In a continuum modeling approach of a multiphase flow, each transport phase is regarded as an individual pseudo-continuum fluid and all these “fluids” co-share the same space and time domains. Chapter 6 delineates the volume-averaging method to construct the pseudo-continuum fluids over which the volume-averaged Eulerian modeling approach is developed. The key concepts and formula of volume-averaged continuum modeling include the definitions of intrinsic and phase averages and their relationship; the volume-averaging theorems; the general form of volume-averaged transport equations and the individual equations of mass, momentum, and energy; the volume and mass balance conditions of all phases; the constitutive relations of the volume-averaged tensors by individual volume-averaged parameters; and the formulation of interfacial transport between phases. The effect of turbulence on phase transport is also handled via Reynolds decomposition and time-averaging over the volume-averaged equations. The detailed formulation of various turbulent transport coefficients and the salient behavior of the turbulence modulation from various interactions among phases are presented.

The second chapter deals with the fundamental of continuum mechanics. It describes stresses and strains in a continuum. This is followed by fundamental laws of mechanics and constitutive relations. Elasticity and Hooke's law are described. Next, the chapter describes poroelasticity as well as poroplasticity, followed by fracture mechanics. Finally, rock physics models as applied to pore pressure detection are described.