# Modeling Travel Time Distributions of Preferential Subsurface Runoff, Deep Percolation and Transpiration at A Montane Forest Hillslope Site

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Site

^{2}with an average altitude of 820 m above sea level. The climate is cool and humid with mean annual precipitation 1380 mm, and mean annual temperature 4.7 °C. Catchment hillslopes are covered with young spruce forest (Picea abies). Soils on the catchment hillsides are mostly Cryptopodzols overlying weathered bedrock, which turns into compact porphyritic biotite granite bedrock at a depth of 5–10 m [43]. The soil profile at the experimental site is approximately 70 cm deep consisting of three soil layers. The upper boundary of weathered bedrock is further on referred to as the soil–bedrock interface. The average slope of the soil surface at the Tomsovka site is 14%.

^{2}. Each of the two trench sections drains about half of the hillslope area—further referred to as the trench section microcatchment (Figure 1a). Subsurface hillslope discharge is collected separately for each trench section at a depth of 75 cm. The respective discharge rates Q

_{A}and Q

_{B}are measured by tipping bucket gauges. Details about the Tomsovka site instrumentation and measurement protocols can be found in Sanda and Cislerova [43].

_{A}(t) and Q

_{B}(t) can be attributed to the unaccounted-for dissimilarities in geometric, material, and vegetation properties of the two trench section microcatchments. Based on our previous research, the spatial variability of preferential pathways and their lateral connectivity are the most probable causes of the observed differences.

#### 2.2. Mean Residence Time

_{r}is the mean water storage (m

^{3}), and Q is the mean discharge (m

^{3}s

^{−1}).

#### 2.3. Travel Time Distributions

_{in}is the tracer mass inflow rate (kg s

^{−1}), J

_{out}is the tracer mass outflow rate (kg s

^{−1}), M is the total applied mass (kg), δ(t) is the Dirac delta function (s

^{−1}), c is the tracer concentration (kg m

^{−3}), q is the soil water flux (m s

^{−1}) perpendicular to the boundary, t is time (s),

**x**is the vector of spatial coordinates (m), Ω

_{out}is the outflow boundary, and g is the travel time distribution function (s

^{−1}).

_{1}, g

_{2}, and g

_{3}are the partial travel time distributions of stormflow, deep percolation, and transpiration, respectively, M

_{1}, M

_{2}, and M

_{3}are the effluent masses (kg) associated with the respective discharge processes, and S is the sink term representing the local intensity of root water uptake (s

^{−1}). The overall (aggregate) travel time distribution g is composed of the three partial travel time distributions (for stormflow, deep percolation, and transpiration). Note that g as well as g

_{1}, g

_{2}, and g

_{3}, as defined in Equation (4), satisfy the basic condition on probability density functions—to integrate to unity.

_{1}and Ω

_{2}represent two-dimensional interfaces separating the hillslope soil from the experimental trench (the seepage face) and the deepest soil layer from the weathered bedrock (see Figure 1), while Ω

_{3}is the three-dimensional domain occupied by the root zone of the vegetation cover.

_{m}) can be evaluated as travel times corresponding to the tracer effluent mass of M/2. Specific median travel times can be determined for the three partial travel time distributions (g

_{1}, g

_{2}, and g

_{3}) as well as for the aggregate and master travel time distributions.

#### 2.4. Two-Dimensional Flow and Transport Model

^{−1}), h is the pressure head (m),

**K**is the hydraulic conductivity tensor (m s

^{−1}), z is the vertical coordinate (m) assumed positive upward, Γ

_{w}is the soil water transfer term describing the interdomain exchange of water (s

^{−1}), and w

_{f}and w

_{m}are the relative volumetric fraction of the preferential flow domain and the soil matrix domain, respectively.

^{3}m

^{−3}),

**q**is the vector of soil water flux (m s

^{−1}),

**D**is the hydrodynamic dispersion coefficient tensor (m

^{2}s

^{−1}), and Γ

_{s}is the solute transfer term describing interdomain tracer mass transfer (kg m

^{−3}s

^{−1}). The values of θ,

**q**, and

**D**are taken from the solution of Richards’ equations. Hydrodynamic dispersion coefficient tensor

**D**is a second rank tensor, composed of molecular diffusion and mechanical dispersion. The components of the tensor

**D**depend on the local magnitude and orientation of soil water fluxes and can be evaluated using the approach of Bear [51].

