Effective porosity plays an important role in contaminant management.
However, the effective porosity is often assumed to be constant in space and hence
heterogeneity is either neglected or simplified in transport model
calibration. Based on a calibrated highly parametrized flow model, a
three-dimensional advective transport model (MODPATH) of a 1300 km

The age of groundwater, i.e. the time elapsed since the water molecule entered the groundwater (Cook and Herczeg, 2000; Kazemi et al., 2006), is useful (i) to infer recharge rates (e.g. Sanford et al., 2004; Wood et al., 2017) and hence to sustainably exploit groundwater resources; (ii) to evaluate contaminant migration, fate and history (Bohlke and Denver, 1995; Hansen et al., 2012) and predict spread of pollutants and timescales for intrinsic remediation (Kazemi et al., 2006); (iii) to analyse aquifer vulnerability or protection to surface-derived contaminants (e.g. Manning et al., 2005; Bethke and Johnson, 2008; Molson and Frind, 2012; Sonnenborg et al., 2016) and indicate the advance of modern contaminated groundwater (Hinsby et al., 2001b; Gleeson et al., 2015; Jasechko et al., 2017) and groundwater quality in general (Hinsby et al., 2007); and (iv) to contribute to the understanding of the flow system, e.g. in complex geological settings (Troldborg et al., 2008; Eberts et al., 2012).

The groundwater science community (de Dreuzy and Ginn, 2016) has a continued interest in the topic of residence time distributions (RTDs) in the subsurface. Turnadge and Smerdon (2014) reviewed different methods for modelling environmental tracers in groundwater, including lumped parameter models (e.g. Maloszewski and Zuber, 1996), mixing-cell models (e.g. Campana and Simpson, 1984; Partington et al., 2011) and direct age models (e.g. Cornaton, 2012; Goode, 1996; Woolfenden and Ginn, 2009). Here, we focus on three different approaches with specific benefits and disadvantages that are commonly applied to simulate groundwater age in 3-D distributed groundwater flow and transport models (Castro and Goblet, 2005; Sanford et al., 2017). Particle-based advective groundwater age calculation utilizing travel time analysis is computationally easy, but neglects diffusion and dispersion. The full advection–dispersion transport simulation of a solute or an environmental tracer is computationally expensive and limited to the specific tracer characteristics (McCallum et al., 2015; Salmon et al., 2015), but accounts for diffusion, dispersion and mixing. The tracer-independent direct simulation of groundwater mean age (Goode, 1996; Engesgaard and Molson, 1998; Bethke and Johnson, 2002) includes advection, diffusion and dispersion processes and yields a spatial distribution of mean ages. A comparison of ages simulated using any of these methods with ages determined from tracer observations, referred to as apparent ages, is desirable as it can improve the uniqueness in flow model calibration and validation (Castro and Goblet, 2003; Ginn et al., 2009) and it potentially informs about transport parameters such as effective porosity, diffusion and dispersion that are otherwise difficult to estimate. However, the approach is far from straightforward as environmental tracers undergo non-linear changes in their chemical species (McCallum et al., 2015) and groundwater models only represent a simplification and compromise on structural and/or parameter heterogeneity. In a 2-D synthetic model, McCallum et al. (2014) investigated the bias of apparent ages in heterogeneous systems systematically. McCallum et al. (2015) applied correction terms, e.g. diffusion correction for radioactive tracers, on apparent ages to improve the comparability to mean advective ages. They concluded that with increasing heterogeneity the width of the residence time distribution increases and that apparent ages would only represent mean ages if this distribution is narrow and has a small variance. It is important here to distinguish between mean and radiometric ages, as defined by Varni and Carrera (1998) for example. The only way they can be directly compared in reality is if no mixing is taking place, i.e. if the flow field can be regarded as pure piston flow, which will give the kinematic age.

