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The increasing complexity of oil and gas reservoirs has led to the need of a better understanding of the processes governing the rock properties. Traditional theoretical and empirical models often fail to predict the behavior of carbonates, tight gas sands and shale gas, for example. An essential part of the necessary investigation is the study of the phenomena occurring at the pore scale. In this direction, the so-called digital rock physics is emerging as a research field that offers the possibility of imaging the rock pore space and simulating the processes therein directly. This report describes our work on developing algorithms to simulate viscous and electric flow through a three dimensional Cartesian representation of the porous space, such as those available through X-ray microtomography. We use finite differences to discretize the governing equations and also propose a new method to enforce the incompressible flow constraint under natural boundary conditions. Parallel computational codes are written targeting performance and computer memory optimization, allowing the use of bigger and more representative samples. Results are reported with an estimate of the error bars in order to help on the simulation appraisal. Tests performed using benchmark samples show good agreement with experimental/theoretical values. Example of application on digital modeling of cement growth and on multiphase fluid distribution are also provided. The final test is done on Bentheimer, Buff Berea and Idaho Brown sandstone samples with available laboratory measurements. Some limitations need to be investigated in future work. First, the computer potential fields show anomalous border effects at the open boundaries. Second, a minor problem arises with the decreased convergence rate for the velocity field due to the increased number of operations, leading to the need of a more sophisticated preconditioner. We intend to expand the algorithms to handle microporosity (e.g. carbonates) and multiphase fluid flow. / text

Pore-scale simulation has become increasingly important in recent years as a tool to understand multiphase flow behavior. Wettability affects aspects of flow such as capillary-pressure saturation curves, residual saturation of each phase, and relative permeability. Simulation of wettability at the pore-scale is still a non-trivial problem, and many different approaches exist to model it. In this work, we implement a variational level set formulation to impose different contact angles at the solid-fluid-fluid contact line for two-phase flow in simple rhomboidal pore geometries, and calculate the maximum mean curvature (equivalently capillary pressure) for each case. We compare our results with a detailed set of analytical and experimental results in a range of pore geometries of varying wettability from Mason and Morrow (1994), and demonstrate the accuracy of this method. While the simulations shown are for relatively simple geometries, the method has the ability to handle arbitrarily complex geometry (such as input from X-ray microtomography imaging). / text

Accurate predictions of macroscopic multiphase flow properties, such as relative permeability and capillary pressure, are necessary for making key decisions in reservoir engineering. These properties are usually measured experimentally, but pore-scale network modeling has become an efficient alternative for understanding fundamental flow behavior and prediction of macroscopic properties. In many cases network modeling gives excellent agreement with experiment by using models physically representative of real media. Void space within a rock sample can be extracted from high resolution images and converted to a topologically equivalent network of pores and throats. Multiphase fluid transport is then modeled by imposing mass conservation at each pore and implementing the Young-Laplace equation in pore throats; the resulting pressure field and phase distributions are used to extract macroscopic properties. Advancements continue to be made in making network modeling predictive, but one limitation is that artificial (e.g. constant pressure gradient) boundary conditions are usually assumed; they do not reflect the local saturations and pressure distributions that are affected by flow and transport in the surrounding media. In this work we demonstrate that flow behavior at the pore scale, and therefore macroscopic properties, is directly affected by the boundary conditions. Pore-scale drainage is modeled here by direct coupling to other pore-scale models so that the boundary conditions reflect flow behavior in the surrounding media. Saturation couples are used as the mathematical tool to ensure continuity of saturations between adjacent models. Network simulations obtained using the accurate, coupled boundary conditions are compared to traditional approach and the resulting macroscopic petrophysical properties are shown to be largely dependent upon the specified boundary conditions. The predictive ability of network simulations is improved using the novel network coupling scheme. Our results give important insight into upscaling as well as approaches for including pore-scale models directly into reservoir simulators. / text

