Introduction

Coastal aquifers are an important source of freshwater at many locations globally (Michael et al. 2017; Jiao and Post 2019). However, climate-change-induced sea-level rise (SLR) poses a serious threat via groundwater salinization and water-table rise (Ferguson and Gleeson 2012; Lu et al. 2013). SLR can impact groundwater in coastal aquifers through the inland subsurface movement of seawater, a process referred to as seawater intrusion (SI). SI threatens the water quality of supply wells and reduces freshwater volumes in coastal aquifers, leading to reduced water security (Morgan et al. 2012; Beebe et al. 2016). SI associated with SLR has been studied extensively in recent decades, including the underlying physical processes and appropriate models to use for assessment and prediction (e.g., Werner and Simmons 2009; Ataie-Ashtiani et al. 2013; Michael et al. 2013; Werner et al. 2013; Ketabchi et al. 2016; Jiao and Post 2019).

Sea-level rise can also impact groundwater by causing the water table to rise within the subsurface, a process referred to as shoaling (Befus et al. 2020). If the water table rises high enough to intersect the land surface, then groundwater flooding will occur. The impact that groundwater shoaling and flooding can have on agriculture and infrastructure is being increasingly recognised and observed (Tully et al. 2019). Despite this, compared to SLR-induced SI, there has been relatively little research on the topic of SLR-induced groundwater rise (Bosserelle et al. 2022).

Groundwater flooding is a lived experience for residents in the coastal low-lying city of Ōtautahi Christchurch (forthwith Ōtautahi). Ōtautahi is the largest city in Te Waipounamu the South Island of New Zealand with a population of ~400,000 people (Stats 2020). Following two earthquakes, in 2010 and 2011, large areas in the east of the city (near the coast), which is built on drained swampland (Wilson 1989), experienced subsidence in the order of ~0.5 m (Hughes et al. 2015; Quigley et al. 2016). Following this, an annual rate of land subsidence of up to 6 mm/year has been estimated for the same areas (Pearson et al. 2021). The relative SLR caused by subsidence and SLR has resulted in recurrent groundwater flooding (Bosserelle et al. 2024) and seawater encroaching up the city’s main surface-water feature, the Ōtākaro Avon River, altering its ecology (Orchard et al. 2020). As such, Christchurch and its inhabitants are likely living through what other coastal cities will face under future SLR.

The lived experience of relative SLR-induced groundwater flooding has made this a topic of interest for both undergraduate and postgraduate groundwater hydrology students at Te Whare Wānanga o Waitaha University of Canterbury, in Christchurch. Subsequent discussions and student research projects relating to the impacts of SLR to groundwater have highlighted a number of recurring misconceptions. It is important to address misconceptions because they interfere with further learning when new information is linked to the existing misconception and they impede the application of science principles to everyday life (Engelmann and Huntoon 2011). This article aims to highlight and address these misconceptions. The misconceptions detailed below are ordered in such a way that they allow for a logical progression through the underlying physics and theory. Simple analytic models are used to aid in the development of critical insights relating to the physical processes (Haitjema 2006). A Jupyter Notebook is available to support readers’ interaction with the concepts and models, as detailed below under Open Research. It is hoped that this work proves useful to students, educators, science communicators and practitioners interested in impacts of SLR to groundwater.

Misconception 1: water-table rise will equal sea-level rise (SLR)

It is well accepted that under SLR, the water table will rise, but by how much? When asked this question, students commonly respond by saying that the water table will rise by the same amount as SLR. This, on the face of it, is a reasonable answer. A number of studies looking at SLR impacts to groundwater have made this assumption and included conceptual diagrams detailing this relationship (e.g., Chesnaux 2015). This assumption does work in some situations, but not all, and generally not for a low-lying coastal city like Christchurch.

In coastal aquifers, fresh groundwater (derived from terrestrial recharge) interacts with seawater. The location where the fresh groundwater and the seawater interact is commonly referred to as the interface. As seawater is denser than freshwater, the seawater tends to form a wedge under the freshwater. A conceptual model is shown in Fig. 1.

