This chapter teaches you how to simulate incompressible, two-phase flow using a sequential formulation that splits the equation system into an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. We discuss fluid objects, the sequential solution procedure, and explicit and implicit transport solvers in some detail. The second part of the chapter is devoted to a number of simulation examples that highlight typical flow behavior. Examples include gravity segregation, homogeneous quarter five-spots, heterogeneous quarter five-spots with viscous fingering, and buoyant migration of CO2 in a sloping aquifer. Furthermore, we discuss water coning, gravity override, capillary fringes, and a simplified simulation of the Norne field model. We end the chapter by a discussion of various sources of numerical errors, including splitting and grid-orientation errors.

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 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 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.

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 Appendix gives you a quick introduction to MRST. We start by explaining how to download, install, and get started using the software. We also discuss the release policy, terms of use, and how the software has been organized into a small core that offers basic functionality and a large set of add-on modules that implement specific solvers, simulators, and tools. We briefly review all modules that are part of the 2018a release. The final two sections discuss rapid prototyping using MATLAB and MRST, present some powerful constructs you can use to speed up your MATLAB code, and explain the key ideas of automatic differentiation and how it has been implemented in MRST. Altogether, this Appendix should provide you with enough knowledge of MRST so that you can effectively use the software to further your understanding of the material presented in the main part of the book.