Tesis sobre el tema "Immiscible two-Phase flow"

Thesis (M. S.)--Mechanical Engineering, Georgia Institute of Technology, 2009.
Committee Chair: Cyrus K. Aidun; Committee Member: Marc K. Smith; Committee Member: S. Mostafa Ghiaasiaan. Part of the SMARTech Electronic Thesis and Dissertation Collection.

The research is a study on the microscopic scale of the immiscible displacement of oil by water in a porous medium such as sandstone. Of particular interest (with application to the oil industry) are the residual saturation of oil, the permeability to water at residual oil saturation and the maximum trapped blob size. Initially the effects of gravity, surface tension and distribution of pore sizes were studied in a computer simulation of a buoyancy driven, quasi-static invasion. The rock was modelled as a three-dimensional lattice of spherical pores connected by narrow cylindrical throats. With the rock water-wet, the tendency of the surface tension to favour the invasion of smaller pores led to a larger residual oil saturation by pore volume than by pore numbers. Also bourne out were some scaling arguments based on percolation theory for the maximum trapped blob size as a function of the relative strength of buoyancy and surface tension forces. The second part of the research investigated the interaction of viscous and surface tension forces. As this is a much more complicated problem, involving the solution of flow equations, the invasion process was first simulated with exact equations of motion on small networks (up to 10x10), where surface tension effects dominate. From these simulations a simplified set of rules was developed to determine which pore in a locality on the oil-water interface is invaded and how long the invasion takes. These rules include a viscous correction to the dominant surface tension forces. Finally, some theory has been developed for the inclusion of the small-scale analysis into a larger model, allowing a full simulation of the viscous dominated invasion to be performed.

This thesis focuses primary on two-phase displacements under capillary-controlled flow conditions at relatively large scales, considering solution techniques that capture the dynamics of two-phase displacements for homogeneous flow domains, and deriving representative averages for heterogeneous systems with strong spacial variations in two-phase properties. First, we review main flow mechanisms encountered at large scales when capillary forces dominate the displacement process, where we present main solution techniques for homogeneous flow domains and introduce analytical treatments for other flow mechanisms that do not follow standard time-scaling. We also present a comprehensive investigation of spontaneous imbibition processes in porous rocks both numerically and analytically, and propose a simplified but accurate analytic approximation using perturbation theory, that considerably improves the implementation process, as compared with the original analytical solution. After that, an investigation of the impact of capillary backpressure on counter-current flow is performed, as this is considered one of the main drawbacks in using the continuum modelling approach. We then apply steady-state capillary-controlled upscaling in heterogeneous environment, where large-scale invasion percolation is coupled with a conventional Darcy solver to identify large-scale trapping due to capillary forces. In other words, a phase may fail to form a connected path across a given domain at capillary equilibrium, and some regions therefore may produce disconnected clusters. In such cases, conventional upscaling processes might not be accurate since identification and removal of these isolated clusters are extremely important to the global connectivity of the system and the stability of the numerical solvers. We present a comprehensive investigation using random absolute permeability fields, for water-wet, oil-wet and mixed-wet systems, where we show that in oil-wet and mixed-wet media, large-scale trapping of oil controlled by variations in local capillary pressure, may be more significant than the local trapping controlled by pore-scale displacement.

Fluid flow and transport in fractured geological formations is of fundamental socio-economic importance, with applications ranging from oil recovery from the largest remaining hydrocarbon reserves to bioremediation techniques. Two mechanisms are particularly relevant for flow and transport, namely spontaneous imbibition (SI) and hydrodynamic dispersion. This thesis investigates the influence of SI and dispersion on flow and transport during immiscible two-phase flow. We make four main contributions. Firstly, we derive general, exact analytic solutions for SI that are valid for arbitrary petrophysical properties. This should finalize the decades-long search for analytical solutions for SI. Secondly, we derive the first non-dimensional time for SI that incorporates the influence of all parameters present in the two-phase Darcy formulation - a problem that was open for more than 90 years. Thirdly, we show how the growth of the dispersive zone depends on the flow regime and on adsorption. To that end we derive the first known set of analytical solutions for transport that fully accounts for the effects of capillarity, viscous forces and dispersion. Finally, we provide numerical tools to investigate the influence of heterogeneity by extending the higher order finite-element finite-volume method on unstructured grids to the case of transport and two-phase flow.

