Mathematical theory
For HS, the Kostiakov (Hasan et al., 2015), Philip (Philip, 1992), and Green–Ampt (Selker et al., 2017) models are commonly used to describe infiltration. The Kostiakov water infiltration model is derived as follows.
KF/PKF infiltration modelKostiakov (1932) suggested the following equation for IR \(i(t)\ \)as a function of time:
\(i(t)=ant^{n-1}\) (1)
The piecewise IR function is as follows:
\(i(t)=\left\{\par \begin{matrix}a_{f}n_{f}t^{n_{f}-1},\ \ \&0<t<t_{p}\\ a_{b}n_{b}t^{n_{b}-1},\ \ \&t\geq t_{p}\\ \end{matrix}\right.\ \) (2)
where\(t\) is time (min); \(i(t)\ \)is the IR at time t (cm/min); \(a\ \)and \(n\) are empirical constants that have no physical meaning; \(a_{f}\) and \(n_{f}\)are parameters before the inflection point, \(a_{b}\) and \(n_{b}\) are parameters after the inflection point.
Because the CI curves of W-RS have a double slope (Moret-Fernández et al., 2019) and the IR is a single-peak curve (Fig. 5(b)), the traditional water infiltration model does not provide a good fit (Moret-Fernández et al., 2019). Thus, this study attempts to fit the IR using GF, PGF, FSF, GMF, BF, and PBF. The water infiltration formulas for each function are derived in the following sections.
GF/PGF infiltration model
Gauss mentioned several concepts related to the Gaussian distribution in his famous book, “Theoria motus corporum coelestium in sectionibus conicis solem ambientium” (Perthes, 2010). The probability density function (PDF) of the GF is shown in Fig. 3(a), and the form of the GF and PGF (Ren et al., 2018) are as follows:
GF:\(i\left(t\right)=k\bullet\exp\left(-\left(\frac{t-t_{p}}{\delta}\right)^{2}\right)+\psi\)(3)
PGF: \(i(t)=\left\{\par \begin{matrix}k_{f}\bullet\exp\left(-\left(\frac{t-t_{p}}{\delta_{f}}\right)^{2}\right)+\psi_{f},\ \ \&0<t<t_{p}\\ k_{b}\bullet\exp\left(-\left(\frac{t-t_{p}}{\delta_{p}}\right)^{2}\right)+\psi_{b},\ \ \&t\geq t_{p}\\ \end{matrix}\right.\ \) (4)
where and\(\ t_{p}\ \)are the coefficient time average and standard deviation of GF, respectively; \(\psi\) is a constant;\(k_{f},\ t_{f},\ \delta_{f}\), and \(\psi_{f}\) are the parameters of the PGF before the inflection point;\(k_{b}\),\(\ t_{b}\),\(\delta_{b}\), and\(\ \psi_{b}\) are the parameters of the PGF after the inflection point.\(\ \)
FSF infiltration modelFourier (2009) proposed that any function can be expanded into an infinite series of trigonometric functions. In this paper, we presume that the IR \(i(t)\ \)has the following form:
\(i\left(t\right)=a_{0}+\sum_{i=1}^{n}\left(a_{i}\sin{(t\bullet\omega)}\operatorname{+cos}{(t\bullet\omega)}\right)\)(5)
where \(n\ \)is the order number; \(a_{i}\) is the FSF sinusoidal coefficient; \(b_{i}\) is the FSF cosine function coefficient; \(a_{0}\)is a constant; and \(\omega\) is the angular frequency.
GMF infiltration model
GMFs were derived by Euler to solve the problem of extending factorials to the set of real numbers (Davis, 1959). The PDF of a typical GMFs is shown in Fig. 3(b); the general formula of the PDF is as follows:
\(f\left(t\middle|s\right)=\left\{\par \begin{matrix}\frac{t^{s-1}e^{-t}}{\Gamma\left(s\right)}\ \ \ t\geq 0\\ \ \ \ \ \ 0\ \ \ \ \ \ \ t<0\\ \end{matrix}\right.\ \) (6)
Substituting \(t=\mu x\) into Eq. (6), the PDF of the GMF can be rewritten as:
\(f\left(t\middle|s,\mu\right)=\left\{\par \begin{matrix}\frac{{\mu^{s}t}^{s-1}e^{-\mu t}}{\Gamma\left(s\right)}\ \ \ t\geq 0\\ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ t<0\\ \end{matrix}\right.\ \) (7)
where \(f\left(t\middle|s,\mu\right)\) is a PDF; \(s\) is the shape parameter that controls the amplitude of the curve; and \(\mu\) is a scale parameter that controls the width of the curve.
The IR of W-RS gradually decreases in the initial stage, and then forms a single-peak curve that first increases and then decreases, that is, the IR is a U-shaped curve from the beginning to the peak (see Fig. 3(b)). To reflect this change, we introduce the critical time \(t_{0}\)into Eq. (7) to obtain the following modified formula for the IR\(\ i(t)\):
\(i\left(t\right)=\eta\frac{{\mu^{s}\left|t-t_{0}\right|}^{s-1}}{\Gamma\left(s\right)}e^{-\mu\left|t-t_{0}\right|}+\sigma\)(8)
where \(t_{0}\) is the time at which the IR reaches the bottom of the valley in the U-shaped curve (min); \(\eta\) is a model coefficient; and σ is a constant.
BF/PBF infiltration model
The Beta distribution refers to a set of continuous probability distributions defined in the interval (0,1). A random variable \(x\)obeys the Beta distribution with parameters \(p,q\) (Eugene et al., 2002). The BF (Gupta et al., 2004) in the interval\(\left(0,\infty\right)\) is given by:
\(B\left(p,q\right)=\int_{0}^{+\infty}\frac{x^{p-1}}{\left(1+x\right)^{p+q}}\text{dx}=\frac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(P+q\right)}\)(9)
The PDF of Eq. (9) is (Kipping, 2013):
\(f\left(t\middle|p,q\right)=\frac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(P+q\right)}t^{p-1}\left(1-t\right)^{q-1}\) (10)
and its PDF properties are shown in Fig. 3(c). We assign Eq. (10) the coefficient μ and the constant φ, and use it to calculate the IR \(i(t)\ \)of W-RS in the following form:
\(i\left(t\right)=\mu\frac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(P+q\right)}t^{p-1}\left(1-t\right)^{q-1}+\varphi\)(11)
Figure 3(c) shows that the PDF of the BF is in the range\(\left[0,1\right]\). Because water infiltration lasts a long time (\(t>1\)), it is necessary to normalize the infiltration time \(t\). We define \(t_{s}=\frac{t}{t_{e}}\) so that the IR\(i(t)\ \)of the BF and PBF can be written in normalized form as:
\(i\left(t_{s}\right)=\mu\frac{\Gamma\left(p\right)\Gamma\left(q\right)}{\Gamma\left(P+q\right)}{t_{s}}^{p-1}\left(1-t_{s}\right)^{q-1}+\varphi\)(12)
\(i\left(t_{s}\right)=\left\{\par \begin{matrix}\end{matrix}\right.\ \) (13)
where \(t_{e}\) is the end of the infiltration time (min); \(t_{s}\) is the normalization time; p and \(q\) are the shape and scale parameters; μ is the model coefficient; and \(\varphi\) is a constant.\(\ \mu_{f},p_{f},q_{f}\), and\(\text{\ φ}_{f}\) are the model parameters before the inflection point, \(\mu_{b}\), \(p_{b}\),\(q_{b}\), and \(\text{\ φ}_{b}\) are the parameters after the inflection point.