Path integral approach to Darcy flow
File(s)
Author(s)
Westbroek, Marise
Type
Thesis
Abstract
We explore a path integral approach to Darcy flow through a stochastic permeable medium. In one dimension, Darcy's law can be solved exactly. We give a derivation of the path integral used to obtain the Darcy pressure statistics. We also outline the computational setup for the conventional finite-volume method and the implementation of a stochastic field generator. We provide a detailed user's guide to the calculation of path integrals on a lattice, including an explicit computational setup and corresponding pseudocode.
The higher-dimensional form of Darcy's law lacks an analytic solution. We show that the simulated annealing algorithm provides a viable alternative to simulating a path integral for Darcy's law. We compare the results for the path integral and simulated annealing methods to those for the finite-volume method. All comparisons pass a Kolmogorov-Smirnov test at the 95% confidence level.
We discuss log-normal and Gaussian fits to the pressure statistics. Finally, we make a number of suggestions for future work, such as the use of the renormalization group and the extension of Darcy's law to multiphase flow.
The higher-dimensional form of Darcy's law lacks an analytic solution. We show that the simulated annealing algorithm provides a viable alternative to simulating a path integral for Darcy's law. We compare the results for the path integral and simulated annealing methods to those for the finite-volume method. All comparisons pass a Kolmogorov-Smirnov test at the 95% confidence level.
We discuss log-normal and Gaussian fits to the pressure statistics. Finally, we make a number of suggestions for future work, such as the use of the renormalization group and the extension of Darcy's law to multiphase flow.
Version
Open Access
Date Issued
2019-01
Date Awarded
2019-03
Copyright Statement
Creative Commons Attribution NonCommercial NoDerivatives Licence
Advisor
King, Peter R.
Vvedensky, Dimitri D.
Sponsor
Imperial College London
Engineering and Physical Sciences Research Council
Grant Number
EP/L015579/1
Publisher Department
Earth Science & Engineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)