Simulations d´écoulements inertiels en milieu poreux

L’écoulement des fluides en milieux poreux concerne de nombreux domaines comme le stockage/déstockage du gaz en réservoir en génie gazier, les écoulements en colonnes de réacteur en génie chimique et dans la récupération pétrolière pour les écoulements autour des puits d’injection et de production en génie pétrolier. En régime visqueux, i.e. pour un Reynolds typiquement inférieur à 1, la physique est bien comprise et les modèles mathématiques associés sont validés. Par contre, Les écoulements caractérisés par un Reynolds non-négligeable, où les effets d’inertie sont significatifs ne sont toujours pas bien compris. Leur modélisation reste encore à compléter. Tout autant que le régime d’écoulement, l’échelle considérée est un paramètre important dans l’étude des écoulements en milieux poreux. En régime visqueux, la quantité de mouvement est décrite à l’échelle microscopique par l’équation de Stokes et à l’échelle macroscopique par l’équation de Darcy obtenue empiriquement puis démontrée par la suite théoriquement. En régime inertiel les équations de quantité de mouvement sont l’équation de Navier-Stokes à l’échelle microscopique et l’équation de Forchheimer à l’échelle macroscopique qui est l’équation de Darcy dans laquelle une correction vectorielle a été introduite pour tenir compte des effets d’inertie. L’équation de Forchheimer a été obtenue théoriquement en utilisant la méthode de prise de moyenne volumique avec fermeture. La dépendance de la correction de Forchheimer par rapport aux propriétés du milieu poreux (forme des grains, désordre…) est étudiée dans ce travail en simulant, en régime inertiel, des écoulements mono- et diphasiques de fluides newtoniens incompressibles dans des géométries de complexité croissante. Ces résultats seront comparés ultérieurement à des expériences de laboratoire sur des micro-réseaux pour étudier la validité des équations macroscopiques en régime inertiel

Modeling transport of charged species in pore networks: solution of the Nernst-Planck equations coupled with fluid flow and charge conservation equations

A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations. The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5\%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation. This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface

Inertial flow in porous media: A numerical investigation on model structures

The aim of this work is to study the correction to Darcy's law for inertial flow in porous media. In many situations encountered in industrial applications such as flow in column reactors, gas flow near wells for hydrocarbon recovery and CO2 sequestration, flow in filters... , Reynolds numbers are large enough to lead to a non-linear relationship between the filtration velocity and the pressure gradient. In this work, a numerical analysis of the non linear -inertial- correction to Darcy's law is carried out for the stationary inertial flow of a one-phase Newtonian incompressible fluid on model 2D and 3D structures. Effective properties appearing in the macroscopic model resulting from the volume averaging of the mass and momentum (Navier-Stokes) equations at the pore scale are determined using the microscopic flow fields and solving the closure problems resulting from up-scaling. From the numerical simulations, the dependence of the correction to Darcy's law on the geometrical properties of the 3D structure is studied. These properties are the shape of the solid grains which may be cubic or spherical and the degree of disorder in their arrangement in the domain. Weak disorder corresponds to a random placement of the grains of identical shape and size within each cell of a regular 3D lattice, while for strong disorder, grain size is also randomly distributed

Numerical investigation of Herschel–Bulkley fluid flows in 2D porous media: Yielding behaviour and tortuosity

Hydraulic tortuosity is commonly used as an input to macroscopic flow models in porous media, accounting for the sinuosity of the streamlines. It is well known that hydraulic tortuosity does not depend on the applied pressure gradient for Newtonian creeping flows. Nevertheless, this is not necessarily the case for yield stress fluids flows, given the directional nature of both yielding and shear-thinning behaviour. This study aims at a breakthrough on the relationship between the hydraulic tortuosity and the level of yielding. To do so, the hydraulic tortuosity of the flow paths is evaluated in 2D porous media by means of direct numerical simulations and subsequently put in relation with the morphological information of the medium provided by pore-network modelling. Moreover, the effects of pore dimensions, spatial disorder and rheological parameters on yielding behaviour are examined. In most situations, the reported tortuosity values are lower than those obtained for Newtonian fluids

