Constraining a density-dependent flow model with the transient electromagnetic method in a coastal aquifer in Mexico to assess seawater intrusion

Electrical resistivity is affected by many parameters, i.e., mostly water content, permeability, composition, temperature, pressure and salinity; thus, electric and electromagnetic methods are widely used to obtain a better insight into aquifer systems. The transient electromagnetic method is typically used to infer the subsurface electrical resistivity distribution at relatively shallow depths (i.e., less than 1 km). A stable current is circulated through a cable loop (transmitter). When the current is abruptly interrupted, a magnetic field is induced, which is damped and disappears quickly. The rapid variation in the magnetic field generates eddy currents into the subsurface that responds to different electrical properties at different depths, and its temporal evolution can be measured by a receiver, usually using the same primary circuit that originated the magnetic field (a cable coil) or another device (e.g. induction magnetometer). The process is repeated several times, applying a stacking process to generate a single mean curve showing the evolution of the induced potentials as a function of time. Finally, this master curve is inverted to establish the geoelectrical model (i.e., true resistivity values at different depths) that fits the subsurface response.

In the La Paz area, a TEM survey was carried out using Sirotem-S equipment, with a single loop configuration (i.e., the same coil was used as transmitter and receiver) with dimensions of 150 m × 150 m, reaching a maximum interpretation depth of 500 m. In total, 50 soundings were acquired, measuring eight curves per location across the study area and complemented with 20 TEM soundings measured during a previous survey (CIGSA 2001). The TEM data were inverted to a 1D model employing an Occam regression, fitting the model responses with a root mean square error (RMSE) of less than 5%. Regarding the depth resolution, it was performed following the proposal by Flores et al. (2013) in order to assure the reliability of the models at depth. The subsurface electrical resistivity distribution was inferred by interpolating the information provided by all the 1D models at different depths using a Kriging scheme. Nevertheless, all the interpretation of the geoelectrical images was performed using only the covered area by the TEM soundings. Geoelectrical anomalies that were not supported by at least two or more soundings were ignored.

Based on the geophysical interpretation, the hydrogeological units were identified, and the geometry of the 3D geomodel was built. In aquifers with seawater intrusion problems, a variable-density groundwater flow may be simulated by a dispersive solute-transport approach. In such an approach, fluid density can vary continuously from cell to cell in a model domain (De Filippis et al. 2016). Thus, the finite element density-dependent subsurface flow code FEFLOW (Diersch 2014) was used for modeling the coastal aquifer of La Paz under the assumption that: (1) the porous medium is considered rigid and incompressible (Boussinesq approximation), (2) fluid motion can be described by Darcy’s law, and (3) dispersion follows the Fickian dispersion law (Diersch 1988; Kourgialas et al. 2016).

Groundwater movement through porous materials of constant density in transient can be described in three dimensions using the general groundwater-flow equation, which is based on Darcy’s law (Eq. 1):where k is hydraulic conductivity (LT⁻¹), h is hydraulic head (L), W is volumetric flux per volume unit (representing sources and/or sinks) in the coordinates x, y (horizontal plane) and z (vertical), S_s is specific storage and t is time (T) (Delleur 2016; Khayyun 2018). The generalized form of Darcy’s law for variable-density fluid flow is derived as follows (Eq. 2):where ρ and ρ^w are the macroscopic mass density of the fluid and the mass density of water (M L⁻³), respectively, g is the vector of gravity acceleration (L T⁻²), p the dynamic pressure (P) and μ is the dynamic viscosity of the fluid (P T⁻¹; Kolditz et al. 1998).

In the La Paz area, the model domain area covers an extension of approximately 475 km² and a depth of 500 m below ground level. As a first approximation, the discretization in the horizontal plane consisted of finite triangular elements with refining zones of approximately 100 m around pumping wells, verifying the Delaunay criterion violations (Weatherill 1992; Rodríguez and Silveira 2017). The vertical discretization of the model area consisted of dividing the model area into four uniform hydrogeological layers (some of these units also have subunits due to the presence of water). The properties of the hydrogeological units were obtained, linking the resistivity maps with well log information from exploration wells and from geological sections defined in previous studies of the area (Hausback 1984; Aranda-Gómez and Pérez-Venzor 1988; SGM 1995; CIGSA 2001; Cruz-Falcón 2007). For the top elevation, a digital elevation model (DEM) with a pixel resolution of 15 m × 15 m and an RMSE of 4.8 m was used (INEGI 2013). To improve the calibration in the north and south areas of the model domain, the mesh was refined. The final configuration resulted in 51,375 nodes and 80,972 elements (Fig. 2).

