Estimations of Regional Stress Based on Measured Local Stress

Abstract

Estimations of regional stress are demonstrated in this paper. Firstly, regional stress is defined and the characteristics of regional stress are then discussed based on the local stresses measured by the Compact Conical-ended Borehole Overcoring (CCBO) technique and the results from the earthquake focal mechanism. Secondly, the regional stresses are estimated by a back analysis of three-dimensional finite element models, using the local stresses measured by the CCBO and hydraulic fracturing.

Introduction

Phenomena relating to rock and rock mass, such as deformation, fracture, earthquake, etc., are associated with rock stress. The measurement of rock stress plays an important role in understanding these phenomena. For the purpose of rock stress measurement, many methods have been proposed and used worldwide. Recent topics about rock stress and rock stress measurements have been summarized in a special issue on rock stress estimation in the International Journal of Rock Mechanics and Mining Sciences (2003).

The term “rock stress” can be used to refer to many types of stress states in a rock mass. The choice of terminology describing the state of stress in a rock mass was recommended by the ISRM suggested method (Hudson et. al., 2003). In this recommendation, local stress is defined as the stress state in a small domain, and regional stress is the stress state in a relatively large geological domain. It is considered that regional stress is not measured directly but local stress can be measured by various methods for rock stress measurement. Accordingly, regional stress that involves large rock volume is able to be estimated from local stresses measured in a small rock volume. Simple statistical procedures such as interpolation or extrapolation may be adopted in the stress estimation approach (Hudson et al., 2003). Also numerical back analysis has been used as an effective tool to estimate from local stress to regional stress (Sakurai and Shimizu, 1986).

In this paper, regional stress is firstly defined and the characteristics of regional stress are then discussed on the basis of local stresses measured by the Compact Conicalended Borehole Overcoring (CCBO) technique (Sugawara and Obara, 1999; Obara and Ishiguro, 2004) and the results obtained from the earthquake focal mechanism. Finally, a back analysis is carried out to estimate regional stresses using a three-dimensional finite element method (3-D FEM) and local stresses measured by the CCBO and hydraulic fracturing.

 

CCBO technique

The state of rock stress is important in the estimation of the stability of rock structures. Various methods for rock stress measurement have been suggested in the present study. The compact conical-ended borehole overcoring (CCBO) technique was one of the methods developed for this purpose (Sugawara and Obara, 1999; Obara and Sugawara, 2003). The CCBO technique can determine the complete three-dimensional stress tensor from a measurement in a single borehole, and can perform a series of measurements in a short period of approximately 2 hours. Furthermore, this technique can carry out stress measurements in a rock mass with a narrow joint spacing and where there is a damaged region around a tunnel. The measurement procedure of the CCBO technique in the field is shown in Fig. 1 and its explanation is well represented in a paper reported by Obara and Sugawara (2003).

Fig. 1.The measurement procedure of the compact conical-ended borehole overcoring (CCBO) technique in the field: 1 drilling a borehole of φ 76 mm, 2 creating a conical borehole socket, 3 borehole socket cleaning, 4 gluing the strain cell, 5 compact overcoring (Obara and Sugawara, 2003).

 

Rock volume for rock stress measurement and regional stress

Ljunggren et al. (2003) presented an overview of methods for rock stress measurement. They discussed the rock volume involved each method for rock stress measurement. The classified representative rock volumes from 10-4 m3 to 1012 m3 are summarized in Fig. 2. Rock stress is measured by widely used methods such as jacking methods, core disking, acoustic borehole breakouts, and overcoring. Hydraulic fracturing involves a small rock volume of less than 102 m3 and these stresses are defined as local stresses (Hudson et al., 2003). On the other hand, the tectonic stress estimated by methods such as the earthquake focal mechanism and fault slip analysis involves large rock volume and these stresses are defined as regional stresses (Amadei and Stephansson, 1997). The typical rock volumes in the methods for stress measurement are represented in Fig. 3. The state of stress within the volume/region corresponding to the dimension of rock structures, such as an open pit mine, an underground powerhouse, a cavern for oil storage or nuclear waste disposal and so on, may be also defined as regional stress. Accordingly, the rock volume as the regional stress can be considered to be larger than 106 m3 (Fig. 2).

