Upscaling for Two-phase Flows in Porous Media

The 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 the oil so that it can be collected. In groundwater contaminant studies the transport of dissolved material, such as toxic metals or radioactive waste, and how it affects drinking water supplies, is of interest. Numerical simulation of these flow are generally difficult. The principal reason for this is the presence of many different length scales in the physical 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 large scale features of interest and model the effect of the small scale 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 flow problem in two distinct stages: the first is the determination of the velocity field which gives rise to an elliptic partial differential equation (PDE) and the second is a transport problem which gives rise to 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 et al. to solve for the velocity field on a coarse grid, and yet still be able to obtain fine scale information through a special means of interpolation. The 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 have a periodic structure. With this assumption, we are able to derive a coupled set of equations for the large scale average and the small scale fluctuations about this average. This is done by means of a special averaging, which is done along the fine scale streamlines. This v coupled set of equations provides better starting point for both the modeling of the largescale small-scale interactions and the numerical implementation of any scheme. We derive an upscaling scheme from this by tracking only a sub-set of the fluctuations, which are used to approximate the scale interactions. Once this model has been derived, we discuss and present a means to extend it to the case where the fluctuations are more general than periodic. In the sections that follow we provide the details of the numerical implementation, which is a very significant part of any practical method. Finally, we present numerical results using the new scheme and compare this with both resolved computations and some existing upscaling schemes.