Non-Darcian Displacement of Oil by a Micellar Solution in Fractal Porous Media

A Buckley–Leverett analysis with capillary pressure to model the oil displacement in fractal porous media is herein presented. The effective permeability for a non-Newtonian micellar fluid is calculated by a constitutive equation used to describe the rheological properties of a displacement fluid. The main assumption of this model involves a bundle of tortuous capillaries with a size distribution and tortuosity that follow fractal laws. The BMP model predicts two asymptotic (Newtonian) regions at low and high shear and a power-law region between the two Newtonian regions corresponding to a stress plateau. Both the stress at the wall and the fluidity are calculated using an imposed pressure gradient in order to determine the mobility of the solution. We analyze different mobility ratios to describe the behavior of the so-called self-destructive surfactants. Initially, the viscosity of the displacing fluid (micellar solution) is high; however, interactions with the porous media lead to a breakage process and degradation of the surfactant, producing low viscosity. This process is simulated by varying the applied pressure gradient. The resulting equation is of the reaction–diffusion type with various time scales; a shock profile develops in the convective time scale, as in the traditional Buckley-Leverett analysis, while at longer times diffusion effects begin to affect the profile. Predictions include shock profiles and compressive waves. These results may find application when selecting surfactants for enhanced oil recovery processes in oilfields.