eventually development of spurious oscillation in areas where the solution gradient is steep. This can be avoided by a careful reconstruction step that ensures that the method is only ﬁrst order accurate near shocks. Well known higher order methods are of TVD, ENO-, or WENO-type, we refer to the reviews [206, 211] and the references therein. Applications to porous media ﬂow can be found in e.g. [166, 198, 212]. Multi-Dimensional Upstreaming Most control volume methods are essentially designed for one spatial dimensional, although multi-dimensional problems can be solved by dimensional splitting. However, Hurtado et.al. [119] presented a methodology with strong similarities to the framework presented here. They deﬁne edge mobilities based on forward tracing on a dual grid. However, the pressure equation is solved by a control volume ﬁnite element method, and this also mean the concepts of primal 5.4 Other Approaches 73 and dual grid is ﬂipped compared to the present method. Another genuinely multi-dimensional approach which is quite similar to the one outlined in this chapter was presented by Edwards in [83, 84], and further developed by Lamine and Edwards in [150, 151]. The main focus of the work is to develop higher order methods for transport. Their methods combine an MPFA pressure solver with multi-dimensional transport. However, the upstream quantities are either ﬂuxes or saturations, and they do not explicitly introduce an interaction region framework for transport. Moreover, the transport method is based on the fractional ﬂow formulation instead of phase velocities. So far, no extension to problems with gravity included (i.e. fractional ﬂow functions with sonic points) has been published. Central Schemes All the schemes mentioned above can be considered as upstreaming methods. An alternative is the class of central schemes. The simplest method is the ﬁrst order Lax-Friedrichs scheme, which suffers from high levels of numerical viscosity. Nessyahu and Tadmor proposed to use higher order central methods, and showed that the higher accuracy reduce the artiﬁcial smearing [174]. The scheme was further improved in e.g. [125, 144]. Multi-dimensional problems can be handled without dimensional splitting [126]. An application to multiphase ﬂow is presented in [18]. A major concern for these methods is how much artiﬁcial smearing they will introduce in areas with sharp discontinuities in the permeability. Discontinuous Galerkin Methods The higher order methods mentioned above construct a polynomial from the function value in multiple cells, rendering a system of equations with some bandwidth. Moreover, extensions to multiple dimensions can be difﬁcult if the grid is irregular, even if dimensional splitting is applied. In the Discontinuous Galerkin (DG) framework, a higher order ﬁnite element method is applied in the interior of the elements. The ﬁnite element framework allows for a high resolution in the interior of the cells, and since the approximation on the interior is decoupled from other cells, parallelising is easy. We let Vh (Ωi ) denote the approximation space on an element Ωi . A DG discretisation of Equation (5.11) on Ωi reads: Find uh ∈ Vh (Ωi ) such that for all vh ∈ Vh (Ωi ) ∂uh vh − f (uh ) · ∇vh dV + (f · n)vh dS = q . (5.46) Ωi ∂t ∂Ωi · n denotes the Here, vh are basis functions in a ﬁnite element space Vh (Ωi ), and f numerical ﬂux over the cell boundary. A ﬁnite element solution will in general not 74 Hyperbolic Discretisation Principles preserve mass over the element edges. Therefore, the interaction between the elements is approximated by a numerical method, usually a lower order method such as Godunov, Enquist-Osher, or Lax-Friedrichs. To prevent artiﬁcial oscillations, some slope limiting is necessary. Time integration is commonly done by RungeKutta methods. The ﬂux approximation is one-dimensional, however, the high resolution on the interior means the method is able to capture multi-dimensionality better than traditional control volume method. For more information, confer [60] and the references therein. Usage of DG in porous media ﬂow is reported in e.g. [88, 117]. 5.4.3 Lagrangian Methods The control volume methods considered in this chapter can be considered Eulerian, in the sense that the grid is constant in time, and ﬂuid is transported between grids cells. Explicit time stepping is computationally cheap, but for stability the schemes must honour a CFL-criterion that limits the size of the time step [61]. In some sense, this is an artiﬁcial constraint imposed by the computational grid. If the transport scheme is implicit in time, the time step can be taken larger, but the computational overhead from solving linear systems can be severe. Moreover, control volume methods suffer from grid orientation effects. By employing a Lagrangian frame of reference that follows the path of ﬂuid particles, the time step restriction can be relaxed . If capillary