The infiltration problem with large constant surface flux in partially saturated porous media

Infiltration problem with two kinds of degenerate cases is considered. After proving the existence of weak solution, we study when water contentc(u(·,t)) has compact support and how the solution behaves ast. Moreover, we get existence and regularity of two kinds of free boundaries under some conditions on the initial date.Tsinghua University     

ONE DIMENSIONAL FILTRATION PROBLEM IN PARTIALLY SATURATED LAYERED POROUS MEDIA

ＯＮＥＤＩＭＥＮＳＩＯＮＡＬＦＩＬＴＲＡＴＩＯＮＰＲＯＢＬＥＭＩＮＰＡＲＴＩＡＬＬＹＳＡＴＵＲＡＴＥＤＬＡＹＥＲＥＤＰＯＲＯＵＳＭＥＤＩＡ￥ＸＩＡＯＳＨＵＴＩＥ（萧树铁）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，Ｔｓｉｎｇｈｕａ...     

FILTRATION IN PARTIALLY SATURATED AND PARTIALLY DRIED LAYERED POROUS MEDIA

1.IntroductionThefiltrationprobleminInferedporousmediaarisesfromthestudiesofwatermovementduringirrigationandofthesalinizationofsoil.Thisproblemhasbeenelaboratelyinvestigatedforthesaturatedcase,whileforthegeneralcase,worksseemtoconcentrateonlyonexperimentalandnumericalaspects.Theseworksrevealsomeinterestingtheoreticalquestions.FOrexample,HillandParlange[2]foundin1972thattheverticalinfiltrationofwaterintwo-layeredsandconstitutedwithfinerupperlayerandcoarserlowerisunstable.Afterthemathematicali…     

Some properties of the solution of a filtration problem in partially saturated porous media

A filtration problem with second initial-boundary value in partially saturated porous media is considered, In addition to discussion of the existence and uniqueness of weak solution of the problem, it is demonstrated that the interface between the saturated and unsaturated regions exists and continues under certain conditions and the solution possesses some properties, e.g., the balance of water content, the time-limit existence of weak solution ect. which differ from those of the solutions of the first initial-boundary value problem.     

The boundary value problems for the filtration equation in partially saturated porous media

In this paper we considered the first and the second boundary value problems for one-dimensional filtration in partially saturated porous media. The existence, uniqueness and regularity of weak solutions were obtained. We also studied the interface problems and discussed the asymptotic behavior of weak solutions.     

Exponential stability in porous media problem saturated by multiple immiscible fluids

This paper concerns a continuum theory of porous media saturated by multiple immiscible fluids. The case of a porous media saturated by two immiscible fluid proposes some new mathematical difficulties. We study the exponential stability of the one-dimensional problem when the nonwetting fluid is trapped in the wetting fluid and the exponential stability of the anti-plane shear deformations when the two fluids saturate the elastic media.     

Longitudinal waves in partially saturated porous media: the effect of gas bubbles

The problem of the propagation of longitudinal waves in a liquid-saturated porous medium when there are gas bubbles present is considered. The decay factor and the phase velocity of Frenkel–Biot waves of the first and second kind are found as a function of the frequency in the linear approximation. It is shown that, in the neighbourhood of the resonance frequency of the bubbles, longitudinal Frenkel–Biot waves change their form. A wave of the first kind is transformed from a fast wave at low frequencies into a slow wave at high frequencies. The dispersion curve of a wave of the second kind consists of two branches – a “low-frequency” branch, the oscillations of which possess the classical properties, and a “high-frequency” branch, which is a weakly decaying high-velocity mode. The frequency dependences of the ratio of the mass velocities of a gas-liquid mixture and of a porous matrix, and also of the perturbations of the stress in the matrix and the pressure in the mixture, are constructed. It is shown that the “high-frequency” branch of a wave of the second kind is characterized by the in phase motion of the gas-liquid mixture and of the porous matrix, while their mass velocities are close, which explains the weak decay of this mode of oscillations. An analytical expression is obtained for the “boundary frequency”, which determines the offset of the “high-frequency” branch of the dispersion curve of the wave of the second kind.     

