## Abstract

The main contribution of this paper is to develop a method to determine the *in situ* stress on an engineering scale by modifying the elasto-static thermal stress model (Sheorey’s model). The suggested method, firstly, introduces correction factors for the local tectonism to reflect the stress distribution difference caused by local tectonic movements. The correction factors can be determined by the least-squares approach based on laboratory tests and local *in situ* stress measurements. Then, the rock elastic modulus is replaced by rock mass elastic modulus so as to show the effect of rock discontinuities on the *in situ* stress. Combining with elasticity theory, equations for estimating the major and minor horizontal stresses are obtained. It is possible to reach satisfactory accuracy for stress estimation. To show the feasibility of this method, it is applied to two deep tunnels in China to determine the *in situ* stress. Field tests, including *in situ* stress measurements by conventional hydraulic fracturing (HF) and rock mass modulus measurements using a rigid borehole jack (RBJ), are carried out. It is shown that the stress field in the two deep tunnels is dominated by horizontal tectonic movements. The major and minor horizontal stresses are estimated, respectively. Finally, the results are compared with those derived from the HF method. The calculated results in the two tunnels roughly coincide with the measured results with an average of 15% allowable discrepancy.

## 1 Introduction

Knowledge of the *in situ* stress is a basic requirement for the design and construction of underground projects. Especially for those involving underground excavation projects, an understanding of the initial state of stress, *i.e*., that prior to any excavation or construction, is essential.

Generally speaking, the *in situ* stress values in the three mutually perpendicular directions of the Earth’s crust are unequal. Vertical stress can be obtained easily based on the overburden pressure without causing much error in most instances [1]. However, the horizontal stresses can be affected significantly by plate tectonics, major geological features, topography, *etc*., which are difficult to estimate. Karl and Richart [2] performed many studies on the distribution characteristics of *in situ* stress in sedimentary rocks. Li [3] proposed a method for estimating *in situ* stress in coal and soft rock masses. Brown and Hoek [4] summarised the relationship between horizontal-to-vertical *in situ* stress ratio *k* with depth of cover by analysing a large amount of measured data. Furthermore, González de Vallejo and Hijazo [5] plotted a large dataset of stress magnitudes *versus* depth on a global scale. Sheorey [6] presented an elasto-static thermal stress model of the Earth to estimate crustal stress, but did not consider the main factors (*e.g*., local tectonic movements) affecting the state of stress. Other methods such as the geological (tectonic structure analysis), or seismic (focal mechanisms), can only determine the orientations of principal stresses rather than their magnitude. To reflect the influence of geological and geophysical factors affecting stress magnitude, González de Vallejo and Hijazo [5] proposed a new method for estimating the ratio *k* of the major horizontal stress to vertical stress based on the decision tree probabilistic method and the empirical relationship between the Tectonic Stress Index and *k* values. In addition, the increase of *in situ* stress due to local factors was expressed by the Stress Amplification Factor, which could provide an estimate of structural stresses in rock masses for rock excavations [7, 8]. Although many scholars have undertaken much valuable research into stress estimation, the major and minor horizontal stresses on an engineering-scale remain hard to determine.

Field testing is a direct method used to obtain the orientations and magnitude of *in situ* stresses. In recent decades, *in situ* stress testing equipment and relative methods have made significant progress. In 2003, the International Society for Rock Mechanics (ISRM) published some suggested methods for determining rock stress [9–12]. These methods cover the basic principles of over-coring and hydraulic fracturing/hydraulic testing of preexisting fracture methods. However, conventional methods of measurement are only available for hard rock ( *e.g*.granite, marble, and similar rocks with high strength), which could not be applied to soft or broken rock (*e.g*. such as is found in fault fracture zones). Besides, for some high *in situ* stress regions, or at great depth, core disking renders the over-coring method inapplicable, and reduces the success rate of the hydraulic fracturing method [13].

Here, a method for estimating the major and minor horizontal stresses in an elastic rock mass at engineering-scale, is presented. Unlike the general Sheorey’s model, the improved method takes into account not only the variation of elastic constants, density, and thermal expansion coefficient of the crust and mantle, but also the stress distribution difference caused by local tectonic movements. In addition, the rock mass elastic modulus, which can reflect the effect of rock discontinuities on the *in situ* stress, especially at shallow crustal depths, is adopted to replace the rock elastic modulus. The improved method can be much better applied to local, shallow rock masses rather than at a global, or regional, scale. Finally, the results of the application of this method to the determination of *in situ* stress around two deep tunnels in China are presented.

## 2 Estimation method of *in situ* stress based on Sheorey’s model

### 2.1 Overview of methods

So far, the formation mechanism of crustal stress is not yet clear. As a result, the methods for its estimation always involve some simplifying assumptions. The most common practice is to assume lateral confinement (*i.e*., no horizontal displacement anywhere due to gravitational loading). Based on the mentioned assumption, the following equations are often used to determine the initial state of *in situ* stress in a relatively uniform soil or rock mass beneath a horizontal free surface [14]:

Where, *γ* is the bulk unit weight of the material, is Poisson’s ratio, and *H* is the depth from the free surface. The terms σ_{xx} and σ_{yy} are two horizontal stresses, and σ_{zz} symbolises the vertical stress at depth *H*.

However, a large number of field measurements show that the horizontal stresses commonly do not follow Equation (2), and in many places are several times larger than the vertical stress. So, assumption (2) is likely to be inapplicable in many cases.

