A Preliminary Analysis of In-Situ Stress at Mount Meager by Displacement Discontinuity Method with Topography and Tectonics Considered

Abstract

Geothermal energy is one of the most stable and clean solutions to replace the traditional fossil fuel energy resource. The South Meager geothermal prospect, located in southwestern British Columbia, contains large geothermal energy resources due to recent volcanic activities. The in-situ stress state in the prospective area that influences the characteristics of fractures, thus affecting productivity, remains unknown. In this paper, we present a preliminary analysis of in-situ stress induced by gravitational load at Mount Meager, with tectonics considered. The in-situ stress model was constructed with 3D displacement discontinuity method based on the site-specific topography. The 3D model reveals that the impact of topography is more prominent in shallow and deep regions, while the impact of tectonics is prominent in an intermediate depth range. With the implementation of inferred tectonic stress state, normal faulting regime takes place at shallow depth (<800 m); at the intermediate depth (800–1600 m), the transition of faulting regime from normal to strike-slip and then to thrust occurs; at deeper depth (>1800 m), the fault type becomes normal again. The orientations of major and minor induced horizontal stresses transition from varying with local topography to perfectly aligned with the orientations of major and minor horizontal tectonic stresses at NWN-SES 330° and NEE-SWW 60°, respectively, as depth increases.

1. Introduction

Geothermal energy has long been recognized as one of the most efficient, clean, and stable sources of renewable energy. However, its great potential comes with great risks and challenges, especially in the exploration stage due to the high costs of drilling operations. The Meager Mountain geothermal prospect, in the southern area, is Canada’s most advanced, volcano-hosted geothermal energy prospect that is stillin its early resource development stage [1]. Numerous reports and publications have confirmed its great potential with zones of high permeability residing in the fractured basement reservoir [1,2]. The abundant geothermal resource could be a result of water either being heated at great depths or by a cooling intrusive phase due to the volcanic activity [3]. The geothermal reservoir in southern Mount Meager is shallow; the temperature can reach 200 °C at 500 m depth, and elevation varies greatly over short distances. More detailed information on geothermal gradient and heat flow can be found in the recent publications by Chen et al. [4] and Liu et al. [5].

With the validated capacity for geothermal resource at south Meager area, it is now vital for the researchers to understand the in-situ stress field. It is one of the most important prior information needs and could greatly impact the production performance during the exploration and exploitation stage of the geothermal extraction. The in-situ stress field, or the original stress condition before excavation or perturbation occurs, can be described by three principal stresses that intersect with each other at 90°. It is found that the applied in-situ stress is closely related to the spatial and temporal variations of fracture aperture and flow rates which eventually affect the heat transfer and preferred direction of fluid flow [6,7]. For the shallow region closer to the topographic surface, the stress-related mechanisms are dominated by the orientation and magnitude of the major horizontal compressive stress [8]. Despite its critical importance, the acquisition of a complete understanding of the in-situ stress state has not been commonly considered quantitatively. Field measurement methods such as hydraulic fracturing test, diagnostic fracture injection test, and casing shoe leak off test are all somewhat subject to local variations due to topography, lithology, geological structures, and stratifications [9,10]. Thus, many indirect attempts have been made to estimate the in-situ stress by inference from field logs and rock mechanical properties such as establishing the relation between formation breakdown pressure and horizontal in-situ stress magnitude and orientation [11], utilizing artificial neural networks and genetic algorithms to correlate the in-situ stress with the displacements of a borehole wall [12,13,14], and using four-arm caliper log borehole deformation data [15,16]. The lack of complete or partial information in the above-mentioned methods is compensated by either making assumptions, using indirect stress-related information, or adopting the stress state in nearby regions. These approaches inevitably assume relatively homogenous and isotropic strata, which is not always an appropriate assumption. For a near-surface region where the depth is of the same order of magnitude as the topographic relief, the perturbation due to the topography is at its greatest and the principal stress tensor components can be extremely heterogeneous [8].

