Study Of Fully Implicit Algorithms For Reaction Transport Problems In Heterogeneous Media

Flow problems in porous media are widespread in daily life,such as groundwater management,oil and gas field development,natural gas collection,coal mining,and so on.In the petroleum industry,in order to assist the exploitation programmes of oil fields and improve the efficiency of oil extraction,mathematical modelling simulating the reactive transport behaviour of fluids in subsurface rock are used.Due to the rapid advancement of research on fluid mechanics,many important problems of multi–physical models,high precision,long time span simulation and high complexity of rock formations emerging in the industrial production,which bring serious challenges to numerical simulation.Based on these difficulties,the development of highly robustness and stability algorithms for coupled reactive transport problems in heterogeneous media is becoming an important direction to solve these problems.In the work,in order to overcome the CFL stability condition and fit the fluid flow pattern,the spatial term and the convection term are discretized by fully implicit finite volume method and upwind scheme in the coupled reaction transport model,respectively.For the time term,two discretization schemes,the fully implicit first–order backward difference scheme(BDF–1)and the second–order backward difference scheme(BDF–2),are used to improve the accuracy of numerical simulation while ensuring the correctness of time discretization.To solve the large–scale nonlinear systems formed after discretization,a fully implicit Newton–Krylov method with high parallelism and scalability is investigated and applied in depth.For each time step,a non–exact Newton method with backtracking is used to process the nonlinear system of equations,and the Krylov subspace iteration method is used as linear solution algorithm to get the Newton iteration direction.Since Jacobi systems corresponding to the Newton iteration directions are usually large,sparse and pathological,the design of the algorithm for solving Jacobi systems will directly determine the performance of the whole algorithm.In response to this issue,based on the idea of “ divide and conquer” domain decomposition,the entire computational domain is decomposed and integrated from a geometric point of view,a class of overlapping restricted additive Schwarz preconditioners are constructed and applied,which can guarantee the robustness and parallel scalability of the simulator.Finally,several numerical results pertaining to the problems in two dimension are presented to illustrate the efficiency,robustness,and the overall performance of the fully implicit simulator.We run test cases on PETSc and focus on the following aspects:(a)verifying that the fully implicit algorithm can achieve theoretical convergence accuracy with respect to time and space discretization;(b)comparing the effects of different temporal discretization methods;(c)exploring the role of different approximation methods in the discretization of convection terms;and(d)exploring the effects of different preconditioners and subdomain processors in the parallel framework of Newton–Krylov algorithm.