We set up generic 3-D models with model dimensions, rock properties, and an initial stress state that are like the ones from a 3-D geomechanical–numerical model of a potential siting area for a high-level radioactive waste disposal site in northern Switzerland, presented by . For implementation in the model, faults are represented by contact elements, which allow an offset, or 3-D elements, which are elastically or plastically weaker than the surrounding rocks (Fig. ).

The two key components of a static stress state are a result of volume forces due to gravity and surface forces from plate tectonics. Neglecting acceleration, the resulting partial differential equation is the equilibrium of forces. For the upper crust assuming linear isotropic elasticity is a good approximation to describe the stress–strain relation e.g.. Thus, for simplicity the three key model parameters in our study are density (ρ), Young's modulus (E), and Poisson's ratio (ν). Additionally, the Mohr–Coulomb criterion, using the friction (μ) and the cohesion (C), will be used for some models. As we introduce a fault in our model with different techniques, we solve the problem numerically using the FEM.

The reference model has an extent of 10 km in each horizontal direction and 3 km in the vertical direction (Fig. ). The model is intersected in its entirety in the centre by a 60∘ inclined fault, represented by cohesionless contact elements with a friction coefficient of μ=0.4 (friction angle ϕ=21.8∘). The main shortening direction is perpendicular to the strike of the fault. Homogeneous linear elastic and isotropic material properties are assigned to the reference model, having a Young's modulus of E=15 GPa, a Poisson's ratio of ν=0.27 and a density of ρ=2550 kg m-3. The FE mesh for the reference model has a resolution of 50 m in the X and Z direction and 500 m in the Y direction. The mesh was created with HyperMesh 2017 and 2019, respectively; the solver used is Abaqus 6.14.1.

The scope of the study is to investigate factors that affect the stress state in the broader vicinity of faults. These include the element resolution (pre-tests), the representation of the fault by contact elements with a variable friction coefficient, representation of the fault by elastic weaker elements or by elements with elasto-plastic rheology, the inclination of the fault, the strike direction relative to the shortening direction, the variation of the rock stiffness (Young's modulus), and the size of the fault and model itself. In order to allow good readability of the study, specific variations of the model are always briefly explained before presenting the modelling results.

We implement an initial stress state of the model that is in equilibrium with the gravitational forces without resulting in any significant displacement along the fault and the model geometry. We follow the technical procedure as explained in . In a second step we apply along the model lateral boundaries displacement boundary conditions that result in tectonic stresses throughout the model volume. The main shortening of the reference model is perpendicular to the fault (X direction) of the order of -10 m (ϵ=1×10-3), which then corresponds to the SHmax orientation. Parallel to the fault strike (Y direction), the model undergoes a slight dilation of 2 m (ϵ=2×10-4), which is then the orientation of Shmin (Fig. ). The stress magnitudes resulting from the boundary conditions are shown in Fig.  along a vertical synthetic well path in the centre of the model. This stress state is in general agreement with stress magnitude data that were derived from a measurement campaign in northern Switzerland using >150 micro-hydraulic fracturing and sleeve reopening tests .

The 3-D stress state of the Earth's crust is described by a second-rank tensor σ, with nine components, but due to its symmetry only six components are independent from each other. As is common in geoscience, compressive stress magnitudes are positive and tensile stresses are negative. The stress state can also be described with the magnitudes and orientations of the three principal stresses. These principal stresses are named from the largest to the smallest as σ1>σ2>σ3.

Furthermore, in our model the vertical stress (SV) is a principal stress (Eq. ). As a result, the two other principal stresses are in the horizontal plane and are labelled as the minimum and maximum horizontal stresses (Shmin and SHmax). 1SV=∫0zρgz The relative ratio of these three principal stresses defines the stress regime :

Additionally, we use the differential stress (σD) and its 3-D equivalent, the von Mises stress σvM;, to visualize the stress state (Eqs.  and ). 2σD=σ1-σ33σvM=12σ1-σ22+σ2-σ32+σ3-σ12 The model results are presented here in the same way whenever possible. Both the stress components (SHmax, Shmin, SV) and the von Mises stress are used to visualize the influence of a fault on the stress state. The results of the models are plotted along a horizontal path at a depth of -660 m (Figs.  and ). This path is always parallel to the main shortening direction (X), except for the models with a variable fault strike. The visualization extends from the footwall block at -3000 m through the fault at 0 to +3000 m in the hanging wall block.

