Geometric structure of parameter space in immiscible two-phase flow in porous media

In a recent paper, a continuum theory of immiscible and incompressible two-phase flow in porous media based on generalized thermodynamic principles was formulated (Transport in Porous Media, 125, 565 (2018)). In this theory, two immiscible and incompressible fluids flowing in a porous medium are treated as a single effective fluid, substituting the two interacting subsystems for a single system with an effective viscosity and pressure gradient. In assuming Euler homogeneity of the total volumetric flow rate and comparing the resulting first-order partial differential equation to the total volumetric flow rate in the porous medium, one can introduce a novel velocity that relates the two pairs of velocities. This velocity, the co-moving velocity, describes the mutual co-carrying of fluids due to immiscibility effects and interactions between the fluid clusters and the porous medium itself. The theory is based upon general principles of classical thermodynamics and allows for many relations and analogies to draw upon in analyzing two-phase flow systems in this framework. The goal of this work is to provide additional connections between geometric concepts and the variables appearing in the thermodynamics-like theory of two-phase flow. In this endeavor, we will encounter two interpretations of the velocities of the fluids: as tangent vectors (derivations) acting on functions or as coordinates on an affine line. The two views are closely related, with the former viewpoint being more useful in relation to the underlying geometrical structure of equilibrium thermodynamics and the latter being more useful in concrete computations and finding examples of constitutive relations. We apply these relatively straightforward geometric contexts to interpret the relations between velocities and, from this, obtain a general form for the co-moving velocity.