Fig. 2.Representative rock volume involved in the method for rock stress measurement and regional stress.

Fig. 3.Typical representative rock volumes in the methods for stress measurement.

 

Estimation of regional stress based on rock stress measured at several points

Rock stress measurements were performed by the CCBO in the Kamaishi mine within a granite body (Obara and Sugawara, 2002, 2003) which is located at Kitagami district in Japan. The geology around the mine mainly consists of sedimentary rock and granodiorite/diorite as shown in the plane of Fig. 4. Rock stress measurements are carried out from six stations located at Ga-1, Ga-2 and Ga-3 within the Ganidake granodiorite at 550 m above sea level and at Ku-2, Ku-3 and Ku-4 within the Kurihashi granodiorite at 250 m above sea level. These measurement stations are distributed in a region that is within a rectangular prism with a basement of 4 km long, 3 km wide and 300 m high (volume: 3.6 × 109 m3). The lines DD’ of east to west and FF’ of north to south in the figure illustrate the peak of a mountain. The directions of the maximum horizontal compression (principal stress having larger absolute value) at six stations are almost the same as approximately N-S (Ku-3, Ga-2, Ga-3) or NNW-SSE (Ku-2, Ku-4, Ga-1). Nevertheless, the location of measurement distributes over a wide region. These directions are a remarkable feature in this region. The principal horizontal stresses σH1 and σH2 with depth are shown in Fig. 5. The stresses σH1 and σH2 linearly increase with increasing depth. The gradient against depth of σH2 is greater than that of σH1 and the differential stress σH1-σH2 increases with increasing depth. It is considered that the state of stress is almost uniform, since the distributions of σH1 and σH2 are smooth.

Fig. 4.Magnitude and direction of principal stresses in horizontal plane measured by the CCBO on the geological map of 550 mL and 250 mL near the Kamaishi mine (Obara and Sugawara, 2002, 2003).

Fig. 5.Distribution of the principal stresses with depth in the horizontal planes σH1 and σH2 (Obara and Sugawara, 2002).

The direction of compressive axes in Kitagami district was calculated by a micro-earthquake focal mechanism. Fig. 6 shows the distribution of hypocenters and compressive axes for micro-earthquakes with a magnitude greater than 3.0 at Kitagami district (Faculty of Science, Tohoku University, 1994). The open and closed circles in the figure indicate hypocenters. The directions of the maximum compressive axis on the closed circles range from N30°W to N30°E and the most dominant direction of those on the open circles are mainly oriented to the E-W direction. It is considered that the open circles for the E-W direction might be caused by plate subduction occurring near the boundary between the upper crust and the Pacific plate in the vertical section. The hypocenters are distributed in the upper crust with a depth shallower than 20 km. The micro-earthquakes of the N-S direction occur near the Kamaishi mine.

Fig. 6.Micro-earthquakes at Kitagami district, observed 1964-1975/11, 1983-1986/8 and 1989/11-1993/10; (a) map of epicenters and the horizontal projection of compressive axis direction of magnitude greater than 3.0, (b) projection of epicenters on the vertical cross section (Faculty of Science, Tohoku University, 1994).

The direction of maximum compressive stress obtained by the CCBO and the micro-earthquake focal mechanism are compared in Fig. 7. The data for the micro-earthquake focal mechanism existing within a radius of 30 km around the Kamaishi mine are used in order to compare with those of the CCBO. The mean direction of maximum compressive stress on the micro-earthquake focal mechanism is estimated to be oriented to N11°W. This result is in good agreement with that of N14°W, as measured by the CCBO. Furthermore, the rock volume of regional stress estimated by the local stress is 3.6 × 109 m3 and that estimated by the focal mechanism is 108 ~ 1010 m3. Considering the above results, the regional stress estimated by the local stress of the CCBO and the micro-earthquake focal mechanism concur and the volume is also almost the same. It suggests that both stresses will be able to be applied to estimate a regional stress with a volume greater than 106 m3.

