ONE DIMENSIONAL FILTRATION PROBLEM IN PARTIALLY SATURATED LAYERED POROUS MEDIA

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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.     

The infiltration problem with constant rate in partially saturated porous media

In the present paper we study the infiltration problem with constant rate in partially saturated porous media. The main results are: (i) There exists an unique weak solution to the infiltration problem with two degenerate situations. (ii) When the infiltration rate is not large enough, the partially saturated state will fall down to completely unsaturated state with the increase of time.     

FILTRATION IN PARTIALLY SATURATED AND PARTIALLY DRIED LAYERED POROUS MEDIA

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

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     

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     

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.     

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.     

On the uniqueness of the solution of a nonlinear filtration problem through a porous medium

The purpose of this work is to study a fluid flow through a porous medium governed by a nonlinear Darcy's law. We also impose a nonlinear semi-permeability condition on some part of the boundary of this medium. The main results are the continuity of the free boundary and the uniqueness of the solution. Received May 5, 1996 / In a revised form November 16, 1996 / Accepted December 17, 1996     

Influence of the inhomogeneity of porous media on the instantaneous compactification of the support of solution of the filtration problem

We study the phenomenon of instantaneous compactification and the initial behavior of the support of solution of the filtration equation for inhomogeneous porous media. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1035–1044, August, 2006.     

On a filtration problem through a porous medium

Summary We prove the existence of a unique solution for a free boundary problem relative to the stationary flow between two water reservoirs of different levels separated by a dam of a non-homogeneous porous medium. Entrata in Redazione il 5 maggio 1973.     

Linear waves at the free surface of a saturated porous media

The principal aim of this paper is to study the propagation of linear waves at the free surface of a saturated porous media. This problem is formulated as an eigenvalue problem with complex eigenvalues, and the solution is given in term of an orthogonal eigenfunction expansion, whose completeness has been taken for granted in the literature regarding the problem as a classical Sturm-Liouville problem, which is not the case due to the complex nature of the eigenvalues. The main purpose of the present work is to prove the completeness of the eigenfunctions for all possible physical values of the parameters involved, even for some values of the parameters, where previous numerical works have found abnormal behavior of the eigenvalues. In those cases if we mistakenly consider the problem as a Sturm-Liouville one, as has been done before, the eigenfunction expansion will not hold, but indeed we will prove that it does.On leave from Institute de Mecánica de los Fluidos, Universidad Central de Venezuela, Caracas, Venezuela.     

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.     

Two-dimensional filtration problem for solutions in a nonhomogeneous porous medium

We present a mathematical model of underground leaching by solutions filtering through a porous medium. The model describes the motion of solutions from injection to extraction boreholes, as well as dissolution and secondary deposition in reduced-pH regions. A numerical algorithm has been developed for solving the problem on a plane in the general case of a homogeneous medium that contains regions with various nonhomogeneities. The algorithm combines triangulation of the region with the finite element method. The model also allows slow variation over time of some of the process parameters, such as porosity and the filtration coefficient. Numerical results are reported for various cases. The model qualitatively describes the main regularities of underground leaching and can be used to study and understand the detailed dynamics of these processes. The model also fills gaps in geological prospecting data, and extraction curves constructed for different wells can be applied to determine the approximate location of a particular nonhomogeneity. Mathematical modeling can help to optimize mineral extraction by underground leaching. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 29–55, 2008.     

A filtration problem in a porous medium with variable permeability

Sunto Un problema di frontiera libera (moto di un fluido attraverso una diga in terra non omogenea) è ricondotto allo studio di una disequazione variazionale. Si danno condizioni sul coefficiente di permeabilità sufficienti ad assicurare che la parte bagnata della diga è un sottografico.

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Entrata in Redazione il 30 giugno 1976.

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This paper is partially supported by National Science Foundation Grant MPS 72-04959 A02, and by C.N.R. of Italy through L.A.N. of Pavia.