The impact of the mesh resolution and sufficiency is investigated by varying the mesh size using elastic material properties only, like the reference model. A mesh resolution of 1000, 500, 250, and 100 m in all directions is tested; a finer resolution has been used with an element size of 50 m in the main shortening and depth direction (X and Z), for which the resolution parallel to the fault (Y) is 500 m.

The model with the coarsest resolution (1000 m) provides stress magnitudes that deviate significantly from the other models (red line in Fig. ). Even for the model with a resolution of 500 m (magenta line in Fig. ), the deviation from the higher-resolution models at a distance greater than 1000 m is clearly visible. All finer-resolution models (≤250 m) have only small differences close to the fault (<1000 m; Fig. ). This shows that all models with a resolution of 250 m and finer have a sufficient mesh resolution. A finer mesh is only useful if the stress changes close to the fault are of interest, which is not the case in this study.

Within the reference model, the fault is represented by a contact surface (μ=0.4, C=0). As a result, the components of the reduced stress tensor increase in the footwall close to the fault and decrease in the hanging wall (Fig. ). SV and Shmin rise to a similar level (+3 and +1 MPa) within the footwall block near the fault. An opposite behaviour is observed for the von Mises stress. SHmax, however, increases only slightly close to the fault (<1 MPa), which is the reason for the decrease in the von Mises stress near the fault. Corresponding to these changes, the stress magnitudes decrease next to the fault within the hanging wall block; the largest amount is for SV, resulting in a slight increase in the von Mises stress. Significant stress changes of more than 1 MPa occur within a distance of 1000 m from the fault. The SHmax orientation is not affected by the fault.

The results of all other models presented subsequently are displayed on a horizontal path at the same depth. For the reference model, the stress variation around the fault for different depth ranges is also shown in Fig. . It remains unchanged that stress variations > 1 MPa are limited to a distance of about 1000 m from the fault. Relatively large variations can be seen at shallow depths (blue, -200 m) in contrast to greater depths (red, -2800 m). The general patterns of stress variation are similar, except for the vertical stress component. SV is smaller in the footwall block close to the fault and larger in the hanging wall block at a depth of -2800 m, in contrast to observations at shallower depths (<2000 m). The reason is that SV becomes σ1 (normal faulting regime) for a depth greater than 2000 m, while at shallower depths a transition from a thrust faulting to a strike-slip regime occurs (Fig. ).

In geomechanics and seismology faults are usually parameterized using the friction coefficient and the cohesion e.g.. Commonly, a friction coefficient between 0.6 and 0.85 is assumed but examples exist of significantly smaller friction coefficients . However, to investigate the influence of the frictional properties of a fault based on the reference model, the friction coefficient is varied from very low (μ=0.1) to very large (μ>1).

Using a very large friction coefficient (μ>1), there is no visible influence by the fault on the stress magnitudes (Fig. ), and the stress magnitudes are identical to a continuous mesh without a contact surface (dashed line in Fig. ). In contrast, for a low-friction case (μ=0.1), stress variation is significant near the fault. The general pattern is similar as for the reference model, but the increase (footwall: +8 MPa) and decrease (hanging wall: -5 MPa) in SV are much larger. Similar but not large Shmin changes of +2 MPa are observed for the footwall and -2 MPa for the hanging wall block. The drop of SHmax in the hanging wall block is significant (-4 MPa), whereas the increase in the footwall block next to the fault is negligible. However, a SHmax decrease of about -1 MPa is visible in both the footwall and hanging wall block, even between 1000 and 3000 m away from the fault. This is a result of stress dissipation due to larger fault offset in the case of low friction. Variation of the von Mises stress is mainly driven by the variation of SV. It is mostly σ3, determined by the fact that SV becomes significantly larger than Shmin in the footwall block about 500 m next to the fault for the models with low-friction contact definition.

