the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Groundwater withdrawal in randomly heterogeneous coastal aquifers

### Martina Siena

### Monica Riva

We analyze the combined effects of aquifer heterogeneity and pumping operations on seawater intrusion (SWI), a phenomenon which is threatening coastal aquifers worldwide. Our investigation is set within a probabilistic framework and relies on a numerical Monte Carlo approach targeting transient variable-density flow and solute transport in a three-dimensional randomly heterogeneous porous domain. The geological setting is patterned after the Argentona river basin, in the Maresme region of Catalonia (Spain). Our numerical study is concerned with exploring the effects of (a) random heterogeneity of the domain on SWI in combination with (b) a variety of groundwater withdrawal schemes. The latter have been designed by varying the screen location along the vertical direction and the distance of the wellbore from the coastline and from the location of the freshwater–saltwater mixing zone which is in place prior to pumping. For each random realization of the aquifer permeability field and for each pumping scheme, a quantitative depiction of SWI phenomena is inferred from an original set of metrics characterizing (a) the inland penetration of the saltwater wedge and (b) the width of the mixing zone across the whole three-dimensional system. Our results indicate that the stochastic nature of the system heterogeneity significantly affects the statistical description of the main features of the seawater wedge either in the presence or in the absence of pumping, yielding a general reduction of toe penetration and an increase of the width of the mixing zone. Simultaneous extraction of fresh and saltwater from two screens along the same wellbore located, prior to pumping, within the freshwater–saltwater mixing zone is effective in limiting SWI in the context of groundwater resources exploitation.

Groundwater resources in coastal aquifers are seriously threatened by seawater intrusion (SWI), which can deteriorate the quality of freshwater aquifers, thus limiting their potential use. This situation is particularly exacerbated within areas with intense anthropogenic activities, which are associated with competitive uses of groundwater in connection, e.g., with agricultural processes, industrial processes and/or high urban water supply demand. Critical SWI scenarios are attained when seawater reaches extraction wells designed for urban freshwater supply, with severe environmental, social and economic implications (e.g., Custodio, 2010; Mas-Pla et al., 2014; Mazi et al., 2014).

The development of effective strategies for sustainable use of groundwater
resources in coastal regions should be based on a comprehensive
understanding of SWI phenomena. This challenging problem has been originally
studied by assuming a static equilibrium between freshwater (FW) and seawater
(SW) and a sharp FW–SW interface, where FW and SW are considered as
immiscible fluids. Under these hypotheses, the vertical position of the FW–SW
interface below the sea level, *z*, is given by the Ghyben–Herzberg solution,
$z={h}_{\mathrm{F}}{\mathit{\rho}}_{\mathrm{F}}/\mathrm{\Delta}\mathit{\rho}$, where *h*_{F} is
the FW head above sea level and $\mathrm{\Delta}\mathit{\rho}={\mathit{\rho}}_{\mathrm{S}}-{\mathit{\rho}}_{\mathrm{F}}$ is the density contrast, *ρ*_{S} and
*ρ*_{F}, respectively, being SW and FW density. Starting from these
works, several analytical and semi-analytical expressions have been developed
to describe SWI under diverse flow configurations (e.g., Strack, 1976; Dagan
and Zeitoun, 1998; Bruggeman, 1999; Cheng et al., 2000; Bakker, 2006;
Nordbotten and Celia, 2006; Park et al., 2009). In this broad context,
Strack (1976) evaluated the maximum (or critical) pumping rate to avoid
encroachment of SW in FW pumping wells. All of these sharp-interface-based
solutions neglect a key aspect of SWI phenomena, i.e., the formation of a
transition zone where mixing between FW and SW takes place and fluid density
varies with salt concentration. As a consequence, these models may
significantly overestimate the actual penetration length of the SW wedge,
leading to an excessively conservative evaluation of the critical pumping
rate (Gingerich and Voss, 2005; Pool and Carrera, 2011; Llopis-Albert and
Pulido-Velazquez, 2013).

A more realistic approach relies on the formulation and solution of a variable-density problem, in which SW and FW are considered as miscible fluids and groundwater density depends on salt concentration. The complexity of the problem typically prevents finding a solution via analytical or semi-analytical methods, with a few notable exceptions (Henry, 1964; Dentz et al., 2006; Bolster et al., 2007; Zidane et al., 2012; Fahs et al., 2014). Henry (1964) developed a semi-analytical solution for a variable-density diffusion problem in a (vertical) two-dimensional homogeneous and isotropic domain. Dentz et al. (2006) and Bolster et al. (2007) applied perturbation techniques to solve analytically the Henry problem for a range of (small and intermediate) values of the Péclet number, which characterizes the relative strength of convective and dispersive transport mechanisms. Zidane et al. (2012) solved the Henry problem for realistic (small) values of the diffusion coefficient. Finally, Fahs et al. (2014) presented a semi-analytical solution for a square porous cavity system subject to diverse salt concentrations at its vertical walls. Practical applicability of these solutions is quite limited, due to their markedly simplified characteristics. Various numerical codes have been proposed to solve variable-density flow and transport equations (e.g., Voss and Provost, 2002; Ackerer et al., 2004; Soto Meca et al., 2007; Ackerer and Younes, 2008; Albets-Chico and Kassinos, 2013). Numerical simulations can provide valuable insights into the effects of dispersion on SWI, a feature that is typically neglected in analytical and semi-analytical solutions. Abarca et al. (2007) introduced a modified Henry problem to account for dispersive solute transport and anisotropy in hydraulic conductivity. Kerrou and Renard (2010) analyzed the dispersive Henry problem within two- and three-dimensional randomly heterogeneous aquifer systems. These authors relied on computational analyses performed on a single three-dimensional realization, invoking ergodicity assumptions. Lu et al. (2013) performed a set of laboratory experiments and numerical simulations to investigate the effect of geological stratification on SW–FW mixing. Riva et al. (2015) considered the same setting as in Abarca et al. (2007) and studied the way quantification of uncertainty associated with SWI features is influenced by lack of knowledge of four key dimensionless parameters controlling the process, i.e., gravity number, permeability anisotropy ratio and transverse and longitudinal Péclet numbers. Enhancement of mixing in the presence of tidal fluctuations and/or FW table oscillations has been analyzed by Ataie-Ashtiani et al. (1999), Lu et al. (2009, 2015) and Pool et al. (2014) in homogeneous aquifers and by Pool et al. (2015) in randomly heterogeneous three-dimensional systems (under ergodic conditions). Recent reviews on the topic are offered by Werner et al. (2013) and Ketabchi et al. (2016).

