The knowledge of the contemporary in situ stress state is a key issue for
safe and sustainable subsurface engineering. However, information on the
orientation and magnitudes of the stress state is limited and often not
available for the areas of interest. Therefore 3-D geomechanical–numerical
modelling is used to estimate the in situ stress state and the distance of
faults from failure for application in subsurface engineering. The main
challenge in this approach is to bridge the gap in scale between the widely
scattered data used for calibration of the model and the high resolution in
the target area required for the application. We present a multi-stage 3-D
geomechanical–numerical approach which provides a state-of-the-art model of
the stress field for a reservoir-scale area from widely scattered data
records. Therefore, we first use a large-scale regional model which is
calibrated by available stress data and provides the full 3-D stress tensor at
discrete points in the entire model volume. The modelled stress state is used
subsequently for the calibration of a smaller-scale model located within the
large-scale model in an area without any observed stress data records. We
exemplify this approach with two-stages for the area around Munich in the
German Molasse Basin. As an example of application, we estimate the scalar values
for slip tendency and fracture potential from the model results as measures
for the criticality of fault reactivation in the reservoir-scale model. The
modelling results show that variations due to uncertainties in the input data
are mainly introduced by the uncertain material properties and missing

The contemporary in situ upper crustal stress field is of key importance for
our understanding of geodynamic processes such as natural and induced
seismicity

The main focus of current research is to quantify stress changes due to
anthropogenic underground usage

Stress map of the Bavarian Molasse with 172 A-C quality data records
based on the World Stress Map database release 2008

Different types of modelling approaches.

The 3-D in situ stress state can be described with a symmetric tensor of
second degree with six independent components

Figure

More important for assessing criticality is the estimation of the
differential stress between the magnitudes of the largest and smallest
principal stresses and their changes during stimulation and production. The

To summarize, our knowledge of the 3-D in situ stress state is based on
sparsely distributed and incomplete information. Only the orientation of the
reduced-stress tensor and, to a lesser extent, information about the stress regime
are relatively well estimated from stress indicators. The crustal in situ
stress magnitudes are underdetermined, since they vary laterally and
vertically. To determine the full stress tensor for every point in a volume, a
3-D geomechanical–numerical model workflow that uses the
available stress information for model-independent constraints for
calibration is essential. Moreover, at reservoir scale, often no stress information is
available for model calibration (Fig.

In this paper we demonstrate the applicability of the multi-stage nesting
workflow for the 3-D geomechanical modelling of the stress tensor. We
exemplify our approach with a 3-D model of the Greater Munich area in the
northern Alpine Molasse Basin and a generic reservoir model (Fig.

The northern Alpine Molasse Basin is a typical asymmetric foreland basin
which extends over 1000

For our model geometry we use the upper 9

The stratigraphic units, their discretization, and according rock properties, which are present in the root and branch models. Units which are only preserved in parts of the root model are marked with *.

Within the root model area (Fig.

The magnitude of

However, information on the horizontal stress magnitudes is sparse even
within the root model area. The magnitude of

The direct estimation of the

In areas with a low number of magnitude stress data records, the stress regime
provides information on the relative magnitudes of

Information from structural geology observing steeply dipping faults in the
Bavarian Molasse Basin

Both the regional-scale root model and the reservoir-scale branch model are
based on the same modelling assumptions. Assuming that accelerations other
than gravity can be neglected, the models solve the partial differential
equation of the equilibrium of forces. Furthermore, we assume a linear
elastic rheology and solve for absolute stresses (no pore pressure). The
general model procedure follows the technical workflow explained in detail by

The root model extends 70

An exact fit of the overburden

Due to the complex topology of the stratigraphy and inhomogeneous rock
properties of the different units the finite element method (FEM) that allows
unstructured meshes is used to solve the partial differential equation of the
equilibrium of forces at discrete points. Thus, both models are discretized
into finite element meshes. The root model is constructed with approximately
10

The root and branch model discretized with 10

The calibration of the root model with stress magnitude data is achieved by
applying two Dirichlet boundary conditions, each on one of the
perpendicular sidewalls of the model (Fig.

