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

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## Abstract

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## 1. Introduction

## 2. Regional Tectonics

## 3. Methods

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

#### 3.2. Formulations of the 3D Displacement Discontinuity Method

**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:

## 4. Verification

- 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).

- 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.

## 5. Results

#### 5.1. Base Case with Zero Tectonic Stress

^{3}, 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.

#### 5.2. Induced Stress State with Tectonic Stress

## 6. Discussion

## 7. Conclusions

- 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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Witter, J. South Meager Geothermal Project: New Perspectives from Recently Unearthed Data; Innovate Geothermal Ltd.: Vancouver, BC, Canada, 2019. [Google Scholar]
- Grasby, S.E.; Ansari, S.M.; Barendregt, R.W.; Borch, A.; Calahorrano-DiPatre, A.; Chen, Z.; Craven, J.A.; Dettmer, J.; Gilbert, H. Garibaldi Geothermal Energy Project—Phase 1; Geoscience BC: Vancouver, BC, Canada, 2021. [Google Scholar]
- Lewis, T.J.; Souther, J.G. Meager Mountain, B.C.: A Possible Geothermal Energy Resource; Geothermal Series; Energy, Mines and Resources Canada, Earth Physics Branch: Ottawa, ON, Canada, 1978. [Google Scholar]
- Chen, Z.; Grasby, S.E.; Yuan, W.; Liu, X. Ground Surface Temperature Monitoring Data Analysis and Applications to Geothermal Exploration in Volcanic Areas, Mount Meager, Western Canada. Geothermics
**2023**, 108, 102610. [Google Scholar] [CrossRef] - Liu, X.; Chen, Z.; Grasby, S.E. Using Shallow Temperature Measurements to Evaluate Thermal Flux Anomalies in the Southern Mount Meager Volcanic Area, British Columbia, Canada; Geological Survey of Canada: Ottawa, ON, Canada, 2022; p. 8890. [Google Scholar]
- Meixner, J.; Schill, E.; Gaucher, E.; Kohl, T. Inferring the in Situ Stress Regime in Deep Sediments: An Example from the Bruchsal Geothermal Site. Geotherm. Energy
**2014**, 2, 7. [Google Scholar] [CrossRef] [Green Version] - Sun, Z.; Jiang, C.; Wang, X.; Zhou, W.; Lei, Q. Combined Effects of Thermal Perturbation and In-Situ Stress on Heat Transfer in Fractured Geothermal Reservoirs. Rock Mech. Rock Eng.
**2021**, 54, 2165–2181. [Google Scholar] [CrossRef] - Griffith, W.A.; Becker, J.; Cione, K.; Miller, T.; Pan, E. 3D Topographic Stress Perturbations and Implications for Ground Control in Underground Coal Mines. Int. J. Rock Mech. Min. Sci.
**2014**, 70, 59–68. [Google Scholar] [CrossRef] - Haimson, B.C. A Simple Method for Estimating Stresses at Great Depths. In Proceedings of the Field Testing and Instrumentation of Rock; ASTM International: West Conshohoncken, PA, USA, 1974; pp. 156–182. [Google Scholar]
- Haimson, B.; Fairhurst, C. Initiation and Extension of Hydraulic Fractures in Rocks. Soc. Pet. Eng. J.
**1967**, 7, 310–318. [Google Scholar] [CrossRef] - Aadnoy, B.S. Inversion Technique to Determine the In-Situ Stress Field from Fracturing Data. J. Pet. Sci. Eng.
**1990**, 4, 127–141. [Google Scholar] [CrossRef] - Zhang, S.; Yin, S. Determination of in Situ Stresses and Elastic Parameters from Hydraulic Fracturing Tests by Geomechanics Modeling and Soft Computing. J. Pet. Sci. Eng.
**2014**, 124, 484–492. [Google Scholar] [CrossRef] - Zhang, S.; Yin, S. Determination of Horizontal In-Situ Stresses and Natural Fracture Properties from Wellbore Deformation. Int. J. Oil
**2014**, 7, 1–28. [Google Scholar] [CrossRef] - Zhang, S.; Yin, S. Determination of Earth Stresses Using Inverse Analysis Based on Coupled Numerical Modelling and Soft Computing. Int. J. Comput. Appl. Technol.
