Percolation thresholds on elongated lattices

We investigate the percolation thresholds of both random and invasion percolation in two and three dimensions on elongated lattices; lattices with a geometry of Ld−1 × nL in d dimensions, where n denotes the aspect ratio of the lattice. Scaling laws for the threshold and spanning cluster density for random percolation are derived and simulation confirms the behaviour. A direct relationship between thresholds obtained for random percolation and invasion percolation is given and verified numerically. Important contributions to understanding two-phase flow observations in porous media and rock have been made using percolation theory [1–3]. Random percolation (RP) is relevant to two-phase displacement if the flow is very slow and the invading fluid is completely wetting. Invasion percolation (IP) is relevant when the invading fluid is completely nonwetting. Both variants of percolation have been used to explain the structure and the amounts of fluids in a two-phase displacement at breakthrough. The fractal structure of the invading fluid paths have been analysed and the properties of IP are believed to be consistent with RP. In spite of this the spanning clusters are not precisely the same, and no relationship between the cluster density at spanning in RP and the breakthrough threshold in IP is known. Estimates of IP thresholds are neglected, particularly in three dimensions, due to the large computational effort required when compared with RP. In most studies of percolation theory a simple square or cubic geometry is considered. In many applications one must consider systems with nonquadratic and noncubic geometries. For example, in the petroleum industry, laboratory measurements (e.g., residual saturations, capillary pressure) are performed on rock cores of high aspect ratio. These measurements are then used as input to reservoir simulation models. The crucial parameter measured is the value of the critical thresholds. Using scaling arguments and small-scale numerical simulations, Monetti and Albano [4] presented scaling laws for the percolation probability in an elongated geometry that depend on the aspect ratio n of the lattice. In this paper we derive new scaling laws for percolation properties of elongated lattices (ELs) in both two and three dimensions, and present simulation data to confirm the theoretical results. We also derive relationships between thresholds observed in RP and IP for ELs and verify the relationships numerically. Using finite-size scaling arguments, Monetti and Albano assumed the expectation value of the percolation threshold 〈pc(L)〉 ∝ L−1/νn−1/ν where ν is the critical exponent for the correlation length. We define a lattice in d dimensions of size L as a simple lattice (SL). 0305-4470/99/440461+06$30.00 © 1999 IOP Publishing Ltd L461 L462 Letter to the Editor Consider an EL consisting of n L SLs linked together in series. The probability Pn(p,L) that an EL of aspect ratio n percolates below p is given by the product of independent probabilities: Pn(p,L) = {P(p,L)}n×{C(p,L)}n−1, whereP(p,L) is the probability of having a spanning cluster on a SL at p, and C(p,L) is the connection probability that the spanning clusters of two SLs are connected at the (d− 1)-dimensional interface. We have measured the magnitude of the connection probability via extensive simulations and found that C(p,L) ' P(p,L) in both two and three dimensions [5]. We therefore approximate the probability by Pn(p,L) = [P(p,L)]2n−1. (1) Assuming the distribution of percolation thresholds on a SL can be accurately described by a distribution of the form ce(−x ) [6,7] where x = (pc−〈pc(L)〉)/b, a, b, and c being constants, we can write the probability of having a spanning cluster in a SL at pc < p as