Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation

Cao, Huibo; Wallis, John; Yardumian, Hrant E.

doi:10.2523/96809-ms

Cited by 37 publications

(51 citation statements)

References 4 publications

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“…O código do CPR foi implementado em MATLAB seguindo [3], aúnica alteração feita foi a utilização do backslash para a resolução do primeiro estágio, uma vez que o MATLAB não possui codificação nativa. O solver linear utilizado foi o GMRES [4], usando precondicionamento pela esquerda, restart = 30, a tolerância usada foi 10 −4 .…”

Section: Resultsunclassified

Método de Precondicionamento CPR em Simulações de Reservatório de Petróleo

Zanardi¹,

Carvalho²,

Goldfeld³

et al. 2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Este trabalho apresenta a implementação, em MATLAB, de um esquema de dois estágios do tipo CPR (do inglês Constrained Pressure Residual) [5], para a resolução de sistemas lineares de grande porte oriundos de simulações de extração de reservatórios de petróleo. Vamos descrever o método CPR, seus dois estágios e apresentar resultados para matrizes de problemas reais, comparando-os com resultados utilizando precondicionadores clássicos.Palavras-chave. Precondicionadores, CPR, Sistemas Lineares. IntroduçãoNos problemas de simulação de reservatórios temos que repetidamente resolver sistemas de equações lineares com matrizes Jacobianas resultantes da linearização (em geral pelo método de Newton) das equações de balanço e volume. Estes sistemas lineares apresentam diferentes tipos de variáveis como, por exemplo, pressão e saturação, e resolvê-los numericamente de maneira rápida e robusta aindaé um grande desafio naárea computacional.Tradicionalmente a maioria dos solvers atuais utilizam métodos de fatoração incompleta dos fatores LU (ILU) [2] aplicada ao sistema completo ou aplicam o método CPR. O CPRé uma técnica de precondicionamento de dois níveis que privilegia a solução da pressão, reconhecendo a sua natureza elítica e, portanto, a dificuldade de convergir as frequências mais baixas do erro. Particularmente quando combinado com o multigrid, o algoritmo tem recebido muita atenção na literatura e vem sendo aplicado em diversos 1 jpzanardi@gmail.com 2

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Section: Resultsunclassified

Método de Precondicionamento CPR em Simulações de Reservatório de Petróleo

Zanardi¹,

Carvalho²,

Goldfeld³

et al. 2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

View full text Add to dashboard Cite

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“…7. Both the left and the right preconditioners are discussed in detail in the literature 1,2,[8][9][10][11] and are not repeated here. (7) In this article, the mathematical modeling used to generate the linear system, Eqn.…”

Section: (3) (4)mentioning

confidence: 99%

Studies of Robust Two Stage Preconditioners for the Solution of Fully Implicit Multiphase Flow Problems

et al. 2009

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The solution of the linear system of equations for a large scale reservoir simulation has several challenges. Preconditioners are used to speed up the convergence rate of the solution of such systems. In theory, a preconditioner defines a matrix M that can be inexpensively inverted and represents a good approximation of a given matrix A. In this work, two-stage preconditioners consisting of the approximated inverses M 1 and M 2 are investigated for multiphase flow in porous media. The first-stage preconditioner, M 1 , is approximated from A using four different solution methods: (1) constrained pressure residuals (CPR), (2) lower block Gauss-Seidel, (3) upper block Gauss-Seidel, and (4) one iteration of block Gauss-Seidel. The pressure block solution in each of these different schemes is calculated using the Algebraic Multi Grid (AMG) method. The inverse of the saturation (or more generally, the nonpressure) blocks are approximated using Line Successive Over Relaxation (LSOR). The second stage preconditioner, M 2 , is a global preconditioner based on LSOR iterations for the matrix A, that captures part of the original interaction of different coefficient blocks. Several techniques are also employed to weaken the coupling between the pressure block and the nonpressure blocks. Effective decoupling is achieved by: (1) an Implicit Pressure Explicit Saturation (IMPES) like approach designed to preserve the integrity of pressure coefficients, (2) Householder transformations, (3) the alternate block factorization (ABF), and (4) the BDS based on least squares. The fourth method is a new technique developed in this work. The aforementioned preconditioning techniques were implemented in a parallel reservoir simulation environment, and tested for large-scale two-phase and three-phase black oil simulation models. This study demonstrates that a two-stage preconditioner based on BDS or ABF combined with Gauss-Seidel sweeps, that also incorporate nonpressure solutions for M, delivers both the fastest convergence rate and the most robust option overall without compromising parallel scalability.