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO
Esta tese propõe um método meshfree adaptativo de advecção para problemas de fluxo bifásico de fluidos incompressíveis e imiscíveis em meios porosos heterogêneos tridimensionais. Este método se baseia principalmente na combinação do método Semi-Lagrangeano adaptativo com interpolação local meshfree usando splines poliharmônicas como funções de base radial. O método proposto é uma melhoria e uma extensão do método adaptativo meshfree AMMoC proposto por Iske e Kaser (2005) para modelagem 2D de reservatórios de petróleo. Inicialmente este trabalho propõe um modelo em duas dimensões, contribuindo com uma melhoria significativa no cálculo do Laplaciano, utilizando os métodos meshfree de Hermite e Kansa. Depois, o método é ampliado para três dimensões (3D) e para um meio poroso heterogêneo. O método proposto é testado com o problema de five spot e os resultados são comparados com os obtidos por sistemas bem conhecidos na indústria de petróleo.
This thesis proposes an adaptive meshfree advection method for two-phase flow problems of incompressible and immiscible fluids through three-dimensional heterogeneous porous media. This method is based mainly on a combination of adaptive semi-Lagrangian method with local meshfree interpolation using polyharmonic splines as radial basis functions. The proposed method is an improvement and extension of the adaptive meshfree advection scheme AMMoC proposed by Iske and Kaser (2005) for 2D oil reservoir modeling. Initially this work proposes a model in two dimensions, contributing to a significant improvement in the calculation of the Laplacian, using the meshfree methods of Hermite and Kansa. Then, the method is extended to three dimensions (3D) and a heterogeneous porous medium. The proposed method is tested with the five spot problem and the results are compared with those obtained by well-known systems in the oil industry.

Cette thèse est centrée autour du développement et de l'analyse des schémas volumes finis robustes afin d'approcher les solutions du modèle diphasique compressible en milieux poreux hétérogènes et anisotropes. Le modèle à deux phases compressibles comprend deux équations paraboliques dégénérées et couplées dont les variables principales sont la saturation du gaz et la pression globale. Ce système est discrétisé à l'aide de deux méthodes différentes (CVFE et DDFV) qui font partie de la famille des volumes finis. La première classe à laquelle on s'intéresse consiste à combiner la méthode des volumes finis et celle des éléments finis. Dans un premier temps, on considère un schéma volume finis upwind pour la partie convective et un schéma de type éléments finis conformes pour la diffusion capillaire. Sous l'hypothèse que les coefficients de transmissibilités sont positifs, on montre que la saturation vérifie le principe du maximum et on établit des estimations d'énergies permettant de démontrer la convergence du schéma. Dans un second temps, on a mis en place un schéma positif qui corrige le précédent. Ce schéma est basé sur une approximation des flux diffusifs par le schéma de Godunov. L'avantage est d'établir la bornitude des solutions approchées ainsi que les estimations uniformes sur les gradients discrets sans aucune contrainte ni sur le maillage ni sur la perméabilité. En utilisant des arguments classiques de compacité, on prouve rigoureusement la converge du schéma. Chaque schéma est validé par des simulations numériques qui montrent bien le comportement attendu d'une telle solution. Concernant la deuxième classe, on s'intéressera tout d'abord à la construction et à l'étude d'un nouveau schéma de type DDFV (Discrete Duality Finite Volume) pour une équation de diffusion non linéaire dégénérée. Cette méthode permet d' avantage de prendre en compte des maillages très généraux et des perméabilités quelconques. L'idée clé de cette discrétisation est d'approcher les flux dans la direction normale par un schéma centré et d'utiliser un schéma décentré dans la direction tangentielle. Par conséquent, on démontre que la solution approchée respecte les bornes physiques et on établit aussi des estimations d'énergie. La convergence du schéma est également établie. Des résultats numériques confirment bien ceux de la théorie. Ils exhibent en outre que la méthode est presque d'ordre deux
The objective of this thesis is the development and the analysis of robust and consistent numerical schemes for the approximation of compressible two-phase flow models in anisotropic and heterogeneous porous media. A particular emphasis is set on the anisotropy together with the geometric complexity of the medium. The mathematical problem is given in a system of two degenerate and coupled parabolic equations whose main variables are the nonwetting saturation and the global pressure. In view of the difficulties manifested in the considered system, its cornerstone equations are approximated with two different classes of the finite volume family. The first class consists of combining finite elements and finite volumes. Based on standard assumptions on the space discretization and on the permeability tensor, a rigorous convergence analysis of the scheme is carried out thanks to classical arguments. To dispense with the underlined assumptions on the anisotropy ratio and on the mesh, the model has to be first formulated in the factional flux formulation. Moreover, the diffusive term is discretized by a Godunov-like scheme while the convective fluxes are approximated using an upwind technique. The resulting scheme preserves the physical ranges of the computed solution and satisfies the coercivity property. Hence, the convergence investigation holds. Numerical results show a satisfactory qualitative behavior of the scheme even if the medium of interest is anisotropic. The second class allows to consider more general meshes and tensors. It is about a new positive nonlinear discrete duality finite volume method. The main point is to approximate a part of the fluxes using a non standard technique. The application of this ideato a nonlinear diffusion equation yields surprising results. Indeed,not only is the discrete maximum property fulfilled but also the convergence of the scheme is established. Practically, the proposed method shows great promises since it provides a positivity-preserving and convergent scheme with optimal convergence rates

