Coarse graining equations for flow in porous media: a HaarWavelets and renormalization approach

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Title: Coarse graining equations for flow in porous media: a HaarWavelets and renormalization approach
Authors: Pancaldi, Vera
Item Type: Thesis or dissertation
Abstract: Coarse graining of equations for flow in porous media is an important aspect in modelling permeable subsurface geological systems. In the study of hydrocarbon reservoirs as well as in hydrology, there is a need for reducing the size of the numerical models to make them computationally efficient, while preserving all the relevant information which is given at different scales. In the first part, a new renormalization method for upscaling permeability in Darcy’s equation based on Haar wavelets is presented, which differs from other wavelet based methods. The pressure field is expressed as a set of averages and differences, using a one level Haar wavelet transform matrix. Applying this transform to the finite difference discretized form of Darcy’s law, one can deduce which permeability values on the coarse scale would give rise to the average pressure field. Numerical simulations were performed to test this technique on homogeneous and heterogeneous systems. A generalization of the above method was developed designing a hierarchical transform matrix inspired by a full Haar wavelet transform, which allows us to describe pressure as an average and a set of progressively smaller scale differences. Using this transform the pressure solution can be performed at the required level of detail, allowing for different resolutions to be kept in different parts of the system. A natural extension of the methods is the application to two-phase flow. Upscaling mobility allows the saturation profile to be calculated on the fine or coarse scale while based on coarse pressure values. To conclude, an alternative approach to upscaling in multi-phase flow is to upscale the saturation equation itself. Taking its Laplace transform, this equation can be reduced to a simple eigenvalue problem. The wavelet upscaling method can now be applied to calculate the upscaled saturation profile, starting with fine scale velocity data.
Issue Date: 12-Mar-2008
Date Awarded: Jan-2008
Format Info.: 3105953 bytes
Supervisor: King, Peter
Christensen, Kim
Author: Pancaldi, Vera
Department: Earth Science and Engineering
Publisher: Imperial College London
Qualification Level: Doctoral
Qualification Name: Doctor of Philosophy (PhD)
Appears in Collections:Earth Science and Engineering PhD theses

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