2.1 First guess – simulations without considering the river water level

The initial dataset is made of hard data and soft data. The hard data consist of UZD values measured during snapshot campaigns. The soft data are dry well depths. The dataset is characterized in terms of spatial statistics in order to justify the use of an appropriate geostatistical tool. UZD is defined in terms of a non-Gaussian probability density function conditioned with a non-negativity constraint.

Water table and UZD variables may show some directional non-stationarities at the local scale, especially looking in the same direction as the directional gradients. Nevertheless, the directional gradients in UZD are much less pronounced than the water table gradients, making it more amenable to treatment with stationary geostatistics. In the cases where a significant trend in the data is identified, the interpolation must be carried out using other geostatistical approaches, such as universal kriging, which deal with the non-stationarity of the variable (Goovaerts, 1999).

2.2 Water table mapping accounting for uncertain SW-GW connectivity

The second part of the mapping methodology is the final mapping of the water table, with the consideration of the SW-GW connection status: the connection status is evaluated for each cell located below the river network using a new disconnection criteria.

2.2.1 Defining a disconnection criteria at the reach scale

Stream–aquifer systems fluctuate from a hydraulically connected to a disconnected state due to the development of an unsaturated zone below the stream bed. During the switching between connection status, the SW-GW connection status is considered as a transitional state; this condition can occur when the capillary zone intersects the riverbed (Brunner et al., 2009). The disconnected SW-GW condition can occur under different settings such as in the case of high hydraulic conductivity contrast between the clogging layer and the aquifer (Brunner et al., 2009; Peterson and Wilson, 1988), the lowering of the water table (Dillon and Liggett, 1983; Fox and Durnford, 2003; Osman and Bruen, 2002; Rivière et al., 2014; Wang et al., 2011), or the biological clogging of the riverbed (Newcomer et al., 2016, 2018; Xian et al., 2019). Considering a constant river water level and river width, the disconnection occurs when any further increase of the hydraulic head difference between the water table and the river water level does not affect the infiltration rate from the stream to the underlying aquifer, which remains constant. Wang et al. (2011) and Rivière et al. (2014) proved that the disconnected state is reached when the saturation profile between the riverbed and the water table is stabilized. The saturation profile fills the space between an inverted area below the riverbed and a capillary fringe above the water table (Rivière et al., 2014; Wang et al., 2011). In the methodology, we assume that the disconnection state is reached when these two capillary fringes are separated without the occurrence of a clogging layer. The thickness of these two areas is controlled by the capillary effect, which mainly depends on the lithology of both the riverbed and the aquifer. Gillham (1984) proposed values for capillary fringe heights for several lithologies resulting from experimental measurements (Table 1). The disconnection criteria is defined as the distance between the riverbed and water table above which the river water and the groundwater are disconnected. It means that for higher distances, a saturation profile develops between the inverted area below the riverbed and the capillary fringe overlying the water table. Accordingly, the disconnection state is identified for a given lithology at each river cell of the estimation grid, when the difference between the first-guess water table and the riverbed elevation equals or exceeds an empirical disconnection criteria. The methodology therefore requires either an explicit bathymetric description of the river or an estimation of the riverbed elevation.

Starting from the knowledge of the riverbed lithology, the disconnection criteria can be estimated from Gillham (1984), and this first guess is uncertain given that the distribution of sedimentary heterogeneities into the alluvial plain induces important lithological contrasts (Jordan and Pryor, 1992; Flipo et al., 2014), and characterizing such heterogeneities requires important geophysical surveys that are out of reach for the development of our methodology. At a station, the lithology is hence uncertain and a fortiori even more uncertain along a river reach. However, the disconnection criteria is defined as a threshold difference value between measured UZD and river water level from which the SW-GW connection status switches. In the absence of such criteria in the literature, an optimization procedure is proposed along the Seine River network given that piezometers are available in the vicinity of the river and that both in-river water level and water table in the piezometers are recorded synchronously. The optimization procedure is described in the application section since it is based on the use of temporal data that are not directly required for the mapping methodology.

If the two signals are correlated it indicates that the river and the aquifer are connected. Contrarily, a very low correlation indicates a disconnection. At the reach scale, many piezometers are available. The standardized and normalized hydraulic head and river water level are compared to assess the local connection status of SW-GW. On a scatter plot, the disconnection appears below a given slope of the regression line.

At a reach scale it is therefore possible to inform the connection status locally (at a limited amount of stations). Along the river, the distance between the riverbed and the water table is evaluated from the first-guess map. The disconnection criteria is evaluated within a range defined by Gillham (1984) as the one that reproduces the most of the locally assessed connection status. In the absence of data, the disconnection criteria defined in our study can be used as a first guess.

