Improved gravity splitting for streamline and reordering methods

For heterogeneous reservoirs, fast, stable and accurate methods are hard to obtain. Large changes in the velocity field leads to severe time-step restrictions in explicit schemes or expensive time steps in implicit schemes. In the absence of gravity, the exact velocity field will be loop-free in the sense that there are no closed integral curves. A sequential splitting strategy with streamline or reordering methods utilize this feature to make fast transport solves for advection-dominated flow. For streamline methods, the absence of closed integral curves ensures that all streamlines have end points in wells. Likewise, this property implies that the Jacobi matrix of an implicit scheme with an upwind flux approximation can be reduced to a triangular matrix by a permutation, or reordering, of the unknowns. For the above methods the effect of gravity is usually handled by operator splitting in the transport equation. Gravity will introduce rotation in the total velocity, which may yield closed integral curves. Even though the effect of gravity in many cases will be very small, it can limit the efficiency and the robustness of streamline and reorder methods. To overcome this problem, we split the total velocity field in a loop-free part without the effect of gravity, and a part driven by gravity only. This is done by solving the pressure equation two times with different right-hand sides. We demonstrate that the new splitting strategy will increase the efficiency and applicability of reordering methods and make streamline tracing more robust.