# A Simple Modelling Framework for Shallow Subsurface Water Storage and Flow

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## Abstract

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## 1. Introduction

## 2. Site Description and Data

## 3. Development of the Shallow Subsurface Modelling Framework

#### 3.1. Model Requirements

- Its results should be comparable to the Richards equation solution;
- It should be computationally simple and parsimonious, with a minimum number of calibrating parameters;
- It should be computationally flexible, allowing future accommodation of any type of soil or vegetation process, yet able to compute relatively short-term depth-averaged volumetric soil water content θ
_{d}(m^{3}/m^{3}) as well as soil water flux L_{d}(e.g., m/d) across a shallow depth ${z}_{r}$ (m).

#### 3.2. Model Conceptualization and Development

_{r}, θ

_{s}, n, l, K

_{s}and α were also obtained from the literature values provided in HYDRUS by the ROSETTA pedotransfer model [29], for six generic soil textures (a loam, a loamy sand, a sandy clay loam, a sandy loam, a silt loam and a silty clay loam), in addition to the CZO clay loam soil. This procedure is described in the Supplementary Material.

_{ls}(z = 0 m) to a relatively shallow depth, ${z}_{r}$ (m) so that the dominance of the top surface soil to hydrological behaviour can be adequately represented. Averaging over greater depths, as is common practice in hydrological modelling, would not simulate the strong depth dependent changes in water content observed at the Sunjia CZO with soil sensors.

_{e}(m

^{3}/m

^{3}), above which a negative surface water balance (P − E < 0) generates an upward flux across the boundary of the soil layer of interest at depth ${z}_{r}$. This process is a simple representation of a non-gravity driven flow process, triggered by a negative water balance at the surface, which may be important for relatively thin layers of soil. If the soil is near saturation in ${z}_{r}$, both upwards and downwards water flux across ${z}_{r}$ occurs, balanced between evaporative forcing and precipitation input into the system. The resulting total flux over a given integration time is therefore the sum of evaporative forcing and precipitation, with each having an opposite sign because of the flux direction. ${\theta}_{e}$ is also indirectly related to the soil water content in the soil below ${z}_{r}$, with hydraulic conductivity in both layers being positively correlated at any given time. To restrict the complexity of the SSMF to a minimum, the upwards flux was set to the surface evaporative forcing, and does not account for capillary forces that may play a role. The values of θ

_{e}are expected to be mostly a function of the soil texture.

^{3}/m

^{3}) is the soil water content of the previous day, as calculated in the following routine (Equation (8)), ${\theta}_{r}$ is the residual water content (m

^{3}/m

^{3}), ${\theta}_{s}$ is the saturated water content (m

^{3}/m

^{3}) and c is an empirical calibrated parameter (no units). This relationship conforms to Laio et al. [36], but does not include a plant transpiration component, where the evapotranspiration is related to the soil moisture thresholds that trigger stomatal closure. In addition, the empirical parameter, c, was added to the original linear equation presented by Laio et al. [36] to gain in generality, and to compensate for the fact that the parameter, E

_{w}, (evapotranspiration at the wilting point) from the original publication is omitted here for simplicity. Rather than using this method to obtain $E{T}_{d}$, the SSMF used Penman–Monteith to estimate $E{T}_{0,d}$.

^{3}/m

^{3}), as defined in Equation (4):

_{tot}(equal to 24 h in the case of the daily time step in this study). The vertical downwards flux across ${z}_{r}$ for this time step ${q}_{t,down}^{{z}_{r}}$ (m/d) was then calculated from the constitutive relationship between soil water content and hydraulic conductivity ${K}_{t}$ (m/d). In our study, we used the Mualem–van Genuchten relationship to evaluate ${K}_{t}$, with the Mualem approximation (m = 1 − 1/n) [37]. For the application to the Sunjia study site, we also considered a modified air-entry value of −0.02 m to represent conditions for the CZO clay loam. Such a small negative air entry value was shown to reduce the effect of non-linearity of the hydraulic conductivity function close to saturated conditions, while still preserving the S-shaped curve typical for the van Genuchten model [38]. This resulted in the relationship presented in Equation (6):

^{3}/m

^{3}) is the saturated water content, ${\theta}_{r}$ (m

^{3}/m

^{3}) is the residual water content, l (no units), and m (no units) are empirical parameters, and ${K}_{s}$ (m/d) is the saturated hydraulic conductivity.

