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

Principles of optimality provide an interesting alternative to modeling hydrological processes in detail on small scales. However, the concepts still seem to be on a visionary level except for the theory of minimum energy expenditure for river networks. Inspired by this approach, we present a theory of minimum energy expenditure in 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 that they minimize the total energy expenditure at given recharge under the side condition 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 seem to be similar in both cases.