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.

当包括毛细管扩散时，通常用Buckley-Leverett方程或Rapoport-Leas方程描述多孔介质中的一维水油驱替。直线几何形状不能代表近井油驱替问题。因此，有兴趣在圆柱几何中描述径向对称的Buckley-Leverett或Rapoport-Leas方程（径向Buckley-Leverett问题）。我们可以证明，在适当的条件下，可以应用相似性变换\（（r，t）\ rightarrow \ eta = r ^ {2} /（2t）\）即使在包括毛细管扩散的情况下，也可以将径向几何结构中的PDE降低为ODE（与直线几何结构中的情况相反（Yortsos，YC（Phys。Fluids 30（10），2928–2935 1987））。情况（1）的毛细管扩散与饱和度无关，而（2）毛细管的扩散与饱和度有关事实证明，具有恒定的毛细管扩散系数的溶液与具有饱和度的毛细管的溶液根本不同我们的分析方法使我们能够观察到分散锋面行为的以下明显差异：在恒定扩散情况下，我们获得了平滑分散的锋面，围绕饱和度的毛细管扩散得到了幂律行为。我们将用有限元软件包获得的与饱和有关的毛细管扩散情况下的初始值问题的数值解与根据相似性变量对该问题的部分解析解进行了比较η。