_{r}. This exponential function was chosen to approach the theoretical water-retention relationship between the hydraulic conductivity and the matric potential. Following Equation (7), a low integral value ${{\displaystyle \int}}_{k=0}^{k=t-1}{q}_{k,down}^{{z}_{r}}dk$ of the previous vertical fluxes, which indicates very dry conditions in the soil below ${z}_{r}$, results in an exponentially-related small vertical flux ${q}_{t,down}^{{z}_{r}}$. In this sense, a can be viewed as a proxy parameter for non-unit hydraulic gradient conditions.

_{end}) and the sum of the downwards flux across ${z}_{r}$ over the sub-daily routine, respectively, calculated as:

_{tot}was reached, the excess water constituted the overland flow ${R}_{d}$ (m/d),

#### 3.3. Model Evaluation

_{d},

_{HYDRUS}and soil water flux dynamics L

_{d,HYDRUS}(m/d) at eight depths of ${z}_{r}$: 0.10, 0.15 0.20, 0.25, 0.30, 0.40, 0.50 and 0.70 m, which were chosen so that the upper soil was more finely represented than the lower soil, where the assumption of a unit gradient is more likely to be valid. Since the CZO clay loam properties are vertically heterogeneous (see Supplementary Material), a weighted arithmetic average as proposed by Destouni [39] was calculated for the calibrated parameters. These averaged soil property parameters were then used as inputs to the SSMF, whereby the initial depth-averaged value over ${z}_{r}$ of the soil water content was ${\theta}_{0}={\theta}_{d=1,HYDRUS}$. Next, the SSMF was calibrated against the soil water content θ

_{d},

_{HYDRUS}through a Monte–Carlo simulation (10,000 runs) and its results L

_{d,mod}were compared with L

_{d,HYDRUS.}Figure 2 provides a summary of this procedure and Table 1 provides the input soil parameters θ

_{r}, θ

_{s}, n, l, K

_{s}, α and SSMF parameter calibration ranges for dt

_{dry}, dt

_{sat}and ${\theta}_{e}$ (as a value relative to the saturated water content ${\theta}_{s}$, a and c.

_{dry}and dt

_{sat}, the SSMF was computed with a constant hourly timestep (dt

_{t}= dt = 1 h, Equation (5)).

_{d}could be implemented, if those fluxes were the focus of interest.

## 4. Results

#### 4.1. Evaluation of SSMF against Data and HYDRUS Simulations

#### 4.2. SSMF Parameters Influence

_{r}= 0.30 m. Whilst the impact during the wet period was minimal, altering any of these parameters had a significant influence during the long dry period (July to November). During other times, ${\theta}_{d}$ remained high (around 50% higher than when ${\theta}_{e}$ was included in the SSMF) while a resulted in relatively faster and stronger drying of the soil between rainfall events. The SSMF set with a constant dt of 1 h performed similarly regarding the representation of ${\theta}_{d}$ dynamics to when dt was variable in time, within the dt

_{dry}; dt

_{sat}range. The influences of the parameters on ${R}_{d}$(Figure 4b) and ${L}_{d}$ (Figure 4d) were not as marked as on ${\theta}_{d},$ and all setups performed relatively well, with their values and time dynamics remaining close to the values modelled by HYDRUS.

