In this thesis, PN simulations of drying are compared with experimentally obtained data fromdrying of a representative 2D microfluidic network in SiO2 under varying thermal conditions withthe aim to identify governing physical pore scale effects. Gravity and viscous effects aredisregarded in this thesis. Instead drying with slight local temperature variation and drying withimposed thermal gradients are studied. Based on this investigation, a powerful non-isothermalPNM is developed. This model incorporates i) the phenomena associated with the temperaturedependency of pore scale invasion, namely thermally affected capillary invasion and vapor flow aswell as ii) the secondary effects induced by wetting liquid films of different morphology. This studyclearly evidences that the macroscopic drying behavior is fundamentally dictated by thetemperature gradient imposed on the PN and moreover by the secondary capillary invasion aswell. In agreement with literature, invasion patterns as in invasion percolation with progressiveevaporation of single clusters are observed in drying with negligible local temperature variation;gradients with temperature decreasing from the surface (negative temperature gradient) canstabilize the drying front, evolving between the invading gas phase and the receding liquid phase,whereas temperature increasing from the surface (positive temperature gradient) leads todestabilization of the liquid phase with early breakthrough of a gas branch and initiation of asecond invasion front migrating in opposite direction to the evaporation front receding from theopen surface of the PN. Special attention is paid on the distinct drying regimes found in thesituation of a positive gradient because they are associated with different pore scale invasionprocesses. More precisely, temperature dependency of surface tension dictates the order ofinvasion as long as the liquid phase is connected in a main liquid cluster (usually found during thefirst period of drying). In contrast to this, detailed study of the vapor transfer mechanismsemphasizes that vapor diffusion through the partially saturated region can control the pore leveldistributions of liquid and gas phase during the period of drying when the liquid phase isdisconnected into small clusters. This is also related to the cluster growth induced by partialcondensation of vapor. It is shown and discussed in detail in this thesis that this effect not onlydepends on direction and height of the temperature gradient for a given pore size distribution butthat moreover the overall evaporation rate influences the cluster growth mechanism. This indicatesthat liquid migration during drying of porous media might be controlled by the interplay of thermalgradients and drying rate. In summary, the study of thermally affected drying of the 2-dimensionalPN reveals complex pore scale mechanisms, usually also expected in drying of real porous media.This leads to the development of a strong mathematical pore scale model based on experimentalfindings. It is demonstrated how this model might be applied to understand and develop moderndrying processes based on the simulation of thermally affected pore scale mass transfer

Dans cette thèse, des simulations PN de séchage sont comparées à des données expérimentales obtenues dans le séchage d´un réseau de micro-fluidique 2D représentatif dans du SiO2 soumis à des conditions thermiques variables dans le but d’identifier les phénomènes physiques à l´échelle des pores qui sont les plus influents. A partir de cette étude, un PN efficace non-isotherme est développé. Ce modèle incorpore i) les phénomènes associés à la dépendendence en température de l´invasion à l´échelle des pores, c´est à dire l´invasion capillaire sous effet thermique et le flux de vapeur ainsi que ii) le transport secondaire induit par d´épais films liquides observé dans les expériences de microfluidique. Cette étude prouve clairement que le comportement macroscopique du séchage est fondamentalement dirigé par le gradient de température imposé sur le PN ainsi que par le transport capillaire secondaire. En accord avec la littérature, les schémas d´invasion que l´on trouve dans l´invasion percolatrice avec l´évaporation progressive d´amas individuels sont observés dans le séchage à variation de température locale négligeable;des gradients où la température diminue à partir de la surface (gradient de température négatif)peut stabiliser le front de séchage, qui évolue entre la phase gazeuse invasive et la phase liquide qui recule, alors qu´une température qui augmente à partir de la surface (gradient de température positif) amène à la déstabilisation de la phase liquide avec une percée prématurée de la branche gazeuse et l’initiation d´un deuxième front de séchage qui migre dans la direction opposé de celle du front de séchage original. Une attention particulière est prêtée aux régimes distincts que l´on trouve dans le second cas (gradient positif) parce qu´ils sont associés à différents procédés d´invasion à l´échelle des pores. Plus précisément, la dépendance en température de la tension de surface établit l´ordre d´invasion tant que la phase liquide est connectée au groupe liquide principal (que l´on trouve généralement pendant la première période de séchage). En revanche,l´étude détaillée des mécanismes de transfert de la vapeur met l´accent sur le fait que la diffusion de la vapeur à travers la région partiellement saturée peut contrôler les distributions des phases gazeuses et liquides à l´échelle des pores pendant la période de séchage lorsque la phase liquide est déconnectée en petits groupes. Cela est aussi lié à la croissance des amas induite par la condensation partielle de la vapeur. Cette thèse montre et discute en détail que cet effet ne dépend pas seulement de la direction et magnitude du gradient de température pour une distribution de tailles de pores donnée mais qu’en plus le taux d´évaporation influence le mécanisme de croissances des amas. Cela indique que la migration du liquide pendant la phase de séchage de milieux poreux peut être contrôlé par l’interaction des gradients thermiques