The Earth can be treated as a concentric geoid structure: from the surface to the centre, the Earth is divided into crust, mantle, and core. The widely accepted cross-section of the Earth is shown in Figure 1. Displacements should be zero at the mantle-core interface (Gutenberg discontinuity) rather than at the crust-mantle interface (Mohorovičić discontinuity).

Supposing that the Earth’s crust is taken as a solid spherical shell filled with an incompressible liquid, the equilibrium equation is [15]:

Here σ_{r} denotes the radial (vertical) stress, and *σθ* is the tangential (horizontal) stress in polar coordinates (*r*, *θ*). The relationship between σ_{r} (*σ*_{θ}) with the radial displacement *u* can be described as follows [15]:

Where *E* is rock elastic modulus.

To determine crustal stress, Sheorey treated the Earth as a solid spherical shell, and presented an elasto-static thermal stress model (Sheorey’s model) by considering the variation of elastic constants, density and thermal expansion coefficient of the crust and mantle. The simplified spherical shell model of the Earth is shown in Figure 2, which consists of a total of 12 slices. Some parameters of each slice are given in Table 1. Here *β* is linear thermal expansion coefficient and *R*_{i} refers to the inner radius of spherical slice *i*. The calculations are then carried out by putting the parameters in Table 1 into Sheorey’s model, and the *in situ* stress ratio *k* of mean horizontal stress to vertical stress can be obtained [6]:

Location | Slice No. | R (10^{3} km) | E (GPa) | γ (MPa/m) | β(10^{−5}/^{∘}C) |
---|---|---|---|---|---|

mantle | 1 | 3.47 | 760 | 0.052 | 2.4 |

2 | 3.87 | 700 | 0.048 | 1.9 | |

3 | 4.37 | 610 | 0.045 | 1.6 | |

4 | 4.87 | 520 | 0.043 | 1.35 | |

5 | 5.37 | 360 | 0.040 | 1.25 | |

6 | 5.958 | 200 | 0.037 | 1.2 | |

crust | 7 | 6.335 | 20 | 0.027 | 0.77 |

8 | 6.34 | 30 | 0.027 | 0 | |

9 | 6.346 | 40 | 0.027 | 2.2 | |

10 | 6.352 | 45 | 0.027 | 1.5 | |

11 | 6.358 | 50 | 0.027 | 0.9 | |

12 | 6.364 | 50 | 0.027 | 0.6 |

Based on Equation (5), the *k* -depth relationships with different rock modulus are shown in Figure 3. This figure also shows the minimum and maximum envelope lines according to Hoek and Brown (1980) as well as González de Vallejo and Hijazo (2008). It can be seen that the maximum or minimum envelope lines are generally similar, with less data scatter for depths greater than 1000 m. The curve obtained by Sheorey’s model is a good mean fit to the measured data. So, it is reasonable to apply the model to estimate *in situ* stress.

In most cases, Equation (6) provides a more general (though not mathematically exact) form of the *in situ* stress ratio *k* for the Earth’s crust [6]:

Where *G* is thermal gradient, and is linear thermal expansion coefficient.

### 2.2 Comments on Sheorey’s model

Although the *in situ* stress ratio *k* for Sheorey’s model has a tendency to be consistent with the field data, some key problems still exist in Equation (6), which can be listed as follows:

The elastic modulus applied to the model is only for rocks rather than rock masses. As a result, the calculated results will neglect the effect of rock discontinuities (

*e.g*., joints or faults) on the*in situ*stress. In practice, the calculated results are proved to be significantly larger than the measured results in the shallow crust. So, it is necessary to determine the relationship between rock elastic modulus and rock mass elastic modulus.Based on Sheorey’s model, the major and minor horizontal stresses cannot be obtained, respectively. In most cases, we pay more attention to the major horizontal stress and shear stress rather than the mean horizontal stress. Sheorey was also aware of the issue, and conducted further research thereof [16]; however, his work still could not solve the problem perfectly.

The distribution of horizontal stress is quite uneven in the shallow crust due to the great differences in local tectonic movements. Although we can predict the direction of the major horizontal stress caused by geological phenomena (

*e.g*., faults or folds), it is still difficult to determine the influence of the tectonism on the horizontal stress. So, we should consider the effect of local tectonic movements on the*in situ*stress values in engineering applications.

### 2.3 The main factors affecting the distribution of *in situ* stress at engineering-scale

When it comes to the factors affecting the orientation and magnitude of *in situ* stresses, there is the problem of scale. Compared with the continental or global scale, engineering projects rarely occur at more than 2000 m in depth. In that case, local stress variations are too small to be represented on a global scale [17]. The distributions and orientations of stresses on continental and regional scales can be found in the World Stress Map (WSM) [18]. However, the local stress distribution may vary considerably due to heterogeneity and anisotropy of the rock mass as well as geological structures therein. Current research shows that tec-tonic stresses, caused by a pervasive force field imposed by active tectonics or past tectonic events, are the main causes of stress in the lithosphere. Zang and Stephansson [19] summarised the influence of tectonic stresses on the rock engineering and divided it into three levels: the first-order is global patterns of tectonic stress due to the relative displacement of plates or isostasy; the second-order is mountain scale tectonic stress, which can vary significantly over short distances; and the third-order is fault-scale stress. Faults are one of the main tectonic structures which can affect stress magnitude and direction [20]. So, different order tectonic stresses are scaled based on their coherent domain over the region in which a stress component can be supposed to be uniform.

