Mathematical analysis of variable density flows in porous media (original) (raw)
Related papers
Fluids
The viscous fluid flow past a semi-infinite porous solid, which is proportionally sheared at one boundary with the possibility of the fluid slipping according to Navier’s slip or second order slip, is considered here. Such an assumption takes into consideration several of the boundary conditions used in the literature, and is a generalization of them. Upon introducing a similarity transformation, the governing equations for the problem under consideration reduces to a system of nonlinear partial differential equations. Interestingly, we were able to obtain an exact analytical solution for the velocity, though the equation is nonlinear. The flow through the porous solid is assumed to obey the Brinkman equation, and is considered relevant to several applications.
On the stability and uniqueness of the flow of a fluid through a porous medium
Zeitschrift für angewandte Mathematik und Physik, 2016
In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman's equation to include inertial effects. Such flows could have relevance to enhanced oil recovery and also to the flow of dense liquids through porous media. In any event, one cannot ignore the fact that flows through porous media are inherently unsteady, and thus, at least a part of the inertial term needs to be retained in many situations. We study the stability of the rest state and find it to be asymptotically stable. Next, we study the stability of a base flow and find that the flow is asymptotically stable, provided the base flow is sufficiently slow. Finally, we establish results concerning the uniqueness of the flow under appropriate conditions, and present some corresponding numerical results.
Transactions of the American Mathematical Society, 1983
The one-dimensional porous media equation u, = (um)xx, m > 1, is considered for x 6E R, r>0 with initial conditions u(x,Q) = u0(x) integrable, nonnegative and with compact support. We study the behaviour of the solutions as l-» oo proving that the expressions for the density, pressure, local velocity and interfaces converge to those of a model solution. In particular the first term in the asymptotic development of the free-boundary is obtained. 0. Introduction. Suppose we have a certain distribution of gas whose density at time t = 0 is given by a function u0(x) of one spatial direction (x E R). If the gas flows through a homogeneous porous medium the density u = m(x, t) at time t > 0 is governed by the equation (0.1) u,= iu">)xx for x E R and t > 0; m is a. physical constant, m > 1, and we have scaled out other physical constants (see [1] for a physical derivation), u satisfies the initial condition (0.2) k(x,0) = «"(*) where u0 satisfies the following assumptions: (0.3) w0GF'(R), «">0,ii"z0, and u0 is compactly supported, i.e. if ß0 = {x E R: u0ix) > 0} we have (0.4) a, = essinf ß0 >-oo, a2-esssupß0 < oo. Sticking to the above application we define the pressure by v = mum~x/im-1) on Q = R X (0, oo) and the local velocity by V =-vx on the domain of dependence (0.5) ß = ß[w] = {(x,t) E Q: u{x,t)>0}. The total mass at time i>0 is Af(i) = / w(x, t) dx and the center of mass is xcit)-Mit)'1 j w(x, t)x dx. Set M0 = / u0ix) dx and x0-M0~'/ u0ix)x dx: M0 > 0 and ax < x0< a2. l0 = a2-ax measures the dispersion of the initial data. Much is already known for problem (0.1)-(0.4); see [19] for a survey of results up to 1980, where the «-dimensional case is considered, n > 1. In particular (0.1)-(0.4)
MATHEMATICAL MODELS OF A DIFFUSION-CONVECTION IN POROUS MEDIA
Mathematical models of a diffusion-convection in porous media are derived from the homogenization theory. We start with the mathematical model on the microscopic level, which consist of the Stokes system for a weakly compressible viscous liquid occupying a pore space, coupled with a diffusionconvection equation for the admixture. We suppose that the viscosity of the liquid depends on a concentration of the admixture and for this nonlinear system we prove the global in time existence of a weak solution. Next we rigorously fulfil the homogenization procedure as the dimensionless size of pores tends to zero, while the porous body is geometrically periodic. As a result, we derive new mathematical models of a diffusion-convection in absolutely rigid porous media.
On the effect of boundaries in two-phase porous flow
Nonlinearity, 2015
In this paper we study a model of an interface between two fluids in a porous medium. For this model we prove several local and global well-posedness results and study some of its qualitative properties. We also provide numerics.
The Discontinuous Solutions of the Transport Equations for Compositional Flow in Porous Media
In the present paper we consider a multicomponent multiphase isothermal flow in porous media with mass exchange between phases. The system of equations of multiphase multicomponent flow has discontinuous solutions, but is not hyperbolic, except some particular cases. For this general, non-hyperbolic system, we propose a free energy condition to select unique physically admissible discontinuous solutions. We also develop a geometrical procedure which provides a tool to analyze the free energy condition. For a two-component mixture, analytical formulae are obtained for the allowed discontinuities.
Chemical Engineering Science, 2007
A challenging problem for diffusive mass transport is to describe and model the phenomena concerning the fluid-porous medium inter-region. Volume averaging techniques that provide a framework for rigorously addressing the issue of obtaining macroscopic models from pointwise models at fluid and porous medium scales have been used to attend the problem. The efforts have resulted in two modeling approaches. The first one, known as the one-domain approach (ODA), considers the system as a continuum where the geometrical (e.g., porosity) and transport parameters (e.g., diffusivity) display rapid spatial changes in the inter-region. The second one, known as the two-domain approach (TDA), uses different models for the fluid and the porous medium scales, and matches them via the development of corresponding jump conditions at the dividing surface. transfer between a microporous medium and an homogeneous fluid: jump boundary conditions. Chemical Engineering Science 61, 1692-1704] have shown that the coefficients involved in the jump conditions can be computed by solving the associated closure problems. However, in the development of the jump conditions some complications arise due to the difficulty of modeling some of the "surface excess" transport mechanisms that take place in the inter-region. To address this problem, an implicit formulation based on the ODA and TDA is proposed. Although the ODA seems to be more suitable for modeling, it requires the knowledge of the spatial variations of the transport parameters. Heuristic interpolations between the fluid and the porous medium parameters have been commonly used; however, there is no guarantee that such models can provide an accurate description of the mass flux. Within a ODA framework, the aims of this paper are: (i) to show that the effective diffusivity coefficient for the case of passive diffusion in a fluid-porous medium inter-region can be posed as a closure problem derived from volume averaging techniques and (ii) to use a simple one-dimensional model to show that a complete knowledge of spatial variations of diffusivity and porosity are necessary for an accurate description of the mass transport phenomena in the entire fluid-porous medium system. The analysis has allowed us to identify a new contribution to the jump at the dividing surface. This contribution consists of the accumulation that occurs at the dividing surface even when there is no chemical reaction or adsorption taking place. ᭧