The aim of this thesis is to replace the intractable problem of using discrete flow models within large vascular networks with a suitably parameterised and tractable continuum perfusion model. Through this work, we directly address the hypothesis that discrete vascular data can be incorporated within continuum perfusion models via spatially-averaged parameterisation techniques. Chapter 1 reviews biological perfusion from both clinical and computational modelling perspectives, with a particular focus on myocardial perfusion. In Chapter 2, a synthetic 3D vascular network was constructed, which was controllable in terms of its size and properties. A multi-compartment static Darcy perfusion model of this discrete system was parameterised via a number of techniques. Permeabilities were derived using: (i) porosity-scaled isotropic (ϕI); (ii) Huyghe and Van Campen (HvC); and (iii) projected-PCA parameterisation methods. It was found that HvC permeabilities and pressure-coupling fields derived from the discrete data produced the best comparison to the spatially-averaged Poiseuille pressure. In Chapter 3, the construction and analysis of high-resolution anatomical arterial vascular models was undertaken. In Chapter 4, various anatomically-derived vascular networks were used to parameterise our perfusion model, including a microCT-derived rat capillary network, a single arterial subtree, and canine and porcine whole-organ arterial models. Allowing for general-connectivity (as opposed to strictly-hierarchical connectivity) yielded a significant improvement on the continuum model pressure. For the whole-organ model however, it was found that the best results were obtained by using porosity-scaled isotropic permeabilities and anatomically-derived pressure-coupling fields. It was also discovered that naturally occurring small length but relatively large radius vessels were not suitable for the HvC method. In Chapter 5, the suitability of derived parameters for use within a dynamic perfusion model was examined. It was found that the parameters derived from the original static network were adequate for application throughout the cardiac cycle. Chapter 6 presents a concluding discussion, highlighting limitations and future directions to be investigated.

This thesis presents a methodology based on Proppant Number approach for optimal fracture treatment design of natural gas wells considering non-Darcy flow effects in the design process. Closure stress is taken into account, by default, because it is the first factor decreasing propped pack permeability at in-situ conditions. Gel damage was also considered in order to evaluate the impact of incorporating more damaging factors on ultimate well performance and optimal geometry. Effective fracture permeability and optimal fracture geometry are calculated through an iterative process. This approach was implemented in a spreadsheet. Non-Darcy flow is described by the β factor. All β factor correlations available in the literature were evaluated. It is recommended to use the correlation developed specifically for the given type of proppant and mesh size, if available. Otherwise, the Pursell et al. or the Martins et al. equations are recommended as across the board reliable correlations for predicting non-Darcy flow effects in the propped pack. The proposed methodology was implemented in the design of 11 fracture treatments of 3 natural tight gas wells in South Texas. Results show that optimal fracture design might increase expected production in 9.64 MMscf with respect to design that assumes Darcy flow through the propped pack. The basic finding is that for a given amount of proppant shorter and wider fractures compensate the non-Darcy and/or gel damage effect. Dynamic programming technique was implemented in design of multistage fractures for one of the wells under study for maximizing total gas production. Results show it is a powerful and simple technique for this application. It is recommended to expand its use in multistage fracture designs.

Hydraulic fracturing has recently been the completion of choice for most tight gas bearing formations. It has proven successful to produce these formations in a commercial manner. However, some considerations have to be taken into account to design an optimum stimulation treatment that leads to the maximum possible productivity. These considerations include, but not limited to, non-Darcy flow and multiphase flow effects inside the fracture. These effects reduce the fracture conductivity significantly. Failing to account for that results in overestimating the deliverability of the well and, consequently, to designing a fracture treatment that is not optimum. In this work a thorough investigation of non-Darcy flow and multi-phase flow effects on the productivity of hydraulically fractured wells is conducted and an optimum fracture design is proposed for a tight gas formation in south Texas using the Unified Fracture Design (UFD) Technique to compensate for the mentioned effects by calculating the effective fracture permeability in an iterative way. Incorporating non-Darcy effects results in an optimum fracture that is shorter and wider than the fracture when only Darcy calculations are considered. That leads to a loss of production of 5, 18 percent due to dry and multiphase non-Darcy flow effects respectively. A comparison between the UFD and 3D simulators is also done to point out the differences in terms of methodology and results. Since UFD incorporated the maximum dimensionless productivity index in the fracture dimensions design, unlike 3D simulators, it can be concluded that using UFD to design the fracture treatment and then use the most important fracture parameters outputs (half length and CfDopt) as inputs in the simulators is a recommended approach.

