# Robustness of a parsimonious subsurface drainage model at the French national scale

Alexis Jeantet, Hocine Henine, Cédric Chaumont, Lila Collet, Guillaume

Thirel, Julien Tournebize

► To cite this version:

Alexis Jeantet, Hocine Henine, Cédric Chaumont, Lila Collet, Guillaume Thirel, et al. Robustness of a parsimonious subsurface drainage model at the French national scale. Hydrology and Earth System Sciences, 2021, 25 (10), pp.5447 - 5471. 10.5194/hess-25-5447-2021. hal-03461421

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# Robustness of a parsimonious subsurface drainage model at the French national scale

Alexis Jeantet<fnref n="1" />, Hocine Henine<fnref n="1" />, Cédric Chaumont<fnref n="1" />, Lila Collet<fnref n="1" />,<fnref n="2" />, Guillaume Thirel<fnref n="1" />, and Julien Tournebize<fnref n="1" />

**Correspondence:** Alexis Jeantet (alexis.jeantet@inrae.fr)

Received: 23 March 2021 – Discussion started: 21 April 2021

Revised: 14 September 2021 – Accepted: 15 September 2021 – Published: 14 October 2021

**Abstract.** Drainage systems are currently implemented on agricultural plots subjected to temporary or permanent waterlogging issues. Drained plots account for 9 % of all arable soils in France. As such, the need for accurate hydrological modeling is crucial, especially in an unstable future context affected by climate change. The aim of this paper is to assess the capacity of the SIDRA-RU hydrological drainage model to represent the variability in pedoclimatic conditions within French metropolitan areas and to demonstrate the utility of this model as a long-term management tool. The model is initially calibrated using the KGE' criterion as an objective function (OF) on a large and unique database encompassing 22 plots spread across France and classified according to three main soil textures (silty, silty–clay, and clayey). The performance of SIDRA-RU is evaluated by monitoring both the set of KGE' calibration values and the quality of simulations on each plot with respect to high and low discharges, as well as the annual drained water balance. Next, the temporal robustness of the model is assessed by conducting, on selected plots, the split-sample test capable of satisfying the data requirements. Results show that the SIDRA-RU model accurately simulates drainage discharge, especially on silty soils. The performance on clayey soils is slightly weaker than that on silty soils yet remains acceptable. Similarly, the split-sample test indicates that SIDRA-RU is temporally robust on all three soil textures. Consequently, the SIDRA-RU model closely replicates the diversity of French drained soil and could be used for its long-term management potential.

## 1 Introduction

Subsurface drainage is an agricultural soil management technique that controls soil water content and increases aeration in soils subjected to temporary or permanent water saturation issues into the soil depth (Jamagne, 1968; Baize and Jabiol, 2011). Plot water conditions are stabilized, thus ensuring better crop yields (Broadhead and Skaggs, 1982; Armstrong et al., 1988; Nijland et al., 2005; Ibrahim et al., 2013), while reducing the flood risk on plots (Henine et al., 2014; Tuohy et al., 2018b). Drained soils often belong to the hydromorphic soil category and sometimes, in the French context, lie on a shallow and impervious layer that reduces the deep infiltration (Thompson et al., 1997; Lange et al., 2011).

In France, all artificially drained soils comprise more than 2.7 million ha of arable soils (RGA – Agreste, 2010), i.e., close to 10 % of all arable land, corresponding to about 20 % for cereal-type field crops. In practice, several techniques exist to drain soils, such as subsurface drainage and open ditch. However, in France, over 80 % of drainage practices are conducted by introducing perforated pipes lying on the impermeable layer. The drain depth, spacing, slope, and diameter of these pipes constitute the main characteristics of each design; they are constrained by the local study site conditions, such as soil characteristics and climate (Mulqueen, 1998).

Since the economic and environmental consequences of climate change are of increasing concern to stakeholders, proper drainage practices have become a major issue. Predicting the long-term behavior of these systems is even more crucial in this context of water resource protection and restoration, since drainage has an impact on water quality (Tournebize et al., 2012, 2017, 2020). The literature contain studies targeting the impact of climate change on drainage

<footnote n="1">¹INRAE, University of Paris Saclay, UR HYCAR, 1 rue Pierre Gilles de Gennes, Antony, France</footnote>

<footnote n="2">²now at: EDF R&D, OSIRIS Department, 7 boulevard Gaspard Monge, 91120 Palaiseau, France</footnote>
practices, with an emphasis on either the increase in annual drained water balance (Pease et al., 2017) or agricultural productivity on drained plots (Jiang et al., 2020b). These topics raise concerns over the sustainability of existing drainage systems, and their need to be redesigned has come to the fore (Deelstra, 2015; Abd-Elaty et al., 2019). A common theme across all these studies is the need to properly represent drainage systems within each study area.

In this context, hydrological modeling offers a widespread tool for predicting drainage discharge, with several models currently in use, e.g., DRAINMOD (Skaggs, 1981; Skaggs et al., 2012) in the USA. This spatially distributed model operates on various spatial scales (Konyha and Skaggs, 1992; Brown et al., 2013) and integrates many modules in order to represent different hydrological processes and solute transports (Breve et al., 1997). In Europe, the MACRO model (Larsbo et al., 2005; Jarvis and Larsbo, 2012) is currently used by the FOCUS group (Adriaanse et al., 1996; Boesten et al., 1997) to evaluate drainage system performance and contaminant transport (Jarvis et al., 1997; Beulke et al., 2001). These two models, despite demonstrating their effectiveness, have been designed using physically based modeling strategies with a large number of parameters. Their calibration on several study sites becomes difficult and time consuming (Beven, 1989). Given this complexity, the SIDRA-RU model offers an interesting alternative. The model is semi-conceptual (Beskow et al., 2011), being composed of one physically based part (the SIDRA module) coupled with a conceptual part (the RU module), and parsimonious, by virtue of requiring the calibration of only six parameters, hence making it easy to configure (Perrin et al., 2003). Initially intended to simulate drained discharge during flood periods (Lesaffre and Zimmer, 1987), the SIDRA model converts weather-dependent soil recharge into drainage discharge by solving a semi-analytical formula derived from the Boussinesq equation (Boussinesq, 1904). Various modules have been integrated so as to better represent infiltration (Kao et al., 1998), water flux in the unsaturated zone (Bouarfa and Zimmer, 2000), or pesticide leaching (Branger et al., 2009). The RU module was recently integrated in order to model water transfer in the unsaturated zone (Henine et al., 2021). This new version inputs a continuous recharge term into the SIDRA module, which then allows for the simulation of drainage discharge over the entire hydrological cycle.

Due to soil diversity within French drained areas, a model used for management purposes must initially be as general as possible and correctly calibrated to ensure that the model behavior matches the behavior of each studied site as closely as possible (Perrin, 2000). As such, a reliable calibration protocol often depends on the choice of objective function (OF), which serves as the numerical criterion to be optimized so that the simulation more accurately resembles reality. Many OFs can be used to calibrate a model, such as the Nash–Sutcliffe efficiency (NSE; Nash and Sutcliffe, 1970)

or the Kling–Gupta efficiency (KGE; Gupta et al., 2009), depending on the purpose of the particular study. The model here is intended for use on future prediction data under a long-term management scenario. From this perspective, the model must be temporally robust, i.e., its performance and parameters must remain independent of the period chosen for calibration (Klemeš, 1986). Such an evaluation can be performed by means of various tests, which tend to depend on the model structure (Refsgaard and Storm, 1996; Refsgaard, 2001; Henriksen et al., 2003; Daggupati et al., 2015). Since SIDRA-RU is a simple model, the split-sample test (Klemeš, 1986) is considered to be sufficient (Refsgaard, 1997). However, the national-scale evaluation of a hydrological model requires a large database, which is not readily available in the drainage hydrology field. This lack of data is the reason for the paucity of studies in the current literature.

The aim of this study is to assess the ability of a hydrological drainage model to simulate observed drainage discharge across several representative sites spread out in France. An exhaustive database, composed of 22 experimental sites and encompassing the main drained regions in France, has been built to account for the large diversity of French drained soils on which the model was tested. Database completeness is one of this paper's main strengths and allows the generalization of our results on soil diversity. The hydrological model chosen for this work is SIDRA-RU, a parsimonious model that yields continuous simulations and can easily be run on the database. In addition, the SIDRA-RU model offers a novel tool for the hydrological drainage modeling community; this study, therefore, provides an opportunity to test its performance at the national scale, which raises another point of interest regarding this study. Moreover, the temporal robustness of the SIDRA-RU model is assessed in the aim of asserting whether or not the model can be used within the scope of a long-term management tool, i.e., one capable of incorporating climate change.