Application of Non-toxic Yield Stress Fluids Porosimetry Method and Pore-Network Modelling to Characterize the Pore Size Distribution of Packs of Spherical Beads

With X-ray computed tomography still being flawed as a result of limitations in terms of spatial resolution and cost, toxic mercury intrusion porosimetry (MIP) is nowadays the prevailing technique to determine PSDs of most porous media. Recently, yield stress fluids porosimetry method (YSM) has been identified as a promising clean alternative to MIP. This technique is based on the particular percolation patterns followed by yield stress fluids, which only flow through certain pores when injected at a given pressure gradient. In previous works, YSM was used to characterize natural and synthetic porous media, and the results were compared with MIP showing reasonable agreement. However, considerable uncertainty still remains regarding the characterized pore dimension with each method arising from the highly complex geometry of the interstices in real porous media. Therefore, a critical stage for the validation of YSM consists in achieving successful characterization of model porous media with well-known pore morphology and topology. With this objective in mind, a set of four packs of glass beads each with a given monodisperse bead size were characterized in the present work using different porosimetry methods: experimental YSM, numerically simulation of MIP and pore-network extraction from a 3D image. The results provided by these techniques were compared, allowing the identification of the pore dimensions being characterized in each case. The results of this research elucidate the causes of the discrepancies between the considered porosimetry methods and demonstrate the usefulness of the PSD provided by YSM when predicting flow in porous media

Numerical porosimetry: Evaluation and comparison of yield stress fluids method, mercury intrusion porosimetry and pore network modelling approaches

Mercury Intrusion Porosimetry (MIP) is still today the reference porosimetry technique despite its environmental health and safety concerns. As a safe alternative, the Yield Stress fluids Method (YSM) consists in computing the Pore Size Distribution (PSD) of a given material from the pressure drop vs. flow rate measurements during injection of a yield stress fluid. However, the question arises whether the PSDs provided by YSM are representative of the actual pore dimensions. To answer this question, three numerical methods to obtain the PSD from digital images are proposed and compared in the present work. First, direct numerical simulations of YSM tests are performed in the considered media. Then, realistic PSDs are extracted from the images by using pore Network Modelling (NM). Furthermore, the obtained networks are also used to simulate MIP tests. The quantitative numerical results allow the evaluation of the relevance of YSM as an alternative to toxic MIP

Dual network extraction algorithm to investigate multiple transport processes in porous materials: Image-based modeling of pore and grain scale processes

The final publication is available at Elsevier via https://doi.org/10.1016/j.compchemeng.2018.12.025 © 2018 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Image processing of 3D tomographic images to extract structural information of porous materials has become extremely important in porous media research with the commoditization of x-ray tomography equipment to the lab scale. Extracted pore networks from images using image analysis techniques enable transport properties calculation for bigger domains at a low computational cost, allowing pore-scale investigation of porous media over meaningful macroscopic length scales. The present study reports a pore network extraction algorithm to simultaneously extract void and solid networks from tomographic images of porous materials using simple image analysis techniques. Crucially, it includes connectivity and geometrical information of both void and solid phases as well as the interlinking of these phases with each other. Validation was obtained on networks extracted from simple cubic and random sphere packings over a range of porosities. The effective diffusivity in the void phase and thermal conductivity in the solid phase was then calculated and found to agree well with direct numerical simulation results on the images, as well as a range of experimental data. One important outcome of this work was a novel and accurate means of calculating interfacial areas between grains and voids directly from digital images, which is critical to many phenomena where phase interactions occur. The efficient ‘dual network’ algorithm is written in PYTHON using open source tools and provides a new way to study critical processes that depend on transport in both void and solid phase such as catalytic reactors and electrochemical systems.University of Engineering and Technology Lahore, PakistanNatural Sciences and Engineering Research Council of Canad

On the use of physical boundary conditions for two-phase flow simulations: Integration of control feedback