Previous studies have examined the influence of topographic stress on underground stress distribution, either two dimensional with the assumption of plane strain for the older studies [17,18], or three dimensional for the more recent studies [19,20]. However, most of these studies are project site-specific, and subject to various constraints; there are, therefore, not applicable to Mount Meager. In this paper, a numerical model utilizing 3D displacement discontinuity method (DDM) is proposed to determine the in-situ stress at Mount Meager, with a particular focus on the perturbations due to topography and tectonic stress fields. In the following sections, we will first briefly discuss the regional geology of the study area; then outline the numerical and computational methodology adopted to determine the in-situ stress magnitude and orientation; followed by a section to validate the approach using a comparison with published results by Yin [21] simulating an underground excavation. Next, the method discussed is used to model the project site in two scenarios for contrast, one with the implementation of tectonic stress and the other without, to demonstrate how it will influence the principal stress state. Finally, a section discussing the implications and application of the results on fracture opening mechanisms will be provided.

2. Regional Tectonics

The study area is in the western margin of the North American Plate in southwest British Columbia, where the Juan de Fuca Plate subducts under the North American Plate with a rate of approximately 40 mm/year and produces the Cascades Volcanic Arc that extends from northern California to southern British Columbia (Figure 1a). The Cascades Volcanic Arc is subdivided into the High Cascades in the south and the Garibaldi Volcanic Belt (GVB) to the north. The Mount Meager Volcanic Complex (MMVC) is close to the northern end of the GVB where the Juan de Fuca Plate terminates against the Explorer Plate [22]. The tectonically related regional stress field in the western margin of the North America Plate is complicated and the orientation and sources of stress vary considerably. A regional stress study based on the inversion of focal mechanism data by Balfour et al. [23] indicates that the orientation of maximum horizontal stress changes with distance to the subduction trench, from approximately margin-normal along the coast associated with subduction induced strain to approximately margin-parallel 100–150 km inland from the coast related to the northward push of the Oregon Block [23] (Figure 1b).

3. Methods

To further reduce the expense and prolonged waiting time associated with field measurements and estimation methods, extensive studies are conducted to determine the in-situ stress using analytical and numerical methods, particularly in the fields of civil, mining and petroleum engineering [13,19,25,26]. Before the advancement of computation, the analytical method, such as the closed-form solutions [27,28], exact conformal mapping method [29,30], and perturbation method [31], were more prominent. The in-situ stress perturbations subject to heterogeneity and various influence factors have been studied more frequently using numerical methods in the recent decades. The numerical models are typically carried out with the finite element method [32,33], distinct element method [34,35], boundary element method [17,18,36], and displacement discontinuity method [37,38,39].

Among the above-mentioned numerical methods, the boundary element method (BEM), with discretization that can be conducted at a reduced spatial dimension compared to the finite element method (FEM) and finite difference method (FDM), has long been recognized for its computational efficiency and proven accuracy [40]. The BEM is based on the elasticity boundary integral equations and can be used in solving crack problems [41]. On the other hand, Crouch and Starfield [42] have proposed the two-dimensional displacement discontinuity method (DDM) to study the induced stresses by simulating the crack as a continuous, traction-free surface in an elastic body of rock. For a crack problem, the equivalence between BEM and DDM has been established explicitly by Liu [43] by discretizing the displacement and traction boundary integral equations.

Despite the advantage of reduced dimensions of the problems, there are still some downsides to the current approach of DDM. Firstly, due to the complexity of a 3D model, most of the research has made the assumption of either 2D plane strain or plane stress for simplification [17,37,44]. However, a 2D analysis is incapable of presenting the entirety of the target area and often neglects some parameters that can greatly influence the results. Secondly, only a few studies have included topographic stress perturbations due to the complicated geometry, either analytically [19,45], or numerically [8,17,18], and they are mostly in 2D.

In this paper, we introduce a 3D DDM model for the determination of the in-situ stress near Mount Meager, with a particular focus on the linear elastic gravitational stresses and tectonic stresses. To systematically present the solutions for better understanding by the audiences, this section is divided into the following steps: first, we will present the methodology used to determine induced stresses by gravitational load, superimposed by tectonic stresses, on and below the ground surface; next, we will derive the basic formulation of 3D DDM and show how it can be used to calculate stresses at any interior point; finally, the numerical method utilized to solve for the unknowns using the boundary conditions and law of superposition will be discussed.