## 2 Materials and methods

### 2.1 French classification of drained soils

A multitude of materials constitute French soils, as defined by their geological context, textural evolution, and regional climate. All of the above characteristics serve to determine the uniqueness of a soil. Making generalizations about soil diversity then becomes a necessary step. Indeed, grouping them by soil category facilitates their modeling. Several official classifications serve to group soil types (FAO, 1988; Krogh and Greve, 1999; Driessen et al., 2000). In this study, we are proposing to classify them by texture, thus making it possible to sort the database into three categories (see Fig. 1 and Table 1). Let us note that we do not consider the geological context or the regional climate to classify soils here.
<table><caption>**Table 1.** Main characteristics of the 22 plots with observed discharges and associated KGE' from the calibration process and split-sample tests.</caption><thead><tr><th rowspan="2">Texture</th><th rowspan="2">Plot name</th><th rowspan="2">Site location</th><th rowspan="2">Data period (years)</th><th rowspan="2">Surface area (ha)</th><th rowspan="2">Tile depth (m)</th><th rowspan="2">Space between drain and mid-drain*</th><th rowspan="2">Use</th><th rowspan="2">KGE' from total calibration</th><th colspan="4">KGE' from the split-sample test</th><th rowspan="2">Index (Fig. 10)</th></tr><tr><th>In calibration P1</th><th>P2</th><th>In evaluation P3</th><th>P4</th></tr></thead><tbody><tr><td rowspan="4">Clayey soil</td><td>Bray_P1</td><td>Bray</td><td>6</td><td>46</td><td>0.9</td><td>5.0</td><td>Calibration only</td><td>0.52</td><td>-</td><td>-</td><td>-</td><td>-</td><td>S1</td></tr><tr><td>Courcival_P3</td><td>Courcival</td><td>10</td><td>1.3</td><td>0.9</td><td>6.0</td><td>Calibration and evaluation</td><td>0.44</td><td>0.42</td><td>0.48</td><td>0.36</td><td>0.34</td><td>C1</td></tr><tr><td>La_Bouzule_P1</td><td>La Bouzule</td><td>13</td><td>1.9</td><td>0.9</td><td>12.0</td><td>Calibration and evaluation</td><td>0.52</td><td>0.58</td><td>0.52</td><td>0.42</td><td>0.31</td><td>C2</td></tr><tr><td>Saint_Laurent_P2</td><td>Saint Laurent</td><td>15</td><td>1.2</td><td>0.9</td><td>10.0</td><td>Calibration and evaluation</td><td>0.76</td><td>0.75</td><td>0.75</td><td>0.58</td><td>0.63</td><td>C3</td></tr><tr><td rowspan="10">Silty soil</td><td>Arrou_P6</td><td>Arrou</td><td>19</td><td>2.0</td><td>0.8</td><td>5.0</td><td>Calibration and evaluation</td><td>0.65</td><td>0.57</td><td>0.78</td><td>0.75</td><td>0.55</td><td>S1</td></tr><tr><td>Arrou_P8</td><td>Arrou</td><td>19</td><td>2.0</td><td>0.8</td><td>10.0</td><td>Calibration and evaluation</td><td>0.65</td><td>0.54</td><td>0.72</td><td>0.72</td><td>0.53</td><td>S2</td></tr><tr><td>Courcelles_P1</td><td>Courcelles</td><td>3</td><td>2.4</td><td>0.9</td><td>5.0</td><td>Calibration only</td><td>0.7</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Courcelles_P2</td><td>Courcelles</td><td>3</td><td>2.4</td><td>0.9</td><td>5.0</td><td>Calibration only</td><td>0.68</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>La_Jaillière_P4</td><td>La Jaillière</td><td>16</td><td>1.0</td><td>0.9</td><td>5.0</td><td>Calibration and evaluation</td><td>0.83</td><td>0.82</td><td>0.87</td><td>0.79</td><td>0.82</td><td>S3</td></tr><tr><td>Longnes_P2</td><td>Longnes</td><td>6</td><td>1.0</td><td>0.9</td><td>6.0</td><td>Calibration only</td><td>0.66</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Longnes_P3</td><td>Longnes</td><td>6</td><td>1.0</td><td>0.9</td><td>7.5</td><td>Calibration only</td><td>0.58</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Melarchez_GP</td><td>Melarchez</td><td>39</td><td>700.0</td><td>0.9</td><td>6.0</td><td>Calibration and evaluation</td><td>0.77</td><td>0.77</td><td>0.75</td><td>0.61</td><td>0.61</td><td>S4</td></tr><tr><td>OS2_P1</td><td>OS2</td><td>3</td><td>2.0</td><td>1.0</td><td>5.0</td><td>Calibration only</td><td>0.54</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Parisot_P2</td><td>Parisot</td><td>4</td><td>0.3</td><td>0.9</td><td>6.0</td><td>Calibration only</td><td>0.81</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rambouillet_P1</td><td>Rambouillet</td><td>3</td><td>1.0</td><td>0.9</td><td>5.0</td><td>Calibration only</td><td>0.77</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Rampillon_GP</td><td>Rampillon</td><td>9</td><td>355</td><td>0.9</td><td>6.0</td><td>Calibration only</td><td>0.77</td><td>0.8</td><td>0.76</td><td>0.71</td><td>0.73</td><td>S5</td></tr><tr><td>Saint_Juvin_P1</td><td>Saint Juvin</td><td>3</td><td>1.4</td><td>0.9</td><td>1.5</td><td>Calibration only</td><td>0.56</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Scaer_P2</td><td>Scaer</td><td>5</td><td>1.4</td><td>0.9</td><td>5.0</td><td>Calibration only</td><td>0.61</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Viry_P1</td><td>Viry</td><td>4</td><td>0.8</td><td>0.9</td><td>7.5</td><td>Calibration only</td><td>0.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Silty-clay soil</td><td>Archigny_P1</td><td>Archigny</td><td>2</td><td>1.0</td><td>0.9</td><td>4.0</td><td>Calibration only</td><td>0.76</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Archigny_P2</td><td>Archigny</td><td>2</td><td>1.0</td><td>0.9</td><td>6.0</td><td>Calibration only</td><td>0.76</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>La_Bouzule_P2</td><td>La Bouzule</td><td>13</td><td>2.9</td><td>0.9</td><td>6.0</td><td>Calibration and evaluation</td><td>0.54</td><td>0.5</td><td>0.6</td><td>0.45</td><td>0.56</td><td>SC1</td></tr></tbody></table>

<footnote>* The midpoint between consecutive drains (meters).</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The pedological distribution of French soils, produced by Jamagne et al. (1977) and Lagacherie and Favrot (1989), overlain on the spatial distribution of French drainage (RGA, 2010; top panel). The spatial distribution of sites aggregated by texture with observed flow data is shown in the bottom panel.</figcaption></figure>

The Lagacherie and Jamagne classification (Jamagne et al., 1977; Lagacherie and Favrot, 1987; Richer-de-Forges et al., 2008) has been used to evaluate this strategy. According to Fig. 1 (top), three distinct soil types occupy most of the regions with the highest drainage ratio (i.e., percentage of a land area that has been drained, with drainage ratio values above 50% of total arable area). First, the Glossic and Planolosic soils, belonging to the Luvisols and mainly located around the Paris Basin and in the Allier region (see

Fig. 1), are defined by textural differentiation between the surface horizon, which is often silty and sometimes sandy–loamy, and a deep clayey horizon. Second, brown acidic and leached soils, mostly distributed in the western part of France (Fig. 1), lie on a magmatic and metamorphic substratum; they are often characterized by a silty–clay or loamy texture. Third, the Pseudogley soils are also substantially drained, yet they remain only slightly correlated with any specific soil texture. We assume here that, among the three studied
soil textures, they are more related to silty–clay soils (Lévy, 1972). Figure 1 (bottom) shows the drainage ratio of arable soil in France. Most drainage systems (approximately 80%) lie on a loamy texture, according to Lagacherie and Favrot (1987), except in the eastern part of France (Fig. 1), where drained soils are either predominantly silty–clay or heavy clayey soils.

## 2.2 Input data

A representative database of drainage discharge across France was specially assembled for this study. The data originate from the following sources: (1) the ORACLE research project (Tallec et al., 2015) and artificial Rampillon wetland (Tournebize et al., 2012, 2017; Lebrun et al., 2019), (2) the partnership with the ARVALIS Institute, which monitors the La Jaillière experimental site, and (3) data from reference drainage sites dating between the 1960s and 1980s (Lagacherie and Favrot, 1987; Jannot, 1988).

The data from this last source stem from monitoring experiments managed by INRAE (formerly Cemagref) that test drainage modalities, i.e., what depth, space, or pipes best fit to the field conditions. The combination of these sources yields a database of nearly 200 years of cumulative hydrological records on drained plots in diverse pedoclimatic contexts over a broad plot scale range (e.g., 0.8–700 ha). The resulting extensive data set compiled for hydrological modeling purposes encourages the transferability of this study's findings.

