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On the Time Validity of John Philip's Two-Term Rainfall Infiltration Model


Rainfall infiltration, the process wherein water enters the soil surface and replenishes moisture in the vadose zone, is an important component of the water balance and hydrologic cycle. Infiltration guarantees a continued availability of moisture to sustain root water uptake, plant growth, groundwater recharge and soil structure. There are several ways to estimate rainfall infiltration rates and volumes. The most rigorous approach would use a partial differential equation (Richards' equation), coupled, if necessary, with a surface water routine and groundwater model (Darcy's law), to describe infiltration into variably-saturated soils. Analytic solutions of Richards' equation and/or Darcy's law and empirical infiltration functions may work well under certain conditions (deep-drained soils with uniform initial moisture content) and/or single rainfall events. Among all research conducted, the Philip's two-term infiltration model, $I = S\sqrt{t} + c K_\text{s}t$, where $I$ [\,\si{L}\,] is the cumulative infiltration, $S$ [\,L\cdot \text{T}$^{-1/2}$\,] signifies the sorptivity, $K_\text{s}$ [\,L\cdot \text{T}$^{-1}$\,] denotes the saturated soil hydraulic conductivity, $c$ is a unitless curve fitting coefficient and $t$ denotes time in units of length, has found widespread use and applicability. This model is particularly easy to use as (1) it only has three unknown parameters, (2) the least squares parameter values are easily determined from experimental data using linear regression, and (3) two of the estimated parameters, $S$ [\,L\cdot \text{T}$^{-1/2}$\,] and $K_\text{s}$ [\,L\cdot \text{T} $^{-1}$\,], have a clear physical significance. In favor of this simplicity, Philip's two-term infiltration model eliminates higher-order terms of a polynomial series of time that account for the effect of gravity on infiltration. This effect becomes more important at later times as a larger proportion of the soil reaches saturation and the soil water pressure head gradient becomes negligible. As a result, Philip's two-term infiltration model, $I = S\sqrt{t} + c K_\text{s}t$, has a limited time validity, $t_{\text{valid}}$ [\,\si{T}\,]. In his work, Philip provides theoretical guidance on the time validity of his two-term infiltration model. This time validity is of great importance as it determines the time span of experimental infiltration data to use for parameter estimation.

In this research, we explore the time validity of Philip's two-term infiltration model using Bayesian inference and the , Soil Water Infiltration Global (SWIG) database. This database consists of a large ensemble of measured cumulative infiltration curves of a wide variety of soils worldwide. Essentially, we test, benchmark and evaluate the approach of Jaswal et al. (2020) on measured data rather than synthetic infiltration data simulated with HYDRUS-1D. The methodology consists of two parts. First, we determine the values of the parameters $S$ [\,L\cdot \text{T}$^{-1/2}$\,] and $K_\text{s}$ [\,L\cdot \text{T} $^{-1}$\,] via Bayesian inference of the Haverkamp infiltration equation using the \textbf{D}iffe\textbf{R}ential \textbf{E}volution \textbf{A}daptive \textbf{M}etropolis (DREAM) algorithm. As semi-implicit solution of Richards' equation, the Haverkamp model is valid for the entire duration of the infiltration experiment. Then, the posterior distribution of the sorptivity and saturated soil hydraulic conductivity of each measured infiltration curve are used in Philip's two-term infiltration model to determine the optimal value of the coefficient $c$ via linear regression. We implement the Bayesian information criterion (BIC) to return, as byproduct of our analysis, the optimal time validity of Philip's two-term infiltration model. The uncertainty of the time validity, $t_\text{valid}$ [\,T\,], can be estimated by evaluating the different posterior samples of $S$ [\,L\cdot \text{T}$^{-1/2}$\,] and $K_\text{s}$ [\,L\cdot \text{T} $^{-1}$\,]. We particularly focus on the ``best'' samples of each soil type in the SWIG database as results confirm that the temporal resolution of the infiltration data plays a critical role. Results demonstrate that coarse textured soils (e.g. sand, loamy sand, sandy loam) have a rather small value of $t_\text{valid}$ [\,T\,] ranging between 0.10 hour to 1.00 hour. Medium textured soils (sandy clay loam, loam, clay loam) exhibit somewhat larger values of the time validity ranging between 1.00 hour to 4.76 hours. Unfortunately, the measured infiltration curves in the SWIG database did not allow us to determine adequate values of the time validity for fine textured soils. The time validity, $t_\text{valid}$ [\,T\,] of clay loam, silty clay loam, silty clay, and clay soils was simply equal to the time of the last infiltration measurement. In other words, the experiments did not last long enough to determine accurately their respective time validity.

All results were compared to those of Jaswal et al. (2020) using synthetic infiltration data. This analysis made evident that (1) the measurement errors of the infiltration data increase the uncertainty of $t_\text{valid}$ [\,T\,]; (2) The much poorer measurement (time) resolution of the infiltration data in the SWIG database makes it difficult to accurately determine the time validity of Philip's two-term infiltration model; (3) For fine textured soils, the infiltration experiments were of insufficient length to reliably estimate the value of $t_\text{valid}$ [\,T\,]. Altogether, we conclude that it is not particularly easy to estimate the time validity of Philip's two-term infiltration model from measured cumulative infiltration data. A large cohort of the infiltration experiments in the SWIG database lack the temporal resolution and necessary length of the experiment to warrant an accurate determination of the time validity. Thus, we recommend using synthetic infiltration data simulated derived from numerical solution of Richards' equation to determine an approximate time validity for each soil type. The resulting estimates of $t_\text{valid}$ [\,T\,] can then serve as guidelines for analysis of real-world infiltration experiments.

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