The objective of this dissertation is to investigate the non-Darcy flow effects on field base data by considering gas viscosity, gas deviation factor and gas density as variables. To achieve it, different correlations from the literature and field data have been combined to Sawyer-Brown Method, thus a contribution has been achieved. Production history of selected gas field has been implemented to a numerical simulator. To find out non-Darcy effects quantitatively, Darcy flow conditions have also been run in the simulator for each scenario in addition to non-Darcy flow correlation runs. Extracted data from simulation runs have been analyzed on the basis of Sawyer-Brown Method by introducing several correlations to consider gas viscosity, gas deviation factor and gas density as variables. Engineering and scientific research on non-Darcy flow is still being conducted in order for better understanding the nonlinear flow behavior of fluids through porous media. The deviations from Darcy&rsquo / s Law are attributed to the occurrence of all or alternating combinations of factors that can be categorized as the anisotropy of porosity and permeability, multi-phase flow of fluids in varying phases, magnitude of pressure drop and the subsequent phase change in fluids, and the change in flow regime at elevated rates of flow in porous media. Throughout this dissertation, the factors causing deviations from Darcy flow behavior have been investigated.

In this study, an efficient coupling between Monte Carlo (MC) molecular simulation and Darcy-scale flow in porous media is presented. The cell-centered finite difference method with a non-uniform rectangular mesh were used to discretize the simulation domain and solve the governing equations. To speed up the MC simulations, we implemented a recently developed scheme that quickly generates MC Markov chains out of pre-computed ones, based on the reweighting and reconstruction algorithm. This method astonishingly reduces the required computational time by MC simulations from hours to seconds. In addition, the reweighting and reconstruction scheme, which was originally designed to work with the LJ potential model, is extended to work with a potential model that accounts for the molecular quadrupole moment of fluids with non-spherical molecules such as CO2. The potential model was used to simulate the thermodynamic equilibrium properties for single-phase and two-phase systems using the canonical ensemble and the Gibbs ensemble, respectively. Comparing the simulation results with the experimental data showed that the implemented model has an excellent fit outperforming the standard LJ model. To demonstrate the strength of the proposed coupling in terms of computational time efficiency and numerical accuracy in fluid properties, various numerical experiments covering different compressible single-phase flow scenarios were conducted. The novelty in the introduced scheme is in allowing an efficient coupling of the molecular scale and Darcy scale in reservoir simulators. This leads to an accurate description of the thermodynamic behavior of the simulated reservoir fluids; consequently enhancing the confidence in the flow predictions in porous media.

