Reservoir Simulation Using an Adaptive Implicit Method

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Abstract
This paper deals with a new implicit method for reservoir simulation. Rather than provide a fixed degree of implicitness in every gridblock at every time step or iteration, the adaptive implicit method operates with different levels of implicitness in adjacent gridblocks. These levels shift in space and time as needed to maintain stability. Shifting is accomplished automatically without user intervention. The technique can be applied to any simulation problem involving N unknowns. The advantage is substantial reduction in computing time and storage requirements compared to fully implicit formulations, while still yielding unconditionally stable solutions. The mathematical procedure involves labeling the implicit/explicit mix of unknowns and then composing the matrix problem. The latter is reducible as long as there is one or more unknowns to be computed explicitly; consequently, appropriate operations are performed to put it in reduced form. This leads to matrix equations of lower order than the original problem that are solved at less cost. We demonstrate each step of the mathematical procedure with an example. Finally, applications to a three-phase coning problem and a three-dimensional (3D) Cartesian problem are presented. By using special displays we demonstrate the degrees of implicitness in each cell and how they shift in space and time during simulation. We also present information regarding the savings in computer time and storage compared with a fixed, fully implicit procedure.