As most of the landscape in northern Germany, the surface morphology is dominated by the Pleistocene ice-age cycles. The repeated advance and retreat of the Fennoscandian Ice Sheet carried glacial and peri-glacial sediments, which form the main features in the landscape: Münsterdorf sits on a geest ridge, a sandy deposit from deposition below the ice, about 20–30 m above sea level in the region. Beyond this geest island, the landscape is lower, between 5–15 m above sea level, and dominated by swampy marshland. The geest ridge was deposited during the Elsterian (455–320 ka) and Saalian (300–125 ka) glacial cycles, while during the last glacial cycle, the Weichselian, the ice did not reach the area around Münsterdorf.

The geest ridge consists of unconsolidated and consolidated peri-glacial sand, interbedded with glacial till, in parts composed of clay lenses.

The entire region is tectonically controlled by the Krempe–Lägerdorf salt ridge (e.g. ), a salt diapir, with salt sequences from the Permian Zechstein period. These Zechstein rocks can usually be found in a depth range between 3–5 km underneath northern Germany, but as the salt layers in the Zechstein evaporite they can flow plastically under stress; numerous salt structures have been pushed up by the large stress imposed by the overburden. The vertical uplift of these salt domes has pushed up the overburden by several kilometres, often reaching the surface. In the Münsterdorf region, the Krempe–Lägerdorf salt ridge has pushed up Cretaceous chalk from its original depth of 0.5–1 km to a position close to the surface (e.g. ).

Just 1.5 km south-west of Münsterdorf, the chalk is just below the surface (6 m below the ground), and several large open-pit mines exploit the material. The deepest open-pit mine reaches down to 90 m below sea level, which requires a substantial effort to dewater the mines (e.g. ). The pumped groundwater from the open-pit mines is routed via an artificial channel in the east towards the river Stör. In the mines, numerous karst features can be observed, such as dissolutionally enlarged fissures and bedding planes and small collapse sinkholes.

The local sports field is located on the southern rim of the village of Münsterdorf. The chalk can be found here at around 20 m depth below the sports field (see Fig. ), confirmed by several boreholes (see ; and references therein). Above the Cretaceous chalk, we find mainly peri-glacial sand, inter-bedded with glacial till.

From the geological and morphological set-up, we identify three key aspects responsible for the presence of karst structures in the Münsterdorf region:

salt diapirism responsible for local tectonics and uplift of Permian and Cretaceous sequences;

Since 2004, collapse sinkholes have occurred regularly on the sports field (Fig. ). These collapse sinkholes are about 2–3 m in diameter and about 3–5 m deep, exposing the peri-glacial sand. The sinkholes occur suddenly, often underneath the grass of the sports field, and are initially steep-walled, with near-vertical sides. Most of the sinkholes are filled with artificial material immediately but, if left open (as in Fig. ), develop towards a mechanically more stable funnel shape. No groundwater can be found in the sinkholes, as the groundwater table nowadays is lower, at about 5–7 m depth, when compared to previous decades (1–3 m depth). Fluctuations due to rainfall are in the sub-metre range. The sinkholes are aligned along a narrow band about 50–70 m wide and 300 m long, striking in a west–easterly direction.

The sudden onset of the sinkhole occurrence and the limitation to the narrow band suggest a relation to a lithological feature in the subsurface and a possible inception by a change in hydraulic conditions (e.g. ). Thus, we regard these two aspects as important constraints for the explanation of the collapse sinkholes along the sports field in Münsterdorf: (i) sudden onset in around 2004, (ii) restriction to the narrow west–east band.

From the numerous geophysical methods used (gravity, geoelectrics, electro-magnetics, georadar, seismics, borehole-based spectrally induced polarisation; see Table  for a summary), especially geoelectrics and georadar have been successful in delineating the glacial till from the peri-glacial sand. In almost all profiles (taken in a north–south direction), the high-resistivity peri-glacial sand is thin (1–2 m) in the north and underlain by more conductive glacial till, while the peri-glacial sand becomes thicker (5–8 m) towards the south, and the glacial till seems to be thinner and more isolated in the south. These results can be explained by the geest island, on which Münsterdorf rests, which peters out towards the south (Fig. ). Cored boreholes confirm this set-up locally (e.g. ).

The chalk layer at about 20 m depth has not been identified in the geoelectric and georadar measurements because electrical conductivities between the chalk and wet till are too similar for the former method and because of depth restrictions for the latter method. However, in seismic p-wave profiles, the chalk layer can be seen.

From the geophysical and borehole measurements, we have derived a simplified lithological set-up underneath the sports field:

three layers – peri-glacial sand, glacial till, soluble chalk;

We argue that the soluble chalk at shallow depth is accessible to groundwater and thus has the potential to be dissolved, resulting in karstified structures. However, as stated below in our first hypothesis, there is also the possibility that voids are present in the sand layers.

