Numerical Simulations And Analysis For The Flow In Fractured Porous Media

The simulation of fluid flow in fractured porous media is of great significance for petroleum and gas reservoirs,water resource management,and safe storage of atomic waste or carbon dioxide.Different filling materials in fractures can lead to differences in permeability relative to the surrounding porous media,making the fracture act as a conduit or barrier,accelerating or blocking fluid flow.Due to the complexity and specificity of fracture distribution,accurate and efficient numerical simulation methods have been a challenging research topic in recent years.This article mainly considers the hybrid-dimensional fracture model,in which the fracture is regarded as(n-1)-dimensional interface in n-dimensional porous medium.It mainly involves three numerical methods,weak Galerkin method,conforming discontinuous Galerkin method and the immersed finite element method.Finally,we use the idea of normal direction averaging the fracture to derive the hybrid-dimensional miscible displacement problem.In Chapter 2 of this paper,we consider the weak Galerkin coupled conforming finite element method for solving the hybrid-dimensional fracture model,where the control equations for the porous matrix and the(n-1)-dimensional interface satisfy Darcy’s law and the mass conservation equation.The weak Galerkin finite element method is an extension of the traditional finite element method,introducing new concepts such as weak functions,weak gradient operators,discrete weak functions,and discrete weak gradients.The advantage of coupled weak Galerkin method for solving the fracture model is its compatibility with any polygonal or polyhedral mesh,which can solve the problem of limited computational mesh in practical geophysical problems.Moreover,for the interface elements where fractures exist,we only need to connect the intersection points of fractures and elements to update the mesh,so this method is also feasible to non-matching mesh.In addition,the discrete system of the coupled weak Galerkin method is symmetric and positive definite,and does not require adjusting problem-related penalty factors.In Chapter 3,we present a conforming discontinuous Galerkin method for solving the fracture model,which is based on the modified weak gradient operator.By increasing the polynomial order of the approximated discrete weak gradient,this method can also be extended to general polygonal or polyhedral meshes.In contrast to the previously discussed method,this method offers the following advantages,firstly,there are no need for any penalty term or stabilizer,making the discrete format as simple as the classical finite element method.Secondly,there are fewer degrees of freedom.Specifically,we introduce a two-grid algorithm for the conforming discontinuous Galerkin method,which calculates the solution in the fracture using the coarse mesh of the original coupled system and then decouples the solution on the fine mesh,significantly reducing the computation time.For above two numerical algorithms,we provide optimal estimates for L2 norm and energy norm,then demonstrate their effectiveness through numerical and benchmark examples.Chapter 4 of this paper addresses the hybrid-dimensional fracture model as an elliptic interface problem with coupled interface conditions.We propose a new discrete fracturematrix approach method on non-matching meshes that is based on the Petrov-Galerkin immersed finite element method(PG-IFE).Unlike traditional PG-IFE methods,the local stiffness matrix of the local functions in interface elements are coupled with degrees of freedom in fracture.Furthermore,we extend this method to cases where the interface intersection points do not coincide with the fracture degrees of freedom.This marks the first instance of extending the PG-IFE method to the hybrid-dimensional fracture model.The numerical scheme has the following advantages:(1)work well on non-matching mesh;(2)the numerical scheme is the simpler than exsiting various numerical methods;(3)it can capture the continuity or discontinuity of pressure at the interface,i.e.,it is effective for numerical simulation of conductive fractures and blocking fractures;(4)compared with the extended finite element methods,it does not require additional degrees of freedom in the interface element.Extensive numerical examples,including numerical examples and some well-known benchmark examples,demonstrate the feasibility and effectiveness of this method.All the numerical results indicate that this method can effectively simulate fluid flow in fracture media.Chapter 5 derives a hybrid-dimensional model for miscible displacement of incompressible fluids.First,we simplify the diffusion-dispersion tensor in the fracture region as a diagonal matrix.Then,using the averaging principle and the mass conservation principle at the(n-1)dimensional fracture,we derive the hybrid-dimensional model for convection-diffusion mode.The interface condition ensures the uniqueness of the numerical solution and can capture both the continuity and discontinuity of concentration.Numerical experiments based on the finite element method show that when the permeability at the fracture is higher than that in the porous matrix,the fluid tends to flow through the fracture,leading to changes in concentration and a higher concentration distribution at the fracture than in the porous matrix.On the other hand,when the permeability at the fracture is lower than that in the porous matrix,the fracture acts as a blocking fracture,and the fluid does not flow through the fracture but instead flows along the path defined by the fractures.In this paper,the fluid flow problem in fractured porous media is studied from the aspects of model derivation and numerical simulations.The error analysis of relevant problems is given and the feasibility and effectiveness of the proposed method are verified by a large number of numerical examples.