In this study we describe and compare eight different strategies to predict the depth variation of stress within a layered rock formation. This reveals the inherent uncertainties in stress prediction from elastic properties and stress measurements, as well as the geologic implications of the different models. The predictive strategies are based on well log data and in some cases on in situ stress measurements, combined with the weight of the overburden rock, the pore pressure, the depth variation in rock properties, and tectonic effects. We contrast and compare stresses predicted purely using theoretical models with those constrained by in situ measurements. We also explore the role of the applied boundary conditions that mimic two fundamental models of tectonic effects, namely the stress- or strain-driven models. In both models, layer-to-layer tectonic stress variations are added to initial predictions due to vertical variation in rock elasticity, consistent with natural observations, yet describe very different controlling mechanisms. Layer-to-layer stress variations are caused by either local elastic strain accommodation for the strain-driven model, or stress transfers for the stress-driven model. As a consequence, stress predictions can depend strongly on the implemented prediction philosophy and the underlying implicit and explicit assumptions, even for media with identical elastic parameters and stress measurements. This implies that stress predictions have large uncertainties, even if local measurements and boundary conditions are honored.
Knowledge of in situ stress magnitudes and their spatial variability is critical to understand the upper crust stress and strain (Zang and Stephansson, 2010; Zoback, 2010; Schmitt et al., 2012; Reiter et al., 2014; Reiter and Heidbach, 2014), which in turn has strong implications for seismotectonics (e.g., earthquake magnitudes; Davies et al., 2012; Langenbruch and Shapiro, 2014; Busetti and Reches, 2014; Scholz, 2015) and their locations (Sibson, 1982; Zoback and Gorelick, 2012; Zakharova and Goldberg, 2014), structural geology (e.g., fault behavior and slip tendency; Gross et al., 1997; Gudmundsson, 2011), and volcano-tectonics (e.g., prediction of dike paths and eruption forecasting; Gudmundsson, 2006). It is also key for the civil engineering, mining, and energy industries, covering topics as diverse as hydraulic fracturing, rock stability, and fluid circulation (Simonson et al., 1978; Van Eekelen, 1982; Warpinski et al., 1985; Hopkins, 1997; King, 2010; Fisher and Warpinski, 2011; Davies et al., 2012). Furthermore, it is important for determining the likelihood of felt seismicity due to human activities, for instance, because of hydraulic fracturing or saltwater disposal (Frohlich, 2012; Ellsworth, 2013; Weingarten et al., 2015; Atkinson et al., 2016).
Information on the in situ stresses, like the magnitude of the minimum principal stress or the orientation of maximum horizontal stress, may be assessed using different techniques such as extended leak-off tests, hydraulic fracturing treatments, borehole breakouts, earthquake source mechanisms, geological indicators, etc. (Terzaghi, 1962; Hast, 1967; Zoback, 2010; Zoback et al., 1985; Amadei and Stephansson, 1997; Zang and Stephansson, 2010; Schmitt et al., 2012; Reiter et al., 2014; Reiter and Heidbach, 2014). Maximum horizontal stress cannot be measured directly. In rocks composed of layers with different elastic properties, like in sedimentary or volcanic areas, layer-to-layer variations in horizontal stresses may arise. This phenomenon occurs in several lithology types such as coal, mudstone, siltstone, sandstone, limestone, shales, lava flows, intrusions, and pyroclastics (Haimson and Rummel, 1982; Warpinski et al., 1985; McLellan, 1987; Evans et al., 1989; Warpinski and Teufel, 1991; Cornet and Burlet, 1992; Gunzburger and Cornet, 2007; Cornet and Röckel, 2012). Such stress variations are well documented, but local information on the stress field is often sparse and incomplete; a continuously sampled stress profile is rarely available because extended leak-off tests often concentrate only on the target formations and are rarely published. In this case, analytic formulations (McGarr, 1988; Gunzburger and Cornet, 2007; Jaeger et al., 2009), or numerical models (Teufel and Clark, 1984; Gudmundsson, 2006; Roche and Van der Baan, 2015), have to be used to assess the stress variations.
This paper focuses on the 1-D depth variation in stress magnitude, and more specifically on the local variation in minimum horizontal stress in an extensional regime.
