Introduction

The Central European Basin System (CEBS) covers a large area from the North Sea to Poland and from Denmark to North Germany. It is composed of two main basins, the Southern Permian Basin and the Northern Permian Basin. These two basins are superimposed on different continental crustal domains with Precambrian to Variscan consolidation ages. The CEBS includes a number of sedimentary sub-basins, among which the major ones are: the Norwegian-Danish Basin (NDB), the North German Basin (NGB), and the Polish Trough (PT) (Fig. 1). The formation and further development of these sub-basins are similar and can be summarized in a three step process (Ziegler 1990; Scheck-Wenderoth and Lamarche 2005): after basins formation in the Late Carboniferous–Early Permian time, accompanied by intense magmatism, a period of thermal subsidence, as predominantly assumed, dominated during latest Late Paleozoic and earliest Triassic. The Late Triassic to Late Jurassic evolution was characterized by a rifting phase followed by Cretaceous–Early Cenozoic inversion and a final subsidence stage in the Cenozoic. Studies from different parts of the CEBS (Vejbaek 1997; Berthelsen 1998; Scheck-Wenderoth et al. 2002) have suggested that basin formation and evolution was mainly controlled by the presence of deep reaching zones of lithospheric weakness. Bayer et al. (1997) have related these zones of observed stress and strain localization with the underlying structure of the lower crust or the lithospheric mantle, where areas of reduced viscosity or elevated temperature can play a key role in focusing deformation. In addition, basins evolution was also strongly influenced by the growth of salt structures (Maystrenko et al. 2005). For example, local salt movements strongly affect and even determine the stress pattern in the overburden (Bayer et al. 1999; Marotta et al. 2000). Moreover, the CEBS is located over a variety of crustal structures which have been tectonically reactivated through time. Dominant are NW-SE striking elements (Fig. 1): (1) the Tornquist Zone (STZ and TTZ) in the north-eastern part of the study area, (2) the Ringkøbing-Fyn-High (RFH), a structural high consisting of faulted blocks separating the Norwegian-Danish Basin from the North German Basin; (3) the Elbe Line (EL) which most probably acts as the southernmost border of the Baltic domain; and (4) the Elbe Fault System (EFS), a weak upper crustal domain active since Late Carboniferous times where strain localization was repeatedly observed and may be ongoing (Mattern 1996; Scheck-Wenderoth et al. 2002). Other important structures are N-S striking grabens which formed during the Mesozoic differentiation phase of the CEBS. The largest of these are the Central Graben (CG), the Glückstadt Graben (GG) and the Horn Graben (HG). Due to the superposition of all these distinctive elements, the CEBS shows a very complex structural setting. Regarding basin evolution, new information is now available concerning crustal and shallow mantle structures as derived from gravity modelling (Yegorowa et al. 2007), deep seismic experiments (MONA-LISA Group 1997; DEKORP-BASIN Group 1999; Guterch et al. 1999), seismic tomography (Gregersen et al. 2002), structural crustal modelling (Gemmer and Nielsen 2001; Scheck-Wenderoth and Lamarche 2005), and stress field modelling (Kaiser et al. 2005). Based on all these results, we used a thin sheet integral modelling approach to investigate the influence of the deep crustal structures and their related rheologies. We tested our results on two different independent observable parameters, the deformation pattern and stress field variations. In order to understand the interconnections between these elements we did not introduce any pre-defined faults in the models as in other studies (Kaiser et al. 2005). However, we relied on previous numerical results for choosing the material parameters and the boundary conditions (Marotta and Sabadini 2003; Marotta et al. 2004; Marotta 2005; Yegorova et al. submitted; Scheck-Wenderoth et al. 2002; Scheck-Wenderoth and Lamarche 2005).

Fig. 1
figure 1

Structural map of the Central European Basin System (Vejabek 1997; Lockhorst 1998; NW European Gas Atlas; Scheck-Wenderoth et al. 2002). The major elements shown are: (1) the Tornquist Zone (TZ) consisting of the Sorgenfrey-Tornquist-Zone (STZ) and the Teysserie-Tornquist-Zone (TTZ). (2) The Ringkøbing-Fyn-High (RFH). (3) The Elbe Fault System (EFS). (4) The N-S-striking Mesozoic Grabens: the Central Graben (CG), the Horn Graben (HG) and the Glückstadt Graben (GG)

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Horizontal deformation velocity field and strain rates

