Understanding and modeling of flow through porous media is an important issue in several branches of engineering. In petroleum engineering, for instance, one wishes to model the "enhanced oil recovery" process, whereby water or steam is injected into an oil saturated porous media in an attempt to displace and collect the oil.; Numerical simulation of these flows are difficult. The principal reason for this is the presence of many different length scales in the problem, and resolving all these is computationally expensive. To circumvent these difficulties a class of methods known as upscaling methods has been developed where one attempts to solve only for average features.; In this thesis, we review some of the previous efforts in upscaling and introduce a new scheme that attempts to overcome some of the existing shortcomings of these methods. In our analysis, we consider the problem in two distinct stages: first is the determination of the velocity field which gives rise to an elliptic partial differential equation (PDE) and second is a transport problem which gives a hyperbolic PDE.; For the elliptic part, we make use of existing upscaling methods for elliptic equations. In particular, we use the multi-scale finite element method of Hou to solve for the velocity field on a coarse grid.; Analysis of the hyperbolic part forms the main contribution of this thesis. We first analyze the problem by restricting ourselves to the case where the small scales are periodic. With this assumption, we derive a coupled set of equations for the average and the small scale fluctuations about this.; This is done by means of a special averaging, done along the fine scale streamlines. We derive an upscaling scheme by tracking only a sub-set of the fluctuations, which are used to approximate the scale interactions. Once this has been derived, we present a means to extend it to the case where the fluctuations are non-periodic.; In the sections that follow we provide the details of the numerical implementation. Finally, we present numerical results using the new scheme and compare this with both resolved computations and some existing upscaling schemes.
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