Simulation of flow through a heterogeneous porous medium with fine-scale features can be computationally expensive if the flow is fully resolved. Coarsening the problem gives a faster approximation of the flow but loses some detail. We propose an algorithm that obtains the fully resolved approximation but only iterates on a sequence of coarsened problems. The sequence is chosen by optimizing the shapes of the coarse finite element basis functions. As a stand-alone method, the algorithm converges globally and monotonically with a quadratic asymptotic rate. Computational experience indicates the number of iterations needed is independent of the resolution and heterogeneity of the medium. However, an externally provided error estimate is required; the algorithm could be combined as an accelerator with another iterative algorithm. A single "inner" iteration of the other algorithm would yield an error estimate; following it with an "outer" iteration of our algorithm would give a viable method. / text

Researchers using Darcy's law to model flow in porous media must satisfy the requirement for sufficient scale separation between the pore scale and the model scale. This requirement is analogous to that for any continuum model, where application is restricted to scales larger than the underlying discrete structure. In the case of Darcy's law when the model scale becomes too small, the measurement of the permeability - the material property required to close the relationship - becomes polluted by the boundary conditions, either physical or numerical. The requirements for adequate scale separation to obtain permeability measurements (also known as satisfying the conditions for a representative elementary volume, or REV, for permeability) have not been previously reported. Likewise, the behavior of Darcy models when applied at sub-REV length scales has not been reported. Here, the results of Stokes simulations of transverse flow in 90,000 sequential random packings of monodisperse cylinders at a variety of liquid fractions and averaging-volume sizes show that approximately 200 cylinders must be present in an averaging volume before the effects of periodic boundary conditions on the Stokes simulations (the conventional choice for permeability measurements using Stokes flow) are no longer evident in the measured permeability. Direct comparisons between flow predictions from a two-dimensional, tensor-based Darcy model and a Stokes model for additional 10,000 domains show that the Darcy model is an unbiased predictor of the flow distribution in the system, even when the permeability is expected to contain boundary-condition artifacts. Though unbiased, the Darcy models do show considerable reduction in accuracy as the model scale shrinks toward the pore scale, with significant declines observed after the side length of a square averaging volume reaches 10 times the cylinder diameter. Finally, a novel approach for visualizing flows using the linear properties of the Stokes equations shows how the periodic boundary conditions affect the flow, and motivates the development of a generalized approach for obtaining permeability that does not require periodic boundary conditions. Modest improvements in the Darcy model relative to the actual Stokes flow result when the new approach is used to obtain permeability at small averaging volumes.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Todd J. Arbogast. Vita. Includes bibliographical references. Also available from UMI.

Thesis (M.S.)--Oregon State University, 2007. / Printout. Includes bibliographical references (leaves 78-79). Also available on the World Wide Web.

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Numerical models for the simulation of longitudinal and transverse Stokes flow in cylindrical periodic porous media are presented. The models, which are based on a finite-volume formulation in primitive variables, utilize digital image representations of the geometries to simulate, making them particularly well-suited for the rapid automated analysis of creeping flow in porous media with complex morphologies. Complete details of the model formulations are given, including extensive treatment of the pressure boundary conditions at the solid-liquid interface needed to guarantee convergence with all possible geometries. The convergence behavior of both models is tested, and the models are shown to be second-order accurate.The models are used to simulate flow over the whole range of volume fractions of liquid in several regular geometries. The longitudinal model is used to simulate flow in square arrays of circular and square ducts, and both models are used to simulate flow in square and hexagonal arrays of circular cylinders and square arrays of square cylinders rotated by varying amounts. For each of the geometries, accurate empirical expressions for the Darcy permeability as a function of volume fraction solid are presented. Where applicable, model predictions of permeability are compared to existing analytical results.Subsequently, the models are used to simulate Stokes flow in random domains over a wide range of fractions liquid. The sequential random packing algorithm is used to generate 1,000 random packings of circular cylinders at each of 14 fractions of liquid, and longitudinal and transverse flow simulations are performed for each geometry. Histograms and summary statistics are computed for the permeability for each fraction liquid, and empirical expressions for mean permeability as a function of fraction liquid are given. The autocorrelation structure of the geometry and of the fluid velocity is analyzed, and an analysis of the scaling of longitudinal permeability variance is presented. In transverse flow at high packing densities, it is found that lightning-like patterns emerge in the fluid velocity. It is also found that the details of flows in such geometries are strongly sensitive to the placement of individual solid obstacles.