A common methodology to predict stresses in the crust beyond individual measurements consists of applying a stress update to a pre-chosen initial stress model. The stress update can represent processes such as tectonic effects, uplift or subsidence (Haxby and Turcotte, 1976; McGarr, 1988; Gunzburger and Cornet, 2007), temperature change (Voight and St. Pierre, 1974; Haxby and Turcotte, 1976; McGarr, 1988; Blanton and Olson, 1999), and viscous behavior (Gunzburger and Cornet, 2007; Cornet and Röckel, 2012). In most of the cases, the stress update depends on the local rock behavior and can be assessed using the elastic properties of the rock. Different models can be used as an initial state of stress. The simplest model, called the lithostatic model, assumes an isotropic state of stress equal to the vertical stress, as calculated from the weight of the overburden rock (Jaeger et al., 2009). Otherwise, in the uniaxial strain model, stress depends on the Poisson's ratio (Warpinski et al., 1985; McLellan, 1987; Savage et al., 1992; Addis et al., 1996; Jaeger et al., 2009).
An alternative prediction strategy is to assume that the state of stress in the crust is close to the maximum strength that rocks can support at the large scale (i.e., scales of a kilometer or more). The critical stress model then becomes applicable. In this model, depending on the frictional strength of pre-existing fractures and faults, the magnitude of the minimum and maximum principal stresses can be assessed if the magnitude of one of these is known (Zoback, 2010). Since the earth may not be critically stressed everywhere, this model can provide bounds for the minimum and maximum principal stresses (Brace and Kohlstedt, 1980; Townend and Zoback, 2000; Zoback, 2010; Konstantinovskaya at al., 2012; Meixner et al., 2014). The rationale for this approach is that faults and fractures are the weakest structures in rocks. Therefore, in theory, the stress state cannot exceed the stress-state-inducing slip on an optimally oriented fault. Any excess stress will lead to rupture, resetting the stresses to below the critical stress.
The eight predictive strategies. Predictive strategy 1 is commonly used in literature.
The use of the critical model to obtain bounds for the upper and lower stresses was advocated by Brace and Kohlstedt (1980) since stress prediction from the elastic properties of rock is prone to large uncertainty, in particular for areas where the crust has a long and complex tectonic history, possibly resulting in nonelastic behavior. However, stress predictions based on elasticity are commonly used, notably because elastic properties are easier to obtain from well logs than strength properties and in situ stress measurements. In addition, numerous applications in seismotectonics, volcano-tectonics, structural geology, and hydrocarbon exploitation require more detailed stress information than upper and lower bounds.
The objective of this work is to analyze different methodologies for 1-D stress prediction. We explore the roles of two different boundary conditions. They correspond either to a regional strain or stress perturbation, referred to as the strain- and stress-driven models, respectively. In the first case, horizontal strain results from a displacement caused by plate tectonics (Savage et al., 1992; Blanton and Olson, 1999; Beaudoin et al., 2011; Song and Hareland, 2012; Reiter and Heidbach, 2014). In the second case, regional tectonic forces impose external stress instead of strain (Teufel and Clark, 1984; Bourne, 2003; Roche et al., 2013). Then, we assess the inherent uncertainty in stress predictions due to lack of information and/or the use of different predictive strategies. Eight different predictive strategies (Table 1) are carried out in order to explore the role of (1) the initial state of stress (uniaxial or lithostatic), (2) the implemented boundary conditions (strain-driven or stress-driven), and (3) the benchmarking and updating procedure (based entirely on the conceptual critical stress model or available in situ local stress measurements) in final predicted stress profiles. For reference, the predictive strategy using an initial uniaxial strain model, in situ stress measurements, and the strain-driven model is the model most commonly used in the literature (Savage et al., 1992; Blanton and Olson, 1999; Beaudoin et al., 2011; Song and Hareland, 2012). For illustration, all modeling strategies are applied to a field study case, where the resulting stress predictions are compared and contrasted.
For models depending on elastic rock behavior, prediction of the stress
state is based on an initial stress model that is then modified by adding
stress corrections (increments or decrements) to the horizontal stress
components, that is
Various approaches exist to estimate the desired stress corrections
Alternatively, one could assume that the magnitude of the tectonic stress
corrections
Conceptual representation of the stress- and strain-driven
models.
