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) rightarrow eta = r^{2}/(2t) 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.
Highlights
Water drive recovery of oil is one of the most important secondary recovery methods
We consider a fully coupled, implicit numerical solution approach based on finite elements, which is solved with the mathematical module of COMSOL to solve the model equations in weak form
The solution consists of a rarefaction wave, followed by a shock to the constant initial state
Summary
In the absence of diffusion, the radial problem is similar to the 1D problem, where the solution again consists of a rarefaction wave and a shock connected to the constant initial state. Contrary to the 1D case, the model equations including the capillary diffusion term can be expressed entirely in terms of η This leads to a system of coupled ODEs for the saturation and pressure. Yortsos [15] did not consider explicit solutions for general cases of saturation-dependent capillary diffusion in a radially symmetric Buckley-Leverett problem. In this manuscript, we carry out an extension of the work by Yortsos in the sense that we consider generic saturation-dependent capillary diffusion. We note that the current manuscript is meant to have a descriptive nature, rather than being mathematically rigorous
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