**2015**, 52, 18–28. [Google Scholar] [CrossRef] - Han, H.; Yin, S. Determination of In-Situ Stress and Geomechanical Properties from Borehole Deformation. Energies
**2018**, 11, 131. [Google Scholar] [CrossRef] - Han, H.; Yin, S. In-Situ Stress Inversion in Liard Basin, Canada, from Caliper Logs. Petroleum
**2020**, 6, 392–403. [Google Scholar] [CrossRef] - Martel, S.J.; Muller, J.R. A Two-Dimensional Boundary Element Method for Calculating Elastic Gravitational Stresses in Slopes. Pure Appl. Geophys.
**2000**, 157, 989–1007. [Google Scholar] [CrossRef] - Slim, M.; Perron, J.T.; Martel, S.J.; Singha, K. Topographic Stress and Rock Fracture: A Two-Dimensional Numerical Model for Arbitrary Topography and Preliminary Comparison with Borehole Observations: Topographic Stress and Rock Fracture. Earth Surf. Process. Landf.
**2015**, 40, 512–529. [Google Scholar] [CrossRef] [Green Version] - Pan, E.; Amadei, B.; Savage, W.Z. Gravitational and Tectonic Stresses in Anisotropic Rock with Irregular Topography. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1995**, 32, 201–214. [Google Scholar] [CrossRef] - Moon, S.; Perron, J.T.; Martel, S.J.; Holbrook, W.S.; St. Clair, J. A Model of Three-Dimensional Topographic Stresses with Implications for Bedrock Fractures, Surface Processes, and Landscape Evolution. J. Geophys. Res. Earth Surf.
**2017**, 122, 823–846. [Google Scholar] [CrossRef] - Yin, S. Geomechanics-Reservoir Modeling by Displacement Discontinuity-Finite Element Method; University of Waterloo: Waterloo, ON, Canada, 2008. [Google Scholar]
- Venugopal, S.; Moune, S.; Williams-Jones, G.; Druitt, T.; Vigouroux, N.; Wilson, A.; Russell, J.K. Two Distinct Mantle Sources beneath the Garibaldi Volcanic Belt: Insight from Olivine-Hosted Melt Inclusions. Chem. Geol.
**2020**, 532, 119346. [Google Scholar] [CrossRef] - Balfour, N.J.; Cassidy, J.F.; Dosso, S.E.; Mazzotti, S. Mapping Crustal Stress and Strain in Southwest British Columbia. J. Geophys. Res.
**2011**, 116, 00B03314. [Google Scholar] [CrossRef] [Green Version] - Natural Resources Canada Plate Tectonics Shape (and Shake) British Columbia. Available online: https://earthquakescanada.nrcan.gc.ca/pprs-pprp/pubs/GF-GI/GEOFACT_plate-tectonics_e.pdf (accessed on 18 January 2023).
- Dusseault, M.B.; Bruno, M.S.; Barrera, J. Casing Shear: Causes, Cases, Cures; OnePetro: Richardson, TX, USA, 1998. [Google Scholar]
- Yong, R.; Wu, J.; Huang, H.; Xu, E.; Xu, B. Complex in Situ Stress States in a Deep Shale Gas Reservoir in the Southern Sichuan Basin, China: From Field Stress Measurements to in Situ Stress Modeling. Mar. Pet. Geol.
**2022**, 141, 105702. [Google Scholar] [CrossRef] - Teraghi, K.; Richart, F.E. Stresses in Rock about Cavities. Géotechnique
**1952**, 3, 57–90. [Google Scholar] [CrossRef] - Amadei, B.; Pan, E. Gravitational Stresses in Anisotropic Rock Masses with Inclined Strata. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1992**, 29, 225–236. [Google Scholar] [CrossRef] - Perloff, W.; Baladi, G.; Harr, M. Stress Distribution within and under Long Elastic Embankments: Research Paper; Joint Highway Research Project; Indiana Department of Transportation and Purdue University: West Lafayette, IN, USA, 1967. [Google Scholar]
- Savage, W.Z.; Swolfs, H.S.; Powers, P.S. Gravitational Stresses in Long Symmetric Ridges and Valleys. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1985**, 22, 291–302. [Google Scholar] [CrossRef] - McTigue, D.F.; Mei, C.C. Gravity-Induced Stresses near Topography of Small Slope. J. Geophys. Res. Solid Earth