It is important to distinguish here between the regional stress or strain corrections (boundary conditions) and the resulting local stress and strain variations (internal conditions). The local stress corrections, calculated with Eqs. (2), (3), (4), or (5), are defined for a specific layer. It is not possible to apply them to all other layers because in layered rocks stresses are prone to change as a function of depth since the stress corrections depend on the elastic rock properties. To calculate the local stress corrections for each layer, one needs to find the regional (external) tectonic perturbations applied to the rock formation that lead to the actual in situ local stresses in all layers. For convenience, we will refer to the regional stress or strain perturbations as the regional perturbations and the resulting local stress changes as the local stress corrections.
The regional perturbations are assessed according to strain- and stress-driven models. These two fundamental models are detailed in the next subsections.
For the strain-driven model a biaxial strain model is appropriate in order
to provide the magnitude of the local stress corrections since no variation
in the overburden stress is assumed (Jaeger et al., 2009). In an isotropic
elastic rock, the local stress corrections
Assuming an average Poisson's ratio of 0.3 or less, the stress correction in
the perpendicular direction is merely 30 % of the one in the direction of
the strain perturbation. Consequently, in cases where the regional strain
correction occurs mainly in one direction, the second term in Eq. (6) is
sometimes neglected, producing
Examples of stress predictions,
Figure 1a and b conceptually illustrate the behavior of the strain-driven
model. For such a model, the local strain corrections
Conversely, the local stress corrections
Figure 2a shows the final local stresses,
In the stress-driven model, the tectonic perturbations are driven by
constant tectonic stresses,
If the layers are coupled together, slip cannot occur at the interface
between the layers, and the strain has to be continuous throughout the
medium. The regional stresses,
Numerical modeling is commonly used to compute the resulting stress profile for more complex media (Teufel and Clark, 1984; Bourne, 2003; Roche et al., 2013; Roche and Van der Baan, 2015). Appendix B contains the analytic formulation for this model for the simple case of two layers submitted to a tectonic force in a single direction.
Figure 2b and c show the profiles of the final local stresses,
For the numerical solution, the modeling is performed according to the same
methodology as briefly described in Sect. 3.4 and in detail in Roche and
Van der Baan (2015). The local stress corrections
In the coupled stress-driven model, the magnitude of the local stress correction only depends on the contrast in Young's moduli, not their absolute values. The layer-to-layer local stress variation increases with increasing contrast in Young's moduli. For instance, in Fig. 2b, the local stress correction and the layer-to-layer stress variations are nearly constant for a ratio of 2 for different combinations of Young's moduli. They are significantly higher for a ratio of 5, despite similarity in Young's moduli in the different configurations. This mechanism is thus different from the strain-driven model, where the local stress corrections depend only on the absolute values of the individual Young's moduli, and not their ratios (Fig. 2a). The stress transfer effect also depends on the thickness of the individual layers (see Eqs. B8 and B9), as confirmed by numerical studies (Roche et al., 2013), contrary to the strain-driven model.
Flow chart detailing the different prediction strategies.
See Sect. 3 for details. Rock properties used in the different models are
The four steps of calculation for the eight predictive strategies are presented in a synthesis view in Fig. 3 and are detailed in the next subsections (Table 1). The first step corresponds to the definition of the initial state of stress used as a base for the stress prediction. The second step is the definition of the calibration stresses. The third step incorporates the regional stress perturbations by combining the initial stresses and the calibrations (stress measurements or critical stress model). In the final step, the initial stresses are updated by including the tectonic effects, using either the stress-driven or strain-driven model.
The stress predictions are based on two initial models with both horizontal
stresses equal to either the horizontal stress of the uniaxial strain model
or the lithostatic stress (Fig. 3). The average density is used to
calculate the lithostatic stresses
To assess the magnitude of the tectonic effects, we compare the initial stress profile with a calibration stress model that is either the critical stress model or locally measured stresses. The choices for initial and calibration stresses produce four base predictions. These different choices for the calibration stress profiles allow us to compare stress predictions based entirely on conceptual models with those using in situ stress measurements.