The scope of the current thesis is the comprehensive understanding of the droplet impact and spreading dynamics on flat and curved surfaces with the aim of simulating high density ratio immiscible two phase flows in porous media. Understanding the dynamic behavior of droplet impingement onto solid substrate can provide significant information about the fluid flow dynamics in porous structures. The numerically study process will be realized by using a high density ratio multi-phase lattice Boltzmann model which is able to simulate multi-phase flows in complex systems. The interfacial information between the two immiscible phases can be captured without tracking or constructing the vapour-liquid interface. A three dimensional lattice Boltzmann model is applied on the study of the impaction of a liquid droplet on a dry flat surface for a liquid-gas system with large density ratio. The impaction of liquid droplet on a curved surface for the liquid-gas system with large density ratio and low kinematic viscosity of the fluid is computed by a two-dimensional multi-relaxation-time (MRT) interaction-potential-based lattice Boltzmann model based on the improved forcing scheme. The dynamics behaviors of the spreading of the liquid droplet on the flat surface as well as the impaction of the liquid droplet on a curved surface are computed, followed by their dependence on the Reynolds number, Weber number, Galilei number and surface characteristics. Moreover, an improved force scheme is proposed for the three-dimensional MRT pseudopotential lattice Boltzmann model which is based on the improved force scheme for the Single relaxation time (SRT) pseudopotential lattice Boltzmann model and the Chapman-Enskog analysis. The validation for the new developed three-dimensional multi-relaxation time lattice Boltzmann model is carried out through Laplace’s law ad by achieving thermodynamic consistency. In addition, the relationship between the fluid-solid interaction potential parameter Gw and the contact angle is investigated for the new developed three-dimensional MRT lattice Boltzmann model. The immiscible two-phase flow in porous media is carried out by a two dimensional MRT lattice Boltzmann model. The porous media structures with different geometrical properties are artificially generated by a Boolean model based on a random distribution of overlapping ellipses/circles. Furthermore, the impact of geometrical properties on the immiscible two-phase flows in porous media is investigated in the pore scale. The lattice Boltzmann model results provide significant information i on the interface between the two immiscible phases in complex systems, it is easy to apply for complex domains with bounce back boundary wall condition and be able to handle multi-phase and multi-component flows without tracing the interfaces between different phases.

