On the rapid estimation of permeability for porous media using Brownian motion paths

Abstract

We describe two efficient methods of estimating the fluid permeability of standard models of porous media by using the statistics of continuous Brownian motion paths that initiate outside a sample and terminate on contacting the porous sample. The first method associates the “penetration depth” with a specific property of the Brownian paths, then uses the standard relation between penetration depth and permeability to calculate the latter. The second method uses Brownian paths to calculate an effective capacitance for the sample, then relates the capacitance, via angle-averaging theorems, to the translational hydrodynamic friction of the sample. Finally, a result of Felderhof is used to relate the latter quantity to the permeability of the sample. We find that the penetration depth method is highly accurate in predicting permeability of porous material.

Analysis and comparison of Green’s function first-passage algorithms with “Walk on Spheres” algorithms

“…It has been empirically observed that GFFP algorithms are more efficient than WOS algorithms when high accuracy is required [2][3][4]. Additionally, it has been observed that there is always an optimal distance from the surface of the absorbing boundary, δ I , for a GFFP algorithm within which a FP surface can be permitted to intersect the boundary [2][3][4]. In this paper, we will provide a rigorous complexity analysis consistent with these observations.…”

“…However in practice and for efficiency, another layer, called the δ I -layer, is introduced such that we use WOS outside the δ I -layer and GFFP inside the δ I -layer. 2 Note that a δ h -layer can be used when WOS is used outside a δ I -layer. This entails a finite probability of terminating Brownian trajectories when WOS is used outside the δ I -layer (see Fig.…”

“…Two properties that have been observed in numerical calculations are that GFFP algorithms are more efficient than WOS algorithms when enough accuracy is required, and for a GFFP algorithm there is always an optimal choice for δ I , the distance from the surface of the absorbing boundary within which a FP surface can intersect the boundary [1][2][3].…”

“…Using the known Green's functions for the electrostatic problem of two conducting intersecting spheres, and the method of inversion for other geometries, Given, Hubbard, and Douglas [1] tabulated the necessary FP probability density functions to exactly deal with absorbing boundaries that are either flat or spherical. This enhancement to WOS was called the Green's function first-passage (GFFP) algorithm [1][2][3][4]. Using GFFP leaves only the statistical sampling error in this Monte Carlo diffusion method.…”

“…The running times for these new first-passage (FP) algorithms [1][2][3][4], which use exact Green's functions for the Laplacian to eliminate the absorption layer in the "walk on spheres" (WOS) method [5][6][7][8][9], are compared with those for WOS algorithms. It has been empirically observed that GFFP algorithms are more efficient than WOS algorithms when high accuracy is required [2][3][4]. Additionally, it has been observed that there is always an optimal distance from the surface of the absorbing boundary, δ I , for a GFFP algorithm within which a FP surface can be permitted to intersect the boundary [2][3][4].…”