et du taux de séchage. En somme, l´étude du séchage sous effet thermique des réseaux de pores 2D révèle des phénomènes complexes à l´échelle des pores, généralement aussi anticipés dans le séchage des milieux poreux réels. Cela mène au développement d´un modèle mathématique efficace au niveau des pores basés sur des découvertes expérimentales. Cette thèse démontre la manière dont ce modèle peut être appliqué afin de comprendre et développer des procédés de séchage modernes basés sur la simulation du transfert de masse sous effet thermique à l´échelle des pores / In this thesis, PN simulations of drying are compared with experimentally obtained data fromdrying of a representative 2D microfluidic network in SiO2 under varying thermal conditions withthe aim to identify governing physical pore scale effects. Gravity and viscous effects aredisregarded in this thesis. Instead drying with slight local temperature variation and drying withimposed thermal gradients are studied. Based on this investigation, a powerful non-isothermalPNM is developed. This model incorporates i) the phenomena associated with the temperaturedependency of pore scale invasion, namely thermally affected capillary invasion and vapor flow aswell as ii) the secondary effects induced by wetting liquid films of different morphology. This studyclearly evidences that the macroscopic drying behavior is fundamentally dictated by thetemperature gradient imposed on the PN and moreover by the secondary capillary invasion aswell. In agreement with literature, invasion patterns as in invasion percolation with progressiveevaporation of single clusters are observed in drying with negligible local temperature variation;gradients with temperature decreasing from the surface (negative temperature gradient) canstabilize the drying front, evolving between the invading gas phase and the receding liquid phase,whereas temperature increasing from the surface (positive temperature gradient) leads todestabilization of the liquid phase with early breakthrough of a gas branch and initiation of asecond invasion front migrating in opposite direction to the evaporation front receding from theopen surface of the PN. Special attention is paid on the distinct drying regimes found in thesituation of a positive gradient because they are associated with different pore scale invasionprocesses. More precisely, temperature dependency of surface tension dictates the order ofinvasion as long as the liquid phase is connected in a main liquid cluster (usually found during thefirst period of drying). In contrast to this, detailed study of the vapor transfer mechanismsemphasizes that vapor diffusion through the partially saturated region can control the pore leveldistributions of liquid and gas phase during the period of drying when the liquid phase isdisconnected into small clusters. This is also related to the cluster growth induced by partialcondensation of vapor. It is shown and discussed in detail in this thesis that this effect not onlydepends on direction and height of the temperature gradient for a given pore size distribution butthat moreover the overall evaporation rate influences the cluster growth mechanism. This indicatesthat liquid migration during drying of porous media might be controlled by the interplay of thermalgradients and drying rate. In summary, the study of thermally affected drying of the 2-dimensionalPN reveals complex pore scale mechanisms, usually also expected in drying of real porous media.This leads to the development of a strong mathematical pore scale model based on experimentalfindings. It is demonstrated how this model might be applied to understand and develop moderndrying processes based on the simulation of thermally affected pore scale mass transfer

This thesis presents an investigation of reactive transport of species in porous media, with the aim to understand better and predict the fate of radionuclide in engineered and natural barriers of future deep geological disposal facilities for nuclear waste. The work involves developments of several pore-scale models for simulating reactive transport by coupling convective, adsorptive and diffusive processes. Pore network models (PNM) are amongst the appealing approaches that provide a suitable description for dealing with mutable pore space structures. Such models have been used to describe conservative as well as reactive transport in saturated and unsaturated porous media. In the present thesis, pore network models based on a regular tessellation of truncated octahedral cells are proposed and developed to simulate mass transport in porous media with incomplete pore space information due to limitation of existing characterisation techniques. Bentonite and Opalinus Clay are selected to illustrate the methodology. The micro- and meso-structure of these clays and their effects on the transport behaviour are investigated. The research shows that the clays are anisotropic and heterogeneous with fast diffusion parallel to the bedding plane and slow diffusion perpendicular to the bedding plane. In addition, different types of species have different accessible porosity and macroscopic diffusion coefficients. The anisotropy and heterogeneity of clays are achieved by different length scales and percentage of pores in different directions in the pore network models. The transport behaviour of various species, including sorption and anion exclusion, is simulated and analyzed. The effect of sorption is simulated via changing the pore radii by a coarse grained mathematical formula or by a formula directly in each pore. The results are in good agreement with experimentally measured macroscopic (bulk) diffusivities for the materials studied, including anisotropic diffusion coefficients. This lends strong support to the physical realism of the proposed models. The developed methodology can be used for any micro and meso-porous material with known distribution of pore sizes. It can be extended to other pore space changing mechanisms, in addition to sorption, to derive mechanism-based evolution laws for the transport parameters of porous media.

