1 Introduction

[page 141, §1]   
[141.1.1] A predictive macroscopic theory of two phase fluid flow inside a rigid porous medium is a longstanding and unsolved problem in statistical and computational physics of fluids, soft matter, and disordered systems [1, 2, 3, 4, 5, 6]. [141.1.2] Describing and predicting the flow of two immiscible and incompressible fluids through a complex disordered geometry is not only of interest for statistical physics, but also of significant practical importance for many applied sciences such as hydrology, petroleum engineering and other applied fields [7, 8, 9, 10].

[141.2.1] Many authors have proposed microscopic models (e.g. network models) to predict macroscopic immiscible displacement in porous media. [141.2.2] An important motivation for these investigations are the longstanding unsolved problems with the traditional macroscopic equations that are based on the concepts of capillary pressure and relative permeabilities (see e.g. [7, 9]). [141.2.3] In particular, one fundamental problem is the nonuniqueness of the capillary pressure as function of saturation. [141.2.4] Other problems with the capillary pressure are its hysteresis, process dependence and dynamic effects such as dependence on velocities or rates of saturation change. [141.2.5] Residual saturations are not constant parameters as it is assumed in the traditional macroscopic theory. [page 142, §0]    [142.0.1] Experimental observations show instead that residual saturations vary as functions of position and time. [142.0.2] Most practitioners ignore these problems and continue to use the traditional set of equations, and many physicists are following their lead by calculating effective relative permeabilities pertaining to the traditional theory.

[142.1.1] Droplets or ganglia of one fluid phase hinder the flow of the other fluid phase. [142.1.2] Experimental observations of this basic phenomenon abound [11, 12, 13, 14, 15, 16, 17, 18]. [142.1.3] It is therefore surprising that the importance of hydraulic percolation for theoretical modeling of two phase flow seems to have remained unnoticed until recently [19, 20, 21, 22, 23, 24, 25].

[142.2.1] Given that a basic constitutive approach to hydraulic percolation for macroscopic capillarity has been presented in [22, 23, 24] my objective in this paper is to extend that basic approach towards a more realistic description. [142.2.2] Let me begin the discussion with the mathematical ingredients of the hydraulic percolation approach. [142.2.3] One ingredient is the introduction of four phases instead of two, a second are balance laws for volume mass and momentum, and the third are the constitutive assumptions. [142.2.4] It is the third ingredient where extensions and modifications of the assumptions in [22, 23, 24] will be made. [142.2.5] An extensive discussion of the modified constitutive assumptions as well as their consequences and comparison to previous theories cannot be presented due to the page limit. [142.2.6] More details will be given elsewhere.