Crossover length from invasion percolation to diffusion-limited aggregation in porous media

We model fluid-fluid displacement in d=2 by a diffusion-limited-aggregation (DLA) algorithm which takes random capillary forces into account. Interpore surface tension is neglected. The invading fluid is nonviscous. We find a crossover length ${\mathit{L}}_{\mathit{c}}$. On length scales much smaller (larger) than ${\mathit{L}}_{\mathit{c}}$, invasion percolation (DLA) patterns are obtained. We argue by scaling, and check by simulations, that ${\mathit{L}}_{\mathit{c}}$∼(Δp̃/Ca${)}_{\mathit{s}}^{2/(2+\mathit{D}}$), Δp̃ stands for a measure of spatial variations of the capillary pressure, Ca is the capillary number, and ${\mathit{D}}_{\mathit{s}}$ is the interface fractal dimension on small length scales (we find ${\mathit{D}}_{\mathit{s}}$=1.3).