New Analytic Solution of Boltzmann Transform for Horizontal Water Infiltration into Sand

Abstract

The use of the Boltzmann transform function, λ(θ), to solve the Richards equation when the diffusivity, D, is a function of only soil water content, θ, is now commonplace in the literature. Nevertheless, a new analytic solution of the Boltzmann transform λ(h) as a function of matric potential for horizontal water infiltration into a sand was derived without invoking the concept or use of D(θ). The derivation assumes that a similarity exists between the soil water retention function and the Boltzmann transform λ(θ). The solution successfully described soil water content profiles experimentally measured for different infiltration times into a homogeneous sand and agrees with those presented by Philip in 1955 and 1957. The applicability of this solution for all soils remains open, but it is anticipated to hold for soils whose air‐filled pore‐size distribution before wetting is sufficiently narrow to yield a sharp increase of water content at the wetting front during infiltration. It also improves and provides a versatile alternative to the well‐known analysis pioneered by Green and Ampt in 1911.

The recent similarity hypotheses to describe water infiltration into homogeneous soils

“…[3, 4 and 5] presented in Figures 2 through 4 illustrate the comparisons of the Prevedello similarity solutions to measured observations and classical solutions of Richards' equation for three different infiltration times. From Figure 2, it is readily apparent that all of the similarity calculations for the Marine sand are nearly identical to those stemming from the classical solution of Philip, as stated in Prevedello et al (2008).…”

“…The similarity solutions of Prevedello et al (2008 and2009) are here tested with measurements of water infiltrating into laboratory columns of Columbia silt loam and Hesperia sandy loam (Davidson et al, 1963). Their measured soil water properties and wetting parameters are given in Table 1.…”

“…Recently, Prevedello et al (2008) focused on soil water content distribution [θ(x,t)] during horizontal infiltration to estimate infiltration into homogeneous soils using measured values of K for specific values of θ and the soil water retention curve θ(h), Richard's equation and an extension of the historic 1894 Boltzmann transform function. As illustrated in Figure 1A, this similarity hypothesis…”

“…Recognizing that these similarity solutions are approximations of classical solutions of Richard's equation, Prevedello et al (2008Prevedello et al ( , 2009) specifically stated that the use and acceptance of the hypotheses remained open for future research. Here, without deriving additional mirror image hypotheses, we merely explore the potential use and associated restrictions of the two recent similarity hypotheses illustrated in Figure 1.…”

Comparison of methods for determining unsaturated hydraulic conductivity in the wet range to evaluate the sensitivity of wetting front detectors

“…The laboratory procedure of Bruce and Klute (1956) 68 is a rapid method for estimating the soil water diffusivity function, D(θ) or the hydraulic conductivity K(h) if water retention data are available. The original method has subsequently been improved by several groups (Clothier et al, 1983;Tyner and Brown, 2004;Prevedello et al, 2008). However, it is difficult to establish steady or transient-state flow conditions under laboratory conditions and the method does not fully represent actual field conditions (Nandagiri and Prasad, 1996).…”