Numerical study of liquid-liquid displacement in homogeneous and heterogeneous porous media

Immiscible liquid-liquid displacement is a fundamental problem in fluid mechanics and has many applications like, for instance, enhanced oil recovery. One of the most valuable methods to increase oil recovery is injecting a fluid into the reservoir to displace the oil, and the primary motivation of this work is to evaluate the liquid-liquid displacement in homogeneous and heterogeneous porous media. We carried out a direct numerical simulation of liquid-liquid displacement in three different porous media: medium I, II, and III. The medium I is homogeneous. Medium II is heterogeneous, in which the diameter of the spheres reduces from the bottom to the top boundary. Finally, in medium III, the diameter of the spheres is randomly distributed all over the domain. The three media were composed of spheres and had the same mean porosity. We tracked the liquid interface front over time until the breakthrough to compute the mass entrapped as a function of viscosity ratio and capillary number. The sweep efficiency increases when the displacing liquid viscosity also increases. The role played by the capillary number is more complex. In medias I and II, the entrapped mass rises with the capillary number. For a specific condition in medium III, when the displacing liquid is more viscous than the displaced one, the opposite can occur, i.e., the entrapped mass decreases with the capillary number. Beyond the capillary number and viscosity ratio, the sweep efficiency strongly depends on the kind of medium. Mainly, the entrapped liquid is smaller in the homogeneous medium.