Darcy Problem Research Articles

Low-order Raviart–Thomas approximations of axisymmetric Darcy flow

Abstract In this paper, we study the lowest-order mixed finite element method for the axisymmetric Darcy problem using Raviart–Thomas elements. In contrast to the Cartesian setting, the method is non-conforming in the sense that the discretely divergence-free functions are not solenoidal. We derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions. We show that if the domain is convex, then the errors converge with optimal order modulo logarithmic terms. Numerical experiments are presented, and they indicate that the estimates are sharp.

Mixed-Dimensional Geometric Multigrid Methods for Single-Phase Flow in Fractured Porous Media

This paper deals with the efficient numerical solution of single-phase flow problems in fractured porous media. A monolithic multigrid method is proposed for solving two-dimensional arbitrary fracture networks with vertical and/or horizontal possibly intersecting fractures. The key point is to combine two-dimensional multigrid components (smoother and intergrid transfer operators) in the porous matrix with their one-dimensional counterparts within the fractures, giving rise to a mixed-dimensional geometric multigrid method. This combination seems to be optimal since it provides an algorithm whose convergence matches the multigrid convergence factor for solving the Darcy problem. Several numerical experiments are presented to demonstrate the robustness of the monolithic mixed-dimensional multigrid method with respect to the permeability of the fractures, the grid size, and the number of fractures in the network.

A Crank–Nicolson discontinuous finite volume element method for a coupled non-stationary Stokes–Darcy problem

In this paper, we propose a fully discrete scheme which is based on discontinuous finite volume element methods in space and a Crank–Nicolson discretization in time, and we obtain the solution at the first time step by applying a first order backward Euler method to solve the coupling of time-dependent Stokes–Darcy problem. The proposed numerical method is established on a baseline finite element family of discontinuous linear elements for the approximation of the velocity and hydraulic head, whereas the pressure is approximated by the piecewise constant elements. Then, the unique solvability of the approximate solution for the discrete problem is derived. In addition, the optimal error estimates of the full discretization in standard L2− norm and broken H1− norm are obtained. Finally, a series of numerical experiments are provided to illustrate the features of the proposed method, such as the optimal accuracy orders, adaptive mesh refinement, mass conservation, capability to conveniently deal with complicated geometries with different types of boundary conditions, and the applicability to the problems with realistic parameters. In particular, the last numerical experiment shows the capability of the method to deal with complicated networks of fractures in porous media domain, it encompasses both fractures with high permeability which act as the prior flow conduit, and fractures with low permeability as a flow barrier.

Spectral discretizations of the Darcy's equations with non standard boundary conditions

This paper is devoted to the approximation of anonstandard Darcy problem, which modelizes the flow in porous media, byspectral methods: the pressure is assigned on a part of the boundary.We propose two variational formulations, as well as three spectraldiscretizations. The second discretization improves the approximation of thedivergence-free condition, but the error estimate on the pressure is notoptimal, while the third one leads to optimal error estimate with adivergence-free discrete solution, which is important for someapplications. Next, their numerical analysis is performed in detailand we present some numerical experiments which confirm the interestof the third discretization.

Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems

In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.

A divergence free weak virtual element method for the Stokes–Darcy problem on general meshes

This paper presents a weak virtual element method on general meshes for the Stokes–Darcy problem with the Beavers–Joseph–Saffman interface condition. The velocity is discretized by the H(div) virtual element. The pressure is approximated by discontinuous piecewise polynomials. Besides, a polynomial space on the element faces is introduced to approximate the tangential trace of the velocity in the Stokes equations. The velocity on the discrete level is exactly divergence free and thus the exact mass conservation is preserved in the discretization. The well-posedness of the discrete problem is proved and an a priori error estimate is derived that implies the error for the velocity in a suitable norm does not depend on the pressure. A series of numerical experiments are reported to illustrate the performance of the method.