forces are included, we must consider advection-diffusion equations on the form ∂c + ∇ · (vf (c)) − ∇ · (D∇c) = q , ∂t (5.47) where D is a diffusion tensor, which can be a function of c. For simplicity, we will sometimes consider a simpliﬁed model problem (on non-conservative form) instead; ∂c + v · ∇c − ∇ · (D∇c) = q , ∂t (5.48) where we have ignored compressibility effects (∇ · v = 0), and assumed a linear ﬂux function f . We go to a Lagrangian frame of reference by introducing the material derivative D ∂ = + v · ∇. Dt ∂t (5.49) Now, Equation (5.48) can be written Dc − ∇ · (D∇c) = q , Dt (5.50) 5.4 Other Approaches 75 which is a parabolic equation without an advective ﬁrst order term. Thus, in a Lagrangian frame of reference there are no steep fronts, and the resulting solution is much smoother than in an Eulerian framework; hence the stability restrictions on time step size is correspondingly less restrictive [91]. Also, Lagrangian methods will be far less sensitive to grid orientation errors than Eulerian techniques. Application of Lagrangian methods to complex recovery processes (tertiary production) is in an early stage of development compared to the control volume framework primarily studied in this work, and thus there are many issues that should be investigated further. Also, implementation of Lagrangian methods is non-trivial compared to Eulerian methods. However, if these methods can be improved on, Lagrangian transport schemes may offer a robust and computationally cheap alternative to control volume approaches [97]. We therefore brieﬂy review some methods that have shown promising results. Characteristic Methods In an advection dominated process, information is mainly transported along characteristic curves, and the characteristic methods are based upon tracing these lines. The classical approach would be to trace characteristics forward with a ﬁnite difference method. However, if we trace forward from time tn , there may be cells that contain no characteristics at time tn+1 , thus some interpolation scheme must be applied. Therefore backward tracing is often preferred. A technique well known in reservoir simulation is the method of modiﬁed characteristics (MMOC) presented in [77], and extended in i.e. [89, 202], with emphasis on porous media ﬂow. An MMOC discretisation of Equation (5.48) read Ω c(x, tn+1 )(¸x∗, tn ) w(x)dV Δt n+1 ∇w(x)D∇c(x, t + Ω )dV = q(x, tn+1 )dV , (5.51) Ω where w is a ﬁnite element method basis function. The point (x∗ , tn+1 ) is deﬁned by back tracing from (x, tn ) along the path described by the material derivative Dc . Dt Even though MMOC outperforms forward tracing, it does in general not conserve mass, since it is based on backward tracing. An adjusted version of MMOC that preserves mass is presented in [75]. To maintain conservation of mass and volume in backward tracing is a challenge for characteristic methods in general, see [22]. The advection and diffusion terms are handled by operator splitting, which may lead to artiﬁcial diffusion near shocks, as shown in [134], where a 76 Hyperbolic Discretisation Principles remedy is also presented. Moreover, there are also issues with handling of boundary conditions. The Euler-Lagrange localised adjoint method (ELLAM), ﬁrst introduced in [51, 111] can be viewed as an improvement of MMOC. Again, we consider the linear problem (5.48), for non-linear problems confer [64]. For backward tracing from time tn+1 , we let wADJ be a test function satisfying the system of equations ∂wADJ − v · ∇wADJ = 0 , ∂t ∇ · (D∇wADJ ) = 0 . (5.52) (5.53) This is a splitting of the adjoint equation of (5.48), see e.g. [200]. Note that in general, both v and D can be time dependent, thus the basis functions must be redeﬁned for each time step. Equation (5.53) is elliptic, suggesting that a ﬁnite element basis is appropriate for wADJ . To honour Equation (5.52), wADJ should be deﬁned as constant along the characteristics ∂x = v. This will also assure a ∂t Lagrangian treatment of transport, allowing for large time steps. Then, a discretisation of Equation (5.48) will result in an equation n+1 c(x, t (∇wADJ D∇c)(x, tn+1 ) dV Ω n n c(x, t )wADJ (x, t+ ) dV + qwADJ dV . (5.54) = ) dV + Δt Ω Ω Ω Here, wADJ (x, tn+ ) is interpreted as the limit value when t → tn from above. ELLAM methods have been formulated for a large range of problems with promising results, we refer to [92, 200] and the references therein for an overview. Other methods such as Characteristic mixed ﬁnite elements can also be considered ELLAM methods [23]. Streamline Methods The streamline methods represent the ﬂow pattern by tracing a set of streamlines starting at different spatial points. In each time step, the pressure equation is solved implicitly. After the pressure solve, the velocity ﬁeld is used to trace streamlines. Along the streamlines, we must solve a one-dimensional hyperbolic problem of the form ∂c + ∇ · (vf (c)) = q . ∂t (5.55) The streamlines usually varies on a much slower time scale than the transport along the streamlines. Therefore, streamline methods allow for time step sizes 5.4 Other Approaches 77 that can be several times the CFL-restriction. After the transport solve, the mass variables must be mapped back to the grid for a new pressure update. A major advantage of streamline methods is that problems in multiple dimensions are reduced to a set of one dimensional problems. These can be solved by for instance front tracking, or higher order methods. Thus, streamlines offer a way to reduce the dimensionality without applying dimensional splitting. However, there are some issues related to the methods. The mapping from the grid to the streamline, known as streamline tracing, is non-trivial, see [104]. Also, in the mapping from streamlines back to the grid, mass is usually not conserved. Moreover, gravity is treated by operator splitting [39], and it is an open question how this can be done properly. For more information of streamline methods, we refer to [67, 103, 165] and the references therein. Chapter 6 Summary of the Papers As a part of the work with this thesis, a number of scientiﬁc papers have been produced. The previous chapters have provided theory and background for the investigations. Here, we further outline the process leading to the papers, and present the main results. 80 Summary of the Papers Paper A: Sufﬁcient criteria are necessary for monotone control volume methods Eirik Keilegavlen, Jan Martin Nordbotten, and Ivar Aavatsmark Published in Applied Mathematics Letters, Vol. 22 (8), Pages 1178-1180, 2009 In Section 4.5, we gave background information on monotonicity for elliptic equations. Sufﬁcient conditions for monotonicity of control volume methods on quadrilateral grids were found in [177]. These criteria were derived by a global analysis, requiring inverse monotonicity of the entire system matrix. However, as stated in Section 4.5, the maximum principle is valid also on subgrids. The present paper considers monotonicity on subdomains consisting of no more than 3 cells. These considerations show that most of the criteria from [177] are necessary as well as sufﬁcient. The importance of the subdomains are reﬂected in a deﬁnition of a discrete maximum principle more precise than what stated in [177]. The main conclusion from this paper is that no linear nine-point control volume method can be constructed that satisﬁes a discrete maximum principle for all media and quadrilateral grids. Referring to the special case of parallelogram grids in homogeneous media shown in Figure 4.7(c), no methods can be monotone below the line b < a. 81 Paper B: Monotonicity for MPFA methods on triangular grids Eirik Keilegavlen and Ivar Aavatsmark Submitted to Computational Geosciences, April 2009 This paper attempts to establish monotonicity properties for MPFA methods on unstructured grids. The aim of the work was to follow up the results in [177] with a similar analysis as for quadrilateral grids. However, as the work in Paper F shows, the techniques applied to quadrilaterals come cannot provide sharp results for triangular grids. The topology of the grid makes splitting of the system matrix inappropriate. Therefore, it seems out of reach to obtain results for general methods on general grids based on a global analysis, i.e. which focus on properties on the entire grid. We therefore have to restore to considering speciﬁc methods, and rely heavily upon numerical experiments. For the special case of uniform triangular grids in homogeneous media, Paper B provides a characterisation of the monotonicity properties of some MPFA methods, namely the O(η)-method for η = 0, 1/3, and 0.5, and the L-method. The regions cover quite a large part of the entire parameter space, apart from the O(0)-method, which has considerably worse properties than the other methods. On genuinely unstructured grids, the failure to obtain sharp sufﬁcient monotonicity constraints limits the analytical results. Numerical simulations on several realisations of randomly perturbed grids are used to give an indication of the properties of the methods. Based on the insight from Paper A, special attention is given to subgrids composed by clusters of cells sharing a vertex. Indeed, for a genuinely unstructured grid, subgrids consisting of all but one cell sharing a vertex prove to yield the most restrictive monotonicity constraints. The main result of these test is that an MPFA discretisation on an unstructured grid cannot be expected to be monotone. The methods prove even more vulnerable when anisotropies are introduced in the permeability tensor. If an MPFA method still is to be applied, the O(1/3)-method seems to be the best choice. In the last part of the paper, simple two-phase experiments are performed. This serves as a test of the role of monotonicity for non-linear problems. Even though the pressure discretisation is not monotone, the saturation proﬁles produced bear no signs of irregularities. 