The oil production from well fractured carbonate reservoirs is a considerable part of the total oil production in the world. The petroleum resource base in naturally fractured reservoirs is estimated to be in the range of billions of barrels in the U.S and in addition, a multibillion- barrel international oil resource base exists in naturally fractured reservoirs. Gravity drainage is important in some of oil recovery processes, either acting as the driving force in processes using horizontal wells or altering the displacement patterns during water-flooding, chemical flooding, CO2 flooding and other EOR methods. The gravity drainage process has a major effect on oil recovery from oil reservoirs. Gravity drainage driven oil production in naturally fractured and other complex reservoirs falls into two regimes: the balk flow regime and the film flow regime. Oil recovery by gravity drainage in a fractured reservoir strongly depends on the capillary height of the porous medium. Capillarity and gravity forces are usually the major driving forces in fractured reservoirs. This PhD thesis consists of two main parts namely: 1) Experimental works on gravity drainage, and 2) Modeling and simulation of the gravity drainage processes using COMSOL® software. An appropriate design of experiment (DOE) method was selected to find the most important parameters contributing in gravity drainage and then conduct the experiments in a useful as well as economic manner. A two-dimensional experimental setup was employed to investigate free fall gravity drainage (FFGD) and controlled gravity drainage (CGD) using unconsolidated glass beads fractured porous media having various fractures configurations. Flow visualization measurements were carried out. Following the flow visualization experiments, parametric sensitivity analysis was performed considering the effects of different system parameters such as fracture aperture, matrix height, permeability, and fluid properties on the dependent variables including drainage rate, critical pumping rate, maximum drainage rate, recovery factor and so on. These experiments enabled us to capture some aspects of the recovery mechanism and the flow communication between matrix block and fracture during gravity drainage. After analyzing the experimental data for the FFGD test runs, it was found that the rate of liquid flowing from matrix to fracture is proportional to the difference of liquid levels in the matrix and in the fracture. In addition, the characteristic rate and the maximum liquid drainage rate from the fractured models were determined for such a stable gravity-dominated process. The experiments showed that the presence of fracture is more influential in lower matrix permeability systems. For a given fracture-matrix system with different initial liquid saturation conditions, it was seen that the production history can be correlated by plotting the fraction of recoverable liquid as a function of time. Furthermore, the recovery factor can be correlated using dimensionless numbers such as the Bond number and the dimensionless time. For the controlled gravity drainage (CGD) test runs conducted, the experimental results indicated that higher pumping rates cause a higher difference between the liquid levels in the fracture and in the matrix, thus the gas breakthrough happens sooner. Moreover, it was found that as long as the porous medium is drained with a constant liquid pumping rate but lower than critical rate, the height difference between the G-L interfaces in matrix and fracture remains constant. In this study, a new concept of “Critical Pumping Rate” (CPR) was defined at which each particular porous medium has recovery factor equal to the recovery factor for higher rates just before gas breakthrough. The difference between liquid levels in fracture and matrix remains unchanged at rates higher than CPR. Known this particular withdrawal rate, there are two main advantages, namely: 1) choosing a pumping rate lower than it to drain the reservoir without getting gas breakthrough; and 2) understanding the physics of pumping behaviour from fractured media and extending the concept to the real cases. In addition, the maximum liquid pumping rate from each physical model was studied and it was found that the rate depends strongly on the storage capacity of the fractures, petrophysical properties of each model as well as physical properties of test fluids. The critical rate, maximum rate, recovery factor at gas breakthrough and difference of gas liquid interface positions in matrix and fracture were correlated by dimensionless numbers such as Bond number, Capillary, and the ratio of permeabilities. Linear regression correlations presented in this study can predict production history and flow behaviour in the fractured porous media for a wide range of dimensionless numbers. The COMSOL® software was used to numerically simulate the gravity drainage processes in the two-dimensional flow experiments for fractured porous media. The parameters of the model were based on theory, as well as on the results of the two-dimensional gravity drainage experiments. The simulation results for the gravity drainage processes compared favourably with the experimental results, as a good match between the numerical solution and the experimental data was found. The simulation model developed provides a basis for further modeling of gravity drainage process in more complicated porous media.