Fig. 1
figure 1

Conceptual diagram of a coastal aquifer

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The problem domain is bounded by the ocean (to the right) and an inland boundary (to the left). While the problem domain is theoretically infinite in the landward direction, the cross section is assumed to occur in a coastal fringe between 1 and 5 km from the coast. The depth to the interface below sea level z [L] is given by the well-known Ghyben-Herzberg relation (Ghyben 1889; Herzberg 1901): z = h/δ, where h [L] is the hydraulic head and δ is the density difference ratio given by δ = (ρs – ρf)/ ρs. Here ρs [M/L3] is the density of the seawater (usually ~1,025 kg/m3) and ρf [M/L3] is the density of the freshwater (usually ~1,000 kg/m3). As such, the Ghyben-Herzberg relation tells us that, at any point in a coastal aquifer, the depth to the interface below sea level will be 1/δ (~40) times the hydraulic head at that location. The inland location where the interface intersects the aquifer base (referred to as the interface toe; xT [L]) is used to separate the problem domain into two regions: a region without the interface (zone 1) and a region containing the interface (zone 2).

The water budget for the problem domain shown in Fig. 1 is comprised of freshwater discharge to the sea (q0 [L2/T]), lateral flow from aquifers inland of the landward boundary (qb [L2/T]), and net recharge (W [L/T], accounting for infiltration, evapotranspiration and distributed pumping). If the distance from the coast to the inland boundary of the problem domain is xb, the flux at the coast under steady-state conditions can additionally be defined as the sum of the recharge over the problem domain plus the flux into the problem domain from inland aquifers i.e., q0 = Wxb + qb.

The Strack (1976, 1989) approach to conceptualising and quantifying physical processes in coastal aquifers is widely used and well accepted (e.g., Werner et al. 2012; Morgan et al. 2013; Beebe et al. 2016). Following Strack (1976, 1989), q0 is defined as a function of the aquifer parameters defined in the preceding, as well as hydraulic conductivity K [L/T] and a measured hydraulic head hb [L] a distance xb [L] from the coast.

In zone 1:

$$q_0=\frac{K\left[\left(h_\text{b}+z_0\right)^2-\left(1+\delta\right)z_0^2\right]+Wx_\text{b}^2}{2x_\text{b}}$$
(1)

In zone 2:

$${q}_{0}=\left(\frac{1+\delta }{\delta }\right)\frac{K}{2{x}_{\text{b}}}{h}_{\text{b}}^{2}+\frac{W{x}_{\text{b}}}{2}$$
(2)

The choice of Eqs. (1) or (2) depends on whether hb has been measured inland of the interface (in zone 1) or above the interface (in zone 2). This can be determined by considering that if hb is greater than δz0 (i.e., greater than head at the interface toe location) it is in zone 1 and if hb is less than δz0 it is in zone 2.

The position of xT is given by:

$${x}_{\text{T}}=\frac{{q}_{0}}{W}-\sqrt{{\left(\frac{{q}_{0}}{W}\right)}^{2}-\frac{K\delta \left(1+\delta \right){z}_{0}^{2}}{W}}$$
(3)

And the distribution of hydraulic head hf [L] is, in zone 1:

$${h}_{\text{f}}=\sqrt{\frac{2{q}_{0}x-W{x}^{2}}{K}+\left(1+\delta \right){z}_{0}^{2}}-{z}_{0}$$
(4)

In Zone 2:

$${h}_{\text{f}}=\sqrt{\left(\frac{\delta }{1+\delta }\right)\frac{2{q}_{0}x-W{x}^{2}}{K}}$$
(5)

Thus, in this way, an analytic modelling framework that accounts for the different density of freshwater and seawater, as well as aquifer parameters, including sea level (i.e., z0), has now been defined. For a base case aquifer with the following parameters hb = 3 m, xb = 3,500 m, K = 10 m/day, W = 20 mm/year, z0 = 30 m, and δ = 0.025, it can be determined that the observation data are in zone 1 and q0 equals 0.33 m2/day. The location of the water table, interface and wedge toe can be seen in Fig. 2. Within the Jupyter Notebook, sliders can be used to alter aquifer parameters and observe the impact on the water table and interface position as well as values of q0 and xT.