82 Summary of the Papers Paper C: Non-hydrostatic pressure in sigma-coordinate ocean models. Eirik Keilegavlen and Jarle Berntsen Published in Ocean Modelling, Vol 28 (4), Pages 240-249, 2009 This paper explores computation of pressure in an ocean modelling setting. Numerical ocean modelling is computationally very expensive, and traditionally, many approximations are done to ease the computational burden. One of these simpliﬁcation is to neglect the non-hydrostatic pressure, stemming from dynamical processes. For coarse grids, this is a good approximation, however, it becomes questionable with the increasing grid resolutions now available. Processes such as mixing of water masses take place on a rather small scale, and is greatly inﬂuenced by non-hydrostatic pressure. Therefore accurate modelling of these effects is vital. For Cartesian grids, equations governing the non-hydrostatic pressure was given in [168]. These were transformed to the terrain-following σ-coordinate system in [132]. However, the σ-coordinate system is not orthogonal unless the bottom is ﬂat. Thus, the classical 5-point stencil used to solve elliptic equations is transformed into a 9-point stencil in terrain following coordinates. In three dimensions, we get a stencil of 15 points instead of 7 [132]. The increased bandwidth can result in a signiﬁcant overhead when solving the linear system. In [36], a new set of governing equations for the non-hydrostatic pressure was proposed, modelling the pressure directly in the non-orthogonal σ-coordinate system. This leads to a 5 or 7 point stencil in 2 and 3 dimensions, respectively. Essentially, the approach of [36] is to replace a multi-point approximation of the ﬂux with a two-point estimate. Paper C explores the differences between the two systems presented in [132] and [36]. It is shown that except from the trivial case of a ﬂat bottom (i.e. the grid is orthogonal) the two approaches are not equal. To investigate the differences by analytical means is out of reach because of the complexity of the systems of equations. Instead, the two approaches are explored by numerical experiments. The differences due to computation of the non-hydrostatic pressure are inﬂuenced by feedback from the rest of the simulation procedure. However, the difference in the primary variables obtained with the two approaches is of comparable size with other well-known sources of errors in ocean modelling. The test cases involve a broad range of physical regimes. Thus, in this application, it seems justiﬁable to use a TPFA-method instead of the more expensive multi-point estimates. 83 Paper D: Simulation of Anisotropic Heterogeneous Near-well Flow using MPFA Methods on Flexible Grids Sissel Mundal, Eirik Keilegavlen, and Ivar Aavatsmark Accepted for publication in Computational Geosciences. In this work, we explore different strategies for local grid reﬁnement in nearwell regions. The local grids are radial-like, and they (mostly) apply logarithmic reﬁnement towards the well to adapt to the expected behaviour of the pressure. The cells are both triangular and quadrilateral, and we test discretisation both in physical and reference space. Simple tests for a single well in an anisotropic medium show that a two-point ﬂux approximation is inadequate to get a convergent solutions; thus multi-point methods must be applied. Further, if the grid spacing in radial direction is not logarithmic, the convergence rate deteriorates. A comparison between the two different cell types shows that triangles yield a much smaller error than quadrilateral. Moreover, discretisation in physical space is sufﬁcient, the logarithmic behaviour of the solution is well captured by the adjustments in the grid. For a heterogeneous test case, the convergence rate is reduced, in agreement with ﬁnite element theory. If more than one well is present, or if the near-well grid do not cover the entire domain, a transition grid is needed in the zone between the near-well grids and the boundary of the domain. In this paper, we apply a triangulation algorithm to construct the transition grid. A test case with three wells indicate a smooth transition between the near-well region and the surrounding grid. To support the single phase test, we also do two-phase simulations for a horisontal well. There are only minor differences in saturation proﬁles and water cut curves for the different discretisation choices. In the two-phase simulation, there is a kink in the saturation in the transition from the near-well grid to the surrounding grid. However, the size of the kink is reduced as the grids are reﬁned. We also test the sensitivity of the solutions with respect to grid resolution in the near-well region. These simulations show that a fairly large number of cells is needed to capture the ﬂow accurately. 