The drainage network of the various study plots is based on similar technical characteristics (see Table 1) composed of PVC perforated and corrugated pipes lying at a depth of 0.85 to 1 m, with an inter-drain spacing from 8 to 24 m. The most widely used method for monitoring drainage discharge consists of measuring the corresponding water level at the drainage collector outlet using a calibration curve fitted at each measurement site, by designing a control section where flow is hydraulically managed. Before the 1980s and 1990s, data were recorded on a paper sheet that followed the motion of a floater linked to the water level. Nowadays, water level sensors (floating systems equipped with ultrasonic measurements) are used, and the data are digitally recorded. To ensure data homogeneity, observed data have been manually assessed by expert judgment in order to highlight periods of suspect data quality; as deemed necessary, the data have been corrected or deleted. The 22 study plots are distributed over three distinct soil textures, namely silty, silty–clay, and clayey. A total of 15 of them are characterized by a silty soil texture that covers most French regions (Fig. 1). The database is more limited with regard to clayey soils, characterized by sites like Saint_Laurent_P2 or Courcival_P3. Some regions, e.g., eastern France, which are strongly characterized by a clayey texture yet with just one clayey site, are not well represented. The SIDRA-RU performance in this region will be estimated from the SIDRA-RU global

performance on clayey soils from the database, comprising 44 years of observed discharges (Table 1). Moreover, some regions with a high drainage ratio, e.g., southwestern France, have no observation points and are therefore not covered by this study. Lastly, the Pseudogley soils are mostly correlated with the silty–clay soils, yet the database does not provide any relevant silty–clay plots. Model performance will thus be estimated by the global performance for all such sites. Each site was defined by the aforementioned technical characteristics (drain depth, mid-drain spacing corresponding to the half-space between two successive drains, and surface area), plus the length of available observed discharge logs and suitability to the split-sample test (Table 1).

Due to a lack of agronomic data, we assume here that growing practices do not affect the subsurface drainage hydrology on the study plots, except in the absence of a tillage technique (Dairon et al., 2017), a situation that is not widespread in France. This assumption is supported by four observations, namely that (1) the subsurface drainage is mainly effective during fall and winter, when the actual evapotranspiration is low; (2) our study investigates drained soils in winter primarily used for winter crops, such as wheat and maize (Zimmer, 1996), so we assume that the studied plots are cultivated every year throughout this season without a fallow period; (3) these crop types impact the subsurface drainage hydrology in a similar manner; hence, annual crop rotation does not add significant bias to the model calibration; and (4) the effect of cover crops (Meyer et al., 2018) has been neglected due to the fact that they were not widely used before 2012 and, for 19 of the 22 sites, the corresponding study periods ended before 2012.

The meteorological data were provided by the SAFRAN database (Vidal et al., 2010), a meteorological reanalysis covering France and supplying both precipitation and potential evapotranspiration (PET; based on the FAO-56 Penman–Monteith PET; Córdova et al., 2015) data on all database-referenced plots. These data are available from 1959 to 2019 at a daily time step and a spatial resolution of 8 km, which is 1000 times greater than the scale of the studied plots. This difference may introduce errors on model outputs, but all such errors are considered to be negligible.

## 2.3 The SIDRA-RU model

The SIDRA-RU model is a semi-conceptual, lumped model that describes the hydrological processes of artificial drainage systems. This model is based on the principle of rainfall–drainage discharge conversion and uses the rainfall, *P*, and potential evapotranspiration, PET, to predict water table height and drainage discharge at the drainage network outlet. In total, three modules have been integrated into the current version of SIDRA-RU (see Fig. 2).

First, an evapotranspiration module converts PET into an approximate value of actual evapotranspiration, called corrected evapotranspiration (CET), from the available water
<figure><img src="image_4.png" /><figcaption>Figure 2. Diagram presenting the various modeling stages of the SIDRA-RU model.</figcaption></figure>

level $S(t)$ in storage to satisfy the evapotranspiration constraint. A $S_{RFU}$ threshold is set, thereby assigning the minimum water level to fully satisfy PET as follows (see Eq. 1):

$$CET(t) = \begin{cases} \text{PET}(t) \cdot e^{-\frac{S_{RFU}-S(t)}{S(t)}} & \text{if } S(t) < S_{RFU} \\ \text{PET}(t) & \text{if } S(t) \geq S_{RFU} \end{cases}. \quad (1)$$

The net infiltration $P_{net}(t)$ is calculated by subtracting CET(t) from P(t).

Second, the RU module, a conceptual storage, calculates the water table recharge term $R(t)$ (millimeters) from the meteorological input and water storage capacity of the soil reservoir. There are two main parameters that control this RU module. On the one hand, the $S_{inter}$ (millimeters) parameter serves as an intermediate threshold of the soil reservoir, defining the water quantity required to generate flow in the reservoir before saturation of the storage as follows (see Eq. 2):

$$S_{RFU} = a \cdot S_{inter}. \quad (2)$$

Here, the parameter $a$ is set to 0.4 due to the water capacity being easily available for use by the crops (RFU is for Réserve Facilement Utilisable), representing approximately 60% of $S_{inter}$ (approximate concept of water-holding capacity) in French drained soils (Tournebize et al., 2015). On the other hand, the $S_{max}$ (millimeters) parameter represents the maximum capacity of the soil reservoir from which the net infiltration is fully converted into $R(t)$. These two parameters constitute an approximate concept of the water-holding capacity of a soil. A total of three stages are to be considered, as follows (Fig. 2):

- Stage 1: $S(t) < S_{inter}$, the water level is too low to allow for the generation of subsurface flows to the drains, i.e., Eq. (2) as follows:

$$R(t) = 0; \quad S(t) = S(t-1) + P_{net}(t). \quad (3)$$

- Stage 2: $S(t) \in [S_{inter}; S_{max}]$, the water level is high enough to partially allow for the generation of water table recharge $R(t)$. The parameter $\alpha$ defines the proportion of $P_{net}(t)$ being converted to recharge $R(t)$, while the remainder updates the water level, i.e., Eq. (3) as follows:

$$R(t) = \alpha \cdot P_{net}(t); \quad S(t+1) = (1-\alpha)P_{net}(t)+S(t). \quad (4)$$

A sensitivity analysis on the SIDRA-RU model has revealed that $\alpha$ is not sensitive to the KGE' criterion (Henine et al., 2021), used in this study as OF (see Sect. 2.4.1) and, moreover, can be set at one-third. Hence, to limit uncertainties relative to the calibration process for a non-sensitive parameter, this approach has been conserved herein.

- Stage 3: $S(t) \geq S_{max}$, water storage is full, i.e., Eq. (4) as follows:

$$R(t) = P_{net}(t) = P(t) - CET(t). \quad (5)$$

Third, the calculated water table recharge $R(t)$ feeds the original SIDRA module (Lesaffre and Zimmer, 1987; Bouarfa and Zimmer, 2000) in order to calculate the water table level $h(t)$ and drainage discharge $Q(t)$ (see Eqs. 5 and 6), in solving a semi-analytical formula derived from the Boussinesq
equation (Boussinesq, 1904). This physically based module is mainly controlled by two parameters, namely the horizon- tal hydraulic conductivity K (meters per day; m d⁻¹), and drainage porosity µ (–).

$$
\frac{dh(t)}{dt} = \frac{R(t) - K \frac{h(t)^2}{L^2}}{A_2 \mu}; \quad h(t+1) = h(t) + \frac{dh(t)}{dt} \quad (6)
$$

$$
Q(t) = A K \frac{h(t)^2}{L^2} + (1 - A) R(t). \qquad (7)
$$

It can be noted that surface runoff is considered to be neg-
ligible in the model, only contributing slightly to total flow
(Kuzmanovski et al., 2015). Furthermore, one of the assump-
tions made in the SIDRA module was to consider that pipes
lie on an impervious layer; thus, all excess water is fully
released through the pipes (Lesaffre, 1989). This assump-
tion seems rather reasonable since a large majority of French
drained sites lie on such soils, according to the studies car-
ried out on the aforementioned drainage reference areas (La-
gacherie and Favrot, 1987; Jannot, 1988; Tournebize et al.,
2012, 2017; Tallec et al., 2015; Lebrun et al., 2019). To be
completely operational, SIDRA-RU requires information on
technical characteristics, such as drain depth P (meters) and
mid-drain spacing L (meters) between the drain and inter-
drain. Furthermore, due to the aforementioned assumptions
dealing with the parameters a and α, a calibration process
is only necessary for the following four parameters: K and
µ, with the K/µ ratio describing the responsiveness of the
system, and Sinter and Smax from the RU module.