As plantações florestais, historicamente, estão associada a intensos debates sobre o seu impacto na dinâmica da água do solo ou subterrânea. Nesse contexto, este trabalho buscou avaliar os possíveis impactos da mudança de cobertura do solo, de pastagem para eucalipto, sobre o aquífero freático em zona de afloramento do Sistema Aquífero Guarani. A área de pesquisa está localizada na bacia do Ribeirão da Onça, situada no município de Brotas, no centro-leste do Estado de São Paulo. Essa é monitorada desde 2004. Foram utilizados dois métodos para avaliar o impacto da mudança de uso do solo, o método iWTF e o método de Darcy. A partir desses métodos, estimou-se as taxas de recarga subterrânea, fluxo lateral, variação de armazenamento e percolação profunda. Além disso foram obtidas a variação do nível freático, espessura saturada e gradiente hidráulico horizontal para o aquífero na área de estudo. Durante os anos hidrológicos de 2005 a 2011, a recarga direta variou de 255 e 876 mm (iWTF), equivalente a 21% e 51% da precipitação anual correspondente. Após 2011, com a mudança de cobertura do solo de pastagem para eucalipto, não obstante os anos hidrológicos de 2012 e 2013 apresentarem precipitação próxima da média de longo período, a recarga direta foi estimada em cerca de 210 mm (2012) e 147 mm (2013), equivalentes a 13% e 10% da precipitação anual. A diminuição na recarga direta levou à redução da espessura saturada, variação do nível freático, gradiente hidráulico horizontal, fluxo lateral e armazenamento anual. Além disso, pode-se perceber que a plantação florestal está exercendo influência sobre a área de pastagem adjacente à plantação. Tem-se observado essa influência através da existência de fluxo transversal em direção à plantação florestal, o qual tem elevado o armazenamento no período de estiagem. Os resultados indicam que a plantação de eucalipto afetou significativamente o comportamento sazonal e anual do balanço hídrico na área de estudo. / Historically forest plantations are associated with intense debate about its impact on soil water dynamics or groundwater. Thus, the objective of this study was to evaluate the potential impacts of land cover change of pasture to eucalyptus on the groundwater in an outcrop zone of the Guarani Aquifer System. The study area is located in the Ribeirão da Onça watershed, located in the municipality of Brotas, in the center-east of the state of São Paulo. To assess the impact of land cover change, I used two methods: i. iWTF and ii. Darcy. From these methods, we estimated groundwater recharge rates, lateral flow, water storage variation and deep percolation. Furthermore, I obtained the water table variation, saturated thickness and horizontal hydraulic gradient into the aquifer in the study area. During the hydrological years 2005-2011, the direct recharge through the results of iWTF method varied between 256 and 876 mm, equivalent to 21% and 51% of the corresponding annual precipitation. After 2011, with the land cover change of pasture to eucalyptus, the direct recharge was computed at about 210 mm (2012) and 147 mm (2013), equivalent to 13% and 10% of the annual precipitation. I did not find significant difference in the annual precipitation in these years compared to the historical mean. Therefore, my findings have shown that the decrease of direct recharge are associated to the land cover change of pasture to eucalypts. The decrease in direct recharge leads to the reduction of saturated thickness, the water level variation, horizontal hydraulic gradient, lateral flow and annual storage. Moreover, it can be seen that the forest plantation has influenced the pasture area, adjacent to the plantation. It has been observed that influence through the existence of cross flow toward the forest plantation, which has high storage, especially in the dry season. The results indicate that the eucalyptus plantation significantly affected the seasonal and annual behavior of the water balance in the study area.

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