This is the standard porous media problem called the quarter five spot problem. Initially, the well is filled with oil (zero saturation in blue) throughout the reservoir. Water (saturation 1) is injected through a well at the lower right corner of the computational domain at a constant rate, and fluid is produced at the same rate through a well in the upper left corner.

The grid is composed of rectangles divided into right triangle pairs. For the computation, we used a mixed finite element method using lowest order Raviart-Thomas elements for the approximation of the pressure system followed by a higher-order Godunov method. The finite element velocity space is made up of piecewise linear functions with continuous normal components on the triangle edges, while the pressure space consists of piecewise constant functions.