In the case of observed data, the local stress corrections
In order to compute the regional stress or strain perturbation, we compare
the initial stress profiles to the calibration stress profiles. This
comparison can be done in the directions of both horizontal principal
stresses if the relevant calibration stresses are available. No stress
measurements are available for the maximum horizontal principal stress in
the following case study. In order to simplify the problem, avoid extra
assumptions, and be able to compare stress predictions obtained with both
calibration stresses independently, we postulate that only the minimum
regional stress
To get the regional stress perturbation
The final step consists of modifying the initial stress profile obtained in step 1 in each layer as a function of the rock properties, depending on the regional tectonic stress or strain perturbation obtained in step 3, in order to get the final stress profile that takes tectonic effects into account. We focus on the minimum principal stresses, and the calculation of the intermediate horizontal principal stress is disregarded because the maximum and minimum principal stresses are more critical to assessing fracturing than the intermediate stress. Likewise, according to our assumption, i.e., a normal faulting regime and no tectonic perturbation in the direction of the intermediate principal stress, the magnitude of the layer-to-layer stress variation is potentially greater in the direction of the minimum principal stress.
We explore both fundamental tectonic models, namely the coupled
stress-driven and the strain-driven models, as applied to the four base
predictions developed so far. We therefore obtain a total of eight different
predictive strategies, leading potentially to eight different predictive stress
profiles (Fig. 3). We use the regional stress
Petrophysical and mechanical properties of the study
case.
For the strain-driven model, the local stress corrections
For the stress-driven model, the local stress corrections
Finally, the critical stress model can be used to obtain upper and lower bounds for all predicted stresses (Brace and Kohlstedt, 1980). In the case study below, we assume a normal faulting regime such that only a lower stress bound is required since the maximum stresses are determined by the overburden weight. We will also assume a uniform coefficient of friction of 0.6.
The field example is located northeast of Fort Nelson, British Columbia, in
the Western Canada Sedimentary Basin (Mossop and Shesten, 1994). The
hydrocarbon reservoir (gas) is stimulated by hydraulic fracturing
treatments. The studied section includes the Evie, Muskwa, and Otter Park
low-porosity shale members of the Horn River Formation (Fig. 4a). They lie
above the carbonatic Keg River Formation and under the shale of the Fort
Simpson Formation (Curtis et al., 2010; Chalmers et al., 2012). The shales
are overpressurized with a pore-pressure gradient of 11 to 16 MPa km
The rock densities and the compressional and shear wave velocities are
obtained from well data (Fig. 4b). The depth dependence of the dynamic
elastic parameters, Young's modulus
For practical purposes, in rocks mainly composed of shale like in this
study, there is no difference between the dynamic and static Poisson's ratio
(Mullen et al., 2007). The static Young's modulus
The Western Canada Sedimentary Basin is stressed tectonically because of the Rocky Mountains located to the southwest, which results in a difference in the horizontal stresses (Beaudoin et al., 2011; Reiter et al., 2014). The orientation of the maximum horizontal stress is well defined all over the basin and trends NW–SW (Bell and Gough, 1979; Bell and Bachu, 2003; Reiter et al., 2014), which is normal for the Rocky Mountain trench and folding. However, the tectonic regime is not well delineated by stress measurements (Reiter et al., 2014). Studies highlight spatial variation in stress regime, with thrust faulting in the foothills, strike slip within the basin, and a normal faulting regime in the eastern part of the basin (Bell and Gough, 1979; Bell and Bachu, 2003; Reiter and Heidbach, 2014).
Likewise, depth variations in stress regime are described, with thrust
faulting at shallow depth (i.e., < 350–600 m), strike slip at
intermediate depth 500–2500 m, and normal faulting at greater depths
> 2500m (Fordjor et al., 1983; McLellan, 1987; Reiter and
Heidbach, 2014). In most of the measurements performed in the basin, the
minimum principal stresses are lower than the vertical stresses calculated
from the weight of the overburden rock (Bell and Grasby, 2012). This
precludes a thrust-fault regime. In the whole basin, the minimum principal
stress gradient ranges from 12 to 27 MPa km
More locally in the studied area, the bottomhole instantaneous shut-in
pressures do not show consistent variation from the toe to the heel, with
minimum principal stresses equal to 26
The minimum principal stress is thus lower than the vertical stress. In addition, the ratio of the minimum principal stress and the vertical stress is very close to the ratio for a critical state of stress. Thus, it is very unlikely that the maximum horizontal stress is higher than the vertical stress because it would involve an overcritical state of stress. Our data therefore indicate that a normal or transtensional stress regime is more likely than a strike-slip regime. This may be due to regional strike-slip faults that induce significant stress perturbations responsible for local normal faulting.
