Site-bond invasion percolation with fluid trapping

We study fluid invasion in porous media using a novel site-bond invasion percolation model. The defending fluid is incompressible, and therefore its trapping by the invading fluid is possible. We carry out flow calculations in order to simulate fluid invasion. The flow patterns indicate that the sample-spanning cluster of the invading fluid contains no closed loops. Moreover, there is only one path of the invading fluid from the inlet to the outlet of the pore space, and therefore the backbone of the cluster is just a long strand. The fractal dimensions Df and Db of the sample-spanning cluster and its backbone are also estimated. We obtain Df≃1.8, in agreement with that of site invasion percolation with trapping, and Db≃1.14, completely different from Db≃1.64 for two-dimensional percolation. Also considered is the scaling law for the conductivity and permeability of the model near the percolation threshold. We propose that the critical exponent t of the conductivity or permeability of this model in d dimensions is given exactly by t=(d−2)ν+1, where ν is the correlation length exponent. This equation, which is the first exact relation between t and ν for a percolation model, had been previously suggested by P.G. de Gennes for random percolation, but had turned out to be only a lower bound to t for that problem.