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

TL;DR

This paper establishes theoretical bounds for the power-law exponent in surface flow advance-time curves by linking it to percolation theory, providing a better understanding and prediction of irrigation dynamics.

Contribution

It introduces a novel theoretical framework connecting the exponent to percolation backbone fractal dimension, improving predictions of surface flow advance curves.

Findings

Theoretical bounds of the exponent match experimental data.

Percolation theory accurately predicts advance-time curves.

Excellent agreement between estimated and observed curves.

Abstract

A fundamental and widely applied concept used to study surface flow processes is the advance-time curve characterized by an empirical power law with an exponent r and a numerical prefactor p (i.e., x = p*t^r). In the literature, different values of r have been reported for various situations and types of surface irrigation. Invoking concepts from percolation theory, we related the exponent r to the backbone fractal dimension Db, whose value 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 showed that the theoretical bounds of Db are in well agreement with experimental ranges of r reported in the literature for two furrow and border irrigation systems. We also used 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. Excellent agreement was obtained between the estimated and observed curves.