Travelling Waves for Gas-Solid Reactions

Sin resume

Traveling waves in a finite condensation rate model for steam injection

Steam drive recovery of oil is an economical way of producing oil even in times of low oil prices and is used worldwide. This paper focuses on the one-dimensional setting, where steam is injected into a core initially containing oil and connate water while oil and water are produced at the other end. A three-phase (oil, water, steam) hot zone develops, which is abruptly separated from the two-phase (oil + water) cold zone by the steam condensation front. The oil, water and energy balance equations (Rankine–Hugoniot conditions) cannot uniquely solve the system of equations at the steam condensation front. In a previous study, we showed that two additional constraints follow from an analysis of the traveling wave equation representing the shock; however, within the shock, we assumed local thermodynamic equilibrium. Here we extend the previous study and include finite condensation rates; using that appropriate scaling requires that the Peclet number and the Damkohler number are of the same order of magnitude. We give a numerical proof, using a color-coding technique, that, given the capillary diffusion behavior and the rate equation, a unique solution can be obtained. It is proven analytically that the solution for large condensation rates tends to the solution obtained assuming local thermodynamic equilibrium. Computations with realistic values to describe the viscous and capillary effects show that the condensation rate can have a significant effect on the global saturation profile, e.g. the oil saturation just upstream of the steam condensation front

Rigorous derivation of a hyperbolic model for Taylor dispersion

In this paper we upscale the classical convection-diffusion equation in a narrow slit. We suppose that the transport parameters are such that we are in Taylor's regime i.e. we deal with dominant Peclet numbers. In contrast to the classical work of Taylor, we undertake a rigorous derivation of the upscaled hyperbolic dispersion equation. Hyperbolic effective models were proposed by several authors and our goal is to confirm rigorously the effective equations derived by Balakotaiah et al in recent years using a formal Liapounov - Schmidt reduction. Our analysis uses the Laplace transform in time and an anisotropic singular perturbation technique, the small characteristic parameter " being the ratio between the thickness and the longitudinal observation length. The Peclet number is written as Ce¿a, with a<2. Hyperbolic effective model corresponds to a high Peclet number close to the threshold value when Taylor's regime turns to turbulent mixing and we characterize it by supposing 4/3 <a <2. We prove that the difference between the dimensionless physical concentration and the effective concentration, calculated using the hyperbolic upscaled model, divided by e2¿a (the local Peclet number) converges strongly to zero in L2-norm. For Peclet numbers considered in this paper, the hyperbolic dispersion equation turns out to give a better approximation than the classical parabolic Taylor model

A free boundary problem involving a cusp : breakthrough of salt water

In this paper we study a two-phase free boundary problem describing the stationary flow of fresh and salt water in a porous medium, when both fluids are drawn into a well. For given discharges at the well ($Q_f$ for fresh water and $Q_s$ for salt water) we formulate the problem in terms of the stream function in an axial symmetric flow domain in {Bbb R^n(n = 2,3). We prove existence of a continuous free boundary which ends up in the well, located on the central axis. Moreover we show that the free boundary has a tangent at the well and approaches it in a $C^1$ sense. Using the method of separation of variables we also give a result about the asymptotic behaviour of the free boundary at the well. For given total discharge ($Q := Q_f + Q_s$) we consider the vanishing $Q_s$ limit. We show that a free boundary arises with a cusp at the central axis, having a positive distance from the well. This work is a continuation of [AD2,3]

Large time behaviour of solutions of the porous medium equation with convection

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Uniqueness conditions in a hyperbolic model for oil recovery by steamdrive

In this paper we study a one-dimensional model for oil recovery by steamdrive. This model consists of two parts: a (global) interface model and a (local) steam condensation/capillary diffusion model. In the interface model a steam condensation front (SCF) is present as an internal boundary between the hot steam zone (containing water, oil and steam) and the cold liquid zone (containing only water and oil). Disregarding capillary pressure away from the SCF, a 2x2 hyperbolic system arises for the water and steam saturation. This system cannot be solved uniquely without additional conditions at the SCF. To find such conditions we make a blow up of the SCF and consider a parabolic transition model, including capillary diffusion. We study in detail the existence conditions for travelling wave solutions. These conditions translate into the missing matching conditions at the SCF in the hyperbolic limit and thus provide uniqueness. We show that different transition models yield different matching conditions, and thus different solutions of the interface model. We also give a relatively straightforward approximation and investigate its validity for certain ranges of model parameters

A free boundary problem involving a cusp

We consider a stationary free boundary problem describing the stationary flow of fresh and salt water in a porous medium. The salt water is supposed to be stagnant, while the fresh water on top of it is drawn into wells. In a previous work it has been shown, that for pumping rates Q < Q_{cr a solution with smooth interface exists. In this part we study the case Q=Q_{cr in two dimensions. We show that the interface has isolated singularities. At each singularity the free boundary develops a cusp or becomes vertical. By means of local analysis techniques we obtain the asymptotic behaviour of the free boundary at these singularities