Infiltration

At the onset of rainfall or snow melt, we calculate event-specific parameters (e.g. soil moisture deficit \(\Delta \theta\)). For each event, we use two wetting fronts (\(wf1\) and \(wf2\)). The second wetting front is active after a rainfall pause (i.e. calculation of event-specific parameters of \(wf2\)). \(wf2\) is active while wetting front depth of \(wf2\) is less than wetting front depth of \(wf1\). In the following, the equations are applied to a dual-wetting front approach.

Total infiltration \(INF\) at time step \(t\) (mm \(\Delta t^{-1}\)):

\[INF=INF_{mat }+INF_{mp}+INF_{sc}\]

Matrix infiltration

Determining interval (\(i_s\)) when rainfall exceeds infiltrability within the current event:

\[\begin{split}i_s=\left\{\begin{array}{lr} no value & (PREC(i)-k_s \cdot \Delta t) \cdot \sum_{i=1}^i PREC(i) \leq k_s \cdot \Delta t \cdot \Delta \theta \cdot \psi_f \\ i & (PREC(i)-k_s \cdot \Delta t \cdot \sum_{i=1}^i PREC(i)>k_s \cdot \Delta t \cdot \Delta \theta \cdot \psi_f \end{array}\right.\end{split}\]

where \(PREC\) is precipitation (mm \(\Delta t^{-1}\)), \(k_s\) is the saturated hydraulic conductivity of the soil matrix, \(\Delta \theta\) is the soil moisture deficit (-) and \(\psi_f\) is the wetting front suction (mm)

Threshold rainfall intensity \(PREC_{gr}\) (mm \(\Delta t^{-1}\)):

\[PREC_{gr}=k_s \cdot \Delta t \cdot\left(\frac{\Delta \theta \cdot \psi_f}{\sum_{i=1}^{i_s-1} PREC(i)}+1\right)\]

\[\begin{split}i_s=\left\{\begin{array}{lr} (i_s - 1) \cdot \Delta t & PREC(i_s) < PREC_{gr} \\ (i_s-1) \cdot \Delta t+\frac{k_s \cdot \Delta t \cdot \Delta \theta \cdot \psi_f}{PREC(i_s) \cdot(PREC(i_s)-k_s \cdot \Delta t)}\frac{\Delta t}{PREC(i_s)} \sum_{v=1}^{i_s-1} PREC_v & PREC(i_s) \geq PREC_{gr}\end{array}\right.\end{split}\]

Infiltration at time step of saturation \(F_s\) (mm \(\Delta t^{-1}\)):

\[F_s=\frac{k_s \cdot \Delta t \cdot \theta_d \cdot \psi_f}{PREC(i_s)-k_s \cdot \Delta t}\]

Matrix infiltration \(INF_{mat}\) at time step \(t\) (mm \(\Delta t^{-1}\)):

\[\begin{split}INF_{mat}=\left\{\begin{array}{lr} z_0 \cdot & z_0 \leq INF_{mp-pot} \\ INF_{mat-pot} & z_0 > INF_{mp-pot} \\ \end{array}\right.\end{split}\]

where \(z_0\) is the surface ponding (mm; i.e. residual rainfall after interception or snow melt).

with potential matrix infiltration at time step \(t\) \(INF_{mat-pot}\) (mm \(\Delta t^{-1}\)):

\[\begin{split}INF_{mat-pot}=\left\{\begin{array}{lr} PREC(t) & t_s \geq t \\ PREC(t) \cdot(t_s-t-\Delta t)+\frac{k_s}{2}(1+\frac{1+\frac{2B}{A}}{\sqrt{1+\frac{4B}{A}+\frac{4 F_s^2}{A^2}}}) & t - \Delta t<t_S<t \\ \frac{k_s}{2}(1+\frac{1+\frac{2 B}{A}}{\sqrt{1+\frac{4 B}{A}+\frac{4 F_s^2}{A^2}}}) & t_s < t \end{array}\right.\end{split}\]

with auxiliary variables:

\[A=K_S \cdot\left(t-t_s\right)\]

\[B=F_s+2 \cdot \Delta \theta \cdot \psi_f\]

Wetting front depth \(z_{wf}\) (mm):

\[z_{wf}=\frac{\sum_i^i INF_{mat}(i)}{\Delta \theta}\]

where \(i_e\) is interval of the event start.

