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In this paper, we present a discontinuous finite volume method for a hybrid-dimensional coupled fracture model on triangular meshes. In matrix and fracture, the pressure is approximated by the discontinuous and continuous piecewise linear elements, and in order to retain local conservation property, the velocity is sought in the discontinuous lowest-order Raviart–Thomas element space and piecewise linear element space. The proposed numerical scheme is only associated with the pressure, so we avoid solving an indefinite saddle-point problem. In addition, the discontinuous method eliminates the continuity of approximate functions across the interelement boundary, so it has the advantage of high localizability and easy handling of complicated geometries. Optimal order error estimates for the pressure and velocity are proved, and numerical experiments are carried out to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a 4-point finite volume method for a fracture model coupling one-dimensional equations in the fracture with two-dimensional equations in surrounding domains. The pressure is approximated in the piecewise constant spaces, whereas the velocity is calculated by the lowest order Raviart-Thomas elements and piecewise constants in matrix and fracture, respectively. Optimal order error estimates are proved on nonuniform triangular meshes for both the pressure and velocity. Beside, we extend the 4-point finite volume method to nonmatching grids between the fracture and matrix without loss of any accuracy. Numerical experiments on matching and nonmatching meshes are tested for models with higher, lower and anisotropic fracture permeability, and results confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a block-centered finite difference method is derived for the Darcy-Forchheimer compressible wormhole propagation in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration, and auxiliary flux in different discrete norms are established rigorously and carefully on nonuniform grids. Finally, some numerical experiments are demonstrated to verify the theoretical analysis and effectiveness of the given scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze a finite volume method based on the Crouzeix–Raviart element for the coupled fracture model, where the fluid flow is governed by Darcy's law in the one‐dimensional fracture and two‐dimensional surrounding matrix. In the numerical scheme, the pressure in the matrix and fracture is respectively approximated by the Crouzeix–Raviart elements and piecewise constant functions, and then the velocity is calculated by piecewise constant functions element by element. The existence and uniqueness of the numerical solution are discussed, and optimal order error estimates for both the pressure p and the velocity u are proved on general triangulations. We finally carry out numerical experiments, and results confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a node-centered finite volume method based on triangulations for a fracture model is presented, in which we restrict the pressure to the linear finite element space while the velocity can be approximated by constant vectors element by element. The numerical scheme is established just associated with the pressure to avoid the saddle-point problem. Error estimates of O ( h ) accuracy for the discrete H 1 semi-norm and the discrete L 2 norm of pressure p and the ( L 2 ) 2 norm of velocity u are developed on general triangulations. Under an additional assumption about essentially symmetric control volumes, the error estimates for the pressure p can be improved to O ( h 3/2 ). Finally, numerical experiments are carried out to verify the accuracy and convergence rates for the proposed finite volume scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mortar finite volume method for a fractured model of flow in porous media. In this model, the permeability coefficients are variable between the fracture and the surrounding porous media. A finite volume method based on Raviart–Thomas elements combined with the mortar technique of domain decomposition is presented, in which sub-domains are triangulated independently and the meshes do not match at interfaces. The great advantage of the method is avoiding solving the saddle-point problem, since the numerical scheme is just related to the pressure p , and the velocity u can be expressed by p . We also prove error estimates of order h on the discrete H 1 norm between the exact solution p and the mortar finite volume solution P and the ( L 2 ) 2 norm between u and U . Finally, numerical experiments have been performed to show the consistency of the convergence rates with the theoretical analysis. [ABSTRACT FROM AUTHOR]

This study concerns the Darcy flow problem in a two-dimensional fractured porous domain, in which the fracture is regarded as a one-dimensional interface, interacting with the surrounding media. In this paper, a finite volume element method (FVEM) is first proposed for the multi-dimensional fracture model, and error estimates for the pressure with optimal convergence are discussed. On this basis, a two-grid FVEM is developed for decoupling the multi-domain fracture model by a coarse grid approximation to the interface coupling conditions, and theoretical analysis demonstrates that approximation accuracy does not deteriorate under the two-grid decoupling technique. Finally, numerical experiments for FVEM and two-grid FVEM are presented to confirm the accuracy of theoretical analysis. [ABSTRACT FROM AUTHOR]