## 3. Model Application

#### 3.1. Model Representation of Stormflow, Deep Percolation, and Transpiration

#### 3.2. Geometric, Material and Boundary Conditions for the Soil Water Flow Model

_{f}was set to 7% at the soil surface and 5% at the depth of 70 cm, with a linear variation between the two endpoints.

_{x’x’}/K

_{z’z’}of the hydraulic conductivity tensor of the preferential flow domain in respect to principal directions x’ and z’ was set equal to ten. Although no direct determination of the lateral hydraulic conductivity was made, this assumption was confirmed by a sensitivity study of lateral conductivity, carried out using a diffusion wave model [59,60], and by a comparative study between runoff predictions obtained with the diffusion wave model and the two-dimensional model [40].

#### 3.3. Simulation Scenarios

^{2}d

^{−1}; this value represents the self-diffusion of water considered for transport of stable isotope O-18 in our previous studies. Longitudinal and transversal dispersivities of 20 cm and 5 cm, respectively, were applied [42].

#### 3.4. Episodal Simulations

^{3}, i.e., <9 mm of the equivalent runoff height), while the remaining six episodes (i.e., 1, 4, 5, 6, 7, and 9) were characterized by at least 2.9 m

^{3}(29 mm) stormflow volume (Figure 2). The largest stormflow volume was observed during Episode 9, when trench sections A and B showed hillslope discharge of 13.0 m

^{3}(130 mm) and 19.8 m

^{3}(198 mm), respectively.

_{1}sums to unity by the end of each episode. The travel time distributions g

_{2}and g

_{3}sum to unity much later as the respective discharge processes continue beyond the end of the event. Due to this, the episodal tracer masses M

_{2}and M

_{3}for deep percolation and transpiration were estimated based on the seasonal simulations. When calculating g

_{2}and g

_{3}, the tracer masses M

_{2}and M

_{3}in Equation (4) were determined by dividing the remaining tracer mass at the end of the episode (i.e., the amount M − M

_{1}) between deep percolation and transpiration in a ratio obtained from the respective seasonal mass balance.

#### 3.5. Seasonal Simulations

## 4. Results and Discussion

#### 4.1. Episodal Travel Time Distributions

_{1}peaks within 20% of stormflow ratio. Multi-peak rainfall loadings resulted in multi-peak travel time distributions. It is obvious that the flow-corrected travel time projection helped to reduce the variability of the travel time distributions (Figure 3b), allowing the estimation of the master travel time distribution for stormflow.

_{2}were below 0.5 at the end of all episodes except for Episode 1 (Figure 4b). As expected, the cumulative travel time distributions for transpiration (g

_{3}) remained lower than 1 at the end of all episodes (Figure 4c). Unlike for stormflow and deep percolation, no significant effect of the shape of the rainfall signal occurs in the case of transpiration. The aggregate cumulative travel time distributions, combining functions g

_{1}, g

_{2}, and g

_{3}according to Equation (4), are shown in Figure 4d. The shape of the aggregate travel time distributions indicates that the effect of episodal stormflow is less pronounced than the combined effects of the more continuous processes—deep percolation and transpiration.

_{m}for stormflow determined for the nine rainfall–runoff episodes. The t

_{m}values ranged from 1.4 to 17.2 days. The stormflow t

_{m}values exhibited great variability among the episodes, caused by the temporal variations of rainfall intensities as well as antecedent soil water content conditions. The t

_{m}values for transpiration and deep percolation as well as the t

_{m}values associated with the aggregate travel time distributions were not evaluated since the episodal simulations were discontinued at the end of episodes, before the cumulative travel time distribution for transpiration and deep percolation could reach the value of 0.5 (in most of the cases except g

_{2}in Episode 1).

_{m}values (<3 days). This was caused by the intense rainfall at the beginning of the episode, triggering the rapid initiation of both stormflow and deep percolation responses. Episode 5 was characterized by a subsurface runoff peak occurring 14 days after the tracer application, hence t

_{m}is relatively long. For five episodes (Episodes 2, 4, 6, 7, and 8), the advective component of transport dominated over the dispersive one. These episodes were characterized by rapid arrival times of the tracer to the hillslope trench, thus shorter median travel times for stormflow (<4 days). For the long-duration episodes with more complex rainfall and runoff patterns, the dispersive component caused enhanced tracer mixing in the hillslope profile leading to longer travel times.