2.4 Methods

2.4.1 Calibration method

Parameter optimization (i.e., calibration) is commonly per-
formed in a hydrological modeling context in order to adapt
the hydrological model parameters to the specific study area
context. This process is relevant for conceptual parameters
since, by construction, they can neither be measured directly
nor be easily correlated with any physical characteristics of
the studied system. Some of the physically based parameters
that are too difficult to measure can also be calibrated.

The model calibration herein has been based on the al-
gorithm implemented in the airGR R package (Coron et al.,
2017, 2020) and is composed of two parts. First, a systematic
examination of the parameter space provides the most likely
zone of convergence, on the basis of a grid-screening algo-
rithm (Mathevet, 2005), according to a given performance
criterion (i.e., OF). Each parameter space is defined by its
specific distribution and intrinsic statistical characteristics
with respect to the soil texture. Second, a steepest-descent
local search procedure (Michel, 1991) seeks to improve the
OF, beginning with the grid-screening part, and find a more
accurate estimate of the parameter set, i.e., with higher model
performance.

The hydraulic conductivity K and drainage porosity µ fol-
low a lognormal distribution (Rousselot and Peyrieux, 1977;
Kosugi, 1994, 1996, 1999; Rousseva et al., 2017; Ren and
Santamarina, 2018). Parameters Sinter and Smax are concep-
tual and, thus, not defined by an intrinsic distribution. How-
ever, they are similar to the water-holding capacity of a soil,
which follows a normal distribution (Vachaud et al., 1985;
Brocca et al., 2007; Biswas et al., 2012; Biswas, 2019); con-
sequently, in this study, both Sinter and Smax are described as
following a normal distribution. The mean and standard de-
viation of each parameter are available in Appendix A. The
ones for K and µ were extracted per soil texture from the
aforementioned reference drainage areas. The ones for Sinter
and Smax were numerically fixed after many calibration tests.

Various OFs are commonly used in hydrological calibra-
tion processes, depending to a great extent on the primary
aim of the study. The most widespread OFs are RMSE (An-
derson and Woessner, 1992), MSE (mean squared error; Ye
et al., 2020), NSE (Nash and Sutcliffe, 1970), and, more re-
cently, KGE (Gupta et al., 2009). Our goal here is to eval-
uate model performance in order to simulate an entire hy-
drological cycle and represent the interannual variations in
the study plot to produce long-term projections about future
drainage hydrology. We have thus introduced the KGE' cri-
terion (Kling et al., 2012), an evolution of KGE that is more
relevant than NSE in reproducing internal flow rate variabil-
ity (Santos et al., 2018). KGE' is defined by three modeling
error components, as combined in Eq. (8), as follows:

$$
KGE' = 1 - \sqrt{(r-1)^2 + (\beta-1)^2 + (\gamma-1)^2}, \quad (8)
$$

together with the following:

- r = cov(Qobs, Qsim) / σobsσsim is the Pearson correlation coefficient, which serves to evaluate the error in both shape and timing between observed and simulated flows, with cov being the covariance between observed and simulated flows and σ their standard deviation.

- β = μsim/μobs is the bias term, which evaluates the bias between observed and simulated flows, with μ being the mean of observed and simulated discharges, respectively.

- γ = μobsσsim/σobsμsim is the ratio between observed and simulated coefficients of variation, which serves to evaluate the flow variability bias.

KGE' values range from -∞ to 1. The model performance
improves as KGE' increases towards 1. If the reader intends
to use the mean flow benchmark as a reference (correspond-
ing to NSE = 0) in order to assess KGE', the target value is
KGE' = -0.41 (Knoben et al., 2019). During the model cal-
ibration step, the data series over the whole time period was
used for each studied plot (Table 1).
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Functional diagram of the split-sample test.</figcaption></figure>

## 2.4.2 Split-sample test

The temporal robustness of the model proves to be of use when facing a time period different from that chosen for calibration, i.e., enhancing the model's capacity to perform equally well over different and contrasting time periods (Li et al., 2012). This point is particularly important when the model is intended for application under future climate change scenarios (Thirel et al., 2015b).

The choice of evaluation strategy mainly depends on model structure. For lumped conceptual models, a simple split-sample test (Klemeš, 1986) is sufficient to assess the robustness of such a model (Refsgaard and Storm, 1996; Daggupati et al., 2015). The split-sample test, as illustrated in Fig. 3, consists of splitting the data period into two sub-periods (P1 and P2) and then calibrating the model over both of them independently.

Thus, two optimal parameter sets are obtained (one covering P1 and the other P2), with each being tested over the other sub-period (e.g., evaluation sub-period P2 for a calibration over sub-period P1). If the KGE' scores from the evaluation sub-period lie close to the calibration KGE' score, then the model calibration is considered as being temporally robust and independent of the chosen time period.

This test was performed on the records from nine plots showing at least 10 years of time series data (Table 1). These series were split into two equal length periods, and the KGE' scores produced over the calibration and evaluation sub-periods were assessed and compared.

## 2.4.3 Numerical evaluation criteria

Evaluating and highlighting model performance limitations during the calibration process using numerical scores, such as NSE, RMSE, or KGE criteria, are often recommended, in addition to graphical analysis (Moriasi et al., 2015). However, understanding their evolution is indeed difficult. To overcome this difficulty, a model performance classification by means of a range of numerical score values will help assess the calibration quality.

In subsurface drainage modeling, the bibliography is not exhaustive with respect to KGE' score ranges. The few articles published deal instead with catchment hydrology mod-

eling (Crochemore et al., 2015; Poncelet et al., 2017). Use of the NSE criterion has been detailed more extensively (Moriasi et al., 2007; Ritter and Muñoz-Carpena, 2013; Moriasi et al., 2015); moreover, studies focusing on model calibration using the NSE score in subsurface drainage modeling state that values above 0.5 are considered to be acceptable (Helwig et al., 2002; Tuohy et al., 2018a). Even if a comparison between the NSE and KGE' scores is theoretically incorrect and not unequivocal (Criss and Winston, 2008; Knoben et al., 2019), we are assuming herein that the score ranges using the NSE criterion can be transposed to those using the KGE' criterion. We have decided to qualify the KGE' values as follows: the model calibration using KGE' values greater than or equal to 0.5 leads to “acceptable” model performance. KGE' values ranging from 0.6 to 0.7 are considered to reflect “good” performance, while a KGE' greater than or equal to 0.7 is deemed “very good” performance (Table 2). All KGE' values below 0.5 are considered to reflect unsatisfactory performance.

The model has also been evaluated in terms of its capacity to reproduce annual cumulative discharges, with a direct comparison conducted between observed and simulated flow rates using both the linear correlation coefficient $R^2$ (Bailly and Carrère, 2015) and the associated linear regression equation.

# 3 Results

## 3.1 Model performance after calibration

Table 1 lists the performance over the entire calibration period obtained from all 22 sites, and Table 2 classifies the model performance from each soil texture according to the score ranges. Performance varies across the three soil textures, with both unsatisfactory KGE' values, e.g., for the Courcival_P3 site, and some very good KGE' values, e.g., Parisot. For 21 of the 22 referenced plots, the calibration KGE' lies above 0.5, thus revealing at least acceptable KGE' values. The silty plots show values ranging from 0.54 to 0.83, including the best model performances, such as the La_Jaillière_P4 plot, with a KGE' of 0.83. They include three acceptable scores, reaching good levels for six of them and very good levels for another six. The silty-clay plots exhibit relatively homogenous KGE' values, ranging from 0.54 to 0.76. As regards the clayey plots, KGE' values display a wider range than on the silty-clay plots, i.e., from 0.44 at the Courcival_P3 plot to 0.76 at Saint_Laurent_P2, but the model performance remains at least acceptable on most of them (including one being very good). Courcival_P3 is the only one indicating an unsatisfactory KGE' value.

The La_Jaillière_P4 plot is used as an example to illustrate the temporal comparison between observed and simulated discharges over 16 years (see Fig. 4). These same graphs are available in Appendix B to illustrate the case of a silty-clay
<table><caption>**Table 2.** KGE' calibration scores.</caption><thead><tr><th>KGE' value range</th><th>Scores</th><th>Number of plots</th><th>Silty soils</th><th>Silty-clay soils</th><th>Clayey soils</th></tr></thead><tbody><tr><td>&lt; 0.50</td><td>Unsatisfactory</td><td>1</td><td>-</td><td>-</td><td>1</td></tr><tr><td>[0.50-0.60]</td><td>Acceptable</td><td>6</td><td>3</td><td>1</td><td>2</td></tr><tr><td>[0.60-0.70]</td><td>Good</td><td>6</td><td>6</td><td>-</td><td>-</td></tr><tr><td>≥ 0.70</td><td>Very good</td><td>9</td><td>6</td><td>2</td><td>1</td></tr><tr><td>Total</td><td>-</td><td>22</td><td>15</td><td>3</td><td>4</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Hydrograph on La_Jaillière_P4 after calibration over the entire available record, plus a magnification of hydrological year 2002–2003 and the associated temporal and direct comparisons between observed (blue curve) and predicted (red curve) cumulative discharges.</figcaption></figure>

soil at La Bouzule_P2 and in Appendix C for the case of a clayey soil at Saint-Laurent_P2.

