Optimal Ordering for Transport Equations in Immiscible Two-Phase Countercurrent Flow

For the two-phase flow problem in porous media, many researchers have conducted in-depth studies on the sequential solution format. These studies mainly use topological sorting algorithms to determine the solution order of saturation to meet the requirements of upwind schemes. However, conventional topological sorting can only solve cocurrent flow problems. When considering countercurrent flow caused by gravity and capillary forces, the topological sorting algorithm will fail. In this paper, we leverage graph theory knowledge and use the strong connected component algorithm to compress the coupled grids, making the topological sorting algorithm still effective for countercurrent flow problems.