_{r}(from NRMSD = 0.09 at z

_{r}= 0.10 m to NRMSD = 0.06 at z

_{r}= 0.7 m, across all soil profiles), except in the CZO clay loam (CL), where the values of NRMSD slightly increased at 0.30 m. More specifically, where${\theta}_{e}$ was set to 0, the NRMSD was consistently poorer, leading to worse performance across all soils and depths (NRMSD = 0.12 on average). If the hydraulic gradient was assumed to be uniform, a = 0 or dt fixed to an 1 h interval, a better NRMSD (NRMSD = 0.08 on average) was observed, almost as good as the complete SSMF (NRMSD = 0.07). Generally, the complete SSMF performed best over individual z

_{r}regardless of soil, particularly as z

_{r}increased.

_{r}. For ${L}_{d}$ and up to 0.3 m, the setup where a = 0 performed better (NRMSD = 0.03 on average) than the SSMF overall (NRMSD = 0.04 on average). For the deeper depths, the SSMF and the setups where dt fixed to an 1 h interval or with a = 0 performed slightly better (NRMSD = 0.06 on average) than the setup with ${\theta}_{e}$ = 0 (NRMSD = 0.07 on average).

## 5. Discussion

#### 5.1. General Performance of the SSMF

_{r}. This is consistent with changes in the validity of the underlying assumptions of gravity driven flow and unit hydraulic gradient, which hold better with increasing depth and thickness of a studied soil layer, making the simulations with the SSMF less sensitive towards its specific parameters. That highlights the benefits of the SSMF, particularly in shallow soils. For the CZO clay loam, performance of the SSMF slightly decreased at 0.30 m, and increased again for deeper depths. That reflects the change in soil properties in the CZO clay loam at around 0.3 m, driven by its long-term agricultural management (Table SM2 in the Supplementary Material). Therefore, in the presence of such vertical sharp heterogeneity, the SSMF can be expected to perform less well at said vertical interface. This is a direct result of the proxy parameter a being calibrated with a set of soil hydraulic parameters encountered in above z

_{r}and that may not represent accurately the properties lower than z

_{r}.

#### 5.2. Evaluation of the Influence of the SSMF Parameters

_{r}is therefore overestimated, so ${\theta}_{d}$ remains relatively high. This impact was greater than using a fixed time step or setting a to 0 (Figure 7), highlighting the value of the ${\theta}_{e}$ parameter.

_{dry}, dt

_{sat}, and a had a more marginal influence than ${\theta}_{e}$ on the overall performance for ${\theta}_{d}$ predictions. Their influence was noticeable when daily dynamics were examined. For the CZO clay loam at 0.30 m presented in this study (Figure 6) a representative pattern of strong and fast decreases in soil water contents resulted when a = 0 and dt was set to a 1-h interval. As stated previously, equating the exponential term in Equation (7) to 1 (a = 0), neglects the conditions in the soil below z

_{r}(Equation (7)), so the vertical flux equals the hydraulic conductivity corresponding to the conditions in the upper zone z

_{r}(unit-hydraulic gradient condition)

_{.}This causes the differential in soil water content between the upper zone and the soil below z

_{r}to be large and positive, producing wetter conditions in the upper soil than in the soil below z

_{r}when rain infiltrates through the soil surface. As the simulated vertical flux is overestimated, water subsequently drains too rapidly from the upper zone. A constant value of dt had a similar effect on the water content dynamics as this also resulted in an overestimation of the vertical flux through z

_{r}(Equation (7)) for wet water contents.

#### 5.3. Overland Flow

#### 5.4. The SSMF Concept Application in the Context of Soil Hydraulics Modelling

_{dry}and dt

_{sat}, a, and ${\theta}_{e}$) together lead in almost all the soil profiles and all values of z

_{r}with the best performance, ${\theta}_{e}$ remained the most critical. A relatively short but constant value of dt could suffice to obtain overall good results when the absolute daily values are not of primary importance. Finally, we have provided an approach which is computationally flexible. The SSMF Matlab implementation is provided in the Supplementary Material. It is light and flexible, and its set of equations, presented in this study, can easily be included in any numerical model. The running time is also very fast (two orders of magnitude faster than HYDRUS 2D). In addition, while more simple than other approaches like HYDRUS, the SSMF is parameterised using measurable physical properties, so that their dynamic nature, and the role these play on hydrological processes, although not specifically addressed here, can still be considered.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic representation of the general study methodology; the goal of the Shallow Subsurface Modelling Framework (SSMF) is to use widely available input data of soil water content θ

_{d}; i.e., soil water retention properties and climate variables (precipitation P

_{d}and evapotranspiration ET

_{d}), to predict future soil water content, θ

_{d,mod}, and the dynamics of water fluxes, L

_{d,SSMF}. To assess the performance of the SSMF, we compared the model outputs with Richards equation solution values, θ

_{d,HYDRUS}and L

_{d,HYDRUS}.