3.1. Stresses Induced by the Gravitational Load in an Elastic Body

For a 2D analysis, the topographic surface is modelled as a long, traction-free crack aerially exposed due to the removal of overburden materials [18]. For a 3D analysis such as the one discussed in this research, we extend this definition to manifest the topographic surface with a group of connected, rectangular-shaped displacement discontinuity (DD) elements with traction-free boundary conditions under a gravity-induced stress state [17,42]. As shown in Figure 2, the global coordinate system $[X,Y,Z]$ is pointing towards east, south, and vertically upwards, respectively. The normal and shear ambient stresses with respect to the global coordinate system on and below the surface within the domain can be expressed as:

$$\begin{gathered} {\sigma }_X^a=\frac{\nu }{1-\nu }\rho gZ-p_X \\ {\sigma }_Y^a=\frac{\nu }{1-\nu }\rho gZ-p_Y \\ {\sigma }_Z^a=\rho gZ \\ {\tau }_{XY}^a={\tau }_{XZ}^a={\tau }_{YZ}^a=0 \end{gathered}$$

(1)

where $\frac{\nu }{1-\nu }$ is the proportion between the vertical gradient in the ambient vertical normal stress and the ambient horizontal normal stress, where $\nu$ is the Poisson’s ratio, $\rho$ is the rock density, $g$ is gravitational acceleration, $Z$ is the elevation above the datum point where the topography flattens out, and $p_i$ is tectonic stress that is re-oriented to align with the global axes, which is only in effect below the perturbed topography.

On the right-hand side of Figure 2, it is shown that for an individual DD element, a local coordinate system [$x,y,z]$ is established given that the z-axis is normal, while x and y axes are horizontal to the locally flattened topographical element surface. A zero-traction condition is applied to each boundary element to represent the topographical plane, implying that both normal and shear stresses acting on the z-plane will vanish on the exposed surface. It is important to emphasize that this traction-free condition is only valid with respect to the element’s local reference frame—the global vertical stress is not necessarily zero when the topographic surface is not parallel to the horizontal flat plane. Subtracting the ambient stresses induced by gravity from the zero-traction condition will give the boundary condition, denoted as b, needed for the calculation of induced stress at any interior point.

3.2. Formulations of the 3D Displacement Discontinuity Method

Using the elastic law as the foundation for the numerical computations of DDM, Crouch and Starfield [42] have derived a system of equations that relate the induced stresses at any given point in the interior body with the DD components on the boundary. As shown on the right-hand side of Figure 2, the DD element with a negligible thickness can be treated as two identical surfaces opposite to each other. The surface on top is denoted as $S^{+}$ and the surface at the bottom is denoted as $S^{-}$, as illustrated in Figure 3.

The figure depicts the general setting of 3D DDM, where the three DD components between surfaces $S{}^{+}$ and $S{}^{-}$ with respect to each local axis, $D_x,D_y,D_z$, are defined as the displacement differences which can be expressed as:

$$\begin{gathered} D_x=D_x^{-}-D_x^{+} \\ {D}_y=D_y^{-}-D_y^{+} \\ {D}_z=D_z^{-}-D_z^{+} \end{gathered}$$

(2)

Using the expressions of DD, the magnitudes of the induced stresses can be calculated using the general form solution [36,42]:

$$\begin{gathered} {\sigma }_x=2G\left\{D_x\left(2{\varphi }_{,xz}-z{\varphi }_{,xxx}\right)+D_y\left(2\nu {\varphi }_{,yz}-z{\varphi }_{,xxy}\right)+D_z\left[{\varphi }_{,zz}+\left(1-2\nu \right){\varphi }_{,yy}-z{\varphi }_{,xxz}\right]\right\} \\ {\sigma }_y=2G\left\{D_x\left(2\nu {\varphi }_{,xz}-z{\varphi }_{,xyy}\right)+D_y\left(2{\varphi }_{,yz}-z{\varphi }_{,yyy}\right)+D_z\left[{\varphi }_{,zz}+\left(1-2\nu \right){\varphi }_{,xx}-z{\varphi }_{,yyz}\right]\right\} \\ {\sigma }_z=2G\left[D_x\left(-z{\varphi }_{,xzz}\right)+D_y\left(-z{\varphi }_{,yzz}\right)+D_z\left({\varphi }_{,zz}-z{\varphi }_{,zzz}\right)\right] \\ {\tau }_{xy}=2G{D}_x\left[\left(1-\nu \right){\varphi }_{,yz}-z{\varphi }_{,xxy}\right]+D_y\left[\left(1-\nu \right){\varphi }_{,xz}-z{\varphi }_{,xyy}\right]+D_z\left[-\left(1-2\nu \right){\varphi }_{,xy}-z{\varphi }_{,xyz}\right]; \\ {\tau }_{xz}=2G\left[D_x\left({\varphi }_{,zz}+\nu {\varphi }_{,yy}-z{\varphi }_{,xxz}\right)+D_y\left(-\nu {\varphi }_{,xy}-z{\varphi }_{,xyz}\right)+D_z\left(-z{\varphi }_{,xzz}\right)\right] \\ {\tau }_{yz}=2G\left[D_x\left(-\nu {\varphi }_{,xy}-z{\varphi }_{,xyz}\right)+D_y\left({\varphi }_{,zz}+v{\varphi }_{,xx}-z{\varphi }_{,yyz}\right)+D_z\left(-z{\varphi }_{,yzz}\right)\right] \end{gathered}$$