Several numerical modeling studies have been performed with the aim of identifying the most effective strategy for the exploitation of groundwater resources in domains mimicking the behavior of specific sites. Dausman and Langevin (2005) examined the influence of hydrologic stresses and water management practices on SWI in a superficial aquifer (Broward County, USA) by developing a variable-density model formed by two homogeneous hydrogeological units. Werner and Gallagher (2006) studied SWI in the coastal aquifer of the Pioneer Valley (Australia), illustrating the advantages of combining hydrogeological and hydrochemical analyses to understand salinization processes. Misut and Voss (2007) analyzed the impact of aquifer storage and recovery practices on the transition zone associated with the salt water wedge in the New York City aquifer, which was modeled as a perfectly stratified system. Cobaner et al. (2012) studied the effect of transient pumping rates from multiple wells on SWI in the Gosku deltaic plain (Turkey) by means of a three-dimensional heterogeneous model, calibrated using head and salinity data.

All the aforementioned field-scale contributions are framed within a deterministic approach, where the system attributes (e.g., permeability) are known (or determined via an inverse modeling procedure), so that the impact of uncertainty of hydrogeological properties on target environmental (or engineering) performance metrics is not considered. Exclusive reliance on a deterministic approach is in stark contrast with the widely documented and recognized observation that a complete knowledge of aquifer properties is unfeasible. This is due to a number of reasons, including observation uncertainties and data availability, i.e., available data are most often too scarce or too sparse to yield an accurate depiction of the subsurface system in all of its relevant details. Stochastic approaches enable us not only to provide predictions (in terms of best estimates) of quantities of interest, but also to quantify the uncertainty associated with such predictions. The latter can then be transferred, for example, into probabilistic risk assessment, management and protection protocols for environmental systems and water resources.

Only a few contributions studying SWI within a stochastic framework have been published to date. The vast majority of these works considers idealized synthetic showcases and/or simplified systems. To the best of our knowledge, only two studies (Lecca and Cau, 2009; Kerrou et al., 2013) have analyzed the transient behavior of a realistic costal aquifer within a probabilistic framework. Lecca and Cau (2009) evaluated SWI in the Oristano (Italy) aquifer by considering a stratified system where the aquitard is characterized by random heterogeneity. Kerrou et al. (2013) analyzed the effects of uncertainty in permeability and distribution of pumping rates on SWI in the Korba aquifer (Tunisia). Both works characterize the uncertainty of SWI phenomena in terms of the planar extent of the system characterized by a target probability of exceedance of a given threshold concentration.

Our stochastic numerical study has been designed to mimic the general behavior of the Argentona aquifer, in the Maresme region of Catalonia (Spain). This area, as well as other Mediterranean deltaic sites, is particularly vulnerable to SWI (Custodio, 2010). We note that the objective of this study is not the quantification of the SWI dynamics of a particular field site. Our emphasis is on the analysis of the impact of random aquifer heterogeneity and withdrawals on SWI patterns in a realistic scenario. Key elements of novelty in our work include the introduction and the study of a new set of metrics, aimed at investigating quantitatively the effects of random heterogeneity on the three-dimensional extent of the SW wedge penetration and of the FW–SW mixing zone. The impact of random heterogeneity is assessed in combination with three diverse withdrawal scenarios, designed by varying (i) the distance of the wellbore from the coastline and from the FW–SW mixing zone and (ii) the screen location along the vertical direction. We frame our analysis within a numerical Monte Carlo (MC) approach.

Section 2 provides a general description of the field site, the mathematical model adopted to simulate flow and transport phenomena in three-dimensional heterogeneous systems and the numerical settings. Section 3 illustrates the key results of our work and Sect. 4 contains our concluding remarks.

## 2.1 Site description

To consider a realistic scenario that is relevant to SWI problems in highly exploited aquifers, we cast our analysis in a setting inspired from the Argentona river basin, located in the region of Maresme, in Catalonia, Spain (see Fig. 1a). This aquifer is a typical deltaic site, characterized by shallow sedimentary units and a flat topography. As such, it has a strategic value for anthropogenic (including agricultural, industrial and touristic) activities. The geological formation hosting the groundwater resource is mainly composed of a granitic Permian unit. A secondary unit of quaternary sediments is concentrated along the Argentona river.

Rodriguez Fernandez (2015) developed a conceptual and numerical model to
simulate transient two-dimensional (horizontal) constant-density flow in the
Argentona river basin across the area of about 35 km^{2} depicted in
Fig. 1b. The aquifer is heavily exploited, as shown by the large number of
wells included in Fig. 1b, mainly located along the Argentona river. The
latter is a torrential ephemeral stream, in which water flows only after
heavy rain events. On these bases, it is assumed that the only water intake
to the river comes from surface runoff of both granitic and quaternary units.
On the basis of transient hydraulic head measurements available at 21
observation wells (not shown in Fig. 1b) in the period 2006–2013, Rodriguez
Fernandez (2015) characterized the site through a uniform permeability, whose
value was estimated as
*k*_{B} = 1.77 × 10^{−11} m^{2}, and provided
estimates of temporally and spatially variable recharge rates, according to
land use or cover (see Fig. 1c).