84 Summary of the Papers Paper E: Multi-dimensional upstream weighting on general grids for ﬂow in porous media Eirik Keilegavlen, Jeremy Kozdon, and Bradley Mallison Draft manuscript In Section 5.3 a framework for multi-dimensional upstream weighting was presented. The methods have only been formulated for Cartesian grids, though. This paper addresses extensions of the methodology to more general grids. The edge mobilities are deﬁned according to the description in Section 5.3. Since the grid no longer is Cartesian, the deﬁnition of edge parameters should be reconsidered. On uniform parallelogram grids, the linear advection problem can be studied by a modiﬁed equation analysis. The schemes considered for Cartesian grids seem to be fairly well behaved also on more general grids. The usefulness of the analysis is however limited due to a lack of understanding of how performance for linear problems are carried over to non-linear cases. Some possible guidelines for adapting the interpolation parameters to grid geometry are also given. Since the grids are no longer K-orthogonal, TPFA methods can no longer be applied. Due to the similarities between the multi-dimensional framework presented in Chapter 5 and MPFA methods, we apply the latter schemes to discretise the elliptic parts of the problem. An MPFA discretisation provides ﬂuxes for each half-edge. In general, the ﬂuxes over the two halves of an edge are not equal. The trace forward in the hyperbolic scheme can be based on either the half-edge ﬂuxes, or the total edge ﬂuxes. The latter option will, however, lead to a prohibitively large cell molecule, and we are forced to consider the half-edge ﬂuxes. To assess the applicability and robustness of the multi-dimensional schemes, numerical tests are performed on a series of grids. Both parallelogram grids and general perturbed quadrilaterals are considered, as well as triangular and dual grids. Overall, the new schemes reduce the bias in the solutions compared to SPU. When multi-dimensional upstream weighting is applied in combination with fully implicit time stepping, the number of non-linear iterations decreases signiﬁcantly. Chapter 7 Conclusions and Future Directions In this ﬁnal chapter of the background, we draw conclusions, and point out directions for further work. 7.1 Control Volume Methods for Elliptic Problems An major part of this work has been devoted to preservation of maximum principles for elliptic equations when constructing control volume discretisations. For quadrilateral grids, sufﬁcient and necessary conditions for monotonicity are now known. These allow us to tell whether a method is monotone based on the discretisation coefﬁcients. For unstructured grids, however, we have no means to tell if a discretisation fulﬁls the discrete maximum principle without inverting a large number of matrices. The main problem is a lack of understanding of which parameters control monotonicity. This issue should be investigated further. In principle, it should be possible to derive the sufﬁcient conditions for monotonicity found in [177] starting with subdomains similar to those studied in Paper A. If such a link between the local and global criteria can be found, this may also inform the analysis for triangular grids. In real ﬁeld applications, much of the computational efforts are spent on solving non-linear, and thereby linear systems. Development of fast solvers for MPFA methods is of major importance for the schemes to gain popularity outside academia. Of special importance is cases where the system matrix is not an M-matrix. This is again related to monotonicity issues. Another open issue regarding MPFA methods is to formulate a proper mathematical framework for analysis of the methods. So far, convergence analysis has 86 Conclusions and Future Directions been based on relations between MPFA methods and MFE or MFD discretisations. However, there are indications that the estimates obtained in this way are too conservative, see e.g. [139]. Further research is thus needed to know when the methods are applicable. As we have seen in Chapter 4, the transmissibilities in control volume methods are determined by both the permeability ﬁeld, and the cell geometry. Hence, the methods may be improved by appropriate adjustments in the grid. This way of thinking has led to development of state of the art triangulation algorithms that are optimised with respect to convergence of ﬁnite element simulations. In the same way, grid generation algorithms that pay special attention to control volume methods should be explored. An example of grid optimisation with respect to monotonicity constraints can be found in [169]. A more radical option is to put more weight on discretisation techniques when the geo-model is built and upscaled. Today, grids are constructed with representation of geology in mind, more than discretisation of ﬂow equations. However, to change this would require a major change in the work ﬂow, which might