**1981**, 86, 9268–9278. [Google Scholar] [CrossRef] - Zhang, H.; Yin, S.; Aadnoy, B.S. Poroelastic Modeling of Borehole Breakouts for In-Situ Stress Determination by Finite Element Method. J. Pet. Sci. Eng.
**2018**, 162, 674–684. [Google Scholar] [CrossRef] - Lobatskaya, R.M.; Strelchenko, I.P.; Dolgikh, E.S. Finite-Element 3D Modeling of Stress Patterns around a Dipping Fault. Geosci. Front.
**2018**, 9, 1555–1563. [Google Scholar] [CrossRef] - Homberg, C.; Hu, J.C.; Angelier, J.; Bergerat, F.; Lacombe, O. Characterization of Stress Perturbations near Major Fault Zones: Insights from 2-D Distinct-Element Numerical Modelling and Field Studies (Jura Mountains). J. Struct. Geol.
**1997**, 19, 703–718. [Google Scholar] [CrossRef] - Hazeghian, M.; Soroush, A. Numerical Modeling of Dip-Slip Faulting through Granular Soils Using DEM. Soil Dyn. Earthq. Eng.
**2017**, 97, 155–171. [Google Scholar] [CrossRef] - Brady, B.H. Boundary Element Method for Mine Design; University of London: London, UK, 1979. [Google Scholar]
- Ritz, E.; Mutlu, O.; Pollard, D.D. Integrating Complementarity into the 2D Displacement Discontinuity Boundary Element Method to Model Faults and Fractures with Frictional Contact Properties. Comput. Geosci.
**2012**, 45, 304–312. [Google Scholar] [CrossRef] - Li, K. Numerical Analysis of Undersea Geostress Field around Fault. Electron. J. Geotech. Eng.
**2015**, 20, 1887. [Google Scholar] - Chai, Y.; Yin, S. 3D Displacement Discontinuity Analysis of In-Situ Stress Perturbation near a Weak Fault. Adv. Geo-Energy Res.
**2021**, 5, 286–296. [Google Scholar] [CrossRef] - Cheng, A.H.-D.; Cheng, D.T. Heritage and Early History of the Boundary Element Method. Eng. Anal. Bound. Elem.
**2005**, 29, 268–302. [Google Scholar] [CrossRef] - Rizzo, F.J. An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics. Q. Appl. Math.
**1967**, 25, 83–95. [Google Scholar] [CrossRef] - Crouch, S.L.; Starfield, A.M. Boundary Element Methods in Solid Mechanics: With Applications in Rock Mechanics and Geological Engineering; Allen & Unwin: Crows Nest, Australia, 1983; ISBN 978-0-04-620010-7. [Google Scholar]
- Liu, Y. On the Displacement Discontinuity Method and the Boundary Element Method for Solving 3-D Crack Problems. Eng. Fract. Mech.
**2016**, 164, 35–45. [Google Scholar] [CrossRef] - Brady, B.H.G.; Bray, J.W. The Boundary Element Method for Elastic Analysis of Tabular Orebody Extraction, Assuming Complete Plane Strain. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1978**, 15, 29–37. [Google Scholar] [CrossRef] - Pan, E.; Amadei, B. Three-Dimensional in Situ Stress Analysis of Rock Masses Perturbed by Irregular Topographies and Underground Openings; Colorado University at Boulder Department of Mechanical Engineering: Boulder, CO, USA, 1998. [Google Scholar]
- Hetherington, R.M. Slope Stability Analysis of Mount Meager, Southwestern British Columbia, Canada. Master’s Thesis, Michigan Technological University, Houghton, MI, USA, 2014. [Google Scholar]
- Sibson, R.H. Fault Rocks and Fault Mechanisms. J. Geol. Soc.
**1977**, 133, 191–213. [Google Scholar] [CrossRef] - Mendoza, W. Dips: Plotting, Analysis and Presentation of Structural Data Using Spherical Projection Techniques; Rocscience Inc.: Toronto, ON, Canada, 2002. [Google Scholar]
- Ristau, J.; Rogers, G.C.; Cassidy, J.F. Stress in Western Canada from Regional Moment Tensor Analysis. Can. J. Earth Sci.
**2007**, 44, 127–148. [Google Scholar] [CrossRef] - Chen, Z.; Grasby, S.E.; Liu, X. Fracture System Analyses of the Mount Meager Area; Natural Resources Canada: Ottawa, ON, Canada, 2021. [Google Scholar]
- Chen, Z.; Grasby, S.; Deblonde, C.; Liu, X. AI-Enabled Remote Sensing Data Interpretation for Geothermal Resource Evaluation as Applied to the Mount Meager Geothermal Prospective Area; Natural Resources Canada: Ottawa, ON, Canada, 2022. [Google Scholar]