(3)

where ${\varphi }_{,ij}$ and ${\varphi }_{,ijk}$ ($i,j,k$ = x, y, z) are the second and third derivatives of the kernel function, respectively:

$$\varphi \left(x,y,z\right)=\frac{1}{8\pi (1-\nu )}{\iint }_D^{}\frac{1}{{\left(x-\xi \right)}^2+{\left(y-\eta \right)}^2+z^2}d\xi d\eta$$

(4)

where $G$ is the shear modulus, $D$ is the domain of the model, ($x,y,z$) is the location of the interior point with respect to the local reference frame, and ($\xi ,\eta$, 0) is the local coordinate of the boundary element.

The above equation for the induced stresses in local coordinate systems, ${\sigma }_{xyz}$, can be simply written as:

$$\mathit{A}\mathit{D}=\mathit{\sigma }$$

(5)

where A is the matrix composed of the coefficients that reflect the influence of DD at each boundary element on the observation point, or any arbitrary point within the domain where induced stresses are of interest. This matrix is solely dependent on the geometry and relative location of the elements from Equation (3). $D$ is the vector composed of the three DD for each element that can be written as $\left[D_x;D_y;D_Z\right]$, and ${\sigma }_{xyz}$ is the stress component expressed with Voigt’s notation, namely $[{\sigma }_x;{\sigma }_y;{\sigma }_z;{\tau }_{xy};{\tau }_{xz};{\tau }_{yz}]$. To satisfy the boundary condition that the normal and shear stresses perpendicular to the topographic surface caused by the overlying material equal zero, the following equation is established:

$${\mathit{\sigma }}^0-{\mathit{\sigma }}^a=\mathit{b}$$

(6)

where ${\mathit{\sigma }}^0$ is the stress vector of original boundary conditions along the crack surface representing the ground surface, or in other words the traction-free condition for the exposed region; ${\mathit{\sigma }}^a$ is the vector of ambient stresses as shown in Equation (1).

The equality between the induced stresses on each boundary DD element in the local coordinate system, ${\sigma }_{xyz},$ and the stress boundary conditions in the global coordinate system, $\mathit{b}$, can be established given that they are under the same reference frame. Therefore, a stress transformation is required to transform the boundary conditions, or in other words the tectonic stresses, from the global coordinate system, i.e., (X, Y, Z), to the local coordinate system, $\mathrm{i}.\mathrm{e}.,\left(x,y,z\right)$. Equating the resolved boundary conditions with $\mathit{\sigma }$, the only set of unknowns, $\mathit{D},$ can be solved for each boundary element on the topographic surface. Finally, superimposing the ambient stresses induced by gravity and tectonic stresses from Equation (1) onto the stresses at the observation point from Equation (4) will result in a complete stress state along and below the topographic surface with gravitational stresses in effect.

The numerical computation of 3D DDM was carried out using MATLAB. The user-defined inputs implemented for computation include the Poisson’s ratio, density, tectonic stress magnitudes and orientations, shear modulus, friction angle, surface topography, discretization of boundary element, and the grid of observation points where the induced stress state is calculated from. According to Crouch and Starfield [42], the induced stresses are not reliable for observation points placed within an element length from the center of each element with a constant displacement discontinuity, unless the observation point is placed precisely at the element center. Therefore, some minor deviations can be observed at the near-surface region; as the depth increases, this type of discrepancy will vanish.