Calculated stresses for the various models and predictive
strategies.
The uniaxial stress
For the calibration stresses (see Sect. 3.2), the minimum horizontal
critical stress
By comparing the initial and calibration values of stress, we obtain the
different regional tectonic perturbations
The lithostatic stress
The uniaxial stress
For nearly all the predictive strategies, the regional tectonic perturbations
Next we compute the local stress corrections
The local stress corrections
The predicted minimum principal stress
The predicted minimum stress
Comparison of the stress- and strain-driven models. Stress
predictions for the stress-driven and the strain-driven models for
For the predictive strategies 1 and 3, based on the uniaxial strain model
(Table 1), layer-to-layer stress variations are initially present before the
tectonic perturbation due to the vertical variation in the Poisson's ratio
(Fig. 5b). Depending on the rock properties, the local stress corrections
For the stress-driven model, similar local stress corrections
The predicted magnitudes of the layer-to-layer stress variations are also very similar. The maximum values are obtained for prediction strategy 8 (Table 1), where layer stress variations reach 11 MPa between the Horn River and Keg River formations, 8 MPa between the Horn River and Fort Simpson formations, and 5 MPa between the Otter Park Member and the surrounding Muskwa or Evie members (light gray curves in Fig. 5c). In this case, however, predicted stress changes are due to stress transfer between layers and depend on the contrast in Young's moduli between layers and their thicknesses (see Appendix B). This causes important differences between the stresses predicted with the stress- or strain-driven models. This is explored in the next subsection.
Almost no regional tectonic perturbations,
For the other strategies, the results obtained with the stress- and
strain-driven models show similar trends (Fig. 6). For both models, the
magnitude of the local stress corrections depends on the magnitude and
polarity (i.e., compression or extension) of the tectonic perturbations
First, for the strain-driven model, the local stress corrections
Second, for the stress-driven model, the local stress corrections
Conversely, for the strain model, the average magnitude varies as a function
of the elastic properties of the studied section. For instance, very
different average local stress corrections are found for strategy 2 (Table 1), i.e.,
As a consequence, a strain-driven model may exhibit lower stress corrections
than a stress-driven model in the compliant layers, and higher local stress
corrections occur for the strain-driven model in the stiffer lithologies. In
the case of an extensional perturbation, like in the study case, the
predicted minimum horizontal stress
On average, the predicted minimum principal stress
A closer scrutiny of the various stress profiles in Fig. 5 shows that the strain-driven tectonic models (strategies 1–4, Table 1) predict that the largest supra-critical stresses occur in the Keg River Formation, whereas the stress-driven models (strategies 5–8, Table 1) also show supra-critical stresses above this formation. A critically stressed Keg River Formation is more likely given that the induced seismicity occurred within or below this formation.
Finally, predictions 2 and 6 (Table 1), based on an initial lithostatic stress and the in situ stress measurements, show the most supra-critical stresses independent of the chosen tectonic model. This can be explained by the fact that the in situ stress measurements are substantially lower than the critical stresses. Furthermore, using the lithostatic stress as the initial state induces higher tectonic effects than using any other initial stress state. As a consequence, stress predictions based on the lithostatic stress and in situ stress measurements creates strongly supra-critical predictions in this case.
