Solving global problem by considering multitude of local problems: Application to fluid flow in anisotropic porous media using the multipoint flux approximation

• A new methodology is introduced for solving the governing equations of flow and transport.
• Global problem is divided into a set of local problems which are easy to solve.
• The global system of equations are assembled automatically in the solver routine.
• This technique uses the equations as suggested by the physics without extra manipulations.
• The result is simple, easy to run, update, and maintain algorithms.

In this work we apply the experimenting pressure field approach to the numerical solution of the single phase flow problem in anisotropic porous media using the multipoint flux approximation. We apply this method to the problem of flow in saturated anisotropic porous media. In anisotropic media the component flux representation requires, generally multiple pressure values in neighboring cells (e.g., six pressure values of the neighboring cells is required in two-dimensional rectangular meshes). This apparently results in the need for a nine points stencil for the discretized pressure equation (27 points stencil in three-dimensional rectangular mesh). The coefficients associated with the discretized pressure equation are complex and require longer expressions which make their implementation prone to errors. In the experimenting pressure field technique, the matrix of coefficients is generated automatically within the solver. A set of predefined pressure fields is operated on the domain through which the velocity field is obtained. Apparently such velocity fields do not satisfy the mass conservation equations entailed by the source/sink term and boundary conditions from which the residual is calculated. In this method the experimenting pressure fields are designed such that the residual reduces to the coefficients of the pressure equation matrix.