Localized Solvers for General Full-Resolution Implicit Reservoir Simulation
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AbstractIn the solution of implicit reservoir simulation timesteps, the Newton iteration updates are often very sparse; this sparsity can be as high as 95% and can vary dramatically from one iteration to the next. We develop, implement and demonstrate a mathematically sound adaptive framework to predict this sparsity pattern before the system is solved. The development first mathematically relates the Newton update in functional space to that of the discrete system. Next, the Newton update formula in functional space is homogenized and solved in such a way that it results in conservative estimates of the numerical Newton update. The cost of evaluating the estimates is linear in the number of nonzero components. The estimates are used to label the components of the solution vector that will be nonzero, and the corresponding submatrix is solved. The computed result is guaranteed to be identical to the one obtained by solving the entire system.When applied to various simulations of three-phase flow recovery processes in the SPE 10 geological model, the observed reduction in computational effort ranges between four to tenfold depending on the level of total compressibility in the system, the time step size and on the degree of complexity in the underlying physics. We show the extensions to the case of flow and multicomponent transport where the reduction in computation effort ranges between four to tenfold. The improvement in computational speed scales strongly with the number of transport components, and to a lesser degree with problem size. The results of the localized and full simulations are identical, as is the nonlinear convergence behavior.