Overall comparison of the models with a different friction in Fig.  shows that the stress perturbations gradually decrease with an increase in the friction coefficient. A stress variation of >1 MPa is limited to a distance of ≈1 km, except for SHmax in the hanging wall block. None of the variations result in a visible change in the SHmax orientation, and it is always parallel to the maximum displacement (X direction).

The representation of a fault by a 2-D plane is not realistic for the immediate vicinity of the fault where a zone of damaged rock is expected. A more realistic approach seems to be the representation by a layer of elements with an elastic rheology of reduced stiffness (Fig. ). This simulates the damage zone around the fault core e.g..

Herein, the fault zone has a width of 10 m represented by three elements normal to the fault (Fig. a). A Young's modulus of E=5, 1, and 0.25 GPa is tested, while the stiffer surroundings have E=15 GPa. The element resolution outside the fault area is 50 m in the X and Z direction and 500 m in the Y direction.

The stress magnitudes along the profile (Fig. ) do not show a significant stress variability in the vicinity of the fault resulting from three of the less stiff elements. Stress changes are restricted to a very narrow domain, which are not visible; they are visually hidden behind the fault line. SHmax decreases depending on the decreasing stiffness. For the model with E=250 MPa fault representation, SHmax is always around 1 MPa lower because of stress dissipation by the low-stiffness fault domain. Therefore, the von Mises stress drops by the same amount. Stress dissipation also effects Shmin, but by a much lower amount; for SV no effect is visible.

Another model version has a thicker fault of 30 m, represented by nine elements normal to the fault (Fig. b). Like the 10 m models, SHmax drops, especially for the model with the least stiff fault domain (E=250 MPa), by around -3 MPa (Fig. ), again an effect of the stress dissipation. Shmin decreases by almost 1 MPa, whereas SV is stable. Near the fault, SHmax, Shmin, and SV decrease significantly, limited to a region narrow to the fault (<100 m). The von Mises stress variation is mainly driven by the reduction of SHmax because of the less stiff fault parts. There is no change in the SHmax orientation to observe and it remains parallel to the X direction.

As purely elastic elements do not allow failure, they cannot dissipate stresses such as a contact surface is able to do. To accommodate both the ability to dissipate stresses and the representation of a damage zone, elements with elasto-plastic rheology within the fault zone are now used. Out of a continuous mesh, elements close to the fault location were selected in a staircase-like manner, which have a specific plastic yield criterion. These fault elements laterally have a range of one (Fig. a) to eight elements (Fig. b). These elements have a friction angle of ϕ=30∘ (friction coefficient μ=0.58) and a low cohesion of C=0.1 kPa. The dilation angle used is ψ=25∘. In contrast to that, the non-fault elements have a much larger cohesion (C=500 kPa) but the same friction and dilation angle. The element resolution in the vicinity of the fault is 100 m in the X and Z direction and 500 m in the Y direction. The elastic material properties are the same as used by the reference model (E=15 GPa, ν=0.27, and ρ=2550 kg m-3).

The representation by means of staircase-like elements with elasto-plastic properties (Fig. ) shows that the impact on the stress components is nearly independent from the number of laterally used elements that allow plastification. SHmax, Shmin, and SV slightly increase in the footwall block near the fault domain and slightly decrease in the hanging wall block, again near the fault domain. The overall variation of SHmax, Shmin, SV, and the von Mises is <1 MPa. Stress magnitudes do not show any discontinuous behaviour at the fault zone, as the reference model do. Stress variations are restricted to a zone of about 1000 m next to the fault domain. Again, the SHmax orientation is not disturbed as a result of the fault.

The model having four weak elements laterally is used again to investigate the impact of the friction. Friction angles of ϕ=30, 25, 20, and 15∘ are applied. The ϕ=30∘ model has already been used for the variation of the number of lateral elements (Fig. : four elements). Modelling results in Fig.  show that a decreasing friction angle increases the stress variation near the fault. SHmax, Shmin, and SV increase in the footwall block near the fault, while a slight decrease can be seen in the hanging wall block. However, swing-in effects can be observed on both sides of the fault. The largest-magnitude changes are about +4.5 MPa for SV, +5 MPa for SHmax, and +2.5 MPa for Shmin. At a distance of >1400 m to the fault centre, the variation of the stresses is <1 MPa. The SHmax orientation remains unaffected.

