Results from numerical simulations on two-dimensional networks used as an analog to pore space in porous rocks are presented to emphasize the existence of preferential paths for transport processes in heterogeneous media. The simulations show that hydraulic flow and electrical current are mostly driven in the so-called ''critical paths'' when the pore size distribution has a decreasing exponential-like shape in opposition to nearly homogeneous or uniform like distributions. The geometry of these flow paths is controlled by spatially correlated large conductances and can be described by a tortuosity factor, &tgr;2. The amount of flow carried by the critical paths is higher than that predicted by statistical or effective medium models which do not take into account the role of spatial correlation. Another interesting result is that the geometry of the critical paths may change in response to alterations in the properties of the pore space, like, for example, the reduction in connectivity due to pressure-induced pore closure. The simulations also show that the critical paths for hydraulic flow are different from those for electrical current. A statistical analysis shows that the hydraulic tortuosity is higher than the electrical tortuosity by a factor of 1.5 on the average which implies that one must be cautious in using the electrical tortuosity by a factor of 1.5 on the average which implies that one must be cautious in using the electrical tortuosity in the equivalent channel model for determining the permeability of rocks. As the numerical simulations have been done on distributions of pore or crack geometries typically encountered in rocks, the existence of critical paths highlighted in this paper is likely to play an important role in the phenomenology of transport processes in the Earth's crust. ¿ American Geophysical Union 1993