Fig. 2
figure 2

Water-table and interface position for a typical coastal aquifer

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The next step is to change the sea level by ∆z0 in the model and see what happens to the water table. For this purpose, the SLR-induced change in water-table height Δh [L] relative to the base of the aquifer can be defined as (Morgan and Werner 2016):

$$\Delta h={h}_{\text{f}}{^\prime}-{h}_{\text{f}}+\Delta {z}_{0}$$
(6)

Here hf′ [L] is the new hydraulic head under SLR. The ∆z0 term in Eq. (6) is required because the datum from which hf and hf′ are measured (i.e., sea level) changes and this change needs to be accounted for when calculating the change in water-table height.

When solving problems of this type, the conditions at the boundaries of the problem domain need to be considered. At the coastal boundary, a change in head (of ∆z0) is being imposed. However, what about at the inland boundary? Two end-member conditions will be considered, termed flux-controlled and head-controlled.

Under the flux-controlled condition, the flux at the inland boundary (qb) is constant and independent of SLR. Thus, assuming that both W and xb do not change with SLR, q0 will not change either. Equations (4) and (5) can then be used to calculate hf′ by simply replacing z0 with z0 + ∆z0. The value of q0 is unchanged and is obtained from Eqs. (1) or (2). Substituting into Eq. (6) then gives the change in water table height as follows:

In zone 1:

$$\Delta h=\sqrt{\frac{2{q}_{0}x-W{x}^{2}}{K}+\left(1+\delta \right){\left({z}_{0}+{\Delta z}_{0}\right)}^{2}}-\sqrt{\frac{2{q}_{0}x-W{x}^{2}}{K}+\left(1+\delta \right){z}_{0}^{2}}$$
(7)

In Zone 2:

$$\Delta h=\Delta {z}_{0}$$
(8)

It can be seen from Eq. (8) that in zone 2 the change in the water-table height equals SLR. However, in zone 1, a more complex relationship exists, as shown by Eq. (7). The physical explanation for this is that in zone 1, the rise in the water table increases the aquifer transmissivity, which in turn reduces the hydraulic gradient required to transmit the same flow rate through the aquifer. This means that the water-table rise is less than the SLR in zone 1. In zone 2, the water table and interface rise in unison with SLR (the freshwater lens is behaving like a bubble) and transmissivity is not altered.

Under head-controlled conditions, the head at the inland boundary is maintained (as might occur due to pumping, surface-water features or drains) despite SLR. As such, the water table rises with SLR at the coast, but this rise diminishes landward, with no rise occurring at the inland boundary. The hydraulic gradient changes with SLR and the flux toward the coast also changes. The post-SLR flux at the coast q0′ [L2/T] is determined from Eqs. (1) or (2) by replacing z0 with z0 + ∆z0 and hb with hb—∆z0. Hydraulic head under SLR hf′ is then determined from Eqs. (3) and (4) with z0 replaced by z0 + ∆z0 and q0 replaced with q0′. For the sake of brevity, the resulting equations for Δh are not included here; however, it is easy to see conceptually that in both zones 1 and 2 Δh ≠ ∆z0.

The aforementioned theory can be used to visualise the change in water-table and interface location under both flux-controlled and head-controlled conditions for a SLR of 2 m and other parameter values as previously listed for the base case (Fig. 3). The accompanying Jupyter Notebook can be used to further explore this under varying aquifer properties and SLR values. The simplified analytic model, although providing insight at the theoretical level, should not be used as an analogue of reality without due consideration of limitations and assumptions including homogeneous, isotropic, constant-recharge and steady-state conditions, and a sharp freshwater–seawater interface. More detailed modelling is generally required for site-specific studies and management recommendations.