To study the impact of the fault dip angle, several models with different fault inclination are prepared. These models have a dip angle of 30, 40, 50, 70, and 80∘, in contrast to the reference model (60∘, Figs.  and ). Elastic material properties are the same as used in the reference model: E=15 GPa, ν=0.27, and ρ=2550 kg m-3 with fault representation by contact elements μ=0.4 and C=0.

In Fig. , it can be seen that the stress perturbation pattern is similar compared to the reference model. With increasing dip angle from 60∘ (reference model) to 70 and 80∘, the stress perturbation slightly decreases. The reduction is most significantly visible for the SV magnitude in the footwall block next to the fault. Stress magnitudes at a distance from the fault increase slightly for the large-dip-angle models as the stress dissipation by the fault decreases.

A decrease in dip angle of the fault results in a significantly more pronounced increase in the stress perturbation near the fault (Fig. ). This results in an increase in SHmax by >4 MPa in the footwall block and a decrease of about -4 MPa in the hanging wall block using a fault inclination of 30∘. An increase in the SV and Shmin magnitudes in the footwall block and a decrease in the hanging wall block are clearly visible. The influence of the fault on the SHmax magnitude on both the footwall and hanging wall block is at a distance to the fault of about 1500 and 2000 m. However, the large distance is an effect of the small fault dip; the real distance is half of the values for the 30∘ model. There is no perturbation of the SHmax orientation.

In addition to the influence of the dip, the influence of the SHmax orientation with respect to the fault strike is investigated. Thus, a strike angle of 90∘ as the reference model is compared with other models where the fault is striking with an angle of 75, 60, 45, 30, and 15∘. To geometrically allow such strike angles, the models are extended in the X direction from 10 to 20, 30, and 50 km for the models with a fault strike of 45, 30, and 15∘, respectively. The resulting boundary conditions are adjusted to ensure comparability.

Results of the strike angle variation (Fig. ) are shown perpendicular to the strike direction of the fault. The impact of the fault strike variation on the SHmax and Shmin magnitude is minimal. Clear deviations are only observed for SV in the footwall block, where SV is smaller compared to the reference model. As a result, the von Mises stress is also less variable in the footwall block next to the fault. The variation of the SHmax orientation varies with distance to the fault but does not exceed 1.5∘, which is significantly smaller than the uncertainties of orientation data records . Therefore, no visualization of that is shown.

Since the models with the fault strike variation and the friction coefficient of 0.4 only cause small SHmax rotations, the influence of a lower friction (μ=0.1) is also investigated. The plot of the stress magnitudes (Fig. ) shows a visible variation of the magnitudes for the different orientations of the fault. The general pattern is similar to the reference model. For SHmax, significant variations in stress magnitude are observed between the models due to stress dissipation resulting from low friction at the fault. The largest magnitudes are for the reference model (90∘) as well as the 15∘ model. In contrast, the 45, 30, and 60∘ models have the largest Shmin magnitudes. As a result of the largest variation of the SHmax magnitudes, the lowest von Mises stresses are observed for 45, 30, and 60∘ models.

For the first time in the model series, a significant variation in the orientation of SHmax is clearly visible with a fault strike variation using a friction coefficient of μ=0.1 (Fig. ). The deviation of the orientation reaches up to about 14∘ in the hanging wall block for the model with a fault strike of 30∘, closely followed by the 45∘ model. The SHmax rotation for the 30, 45, and 15∘ models is clockwise, parallel to the strike of the fault, while in the models with a strike of the fault of 60∘ as well as 75∘, SHmax orientation is counterclockwise, i.e. tends to be perpendicular to the orientation of the fault. In the footwall block, the rotation of SHmax is also visible but less so than in the hanging wall block, with a maximum of about 6∘.

