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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.

Relative permeabilities for three-phase flow are commonly predicted from two-phase flow measurements using empirical models. These models are usually tested against available steady state data. However, the oil flow is unsteady state during various production stages such as gas injection after water flood. Accurate measurement of oil permeability([subscript ro]) during unsteady tertiary gas flood is necessary to study macroscopic oil displacement rate under micro scale events including double drainage, coalescence and reconnection, bulk flow and film drainage. We measure the three-phase oil relative permeability by conducting unsteady-state drainage experiments in a 0.8m water-wet sandpack. We find that when starting from capillary-trapped oil, k[subscript ro] starts high and decreases with a small change in oil saturation, and shows a strong dependence on both the flow of water and the water saturation, contrary to most models. The observed flow coupling between water and oil is stronger in three-phase flow than two-phase flow, and cannot be observed in steady-state measurements. The results suggest that the oil is transported through moving gas/oil/water interfaces (form drag) or momentum transport across stationary interfaces (friction drag). We present a simple model of friction drag which compares favorably to the experimental data. We also solve the creeping flow approximation of the Navier-Stokes equation for stable wetting and intermediate layers in the corner of angular capillaries by using a continuity boundary condition at the layer interface. We find significant coupling between the condensed phases and calculate the generalized mobilities by solving co-current and counter-current flow of wetting and intermediate layers. Finally, we present a simple heuristic model for the generalized mobilities as a function of the geometry and viscosity ratio. To identify the key parameter controlling the measured excess oil flow during tertiary gasflood, we also conduct simultaneous water-gas flood tests where we control water relative permeability and let water saturation develop naturally. The measured data and pore scale calculations indicate that viscous coupling can not explain completely the observed flow coupling between oil and water. We conclude that the rate of water saturation decrease, which controls the pore scale mechanisms including double drainage, reconnection, and film drainage significantly influences the rate of oil drainage during tertiary gas flood. Finally, we present a simple heuristic model for oil relative permeability during tertiary gas flood, and also explain how Stone I and saturation-weighted interpolation should be used to predict the permeability of mobilized oil during transient tertiary gasflood. / text

When CO₂ is injected in deep saline aquifers on the scale of gigatonnes, pressure buildup in the aquifer during injection will be a critical issue. Because fracturing, fault activation and leakage of brine along pathways such as abandoned wells all require a threshold pressure (Nicot et al., 2009); operators and regulators will be concerned with the spatial extent of the pressure buildup. Thus a critical contour of overpressure is a convenient proxy for risk. The location of this contour varies depending on the target aquifer properties (porosity, permeability etc.), the geology (presence of faults, abandoned wells etc.), and boundary conditions. Importantly, the extent also depends on relative permeability (Burton et al., 2008). First we describe ways of quantifying the risk due to pressure buildup in an aquifer with a constant pressure boundary, using the three-region injection model to derive analytical expressions for a specific contour of overpressure at any given time. All else being the same, the two-phase-region mobilities (and hence relative permeability characteristics) provide a basis for the ranking of storage formations based on risk associated with pressure elevation during injection. The pressure buildup during CO₂ injection will depend strongly upon the boundary conditions at the boundary of the storage formation. An analytical model for pressure profile in the infinite-acting aquifer is developed by combining existing water influx models in traditional reservoir engineering (Van-Everdingen and Hurst model, Carter-Tracy model) to the current problem for describing brine efflux from the storage aquifer when CO₂ injection creates a "three-region" saturation distribution. We determine evolution of overpressure with time for constant pressure, no-flow and infinite-acting boundary conditions, and conclude that constant pressure and no-flow boundary conditions give the most optimistic and pessimistic estimates of risk respectively. Compositional reservoir simulation results, using CMG-GEM simulator are presented, to show the effect of an isolated no-flow boundary on pressure buildup and injectivity in saline aquifers. We investigate the effect of multiple injection wells on single-phase fluid flow on aquifer pressure buildup, and demonstrate the use of an equivalent injection well concept to approximate the aquifer pressure profile. We show a relatively inexpensive method of predicting the presence of unanticipated heterogeneities in the formation, by employing routine measurements such as injection rate and injection pressure to track deviation in the plume path. This idea is implemented by combining Pro-HMS (probabilistic history matching software, that carries out geologically consistent parameter estimation), and a CMG-GEM model which has been tuned to the physics of the CO₂-brine system. / text

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Carbon capture and sequestration (CCS) is an engineering-based approach for mitigating excess anthropogenic CO2 emissions. Deep brine aquifers and basalt reservoirs have shown outstanding performance in CO2 storage based on their global widespread distribution and large storage capacity. Capillary trapping and mineral trapping are the two dominant mechanisms controlling the distribution, migration, and transportation of CO2 in deep brine aquifers and basalt reservoirs. Understanding the behavior of CO2 in a storage reservoir under realistic conditions is important for risk management and storage efficiency improvement. As a result, numerical simulations have been implemented to understand the relationship between fluid properties and multi-phase fluid dynamics. However, the physics-based simulations that focus on the uncertainties of fluid flow dynamics are complicated and computationally expensive. Machine learning method provides immense potential for improving computational efficiency for subsurface simulations, particularly in the context of parametric sensitivity. This work focuses on parametric uncertainty associated with multi-phase fluid dynamics that govern geologic CO2 storage. The effects of this uncertainty are interrogated through ensemble simulation methods that implement both physics-based and machine learning modeling strategies. This dissertation is a culmination of three projects: (1) a parametric analysis of capillary pressure variability effects on CO2 migration, (2) a reactive transport simulation in a basalt fracture system investigating the effects of carbon mineralization on CO2 migration, and (3) a parametric analysis based on machine learning methods of simultaneous effects of capillary pressure and relative permeability on CO2 migration. / Doctor of Philosophy / Carbon capture and sequestration (CCS) has been proposed as a technological approach to mitigate the deleterious effects of anthropogenic CO2 emissions. During CCS, CO2 is captured from power plants and then pumped in deep geologic reservoirs to isolate it from the atmosphere. Deep sedimentary formations and fractured basalt reservoirs are two options for CO2 storage. In sedimentary systems, CO2 is immobilized largely by physical processes, such as capillary and solubility trapping, while in basalt reservoirs, CO2 is transformed into carbonate minerals, thus rendering it fully immobilized. This research focuses on how a large range of capillary pressure variabilities and how CO2-basalt reactions affect CO2 migration. Specifically, the work presented utilizes numerical simulation and machine learning methods to study the relationship between capillary trapping and buoyancy in a sandstone formation, as well as the combined effects of capillary pressure and relative permeability on CO2 migration. In addition, the work also identifies a new reinforcing feedback between mineralization and relative permeability during reactive CO2 flow in a basalt fracture network. In aggregate, the whole of this work presents a new, multi-dimensional perspective on the multi-phase fluid dynamics that govern CCS efficacy in a range of geologic formations.

