Pressure statistics from the path integral for Darcy flow through random porous media

Abstract

The path integral for classical statistical dynamics is used to determine the properties of one-dimensional Darcy flow through a porous medium with a correlated stochastic permeability for several spatial correlation lengths. Pressure statistics are obtained from the numerical evaluation of the path integral by using the Markov chain Monte Carlo method. Comparisons between these pressure distributions and those calculated from the classic finite-volume method for the corresponding stochastic differential equation show excellent agreement for Dirichlet and Neumann boundary conditions. The evaluation of the variance of the pressure based on a continuum description of the medium provides an estimate of the effects of discretization. Log-normal and Gaussian fits to the pressure distributions as a function of position within the porous medium are discussed in relation to the spatial extent of the correlations of the permeability fluctuations.

Path Integral Renormalization of Flow through Random Porous Media

“…Correlations have a pronounced effect of the variations of the conductivity [32], so in this section, we introduce a finite correlation length to the fluctuations to determine its effect on the RG transformations. We consider a system with a finite isotropic correlation length l by defining the correlation function as…”

“…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.…”

Pressure and flow statistics of Darcy flow from simulated annealing

“…The foregoing scenarios span several areas of science and engineering and have led to the implementation of a broad range of theoretical and computational methods [1]. Foremost among these are molecular dynamics [15], lattice Boltzmann methods [16][17][18], fluid dynamics (computational and theoretical) [16,19], path integral [20][21][22], finite-volume [23] and finiteelement [24] methods, and continuous-time random walk theory [25][26][27][28]. These techniques model flow through porous media at various levels of spatial and temporal resolution, ranging from the scale of individual pores to macroscopic pressure and flow rates.…”

“…In previous work [22,35], we formulated the solution to one-dimensional Darcy flow in a random porous medium as a path integral over pressure paths within the medium. In discrete form, the path integral is a tool to simulate Darcy pressure paths {p i } on a spatial lattice using Markov chain Monte Carlo methods according to the weighting e −S , where the 'action' S = S({p i }) restricts the paths to follow Darcy's law in the presence of quenched stochastic permeability.…”