Since the elastic material properties have a significant influence on the deformation on the rock on both sides of the fault, the Young's modulus of the host rock is varied. In addition to the Young's modulus of the reference model (E=15 GPa), stiffnesses of 5, 20, 30, 40, 60, 80, and 100 GPa are tested. In order to keep the model comparable, the boundary conditions were adapted (Table ) so that the far-field stress magnitudes of the different models were equal.

The variation of the Young's modulus has a limited effect on SHmax in the footwall block (Fig. ), where in the hanging wall block SHmax decreases by up to -4 MPa with increasing Young's modulus next to the fault. Shmin increases slightly with the Young's modulus in the footwall block and decreases in the same way in the hanging wall block slightly by up to -2 MPa. The SV magnitude shows the same pattern, but the stress deviation is much larger near the fault: up to +7 MPa in the footwall and -4.5 MPa in the hanging wall block. The von Mises stresses decrease with increasing Young's modulus in the footwall block next to the fault and increase in the hanging wall block next to the fault.

In general, the stress perturbation increases due to a larger Young's modulus as stress dissipates on the fault. The lateral influence of the fault on the stress components, producing a stress variation of more than 1 MPa, is limited to a range from -1000 to +1000 m next to the fault. Again, the SHmax orientation is always parallel to the direction of principal shortening.

It is obvious that the influence of the fault on the stress state also depends on the size of the fault surface or on the overall size of the model. For this purpose, the size of the active fault surface using the reference model geometry is reduced to 4000×1000 and 4000×3000 m2 (Fig. ). Also, the reference model with the full fault surface is doubled and quadrupled in size. The resulting models then have dimensions of 20×20×6 and 40×40×12 km3, respectively. The resulting mesh resolution is then 100 and 200 m in the X and Z directions, respectively, and 1 and 2 km in the strike direction (Y) of the fault, which is parallel to Shmin. The boundary conditions were adjusted accordingly to generate a similar stress state.

The comparison of the results in Fig.  shows that as the size of the fault increases, the magnitude deviation near the fault increases. Thus, in the hanging wall block SHmax is reduced by almost -3 MPa, while SV in the footwall block increases by more than +5 MPa for the model with a side length of 40 km. As a result, the von Mises stress in the footwall block decreases more significantly close to the fault. However, the increase in the fault surface area does not have a significant influence on the far-field stress pattern. Significant stress changes (>1 MPa) occur up to about 1000 m from the fault. No rotation of the SHmax orientation can be observed.

The effect of stress anisotropy is studied by defining variable lateral boundary conditions. The shortening perpendicular to the fault strike (X direction) is tested from 1, 2, 3, 4, 6, 8, 10 (reference model), 12, 14, 16, and 20 m (ϵ=-1×10-4 to -2×10-3), where the dilation to the fault (Y direction) remains identical to the reference model of -2 m (ϵ=2×10-4). Everything else is identical to the reference model.

The different SHmax magnitudes directly result from the variable shortening applied to the model boundaries (Fig. ). The overall pattern is like the reference model. The observed variation is low for low strain, where variation is larger for higher strain. SHmax is smaller for larger strain away from the fault and increases a bit next to the fault. In the footwall block, the pattern is clear: the closer to the fault, the smaller the SHmax.

The variation of Shmin is similar to SHmax: variation is small for less shortening and increases with increasing shortening of the model (Fig. ). Shmin increases in the footwall block next to the fault and is smaller next to the fault in the hanging wall block.

Larger variation can be seen for SV, with an increase in the footwall bock and a decrease in the hanging wall block. The SV magnitude variation in the footwall block increases from +0.4 MPa for 2 m of shortening to +2.3 MPa for 20 m of shortening. Nearly the similar amount of decrease happens in the hanging wall block.

The von Mises stress variation (Fig. ) increases with the increase in shortening compared to the reference model. For the model with little strain (<4 m) the observed variation of the von Mises stress displays another pattern. The von Mises stress increases in the footwall block and decreases in the hanging wall block next to the fault. Again, major stress variations are limited to a distance of less than 1000 m next to the fault. The SHmax orientation is not affected for larger shortening perpendicular to the fault. For the models with a shortening of 1 and 2 m in the X direction, the stress magnitudes are horizontally isotropic (Shmin=SHmax) and the SHmax orientation is not clearly defined.

