Optimality principles have been used to investigate physical processes in different areas. This work applied an optimal principle (that water flow resistance is minimized for the entire flow domain) to steady-state unsaturated flow processes. Based on the calculus of variations, under optimal conditions, hydraulic conductivity for steady-state, gravity-dominated unsaturated flow is proportional to a power function of the magnitude of water flux. This relationship is consistent with an intuitive expectation that for an optimal water flow system, locations where relatively large water fluxes occur should correspond to relatively small resistance (or large conductance). This theoretical result was also consistent with observed fingering-flow behavior in unsaturated soils and an existing model.
This work presents a new theory that hydraulic conductivity for steady-state unsaturated flow is proportional to a power function of water flux when water flow resistance is minimizedfor the entire flow domain. This result was also demonstrated to be consistent with observed fingering-flow behavior.
Optimality principles refer to that state of a physical process that is controlled by an optimal condition that is subject to physical or resource constraints. For example, Eagleson (2002) demonstrated that under natural conditions and in water-limited areas, vegetation tends to grow under maximum-productivity and unstressed conditions. He called the function and forms of vegetation, following the optimality principle, the results of “Darwinian expression.” After studying a variety of natural phenomena characterized by tree-like structures, Bejan (2000) proposed a “constructal theory,” which states that “for a finite-size open system to persist in time (to survive) it must evolve in such a way that it proves easier and easier access to the currents that flow through it.” While the definition of “easy access” is not always clear, Bejan (2000) demonstrated that tree-like structures are the direct results of the minimization of flow resistance across whole flow systems under consideration. During the past 30 yr, the maximum entropy production (MEP) principle has been successfully applied, in a heuristic sense, to the prediction of steady states of a wide range of systems (Niven, 2010; Kleidon, 2009). The MEP principle states that a flow system subject to various flows or gradients will tend toward a steady-state position of maximum thermodynamic entropy production (Niven, 2010). The theoretical connections between these optimality principles and the currently existing fundamental laws, however, are not well established. The alternative point of view is that these principles are actually self-standing and do not follow from other known laws (Bejan, 2000).
The role of optimality principles in forming complex natural patterns has been recognized for many years in the surface hydrology community (Leopold and Langbein, 1962; Howard, 1990; Rodriguez-Iturbe et al., 1992; Rinaldo et al., 1992; Liu, 2010). For example, Leopold and Langbein (1962) proposed a maximum entropy principle for studying the formation of landscapes. Rodriguez-Iturbe et al. (1992) postulated principles of optimality in energy expenditure at both local and global scales for channel networks. While previous studies mainly use spatially “discrete” approaches as a result of considering energy dissipation through channel networks only, Liu (2010) developed a group of (partial differential) governing equations for steady-state optimal landscapes (including both channel networks and associated hillslopes) using the calculus of variations.
The importance of optimality principles has also been intuitively recognized in the vadose zone hydrology community for a long time. For example, it seems to be well known that fingering flow is due to the fact that unsaturated water tends to form flow paths corresponding to the minimized flow resistances. (Note that for a given water flux, fingering flow always gives lower flow resistance [or higher conductance] compared with uniform flow because fingering flow paths generally have higher local water saturations that correspond to larger unsaturated conductivities.) Rigorous applications of this optimality principle have not been fully explored, however. Because of fingering flow, water propagates quickly to significant depths while bypassing large portions of the vadose zone, and solute travel times from a contamination source (located on the soil surface or in the vadose zone) to the groundwater are shorter than a priori expected. As a result of the important effects of this flow process on groundwater contamination (an important issue for water resources management), preferential flow has been a major research area in the vadose zone hydrology community for a number of years and considered probably the most frustrating processes in terms of hampering accurate predictions of contaminant transport in the vadose zone (e.g., Glass et al., 1988; Flury and Flühler, 1995; Liu et al., 2003; Šimůnek et al., 2003; Nimmo, 2010).
This study developed a conductivity relationship for gravity-dominated unsaturated flow derived from a principle that the energy dissipation rate (or flow resistance) is minimized for the entire flow system. Preliminary evaluation of this relationship was conducted by comparing it with relevant experimental observations and the currently existing models. The potential limitations and further improvements of this work are also briefly discussed.
Under optimal flow conditions corresponding to the minimum energy dissipation rate (or flow resistance), the derived conductivity is a power function of water flux (Eq. ) for gravity-dominated unsaturated flow. This result physically makes sense. For the positive power values, the smallest flow resistance occurs within flow paths with the largest water flux. Intuitively, it is easy to understand that this conductivity distribution will result in minimized total flow resistance globally. This finding is also consistent with our daily life experiences. For example, to maximize traffic transportation efficiency, our highways always have more lanes (or higher “conductance”) in locations with high traffic fluxes. (Highway networks may be considered to be analogous to fingering flow paths.)
It is of interest to note that for typical values of β = 4 and γ = 0.7 (Brooks and Corey, 1964; Sheng et al., 2009), a = 0.4 is close to the value of 0.50 given in Eq. . Whether or not a single value for parameter a is valid for different soils needs further research based on experimental observations.
Finally, this work is the first step to incorporate the optimality principle into unsaturated flow. Consequently, some limitation of the current work still exists. For example, detailed pore-scale unsaturated-flow physics is not adequately incorporated yet. This physics requires that the upper limit of K in Eq.  should be F(h), which, however, is not reflected in our theory. This can be approximately accounted for in practice by limiting the K value calculated from Eq.  to the corresponding F(h) value. Nevertheless, the major focus of this study was to highlight the potential for developing new unsaturated water flow theories based on the optimal principle. This principle may hold the key to resolving a number of problems associated with emerging patterns in unsaturated soils.
Based on the calculus of variations, this work showed that under optimal conditions, hydraulic conductivity for steady-state, gravity-dominated unsaturated flow is proportional to a power function of the magnitude of water flux. It is consistent with an intuitive expectation that for an optimal water flow system, locations where relatively large water fluxes occur should correspond to relatively small resistance (or large conductance). Consistence between this theoretical result with observed fingering flow behavior in unsaturated soils and the ARM was also demonstrated. Finally, it is important to note that the classic unsaturated-flow theory is applicable to capillarity-dominated cases while the current work focused on unsaturated flow under gravity-dominated conditions. Whether the optimality principle can be used to develop a general theory (that includes both cases as two special ones) deserves further research.
The initial version of the paper was carefully reviewed by Dr. Jim Houseworth and Dr. Dan Hawkes. The work was performed under DOE Contract DE-AC03–76F00098.