A singular-degenerate free boundary problem arising from the moisture evaporation in a partially saturated porous medium

Summary We consider a moisture evaporation process in a porous medium which is partially saturated by a fluid. The mathematical model is a singular-degenerate nonlinear parabolic free boundary problem. We first transform the problem into a weak form in a fixed domain and then derive some uniform estimates for the proper approximate solution. The existence of a weak solution is established by a compactness argument. Finally, the regularity of the solution and interfaces are investigated.     

Elastic wave damping in thin-layered saturated porous media

Monochromatic wave propagation in thin-layered saturated porous media is examined by averaging differential equations with rapidly oscillating coefficients. Particular attention is given to the transformation mechanism for the damping of such waves. Existing results in this area /1/ are extended and refined.     

Discrete mass conservation for porous media saturated flow

Global and local mass conservation for velocity fields associated with saturated porous media flow have long been recognized as integral components of any numerical scheme attempting to simulate these flows. In this work, we study finite element discretizations for saturated porous media flow that use Taylor–Hood (TH) and Scott–Vogelius (SV) finite elements. The governing equations are modified to include a stabilization term when using the TH elements, and we provide a theoretical result that shows convergence (with respect to the stabilization parameter) to pointwise mass‐conservative solutions. We also provide results using the SV approximation pair. These elements are pointwise divergence free, leading to optimal convergence rates and numerical solutions. We give numerical results to verify our theory and a comparison with standard mixed methods for saturated flow problems. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 625–640, 2014     

An inverse pollution problem in porous media

In this paper, we deal with an inverse source problem of integro-differential parabolic equations, which comes from nonlinear pollution problems in porous media. We study the existence and uniqueness of solutions for the direct problem as well as the existence of quasisolutions of the inverse source problem in an appropriate class of admissible source functions.     

On the filtration through porous media with partially soluble permeable grains

In this paper we consider the incompressible viscous fluid flow through a porous medium whose grains are also permeable and release mass to the flow. In each component porosity and permeability depend on saturation. The flow is modelled with a nonlinear parabolic equation for the pressure, with a degenerate parabolic term, depending not only on the saturations, but also on the space variable and on time averages of the saturation. We generalize the classic approach of Alt and Luckhaus to this situation and establish existence of at least one weak solution and bounds for its absolute value. Received December 1998     

Microscopical investigation of wave propagation phenomena in residual saturated porous media

Wave propagation is used in many fields for measurement and characterization. Corresponding multiphase models usually use a continuous approach. Nevertheless, systems like wetted rocks may be saturated residually in certain situations. In such cases, one fluid is distributed as clusters, each different in size and shape. One single, continuous phase cannot account for a variety of fluid clusters, either disconnected from each other or connected only about thin liquid films. Therefore, we present a model that considers a heterogeneous distribution of disconnected fluid clusters in the form of harmonic oscillators. These oscillators are described and distinguished by their mass, damping and eigenfrequency. Hence, the model allows to characterize different clusters and includes an additional damping mechanism due to oscillations of the fluid clusters. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational homogenization and nonlinear effects in heterogeneous fluid saturated porous media

For the two-scale modelling of deforming fluid-saturated porous structure we apply the asymptotic homogenization approach to the fluid-structure interaction problem involving linear elastic porous skeleton and the Newtonian compressible fluid. The sensitivity analysis of effective coefficients depending on the geometrical configuration is used to introduce a weakly nonlinear formulation which enables to capture influences of the deformation on the material properties. The paper is devoted to the comparison of the linear and the weakly nonlinear models (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

A finite element collocation method for variably saturated flows in porous media

One common formulation of Richard's equation for variably saturated flows in porous media treats pressure head as the principal unknown and moisture content as a constitutive variable. Numerical approximations to this “head-based” formulation often exhibit mass-balance errors arising from inaccuracies in the temporal discretization. This article presents a finite-element collocation scheme using a mass-conserving formulation. The article also proposes a computable index of global mass balance.