Flow and transport parameters such as hydraulic conductivity, conductance of streambeds and drains, recharge, and dispersivities have gained more and more focus in calibration of groundwater models, recently also on large scales, where of head, flow and tracer observations are widely used as targets (McMahon et al., 2010). However, effective porosity has not received nearly as much attention, and its spatial variability in particular is often neglected, except for Starn et al. (2014). The lack of focus on calibrating distributed effective porosity on a regional scale might be related to the common assumption that recharge in humid climates can be precisely estimated and porosity of porous media is relatively well known from the literature (Sanford, 2011). However, for steady-state flow (Ginn et al., 2009) in a layered aquifer system, Bethke and Johnson (2002) concluded that the mean groundwater age exchange between flow and stagnant zones is only a function of the volume of stored water (Harvey and Gorelick, 1995; Varni and Carrera, 1998). Thus, the groundwater age exchange is directly related to the porosity. Yet, the calibration of a spatially distributed porosity field and its application to simulate groundwater ages and infer capture zones has not gained much attention.

The uniqueness of the presented study lies in the calibration of a
three-dimensional, spatially distributed, effective porosity field in a
regional-scale complex multi-layered heterogeneous coastal aquifer system.
The aims are as follows: (i) to use apparent ages inferred from dissolution- and
diffusion-corrected

The 1300 km

Investigation area at the border between Denmark and Germany.
Simulated hydraulic heads are from the shallow aquifer (Meyer et al., 2018a).
Topography,

The aquifer systems are geologically complex and highly heterogeneous, spanning Miocene through Holocene deposits. The bottom of the aquifer system is defined by low-permeability Palaeogene marine clay. The overlying Miocene deposits consist of alternating marine clay and deltaic silt and sand (Rasmussen et al., 2010). The Maade formation, an upper Miocene marine clay unit, with a relatively large thickness in the west while thinning out to the east, is located below the Pleistocene and Holocene deposits. Buried valleys filled with glacial deposits, mainly from the Saalian glaciation, cut through the Miocene and reach depths up to 450 m below the surface. They are important hydrogeological features as they may constitute preferential flow paths and locally connect the Pleistocene and Miocene aquifers.

In our previous studies (Jørgensen et al., 2015; Høyer et al., 2017; Meyer et al., 2018a), the
available geological and geophysical information including borehole
lithology, airborne electromagnetic (AEM) and seismic data were assembled
into a heterogeneous geological voxel model comprising 46 geological units
with raster sizes of 100 m

The age simulation and calibration of effective porosities builds upon the
calibrated regional-scale groundwater flow model (MODFLOW) of a highly
heterogeneous coastal aquifer system by Meyer et al. (2018a). First,
an advective transport simulation using MODPATH (Pollock, 2012) was used for the
calibration of effective porosities of seven different geological units.

During a field campaign in February 2015, 18 groundwater samples were
collected from wells at seven sites with screens at different depths and in
different aquifers (Fig.

The

A modified chemical
correction was applied that takes into account the effect of dissolution as
described by Boaretto et al. (1998). This method was successfully used in
Danish geological settings similar to those investigated in the present study
(Eq. 2; Boaretto et al., 1998; Hinsby et al., 2001a). The initial

Conceptual regional model showing a simplified geology featuring buried valleys, groundwater flow and stagnant zones (as used for the diffusion correction). Arrows indicate general flow field of groundwater. Also shown are the boundary conditions, i.e. density corrected coastal boundary, drained marsh area, rivers.

Subsequently, a diffusion correction was made to take into account diffusion
loss into low-permeability layers (Sanford, 1997). Aquitard diffusion is
sensitive to porosity, diffusion coefficient and the thicknesses of the
active flow (aquifer) and stagnant (aquitard) zones (Sudicky and Frind,
1981). Because of the geological complexity, the sand-to-clay ratio based on
voxel lithology was used to calculate the relative aquifer

Sampling wells, uncorrected and corrected groundwater ages. Bold indicates samples
used for calibration. Note that lower numbers of the
wells indicate deeper locations (m b.s.