**Figure 3.**Model conceptualization of the Shallow Subsurface Modelling Framework (SSMF). The fluxes are P, the precipitation at the land surface, E, the evaporation from the soil, R the overland flow and L, the vertical losses through the bottom boundary of ${z}_{r}$, the upper zone of the soil. θ is the depth-averaged volumetric water content over ${z}_{r}$ and SHP stands for Soil Hydraulic Properties over ${z}_{r}$. The vertical axis is positively oriented from the surface down the soil profile. The values in blue are the inputs to the SSMF; the values in green are the outputs.

**Figure 4.**Daily observed precipitation (

**a**), soil water content ${\theta}_{d}$ (

**b**), overland flow ${R}_{d}$ (

**c**), and vertical flux ${L}_{d}$ (

**d**) in the CZO clay loam profile of the Sunjia catchment at z

_{r}= 0.40 m, from November 2012 until December 2013. In (

**b**–

**d**), the time series are obtained from the calibrated HYDRUS software (red) and from the Shallow Subsurface Modelling Framework (SSMF, blue). For ${\theta}_{d}$ (

**b**) and ${R}_{d}$ (

**c**), the data range and its average value across the 12 sensors are represented respectively by a green shaded area and a green line.

**Figure 5.**Scatter plots of the soil water content ${\theta}_{d}$ values [m

^{3}/m

^{3}] from the Shallow Subsurface Modelling Framework (SSMF) (y-axis) and from HYDRUS (x-axis), across the eight studied depths ${z}_{r}$ (rows), and for the seven soil profiles (columns): CZO clay loam (CL), loam (L), loamy sand (LSa), sandy clay loam (SaCL), sandy loam (SaL), silt loam (SiL) and silty clay loam (SiCL). The black line represents the 1:1 line.

**Figure 6.**Influence of ignoring ${\theta}_{e}$ or a, or setting dt to a relatively fast 1 h interval on SSMF performance versus HYDRUS. The outputs are (

**a**) daily observed precipitation, (

**b**) overland flow ${R}_{d}$, (

**c**) soil water content ${\theta}_{d}$ and (

**d**) vertical flux ${L}_{d}$ for the CZO clay at z

_{r}= 0.30 m. In (

**b**–

**d**), the time series are obtained from the calibrated HYDRUS software (red) and from the Shallow Subsurface Modelling Framework (SSMF, blue), as well as for the SSMF, for which one of those parameters was ignored: ${\theta}_{e}$ (pink line), a (black line), and $d{t}_{dry}$ and $d{t}_{sat}$ (green line).

**Figure 7.**Normalised root mean square deviation (NRMSD) between SSMF and HYDRUS for the daily soil water content ${\theta}_{d}$ (left column) and for the daily vertical flux ${L}_{d}$ (right column). Simulations are for the eight soil texture profiles: CZO clay loam (CL), loam (L), loamy sand (LSa), sandy clay loam (SaCL), sandy loam (SaL), silt loam (SiL) and silty clay loam (SiCL). The values of the studied depths (z

_{r}) are on the x-axis; the NRMSD values on the y-axis. The NRMSD values are presented for the simulations from the SSMF (blue crosses), and for the setups where the individual parameters were omitted: ${\theta}_{e}=0$ (pink circles), a = 0 (black squares) and dt = 1 h (green triangles).