Our numerical analysis focuses on the coastal portion of the Argentona basin (Fig. 1c). This region extends for about 2.5 km along the coast (i.e., along the full width of the basin) and up to 750 m inland from the coast. The offshore extension of the aquifer is not considered. The size of the study area has been selected on the basis of the results of preliminary simulations on larger domains, which highlighted that salt concentration values were appreciable only within a narrow (less than 400 m wide) region close to the coast. The vertical thickness of the domain ranges from 50 m along the coast up to 60 m at the inland boundary, the underlying clay sequence being considered to represent the impermeable bottom of the aquifer. SWI is simulated by means of a three-dimensional variable-density flow and solute transport model based on the well-established finite element USGS SUTRA code (Voss and Provost, 2002) over the 8-year time window 2006–2013. Details of the mathematical and numerical model are discussed in the following sections.

## 2.2 Flow and transport equations

Fluid flow is governed by mass conservation and Darcy's law:

where ** q** (L T

^{−1}) is the specific discharge vector, with components

*q*

_{x},

*q*

_{y}and

*q*

_{z}, respectively, along

*x*,

*y*(see Fig. 1c) and

*z*axes (

*z*denoting the vertical direction);

*C*(–) is solute concentration, or solute mass fraction, expressed as mass of solute per unit mass of fluid;

**(L**

*k*^{2}) is the diagonal permeability tensor, with components

*k*

_{11}=

*k*

_{x},

*k*

_{22}=

*k*

_{y}and

*k*

_{33}=

*k*

_{z}, respectively, along directions

*x*,

*y*and

*z*;

*ϕ*(–) is aquifer porosity;

*ρ*(M L

^{−3}) and

*μ*(M L

^{−1}T

^{−1}), respectively, are fluid density and dynamic viscosity;

*p*(M L

^{−1}T

^{−2}) is fluid pressure;

*g*(L T

^{−2}) is gravity;

*S*

_{S}(L

^{−1}) is the specific storage coefficient, and the superscript T denotes transpose. Equations (1) and (2) must be solved jointly with the advection–dispersion equation:

**D** (L^{2} T^{−1}) is the dispersion tensor defined as

where *D*_{m} (L^{2} T^{−1}) is the molecular diffusion
coefficient, and *α*_{L} and *α*_{T} (L),
respectively, are the longitudinal and transverse dispersivity coefficients.
Closure of the system of Eqs. (1)–(4) requires a relationship between fluid
properties (*ρ* and *μ*) and solute concentration *C*. Fluid viscosity
can be assumed constant in typical SWI settings and the following model has
been shown to be accurate at describing the evolution of *ρ* with *C*
(Kolditz et al., 1998):

where *C*_{S} is SW concentration. Table 1 lists aquifer and fluid
parameters adopted.

## 2.3 Numerical model

The domain depicted in Fig. 1c is discretized through an unstructured
three-dimensional grid formed by 101 632 hexahedral elements. The resolution
of the mesh increases towards the sea, where our analysis requires the
highest spatial detail. The element size along both horizontal directions *x*
and *y* ranges from 10 m (in a 200 m wide region along the coast) to 60 m
(close to the inland boundary). Element size along the vertical
direction is 2 and 4 m, respectively, within the first 10 m from the top
surface and across the remaining 40 m. These choices are consistent with
constraints for numerical stability, namely Λ_{L}≤4*α*_{L}, Λ_{L} being the distance between
element sides measured along a flow line and *α*_{L}=5 m. In
the absence of transverse dispersivity estimates and in analogy with main
findings of previous works (e.g., Cobaner et al., 2012), we set
${\mathit{\alpha}}_{\mathrm{T}}={\mathit{\alpha}}_{\mathrm{L}}/\mathrm{10}$.

A sequential Gaussian simulation algorithm (Deutsch and Journel, 1998) is
employed to generate unconditional log permeability fields, *Y*(** x**)=ln

*k*(

**), characterized by a given mean 〈Y〉=ln**

*x**k*

_{B}and variogram structure. Since no

*Y*data are available, for the purpose of our simulations we assume that the spatial structure of

*Y*is described by a spherical variogram, with moderate variance, i.e., ${\mathit{\sigma}}_{Y}^{\mathrm{2}}=\mathrm{1.0}$, and isotropic correlation scale

*λ*=100 m. This value is consistent with the observation that the integral scale of log conductivity and transmissivity values inferred worldwide using traditional (such as exponential and spherical) variograms tends to increase with the length scale of the sampling window at a rate of about 1∕10 (Neuman et al., 2008). The MC approach allows us to provide a probabilistic description of the SWI processes associated with space–time distributions of flux and solute concentration obtained in each random realization of the permeability field by solving the transient flow and transport Eqs. (1)–(5). Note also that

*λ*>5Δ in our model, Δ being the largest cell size within a 200 m wide region along the coast. A model satisfying this condition has the benefit of (a) yielding a proper reconstruction of the spatial correlation structure of

*Y*and (b) limiting the occurrence of excessive variations between values of aquifer properties across neighboring cells (Kerrou and Renard, 2010) in the region where SWI takes place.

Equations (1)–(5) are solved jointly, adopting the following boundary
conditions. The lateral boundaries perpendicular to the coastline and the
base of the aquifer are impervious; along the inland boundary we set a
time-dependent prescribed hydraulic head, *h*_{F} (with ${h}_{\mathrm{F}}=z+p/({\mathit{\rho}}_{\mathrm{F}}g$) ranging between 1.2 and 3.5 m), and
*C*=*C*_{F}; along the coastal boundary we impose a prescribed head,
*h*_{S}=0 (with ${h}_{\mathrm{S}}=z+p/\left({\mathit{\rho}}_{\mathrm{S}}g\right)$), and

** n** being the normal vector pointing inward along the boundary; i.e.,
the concentration of the water entering and leaving the system across the
coastal boundary is, respectively, equal to

*C*

_{S}and

*C*. The area of interest is characterized by five recharge zones, identified on the basis of the land use (inferred from the SIGPAC, 2005, dataset). The total recharge varies slightly in time (on a monthly basis), with mean value equal to 7.6 L s

^{−1}(see Fig. 1c). Initial conditions are set as

*h*=0 and

*C*=0, as commonly assumed in the literature (e.g., Jakovovic et al., 2016) and flow and transport are simulated for an 8-year period (2006–2013) with a uniform time step Δ

*t*=1 day. Varying the initial conditions (e.g., setting,

*h*=2.4 m, corresponding to the mean value of

*h*set along the inland boundary), did not lead to significantly different results at the end of the simulated 8-year period,

*t*=

*t*

_{w}(details not shown).