To start our discussion on the origin of the subsurface voids, we propose two different competing scenarios for the development of the collapse sinkholes.

Non-karstified. The initial void causing the sinkhole collapse has developed above the lower till layer, at about 6–8 m depth in the peri-glacial sand.

As we know the sinkhole sizes (w=2–3 m diameter, d=2–5 m depth), the initial void should have a volume around V=πw22d∈[6,35] m3, a substantial initial void, which at this depth should possibly be detectable by geophysical methods. We try to estimate the effect on gravity and ERT (electrical resistivity tomography) surveys below.

We note that in this scenario the initial void would be located in insoluble unconsolidated rocks; thus the origin of the void cannot be dissolution of material, only either removal of material by erosion or melting of a block of ice. We then have to ask ourselves why such a void is only present along the narrow band on the sports ground, and not much more widespread, as expected in a landscape dominated by glacial features.

In our summary of the geophysical work done over the years, we mentioned the identification of the lithological set-up of the region around the sports field. However, geophysical exploration was less successful in identifying the subsurface voids causing the collapse sinkholes. Only old sinkholes show a faint signal in georadar and (in parts) in geoelectrical measurements.

We therefore present results from forward modelling of gravity and geoelectrical signals induced by subsurface voids in a simplified set-up (Figs.  and ): a top layer of dry peri-glacial sand (0–10 m), followed by a wet layer of peri-glacial sand (10–20 m), and below the soluble chalk (>20 m). We do not include a glacial till layer, as (i) the model depicts the situation close to the transition zone from thicker glacial till to thin, more isolated glacial till, and (ii) the material properties of the glacial till, especially electrical resistivity, are not too different from the wet chalk.

Depending on the hypothesis, voids are either (i) in the peri-glacial sand (air-filled; see Fig. ) or (ii) in the chalk (water-filled; see Fig. ). Material properties are listed in Table .

The gravity forward model is based on box models for the lithological layers and spheres for the voids (e.g. ). All density values are referenced against the chalk density; thus the calculated Bouguer gravity values will be negative, as peri-glacial sand and voids have lower densities when compared to the chalk.

For the electrical resistivity model, we first create a 2D forward model with res2Dmod (e.g. ) to calculate apparent resistivities for the given set-up. Then, these apparent resistivities are perturbed by random noise (2 %) and then loaded into res2Dinv (e.g. ) to find the resulting resistivity model from inverse modelling.

Following our two hypotheses for the void origin, the non-karstified and the karstified case, we define two forward models with voids at different depths: (i) shallow origin (6–8 m) in insoluble dry sand – air-filled voids, (ii) deeper origin (20–22 m) in soluble chalk – water-filled voids. We locate three voids in these two different depth ranges, with increasing radius from left to right (1.1, 1.7, and 2.0 m) to account for the uncertainty in initial void volume introduced before.

Non-karstified. In the non-karstified set-up (Fig. ), the voids are located in the dry peri-glacial sand. The Bouguer gravity is negative because the density of the peri-glacial sand (1800–19 000 kg m-3) is lower than the density of the chalk (2600 kg m-3), which we took as a reference density. The air-filled voids are modelled with a density of 0 kg m-3. All three initial voids are visible in the Bouguer signal, but the amplitude (less than 1 mGal) would be hard to detect within the accuracy of a typical relative gravimeter.

In the ERT cross section, we used electrical resistivities of 1000 Ωm for the dry peri-glacial sand and 800 Ωm for the air-filled voids. Below, both the wet peri-glacial sand and the chalk are more conductive, with values around 200 Ωm. The resulting resistivity model can distinguish between dry sand and wet sand/chalk but does not reveal the void spaces, as their resistivity is too close to the one of the dry sand.

From this simple forward-modelling exercise, we would expect to find hints for initial voids from geophysical signals (mainly gravity), if these initial voids would be in the shallow peri-glacial sand. For a deeper-seated origin of the voids, e.g. in the chalk, no clear geophysical signal in either gravity or geoelectrics can be expected. As outlined in , we have not found evidence for subsurface voids in the geophysical data.

We first discuss two simple end-members of a situation prone to collapse sinkhole generation (e.g. ): a first case considering buoyancy only and a second case assuming that some of the stresses are supported by a stronger elastic material.

