Percolation theory can be used to find water flow paths of least resistance. Application of percolation theory to drainage networks allows identification of the range of exponent values that describe the tortuosity of rivers in real river networks, which is then used to generate the observed scaling between drainage basin area and channel length, a relationship known as Hack's law. Such a theoretical basis for Hack's law may allow interpretation of the range of exponent values based on an assessment of the heterogeneity of the substrate.

River networks display complex organization as documented in numerous
studies. This work addresses only one of them, in particular, the
relationship between drainage basin area and river length, which is
non-trivial. In Euclidean geometry the basin area,

Hack (1957) (p. 65) asserts that the relationship was a consequence of the
lengthening of drainage basins with increasing size. But Montgomery and
Dietrich (1992) compare straight-line basin length,

Hack's law explanations have been sought in fractal (Tarboton et al., 1988;
Maritan et al., 1996), constructal (Reis, 2006), and “feasible optimality”
(Rigon et al., 1998) theories. Fractal theories produce the required
self-similar drainage basins (Peckham, 1995) as well as increasing stream
sinuosity downstream. I suggest that Hack's law can be understood using
percolation theory (Stauffer and Aharony, 1994) because (1) the fractal
structure of the percolation cluster generates values for

There are two distinct applications of percolation theory to flow or conduction problems, and these two applications are those that provide the bounds to Hack's exponent values. The more familiar application is to a binary system, where, e.g., bonds either connect neighboring sites (which in the simplest case are located on a lattice, or grid), or they do not. If enough such neighboring sites are connected, a continuous path of interconnected bonds spans the system. This is denoted the percolation threshold. The shortest distance across the system within this connected cluster is called the chemical path length (Porto et al., 1997). Since all bonds have equal resistance, the shortest flow path also has the lowest resistance and optimal dissipation.

The second possibility is a system in which bonds of varying resistance
connect each pair of neighboring sites. When the system is strongly
heterogeneous, i.e., when the distribution of the natural logarithm of the
resistances has variance,

The restriction of river networks to the surface of the earth, and the
measurement of stream lengths on 2-D maps, makes the topology of stream
connections and the application of percolation theory two-dimensional. In two
dimensions, the chemical path length scales with the system size (Sheppard et
al., 1999),

One can start from an initially homogeneous landscape, and allow stream incision through random headward erosion, analogous to the processes treated in early landscape evolution models (Willgoose et al., 1991), which generate hierarchical structures from random chance associated with rainfall magnitude variability. A connected path with the lowest dissipation (shortest length) will soon acquire the highest flow, through channel erosion feedbacks. Thus, once a river makes a random choice, the enhanced erosion power from the stream reinforces the initial random choice.

The optimal path exponent describes the tortuosity of a channel, when the channel is determined by a global optimization of the flow path in a heterogeneous substrate, and could not be a simple product of headward erosion, which might produce only a local optimization. In such a case geological constraints from varying erodibility can dominate as channels extend either upward, by headward erosion, or downward (e.g., by overtopping of sills).

Using the above result that

Note that, while, e.g., Willemin (2000) found a wider range of

Percolation predictions generate a range of exponents consistent with those reported in Hack's law, including the tendency for the largest exponent values to occur in geologically heterogeneous environments.

The statistical nature of percolation theory is in accord with the tendency of the spread in Hack's exponent values to diminish with increasing sample size.

The source of the tortuosity in the “optimal paths” of lowest energy dissipation is in general accord with the “feasible optimality” (Rigon et al., 1998) proposed to explain Hack's law.

I appreciate the dialogue with the editor, Daniel Schertzer. Edited by: D. Schertzer Reviewed by: B. Watson and one anonymous referee