be difﬁcult to achieve in practise. Control volume methods are of great interest for industrial applications due to their the local ﬂux expressions and applicability to general grids. In Chapter 4, we mentioned three methods or classes of methods; TPFA, non-linear TPFA, and MPFA. Of these, TPFA is not consistent, and will in many cases not be sufﬁcient to capture the ﬂow pattern accurately. MPFA methods will often render a better representation of the ﬂow, however, they suffer from spurious oscillations in the pressure solution. The non-linear TPFA requires iterations on the numerical scheme even for linear problems. Methods with larger cell stencil can be expected to have better properties due to their greater ﬂexibility, but to the price of an increased computational cost when solving linear systems. The question then becomes how to deal with problems where we cannot preserve both convergence, monotonicity, and linearity of the solution scheme. In some cases, we may be forced to let go of one of the properties, thus guidelines for what to do should be developed. Another possibility is to apply ’hybrid’ methods, that attempt to achieve a compromise between robustness and accuracy, see [149] for an example. Which method is best, or more generally, which schemes provide reasonable results, is probably case dependent, and theoretical analysis of simpliﬁed cases can only give limited insight. It may be necessary to perform systematic studies of the performance of control volume schemes for elliptic problems under different physical circumstances, as was done in Paper C. Due to the broad range in processes of interest (cf Chapter 1), this is an enormous task. 7.2 Simulation in Near-Well Regions 7.2 87 Simulation in Near-Well Regions In Paper D, we considered grids and discretisations adapted to the ﬂow pattern in near-well regions. The single phase simulations indicated that the near-well grid should be triangular, and that the transmissibilities can be computed in physical space, provided the grid has logarithmic spacing in the radial direction. Further, the methods are applicable also for multiphase ﬂow. However, the presented numerical simulations are far from realistic, thus there is a long way to go before the methods can be applied to real cases. We now discuss two possible bottlenecks; extensions to three dimensions and reduction of the computational cost of the methods. A non-conventional well can have several branches, each of which can be horisontal or skew. The well can also perforate multiple geological layers. To create a grid that adapts to the logarithmic pressure variation and also honour the main geological features near the well is a challenging task. The results in Paper D suggest that the grid should be based upon triangles, possibly by applying a prismatic extension to three dimensions. However, creating a 2 1/2-D grid that can adapt to a realistic well trajectory and reservoir geometry will be difﬁcult. On the other hand, simulations will be computationally more expensive on tetrahedral grids than on prismatic extensions of triangles. Also, the coupling between the near-well grid and the global grid becomes much more complicated when going from 2D to non-conventional 3D wells. Some ideas on how this can be done can be found in e.g. [34, 94, 124]. Even though the two-phase experiments in Paper D were far from realistic, a fairly large number of grid cells was needed near the well to capture the dynamics somewhat accurately. For a realistic ﬁeld, with at least tens of (non-conventional) wells, each with a length of hundreds of meters, the number of grid cells needed can be severely high. Moreover, to enhance recovery, wells are frequently opened and shut down. To apply local grid reﬁnement around a well that has been closed for a while may be a waste of computational resources. These topics need to be addressed before the techniques developed in Paper D are applicable to ﬁeldcases. One possible solution is to apply a local time stepping in the near-well region. The mapping between the local and global grid can be handled by the windowing technique presented in [70, 170]. However, conservation of mass in the mapping might be an issue. Another option would be to use the ﬁne-scale solution obtained by the methods presented in Paper D as a basis for upscaling to a coarser grid. This will be similar to the upscaling techniques mentioned in Section 4.6.2. An alternative to compute a well index based on upscaling might be to apply the MSFV method presented in Section 4.7.3 Treatment of wells in the MSFV method is based on introducing an extra basis function to capture the singularity 88 Conclusions and Future Directions in the solution, see [164, 231]. Another possible strategy is to consider the MSFV method as a domain decomposition method as outlined in [176], and assign a separate subdomain for each near-well region. This can allow for a good resolution of ﬂow in the vicinity of the well, and still ease the computational burden of solving the pressure equation for large system matrices. Similar considerations can be found in [21]. 