The goal is to investigate the impact of faults on the far-field stress state (>100 m). The model design does not allow estimations of the stress state or stress perturbations close to a fault (<100 m). Investigating that, a much finer mesh resolution would be needed. It is also questionable whether and which methods of fault implementation are suitable for this purpose.

Like all generic models, those used here are a significant simplification of rock physics, geological structures, and the fault representation itself. Except for two scenarios, only linear elastic material properties are used to represent the rock volume. This neglects various rheological processes within the Earth’s crust. But Hooke's law seems to be a proper approximation for the major mechanical behaviour of rocks in the upper crust, as the elastic thickness of the crust (Te) is usually much larger than the models used here . According to field investigations by , most brittle deformation can be explained using linear elastic material properties. Furthermore, the focus is not on stress changes during the co-seismic phase e.g. or deformation over several seismic cycles. The focus is on the quasi-static stress state in the inter-seismic phase.

The reference geometry is a normal faulting structure with a fault dip of 60∘, but the applied boundary conditions result in a thrust to strike-slip faulting regime at the depth, where stresses are plotted, usually at -660 m. Even if most models use specific structures and specific stress regime conditions, other structural settings or faulting regimes are covered by some of the models or specific result presentations. These are the variation of the dip angle (Figs.  and ), the variation of the strain (Fig. ), and the variation of the depth for the reference model (Fig. ). Therefore, results for all stress regimes and faulting structures are provided. However, the overall behaviour remains unchanged.

The specific objective was to investigate how faults can lead to stress rotations since this has been claimed to be the reason for observed stress rotations on scales of tens of kilometres. However, for most scenarios only stress magnitudes are shown here. This is of course due to the fact that many models do not show SHmax rotations. Visualizing the stress magnitudes gives a much broader insight into the effect of faults on the stress state. And, if the stress magnitudes change, stress rotation is possible, but if the magnitudes do not change, rotation can be ruled out. Therefore, the stress magnitude visualization used also acts as a proxy for potential stress rotation.

To allow good comparability of modelling results, constant boundary conditions have been used, with a few exceptions. The models with different strain have different stress magnitudes as a result. For models having a different extent or a variable Young's modulus, the boundary conditions were scaled accordingly to ensure comparability. The models with a lower Young's modulus in the fault zone and low-friction contact faults dissipate localized stresses, which has not been corrected, as the influence on the result is small.

Several of the model scenarios use contact elements to represent a fault within the model. This is the case for the reference model, the variation of the friction, the fault dip and fault strike angle, the Young’s modulus variation in the host rock, the model size, and the boundary conditions. The overall observation is an increase in the stress components (SHmax, Shmin, and SV) in the footwall block and a decrease within the hanging wall block, both next to the fault (Fig. a–d). In contrast, the von Mises stresses decrease in the footwall block and increase in the hanging wall block. This is the case as SV varies more than the other stress components.

For these contact surfaces, no cohesion (C=0) is used, which is nevertheless a reasonable and conservative simplification in particular for pre-existing faults or fault zones, as granular material has a very low cohesion: C<1 kPa . On the other hand, cohesion strengthening can increase the cohesion to C>1 MPa , C=8 MPa , or C=35 MPa for very high temperatures . According to , cohesion will reach 3 MPa for a 100-year earthquake recurrence interval at a depth of about 2 km.

The friction coefficients used for the contact surfaces reach from 0.1 over 0.4 (reference model) to 1.0 and larger. In the past, it was assumed that the friction coefficient of faults is about 0.6 to 0.85 . But the friction can be much smaller if clay minerals dominate , in the case of dynamic offset , or for high pore pressures .

Low friction is also expected for large fault (zones) or subduction zones . The friction coefficient is of the order of 0.08 for the 2011 Tohoku–Oki earthquake , μ=0.12–0.25 or 0.05–0.2 for the San Andreas Fault , and μ=0 to 0.1 for tremors in general . assumes a friction coefficient of 0.01 to 0.07 for large-scale plate boundaries. However, the investigated range of friction covers this variation well, except for μ<0.1.