In order to test and quantify the consistency of the numerical model, the obtained results were compared with different types of observable data sets. The first one is the horizontal deformation velocity field. This data set is based on the last ten years of GPS observations which were used to define the ITRF2000 velocity solution database (Altamini et al. 2002). In order to compare the results of our model with geodetic data, the deformation is represented by the horizontal eigenvectors of the strain rate tensor. As already observed in previous work by Marotta (2005), the comparison between the geodetic and the model eigenvalues allows to quantify the quality of the model. Figure 2 shows the horizontal strain-rate eigenvectors derived from ITRF2000 solutions in Central Europe (black indicating extension, grey compression) computed for triangular domains following the approach of Devoti et al. (2002). The main features are (1) a general SW-NE-directed compression in opposition to a SE-NW-directed extension, and (2) while at high latitudes extension dominates over compression, at middle latitudes the two types of deformation styles are comparable.

Fig. 2
figure 2

Triangular horizontal strain-rate eigenvectors derived from the ITRF2000 solutions in Central Europe (Black indicates extension, and grey compression). The thinner black and grey coloured lines in the left angle indicate non-significant strain rates (modified from Marotta 2005)

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Stress field pattern

The second data set used to test the quality of the results is represented by the stress field as obtained from the “World Stress Map Project” (Reinecker et al. 2005). This database provides the direction of the maximum horizontal stress and the related regime as derived from several sources, such as earthquake focal mechanisms, young geological indicators and borehole breakouts (Fig. 3). These data can be compared to the maximum horizontal component of the stress tensor derived from the model. As usually assumed in any continuum approach to model continental deformation, the lithosphere is regarded as an isotropic medium. This reasonable assumption is reflected by a rheological law which explicitly links strain rates with stress components. Consequently, directions of stresses and strain rate eigenvectors coincide. Thus, after having obtained the horizontal velocity deformation field, the largest horizontal component of the stress tensor is easily estimated from the horizontal gradients of the velocity field and from the nodal values of the viscosity implemented in the thin sheet calculations (Ranalli 1995; Turcotte and Schubert 2002).

Fig. 3
figure 3

Map showing the direction of the largest horizontal stress for the area under investigation (Reinecker et al. 2005, “The release 2005 of the World Stress Map”)

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Model set up

A spherical thin sheet approach was adopted to calculate the deformation and the stresses inside the Central European Basin System, based on Marotta et al. (2004) where a detailed description of the mathematical formulation of the method is given. The domain of the CEBS was discretized by 2,000 spherical finite triangular elements, Fig. 4a. Our mesh was obtained by refining of the original grid of Marotta et al. (2004), and then it was adapted to the local frame. The model area covers a region extending from the 50°N to the 60°N parallel and from the 0°E to the 25.5°E meridian. We considered the asthenosphere as an inviscid element thus assuring a stress-free condition at the base of the plate. The lithosphere consists of three main vertical structural layers, the sediments which cover the basin area, the crust (further divided in two layers) and the mantle. Each domain may behave in a brittle, ductile or elastic manner in relation to the thermal state and the local strain rate. For brittle failure plastic deformation is used, therefore all properties depend on pressure but not on temperature or rock type (Byerlee 1978; Cloetingh and Burov 1996). Due to the thin sheet assumption of plain stress, it follows that one of the principal stresses coincides with the vertical stress. Under this condition, known as “Andersonian stress regime”, the brittle yield strength, σ B , follows Byerlee’s law (Ranalli 1995; Turcotte and Schubert 2002):

$$ \sigma _{B} = {\left( {\sigma _{H} - \sigma _{V} } \right)}_{B} = \beta \cdot z; $$

where σ H and σ V represent the maximum and the minimum principal stress respectively, z is the depth while β is the brittle failure coefficient depending on the type of fault, the angle of fracture and the pore pressure. Its value amounts 16 Mpa km−1 for extensional setting and 40 Mpa km−1 for compression (Lynch and Morgan 1987; Jiménez-Munt et al. 2005). At high temperatures deformation within the lithosphere is dominated by creep mechanisms. Laboratory studies have demonstrated that the ductile yield strength is a function of a power of the strain rate and depends strongly on temperature and rock type (e.g. Goetze and Evans 1979; Ranalli 1995). Following these suggestions, to model ductile behaviour a power law steady-state flow creep relation is assumed (e.g. Ranalli 1995; Turcotte and Schubert 2002):

