The Study On Highly Scalable Linear And Nonlinear Preconditioners For Fractured Reservoir Problems

Reservoir simulation has been widely used in the field of petroleum engineering and is an indispensable tool in current oil and gas exploitation.By modeling and simulating the flow process of underground porous media,the dynamic prediction and evaluation of distinct oil recovery projects can help to improve oil recovery and reduce the overall production cost.With the development of complex reservoir exploration,the numerical simulation often face severe challenges brought by multiple physical models,high resolution,long time size and complex grid structure.Therefore,the parallelization of reservoir simulation becomes an important means to solve these problems.Generally,the discreted nonlinear system has unbalanced nonlinear or linear properties,which makes it difficult for the traditional algorithms to achieve the ideal convergence speed or even no convergence.In large-scale engineering computing,it is often necessary to build suitable preconditioners to improve the convergence and speed for the iterative algorithm.Domain decomposition method is a divide and conquer strategy algorithm,which decomposes the original problem into sub-problems equal to the number of cores.Each sub-problem completes the calculation independently on the corresponding processor,and it is very suitable for large-scale scientific computing due to high parallelism.This paper focuses on a class of overlapping restricted additive Schwarz preconditioner,and designs a robust and scalable simulator for three reservoir problems based on the high performance computing platform.The main contents of this dissertation are as follows:Firstly,a highly scalable multilevel restricted additive Schwarz(RAS)method is proposed for the high-resolution simulation of single-phase compressible gas reservoirs.The second-order fully implicit scheme is used to get rid of the limitation of CFL condition on the stability of time step.Several types of multilevel preconditioners are constructed flexibly by adopting additive and multiplicative strategies as well as various choices of interpolation and restriction operators.We focus on the application of the different types of preconditioners to the linear systems generated by the inexact Newton method.And it is found that the V-Cycle restricted Schwarz method based on the second-order scheme is very beneficial to our problems.Numerical experiments show that the proposed algorithm has good robustness and scalability for the large-scale solutions of standard benchmarks and realistic problems as well as problems with highly heterogeneous media.Secondly,the dual-porosity and dual-permeability model is coupled with the Peng-Robinson equation of state to solve the gas flow problem in fractured porous media.To simulate this dual continuous mathematical model,we introduce and study a highly parallel and scalable fully implicit Newton-Krylov method.The nonlinear systems are solved by inexact Newton method with the line search at each time step,as well as the Jacobi system by the Schwarz preconditioned Krylov method at each Newton step.Then,an explicit-first-step,single-diagonalcoefficient,diagonally implicit Runge-Kutta is proposed for the fully implicit discretization,which is second-order and L-stable.To accelerate the convergence and improve the scalability of the solver,we extend a class of multilevel additive Schwarz methods.Experimental results show that the fully implicit multilevel Schwarz algorithm is highly efficient for solving both standard benchmarks as well as realistic problems with several hundreds of millions of unknowns and scalable to 8192 processors.Finally,a nonlinearly constrained pressure residual algorithm is proposed to solve the two-phase flow problem with double porosity and double permeability media.The second-order fully implicit scheme of the algorithm is used to release the limitation of time step size in the simulation at an extreme scale.For the fully implicit two-phase flow problem,the traditional iterative methods,such as the Newton-type method,often have long-time stagnation or no convergence due to the high nonlinearity of the system or the violation of the saturation boundedness.Then,the variational inequality is solved by a preconditioned active-set reduced space method.A nonlinear preconditioning technology based on pressure residual is mainly studied,which effectively eliminates the highly nonlinear components leading to iterative failure,and introduces a class of multi-level Schwarz preconditioners to reduce the condition number for the Jacobian system in the reduced space method.2D and 3D numerical experiments show that the proposed algorithm has good robustness and scalability.