Predicting oil recovery using percolation

One particular practical problem in oil recovery is to predict the time to breakthrough of a fluid injected in one well and the subsequent decay in the production rate of oil at another well. Because we only have a stochastic view of the distribution of rock properties we need to predict the uncertainty in the breakthrough time and post-breakthrough behaviour in order to calculate the economic risk. In this paper we use percolation theory to predict (i) the distribution of the chemical path (shortest path) between two points (representing well pairs) at a given Euclidean separation and present a scaling hypothesis for this distribution which is confirmed by numerical simulation, (ii) the distribution of breakthrough times which can be calculated algebraically rather than by very time consuming direct numerical simulation of large numbers of realisations.