Unsaturated porous media flow with thermomechanical interaction

We propose a model for unsaturated poro‐plastic flow derived from the thermodynamic principles. For the isothermal case, the problem consists of a degenerate coupled system of two PDEs with two independent hysteresis operators describing hysteresis phenomena in both the solid and the pore fluids. Under natural hypotheses, we prove the existence of a global strong solution for this system. Copyright © 2015 John Wiley & Sons, Ltd.     

Travelling waves of phase transitions in porous media

Liquid–vapour phase changes for a fluid flow through a porous medium are considered; in particular, the model allows for phase mixtures and includes an equilibrium pressure. Existence and uniqueness of travelling waves is established in a wide range of situations; the end states may be formed either by pure phases or mixtures; in the latter case the pressure equals the equilibrium pressure. A formal asymptotic analysis for vanishing relaxation time is carried out to show that the friction and reaction source terms have smoothing effect when the pressure is close to the equilibrium pressure and pure phases are avoided.     

A multicomponent flow model in deformable porous media

We propose a model for multicomponent flow of immiscible fluids in a deformable porous medium accounting for capillary hysteresis. Oil, water, and air in the soil pores offer a typical example of a real situation occurring in practice. We state the problem within the formalism of continuum mechanics as a slow diffusion process in Lagrange coordinates. The balance laws for volumes, masses, and momentum lead to a degenerate parabolic PDE system. In the special case of a rigid solid matrix material and three fluid components, we prove under further technical assumptions that the system is mathematically well posed in a small neighborhood of an equilibrium.     

Unfolding homogenization in doubly periodic media and applications

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.     

On unsteady dispersion flow in porous media

The generalising dispersion equations of flow through porous media have been investigated. The Laplace transform has been applied to obtain the solution to dispersion problem as a result of adsorption. The generalised closed form solution for dispersion has been presented and the different types of variations in concentration have been graphically discussed. When the steady state occurs, the concentration becomes constant but for small value of time (say 0.5) the concentration tends to zero as distance increases.     

Bingham flow in porous media with obstacles of different size

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability of reaction-fronts in porous media

The stability of reaction-fronts in porous media is studied with analytical and numerical methods. A stability criterion has been derived using linear stability analysis assuming a sharp font. The sharp front assumption is an approximation of the mathematical model in the limit of an infinite rapid reaction. The criterion shows that the stability of a sharp reaction front is dependent on the permeability that develops behind it. The sharp front is unstable for perturbations of any wave-length if the permeability increases behind the front. The criterion shows that short wave-length perturbations are more unstable than long wave-length perturbations. The sharp front is labile when the permeabilities are the same at both sides of the front. This means that the perturbed front moves unchanged forward. Finally, perturbations will die out in case the permeability decreases behind the sharp front. The stability of non-sharp fronts are simulated numerically when dissolution is by first order kinetics, the transport is by convection and diffusion and when the permeability and specific reactive surface depends on the porosity. The numerical experiments behave according to the stability criterion.     

Traveling waves in porous media with phase-dependent damping

We consider a simple model for the fluid flow in a porous medium. The model consists of a hyperbolic system of balance laws, which take into account phase changes and allow for metastable states thanks to the introduction of an equilibrium pressure. A damping term is included as well, which depend not only on the velocity but also on the phase of the fluid; in particular, it vanishes in the vapor phase. The existence and uniqueness of traveling waves is proved in several important cases.     

On the well-posedness of incompressible flow in porous media with supercritical diffusion

In this article we study the heat transfer equation with a supercritical diffusion term of an incompressible fluid in porous media governed by Darcy's law. We obtain the global well-posedness for small initial data belonging to critical Besov spaces and the local well-posedness for arbitrary initial data. We further show the pointwise blowup criterion.     

Compressible multicomponent flow in porous media with Maxwell-Stefan diffusion

We introduce a Darcy-scale model to describe compressible multicomponent flow in a fully saturated porous medium. In order to capture cross-diffusive effects between the different species correctly, we make use of the Maxwell–Stefan theory in a thermodynamically consistent way. For inviscid flow, the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell-Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore, it is proven by relative entropy techniques that solutions of the Darcy-scale model tend in a certain long-time regime to solutions of a parabolic limit system.     

FILTRATION IN PARTIALLY SATURATED AND PARTIALLY DRIED LAYERED POROUS MEDIA

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

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

A stochastic model for fronts in porous media

We analyze a stochastic model for the motion of fronts in two-phase fluids and derive upscaled equations for the capillary pressure. This extends results of [11], where the same law for the capillary pressure was derived under an assumption on typical explosion patterns. With the work at hand we remove that assumption and show that in the stochastic case the upscaled equations hold almost surely.     

Application of homogenization theory related to Stokes flow in porous media

We consider applications, illustration and concrete numerical treatments of some homogenization results on Stokes flow in porous media. In particular, we compute the global permeability tensor corresponding to an unidirectional array of circular fibers for several volume-fractions. A 3-dimensional problem is also considered.     

Bifurcation of nonlinear problems modeling flows through porous media

This paper deals with a class of nonlinear boundary value problems which appears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems. This work is supported in part by NSF of Shandong Province and NNSF of China     

Derivation and analysis of an effective model for biofilm growth in evolving porous media

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore‐scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore‐scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm‐liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.     

On the dimension of global attractors for porous media equations

We provide a short proof on the infinite dimensionality of global attractors for a class of porous media equations. The proof is mainly based on the Z₂ index theory and proper use of energy functions and is completely different from the approaches in the existing literatures (M. Efendiev and S. Zelik, Finite and infinite dimensional attractors for porous media equations, Proc. London Math. Soc. 2008, 96:51–77; M. Efendiev, Infinite dimensional attractors for porous medium equations in heterogeneous medium, Math. Meth. Appl. Sci. 2012, DOI: 10.1002/mma.2619). Copyright © 2013 John Wiley & Sons, Ltd.     

On some local property of the shape of interfaces for the porous media equation

We consider the porous media equation with absorption for various conditions and prove that the shape of ist interface never becomes strongly upward convex. For this sake we derive an improperly posed estimate for solutions of the porous media equation for the non‐characteristic Cauchy problem (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indecomposable quasi‐characteristics scheme on pyramidal stencil and its application for numerical simulation of two‐phase flows through heterogeneous porous medium

A new high‐resolution indecomposable quasi‐characteristics scheme with monotone properties based on pyramidal stencil is considered. This scheme is based on consideration of two high‐resolution numerical schemes approximated governing equations on the pyramidal stencil with different kinds of dispersion terms approximation. Two numerical solutions obtained by these schemes are analyzed, and the final solution is chosen according to the special criterion to provide the monotone properties in regions where discontinuities of solutions could arise. This technique allows to construct the high‐order monotone solutions and keeps both the monotone properties and the high‐order approximation in regions with discontinuities of solutions. The selection criterion has a local character suitable for parallel computation. Application of the proposed technique to the solution of the time‐dependent 2D two‐phase flows through the porous media with the essentially heterogeneous properties is considered, and some numerical results are presented. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 44–55, 2002