A first-type flow boundary (Dirichlet) was set along the coastline to the north of the study area (zone 1, Fig. 2). For the first layer, the boundary was set to zero, and for the rest of the layers, the boundary was set to the equivalent freshwater head assuming a seawater density of 1,025 kg m^–3. Second-type flow boundaries (Neumann) were set along lateral limits (zone 2, Fig. 2): in the eastern flank of the model due to the presence of relatively impermeable bedrock (mainly granite and gabbro) along a fault (La Paz fault) which creates a physical boundary condition (acting as a barrier) to the alluvial aquifer; in the western flank of the model, groundwater flows parallel to topographic lows which forms an intermittent streamline (Fig. 1) which, added to the lack of wells close the aquifer limit, serves as groundwater divides, forming a hydraulic no-flow boundary. A first-type flow boundary was set along the southern flanks of the study area where inflow is expected (zone 3), with initially estimated at 70% of the aquifer recharge (Cruz Falcón et al. 2013). All the 147 pumping wells were included in the model using well depths (7–201 m), extraction rates (0.3–50 L/s), and tube radius (0.07–0.17 m). Table 1 shows the properties of the different hydrogeologic units used in the model domain such as the initial values for hydraulic conductivity, the specific storage, and the porosity.

Spatial variable recharge rates were assigned as the annual inflow parameter on the top layer of the model. The different recharge sources included the following: recharge by rainwater infiltration, irrigation return flow in agricultural lands, and leaks from the water system in urbanized areas. The recharge values from precipitation were obtained from an annual hydrological balance (Torres-Martínez 2017). The use of monthly recharge rates was discarded due to the availability of groundwater extractions only at the annual level—for example, for 1997 (Fig. 3), which was an average year, the recharge values varied from 14 mm in the valley floor to 46 mm in the mountain ranges. The annual recharge values were assigned by zones according to the different land uses and soil types in the study area, adding an extra recharge flux in the urban areas caused by water leaks from the obsolete urban water supply system, which has been estimated by the local authority at 31.8%, on average, of the total water supplied to La Paz (SAPA 2015). The additional recharge flux assigned in the irrigated agriculture area was between 15 and 35% of the total water used for irrigation, according to the different crop type in areas with similar climatic characteristics. Groundwater inflow coming from the western flank of the aquifer was estimated according to values reported by Cruz Falcón et al. (2013), who estimated the horizontal inflows through pumping tests.

A two-stage calibration was performed. First, the flow model was calibrated for the steady-state mode by matching the hydraulic-heads calculated by the model to the measured groundwater levels of 42 wells, which belong to the monitoring network of CONAGUA. It is important to mention that water level measurements have been performed approximately once a year by CONAGUA using water level meters (manual method) and taking the reference level elevation derived from differential GPS surveys. Although there are temporal data about the hydraulic-heads in the majority of these wells, CONAGUA has not been used to perform a rigorous seasonal periodicity regarding these water-table measurements. Thus, the selected year for steady-state conditions was 1997 because it was the year with the most hydraulic head coverage data. The hydraulic conductivity and lateral influx rates in the south were used as calibration parameters for the flow model. Second, based on the outcomes of the first stage, the flow model was calibrated in transient-state mode considering 20 years (1990–2010), featuring an automatic time-step control scheme on behalf of the stability given to the model and natural convergence.

For the purpose of evaluating the effectiveness of the model, a qualitative and quantitative assessment was performed. The qualitative assessment (calibration process) consisted of comparing measured versus simulated hydraulic heads of wells to identify outliers and bias of the model by comparing it graphically against a 1:1 line (Piñeiro et al. 2008); furthermore, time series allow us to identify where the model is more accurate (Vaze et al. 2011). Once the hydraulic heads were adjusted from the steady-state simulation, a quantitative assessment was carried out comparing measured versus calculated hydraulic-heads using statistical techniques such as the coefficient of correlation (r²), the RMSE and the Nash-Sutcliffe Goodness of fit (NSE-GOF; Eq. 3), which are useful for describing the degree of collinearity between simulated and observed values (r²), identifying the error measured in the units of interest (RMSE), and evaluating the prediction of the model (NSE-GOF; Moriasi et al. 2007; Ritter and Muñoz-Carpena 2013; Chai and Draxler 2014). The term SD in Eq. (3) denotes the standard deviation of the observed values.

