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.

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.