$$ \sigma _{D} = {\left( {\sigma _{H} - \sigma _{V} } \right)}_{D} = {\left( {\frac{{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }}} {{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{0} }}} \right)}^{{\frac{1} {n}}} \cdot \exp {\left( {\frac{E} {{nRT}}} \right)} $$

where \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon } \) is the effective strain rate given by the second invariant of the strain tensor, R is the gas constant, T is the absolute temperature and \( \ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }_{0} , \) n, E are material constants depending on the rock type. The steady-state power law flow creep equation indicates that the critical differential stresses decrease exponentially with increasing temperature. However, above a critical stress ductile deformation becomes to be governed by dislocation processes. Following previous works (e.g. Weertman and Weeertman 1975; Goetze and Evans 1979), it is commonly assumed for olivine that for applied stresses higher than 200 MPa in magnitude, the governing equation takes the form of the so-called Dorn law (Weertman and Weertman 1975; Ranalli 1995; Turcotte and Schubert 2002):

$$ \sigma _{D} = \sigma _{0} {\left\{ {1 - {\sqrt {{\left[ {{\left( {\frac{{RT}} {{E_{A} }}} \right)} \cdot \ln {\left( {\frac{{\varepsilon _{D} }} {{\ifmmode\expandafter\dot\else\expandafter\.\fi{\varepsilon }}}} \right)}} \right]}} }} \right\}}\begin{array}{*{20}c} {;} & {{\sigma \ge 2000\,{\text{MPa}}}} \\ \end{array} $$

where σ 0, E A and ε D are flow parameters. The yield strength envelope,σ Y , is given by a contour function assuming the form:

$$ \sigma _{Y} = \pm \min \left\{ {{\left| {\sigma _{B} } \right|};\left. {{\left| {\sigma _{D} } \right|}} \right\}} \right., $$

which is positive for extension and negative for compression. The regional strain rate ranges between 10−16 and 10−14 s−1 for both compressional and extensional settings (Molnar and Tapponier 1981; Cloetingh and Burov 1996). As demonstrated in previous studies (Lankreijer et al. 1997; Marotta et al. 2000), the results are not strongly affected by changes in strain rate values, because strain rates within one order of magnitude provide changes less than 10%. The effective viscosity (η) is calculated from the depth integral of the yield strength envelope and is a function of the temperature and local strain rate. The boundary conditions imposed on the model are given in terms of crustal thickness and horizontal velocity (Fig. 4a, b). For crustal thickness, the European Moho contour map published by Ziegler and Dèzes (2005) was adapted to the size of the study area and to the nodes of our triangular mesh. Although different boundary velocity conditions play an important role in terms of stresses and strains, a detailed analysis of their influences is beyond the present purpose of the paper. Moreover, the results of Marotta et al. (2004) suggest that amplitudes and directions of predicted horizontal velocities at sites located well away from plate boundaries are more sensitive to the adopted modelling parameters than to slightly changeable boundary conditions. Consequently, in order to better constrain the role of lithospheric structures, all the calculations were performed imposing one set of fixed velocity boundary conditions to the integral grid. To determine the suitable boundary conditions, the nodal values of horizontal deformation velocity as obtained from the reference model of Marotta et al. (2004) were adjusted to our new grid. The applied velocity boundary conditions within the study performed by Marotta et al. (2004) were modelled relative to an assumed fixed Eurasian plate and comprised (Fig. 5):

  1. 1.

    Ridge push forces acting along the North Atlantic Ridge. These forces were parameterized in terms of velocity boundary conditions applied along the ridge. In this way, horizontal velocities simulate the line forces acting along the plate boundary, thick grey arrows of Fig.  5. The line force normal to the ridge was calculated from the eigenvalues of the stress tensor within the elements used to define the ridge (Richardson et al. 1979). Velocities at the boundaries range between a minimum of about 1 mm/year to a maximum value of 5 mm/year. Accordingly, the range of predicted push forces along the westernmost part of the Atlantic Ridge varies from about 1012 N/m for an imposed velocity boundary condition of about 1 mm/year up to a maximum of about 1013 N/m for an imposed velocity boundary condition of 5 mm/year.

  2. 2.

    Effects related with the continental convergence between the African plate and the Eurasian plate, thin black arrows of Fig.  5. The velocity of Africa relative to Eurasia was constrained from the data derived from NUVEL-1A whose patterns resemble a counter-clockwise rotation of Africa with respect to Europe superimposed on a continental convergence between the two plates of the order of 1 cm/year.

  3. 3.