Generally speaking, the main factors affecting stress distribution on an engineering scale may include geological and structural anisotropies, sedimentary loads, relief effects, glacial rebound, loads produced by submarine elevations (or the convexity of the oceanic lithosphere), rock composition, and geomechanical behaviour [5]. The magnitude of the *in situ* stress turns out to be non-uniform owing to the local influence of discontinuities, faults, dikes, heterogeneities, intrusive bodies, and folds [21]. Lithological heterogeneities and structural anisotropies may also lead to stress concentrations. The process of erosion and denudation may cause high horizontal stresses, especially for deep-cut river valleys. The geological history and behavioural evolution of rocks may exert a significant influence on the long-term state of stress. In addition, as affected by the disturbance of excavations, the orientation and magnitude of *in situ* stress in the vicinity of the excavation section vary considerably.

## 3 Improved Sheorey’s method

For a uniform soil or rock mass, Equation (6) can be expressed as:

Where

In Equation (7), the relationship between the *in situ* stress ratio *k* and depth is inversely proportional. For a certain depth, the value of *k* is mainly dependent on the two parameters *a* and *b*. Moreover, when the buried depth changes slightly in the shallow crust, the thermal expansion coefficient *β*, and thermal gradient *G*, can be treated as constants. Then, the *in situ* stress ratio *k* is taken to consist of a constant term and a hyperbolic term. In addition, the two parameters *a* and *b* are only dependent on rock properties, which reflect the intensity of tectonic movements. However, tectonic stress (including magnitude and direction) often changes significantly in different parts of the study area. As a result, it is not easy to determine the parameters *a* and *b* accurately.

To reflect the influence of local tectonic movements on the *in situ* stress, we introduce the local tectonic correction factors *ξ* and *η*. The following equation provides a more common form of the *in situ* stress ratio *k* for the Earth’s crust:

Where *E*_{rm} stands for rock mass elastic modulus. The two parameters *ξ* and *η* are local tectonic correction factors which will be determined by the least-squares approach based on local stress measurements and laboratory tests. For a certain project, the parameters *β*, *G*, *γ*, *ν* and *E*_{rm} in Equation (8) can be easily determined based on laboratory tests and empirical formulas. Then, the *in situ* stress ratio *k* in Equation (8) can be taken as the function of *ξ* and *η*. Furthermore, the least-squares fitting analysis is carried out to determine the initial values of *ξ* and *η* based on local stress measurements. If some stress values deviate considerably, they should be neglected so as to obtain the optimum tectonic correction factors. It is noted that if the study area suffers the same tectonic movements, the parameters *ξ* and *η* can be treated as constants.

When the rock mass elastic modulus is unknown, we can indirectly determine it based on laboratory tests and rock quality information. Guo, *et al*. [22] provided the following relationship between the factors *E*_{rm}/*E*_{i} and *RMDI* through analysing many measurements:

Where *E*_{i} is the intact rock elastic modulus, and *RMDI* is a rock mass integrity index, which can be determined by using Equations (10) and (11).

The indicator *RMDI* can be defined as a ratio of intact rock to entire rock mass in certain range. It can reflect (quantitatively) the effect of discontinuities (*e.g*., joints or faults) on rock integrity based on drilling images. Drilling images, from the use of a borehole camera, is often used to obtain the structural characteristics of fractured rock. By using such imaging with acoustic and optical televiewers, continuous and oriented 360^{∘} views of the borehole wall are available, from which the character, inter-relationship, and orientations of lithologic and structural planar features can be easily defined [23, 24]. For a given depth range [*h*_{1}, *h*_{2}], the *RMDI* can be expressed as [25]:

Here, *z* is the depth. *f*(*z*) is an integrity index density function of the rock mass which can be determined by Equation (11):

Where, *δ* is an impact factor for rock fragmentation which depends on filling type and rock mass size. For filling conditions, taking clay filling as an example, *δ* is taken as zero. Otherwise, the value of *δ* is dependent on rock mass size. Based on the published literature [25], as the rock mass changes from intact, massive, bedded, fractured, to non-compact states, the value of *δ* changes from 1.0, 0.8, 0.5, 0.2, to 0.1, respectively.

In addition, studies [26, 27] show that horizontal stress mainly consists of two parts: one is the horizontal component of gravitational stress, and the other is the horizontal tectonic stress. When the stress field is mainly caused by gravity, horizontal stresses can be estimated by Equation (2). However, as the horizontal tectonic movement predominates, the following equations provide the basic relationship for elastic rock behaviour, with the constraints that one of the principal stresses is vertical and varies linearly with depth [3]:

Where σ_{H} and σ_{h} are the major and minor horizontal stresses, respectively, σ_{T} is the horizontal tectonic stress, and *λ* is the lateral pressure coefficient due to gravitational loading with *λ* = ν/(1−ν).

As mentioned before, the ratio *k* = (σ_{H} + σ_{h})/(2σ_{V}). Then, Equation (8) can be rewritten as:

In Equation (12), both σ_{H} and σ_{h} have the common part in σ_{T}. Then, we can get the relationship between σ_{H} and σ_{h} by removing σ_{T}.

Based on Equations (13) and (14), we can obtain:

According to Equation (15), the *in situ* stress at an arbitrary depth can be estimated. Figure 4 illustrates the flowchart of the improved method.

To verify the reliability of this method in estimation of *in situ* stress, it is necessary to compare the calculated results with field measurements.

