An engineering approach to study the effect of saturation-dependent capillary diffusion on radial Buckley-Leverett flow

Abstract

1D water oil displacement in porous media is usually described by the Buckley-Leverett equation or the Rapoport-Leas equation when capillary diffusion is included. The rectilinear geometry is not representative for near well oil displacement problems. It is therefore of interest to describe the radially symmetric Buckley-Leverett or Rapoport-Leas equation in cylindrical geometry (radial Buckley-Leverett problem). We can show that under appropriate conditions, one can apply a similarity transformation (r, t) → η= r²/ (2 t) that reduces the PDE in radial geometry to an ODE, even when capillary diffusion is included (as opposed to the situation in the rectilinear geometry (Yortsos, Y.C. (Phys. Fluids 30(10),2928–2935 1987)). We consider two cases (1) where the capillary diffusion is independent of the saturation and (2) where the capillary diffusion is dependent on the saturation. It turns out that the solution with a constant capillary diffusion coefficient is fundamentally different from the solution with saturation-dependent capillary diffusion. Our analytical approach allows us to observe the following conspicuous difference in the behavior of the dispersed front, where we obtain a smoothly dispersed front in the constant diffusion case and a power-law behavior around the front for a saturation-dependent capillary diffusion. We compare the numerical solution of the initial value problem for the case of saturation-dependent capillary diffusion obtained with a finite element software package to a partially analytical solution of the problem in terms of the similarity variable η.