Pressure and flow statistics of Darcy flow from simulated annealing

Abstract

The pressure and flow statistics of Darcy flow through a random permeable medium are expressed in a form suitable for evaluation by the method of simulated annealing. There are several attractive aspects to using simulated annealing: (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 the case of multiphase flow, and (iii) the action used for simulated annealing is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. In this paper, we show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with the more conventional finite-volume calculations.

Path Integral Renormalization of Flow through Random Porous Media

“…We have at our disposal calculations in 1, 2, and 3 dimensions [32,33], so we can investigate the dimensional dependence of the accuracy of our calculations. As noted in the Introduction, RG equations in different spatial dimension have the same diagrammatic structure, but the methods used for the numerical valuation of the path integral are different in d = 1 [32] and d = 2 and d = 3 [33]. This should provide an additional point of interest for our comparisons.…”

“…The path integral for Darcy's law can be evaluated numerically [32,33] to obtain the pressure statistics of the flow for stochastic conductivities with various correlation lengths and boundary conditions. These results provide benchmarks for the results reported here, but the analytic character of our RG analysis offers several advantages over the numerical approach: (i) different spatial dimensions are handled by the same basic procedure, whereas numerical evaluation necessitates different methodologies, (ii) changes to the initial distribution and correlation length of the conductivity along the RG trajectory determine the appropriate choices of these quantities at various length scales, and (iii) the extension to more complex settings, e.g., multiphase flow, can, with suitable modifications, be carried out within the same framework.…”