Optimizing proppant pack conductivity and proppant-transport and -deposition patterns in a hydraulic fracture is of critical importance to sustain effective and economical production of petroleum hydrocarbons. In this research, a numerical modeling approach, combining the discrete element method (DEM) with the lattice Boltzmann (LB) simulation, was developed to provide fundamental insights into the factors regulating the interactions between reservoir depletion, proppant-particle compaction and movement, single-/multiphase flows and non-Darcy flows in a hydraulic fracture, and fracture conductivity evolution from a partial-monolayer proppant concentration to a multilayer proppant concentration. The potential effects of mixed proppants of different sizes and types on the fracture conductivity were also investigated. The simulation results demonstrate that a proppant pack with a smaller diameter coefficient of variation (COV), defined as the ratio of standard deviation of diameter to mean diameter, provides better support to the fracture; the relative permeability of oil was more sensitive to changes in geometry and stress; when effective stress increased continuously, oil relative permeability increased nonmonotonically; the combination of high diameter COV and high effective stress leads to a larger pressure drop and consequently a stronger non-Darcy flow effect. The study of proppant mixtures shows that mixing of similar proppant sizes (mesh-size-20/40) has less influence on the overall fracture conductivity than mixing a very fine mesh size (mesh-size-100); selection of proppant type is more important than proppant size selection when a proppant mixture is used. Increasing larger-size proppant composition in the proppant mixture helps maintain fracture conductivity when the mixture contains lower-strength proppants. These findings have important implications to the optimization of proppant placement, completion design, and well production. In the hydraulic-mechanical rock-proppant system, a fundamental understanding of multiphase flow in the formation rock is critical in achieving sustainable long-term productivity within a reservoir. Specifically, the interactions between the critical dimensionless numbers associated with multiphase flow, including contact angle, viscosity ratio, and capillary number (Ca), were investigated using X-ray micro computed tomography (micro-CT) scanning and LB modeling. The primary novel finding of this study is that the viscosity ratio affects the rate of change of the relative permeability curves for both phases when the contact angle changes continuously. Simulation results also indicate that the change in non-wetting fluid relative permeability was larger when the flow direction was switched from vertical to horizontal, which indicated that there was stronger anisotropy in larger pore networks that were primarily occupied by the non-wetting fluid. This study advances the fundamental understanding of the multiphysics processes associated with multiphase flow in geologic materials and provides insight into upscaling methodologies that account for the influence of pore-scale processes in core- and larger-scale modeling frameworks. During reservoir depletion processes, reservoir formation damage is an issue that will affect the reservoir productivity and various phases in fluid recovery. Invasion of formation fine particles into the proppant pack can affect the proppant pack permeability, leading to potential conductivity loss. The combined DEM-LB numerical framework was used to evaluate the role of proppant particle size heterogeneity (variation in proppant particle diameter) and effective stress on the migration of detached fine particles in a proppant supported fracture. Simulation results demonstrate that a critical fine particle size exists: when a particle diameter is larger or smaller than this size, the deposition rate increases; the transport of smaller fines is dominated by Brownian motion, whereas the migration of larger fines is dominated by interception and gravitational settling; this study also indicates that proppant packs with a more heterogeneous particle-diameter distribution provide better fines control. The findings of this study shed lights on the relationship between changing pore geometries, fluid flow, and fine particle migration through a propped hydraulic fracture during the reservoir depletion process. / Doctor of Philosophy / Hydraulic fracturing stimulation design is required for unconventional hydrocarbon energy (e.g., shale oil and gas) extraction due to the low permeability and complex petrophysical properties of unconventional reservoirs. During hydrocarbon production, fractures close after pumping due to the reduced fluid pressure and increased effective stress in rock formations. In the oil and gas industry, proppant particles, which are granular materials, typically sand, treated sand, or man-made ceramic materials, are pumped along with fracturing fluids to prevent hydraulic fractures from closing during hydrocarbon extraction. In order to relate the geomechanical (effective stress), geometric (pore structure and connectivity), and transport (absolute permeability, relative permeability, and conductivity) properties of a proppant assembly sandwiched in a rock fracture, a geomechanics-fluid mechanics framework using both experiment and simulation methods, was developed to study the interaction and coupling between them. The outcome of this research will advance the fundamental understanding of the coupled, multiphysics processes with respect to hydraulic fracturing and benefit the optimization of proppant placement, completion design, and well production.

