**4. Conclusions**

A sensitivity analysis of the instantaneous infiltration rates in the Talbot–Ogden model is developed with rigorous mathematical deduction and reasonable physical assumptions. This analysis starts from the two-bin case to the *n*-bin case. It is concluded that the infiltration rate increases with finer discretization in the moisture content domain. Numerical experiments on the overall infiltration rates have confirmed this theoretical conclusion. Therefore, the choice of the number of bins is very important for di fferent soil textures. When θ*d* approaches θ*<sup>e</sup>*, the Talbot–Ogden model always generates higher infiltration fluxes than the Green–Ampt model, where *K* = *Ks* and *Hc* = −ψ*<sup>b</sup>* are set.

An asymptotic analysis is also made on estimating the largest infiltration rates. Using the loose assumption of a line-shaped wetting front, the asymptotic analysis provides an upper bound of the infiltration rate di fference resulting from two arbitrarily di fferent moisture content discretizations. The numerical experiments then illustrate this result. These upper bounds for di fferent soil textures can determine the accuracy of the Talbot–Ogden model in predicting the fluxes in various environments. Note that if the wetting front is apparently nonlinear, which is possible in the Talbot–Ogden model especially when precipitations are intermittent, then this asymptotic analysis may be adjusted to be piecewise for di fferent intervals of the moisture content domain. The nonlinearity can be handled using exponential functions (i.e., *<sup>z</sup>*α, α > 1) to approximate the depths of the wetting fronts.

There are basically two factors determining the infiltration rates in the Talbot–Ogden model. One is the number of bins by the *n*-bin case analysis and the other is the front depth discrepancy between the leftmost bin and the rightmost bin in our asymptotic analysis. Moreover, it is intrinsically mass conservative and is always computable.

The analysis will be extended to predicting the infiltration rate variation as a function of time but not only restricted to the instantaneous moment. Meanwhile, bugs and cracks can be easily incorporated into this model, although this will make the flux analysis more complicated. This model can also be extended to heterogeneous soil textures by solving stacked homogeneous soil layers, which is future work.

**Author Contributions:** In this study, L.L. has made contributions to conceptualization, formal analysis, data curation, investigation and writing-review and editing. H.Y. has made his contributions to methodology, software, formal analysis, resources, and writing-original draft. Both authors have acquired funds for this research.

**Funding:** This research was sponsored by the Natural Science Foundation for Young Scientists of Jiangsu Province (grant no. BK20180450), the Natural Science Foundation of Nanjing University of Posts and Telecommunications NUPTSF (grant no. NY219080) and the China Postdoctoral Science Foundation (grant no. 2018T110531).

**Acknowledgments:** The authors gratefully acknowledge the assistance of Talbot, Ogden and Craig Douglas for their extraordinary insights and comments in the development of this paper. The authors also thank King Abdullah University of Science and Technology for the support on computational resources.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Physical Notations and Units**