The oil production from well fractured carbonate reservoirs is a considerable part of the total oil production in the world. The petroleum resource base in naturally fractured reservoirs is estimated to be in the range of billions of barrels in the U.S and in addition, a multibillion- barrel international oil resource base exists in naturally fractured reservoirs. Gravity drainage is important in some of oil recovery processes, either acting as the driving force in processes using horizontal wells or altering the displacement patterns during water-flooding, chemical flooding, CO2 flooding and other EOR methods. The gravity drainage process has a major effect on oil recovery from oil reservoirs. Gravity drainage driven oil production in naturally fractured and other complex reservoirs falls into two regimes: the balk flow regime and the film flow regime. Oil recovery by gravity drainage in a fractured reservoir strongly depends on the capillary height of the porous medium. Capillarity and gravity forces are usually the major driving forces in fractured reservoirs. This PhD thesis consists of two main parts namely: 1) Experimental works on gravity drainage, and 2) Modeling and simulation of the gravity drainage processes using COMSOL® software. An appropriate design of experiment (DOE) method was selected to find the most important parameters contributing in gravity drainage and then conduct the experiments in a useful as well as economic manner. A two-dimensional experimental setup was employed to investigate free fall gravity drainage (FFGD) and controlled gravity drainage (CGD) using unconsolidated glass beads fractured porous media having various fractures configurations. Flow visualization measurements were carried out. Following the flow visualization experiments, parametric sensitivity analysis was performed considering the effects of different system parameters such as fracture aperture, matrix height, permeability, and fluid properties on the dependent variables including drainage rate, critical pumping rate, maximum drainage rate, recovery factor and so on. These experiments enabled us to capture some aspects of the recovery mechanism and the flow communication between matrix block and fracture during gravity drainage. After analyzing the experimental data for the FFGD test runs, it was found that the rate of liquid flowing from matrix to fracture is proportional to the difference of liquid levels in the matrix and in the fracture. In addition, the characteristic rate and the maximum liquid drainage rate from the fractured models were determined for such a stable gravity-dominated process. The experiments showed that the presence of fracture is more influential in lower matrix permeability systems. For a given fracture-matrix system with different initial liquid saturation conditions, it was seen that the production history can be correlated by plotting the fraction of recoverable liquid as a function of time. Furthermore, the recovery factor can be correlated using dimensionless numbers such as the Bond number and the dimensionless time. For the controlled gravity drainage (CGD) test runs conducted, the experimental results indicated that higher pumping rates cause a higher difference between the liquid levels in the fracture and in the matrix, thus the gas breakthrough happens sooner. Moreover, it was found that as long as the porous medium is drained with a constant liquid pumping rate but lower than critical rate, the height difference between the G-L interfaces in matrix and fracture remains constant. In this study, a new concept of “Critical Pumping Rate” (CPR) was defined at which each particular porous medium has recovery factor equal to the recovery factor for higher rates just before gas breakthrough. The difference between liquid levels in fracture and matrix remains unchanged at rates higher than CPR. Known this particular withdrawal rate, there are two main advantages, namely: 1) choosing a pumping rate lower than it to drain the reservoir without getting gas breakthrough; and 2) understanding the physics of pumping behaviour from fractured media and extending the concept to the real cases. In addition, the maximum liquid pumping rate from each physical model was studied and it was found that the rate depends strongly on the storage capacity of the fractures, petrophysical properties of each model as well as physical properties of test fluids. The critical rate, maximum rate, recovery factor at gas breakthrough and difference of gas liquid interface positions in matrix and fracture were correlated by dimensionless numbers such as Bond number, Capillary, and the ratio of permeabilities. Linear regression correlations presented in this study can predict production history and flow behaviour in the fractured porous media for a wide range of dimensionless numbers. The COMSOL® software was used to numerically simulate the gravity drainage processes in the two-dimensional flow experiments for fractured porous media. The parameters of the model were based on theory, as well as on the results of the two-dimensional gravity drainage experiments. The simulation results for the gravity drainage processes compared favourably with the experimental results, as a good match between the numerical solution and the experimental data was found. The simulation model developed provides a basis for further modeling of gravity drainage process in more complicated porous media.