As a free surface, or a fault with a very low friction coefficient, is unable to build up shear stresses , principal stresses will be parallel and perpendicular to the surface . A classic example is the San Andreas Fault , where the interpretation of borehole breakouts and drilling-induced tensile fractures from nearby borehole indicates SHmax orientations that are almost perpendicular to the fault . However, the distance of these boreholes is >1000 m from the fault core in most cases and it is thus questionable whether the derived SHmax orientations can be used as an observable for the fault strength. show in their analysis of borehole breakouts and drilling-induced tensile failures of the SAFOD borehole through the San Andreas Fault that significant SHmax rotations can only be resolved in the near field of the fault.

A decrease in horizontal stresses near the faults in the hanging wall and an increase in the footwall are reported for the Forsmark DBT 1 well Sweden,. Fewer borehole breakouts in the hanging wall block and more in the footwall block are observed from the KTB well Germany,. A reduction of σ3 by about 5 MPa has been observed within less than 10 m near a tunnel at the Grimsel test site Switzerland,. All these observations fit the results of the models having a fault representation by contact elements, where the horizontal stresses are smaller above the fault (Fig. ), and the horizontal differential stress is smaller in the hanging wall block (Fig. ). The latter would make the occurrence of borehole breakouts less likely in the hanging wall.

In contrast to that, larger horizontal stresses above a fault have been observed for the Lansjärv well Sweden,. The maximum horizontal stresses are observed about 100 m above the fault in the hanging wall block, which also points to other causes. One possible explanation is the lithological variation in that well, where several pegmatites and amphibolites in that depth range have been observed , which eventually provide larger magnitudes as a result of a larger Young's modulus.

According to the magnitude variation is positively correlated with the stress ratio and negatively correlated with the friction. This can be clearly confirmed by this study (Figs.  and ), where the stress variation near faults is largest for low-friction models and models with a larger strain variation. Observations indicate that stresses decrease near a fault after an earthquake . This can be confirmed by the models for the hanging wall, but not for the footwall block. Either the observations are from the hanging wall block only, or other factors, like the 3-D structure, are responsible, which are not represented by the models used here.

All models analyse the variation of stress components and the orientation towards generic models with only one homogeneous fault. The extent to which the results can be applied to other scenarios remains questionable. There are some scenarios where we assume that other factors could have a greater influence on the stress state. These include extensive settings such as horst and graben structures, listric faults, or step-over zones. In such cases, whole blocks may be completely decoupled, either by faults or by any kind of decoupling horizon (salt, wet clay, or pore over-pressure). The stress state in such a block is then dominated by gravity only. One potential example of this is the Arches National Park in Utah, USA, where the joints are almost perpendicular to the normal faults and are constant over several hundred metres . Secondary faulting also provides a possible explanation for the complex stress pattern within the Viking Graben North Sea,. According to , large stress perturbations can be caused by a fault step-over structure in hydrothermal systems over a distance of more than 1000 m in the Great Basin, western United States.

Faults or fault zones in nature are never planar structures, as assumed by the presented models. Roughness plays a role, but the roughness in the direction of previous slip is much less than in other directions . The geometrical complexity is a result of non-planarity (bending, listric, bifurcation), combination or coalesce of faults (step-over or relay zones), or other factors e.g.. Fault zones can exist out of several single parallel faults, which would probably produce a more widely distributed area of stress perturbation. Pore pressure, especially above hydrostatics, has a significant impact on effective fault normal stresses . Despite the large number of models presented, such complex structures or properties have not been tested.

Stress changes near the fault tip (e.g. horsetail fault terminations) lead to a complex stress pattern . To model that, using only linear elastic material properties would result in unrealistic local stress peaks as elastic energy would not be dissipated by plastic deformation. Therefore, such structures are not considered here. However, it can be assumed that stress changes induced by fault tips are negligible at distances of a few kilometres from the fault .