Meyer et al. (2018a) simulated the 3-D steady-state regional groundwater flow
using MODFLOW 2000 (Harbaugh et al., 2000). A brief description of the model
set up and calibration results are presented here; further details can be
found in Meyer et al. (2018a). The model was discretized horizontally by
200 m

Horizontal hydraulic conductivities (one for each hydrogeological unit), two
anisotropy factors (

The steady-state MODFLOW flow
solution (calibration results summarized in Fig.

Calibration results of steady-state
groundwater flow model that forms the basis for the advective transport model
(modified after Meyer et al. (2018a). Panel

Advective transport simulation was performed using MODPATH (Pollock, 2012) in
particle back-tracking mode. Hereby, the travel time of a particle, released
in a cell, is calculated based on the MODFLOW cell-by-cell flow rates (

The groundwater age can be seen as the backward integration of travel times along the travel path back to its recharge location. Hence, the simulated groundwater age is a function of the ratio of flux to effective porosity and the travel distance. In this study, the total flux is controlled by prescribed recharge and heterogeneous distribution of hydrogeological parameters (e.g. hydraulic conductivity, porosity).

In order to ensure stability (Konikow et al., 2008), 1000 particles were
distributed evenly in the cell of the well screen and their average simulated
particle age was compared with apparent groundwater ages (derived from
Eq.

The corrected

The flow solution of the calibrated flow model (Meyer et al., 2018a)
constitutes the base for the 3-D advective transport model. Depending on the
depositional environment and clay or sand content, effective porosities of seven
units corresponding to two Pleistocene sand, two Pleistocene clay, one
Miocene sand and two Miocene clay units were estimated using a regularized
(Tikhonov) inversion with PEST (Tikhonov and Arsenin, 1977; Doherty, 2016).
As the calibration approach is similar to the one of Meyer et al. (2018a),
only additional characteristics are described in the following. Average
corrected

When Tikhonov regularization is applied, a
regularized objective function (

Calibration settings such as
initial and preferred values and final parameter estimates are shown in
Table

To visualize the mean groundwater age pattern in the regional 3-D aquifer
system, direct simulation of mean groundwater age was performed with MT3DMS
(standard finite difference solver with upstream weighting) and the chemical reaction
package using a zeroth-order production term (Goode, 1996; Bethke and
Johnson, 2008). Hereby, mean groundwater age is simulated in analogy to
solute transport as an “age mass” (Bethke and Johnson, 2008). For each
elapsed time unit (day) the water “age mass” increases by 1 day in each
cell. Increases or decreases in ages are a result of diffusion, dispersion and
advection (Bethke and Johnson, 2008). The transient advection–dispersion
equation of solute transport of “age mass” in three dimensions and with
varying density and porosity is given by Goode (1996):

Well capture zones are used in water management to define areas of
groundwater protection, where human actions, such as agricultural use, are
restricted. Simulated by the uniform effective porosity and distributed
effective porosities model, the capture zones of one existing well (Abild,
abstraction rate 27 m

Figure

Apparent groundwater

The match between the average of simulated groundwater ages (particle
tracking with MODPATH) and corrected

Calibration results:

Results from the calibrated distributed effective porosities model were
compared to those from a uniform effective porosity model with an effective
porosity of 0.3, which is a typical textbook value for porous media (Hölting
and Coldewey, 2013; Anderson et al., 2015) and often used in groundwater
modelling studies (e.g. Sonnenborg et al., 2016). The calibrated distributed
effective porosity model is able to match all the observations reasonably.
This is not the case for the single effective porosity model, where
one sample in particular is poorly simulated with an estimate of more than
5500 years, whereas the corresponding observation only reach about 1200 years. The ME and
RMS of the calibrated distributed effective porosity model were

Calibration settings and results: parameters with initial, preferred and estimated values for effective porosity.

The estimated effective porosities of the seven hydrogeological units are
listed in Table

The parameter identifiability (Fig.

Pleistocene sand 2 represents less than 5 % of the total amount of
cells (Table

Horizontal geological cross section at an elevation of

Figure

Results of the analysis of particle age distributions and path lengths. Bold indicates well screens used for calibration.