Fig. 7.Comparison of the horizontal maximum compressive direction; (a) estimation by the shallow microearthquake, using the data within a circle of radius 30 km from the Kamaishi mine, (b) the maximum principal direction measured by the CCBO in the Kamaishi mine (Obara and Sugawara, 2003).

 

Estimation of regional stress by the 3-D finite element method

Homogeneous model

The study area for a homogeneous model is Mt. Torigata, which is in the Kochi Prefecture of Shikoku Island, Japan, mainly consists of limestone with 1459 m above sea level. The Torigata open-pit limestone mine is located at the top of Mt. Torigata. The size of this mine is now approximately 2.5 km in the east-west direction and 1.0 km in the north-south direction and its current excavation level ranges from 1205 m above sea level at location No. 1 and 1220 m above sea level at location No. 2 as shown in Fig. 8. The local stress is measured by the CCBO at a horizontal gallery excavated at 960 m above sea level of location No.1 (Obara et al., 2000). A back analysis using data obtained from the CCBO was performed using the 3-D FEM and its procedures are well described by Kang et al. (2003). The 3-D FEM model for the analysis assumes linear and homogeneous measurements. The model domain is 10.1 km long, 4.1 km wide and 3.3 km high, which is an approximately volume of 1.4 × 1011 m3 (Fig. 9). The X-axis and Y-axis are orientated at N25.8°W and N64.2°E respectively, and the Z-axis is vertical. The Y-axis coincides with the orientation of the ridge of Mt. Torigata. The intersection of the lines a-a’ and b-b’ represents the location of No. 1 in Fig.8. The model domain is discretized into 224,128 tetrahedral elements and 187,654 nodes.

Fig. 8.Topographical map of the Torigata mine and measuring station of rock stress by the CCBO (Kang et al., 2003).

Fig. 9.Homogeneous finite element model (Kang et al., 2003).

Fig. 10 shows the magnitudes and directions of the regional stress estimated by a back analysis in the case of Poisson’s ratio ν = 0.25 (Fig. 10 (a)) and the local stress measured by the CCBO (Fig. 10 (b)). In the figure, the regional stress is in a more compressive state than the local stress, and the direction of maximum principal regional stress is rotated to about 1 degree clockwise with respect to maximum principal local stress. It means that the horizontal stress due to gravity is not considered in the analyzed regional stress, but is included in the local stress measured by the CCBO. Fig. 11 illustrates the three-dimensional distribution of the measured and analyzed local stresses on a vertical line passing through a borehole of location No.1. The measured local stresses are plotted at a height of 960 m above sea level. The normal stresses are compressive and their magnitudes increase with depth. On the other hand, the shear stresses are mostly positive and their magnitudes are almost uniform. The measured local stresses in the horizontal plane are in good agreement with the analyzed results. The stresses at the present excavation level are not reduced and are the same as those at 960 m above sea level. It is clear that the ratio of the horizontal to vertical stress components is large near the top of the mountain, and then becomes small.

Fig. 10.Comparison of principal direction; (a) estimation of regional stress by a back analysis in the case of Poisson' ratio 0.25, (b) estimation of local stress by the CCBO (Kang et al., 2003).

Fig. 11.Distribution of the analyzed stresses on the vertical line through the location of borehole No.1 and the measured stresses by the CCBO (large marks) (Kang et al., 2003).