The understanding of multiphase flow properties is essential for the exploitation of hydrocarbon reserves in a reservoir; these properties in turn are dependent on the geometric properties and connectivity of the pore space. The determination of the pore size distribution in carbonate reservoirs remains challenging; carbonates exhibit complex pore structures comprising length scales from nanometers to several centimeters. A major challenge to the accurate evaluation of these reservoirs is accounting for pore scale heterogeneity on multiple scales. This is the topic of this thesis. Conventionally, this micron scale information is achieved either by building stochastic models using 2D images or by combining log and laboratory data to classify pore types and their behaviour. None of these capture the true 3D connectivity vital for flow characterisation. We present here an approach to build realistic 3D network models across a range of scales to improve property estimation through employment of X-ray micro-Computed Tomography (μCT) and Focussed Ion Beam Tomography (FIBT). The submicron, or microporous, regions are delineated through a differential imaging technique undertaken on x-ray CT providing a qualitative description of microporosity. Various 3-Phase segmentation methods are then applied for quantitative characterisation of those regions utilising the attenuation coefficient values from the 3D tomographic images. X-ray micro-CT is resolution limited and can not resolve the detailed geometrical features of the submicron pores. FIB tomography is used to image the 3D pore structure of submicron pores down to a scale of tens of nanometers. We describe the experimental development and subsequent image processing including issues and difficulties resolved at various stages. The developed methodology is implemented on cores from producing wackstone and grainstone reservoirs. Pore network models are generated to characterise the 3D interconnectivity of pores. We perform the simulations of petrophysical properties (permeability and formation resistivity) directly on the submicron scale image data. Simulated drainage capillary pressure curves are matched with the experimental data. We also present some preliminary results for the integration of multiscale pore information to build dual-scale network models. The integration of multiscale data allows one to select appropriate effective medium theories to incorporate sub-micron structure into property calculations at macro scale giving a more realistic estimation of properties.

The understanding of multiphase flow properties is essential for the exploitation of hydrocarbon reserves in a reservoir; these properties in turn are dependent on the geometric properties and connectivity of the pore space. The determination of the pore size distribution in carbonate reservoirs remains challenging; carbonates exhibit complex pore structures comprising length scales from nanometers to several centimeters. A major challenge to the accurate evaluation of these reservoirs is accounting for pore scale heterogeneity on multiple scales. This is the topic of this thesis. Conventionally, this micron scale information is achieved either by building stochastic models using 2D images or by combining log and laboratory data to classify pore types and their behaviour. None of these capture the true 3D connectivity vital for flow characterisation. We present here an approach to build realistic 3D network models across a range of scales to improve property estimation through employment of X-ray micro-Computed Tomography (μCT) and Focussed Ion Beam Tomography (FIBT). The submicron, or microporous, regions are delineated through a differential imaging technique undertaken on x-ray CT providing a qualitative description of microporosity. Various 3-Phase segmentation methods are then applied for quantitative characterisation of those regions utilising the attenuation coefficient values from the 3D tomographic images. X-ray micro-CT is resolution limited and can not resolve the detailed geometrical features of the submicron pores. FIB tomography is used to image the 3D pore structure of submicron pores down to a scale of tens of nanometers. We describe the experimental development and subsequent image processing including issues and difficulties resolved at various stages. The developed methodology is implemented on cores from producing wackstone and grainstone reservoirs. Pore network models are generated to characterise the 3D interconnectivity of pores. We perform the simulations of petrophysical properties (permeability and formation resistivity) directly on the submicron scale image data. Simulated drainage capillary pressure curves are matched with the experimental data. We also present some preliminary results for the integration of multiscale pore information to build dual-scale network models. The integration of multiscale data allows one to select appropriate effective medium theories to incorporate sub-micron structure into property calculations at macro scale giving a more realistic estimation of properties.