    Effects related to subduction forces along the Aegean trench, thin white arrows of Fig.  5. Velocities values used were the same as geodetically determined by Mc Clusky et al. (2000).

The eastern boundary of the model was kept fixed imposing a shear stress free condition, black circles of Fig.  5.

Fig. 4
figure 4

a Finite element grid adopted fort he tectonic predictions in the study. b Crustal thickness variation used in the analysis. Adapted to the domain from the European Moho base-map, Ziegler and Dèzes (2005)

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Fig. 5
figure 5

Finite element grid adopted for the tectonic predictions as described in the modelling study of Marotta et al. (2004). The thick grey arrows along the western and northern boundaries represent ridge push forces from the North Atlantic. The counter-clockwise rotation of the African plate with respect the Eurasian plate is reflected by the thin black arrows along the south-western boundary. The velocities at the Aegean Trench, thin white arrows, were geodetically determined by McClusky et al. (2000) and reflect the horizontal component of the trench subduction forces. The southern border between the model domain and the Arabian region is held fixed (black triangles), while the eastern boundary is assumed to be shear stress free (black dots)

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In choosing the set of velocity boundary conditions to impose on our study area, the effects related to the Middle Atlantic spreading were not taken into account, whereas those related with Africa-Eurasia collision as well as with the trench Aegean subduction were inserted. Our choice is based on the fact that (1) this type of boundary setting revealed to give the best fit in the reference model of Marotta et al. (2004), and (2) the present-day deformation pattern of the CEBS was mainly caused by collision-induced compressive stresses.

The velocity field was then approximated by linear polynomial interpolating functions and numerical integration was performed by Gaussian quadrature with 7 integration points for each element. In relation to the reference model, the modifications can be summarized as follows: (1) we avoided any oversimplification which can limit the capability of reproducing the effects due to variation in crustal and mantle thickness; (2) lateral contrasts were inserted in the most realistic manner following the information derived from different previous studies; (3) a coupled set of thermo-mechanical equations were used as governing ones.

Crustal and lithospheric mantle thickness variations

In the thin sheet approach the lithosphere is generally treated as one viscous layer or as two layers, mantle and crust, with constant or slightly variable thickness and vertically-averaged rheological properties. Despite the successes reached in reproducing the general stress trend in Europe (e.g. Marotta et al. 2004), the oversimplifications included in these models turned out to be a limit in modelling tectonic scenarios in which depth related processes play an important role. Considering the Moho map (Ziegler and Dèzes 2005), it is evident that the CEBS is characterized by variable crustal thickness ranging from currently believed 25 km in its central part down to 48–50 km in the Baltic and East European Craton (EEC). Furthermore, the internal crustal structure is complicated by a thick sequence of sedimentary layers of different age (Ziegler 1990; Bayer et al. 1999; Krawczyk et al. 2002; Scheck-Wenderoth and Lamarche 2005; Majdanski et al. 2006). Going deeper, the asthenosphere–lithosphere thermo-mechanical boundary seems to behave almost similar to the Moho discontinuity, deepening below the EEC domain (Gregersen et al. 2002). As well documented from large seismic experiments (MONA-LISA Group 1997; DEKORP-BASIN Group 1999) and gravity modelling (Yegorowa et al. 2007), all these features are related to both lateral and vertical changes in geophysical properties. We performed several models for reproducing all the elements described in a stepwise approach in order to quantify their role in deformation.

Lateral rheological contrasts and thermal effects

Lateral structural contrasts have been incorporated in some of the previous models in terms of lateral variations of the Argand number, A r (England and Houseman 1985; Marotta et al. 2002; Jimènez-Munt et al. 2005). This number provides a measure of the relative importance of viscous and buoyancy forces, and includes the vertically averaged dependence of the viscosity on temperature and lithospheric composition. Its value varies between 0 (for a strong lithosphere) and 50 (for a very soft one), but the parameter is not always consistent with experimentally observed rock properties (England and Houseman 1989; Jimènez-Munt et al. 2005). Moreover, as widely debated by several authors in the past (England and Houseman 1985; Tomasi et al. 1995; Ranalli 1995) the viscosity has a strong dependence on the temperature profile, which controls the rheology of the Earth. Several studies (Bijwaard et al. 1998; Goes et al. 2000) have demonstrated the presence of 100–300°C fluctuation in the temperature under north-western Europe due to lithospheric mantle thickness variations. This inferred anomalous large scale mantle structure strongly influences the intraplate tectonic activity and may also cause variations in stress and tectonic style. Despite of this fact, most of the previous thin sheet models did not include any thermal calculations or tried to incorporate the temperature-related effects more or less arbitrarily. In order to avoid all these limitations, we implemented a three dimensional thermal conductive model which allows to solve for the temperature field with depth. The most important rock properties, such as radiogenic heat production, thermal conductivity and volumetric expansion coefficient, as derived from several localized studies for the different sub-parts of the study area were also taken into account. The temperature field is then used as input for the rheological model in order to derive the best fit for the CEBS.