Les expériences de déplacement en milieu poreux sont la méthode habituellement utilisée pour étudier l'écoulement biphasique immiscible. Cependant, malgré les aspects de reproductibilité, un inconvénient majeur est que ces expériences de type "boîte noire" ne permettent pas d'observer et de capturer les phénomènes clés à l'échelle des pores, y compris les interactions interfaciales et les détails sur la mobilisation de l'huile piégée (par exemple, la taille et la distribution des ganglions résiduels). C'est pourquoi les dispositifs micromodèles microfluidiques sont désormais largement utilisés dans les expériences de récupération assistée d'huile (EOR) en laboratoire. Ils préservent les détails structurels de la roche tout en offrant des avantages tels que la facilité de nettoyage et la répétabilité. Le suivi visuel du déplacement des fluides est particulièrement important car il peut fournir plus de détails sur le comportement des phases mouillantes et non mouillantes dans les milieux poreux, aidant à élaborer des stratégies ciblées pour améliorer les taux de récupération du pétrole. Cette thèse explore la dynamique complexe des écoulements biphasiques immiscibles en combinant des modèles de milieux poreux microfluidiques, souvent appelés « réservoir-sur-puce », avec des simulations numériques.Dans nos expériences, nous avons utilisé des techniques morphologiques pour surveiller et enregistrer le comportement de déplacement dans un écoulement biphasique, en étudiant systématiquement les effets de différents nombres capillaires (Ca) et rapports de viscosité (M) sur les mécanismes d'écoulement et la mobilisation de l'huile résiduelle. Les résultats ont indiqué que pendant l'inondation par l'eau, le déplacement présentait des caractéristiques de doigté visqueux à des valeurs plus basses de Ca et M. En augmentant le débit pour améliorer Ca de dix fois, l'huile résiduelle montrait une invasion latérale et même arrière des chemins de flux sans changements significatifs dans la taille des grappes. Avec l'augmentation de M, la taille des grappes et la taille maximale des grappes ont diminué, conduisant à une distribution plus uniforme de l'huile résiduelle et à une Sor plus faible. Le mécanisme de mobilisation de l'huile résiduelle s'est manifesté par la rupture des ganglions, les nouveaux petits ganglions formés étant mobilisés sous des pressions plus élevées. La distribution des grappes d'huile résiduelle est conforme à la théorie de percolation, où l'exposant de mise à l'échelle τ est de 2,0. Tous les résultats expérimentaux pour Sor et les valeurs de Ca correspondantes se sont regroupés sur la courbe classique de désaturation capillaire (CDC).Les résultats expérimentaux ont servi de fondement pour développer un modèle numérique utilisant une approche de champ de phase. Ce modèle, basé sur le système d'équations de Cahn-Hilliard-Navier-Stokes, capture efficacement le comportement d'écoulement biphasique de fluides immiscibles dans des domaines confinés. Il intègre les équations de conservation de la masse et de la quantité de mouvement, enrichies par la dynamique de séparation de phase et les considérations d'énergie interfaciale. Les simulations numériques, exécutées sur la plateforme d'éléments finis en source ouverte Fenics, s'alignent qualitativement et quantitativement avec les observations expérimentales, confirmant la précision du modèle pour prédire les comportements fluidiques sous diverses conditions physiques, et avançant notre compréhension de la dynamique des fluides à l'échelle des pores. Les simulations se concentrent sur l'analyse de l'influence des propriétés des fluides et des conditions opérationnelles sur les mécanismes de déplacement à l'échelle des pores
The core-flood experiments are the usual method used to study the immiscible biphasic flow. However, beside reproducibility aspects, a significant drawback is that with these black box experiments, we cannot observe and capture key phenomena at the pore scale, including interfacial interactions and details about mobilization of the trapped oil (e.g. size and distribution of residual ganglia). This is why microfluidic micromodel devices are now extensively used in lab EOR experiments. They preserve the structural details of the rock while offering advantages such as easy cleaning and repeatability. Visual tracking of fluids displacement is particularly important as it can provide more details about the behavior of wetting and non-wetting phases in porous media, aiding in targeted strategies to enhance oil recovery rates. This thesis explores the intricate dynamics of immiscible two-phase flows combines microfluidic porous medium models, often referred to as “reservoir-on-a-chip”, with numerical simulations.In our experiments, we used morphological to monitor and record displacement behavior in biphasic flow, systematically studying the effects of different capillary numbers (Ca) and viscosity ratios (M) on the flow mechanisms and the mobilization of residual oil. The results indicated that during waterflooding, displacement exhibited characteristics of viscous fingering at lower Ca and M values. By increasing the flow rate to enhance Ca tenfold, the residual oil showing lateral and even backward invasion of flow paths without significant changes in cluster size. With increasing M, both the cluster size and the maximum cluster size decreased, leading to a more uniform distribution of residual oil and lower Sor. The mobilization mechanism of residual oil manifested as ganglia breakup, with newly formed smaller ganglia being mobilized under higher pressures. The distribution of residual oil clusters is consistent with percolation theory, where the scaling exponent τ is 2.0. All experimental results for Sor and corresponding Ca values collapsed onto the classical Capillary Desaturation Curve (CDC).The experimental findings served as a foundation for developing a numerical model using a phase-field approach. This model, based on the Cahn-Hilliard-Navier-Stokes system of equations, effectively captures the bi-phasic flow behavior of immiscible fluids within confined domains. It incorporates conservation of mass and momentum equations, enhanced by phase separation dynamics and interfacial energy considerations. The numerical simulations, executed on the open-source finite element platform Fenics, align qualitatively and quantitatively with experimental observations, affirming the accuracy of model in predicting fluid behaviors under varied physical conditions, advancing our understanding of pore-scale fluid dynamics. Simulations focus on dissecting the influence of fluid properties and operational conditions on the displacement mechanisms at the pore scale