The various inputs and/or models give rise to inherent uncertainty in stress prediction. In the study case, the maximum differences in final stress obtained between all the stress predictions reach 20, 18, and 28 MPa in the Fort Simpson, Horn River, and Keg River formations, respectively (Fig. 5). For the different stress predictions that are based on a uniaxial stress, the maximum difference reaches 7, 6, and 8 MPa in the Fort Simpson, Horn River, and Keg River Formations. For an initial lithostatic stress, they reach 20, 15, and 28 MPa in the same formations. When using the critical stress as calibration stress, the maximum changes between the various stress predictions are 6, 9, and 16 MPa in the Fort Simpson, Horn River, and Keg River Formations. They are 15, 4, and 16 MPa in the same formations when using the in situ stress measurement as calibration stresses. Assuming a strain-driven model, they reach 8, 8, and 18, and they reach 14, 18, and 12 MPa assuming a stress driven model. Likewise, the range of predicted stress discontinuities between layers is 1–7 and 1–12.5 MPa between the Fort Simpson and Horn River formations and between the Horn River and Keg River formations, respectively. These uncertainties in stress prediction have fundamental implications on fracturing and containment capacity.
In situ stress depends on several effects, including pore pressure, rock viscosity, rock anisotropy, preexisting fracturing, thermal effects, etc. (Cornet and Burlet, 1992; Addis et al., 1996; Khan et al., 2011; Meixner et al., 2014; Roche and Van der Baan, 2015). In this paper we focus on tectonic forces in combination with the vertical variation in rock properties while disregarding other potential effects. First, we discuss the choice of the initial and the calibration stress, then the stress- and the strain-driven model, and finally uncertainties in stress prediction. The choice of the initial regional tectonic regime and whether tectonic perturbations in one direction or in both horizontal directions are taken into account are also critical, but these are beyond the scope of the paper.
Stress predictions are based on an initial stress model that is modified to account for the tectonic effects. We used two models for the initial stress: the lithostatic and uniaxial strain models. In the literature most stress predictions are based on an initial uniaxial strain model (Voight and St. Pierre, 1974; Haxby and Turcotte, 1976; Savage, 1992; Blanton and Olson, 1999), and few are based on an initial lithostatic model (McGarr, 1988; Roche et al., 2014; Roche and Van der Baan, 2015). The choice of the initial stress model influences the final stress predictions for several reasons.
For a uniaxial strain model, initial layer-to-layer stress variations occur due to the Poisson's effect. These variations are added to those created by tectonic effects. In the study case, the tectonic effects dominate Poisson's effects. Still, the Poisson's effect may change the magnitude and direction of the final layer-to-layer stress variations if the tectonic effects are low, the variation in the Poisson's ratio is significant, and the pore pressure is low.
The magnitude of the regional tectonic perturbation depends on the chosen initial stress model. For instance, the initial stresses calculated with a uniaxial strain model are lower than those calculated with a lithostatic model. Hence, for an in situ stress measurement that is higher than the lithostatic stresses (i.e., compressive regime), the magnitude of the tectonic perturbations is greater for an initial uniaxial strain model than for a lithostatic model. As a consequence, stress predictions based on a uniaxial strain model tend to underestimate the layer-to-layer stress variations. For an extensional regime, the tectonic perturbation may be either greater or smaller for the lithostatic model, if the in situ stress measurement is closer to the lithostatic state of stress or to the uniaxial stress, respectively.
The choice of the initial model is potentially more critical when looking at its effect on the polarity of the regional tectonic perturbations. For instance, perturbations that are in compression or in extension may be obtained if the in situ stress measurements are comprised between the lithostatic and uniaxial stresses. In such a case, by assuming a tectonic perturbation in one direction, final stress predictions based on an initial uniaxial stress are likely to overestimate and underestimate the minimum stress in the stiff and compliant layers, respectively. Consequently, tensile and shear failures are more likely to appear inhibited in the stiff layer and promoted in the compliant layer.
This discussion raises the question: which model is more likely to dominate? The stress state predicted by the uniaxial strain model has rarely been observed in field data (Jaeger et al., 2009; McGarr, 1987, 1988). Also, it creates a bias toward highly extensional tectonic regimes and excludes thrust and strike-slip regimes (McGarr, 1987, 1988). However, the lithostatic model also appears implausible. There is little experimental evidence to support this model because the time-dependent failure mechanisms that could remove all the deviatoric stresses for indefinite periods of time do not seem to exist in the upper crust (Kirby and McCormick, 1984; McGarr, 1988). Nevertheless, the lithostatic stress state appears to be the more realistic calibration stress (McGarr, 1988).