A simple embedded discrete fracture–matrix model for a coupled flow and transport problem in porous media

Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

An object-oriented framework for multiphysics problems combining different approximation spaces

An object-oriented framework is developed to implement discrete models in a monolithic solution of coupled systems of partial differential equations by combining different kinds of finite element approximation spaces chosen for each field. In this sense, we aim at contributing to applications of what is currently classified as “multiphysics simulation”, by composing numerical techniques where each physical phenomenon or scale component is approximated by its most appropriate numerical scheme. The integration of the methods is discussed, in a systematic and generic manner, based on an existing object oriented finite element computational framework, allowing any of the usual kinds of affine and/or curved element geometry (point, segment, triangle, quadrilateral, tetrahedral, hexahedral, prismatic or pyramidal). They can be used in several finite element formulations that require continuous, discontinuous, H(div)-conforming functions, or interactions with lower dimensional approximations, which typically occurs in hybrid methods or reduced models. Furthermore, to improve accuracy and/or efficiency, when necessary and/or allowed by the adopted mathematical formulation, different levels of mesh refinements can be adopted for each field, with different refinement configurations (h, p, hp, and directional refinements), as long as the meshes are nested. The generality and flexibility of the proposed framework is verified against a particular set of two dimensional test problems modeled by different formulations - Darcy's problem coupled with tracer transport, fluid flow coupled with geomechanical interaction, multiscale hybrid formulation for linear elasticity, and a hybrid mixed formulation for discrete fracture networks. The main implementation ideas can also be applied to three dimensional problems, as well as to other kinds of approximation space combinations.

Partitioned penalty methods for the transport equation in the evolutionary Stokes–Darcy‐transport problem

There has been a surge of work on models for coupling surface‐water with groundwater flows which is at its core the Stokes–Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes–Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algorithm development for the evolutionary transport problem has, however, focused on a quasi‐static Stokes–Darcy model and a single domain (fully coupled) formulation of the transport equation. This report presents a numerical analysis of a partitioned method for contaminant transport for the fully evolutionary system. The algorithm studied is unconditionally stable with one subdomain solve per step. Numerical experiments are given using the proposed algorithm that investigates the effects of the penalty parameters on the convergence of the approximations.

Stability analysis of MacCormack rapid solver method for evolutionary Stokes–Darcy problem

A numerical method based on the explicit–implicit scheme of MacCormack finite difference scheme was applied to the solution of Parabolized Navier–Stokes equations. Partitioned methods, which solve the coupled problem by successively solving the sub-physics problems, have recently been studied for the full evolutionary groundwater–surface water flow with convergence established over both bounded and long time intervals. A MacCormack rapid solver method based on interface approximation via temporal extrapolation is proposed for devising decoupled marching algorithms for the mixed model. Under a time step limitation of the form C1h≤Δt≤C2h (where h, Δt are mesh size and time step, respectively, and Cj, j∈{1,2}, are two physical parameters) we prove both uniform and asymptotic stabilities over long time intervals. Some numerical experiments are presented and discussed.

Dual virtual element method in presence of an inclusion

We consider a Darcy problem for saturated porous media written in dual formulation in presence of a fully immersed inclusion. The lowest order virtual element method is employed to derive the discrete approximation. In the present work we study the effect of cells with cuts on the numerical solution, able to geometrically handle in a more natural way the inclusion tips. The numerical results show the validity of the proposed approach.

A multi-grid technique for coupling fluid flow with porous media flow

In this paper, we consider the coupling of fluid flow with porous media flow. A multi-grid finite element method for the coupled Stokes–Darcy problem with the Beavers–Joseph interface condition is proposed and discussed. The optimal error estimates are obtained. Numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi-grid method.

Long time stability and convergence rate of MacCormack rapid solver method for nonstationary Stokes–Darcy problem

We propose and study a combination of two second-order implicit–explicit (IMEX) methods for the coupled Stokes–Darcy system that governs flows in karst aquifers. The first is a second-order explicit two-step MacCormack scheme and the second is a second-order implicit Crank–Nicolson method. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. This combination so called the MacCormack rapid solver method is very efficient (faster, at least of second order accuracy in time and space) and can be easily implemented using legacy codes. Under time step limitation of the form Δt≤Ch (where h,Δt are mesh size and time step, respectively, and C is a physical parameter) we prove both long time stability and the rate of convergence of the method. Some numerical experiments are presented and discussed.