_{m}values are strongly affected by temporal rainfall patterns and antecedent soil moisture distributions, in addition to overall hillslope storage and cumulative input of water. The episode-based stormflow t

_{m}values fell into two groups characterized by t

_{m}< 7 days and t

_{m}> 7 days. Longer values of t

_{m}were obtained for the episodes with a complex temporal rainfall distribution causing multi-peak stormflow responses. It can be seen that a greater subsurface storage is not uniquely associated with longer stormflow t

_{m}values (Figure 5).

_{m}for stormflow. Dusek and Vogel [41] demonstrated a nonlinear relationship between initial hillslope storage and stormflow. Furthermore, the initial distribution of soil water within the hillslope, specifically, the extent of soil saturation near the soil–bedrock interface, was a key factor in stormflow generation. Dusek and Vogel [41] reported that deep percolation increased with increasing initial hillslope storage. However, in the present study, highly variable t

_{m}values of stormflow were found for similar hillslope storages (Figure 5). This again signifies the importance of temporal rainfall patterns and initial soil moisture distributions.

#### 4.2. Seasonal Soil Water Balance, Residence Times, and Travel Time Distributions

_{m}, were found for different discharge processes (Table 5). The shortest median travel times were determined for transpiration. As expected, the stormflow t

_{m}values showed the greatest interseasonal variability compared to the other two discharge processes.

_{m}and T are very similar. The differences between t

_{m}and T values for the drier seasons 2007 and 2008 are greater.

#### 4.3. Tracer Mass Partitioning

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Two-dimensional flow domain representing the hillslope segment at the Tomsovka experimental site: (

**a**) schematics of the hillslope segment with the experimental trench for collecting subsurface hillslope discharge (

**b**) flow domain with the detail of the subsurface trench.

**Figure 2.**Observed (trench sections A and B) and simulated hillslope discharge hydrographs during selected rainfall–runoff episodes recorded over the period of three years. The rainfall–runoff episodes are marked with numbers. Adopted from Dusek and Vogel [42].

**Figure 3.**Episodal travel time distributions of stormflow for the selected rainfall–runoff episodes plotted against travel time since tracer application (

**a**) and against flow-corrected travel time obtained as the cumulative volume of stormflow divided by the final volume of rainfall (

**b**). The master travel time distribution is shown in the flow-corrected time projection.

**Figure 4.**Cumulative travel time distributions of stormflow (

**a**), deep percolation (

**b**), and transpiration (

**c**), as well as aggregate distributions combing all discharge processes (

**d**) for the selected rainfall–runoff episodes. Median travel times t

_{m}for stormflow correspond to the value of cumulative travel time distribution equal to 0.5.

**Figure 5.**The relationships between net water input (rainfall minus transpiration), hillslope storage and stormflow median travel times, t

_{m}, for the selected rainfall–runoff episodes. The circles are labeled with the episode numbers. The magnitude of the respective median travel time (cf. Table 2) is expressed by the circle diameter.

**Figure 6.**Cumulative travel time distributions of stormflow, deep percolation and transpiration, together with the aggregate travel time distributions for growing seasons 2007, 2008, and 2009.

**Table 1.**The soil hydraulic parameters † used for the two-dimensional dual-continuum model. SM and PF refer to the soil matrix and preferential flow domain, respectively.