Table 1Values for the capillary fringe height, regarding the lithology, after Gillham (1984).

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2.2.2 Final step of the mapping methodology

Disconnected portions of river are deduced from the preliminary water table map with the application of the disconnection criteria. A final dataset of UZD is then created from the UZD first guess at each sampling location, to which connected river sections are added with a nil UZD value. An ordinary kriging is performed with this final UZD dataset (Fig. 1, step 5) for which a variogram model is fitted using the UZD data that are not affected by permanent pumping, as it is described in Sect. 2.1.2 (Fig. 2b and d). The kriging methodology consists of solving the following equation system:

$$\left\{\begin{array}{l}{Z}^{\ast }=\sum _{\mathit{\alpha }}{\mathit{\lambda }}_{\mathit{\alpha }}{Z}_{\mathit{\alpha }}+m{\mathit{\lambda }}_{\mathrm{m}},\\ \mathrm{Var}R=C_{\mathrm{00}}-\sum _{\mathit{\alpha }}{\mathit{\lambda }}_{\mathit{\alpha }}{C}_{\mathit{\alpha }\mathrm{0}},\end{array}\right.$$

where Z^∗ is the estimation, α is the observation point, λ_α and λ_m are the weights for observation point and mean value, R is the residual value (i.e., absolute error), and C₀₀ and C_0α are the covariance function values for the origin and the α point. Therefore, the value of R is not determined by solving this system, only the variance of R is calculated.

The final water table map is obtained using the reference DEM, from which the UZD kriged map is deduced.

Figure 2Experimental variogram and fitted variogram model of unaffected UZD data and Gaussian score, for the LWC dataset (a, b) and HWC dataset (c, d).

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3.1 Paris urban area

The Seine and Marne rivers constitute a meandering fluvial system flowing from the southeast to the northwest (Fig. 3). The confluence between the Marne and the Seine River is located in the southeast of the study area.

Figure 3Alluvial plain and regional substratum of the Paris urban area. Piezometers location for the two campaigns: low water campaign (LWC) and high water campaign (HWC).

The alluvial plain of the Seine and Marne rivers is overlying incised valleys of Eocene to Oligocene sedimentary series, exposing late Lutetian limestones in the south of the study area and early Bartonian limestones in the north (Fig. 3). Alluvial sediments constitute the alluvial aquifer and substratum of the Seine and Marne rivers. Lutetian and Bartonian limestones are underlying aquifers which are separated by thin heterogeneous and discontinuous formations with low hydraulic conductivity. The high proportion of soil sealed areas and the proliferation of pumping wells due to the urbanization reduce the infiltration of rainfall, making the anthropogenic pressure the main controlling factor of the water table and SW-GW connection status.

Water tables have been monitored by water managers since the 1970s in central Paris area and since the 2000s in suburban areas. Water managers noticed that the water table of alluvial aquifer in the central area is usually stable at very low levels such that drying out of superficial aquifers may occur. The water table of peripheral areas remains unaffected by such water table drawdown.

Figure 4Temporal analysis charts for SW-GW connection status determination: (a) recorded time series for river, disconnected piezometer and connected piezometer; (b) relationship between standardized river level and UZD for the disconnected piezometer and connected piezometer.

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Regardless of the groundwater context, the Seine River is fully embanked, and the river bottom is periodically dredged for navigation purposes along the Paris city crossing. Given those anthropogenic forcings, water managers suspect that the Seine River may be disconnected from its underlying alluvial aquifer in some parts of the Paris central area. The interpretations of the joint response of the alluvial aquifer to hydrological events during the 1990–2018 period described further are supported by monthly records for UZD at monitored piezometers located near the Seine River. Seine River water levels vary under two different hydrological regimes (Fig. 4a): one nominal hydrological regime, corresponding to the low-flow periods during which the river flow and water level are artificially regulated for navigation and water management purposes, and one flood regime during which the river water level reaches the flood peak (eventually causing flood damages). The UZD can vary in relation with the Seine River water level or not. In the case of no variation of UZD, it is assumed that UZD is regulated by the GW pumpings, inducing a disconnection between the water table and river water level.