Elemental sulphur (S0) is produced from processing bitumen from the oil sands region, Alberta. Long term storage of this S0 is under consideration. The objective of the current study was to determine the controls on water migration in variably saturated S0 blocks. Based on visual observations of S0 blocks, they were characterized as a hydrophobic fractured porous media. Thus, measurements of the hydraulic characteristics, including porosity (n) and hydraulic conductivity (K) of the matrix and the fractured media, were undertaken. These data were used to create characteristic relationships of unsaturated K (Kunsat) and volumetric moisture content (è) change with change in positive injection pressure (Ø).<p>Analyses showed that the mean total matrix n (nm) was 0.094 ± 0.035 (n = 280), the mean n available for water migration (na) was 0.065 ± 0.044 (n = 8) and the mean (geometric) K for the matrix was 2.0 x 10-6 ± 2.1 x 10-6 ms-1. In the case of vertical fractures, the aperture frequencies were measured to be 2.5, 10.0 and 21.0 m-1 for fractures with apertures > 1.4, 1.4 to 0.6 and < 0.6 mm respectively while the frequency of horizontal fractures, were measured to be 1.7 and 3.7 m-1 for with apertures > 1.4, and < 1.4 mm respectively. The fracture n (nf) was determined to be 0.0135. è Ø relationships were determined for both the fractured and non fractured media. From these plots, water entry values of 9 mm and 1 m were determined for the fracture pore space and the matrix pore space, respectively.<p>Simulations of packer tests resulted in a bulk saturated K (Kb) values ranging from 8.5 x 10-5 to 2 x 10-4 ms-1 above 9 m depth and 3 x 10-6 to 1.5 x 10-5 ms-1 below 9 m depth. Coupled Kunsat Ø and è Ø relationships were used to conceptually describe water migration in S0 blocks under different precipitation and mounding conditions. These plots also showed that the Kb is dominated by the fractures.

Elemental sulphur (S0) is produced from processing bitumen from the oil sands region, Alberta. Long term storage of this S0 is under consideration. The objective of the current study was to determine the controls on water migration in variably saturated S0 blocks. Based on visual observations of S0 blocks, they were characterized as a hydrophobic fractured porous media. Thus, measurements of the hydraulic characteristics, including porosity (n) and hydraulic conductivity (K) of the matrix and the fractured media, were undertaken. These data were used to create characteristic relationships of unsaturated K (Kunsat) and volumetric moisture content (è) change with change in positive injection pressure (Ø).<p>Analyses showed that the mean total matrix n (nm) was 0.094 ± 0.035 (n = 280), the mean n available for water migration (na) was 0.065 ± 0.044 (n = 8) and the mean (geometric) K for the matrix was 2.0 x 10-6 ± 2.1 x 10-6 ms-1. In the case of vertical fractures, the aperture frequencies were measured to be 2.5, 10.0 and 21.0 m-1 for fractures with apertures > 1.4, 1.4 to 0.6 and < 0.6 mm respectively while the frequency of horizontal fractures, were measured to be 1.7 and 3.7 m-1 for with apertures > 1.4, and < 1.4 mm respectively. The fracture n (nf) was determined to be 0.0135. è Ø relationships were determined for both the fractured and non fractured media. From these plots, water entry values of 9 mm and 1 m were determined for the fracture pore space and the matrix pore space, respectively.<p>Simulations of packer tests resulted in a bulk saturated K (Kb) values ranging from 8.5 x 10-5 to 2 x 10-4 ms-1 above 9 m depth and 3 x 10-6 to 1.5 x 10-5 ms-1 below 9 m depth. Coupled Kunsat Ø and è Ø relationships were used to conceptually describe water migration in S0 blocks under different precipitation and mounding conditions. These plots also showed that the Kb is dominated by the fractures.

