Invasion percolation in a destabilizing gradient

Extensive two- and three-dimensional computer simulations have been carried out to investigate the effect of a destabilizing external field (such as gravity) on invasion percolation processes. The displacement patterns found under these conditions are dominated by the growth of a single branch. This branch can be described in terms of a connected string of blobs of size ${\xi }_{\mathit{w}}$, which form a directed random walk along the direction of the field. On length scales smaller than ${\xi }_{\mathit{w}}$, the displacement figures have the structure of invasion percolation clusters without a destabilizing field. The dependence of the correlation length ${\xi }_{\mathit{w}}$ on the magnitude of the field gradient g is given by ${\xi }_{\mathit{w}}$∼‖g${\mathrm{\Vert }}^{\mathrm{-}\nu /(\nu +1)}$ (where ν is the ordinary percolation correlation length exponent) in accord with the theoretical arguments of Wilkinson [Phys. Rev. A 30, 520 (1984); 34, 1380 (1986)]. For continuous threshold distributions the exponent relating ${\xi }_{\mathit{w}}$ to g does not depend on the shape of the distribution. For discontinuous distributions the behavior is quite different.