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“…More information on multigrid methods can be found in e.g. [221], applications to reservoir simulation are considered in [49,210]. We point out that the classical algebraic multigrid approaches require the system matrix (for the restricted pressure problem) to be an M-matrix.…”

Section: Linear Solversmentioning

confidence: 99%

“…For transport equations, local methods like incomplete LU-factorisation can be expected to perform well. Different strategies for decoupling the system into pressure and transport parts are considered in [49,148,210,225]…”

Section: Linear Solversmentioning

confidence: 99%

Sufficient criteria are necessary for monotone control volume methods

Keilegavlen

Nordbotten

Aavatsmark

2009

Applied Mathematics Letters

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The dissertation consists of three parts. The first part is devoted to background theory, and is structured as follows. Chapter 1 gives a high-level overview over important concepts and challenges in reservoir simulation. In Chapter 2, mathematical models for flow in porous media are presented. The next three chapters are devoted to numerical techniques for solving the equations. In Chapter 3, we consider various aspects of reservoir simulation. Discretisation techniques for elliptic and hyperbolic equations are discussed in Chapter 4 and 5, respectively. Summary of papers produced are given in Chapter 6, while in the last chapter, conclusions are drawn, and future research directions are pointed out. The second part consists of in total five papers and manuscripts that have been produced during the work with the thesis. Three of these are published in journals or accepted for journal publication, one is submitted, and one is still in a draft stage. The third part contains a conference proceedings paper that is not considered part of the thesis, however, some of the results therein is useful for the understanding of the rest of the work. III Related publications F Monotonicity of Control volume methods on triangular grids Part I Background Chapter 1 A petroleum reservoir consists of rock perforated by small channels, or pores, that are filled with hydrocarbons. To act as a reservoir, the pores must be connected, so that the fluids can flow. The rock in a reservoir has originated from sedimentary deposition processes that took place millions of years ago. It therefore has a layered structure, where each layer consists of a different type of rock, as seen in 1.1 Reservoir Characteristics Seal Impermeable rock GAS OIL WATER Figure 1.2: Schematic overview of a petroleum reservoir. The hydrocarbons are trapped by a sealing rock. The fluids are in hydrostatic equilibrium. of average parameters, the geo-model can consist of up to 10 7 − 10 8 grid cells. This is at least one to two orders of magnitude more than what is possible for a traditional reservoir simulator to handle. Therefore, a coarser grid is constructed, and the rock properties are represented on the new cells. The coarsening process is known as upscaling. Ideally, the upscaling should not be detrimental to the quality of the reservoir model. It is therefore crucial to construct the coarser grid so that the rock parameters can be represented accurately in each cell. 1.1.2 Fluid Characterisation Together with the sediments that later became the reservoir rock, organic material was also deposited. This material has since evolved into oils and gasses by chemical reactions. It is commonly believed that the hydrocarbons are not formed in the reservoir, but in some source rock. Later, the hydrocarbons migrate through the reservoir. The main driving forces of the migration are buoyancy effects, but also capillary forces can affect the movement. Eventually, the hydrocarbons may reach a low conductivity rock formation, a seal, which prevents them from propagating any...

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Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation

Cited by 37 publications

References 4 publications

Método de Precondicionamento CPR em Simulações de Reservatório de Petróleo

Método de Precondicionamento CPR em Simulações de Reservatório de Petróleo

Studies of Robust Two Stage Preconditioners for the Solution of Fully Implicit Multiphase Flow Problems

Sufficient criteria are necessary for monotone control volume methods

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