**Table 1.**Input soil parameter values and the range of the calibration parameters used in the Shallow Subsurface Modelling Framework (SSMF). The procedure to determine the value of the soil parameters is described in the Supplementary Material. For the CZO clay loam profile, the soil parameters θ

_{r}, θ

_{s}, n, l, K

_{s}and α reported in this table were obtained from the calibration of the Richards equation with HYDRUS on soil data from the Sunjia catchment. For the six generic soil texture profiles, they are the literature values provided in HYDRUS by the ROSETTA pedotransfer model [29]. For all soil profiles, dt

_{dry}, dt

_{sat}a, θ

_{e}and c are the SSMF parameters. The ranges that were used for the calibration of the SSMF are given in the table.

${\mathit{\theta}}_{\mathit{r}}\text{}({\mathbf{m}}^{3}/{\mathbf{m}}^{3})$ | ${\mathit{\theta}}_{\mathit{s}}\text{}({\mathbf{m}}^{3}/{\mathbf{m}}^{3})$ | $\mathit{n}$ (-) | $\mathit{l}$ (-) | ${\mathit{K}}_{\mathit{s}}\text{}(\mathbf{m}/\mathbf{d})$ | α (1/m) | $\mathit{d}{\mathit{t}}_{\mathit{d}\mathit{r}\mathit{y}}\text{}\left(\mathbf{hr}\right)$ | $\mathit{d}{\mathit{t}}_{\mathit{s}\mathit{a}\mathit{t}}\text{}\left(\mathbf{hr}\right)$ | $\mathit{a}$ (-) | $\mathit{c}$ (-) | ${\mathit{\theta}}_{\mathit{e}}/{\mathit{\theta}}_{\mathit{s}}$ (-) | |
---|---|---|---|---|---|---|---|---|---|---|---|

CZO clay loam 0–0.3 m | 0.050 | 0.390 | 1.256 | 0.5 | 0.035 | 0.05 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

CZO clay loam 0.3–3 m | 0.050 | 0.385 | 1.438 | 0.5 | 0.029 | 0.086 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Loam 0–3 m | 0.078 | 0.430 | 1.56 | 0.5 | 0.250 | 3.6 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Loamy sand 0–3 m | 0.035 | 0.437 | 1.5 | 0.5 | 1.466 | 4.85 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Sandy clay loam 0–3 m | 0.1 | 0.39 | 1.48 | 0.5 | 0.314 | 5.9 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Sandy loam 0–3 m | 0.041 | 0.453 | 1.378 | 0.5 | 0.621 | 3 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Silt loam 0–3 m | 0.067 | 0.45 | 1.41 | 0.5 | 0.108 | 2 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

Silty clay loam 0–3 m | 0.089 | 0.430 | 1.23 | 0.5 | 1.466 | 1 | [0;24] | [0;$d{t}_{dry}$] | [0;100] | [0;10] | [0,1] |

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## Share and Cite

**MDPI and ACS Style**

Verrot, L.; Geris, J.; Gao, L.; Peng, X.; Oyesiku-Blakemore, J.; Smith, J.U.; Hodson, M.E.; Zhang, G.; Hallett, P.D.
A Simple Modelling Framework for Shallow Subsurface Water Storage and Flow. *Water* **2019**, *11*, 1725.
https://doi.org/10.3390/w11081725

**AMA Style**

Verrot L, Geris J, Gao L, Peng X, Oyesiku-Blakemore J, Smith JU, Hodson ME, Zhang G, Hallett PD.
A Simple Modelling Framework for Shallow Subsurface Water Storage and Flow. *Water*. 2019; 11(8):1725.
https://doi.org/10.3390/w11081725

**Chicago/Turabian Style**

Verrot, Lucile, Josie Geris, Lei Gao, Xinhua Peng, Joseph Oyesiku-Blakemore, Jo U. Smith, Mark E. Hodson, Ganlin Zhang, and Paul D. Hallett.
2019. "A Simple Modelling Framework for Shallow Subsurface Water Storage and Flow" *Water* 11, no. 8: 1725.
https://doi.org/10.3390/w11081725