The dynamic solution of multiphase flow through porous media is of special interest to several fields of science and engineering, such as petroleum, geology and geophysics, bio-medical, civil and environmental, chemical engineering and many other disciplines. A natural application is the modeling of the flow of two immiscible fluids (phases) in a reservoir. Others, that are broadly based and considered in this work include the hydrodynamic dispersion (as in reactive transport) of a solute or tracer chemical through a fluid phase. Reservoir properties like permeability and porosity greatly influence the flow of these phases. Often, these vary across several orders of magnitude and can be discontinuous functions. Furthermore, they are generally not known to a desired level of accuracy or detail and special inverse problems need to be solved in order to obtain their estimates. Based on the physics dominating a given sub-region of the porous medium, numerical solutions to such flow problems may require different discretization schemes or different governing equations in adjacent regions. The need to couple solutions to such schemes gives rise to challenging domain decomposition problems. Finally, on an application level, present day environment concerns have resulted in a widespread increase in CO₂capture and storage experiments across the globe. This presents a huge modeling challenge for the future. This research work is divided into sections that aim to study various inter-connected problems that are of significance in sub-surface porous media applications. The first section studies an application of mortar (as well as nonmortar, i.e., enhanced velocity) mixed finite element methods (MMFEM and EV-MFEM) to problems in porous media flow. The mortar spaces are first used to develop a multiscale approach for parabolic problems in porous media applications. The implementation of the mortar mixed method is presented for two-phase immiscible flow and some a priori error estimates are then derived for the case of slightly compressible single-phase Darcy flow. Following this, the problem of modeling flow coupled to reactive transport is studied. Applications of such problems include modeling bio-remediation of oil spills and other subsurface hazardous wastes, angiogenesis in the transition of tumors from a dormant to a malignant state, contaminant transport in groundwater flow and acid injection around well bores to increase the permeability of the surrounding rock. Several numerical results are presented that demonstrate the efficiency of the method when compared to traditional approaches. The section following this examines (non-mortar) enhanced velocity finite element methods for solving multiphase flow coupled to species transport on non-matching multiblock grids. The results from this section indicate that this is the recommended method of choice for such problems. Next, a mortar finite element method is formulated and implemented that extends the scope of the classical mortar mixed finite element method developed by Arbogast et al [12] for elliptic problems and Girault et al [62] for coupling different numerical discretization schemes. Some significant areas of application include the coupling of pore-scale network models with the classical continuum models for steady single-phase Darcy flow as well as the coupling of different numerical methods such as discontinuous Galerkin and mixed finite element methods in different sub-domains for the case of single phase flow [21, 109]. These hold promise for applications where a high level of detail and accuracy is desired in one part of the domain (often associated with very small length scales as in pore-scale network models) and a much lower level of detail at other parts of the domain (at much larger length scales). Examples include modeling of the flow around well bores or through faulted reservoirs. The next section presents a parallel stochastic approximation method [68, 76] applied to inverse modeling and gives several promising results that address the problem of uncertainty associated with the parameters governing multiphase flow partial differential equations. For example, medium properties such as absolute permeability and porosity greatly influence the flow behavior, but are rarely known to even a reasonable level of accuracy and are very often upscaled to large areas or volumes based on seismic measurements at discrete points. The results in this section show that by using a few measurements of the primary unknowns in multiphase flow such as fluid pressures and concentrations as well as well-log data, one can define an objective function of the medium properties to be determined, which is then minimized to determine the properties using (as in this case) a stochastic analog of Newton’s method. The last section is devoted to a significant and current application area. It presents a parallel and efficient iteratively coupled implicit pressure, explicit concentration formulation (IMPEC) [52–54] for non-isothermal compositional flow problems. The goal is to perform predictive modeling simulations for CO₂sequestration experiments. While the sections presented in this work cover a broad range of topics they are actually tied to