In this study, the method of inference of macroscale thermodynamic potentials, forces, and exchange coefficients for variably saturated groundwater flow is outlined based on the entropy balance. The theoretical basis of the method of inference is the explicit calculation of the internal entropy production from microscale, thermodynamic flux–force relationships using, e.g., hyper-resolution variably saturated groundwater flow models. Emphasis is placed on the two-scale nature of the entropy balance equation that allows simultaneously incorporating movement equations at the micro- and macroscale. The method is illustrated with simple hydrologic cross sections at steady state and periodic sources/sinks at dynamic equilibrium, and provides a thermodynamic point of view of upscaling in variably saturated groundwater flow. The current limitations in the connection with observable variables and predictive capabilities are discussed, and some perspectives for future research are provided.

The current earth science literature indicates that entropy balance considerations have been mainly applied in the context of optimality and self-organization. Theories of optimality and self organization are appealing when dealing with complex nonlinear systems, because of their apparent usefulness in interpreting interactions of gradients and fluxes and in quantifying (predicting) systems' states and uncertainties. In this context, the entropy and energy balance received attention, because of its physics-based foundation in nonequilibrium thermodynamics and potential connection with information theory (e.g., Dewar, 2003; Koutsoyiannis, 2014). The entropy balance appears to be useful in applications to hydrologic (e.g., Zehe et al., 2013; Ehret et al., 2014), ecohydrologic (e.g., Dewar, 2010; Miedziejko and Kedziora, 2014; del Jesus et al., 2012), and atmospheric sciences (e.g., Paillard and Herbert, 2013), and in general to open complex nonlinear thermodynamic systems (Abe and Okuyama, 2011).

The entropy balance states that in an open system, the change in entropy equals the internal production of entropy minus the divergence of the entropy current. A dynamic equilibrium or steady state is obtained, when entropy production inside (due to, e.g., flow processes of heat or matter) equals the divergence of the entropy current, i.e., the entropy exchange with the outside. Note also, dynamic equilibrium refers to a state of stationarity in the statistical sense. Optimality of the dynamic equilibrium may be achieved, because the gradient, which drives the flux and, thus, the production of entropy, is reciprocally depleted by the same flux (Kleidon et al., 2013).

In hydrology, the entropy balance has been applied to conceptual problems based on the overarching rationale that entropy production is maximized (maximum entropy production, MEP) in obtaining a state of dynamic equilibrium by optimizing the fluxes and gradients in competition via an adjustment of some (non)linear exchange coefficient. There have been some studies demonstrating how entropy production can be maximized by optimizing an exchange coefficient to obtain an optimal system's state. In hydrology, there are quite a few examples of the application and discussion of the MEP principle (e.g., Ehret et al., 2014; Westhoff et al., 2014; Kleidon and Schymanski, 2008) also in connection with data (e.g., Zehe et al., 2013). However, its validity and applicability to hydrologic systems is still in question (Westhoff and Zehe, 2013).

Often the entropy balance has been applied at steady state with simple bucket models, which are well mixed (i.e., without internal gradients). For example, Porada et al. (2011) performed a detailed entropy production analysis of the land surface hydrologic cycle including the shallow vadose zone assuming vertical equilibrium of the soil bucket model. Applying linear bucket models without considering internal gradients, Kleidon and Schymanski (2008) showed that if the natural system possesses enough degrees of freedom, in case of steady state, the system will tend towards a certain exchange coefficient, when entropy production is maximized. For similar bucket models, Westhoff et al. (2014) demonstrated the impact of periodic boundary forcing on entropy production, which may result in more than one maximum for unique values of the exchange coefficient at dynamic equilibrium. Interestingly, these studies did not calculate the internal entropy production explicitly. Instead, entropy production was estimated indirectly from the exchange with the outside (i.e., the divergence of the entropy current).

In order to optimize effective values of a simple two-box model, Schymanski et al. (2010) recognized the potential of explicitly estimating the internal entropy production using a simple distributed model of the water and carbon balance (Klausmeier, 1999), which is based on coupled equations of moisture and biomass and is able to produce vegetation patterns. This study highlights an interesting aspect of entropy balance considerations that is the inference of upscaled effective parameters and state variables to represent subgrid scale variability in coarse scale (macroscale) models. Thus, ultimately, the appeal of the entropy balance maybe the inference of upscaled or effective exchange coefficients and forces/gradients, which may be used to quantitatively describe the complex system without the explicit knowledge about microscopic details (Dewar, 2009). In this context, a popular example is gas diffusion, which can be captured by an inferred, macroscopic diffusion coefficient and gradient instead of honoring the motion and interactions of individual molecules.

In this study, the method of inference of effective hydrologic exchange coefficients, potentials and forces is outlined using the entropy balance equation in applications to simple hydrologic cross sections. The purpose of this study is to direct attention to the potential insights gained from a new branch of theoretical hydrology combining modern thermodynamic principles with numerical experiments. While the thermodynamic principles constitute the link between different spatial scales that may be useful in upscaling hydrologic process across a hierarchy of scales, the numerical experiments constitute the methodological pillar to obtain explicitly the internal entropy production or dissipation required in the upscaling, equivalent to ab initio simulations in molecular dynamics (Kresse and Hafner, 1994). The following sections provide the basic theory with an emphasis on the two-scale nature of the entropy balance, and the application to the hydrologic cross sections with ensuing discussion and conclusions.