Numerical modelling

Many studies in the past have related large-scale deformations to geodynamical causes. Failing to reproduce local features has led to a large number of studies on modes of lithospheric deformation as a function of its rheological structure. We performed several numerical “tectonic deformation experiments” in order to explore the consequences of compositional and thermal parameters. Our goal is the quantification of the impact of rheology on the tectonic deformation and stress setting. Here follows a brief description of the procedure followed and then the main results are discussed.

Case 1: Impact of the sedimentary fill

Model assumptions

To consider the impact of the thermal blanketing of the sedimentary fill, the sediments were modelled as a single vertical layer with a rheology very close to salt. Table 1 illustrates the model creep parameters, while Fig. 6 shows the sediment thickness map used in all calculations. It was adapted to our study area from the depth-maps published by Scheck-Wenderoth and Lamarche (2005). Its main features can be summarized as follows: there are two directions of depocentre axes, (1) NW-SE-trending with maxima under the Permian Basins and (2) N-S-trending just above the Mesozoic grabens. It images a complex pattern of lows and highs. The basin is framed by the Tornquist Zone in the north and the EFS in the south. The presence of the Ringkøbing-Fyn-High is also represented in the map. In this case study, the other structures comprise a single layered crust overlying a mantle of constant depth of 100 km. The Moho boundary is shown in Fig. 4b. Different models have been performed with creep parameters representing granite, granodiorite, felsic-granulite, diabase and quartzite crustal rocks, while the mantle was imagined with an olivine or a dunite dominated rheology (see Table 1 for the reference parameters).

Table 1 Creep parameters of the rocks used in the modelling study (Ranalli 1995; Chopra and Peterson 1981)
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Fig. 6
figure 6

Map showing the depth of the sedimentary layer used in the thermal and rheological models. It is adapted to the study area from the base-map of the Zechstein published by Scheck-Wenderoth and Lamarche (2005). The main elements presented are: (1) East European Craton (EEC). (2) Sorgenfrey-Tornquist-Zone (STZ). (3) Teysserie-Tornquist-Zone (TTZ). (4) Polish Trough (PT). (5) Ringkøbing-Fyn-High (RFH). (6) Norwegian-Danish Basin (NDB). (7) Central Graben (CG). (8) Horn Graben (HG). (9) Glückstadt Graben (GG). (10) North German Basin (NGB). (11) Elbe Fault System (EFS)

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Results

Despite the differences in rocks properties, the models showed the same general pattern, and therefore we illustrate and discuss only one example. In this case the crust has a quartzite dominated rheology, while the mantle is characterized by olivine creep behaviour (see Table 2 for the thermal parameters used). Figure 7 illustrates the results obtained in terms of strain rates and the deformation velocity field. The model resembles the reference deformation pattern shown in Fig. 2. An overall extensional setting oriented, mainly NW-SE characterizes the north-eastern part of the domain where almost no sediments are presented, while a compressional deformation style, NE-SW-oriented, becomes comparable to extension or even dominant in the south-west part of the study domain. The inclusion of the sedimentary fill proves to induce higher strain-rate eigenvalues without affecting the deformation style. As major effect the sediments weaken the lithosphere. This is due to their low value of thermal conductivity causing high temperatures in the upper part of the plate (Ziegler et al. 1998; Cloetingh and Van Wees 2005). This weakening effect leads to the formation of wide areas prone for strong deformation, (see Fig. 7) with increased strain rate values in the region covered by sediments.

