Path integral approach to Darcy flow

Marise Westbroek, Engineering And Physical Sciences Research Council, Imperial College London, Dimitri D. Vvedensky, Peter R. King
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
more » ... utational 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.
doi:10.25560/68436 fatcat:scynqhk7t5bznodkgc4dcbwrfy