An Efficient Numerical Method For A Class Of Coupled Energy Problems With Multiple Physical Fields

The energy problem of multi physical field coupling is widely used in engineering field.The most common one is the coupling of Navier-Stokes equation and Darcy equation.Navier-Stokes equation is a motion equation describing momentum conservation of viscous incompressible fluid,which reflects the basic mechanical law of viscous fluid flow.Darcy equation describes the basic law of fluid seepage in rock and soil pores,and reflects the law of water seepage in rock and soil pores,that is,the relationship between seepage energy loss and seepage velocity.However,in many practical applications,the Navier-Stokes-Darcy model has some limitations in describing the heterogeneity of porous media.It is worth noting that there is not only one kind of pore in the natural fractured reservoir,but two kinds of coexisting and interacting media,namely dense matrix and microfracture.In addition,it is more important to characterize the intrinsic characteristics and accurately simulate the flow interaction between various media.The existing Navier-Stokes-Darcy model does not consider dual porous media when coupling porous media flow with free flow.Therefore,in order to improve the reliability of the simulation,dual porosity Navier-Stokes model is used to describe the shale oil acquisition process.In order to reduce the scale of solution and save storage space,we need to build an efficient numerical algorithm to decouple the velocity and pressure,and use the fully overlapping domain decomposition parallel algorithm to realize the numerical simulation of the coupling problem,so as to have a deeper understanding of the law of fluid motion.On the basis of previous studies,this thesis makes study on the coupling problem of multiple physical fields:(1)Based on the traditional coupled finite element method,the coupling and decoupling stabilization finite element method of dual porosity Navier-Stokes model which is coupled between free flow region and porous medium region is studied by using four interface conditions on the interface.The first stage decoupling is based on the idea of block time step technology based on four interface conditions to decouple free flow region and double pore flow zone.The second stage decoupling is to decouple the double pore equation by mass exchange term,which makes the coupling problem be decomposed into three sub problems in a non-iterative way.In order to improve the efficiency of the calculation,a new parallel decoupling finite element method is proposed,which combines the decoupling stabilization finite element method with the finite element parallel algorithm of fully overlapping region decomposition,and presents a basic scheme for solving coupled dual porosity Navier-Stokes equations.The time is in the first order Euler lattice,and the calculation of the three linear terms is in the Oseen form of space non iteration.In addition,the stability and convergence of the coupling scheme and the decoupling scheme are compared.Finally,the theoretical results are verified by numerical experiments with real solutions.Comparing the accuracy and calculation time between parallel and serial coupled finite element methods and decoupling methods,we find that compared with the coupled finite element method,the serial decoupling finite element method has better convergence and the parallel decoupling finite element method is the most effective.(2)Based on the dual porosity Stokes coupling model,the Lagrange multiplier method of dual porosity Stokes model is derived.Specifically,the research method uses Lagrange multipliers to impose interface conditions.Therefore,it can be used in the process of non-uniform domain decomposition.In addition,with the development and optimization of the code for porous media flow,each subproblem is solved alternately or simultaneously.Secondly,the existence of weak solution is proved by using the derived variational formula,which is the basis of approximate solution of domain decomposition strategy.Finally,the error estimates of discretization are given.(3)Combining Navier-Stokes equation with Darcy equation,a mathematical model of seepage in heterogeneous porous media is established.With appropriate parameters,the equation can be used to model free flow or porous media flow without detailed understanding of the interface between the two regions.Therefore,the Navier-Stokes-Brinkman equation provides an alternative for the coupling of Darcy equation and Navier-Stokes equation.In this chapter,after a brief review of the Navier-Stokes-Brinkman problem and its discretization process,a posteriori error estimation method based on residuals is proposed.