The chapter starts by explaining how petroleum reservoirs are formed and gives a brief introduction to various concepts from geology to non-geologists. Next, we discuss the continuum hypothesis and how flow through subsurface porous media is modeled on different spatial scales. An essential part is to develop a description of petrophysical properties like porosity and permeability. We explain how this is achieved in MRST, and outline a few examples of models that give realistic representations of reservoir rocks. This includes the popular SPE10 benchmark and a model of a shallow-marine formation.

The preface briefly describes the purpose of the book and how it differs from many other books.

The chapter explains the need for modeling subsurface flow to solve important societal challenges. We introduce the basic processes involved in primary, secondary, and tertiary petroleum recovery, and explain the ingredients used in reservoir simulation. Finally, we outline the scope of the book and introduce the companion software MRST, which is used widely throughout.

The two-point flux-approximation (TPFA) scheme is robust in the sense that it generally gives a linear system that has a solution regardless of the variations in K and the geometrical and topological complexity of the grid. The resulting solutions will also be monotone, but the scheme is only consistent for certain combinations of grids and permeability tensors K. This implies that a TPFA solution will not necessarily approach the true solution when we increase the grid resolution. It also means that the scheme may produce different solutions depending upon how the grid is oriented relative to the main flow directions. In this chapter, we first explain the lack of consistency for TPFA, before we introduce a few consistent schemes implemented in MRST, including the mimetic finite-difference method and one example of a multipoint flux approximation method (MPFA-O). These can all be written on a general mixed hybrid form, which is motivated by mixed finite-element methods. We explain how you can specify different methods that reduce to known methods on simple grids by adjusting the inner product in the mixed hybrid formulation.

This chapter presents flow diagnostics methods you can use to delineate volumetric communications and improve your understanding of how flow patterns in the reservoir are affected by geological heterogeneity and respond to engineering controls. Using these methods, you can answer questions such as: to what region does a given injector provide pressure support? Which injection and production wells are in communication? Which parts of the reservoir affect this communication? How much does each injector support the recovery from a given producer? Do any of the wells have backflow? What is the sweep and displacement efficiency within a given drainage, sweep, or well-pair region? Which regions are likely to remain unswept? Flow diagnostics also provide several measures of the dynamic heterogeneity of a reservoir model, i.e., the variation in flow paths and their associated travel or residence times. We present several examples that demonstrate how you can use flow diagnostics to analyze interwell communication, improve well placement and sweep efficiency, and pre- and postprocess multiphase flow simulations.

Generating a coarser volumetric description of the reservoir rock is a common task in reservoir engineering. This chapter discusses how to partition a fine grid model into a smaller set of coarse blocks. After the partition, the coarse blocks will each consist of a finite collection of cells from the underlying fine model. Through a series of examples, we demonstrate a variety of different partition methods. Whereas the simplest methods only utilize the geometry or topology of the grid, the more advanced methods can compute partitions that adapt to petrophysical properties, fluid contacts, flow fields, near-well regions, or underlying geological properties like depositional environments, flow units, rock types, etc.

The chapter introduces you to mathematical modeling of flow in porous media. We start by explaining Darcy's law, which together with conservation of mass comprises the basic models for single-phase flow. We then discuss various special cases, including incompressible flow, constant compressibility, weakly compressible flow, and ideal gases. We then continue to discuss additional equations required to close the model, including equations of state, boundary and initial conditions. Flow in and out of wells take place on a smaller spatial scale and is typically modeled using special analytical submodels. We outline basic inflow–performance relationships for the special cases of steady and pseudo-steady radial flow, and develop the widely used Peaceman well model. We also introduce streamlines, time-of-flight, and tracer partitions that all can be used to understand flow patterns better. Finally, we introduce basic finite-volume discretizations, including the two-point flux approximation method, and show how such schemes can be implemented very compactly in MATLAB if we introduce abstract, discrete differentiation operators that are agnostic to grid geometry and topology.

A reservoir simulator consists of a large set of models and parameters to describe the geology, fluid behavior, wells and surface facilities, and rules and conditions that describe and control the production process. On the numerical side, you have a combination of discretization methods, nonlinear solvers, linearizations, linear solvers, preconditioners, stability and convergence checks, and algorithms for automated time-step selection. The AD-OO framework in MRST is developed to encapsulate all these details, such that on one side it offers industry-grade simulator capabilities, and on the other side a flexible framework for rapid development of new proof-of-concept implementations. This chapter outlines the design philosophy behind AD-OO and explains many of its ingredients in detail. By reading this chapter, you will get an in-depth introduction to all the details you need to make a full-fledge reservoir simulator. The chapter ends with a discussion of three cases: a simple pressure depletion, multisegment representation of instrumented wells, and the full SPE 9 benchmark.

The chapter explains how you can generate grid models to represent subsurface reservoirs. We outline a number of elementary grid types: structured/rectilinear grids, fictitious domains, Delaunay triangulations, and Voronoi grids. We then explain stratigraphic grids that are commonly used to model real subsurface formations, including in particular corner-point and perpendicular bisector (PEBI) grids. We explain how such grids are represented in MRST using a data structure for general unstructured grids, and we discuss how to compute geometric properties like volumes, face areas, face normals, etc. We end the chapter by presenting an overview of alternative gridding techniques, including composite grids, multiblock grids, and control-point and boundary conformal grids.

The black-oil equations constitute the industry-standard approach to describe compressible three-phase flow. Black-oil models generally have stronger coupling between fluid pressure and the transport of phases/components than the two-phase, incompressible flow models discussed in the previous chapter. For this reason it is common to use a fully coupled solution strategy, in which the whole system of equations is discretized implicitly and all primary unknowns are solved for simultaneously. This chapter introduces you to the underlying physics and describes the various rock-fluid and PVT properties that enter these models, like formation-volume factors, dissolution and vaporization ratios, bubble-point pressures, saturated and undersaturated states, etc. We also explain the basics of how the resulting models are discretized and implemented in MRST. Our implementation will rely heavily on the discrete operators discussed earlier in book. We end the chapter simulating the SPE 1 benchmark case in MRST and a discussion of limitations and potential pitfalls for black-oil models.

The chapter discusses numerical discretization of first-order quasilinear hyperbolic PDEs, so-called conservation laws. We start by briefly reviewing some of the theory for these equations, including weak solutions, discontinuities, and entropy conditions. We then present a general family of conservative finite-volume methods that includes centered as well as upwind and Godunov-type schemes. We demonstrate typical deficiencies in classical schemes including smearing of discontinuities and creation of nonphysical oscillations. We end the chapter by presenting the implicit, upstream-mobility scheme, which is the most widespread method in reservoir simulation.

This chapter explains how you can discretize the basic equations for single-phase, compressible flow by use of the discrete differential and averaging operators introduced in Chapter 4. These operators enable you to implement the flow equations in a compact form similar to the continuous mathematical description. By using automatic differentiation, you can automatically linearize and assemble the corresponding linear system without having to explicitly derive and implement expressions for partial derivatives in the Jacobian matrix. The combination of discrete operators and automatic differentiation with a flexible grid structure, a highly vectorized and interactive scripting language, and a powerful graphical environment, is the main reason MRST has proven to be an efficient tool for developing new proof-of-concept codes. To demonstrate this, we first develop a compact solver for compressible flow, and then extend the basic single-phase model to include pressure-dependent viscosity, non-Newton fluid behavior, and temperature effects.