We are now ready to introduce magnetic fields, which are generated by electrical currents and which apply forces on moving charges and current-carrying wires. Historically, magnetic effects in lodestones, an iron ore that can be magnetized, have been known for a long time. The first magnetic compasses date back to about 1000 BCE, and the ancient Chinese are believed to have used such devices for navigation as early as 1100 CE. The properties of magnetic fields can be derived from a number of observations of magnetic effects that have been recorded over many years. One of the earliest such observations, by Hans Christian Oersted in 1820, was that a current-carrying wire exerts a torque on a permanent magnet (such as a compass). Current-carrying wires can also exert forces on each other, as first observed by Biot and Savart and more fully characterized by Ampère. Finally, beams of charged particles, such as electrons in a cathode ray tube (see TechNote 3.4), are deflected when in the presence of current-carrying wires. Each of these phenomena can be described quantitatively in terms of a magnetic field produced by current distributions, as we will discuss throughout this chapter.

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.