The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.compchemeng.2018.08.012 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/The sensitivity of two-phase flow simulations using the Euler–Euler model on the inlet boundary conditions (BCs) is studied. Specifically, the physical relevance of Dirichlet uniform inlet velocity BCs is studied which are widely used due their simplicity and the lack of a priori knowledge of the slip velocity between the phases. It is found that flow patterns obtained with the more physically realistic uniform inlet pressure BCs are radically different from the results obtained with Dirichlet inlet velocity BCs, refuting the argument frequently put forward that Dirichlet uniform inlet velocity BCs can be interchangeably used because the terminal slip velocity is reached after a short entrance region. A comparison with experimental data is performed to assess the relevance of the flows obtained numerically. Additionally, a multivariable feedback control method is demonstrated to be ideal for enforcing desired flow rates for simulations using pressure BCs.Natural Sciences and Engineering Research Council of Canad

Inertial one and two phase flow in porous media, a numerical Investigation

: Ce travail concerne l'écoulement inertiel en milieu poreux rencontré dans diversessituations telles que les écoulements autour des puits pour la récupération pétrolière, lesécoulements dans les réacteurs catalytiques, etc. En régime stationnaire, les différents modèlesmacroscopiques pour décrire ces écoulements inertiels (non-linéaires) demeurent encore sujetsà débat. Ces modèles consistent en une loi de Darcy corrigée de termes dont la dépendancevis à vis de la vitesse de filtration relève du régime d'écoulement. Dans ce travail, une attentionparticulière est portée tout d'abord à l'étude numérique (DNS), sur des structures modèles, de lalimite de stationnarité de l'écoulement monophasique newtonien qui correspond à la premièrebifurcation de Hopf, caractérisée par un nombre de Reynolds critique. La connaissance de cettelimite est cruciale puisqu'elle détermine le domaine de validité des modèles macroscopiquesstationnaires pertinents. Dans un deuxième temps, la dépendance de la déviation (inertielle) àla loi de Darcy par rapport aux propriétés de la structure poreuse (forme des grains, désordre)et à l'orientation de l'écoulement est étudiée dans le cas de structures 2D et 3D. Les propriétéseffectives de la structure à l'échelle macroscopique sont déterminées à partir de la résolutionnumérique des problèmes de fermeture associés au modèle macroscopique obtenu par prisede moyenne des équations de Navier-Stokes. Afin de déceler l'origine de cette déviation et sesdifférentes formes, l'évolution de la structure microscopique de l'écoulement en fonction dunombre de Reynolds est analysée. Plus particulièrement, le rôle des zones de recirculation, etles corrélations avec la courbure des lignes de courant multipliée par l’énergie cinétique localeet la variation de l’énergie cinétique le long de ces lignes sont étudiés. La dernière partie dutravail est consacrée à une étude numérique, toujours dans des situations modèles, de ladéviation à la loi de Darcy généralisée dans le cas de l'écoulement diphasique inertiel.This work focuses on inertial flow in porous media encountered in differentindustrial situations such as flow around wells in oil recovery, flow in filters and in columns ofreactors for chemical engineering, etc. In stationary flow regime, the different macroscopicmodels describing inertial (non-linear) flow are still discussed. These models consist in theDarcy’s law with correction extra terms whose dependence upon the filtration velocity is afunction of the flow regime. In this work, a particular attention is attributed first to the numericalinvestigation (DNS), on model structures, of the limit of one phase Newtonian stationary flowwhich corresponds to the first Hopf bifurcation, characterized by a critical Reynolds number.The knowledge of this limit is crucial since it establishes the ranges of validity of the relevantmacroscopic stationary models. In a second step, the dependence of the deviation (inertial)from Darcy’s law on the properties of the porous structure (grains shape, disorder) and on theorientation of the flow is analyzed in 2D and 3D situations. The effective properties of thestructure and the flow at the macroscopic scale are obtained from the numerical resolution ofthe closure problems associated to the macroscopic model obtained from an up-scalingprocedure (volume averaging) of the Navier-Stokes equations. In order to identify the origin ofthe deviation and its different forms, the variation of the microscopic flow structure with theReynolds number is analyzed. More specifically, the role of the recirculation zones, and thecorrelations with flow streamlines curvature multiplied by the local kinetic energy and thevariation of the kinetic energy along these lines are studied. The last part of the work isdedicated to a numerical investigation of the deviation from the generalized Darcy’s law in thecase of two phase inertial flow