The results show a wide variety of mean particle ages
(Table

Particle age distributions at sampling wells A–G (see
Fig.

The mean distance that particles travel from their recharge points to the
sampling well (Table

Particle tracking time over path length for the different well
locations (see Fig.

Figure

Mean error (ME) and root mean
square (RMS) between corrected

Minimum ME and RMS values are achieved for longitudinal dispersivities

Directly simulated mean ages:

The directly simulated mean age distribution on a regional scale
(Fig.

The steady-state distribution of direct simulated mean groundwater age was
reached after 26 000 years. Over this time span the system has been exposed
to transient stresses from human activity and climatic changes (glacial
cover, sea level, etc.). Therefore, the steady-state assumption is a
notable
simplification, which is further discussed in
Sect.

Frequency distributions (bin size

In Fig.

The comparison of the advective ages with the direct simulated ages at the
sampling well locations shows a good match for advective ages with a small
variance and worsens when the variance increases
(Fig.

Mean advective age (MODPATH (MP) particle backtracking) compared to directly simulated mean groundwater age at sampling well locations; error bars on advective age represent 1 standard deviation.

Figure

Uncertainties in model results originate partly from simplifications in boundary conditions and geological heterogeneities that are not resolved at the grid scale. Groundwater recharge, drain levels, well abstractions and sea levels were assumed to be constant over time for practical reasons and to reduce computational time. However, Karlsson et al. (2014) showed that recharge has changed significantly in Denmark during the last centuries. Changes in recharge could result in different age patterns (Goderniaux et al., 2013). Similarly, sea level changes that were disregarded in this study would have an effect on the groundwater age distribution in the coastal areas (Delsman et al., 2014). Prescribing a vertical coastal age boundary of zero years is another simplification that neglects the vertical mixing and dispersion, which would result in an increase of age with depth (Post et al., 2013). However, since these physical processes were difficult to quantify, estimating age at this boundary would be highly uncertain. Thus, a constant age of zero years was applied.

Capture zones at a well in Abild and a virtual well (AW) with a comparison of capture zones for a model with homogeneous porosity of 0.3 in all geological units (solid lines) and one with seven different porosities (dashed lines).

The area close to the coast has not only been affected by changing sea levels
during the past thousands of years, but also by saltwater intrusion. In this
study, the density effects on flow were accounted for in a simplified way by
using a density-corrected constant head boundary at the coast. Both sea
level changes and density effects would also have affected the age
distribution. The impact on age calculations due to density effects would be
largest close to the coast. However, most of the groundwater samples used for
age estimations were collected several tens of kilometres inland and are
therefore expected to be affected to a minor extent. To quantify the impact
of boundary conditions and saltwater intrusion on the particle tracking, the
differences of particle travel path lengths for a 200-year period,
investigated based on the present model and a preliminary density-driven
model (SEAWAT) accounting for non-stationary and density effects (similar to
the one presented in Meyer, 2018), are computed. The relative differences are
below 10 % (except at location A and B). Also, the uncertainties introduced
by simplifying the density boundary effects are likely to be less important
compared to other uncertainties associated, for example, with estimating the
groundwater age by the procedures for correcting

Uncertainties in the use of

As mentioned in the introduction, the apparent age (or radiometric age) is
not equal to the mean particle-based kinematic age. This introduces
additional, but unknown uncertainties. Ideally, one could develop an
advection–dispersion equation for the second moment and solve for the variance of
ages (Varni and Carrera, 1998) and use that together with the directly
simulated mean age (or first moment) to establish a relation between
radiometric and mean ages. This has not been pursued as we believe the
benefits from this would be masked by uncertainty in age dating

The comparison of groundwater ages, estimated from tracer concentration in a water sample, and simulated groundwater ages, either derived by particle tracking or direct age modelling, bears the problem of commensurability, the comparison of a point measurement relative to the modelling scale. The water sample represents the age distribution in the direct surrounding of the well screen, which only makes up a few percent of the water in one model cell.