Mixing of miscible gas with oil in a reservoir decreases the effective strength of the gas, which can adversely affect miscibility and recovery efficiency. The mixing that occurs in a reservoir, however, is widely debated and often ignored in reservoir simulation, where very large grid blocks are used. Large grid blocks create artificially large mixing that can cause errors in predicted oil recovery. Reservoir mixing, or dispersion, is caused by diffusion of particles across streamlines of varying velocities. Mixing is enhanced by any mechanism that increases the area of contact between the gas and the oil, thereby allowing the effects of diffusion to be magnified. This is, in essence, the cause of scale-dependent dispersion. The contact area grows primarily because of variations in streamlines and their velocities around grains and through layers of various permeabilities (heterogeneity). Mixing can also be enhanced by crossflow, such as that caused by gravity and by the effects of other neighboring wells. This dissertation focuses on estimation of the level of effective local mixing at the field scale and its impact on oil recovery from miscible gas floods. Pore-level simulation was performed using the Navier-Stokes and convection-diffusion equations to examine the origin of scale dependent dispersion. We then estimated dispersivity at the macro scale as a function of key scaling groups in heterogeneous reservoirs. Lastly, we upscaled grid blocks to match the level of mixing at the pattern scale. Once the contact area ceases to grow with distance traveled, dispersion has reached its asymptotic limit. This generally occurs when the fluids are well mixed in transverse direction. We investigated a variety of pore-scale models to understand the nature of scale dependency. From the pore-scale study, we found that reservoir mixing or dispersion is caused by diffusion of particles across streamlines. Diffusion can be significantly enhanced if the surface area of contact between the reservoir and injected fluid are increased as fluids propagate through the reservoir. Echo and transmission dispersivities are scale dependent. They may or may not reach an asymptotic limit depending on the scale of heterogeneities encountered. The scale dependence results from an increase in the contact area between solute (gas) and resident fluid (oil) as heterogeneities are encountered, either at the pore or pattern-scale. The key scaling groups for first-contact miscible (FCM) flow are derived and their impact on mixing is analyzed. We examine only local mixing, not apparent mixing caused by variations in streamline path lengths (convective spreading). Local mixing is important because it affects the strength of the injected fluid, and can cause an otherwise multicontact miscible (MCM) flood to become immiscible. We then showed how to upscale miscible floods considering reservoir mixing. The sum of numerical dispersion and physical dispersion associated with the reservoir heterogeneities, geometry and fluid properties must be equal at both the fine- and large-scales. The maximum grid-block size allowed in both the x- and z-directions is determined from the scaling groups. Small grid-blocks must be used for reservoirs with uncorrelated permeabilities, while larger grid blocks can be used for more layered reservoirs. The predicted level of mixing for first-contact miscible floods can be extended with good accuracy to multicontact miscible (MCM) gas floods. / text

NAPL (non-aqueous phase liquid) blob mobilization and solubilization models were developed to predict residual NAPL fate and describe flow dynamics of various displacing phases (water and surfactant foam). The models were achieved by pore-scale mass and force balances and were focused on the understanding of the physico-chemical interactions between NAPL blobs and the displacing phases. The pore-level mass balance indicated changes in NAPL saturation instead of mass reduction occurring with blob solubilization. The force balance was used to explain the complex flow configurations among NAPL blobs and the displacing phases. Some factors such as the wettability and the spreading/entering coefficients were useful in determining flow configurations. From the models developed in this study, dimensional analysis was performed to identify NAPL blob motion during water or surfactant foam flooding. In non-dimensionalized forms, a Trapping number employed as an indicator of blob displacement performance was modified to quantify the onset of blob mobilization. Its value for water flooding was nearly 2-3 orders of magnitude greater than that of surfactant foam flooding. Next, to investigate the blob flow regime in porous media, a blob velocity was computed. Regardless of the displacing phases, a blob’s velocity increased with increasing blob sizes after commencement of blob motion, and the velocity of DNAPL (dense non-aqueous phase liquid) blobs was greater than that of LNAPL (light non-aqueous phase liquid) blobs. From this investigation, it is expected that the pore-scale solubilization and mobilization models would provide better understanding leading to a predictive capability for the flow behavior of NAPL blobs removed by various displacing phases in a porous medium. Additionally, the models based on newly approached concepts and modified governing equations would be useful in conceptualization, as well as the model prediction of other immiscible or miscible fluids flowing through a porous medium. Further, the models developed in our study would be a useful contribution to the study of small-scale contaminants or substances such as particle and bacterial transport in porous media.

