Abstract
Adaptive implicit petroleum reservoir simulations result in huge, often very ill-conditioned linear systems of equations. The full system contains characteristics of both hyperbolic and nearly elliptic sub-systems. Traditional single stage solvers, such as variable degree ILU, are often not efficient at converging nearly elliptic, such as pressure, components. Therefore, multi-stage preconditioning methods, such as the constrained pressure residual (CPR) method, are a popular approach to “divide and conquer” coupled systems. An algebraic multigrid (AMG) method provides a technique to efficiently solve suitably extracted, nearly elliptic sub-systems as a first stage of CPR. An ILU technique can then be used as a second stage.
The primary objective of an efficient two-stage preconditioning strategy consists of extracting nearly elliptic sub-systems that are suitable for an efficient AMG solution while simultaneously ensuring a fast overall convergence of the two-stage approach. This research aims to modify and apply the dynamic row sum preconditioner (DRS) to adaptive-implicit black oil simulation. This extends the original DRS preconditioner (Gries, et al., 2014), which was developed in consideration of individual matrices.
Physical weighting is incorporated into the DRS technique using additional information determined during Jacobian building. The adaptive-implicit formulation produces explicit saturation terms that are fully coupled to pressure from the same location. The DRS method is modified to approximately decouple such saturations so that they may be excluded from the iterative matrix solution.
The multi-coloured ordering of variables is also considered. Such methods have proved efficient by reducing the number of unknowns from the iterative matrix solution. A parallel algorithm using domain decomposition and red-black ordering for the two-stage approach is examined. The effect of increasing the degree of ILU on the boundaries between domains selectively is investigated for the two-stage approach and a single-stage approach. The results of full simulation show significant acceleration using the preconditioned two-stage approach.