The differences between mean advective ages and directly simulated mean ages as described in Sect. 4.4 can be related to the simulation methods. While particle tracking neglects dispersion, but allows an age distribution in a cell to be simulated (by perturbing the measurement location so to speak), direct age modelling allows for dispersion/diffusion to be accounted for, resulting in only the mean age at a cell. The mismatches between advective and direct age can be related to the diffusion and dispersion processes (here represented by numerical dispersion as dispersivity was set to zero), which are included in the direct age approach, but neglected in simulating advective ages.

The analysis of the advective age and travel distance distributions
(Figs.

The regional age pattern derived from direct age simulation is consistent with the findings of Meyer et al. (2018a) about the flow system. The two-aquifer system is separated by a confining aquitard in the west. The shallow aquifer system consisting of glaciotectonically disturbed Pleistocene sands mixed with clays is dominated by local and intermediate flow regimes and contains water of younger ages. The confining aquitard (Maade formation) shows older waters and a positive relation between ages and aquitard thickness what agrees with Bethke and Johnson (2008). In the deep Miocene sand aquifer that is interbedded with Miocene clay, regional flow regimes dominate and groundwater ages vary from young waters in the recharge areas in the east, where the overlying confining aquitard does not exist, to very old waters (up to 10 000 years) in the west. The confining Miocene aquitard (Maade formation) influences the age distribution pattern in the underlying Miocene sand in two ways. First, it limits the ability of deeper groundwater to seep upward and mix with the younger waters in the shallow aquifer. Secondly, the age flux from the aquitard to the aquifer shows a positive correlation with the ratio between aquitard thickness and aquifer thickness (Bethke and Johnson, 2008).

At the buried valleys, groundwater exchange and hence age mixing occurs. Upwelling of the older groundwater from the deeper aquifer happens preferentially through these buried valleys. The dense drainage network in the west close to the coast acts as a regional sink, with younger groundwater flowing horizontally and older water flowing vertically and discharging to the drains. At the coastal boundary in the west, where a constant concentration of an “age mass” zero was assigned to the density-corrected constant head boundary, an age wedge characterized by waters of contrasting ages is established as a result of intruding young ocean water that meets old waters in the transition zone. This agrees with the findings by Post et al. (2013) based on simulation of synthetic groundwater age patterns in coastal aquifers using density-driven flow.

The results of our study differ significantly from findings by Sonnenborg et al. (2016), who investigated a regional aquifer system with a similar geological setting located a few hundred kilometres north of the present study area. Their direct simulation of groundwater ages shows a pattern of much younger water than here, rarely exceeding 700 years even in the deepest aquifers, while in our study ages exceeding 10 000 years occur. The discrepancies may arise from differences in the geological models. In the area of Sonnenborg et al. (2016) the thickness of the Miocene sand units decreases towards the west and disappears before reaching the west coast. Sonnenborg et al. (2016) conclude that rivers control the age distribution even in deep aquifers. Based on particle tracking they found that the flow regimes were dominated by local and intermediate flow (see Tóth, 1963), with flow lengths not exceeding 15 km. In contrast, in the study presented here, the Miocene sand extends to the coast and probably beyond. While the age pattern in the shallow aquifers is controlled by rivers and drains (similarly to Sonnenborg et al., 2016), the age pattern in the deep aquifers is dominated by the extent and thickness of the Maade formation, the marsh area as a location of preferred discharge and the occurrence of buried valleys as locations of groundwater exchange, especially upwelling of old groundwater. Particle path lengths reach up to 30 km and regional flow dominates in the Miocene aquifer.

