Pressure and flow statistics of Darcy flow from simulated annealing

Abstract

The pressure and flow statistics of Darcy flow through a three-dimensional random permeable medium are expressed as a path integral in a form suitable for evaluation by simulated annealing. There are several advantages to using simulated annealing for this problem: (i) any probability distribution can be used for the permeability, (ii) there is no need to invert the transmissibility matrix which, while not a factor for single-phase flow, offers distinct advantages for multiphase flow, and (iii) the action used for simulated annealing, whose extremum yields Darcy’s law, is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. We show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with those obtained from standard finite-volume calculations.