Summary
Implicit-reservoir-simulation models offer improved robustness compared with semi-implicit or explicit alternatives. The implicit treatment gives rise to a large nonlinear algebraic system of equations that must be solved at each timestep. Newton-like iterative methods are often used to solve these nonlinear systems. At each nonlinear iteration, large and sparse linear systems must be solved to obtain the Newton update vector. It is observed that these computed Newton updates are often sparse, even though the sum of the Newton updates over a converged iteration may not be. Sparsity in the Newton update suggests the presence of a spatially localized propagation of corrections along the nonlinear iteration sequence. Substantial computational savings may be realized by restricting the linear-solution process to obtain only the nonzero update elements. This requires an a priori identification of the set of nonzero update elements. To preserve the convergence behavior of the original Newton-like process, it is necessary to avoid missing any nonzero element in the identification procedure. This ensures that the localized and full linear computations result in the same solution. As a first step toward the development of such a localization method for general fully implicit simulation, the focus is on sequential implicit methods for general two-phase flow. Theoretically conservative, a priori estimates of the anticipated Newton-update sparsity pattern are derived. The key to the derivation of these estimates is in forming and solving simplified forms of infinite-dimensional Newton iteration for the semidiscrete residual equations. Upon projection onto the discrete mesh, the analytical estimates produce a conservative indication on the update's sparsity pattern. The algorithm is applied to several large-scale computational examples, and more than a 10-fold reduction in simulation time is attained. The results of the localized and full simulations are identical, as is the nonlinear convergence behavior.
In 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.
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