The initial stresses influence the computed tectonic perturbations. Their magnitudes and polarities also depend on the calibration stresses. Two types of reference are used here: in situ stress measurements and critical stresses. In situ stress measurements are a better choice because they provide a direct measurement of the local stress. However, such data are often not available, have uncertainty, or are limited to specific depths. Thus, it is useful to be able to predict stress based only on models. This can be done using the critical state of stress. In the latter case, we assume that the stress is controlled by friction on preexisting cohesionless faults. Wrong estimation of this behavior will lead to a change in the critical stress. For instance, in clay-rich formations, a lower coefficient of friction may be expected. Likewise, it has been shown that the stress observed close to an active fault may imply a plastic behavior for the fault gouge that does not satisfy Coulomb failure criterion, but rather obeys a nonassociative plastic flow rule (Sulem, 2007). In this case, we may expect a lower critical stress than the one used in this paper.
If computation of the tectonic perturbation is based on stress measurements, then likely only a few points will determine their magnitudes. This is not the case if the critical stress model acts as a reference since then local differences are available for all layers. This points to a likely trade-off in the final predictions between paucity of information (lacking and uncertain measurements) versus potential for systematic errors due to an incorrectly assumed calibration state of critical stress. Likewise, uncertainty in the magnitude of the stress measurements is likely to be responsible for significant variation in the predicted stresses.
The magnitude of the maximum horizontal stress cannot be measured accurately or precisely (Schmitt et al., 2012). Hence, although we provide formulations involving the measurement of the maximum horizontal stress as a theoretical possibility, the use of in situ stress measurements as calibration stresses is only possible for a normal faulting regime since maximum horizontal stress measurements would be needed in case of a strike-slip or thrust faulting regime.
In this study, the difference between the calibration stresses, i.e., in situ measurement and critical stress, is relatively small compared to the difference between the initial stresses, i.e., lithostatic and uniaxial stresses. Hence, the change in tectonic perturbation, obtained for a similar initial stress but different calibration stresses, is only 25 % of the change obtained for similar calibration stress but different initial stresses. In our case, the stress predictions obtained using in situ measurements and critical stresses are thus similar.
For similar initial and calibration stresses, the stress predictions obtained with the strain-driven and the stress-driven models share a similar trend, consistent with in situ stress measurements, as a first approximation. For both models, stresses predicted in one specific layer depend on the tectonic perturbation and the elastic parameters. Nevertheless, significant differences occur in terms of stress magnitude and stress gradients (see Sect. 5.4). It is therefore fundamental to know which model dominates in nature, or if both models occur.
The strain-driven model has been widely used, notably as a standard method for oil and gas reservoir exploration (Thiercelin and Plumb, 1994; Blanton and Olson, 1999; Beaudoin et al., 2011; Song and Hareland, 2012). The stress-driven model may involve discontinuous strain across layer boundaries if the layers are not coupled together. Such a behavior may appear as a nonphysical consequence of the model, leading authors to disregard this model in favor of the strain-driven model (Blanton and Olson, 1999). This explains why the stress-driven model is scarcely used (Teufel and Clark, 1984; Bourne, 2003; Roche et al., 2013). However, if the layers are coupled together, both the regional stress and regional strain are continuous throughout the layering. Likewise, the possible occurrence of strain discontinuities does not appear decisive in disregarding the stress-driven model because strain decoupling likely exists in natural rocks, notably in the case of detachment faults (Cornet and Burlet, 1992; Meixner et al., 2014). Also, a recent study shows that fracturing depends on the contrast in elasticity between layers, as well as the layer thicknesses, rather than solely on the properties of the layer in which fracturing develops (Roche et al., 2014). This tends to support the stress-driven model rather than the strain-driven model. Lastly, bed-parallel faults occurring in tabular rocks have been described (Roche et al., 2012a, b). Such a structure may highlight strain discontinuities. This is also in accordance with the stress-driven model. Otherwise, for the strain-driven model, an additional mechanism must be involved to create these structures.