In stage 5, we define

Where σ_{ci} and σ_{mi} are the calculated and measured stresses at point *i*, respectively. According to Equation (16), if α ≤ 0.2, the process is terminated and we will have the equations for estimating *in situ* stresses at an arbitrary point; otherwise, it is used to recalculate the parameters *ξ* and η by Equation (8).

However, if the precision is not satisfied after 10 times, the user should decrease the accuracy of the estimation (*i.e*., increase the value of α). Also, another solution is to increase the number of *in situ* stress measurements.

It is noted that if at a location the *in situ* stresses are measured in competent parts of the same formation, then the corresponding stresses in an adjacent location can be estimated by the relationship:

Where the suffix 1 denotes the hard (or soft) rock in which stress measurements have been carried out: suffix 2 refers to the hard (or soft) rock at an adjacent location where the *in situ* stresses are to be estimated.

## 4 Engineering application – a case study

To show the feasibility of this improved method, it is applied to Meihuashan Tunnel to determine the *in situ* stresses around it.

### 4.1 Project overview

Meihuashan Tunnel is located in Longyan City, Fujian Province, southeast of China. It is 15,780 m in length with a maximum depth of 688.21 m. The location of the tunnel is shown in Figure 5(a).

According to tectonics, the tunnel lies in the depression belt of southwest of Fujian. Figure 5(b) shows the geological cross-section along the longitudinal axis of the tunnel. There are seven faults exposed along the tunnel. In addition, NE-NEE and NNE trending faults are well developed, and control the distribution of strata and intrusive bodies in this area. The rocks around the tunnel are sandstone, siltstone of the Permian Longtan formation, early Jurassic Yanshanian granite, and Devonian quartz sandstone. At the test site, the rock is mainly moderate weathered granite with a medium to coarse-grained structure.

### 4.2 Field tests

Within the project area, the *in situ* stress and rock mass elastic modulus measurements were carried out. The implementation process of the field tests is described as follows.

#### 4.2.1 *In situ* stress test

*In situ* stress testing by the hydraulic fracturing method was carried out. The test site in Figure 5(b) is located in the tunnel area, and two testing boreholes have been placed. One is the horizontal with a maximum depth of 33.9 m. The other is the vertical with its depth ranging from 3.9 m to 41.9 m. The *in situ* stress measurements in the horizontal and vertical borehole are shown in Tables 2 and 3, respectively. Also, Figure 6 shows the variation of the *in situ* stress measurements with borehole depth.

Based on Table 2 and Figure 6(a), we can conclude that the maximum principal stress is horizontal. When the horizontal borehole depth exceeds 21.9 m, the minimum horizontal stress approximately equals the vertical stress. In addition, by neglecting the effect of excavation on the *in situ* stress in the vertical borehole (Figure 6(b)), the stresses were such that σ_{H} > σ_{V} > σ_{h}. That is, the stress field at the test site is dominated by horizontal tec-tonic movements.

Measuring point No. | Borehole depth/m | σ_{1} (MPa) | σ_{3} (MPa) | σ_{V} (MPa) |
---|---|---|---|---|

1 | 1.9 | 7.33 | 5.71 | 11.41 |

2 | 3.9 | 13.75 | 7.46 | 11.41 |

3 | 5.9 | 18.18 | 10.2 | 11.41 |

4 | 7.9 | 16.8 | 10.45 | 11.41 |

5 | 9.9 | 21.08 | 11.89 | 11.41 |

6 | 11.9 | 15.57 | 9.63 | 11.41 |

7 | 13.9 | 12.71 | 8.4 | 11.41 |

8 | 17.9 | 15.13 | 9.8 | 11.41 |

9 | 21.9 | 16.45 | 11.24 | 11.41 |

10 | 25.9 | 15.42 | 11.40 | 11.41 |

11 | 29.90 | 16.26 | 11.03 | 11.41 |

12 | 33.90 | 15.71 | 11.41 | 11.41 |

Note: rock mass unit weight is 26 kN/m^{3}, the below is same.

#### 4.2.2 Rock mass elastic modulus test

Rock mass elastic modulus tests in the vertical borehole were carried out using a rigid borehole jack (type of BJ-91A) [28]. The rigid borehole jack was developed by Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, and has been widely used in many projects. Figure 7 shows a typical pressure *versus* deformation history recorded at a depth of 19.2 m.

Based on the measured data, the rock mass elastic modulus at the measurement points can be obtained by Equation (18).

Where *E*_{rm} is the measured rock mass elastic modulus, *K* is the influence coefficient under three-dimensional test conditions with *K* = 0.93, *B* is the pressure modified coefficient with *B* = 0.98, and *D* is the borehole diameter. The parameters *ΔP* and *ΔD* refer to the incremental pressure and deformation, respectively and *T* (ν, ψ) is a coefficient related to bearing plate arc and rock Poisson’s ratio *ν* with *T* (ν, ψ) = 2.547.

The rock mass elastic modulus measurements in the vertical borehole depth are shown in Table 4. Comparing Table 3 with Table 4, we find that the depth of the *in situ* stress at the point of measurement is inconsistent with that of the measured rock mass elastic modulus. To obtain the rock mass elastic modulus at the measuring points for *in situ* stress, we should calculate the rock mass elastic modulus at the required depth based on the measured data. As the depth of the adjacent measuring points changes by no more than 3 m, a two-dimensional linear interpolation method can be used to obtain *E*_{cm} where desired. Thereafter, the rock mass elastic modulus *E*_{cm} at the measuring points for *in situ* stresses is obtained (Table 4).

