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

Abstract

Abstract. 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.

Minimum dissipation of potential energy by groundwater outflow results in a simple linear catchment reservoir

“…as e.g., proposed by Hergarten et al (2014) and Savenije (2017). This interpretation would imply that the linear reservoir behaviour is probably less likely to be observed in young catchments, in arid regions as groundwater reservoirs may not be connected, and in mountainous catchments where steep slopes result in a shorter timescale for groundwater outflow.…”

“…It remains a riddle why complex and heterogeneous systems often demonstrate surprisingly simple behavior (Savenije, 2001). In an attempt to explain this apparently simple behavior, Fenicia et al (2006) hypothesized that the process of self-organization forming river basin landscapes and drainage networks (Rodriguez-Iturbe and Rinaldo, 1997) is mirrored underground and that the groundwater drainage system co-evolved with the geological formation 20 and erosion processes, leading to efficient dissipation of the available head, that is, the associated free potential energy (see also Hergarten et al, 2014;Savenije, 2017).…”

Energy states of soil water – a thermodynamic perspective on storage dynamics and the underlying controls

“…Other studies investigated the role of connected flow networks such as river networks 56 or rill systems and suggested that they increase the power in coupled water and sediment 57 fluxes (Howard, 1990; Favis-Mortlock et al, 2000; Paik and Kumar, 2010; Kleidon et al, 58 2013). This is because these networks minimize local dissipative losses for instance in the 59 river network (Rinaldo et al, 1996) or in subsurface preferential flow paths (Zehe et al, 2010; 60 Hergarten et al, 2014). Recent studies employed thermodynamic optimality approaches to 61 predict partitioning of net short wave radiation into long wave outgoing radiation and 62 turbulent fluxes of latent and sensible heat (Kleidon et al, 2014;Renner et al, 2016), to 63…”

Maximum energy dissipation to explain velocity fields in shallow reservoirs

“…Similar limits are also present in other system settings, with different forms of energy or more degrees of freedom for a system to adapt. For example, it has been shown that the yearly mean atmospheric heat transport appears to be such that the dissipative process of heat transport maximizes entropy production (Lorenz, Lunine, Withers, & McKay, 2001;Paltridge, 1979); the statistical nature of fractal river networks can be reproduced by stating that energy dissipation of flow through the river network is minimized (Hergarten, Winkler, & Birk, 2014;Howard, 1990;Rinaldo et al, 1992; Rodriguez-Iturbe, Rinaldo, Rigon, Bras, Figure 1 Main flow patterns observed in rectangular shallow reservoirs with the inlet and outlet channels along the reservoir centre line Ijjasz-Vasquez et al, 1992;Rodríguez-Iturbe, Rinaldo, Rigon, Bras, Marani et al, 1992); river meanders can be predicted by minimizing the variance of shear and the friction factor, leading to the most probable form of channel geometry (Langbein and Leopold, 1966); the maximum power principle can be used to predict vertical turbulent heat fluxes (Kleidon & Renner, 2013) or the development of preferential river flow structures at the continental scale ; while enhanced infiltration of rainwater by preferential macropore structures is explained by the principle of maximum free energy dissipation . Whilst these extremum principles appear to be contradictory at first sight, they are merely two sides of the same coin.…”