Oil reservoirs and soil can be homogeneously wet (water-wet, oil-wet, neutralwet) or heterogeneously wet (mixed wet or fractionally wet). The goal of this research is to model the detailed configuration of wetting and non-wetting phases within homogeneously and heterogeneously wet porous media. We use a dense random pack of equal spheres as a model porous medium. The geometry of the sphere pack is complex but it is known. In homogeneously wet porous media we quantify the effect of low saturations of the wetting phase on the non-wetting phase relative permeability by solving analytically the geometry of the wetting phase. At low saturations (at or near the drainage endpoint) the wetting phase exists largely in the form of pendular rings held at grain contacts. Pore throats correspond to the constriction between groups of three grains, each pair of which can be in contact. Thus the existence of these pendular rings decreases the void area available for the flowing non-wetting phase. Consequently, the existence of the pendular rings decreases the permeability of non-wetting phase. Our model explains the significant permeability reduction of the non-wetting phase with a small change in the wetting phase in a low permeability porous medium. To model heterogeneously wet porous medium, we assume that the porous medium is fractionally wet where each grain is either oil-wet or water-wet. These waterwet or oil-wet grains are distributed randomly within the porous medium. We calculate analytically the stable fluid configuration in individual pores and throats of a fractionally wet medium. The calculation is made tractable by idealizing the configurations as locally spherical (menisci) or toroidal (pendular rings.) Because the calculation of the interface position is entirely local and grain-based, it provides a single, generalized, geometric basis for computing pore-filling events during drainage as well as imbibition. This generality is essential for modeling displacements in fractionally wet media. Pore filling occurs when an interface becomes unstable in a pore throat (analogous to the Haines condition for drainage in a uniformly wet throat), when two or more interfaces come into contact and merge to form a single interface (analogous to the Melrose condition for imbibition in uniformly wet medium), or when a meniscus in a throat touches a nearby grain (a new stability criterion). The concept of tracking the fluid/fluid interfaces on each grain means that a traditional pore network is not used in the model. The calculation of phase saturation or other quantities that are conveniently computed in a network can be done with any approach for defining pore bodies and throats. The fluid/fluid interfaces are mapped from the grain-based model to the network as needed. Consequently, the model is robust as there is no difference in the model between drainage and imbibition, as all criteria are accounted for both increasing and decreasing capillary pressure. / text

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Thesis (Ph.D.)--University of Oklahoma, 2004. / Title from title screen (viewed on Dec. 7, 2007). Title from document title page. Includes bibliographical references. Available in PDF format via the World Wide Web.

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 energy analysis of the wave motion in the Lamb’s problem for a poroelastic half-space in the framework of Biot’s theory is presented. The results for the energy velocity and quality factor of poroelastic waves are revisited. In the case of no dissipation the approach originally established for perfectly elastic media by Miller & Pursey is generalized herein to include poroelastic waves. Special cases of the resonant excitation of the Rayleigh wave and the absence of the Rayleigh wave beyond the cut-off frequency are discussed in detail. Directional diagrams for the volumetric waves are presented. A quantitative picture of the energy partition among the traveling waves is provided for several driving configurations. In the general case of dissipative media the analysis is based on the semi-analytic solution of the Lamb’s problem. In the near field, the surface load generates three wavetrains corresponding to the bulk modes. These wavetrains consist of waves which are longer and exhibit greater viscous attenuation than the corresponding volumetric modes, so that, P1, P2 and S modes emerge from the corresponding wavetrains at a certain distance from the source. For the far field, asymptotic expressions have been obtained and clearly indicate that it is only in the far field that the wave motion represents the superposition of the P1, P2, S and Rayleigh waves characterized by their corresponding wavelengths and attenuations. Moreover, these waves also exhibit geometric attenuation x^3/2 (similar to the waves in a perfectly elastic half-space). To analyze the energy partition the total input power supplied by the source is decomposed into the contributions associated with the wavetrains and the Rayleigh wave. These results provide the means for controlling the excitation of the various wave modes via changes to the driving configuration. Biot’s theory is a particular example of a non-conservative Lagrangian system with a Rayleigh dissipation function. The group velocities of poroelastic waves are complex and do not provide any information about the velocity of the energy transport. Moreover, in general the precise physical meaning of the complex group velocity is unclear. The analysis based on the detailed study of the coupled system of the damped Klein-Gordon equations (Biot’s theory yields such a formalism in the low frequency limit) suggests that both precise and approximate physical interpretations of the complex group velocity are possible. Moreover, these considerations further allow the derivation of exact closed form expressions for the energy velocity and Q factor for both longitudinal and shear poroelastic waves from energy principles. Most notably, the analysis of the resulting expressions reveals that the energy velocity of both longitudinal and shear waves equals (exceeds) the corresponding phase velocity in the case of the low (full) frequency range Biot’s theory. The exact expression for the Q factor contains an additive correction due to viscoelastic interphase interaction in the higher frequency range.

The mathematical modeling and simulation of the flow of fluid through porous media are important in many areas. Relative permeability and capillary pressure functions are macroscopic properties that are defined within the mathematic model. Accurate determinations of these functions are of great importance. An established inverse methodology provides the most accurate estimates of the unknown functions from the available data. When the inverse method is used to determine the flow functions, the media properties, absolute permeability and porosity are typically represented by single average values for the entire sample. Fortunately, an advanced core analysis tools utilizing nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI) to determine complete distributions of porosity and permeability has been developed. The process for determining multiphase properties from experimental data is implemented with the computer program SENDRA. This program is built around a two-dimension, two-phase simulator. In this thesis, the computer code is extended to represent all three spatial coordinate directions so that the porosity and permeability distributions in three-dimensional space can be taken into account. Taking the sample's heterogeneity into account is expected to obtain more accurate multiphase property. Three synthetic experiments are used to show the erroneous estimation of flow functions associated with the homogeneity assumption. A proposal approach is used to predict the relative permeability of wetting phase using NMR relaxation data. Several sets of three-dimensional NMR experiments are performed. Three-dimensional saturation distribution and relaxation are determined. Relative permeability of wetting phase are calculated by applying an empirical relation. This approach provides a in situ measurement of relative permeability of wetting phase from NMR data.

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.

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