A statistical structural model was used in the context of local volume averaging to examine the effects of interfacial tension, the interfacial viscosities, and wetting upon the capillary pressure and relative permeability functions in an unsteady-state displacement. The statistical structural model is a highly simplified idealization of the local pore structure, in which pores are randomly oriented in pore structure, in which pores are randomly oriented in space, but no pore interconnections are recognized.
Whether we are concerned with an imbibition or a drainage, it is not important to distinguish between steady-state and unsteady-state relative permeabilities when the local capillary number Nca (r) 1. Unless the interfacial tension is very small everywhere or the overall pressure gradient very large, it may be satisfactory to use steady-state representations for the relative permeabilities everywhere except in the immediate neighborhood of a displacement front. The local capillary number Nca will always be largest in the immediate neighborhood of the displacement front. Within the immediate neighborhood of a displacement front, simulations, especially of inhibitions, should employ unsteady-state relative permeabilities.
The configuration of the pore space in a permeable rock, in a bed of sand, or in an irregular bed of spheres will normally be at least in part a random function of position in space. Although the usual equations of motion are believed applicable to each phase moving through an individual pore, they can not phase moving through an individual pore, they cannot be solved, because an a priori description of the necessary boundary conditions is not possible.
The usual approach in avoiding this difficulty is to speak in terms of averaged variables. Formally, local volume averages of the equations of motion for each phase can be written that are valid for each point within a multiphase flow through a permeable structure. The advantages are that all local volume averaged variables are continuous functions of position in space and that detailed configurations for all of the various phase interfaces are no longer required.
The disadvantage is that in place of this lost information we have several integrals, descriptions for which must be given before we can proceed further. For a two-phase flow through a proceed further. For a two-phase flow through a porous medium in the absence of interphase mass porous medium in the absence of interphase mass transfer, these integrals are the permeability to a single phase, the relative permeability for each phase, and the capillary pressure. phase, and the capillary pressure. In describing these integrals, one can rely strictly on empiricism or correlations of experimental data. This is not entirely satisfactory, since there are many interacting physical phenomena that are difficult to distinguish phenomena that are difficult to distinguish without a theoretical structure upon which to base data correlations.
Lin and Slattery (1982) adopted another approach in which these integrals are model led through the introduction of an idealized structural model for the pore space bounded by the local averaging surface defined at each point within the porous medium. As their idealized model, they adopted a three-dimensional, randomized network of sinusoidal pores. At each node in the network, they required mass conservation and continuity of pressure for each phase. A beta probability density function was phase. A beta probability density function was used to describe the distribution of the pore neck radii.
There have been many other structural models described in the literature, but, although in retrospect we can make the connection, none were developed in the context of local volume averaging. When the corresponding capillary pressure and relative permeability functions were pressure and relative permeability functions were computed, the implication was that these were integral averages for some macroscopic region such as a sandstone core or a tube packed with beads rather than the pointwise properties required in the local volume averaged equations of motion.