Figure 4 shows that the simulated discharges are in good agreement with observations in terms of both seasonal dynamics (dry and wet season alternation) and cumulative distribution. Rainfall series are not directly correlated with discharge, as some rainfall events appearing from September to November do not systematically lead to subsurface flow. However, winter rains typically turn into discharge after a period of 1 or 2 days. A graphical analysis shows that simulated drainage discharges generally start in the same period as the observed discharge, with various delays depending on the soil type.

During hydrological year 2002–2003, the drainage season starts on the same day for both simulation and observation (i.e., 1 November). In 2002–2003, the maximum observed drainage discharge lies close to 15 mm d⁻¹, versus a lower simulated peak of 12 mm d⁻¹. The SIDRA-RU model

correctly predicts the temporal evolution and magnitude of drainage discharges while accurately delimiting the drainage seasons. The simulated peak flows closely match the observed ones. Peak flows often tend to be underestimated in simulation by a few millimeters per day (hereafter mm d⁻¹), although the timing is usually well estimated. On the whole, the dry periods are well represented by the model. Drying times are a bit longer for the simulations but typically lie within a few days of observations. Note that spring flows are sometimes not well simulated, as was the case in February 2001 and July 2006.

The capacity of the SIDRA-RU model to represent the water balance has also been assessed. Figure 4 shows that the simulated cumulative drainage discharge over March 2002 lies close to the observed discharge, with a slight underestimation of 10 mm, i.e., below 3% of the annual drained water balance relative to 350 mm. The linear regression between observed and simulated cumulative discharges yields
an equation close to a 1 : 1 equation, with an $R^2$ of 1, leading to the assessment that the water balance is fully respected over this year on the La_Jaillière_P4 plot, in terms of both time and total quantity.

Unlike at La_Jaillière_P4, the SIDRA-RU model at the Courcival_P3 plot (see Fig. 5), which lies on a swelling clayey soil, shows larger discrepancies between observed and simulated discharges on the plots. The red and blue curves do not coincide on a significant portion of the logs. From 1985 to 1995, simulated discharges often started later than the observed ones, with delays ranging approximately from 2 weeks to 2 months. The start of the drainage season is defined here when significant discharges appear.

The plots for the cumulative drained discharge in 1992–1993 reveal that the annual drained water balance diverges by +3 mm from simulations, but the linear regression indicates a slope equal to 1.1, which is quite high and, moreover, shows that the cumulative discharges have not been well simulated.

Figure 6 provides a comparison between the predicted and observed total cumulative discharges on each plot and for each hydrological year, as classified by soil texture. The linear regressions lie close to the 1 : 1 equation, with an $R^2$ above 0.9 for all three textures, thereby indicating that SIDRA-RU is representing the water balance at nearly all times. However, Fig. 6 does show a few discrepancies between prediction and observation, especially on the silty plots, with a deviation in the simulated cumulative discharge of 300 mm. These same observations are drawn on clayey soils, with the same discrepancies and a smaller data set.

In order to use the SIDRA-RU model for a long-term prediction of agricultural drainage management in France, the model must first be able to reproduce both high and low drainage discharges. Figure 7a depicts the differences in the Q05 quantile between observed and simulated drainage discharge. This quantile represents the values under which the annual drainage discharge occurs 5 % of the time; it is used in order to evaluate low flows. The Q95 analysis (Fig. 7b) result, under which the annual drainage discharge occurs in 95 % of the cases, serves to evaluate high flows. Figure 7c displays an analysis of the average discharges (Qmean). This Qmean analysis is then applied to each hydrological year of each plot on the nonzero flows, thus reducing the predicted drainage discharges by the observed ones; results are classified by texture, using box plots (Tukey, 1977). Regarding the Q05 quantiles (Fig. 7a), results show that, for the three textures, bias between simulated and observed Q05 ranges from −0.020 to 0.030 mm d⁻¹, with some extreme points (mainly on the silty texture). The medians of biases all lie close to zero as well (from −0.002 to 0.002 mm d⁻¹), thus revealing that the model correctly predicts low flows. Regarding the Q95 quantiles (Fig. 7b), the median values are once again close to zero (from −0.247 to −0.040 mm d⁻¹); however, the ranges of the Q95 biases lie above those of the Q05 quantiles. On silty soils, the box plot limits of Q95 biases range from −1 to +0.5 mm d⁻¹, and the whiskers range

from −3 to +3 mm d⁻¹. Similarly, for silty–clay soils, the Q95 biases vary from −3 to −2 mm d⁻¹; the discrepancies are larger on clayey soils, where the Q95 biases vary from −4 to +4 mm d⁻¹.

Figure 7c shows that the box plot medians for the Qmean biases also lie close to zero (from 0.007 to 0.057 mm d⁻¹). The Qmean biases range from −0.5 to +0.5 mm d⁻¹ for silty soils, from −0.3 to +0.6 mm d⁻¹ for silty–clay soils, and from −0.8 to +0.9 mm d⁻¹ for clayey soils. SIDRA-RU performs at a level of good agreement with respect to the average drainage discharges. The deviation on Qmean biases is higher on clayey soils, thus reflecting the greater difficulties of the SIDRA-RU model in simulating Qmean on this texture.

## 3.2 Model robustness

The KGE' values obtained during the evaluation period were then compared to those found during the calibration period, as illustrated in Fig. 8.

For starters, this figure shows that all points are located under the line $y = x$, informing, therefore, that all evaluation KGE' values from a specific period are always less than the calibration KGE' values from the same period. Moreover, Fig. 8 indicates that the deviations differ according to soil texture.

On silty soils, the maximum variation is observed at Melarchez (the largest studied site, with 700 ha), with KGE' values varying from 0.66 to 0.55 over the second sub-period (Table 1), yet the evaluation and calibration KGE' values are similar on four of the silty plots over both sub-periods. Furthermore, the evaluation KGE' values on silty soils remain at least acceptable. Results are similar for the silty–clay soils, which feature validated KGE' values close to the calibration values. The deviations in KGE' values are more significant on clayey soils. Indeed, KGE' varies from 0.75 to 0.58 at the Saint_Laurent_P2 site, i.e., a score going from very good to acceptable. Saint_Laurent_P2 is the only clayey plot that remains at least acceptable, according to Table 2. On the La_Bouzule_P1 plot, the KGE' value is reduced from 0.52 in calibration to 0.31 in validation. However, the results of Fig. 8 show that KGE' deviations on clayey soils are less than 0.21.

A graphical comparison between the predicted drainage discharges, calculated using the evaluation parameter set, and the observed discharges helps assess model robustness, as depicted in Fig. 9 at La_Jaillière_P4 during the 2002–2003 hydrological year. The calibration simulations use the parameter set calibrated over sub-period P2 (including the 2002–2003 season), while the evaluation simulations use the parameter set calibrated over sub-period P1. Figure 9 indicates that both calibration and evaluation drainage discharges lie close to the observed levels. The peak flows from both simulation curves have been superimposed on the main part of the drainage season; hence, the evaluation parameter set performs well over the studied time period.
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Hydrograph on Courcival_P3 after calibration over the entire available record, plus a magnification of the hydrological year 1992–1993 and the associated temporal and direct comparisons between observed (blue curve) and predicted (red curve) cumulative discharges.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Comparison of the drained annual discharges between simulation and observation – each point represents the drained water quantity from a hydrological year for a given site, and all points have been classified by texture.</figcaption></figure>

Differences between the two simulations only exist at the beginning of the drainage season, as strongly controlled by the RU module. The drainage season appears to start on the same day, according to both simulation curves, but the discharge magnitude during the first few days differs from one simulation to another. This same graphical approach has been used on a clayey soil available in Appendix D for Saint_Laurent_P2 during the 1982–1983 season. These observations are similar to the previous ones on La_Jaillière_P4.

