Problem Description

Let us consider a polygonal shaped mass conserving cell which arises in the dis- cretization of some FV scheme. The Galerkin FE scheme can be treated in the same way as CVFE scheme just by introducing control volumes. The numerical solution of the flow equations will provide fluxes on the cell interfaces. This flux information is global because the cell interface fluxes satisfy conservation for each mass conserving cell and continuity across neighboring cells. The velocity interpolation methods, on the other hand, only work locally within each mass conserving cell, consistent with these fluxes. This means that the global cell face flux information serves as known boundary conditions for the velocity interpolation of a particular mass conserving cell.

As just mentioned, the boundary conditions provide knowledge of the face fluxes. When we utilize lower order boundary conditions, we mean that flux along the bound- ary face is uniform, i.e. the normal component of the velocity is characterized by a single average value. When we utilize higher order boundary conditions, the normal component of the velocity varies linearly, i.e. it is characterized by two variables: the average and the slope of variation. (Note that in 3D there is a distinction between

the same boundary conditions in the physical cell and in the isoparametric reference cell, but in 2D they are equivalent (Hægland et al., 2007).) In particular, a lower order boundary condition can be viewed as a constrained higher order boundary con- dition with zero slope of the linear variation. Thus a general discussion can be made assuming higher order boundary conditions; the results for lower order boundary conditions arise as a special case.

The boundary conditions can be equivalently expressed by corner velocities in- stead of fluxes for the non-degenerate cases. As shown in Figure 2.2, if the angle between two adjacent faces is different from π, the face fluxes can be used to uniquely define the corner velocities. When the angle between two adjacent faces is close to π (degenerate case), the corner velocity will not be well defined. The degenerate case may occur for grids with local grid refinement and faults, as well as CVFE grids. The unrobustness caused by the degeneracy of the physical cell will be addressed in section 2.5.2 and 2.5.3.

It is worth mentioning that Matringe and Gerritsen (2004) have proposed a local refinement tracing method where the boundary conditions on the subgrid are deter- mined using the fluxes of the neighboring coarse blocks with a “slope limiter”. This will generate a higher order boundary condition for the coarse blocks without using a higher order numerical scheme for the fluxes. If a similar method was developed for unstructured grids, it would be immediately applicable to our analysis since we start with the known boundary conditions.

The various situations that are frequently encountered in practice are depicted in Figure 2.1 and explained as follows. As mentioned earlier, TPFA calculates one av- erage flux for each interface between two cells. This is most often seen in simulations

i− 1 i i + 1 vi−1,1 vi−1,2 vi,1 vi,2 vi+1,1 vi+1,2 fi fi+1

Figure 2.2: The degrees of freedom for a polygonal cell. The degrees of freedom may be associated with the average flux and the slope of linear variation of flux for each face (blue). Or equivalently they may be associated with the components of corner velocities (red).

require MPFA, which is able to calculate two half face fluxes. As shown in Figure 2.1(d), the interface between two mass conserving cells is part of a non-convex poly- gon and consists of two straight lines. Thus for the polygonal cell, only one average flux is known for each edge of the polygon. The situation of the CVFE schemes are similar to that of the point-distributed FV schemes as shown in Figure 2.1(e). The only difference is that the CVFE scheme is based on heterogeneous rock properties within the polygonal cell. Note that all the situations depicted in Figure 2.1 involve only lower order boundary conditions. Thus for practical purposes and simplicity, the test cases and examples in this section assume lower order boundary conditions. However, the theory and the methods developed in this section are not restricted to lower order boundary conditions.

Now that we have discussed the boundary conditions, the problem we are trying to solve becomes easier to state. The local velocity field within a cell is not deter- mined uniquely by the boundary conditions. It also depends on the local velocity interpolation method used, which will make assumptions of the functional form of the local velocity field as well as certain properties that it should satisfy. As a basic requirement, the velocity interpolation method should provide sufficient degrees of freedom in order to be able to honor the boundary flux conditions; on the other hand, it should not introduce too many additional degrees of freedom without physically meaningful constraints on the velocity field to obtain a robust solution.

The following discussion will focus on 2D situations. The extension to 3D is straightforward in some cases but may require additional study for some classes of 3D unstructured grids.

A B C D vA1 vA2 vB1 vB2 vC1 vC2 vD1 vD2 fAB fBC fCD fDA

Figure 2.3: The degrees of freedom for the unit square.