Transferring the concept of minimum energy dissipation from river networks to subsurface flow patterns

Principles of optimality provide an interesting alternative to modeling hydrological processes in detail on small scales and have received growing interest in the last years. Inspired by the more than 20 years old concept of minimum energy dissipation in river networks, we present a corresponding theory for subsurface flow in order to obtain a better understanding of preferential flow patterns in the subsurface. The concept describes flow patterns which are optimal in the sense of minimizing the total energy dissipation at a given recharge under the constraint of a given total porosity. Results are illustrated using two examples: two-dimensional flow towards a spring with a radial symmetric distribution of the porosity and dendritic flow patterns. The latter are found to be similar to river networks in their structure and, as a main result, the model predicts a power-law distribution of the spring discharges. In combination with two data sets from the Austrian Alps, this result is used for validating the model. Both data sets reveal power-law-distributed spring discharges with similar scaling exponents. These are, however, slightly larger than the exponent predicted by the model. As a further result, the distributions of the residence times strongly differ between homogeneous porous media and optimized flow patterns, while the mean residence times are similar in both cases.