Theoretical bounds for the exponent in the empirical power-law advance-time curve for surface flow

A fundamental and widely applied concept used to study surface flow processes is the advance-time curve, which indicates the time at which water arrives at any given distance along a field length. The advance-time curve is typically characterized by an empirical power law with an exponent r and a numerical prefactor p (i.e., x=ptr). In the literature, different values of r have been reported for various situations and types of surface irrigation. Invoking concepts from percolation theory, a theoretical approach from statistical physics for quantifying transport properties in complex systems, we relate the exponent r to the backbone fractal dimension Db (r = 1/Db). The backbone, through which transport occurs predominantly in a system, is formed by multiply connected loops composed of several paths that bring fluid from an initial point to a final point. The backbone fractal dimension Db, characterizing the complex structure of the backbone, depends on two factors: dimensionality of the system (e.g., two or three dimensions) and percolation class (e.g., random or invasion percolation with/without trapping). We show that the theoretical bounds of Db are well in agreement with experimental ranges of r reported in the literature for two furrow and border irrigation systems. We further use the value of Db from the optimal path class of percolation theory to estimate the advance-time curves of four furrows and seven irrigation cycles (i.e., 28 experiments). The optimal path is the most energetically favorable path through a system. Excellent agreement was obtained between the estimated and observed curves. We also discuss the indirect effects of initial water content, inflow rate, surface slope, and infiltration rate (soil texture) on system dimensionality, percolation class and, consequently, the backbone fractal dimension value. More specifically, we postulate that for closely-spaced furrows with steep slopes, low infiltration rates, and relatively high inflow rates, the wetting front advance will be mostly quasi one-dimensional. Since the backbone fractal dimension in one dimension is 1, one should expect the exponent r to be near 1.