4. Verification

In this section, an underground excavation is modelled for verification purposes. A comparison is made between the induced vertical stress calculated with the above-mentioned 3D displacement discontinuity model and the results using a boundary element model from the existing literature [21]. The following configurations are applied to each model:

For 3D DDM:

• Domain: semi-infinite;
• 20 m × 20 m area of excavation at 30 m deep below the topographic surface. The excavation volume is 2 m in height, which occupies from 29 m to 31 m underground;
• The boundary area that contains rectangular DD elements includes the horizontal ground surface and a horizontal plane with top and bottom surfaces that represent the underground excavation;
• A constant 1 m displacement discontinuity in the surface-normal direction is prescribed on each side of the DD plane;
• The displacement discontinuities in the horizontal directions are negligible;
• Poisson’s ratio is 0.333, Elastic modulus is 1 GPa;
• The observation points are placed along the X-axis at 28.5 m deep (0.5 m above the top surface of the excavation).

For 3D BEM [21]:

• Domain: semi-infinite;
• The boundary element plane covers the same area as DDM;
• 20 m × 20 m × 2 m excavation volume at 30 m deep;
• U_z = −0.5427 m, U_z = 0.4573 m are prescribed on top and bottom surface, respectively;
• Traction-free surface is applied;
• Poisson’s ratio is 0.333, elastic modulus is 1 GPa;
• Discretized with 4-node rectangular elements.

Figure 4 illustrates the vertical induced stress calculated using both methods. The vertical stresses along the X-axis at 0.5 m above the excavation surface calculated using both methods show an average difference of 9%. The largest variations are observed near the boundaries of the excavation, despite the vertical stress being zero on each edge of the excavation for both methods. As the observation points move further away from the excavation area, the induced vertical stress gradually converges to zero, for both DDM and BEM.

5. Results

For the numerical simulations described in the following section, the rectangular DD plane modelled is highlighted in Figure 5b, where the corners of the DD plane are pinned as in Figure 5a. A total of 900 DD points—30 N-S points × 30 E–W points—are used to construct the model to keep to a reasonable computational time. To be consistent, each layer that contains the observation points covers the same area as the DD plane, except that only 100 points will be used on each layer, with a total of 9 layers at varying depth or elevation.

Since 3D DDM is utilized in this research, all results are captured in 3D so we can examine the in-situ stress condition in 3D view, which is one of the main advantages of 3D analysis compared to a 2D analysis. However, to better visualize the orientations and magnitudes of the in-situ stress state, most of the figures in this section will be presented with either 2D plots or stereographical projections.

5.1. Base Case with Zero Tectonic Stress

A base case with zero tectonic stress in all directions will be presented in this section to better understand how the in-situ stress state is affected with and without the implementation of tectonic stress. In the coastal mountains in southwest British Colombia where Mount Meager is located, the volcanic rocks are composed of basalt to rhyolite [46]. The rock properties used for both study cases are as follows: Poisson’s ratio of 0.25, Young’s Modulus of 70 GPa, the density of rock is 3000 kg/m³, and a friction angle of 30°. Previous research has shown that the frictional strength is negligible up to approximately 8 km deep. When depth goes beyond 8 km, a magnitude of 50 MPa should be implemented as the frictional strength, which increases with respect to the depth [37,47]. Since this analysis focuses on the shallower region, this parameter will not be taken into consideration.

Figure 6 shows the principal stress directions on the topographic surface in 3D. It can be observed that the minimum principal stress, S3, is roughly perpendicular to the ground surface, which is consistent with the boundary condition of traction-free surface. No apparent trend is recognizable for the orientations of maximum ($S_H$) and minimum ($S_h$) horizontal stress on the surface. Note that this figure applies to both scenarios (with and without tectonic stress), since tectonic stresses are assumed to have no effect on the exposed elevated surface in the mountain range.

To view the variations of the modelled stress in a vertical direction associated with topographic features, the orientations of principal stresses are plotted in cross-section views from four different azimuths in Figure 7. As previously, the vertical stress perpendicular to the ground surface is the minimum principal stress on the surface; the maximum principal stress, S1, becomes the vertical stress as the depth increases. On the other hand, S2 and S3 show a higher degree of horizontality as the depth increases, which is better visualized in Figure 8. In Figure 8, the orientations of the three principal stresses at each given depth are plotted in stereographic projection, with the contour illustrating the density of concentration. As above, the vertical stress switches from being S3 on the surface to S1 below the surface. In the absence of tectonic stresses, the remaining two major stresses primarily reflect the effects of topography-related gravitational variations. As the depth increases, the topographic difference decreases in all directions, the plunges of the two gradually turn to horizontal and the directions of the two become either parallel or normal to the slope.