Similar observations have also been recorded on the cumulative discharges (Fig. 9), which display an identical behavior, except for a gap between two predicted flows appearing at the start of the drainage season, and have tended to remain so throughout the year. A direct comparison of the cumulative flows between observation and simulation yields linear regressions near the 1:1 equation, with an $R^2$ of 0.999 in evaluation and 1 in calibration. This result attests to the water balance in the calibration being close to that in evaluation. These same results are listed in Appendix D for
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Differences between prediction and observation of low (*Q*05), high (*Q*95), and average (*Q*mean) flows. Each hydrological year from each site has been independently considered. Results are compiled by soil texture as follows: 142 points for silty soils, 17 for silty-clay soils, and 44 for clayey soils.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Comparison between the KGE' values assessed over the calibration and evaluation periods on the nine plots used for the split-sample test, where U is unsatisfactory, A is acceptable, G is good, and VG is very good.</figcaption></figure>

Saint_Laurent_P2. With the exception of the deviation at the beginning of the curves due to an early evaluation start, the curves are parallel. The water balances are similar when using either parameter set.

The use of the split-sample test for model calibration raises the question of parameter set similarity between the two calibration periods across all sites studied. Figure 10 illustrates how each of the four parameters (K, μ, Sinter, and Smax) evolves, depending on the two calibration periods. Each study site has been labeled in Fig. 10 with the avail-

able corresponding index from Table 1. Results indicate that hydraulic conductivity K is the least-changing parameter between the two periods.

One of the two extreme outlier points corresponds to Melarchez (point S4), where K changes from 1.46 m d⁻¹ for sub-period P1 to 1.75 m d⁻¹ for sub-period P2, thus demonstrating that hydraulic conductivity values remain within the same order of magnitude. Similar results were found for drainage porosity μ, with seven of the nine sites lying very close to the 1 : 1 line, thus confirming the similarity of both calibrated μ values. La_Bouzule_P1 (i.e., C1; Fig. 10) displays the strongest deviation, from 0.03 in P1 to 0.01 in P2, which is nonetheless a small variation and acknowledges that μ is conserved between both sub-periods.

Soil reservoir parameters Sinter and Smax are those exhibiting the greatest change between the two calibrated parameter sets, and in a very similar pattern. Among the nine sites, six show a deviation of less than 40 mm on Smax, which seems to be acceptable. However, for the three other sites, i.e., Melarchez (S4), La_Bouzule_P1 (C2) and Saint_Laurent_P2 (C3), deviations range from 50 to 150 mm, indicating larger deviations for both conceptual parameters, especially on clayey soils. These observations are consistent with those derived from Fig. 8 for La_Jaillière_P4 and from Appendix D.

# 4 Discussion

## 4.1 A representative database

The literature (Perrin, 2000; Coron et al., 2012; Montanari et al., 2013; Thirel et al., 2015a) suggest that, when assessing generalizable hydrological models, the aim typically consists of determining the ability of a model to reproduce the hydrological behaviors of various study areas. The larger the database, the more reliable the study because the model is being evaluated over a wider diversity of geological and cli-
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Hydrographs of observed and predicted discharges on calibration (parameter set derived from sub-period P2) and evaluation (parameter set from sub-period P1) recorded at La Jaillière in 2002–2003. Predicted cumulative discharges are directly compared to observations.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Comparison of parameter sets from both calibration periods (sites are referenced by their associated index, as shown in Table 1).</figcaption></figure>
matic contexts (Gupta et al., 2014). To the best of our knowl-
edge, however, in drainage modeling, the models are often
evaluated on a short-term database, with just a few years on a
few sites, e.g., DRAINMOD (Skaggs et al., 2012), MACRO
(Jarvis and Larsbo, 2012), ADAPT (Gowda et al., 2012), and
SWAT (Arnold et al., 2012). This lack of data limits the op-
portunities to test models in various soil types and, moreover,
prevents the assessment of the relevance of models over a
broad array of spatial scales.

The database used in this study was specifically built to
assess the performance of a drainage discharge model on a
larger data set. In total, 22 experimental sites were compiled,
accounting for nearly 200 hydrological years spread over the
main drainage areas in France, on contrasting soil types and
in different pedoclimatic settings, featuring an extensive plot-
scale range. This database has been classified according to
three main soil textures (silty, silty–clay, and clayey) using
available drainage discharge data. Not all regions with a high
drainage rate are well represented; however, 80 % of French
drained soils do have silty soil textures, as represented by
the majority (15 of the 22 plots) in the database. The real
advantage of this database is the ability it offers to apply the
model to referenced sites that represent a large majority of
France’s drainage diversity. This topic accounts for both the
originality and key contribution of this study.

## 4.2 A parsimonious hydrological model

The diversity of the database introduced requires a model that operates correctly and in accordance with each site's specific conditions, i.e., as generalizable as possible. In this context, a physically based model is theoretically better suited. However, this kind of model is generally composed of many parameters representing the complexity of a study site, such as current crop, root depth, saturated water content (i.e., unsaturated one), hydraulic parameters, or water-holding capacity. Using such a model on a large database requires providing all these characteristics for each site, which is in practice very difficult because the measurement technics are globally expensive. The calibration might be useful, but it is time consuming due to the large number of parameters. Conversely, a simple model like SIDRA-RU offers significant advantages, as it requires little information and faster calibration, thus becoming in practice suitable for generalizing. The model is lumped and, initially requiring the calibration of six parameters, the assumptions made in this study allow us to reduce this number to four parameters. Compared to the original SIDRA model (Lesaffre and Zimmer, 1987), the current version simulates continuous discharges over several hydrological years, thus simulating both wet and dry periods. The calibration process only necessitates brief rudimentary knowledge of the soil texture, used here as a priori input data to establish the model parameter distributions. Regarding input variables, the model requires rainfall and PET data at a daily time step to predict the drainage discharge. Managed in con-

junction with the SAFRAN climate database to satisfy data
needs across France, SIDRA-RU can easily be launched on
all drained areas throughout the territory.

4.3 SIDRA-RU model performance

Once calibrated, the model shows a performance ranging
from acceptable to very good on all sites, judging from the
KGE' value, except for clayey soil. The model provides accu-
rate simulations on small plots (0.8 ha at Parisot_P2) and on
large plots (700 ha at Melarchez). In comparing the SIDRA-
RU performance with that of other models tested in more lo-
cal studies (Gowda et al., 2012; Skaggs et al., 2012; Muma et
al., 2017; Revuelta-Acosta et al., 2021), SIDRA-RU exhibits
an equivalent simulation quality. For a vast majority of sim-
ulations carried out by the model, the observed annual water
balance is indeed reproduced with a good agreement. These
performances are close to those obtained using pure physical
approaches, like the models RZWQM (Ma et al., 2007) and
ADAPT (Sands et al., 2003). Moreover, this congruence in
annual cumulative water quantity between observation (inte-
grating drained discharge and runoff) and simulation (with-
out runoff) validates the assumption that considers runoff to
be negligible, according to the conclusions of Kuzmanovski
et al. (2015). Similarly, this result validates the assumption
made regarding the fact that a large majority of drained soils
lie on an impervious layer, hence authorizing us to neglect
the recharge to groundwater. This result also supports the as-
sumption made regarding the fact that the effect of the current
crop growth is neglected as long as subsurface drainage oc-
curs mainly in winter on non-fallow soil. Also, the SIDRA-
RU model allows the simulation of the temporal variation
in drainage discharge for both dry and wet periods; more-
over, flood peaks are simulated on time, as are recession peri-
ods, thus offering a performance comparable to that of more
complex models, e.g., DRAINMOD (Skaggs et al., 2012),
CATHY (Muma et al., 2017), and RZWQM (Jiang et al.,
2020a). Overall, the simple design of the SIDRA-RU model
allows the achievement of a performance at least as good as
that found in current models, like MACRO, on the La Jaili-
lière plot (Kuzmanovski et al., 2015), or WEPP (Revuelta-
Acosta et al., 2021).

Let us note, however, that some peak flows have been
slightly underestimated. This behavior is generalizable and
has been observed on most database sites. However, the phe-
nomenon varies from one site to another without any clear
relationship with any soil characteristic and mostly depend-
ing on the calibration quality. One explanation might be that
the SIDRA-RU model is calibrated on both high and low
drainage discharges, as represented by a unique parameter
set, which consequently introduces biases. These biases be-
come more significant on extreme discharges. Furthermore,
the missed variations sometimes lead to missing the be-
ginning of the drainage season. During some hydrological
years, the observed drainage discharge starts before water
level in the soil reservoir reaches the $S_{inter}$ threshold during simulation. Indeed, a rainfall event appearing from August to October is not specifically converted to discharge every time, mainly due to the nonlinear processes controlling the precipitation–discharge transformation. Furthermore, the assumptions related to SIDRA-RU, e.g., neglecting lateral communication with other plots of land, might also be responsible for this phenomenon. This early start might prevent the complete filling of the annual water balance. However, deviations in this indicator remain minor in the large majority of the cases, i.e., from 5% to 10%, and produce no significant consequences on long-term studies.