Macropore infiltration

Macropore infiltration \(INF_{mp}\) at time step \(t\) (mm \(\Delta t^{-1}\); Weiler, 2005):

\[\begin{split}INF_{mp}=\left\{\begin{array}{lr} z_0 \cdot (1 - e^{-(\frac{\rho_{mpv}}{82})^{0.887}}) & 0 < z_0 \cdot (1 - e^{-(\frac{\rho_{mpv}}{82})^{0.887}}) \leq INF_{mp-pot} \\ INF_{mp-pot} & z_0 \cdot (1 - e^{-(\frac{\rho_{mpv}}{82})^{0.887}}) > INF_{mp-pot} \\ \end{array}\right.\end{split}\]

where \(z_0\) is the surface ponding (mm; i.e. matrix infiltration excess).

with potential macropore infiltration \(INF_{mp-pot}\) at time step \(t\) (mm \(\Delta t^{-1}\))

\[INF_{mp-pot}=\pi \cdot(y_{mp}(t)^2-y_{mp}(t-\Delta t)^2) \cdot \rho_{mpv} \cdot \frac{\Delta z_{mp} \cdot \Delta \theta}{\Delta t}\]

where \(\rho_{mpv}\) is density of vertical macropores (\(m^2\)) and \(\Delta z_{mp}\) depth of non-saturated macropore (mm)

Radial distance of the macropore wetting front \(y_{mp}\) (mm):

\[y_{mp}=\frac{1}{2} \cdot \frac{b^{(1 / 3)}}{\Delta \theta}+\frac{1}{2} \cdot \frac{a}{b^{(1 / 3)}}+\frac{1}{2} \cdot r_{mp}\]

\[a=\Delta \theta \cdot r_{mp}^2\]

\[b=r \cdot \Delta \theta \cdot(12 c-a+2 \sqrt{6} \cdot \sqrt{c \cdot(6 c-a)}\]

\[c=t_{mp} \cdot k_s \cdot \psi_s\]

Duration of macropore infiltration \(t_{mp}\) (\(y_{mp}=r_{mp}\) at time t=0)

\[t_{mp}=\frac{\Delta \theta}{k_s \cdot \Psi_s \cdot r_{mp}} \cdot(\frac{y_{mp}^3}{3}-\frac{y_{mp}^2 r}{2}-\frac{r_{mp}^3}{6})\]

where \(r_{mp}\) is the radius of the macropore (mm; \(r_{mp}`=2.5). Macropore infiltration stops if :math:`z_{wf}\) is greater than \(l_{mpv}\).

Shrinkage crack infiltration

Shrinkage crack infiltration \(INF_{cs}\) at time step \(t\) (mm \(\Delta t^{-1}\); Steinbrich et al., 2016):

\[\begin{split}INF_{sc}=\left\{\begin{array}{lr} z_0 & z_0 \leq INF_{sc-pot} \\ INF_{sc-pot} & z_0 > INF_{sc-pot} \\ \end{array}\right.\end{split}\]

where \(z_0\) is the surface ponding (mm; i.e. macropore infiltration excess).

Potential shrinkage crack infiltration \(INF_{sc-pot}\) at time step \(t\) (mm \(\Delta t^{-1}\); Steinbrich et al., 2016):

\[INF_{sc-pot}=2 \cdot l_{sc} \cdot(y_{sc}(t)-y_{sc}(t-\Delta t)) \cdot \frac{\Delta z_{sc} \cdot \Delta \theta}{\Delta t}\]

where \(l_{sc}\) is the horizontal length of shrinkage cracks (mm \(m^{-2}\)) and \(\Delta z_{sc}\) is the depth of non-saturated shrinkage crack (mm)

Horizontal distance of the shrinkage crack wetting front \(y_{sc}\) (mm):

\[y_{sc}(t)=\sqrt{\frac{2 \cdot k_s \cdot \Psi_s \cdot t_{sc}}{\Delta \theta}}\]

\[t_{sc}=\frac{y_{sc}(t-\Delta t)^2 \cdot \Delta \theta}{2 \cdot k_s \cdot \psi_s}\]

Calculation of depth of shrinkage cracks \(z_{sc}\) at beginning of event:

\[\begin{split}z_{sc} = \begin{cases} 700 \cdot clay & \theta_{rz} < \theta_{4} \\ 700 \cdot clay \cdot (1 - \frac{\theta_{rz}}{\theta_{27} - \theta_{4}}) & \theta_{4} \leq \theta_{rz} \leq \theta_{27} \\ 0 & \theta_{rz} > \theta_{27} \end{cases}\end{split}\]

with clay content of soil \(clay\) (-)

\[clay=\frac{clay_{max} \cdot (\theta_6 - clay_{min})}{0.3}\]

where \(clay_{min}\) is the lower limit of clay content (-; \(clay_{min}`=0.01) and :math:`clay_{max}\) is the upper limit of clay content (-; \(clay_{max}`=0.71). :math:`INF_{sc}\) occurs only if shrinkage cracks are available and stops if \(z_{wf}\) is greater than \(z_{sc}\).