Abstract
This paper deals with a new implicit method for reservoir simulation. Rather than provide a fixed degree of implicitness in every grid block at every time step or iteration, the adaptive implicit method operates with different levels of implicitness in adjacent grid blocks. These levels shift in space and tithe as needed to maintain stability. Shifting is accomplished automatically without user intervention. The technique can he 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 impplicit/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. 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 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.
Introduction
With the growth of reservoir simulation technology, the tendency has been to develop simulators that offer implicit or near-fully implicit capabilities. Implicit calculation are often necessary to maintain stability when one or more of the computed variables (pressure. saturation, temperature, composition, etc.) undergoes large surges over a time step. An advantage one accrues from a highly implicit simulator, besides stability, is that large timestep sizes can be tolerated. However, this advantage is offset by larger time-truncation errors, and substantially higher processor times and storage requirements. Moreover, most commercial reservoir simulators provide only a fixed level of implicitness in every grid block at every timestep. Typically, however, only a small fraction of the total grid blocks in a reservoir model undergo rapid changes in the computed variables to justify high levels of implicitness. For example, in a well undergoing gas and water coning, rapid changes in saturation and pressure occur only in the grid blocks in the near-well region, while farther away the changes are more subdued. A fully implicit model similar to that discussed in Ref. 1 adequately handles such a situation with guaranteed stability. But obviously, some overkill is involved in those cells where the changes are modest. On the other hand, an implicit pressure explicit saturation (IMPES) treatment of such a problem can result in serious underkill and instability in the blocks near the wellbore, unless the simulator is especially modified.
The overkill problem cited above is substantially worse in large global reservoir simulations where one or more intervening grid blocks separate the well blocks.
SPEJ
P. 759
A Neural Network-Based Approach for Approximating Arbitrary Roots of Polynomials