Constrained Residual Acceleration of Conjugate Residual Methods

Wallis, J.R.; Kendall, Richard P.; Little, T E

doi:10.2118/13536-ms

Cited by 190 publications

(78 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The default linear solver employs GMRES, and for preconditioning we use the two-stage Constrained Pressure Residual (CPR) strategy presented in Wallis (1983) and Wallis et al (1985). An algebraic reduction step is used to construct the pressure equation in the first stage, which is then solved using an AMG solver.…”

Section: Numerical Framework Descriptionmentioning

confidence: 99%

Modeling of Multi-Component Flow in Porous Media with Arbitrary Phase Changes

Voskov

2011

All Days

View full text Add to dashboard Cite

I present a novel computational framework for simulating flow in porous media for mixtures having an arbitrary number of phases. This framework is designed to keep a unified computational structure for different nonlinear formulations and complex physical models. That includes generic phase appearance/disappearance treatment, efficient linear solver typical for mixed types of unknowns, finite volume discretization using flexible spatial stencil etc. The behavior of components in any phase is computed using mixed phase equilibrium assumptions. This type of simulation is very important for modeling of natural and industrial processes, such as development of natural hydrocarbon resources, including gas-hydrates, CO2 injection into hydrocarbon reservoirs and saline aquifers, and for modeling of thermal processes. In this framework I use a Fully Implicit (FI) time approximation with general extension to the Adaptive Implicit Method (AIM) using flexible algebraic reduction. The framework is built on top of an Automatic Differentiation with Expression Templates Library (ADETL) which generates the corresponding derivatives for any nonlinear relation and helps to construct Jacobian matrix for the nonlinear solver. Phase behavior for the new framework is calculated using Compositional Space Parameterization (CSP) approach, which helps to improve both efficiency and robustness of the standard Equation of State (EoS) computations. Within this simulation framework, the nonlinear behavior of different variable sets, including both natural and mass-type variables is investigated. The general extension of two-phase variable substitution is used to handle the phase appearance and disappearance for systems with arbitrary numbers of phases. Examples will be presented including different types of CO2 injection in a hydrocarbon reservoir, such as miscible and immiscible two-phase CO2 injection, thermal injection of CO2+steam (three-phase), and cold CO2 injection (three phase with the second liquid CO2-rich phase).

show abstract

Section: Numerical Framework Descriptionmentioning

confidence: 99%

Modeling of Multi-Component Flow in Porous Media with Arbitrary Phase Changes

Voskov

2011

All Days

View full text Add to dashboard Cite

show abstract

“…Consequently the Constrained Pressure Residual (CPR) method [33] [34] aims at preconditioning the one-level method by computing a pressure approximation, e.g. by multigrid, in advance to applying a one-level method.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations

Gries

Stüben

Brown³

et al. 2013

All Days

View full text Add to dashboard Cite

Fully implicit petroleum reservoir simulations result in huge, often very ill-conditioned linear systems of equations to solve for different unknowns, for example, pressure and saturations. It is well known that the full system matrix contains both hyperbolic as well as nearly elliptic sub-systems. Since the solution of the coupled system is mainly determined by the solution of their elliptic (typically pressure) components, (CPR-type) two-stage preconditioning methods still belong to the most popular approaches to tackle such coupled systems. After a suitable extraction and decoupling, the numerically most costly step in such two-stage methods consists in solving these elliptic sub-systems. It is known that algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. The main advantage of AMG-based solvers -their numerical scalability -makes them particularly efficient for solving huge linear systems.Depending on the application, the system's properties range from simple to highly indefinite. Unfortunately decoupling pressure and saturation related parts may introduce further difficulties. Consequently, in complex industrial simulations, the application of AMG to elliptic sub-systems might not be straightforward. In fact, an important goal in defining an efficient two-stage preconditioning strategy consists in extracting elliptic sub-systems that are suitable for an efficient AMG solution and, at the same time, ensure a fast overall convergence of the two-stage approach.The importance of this will be demonstrated for several industrial cases. In particular, some of these cases are very hard to solve by AMG if applied in a standard way.Preliminary results for a CPR-type coupling of SAMG to CMG's PARASOL, a variable degree variable ordering ILU preconditioner using FGMRES, are compared to using PARASOL by itself. Alternative preconditioning operators will be presented giving elliptic sub-systems which are not only more suitable for applying AMG efficiently but also help accelerate the CPR-type process. Comparisons with one-level iterative methods will show the acceleration by AMG is highly superior. Finally, a strategy is presented that combines all linear solver parts in one single AMG-iteration. In this sense CPR can be seen as a special case of AMG for systems. This, in turn, yields a -formally -very simple but simultaneously very flexible solution approach.

show abstract

“…The adjoint equation with the transpose of the Jacobian matrix, unfortunately, cannot be solved by a linear solver using a preconditioner designed and optimized specifically for reservoir simulation, such as CPR (Wallis 1983;Wallis et al 1985) or the nested-factorization method (Appleyard 1983). This is because the preconditioners exploit properties of the original Jacobian matrix to accelerate convergence.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2013

View full text Add to dashboard Cite

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

show abstract

Constrained Residual Acceleration of Conjugate Residual Methods

Cited by 190 publications

References 4 publications

Modeling of Multi-Component Flow in Porous Media with Arbitrary Phase Changes

Modeling of Multi-Component Flow in Porous Media with Arbitrary Phase Changes

Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations

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

Contact Info

Product

Resources

About