A pumping well is activated at time *t*_{w} and flow and transport are
simulated for 8 additional months. We consider three diverse pumping schemes,
reflecting realistic engineered operational settings. The borehole is located
along the vertical cross-section B–B^{′}, as shown in Fig. 2a. Scheme 1 (S1)
and Scheme 2 (S2) consider the pumping well to be placed 180 m away from the
shoreline, i.e., at *y*^{′} = *y*∕*λ* = 1.8, outside of the mixing
zone which is in place prior to pumping in all MC realizations. In S1
(Fig. 2b) the well is screened in the upper part of the aquifer, the screen
starting from 2 m below the top and extending across a total thickness of
15 m. The double-negative hydraulic barrier is a technology employed to manage (or possibly prevent) SWI. It
relies on the joint use of an FW (*production* well) and an SW
(*scavenger* well) pumping well. We employ this setting in S2
(Fig. 2c), which is designed by adding an additional screen in the lower part
of the aquifer to S1, starting from 34 m below the ground surface and
extending across a total thickness of 12 m. In S2, production and scavenger
wells are located along the same borehole, complying with technical and
economic requirements typically associated with field applications (Pool and
Carrera, 2009; Saravanan et al., 2014; Mas-Pla et al., 2014). This scheme is
also particularly appealing when applied to renewable energy resources, such
as the pressure retarded osmosis, which allows conversion of the chemical
energy of two fluids (FW and SW) into mechanical and electrical energy
(Panyor, 2006). Scheme 3 (S3, Fig. 2d) shares the same operational design of
S2, but the borehole is moved seawards, at *y*^{′} = 0.8, i.e., within the
mixing zone in place before pumping. In all pumping schemes, the withdrawal
rate is constant in time and uniformly distributed along the well screens,
where pumping is implemented by setting a flux-type condition. We set a total
withdrawal rate *Q* = 5 L s^{−1} at the upper well screen in S1, S2
and S3; an additional extraction at the same rate *Q* is imposed along the
lower screen in S2 and S3. During the pumping period, transport is simulated
using a time step Δ*t* = 30 min for the first month and with
increasing Δ*t* progressively up to a maximum value Δ*t* = 120 min for the remaining 7 months, as the system showed
progressively smoother variations while approaching steady state. In Sect. 3
all schemes are compared against a benchmark case, Scheme 0 (S0), where no
pumping wells are active.

## 2.4 Local and global SWI metrics

To provide a comprehensive characterization of SWI phenomena across the whole three-dimensional domain, we quantify the extent of the SW wedge and of the associated mixing zone on the basis of seven metrics, as illustrated in Fig. 3.

For each MC realization and along each vertical cross-section perpendicular
to the coast, we evaluate (i) toe penetration, ${L}_{\mathrm{T}}^{\prime}$, measured
as the distance from the coast of the iso-concentration curve *C*∕*C*_{S} = 0.5 at the bottom of the aquifer; (ii) solute spreading at the toe,
${L}_{\mathrm{S}}^{\prime}$, evaluated as the separation distance along the direction
perpendicular to the coast (*y* axis in Fig. 1c) between the
iso-concentration curves *C*∕*C*_{S} = 0.25 and
*C*∕*C*_{S} = 0.75 at the bottom of the aquifer; (iii) mean width of
the mixing zone, ${W}_{\mathrm{MZ}}^{\prime}$, i.e., the spatially averaged vertical
distance between iso-concentration curves *C*∕*C*_{S} = 0.25 and
*C*∕*C*_{S} = 0.75 within the region $\mathrm{0.2}{L}_{\mathrm{T}}^{\prime}$ ≤ *y*^{′} ≤ $\mathrm{0.8}{L}_{\mathrm{T}}^{\prime}$. All quantities ${L}_{\mathrm{T}}^{\prime}$,
${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$ are dimensionless, corresponding to
distances rescaled by *λ*, the correlation scale of *Y*. We also
analyze the (dimensionless) areal extent of SW penetration and solute
spreading at the bottom of the aquifer by evaluating (iv) ${A}_{\mathrm{T}}^{\prime}$
and (v) ${A}_{\mathrm{S}}^{\prime}$, as obtained by integrating ${L}_{\mathrm{T}}^{\prime}$ and
${L}_{\mathrm{S}}^{\prime}$ along the coastline. Finally, we quantify SWI across the
whole thickness of the aquifer by evaluating the dimensionless volumes
enclosed between (vi) the sea boundary and the iso-concentration surface
*C*∕*C*_{S} = 0.5, here termed ${V}_{\mathrm{T}}^{\prime}$, and (vii) the
isosurfaces *C*∕*C*_{S} = 0.25 and *C*∕*C*_{S} = 0.75,
here termed ${V}_{\mathrm{S}}^{\prime}$. First- and second-order statistical moments of
each of these metrics are evaluated within the adopted MC framework. With the
settings employed here, a flow and transport simulation on a single system
realization is associated with a computational cost of about 12 h (on an
Intel Core^{™} i7-6900K CPU@ 3.20 GHz
processor). Results illustrated in the following sections are based on a
suite of 400 MC simulations (i.e., 100 MC simulation for each scenario,
S0–S3). Details concerning the analysis of convergence of the first and
second statistical moments of the metrics here introduced are provided in
Appendix A.

