Finite Volume Methods For Fracture Models:Analysis And Applications

In recent years,the modeling of fluid flow in porous media with fractures is important for many engineering applications such as petroleum reservoir modeling,nuclear waste storage monitoring,groundwater waste problem,and so on.It has been an important area of research.Compared to the permeability in surrounding domains,fractures are much more permeable or much less permeable due to crystallization.The permeability coefficient may vary over several orders of magnitude within a small region,and there are interactions between fractures and surrounding domains,hence fracture models always are very complicated.We introduce two kinds of fracture models in this paper:the discontinuous per-meability fracture model and the coupled fracture model.In the discontinuous permeability fracture model,the permeability tensor is discontinuous in the fracture and matrix.In general,the width of fracture is very small in contrast to the size of the whole domain,so applying the standard finite element method or finite volume method to this kind of model poses obvious problems.In this paper,we uso the mortar finite volume method to discrete fractures models,in which we can use different and independent partitions in fracture and surrounding domains,and the meshes will not match on interfaces of two-adjoint domains,so the mortar conditions are introduced to replace the continuous conditions in original problem.Another way to solve the problem of small width in fractures is establishing new fractures models,i.e.the coupled fracture model.The fracture is regarded as(n-1)-dimensional interface in n-dimensional domain,and the width of the fracture is ignored.The coupled fracture model is derived us-ing a process of averaging in the fracture.The dimension of fracture is reduced but the features of fluid flow in the fracture are not changed and interactions between fractures and surrounding matrix are taken into account in this model.In our paper,we will apply finite volume methods based on triangulations to the coupled fracture model.As a technique of dicretization,the finite volume method has the following advantages.It is adapt to complex problems and complex computational domains;triangular or quadrilateral meshes,structured or nonstructured meshes are allowed;numerical schemes of finite volume methods are simple to calculate,since test functions are piecewise constant;last but not least,finite volume methods hold local conservation property.For the coupled fracture model,the classical conforming finite volume method is considered,using continuous piecewise linear element to approximate the solution to partial difference equations;the nonconforming finite volume method uses nonconform-ing piecewise linear function to approximate solution;and for the discontinuous finite volume method,the approximation is discontinuous piecewise linear function.The test functions of methods are all piecewise constant,and since trial functions and test functions are from different spaces,two partitions are needed in finite volume method,the primal partition corresponding to trial functions and the dual partition corresponding to test functions.And it is also because the difference between trial and test functions,the theoretical analysis of finite volume methods are much more difficulty than finite element methods.In the following of this paper,we will describe the constructions of dual partitions and control volumes for every kind of finite volume method in detail,analyze the existence and uniqueness of numerical solutions,and prove error estimates.At the end of each chapter,numerical experiments are carried out to illustrate the accuracy and efficiency of finite volume methods.In chapter 1,we briefly introduce the development and research results for finite volume methods,and the historical background and numerical models for fracture problems.Finally,we give the outline of our paper.The content of chapter 2 is based on the following paper.S.Chen,H.Rui,A Mortar Finite Volume Method for a Fractured Model in Porous Media,J.Math.Anal.Appl.448(2017)707-721.In chapter 2,the discontinuous permeability fracture model is solved.The permeability tensor in matrix and fracture is discontinuous.Fluid flow in subdomains satisfies Darcy's law and mass conservation,and the pressure and flux are continuous across the interface of subdomains.The fracture model with Dirichlet boundary condition is described as follows.divui = qi in Qi,i = 1,2,f,Ui =-Ki(?)pi in ?i,i=1,2,f,P_i = Pi on ?i,i = 1,2,f,Pi = Pf on ?if,i=1,2,ui·n = uf·n on ?if,i=1,2,Based on the idea of approximating flux in interior edges by the pressure defined on two sides of every edge and mortar technique,we establish the finite volume method with triangulations for the above fracture model.We integrate the first equation in every triangle,and then by partial integrate,the integral in every triangle is transferred into the flux on the boundary of triangle.The pressure is approximated by a constant,and flux defined in every edge is approx-imated by the difference of the pressure in two triangles sharing the edge.So the numerical scheme is only related to the pressure and we avoid the saddle-point problem compared to the mixed finite element method.The mortar finite element method as a coupling technique is important in solving problems with complicated domains and interacting processes,since it allows us to use independent grids,form different discretization methods,or even solve different mathematic problems in different parts of the domain.We use different partitions in different domains,and meshes on interfaces will not match,so the mortar conditions are introduced to replace the continuous conditions in the original fracture model.We get the approximate pressure from the finite volume scheme,and then the flux on every edge can be expressed.Using the definition of element defined on the Raviart-Thomas finite element space with the lowest order(RT0),we can calculate the velocity in RT0.Besides,the convergence theory is proved.Error estimates show that the discrete H1 seminorm and L2 norm for the error of pres-sure and the(L2)2 norm of velocity on the non-matching triangular grids are all first-order accuracy.Finally,numerical experiments are carried out to verify the performance of the proposed method,and numerical results show the consistency of the convergence rates with the theoretical analysis.In a word,compared to the mixed