3.2 UZD datasets: low- and high-flow campaigns

This study is based on two UZD snapshot campaigns (Table 2) involving measurements in piezometers that are not periodically monitored (Fig. 3): the low water campaign (LWC), which gathered 314 measurements during the low-flow period of October 2015, and the high water campaign (HWC), gathering 202 measurements during the June 2016 flood event. Both campaigns lasted about a week. HWC took place a few days after the flood peak (1750 m³ s⁻¹) was reached at the Parisian Austerlitz gauging station. The two datasets include around 22 % of samples affected by anthropic pressure (pumping wells and underground structures) (Table 2). Most of these samples are located in the Paris central area where the water table is affected by permanent pumping.

Table 2Overview of the samples used for the application of water table mapping.

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3.3 Reference DEM for each campaign

The used DEM is the IGN scan 25 (IGN, 2015). As previously mentioned, the DEM is first merged with hydrological data specific for each campaign. Then it is smoothed using a 325 m research radius for moving-average filtering.

River water levels are deduced from the recorded data of six discharge gauging stations. The distribution of river water levels is interpolated using a constant gradient between each gauging station. During low-flow periods, the average gradient value is 0.01 ‰. The Seine River discharge is regulated through a series of locks and dams for navigation purposes. At each lock station, water levels are maintained at a given elevation below a threshold water flow of 600 m³ s⁻¹ at the Paris Austerlitz station. When a flood occurs as in June 2016, the lock stations are opened, and the water surface returns to its natural 0.2 ‰ gradient. During the LWC, the Seine River discharge was 160 m³ s⁻¹, so that all locks were up, while they were open during the HWC, when the average discharge still reached 1000 m³ s⁻¹ a week after the flood peak.

3.4 Variograms

The experimental variograms and the associated fitted variogram models are depicted in Fig. 2. For both datasets, the shape of the variograms is similar, with a sharp increase in the semi-variance near the origin, followed by a smooth evolution until it reaches the sill value. The range is higher for the LWC than for the HWC. The range of the Gaussian score is 12 km for LWC and 5 km for HWC (Fig. 2a and b). The range of the raw data is 2 km for HWC while it is 6 km for the LWC. It can be noted that the variographic models for unaffected UZD data differ between LWC and HWC datasets in terms of sill value (Fig. 2a and b). The sill value for the variogram model of the unaffected UZD HWC dataset is 8 m² while it is 5 m² for the unaffected UZD LWC dataset. This can be due to either the lower number of samples collected during the HWC, or to variations in the inner structure of the flow propagation process (Chen et al., 2018; Samine Montazem et al., 2019). For both campaigns (LWC and HWC), the variogram models of Gaussian scores have a range larger than the one of unaffected UZD datasets. This is due to the increase in the spatial correlation of the variable once the unaffected UZD data are transformed into Gaussian data.

3.5 Assessing the disconnection criteria

The Gaussian simulations are run on a 25 m × 25 m grid that matches the DEM resolution. The average of the hundred UZD values is subtracted to the smoothed DEM that includes river water levels evaluated for each hydrological context (LWC and HWC). The streambed of the Seine River consists of mixed fine sand and silt. The a priori capillary fringe height is within 0.1 and 1.0 m (Table 1). Therefore, the a priori value for the disconnection criteria is between 0.2 and 2 m. The available observed data are composed of monthly measurements of UZD among 26 piezometers during the 1990–2018 period. These piezometers are distributed along 18 cross sections of the Seine River. For each piezometer, standardized UZD and river water level values are calculated. As described in Sect. 2.2.1, the SW-GW connection status can be deduced from the relation between UZD and river water level. Two classes of piezometers are identified given the linear regression between standardized UZD and standardized river water level: disconnected piezometers and connected piezometers. Please note that during disconnection, the flow rate is still related to the hydraulic head difference. The transition cases are therefore included in the connected piezometer group. In the case of a significant slope of the regression line (>0.57), the piezometer is considered connected. It is disconnected otherwise. A total of 15 piezometers are considered as connected and 11 piezometers are considered as disconnected. Therefore, 9 cross sections along the Seine River are connected and 9 sections are disconnected (Fig. 5a).

Figure 5Graphical representation for disconnection criteria adjustment: (a) map of observed SW-GW status related to estimated SW-GW connection status using the optimal 0.75 m value for disconnection criteria; (b) relative number of valid SW-GW connection status out of nine disconnected cross sections and nine connected cross sections.

As an example, two contrasted situations among the 26 piezometers are displayed (Fig. 4a). The UZD measured in the blue piezometer is linearly related to the river water level, while the UZD measured in the red piezometer remains roughly constant. There is a linear regression between UZD measured in the blue piezometer (Fig. 4b) which confirms that the blue piezometer is connected. In the case of disconnected piezometers, a constant UZD value is measured for most samples. It indicates that UZD is regulated artificially (Fig. 4a).