## 3.1 Effects of three-dimensional heterogeneity on SWI

The effects of heterogeneity on SWI are inferred by comparing the results of
our MC simulations against those obtained for an equivalent homogeneous
aquifer, characterized by an effective permeability, *k*_{ef}. The
latter is here evaluated as ${k}_{\mathrm{ef}}={e}^{\u2329Y\u232a+{\mathit{\sigma}}_{Y}^{\mathrm{2}}/\mathrm{6}}=\mathrm{2.09}\times {\mathrm{10}}^{-\mathrm{11}}$ m^{2} (Ababou, 1996). A first
clear effect of heterogeneity is noted on the structure of the
three-dimensional flow field. This is elucidated in Fig. 4, where we depict
the permeability color map and streamlines (dashed curves) obtained at the
end of the 8-year simulation period along the vertical cross-section B–B^{′}
perpendicular to the coastline. Note that henceforth, results
are depicted in terms of dimensionless spatial coordinates, *y*^{′} = *y*∕*λ* and *z*^{′} = *z*∕*λ*. Flow within the equivalent
homogeneous aquifer (Fig. 4b) is essentially horizontal at locations far from
the zone where SWI phenomena occur. Otherwise, the vertical flux component,
*q*_{z}, is non-negligible throughout the whole domain for the heterogeneous
system (Fig. 4c) and streamlines tend to focus towards regions characterized
by large permeability values. Figure 4 also depicts iso-concentration curves
*C*∕*C*_{S} = 0.25, 0.5 and 0.75 within the transition zone (red
curves). It can be noted that iso-concentration profiles tend to be
sub-parallel to streamlines directed towards the seaside boundary. In the
homogeneous domain the slope of these curves varies mildly and in a gradual
manner from the top to the bottom of the aquifer. Their slope in the
heterogeneous domain is irregular and markedly influenced by the spatial
arrangement of permeability. In other words, the way streamlines are
refracted at the boundary between two blocks of contrasting permeability
drives the local pattern of concentration contour lines. As a consequence,
iso-concentration curves tend to become sub-vertical when solute is
transitioning from regions characterized by high *k* values to zones
associated with moderate to small *k*, a sub-horizontal pattern being
observed when transitioning from low to high *k* values.

Figure 5 depicts isolines *C*∕*C*_{S} = 0.5 at the bottom of the
aquifer (Fig. 5a) and along three vertical cross-sections, selected to
exemplify the general pattern observed in the system (Fig. 5b–d), evaluated
for (i) the 100 heterogeneous realizations analyzed (dotted blue curves),
(ii) the equivalent homogeneous system (denoted as *Hom*; solid red
curve) and (iii) the configuration obtained by averaging the concentration
fields across all MC heterogeneous realizations (denoted as *Ens*;
solid blue curve). These results suggest that iso-concentration curves
exhibit considerably large spatial variations within a single realization.
Comparison of the results obtained for *Ens* and *Hom* reveals
that the mean wedge penetration at the bottom layer is slightly overestimated
by the solution computed within the equivalent homogeneous system. Otherwise,
along the coastal vertical boundary, the extent of the area with (mean)
relative concentration larger than 0.5 is generally underestimated by
*Hom*. These outcomes (hereafter called rotation effects) are
consistent with previous literature findings associated with
(a) deterministic models (Abarca et al., 2007) or (b) stochastic analyses
performed under ergodicity assumptions (Kerrou and Renard, 2010, and Pool et
al., 2015). Abarca et al. (2007) showed that a similar effect can be observed
in a homogenous domain when considering increasing values of the dispersion
coefficient. Kerrou and Renard (2010) and Pool et al. (2015) noted that the
strength of the rotation effect increases with the variance of the
log permeability field.

In the following, we analyze values of ${\mathit{\xi}}^{\prime \prime}={\mathit{\xi}}^{\prime}/{\mathit{\xi}}^{\prime \mathrm{Hom}}$, *ξ*^{′} representing a given SWI metric (as listed in
Sect. 2.4) and *ξ*^{′Hom} being the corresponding metric
obtained on the equivalent homogeneous system. Figure 6 depicts the range of
variability of *ξ*^{′′} across the set of MC realizations (symbols). Each
plot is complemented by the depiction of (i) the (ensemble) average of *ξ*^{′′}, as evaluated through all MC realizations, 〈*ξ*^{′′}〉 (black solid line); (ii) the confidence intervals,
$\u2329{\mathit{\xi}}^{\prime \prime}\u232a\pm {\mathit{\sigma}}_{{\mathit{\xi}}^{\prime \prime}}$ (black dashed lines),
${\mathit{\sigma}}_{{\mathit{\xi}}^{\prime \prime}}$ being the standard deviation of *ξ*^{′′};
(iii) *ξ*^{′′Ens}, evaluated on the basis of the
ensemble-average concentration field (blue line). Inspection of Fig. 6a–c
complements the qualitative analysis of Fig. 5 and suggests that
heterogeneity causes (on average) a slight reduction of the toe penetration,
$\u2329{L}_{\mathrm{T}}^{\prime \prime}\u232a$ and ${L}_{\mathrm{T}}^{\prime \prime \mathrm{Ens}}$ being slightly less than 1, and an enlargement of
the mixing zone, $\u2329{L}_{\mathrm{S}}^{\prime \prime}\u232a$ and
$\u2329{W}_{\mathrm{MZ}}^{\prime \prime}\u232a$ being larger than 1. The
results of Fig. 6a–c also emphasize that, while ${L}_{\mathrm{T}}^{\prime \prime \mathrm{Ens}}$ calculated from the ensemble-average concentration
distributions is virtually indistinguishable from $\u2329{L}_{\mathrm{T}}^{\prime \prime}\u232a$, ${L}_{\mathrm{S}}^{\prime \prime \mathrm{Ens}}$ and ${W}_{\mathrm{MZ}}^{\prime \prime \mathrm{Ens}}$ markedly
overestimate $\u2329{L}_{\mathrm{S}}^{\prime \prime}\u232a$ and
$\u2329{W}_{\mathrm{MZ}}^{\prime \prime}\u232a$, respectively, as they
visibly lie outside of the corresponding confidence intervals of width
${\mathit{\sigma}}_{{\mathit{\xi}}^{\prime \prime}}$. Note that even as Fig. 6a–c have been computed along
cross-section B–B^{′}, qualitatively similar results have been obtained along
all vertical cross-sections, as also suggested by the behavior of the
dimensionless areal extent of toe penetration, ${A}_{\mathrm{T}}^{\prime \prime}$ (Fig. 6d),
and solute spreading, ${A}_{\mathrm{S}}^{\prime \prime}$ (Fig. 6e), as well as of the
dimensionless volumes ${V}_{\mathrm{T}}^{\prime \prime}$ and ${V}_{\mathrm{S}}^{\prime \prime}$
(Fig. 6f–g). Overall, the results in Fig. 6 clearly indicate that the
ensemble-average concentration field can provide accurate estimates of the
average wedge penetration while rendering biased estimates of quantities
characterizing mixing. Our findings are consistent with previous studies
(e.g., Dentz and Carrera, 2005; Pool et al., 2015) in showing that an analysis
relying on the ensemble concentration field tends to overestimate
significantly the degree of mixing and spreading of the solute because it
combines the uncertainty associated with sample-to-sample variations of
(a) the solute center of mass and (b) the actual spreading. Our results also
highlight that the extent of this overestimation decreases whenever integral
rather than local quantities are considered: ${V}_{\mathrm{S}}^{\prime \prime \mathrm{Ens}}$ in Fig. 6g is indeed about half of ${L}_{\mathrm{S}}^{\prime \prime \mathrm{Ens}}$ and ${A}_{\mathrm{S}}^{\prime \prime \mathrm{Ens}}$.