The stereographical diagrams in Figure 8 were plotted using the commercial software DIPS, a program designed to interactively analyze the orientation-based geological data input utilizing stereographic projection [48]. Each symbol on the 2D stereonet in Figure 8 indicates a line in 3D, representing one of the three directions of the principal stresses for each element, described by its trend and plunge.

The contour of magnitudes for all three principal stresses on each layer mimics the topographic surface and remains almost unchanged in shape with decreasing elevation. As an example, the magnitude contour for S1 is plotted in four cross-sections with different directions in Figure 9. Note that compression is taken as negative in this study. Each contour line is roughly parallel with the topography with a constant increase in magnitude as the depth increases. Slight deviations between the magnitude and elevation contours are observed at the shallow depth due to the coarse precision of the numerical simulation. This minor discrepancy can be avoided by utilizing a finer grid of the boundary surface with a larger number of boundary elements. The same trend is observed for S2 and S3; additional plots are omitted for brevity.

5.2. Induced Stress State with Tectonic Stress

Although there is evidence from the field observations such as the orientation of local geological structures, as well as existing literature inferring the far-field stress state near Mount Meager, the exact magnitudes and orientations remain unknown. Without pre-existing tectonic stress tests conducted on-site, the analysis in this section will be performed based on a scenario constructed according to relevant literature [23,49]. Stress inversions for groups of focal mechanisms across SW British Colombia region by Balfour et al. [23] show a change in $S_H$ with distance from the subduction margin, where $S_H$ directions are predominantly NE similar to $e_H$, the direction of maximum horizontal shortening. Further inland from the margin, the stress becomes NWN-SES, which is likely related to the residual strain from the northward push of the Oregon Block. Therefore, the following tectonic stress state is used: major compressive horizontal stress of 20 MPa trending 30$°$ W from the North, and minor compressive horizontal stress of 10 MPa trending 60$°$ E from the North.

The stereographical projections shown in Figure 10 suggest that the tectonic stresses contribute greatly toward the induced stress orientation at deeper depth, as $S_H$ and $S_h$ are roughly parallel with the directions of tectonic stresses. Again, the vertical stress is switching from S3 to S1 starting right below the surface where the stress induced by gravity dominates. Between the depth of approximately 1200 m to 1600 m, the influence of tectonic stress becomes stronger than the gravitational stress. Thus, S1 becomes parallel to either direction of major or minor horizontal stress, and the vertical stress becomes either S2 or S3. At depth deeper than 1800 m, the gravitational stress once again becomes the most dominant driving force of the total induced stresses, with vertical stress being S1. At depth of 3200 m and beyond, S1, S2, and S3 are almost perfectly aligned with the vertical direction and the orientations of major and minor horizontal stresses, respectively.

Similar observations are made for the magnitude contours of S1 shown in cross-section views with a varying azimuth in Figure 11 as compared to the case without tectonic stresses: except for the slight deviations observed at the near-surface region due to computational error, the magnitude contour lines closely trace the elevation contour lines.

6. Discussion

The Meager Mountain Volcanic Complex (MMVC) is a high temperature geothermal system with an identified geothermal reservoir of fractured quartz diorite in the basement [4]. Although efforts have been made to produce the heat energy resource, insufficient flow from the production tests prevented commercial development. A better understanding of the fracture networks and their characteristics, distribution, density and controlling factors is critical to improving geothermal resource development strategies and commercial success in this area. MMVC has a stress field controlled by both regional tectonics and topography. The results from the numerical simulation of topographical impact coupled with tectonic stresses on the in-situ stress field in this study have important implications for understanding fracture development and its controls to potential geothermal reservoirs.

