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.

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.