# Tsallis Entropy Theory for Derivation of Infiltration Equations

Published by the American Society of Agricultural and Biological Engineers, St. Joseph, Michigan www.asabe.org

Citation:  Transactions of the ASABE. 53(2): 447-463. (doi: 10.13031/2013.29585) @2010
Authors:   V. P Singh
Keywords:   Entropy, Infiltration, Green-Ampt equation, Holtan equation, Horton equation, Kostiakov equation, Overton equation, Philip equation, Principle of maximum entropy, Tsallis entropy

An entropy theory is formulated for deriving infiltration equations for the potential rate (or capacity) of infiltration in unsaturated soils. The theory is comprised of five parts: (1) Tsallis entropy, (2) principle of maximum entropy (POME), (3) specification of information on the potential rate of infiltration in terms of constraints, (4) maximization of entropy in accordance with POME, and (5) derivation of the probability distribution of infiltration and its maximum entropy. The theory is illustrated with the derivation of six infiltration equations commonly used in hydrology, watershed management, and agricultural irrigation, including Horton, Kostiakov, Philip two-term, Green-Ampt, Overton, and Holtan, and the determination of the least biased probability distributions underlying these infiltration equations and the entropies thereof. The theory leads to the expression of parameters of the derived infiltration equations in terms of three measurable quantities: initial infiltration capacity (potential rate), steady infiltration rate, and soil moisture retention capacity. In this sense, these derived equations are rendered nonparametric. With parameters thus obtained, infiltration capacity rates are computed using these six infiltration equations and are compared with field experimental observations reported in the hydrologic literature as well as the capacity rates computed using parameters of these equations obtained by calibration. It is found that infiltration capacity rates computed using parameter values yielded by the entropy theory are in reasonable agreement with observed as well as calibrated infiltration capacity rates.