If several

Left: exemplified schematic models with the data records used for
calibration (star:

The calibration workflow for the root and branch model.

The same procedure is applied for the calibration of

Application of this calibration procedure is fast and simple since the
best-fit boundary conditions can be found by combining two
linear slopes based on the calibration data and the displacement boundary
conditions. Therefore, to find the best-fit boundary conditions only
three different models with arbitrary displacement boundary conditions are
required (Fig.

It is assumed that the stress data records used for the calibration are the result of the far-field stress state and its interaction with structural features such as local density and/or strength contrasts represented within the root model. If the measurements were, e.g. the result of an unknown or unimplemented local active fault, the results of the calibration would not be reliable. Thus, in general, the data used for calibration should be representative for a large volume of the individual lithological layer.

Under this assumption the best-fit model simulates the stress state at
discrete points in the entire model volume. Hence, information on the stress
state is now also available in areas of the root model where previously no
observables on the stress state were available. This means that in the
branch model, which does not include any observed stress data records,
simulated information on the stress state is also now available from the root model
and can be used to calibrate the branch model (Fig.

Since the branch model is calibrated in the same way as the
root model (but with a simulated stress state from the root model as
calibration points instead of observed stress data records), a large number of
potential calibration points with

For a successful transfer of the stress state from the root to the branch model, it is critical that the stress state used for the calibration of the branch model is obtained at discrete points of the root model and not in its volume. Otherwise the stress state extracted from the root model is potentially biased due to interpolations from discrete points into the volume, which are performed by the visualization software. Since the large number of possible calibration points can be chosen arbitrarily, their locations need to be considered. We recommend using calibration points close to the border of the branch model but outside the zone prone to boundary effects. Calibration points from the root model in the centre of the branch model are a contradiction of the two-stage approach which aims at finding local stress changes due to high-resolution structural features that are only present in the branch model. Due to the lack of any other stress data in the branch model area, the calibration procedure imposes the root model's basic stress state on the branch model, which prevents such local stress perturbations. Hence, this necessary imposition should be reduced to the boundaries of the branch model that are not used for interpretation. Furthermore, the calibration points should be evenly distributed along the branch model boundary and represented in all stratigraphic units to account for different material properties. Special attention needs to be paid to units which are either only present in the root or the branch model or have a significantly different geometry or rock properties in the two models.

Results of the best-fit root model

The generic branch model results are shown by means of slip tendency
(ST) values

Stratigraphy and model result of the root model along the borehole
of the geothermal project in Sauerlach. Lines show the results of the
best-fit root model: blue for the

In the following two sections we present the results of two model scenarios
for the root model that fit equally well the observed stress data, but with
different stress regimes (Fig.

The best-fit root model of the stress state at discrete points in the
Greater Munich area is calibrated using

Figure

In Fig.

The second row of Fig.

The last row in Fig.

In this section we show the model results of the branch model (Fig.

The high dependence of slip tendency on the orientation, friction, and
cohesion of the fault is displayed in Fig.

Information provided by the branch model is used in an early pre-drilling
stage of a project to assess whether the initial conditions of the reservoir and
its criticality allow safe production; i.e. both slip tendency and fracture
potential have low values as in Fig.

One of the key points in geomechanical modelling is the reliability of the
model results in terms of the predicted processes and the presented
multi-stage simulation of the in situ stress field. As already mentioned in
the result Sect.