The inverted TEM soundings (1D models) were interpolated to generate resistivity slices at different depths, providing total spatial coverage of the subsurface electrical distribution over the aquifer exploitation area. The resistivity maps were used to correlate the original geological model, including the fault system (Table 2), as well as to delineate the seawater intrusion.

As an example, Fig. 4 shows the cross-section A–A′ where nine TEM soundings were interpolated to generate a pseudo 2D model. This cross-section reaches approximately 500 m deep and 22 km inland in the NW–SE direction starting from the coastline (A–A’ location is shown in Fig. 5). The shallowest layer shows a lateral variation with resistivity values ranging from 5 to 20 Ω m in the northwestern portion and more than 20 Ω m in the opposite site of the profile. This behavior could be associated with recent sediments (Unit U1a). Additionally, the lower resistivities measured in these materials may suggest intrusion of the seawater in unit U2 inland (red arrow). Another subunit has been defined as U1b, related to sands and altered rocks with resistivities ranging from 15 to 50 Ω m. Underlying these horizons, unit U3 is defined as a low-medium resistivity layer that is most likely associated with sands and rhyolitic boles in a clay matrix of low permeability. This unit is probably responsible for preventing the inland advance of seawater in unit U2. Underneath this layer, subunit U4a is found in the southern portion, mainly composed of sandy-clayey materials with resistivity values less than 15 Ω m. Although groundwater is currently extracted from subunit U1b (the maximum depth of the wells in this subunit is approximately 200 m), subunit U4a may be considered a potential groundwater source. A different horizon (U4b) is found below 300 m depth, showing resistivities ranging from 15 to 35 Ω m, most likely associated with sandy-clayey materials. Finally, in the southeastern portion of the profile, the granitic intrusive basement is observed as a high resistivity anomaly with values greater than 100 Ω m.

On the horizontal slices, the low-resistivity zone (subunit U1a) is found in the northern area along La Paz Bay. This conductive zone extends to a depth of approximately 75 m (Fig. 5) and most likely indicates the presence of brackish water as a result of seawater intrusion into the sandy material and altered rocks. Between depths of 100 and 200 m, unit U3 is predominant in the central region of the area and can be associated with agglomerates and sandy materials; however, in the southern region, low permeability sands and igneous rocks are inferred (Fig. 5). Below this horizon, between 200 to 300 m, there is a layer characterized by the presence of brackish water in the northern portion and clayey matrix rocks in the middle of the valley. Nevertheless, at the bottom of this horizon (between 250 and 300 m), a sandy-clayey layer appears and extends from the center to the SW. The deepest layer reached by the geoelectrical study (400–500 m) suggests the presence of granitic igneous rocks, which are considered the basement of the aquifer. The results of the horizontal depth slices were compared with borehole information and were used as an input for groundwater flow modeling.

Figure 5 clearly shows the influence of the saline intrusion in the northern portion of the La Paz aquifer in Baja California (<10 Ω m). It is possible that other structures are buried by the volcanic breccia to the NW of the study area, in such a way that the marine intrusion can extend to the SW of the coastline. At greater depths, the media becomes more resistive, whereby the influence of the marine intrusion only developed in the shallower units. Figure 5 also shows high-resistivity values in the shallower layers that cover most of the study area, and could be associated with the low water content in the volcanic rock.

According to the geoelectric study, the system works as an unconfined aquifer consisting of granular materials (gravels and sands) in the groundwater exploitation area, which is underlain by a semiconfined aquifer formed by a layer of fractured volcanic rock. Finally, the basement is identified as granitic intrusive and compact metamorphic complexes; furthermore, these two aquifers (the shallow overexploited aquifer and the deeper unexploited aquifer) have variable depths. The first aquifer is quite horizontally homogenous reaching almost 200 m of thickness, while the second aquifer varies from 200 to 400 m depth, which is in agreement with the results from previous studies (e.g. Cruz-Falcón et al. 2010).

