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.