Dual Virtual Element Method for Discrete Fractures Networks

The accurate description of fluid flow and transport in fractured porous media is of paramount importance to capture the macroscopic behaviour of an oil reservoir, a geothermal system, or a CO2 sequestration site, to name few applications. The construction of accurate simulation model for flow in fractures is challenging due to the high ratios between a fracture's length and width, which makes modeling by lower-dimensional manifolds a natural option. In this paper we present a mixed-dimensional Darcy problem able to describe pressure and Darcy velocity in all the dimensions, i.e. in the rock matrix, in the fractures, and in their intersections. Moreover, we present a mixed-dimensional transport problem which, given the Darcy velocity, describes coupled advection and diffusion of a passive scalar into the fractured porous media. The approach can handle both conducting and blocking fractures. Our computational grids are created by coarsening of simplex tessellations that conform to the fractures surfaces. An accurate choice of the discrete approximation of the previous model, by virtual finite element and finite volume, allows us to simulate complex problem with a good balance in term of accuracy and computational cost. We illustrate the performance of our method by comparing to benchmark studies for two-dimensional fractured porous media, as well as a complex three-dimensional fracture geometry.

A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

A stabilized finite volume method for the evolutionary Stokes–Darcy system

In this paper, we propose a stabilized fully discrete finite volume method based on two local Gauss integrals for a non-stationary Stokes–Darcy problem. This stabilized method is free of stabilized parameters and uses the lowest equal-order finite element triples P1–P1–P1 for approximating the velocity, pressure and hydraulic head of the Stokes–Darcy model. Under a modest time step restriction in relation to physical parameters, we give the stability analysis and the error estimates for the stabilized finite volume scheme by means of a relationship between finite volume and finite element approximations with the lower order elements. Finally, a series of numerical experiments are provided to demonstrate the validity of the theoretical results.

A stabilized finite volume element method for a coupled Stokes–Darcy problem

In this paper, we present a stabilized finite volume element method with the conforming finite element triples P1–P0–P1 and P1–P1–P1 for approximating the velocity, pressure, and hydraulic head of a coupled Stokes–Darcy problem. The proposed method is convenient to implement, computationally efficient, mass conserving, optimally accurate, and able to handle complex geometries; therefore, this method has great potential to be useful for realistic problems involving coupled free flow and porous media flow. To offset the lack of the inf-sup condition of the P1–P0 and P1–P1 elements for the Stokes equation, a parameter free stabilization term is added to the discrete formulation. Stability and optimal error estimates are proved based on a bridge built up between the finite volume element method and the finite element method. An element level implementation of the stabilization term is discussed so that an existing code package can be conveniently modified to handle the stabilization procedures. A series of numerical experiments are provided to illustrate the above features of the proposed method, the theoretical results, and the realistic applications.

A Multi-grid Decoupling Method for the Coupled Fluid Flow with the Porous Media Flow

In this paper, we propose a multi-grid decoupling method for the coupled Navier–Stokes–Darcy problem with the Beavers–Joseph–Saffman interface condition. The basic idea of the method is to first solve a much smaller global problem on a very coarse initial grid, then solve a linearized Newton problem and a Darcy problem in parallel on all the subsequently refined grids. Error bounds of the approximate solution for the proposed method are analyzed, and optimal error estimates are obtained. Numerical experiments are conducted to verify the theoretical analysis and indicate the effectiveness of the proposed method.

A stabilized cut finite element method for the Darcy problem on surfaces

We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three dimensional mesh as trial and test functions. Since we consider a partial differential equation on a surface, the resulting discrete weak problem might be severely ill conditioned. We propose a full gradient and a normal gradient based stabilization computed on the background mesh to render the proposed formulation stable and well conditioned irrespective of the surface positioning within the mesh. Our formulation extends and simplifies the Masud–Hughes stabilized primal mixed formulation of the Darcy surface problem proposed in Hansbo and Larson (2016) on fitted triangulated surfaces. The tangential condition on the velocity and the pressure gradient is enforced only weakly, avoiding the need for any tangential projection. The presented numerical analysis accounts for different polynomial orders for the velocity, pressure, and geometry approximation which are corroborated by numerical experiments. In particular, we demonstrate both theoretically and through numerical results that the normal gradient stabilized variant results in a high order scheme.

Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow

We present an a priori stability and convergence analysis of a new mixed discontinuous Galerkin scheme applied to the instationary Darcy problem. The analysis accounts for a spatially and temporally varying permeability tensor in all estimates. The proposed method is stabilized using penalty terms in the primary and the flux unknowns.