Calibration of the flow model was developed in two phases. In the first stage, flow model calibration under steady-state conditions was performed using the hydraulic head values of 42 observation wells across the domain area recorded by CONAGUA in 1997 (Fig. 3). Once the results were obtained, the performance of the model was evaluated first by a visual inspection through a scatter plot graph between observed and simulated head values looking for an adjustment close to the 1:1 line (Fig. 6), which helped to identify the presence of outliers or bias in the model (Moriasi et al. 2007; Ritter and Muñoz-Carpena 2013). Following this evaluation, a statistical assessment was completed, obtaining a good fit reflected in the values of r² = 0.96, RMSE = 8.11 m, and NSE-GOF = 0.99. Accordingly, the groundwater flow model can be considered a model that successfully reproduces steady-state conditions.

The second-stage calibration of the flow model consisted of calibrating the model in the transient-state mode, in which an annual variability flow boundary condition was considered in the southern model boundary. In this step, the number of wells used was reduced from 42 to 20 due to a lack of information over the period. The results obtained for this calibration showed a lower percentage of adjustment in comparison with the steady-state condition (r² = 0.85, RMSE = 3.53 m, NSE-GOF = 0.75); however, these numbers can be considered satisfactory adjustment values. With this estimation achieved, it is possible to simulate and infer the tendency of the groundwater level fluctuations for each well and, therefore, the model can be further used for forecasting, taking into account the respective uncertainty analysis, which can be computed using different techniques such as null-space Monte Carlo method and linear uncertainty analysis, among others (Herckenrath et al. 2011; Sepúlveda and Doherty 2015).

Figure 7 depicts the observed versus simulated groundwater levels for six selected wells in different zones (urban, agricultural, and coastal zones) along the study area. Overall, almost all the wells located in the urban zone of La Paz showed a small water-table rise between 2000 and 2010. In fact, well 416 (located in the eastern part of the city) showed the largest water-table rise during this period, increasing from 1.7 to 10.0 m asl. Previous work indicated that the chemistry of the groundwater in the urban zone is similar to that of the groundwater in the recharge areas of El Novillo and Las Cruces ranges (Tamez-Meléndez et al. 2016). In addition, the city of La Paz is supplied by groundwater taken from wells located in those recharge zones, which confirms that the increase in the water table in the urbanized zone is due to the infiltration from leaks of the obsolete water supply system, which was estimated on the order of 6.9 million m³ year^–1. On the other hand, the wells located in the agricultural zone showed a general downward trend in the water table from 2000 to 2010 (the water table values of wells 152 and 205 are examples of this trend), with groundwater levels between 2 and 8 m below sea level (m bsl) on average. However, the wells located close to the coastline showed no significant temporal changes in the water table (probably due to seawater recharge), with levels that varied between 3.0 m asl and 1.6 m bsl (wells 368 and 197-A, Fig. 7).

Figure 8a illustrates the hydraulic head contours in slice 1 (50 m depth), which shows the zero head extending from the coastline to the east-central portion of the valley, approximately in the area between the city of La Paz and the agricultural zone. This finding suggests that the hydraulic gradient has been inverted due to the cone of depression caused by intensive pumping in the agricultural area, thereby leading a flow coming from sea to inland. Based on the water-table levels discussed previously and contrasting them with the simulated hydraulic heads in Fig. 8a and the calculation of backward streamlines generated from each pumping well in Fig. 8b, an important flow coming from the southeastern region of the study area—recharge area of El Novillo mountain range (Fig. 1)—could be inferred. This flow has helped to slow down the advancement of the freshwater–saltwater interface between La Paz (towards the south of the city) and the agricultural lands, mitigating the saline intrusion. According to the model calculations, the groundwater residence time extends up to 1,500 years (Fig. 8b). Figure 8a also shows that some local flow lines are originating at the coastline and connecting with the city of La Paz, indicating pathways for saline-water intrusion towards the city. Coinciding with the interpretation of Figs. 7 and 8a,b, the inversion of the hydraulic gradient has caused the wells located in the agricultural area (towards the southwest of the city of La Paz) to extract groundwater with flow patterns come from both the coastline and the southern/southeastern portion of the study area, promoting the mixing of freshwater with seawater and, therefore, increasing the salinity gradient in the aforementioned overexploited areas.

