Distribution of shortest paths at percolation threshold: application to oil recovery with multiple wells

In this work we investigate the distribution of shortest paths in percolation systems at the percolation threshhold in two dimensions (2D). We study paths from one given point to multiple other points. In oil recovery terminology, the given single point can be mapped to an injection well (injector) and the multiple other points to production wells (producers). In the previously studied standard case of one injection well and one production well separated by Euclidean distance r, the distribution of shortest paths l, P(l|r), shows a power-law behavior with exponent gl=2.14 in 2D. Here we analyze the situation of one injector and an array A of producers. Symmetric arrays of producers lead to one peak in the distribution P(l|A), the probability that the shortest path between the injector and any of the producers is l, while the asymmetric configurations lead to several peaks in the distribution. We analyze configurations in which the injector is outside and inside the set of producers. The peak in P(l|A) for the symmetric arrays decays faster than for the standard case. For very long paths all the studied arrays exhibit a power-law behavior with exponent g≃gl.