Measuring point No. | Borehole depth/m | σ_{H} (MPa) | σ_{h} (MPa) | σ_{V} (MPa) | Azimuth σ_{H} degree |
---|---|---|---|---|---|

1 | 3.9 | 3.48 | 2.46 | 11.54 | |

2 | 5.9 | 7.91 | 6.27 | 11.59 | |

3 | 7.9 | 20.48 | 11.78 | 11.65 | |

4 | 9.9 | 24.98 | 12.87 | 11.70 | |

5 | 11.9 | 22.75 | 12.02 | 11.75 | |

6 | 13.9 | 13.63 | 8.93 | 11.80 | |

7 | 15.9 | 19.21 | 10.31 | 11.85 | |

8 | 17.9 | 16.27 | 8.81 | 11.91 | |

9 | 19.9 | 16.87 | 9.08 | 11.96 | |

10 | 21.9 | 17.71 | 9.22 | 12.01 | |

11 | 23.9 | 17.90 | 9.45 | 12.06 | |

12 | 25.9 | 18.11 | 9.77 | 12.11 | 291 |

13 | 29.9 | 19.34 | 10.07 | 12.22 | |

14 | 31.9 | 19.90 | 11.58 | 12.27 | |

15 | 35.9 | 20.76 | 11.67 | 12.37 | 295 |

16 | 37.9 | 21.29 | 11.15 | 12.43 | 292 |

17 | 39.9 | 22.62 | 12.44 | 12.48 | 294 |

18 | 41.9 | 25.39 | 13.73 | 12.53 | 293 |

Note: the overburden depth at the test site is about 440 m.

Measuring | Measured results | Calculated results | ||
---|---|---|---|---|

point No. | Borehole depth (m) | E_{rm} (GPa) | Borehole depth (m) | E_{cm} (GPa) |

1 | 3.2 | 29.54 | 3.9 | 38.53 |

2 | 4.3 | 43.66 | 5.9 | 30.15 |

3 | 5.2 | 31.57 | 7.9 | 37.92 |

4 | 6.3 | 29.34 | 9.9 | 58.68 |

5 | 7.2 | 28.08 | 11.9 | 42.95 |

6 | 8.3 | 43.55 | 13.9 | 55.07 |

7 | 9.2 | 43.14 | 15.9 | 64.45 |

8 | 10.3 | 67.57 | 17.9 | 52.52 |

9 | 11.2 | 45.01 | 19.9 | 43.21 |

10 | 12.3 | 41.77 | 21.9 | 46.12 |

11 | 13.2 | 53.92 | 23.9 | 51.59 |

12 | 14.3 | 55.73 | 25.9 | 49.9 |

13 | 15.2 | 63.07 | 29.9 | 43.94 |

14 | 16.3 | 65.24 | 31.9 | 53.43 |

15 | 19.2 | 42.19 | 35.9 | 55.46 |

16 | 22.3 | 46.7 | 37.9 | 62.55 |

17 | 25.2 | 55.56 | 39.9 | 64.48 |

18 | 28.3 | 30.48 | 41.9 | 66.40 |

19 | 31.2 | 54.87 | ||

20 | 34.3 | 48.5 | ||

21 | 37.2 | 61.11 |

### 4.3 *In situ* stress determination based on the improved method

#### 4.3.1 Selection of the measured results

Due to the excavation of the tunnel, the stress field in the region of the horizontal borehole depth can be divided into three parts, as shown in Figure 8(a). When the borehole depth is less than 5.9 m, the *in situ* stress measurements become smaller, and we can take that region as the stress relief zone. As the borehole depth ranges from 5.9 m to 11.9 m, the *in situ* stress measurements appear to be larger, and we can take it as stress concentration zone. When the bore-hole depth exceeds 11.9 m, the *in situ* stress measurements change slightly, and we take the region as a stress stabilised zone. So, the depth of the stress disturbance zone affected by the excavation of the tunnel is about 11.9 m according to the horizontal borehole.

Similarly, the distribution characteristics of the *in situ* stresses along the vertical borehole can be analysed. Also, the stress field in the region of the vertical borehole depth can be divided into stress relief zone, stress concentration zone, and stress stabilised zone (Figure 8(b)). Unlike the horizontal borehole, the *in situ* stresses in the stress stabilised zone increased quasi-linearly with increasing bore-hole depth. In addition, the depth of the stress disturbance zone, as affected by the excavation of the tunnel, is about 17.9 m according to the vertical borehole. When the depth is less than 17.9 m, it is actually a secondary stress field, which cannot reflect the initial stress state at the test site. Therefore, the measurements from points 1 to 7 should be removed to avoid their undue influence of the accuracy of the *in situ* stress determination.

#### 4.3.2 Determination of the parameters

To quantify the rock mechanical properties of granite, uniaxial, and triaxial, compression tests are carried out. Based on the test data, *v* = 0.25 and *r* = 0.026 MPa/m. The coefficient of linear thermal expansion is not widely reported, but Table 5 provides the values for some rocks available from published literature [29, 30]. From Table 5, we can take the thermal expansion coefficient for granite as 6.0 × 10^{−6}/^{∘}C. For crustal rocks, the thermal gradient *G* is taken as 0.024^{∘}C [16]. In addition, through collecting a large number of *in situ* stress measurements under the same tectonism, we used the least-squares approach to determine the two constants *ξ* and *η* in Equation (8), and the distribution of error can thus be controlled. The best approximate solution for this over-determined problem was found when *ξ* = 0.85 and *η* = 0.11 [31].