each other and serve to achieve the unifying, ultimate goal of developing a complete and robust reservoir simulator. The major results of this work, particularly in the application of MMFEM and EV-MFEM to multiphysics couplings of multiphase flow and transport as well as in the modeling of EOS non-isothermal compositional flow applied to CO₂sequestration, suggest that multiblock/multimodel methods applied in a robust parallel computational framework is invaluable when attempting to solve problems as described in Chapter 7. As an example, one may consider a closed loop control system for managing oil production or CO₂sequestration experiments in huge formations (the “instrumented oil field”). Most of the computationally costly activity occurs around a few wells. Thus one has to be able to seamlessly connect the above components while running many forward simulations on parallel clusters in a multiblock and multimodel setting where most domains employ an isothermal single-phase flow model except a few around well bores that employ, say, a non-isothermal compositional model. Simultaneously, cheap and efficient stochastic methods as in Chapter 8, may be used to generate history matches of well and/or sensor-measured solution data, to arrive at better estimates of the medium properties on the fly. This is obviously beyond the scope of the current work but represents the over-arching goal of this research. / text

Physical lab experiments have shown that the pressure caused by an impinging jet on a granular bed has the potential to form craters. This poses a danger to landing success and nearby spacecraft for future rocket missions. Current numerical simulations for this process do not accurately reproduce experimental results. Our goal is to produce improved simulations to more accurately and effi- ciently model the changes in pressure as gas flows through a porous medium. A two-dimensional model in space known as the nonlinear Porous Medium Equation as it is derived from Darcy’s law is used. An Alternating-Direction Implicit (ADI) temporal scheme is presented and implemented which reduces our multidimensional problem into a series of one-dimensional problems. We take advantage of explicit approximations for the nonlinear terms using extrapolation formulas derived from Taylor-series, which increases efficiency when compared to other common methods. We couple our ADI temporal scheme with different spatial discretizations including a second-order Finite Difference (FD) method, a fourth-order Orthogonal Spline Collocation (OSC) method, and an Nth-order Chebyshev Spectral method. Accuracy and runtime are compared among the three methods for comparison in a linear analogue of our problem. We see the best results for accuracy when using an ADI-Spectral method in the linear case, but discuss possibilities for increased effi- ciency using an ADI-OSC scheme. Nonlinear results are presented using the ADI-Spectral method and the ADI-FD method.

The importance of coronary microcirculatory perfusion is highlighted by the severe impact of microvascular diseases such as diabetes and hypertension on heart function. Recently, highly-detailed three-dimensional (3D) data on ex vivo coronary microvascular structure have become available. However, hemodynamic information in individual myocardial capillaries cannot yet be obtained using current in vivo imaging techniques. In this thesis, a novel data-driven modelling framework is developed to predict tissue-scale flow properties from discrete anatomical data, which can in future be used to aid interpretation of coarse-scale perfusion imaging data in healthy and diseased states. Mathematical models are parametrised by the 3D anatomical data set of Lee (2009) from the rat myocardium, and tested using flow measurements in two-dimensional rat mesentery networks. Firstly, algorithmic and statistical tools are developed to separate branching arterioles and venules from mesh-like capillaries, and then to extract geometrical properties of the 3D capillary network. The multi-scale asymptotic homogenisation approach of Shipley and Chapman (2010) is adapted to derive a continuum model of coronary capillary fluid transport incorporating a non-Newtonian viscosity term. Tissue-scale flow is captured by Darcy's Law whose coefficient, the permeability tensor, transmits the volume-averaged capillary-scale flow variations to the tissue-scale equation. This anisotropic permeability tensor is explicitly calculated by solving the capillary-scale fluid mechanics problem on synthetic, stochastically-generated periodic networks parametrised by the geometrical data statistics, and a thorough sensitivity analysis is conducted. Permeability variations across the myocardium are computed by parametrising synthetic networks with transmurally-dependent data statistics, enabling the hypothesis that subendocardial permeability is much higher in diastole to compensate for severely-reduced systolic blood flow to be tested. The continuum Darcy flow model is parametrised by purely structural information to provide tissue-scale perfusion metrics, with the hypothesis that this model is less sensitive and more reliably parametrised than an alternative, estimated discrete network flow solution.