3.6 Sensitivity analysis of the disconnection criteria

The distribution of SW-GW connection status is constrained by the disconnection criteria. To estimate this criteria, a sensitivity analysis is achieved. The tested values range from 3 to 0.5 m with a 0.1 m step. This analysis shows that it is not possible to validate the connection status for all cross sections. Therefore, we compare the relative numbers of matched connected cross sections and disconnected cross sections. When the relative numbers are equal, the optimal value is reached, maximizing the total number of sections for which the connectivity status is correctly predicted. Different methods to evaluate the best criteria can be used (e.g., only valid disconnected cross sections or valid connected cross sections). In this study the maximization of relative number of connected and disconnected sections is used in order to obtain an average value of the disconnection criteria that does not favor either disconnection or connection. A way to obtain a better validation of cross section would be to spatialize the disconnection criteria. However, the spatialization of the disconnection criteria must be supported by geological arguments. This optimal value is 0.75 m (Fig. 5b). This value is used to obtain the final water table maps. The value of the disconnection criteria impacts the length of disconnected reaches. When the value for disconnection criteria is overestimated, the length of disconnected reach is underestimated. Contrarily, when the value is underestimated, the length of disconnected reaches is overestimated. In the application presented here for LWC, the length of disconnected reach for a 3 m disconnection criteria value is 150 m in the central area, while it reaches a 6 km length when the disconnection occurs for a 0.25 m disconnection criteria. When the optimal value of 0.75 m is applied for disconnection criteria, the length of disconnect reach is 5 km.

Figure 6Final water table maps obtained using ordinary kriging of UZD deduced from the reference DEM for (a) LWC and (b) HWC. The disconnected reaches of the river network are indicated in red for a disconnection criteria of 0.75 m.

Further investigation could be carried out to evaluate the reliability of the estimated disconnection criteria, comparing it with the application of other methodologies such as those described in Lamontagne et al. (2014). Though this would allow for the determination of the SW-GW flow rate and hydrogeological dynamics, it cannot be applied in our case study context given that there are no data about the riverbed hydraulic conductivity. Such development would constitute a supplementary step after the water table mapping toward the description of the hydrological functioning of the study area.

3.7 Final mapping integrating SW-GW connectivity

The most important GW hydraulic gradients (1 %) are located close to the Seine and Marne rivers and the areas with an important topographic gradient (Fig. 6). The lowest values of hydraulic gradient are between 0.1 ‰ and 1 ‰ with an average 0.6 ‰ value in rather flat alluvial plains in the north area and the southeast area. The global flow pattern is therefore driven by SW-GW connection status and topography, with the exception of the central area where permanent pumping generates significant water drawdown and a subsequent SW-GW disconnection. In this area, the difference between riverbed elevation and estimated water table is 4 m. The implementation of disconnected reach during final mapping is a key element to reflect the specificity of urban groundwater such as water drawdown caused by pumping wells. The mapped water table near the disconnected reach is only affected by the observed depletion of the water table in wells and dry wells. All disconnected sections are located in the central area during both campaigns. The rise in river water levels during HWC modifies the water table map significantly, especially in the vicinity of the river.

The SW-GW relation type for connected sections (gaining, losing and asymmetrical) is established regarding the head difference between the river water level and water table at a 50 m distance (two pixels of the map) from the river center-line. In cases where the river water level is higher than the water table for both river banks, the river is losing water to the aquifer. In the opposite case, the river is gaining groundwater. The sections where the river is gaining on one bank and losing on the other were also identified.

The main effect of flood events is to increase the river water level. In connected sections, the increase in hydraulic gradient between river and water table favors the river infiltration toward the aquifer. Comparing Fig. 6a with Fig. 6b, it appears that this infiltration causes the switching from losing to gaining SW-GW relation type during flood events. This is supported by the reduction of the total length of losing sections during LWC (Table 3). The hydrogeologic flow associated with the increase in infiltration induces the reconnection process propagating from the losing sections toward the disconnected sections.

As a consequence, almost the whole river network is reconnected to the GW, leading to a rise of the mapped water table. As pumping is increased during a flood to avoid damages against the buildings and underground infrastructures, a small portion of the Seine River remains disconnected in central Paris (0.75 km; see Fig. 6b).

Table 3Length of disconnected sections, connected gaining sections and connected losing sections, and head difference between the water table and riverbed calculated from preliminary mapping, for LWC and HWC.

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