## 3.2 Effects of pumping on SWI in three-dimensional heterogeneous media

Here, we investigate SWI phenomena in heterogeneous systems when a pumping
scheme is activated as described in Sect. 2.3. Figure 7 collects contour maps
of relative concentration *C*∕*C*_{S} obtained for all schemes along
cross-section B–B^{′} at the end of the 8-month pumping period, with
reference to (i) *Hom* (the equivalent homogeneous system, left
column) and (ii) *Ens* (the ensemble-average concentration field, right column). Contour lines
*C*∕*C*_{S} = 0.25, 0.50 and 0.75 are also highlighted. Extracting
FW according to the engineered solution S1 (Fig. 7c–d) results in a slight
landward displacement of the SWI wedge and in an enlargement of the
transition zone, as compared to S0 (Fig. 7a–b), where no pumping is
activated. This behavior can be observed in the homogeneous as well as (on
average) in the heterogeneous settings, and is related to the general
decrease of the piezometric head within the inland side caused by pumping
which, in turn, favors SWI. One can also note that the partially penetrating
well induces non-horizontal flow (in the proximity of the well) thus
enhancing mixing along the vertical direction. Wedge penetration and solute
spreading are further enlarged in S2 (see Fig. 7e–f), where the total
extracted volume is increased with respect to S1 through an additional
pumping rate at the bottom of the aquifer. However, when the pumping well is
operating within the transition zone (S3, Fig. 7g–h) the SW wedge tends to
recede, approaching the pumping well location. In this case, the lower screen
acts as a barrier limiting the extent of the SWI at the bottom of the
aquifer.

The combined effects of groundwater withdrawal and stochastic heterogeneity on SWI are investigated quantitatively through the analysis of the temporal evolution of the seven metrics introduced in Sect. 2.4.

Figure 8 illustrates mean values of *ξ*^{′} (with ${\mathit{\xi}}^{\prime}={L}_{\mathrm{T}}^{\prime}$, ${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$) computed for all pumping
schemes considered across the collection of all heterogeneous realizations
(black line) along the vertical cross-section B–B^{′} versus dimensionless
time ${t}^{\prime}=\left(t-{t}_{w}\right)Q/{\mathit{\lambda}}^{\mathrm{3}}$. Confidence
intervals $\u2329{\mathit{\xi}}^{\prime}\u232a\pm {\mathit{\sigma}}_{{\mathit{\xi}}^{\prime}}$ are
also depicted (dashed lines). As additional terms of comparison, Fig. 8 also
includes *ξ*^{′Hom} (red line) and *ξ*^{′Ens} (blue line), respectively, evaluated in the equivalent
homogeneous domain and in the ensemble-average concentration field. Figure 8
shows that the toe penetration, ${L}_{\mathrm{T}}^{\prime}$, and the solute spreading,
as quantified by ${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$, do not vary
significantly with time in the absence of pumping (S0) because stationary
boundary conditions are imposed for ${t}^{\prime}>\mathrm{0}$. Pumping schemes S1 and S2 cause
the progressive inland displacement of the toe, together with an overall
increase of spreading at the bottom of the aquifer. This phenomenon is more
severe in S2, where SW and FW are simultaneously extracted. On the other
hand, the vertical width of the mixing zone, e.g., ${W}_{\mathrm{MZ}}^{\prime}$, is not
significantly affected by the pumping within schemes S1–S2. Configuration
S3, in which the pumping well is located within the mixing zone, leads to the
most pronounced changes in the shape and position of the SW wedge. The toe
penetration first decreases rapidly in time and then stabilizes around the
well location. Quantities ${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$ show an
early time increase, suggesting that a rapid displacement of the wedge
enhances mixing and spreading. As the toe stabilizes over time, both
${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$ decrease, reaching values equal to
(or slightly less than) those detected in the absence of pumping.
Consistent with the observations of Sect. 3.1, Fig. 8 supports the findings
that (i) an equivalent homogeneous domain is typically characterized by the
largest toe penetration and the smallest vertical width of the mixing zone,
the only exception being observed in S3 at late times, when ${L}_{\mathrm{T}}^{\prime \mathrm{Hom}}\cong {L}_{\mathrm{T}}^{\prime \mathrm{Ens}}<\u2329{L}_{\mathrm{T}}^{\prime}\u232a$; (ii) basing the characterization of
the mixing zone on ensemble-average concentrations enhances the actual
effects of heterogeneity and yields overestimated mixing zone widths.
Figure 8 also highlights that the extent of the confidence interval
associated with toe penetration is approximately constant in time and does
not depend significantly on the analyzed flow configuration. Otherwise, the
confidence intervals associated with our mixing zone metrics depend on the
pumping scenario and tend to increase with time if FW and SW are
simultaneously extracted, especially in S2.