Regarding the split-sample test, results first showed that evaluation KGE' values were all lower than calibration KGE' values from the same period. This finding seems to be normal and assesses the consistency of the test, since an evaluation parameter set will normally always yield a lower KGE' value than the one obtained from the calibration set, thus corresponding to the best result, depending on the chosen OF (here the KGE' criterion). Second, performances using calibration parameters from both sub-periods are similar across all sites, with the evaluation KGE' values being close to the calibration values and, moreover, remaining quite good, especially on silty and silty–clay soils. Performance is lower on the clayey soils, while still remaining acceptable when considering the difficulties the model experiences on clayey soils during calibration. The drainage discharge behaviors are similar as well, as logs from both evaluation and calibration simulations do merge on most occasions with equal water balances. These results attest to the temporal robustness of the SIDRA-RU model on the studied textures, which represent the largest proportion of drained areas in France.

## 4.4 Calibration consistency

Another important analysis is the consistency of model calibration. The SIDRA module actually solves a simplified formula derived from the Boussinesq equation and requires a good estimation of both hydraulic parameters, namely hydraulic conductivity *K* and drainage porosity *μ*. Thus, model calibration is only reliable if the calibrated *K* and *μ* are probable according to the case study soil type. Figure 11 compares the distribution of the calibrated values of silty plots with the distribution obtained using the measured values of *K* and *μ* during tests conducted on reference silty drainage sites. The theoretical curves have also been drawn according to these distributions, using the mean and standard deviation of each sample while following a lognormal law. The database provided only allows the assessment of the quality of the calibration on silty soils since 15 sites are referenced for this particular texture. On both of the other soil textures, only three sites are available for silty–clay soils and four for clayey soils, which is insufficient to compare the calibrated values to a reference base, hence constituting a limitation of this study.

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Distribution of *K* and *μ* from calibrated values on silty soils vs. values extracted from reference drainage sites placed on silty soils.</figcaption></figure>

Regarding hydraulic conductivity *K*, some divergences exist between the reference base and the calibrated values, especially for the highest hydraulic conductivity values, i.e., close to $2 \text{ m d}^{-1}$. However, over the remaining range of values, the histograms from the calibrated *K* are congruent with the one from the reference drainage tests. The theoretical laws are also similar between the reference base and calibrated values, even though the curve derived from the calibrated value shows a higher standard deviation due to high hydraulic conductivity values. Despite some discrepancies, the distribution of calibrated *K* values for silty soils seems to be reliable on the basis of soil type. As for *μ*, the same method has been applied (Fig. 11); the histograms and theoretical laws do match, thereby concluding that the calibration process reliably estimates *μ* as well. The split-sample test confirms this analysis, showing that *K* and *μ* from silty sites are conserved from both calibration sub-periods. Each *K* and *μ* value seems to be calibrated with a robust and consistent data point, attesting to the relevance of calibration on the physically based module.

As regards $S_{inter}$ and $S_{max}$, these parameters are conceptual, and no observed data can be used as a reference to define their statistical distributions. Furthermore, even if they were to constitute a conceptual approach of the water-holding capacity, soil texture is insufficient as a description, and no distributions can be determined based on the database classification employed. One solution might consist of comparing the $S_{inter}$ and $S_{max}$ of each site with the associated actual value of the water-holding capacity; however, the database
does not provide this information for every experimental site, and measuring it in situ would be very expensive and, thus, infeasible. Nevertheless, it can be noticed that the range of values of $S_{max}$ are consistent with water-holding capacities of those kinds of soils, ranging from 70 to 200 mm for 1 m depth of soil.

## 4.5 Weaker performance on clayey soils

On non-deformable clayey soils, such as Saint_Laurent_P2, SIDRA-RU produces a good performance. However, at Courcival_P3, a plot lying on a deformable swelling clayey soil, the SIDRA-RU performance is significantly poorer. As such, the latter does not constitute an exception in the drainage modeling community. Indeed, the literature identifies clayey soils as being a recurring problem for drainage modeling (Robinson et al., 1987; Snow et al., 2007), especially in mole drainage, as currently practiced on heavy clayey soils and swelling clays (Jarvis and Leeds-Harrison, 1987; Tuohy et al., 2016). This finding is mainly due to a difference in hydraulic characteristics between silty soils, on which the model design is primarily based, and heavy or swelling clayey soils. The latter are characterized by natural pedological deformations, like soil surface fracturing, which lead to preferential flow zones before saturation (Beven and Germann, 1982; Jarvis and Leeds-Harrison, 1987). The horizontal soil profile is no longer homogeneous, which contradicts one of the main hypotheses of SIDRA. Moreover, agricultural practices like plowing exacerbate this phenomenon and, therefore, affect soil porosity. One way to improve the results at Courcival_P3 would be to artificially locate the pipe at a depth of 30 cm instead of the current 90 cm.

Another critical assumption of the SIDRA module is the elliptical shape of the water table; this assumption facilitates the numerical resolution of the Boussinesq equation. As regards heavy clayey soils, this hypothesis is no longer suitable, since the water table shape evolves towards a rectangular structure (Fig. 12). This phenomenon is due to the very low hydraulic conductivity (Robinson and Rycroft, 1999; Skaggs et al., 1999), which, for SIDRA-RU, is difficult to integrate. Furthermore, the water table level drops when approaching the drain because the soil has been turned over on this profile in order to bury the drain, like for the aforementioned natural pedological deformations.

Furthermore, the issues observed on clayey soils are specifically significant at the start of the drainage season. At Saint_Laurent_P2, delays occur more frequently than on silty soil, so the more the plot is defined by heavy or swelling clayey soil, the longer the delay. At Courcival_P3, these delays were of the order of 1 month. The RU module design partially addresses this problem by generating a soil profile recharge before saturation, but this issue remains a major limitation of the model, i.e., the more clayey the soil, the poorer the model performs. A soil is considered to be mainly clayey once the clay fraction exceeds 35% (Richer-

de-Forges et al., 2008), which can be defined as the limit beyond which good model performance is no longer guaranteed.

Some models show their efficiency when simulating drainage discharge on such soils. Among them, there is the MACRO model which showed its efficiency in simulating drainage discharge in Europe, including structured soils like heavy clayey soils (Köhne et al., 2009). However, these models are based on a more physically oriented approach than SIDRA-RU. Building on this type of concept would probably improve the latter in simulating drainage discharges on clayey soils but might undermine its generalist nature.

## 4.6 A less robust RU module

Another limitation of this model is that the RU parameters are slightly less temporally robust than the SIDRA parameters, showing fewer stable values between calibration sub-periods than the SIDRA parameters. This issue might originate from dependencies of the RU module on interannual meteorological variations. The $S_{inter}$ parameter represents the soil storage threshold that allows the model to initiate each drainage season. Compared to a common hydrological period without too many extreme discharge events, a dry calibration period, defined by a larger occurrence of dry events on a same period due to climate conditions, increases $S_{inter}$ values; hence, the model stores a larger quantity of water, delaying the start of the drainage season. Conversely, a wet calibration period will tend to decrease $S_{inter}$ in anticipation of the start of the drainage season. Regarding Courcival_P3, the split-sample test showed that $S_{inter}$ evolved from 175 mm in 1986–1990 to 202 mm in 1990–1995, a dry period (see Fig. 5). Consequently, if the two sub-periods are meteorologically contrasted, $S_{inter}$ (also as $S_{max}$) will differ from the two calibrated parameter sets. One solution might be to consider the $S_{inter}$ parameter as being a variable to be adjusted according to the meteorological conditions of the previous year and to soil parameters, e.g., whereby the actual water-holding capacity is defined as a new parameter. Clayey soils intensify this robustness issue due to weather fluctuations exacerbating the formation of preferential flow zones.

## 4.7 Interpretations bounded by choices

Another point we wish to discuss deals with the dependency of the aforementioned interpretations on the conditions and choices involved in conducting this study. We noted above the lack of data in the database used for certain French regions. Making assumptions, e.g., grouping soils by categories, to address this weaknesses would then offer a good alternative yet still be dependent on arbitrary decisions. Adding reference sites could complete and significantly improve the SIDRA-RU model robustness analysis.

Moreover, the model is based on some rather important assumptions, such as neglecting both the surface runoff and
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Evolution of the water table shape between a silty and a heavy clayey drained soil, adapted from Bouarfa and Zimmer (2000) and Branger et al. (2009).</figcaption></figure>

recharge to groundwater, or the ones made on the parameters *a* and *α*, which limit its field of application. Indeed, even if these assumptions do seem to be acceptable on sites used to represent French subsurface drainage, up to now there is no evidence of their relevance on other sites. This limits the use of SIDRA-RU to the specific conditions outlined herein. Nonetheless, it should be noted that SIDRA-RU may be used without this assumption, in integrating the depth infiltration with Hooghoudt's equation, according to the principle of equivalent depth (Zimmer, 1992). A term, *D*<sub>s</sub>, designating the deep seepage rate, is introduced in the Boussinesq's equation to reduce the recharge rate to the drains.