Depth | θ_{r} | θ_{s} | α | n | K_{s} | h_{s} | |
---|---|---|---|---|---|---|---|

(cm) | (cm^{3} cm^{−3}) | (cm^{3} cm^{−3}) | (cm^{−1}) | (-) | (cm d^{−1}) | (cm) | |

SM | 0–8 | 0.20 | 0.55 | 0.050 | 2.00 | 567 | 0.00 |

8–20 | 0.20 | 0.54 | 0.050 | 1.50 | 67 | −0.69 | |

20–70 | 0.20 | 0.49 | 0.020 | 1.20 | 17 | −1.48 | |

70–75 | 0.20 | 0.41 | 0.020 | 1.20 | 1.3 | −1.88 | |

75–300 | 0.00 | 0.21 | 0.020 | 1.20 | 0.4 | −2.61 | |

PF | 0–70 | 0.01 | 0.60 | 0.050 | 3.00 | 5000 | 0.00 |

70–300 | 0.01 | 0.60 | 0.050 | 3.00 | 0.4 | 0.00 |

_{r}and θ

_{s}are the residual and saturated water contents, K

_{s}is the vertical saturated hydraulic conductivity, h

_{s}is the air-entry value, and α and n are empirical fitting parameters.

**Table 2.**Estimated median travel times t

_{m}of stormflow determined for the selected rainfall–runoff episodes.

Episode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Episode Duration (d) | 27 | 22 | 26 | 18 | 26 | 14 | 32 | 14 | 33 |

t_{m} (d) | 11.2 | 3.9 | 11.5 | 3.2 | 14.2 | 2.9 | 1.4 | 1.9 | 17.2 |

Season | 2007 | 2008 | 2009 |

Season Duration (d) | 171 | 191 | 191 |

Observed Rainfall (mm) | 634 | 724 | 874 |

Stormflow (mm) | 97 | 169 | 301 |

Deep Percolation (mm) | 76 | 125 | 182 |

Transpiration (mm) | 429 | 438 | 396 |

Initial Storage (mm) | 106 | 145 | 145 |

Final Storage (mm) | 138 | 137 | 140 |

**Table 4.**Mass fractions of tracer relative to applied mass. Net water input is rainfall minus actual transpiration.

Season | 2007 | 2008 | 2009 |

Net Water Input (mm) | 205 | 286 | 478 |

Stormflow (%) | 5.0 | 12.8 | 23.5 |

Deep Percolation (%) | 9.9 | 13.5 | 18.8 |

Transpiration (%) | 76.3 | 66.2 | 51.0 |

Residual (%) | 8.8 | 7.5 | 6.7 |

Season | 2007 | 2008 | 2009 |

Stormflow (d) | 120.9 | 78.5 | 33.5 |

Deep Percolation (d) | 95.8 | 75.5 | 35.4 |

Transpiration (d) | 26.4 | 35.4 | 23.5 |

Aggregate (d) | 30.4 | 46.2 | 30.1 |

**Table 6.**Mass fractions of tracer relative to applied mass at the end of rainfall–runoff episodes. Net water input is rainfall minus actual transpiration.

Episode | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Net Water Input (mm) | 121 | 28 | 130 | 91 | 102 | 80 | 77 | 36 | 260 |

Stormflow (%) | 14.9 | 7.2 | 13.6 | 8.7 | 14.5 | 16.4 | 18.8 | 9.4 | 32.5 |

Deep Percolation (%) | 7.0 | 2.5 | 8.9 | 5.1 | 6.0 | 3.5 | 6.3 | 2.5 | 11.6 |

Transpiration (%) | 13.7 | 9.2 | 18.0 | 15.1 | 6.6 | 9.4 | 21.0 | 12.6 | 4.4 |

Residual (%) | 64.4 | 81.2 | 59.5 | 71.0 | 72.8 | 70.7 | 53.9 | 75.5 | 51.5 |

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**MDPI and ACS Style**

Dusek, J.; Vogel, T.
Modeling Travel Time Distributions of Preferential Subsurface Runoff, Deep Percolation and Transpiration at A Montane Forest Hillslope Site. *Water* **2019**, *11*, 2396.
https://doi.org/10.3390/w11112396

**AMA Style**

Dusek J, Vogel T.
Modeling Travel Time Distributions of Preferential Subsurface Runoff, Deep Percolation and Transpiration at A Montane Forest Hillslope Site. *Water*. 2019; 11(11):2396.
https://doi.org/10.3390/w11112396

**Chicago/Turabian Style**

Dusek, Jaromir, and Tomas Vogel.
2019. "Modeling Travel Time Distributions of Preferential Subsurface Runoff, Deep Percolation and Transpiration at A Montane Forest Hillslope Site" *Water* 11, no. 11: 2396.
https://doi.org/10.3390/w11112396