Efficient and Effective Solvers with Preconditions in the Parallel Software of Large-Scale Petroleum Reservoir Simulation

Abstract

文章的主要工作围绕着如何精心构造一个面向大规模油藏数值模拟的并行模拟器来进行.为此,对线性问题的高效预处理求解方法进行了理论分析和技术研究.主要研究内容包括：求解非线性方程和线性方程的迭代方法,基于物理背景和代数背景的预处理技术,并行计算,非线性问题求解性能提高的途径等.文章的研究重点定位于如何提高油藏数值模拟非线性方程组的求解性能.围绕着研究对象,文章对极小剩余类和正交剩余类方法进行理论分析,提出并实现了针对油藏数值模拟求解器的“内迭代-外迭代”双重迭代模式.在预处理技术方面,文章将整体预条件子看作一个“系统”,提出并采用基于大规模分布式并行处理的、面向油藏数值模拟线性系统的、基于区域分解、粗网格、约束剩余、子空间投影校正技术和块不完全分解技术的整体预条件子构造模式.该论文的创造性内容主要包括：1）将油藏数值模拟中Watts校正进行广义化,形成代数粗网格校正预处理算法,并利用代数粗网格校正预处理代替Watts校正;2）提出并使用多角度子空间斜交投影方法来构造并行的线性方程求解方法的整体预条件子,利用两类预条件子形成两种最佳“预条件子+迭代算法”组合;3）非线性问题求解的Broyden快速算法在秩二校正的BFGS算法上推广,形成拟牛顿秩二BFGS校正快速算法;4）基于论文的工作,形成了油藏数值模拟的高效并行线性求解器、非线性求解器和油藏数值模拟并行软件.Petroleum reservoir simulation uses a numerical material balance approach to generate realistic development scenarios. Reservoir description data often comes in finely gridding geological models containing more grid cells than can be efficiently handled. Many large-scale simulations are still limited by the CPU frequency and memory requirements. The objective of our project that lead to the research motivation of this doctoral thesis is to develop a simulator that can be used for solving a variety of practical and challenging reservoir simulation problems. It is widely recognized that the most computationally expensive part is the solution of the sparse linear equations from finite-difference or finite-element discretization. In this dissertation, theoretical analysis and practical techniques of efficient and effective solvers with preconditions are elaborated for large-scale petroleum reservoir parallel simulators. Discussion details involved include methods of solving nonlinear and linear equations, preconditions rooted in both physical and algebraic backgrounds, parallelism, and performance improving approach.This dissertation discusses how to improve the performance of solving nonlinear equations originated from petroleum reservoir numerical simulations. Relating to this subject, minimal residual and orthogonal residual methods are theoretical analyzed and inner-outer iteration algorithms similar to FGMRES or GMRESR are introduced; two types of multipurpose oblique projection correction preconditions are used, one type being based on an incomplete LU decomposition to solve a small linear system of (P~TAP)z = r involved and the other being-based on iterative algorithms to solve that small system; two of best combinations of both Krylov subspace methods and preconditions are provided, numerical tests of petroleum reservoir data on parallel computers show that different Krylov subspace methods combinings with appropriate preconditions are able to achieve nearly optimal performance. In the end of the work, performance-improving approach is given which suggests to use quasi-newton efficient implementations combined with inexact newton methods, quasi-newton algorithm is used to associate linear system preconditions with nonlinear solvers in order to save the expense such as isolated construction of the ILU decomposition, inexact newton is used to associate linear solvers with nonlinear solvers in order to choose dynamic convergence tolerance and to avoid the phenomenon of over-solving. In order to analyze the manners and technical details of the used preconditions, characteristics of partial differential equations are introduced in the second chapter of the thesis. The originality work of the thesis includes : 1) as a generalization of Watts correction method, coarse grid correction preconditioning is considered instead of Watts correction; 2) multipurpose oblique projection correction preconditioning method is brought forward and implemented in the parallel solver,two types of the preconditioning are given in order to be combined with different Krylov subspace methods; 3) qnasi-newton Rank-2 BFGS correction efficient implementations are provided ; 4) parallel software , nonlinear and linear solvers are finished basing on the work of this thesis. Novel viewpoint in the thesis includes : 1) in the view of projection correction, we give out the general theory about Krylov subspace methods basing on minimal residual and orthogonal residual methods; 2) the analyse and efficiency display of decoupling operator; 3) the mathematical explanation of so-called combinative and CRP (constrained residual precondition) preconditions; 4) Numerical comparison and analyse of preconditioned Krylov subspace methods for a typical PDE problem and for petroleum reservoir simulation black oil model; 5) some analyse about nonlinear (damped) newton, quasi-newton algorithm and their interaction associating with linear solvers and preconditions. Other important works include: 1) characteristics of partial differential equations are introduced and analysed ; 2) introduction of iteration methods last century; 3) introduction of preconditions used in petroleum reservoir simulation; 4) introduction about nonlinear (damped) newton, quasi-newton methods; 5) a number of numerical experiments; 6) the work about parallel solvers and parallel simulators