We compute the slip tendency for model scenarios which use the extreme values
of the input parameters range of uncertainties. The model's linear elastic
behaviour allows the individual quantification of the impact of different
model parameter uncertainties on the model's reliability. Therefore we
compute several model scenarios in which sequentially only a single parameter
is changed to an extreme value. This enables us to derive the individual
impact of different parameters and quantify the most important ones.
The results of the slip tendency for each model scenario are subsequently
compared to the best-fit slip tendency values from the best-fit model (Table

The two main sources for the variability of slip tendency can be identified
as the model-independent data for the

Slip tendency proves to be quite robust (

The objective of this work was to demonstrate the multi-stage approach for a
high-resolution 3-D geomechanical–numerical modelling workflow assessing the
criticality in reservoirs. In contrast to a single model, which includes both
stress data records for calibration and high-resolution representation of a
local reservoir structure, we use two models of different sizes. The regional-scale root model is calibrated on stress data records and provides the stress
state for the calibration of the reservoir-scale branch model. This approach
provides a cost-efficient, quick, and reliable state-of-the-art calibration
of geomechanical–numerical models of the contemporary 3-D in situ stress field
across scales. It is used to assess the criticality of reservoirs which can
be quantified by scalar values such as slip tendency. If detailed information
on the fracture behaviour of the rock are known, more elaborate fracture
criteria than Mohr-Coulomb

The expected maximum variations in slip tendency (ST) introduced by
the uncertainties of the model parameters. This comparison is made at 40
locations in the Malm

A single model with the same functionality as the two models in the multi-stage approach needs to account for the required high resolution in the reservoir area and the large model extent to include data for calibration. These two requirements are not contradictory per se but prolong the process of mesh generation, e.g. by needing to harmonize a regional-scale low-resolution and local-scale high-resolution structural model in the area of overlap. Furthermore, the manageability of the model (e.g. logical size) and the available time for computation (number of elements) in most instances requires a variable resolution which is refined only in the target area. Such a change in element size in a single model is possible but the mesh generation is cumbersome and needs a high number of elements. For a THM simulation of production and (re)injection, incrementation over time significantly increases the computation time for each single element. Furthermore, in a single large model, only a very small area is of interest, hence large areas are simulated to no purpose while at the same time the logical size, computation time, and effort are increased.

If a multi-stage approach with two models is applied, each model has its own
fixed resolution with no required variation in element size (Fig.

In addition the application of two models opens further possibilities for improved and safer exploration and drilling. Structural features and stress magnitude measurements recorded during advanced exploration or even initial drillings can be implemented into the model workflow due to the simplified mesh regeneration. Even a change in the target area within the root model can be more easily implemented in the workflow since only a new branch model is required. The calibration of the root model can be updated with new stress data records as soon as they become available. Finally, a large calibrated root model may include several target areas and can be reused and applied for more than one project.

The two models in the presented two-stage approach are calibrated with
different Dirichlet boundary conditions applied to an initial stress state.
This approach follows the modelling procedure using isotropic elastic
materials described by e.g.

Our root model is calibrated with data records which display the stress state as a result of the geologic history and tectonic evolution. In the presented region the stress field is very homogeneous but in other regions significant local lateral variations exist and need to be accounted for. This can be accomplished for example by lateral variations of the material properties or faults. It is crucial to ensure that the data used for the calibration is representative for the regional material and geometry in the root model.

The branch model, however, is calibrated on the stress state simulated in the root model. Both calibration procedures are not limited in the number of calibration points and a weighting of the calibration points according to reliability can be easily realized. An extension of the two-stage approach to include three (or even more) models of different sizes is possible. Furthermore, the calibration procedure allows running several alternating models with different calibration data or differently weighted calibration points as well as variations in rock properties to quantify model-specific variations. This ability was used to quantify the reliability of the model's results. It is also useful for future attempts at statistically determining uncertainties in the model's results.