In order to evaluate the historical evolution of seawater intrusion, different isolines were generated for measured chloride concentrations (250, 500, 750, 1,000 and 1,500 mg L^–1) in the study area from 1996 to 2015 (Fig. 9). The isolines were created by interpolation of point values through the inverse distance weighting method. The chloride data of the years 1996, 2001 and 2005 were provided by CONAGUA, chloride data of 2010 were obtained from CONAGUA and Rosales-Ramírez (2012), whereas the data of 2013 and 2015 were obtained from published (Tamez-Meléndez et al. 2016) and unpublished works performed by our team. The changing monitoring network of chloride concentration over time represents a limitation.

Of these isolines, the concentration of 250 mg L^–1 is an essential value because it is the taste-based recommended concentration of chloride in drinking water established by the World Health Organization (WHO 2011), whereby beyond this level, the taste could usually be affected. The results indicate that from 1996 to 2010, the green area (<250 mg L^–1) tends to cover a wider area of La Paz, whereas the yellow isoline (between 750 and 1,000 mg L^–1) tends to disappear in the city area, which suggests the retreat of the freshwater–saltwater interface in the city during this period. This finding agrees with the water-table increase in the wells located in the city (Fig. 7), which is caused by artificial recharge coming from the water-supply-system leaks. Conversely, the red area (>1,500 mg L^–1) shows an increasing trend in the center of the irrigation district from 1996 to 2001, which agrees with the descending water-table behavior of the wells located in the agricultural zone between the years 1995 and 2000. Thus, the enhancement of the >1,500 mg L^–1 isoline is undoubtedly due to the advancement of the freshwater–seawater interface as a consequence of the heavy groundwater extraction for irrigation purposes. Towards the south, the green area (<250 mg L^–1) has progressed in the same period from Carrizal Valley towards the city of La Paz, oscillating between 6.5 and 8 km from the southern part of La Paz Bay. This isoline is mainly affected by two groundwater regional flows coming from the southern and southeastern parts of the aquifer, as previously described. In fact, this portion of the aquifer hosts the main production wells operated and exploited by the water utility of La Paz, who maintain the chloride concentrations below the taste-based limit mentioned by WHO, thereby satisfying the current water quality conditions for the public drinking water system. Otherwise, the isoline of 1,000 mg L^–1 (concentration considered to be the limit between freshwater and brackish water) showed an advance in 2001 with regard to 1996. However, this isoline has been retreating since 2001, showing a reduction of 2.1 km in the area near El Centenario and 5.3–6.5 km to the east of this community in 2015. Nevertheless, despite the overexploitation of this aquifer system, almost all the chloride isolines have retreated in 2015 in comparison with those in previous years, indicating that there was an efficient aquifer recharge in the last several years that diluted the groundwater.

One of the possible causes of the retreat of the seawater intrusion interphase can be attributed to the fact that in periods prior to those included in this work, some prolonged droughts were present since the 1950s in northern Mexico, as described in other works (CONAGUA 2014; Domínguez 2016). Those droughts could have caused water stress on the aquifer favoring the advancement of the seawater intrusion, and after this stress, the aquifer has been recovering to its natural condition. In addition, another variable to consider is that there was no proper regulation and control of groundwater resources in Mexico until 1992. In that year, the Law on National Waters of Mexico was created, and it was not until 1994 when a national registry of wells was implemented, where the number of wells, the different uses and the allocation of groundwater was reported for each aquifer (CONAGUA 2012; Pineda Pablos et al. 2014). This information leads to an additional consideration of droughts because excessive exploitation was carried out in the years before those considered in this study. However, these policies were enacted according to political and economic needs, sometimes obviating the aquifer overdraft (Hoogesteger 2018; Bhattacharjee et al. 2019). Another factor to take into account in the retreat of the freshwater–saltwater interface, is the occurrence of high precipitation events (hurricanes) in the south of Baja California, which increases precipitation significantly in a short time compared to the monthly average precipitation (Fig. 10). These extraordinary high-precipitation events have been demonstrated to decrease the risk of a water crisis in other semiarid regions of northern Mexico (Sisto et al. 2016). However, the contribution of hurricanes is not a piece of conclusive evidence for the aquifer recharge in this region, because the annual precipitation values do not show an increased tendency over time. Therefore, the contribution of the three mentioned factors (absence of extended drought periods, implementation of a national registry of wells and hurricanes) should be taken into account together as the probable explanation of the recent trend of the freshwater–seawater moving back to the coastline in the La Paz aquifer, despite the continuous overextraction of groundwater.