Explicit infiltration equations and the Lambert W-function

Abstract

10The Green and Ampt infiltration formula, as well as the Talsma and Parlange formula, are two-parameter equations that are 11 both expressible in terms of Lambert W-functions. These representations are used to derive explicit, simple and accurate approx-12 imations for each case. The two infiltration formulas are limiting cases that can be deduced from an existing three-parameter in-13 filtration equation, the third parameter allowing for interpolation between the limiting cases. Besides the limiting cases, there is 14 another case for which the three-parameter infiltration equation yields an exact solution. The three-parameter equation can be 15 solved by fixed-point iteration, a scheme which can be exploited to obtain a sequence of increasingly complex explicit infiltration 16 equations. For routine use, a simple, explicit approximation to the three-parameter infiltration equation is derived. This approxi-17 mation eliminates the need to iterate for most practical circumstances. 20Due to the many circumstances where infiltration 21 into porous media plays a role, theoretical equations for 22 predicting quantities such as infiltration flux and cu-23 mulative infiltration are in widespread use. A subset of 24 these circumstances involves one-dimensional vertical 25 infiltration, a branch of vadose-zone hydrology that has 26 a rich history stretching back to the early part of last 27 century. For a given soil type, the formulas aim to es-28 timate IðtÞ, the cumulative infiltration, I, that enters the 29 soil as a function of time, t. The archetype problem to 30 which infiltration laws apply is infiltration into an ini-31 tially dry, homogeneous soil where the surface of the soil 32 is saturated, but not ponded. It is this situation that is 33 considered below. 34In practice, it is useful to have infiltration laws that 35 are both physically based and easy to implement. The 36 latter feature is inherent in explicit expressions for IðtÞ, 37 whereas the former is a feature of laws that are based on 38 standard soil properties such as the soil-water diffusivity, 39 D, and hydraulic conductivity, K. Physically based in-40 filtration laws for one-dimensional infiltration typically 41 use the sorptivity, S, and particular values of the hy-42 draulic conductivity, e.g., the hydraulic conductivity at 43 saturation, K s , or at the surface moisture content. The 44 sorptivity, we recall, is derived from D and the boundary 45 and initial conditions that pertain [1][2][3][4]. 46 As demonstrated elsewhere [5][6][7][8], infiltration laws 47 have two ''limiting'' behaviors. We remark that they are 48 limits in that they appear to cover the possible range of 49 infiltration behaviour; they are not formal mathematical 50 limits. One limit is represented by the Green and Ampt 51 formula [9], which relies on a soil having a rapidly 52 varying diffusivity and a near-constant hydraulic con-53 ductivity. The other is represented by Talsma and Par-54 lange [10] result relying on proportionality between D 55 and dK=dh (h being...

Infiltration into soils: Conceptual approaches and solutions

“…A generalization of equation (24) that includes ponded conditions was introduced by Parlange et al [1985] and Haverkamp et al [1990], with an explicit form presented by Barry et al [1995]. The implicit form of the Green and Ampt solution (equation (19)) can be inverted to obtain explicit infiltration equations by using the Lambert Wfunctions [Parlange et al, 2002]. The properties of this function have been extensively discussed by Barry et al [2005].…”

Comment on “Recent advances on solving the three-parameter infiltration equation” by Prabhata K. Swamee, Pushpa N. Rathie, Luan Carlos de S.M. Ozelim and André L.B. Cavalcante, Journal of Hydrology 509 (2014) 188–192

“…The two limits of , 0 and 1, correspond to the Green-Ampt and Talsma-Parlange limits, respectively. S and K are used to non-dimensionalise the three parameter equation as (Parlange et al, 2002)…”

“…(6) of Barry et al (2012). Swamee et al (2014) also noted that they could not reproduce the absolute error plots given by Barry et al (2012), i.e., they remark that "a different precision was attested by Barry et al (2012) for Parlange et al (2002) solution". This error was due the taking the value as a percentage and dividing by 100.…”

Comment on “Explicit equations for infiltration” by Prabhata K. Swamee, Pushpa N. Rathie and Luan Carlos de S.M. Ozelim, Journal of Hydrology 426–427 (2012) 151–153

“…Our main purpose is to suggest how they might improve their formula based on an existing approximation for the three-parameter infiltration equation, given by Parlange et al (2002).…”