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 k 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.