Rapid Diffusion Monte Carlo Algorithms for Fluid Dynamic Permeability

In our previous study, [8], we described two efficient methods of estimating the fluid permeability of a porous medium as a function of the medium's porosity. These methods use the statistics of Brownian particles diffusing near a sample of the medium. The Brownian trajectories are constructed by using the Green's function first-passage method. These trajectories are built up as a series of discrete jumps, each jump leaving the center of a first-passage domain.and landing at a point on its first-passage surface. Using transition probabilities that are sampled from the Laplacian Green's function for the geometry of the first-passage domain, each landing position is determined. This is an exact sampling method that is an acceleration of the walk on spheres method. The first of these two firstpassage methods estimates permeability in terms of the fluid-dynamic penetration depth, identifying the latter with a penetration property of Brownian paths. The second method computes the effective electrostatic capacitance of the sample and relates it, via angleaveraging theorems, to the translational hydrodynamic friction, and then uses a mean-field approximation to equate the latter quantity to the permeability of the porous medium. For the sampling of porous media, we exploited our "sharp-boundary" sampling method. We improve on our previous permeability estimates using a refinement of the sharp-boundary sampling algorithm. In our new algorithm, we decrease the number of jumps required to simulate a complete Brownian path. This is accomplished by starting the paths directly on the spherical sharp boundary of the porous medium sample. This is mathematically equivalent to the old method, but both much faster and more numerically precise. In fact, our new method is about three times faster, and the new method provides better permeability estimates at low porosities. The new method produces trajectories with many fewer jumps on average. Since the computational cost of these methods is roughly proportional to number of jumps, this explains the speed up. Moreover, our improvements in precision and execution time are most dramatic when simulating low porosity sample. With our previous method, the low porosity case was the most demanding. In several of the cases studied here, our permeability results are identical to those obtained from detailed deterministic solution of the Stokes equations, to within statistical error. Finally, the reduction in the average number of jumps reduces the effective dimensionality of the problem. This opens the possibility of further acceleration though the use of quasirandom numbers, [2j. "Department of Computer Science, Florida State University, 203 Love Building, Tallahassee, FL 32306-4530, USA, E-mail: chwangOcs.fsu.edu tDepartment of Computer Science, Florida State University, 203 Love Building, Tallahassee, FL 32306-4530, USA, E-mail: mascagniOcs.fsu.edu, URL: http://vtfv.cs.fsu.edu/~mascagni *Angie Inc., 7406 Alban Station Court, Suite A112, Springfield, VA 22150 USA, E-mail: giveno angle inc. com 214 Chi-Ok Hwang, Michael Mascagni and James A. Given