This paper studies a model for flow in a fractured porous medium with intersecting fractures. The fractures are treated as lower-dimensional manifolds and then physical transmission conditions express the pressure jump and the continuity of the flux across the fractures. Specific attention is borne to the conditions when several fractures intersect. The resulting system is discretized with mixed finite element, and the well-posedness of both the continuous and the discrete problems are proved. Then a domain decomposition method is formulated, so that the problem is reduced to the set of fractures, and a simple preconditioner is proposed. Numerical results exemplify the performance of the method.

This article is concerned with a posteriori error estimates for a discrete-fracture, multidimensional, numerical model for flow in a fractured porous medium. Local residual error estimators are defined and upper and lower bounds in terms of these estimators for both the pressure and the Darcy velocity are derived. Numerical examples using these estimates for automatic grid refinement are given.

In this work we introduce a stabilized, numerical method for a multidimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the $$(n-1)$$-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a weakly consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM’s.

This paper is concerned with the numerical solution of porou s-media flow and transport problems, i. e. heterogeneous, advection-di ffusion problems. Its aim is to investigate numerical schemes for these problems in which di fferent time steps can be used in di fferent parts of the domain. Global-in-time, non-overlapping domain-decompo siti n methods are coupled with operator splitting making possible the di fferent treatment of the advection and di ffusion terms. Two domain-decomposition methods are considered: one uses the time-dependent Steklov–Poincaré operator and the other uses optimized Schwarz waveform rela xation (OSWR) based on Robin transmission conditions. For each method, a mixed formulat ion of an interface problem on the space-time interface is derived, and di fferent time grids are employed to adapt to di fferent time scales in the subdomains. A generalized Neumann-Neumann pr econditioner is proposed for the first method. To illustrate the two methods numerical result s for two-dimensional problems with strong heterogeneities are presented. These include both a cademic problems and more realistic prototypes for simulations for the underground storage of n uclear waste.

Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modeled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparing fracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations.

In this work, we present a novel discrete fracture model for single-phase Darcy flow in porous media with fractures of co-dimension one, which introduces an additional unknown at the fracture interface. Inspired by the fictitious domain method, this Lagrange multiplier couples fracture and matrix domain and represents a local exchange of the fluid. The multipliers naturally impose the equality of the pressures at the fracture interface. The model is thus appropriate for domains with fractures of permeability higher than that in the surrounding bulk domain. In particular, the novel approach allows for independent, regular meshing of fracture and matrix domain and therefore avoids the generation of small elements. We show existence and uniqueness of the weak solution of the continuous primal formulation. Moreover, we discuss the discrete inf-sup condition of two different finite element formulations. Several numerical examples verify the accuracy and convergence of proposed method.

This paper is concerned with the numerical solution of porous-media flow and transport problems, i.e. heterogeneous, advection–diffusion problems. Its aim is to investigate numerical schemes for these problems in which different time steps can be used in different parts of the domain. Global-in-time, non-overlapping domain-decomposition methods are coupled with operator splitting making possible the different treatment of the advection and diffusion terms. Two domain-decomposition methods are considered: one uses the time-dependent Steklov–Poincaré operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, a mixed formulation of an interface problem on the space–time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains. A generalized Neumann–Neumann preconditioner is proposed for the first method. To illustrate the two methods numerical results for two-dimensional problems with strong heterogeneities are presented. These include both academic problems and more realistic prototypes for simulations for the underground storage of nuclear waste.

The Optimized Schwarz method has been introduced and analyzed over the last decade, where the convergence speed of the Jacobi iteration is significantly enhanced by using general transmission conditions on the interfaces together with optimized parameters. In particular, Ventcell transmission conditions (see [3–6, 8–10]) have been studied for the primal formulation with different numerical schemes showing that the convergence of the Optimized Schwarz algorithm with Ventcell conditions is improved over that with Robin conditions. Ventcell conditions are second order differential conditions, see [12].

In this paper, we study two different model reduction strategies for solving problems involving single phase flow in a porous medium containing faults or fractures whose location and properties are known. These faults are represented as interfaces of dimension N − 1 immersed in an N dimensional domain. Both approaches can handle various configurations of position and permeability of the faults, and one can handle different fracture permeabilities on the two inner sides of the fracture. For the numerical discretization, we use the hybrid finite volume scheme as it is known to be well suited to simulating subsurface flow. Some results, which may be of use in the implementation of the proposed methods in industrial codes, are demonstrated.

