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Simulating two phase immiscible flow in heterogeneous media using finite elements and multigrid methods

Numerical simulation of enhanced hydrocarbon recovery has always been recognized as a computationally intensive task. Multigrid methods are described for solving the pressure equation arising from a finite element discretization of the equations modeling the flow of two immiscible and incompressible fluids in a heterogeneous porous medium. Comparisons are made between the performance of these methods and a preconditioned conjugate gradient algorithm by solving some two dimensional five spot problems on both homogeneous and heterogeneous media. Multigrid appears more suitable for large, three dimensional problems by virtue of the observation that the number of multigrid cycles required to solve the pressure equation remains constant at the problem size, n, is increased, whereas the number of conjugate gradient cycles is proportional to n{sup 1/3}. An element-by-element implementation is also described that minimizes storage and produces code that can be easily vectorized and parallelized on shared memory computers.