In this thesis a multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed, the Multiscale Robin Coupled Method (MRCM). The procedure is based on a new domain decomposition method in which the local problems are subject to Robin boundary conditions. The method allows for the independent definition of interface spaces for pressure and flux over the skeleton of the decomposition that can be chosen with great flexibility to accommodate local features of the underlying permeability fields. Numerical simulations are presented aiming at illustrating several features of the new method. We illustrate the possibility to recover the multiscale solution of two wellknown methods of the literature, namely, the Multiscale Mortar Mixed Finite Element Method (MMMFEM) and the Multiscale Hybrid-Mixed (MHM) Finite Element Method by suitable choices of the parameter b in the Robin interface conditions. Results show that the accuracy of the MRCM depends on the choice of this algorithmic parameter as well as on the choice of the interface spaces. An extensive numerical assessment of the MRCM is conduct with two types of interface spaces, the usual piecewise polynomial spaces and the informed spaces, the latter obtained from sets of snapshots by dimensionality reduction. Different distributions of the unknowns between pressure and flux are explored. The results show that b, suitably nondimensionalized, can be fixed to unity to avoid any indeterminacy in the method. Further, with both types of spaces, it is observed that a balanced distribution of the interface unknowns between pressure and flux renders the MRCM quite attractive both in accuracy and in computational cost, competitive with other multiscale methods from the literature. The MRCM solutions are in general only global conservative. Two postprocessing procedures are proposed to recover local conservation of the multiscale velocity fields. We investigate the applicability of such methods in highly heterogeneous permeability fields in modeling the contaminant transport in the subsurface. These methods are compared to a standard procedure. Results indicate that the proposed methods have the potential to produce more accurate results than the standard method with similar or reduced computational cost. / Nesta tese é proposto um método misto multiescala visando a aproximação precisa de campos de velocidade e pressão em meios porosos altamente heterogêneos, o método Multiscale Robin Coupled Method (MRCM). Este procedimento é baseado em um novo método de decomposição de domínio no qual os problemas locais são definidos com condições de contorno de Robin. O método permite a definição independente de espaços de interface para pressão e fluxo sobre o esqueleto da decomposição que pode ser escolhida com grande flexibilidade para acomodar características locais dos campos de permeabilidade subjacentes. Simulações numéricas são apresentadas visando ilustrar várias características do novo método. Ilustramos a possibilidade de recuperar a solução multiescala de dois métodos bem conhecidos da literatura, a saber, o Multiscale Mortar Mixed Finite Element Method (MMMFEM) e o Multiscale Hybrid-Mixed (MHM) Finite Element Method por escolhas adequadas do parâmetro b nas condições da interface de Robin. Os resultados mostram que a precisão do MRCM depende da escolha deste parâmetro algorítmico, bem como da escolha dos espaços de interface. Uma extensa avaliação numérica do MRCM é conduzida com dois tipos de espaços de interface, os usuais espaços polinomiais por partes e os espaços informados, este último obtidos a partir da redução de dimensionalidade de conjutos de espaços de snapshots. Diferentes distribuições de incógnitas entre pressão e fluxo são exploradas. Os resultados mostram que b, adequadamente adimensionalizado, pode ser fixado em unidade para evitar qualquer indeterminação no método. Além disso, com ambos os tipos de espaços, observa-se que uma distribuição equilibrada de incógnita entre pressão e fluxo nas interfaces torna o MRCM bastante atraente tanto em precisão quanto em custo computacional, competitivo com outros métodos multiescala da literatura. As soluções MRCM são, em geral, apenas globalmente conservativas. Dois procedimentos de pós-processamento são propostos para recuperar a conservação local dos campos de velocidade multiescala. Investigamos a aplicabilidade de tais métodos em campos de permeabilidade altamente heterogêneos na modelagem do transporte de contaminantes na subsuperfície. Esses métodos são comparados a um procedimento padrão da literatura. Os resultados indicam que os métodos propostos têm o potencial de produzir resultados mais precisos do que o método padrão com custo computacional similar ou reduzido.