The fully three-dimensional nature of the analyzed problem is exemplified in
Fig. 9, where we depict isolines *C*∕*C*_{S} = 0.5 along the
bottom of the aquifer at the end of the pumping period for the equivalent
homogeneous system (Fig. 9a) and as a result of ensemble averaging across the
collection of heterogeneous fields (Fig. 9b). The spatial pattern of
iso-concentration contours associated with pumping schemes S1–S3 departs
from the corresponding results associated with S0 within a range extending
for about 10*λ* around the well, along the direction parallel to the
coastline.

We quantify the global effect of pumping on the three-dimensional SW wedge by evaluating, for each pumping scenario and each MC realization, the temporal evolution of ${\mathit{\xi}}^{\ast}=\left({\mathit{\xi}}^{\prime}-{\mathit{\xi}}_{\mathrm{S}\mathrm{0}}^{\prime}\right)\left(\right)open="/">\phantom{\left({\mathit{\xi}}^{\prime}-{\mathit{\xi}}_{\mathrm{S}\mathrm{0}}^{\prime}\right){\mathit{\xi}}_{\mathrm{S}\mathrm{0}}^{\prime}}{\mathit{\xi}}_{\mathrm{S}\mathrm{0}}^{\prime}$, i.e., the relative percentage variation of areal extent, ${\mathit{\xi}}^{\prime}={A}_{\mathrm{T}}^{\prime},{A}_{\mathrm{S}}^{\prime}$, and volumetric extent, ${\mathit{\xi}}^{\prime}={V}_{\mathrm{T}}^{\prime},{V}_{\mathrm{S}}^{\prime}$, of wedge penetration and mixing zone with respect to their counterparts computed in the absence of pumping ${\mathit{\xi}}_{\mathrm{S}\mathrm{0}}^{\prime}$. Figure 10 shows that the penetration area, ${A}_{\mathrm{T}}^{\ast}$, and the solute spreading, as quantified through ${A}_{\mathrm{S}}^{\ast}$, at the bottom of the aquifer tend to increase with pumping time. Corresponding results for ${V}_{\mathrm{T}}^{\ast}$ and ${V}_{\mathrm{S}}^{\ast}$ are depicted in Fig. 11. The scenario where these quantities are most affected by pumping is S2, where the FW and SW pumping wells are located outside the transition zone (in place prior to pumping). Placing both wells within the transition zone prior to pumping yields a significant decrease of the effect of pumping on both areal and volumetric metrics. We further note that the mean penetration area and volume can be accurately determined by the solution obtained within the equivalent homogeneous domain and is associated with a relatively small uncertainty, as quantified by the confidence intervals. Heterogeneity effects are clearly visible on spreading along the bottom of the aquifer. Figure 10d–f highlight that ${A}_{\mathrm{S}}^{\ast \mathrm{Hom}}$ and ${A}_{\mathrm{S}}^{\ast \mathrm{Ens}}$ are not accurate approximations of $\u2329{A}_{\mathrm{S}}^{\ast}\u232a$. Spreading uncertainty, as quantified by ${\mathit{\sigma}}_{{A}_{\mathrm{S}}^{\ast}}$, is significantly larger than ${\mathit{\sigma}}_{{A}_{\mathrm{T}}^{\ast}}$, especially in pumping scenario S2. On the other hand, the spreading of the SWI volume, as quantified by ${V}_{\mathrm{S}}^{\ast}$ (Fig. 11d–f), can be accurately estimated by the homogeneous equivalent system, as well as by the ensemble mean concentration field, and it is characterized by a relatively low uncertainty.

Our analyses document that for a stochastically heterogeneous aquifer an
operational scheme of the kind engineered in S3 (i) is particularly efficient
for the reduction of SWI maximum penetration (localized at the bottom of the
aquifer) and (ii) is advantageous in controlling the extent of the volume of
the SW wedge, as compared to the double-negative barrier implemented in S2.
However, this withdrawal system may lead to the salinization of the FW
extracted from the upper screen due to upconing effects. This aspect is
further analyzed in Fig. 12 where the temporal evolution of *C*_{T}∕*C*_{S} is depicted, *C*_{T} being the salt concentration
associated with the total mass of fluid extracted by the upper screen
(production well) in S3. Results for the equivalent homogeneous system (red
curve), each of the heterogeneous MC realizations (grey dashed curves) and
the ensuing ensemble-average concentration 〈*C*_{T}〉∕*C*_{S} (blue curve) are depicted. Vertical bars
represent the 95 % confidence interval around the ensemble mean,
evaluated as $\u2329{C}_{\mathrm{T}}\u232a/{C}_{\mathrm{S}}\pm \mathrm{1.96}\phantom{\rule{0.25em}{0ex}}{\mathit{\sigma}}_{{C}_{\mathrm{T}}/{C}_{\mathrm{S}}}/\sqrt{n}$. Here
${\mathit{\sigma}}_{{C}_{\mathrm{T}}/{C}_{\mathrm{S}}}$ is the standard deviation of
*C*_{T}∕*C*_{S} and *n*=100 is the number of MC
realizations forming the collection of samples. The width of these confidence
intervals can serve as a metric to quantify the order of magnitude of the
uncertainty associated with the estimated mean. The equivalent homogeneous
system significantly underestimates the ratio 〈*C*_{T}〉∕*C*_{S}. Moreover, Fig. 12 highlights the marked
variability of the results across the MC space. As such, it reinforces the
conclusion that the mean value 〈*C*_{T}〉
is an intrinsically weak indicator of the actual salt concentration at the
producing well.