Une pile à membrane échangeuse de protons (PEMFC) est un dispositif convertissant l’hydrogène en électricité grâce à une réaction électrochimique appelé électrolyse inverse. Comme chaque pile à combustible ou batterie, les PEMFC sont composées d’une série de couches. Nous nous intéressons à la couche de diffusion (GDL) du côté de la cathode. La GDL est constituée de fibres de carbone traitées pour être hydrophobes. Elle peut être vue comme un milieu poreux mince avec une taille moyenne de pores de quelques dizaines de microns. Une question clé dans ce système est la gestion de l'eau produite par la réaction. Dans ce contexte, le principal objectif de la thèse est le développement d'un outil numérique visant à simuler la formation de l'eau liquide dans la GDL. Une approche réseau de pores est utilisée. Nous nous concentrons sur un scénario où l’eau liquide se forme par condensation dans la GDL. Les comparaisons entre simulations et expériences effectuées grâce à un dispositif microfluidique bidimensionnel, sont d'abord présentées pour différentes conditions de mouillabilité, de distributions de température et d'humidité relative à l’entrée, afin de valider le modèle. Une étude de sensibilité est alors effectuée afin de mieux caractériser les paramètres contrôlant l'invasion de l'eau. Enfin, les simulations sont comparées à des distributions d’eau obtenues in-situ par micro-tomographie à rayons X, ainsi que des distributions expérimentales de la littérature obtenues par imagerie neutronique. / A Proton Exchange Membrane Fuel Cell (PEMFC) is a device converting hydrogen into electricity thanks to an electrochemical reaction called reverse electrolysis. Like every fuel cell or battery, PEMFCs are made of a series of layers. We are interested in the gas diffusion layer (GDL) on the cathode side. The GDL is made of carbon fibers treated hydrophobic. It can be seen as a thin porous medium with a mean pore size of few tens of microns. A key question in this system is the management of the water produced by the reaction. In this context, the main objective of the thesis is the development of a numerical tool aiming at simulating the liquid water formation within the GDL. A pore network approach is used. We concentrate on a scenario where liquid water forms in the GDL by condensation. Comparisons between simulations and experiments performed with a two-dimensional microfluidic device are first presented for different wettability conditions, temperature distributions and inlet relative humidity in order to validate the model. A sensitivity study is then performed to better characterize the parameters controlling the water invasion. Finally, simulations are compared with in situ experimental water distributions obtained by X-ray micro-tomography as well as with experimental distributions from the literature obtained by neutron imaging.

abstract: The understanding of multiphase fluid flow in porous media is of great importance in many fields such as enhanced oil recovery, hydrology, CO2 sequestration, contaminants cleanup, and natural gas production from hydrate bearing sediments. In this study, first, the water retention curve (WRC) and relative permeability in hydrate bearing sediments are explored to obtain fitting parameters for semi-empirical equations. Second, immiscible fluid invasion into porous media is investigated to identify fluid displacement pattern and displacement efficiency that are affected by pore size distribution and connectivity. Finally, fluid flow through granular media is studied to obtain fluid-particle interaction. This study utilizes the combined techniques of discrete element method simulation, micro-focus X-ray computed tomography (CT), pore-network model simulation algorithms for gas invasion, gas expansion, and relative permeability calculation, transparent micromodels, and water retention curve measurement equipment modified for hydrate-bearing sediments. In addition, a photoelastic disk set-up is fabricated and the image processing technique to correlate the force chain to the applied contact forces is developed. The results show that the gas entry pressure and the capillary pressure increase with increasing hydrate saturation. Fitting parameters are suggested for different hydrate saturation conditions and morphologies. And, a new model for immiscible fluid invasion and displacement is suggested in which the boundaries of displacement patterns depend on the pore size distribution and connectivity. Finally, the fluid-particle interaction study shows that the fluid flow increases the contact forces between photoelastic disks in parallel direction with the fluid flow. / Dissertation/Thesis / Doctoral Dissertation Civil and Environmental Engineering 2016