Table 2 Values of the parameters used in the thermal modelling (see text for discussion)
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Fig. 7
figure 7

Strain rates (upper figure) and horizontal velocity field (lower figure) obtained only considering the sedimentary thermal blanketing (light background indicates extension, and dark background compression). There is a general agreement with the reference model depicted in Fig. 2 

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Case 2: Shallow mantle thermal effects

Model assumptions

For reproducing the shallow mantle discontinuities, the lithosphere was no longer considered as a plate of constant thickness. For modelling the mantle we followed the general results derived from the TOR project (Gregersen and Voss 2002; Voss et al. 2006). This large-scale seismic experiment was mainly concentrated on the deep lithosphere and asthenosphere, confirming the presence of large lateral transitions in the lithosphere of the CEBS. From the TOR results, the transition is considered to be sharp and steep in two places: (1) at the northern rim of the Tornquist Zone near the border between Sweden and Denmark where the lithosphere reaches depths of 200–250 km, and (2) near the southern edge of the Ringkøbing-Fyn-High where the difference becomes smaller, from 90–100 km of depth to 120–150 km. Figure 8 illustrates the depth map which we generated for the asthenosphere boundary. The parameters used for reproducing the rheology of the sedimentary layer, the crust and the mantle were the same as in Case 1: salt, quartzite and olivine, in order to isolate and quantify shallow mantle thermal effects. The depth map of the asthenosphere (Fig. 8), illustrates three domains within the study area: (1) a relative weak Variscan domain in the south and south-west; (2) a stiff and strong lithosphere beneath the EEC in the east and north-east, and (3) a transition zone extending between these two areas.

Fig. 8
figure 8

Map showing the lithosphere–asthenosphere isothermal boundary as derived from the TOR experiments results (Gregersen and Voss 2002)

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Results

The obtained results reproduce the overall scenario observed after the Late Cretaceous–Early Tertiary Alpine orogeny, when the region between the Sorgenfrey-Tornquist-Zone and the Elbe Line accommodated the main inversion structures, with normal faults being reactivated as reverse faults. The model now generates strong strain localization concentrated along the transition domain (see Fig. 9). The reproduced rheological heterogeneities affect the direction of the strain-rate eigenvectors leading to changes in the azimuth of the eigenvectors. The presence of a stiffer lithosphere below the transitional domain along the asthenosphere boundary acts as a barrier preventing a release of deformation for the area south of it. Coherent with the boundary conditions imposed along the southern border of the domain, compression is mainly NW-SE-directed providing better agreement with geodetic observations (Fig. 2). Moreover, the results demonstrate that rheological heterogeneities are able to induce changes in the eigenvalues of the strain-rate tensor. As a consequence, the domain between the Elbe Line and the Tornquist Zone is now characterized by higher strain rate eigenvalues. Furthermore, the region of the Polish Trough shows a change in deformation style with a transition from a compressive dominated regime to an extensional one. In order to better constrain the results, the predicted direction of the maximum horizontal stress component, S Hmax, (Fig. 10) was compared with the observed data (Fig. 3). The actual World Stress Map (Reinecker et al. 2005) shows a uniform pattern of WNW-ESE-directed maximum horizontal compressive stress in North Europe with a clockwise rotation toward ENE-WSW in the north-eastern part of Germany and Poland. The calculated largest horizontal stress field reproduces the fan-like structure only within a limited area. The predicted change in the direction of the largest horizontal stress is localized only in the northeast part of the North German Basin and in the southernmost part of the Polish Trough. The results of Fig. 10 show a mismatch between the predicted and the observed S Hmax in the westernmost onshore part of Denmark and below the Danish sector of the Central North Sea Graben, dashed framed region of Fig. 10. Within these areas our results image a NE-SW-oriented maximum horizontal stress component which is almost perpendicular to the observed WNW-ESE-oriented S Hmax. The local mismatch found between the model predictions and the observations may be directly related to the particular composition of the lithosphere in those areas.

Fig. 9
figure 9

Maximum horizontal strain rate as predicted by the model discussed in Case 2. As a consequence of having considered a transition domain in the lithospheric mantle, the deformation, mainly compression, is now focused along this area

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Fig. 10
figure 10

Map showing the direction of the maximum horizontal stress component, S Hmax, as predicted by the model discussed in Case 2. The areas framed by dashed quadrates are those where the strongest disagreement was found

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Case 3: Lateral rheological heterogeneities in the crust

