Simulation of Viscous Fingering in Rectangular Porous Media with Lateral Injection and Two- and Three-Phase Flows

Abstract

Understanding multiphase flow in porous media is of tremendous importance for many industrial and environmental applications at various spatial and temporal scales. The present study consequently focuses on modeling multiphase flows by the volume-of-fluid method (sharp interface) in porous media with a simplified Darcy-scale approach and shows simulations of Saffman–Taylor fingering. The simplification of the Darcy-scale approach is performed by assuming sharp interfaces between pure phases. The volume-of-fluid method with octree mesh refinement is used. It is implemented in the Gerris (Popinet in J Comput Phys 190(2):572–600, 2003. doi: 10.1016/S0021-9991(03)00298-5 ; J Comput Phys 228(16):5838–5866, 2009. doi: 10.1016/j.jcp.2009.04.042 ) code which allows efficient parallel computations. We measure the scaling properties of the fractal viscous fingering patterns that appear in the numerical simulations. One of these properties is the fractal or Hausdorff dimension $$D_\mathrm{F}$$ . The other is the variation of the area A of the viscous fingering cluster with the length L of its perimeter, which varies as a simple power law $$A \sim L^{\alpha }$$ . The injection of an intermediate-viscosity Newtonian fluid as a second step is also simulated. We are thus able to observe an increase of recovery of the high-viscosity fluid behind the fingering front, due to the reduction of the viscosity contrast. Some of these results are compared to waterflooding experiments of extra-heavy oils in quasi-2D square slab geometries of Bentheimer sandstone.