#### 4.3.3 Estimation of *in situ* stress

As described before, *in situ stress* at eleven measuring points are estimated based on Equation (15). To begin with, the relationship between *ξ* and *η* are considered equal to 0.85 and 0.11, respectively, as initial values. The first trial results show that *α* = 0.275 which was greater than 0.2. Therefore, *ξ* and *η* should be modified by removing some measured data which deviate considerably from the mean. When the new values of *ξ* and *η* are equal to 0.72 and 0.08, *α* is decreased to 0.149 (Fig. 9). Then, the precision requirement for estimating the *in situ* stress in the project area is satisfied. Table 6 shows the comparison of measured *in situ* stresses by HF with the finally calculated values by the improved method in Meihuashan Tunnel. This comparison shows the acceptability of the results of the improved method with regard to the estimation of the *in situ* stresses.

Measuring | Borehole | E_{cm} | Measured results | Calculated results | Mean horizontal stress | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

point | depth | (GPa) | (MPa) | (MPa) | (MPa) | ||||||

No. | (m) | σ_{H} | σ_{V} | σ_{h} | σ_{H} | σ_{h} | Measured | Sheorey’s | In the | ||

model | paper | ||||||||||

8 | 17.9 | 52.52 | 16.27 | 11.91 | 8.81 | 19.97 | 9.30 | 12.54 | 18.67 | 14.64 | |

9 | 19.9 | 43.21 | 16.87 | 11.96 | 9.08 | 17.19 | 8.39 | 12.98 | 16.10 | 12.79 | |

10 | 21.9 | 46.12 | 17.71 | 12.01 | 9.22 | 18.10 | 8.70 | 13.47 | 16.95 | 13.40 | |

11 | 23.9 | 51.59 | 17.90 | 12.06 | 9.45 | 19.80 | 9.28 | 13.68 | 18.52 | 14.54 | |

12 | 25.9 | 49.9 | 18.11 | 12.11 | 9.77 | 19.32 | 9.13 | 13.94 | 18.08 | 14.23 | |

13 | 29.9 | 43.94 | 19.34 | 12.22 | 10.07 | 17.59 | 8.58 | 14.71 | 13.56 | 13.09 | |

14 | 31.9 | 53.43 | 19.90 | 12.27 | 11.58 | 20.52 | 9.57 | 15.74 | 16.47 | 15.05 | |

15 | 35.9 | 55.46 | 20.76 | 12.37 | 11.67 | 21.22 | 9.82 | 16.22 | 19.19 | 15.52 | |

16 | 37.9 | 62.55 | 21.29 | 12.43 | 11.15 | 23.43 | 10.57 | 16.22 | 18.06 | 17.00 | |

17 | 39.9 | 64.48 | 22.62 | 12.48 | 12.44 | 24.07 | 10.80 | 17.53 | 19.84 | 17.44 | |

18 | 41.9 | 66.40 | 25.39 | 12.53 | 13.73 | 24.70 | 11.02 | 19.56 | 21.89 | 17.86 |

Furthermore, the comparison of the mean measured horizontal stress and that calculated by Sheorey’s model, as well as those found using the improved method, is shown in Figure 10. It can be seen that the *in situ* stresses estimated by the improved method are closer to the measured results than those found by Sheorey’s model. The calculated results using Sheorey’s model differ significantly from the measured results on the whole.

## 5 Case study

As an example of the use of the improved method, another case study to determine the *in situ* stresses for the Qinling water-conveyance tunnel project in China was performed (Fig. 11). The tunnel is 81.779 km long, and its maximum overburden depth is 2004 m. It lies in the west of Qinling Mountain, Shaanxi Province, northwest of China. The topography is controlled mainly by tectonics, and the faults are well-developed. The rocks around the tunnel are formed in the Cretaceous, Carboniferous, and Devonian systems, and are predominantly: granite, quartz schist, gneiss, granodiorite, sandstone, phyllite, and micaceous schist.

*In situ* stress testing was undertaken by the hydraulic fracturing method, and rock mass elastic modulus testing was carried out using a rigid borehole jack [32]. Boreholes #3 and #6 were arranged in the tunnel site (Fig. 11). At the test site, the rock is mainly moderate weathered granite based on borehole #6. The rock mass is relatively intact. The overburden depth at the test site is 760 m. Two boreholes were opened: one is the horizontal borehole with a maximum depth of 20.5 m, the other is the vertical borehole with a depth ranging from 7.5 m to 19.5 m. For the horizontal borehole, as the horizontal depth exceeds 9.5 m, the minimum principal stress remained almost stable, ranging from 12.8 MPa to 13.0 MPa. The *in situ* stress and rock mass elastic modulus measurements in the vertical borehole #6 are shown in Table 7. Similarly, the rock is mainly moderate weathered gneiss as evinced by arising from borehole #3. The rock mass is intact and hard. The overburden depth at the test site is 1000 m. As the horizontal depth exceeds 11.0 m, the minimum principal stress remained almost stable with an average value of 14.0 MPa. The *in situ* stress and rock mass elastic modulus measured in the vertical borehole (#3) are shown in Table 8.