Furthermore, the choice of calibration process drives the calibration results based on study goals. We have implemented herein a grid-screening algorithm that assigns the best combination of parameters according to their respective distributions, coupled with a step-by-step algorithm. This approach has the advantage of being theoretically entirely automatic and, thus, eliminates the subjective aspect of calibration; however, external decisions influencing results are still necessary.

As an example, in order to track the main purpose of this study, SIDRA-RU is calibrated using the KGE' criterion as its OF, in combining three criteria (Kling et al., 2012), for a relevant approach to properly represent the interannual variability in drainage discharge. However, if this model is to be used for another purpose, like predicting drainage season initiation, then the KGE' criterion might not be the most efficient OF. This statement highlights the fact that the SIDRA-RU model is robust with respect to KGE', yet nothing proves to be the same for other OFs or purposes.

The distribution functions of each model parameter are required for the calibration, which in this case depends on the decision to classify soils based on their texture. Regarding *K* and *μ*, these distributions are established from measured data extracted from the reference drainage tests conducted in the 1980s, as mentioned above. These measurements are subjected to uncertainty, depending on both the number of measurements and the method employed; they constitute a significant source of error. We also mentioned above that

classifying a soil by its texture alone is a major assumption and one that distorts the distributions. Consequently, the calibrated parameters, constrained by the distributions a priori defined, might be biased due to an overly wide range of referenced values that bias the mean and standard deviation, as shown in Fig. 10 on silty sites. This problem is particularly worrisome since both those parameters are physical, making it prohibited to set outliers.

Driving the distribution with realistic value ranges at each site can prove to be a relevant solution. For example, if we consider that, on a specific soil, *K* might be included within a smaller value range, according to information obtained from the study site, then the calibration might lead to a more realistic value of *K*. This strategy reduces the risk of extracting parameters with secondary optima, and moreover, calibration accuracy is refined. However, better knowledge of the soil characteristics is required, with the assumption that by knowing the reliable ranges, calibration becomes semi-automatic, thus introducing an arbitrary decision-making factor into the process. In addition, this supplementary knowledge requires relatively complete databases of drained plots, which, as previously discussed, constitutes a persistent issue.

## 4.8 Coupling with modules to simulate pollutant leaching

In the perspective of long-term management on drained plots, predicting flows in order to better monitor the use of agricultural pollutants is a major concern due to pollutant transfers occurring with drainage flow (Kladivko et al., 2001; Tranjanov et al., 2018). Thus, a good model can be used as a decision-making tool, for example to restrict pollutants' application during flow period for the case of pesticides (Lewan et al., 2009; Zajíček et al., 2018; Kobierska et al., 2020). Indeed, using water content as an indicator to anticipate the start of drainage flow in order to reduce pesticide applications is a recommended strategy instead of restriction timing (Brown and van Beinum, 2009; Lewan et al., 2009). In this context, using SIDRA-RU may be quite relevant. However, the current form of the RU module is not optimal to accu-
rately represent the fate of pollutant in soil profile as it is too
simple to precisely represent the behavior of the water ta-
ble inside the unsaturated zone. To overcome this problem,
this model type is generally coupled with pedotransfer func-
tions (Jury and Roth, 1992; Magesan et al., 1994) to trans-
fer water and pollutant stock from the unsaturated zone to
the saturated zone. Within this framework, the perspective
of the PESTDRAIN module (Branger et al., 2009), coupled
with the SIDRA-RU model, should allow the simulation of
pesticide leaching by including two reservoirs, namely a fast
reservoir to mimic preferential flow above the drain area and
a slow reservoir through the soil matrix. Based on a simi-
lar approach, combining SIDRA-RU with a nitrate leaching
module might also be useful in order to correctly assess water
pollution on French drained plots.

5 Conclusion

The aim of this study has been to implement an large
database characterizing the main drainage areas throughout
France, so as to assess the performance and robustness of
the SIDRA-RU model. A database comprising 22 drained
sites was built to represent French soil diversity and describe
the three main soil textures from France's main drained
parts (i.e., silty, silty-clay, and clayey). Results indicate that
the SIDRA-RU model yields satisfactory drainage discharge
simulations for nearly all studied sites. Moreover, the model
shows especially good performance on silty soils, which ac-
count for 80% of all drained plots in France. Despite a num-
ber of limitations, particularly for clayey soils, the model was
found to be temporally robust at the national scale, which
enables us to conduct long-term impact studies. Once cali-
brated, this model can indeed be used to assess the resilience
of drainage systems under climate change according to cli-
mate scenarios like those from the CMIP5 project (Eyring
et al., 2007; Taylor et al., 2011) or the upcoming CMIP6
project.
Appendix A

<table><caption>Table A1. Table of mean and standard deviation (SD) per soil texture used for the parameter distribution.</caption><thead><tr><th rowspan="2">Texture</th><th colspan="2">K (m d<sup>-1</sup>)</th><th colspan="2">μ (-)</th><th colspan="2">Sinter (mm)</th><th colspan="2">Smax (mm)</th></tr><tr><th>Mean</th><th>SD</th><th>Mean</th><th>SD</th><th>Mean</th><th>SD</th><th>Mean</th><th>SD</th></tr></thead><tbody><tr><td>Clayey</td><td>0.32</td><td>2.53</td><td>0.017</td><td>2.081</td><td>138.39</td><td>53.33</td><td>171.68</td><td>70.19</td></tr><tr><td>Silty-clay</td><td>0.99</td><td>3.52</td><td>0.018</td><td>2.189</td><td>138.39</td><td>53.33</td><td>171.68</td><td>70.19</td></tr><tr><td>Silty</td><td>0.90</td><td>3.17</td><td>0.031</td><td>1.926</td><td>138.39</td><td>53.33</td><td>171.68</td><td>70.19</td></tr></tbody></table>

Appendix B

<figure><img src="image_15.png" /><figcaption>Figure B1. Hydrograph on La_Bouzule_P2 after calibration over the entire available record, plus a magnified plot of hydrological year 1982–1983 and the associated temporal and direct comparisons between observed (blue curve) and predicted (red curve) cumulative discharges.</figcaption></figure>
Appendix C

<figure><img src="image_16.png" /></figure>

**Figure C1.** Hydrograph on Saint_Laurent_P2 after calibration over the entire available record, plus a magnified plot of hydrological year 1982–1983 and the associated temporal and direct comparisons between observed (blue curve) and predicted (red curve) cumulative discharges.

Appendix D

<figure><img src="image_17.png" /></figure>

**Figure D1.** Hydrographs of observed and predicted discharges on the calibration (parameter set derived from sub-period P2) and evaluation (parameter set from sub-period P1) recorded at Saint_Laurent_P2 in 1982–1983. Predicted cumulative discharges are directly compared to observations.
**Code availability.** An intellectual property certificate has been issued by the Agency for Program Protection for the model code, under the reference IDDN.FR.001.260018.000.S.P.2018.000.30100. There is no public repository providing it. However, this paper provides an exhaustive description of the model, enabling readers to easily reproduce it. The code is written in FORTRAN language and then imported in R software, to be computed with the calibration algorithm "airGR" (Coron et al., 2017, 2020) available on CRAN (Comprehensive R Archive Network).

**Data availability.** Climatic data for this work were provided by Météo-France and are not publicly available since they are part of a commercial product. Météo-France can freely provide them on request for research and non-profit purposes. Most of the flow data are available on request. However, some flow data are provided by private partners and are not available.

**Author contributions.** AJ, HH, and JT conceptualized the work. LC and GT provided the methodological guidelines. CC, HH, and JT collected data and rendered them sustainable. AJ drafted the paper. HH, CC, LC, GT, and JT all revised the paper and contributed to its analyses and discussions.

**Competing interests.** The authors declare that they have no conflict of interest.

**Disclaimer.** Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Acknowledgements.** This work benefited from the French state aid managed by the ANR under the Investissements d'avenir programme (grant no. ANR-16-CONV-0003; CLAND project). Our gratitude is also expressed to the Météo-France Meteorological Agency for providing the data used in this work from the SAFRAN database. Moreover, thanks are addressed to both the ARVALIS Plant Sciences Institute for data from the La Jaillière measurement site and the ORACLE Research Observatory for its data. The authors would like to thank the editor (Christian Stamm) and the two reviewers, for their constructive and kind comments and suggestions.

**Financial support.** This research has been supported by the ANR under the Investissements d'avenir programme (grant no. ANR-16-CONV-0003; CLAND project).

**Review statement.** This paper was edited by Christian Stamm and reviewed by Nicholas Jarvis and one anonymous referee.

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