The theory outlined in Kondepudi and Prigogine (2014) is applied to the
problem of variably saturated groundwater flow at constant temperature.
Based on conservation of energy (and the balance equation for
concentrations, which is not required in this analysis) Kondepudi and
Prigogine (2014) write the entropy balance as follows:

In the considered case of variably saturated groundwater flow,

Performing an entropy balance at steady state leads to

Because of the two-scale nature of Eqs. (1) and (2), movement equations are
introduced at the macroscale and microscale. At the macroscale,

At the microscale, the chemical potential

Thus, the two-scale nature of Eq. (2) allows to apply different thermodynamic
flux–force relationships at the different scales that are the conductance
concept at the macroscale (Eq. 7) and essentially Darcy's law or Richards'
equation (Eq. 9) at the microscale. In Eq. (2), the entropy production serves as an
automatic spatial and also temporal integrator of the microscale
fluctuations. These two characteristics are remarkable. Note, the calculation
(integration) of the entropy balance may be performed over the global domain
of volume

The basis of the method of inference is that the internal, microscopic
entropy production

Schematic of a simple profile with Dirichlet boundary conditions
on the right and left

Directed at a heat flow example in Kondepudi and Prigogine (2014), a simple
cross section is considered (Fig. 1) with steady-state, variably saturated
groundwater flow,

In case of this simple example, applying

This example expands example 1 to steady-state groundwater flow including
recharge represented by the mass rate

Schematic of a simple profile with a Dirichlet boundary condition
on the right

Schematic of a simple profile with Dirichlet boundary conditions
on the right and left

The general expression for the macroscopic potential of the cross section is

With Eq. (7) and

For

Schematic of a simple profile with Dirichlet boundary conditions
on the right and left

Schematic of a simple profile with Dirichlet boundary conditions
on the right and left

For a heterogeneous profile and/or

For

In this example, a no-flow boundary condition on the left is considered
resembling a hillslope with a no-flow boundary along a hypothetical ridge on
the left side, and a Dirichlet boundary condition along a hypothetical stream
on the right side. Now, a source/sink

Schematic of the discrete example consisting of three microscale
elements with a Dirichlet boundary condition on the right side (

Note, again

Recognizing that

Apparently, on the right-hand side of Eqs. (35), (36), and (37) all terms may
be calculated from the numerical simulations except the entropy change rate

In this schematic, there are three microscale elements with sources/sinks in
each individual element (

Any changes in the entropy of the system with time are due to transient
effects that cancel out at dynamic equilibrium

A special case may be considered, in which the system depicted in Fig. 6 is
also closed on the right side resulting in a sole exchange with the
surroundings via the periodic source/sink (e.g.,
infiltration/evapotranspiration)

Schematic of a simple profile with a no-flow boundary condition on
the left and right (based on symmetry) and transient, spatially varying
sources/sinks

The major advantage of the proposed inference theory is the estimation of
macroscopic variables that are thermodynamically consistent with the
microscale fluctuations. This is discussed in the context of the simple
example 1 interpreting the entropy current

Equations (12) and (13) have not been applied before in the context of
hydrology. While the equations illustrate the basic idea for the simplest
case of a Darcy experiment, one may argue that the insight gained from this
example is rather limited, because

It is important to emphasize that one can also obtain, in an ad hoc fashion,
the forces and conductance coefficients for any subdomain

Under purely saturated groundwater flow conditions, the estimates of
macroscale variables can be used directly for predictions, because

This also brings up the question of whether one is able to establish a
connection of the proposed theory with observations of real-world systems.
Obviously,

Assuming a time-varying force, i.e., Dirichlet boundary conditions, temporal
integration of Eq. (11) over one full cycle at dynamic equilibrium yields

In this study, the method of inference based on the entropy balance equation was introduced. The theoretical basis is the explicit calculation of the internal microscale entropy production, which is used in the balance equation to solve for macroscale potentials, and thermodynamic forces and fluxes. The proposed method was illustrated with simple hydrologic cross sections of steady-state, variably saturated groundwater flow and a periodic source/sink (infiltration/evapotranspiration) at dynamic equilibrium.

The entropy balance equation is remarkable, because the equation unifies the macro- and microscale in one equation allowing the simultaneous application of two different movement equations, which are the conductance equation at the macroscale and Darcy's law/Richards' equation at the microscale, in this study. The derivations lead to expressions for macroscale variables that are a function of the entropy production (i.e., the internal fluctuations of the microscale flux–force relationships) and provide a thermodynamically consistent link between the two different scales. Therefore, the derivation provides a different theoretical perspective of variably saturated groundwater flow and new approaches for obtaining effective macroscale variables. The discussion suggests that these may be derived consistently for a hierarchy of scales. With the advent of high-performance computing in hydrology, there is strong potential for additional insight from hyper-resolution numerical experiments to explicitly calculate the internal entropy production. For example, existing and new averaging and upscaling laws may be tested and derived using series of numerical experiments with, e.g., varying subsurface heterogeneity configurations and boundary conditions. These experiments may also be useful in deriving new movement equations at the macroscale replacing empirical, calibrated parameterizations, and regionalization approaches.

The study is a contribution to the field of theoretical hydrology, providing a thermodynamic perspective of inference in hydrology. While inference of macroscale variables' necessitates explicit calculation of the entropy production and thus considerable computational resources, these resources are well invested: obtaining previously unknown macroscale parameters is at the center of the ubiquitous challenge of upscaling, and applying the proposed framework may help in finding general upscaling relationships over a hierarchy of scales. The connection to real-world observations needs to be established in the future, also with the help of numerical simulations. In the provided theoretical setting, the usefulness of the method for predictions is evident from the simple examples provide here, however, for real-world predictions this remains to be demonstrated.

I would like to thank the associate editor, Murugesu Sivapalan, and the reviewers Stan Schymanski, Martijn Westhoff, Erwin Zehe, one anonymous reviewer, and Claudius Bürger for their constructive comments and suggestions during the review process. I also would like to thank Elbert Branscomb for the initial discussions and Axel Kleidon for more extensive discussions. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: M. Sivapalan