A theoretical and numerical approach to some free boundary problems

Summary In this paper some free boundary problems related to the flow of fluids in porous media are studied. Using a method due to Baiocchi, for these problems we not only establish theoretical results (existence and uniqueness theorems for the solution) but at the same time develop an algorithm for the numerical approach of the solution. Such an algorithm is rigorous from a mathematical point of view and it competes very well with the ones already known both in simplicity of programming and in speed of execution. Entrata in Redazione il 18 luglio 1973. ? Laboratorio di Analisi Numerica del C.N.R. di Pavia ? and ? Università di Pavia ?. This work was supported by C.N.R. in the frame of L.A.N. at Pavia.     

On the solution of some boundary-value problems arising in elastico-viscous fluid mechanics

Résumé En mécanique des fluides visquex et élastiques, on rencontre souvent des équations différentielles du troisième ordre avec seulement deux conditions aux limites. On voit que la méthode ordinairement employée est approximative, n'est pas uniformément valable et donne des résultats faux. On donne également une autre méthode de résolution.     

A numerical scheme for multi-point special boundary-value problems and application to fluid flow through porous layers

In this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi-point boundary-value problems over adjacent domains. In each subdomain the solution is governed by a different equation. The solutions are required to be smooth across the interface nodes. The approach is based on using finite difference approximation of the derivatives at the interface nodes. Smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations. A modified multi-dimensional Newton’s method is proposed for solving the nonlinear system. The accuracy of the proposed scheme is validated by examples whose exact solutions are known. The proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media.     

On the stochastic versions of Neumann and oblique derivative problems

In some areas, for instance geodesy, one finds the need of suitably defining the solution of certain boundary value problems (BVPs), for instance, the Laplace equation, where boundary data are very irregular and can be described as fields of random variables, with suitable regularity constraints (Rummel and Sansò, Lecture Notes on Earth Sciences, Vol. 65, 1997). This item has been attacked in the literature, although mainly for the case of the Dirichlet problem while much less material is available, for instance, for the Neumann and the Oblique Derivative Problem. In studying these stochastic problems in detail, the authors have found fairly general criteria which provide an automatic translation of a deterministic result into the corresponding stochastic one.     

Free boundary problems in the theory of fluid flow through porous media: Existence and uniqueness theorems

Summary Elliptic free boundary problems in the theory of fluid flow through porous media are studied by a new method, which reduces the problems to variational inequalities: existence and uniqueness theorems are proved. Entrata in Redazione il 3 agosto 1972. Research supported by C.N.R. in the frame of the collaboration between L.A.N. of Pavia and E.R.A. 215 of C.N.R.S. and of Paris University. ? Laboratorio di Analisi Numerica del C.N.R. di Pavia ? and ? Università di Pavia ?. ? Università di Pavia ? and ? G.N.A.F.A. del C.N.R. ?.     

Mathematical problems on the fluid-solute-heat flow through porous media

A degenerate parabolic system arising from the fluid-solute-heat flow through partially saturated porous media is considered. The existence of weak solutions to the initial boundary value problem of this system is established by time discretization, and the continuity of the weak solutions is discussed.     

The fluid flow through porous media. Regularity of the free surface

The fluid flow through an earth dam separating two water reservoirs of different levels gives rise to a free boundary problem. In [1] we have proved the existence of a solution to this problem. In this paper we show that the free boundary is regular.     

Thermomechanical effects in the flow of a fluid in porous media

This paper deals with analysis, by methods of extended thermodynamics, of the thermomechanical effects which arise in the flow of a weakly viscous fluid in a porous medium. Under the hypothesis that the fluid fills all the interstices among the powder and that the size of the powder grains and of the interstices is much lower than a suitable characteristic length, linearized field equations are written, which include, in a natural way, terms which take into account the Dufour, Soret, and virtual mass effects. As a limiting case when the evolution time of the heat flux goes to infinite and no entropy flux is carried, the flow of liquid helium II in a porous medium is obtained.     

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.     

An elliptic-parabolic system arising from the fluid-solute-heat flow through saturated porous media

In this paper an elliptic-parabolic coupled system arising from the fluid-solute-heat flow through a saturated porous medium is considered. The uniqueness and the existence of classical solutions are proved. The asymptotic behavior of solutions for large time is shown, too.     

A conforming mixed finite-element method for the coupling of fluid flow with porous media flow

Salim Meddahi ^{We consider a porous medium entirely enclosed within a fluidregion and present a well-posed conforming mixed finite-elementmethod for the corresponding coupled problem. The interfaceconditions refer to mass conservation, balance of normal forcesand the Beavers–Joseph–Saffman law, which yieldsthe introduction of the trace of the porous medium pressureas a suitable Lagrange multiplier. The finite-element subspacesdefining the discrete formulation employ Bernardi–Raugeland Raviart–Thomas elements for the velocities, piecewiseconstants for the pressures and continuous piecewise-linearelements for the Lagrange multiplier. We show stability, convergenceand a priori error estimates for the associated Galerkin scheme.Finally, we provide several numerical results illustrating thegood performance of the method and confirming the theoreticalrates of convergence.}     

Free boundary fluid flow in a trough: A numerical approach

This paper examines the slow flow of a viscous liquid in an open rectangular container, one side (the base) of which moves steadily along its own plane, thereby providing the driving force the liquid needs. Unlike the two vertical sides that are rigid and stationary, the top side is left open so that the upper part of the liquid is in contact with air and is being controlled by surface tension and gravity. A numerical procedure for obtaining solutions for the cases when the capillary numbers are small is provided and the curves of the free boundaries obtained here are presented for some flow parameters. The deviation of the shape of the free boundary is observed to be strongly dependent on the aspect ratio of the boundary (i.e., the ratio of the vertical to horizontal spread of the liquid) with its curvature changing sign in the interval [1, 1.5].     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

Numerical modeling of fluid flow in anisotropic fractured porous media

A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). A function of fluid exchange between the fractures and porous blocks depending on flow direction is given. The flow function is based on the difference between the pressure gradients. This feature enables one to take into account anisotropic properties of filtration in a more general form. The results of numerical solving a model two-dimensional problem are presented. The computational algorithm is based on a finite-element space approximation and explicit-implicit time approximations.     

Evolution of microstructure in unstable porous media flow: A relaxational approach

We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two‐phase flow in a porous medium leads to a mathematically ill‐posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well‐posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.     

On some questions arising in numerical realization of amplitude-phase methods

For computing rapidly oscillating solutions of certain second order differential equations a new version of amplitude-phase methods has recently been proposed in [11]. Error estimates were given to approximate solutions for large arguments in [15]. One of the most important points in these methods is the introduction of Prüfer transformation modified by auxiliary functions. Their appropriate choice makes the methods applicable and efficient. When implementing and applying the methods to practical problems, we face some further questions. In this paper we describe and try to answer them. Efficiency of the methods is confirmed by numerical experiments on concrete problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

On some singular perturbation problems arising in stochastic control

We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space.

Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. Similar results hold in the affine case under mean square stability of the linear part of the equation.