In this paper we are interested in the fast path fracture and we aim to use global-in-time, nonoverlapping domain decomposition methods to model flow and transport problems in a porous medium containing such a fracture. We consider a reduced model in which the fracture is treated as an interface between the two subdomains. Two domain decomposition methods are considered: one uses the time-dependent SteklovPoincar{e} operator and the other uses optimized Schwarz waveform relaxation (OSWR) based on Ventcell transmission conditions. For each method, a mixed formulation of an interface problem on the space-time interface is derived, and different time grids are employed to adapt to different time scales in the subdomains and in the fracture. Demonstrations of the well-posedness of the Ventcell subdomain problems is given for the mixed formulation. An analysis for the convergence factor of the OSWR algorithm is given in the case with fractures to compute the optimized parameters. Numerical results for two-dimensional problems with strong heterogeneities are presented to illustrate the performance of the two methods.

This article is concerned with the numerical discretization of a model for incompressible two-phase flow in a porous medium with fractures. The model is a discrete fracture model in which the fractures are treated as interfaces of dimension 2 in a 3-dimensional simulation, with fluid exchange between the 2-dimensional fracture flow and the 3-dimensional flow in the surrounding rock matrix. The model takes into account the change in the relative permeabilities and in the capillary pressure curves which occurs at the interface between the fracture and the rock matrix. The model allows for barriers which are fractures with low permeability. Mixed finite elements and advective upstream weighting are used to discretize the problem and numerical experiments are shown.

We consider three-dimensional incompressible two-phase flow in a porous medium with fractures. Interaction between the fractures and the rock matrix is taken into account. The location, the geometry and the rock properties of the fractures are considered known (discrete model) and the fractures are modeled as two-dimensional interfaces (reduced model). The fractures and the rock matrix have different rock types. Two-phase flow is modeled using the global pressure formulation. The model is presented and numerical experiments are shown.

General hexahedral and quadrangular grids present a challenge for mixed finite elements for second-order, elliptic problems. We define and analyze a mixed finite element method for a mesh made up of star-shaped polygons. The scalar unknown is approximated by element-wise constants and the vector unknown is determined by its flux through the edges of the polygons. The elements are composite elements. Each polygon is split into triangles by taking an interior point of the polygon, one for which it is star-shaped, and considering the triangles radiating from that point and having one side as a side of the polygon. Convergence of the method is proven, and numerical experiments are shown to confirm the theoretical results.

Flow and transport problems in porous media are well-known for their high computational cost. In the far field simulation of an underground nuclear waste disposal site, one has to work with extremely different length and time scales, and highly variable coefficients while satisfying strict accuracy requirements. One strategy for tackling these difficulties is to apply a non-overlapping domain decomposition method which allows local adaptation in both space and time and makes possible the use of parallel algorithms. The substructuring method with a Steklov Poincaré operator, which is widely used by engineers for steady problems with strong heterogeneities, is a promising option. The optimized Schwarz waveform relaxation (OSWR) method, which has been developed over the last decade for finite element and finite volume methods, is another potential choice.

We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimer law while that in the surrounding matrix is governed by Darcy's law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimer law is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy's law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimer law in the entire domain when the Forchheimer coefficient in the matrix tends toward zero.

This paper is concerned with global-in-time, nonoverlapping domain decomposition methods for the mixed formulation ofthe diffusion problem. Two approaches are considered: one uses the time-dependent Steklov--Poincaré operator and theother uses optimized Schwarz waveform relaxation (OSWR) based on Robin transmission conditions. For each method, amixed formulation of an interface problem on the space-time interfaces between subdomains is derived, and differenttime grids are employed to adapt to different time scales in the subdomains. Demonstrations of the well-posedness ofthe Robin subdomain problems involved in the OSWR method and a convergence proof of the OSWR algorithm are given forthe mixed formulation. Numerical results for two-dimensional problems with strong heterogeneities are presented toillustrate the performance of the two methods.

A Composite Hexahedral Mixed Finite Element for hexahedral meshes is presented. This composite element is based on the decomposition of the hexahedron into five tetrahedrons. A diffusion equation with an anisotropic diffusion coefficient and a known exact solution is solved on Kershaw meshes. Error charts for the $L^2$-error in both the scalar and vector variables are shown.

The numerical method used here (see [6]) is a mixed finite element method based on the weak formulation of the problem