Even without any additional computations, a first-order assessment of the
impact of individual data records on the model calibration can be made by assessing changes in the boundary conditions. Therefore the
best-fit boundary conditions derived with and without certain data
records are compared. Such a data record could be a newly performed hydraulic
fracturing experiment which provides an additional

The models showed in this work do not include any implicit faults and no strain partitioning is assumed. The calibration of a model including faults and fault-specific behaviour, e.g. strain weakening or hardening or long-term relaxation of the gauge material, is possible as well if sufficient information on the fault properties are available. However, due to the non-linearities introduced by active faults the calibration process requires a regression analysis of a higher degree, hence several more test scenarios. This is beyond the scope of this work.

Apparently the model's reliability is mainly affected by the lack and high
uncertainty of

The uncertainties related to the material properties are another large factor
that limits the model's reliability. This can be mitigated at least partly
by using data from extensive databases

The uncertainties in the strike and dip of faults have a comparably small share in the reliability of the model while being challenging to mitigate due to the general uncertainties in the interpretation of 3-D subsurface structures. The fault parameters cohesion and friction angle which are even more difficult to determine compared to the orientation reduce the model's reliability to a slightly higher degree compared to strike and dip. Increasing the model's reliability through a better understanding of these parameters is possible but requires a detailed understanding of the great variability of in situ fault zone behaviour and extent at depth.

Statistical methods to quantify uncertainties in the subsurface geometry
exist for purely static structural models

This model focusses on the stress tensor in the uppermost part of the crust
and its extent is accordingly chosen. Deep-seated processes at depths
larger than 9 km are, therefore, not represented in the model. However,
as shown by

The model does not include any faults. The inclusion of faults makes sense in situations where detailed information on fault geometry, extent, and parameters are available and a significant impact of the faults on the regional stress field or (re)activation is expected. However, in this example, the available stress data suggests that no faults with a major impact are located neither within the root model nor the branch model area. Therefore the variations introduced by omitting faults is assumed to be small.

Variations of the model results are also introduced by the multi-stage
calibration approach itself and cannot be mitigated due to both models 3-D
stress state with lateral and vertical variations. The model's calibration,
however, depends on the variations of only two independent boundary
conditions. Additionally, small variations may be introduced by the model
assumptions. However, these variations can be disregarded in the light of the
major reasons for variations due to the small amount of stress magnitude data
and rock properties. Table

Hydrocarbon reservoirs are currently exploited on a minor level in the Alpine
Foreland

Within the root model perimeter, several geothermal projects are currently at
the planning stage, namely Bernried, Gräfelfing/Planegg, Königsdorf,
Markt Schwaben, Puchheim/Germering, Raststätte Höhenrain, Starnberg,
Weilheim/Wielenbach, and Wolfratshausen

Furthermore, the two-stage approach could be extended to a three-stage approach which incorporates a global model of the entire Bavarian Molasse Basin. More data for calibration, as well as more potential applications, might be available in such an enhanced area. Thereby the regional or global root model could be established as a community model which provides the stress state for further applications and/or local models for planned projects.

In this work we present a multi-stage 3-D geomechanical–numerical
modelling approach, which provides a cost-efficient, reliable, and fast
way to generate and evaluate the criticality of the stress state in a
small target area where, in general, no stress data for model calibration are
available. The approach uses a large-scale root model which is calibrated on
available stress data and a small-scale branch model which is calibrated on
the root model. We exemplify this in a two-stage approach in the German
Molasse Basin around the municipality of Munich. The discussion of
reliability of the model results clearly shows (1) that variations are large
and (2) that they are mainly introduced by the uncertain material properties
and missing

The stress orientation data used for model set-up and calibration is available from Reiter et al. (2016) and Heidbach et al. (2016).

The research leading to these results has received funding from the European
Community’s Seventh Framework Programme under grant agreement No. 608553
(Project IMAGE). The authors would like to thank François Cornet and an
anonymous reviewer for their comments which helped to clarify the manuscript,
Thomas Fritzer (LFU Augsburg) for his support, Dietrich Stromeyer for
discussing the procedure and the programming of the stress data analysis
tool, and Arno Zang for his comments which significantly improved the
manuscript. The map is prepared with the Generic Mapping Tool GMT