The thesis investigates the flow of non-Newtonian fluids in porous media using network modeling. Non-Newtonian fluids occur in diverse natural and synthetic forms and have many important applications including in the oil industry. They show very complex time and strain dependent behavior and may have initial yield stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. They are generally classified into three main categories: time-independent in which strain rate solely depends on the instantaneous stress, time-dependent in which strain rate is a function of both magnitude and duration of the applied stress and viscoelastic which shows partial elastic recovery on removal of the deforming stress and usually demonstrates both time and strain dependency.The methodology followed in this investigation is pore-scale network modeling. Two three-dimensional topologically-disordered networks representing a sand pack and Berea sandstone were used. The networks are built from topologicallyequivalent three-dimensional voxel images of the pore space with the pore sizes, shapes and connectivity reflecting the real medium. Pores and throats are modeled as having triangular, square or circular cross-section by assigning a shape factor, which is the ratio of the area to the perimeter squared and is obtained from the pore space description. An iterative numerical technique is used to solve the pressure field and obtain the total volumetric flow rate and apparent viscosity. In some cases, analytical expressions for the volumetric flow rate in a single tube are derived and implemented in each throat to simulate the flow in the pore space. The time-independent category of the non-Newtonian fluids is investigated using two time-independent fluid models: Ellis and Herschel-Bulkley. Thorough comparison between the two random networks and the uniform bundle-of-tubes model i is presented. The analysis confirmed the reliability of the non-Newtonian network model used in this study. Good results are obtained, especially for the Ellis model, when comparing the network model results to experimental data sets found in the literature. The yield-stress phenomenon is also investigated and several numerical algorithms were developed and implemented to predict threshold yield pressure of the network. An extensive literature survey and investigation were carried out to understand the phenomenon of viscoelasticity and clearly identify its characteristic features, with special attention paid to flow in porous media. The extensional flow and viscosity and converging-diverging geometry were thoroughly examined as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista-Manero model, which successfully describes shear-thinning, elasticity and thixotropic timedependency, was identified as a promising candidate for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm that employs this model to simu...

We present a method to efficiently simulate coronary perfusion in subject-specific models of the heart within clinically relevant time frames. Perfusion is modelled as a Darcy porous-media flow, where the permeability tensor is derived from homogenization of an explicit anatomical representation of the vasculature. To account for the disparity in length scales present in the vascular network, in this study, this approach is further refined through the implementation of a multi-compartment medium where each compartment encapsulates the spatial scales in a certain range by using an effective permeability tensor. Neighbouring compartments then communicate through distributed sources and sinks, acting as volume fluxes. Although elegant from a modelling perspective, the full multi-compartment Darcy system is computationally expensive to solve. We therefore enhance computational efficiency of this model by reducing the N-compartment system of Darcy equations to N pressure equations, and N subsequent projection problems to recover the Darcy velocity. The resulting 'reduced' Darcy formulation leads to a dramatic reduction in algebraic-system size and is therefore computationally cheaper to solve than the full multi-compartment Darcy system. A comparison of the reduced and the full formulation in terms of solution time and memory usage clearly highlights the superior performance of the reduced formulation. Moreover, the implementation of flux and, specifically, impermeable boundary conditions on arbitrarily curved boundaries such as epicardium and endocardium is straightforward in contrast to the full Darcy formulation. Finally, to demonstrate the applicability of our methodology to a personalized model and its solvability in clinically relevant time frames, we simulate perfusion in a subject-specific model of the left ventricle.

The study of flow of non‐Newtonian fluids in porous media is very important and serves a wide variety of practical applications in processes such as enhanced oil recovery from underground reservoirs, filtration of polymer solutions and soil remediation through the removal of liquid pollutants. These fluids occur in diverse natural and synthetic forms and can be regarded as the rule rather than the exception. They show very complex strain and time dependent behavior and may have initial yield‐stress. Their common feature is that they do not obey the simple Newtonian relation of proportionality between stress and rate of deformation. Non‐Newtonian fluids are generally classified into three main categories: time‐independent whose strain rate solely depends on the instantaneous stress, time‐dependent whose strain rate is a function of both magnitude and duration of the applied stress and viscoelastic which shows partial elastic recovery on removal of the deforming stress and usually demonstrates both time and strain dependency. In this article, the key aspects of these fluids are reviewed with particular emphasis on single‐phase flow through porous media. The four main approaches for describing the flow in porous media are examined and assessed. These are: continuum models, bundle of tubes models, numerical methods and pore‐scale network modeling. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010