The possible benefits of thermal remediation of NAPL-contaminated subsurface were analyzed at pore-scale. Force balance analysis was performed to provide the insight and information on the critical conditions for the blob mobilization. First, the critical blob radius for blob mobilization was calculated in terms of blob radius, temperature, and water velocity. Temperature increase enhanced the blob mobilization along with the decrease of interfacial tension. Water velocity increase also enhanced the blob mobilization. Critical water velocity provided the critical condition for the initiation of blob mobilization to distinguish singlet and doublet in blob size. Second, the terminal (or steady state) blob velocity at the steady state blob motion was determined. Increases of temperature and water velocity raised the terminal blob velocity. When the observation of blob mobilization moved from REV scale (macroscale) to pore-scale, terminal blob velocity showed the different phenomena according to the change of oil saturation. At macro-scale, the terminal blob velocity was smaller than water velocity by an order or two. However, the terminal blob velocity reached to water velocity at pore-scale. This investigation would provide the better understanding on the pore-scale analysis of residual NAPL blob mobilization by thermal remediation. Additionally, the pore-scale analysis developed in this study would be incorporated into a general conservation equation in terms of the accumulation of multiple blobs. It would derive continuumaveraged equations that accurately represent pore-level physics. In conclusion, the study on the critical conditions for the initiation of blob mobilization as a single discrete blob would have some contribution to the transport and fate of NAPL contaminant and the desired subsurface remediation.

NAPL (non-aqueous phase liquid) blob mobilization and solubilization models were developed to predict residual NAPL fate and describe flow dynamics of various displacing phases (water and surfactant foam). The models were achieved by pore-scale mass and force balances and were focused on the understanding of the physico-chemical interactions between NAPL blobs and the displacing phases. The pore-level mass balance indicated changes in NAPL saturation instead of mass reduction occurring with blob solubilization. The force balance was used to explain the complex flow configurations among NAPL blobs and the displacing phases. Some factors such as the wettability and the spreading/entering coefficients were useful in determining flow configurations. From the models developed in this study, dimensional analysis was performed to identify NAPL blob motion during water or surfactant foam flooding. In non-dimensionalized forms, a Trapping number employed as an indicator of blob displacement performance was modified to quantify the onset of blob mobilization. Its value for water flooding was nearly 2-3 orders of magnitude greater than that of surfactant foam flooding. Next, to investigate the blob flow regime in porous media, a blob velocity was computed. Regardless of the displacing phases, a blob’s velocity increased with increasing blob sizes after commencement of blob motion, and the velocity of DNAPL (dense non-aqueous phase liquid) blobs was greater than that of LNAPL (light non-aqueous phase liquid) blobs. From this investigation, it is expected that the pore-scale solubilization and mobilization models would provide better understanding leading to a predictive capability for the flow behavior of NAPL blobs removed by various displacing phases in a porous medium. Additionally, the models based on newly approached concepts and modified governing equations would be useful in conceptualization, as well as the model prediction of other immiscible or miscible fluids flowing through a porous medium. Further, the models developed in our study would be a useful contribution to the study of small-scale contaminants or substances such as particle and bacterial transport in porous media.

