Preferential pathways for fluid and solutes in heterogeneous groundwater systems: self-organization, entropy, work

Abstract

Abstract. Patterns of distinct preferential pathways for fluid flow and solute transport are ubiquitous in heterogeneous, saturated and partially saturated porous media. Yet, the underlying reasons for their emergence, and their characterization and quantification, remain enigmatic. Here we analyze simulations of steady-state fluid flow and solute transport in two-dimensional, heterogeneous saturated porous media with a relatively short correlation length. We demonstrate that the downstream concentration of solutes in preferential pathways implies a downstream declining entropy in the transverse distribution of solute transport pathways. This reflects the associated formation and downstream steepening of a concentration gradient transversal to the main flow direction. With an increasing variance of the hydraulic conductivity field, stronger transversal concentration gradients emerge, which is reflected in an even smaller entropy of the transversal distribution of transport pathways. By defining “self-organization” through a reduction in entropy (compared to its maximum), our findings suggest that a higher variance and thus randomness of the hydraulic conductivity coincides with stronger macroscale self-organization of transport pathways. Simulations at lower driving head differences revealed an even stronger self-organization with increasing variance. While these findings appear at first sight striking, they can be explained by recognizing that emergence of spatial self-organization requires, in light of the second law of thermodynamics, that work be performed to establish transversal concentration gradients. The emergence of steeper concentration gradients requires that even more work be performed, with an even higher energy input into an open system. Consistently, we find that the energy input necessary to sustain steady-state fluid flow and tracer transport grows with the variance of the hydraulic conductivity field as well. Solute particles prefer to move through pathways of very high power in the transversal flow component, and these pathways emerge in the vicinity of bottlenecks of low hydraulic conductivity. This is because power depends on the squared spatial head gradient, which is in these simulations largest in regions of low hydraulic conductivity.

Energy efficiency in transient surface runoff and sediment fluxes on hillslopes – a concept to quantify the effectiveness of extreme events

“…This implies that the system ends in a dead state called thermodynamic equilibrium where all gradients have been depleted, corresponding to minimum free energy and maximum entropy. Open thermodynamic systems may however prevail in an organized state far away from the entropy maximum, if there is an external feedback sustaining a net influx of free energy to perform the necessary work to act against the depletion of gradients and to export the entropy produced during irreversible processes (Zehe et al, 2021). In the following we want to clarify this aspect for surface runoff and related ideas of thermodynamic optimality, which appear to be contradictory at first sight.…”

“…Although energy is conserved and cannot disappear due to the 1 st law of thermodynamics, free energy is not a conserved property, but is dissipated during irreversible processes due to the related production of entropy. Free energy is basically energy without entropy, and the free energy of a flow system is thus equivalent to its capacity to perform work to steepen a concentration gradient (Zehe et al, 2021) or to create motion in form of coupled water and sediment fluxes (Bagnold, 1966). Frictional dissipation during the latter implies production of heat through production of entropy, which increases the average kinetic energy of the molecules in the riverbed or the hillslope surface materials.…”

“…First, there exist at least three forms of physical entropy (not to mention information entropy), (cf. Popovic, 2017) entropy produced by mixing and depletion of chemical potential and geo-potential gradients, and radiation entropy produced by radiative cooling (Kleidon, 2016, Zehe et al, 2021. And second, a proper definition of entropy production requires a clear definition of the system and its boundary, otherwise "Nobody really knows what entropy is" (Von Neumann, cited in Tribus and McIrving, 1971).…”

Flow recession behavior of preferential subsurface flow patterns with minimum energy dissipation

“…However, the description of heterogeneity and its representation in numerical models is one of the major challenges in hydrology. On the other hand, there seems to be an increasing number of studies attempting to derive preferred stated of the atmosphere and the hydrosphere from principles of optimality (e.g., Kleidon and Schymanski, 2008;Kleidon and Renner, 2013;Kleidon et al, 2014;Kleidon and Savenije, 2017;Kleidon et al, 2019;Zehe et al, 2010Zehe et al, , 2013Zehe et al, , 2021Westhoff and Zehe, 2013;Westhoff et al, 2014Westhoff et al, , 2017Zhao et al, 2016). In the context of fluid flow, minimum energy dissipation seems to be a promising concept, which was successfully applied to river networks (Howard, 1990;Rodriguez-Iturbe et al, 1992a, b;Rinaldo et al, 1992;Maritan et al, 1996) and to the cardiovascular system of mammals (West et al, 1997;Enquist et al, 1998Enquist et al, , 1999Banavar et al, 1999;West et al, 1999a, b).…”

Probabilistic identification of Preferential Groundwater Networks

“…Berkowitz and Zehe (2020) underlined the need for additional studies to develop a unified theoretical framework comprising surface and subsurface water networks starting from a comprehensive analysis of fractal behaviors and (power-law based) statistical descriptions that are documented for these systems in diverse contexts. Leveraging on energy concepts, Zehe et al (2021) analyzed fluid flow and solute transport in (synthetically generated) randomly heterogeneous hydraulic conductivity fields and found that high solute concentrations associated with preferential pathways can be correlated with an elevated power in fluid flow therein.…”