The discrete-element method (DEM) is a computational method to simulate the interaction of a large number of particles (e.g. ). Often, the particles are spheres with pre-described radius and some material properties such as density ρ (kg m-3), Young modulus E (Pa), and Poisson ratio ν (–). Spheres are packed into a three-dimensional region to represent a macroscopic material, with packing being either regular or irregular. Neighbouring spheres can interact, with a large choice of mechanical laws defined. Modelling of the mechanical deformation consists of several steps: (i) set-up of modelling domain, (ii) packing with spheres, (iii) definition of material properties and interaction laws, (iv) time-stepping procedure to track position and velocity of each sphere.

We use the open-source DEM model YADE (e.g. ). Our modelling domain is 50 × 50 × 50 m in size, thus 10 times larger than the collapse sinkholes in Münsterdorf (Fig. ). This large modelling domain ensures that the fixed boundary spheres (also shown in red as the chalk) have no influence on the deformation (see , for more details on the implementation). The material properties for the different layers (upper and lower peri-glacial sand, glacial till, chalk) have been taken from the literature (Young modulus, Poisson ratio) and from the direct-push-based methods discussed above (friction angle and cohesion).

The spheres in the initial model (upper and lower peri-glacial sand: light brown; glacial till: orange; chalk: red) are packed, and the model is stable; thus no deformation is observed (Fig.  top). We then initiate a void in the chalk (red spheres) by removing spheres within a given radius (removed spheres marked in blue). The removed spheres represent either a water- or an air-filled initial void. This initial void in the chalk destabilises the model domain mechanically. The material above the void starts to break down into the void until a new mechanical equilibrium is reached.

Two scenarios are shown in Fig.  (middle and bottom row). i.

Buoyant support. In the left column, the subsurface is characterised by a high water table, and the initial void is filled with water. Thus the difference between gravitational force and buoyancy force is smaller, (ρs-ρw)g, and thus the buoyant force compensates the gravitational force significantly. The model experiences breakdown of sand layers above the chalk, but the elastic strength of the till supports the weight of the column above, and the void does not migrate towards the surface. The initial void enlarged by chemical dissolution becomes enlarged by additional breakdown but remains stable. No collapse sinkhole will form (unless the boundary conditions are changed).

By comparing these two models we have shown that a simple change in the hydraulic boundary condition, here a drop in water table, which reduces buoyancy and thus increases the weight of the column hanging above the void, can trigger a collapse sinkhole.

We stress that the timescales of dissolution and mechanical breakdown are completely decoupled: while the void in the chalk, created by chemical dissolution of the material on timescales of 1000–100 000 years, might initiate roof collapse into the insoluble formation, the surface remains stable because the overlying material is partially supported by the mechanical strength of the layers. However, a drop in water table through groundwater withdrawal on timescales of years can destabilise the system and initiate a sudden collapse of the surface, creating a collapse sinkhole.

Of course one can argue that a similar situation might occur for a void located in the peri-glacial sand. However, we have no evidence for a mechanically sound layer in the top part of the insoluble overburden (down to 6–8 m); thus a potential subsurface void in the peri-glacial sand is unlikely to be mechanically meta-stable but will collapse more or less immediately. Thus we argue that the void origin is more likely to be in the chalk layer. We support this argument further in the next section.

From the direct-push-based electrical conductivity measurements, we found values for the electrical conductivity around σe,f≃30–50 mS m-1 where direct-push profiling reached the hydraulically conductive part of the chalk, in parts even higher. We convert this electrical conductivity representative of the fluid, σe,f (µS cm-1), to the total amount of dissolved solids, TDS (mg L-1), using an experimentally determined linear relation (e.g. ) 17TDS=keσe,f, with ke∈[0.55,0.80] (cm mg L-1 µS-1) a correlation factor and the electrical conductivity σe,f (µS cm-1) in this equation referenced to 25 ∘C. Using ke=0.6, we obtain a TDS value around 180–300 mg L-1.

When we argue that a large amount of the TDS value can be explained with dissolution of the chalk, we derive the concentration of calcium c (mol m-3) from the TDS value, using 18c=TDSmr, with mr=0.101 kg mol-1 the atomic mass of calcite, resulting in c∼2–3 mol m-3.

If we compare this calcium concentration c derived from TDS to the calcium equilibrium concentration ceq (mol m-3) for calcite dissolved under closed-system conditions (decoupled from the atmosphere), which is in the range of ceq≃2–5 mol m-3 (e.g. ), we find that the dissolution of chalk is close to or at its equilibrium value, thus at its maximum.

Arguing that the high calcium concentration in the probed water in the chalk has been attained locally, the EC results from direct-push-based methods support the idea of that enlargement of voids in chalk by chemical dissolution has created the initial void space at around 20 m depth. The isolated area of high permeability within the chalk that was identified using the direct-push-based HPT as well as the depth interval with strongly reduced cone pressure (CPT measurement) supports the theory that the high calcium concentration has been generated by the local dissolution, creating a large secondary permeability in the chalk.