Relations Between Seepage Velocities in Immiscible, Incompressible Two-Phase Flow in Porous Media

Abstract

Based on thermodynamic considerations we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.

Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media

“…A recent work [21] has explored a new approach to immiscible two-phase flow in porous media based on elements borrowed from thermodynamics. That is, it is using the framework of thermodynamics, but without connecting it to processes involving heat.…”

“…The Buckley-Leverett equation is based solely on the principle of mass conservation and on the fractional flow rate being a function of the saturation. In the approach of Hansen et al [21], equations are derived that originate from Euler homogeneity as in ordinary thermodynamics. These equations transcend the details of the physics involved in the same way that the equations of thermodynamics are universally applicable as long as a set of simple underlying conditions are met.…”

“…It is the aim of this paper to formulate a description of immiscible two-phase flow in porous media that may form a link between the continuum-level approach of Hansen et al [21] and the pore-level description of the problem -a sort of "statistical mechanics" from which the pseudo-thermodynamics may be derived, but which also describes the flow problem at the pore level. After defining the system and the variables involved in Section II, we will in Section III review the pseudothermodynamic approach [21]. The next Section IV we introduce the central object in the paper, the differential transversal area distribution which corresponds to the Boltzmann distribution in ordinary statistical mechanics, and relate it to the pseudo-thermodynamics relations.…”

Pore Network Modeling of the Effects of Viscosity Ratio and Pressure Gradient on Steady-State Incompressible Two-Phase Flow in Porous Media

“…However, other quantities could also, in principle, be used to convey the same information. One example is the velocities presented in[17].…”

Effective Rheology of Two-Phase Flow in a Capillary Fiber Bundle Model

“…These transient flow patterns were modeled by invasion percolation [8] and diffusion limited aggregation (DLA) models [9]. When steady state sets in after the initial instabilities, the flow properties in are characterized by relations between the global quantities such as flow rate, pressure drop and fluid saturation [10,11]. It has been observed theoretically and experimentally that, in the regime where capillary forces compete with the viscous forces, the two-phase flow rate of Newtonian fluids in the steady state no longer obeys the linear Darcy law [12,13] but varies as a power law with the applied pressure drop [14][15][16][17].…”