Measuring | Borehole | E(GPa)_{rm} | σ(MPa)_{H} | σ(MPa)_{h} | σ(MPa)_{V} | Azimuth σ_{H} |
---|---|---|---|---|---|---|

point No. | depth/m | degree | ||||

1 | 7.5 | 21.1 | 14.0 | 11.9 | 20.7 | |

2 | 10.5 | 25.3 | 19.4 | 12.8 | 20.8 | |

3 | 13.5 | 45.1 | 23.4 | 14.1 | 20.9 | |

4 | 16.5 | 40.5 | 26.3 | 14.9 | 21.0 | |

5 | 19.5 | 47.9 | 27.4 | 15.2 | 21.0 | 276 |

Note: the overburden depth at the test site is about 760 m.

Measuring | Borehole | E(GPa)_{rm} | σ(MPa)_{H} | σ(MPa)_{h} | σ(MPa)_{V} | Azimuth σ_{H} |
---|---|---|---|---|---|---|

point No. | depth/m | degree | ||||

1 | 8.5 | 43.8 | 17.2 | 12.6 | 27.2 | |

2 | 11.5 | 63.2 | 23.9 | 13.7 | 27.3 | |

3 | 14.5 | 59.6 | 26.6 | 14.4 | 27.4 | |

4 | 17.5 | 62.0 | 31.2 | 17.2 | 27.5 | |

5 | 20.5 | 53.6 | 31.4 | 17.8 | 27.6 | 286 |

Note: the overburden depth at the test site is about 1000 m.

For the purpose of evaluating the applicability of the improved method, the measured data in borehole #6 are used to determine the local tectonic correction factors *ξ* and *η*, and the measurements in borehole #3 are mainly used for validation purposes. For borehole #6, we can get *ν* ;= 0.25, *r* = 0.027 MPa/m, and *β* = 7.3 × 10^{−}6/^{∘}C from laboratory tests [32]. The thermal gradient *G* is taken as 0.024^{∘}C/m. In addition, the measured *in situ* stresses are analysed by the least-squares procedure. The best approximate solution for this over-determined problem was found when *ξ* = 0.64 and *η* = 0.18. The parameter α was 0.146, which satisfied the precision requirements for stress determination. According to tectonics, the tunnel belongs to the same tectonic background; the stress state for the elastic rock in the tunnel can then be estimated by use of Equation (19):

In borehole #3:*ν* = 0.28, *r* = 0.027 MPa/m, *β* = 5.1 × 10^{−6}/^{∘}C based on laboratory tests [32]. The thermal gradient *G* was 0.024^{∘}C/m. These parameters are substituted into Equation (19), and the resulting maximum and minimum horizontal stresses are listed in Table 9. The parameter was 0.136, which indicated that the precision for stress determination was satisfactory. The comparison of the mean measured horizontal stresses, and that calculated based on Sheorey’s model, and the improved method are plotted in Figure 12. We can conclude that the *in situ* stresses estimated by the improved method are closer to those measured than those estimated by the use of Sheorey’s model. Therefore, the improved method in this paper is practical with regard to the estimation of the *in situ* stress state.

Measuring point No. | Borehole depth (m) | E_{cm} (GPa) | Measured results (MPa) | Calculated results (MPa) | Mean horizontal stress (MPa) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

σ_{H} | σ_{V} | σ_{h} | σ_{H} | σ_{h} | Measured | Sheorey’s model | In the paper | ||||

2 | 11.5 | 63.2 | 23.9 | 27.3 | 13.7 | 27.20 | 15.15 | 18.8 | 27.68 | 21.17 | |

3 | 14.5 | 59.6 | 26.6 | 27.4 | 14.4 | 27.38 | 15.23 | 20.5 | 27.87 | 21.31 | |

4 | 17.5 | 62.0 | 31.2 | 27.5 | 17.2 | 29.40 | 15.93 | 24.2 | 29.98 | 22.66 | |

5 | 20.5 | 53.6 | 31.4 | 27.6 | 17.8 | 29.91 | 16.12 | 24.6 | 30.50 | 23.01 |

## 6 Conclusions

An improved method for estimating the magnitude of horizontal stresses on an engineering scale based on Sheorey’s model is presented. It considers not only the variation of elastic constants, density, and thermal expansion coefficient of the crust and mantle, but also the stress distribution difference caused by local tectonic movements. So, the general Sheorey’s model, applicable at global or regional scale could be, with this modification, much better applied to local, shallow rock masses, where local tec-tonic movements can be recognised and measured. Besides, the effect of rock discontinuities on the *in situ* stress is taken into account by replacing the rock elastic modulus with the rock mass elastic modulus. Combining this with the theory of elasticity, equations for estimating horizontal stress are obtained. It is visible that the improved method does not obey the principal of Sheorey’s model. On an engineering scale, the improved method can also be used to estimate the *in situ* stress state for elastic rock masses in intraplate regions with hard and more intact rock. The major and minor horizontal stresses can be determined. In addition, the proposed method can also be used to determine *in situ* stress in some places where field tests are hard to carry out.

The suggested method is applied to two deep tunnels in China and the results show that the calculated *in situ* stresses match those measured. Also, it is shown if the local tectonic correction factors are adequate, the stress state will be calculated with the average accuracy of 15% in an unknown location. Moreover, it is possible to achieve an estimated *in situ* stress with the desired error by changing the provisions in the method.