7.3 Multi-Dimensional Upstream Weighting As we have seen in this thesis, there has been a considerable effort to construct robust and accurate schemes for elliptic equations for use in reservoir simulation. In the same way, it is natural to consider more advanced transport schemes that can be applied within today’s framework. The multi-dimensional upstream methods introduced in Chapter 5 can be considered analogous to MPFA methods for hyperbolic equations. Since the upstream schemes are based on a control-volume methodology, they ﬁt naturally into mainstream simulators, and they might yield improvements in accuracy as well as performance. However, several aspects to the methods should be further investigated. In this work, we have only considered two-phase problems. Grid orientation effects are especially severe for adverse mobility ratios. These situations often arise when heavy oil is produced with the help of EOR-techniques. During tertiary production, compositional modelling is appropriate. Therefore, the framework introduced in Chapter 5 should be extended to three phases and compositional ﬂow. Several challenges can be expected for such extensions. The distribution of masses in a two-phase problem can be described by a single variable that fulﬁl a comparison principle. This motivates us to require monotonicity of numerical schemes. For three-phase or compositional problems, we do not have such a property. Therefore, it is not clear what a robust and rigorous scheme should be based on. We have only considered FIM and IMPES time stepping strategies. For a compositional problem, the high number of unknowns makes fully implicit methods too expensive, while the IMPES time step restriction may be severe. Therefore, more ﬂexible methods should be considered. One natural candidate is an adaptive implicit scheme. For AIM, it is an interesting question what are appropriate primary variables; saturations or component masses. Other options are asynchronous time stepping, or maybe multiscale methods. Except from very simpliﬁed cases, the strength of the non-linearities in the pressure equation mean only an approximate solution can be found, and thus the resulting velocity ﬁeld will be inexact. As an consequence, if volume-related variables (i.e. saturations) are transported according to the velocity ﬁeld, unphysical 7.3 Multi-Dimensional Upstream Weighting 89 values may arise. In Chapter 5 and Paper E, we circumvented this problem by using mass variables instead of saturations. The incompressible two-phase simulations performed in [141] and Paper E show no sign of problem with this approach. However, it is an open question how the volume relaxed framework will affect more difﬁcult physical problems such as large differences in component densities, highly compressible ﬂuids, and phase equilibrium calculations. We emphasis that the inexactness of the pressure solution must be treated somehow; the volume relaxed framework has proven to be robust and provide physical values for the mass variables for the tests performed so far. The upstream strategies mainly considered in this work are only ﬁrst order accurate. It is thus natural to consider higher order methods in space, that is, to treat hyperbolic problems with the same order of accuracy as the elliptic equations. A higher order reconstruction of data would require a cell stencil that is larger than the scheme presented in Chapter 5. The reconstruction should be done in a way that minimise bias with respect to grid orientation, furthermore, spurious oscillations should be avoided. All investigations done so far are for two-dimensional ﬂow. An extension of the trace forward methodology to three dimensions should be possible. In practice, however, the hexahedral cells which are prevailing in reservoir grids, most often follow the geological layering, and thus be thin in the vertical direction. As a consequence, all the schemes considered in this work become very similar to the standard SPU-method for transport orthogonal to the layering. The reason why the cells are long and thin is that we expect the ﬂow to a large extent to be conﬁned within the layers. Thus, for hexahedral cells it might be sufﬁcient to apply the multi-dimensional schemes to transport aligned with the geological layers. For other cells, i.e. tetrahedral and their duals, further investigations are needed. On the theoretical side, there is a lack of understanding of what are the crucial properties for solving non-linear problems. For linear problems, the TMU scheme [197] is ideal, whereas this is not the case for non-linear transport. A better understanding can hopefully also help construction of limiters for more general grids. 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