A Fast Algorithm for Invasion Percolation

Abstract

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen-Kopelman algorithm. Computation time to percolation for a 3D system having N total sites and M invaded sites at percolation goes as O(M log M) for the proposed binary-tree algorithm and as O(M N) for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between M and N is M = N D/E , where D is the fractal dimension of an infinite cluster and E is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling M = N 0.852 in 3D (valid through the second decimal place) instead of M = N 0.843 based on the infinite cluster fractal dimension D = 2.53.

On the correlation between material structure and seismic attenuation anisotropy in porous media

“…Wilkinson and Willemson, 1983;Masson and Pride, 2014]. In all these cases, no significant anisotropy was observed across the seismic band of frequencies (1 to 10 4 Hz) despite anisotropy being present in the fluid patch shapes.…”

A fast two-step algorithm for invasion percolation with trapping

“…When using fast algorithms (e.g. Sheppard et al, 1999;Masson and Pride, 2014) for solving the NTIP problem in step 2, the CPU time to percolation for systems having N total sites and M NT invaded sites at percolation is at most (…”

“…This data structure permits us to find the site with the largest invasion potential in ( ) O 1 operation and to insert new neighbor sites in ( ( )) O n log operations, where ≤ n N is the number of active sites in the list (e.g. Schwarzer et al, 1999;Sheppard et al, 1999;Masson and Pride, 2014). I refer to Masson and Pride (2014) for a detailed implementation of NTIP using a perfectly balanced binary tree that strictly guarantees…”

“…Schwarzer et al, 1999;Sheppard et al, 1999;Masson and Pride, 2014). I refer to Masson and Pride (2014) for a detailed implementation of NTIP using a perfectly balanced binary tree that strictly guarantees…”

“…A search through the recent literature (Chen et al, 2012;Yang et al, 2013) shows that less efficient algorithms with execution time O (MN) are still widely used, further, I found no study that provides the reader with a detailed fast TIP algorithm that can easily be turned into code. The present paper together with Masson and Pride (2014) should allow the graduate student or people less familiar with IP to get started quickly with adequate algorithms. The solution I propose for TIP is based on the simple observation that trapping can be treated a posteriori through time-reversal of the invasion sequence obtained in NTIP.…”