Il est important de comprendre les forces motrices qui contrôlent l'écoulement de deux fluides immiscibles dans un milieu poreux. En effet, il existe une large gamme d'applications des écoulements diphasiques en milieux poreux, notamment ceux qui concernent la récupération assistée du pétrole (EOR). Le développement des techniques quantitatives d'imagerie par résonance magnétique (IRM) ouvre de nouvelles possibilités pour étudier et caractériser les flux multiphasiques en milieu poreux. Ce travail s’intéresse précisément à décrire le déplacement de deux fluides immiscibles (eau-huile) au sein d’un milieu poreux en utilisant les techniques d’IRM. Le milieu poreux est initialement saturé d’huile qu’on vient déplacer en injectant de l’eau par le bas, l’huile et l’eau pouvant s’évacuer par le haut. L’objectif général de l’étude est de déterminer le déplacement et la déformation du front (eau-huile) au cours du temps, et de préciser les mécanismes de piégeage des phases. Des expériences sont menées sur deux modèles poreux. L’un mouillant à l’huile consiste en un empilement de petites billes en polystyrène (0,4 mm < dp < 0,6 mm), l’autre mouillant à l’eau est un sable légèrement compacté (0,02 mm < dp < 0,50 mm). Nous avons utilisé un dispositif de micro-imagerie RMN fonctionnant à 14 T (résonance 1H à 600 MHz) pour acquérir des images à haute résolution (0.2 mm) à l’intérieur des milieux poreux au cours du déplacement des deux fluides. Les résultats obtenus ont montré que le profil de saturation en huile est fortement influencé par les propriétés du matériau poreux, telles que la porosité et la perméabilité de l'échantillon, le mouillage des phases, le débit d'injection de l’eau ou encore l’hétérogénéité de la matrice solide. L'influence du débit d’injection d’eau sur la saturation résiduelle en huile a été plus particulièrement étudiée. Les résultats expérimentaux permettent une compréhension fine du déplacement de deux fluides non miscibles pour deux types de milieux poreux, qui se différencient principalement par les effets de la mouillabilité. Dans le même temps, une simulation numérique du déplacement vertical ascendant de l’huile poussée par de l’eau dans une colonne poreuse a été réalisée et les résultats ont été comparés à nos expériences sous IRM
It is important to understand the driving forces that control the flow of two immiscible fluids in a porous medium. Indeed, there is a wide range of applications of two-phase flows in porous media, especially those relating to enhanced oil recovery (EOR). The development of quantitative magnetic resonance imaging (MRI) techniques opens up new possibilities for studying and characterizing multiphase flows in porous media. This work is specifically concerned with describing the displacement of two immiscible fluids (water-oil) in a porous medium using MRI techniques. The porous medium is initially saturated with oil which is displaced by injecting water from below, oil and water can be evacuated from above. The general objective of the study is to determine the displacement and the deformation of the front (water-oil) over time, and to specify the trapping mechanisms of the phases. Experiments are conducted on two porous models. One oil wetting consists of a stack of small polystyrene beads (0.4 mm < dp < 0.6 mm), the other wetting with water is a slightly compacted sand (0.02 mm < dp <0.50 mm). We used a 14 T NMR micro-imaging device (1H resonance at 600 MHz) to acquire high resolution images (0.2 mm) inside the porous media during the movement of the two fluids. The results obtained showed that the oil saturation profile is strongly influenced by the properties of the porous material, such as the porosity and the permeability of the sample, the wetting of the phases, the injection rate of the water or even the heterogeneity of the solid matrix. The influence of the water injection flow rate on the residual saturation of oil has been studied more particularly. The experimental results allow a fine understanding of the displacement of two immiscible fluids for two types of porous media, which mainly differ by the effects of wettability. At the same time, a numerical simulation of the upward vertical displacement of oil pushed by water in a porous column was performed and the results compared to our MRI experiments

