A statistical model of karstic flow conduits, based on statistical physics of random walks, is developed. It allows us to compute the mean depth of flow conduits versus the distance from the inlet and versus the dip. It provides results that are in good qualitative agreement with previous results of other authors: the mean depth increases, slowly, with the distance, and it increases, not in a regular fashion, with the dip. The variability of the depth of the conduits, possibly leading to some conduits far from the water table, and the fact that well developed conduits are scarce or not, is linked to the probability of exploitation of the different fractures, the potentially permeable bedding planes, faults and joints in the karstifiable rock. On the basis of this result, we propose that interesting cavities - from the point of view of caving and cave diving - are found only in a small range of those exploitation probabilities. Finally, we emphasize the non-euclidean properties of flow conduits; especially, that many shortest pathways may exist and that a straight line is not usually the shortest pathway that actually develops between inlet and outlet.