Nonequilibrium transport equations are derived for two types of diffusive systems: (1) viscous fluids made of a single molecular species that support thermal flux and (2) two-component (solute and solvent) miscible fluids that support solute flux and thermal flux. The general statement of energy conservation for any viscous fluid is derived and used to obtain the statement of entropy conservation for each system type. This identifies the irreversible entropy production of each system, which in turn produces linear transport laws relating the nonequilibrium diffusive flux to the gradients in the intensive parameters. The matrix of transport coefficients in the transport laws is proven to be symmetric (Onsager symmetry) using the continuum governing equations and requires the direction of flow to be reversed to obtain symmetry. Capillary physics is treated using Cahn–Hilliard theory that resolves the gradients in concentration across transition layers separating two immiscible, or partially miscible, fluid. The rules of contact-line movement (imbibition and drainage) in conduits are derived from a more macroscopic perspective where the transition layers are modeled as sharp interfaces.

This chapter subdivides the hydrocarbon migration into primary, secondary, and tertiary migrations. These are described as a multiphase fluid flow driven by petroleum fluid potential gradients. The primary migration represents the release of generated hydrocarbon molecules from the kerogen matrix when the sorptive capacity of the matrix is exceeded, often called expulsion by pressure-driven movement through the source rock matrix and transient microfractures. In the case of oil, the secondary and tertiary migrations represent a longer-range flow from source rock to reservoir and remigration from one accumulation to another, respectively. It takes place through a combination of carrier beds, faults, and fractures driven by the balance between fluid potential gradients that are created by buoyancy force, hydraulic gradient, capillary pressure and frictional resistivity force. Description of each force contains mathematical formulations. The secondary migration is described as including separate phase flow, diffusion, solution, and dissolution of gas in oil and water and chemical cracking. The discussion is supported by case studies from the literature.

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.