The groundwater age distribution in aquifers is closely related to the
distribution of physical (e.g. hydraulic conductivity and porosity) and
chemical parameters (e.g. concentrations of contaminants and natural geogenic
elements) of the aquifers and aquitards. Hence, tracer- and model-estimated
groundwater age distributions provide important information for the
assessment of the hydraulic properties of the subsurface as demonstrated in
this study, and as an indicator of groundwater quality and vulnerability
(Hinsby et al., 2001a; Sonnenborg et al., 2016), including contaminant
migration (Hinsby et al., 2001a), contents of harmful geogenic elements such
as arsenic and molybdenum (Edmunds and Smedley, 2000; Smedley and Kinniburgh,
2002, 2017), and the risk of saltwater intrusion (MacDonald et al., 2016;
Larsen et al., 2017; Meyer et al., 2018b). Groundwater age distributions in
time and space are therefore important pieces of information for groundwater status
assessment and the development of proper water management strategies that
consider and protect both water resources quality and quantity (MacDonald et
al., 2016). Water quality issues are often related to human activities such
as contamination or over-abstraction (MacDonald et al., 2016) and are
typically found in waters younger than 100 years to depth of about 100 m
(Seiler and Lindner, 1995; Hinsby et al., 2001a), although deep subsurface
activities may threaten deeper and older resources (Harkness et al., 2017).
Deeper and older water is generally not contaminated or affected by human
activities, but the impact of natural processes and contents of dissolved
trace elements increases with depth and transport times (Edmunds and Smedley,
2000). Similarly, the risk of saltwater intrusion from fossil seawater in
old marine sediments increases with depth in inland aquifers and reduces the
amount of available high-quality groundwater resources (MacDonald et al.,
2016; Larsen et al., 2017; Meyer et al., 2018b). Furthermore, old groundwater
resources that are only slowly replenished are more vulnerable to
over-exploitation, which leads to declining water tables, increasing hydraulic
gradients and long-term non-steady-state conditions that change the regional
flow pattern (Seiler and Lindner, 1995) and potentially result in
contamination of deeper groundwater resources by shallow groundwater leaking
downward. The presented modelling results show that the Miocene sand aquifer
is protected by the overlying Maade formation over a wide area. The Miocene aquifer
bears old waters (

The originality of this study comes from a 3-D multi-layer coastal regional
advective transport model, where heterogeneities are resolved on a grid
scale. The distributed effective porosity field was found by parameter
estimation based on apparent ages determined from

The advective age distributions at the well locations show a wide range of ages from a few hundred to several thousand years. Younger waters show narrower unimodal age distribution with small variances while older waters have wide age distributions and are often multi-modal with large variances. The variances in age distribution reflect the spatial heterogeneity encountered by the groundwater when travelling from the recharge location to the sampling point.

The estimated effective porosity field was subsequently applied in a direct
age simulation that provided insight into the 3-D groundwater age pattern in
a regional multi-layered aquifer system and the probable advance of modern
potentially contaminated groundwater. Large areas in the shallow Pleistocene
aquifer is dominated by young recharging groundwater (

The study clearly demonstrates the governing effect of the highly complex geological architecture of the aquifer system on the age pattern. Even though there are multiple uncertainties and assumptions related to groundwater age and its use in calibration, the results demonstrate that it is possible to estimate transport parameters that contain valuable information for assessment of groundwater quantity and quality issues. This can be used in groundwater management problems in general, as demonstrated in an example of capture zone delineation where a heterogeneous distributed effective porosity field resulted in a 50 % change in the capture zone area compared to the case of homogeneous effective porosity. The adopted approach is easy to implement even in large-scale models where auto-calibration of transport parameters using models based on the advection–dispersion equation might be restricted by computer runtime.

The data are available from the authors.

RM, PE, JAP, KH and TOS contributed to the conception of the work.
RM, PE and TOS designed the modelling approach.
RM and KH planned and conducted the groundwater sampling campaign, and the use and correction of

The authors declare that they have no conflict of interest.

This study stems from the SaltCoast project generously funded by GeoCenter Denmark. The authors extend sincere thanks to all individuals and institutions whose collaboration and support at various stages facilitated completion of this study. The authors thank the editor Graham Fogg as well as Timothy Ginn and one anonymous reviewer for their comments that helped to improve the paper substantially.Edited by: Graham Fogg Reviewed by: Timothy Ginn and one anonymous referee