This chapter introduces the basic equations used to describe multiphase flow. It also introduces key concepts such as saturation, wettability, relative permeability, and capillary pressure. Combining the multiphase extension of Darcy's law with mass conservation of fluid phases or chemical components gives a system of parabolic PDEs. The chapter derives the so-called fractional flow formulation and discusses several special cases of two-phase flow equations. The chapter ends with a discussion of various analytical and semi-analytical 1D solutions, including the classical Buckley–Leverett problem.

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.