No description available.

Pore scale interaction between matrix and fracture during miscible and immiscible CO2 injection was studied experimentally using visual models. Initially, visualization experiments were conducted on 2-D glass bead packed models by injecting n-heptane (solvent) displacing different kinds of processed oil. The focus was on the displacement patterns and solvent breakthrough controlled by matrix-fracture interaction and the pore scale behaviour of solvent-oil interaction for different fracture and injection conditions (rate, vertical vs. horizontal injection) as well as oil viscosity. Besides the visual investigation, effluent was also analyzed to calculate the solvent cut and oil recovery. Next, the process was modeled numerically using a commercial compositional simulator and the saturation distribution in the matrix was matched to the experimental data. The key parameters in the matching process were the effective diffusion coefficients and the longitudinal and transverse dispersivities. The diffusion coefficients were specified for each fluid and dispersivities were assigned into grid blocks separately for the fracture and the matrix. Finally, glass etched microfluidic models were used to investigate pore scale interaction between the matrix and the fracture. The models were prepared by etching homogeneous and heterogeneous micro scale pore patterns on glass sheets bonded together and then saturated with colored n-decane as the oleic phase. CO2 was injected at miscible and immiscible conditions. The focus was on visual pore scale analysis of miscibility, breakthrough of CO2 and oil/CO2 transfer between the matrix and the fracture under different miscibility, injection rate and wettability conditions. / Petroleum Engineering

[pt] A avaliação do potencial de contaminação de capatações de água, por causa das águas residuais provenientes dos sistemas de tanque séptico, é feita a partir da definição da distância de separação mínima que deve existir entre a captação e o local de infiltração do efluente. A determinação dessa distância define a zona de proteção da captação. Existem três metodologias para definir o tamanho dessa zona de proteção: metodologias baseadas em distâncias fixas e tempos de trânsito, metodologias baseadas na vulnerabilidade e metodologias baseadas no risco de infecção. No caso da Costa Rica, as avaliações são feitas através do uso da metodologia baseada no tempo de trânsito. O tempo de trânsito empregado corresponde ao tempo de sobrevivência dos vírus. Nesta análise determina-se a distância máxima percorrida pelos vírus durante esse tempo, e essa distância define a separação mínima. Esse método considera que o transporte ocorre por percolação vertical saturada através da zona não saturada, e por transporte ao longo da interface água-ar na zona saturada segundo o gradiente natural. Neste trabalho apresenta-se um novo procedimento, baseado no risco de infecção, para a determinação da distância de separação considerando os efeitos da saturação variável e o fraturamento. Este procedimento determina a distância máxima percorrida, a partir do cálculo das concentrações de vírus. A distância de separação mínima corresponde à distância entre a fonte de injeção e o ponto aonde a concentração atinge o valor máximo de concentração permitida. Para o desenvolvimento deste novo procedimento foi implementado um código de programação que inclui: fluxo saturado-não saturado e transporte explícito nos poros e nas fraturas, advecção, dispersão, decaimento, sorção na superfície dos sólidos, sorção nas interfaces água-ar e água-sólido, filtração mecânica e exclusão de poros. Foi realizada uma análise comparativa entre as metodologias acima descritas para três geometrias tipo representativas das condições estratigráficas de algumas áreas do Vale Central da Costa Rica. Os resultados obtidos indicaram que a metodologia normalmente empregada na Costa Rica pode ser inadequada para prever na maioria dos casos a possibilidade de contaminação. / [en] Setback distances of wellhead and catchments from septic tanks are establised by three aproaches: methods based on fixed setback distances or fixed travel times; methods based on vulnerability analysis and methods based on infection risk. In Costa Rica, the determination of setback distances is based on fixed travel times. This approach considers that during and specified travel time all microorganisms will be inactivated, and that the distance traveled during this time defines the minimum safe separation. In this approach a unitary hydraulic gradient and saturated hydraulic conductivity are considered for transport in the unsaturated zone and the natural hydraulic gradient and saturated conductivity for transport in the saturated zone. Only advection is considered as the responsible mechanism for virus transport. A new procedure is presented in this document to define the setback distance. This procedure is based on the infection risk approach. According to this approach the minimum required setback distance is defined as the distance between the injection point and the location where the contaminant reaches a maximum allowable concentration. This procedure was implemented in a computer code that considers variable saturated water flow, fractured-porous media, advection, dispersion, dynamic sorption, inactivation and mechanical filtration. A comparative analysis was performed for three hypothetical geometries using the two approaches described. The results indicate the approach normally used in Costa Rica may no reproduce adequately the possibility of catchments and wellhead contamination.