Macroscopic transport properties of porous media depend on textural rock parameters such as porosity, grain size and grain shape distributions, surface-to-volume ratios, and spatial distributions of cement. Although porosity is routinely measured in the laboratory, direct measurements of other textural rock properties can be tedious, time-consuming, or impossible to obtain without special methods such as X-ray microtomography and scanning electron microscopy. However, by using digital three-dimensional pore-scale rock models and physics-based algorithms researchers can calculate both geometrical and transport properties of porous media. Therefore, pore-scale modeling techniques provide a unique opportunity to explore explicit relationships between pore-scale geometry and fluid and electric flow properties. The primary objective of this dissertation is to investigate at the pore-scale level the effects of grain shapes and spatial cement distribution on macroscopic rock properties for improved understanding of various petrophysical correlations. Deposition and compaction of grains having arbitrary angular shapes and various sizes is modeled using novel sedimentation and cementation pore-scale algorithms. Additionally, the algorithms implement numerical quartz precipitation to describe preferential cement growth in pore-throats, pore-bodies, or uniform layers. Subsequently, petrophysical properties such as geometrical pore-size distribution, primary drainage capillary pressure, absolute permeability, streamline-based throat size distribution, and apparent electrical formation factor are calculated for several digital rock models to evaluate petrophysical correlations. Furthermore, two geometrical approximation methods are introduced to model irreducible (connate) water saturation at the pore scale. Consolidated grain packs having comparable porosities and grain size distributions but various grain shapes indicate that realistic angular grain shape distribution gives the best agreement of petrophysical properties with experimental measurements. Cement volume and its spatial distribution significantly affect pore-space geometry and connectivity, and subsequently, macroscopic petrophysical properties of the porous media. For example, low-porosity rocks having similar grain structure but different cement spatial distribution could differ in absolute permeability by two orders of magnitude and in capillary trapped water saturation by a factor of three. For clastic rocks with porosity much higher than percolation threshold porosity, pore-scale modeling results confirm that surface-to-volume ratio and porosity provide sufficient rock-structure character to describe absolute permeability correlations. In comparison to surface-to-volume ratio, capillary trapped (irreducible) water saturation exhibits better correlation with absolute permeability due to weak pore space connectivity in low-porosity samples near the percolation threshold. Furthermore, in grain packs with fine laminations and permeability anisotropy, pore-scale analysis reveals anisotropy in directional drainage capillary- pressure curves and corresponding amounts of capillary-trapped wetting fluid. Finally, results presented in this dissertation indicate that pore-scale modeling methods can competently capture the effects of porous media geometry on macroscopic rock properties. Pore-scale two- and three-phase transport calculations with fast computers can predict petrophysical properties and provide sensitivity analysis of petrophysical properties for accurate reservoir characterization and subsequent field development planning. / text

Polymers used in enhanced oil recovery (EOR) help to control the mobility ratio between oil and aqueous phases and as a result, polymer flooding improves sweep efficiency in reservoirs. However, the conventional wisdom is that polymer flooding does not have considerable effect on pore-level displacement because pressure forces would not be enough to overcome trapping caused by capillary forces. Recently, both coreflood experiments and field data suggest that injecting viscoelastic polymers, such as hydrolyzed polyacrylamide (HPAM), can result in lower residual oil saturation. The hypothesis is that the polymer elasticity provides several pore-level mechanisms for oil mobilization that are generally not significant for purely-viscous fluids. Both experiments and modeling need to be performed to investigate the effect of polymer elasticity on residual oil saturation. Pore-scale modeling and micro-fluidic experiments can be used to investigate pore-level physics, and then used to upscale to the macro-scale. The objective of this work is to understand the effect of polymer elasticity on apparent viscosity and residual oil saturation in porous media. Single- and multi-phase pore-level computational fluid dynamics (CFD) modeling for viscoelastic polymer flow is performed to investigate the dominant mechanisms at the pore level to mobilize trapped oil. Several interesting results are found from the CFD results. First, the elasticity of the polymer results in an increase in normal stress at the pore-level; therefore, the normal stresses exerted on a static oil droplet are significant and not negligible as for a purely-viscous fluid. The CFD results show that viscoelastic fluid exerts additional forces on the oil-phase which may help mobilize trapped oil out of the porous medium. Second, due to the elasticity of polymer, the viscoelastic polymer has some level of pulling effect; while passing above a dead-end pore it can pull out the trapped oil phase and then mobilize it. However, both CFD modeling and micro-fluidic experiments show the pulling-effect is not likely the main mechanism to reduce oil saturation at pore-level. Third, dynamic CFD simulations show less deformation of the oil phase while viscoelastic polymer is displacing fluid compared to purely viscous fluid. It may justify the hypothesis that polymer elasticity resists against snap-off mechanism. As a result, when viscoelastic polymer displaces the oil ganglia, the oil phase does not snap off, and the oil phase remains connected, and therefore easier to move in porous media compared to disconnected oil. For single phase flow, a closed-form flow equation has been developed based on CFD modeling in converging/diverging ducts representative of pore throats. The pore-level equations were substituted into a pore-network model and validated against experimental data. Good agreement is observed. This study reveals important findings about the effect of polymer elasticity to reduce the residual oil saturation; however, more experiments and simulations are recommended to fully-understand the mobilization mechanisms and take advantage of them to optimize the polymer-flooding process in the field. / text