A non-equilibrium model of a porous medium saturated with immiscible fluids

A model of a thermoelastic porous medium, saturated with two immiscible fluids, is considered. It is assumed that there are no phase transitions, the contribution of pulsations to the stress and kinetic energy is small, and that the components of the medium are in thermodynamic equilibrium. The non-equilibrium of the state, related to the finite time of redistribution of the fluids among the pores of the channels due to the presence of surface forces, is taken into account. A general form of the governing relations, necessary and sufficient to satisfy the principles of thermodynamic compatibility and independence of the choice of system of coordinates, is obtained. It is shown that the establishment of equilibrium is accompanied by dissipation due to capillary forces, which does not lead to seepage dissipation or thermal dissipation. For the case when the deformation of the skeleton and the deviation of the mean porous pressure and the temperature from the initial values are small, while the saturation and the non-equilibrium parameter undergo finite changes, an approximation of the potential of the skeleton is proposed in the form of a quadratic expansion in small parameters. A feature of the expansion is the presence of an initial value of the potential, which depends on the saturation and non-equilibrium. The relationship between the thermodynamic potential and the non-equilibrium kinetics, related to the requirement that the dissipation by the capillary forces should be non-negative, is determined. A generalized Darcy's law is formulated, which takes cross terms into account. It is shown that the proposed approximations enable key effects, which accompany the motion of immiscible fluids in a porous medium, to be described.     

Finite element modeling of a leaking third component migration from a heat source buried into a fluid saturated porous medium

A finite element model for a leaking species migration from a heat source buried into a fluid saturated porous medium is demonstrated. A semi-implicit algorithm is coupled with the velocity correction procedure to solve the transient equations of the generalised porous medium model. A parametric study is carried out for different Darcy and Rayleigh numbers and size of the leaking hole. The results show that the leaking hole size has a significant effect on migration of the third component into the porous medium.     

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.     

Stability of magnetic fluid motions in a saturated porous medium

We examine the asymptotic stability of both equilibrium and arbitrary basic flows of a magnetic fluid saturated in a porous medium. In both cases, we determine the stability bounds and determine the conditions when these flows are asymptotically stable. We also establish the uniqueness for an initial boundary value problem of magnetic fluids in the porous medium.     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

Effective acoustic equations for a two-phase medium with microstructure

We study acoustic wave propagation in a two-phase medium in which the solid phase is a linear elastic material, and the fluid phase is assumed to be a compressible Newtonian barotropic fluid. Assuming that properties of the medium change rapidly on the small scale ε, we analyze the microscopic nonlinear Navier-Stokes equations and show that they can be linearized when ε tends to zero. Using a variant of Tartar's method of oscillating test functions, we derive effective acoustic equations which turn out to be viscoelastic. In order to treat disordered materials occurring in nature, we develop a new approach to describing geometry of a nonperiodic medium with length scale separation. Our approach is not based on probabilistic considerations. Instead, we postulate that certain inequalities hold uniformly on the microscale.     

A free boundary problem for a fluid flow in a heterogeneous porous medium

In this paper we study the problem of seepage of a fluid through a porous medium, assuming the flow governed by a nonlinear Darcy law and nonlinear leaky boundary conditions. We prove the continuity of the free boundary and the existence and uniqueness of minimal and maximal solutions. We also prove the uniqueness of theS ₃-connected solution in various situations.     

Parallel CPR preconditioner for a multicomponent fluid flow filtration problem in a porous medium

The constrained pressure residual (CPR) preconditioning method is considered with regard to solution of systems with matrices appearing in discretization of PDE systems describing multicomponent fluid flow in porous media. New versions of algorithms are proposed. Numerical experiments using an actual parallel hydrodynamic simulator were performed for test and actual oil fields in Western Siberia, these experiments confirm the efficiency of the methods.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

Radial displacements of an infinite liquid saturated porous medium with spherical cavity

The components of radial displacements in solid and liquid parts of a liquidsaturated porous medium with spherical cavity subjected to an arbitrary time dependent force have been obtained. Laplace transform technique has been used to solve the problem. Numerical calculation has been performed for two specific models. Variations of displacement in solid and liquid parts of the medium have been shown graphically.     

Adaptive approximation of a filtration problem for a viscous compressible fluid in a fissured porous medium

A filtration problem for a viscous compressible multiphase fluid mixture in a porous medium is considered in the case of high degree anisotropy caused by fissures. A correct spatial discretization is discussed. A comparison between the subgrid method and other multipoint methods is performed.     

The inverse problem for an acoustic equation in a weakly horizontally inhomogeneous medium

The inverse problem of reconstructing the coefficients A and B in the equation {fx379-01} in the half-plane z > 0 is considered. It is assumed that an instantaneous point source at z = 0 generates a wave field U(t, z, x), which is known on the boundary. It is also known that the coefficients A and B can be represented in the form {fx379-02}. Here ε is a small parameter. An algorithm for determinating the coefficients A₀, B₀, A₁, and B₁ with accuracy O(ε²) is constructed. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 81–99.     

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.     

On accelerated flows of an Oldroyd-B fluid in a porous medium

In this work, the problems dealing with unsteady unidirectional flows of an Oldroyd-B fluid in a porous medium are investigated. By using modified Darcy's law of an Oldroyd-B fluid, the equations governing the flow are modelled. Employing Fourier sine transform, the analytic solutions of the modelled equations are developed for the following two problems: (i) constant accelerated flow, (ii) variable accelerated flow. Explicit expressions for the velocity field and adequate tangential stress are obtained in each case. The solutions for Newtonian, second grade and Maxwell fluids in a porous medium appear as the limiting cases of the present analysis.     

Selfsimilar solutions of the problem of heat and mass transfer in a saturated porous medium with a volume heat source

Selfsimilar solutions of a system of stationary equations of heat condunction and filtration of molten material in the presence of a volume heat source generated by absorption of the energy of electromagnetic radiation, are considered. The possibility of the existence of a self-similar solution in the case of various (plane, cylindrical and spherical) spatial symmetries is studied. The existence of a selfsimilar solution is shown for the axisymmetric case when the radiation obeys a prescribed law. The influence of the surface volume heating and convective heat transfer due to filtration is studied. A solution for the case when the filtration of the molten phase is quasistationary is also investigated.     

On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium

We consider a system of nonlinear coupled partial differential equations that models immiscible two-phase flow through a porous medium. A primary difficulty with this problem is its degenerate nature. Under reasonable assumptions on the data, and for appropriate boundary and initial conditions, we prove the existence of a weak solution to the problem, in a certain sense, using a compactness argument. This is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter. We also establish a priori estimates for uniqueness of a solution.     

Exact analytical solution for a thermal boundary layer in a saturated porous medium

The forced convection thermal boundary layer in a porous medium as an analytically tractable special case of a mixed convection problem is considered. It is shown that some general features of the mixed convection solutions reported recently by other authors [B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett. (published online, 2005); M. Guedda, Multiple solutions of mixed convection boundary layer approximations in a porous medium, Appl. Math. Lett. (published online, 2005)] can already be recovered from this exactly solvable case.