Geological storage of carbon dioxide (CO2) in deep saline aquifers is one approach to mitigate release from large point sources to the atmosphere. Understanding of in-situ processes providing trapping is important to the development of realistic models and the planning of future storage projects. This thesis covers both field- and pore-scale numerical modeling studies of such geohydrological processes, with focus on residual trapping. The setting is a CO2-injection experiment at the Heletz test site, conducted within the frame of the EU FP7 MUSTANG and TRUST projects. The objectives of the thesis are to develop and analyze alternative experimental characterization test sequences for determining in-situ residual CO2 saturation (Sgr), as well as to analyze the impact of the injection strategy on trapping, the effect of model assumptions (coupled wellbore-reservoir flow, geological heterogeneity, trapping model) on the predicted trapping, and to develop a pore-network model (PNM) for simulating and analyzing pore-scale mechanisms. The results include a comparison of alternative characterization test sequences for estimating Sgr. The estimates were retrieved through parameter estimation. The effect on the estimate of including various data sets was determined. A new method, using withdrawal and an indicator-tracer, for obtaining a residual zone in-situ was also introduced. Simulations were made of the CO2 partitioning between layers in a multi-layered formation, and parameters influencing this were identified. The results showed the importance of accounting for coupled wellbore-reservoir flow in simulations of such scenarios. Simulations also showed that adding chase-fluid stages after a conventional CO2 injection enhances the (residual and dissolution) trapping. Including geological heterogeneity generally decreased the estimated trapping. The choice of trapping model may largely effect the quantity of the predicted residual trapping (although most of them produced similar results). The use of an appropriate trapping model and description of geological heterogeneity for a site when simulating CO2 sequestration is vital, as different assumptions may give significant discrepancies in predicted trapping. The result also includes a PNM code, for multiphase quasi-static flow and trapping in porous materials. It was used to investigate trapping and obtain an estimated trapping (IR) curve for Heletz sandstone.

On s'intéresse dans ce travail à la modélisation d'un écoulement diphasique gaz-liquide co-courant descendant dans les réacteurs à lit fixe de particules, procédé largement utilisé dans le domaine industriel. En raison de la complexité de l'écoulement, induite par les nombreuses configurations multiphasiques pouvant coexister au sein du lit, les modèles développés directement à l'échelle du réacteur sont généralement issus d'approches semi-empiriques, en considérant l'écoulement homogène. Or, il a été observé que des hétérogénéités locales, géométrique et hydrodynamique, telle qu'une mal-distribution de la phase liquide, entrainaient une diminution du taux de réaction et conduisait les modèles existants à surestimer la productivité d'un réacteur. La nécessité de prendre en compte les phénomènes microscopiques dans un modèle macroscopique à l'échelle du réacteur rend l'utilisation d'approches multi-échelles indispensable. L'écoulement étant cependant d'une nature complexe, le changement d'échelle ne peut se faire de façon directe et nécessite donc la mise en place d'outils de modélisation adaptés à une échelle intermédiaire. Dans une première étape, la méthode de simulation numérique directe ``Volume-Of-Fluid'' (VOF) est validée dans le cas d'un film ruisselant dans un tube capillaire. Cette méthode est ensuite utilisée, à l'échelle microscopique, afin de proposer et de valider des relations de fermeture pour un modèle de type ``réseau de pores'' pouvant être utilisé à une échelle intermédiaire, celle du Volume Elémentaire Représentatif. Ce changement d'échelle est tout d'abord effectué dans le cas d'un lit fixe en deux dimensions, c'est-à-dire un empilement de cylindres entre deux plaques. Cette configuration permet la mise en place d'un dispositif expérimental qui, couplé à des simulations VOF 2D à plus grande échelle, valide l'approche de type "réseau de pores" adoptée. Le modèle réseau est ensuite étendu au cas d'un lit fixe réel, c'est-à-dire en trois dimensions, dont la géométrie est obtenue par micro-tomographie. Les lois de comportement locales sont redéfinies à l'aide de simulations numériques directes à l'échelle microscopique. Les résultats provenant de simulations de type « réseaux de pore » sont ensuite confrontés, dans le cas d'une répartition homogène des phases, aux modèles 1D habituellement utilisés pour les écoulements diphasiques en lit fixe. Enfin, une campagne expérimentale est menée afin d'observer, par imagerie scanner, l'étalement d'un jet de liquide sur un empilement de grains. Une comparaison qualitative est ensuite effectuée entre les observations expérimentales et les simulations numériques réseaux dans le cas spécifique de l'étalement d'un jet de liquide / We study in this work the modelling of two-phase cocurrent downflows in fixed bed reactors, a process widely used in industry. Due to the flow complexity, i.e., the presence of different interface configurations and, therefore, different phase interactions, most models have been developed using empirical approaches, with the assumption of a homogeneous flow in the reactor. However, several studies showed that local heterogeneities, geometric and hydrodynamic, such as the liquid distribution, could