We investigate quantitatively the role of stochastic heterogeneity and
groundwater withdrawal on seawater intrusion (SWI) in a randomly
heterogeneous coastal aquifer through a suite of numerical Monte Carlo (MC)
simulations of transient variable-density flow and transport in a three-dimensional
domain. Our work attempts to include the effects of random heterogeneity of
permeability and groundwater withdrawal within a realistic and relevant
scenario. For this purpose, the numerical model has been tailored to the
general hydrogeological setting of a coastal aquifer, i.e., the Argentona
river basin (Spain), a region which is plagued by SWI. To account for the
inherent uncertainty associated with aquifer hydrogeological properties, we
conceptualize our target system as a heterogeneous medium whose permeability
is a random function of space. The SWI phenomenon is studied through the
analysis of (a) the general pattern of iso-concentration curves and (b) a set
of seven dimensionless metrics describing the toe penetration and the extent
of the mixing or transition zone. We compare results obtained across a
collection of *n*=100 MC realizations and for an equivalent homogeneous
system. Our work leads to the following major conclusions.

Heterogeneity of the system affects the SW wedge along all directions both in the presence and absence of pumping. On average, our heterogeneous system is characterized by toe penetration and extent of the mixing zone that are, respectively, smaller and larger than their counterparts computed in the equivalent homogeneous system.

Ensemble (i.e., across the MC realizations) mean values of linear, ${L}_{\mathrm{T}}^{\prime}$, areal, ${A}_{\mathrm{T}}^{\prime}$, and volumetric, ${V}_{\mathrm{T}}^{\prime}$, metrics representing 50 % of SWI penetration virtually coincide with their counterparts evaluated in the ensemble-average concentration field.

Ensemble mean values of linear, ${L}_{\mathrm{S}}^{\prime}$ and ${W}_{\mathrm{MZ}}^{\prime}$, and areal, ${A}_{\mathrm{S}}^{\prime}$, metrics representing the mixing and spreading of SWI penetration are markedly overestimated by their counterparts evaluated on the basis of the ensemble-average concentration field. Therefore, average concentration fields, typically estimated through interpolation of available concentration data, cannot be employed to provide reliable estimates of solute spreading and mixing.

All of the tested pumping schemes lead to an increased SW wedge volume compared to the scenario where pumping is absent. The key aspect controlling the effects of groundwater withdrawal on SWI is the position of the wellbore with respect to the location of the saltwater wedge in place prior to pumping. The toe penetration decreases or increases depending on whether the well is initially (i.e., before pumping) located within or outside the seawater intruded region, respectively. The water withdrawal scheme that is most efficient for the reduction of the maximum inland penetration of the seawater toe is the one according to which freshwater and saltwater are, respectively, extracted from the top and the bottom of the same borehole, initially located within the SW wedge. This result suggests the potential effectiveness of the so-called “negative barriers” in limiting intrusion, even when considering the uncertainty effects stemming from our incomplete knowledge of permeability spatial distributions.

Salt concentration, *C*_{T}, of water pumped from the producing well
is strongly affected by permeability heterogeneity. Our MC simulations
document that *C*_{T} can vary by more than 2 orders of magnitude
amongst individual realizations, even in the moderately heterogeneous aquifer
considered in this study. As such, relying solely on values of *C*_{T}
obtained through an effective homogeneous system or on the ensemble-average
estimates 〈*C*_{T}〉 does not yield
reliable quantification of the actual salt concentration at the producing
well.

Future development of our work includes the analysis of the influence of the degree of heterogeneity and of the functional format of the covariance structure of the permeability field on SWI, also in the presence of multiple pumping wells.

All data used in the paper will be retained by the authors for at least 5 years after publication and will be available to the readers upon request.

Here, we assess the stability of the MC-based first- and second-order
(statistical) moments associated with the metrics introduced in Sect. 2.4.
Figures A1 and A2, respectively, depict the sample mean, 〈*ξ*^{′′}〉, and standard deviation, *σ*(*ξ*^{′′}), of all metrics evaluated at the end of the 8-year period versus the
number of MC simulations, *n*. The estimated 95 % confidence intervals,
computed according to Eqs. (3) and (8)

of Ballio and Guadagnini (2004), are also depicted. Figures A1 and A2 show that the oscillations displayed by the quantities of interest are in general limited and do not hamper the strength of the main message of our work. As expected, moments of integral quantities (i.e., ${A}_{\mathrm{T}}^{\prime \prime}$, ${A}_{\mathrm{S}}^{\prime \prime}$, ${V}_{\mathrm{T}}^{\prime \prime}$, ${V}_{\mathrm{S}}^{\prime \prime}$, ${W}_{\mathrm{MZ}}^{\prime \prime}$) tend to stabilize faster than their counterparts evaluated for local quantities (${L}_{\mathrm{T}}^{\prime \prime}$ and ${L}_{\mathrm{S}}^{\prime \prime}$).

The authors declare that they have no conflict of interest.

The authors would like to thank the EU and MIUR for funding,
in the frame of the collaborative international consortium (WE-NEED)
financed under the ERA-NET WaterWorks2014 Cofunded Call.
This ERA-NET is an integral part of the 2015 Joint Activities
developed by the Water Challenges for a Changing World Joint Programme Initiative (Water JPI).
We are grateful to Albert Folch, Xavier Sanchez-Vila and
Laura del Val Alonso of the Universitat Politècnica de Catalunya and
Jesus Carrera of the Spanish Council for Scientific Research for sharing data
on the hydrogeological characterization of the Argentona site with
us.

Edited by: Bill X. Hu

Reviewed by: Damiano Pasetto and two anonymous referees

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