Fig. 3
figure 3

Impacts of a 2-m SLR in a coastal aquifer under both flux-controlled and head-controlled inland boundary conditions

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Misconception 2: inland movement of the interface causes the rise in the water table under SLR

A second misconception is that the rise in the water table with SLR is caused by the presence and movement of the interface. That is, it is thought that SLR-induced SI is causing the water table to rise—for example, recently a student wrote: “SI causes groundwater shoaling in coastal aquifers under SLR. In response to SI, groundwater shoaling occurs as dense marine water intrudes, and the freshwater table is forced higher”. This idea has also appeared in the popular media—for example: “Sea level is rising and has been now for well over 100 years…That means that when you do get a storm…, the drainage systems …may not be as able to cope as they used to, because the level of the groundwater is being pushed up by intrusion of seawater because of the sea levels rising.” (Briggs 2022). Another example is “While many coastal areas are focused on overland flooding as a result of sea-level rise, the threat of rising groundwater tables, known as ‘shoaling’, is not as well known or understood. Shoaling occurs when rising seawater pushes inland. The denser marine water underlies shallow freshwater aquifers, pushing them upward. In some low-lying areas, shoaling could force groundwater water to the surface, increasing the likelihood of flood damage” (ScienceDaily 2020).

It is not, however, SI that is causing shoaling, but rather the change in conditions at the boundary of the problem domain, i.e., the SLR, that is causing the water table to rise. To illustrate this point, if the sea was instead a freshwater lake and the level of the lake rose, then the water table would also rise, despite there not being an interface. The exact nature of the rise in the water table would depend on what is happening further inland, at the inland boundary, as was discussed in the previous section and as was shown previously by Werner and Simmons (2009) and Michael et al. (2013), among others. To be clear, the process of water-table rise from SLR is generally understood by hydrogeologists. However, it would appear that in simplifying this rather nuanced concept when speaking with the general public, including reporters, there is a tendency for it to be misunderstood and reported incorrectly.

Inland movement of the interface is caused by a reduction in flux toward the coast and, to a lesser extent, the increase in aquifer transmissivity caused by SLR (e.g., Werner and Simmons 2009; Michael et al. 2013). This is evident from Eq. (3). When SLR occurs under flux-controlled conditions, flux toward the coast does not change and SI is solely due to a change in aquifer transmissivity. The SLR causes aquifer transmissivity to increase by a small amount which in turn causes the interface to rise upward and slightly landward. When SLR occurs under head-controlled conditions, SI is due to both a reduction in flux toward the coast and a change in transmissivity. As such, and as shown in Fig. 3, SI from SLR is smaller under flux-controlled conditions then head-controlled conditions (Werner and Simmons 2009). Conversely, water-table rise is larger under flux-controlled conditions than head-controlled conditions. Clearly then, it is not SI that is causing shoaling.

While it is not SI that causes shoaling, the presence and movement of the interface does mean the rise of the water table is slightly different from what would have occurred without an interface (hereafter termed the no-interface case). For the no-interface case, the same conceptualisation as shown in Fig. 1 can be used, except that at x = 0 the boundary is freshwater. Using equations for steady flow in an unconfined aquifer with recharge and no interface in Fetter (2001), gives the following:

$$q_0=\frac{K\left[\left(z_0^2{+h}_\text{b}^2\right)-z_0^2\right]+Wx_\text{b}^2}{2x_\text{b}}$$
(9)
$${h}_{\text{f}}=\sqrt{\frac{2{q}_{0}x-W{x}^{2}}{K}+{z}_{0}^{2}}-{z}_{0}$$
(10)

Here, all parameters are as defined previously. Equation (6) is used to determine the change in water level for a change in the coastal boundary of Δz0. For a flux-controlled inland boundary condition, hf′ is determined using Eq. (10) by replacing z0 with z0 + ∆z. The value of q0 is unchanged and is obtained from Eq. (9). For a head-controlled inland boundary condition, hf′ is determined using Eq. (10) by replacing z0 with z0 + ∆z and q0 with q0′. Here q0′ is determined using Eq. (9) with z0 replaced by z0 + ∆z and hb replaced by hb – z0.

