The Rain on Underground Porous Media Part I: Analysis of a Richards Model

The Richards equation models the water flow in a partially saturated underground porous medium under the surface. When it rains on the surface, boundary conditions of Signorini type must be considered on this part of the boundary. The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler’s scheme in time and finite elements in space. The convergence of this discretization leads to the well-posedness of the problem.     

A finite-element coarse-grid projection method for incompressible flow simulations

Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure-correction schemes used for the incompressible Navier-Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. The minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions.     

On a semi-coercive problem in a porous journal bearing with cavitation problem as boundary condition and subject to an integral condition

We study a non-linear problem in pressure saturation modelling of a free boundary problem, arising in self-lubricating bearings, with Neumann boundary conditions for the pressure and a non-local constraint on the saturation variable, which indeed is a Lagrange multiplier. We prove an existence theorem by introducing an artificial time dependence and using the pseudo-characteristics discretization method and semi-coercive variational inequalities.     

Solving the homogeneous isotropic linear elastodynamics equations using potentials and finite elements. The case of the rigid boundary condition

In this article, elastic wave propagation in a homogeneous isotropic elastic medium with a rigid boundary is considered. A method based on the decoupling of pressure and shear waves via the use of scalar potentials is proposed. This method is adapted to a finite element discretization, which is discussed. A stable, energy preserving numerical scheme is presented, as well as 2D numerical results.     

Steady incompressible flow around objects in general coordinates with a multigrid solution method

Steady incompressible flow around objects in general coordinates is investigated. First, an overview of the popular approaches to discretize incompressible flow problems in general coordinates is given. It has been chosen to solve the equations on a staggered grid with contravariant flux unknowns and pressure as primitive variables. A solution method multigrid is used, with a line smoother able to deal with stretched cells. For flow problems around objects solved with a single block discretization periodic boundary, conditions are prescribed and adaptations for the discretization and the multigrid method are given. Steady flow around a circular cylinder and around an ellipse are presented. © 1994 John Wiley & Sons, Inc.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.     

A three field stabilized finite element method for the Stokes equationsUne méthode d'éléments finis stabilisée à trois champs pour les équations de Stokes

We consider in this work the boundary value problem for Stokes equations on a two dimensional domain in cases where non-standard boundary conditions are given. We study the cases where pressure and normal or tangential components of the velocity are given in different parts of the boundary and solve the problem with a minimal regularity. We introduce the problem and its variational formulation which is a mixed one. The principal unknowns are the pressure and the vorticity, the multiplier is the velocity. We present the numerical discretization which needs some stabilization. We prove the convergence and the behavior of the a priori error estimates. Some numerical tests are also presented. To cite this article: M. Amara et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 603–608.     

On Finite Displacement of an Elastoviscoplastic Material in a Gap between Two Rigid Coaxial Cylindrical Surfaces

In the framework of the theory of large deformations, we obtain the solution of a boundary value problem on the flow of an elastoviscoplastic material in a gap between two rigid coaxial cylindrical surfaces under pressure drop changing with time. It is assumed that slip of the material is possible on both surfaces. We consider reversible deformation, the development of viscoplastic flow under the increasing and constant pressure drop, deceleration of the flow under the decreasing pressure drop, and the unloading of the medium.     

Approximation of Navier–Stokes incompressible flow using a spectral element method with a local discretization in spectral space

A spectral element technique is examined, which builds upon a local discretization within the spectral space. To approximate a given system of equations the domain is subdivided into nonoverlapping quadrilateral elements, and within each element a discretization is found in the spectral space. The difference is that the test functions are divided into the higher-order polynomials, which have zero boundaries and lower-order polynomials, which are nonzero on one boundary. The method is examined for Navier–Stokes incompressible flow for fluid flow within a driven cavity and for flow over a backstep. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 587–599, 1997     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.     

Error Estimates for Finite-Element Navier-Stokes Solvers without Standard Inf-Sup Conditions

The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C^1 elements for velocity and C^0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.     

Hybrid Finite Element Method for Two-phase Miscible Displacement in Porous Media

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Gas injection into a moist porous medium with the formation of a gas hydrate

The model problem of the formation of a gas hydrate when a gas is injected into a porous medium, filled in the initial state with a gas and water, is considered in the one-dimensional approximation. A detailed pattern of the seepage flow with phase transitions for different modes of gas injection is obtained. Three seepage modes in a porous medium are possible, which differ qualitatively in the temperature and hydrate saturation fields. At low boundary pressures no hydrate is formed and the temperature distribution increases monotonically. As the boundary pressure increases, when the corresponding values of the pressure and temperature on the phase diagram lie in the region of gas-hydrate stability (below the equilibrium curve), a purely frontal pattern of hydrate formation is obtained with a monotonic temperature distribution. When the boundary pressure is increased further, an extended region of hydrate formation appears with a convex temperature profile, where, depending on the values of the boundary pressure, the hydrate saturation may be continuous (at high boundary pressures) or change abruptly at lower boundary pressures.     

Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case

We introduce an approximation procedure and provide existence results for two-phase flow equations in porous media. The medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations. Our main interest is the outflow boundary condition which models an interface with open space. The approximate system introduces standard boundary conditions and can be used in numerical schemes. It allows the derivation of maximum principles. This is the basis for the derivation of the limiting system in the form of a variational inequality.     

Spectral discretization of Darcy equations coupled with Stokes equations by vorticity–velocity–pressure formulations

We propose to make the numerical analysis of a model coupling the Darcy equations in a porous medium with the Stokes equations in the cracks. The coupling is provided by a pressure continuity on the interface. We describe a discretization by spectral element methods. We derive a priori optimal error estimates and we present some numerical experiments which confirm the results of the analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1628–1651, 2017     

A One‐dimensional flow problem in porous media with hydrophile grains

We study a free boundary problem describing the propagation of the wetting front following the injection of a liquid into a porous medium with hydrophile granules. The absorption process produces a non‐local interaction with the flow so that the porosity appearing in the parabolic equation for pressure is a functional of saturation and of the free boundary. Our analysis is confined to the unsaturated regime, which is the first stage of the process. An existence theorem is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

On the penalty-projection method for the Navier–Stokes equations with the MAC mesh

We deal with the time-dependent Navier–Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear’s scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method.     

Algebraic preconditioning for Biot-Barenblatt poroelastic systems

Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.     

Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.     

Homogenization of a two‐phase incompressible fluid in crossflow filtration through a porous medium

We study the homogenization of a slow viscous two‐phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid‐fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two‐phase fluid in the free domain, a coupled Darcy law connecting two‐phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.