[pt] Esse trabalho apresenta um modelo numérico para a análise de processos acoplados de efeitos mecânicos e escoamento bifásico em meios porosos fraturados, com a utilização de diferentes métodos numéricos combinados (Elementos finitos contínuos e descontínuos), e utilizando uma mesma malha de elementos finitos para representar uma célula de modelo de simulação de reservatório. As descontinuidades são inseridas na malha como elementos de nós duplicados colapsados. Empregam-se procedimentos numéricos desenvolvidos em dois grupos distintos. Um primeiro grupo de simulações trata de um procedimento numérico de escoamento bifásico, com ênfase à obtenção de um balanço de volumes verdadeiramente conservativo. Nesta fase, avalia-se uma formulação numérica que emprega um processo em três etapas: o método dos elementos finitos (EF), para a aproximação da equação da pressão; intermediariamente, utiliza-se o método de Raviart-Thomas de mais baixa ordem para aproximação da velocidade; e a aproximação da equação da saturação pelo método dos elementos finitos descontínuos (MEFD), que utiliza um limitador de inclinação multidimensional de modo a evitar oscilações na reconstrução dos dados de saturação. Para fins de validação da formulação desenvolvida, comparam-se os resultados obtidos com simulações utilizando o Método do Volumes Finitos (VF). O segundo grupo de simulações trata de acoplar o módulo mecânico (em EF) ao simulador de escoamento bifásico, de modo que a variação do estado de tensões, decorrente da explotação, seja considerada nas simulações. Essa análise permite uma melhor representação do fenômeno estudado além de proporcionar melhores resultados quanto à chegada da água nos poços produtores, afetando a produtividade do modelo. / [en] This work presents a numerical model for the analysis of coupled processes of mechanical effects and two-phase flow in fractured porous media using different numerical methods (continuous and discontinuous finite elements), and using the same finite element mesh to represent a cell of reservoir simulation model. The discontinuities are inserted into the mesh as elements of collapsed duplicate nodes. Numerical procedures developed in two distinct groups are used. A first group of simulations deals with a numerical two-phase flow procedure, with special emphasis on obtaining a truly conservative volume balance. At this stage, a numerical formulation using a three-step process is evaluated: The Finite Element Method (EF), for the approximation of the pressure equation; the lower order of Raviart-Thomas method is used to velocity approximation; and the approximation of the saturation equation by the discontinuous finite element method (MEFD), using a multidimensional slope limiter in order to avoid oscillations in the reconstruction of the saturation data. For the validation of the developed formulation, the results obtained are compared with simulations using the Finite Volume Method (VF), with focus on the analysis of the conservation of volumes. The second group of simulations couple the mechanical module (in EF) to twophase flow computer program, so the variation of the stress state, due to exploitation, is considered in the simulations. This analysis allows a better representation of the phenomenon. In addition to providing better results regarding the arrival of water in the producing wells, affecting the productivity of the model.

