  Growth of wormholes in porous media can lead to self-organization of flow
networks with an overwhelming geometric complexity. Despite decades of study,
the mechanism by which a dominant wormhole develops its path during growth
remains elusive. Here we show that the trajectory of a growing wormhole can be
predicted by identifying the flowpath with a so-called minimum cumulative
surface. Our theoretical analysis indicates that the cumulative surface
determines the position of the dissolution front. We then show, using numerical
simulation based on greyscale nanotomography data, that the tip of an advancing
pore always follows the migration of the most far reaching dissolution front.
Finally, we show the good accord between theory and the observation with in
situ microtomography. Our result suggests that wormholing is a deterministic
process rather than, as classical theories imply, a stochastic one that can
only be analysed statistically. Our results shed new light on engineering of
artificial flow systems by exploiting self-organization in natural porous
materials.