Inertial one and two phase flow in porous media, a numerical Investigation

: Ce travail concerne l'écoulement inertiel en milieu poreux rencontré dans diversessituations telles que les écoulements autour des puits pour la récupération pétrolière, lesécoulements dans les réacteurs catalytiques, etc. En régime stationnaire, les différents modèlesmacroscopiques pour décrire ces écoulements inertiels (non-linéaires) demeurent encore sujetsà débat. Ces modèles consistent en une loi de Darcy corrigée de termes dont la dépendancevis à vis de la vitesse de filtration relève du régime d'écoulement. Dans ce travail, une attentionparticulière est portée tout d'abord à l'étude numérique (DNS), sur des structures modèles, de lalimite de stationnarité de l'écoulement monophasique newtonien qui correspond à la premièrebifurcation de Hopf, caractérisée par un nombre de Reynolds critique. La connaissance de cettelimite est cruciale puisqu'elle détermine le domaine de validité des modèles macroscopiquesstationnaires pertinents. Dans un deuxième temps, la dépendance de la déviation (inertielle) àla loi de Darcy par rapport aux propriétés de la structure poreuse (forme des grains, désordre)et à l'orientation de l'écoulement est étudiée dans le cas de structures 2D et 3D. Les propriétéseffectives de la structure à l'échelle macroscopique sont déterminées à partir de la résolutionnumérique des problèmes de fermeture associés au modèle macroscopique obtenu par prisede moyenne des équations de Navier-Stokes. Afin de déceler l'origine de cette déviation et sesdifférentes formes, l'évolution de la structure microscopique de l'écoulement en fonction dunombre de Reynolds est analysée. Plus particulièrement, le rôle des zones de recirculation, etles corrélations avec la courbure des lignes de courant multipliée par l’énergie cinétique localeet la variation de l’énergie cinétique le long de ces lignes sont étudiés. La dernière partie dutravail est consacrée à une étude numérique, toujours dans des situations modèles, de ladéviation à la loi de Darcy généralisée dans le cas de l'écoulement diphasique inertiel.This work focuses on inertial flow in porous media encountered in differentindustrial situations such as flow around wells in oil recovery, flow in filters and in columns ofreactors for chemical engineering, etc. In stationary flow regime, the different macroscopicmodels describing inertial (non-linear) flow are still discussed. These models consist in theDarcy’s law with correction extra terms whose dependence upon the filtration velocity is afunction of the flow regime. In this work, a particular attention is attributed first to the numericalinvestigation (DNS), on model structures, of the limit of one phase Newtonian stationary flowwhich corresponds to the first Hopf bifurcation, characterized by a critical Reynolds number.The knowledge of this limit is crucial since it establishes the ranges of validity of the relevantmacroscopic stationary models. In a second step, the dependence of the deviation (inertial)from Darcy’s law on the properties of the porous structure (grains shape, disorder) and on theorientation of the flow is analyzed in 2D and 3D situations. The effective properties of thestructure and the flow at the macroscopic scale are obtained from the numerical resolution ofthe closure problems associated to the macroscopic model obtained from an up-scalingprocedure (volume averaging) of the Navier-Stokes equations. In order to identify the origin ofthe deviation and its different forms, the variation of the microscopic flow structure with theReynolds number is analyzed. More specifically, the role of the recirculation zones, and thecorrelations with flow streamlines curvature multiplied by the local kinetic energy and thevariation of the kinetic energy along these lines are studied. The last part of the work isdedicated to a numerical investigation of the deviation from the generalized Darcy’s law in thecase of two phase inertial flow