Model applicability
We now analyze the conditions under which each of the models should be employed. For W-RS, if the initial infiltration process (stage I of Fig. 7 ) has a short duration, and the stable IR before the inflection point (stage II of Fig. 7 ) is equal to that after the inflection point (stage IV of Fig. 7 ), the IR is a symmetrical bell-shaped curve (ignoring the initial short-duration infiltration), so the infiltration process can be modeled by GF. When the short duration of the initial infiltration process is considered, and the stable IR before the inflection point is equal to after the inflection point, GMF produces better simulation results. When the BPF is used to fit the IR of W-RS, the process can be divided into three patterns. Model 1: a monotonous decrease in infiltration in the first part (stages I and II of Fig. 7 ) with \(0<p\leq 1\) and\(q>1\), and the right-skewed distribution curve infiltration process in the second part (stages III, IV, and V ), with\(1<q<p\). Model 2: U-shaped infiltration in the initial process (stages I, II, and III ) with \(0<p<1\),\(0<q<1\). The initial IR is greater than that at the inflection point when \(0<p<q<1\), and less than that at the inflection point when \(0<q<p<1\). IR decreases monotonously in the second part (stages IV and V ) with \(0<p\leq 1\) and \(q>1\).Model 3: IR with a left-skewed distribution curve when\(p>q>1\), and a right-skewed distribution curve when \(1<p<q\)under the condition that the IR in stage I can be neglected. In this study, the IR is calculated by model 3 with the BF and a left-skewed distribution when \(\theta\)=4.7%, and by model 2 with the PBF when \(\theta\)=6.2% and\(\theta\)=9.6%.
The relationship between the mathematical model proposed in this paper and the traditional water infiltration model is as follows:
For the GMF water infiltration model given by Eq. (8), we define\(t_{0}=0\),\(\ s=1\). As\(\Gamma\left(1\right)=1,\left|t-t_{0}\right|^{1-1}=1,\mu^{1}=\mu\), Eq. (8) degenerates into the following mathematical model:
\(i\left(t\right)=\text{ημ}e^{-\mu t}+\sigma\) (14)
This is similar to Horton’s infiltration model. The coefficients\(\ \eta,\mu\) in Eq. (14) are equivalent to\(i_{c}-i_{0}\), and \(\sigma\) is equivalent to \(i_{0}\) (Beven, 2004; VERMA, 1982). Note that \(i_{c}\) is the maximum IR in the initial stages of Horton’s infiltration model, whereas μ influences the decrease in the infiltration curve and \(\eta\) influences the overall size of the IR in Eq. (14). We consider Eq. (14) to be a variation of Horton’s model. When \(t>0\), IR is a monotonically decreasing function.
When \(t_{0}=0\) and \(\mu=1\), Eq. (8) degenerates into a complex exponential function:
\(i\left(t\right)=\eta\frac{t^{s-1}}{\Gamma\left(s\right)}e^{-t}+\sigma\)(15)
Numerical simulations show that when \(s>4.907\), Eq. (15) is a monotonically decreasing function; for \(0<s<4.907\), Eq. (15) is a single-peak function.
For the BF water infiltration model in Eq. (12), we define \(P=1\). Then,\(\ \Gamma\left(1\right)=1\), and so Eq. (12) can be rewritten as:
\(i\left(t\right)=\mu\frac{\Gamma\left(q\right)}{\Gamma\left(1+q\right)}\left(1-t_{S}\right)^{q-1}+\varphi\)(16)
Because \(\Gamma\left(1+q\right)=q\Gamma\left(q\right)\), this can be expressed as follows:
\(i\left(t\right)=\frac{\mu}{q}\left(1-t_{S}\right)^{q-1}+\varphi\)(17)
When \(q>1\), Eq. (17) is a monotonically increasing function, and when \(0<q<1\), Eq. (17) is a descending function and \(i\left(t\right)>0\). If \(q<0\), then Eq. (17) is a descending function and \(i\left(t\right)<0\).
In other situations, we define \(q=1\) so that\(\text{\ Γ}\left(1\right)=1\) and\(\left(1-t_{S}\right)^{q-1}=1\). Then, Eq. (12) can be rewritten as follows:
\(i\left(t\right)=\mu\frac{\Gamma\left(p\right)}{\Gamma\left(p+1\right)}{t_{S}}^{p-1}+\varphi\)(18)
As above, \(\Gamma\left(p+1\right)=p\Gamma\left(p\right)\), so this can be expressed as:
\(i\left(t\right)=\ \frac{\mu}{p}t^{p-1}+\varphi\) (19)
Equation (19) is similar to Kostiakov’s infiltration model, and the coefficient \(\mu/p\) is equivalent to Kostiakov’s model coefficient (Kostiakov, 1932; Parhi et al., 2007), that is, \(\mu/p=an\). When 0<p<1, Eq. (19) is a monotonically decreasing function, and when p>1, Eq. (19) is a monotonically increasing function; when p=1, Eq. (19) is a constant \(\varphi\).
For \(p=0.5\), Eq. (19) can be rewritten as follows:
\(i\left(t\right)={2\mu t}^{-0.5}\ +\varphi\) (20)
This is similar to Philip’s water infiltration model. The difference is that the coefficient in Philip’s model is 0.5S (Philip, 1957), whereas the coefficient of Eq. (20) is \(2\mu\), which is also a monotonically decreasing function.
From the above, we can assume that Horton’s infiltration model is a special case of Eq. (10), whereas the Kostiakov and Philip infiltration models are special cases of Eq. (20). The mathematical GMF and BF models have an extremely wide range of applications.
CONCLUSION
Unlike HS, in which the CI increases monotonously with time and the IR decreases monotonously, W-RS exhibits the following characteristics: (1) A two-stage CI with an overall growth phenomenon and an IR with a mutation phenomenon. Larger values of ISWC produce an earlier inflection point in the CI and larger values of IR at the inflection point. (2) IR has a single peak when the initial infiltration process (stage I ) is neglected, and the stable post-peak IR is higher than the pre-peak value.
The applicability of KF, PKF, GF, PGF, FSF, GMF, BF, and PBF models was analyzed using HS and W-RS samples. The KF, GMF, and BF functions were found to be suitable for HS. For W-RS, the GMF function not only reflects the monotonous decrease in infiltration (\(s\leq 1\)), but also recreates the complete infiltration process, namely the gradual decline in IR in the initial stage, gradual increase before the inflection point, subsequent gradual decrease after the inflection point, and final stable infiltration (\(s>1\)). The BF model reflects the monotonous decrease process in HS (\(0<p\leq 1\) and \(q>1\)). PBF gives a U-shaped IR that decreases gradually from the initial point and then gradually increases before the inflection point, and also reflects the IR with a right-skewed distribution curve about the left and right inflection points. Therefore, the BF/PBF model offers the better simulation accuracy and has the widest applicability.