Evaluation of the path integral for flow through random porous media

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

Figure 1

Two realizations of the variation of the permeability $K_i$ that are used in Darcy's law (4) and in the path integral (7). For $\xi =0.01X$ (solid blue line), the effect of correlations is small, while for $\xi =0.5X$ (broken red line) correlations throughout the system are evident.

Figure 2

Pressure statistics obtained from the path integral (7) (blue filled circles) and the FVM (red crosses) with (a, b) Dirichlet and (c, d) Neumann boundary conditions for (a, c) $\xi =0.01X$ and (b, d) $\xi =0.05X$. Fewer data points are shown than used for statistical analysis. Red curves are lognormal distributions (in the standard and reverse orientations) and blue curves are normal distributions, each with the same mean and variance as the pressures at the indicated positions. Pressures are obtained from ${10}^4$ simulations on a system of 240 lattice sites with $\mathrm{\Delta }x=0.5$ m by sampling the permeability distribution using (7) and (19). The sparse matrix solver umfpack [36] was used for the FVM simulations. To within statistical uncertainty (the error bars are smaller than the symbol sizes), the path integral and FVM simulations agree to a confidence level of 95%. The pressure gauge invariance of Darcy's equation means that only pressure differences are meaningful; negative pressures are the result of a particular choice of a zero of pressure.