finite element method,we get the same results in two aspects of theory and practice for the two points flux approximation mortar finite volume method.But the established numerical scheme is only related to the pressure,so we avoid the saddle-point problem,and we use different triangulations in different subdomains by the mortar technique,which makes this method more flexible and accurate.The content of chapter 3 is based on the following two papers.S.Chen,H.Rui,A Node-Centered Finite Volume Method for A Fracture Model on Triangulations,Appl.Math.Comput.327(2018)55-69.S.Chen,H.Rui,The finite volume method based on the Crouzeix-Raviart element for a fracture model,Submitted.In chapter 3,we consider the coupled fracture model in 2-dimensional do-main.The whole domain ? is divided by fracture ? into two surrounding domains?i,i = 1,2,and the fracture is regarded as a 1-dimensional interface between 2-dimensional surrounding domains.Fluid flow in the fracture and matrix is governed by Dracy's law and mass conservation,and the interaction between fracture and matrix is taken into account.The classical finite volume methods based on triangulation are used for the coupled fracture model,the conforming finite volume method using continuous piecewise linear element(P1)to approximate pressure,and the non-conforming finite volume method using Crouzeix-Raviart element(CR)and piecewise constant to approximate pressure in matrix and fracture respectively.The primal partitions of those two methods are triangulations,however due to the difference of spaces for the trial functions,we should form difference dual partitions and control volumes for two methods.In the conforming finite volume method,the pressure is approximated by P1 element,and the degrees of freedom are defined on all nodes of primal partition.The dual partition is constructed by connecting the circumcenters of neighboring triangles and control volumes are corresponding to nodes.While in the nonconforming finite volume method,the freedom degrees are defined on all midpoints of edges,and the dual partition is formed as follows:Connecting an interior point and three vertices in any triangle and every triangle is divided into three smaller triangles.Two smaller triangles sharing the same edge is regarded as a control volume.The basic idea of methods are multiplying test functions on both sides of conservative equation and then integrating it in every control volume.By partial integrate,the integral in control volume is transferred into that defined on boundary.Approximate the integral and get the numerical finite volume scheme only about pressure.Then the velocity will be calculated by Darcy's law element by element.We also analysis the existence and uniqueness of solutions to two kinds of finite volume numerical schemes,and error estimates show that the discrete H1 seminorm and L2 norm of pressure and the(L2)2 norm of velocity all hold one-ordcer accuracy.For the conforming finite volume method,if the triangulation is essential symmetric,the discrete H1 semi-norm and L2 norm of pressure can be improved to o(h3/2).And we carry out numerical experiments to test the accuracy of proposed classical finite volume methods for the coupled fracture model.We know one important feature of finite volume mEthods is its local conservative property.So we compare classical finite volume methods with the corresponding finite element method in numerical experiments,and we obtain that finite volume methods hold the same convergence orders for the pressure and velocity as finite element methods,but results show that finite volume methods maintain the local conservation for the coupled fracture model.The coupled fracture model with Dirichlet boundary condition reads as fol-lows:ui =-Ki(?)pi in ?i,i=1,2,div ui = gi in ?i,i = 1,2,uf=-kf,yd(?)pf/(?)y in ?,(?)uf/(?)y= 9f +(u1·n1|?+u2 · n2|?)in ?,??u2·ni+?fpi= ?fpf-(1-?)ui+1 · ni+1 in ?,Pi = Pi on ?i,i=1,2,pf = pf on(?)?.The content of chapter 4 is based on the following paper.S.Chen,H.Rui,The Discontinuous Finite Volume Method for a Fracture Model,Submitted.In chapter 4,we apply the discontinuous finite volume method to the coupled fracture model,and establish a numerical scheme only associated with pressure.The pressure in matrix and fracture are approximated by discontinuous piecewise linear function and continuous piecewise linear function,while the test function-s in matrix and fracture are all piecewise constant.The discontinuous finite volume method is inspired by discontinuous Garlerkin(DG)method.The con-tinuity of the numerical solution across the interelement boundary is eliminated,and a penalty term is added to enforce the connection.DFVM combines the benefits of DG and FVM,such as the flexibility,highly order accuracy of DG,and conservative property,simplicity of FVM.For the conforming finite volume method discussed in chapter 3,the dual partition of the conforming finite volume method is constructed by connecting barycenters of two adjoint triangles,so con-trol volumes are correspondence with nodes of the primal partition.While for the non-conforming finite volume method,the dual partition is constructed by connecting interior points of triangles and vertices,and control volumes consistent of two small triangles sharing a common edge,so control volumes are corresponding with edges of primal partition.Therefore,for nodes and edges located on interface of domains,we cannot form the same complete control volume as interior nodes and edges.It is the reason that the optimal order convergence rate for the error estiCate in L2 norm of pressure is not obtained.In the discontinuous finite volume method,we connect the barycenter of every triangle with three vertices,and then every triangle is divided into three small triangles,and every small triangle is regarded as a control volume.The dual partition consists of all small triangles.Compared to classical finite volume method,the dual partition of the discontinuous finite volume method has the best local property since only one element in the primary partition is required for a control volume.Therefore,the discontinuous finite volume method has the advantage of high localizability and flexibility to handle complicate geometries.In this chapter,optimal order error estimates for the discrete H1 semi-norm and the L1 norm of the pressure are proved.Numerical experiments are completed to test the convergence rates of the proposed scheme,and results confirm our theoretical analysis.