have a great influence on the flow at the reactor-scale and, therefore, on the reactor performance. Consider the microscopic phenomena in a macroscopic model require the use of multi-scale approaches. However, due to the flow complexity, the upscaling cannot be done directly and requires the development of modelling tools suitable for an intermediate scale. In a first step, the direct numerical method \ Volume-Of-Fluid" (VOF) is validated in the case of a two-phase flow in a capillary tube with the presence of a thin film. Then, this method is used, at a microscopic level to propose and validate closure laws for a pore-network model which will be used to simulate the flow at the intermediate scale. This upscaling approach is first tested in a two-dimensional case,i.e., an array of cylinders between two walls. This configuration allows the set up of an experimental approach, coupled with 2D VOF simulations at the intermediate scale, in order to validate the pore-network approach. The pore-network approach is then extended to a real fixed bed, i.e. in three dimensions, whose geometry is obtained by micro-tomography. Local laws of the pore-network model are redefined using direct numerical simulations at a microscopic scale. Pore-network simulations are then compared, for a homogenous phase distribution, with 1D models typically used for two-phase flow in fixed beds. Finally, an experimental campaign was set up to observe, by imaging scanner, the spreading of a liquid jet on a fixed bed pilot. A qualitative comparison is then performed between experimental observations and pore-network simulations in the specific case of the spreading of a liquid jet

La protection et la remédiation des ressources en eau sont un enjeu sociétal majeur, ainsi il est nécessaire de comprendre l’évolution de solutés, tels les polluants, au sein de la zone saturée et non saturée. Dans ce but, de nombreux travaux ont été consacrés à la modélisation du transport réactif en milieux poreux. Son déroulement à l’échelle de Darcy dépend des hétérogénéités microscopiques du milieu. Les modèles de réseau de pores qui simplifient la géométrie en un ensemble de pores reliés par des liens de sections constantes, permettent de se placer à une échelle mésoscopique, faisant le lien entre l’échelle porale et l’échelle de Darcy. Sur de telles formes géométriques, l’écoulement admet un traitement analytique. En ce qui concerne le transport réactif des solutés, nous proposons une solution analytique dans les liens qui permet de calculer le débit de masse entre pores. Le modèle de transport se formule alors comme un système d’équations de Volterra de secondes espèces dont les noyaux de convolution sont des séries d’exponentielles décroissantes (hormis le premier terme qui est constant). Leurs temps de relaxation sont pilotés essentiellement par le temps de dispersion td. Dans la limite où td tend vers 0 à Péclet constant, les termes transitoires des noyaux se réduisent à un Dirac, débouchant sur un premier modèle simplifié à réponse instantanée c'est-à-dire un modèle de transport quasi-statique. Dans le cas où les volumes des pores sont suffisamment grands, les noyaux se réduisent à leur premier terme. Ces formulations du transport généralisent celles de la littérature. En particulier pour des Péclet petit ou grand on retrouve respectivement les modèles usuels en régime dispersif et convectif. Numériquement, la décroissance exponentielle des noyaux permet d’optimiser le calcul des convolutions avec une précision arbitrairement fixée, réduisant drastiquement le temps de résolution. / Protection and remediation of ground water resources are a major societal challenge. It implies to understand the evolution of solutes as pollutant in the saturated and non-saturated zones. For that purpose numerous studies have been conducted for modeling the reactive transport in a porous media. At Darcy scale, the behavior of solutes depends on microscopic heterogeneity for the media. The Pore Network Models (PNM) simplifies drastically its geometry and considers pores linked by straight throats the section of which is constant. They give a description which is in between the macroscopic and the pore descriptions. With such geometry it is possible to use a Poiseuille flow modeling the flux. With respect to the reactiontransport equation, we seek the analytical solution of the CDE in throats, which in turn allows computing the mass flux in pores. The transport solution consists of a Volterra equation system. Its convolution kernels result in a summation of time function which is decreasing exponentially with time (except the first term which still constant). The time constant is driven by the diffusion time td. As td goes to zero, keeping the Peclet number fixed, each term of the summation reduces to a Dirac. The response of the system is then instantaneous. When the volume of the pore is large enough it is possible to neglect all the term of the kernel except the constant one. In the limit where the Peclet number goes to zero, usual models are recovered. Numerically, the exponential time decreasing of the kernel allow to optimize their computational time up to an arbitrary fixed precision.