Adaptation of the CPR Preconditioner for Efficient Solution of the Adjoint Equation

Abstract

It is well known that the adjoint approach is the most efficient approach for gradient calculation, and it can be used with gradient-based optimization techniques to solve various optimization problems, such as the production-optimization problem and the history-matching problem. The adjoint equation to be solved in the approach is a linear equation formed with the "transpose" of the Jacobian matrix from a fully implicit reservoir simulator. For a large and/or complex reservoir model, generalized preconditioners often prove impractical for solving the adjoint equation. Preconditioners specialized for reservoir simulation, such as constrained pressure residual (CPR), exploit properties of the Jacobian matrix to accelerate convergence, so they cannot be applied directly to the adjoint equation. To overcome this challenge, we have developed a new two-stage preconditioner for efficient solution of the adjoint equation by adaptation of the CPR preconditioner (named CPRA: CPR preconditioner for adjoint equation).The CPRA preconditioner has been coupled with an algebraic multigrid (AMG) linear solver and implemented in Chevron's extended applications reservoir simulator (CHEARS V R ). The AMG solver is well known for its outstanding capability to solve the pressure equation of complex reservoir models; solving the linear system with the "transpose" of the pressure matrix is one of the two stages of construction of the CPRA preconditioner.Through test cases, we have confirmed that the CPRA/AMG solver with generalized minimal residual (GMRES) acceleration solves the adjoint equation very efficiently with a reasonable number of linear-solver iterations. Adjoint simulations to calculate the gradients with the CPRA/AMG solver take approximately the same amount of time (at most) as do the corresponding CPR/ AMG forward simulations. Accuracy of the solutions has also been confirmed by verifying the gradients against solutions with a direct solver. A production-optimization case study for a real field using the CPRA/AMG solver has further validated its accuracy, efficiency, and the capability to perform long-term optimization for large, complex reservoir models at low computational cost. Mathematical Derivation of the CPRA PreconditionerDefinition of the Optimization Problem. The optimization problem discussed in the preceding requires a sequence of control vectors u n be found for n ¼ 0, 1, …, N -1, where n is the control-step index and N is the total number of control steps, to maximize (or minimize) a performance measurement J(u 0 , …, u N-1 ) (Sarma et al. 2005). The

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

“…The Newton iterations terminate when the maximum relative norm of the residual is less than 10 −6 (tight convergence criteria are required for the adjoint solution, discussed below). For the solution of the linear system at each Newton iteration we use GMRES preconditioned by the constrained pressure residual method, as described in [20]. Iteration is terminated when the Euclidean norm of the initial residual has decreased by five orders of magnitude.…”

“…For the solution of the linear system in (3.6), we use GM-RES preconditioned by the transpose of the CPR (constrained pressure residual) preconditioner, as described in [20]. In these linear solutions, we require very high accuracy to guarantee that residual errors accumulated over hundreds of time steps do not pollute the gradients (which would influence the computed optimum).…”

“…In subsequent work, the focus was on gradient-based optimization (and in some cases on the optimization of 'smart wells') for water flooding [1,5,34,36,39]. Recent studies have addressed the implementation of adjoint-based procedures into general purpose simulators, the treatment of general constraints, and regularization and other numerical issues [15,20,24,33]. Refer to [21] for a more complete overview of adjoint-based optimization methods.…”

Use of reduced-order models in well control optimization

“…This is computationally a very efficient procedure in comparison with a finite difference approach, which would require one forward base simulation plus q forward perturbed simulations to compute dJ=du. We note that the backward computation may require an adapted version of the pre-conditoning scheme used for the forward computation (see Han et al 2013). Moreover, it is generally necessary to compute the forward simulations with tight nonlinear solver tolerances in order to obtain accurate adjoint gradients.…”

Constraint handling for gradient-based optimization of compositional reservoir flow

“…In subsequent work, the focus was on gradientbased optimization (and in some cases on the optimization of "smart wells") for water flooding [1,4,28,29,32]. Recent studies have addressed the implementation of adjoint-based procedures into general purpose simulators, the treatment of general constraints, and regularization and other numerical issues [12,16,21,27]. Refer to [17] for a more complete overview of adjoint-based optimization methods.…”