Les situations où deux fluides non miscibles sont présents dans un matériau granulaire déformable sont largement rencontrées dans la nature et dans de nombreux domaines de l'ingénierie et de la science. Comprendre l'évolution de tels systèmes multiphases nécessite la connaissance de toutes les phases, leur distribution et interactions. Un modèle micro-hydromécanique couplé est présenté dans cette thèse sur la base de travaux précédents, visant à simuler le drainage quasi-statique de matériaux granulaires déformables. Il combine une approche de type réseau de pores et la méthode des éléments discrets (DEM) pour les fluides et les grains respectivement. Un critère local de mouvement d'interfaces fluides est établi, afin d'approximer au mieux le rôle de la géométrie porale sur les phénomènes capillaires et notamment les forces exercées sur les grains solides à l'intérieur de chaque pore. Une attention particulière est dédiée aux événements de piégeage du fluide drainé et à l'invasion préférentielle le long des bords du domaines. Le modèle est valide par la comparaison avec des résultats expérimentaux (courbes de rétention d'eau). Nous appliquons le modèle pour étudier deux questions: (1) les effets de taille finie et à la question du volume élémentaire représentatif (REV); (2) le paramètre de contrainte effective de Bishop et la relation entre contrainte effective macroscopique contrainte de contact micromécanique. Finalement, une extension du modèle au régime pendulaire est présentée et des premiers résultats sont présentés et discutés
The situation of two immiscible fluids through a deformable granular material is widely encountered in nature and in many areas of engineering and science. To understand the physical evolution of the multiphase system is of great importance for the applications. It requires the knowledge of all component phases, their distribution and interactions. A pore-scale coupled hydromechanical model is presented in this thesis based on previous work, aiming at simulating the quasi-static drainage of a deformable granular materials. The model combines a pore network approach and the discrete element method (DEM) for the fluids and grains, respectively. A local criterion for determining the local movements of the fluids interfaces established to approximate the role of the local pore geometry on capillarity and namely on the forces exerted on the solid grains inside each pore. Special attentions have been paid to the entrapment events of the receding fluid and to the preferential invasion along the boundaries. The model is validated through comparisons with experimental results (water retention curves). We apply the model for examining two issues: (1) finite size effects and the concept of representative elementary volume (REV); (2) Bishop's effective stress parameter and to the relationship between macro-scale effective stress and micro-scale contact stress. Finally, an extension to the pendular regimes is proposed and first results are presented and analyzed

We derive the two-scale limit of a linear or nonlinear saturation equation with a flow-based coordinate transformation. This transformation consists of the pressure and the streamfunction. In this framework the saturation equation is decoupled to a family of one-dimensional nonconservative transport equations along streamlines. This simplifies the derivation of the two-scale limit. Moreover it allows us to obtain the convergence independent of the assumptions of periodicity and scale separation. We provide a rigorous estimate on the convergence rate. We combine the two-scale limit with Tartar's method to complete the homogenization.

To design an efficient numerical method, we use an averaging approach across the streamlines on the two-scale limit equations. The resulting numerical method for the saturation has all the advantages in terms of adaptivity that methods have. We couple it with a moving mesh along the streamlines to resolve the shock more efficiently. We use the multiscale finite element method to upscale the pressure equation because it gives access to the fine scale velocity, which enters in the saturation equation, through the basis functions. We propose to solve the pressure equation in the coordinate frame of the initial pressure and saturation, which is similar to the modified multiscale finite element method.

We test our numerical method in realistic permeability fields, such as the Tenth SPE Comparative Solution Project permeabilities, for accuracy and computational cost.

碩士
國立交通大學
應用數學系所
97
The mathematical model of the waterflood problem which is applied in this paper can be divided into two sections. One is the pressure equation and the other is the saturation equation. And the saturation equation also can be pa- rtitioned into the transport stage saturation and the diffusive stage saturation. However, we will pay more attention to solve the transport stage saturation in this research. Here we construct a meters reservoir system for simu- lation. An efficient numerical method, locally conservative Eulerian-Lagrangian methods (LCELM), is developed to compute the transport equation to improve the conservation of waterflood. From the results of the numerical simulations, we can realize the relation between temporal variation and the flow condition.

碩士
國立交通大學
應用數學系所
100
The primary objective of this thesis is to introduce a multigrid method to solve elliptic equation with strongly discontinuous coefficients. In the beginning, we explain how to use the multigrid method to solve a 3D elliptic equation with strongly discontinuous coefficients, and then show some numerical testing results. Also, we provide some results compared with other numerical methods to show the efficency of the mutigrid method. Furthermore, we apply the multigrid method to solve two mathematical problems, one is for the waterflooding problem and the other is the incompressible Navier-Stokes equations. A locally conservative Eulerian-Lagrangian method (briefly LCELM) is used to compute the transport part of the two models. Some numerical results for the two problems will be presented as well. ii