The water-table elevation for a 2-m SLR for the interface and no interface cases are shown in Fig. 4. The difference in water-table elevation caused by the presence of the interface under pre-SLR and post-SLR (flux and head-controlled) conditions is shown in Fig. 5.

Fig. 4
figure 4

Water-table elevation for the interface and no interface case under pre-SLR and post-SLR (flux and head-controlled) conditions

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Fig. 5
figure 5

Difference in water-table elevation caused by the presence of the interface under pre-SLR and post-SLR (flux and head-controlled) conditions

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Misconception 3: seawater intrusion (SI) caused by SLR is small compared to SI caused by pumping

Another misconception observed among students is that SI from SLR is minimal relative to SI from groundwater pumping. A comparison of SI from SLR and groundwater pumping for the United States, for example, concluded that groundwater pumping is a more important risk (Ferguson and Gleeson 2012). However, that study assumed flux-controlled conditions and therefore likely underestimated the risk posed by SLR (Lu et al. 2013). As shown in Fig. 3, SLR-induced SI under flux-controlled conditions is likely to be minimal compared with head-controlled conditions. Additionally, Lu et al. (2013) highlighted that if changes in saltwater volume (a measure of reduced water security) are used as a quantitative indicator, rather than wedge toe location, impacts of SLR-induced SI (under both flux and head-controlled conditions) will likely lead to more extensive freshwater storage losses than from pumping because, while SLR applies to the entire length of coastline, pumping is generally more localised. Additionally, it is worth noting that groundwater extraction can be managed in a manner that is relatively adaptive to local impacts, while climate-change-induced SLR cannot.

Seawater intrusion associated with groundwater pumping has been classified into three different classes—passive, active and passive-active (Fetter 2001; Werner 2017). These classes help managers predict the severity of SI under different pumping regimes. However, they can also be applied within a SLR-induced SI context for developing insight into tipping points beyond which rapid dispersive aquifer salinization could occur.

Under passive SI, the interface moves landward against the flow of fresh groundwater. This is the class of SI detailed so far in this article. Under active SI, a plume of saltwater moves landward in the same direction as groundwater flow and is highly dispersive and rapid, relative to passive SI. Passive-active SI occurs when the interface reaches the groundwater mound location (i.e., when xT/xn = 1, where xn [L] is the distance to the groundwater mound). In this situation, passive SI occurs on the coastal side of the mound and active SI occurs inland of the mound. Passive-active SI can occur despite heads throughout the coastal aquifer being above sea level. In Fig. 6, for example, one can see that for the base case aquifer described earlier, the threshold for passive-active SI has been reached under head-controlled conditions for a SLR of 2.3 m. The head at the inland boundary was 3 m pre-SLR, so the head when the threshold was reached is 0.7 masl.

Fig. 6
figure 6

Onset of passive-active SI with SLR under flux and head-controlled conditions

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Historically, areas with shallow groundwater have been drained to reduce flood risk and allow urban and agricultural development (Oude Essink et al. 2010; Rasmussen et al. 2012; Vandenbohede 2016). Future land drainage to counter the effects of SLR-induced groundwater shoaling is therefore likely. Land drainage will, in turn, induce spatially distributed head-controlled conditions throughout the aquifer which will increase SI (Fig. 3) and the propensity for a transition from passive to passive-active SI (Fig. 6). This highlights the need to manage SLR-induced groundwater salinisation and shoaling in an integrated manner.

Conclusion

The impacts of SLR to groundwater, including salinisation and shoaling, are widely accepted. However, a number of misconceptions on the topic have been observed. These misconceptions have been explored in this article using relatively simple analytic models with the aid of a Jupyter Notebook. The key takeaways are as follows:

  • Water-table rise is only equal to SLR above the interface under flux-controlled inland boundary conditions.

  • Water-table rise under SLR is not caused by SI, but rather is caused by the change in levels at the coastal boundary.

  • SI caused by SLR is a considerable risk, especially under the head-controlled conditions, which will become more common when land is drained to counter the effects of shoaling.