Accurate prediction of the in situ stresses in the Earth is hampered by epistemic and aleatoric uncertainty. Epistemic (or systematic) uncertainty is caused by lack of knowledge and data; aleatoric (or statistical) uncertainty is caused by the inherent randomness of a phenomenon such as the rolling of dice. The variation in final predicted stresses stems from a range of different sources for uncertainty, including observational and interpolation–extrapolation uncertainty, parameter uncertainty, model uncertainty, and numerical–algorithmic uncertainty (Kennedy and O'Hagan, 2001).
Observational, interpolation, and extrapolation uncertainty arises, for instance, since the stress measurements used in step 2 are prone to observational error, thus influencing the final stress predictions. In addition, the stress measurements are often limited to specific layers and sites, thereby requiring interpolation and/or extrapolation to derive values appropriate for the areas and depth zones under consideration. This introduces uncertainty and spread in the final predicted stress values.
Next, parameter uncertainties are also important, for instance, in assumed elastic parameters (Young's modulus, Poisson's ratio) and friction coefficients. Some of these parameters are derived from well logs and are thus prone to observational error; yet even in the absence of such observational uncertainty, we lack knowledge of how to exactly convert measured parameters (e.g., dynamic moduli) to required modeling parameters (e.g., static moduli), the control of viscoelasticity, or the behavior for the faults, thus creating parameter uncertainty. Furthermore, each predictive strategy depends on different parameters, with their own uncertainties and possible biases, thus introducing further diversity in the final predicted stresses.
Moreover, we have model uncertainty because full knowledge of the actual driving forces and most appropriate geologic boundary conditions is lacking. For instance, both the stress-driven and strain-driven models are geologically plausible. Likewise, lithostatic, uniaxial, and critical states of stress are reasonable assumptions in various circumstances, yet the most appropriate one is rarely known. Also, assumptions implicit in the derivation of the provided equations create model uncertainty. Again, lack of knowledge (epistemic uncertainty) causes diversity in final predictions.
Finally, there is numerical (or algorithmic) uncertainty, caused by numerical errors and approximations when solving for the stress state, e.g., using the discrete-element method in step 4 or in the implementation of any of the provided equations. We assume that these only play a very minor role in our case; for instance, convergence of solutions was closely monitored. Nonetheless, numerical uncertainty remains a possible source of uncertainty in all computer simulations.
Different strategies to predict the vertical variations in the in situ stresses lead to different answers. Such stress predictions take into account the weight of the overburden rock, the pore pressure, the variation of the rock properties, and the tectonic effects. In addition, they assume either stress- or strain-driven models, they assume an initial uniaxial or lithostatic model, and they use a critical model or in situ stress measurements. The different prediction strategies generally lead to similar trends in predicted stresses, yet differences appear both within a layer and between layers due to fundamentally different underlying mechanisms, assumptions, and governing parameters. The spread and diversity in final stress predictions is caused by mostly epistemic uncertainty, expressed as lack of knowledge, data, and/or observational errors in some measurements or variables. Nonetheless, the combined analysis of all eight stress predictions helps reveal the uncertainty, or conversely similarity, in all stress predictions.
The data for the study case were acquired as part of a joint-industry project and are currently proprietary. However, the project partners will consider requests for data access.
In the lithostatic model, the principal stresses
The uniaxial strain model assumes that no regional horizontal strains exist
but horizontal stresses are imposed by Poisson's effect (that is, a
horizontal force due to vertical loading; Savage et al., 1992; Jaeger et al., 2009). The overburden stress
In the critical stress model, the horizontal stresses are assumed equal but
the ratio between horizontal and vertical stress is set such that
preexisting, optimally oriented faults and fractures are at the point of
shear failure. The ratio of effective maximum to minimum critical principal
stresses
In order to analytically describe such a behavior, we assume that only one
regional tectonic perturbation
The authors declare that they have no conflict of interest.
The authors would like to thank the sponsors of the Microseismic Industry Consortium and the Helmholtz-Alberta Initiative for financial support and Itasca for software licensing. We thank the three anonymous reviewers for their comments that helped to significantly improve the paper. We also thank A. Gudmundsson, P. McLellan, A. Zang, and O. Heidbach for helpful discussions. Edited by: B. Grasemann Reviewed by: three anonymous referees