On the effect of boundaries in two-phase porous flow (original) (raw)
Closure Conditions for Two-Fluid Flow in Porous Media
Transport in Porous Media - TRANS POROUS MEDIA, 2002
Modeling of multiphase flow in porous media requires that the physics of the phases present be well described. Additionally, the behavior of interfaces between those phases and of the common lines where the interfaces come together must be accounted for. One factor complicating this description is the fact that geometric variables such as the volume fractions, interfacial areas per volume, and common line length per volume enter the conservation equations formulated at the macroscale or core scale. These geometric densities, although important physical quantities, are responsible for a deficit in the number of dynamic equations needed to model the system. Thus, to obtain closure of the multiphase flow equations, one must supplement the conservation equations with additional evolutionary equations that account for the interactions among these geometric variables. Here, the second law of thermodynamics, the constraint that the energy of the system must be at a minimum at equilibrium, ...
The closure problem for two-phase flow in homogeneous porous media
Chemical Engineering Science, 1994
In a previous study of two-phase flow in homogeneous porous media, the closure problem was presented in terms of a pair of boundary value problems involving four integro-differential equations for second-order tensor fields. In this paper we show how the original closure problem is transformed to one containing Stokes'-like equations that can be solved to determine the two permeability tensors and the two viscous drag tensors. The permeability tensors, Kp and K y ' are symmetric, exhibit a clear dependence on the volume fractions of the two phases, and may depend on the ratio of viscosities. On the basis of order of magnitude analysis, the coupling, or viscous drag tensors, Kpy and Kyp ' are found to be constrained by Kpy • Kyp = 0(1).
Mathematical analysis of variable density flows in porous media
Journal of Evolution Equations, 2015
We consider a simple model describing the motion of a two-component mixture through a porous medium. We discuss well-posedness of the associated initial-boundary value problem, in particular, with respect to the choice of boundary and far-field conditions. The existence of global-in-time solutions is proved in the ideal case when the fluid occupies the whole physical space. Finally, similar results are obtained also for the boundary value problems in the simplified 1-D geometry.
The Interfaces of One-Dimensional Flows in Porous Media
Transactions of the American Mathematical Society, 1984
transactions of the american mathematical society Volume 285, Number 2, October 1984 ñ>l J\x\<R that reduces to (H2) if N = 1, uo > 0. Whenever T* < oo we say that the solution blows up in a finite time. They also prove that for nonnegative solutions a necessary and sufficient condition for global existence, i.e. T* = oo, is
Homogenized model of immiscible incompressible two-phase flow in double porosity media : A new proof
arXiv: Analysis of PDEs, 2016
In this paper we give a new proof of the homogenization result for an immiscible incompressible two-phase flow in double porosity media obtained earlier in the pioneer work by A. Bourgeat, S. Luckhaus, A. Mikeli\'c (1996) and in the paper of L. M. Yeh (2006) under some restrictive assumptions. The microscopic model consists of the usual equations derived from the mass conservation laws for both fluids along with the standard Darcy-Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, i.e. where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium is discontinuous. The important difference with respect to the results of the cited papers is that the global pressure function as well as the saturation are also discontinuous. This makes the init...
Journal of Mathematical Analysis and Applications, 2007
In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.
On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium
Journal of Mathematical Analysis and Applications, 2007
We consider a system of nonlinear coupled partial differential equations that models immiscible twophase 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.
Diffuse-interface Modeling of Two-phase Flow for a One-component Fluid in a Porous Medium
Transport in Porous Media, 2006
The diffuse-interface (DI) model for the two-phase flow of a one-component fluid in a porous medium has been presented by Papatzacos [2002, Transport Porous Media 49, 139-174] and by Papatzacos and Skjaeveland [2004, SPE J. (March 2004), 47-56]. Its main characteristics are: (i) a unified treatment of two phases as manifestations of one fluid with a van der Waals type equation of state, (ii) the inclusion of wetting, and (iii) the absence of relative permeabilities. The present paper completes the presentation by including the implementation of wetting in the general case of a mixed-wet rock. As a result of this implementation, some statements are made about capillary pressure, confirming similar statements by Hassanizadeh and Gray [1993, Water Resour. Res. 29, 3389-3405]. As an application of the model, we show that relative permeabilities depend on the spatial derivatives of the saturation.