Two dimensional simulations of percolation are realized on square networks of pore throats with a random capillary pressure distribution. We analyse the influence of a destabilizing gravity field (g) and of the standard deviation of the distribution of the capillary pressure thresholds (W t). The fragmentation process is not taken into account in this study. For an increase of g or/and when W t decreases, two transitions are analyzed with three different regimes displacement patterns: Invasion percolation, invasion percolation in a gradient, and invasion in a pure gradient. The transitions are controlled both by the ratio g/W t and by the sample size (L). A scaling law between the saturation at the percolation threshold and g/W t allows delineating the three regimes in agreement with theoretical argument of the percolation in a gradient. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 200647.55.Mh Flows through porous media, 61.43.Hv Fractals; macroscopic aggregates (including diffusion-limited aggregates) , 47.55.Kf Particle-laden flows,
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