A summary of classical concepts from thermodynamics concerning phase transitions and a survey of phase transitions for a variety of thermotropic and lyotropic systems. The Landau theory for first- and second-order transitions applied to nematics and smectics is discussed in detail. Classic lattice models (Ising, Heisenberg, XY, Lebwohl–Lasher, …) and critical exponents, often associated with certain liquid crystals transitions, are introduced. Examples of experimental phase diagrams for thermotropic and lyotropic liquid crystals are shown.

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.