An Adaptive Sequential Fully Implicit Domain-Decomposition Solver

Summary Modern reservoir simulation must handle complex compositional fluid behavior, orders-of-magnitude variations in rock properties, and large velocity contrasts. We investigate how one can use nonlinear domain-decomposition preconditioning to combine sequential and fully implicit (FI) solution strategies to devise robust and highly efficient nonlinear solvers. A full simulation model can be split into smaller subdomains that each can be solved independently, treating variables in all other subdomains as fixed. In subdomains with weaker coupling between flow and transport, we use a sequential fully implicit (SFI) solution strategy, whereas regions with stronger coupling are solved with an FI method. Convergence to the FI solution is ensured by a global update that efficiently resolves long-range interactions across subdomains. The result is a solution strategy that combines the efficiency of SFI and its ability to use specialized solvers for flow and transport with the robustness and correctness of FI. We demonstrate the efficacy of the proposed method through a range of test cases, including both contrived setups to test nonlinear solver performance and realistic field models with complex geology and fluid physics. For each case, we compare the results with those obtained using standard FI and SFI solvers. This paper is published as part of the 2021 Reservoir Simulation Conference Special Issue.

Optimization of Multi-Quality Water Networks: Can Simple Optimization Heuristics Compete with Nonlinear Solvers?

Optimal management of water systems tends to be very complex, especially when water quality aspects are included. This paper addresses the management of multi-quality water networks over a fixed time horizon. The problem is formulated as an optimization program that minimizes cost by determining the optimal flow distribution that satisfies the water quantity and quality requirement in the demand nodes. The resulted model is nonlinear and non-convex due to bilinear terms in the mass balance equations of blending multi-quality flow. This results in several local optima, making the process of solving large-scale problems to global optimality very challenging. One classical approach to deal with this challenge is to use a multi-start procedure in which off-the-shelf local optimization solvers are initialized with several random initial points. Then the final optimal solution is considered as the lowest objective value over the different runs. This will lead to a cumbersome and slow solution process for large-scale problems. In light of the above, this study supports using ultra-fast simple optimization heuristics, which despite their moderate accuracy, can still reach the optimum solution when run many times using a multi-start procedure. As such, the final solution from simple optimization heuristics can compete with off-the-shelf nonlinear solvers in terms of accuracy and efficiency. The paper presents a simple optimization heuristic, which is specially tailored for the problem and compares its performance with a state-of-the-art nonlinear solver on large-scale systems.

Comparison of nonlinear solvers within continuation method for steady-state variably saturated groundwater flow modelling

Nonlinearity continuation method, applied to boundary value problems for steady-state Richards equation, gradually approaches the solution through a series of intermediate problems. Originally, the Newton method with simple line search algorithm was used to solve the intermediate problems. In the present paper, other solvers such as Picard and mixed Picard–Newton methods are considered, combined with slightly modified line search approach. Numerical experiments are performed with advanced finite volume discretizations for model and real-life problems.

Computation of Nonlinear Thermoelectric Effects with Adaptive Methods

An implementation for a fully automatic adaptive finite element method (AFEM) for computation of nonlinear thermoelectric problems in three dimensions is presented. Adaptivity of the nonlinear solvers is based on the well-established hp-adaptivity where the mesh refinement and the polynomial order of elements are methodically controlled to reduce the discretization errors of the coupled field variables temperature and electric potential. A single mesh is used for both fields and the nonlinear coupling of temperature and electric potential is accounted in the computation of a posteriori error estimate where the residuals are computed element-wise. Mesh refinements are implemented for tetrahedral mesh such that conformity of elements with neighboring elements is preserved. Multiple nonlinear solution steps are assessed including variations of the fixed-point method with Anderson acceleration algorithms. The Barzilai-Borwein algorithm to optimize the nonlinear solution steps are also assessed. Promising results have been observed where all the nonlinear methods show the same accuracy with the tendency of approaching convergence with more elements refining. Anderson acceleration is the most efficient among the nonlinear solvers studied where its total computing time is less than half of the more conventional fixed-point iteration.