The control of excessive water production in oil and gas producing wells is of increasing importance to the field operator, primarily when trying to maintain the survivability of a mature field from shut in. During the last two decades many chemicals have been studied and applied under the name of relative permeability modifier (RPM) to combat this problem. These chemicals were mostly bullheaded individually into the affected zones, consequently their application resulted in low to medium success, particularly in treating reservoirs suffering from matrix flow. It has been found that the disproportionate permeability reduction depends on the amount of polymer dispersed or absorbed by the porous rock. If single polymers are employed to treat excessive water production in a matrix reservoir they cannot penetrate deep into the formation rock because the polymer will start to build as a layer on the surface of the rock grains. As a result the placement of polymer into the formation will no be piston like and the dispersion over the rock pores will be uneven. To improve water shutoff technology a method of injecting chemicals sequentially is recommended provided that the chemical’s viscosity is increasing successively with the chemicals injected. / Experimentally confirmed, injecting chemicals sequentially provides better results for conformance control. The value of post treatment water mobility is conspicuously lowered by the method of applying injecting chemicals sequentially in comparison with the single chemical injection method. For instance, the residual resistance factor to water (Frrw) at the first cycle of brine flushing for this method is approximately five times higher than the Frrw obtained by injecting only one single chemical. Furthermore, for the second cycle of brine flushing Frrw is still higher by a ratio of about 2.5. In addition to this improvement residual resistance factor to oil Frro for this method is less than two which has been considered as the upper limit for conformance control in matrix reservoir. Accordingly injecting chemical sequentially can be applied for enhancing relative permeability modifier performance in matrix reservoir.

This work provides a semi-analytical development of the pressure-mobility behavior of solution gas-drive reservoir systems producing below the bubble point pressure. Our primary result is the "characteristic" relation which relates normalized (or dimensionless) pressure and mobility functions. This formulation is proven with an exhaustive numerical simulation study consisting of over 900 different cases. We considered 9 different pressure-volume-temperature (PVT) sets, and 13 different relative permeability cases in the simulation study. We also utilized 7 different depletion scenarios. The secondary purpose of this work was to develop a correlation of the "characteristic parameter" as a function of rock and fluid properties evaluated at initial reservoir conditions such as: API density, GOR, formation volume factor, viscosity, reservoir pressure, reservoir temperature, oil saturation, relative permeability end points, corey exponents and oil mobility: We did successfully correlate the characteristic parameter as a function of these variables, which proves that we can uniquely represent the pressure-mobility path during depletion with specific reservoir and fluid property variables, taken as constant values for a particular case. The functional form of our correlation along with all relevant equations are shown on the body of this document.

In this dissertation we discuss some upscaling methods for flow and transport in heterogeneous reservoirs. We studied realization-based multi-phase flow and transport upscaling and ensemble-level flow upscaling. Multi-phase upscaling is more accurate than single-phase upscaling and is often required for high level of coarsening. In multi-phase upscaling, the upscaled transport parameters are time-dependent functions and are challenging to compute. Due to the hyperbolic feature of the saturation equation, the nonlocal effects evolve in both space and time. Standard local two-phase upscaling gives significantly biased results with reference to fine-scale solutions. In this work, we proposed two types of multi-phase upscaling methods, TOF (time-offlight)- based two-phase upscaling and local-global two-phase upscaling. These two methods incorporate global flow information into local two-phase upscaling calculations. A linear function of time and time-of-flight and a global coarse-scale two-phase solution (time-dependent) are used respectively in these two approaches. The local boundary condition therefore captures the global flow effects both spatially and temporally. These two methods are applied to permeability distributions with various correlation lengths. Numerical results show that they consistently improve existing two-phase upscaling methods and provide accurate coarse-scale solutions for both flow and transport. We also studied ensemble level flow upscaling. Ensemble level upscaling is up scaling for multiple geological realizations and often required for uncertainty quantification. Solving the flow problem for all the realizations is time-consuming. In recent years, some stochastic procedures are combined with upscaling methods to efficiently compute the upscaled coefficients for a large set of realization. We proposed a fast perturbation approach in the ensemble level upscaling. By Karhunen-Lo`eve expansion (KLE), we proposed a correction scheme to fast compute the upscaled permeability for each realization. Then the sparse grid collocation and adaptive clustering are coupled with the correction scheme. When we solve the local problem, the solution can be represented by a product of Green's function and source term. Using collocation and clusering technique, one can avoid the computation of Green's function for all the realizations. We compute Green's function at the interpolation nodes, then for any realization, the Green's function can be obtained by interpolation. The above techniques allow us to compute the upscaled permeability rapidly for all realizations in stochastic space.