Le but de cette thèse est de développer et d’analyser les méthodes Hybrides d’Ordre Élevé (HHO: Hybrid High-Order, en anglais) pour des problèmes d’interfaces. Nous nous intéressons à deux types d’interfaces (i) les interfaces diffuses, et (ii) les interfaces traitées comme frontières internes du domaine computationnel. La première moitié de ce manuscrit est consacrée aux interfaces diffuses, et plus précisément aux célèbres équations de Cahn–Hilliard qui modélisent le processus de séparation de phase par lequel les deux composants d’un fluide binaire se séparent pour former des domaines purs en chaque composant. Dans la deuxième moitié, nous considérons des modèles à dimension hybride pour la simulation d’écoulements de Darcy et de transports passifs en milieu poreux fracturé, dans lequel la fracture est considérée comme un hyperplan (d’où le terme hybride) qui traverse le domaine computationnel. / The purpose of this Ph.D. thesis is to design and analyse Hybrid High-Order (HHO) methods on some interface problems. By interface, we mean (i) diffuse interface, and (ii) interface as an immersed boundary. The first half of this manuscrit is dedicated to diffuse interface, more precisely we consider the so called Cahn–Hilliard problem that models the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. In the second half, we deal with the interface as an immersed boundary and consider a hybrid dimensional model for the simulation of Darcy flows and passive transport in fractured porous media, in which the fracture is considered as an hyperplane that crosses our domain of interest.

Cette thèse est consacrée à l'étude mathématique d'un problème inverse en hydrogéologie : le but est d'identifier des fractures en milieu poreux, connaissant des mesures de l'écoulement dans le sous-sol. Le nombre, la localisation et les paramètres physiques des fractures sont recherchés. Ce problème est formulé comme la minimisation au sens des moindres carrés d'une fonctionnelle évaluant l'écart entre les mesures et les résultats du modèle direct. L'écoulement est celui d'un fluide monophasique incompressible (loi de Darcy). Un modèle traitant les fractures comme des interfaces est utilisé. Le problème direct est le modèle de fracture discrétisé par la méthode des éléments finis mixtes hybrides.Pour résoudre ce problème inverse, un nouvel algorithme itératif a été développé, basé sur l’utilisation d’indicateurs de fractures mis au point pendant la thèse. Ces indicateurs donnent une information au premier ordre concernant l'effet de l'ajout d'une nouvelle fracture. Comme ces indicateurs sont peu coûteux, un grand nombre de configurations de fractures sont testées à chaque itération. L’algorithme a été programmé, validé puis testé numériquement dans des situations variées, en utilisant des mesures synthétiques. Il donne des résultats très satisfaisants, bien que ce problème soit réputé difficile.Enfin, l’étude de l’identifiabilité du problème inverse a été amorcée. Pour un modèle simplifié de fractures (failles très perméables, cas le plus courant dans le sous-sol), on a montré que le problème. / This PhD is dedicated to the mathematical study of an inverse problem in hydrogeology: the goal is to identify fractures in porous medium, knowing measurements of the underground flow. The number, the location and the physical parameters of the fracture are looked for. This problem is formulated as the least squares minimization of a function evaluating the misfit between measurements and the result of the direct model. We used a model describing the flow of a monophasic incompressible fluid (Darcy's law), in a porous medium containing some fractures represented by interfaces. The direct problem is the fracture model discretized by the mixed hybrid finite element method. To solve this inverse problem, we developed an iterative algorithm, which is based on the use of fracture indicators that have been developed durig the thesis. These indicators give a first order information concerning the effect of the addition of a new fracture. As these indicators are inexpensive, a large number of configurations of new fractures is tested at each iteration. The algorithm was programmed, validated and tested numerically in various situations, using synthetic measurements. It gives very satisfactory results, although this problem is considered difficult. Finally, an early study of identifiability of the inverse problem of fractures in porous medium has been achieved. It allowed to prove the identifiability for a simplified model (very permeable faults, which is common in the underground). The question of identifiability for the full fracture model remains open.