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Ashoori E.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada | Rossen W.R.,Technical University of Delft
Colloids and Surfaces A: Physicochemical and Engineering Aspects | Year: 2011

In foam EOR, complex dynamics of bubble creation and destruction controls foam properties. Here we reconsider whether and when non-equilibrium effects are important, focusing specifically on the entrance region, where injected gas and liquid are transformed into foam. We solve for water saturation and foam texture in the entrance region using the population-balance foam model of Kam (2008), which features three steady states (no foam, strong foam, and an unstable intermediate state) at some injection rates, as seen in experiments. We derive and solve equations for water saturation and foam properties along the entrance region at steady state. Mathematical conditions on the entrance region itself can control which of the several possible steady states is ultimately taken downstream by foam. For instance, if foam is not pre-generated, and capillary-pressure gradients are neglected, as in many published simulation studies, the final steady state downstream is the one with highest water saturation - the weakest foam. Simulations neglecting capillary pressure therefore may lead to inference of the wrong foam state in the formation or core. In some cases, in the presence of capillary pressure, analysis of the asymptotic dynamic behavior in the vicinity of possible downstream steady states may rule out some possible steady states. We show that the apparent length of entrance region can be quite different if one measures water saturation or pressure gradient. Finally, we fit foam kinetic parameters to the length of the entrance region seen in some experiments; a companion paper [18] investigates the effect of these parameters on the traveling wave at the shock front downstream. © 2011 Elsevier B.V. Source


Ashoori E.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada | Rossen W.R.,Technical University of Delft
Colloids and Surfaces A: Physicochemical and Engineering Aspects | Year: 2011

In foam EOR, complex dynamics of bubble creation and destruction controls foam properties. Recently, there has been consensus that local equilibrium between bubble creation and destruction adequately describes foam displacements. We assume that local equilibrium applies throughout a foam displacement on the field scale, with the exception of an entrance region and at shock fronts, where saturations and texture (bubble size) change abruptly. We find a range of conditions in which the local-equilibrium condition applies even within the shock front. In a waterflood, the width of a shock transition zone is determined by capillary-pressure gradients. For foam, this equation is joined by one for evolving foam texture. One expects that slow foam dynamics widens the traveling wave at the shock considerably. Theory and simulations show that the width of and mobility inside a shock front can affect foam sweep. If there is no gas ahead of the foam, as is common in published simulations, we prove that foam texture is everywhere at local equilibrium within the shock, regardless of the foam model, as previously observed for one dynamic foam model. If there is gas initially in the formation, slow foam generation and coalescence processes can actually narrow the shock from that assuming local equilibrium. In other cases, the dynamics of the traveling wave leads to oscillations near the shock; these are not numerical artifacts, but reflections of the models. Multiple steady states seen in experiment for some injection rates can be predicted by certain foam models. The approach of solving for the traveling wave can rule out some of these states for certain displacements. © 2011 Elsevier B.V. Source


Mailybaev A.A.,Moscow State University | Bruining J.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada
Combustion and Flame | Year: 2011

We study one-dimensional flows, when air is injected into a porous medium filled with inert gas, medium or high viscosity oil and water, giving rise to a combustion wave in a process known as high-temperature oxidation (HTO). In the oil we distinguish three pseudo-components: asphaltenes, medium and light oil. At high temperatures, the heaviest components (" precoke" ) are converted to coke, which undergoes combustion. Medium oil components are cracked at intermediate temperatures releasing gaseous oil. Light oil components and water are vaporized. The oxidation rate of gaseous oil components is negligible. Combustion regimes are described in the form of a sequence of waves. We develop a simple mathematical pathway based on Zeldovich's approach to provide analytical formulae for parameters in these waves. It is shown that there is a combustion regime in which either coke or oxygen are partially consumed in the combustion as well as a regime in which both are consumed completely. Each of the regimes can be subdivided in two regimes, where the reaction is either trailing or leading with respect to the thermal wave. Explicit conditions for each combustion regime are given. The structure of the oil cracking layer is investigated. Stability of the solutions is studied. We analyse our formulae for typical in situ combustion data and compare the results with numerical simulations. © 2010 The Combustion Institute. Source


Reaiche M.M.C.R.,Instituto Nacional Of Matematica Pura E Aplicada
Operations Research Letters | Year: 2016

We derive a lower bound for the sample complexity of the Sample Average Approximation method for a certain class of multistage stochastic optimization problems. In previous works, upper bounds for such problems were derived. We show that the dependence of the lower bound with respect to the complexity parameters and the problem's data are comparable to the upper bound's estimates. Like previous results, our lower bound presents an additional multiplicative factor showing that it is unavoidable for certain stochastic problems. © 2016 Elsevier B.V. All rights reserved. Source


Goncalves P.,Pontifical Catholic University of Rio de Janeiro | Goncalves P.,University of Minho | Jara M.,Instituto Nacional Of Matematica Pura E Aplicada | Jara M.,University of Paris Dauphine
Archive for Rational Mechanics and Analysis | Year: 2014

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture. © 2013 Springer-Verlag Berlin Heidelberg. Source

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