The Geological Survey of Canada conducted a field geological and fracture study in the Mount Meager area in 2019. In the study area, fractures are common in intrusive and metamorphic basement rocks and volcanic rocks that cover them. Figure 12 is a summary map showing the orientation and dip of major fracture groups in the Mount Meager area based on the 2019 GSC field observations and available geological interpretations. Preliminary analysis suggests at least three types of fractures, each with distinct characters, that are likely related to different geological processes [50]. The fractures related to regional tectonic deformation are consistent in character and common in basement rocks. Their strikes are often in good spatial alignments with volcanic eruption centers/vents and observed earthquake events. The interpreted tectonic lineaments from volcanic rupture centers/vents and earthquake centers show the majority of them strike NWN-SES (Figure 12). This observation is consistent with the orientation of the imposed major horizontal compressional stress. Fractures associated with volcanic doming and eruption activities may vary geographically. They are circular/radial segments, and the strikes change spatially depending on their location relative to the eruption center. This is particularly evident from the interpreted fault/fracture zones in Landsat images [51]. Volcanic activity may overprint tectonic fractures. The pre-existing zones of weakness can be further complicated by reactivation and modification during volcanic activity. Gravitational fractures are common in volcanic areas and commonly appear parallel to the slope. This type of fracture causes instability in the mountains’ ridges and peaks and can lead to slides and rock avalanches, such as the 2010 Mount Meager landslide. This is also consistent with the predicted orientation of failure from the numerical model.

Albeit useful, the simulation has limitations, mainly because of the number of assumptions and simplifications made due to the lack of data. First, although tectonic stresses are discussed in the literature, their exact orientations and magnitudes are unknown. Specific assumptions with respect to the orientations and magnitudes were made in the numerical model, which could affect the model outputs as well as applicability. Although change in the magnitudes of the tectonic stresses will not affect the general trend of the orientation of the principal stresses, it will affect the depth of stress regime transition. The second limitation is that linear elastic, homogeneous, and isotropic materials are used for modelling in this research, which sometimes can be unrealistic if the study area has a complicated geology and tectonic history. This deficiency can be overcome to accommodate the elastic-plastic or plastic deformation, heterogeneity, and anisotropic materials by incorporating more complex boundary conditions. This is out of the scope of this study. Third, other potentially influencing factors such as thermal and pore pressure effects are not considered in the current model of in-situ stress though the porosity is low (ranging from <1% to 3% from rock samples) in general for the basement quartz diorite. These limitations will be addressed in future studies.

7. Conclusions

Both topography and tectonics affect the in-situ stress state. The impact of topography is more prominent in the shallow region above the depth of approximately 400 m, while the impact of tectonics is prominent in an intermediate depth range depending on the magnitude of tectonic stress, topographic variation and the density of overlaying rock.

In the absence of tectonic stress, gravity differences caused by topographic variations control the induced in-situ stress field. Except at the ground surface, the S1 is approximately vertical, while the orientations of $S_H$ and $S_h$ are variable with topographic feature. With increasing depth, S1 becomes vertical and $S_H$ and $S_h$ become horizontal. The topography induced stress field is always in a normal faulting regime. At shallow depths near the surface, the orientation of $S_H$ tends to be parallel to the strike of slope and $S_h$ to be perpendicular to the strike of slope.

In the presence of the tectonic stress field where the major compressive horizontal stress from the north 30° W direction is 20 MPa and the minor compressive horizontal stress from the north 60° E direction is 10 MPa, the following observations were made from this study:

• In shallow depths, (<800 m), where the in-situ stress is similar to the tectonic-stress free case, a normal faulting regime ($S_V>S_H$ > $S_h$) prevails. The orientations of $S_H$ and $S_h$ depend on local topographic variation.
• In a depth interval of approximately between 800 m to 1600 m, we see a transition from a normal faulting regime through a strike-slip faulting regime ($S_H$ > $S_V$ > $S_h$) to a thrust faulting regime ($S_H$ > $S_h$ > $S_V$) taking place, due to the interplays between tectonic stresses and topography induced stresses.
• As the gravity load of the rock mass (vertical stress) increases linearly with depth, the vertical stress surpasses the magnitude of tectonic exerted principal stress at depth > 1800 m and the fault type once again becomes normal faulting ($S_V$ > $S_H$ > $S_h$). The orientations of $S_H$ and $S_h$ are converged toward to the orientations of major and minor horizontal stresses at NWN-SES 330° and NEE-SWW 60°, respectively.

On the local scale, the fractures trending in the direction parallel and subparallel to the direction of $S_H$ are more likely to be open and thus improve rock permeability and fluid productivity. Consequently, the placement of horizontal wells should consider the in-situ stress state as an important influential factor.