Model assumptions

In order to reproduce lateral changes in geophysical properties, the crust was split into two layers. At this point the vertical structures comprise: (1) the sediments with a salt-driven rheology, see Fig. 7 for the depth map; (2) the upper crust with a quartzite or granite rheology and a lower boundary at oscillating depth between 20 and 30 km according to seismic results; (3) a lower crust down to the Moho (Fig. 4b), for which several rheological analogues were implemented; and (4) a mantle of variable thickness (Fig. 8) with an olivine dominated behaviour. As second step, for modelling lateral structural contrasts, we subdivided the whole area into four domains (Fig. 11) with different rheologies in the lower crust. Our choices for the rheological analogues were based on results derived from local studies. The range of P-wave lower crustal velocities (6.2–6.5 km s−1) and densities (2,700–2,800 kg m−3) observed in the area along the Elbe Fault System motivated a granitic to a granodioritic dominant composition. The high values for P-wave velocities reaching 6.9–7.0 km s−1 and the range of densities (2,900–3,100 kg m−3) in the lower crust north of the Elbe River and below the Sudetes in Poland suggest a lithology typical for gabbros. Finally, we modelled this area with a felsic-granulite composition. To the north this felsic-granulite composed lower crust is bounded by the Tornquist Zone. For the Baltic region we used a slightly stronger mafic-granulite rheology. For the area south of the Elbe Fault system we considered three cases. First the lower crust was modelled with a granodiorite dominant lithology, resembling a weak crustal body. A quartzite composition and a diabase dominated composition were chosen respectively for the second and the third experiments, reproducing a stronger crust. Table 3 summarizes the different data used for the simulations.

Fig. 11
figure 11

2-D grid used for the integral model as discussed in Case 3. The lateral domains with different rheological properties in the lower crust are shown

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Table 3 List of models considered in the analysis
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Results

We limit the discussion to three examples (Model 2, Model 4 and Model 6 of Table 3) because the best results were obtained by modelling the upper crustal layer with a dominant quartzite rheology. Figures 12, 13 and 14 illustrate the resulting strain rates. Despite different rheologies used for modelling the Variscan domain, the models generated the same trend, therefore we discuss them together. The main feature shown in the figures is a southward shift of the zone of strain localization. Now, the area with highest deformations is found below the EFS, while a minor compression is still visible along the Tornquist Zone. The lithosphere is weaker beneath the EFS than below the surrounding areas due to the granitic composition in the lower crust which provides increased temperatures. This weakness explains the observed strain localization under this region. The introduction of lateral contrasts improves the results in those locations where the strongest mismatch was found before, maintaining at the same time the overall consistency between the model and observations. The kinematical scenario for the North German Basin is now in agreement with its recent compressional phase and the present-day setting. N-S to NNE-SSW-oriented compression is stronger in the weaker crust below the EFS rather than in the northern part where the predicted strain rate pattern indicates low deformation rates. This aspect is in agreement with the observed local aseismicity of the region, suggesting a rather undeformable area in the northern part of the basin. The results of Figs. 12, 13 and 14 show a change in deformation patterns within the area of the Polish Basin which is now characterized by a N-S-oriented to a NWN-SES-oriented compression. This variation in the deformation patterns indicates that lateral rheological heterogeneities can affect the azimuth of the strain-rate eigenvectors as well as the related eigenvalues. Moreover, the presence of the high-density lower crust below the Polish Trough has moved the centre of maximum strain localization from the TTZ southward. This scenario is in agreement with the present-day tectonic setting of this region which is dominated by a collision-induced compressive intra-continental stress pattern with the main depocentres located in its southernmost part. The predicted orientation of the principal horizontal compressive stress (Figs. 15, 16 and 17) shows a general agreement with the present-day stress pattern observed over North Central Europe (Zoback 1992; Reinecker et al. 2005). The modelled stress field images the overall NW-SE-oriented principal axes and also the progressive rotation to S-N/NE in the eastern part of the study region. Focusing on the NGB where many previous studies have failed in predicting the stress field orientation, our results are consistent with the observation showing the fan-like pattern in the direction of S Hmax. Lateral heterogeneities induce changes in the magnitude of the stress tensor as well as in its orientation being able to resemble more local features. Furthermore, the predicted stress regime suggests that the north-easternmost area of the North German Basin is still in a state of horizontal compression, having not yet reached isostatic equilibrium as derived by Marotta et al. (2002). In a regional contest, an overall agreement between the predicted and the observed regional stress field is evident, although there is still a region of mismatch. In fact, the model fails in reproducing the orientation of the S Hmax at its western boundary where it images a NE-SW-oriented maximum stress instead of a NW-SE-oriented one. This can be related to the particular boundary conditions applied along that border, i.e. a pure boundary effect.