The distribution of stresses is dependent on Poisson’s ratio. That is, the value of the stresses analyzed at 500 m above sea level is less than about 10 MPa in the case where Poisson’s ratio ν = 0.15 and more than 10 MPa in the case where Poisson’s ratio ν = 0.35, although the result of the measured stress is consistent with that of the analyzed stress at 960 m above sea level. This means that it is essential for determining the distribution of rock stress with high accuracy in order to estimate Poisson’s ratio. Furthermore, Poisson’s ratio can be determined more precisely by carrying out a back analysis using the same procedure because the rock stress has been measured at deeper levels (Obara et al., 2000).

Inhomogeneous model

An inhomogeneous finite element model for a back analysis consists of three layers as shown in Fig. 12 (Kawasaki et al., 2003). The objective area for the estimation of regional stress is in the Tono region of Japan. The model domain is 1.0 km long, 1.0 km wide and 0.5 km high, which is a volume of 5.0 × 108 m3. The X-axis and Y-axis are parallel to the east and the north respectively, and the Z-axis is vertical to the zenith. Of the three layers, the upper and middle layers consist of sedimentary rock and the lower layer consists of granite. The model assumes linear and inhomogeneous measurement with Poisson’s ratio 0.3. In addition, Young’s modulus and unit weight of the upper layer are 2 GPa and 18 kN/m3 respectively, of the middle layer are 2 GPa and 19 kN/m3 respectively and of the lower layer are 6 GPa and 26 kN/m3, respectively. The stress measurements by hydraulic fracturing were performed at 47 points in the five boreholes drilled in this region. A back analysis was carried out using all stresses measured in these boreholes.

Fig. 12.Inhomogeneous finite element model for a back analysis (Kawasaki et al., 2003).

Fig. 13 shows the distribution of the measured and analyzed stress components in the horizontal plane along the axis of the TM1 borehole, which was vertically drilled at the center of the model. The distribution of analyzed stresses is continuous in the upper and middle layers, but the discontinuity in stress distribution appears on the boundary between the middle and lower layers. The reason for this may be due to the difference in Young’s modulus of each layer. In the result of the measured normal stresses, the absolute values become large at a depth of about 200 m compared with those at a shallow depth. This tendency corresponds with the result of the analyzed stresses. The distribution of minimum principal stress direction is presented in Fig. 14. The directions of the analyzed minimum principal stress are changed on the boundary of layers as shown in Fig. 14 (a). That is, the analyzed directions are changed from counter-clockwise in the middle layer to clockwise in the lower layer, then converge at 45 degrees of the lower layer. The measured directions differ from those obtained from the finite element analysis, because the analyzed results are based on all rock stresses measured at 47 points. However, the measured directions in the middle layer, indicated by the dotted line in Fig. 14 (b), are similar to the analyzed directions. When we consider the above facts, it may be difficult to sufficiently analyze the regional stress with accuracy.

Fig. 13.Comparison between the analyzed and measured stresses in horizontal plane (opened symbols represent measured results by hydraulic fracturing and closed symbols are analyzed results).

Fig. 14.Distribution of the minimum principal direction from east anti-clockwise; (a) analyzed result, (b) measured result.

Combining these results, the regional stress is not always uniform. That is, even though the volume of rock masses is small, the stress state of various features occurs. Furthermore, when there are a number of different layers within rock masses, the stress distribution occurrs discontinuously near the boundary between layers, while the regional stress is uniform and the stress distribution is homogeneously continuous. Accordingly, it is concluded that the uniformity of regional stress is dependent on the type, condition, formation and extent of the rock mass, the shape of the geological boundary, and the rock volume corresponding to the dimension of rock structures.

 

Conclusions

Demonstrating the case examples of rock stress measurement and back analysis of rock stress, the characteristics of regional stress were discussed. In the estimation in this paper of regional stress using numerical models, the rock body was assumed to be linear. However, the estimation of regional stress would become more complex when geological discontinuities such as faults are considered. Therefore, in order to estimate regional stress with a high reliability, many rock stress measurements should be performed over a wide three-dimensional region, and the regional stress field in the volume corresponding to the dimension of rock structure should then be estimated based on those results.