Two-Phase Flow: Structure, Upscaling, and Consequences for Macroscopic Transport Properties

In disordered porous media, two-phase flow of immiscible fluids (biphasic flow) is organized in patterns that some times exhibit fractal geometries across a range of length scales, depending on the capillary, gravitational, and viscous forces at play. These forces, as well as the boundary conditions, also determine whether the flow leads to the appearance of fingering pathways, i.e., unstable flow, or not. We present a short review of these aspects, focusing on drainage and summarizing when these flows are expected to be stable or not, what fractal dimensions can be expected, and in which range of scales. We based our review on experimental studies performed in two-dimensional Hele-Shaw cells or addressing three-dimensional porous media by use of several imaging techniques. We first present configurations in which solely capillary forces and gravity play a role. Next, we review configurations in which capillarity and viscosity are the main forces at play. Eventually, we examine how the microscopic geometry of the fluid clusters affects the macroscopic transport properties. An example of such an upscaling is illustrated in detail: for air invasion in a monolayer glass-bead cell, the fractal dimension of the flow structures and the associated scale ranges depend on the displacement velocity. This controls the relationship between saturation and the pressure difference between the two phases at the macroscopic scale. We provide in this case expressions for dynamic capillary pressure and residual fluid-phase saturations.