Fig. 12
figure 12

Predicted strain-rate eigenvectors for Model 2 of Table 3 (light background indicates extension and dark background compression)

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Fig. 13
figure 13

Strain-rate eigenvectors predicted by Model 4 of Table 3 

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Fig. 14
figure 14

Strain-rate eigenvectors obtained from Model 6 of Table 3 

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Fig. 15
figure 15

Predicted direction of the maximum horizontal compressive stress, S Hmax. The illustration refers to the case in which a quartzite-type lower crust was chosen for modelling the domain extending eastward the EFS (Model 2 of Table 3)

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Fig. 16
figure 16

Direction of S Hmax with a felsic-granulite dominated rheology in the lower crust below the eastern part of the study domain (Model 4 of Table 3)

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Fig. 17
figure 17

Predicted S Hmax direction for a granite dominated rheology along the eastern part of the study region (Model 6 of Table 3)

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Final remarks

A number of models have been presented to test the relative contribution of variations of compositional and thermal parameters to the setting of the regional stress field and deformation pattern in the CEBS. The first sets of models (Case 1 and Case 2) were mainly focused on the evaluation of the effects induced by thermal fluctuations on the predicted stress field and strain pattern. These models resembled the overall features. On the other hand, the introduction of lateral crustal structural heterogeneities is necessary to explain more local features, as shown in Case 3. The presence of lateral contrasts induces changes in the direction of the strain-rate eigenvectors as well as in their related magnitudes. Moreover, lateral rheological heterogeneities affect the direction of the maximum component of the stress tensor. Our results suggest that rheology is a first order driving element for deformations and stresses even in tectonic scenario like the CEBS which are almost dominated by boundary forces.

Conclusions

A viscous thin sheet model has been used to investigate the role of rheological contrasting structures on deformation and stresses focusing on the area of the CEBS. Unlike many previous integral studies, the effects of variations in the thermal regime have been considered in order to determine the best fitting rheologies for different sub-domains. The consistency of the results has been constrained by direct comparisons of the model outcomes and two independent sets of data. For this purpose, the reference parameters are the present-day regional stress field obtained from the “World stress Map Project” (Reinecker et al. 2005) and the strain rate eigenvectors derived by the last ten year GPS observations of the ITRF2000 database (Altamini et al. 2002). The results demonstrate that the observed deformation and stress pattern in the CEBS can be reproduced with reasonable agreement by the model, concerning the overall trend as well as more local features.

Concerning the deformation style, lateral rheological heterogeneities have been proved to induce variations in the azimuth of the strain-rate eigenvectors as already suggested by previous studies (e.g. Marotta 2005). On the other hand, lateral contrasts turn out to affect the strain-rate eigenvalues revealing their key role in stiffening the propagation of tectonic deformation in the CEBS. In a local contest, the obtained results reproduce a weak lower crust below the EFS and suggest a rather undeformable area in the northern part of the NGB, in agreement with previous results (e.g. Marotta et al. 2000).

In contrast to Golke and Coblentz (1996), we have demonstrated that the direction of the principal stress axes is not totally independent of the rheology of the lithosphere. The presence of different structural domains at crustal and shallow mantle level is responsible for the present-day local variations observed in the direction of the regional stress field. Moreover, our results suggest a present-day state of horizontal compression in the easternmost part of the NGB. This feature is in agreement with previous models (Marotta et al. 2000; Marotta et al. 2002), with results from gravity studies (Scheck-Wenderoth et al. 1999), and with observations from deep seismic experiments (MONA-LISA Group 1997; DEKORP-BASIN Group 1999). The results have shown that strong strain localization and major deviation in the stress directions occur without considering pre-defined ‘a-doc’ faults as has been often used in this context (e.g. Kaiser et al., 2005). In contrast, the same results indicate that large inherited structures may play an important role in the evolution of complex basins controlling processes including strain localization which finally leads to failure and to faults formations. The best agreement between the results and the observations has been found when:

  • The sedimentary layer is represented by a salt dominated rheology.

  • Variations in the thickness of the lithosphere are modelled, considering a step-wise transition between the Elbe Line in the south and the Tornquist Zone in the north.

  • The upper crustal layer is modelled with a stronger quartzite-like rheology rather than with a weaker granite dominated one.

  • Lateral structural contrasts at lower crustal depth are considered. These discontinuities comprise the presence of (1) the high density body in the lower crust below north Germany, and (2) a relatively zone of weakness along the EFS.

Our analysis was focused on the influence of large-scale structures on stresses and deformation patterns. It demonstrated that a key factor for the distribution of the compressive stress and strain localization in the CEBS area is the presence of lateral rheological heterogeneities. Strong lateral contrasts are driving the propagation of tectonic deformation and affect the regional tectonic setting.