**Figure 1.**Diagrams showing the plate tectonic background and regional characteristics of the stress field of the Cascades Volcanic Arc in the North American Plate. (

**a**) Map showing tectonic setting of the study area (figure modified from [24]), and (

**b**) cartoon showing the dominant stress orientations (red arrows) in southwest British Columbia. The red crosses in circles indicate motion into the page due to the northward push of the Oregon Block (figure modified from [23]).

**Figure 2.**Schematic diagram of the basic configuration of a DDM model. The global DD plane and the domain are plotted on the left in global axes; an enlarged individual DD element on the topographic surface is plotted on the right in local axes. The traction-free condition on the exposed surface is visualized with the figure at the bottom for such a boundary DD element. Note that the DD element is not drawn to scale as its thickness is considered negligible compared to its lateral extent.

**Figure 5.**Map of Mount Meager area showing (

**a**) a Google Earth Satellite image with the four corner markers of the DD plane for the modelling; (

**b**) elevation contour map of the region with DD plane highlighted in green. The locations of cross sections are indicated with different colors.

**Figure 6.**Principal stress directions on the surface in 3D, applicable to both cases with and without the implementation of tectonic stresses.

**Figure 7.**Orientation of principal stresses without tectonic stress in cross-section views with varying azimuth. (

**a**) E-W; (

**b**) NW-SW; (

**c**) N-S; (

**d**) NW-SE. All plots are drawn with unit vectors to represent the direction; the length is irrelevant to the magnitude. Locations of the cross sections are shown in Figure 5b.

**Figure 8.**Stereographical projection of induced principal stress directions without tectonic stresses in the study area at depth of (

**a**) 0 m; (

**b**) 400 m; (

**c**) 800 m; (

**d**) 1200 m; (

**e**)1600 m; (

**f**) 2000 m; (

**g**) 2400 m; (

**h**) 2800 m; (

**i**) 3200 m.

**Figure 9.**Principal stress magnitude contours without tectonic stress in cross-section views with varying azimuth. (

**a**) E-W; (

**b**) NW-SW; (

**c**) N-S; (

**d**) NW-SE.

**Figure 10.**Stereographical projection of induced principal stress directions with tectonic stresses in the study area at depth of (

**a**) 0 m; (

**b**) 400 m; (

**c**) 800 m; (

**d**) 1200 m; (

**e**) 1600 m; (

**f**) 2000 m; (

**g**) 2400 m; (

**h**) 2800 m; (

**i**) 3200 m.

**Figure 11.**S1 magnitude contour with tectonic stresses in cross-section view with varying azimuth. (

**a**) E-W; (

**b**) NW-SW; (

**c**) N-S; (

**d**) NW-SE.

**Figure 12.**Orientations of major tectonic features interpreted from volcanic eruption center/vents, recorded earthquakes, and known and inferred faults from geophysical data and fracture strike groups. Figure modified from [50].

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**MDPI and ACS Style**

Chai, Y.; Chen, Z.; Yin, S.
A Preliminary Analysis of In-Situ Stress at Mount Meager by Displacement Discontinuity Method with Topography and Tectonics Considered. *Energies* **2023**, *16*, 1397.
https://doi.org/10.3390/en16031397

**AMA Style**

Chai Y, Chen Z, Yin S.
A Preliminary Analysis of In-Situ Stress at Mount Meager by Displacement Discontinuity Method with Topography and Tectonics Considered. *Energies*. 2023; 16(3):1397.
https://doi.org/10.3390/en16031397

**Chicago/Turabian Style**

Chai, Yutong, Zhuoheng Chen, and Shunde Yin.
2023. "A Preliminary Analysis of In-Situ Stress at Mount Meager by Displacement Discontinuity Method with Topography and Tectonics Considered" *Energies* 16, no. 3: 1397.
https://doi.org/10.3390/en16031397