Although we have tried to decrease the error as much as possible, the difference between the measured *in situ* stress and the calculated results remained. On one hand, the rock mass has all the characteristics of inhomogeneity and discontinuity, while the improved method used for *in situ* stress determination is based on the theory of elasticity and cannot reflect the non-linear mechanical behaviour of the rock mass. On the other hand, major geological features will also influence the *in situ* stress measurements. So, it is only a preliminary attempt to estimate the *in situ* stress by using Equation (15) and how to improve the accuracy of the equation remains a topic worthy of future research.

## Acknowledgement

The authors gratefully acknowledge the financial support of the National Science Foundation of China (Nos. 51539002, 51609018, 51379022) and the National key research and development project of China (Nos. 2016YFC0401802, 2016YFC0401804).

## References

[1] Hoek E., Brown E. T., Underground Excavations in Rock. The Institution of Mining and Metallurgy, London, 1980, 93–10110.1201/9781482288926Search in Google Scholar

[2] Karl T., Richart F. E., Stresses in rock about cavities. Geotechnique, 1952, 3(2), 57–9010.1680/geot.1952.3.2.57Search in Google Scholar

[3] Li W. P., A preliminary estimation method of geostresses in coal and soft rock masses. Chin. J. Rock Mech. Eng., 2000, 19(2), 234–237 (In Chinese with English abstract)Search in Google Scholar

[4] Brown E. T., Hoek E., Technical note trends in relationships between measured in–situ stress and depth. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr., 1978, 15(4), 211–21510.1016/0148-9062(78)91227-5Search in Google Scholar

[5] Hijazo, T., González de Vallejo, L.I. In-situ stress amplification due to geological factors in tunnels: The case of Pajares tunnels, Spain[J]. Engineering geology, 2012, 137–138: 13–20.10.1016/j.enggeo.2012.03.007Search in Google Scholar

[6] Sheorey P. R., A Theory for In–Situ Stress in Isotropic and Transversely Isotropic Rock. Int. J. Rock Mech. Min. Sci. and Geomech. Abstr., 1994, 31(1), 23–3410.1016/0148-9062(94)92312-4Search in Google Scholar

[7] González de Vallejo, L.I., Hijazo, T. A new method of estimating the ratio between in situ rock stresses and tectonics based on empirical and probabilistic analyses[J]. Engineering geology, 2008, 101(3): 185–19410.1016/j.enggeo.2008.05.003Search in Google Scholar

[8] González de Vallejo, L.I. Hijazo,T. Galera, J.M. Entezari, Z.A. (2014). In–situ stress amplification in tunnels from Spain, Iran and Chile estimated by TSI and SAF indices. In: Rock Engineering and Rock Mechanics: Structures in and on Rock Masses, 2014, 83: 523–528.Search in Google Scholar

[9] Hudson J. A., Cornet F. H., Christiansson R., ISRM suggested methods for rock stress estimation – part 1: strategy for rock stress estimation. Int. J. Rock Mech. Min. Sci., 2003, 40(7), 991– 99810.1016/j.ijrmms.2003.07.011Search in Google Scholar

[10] Sjöberg J., Christiansson R., Hudson J. A., ISRM suggested methods for rock stress estimation – Part 2: overcoring meth- ods. Int. J. Rock Mech. Min. Sci., 2003, 40(7), 999–101010.1016/j.ijrmms.2003.07.012Search in Google Scholar

[11] Haimson B. C., Cornet F. H., ISRM suggested methods for rock stress estimation – part 3: hydraulic fracturing (HF) and/or hydraulic testing of pre–existing fractures (HTPF). Int. J. Rock Mech. Min. Sci., 2003, 40(7), 1011–102010.1016/j.ijrmms.2003.08.002Search in Google Scholar

[12] Christiansson R., Hudson J. A., ISRM suggested methods for rock stress estimation – part 4: quality control of rock stress estimation. Int. J. Rock Mech. Min. Sci., 2003, 40(7), 1021–102510.1016/j.ijrmms.2003.07.004Search in Google Scholar

[13] Ljunggren C., Chang Y., Janson T., Christiansson R., An overview of rock stress measurement methods. Int. J. Rock Mech. Min. Sci., 2003, 40(7), 975–98910.1016/j.ijrmms.2003.07.003Search in Google Scholar

[14] Amadei B., Stephenson O., Rock Mechanics for Underground Mining. Chapman & Hall, London, 1993Search in Google Scholar

[15] Timoshenko S., Goodier J. N., Theory of Elasticity, Kogakusha, Tokyo, 1951Search in Google Scholar

[16] Sheorey P. R., Mohan G. M., Sinha A., Influence of Elastic Constants on the Horizontal In–situ Stress. Int. J. Rock Mech. Min. Sci., 2001, 38(8), 1211–121610.1016/S1365-1609(01)00069-7Search in Google Scholar

[17] Hoek, E., Marinos, P.G. Tunnelling in overstressed rock. In: Rock Engineering in Diflcult Ground Conditions – Soft Rocks and Karst. Vrkljan, I (ed). London: Taylor and Francis Group, 2009, 49–60.Search in Google Scholar

[18] Zoback M. L., Zoback M. D., Adams J., et al. Global patterns of tectonic stress[J]. Nature, 1989, 341(6240), 291–298.Search in Google Scholar

[19] Zang, A., Stephansson, O. Stress field of the Earth’s crust[M]. Springer, 2010.10.1007/978-1-4020-8444-7Search in Google Scholar

[20] Zoback M. D., Tsukahara H., Hickman S. Stress measurements at depth in the vicinity of the San Andreas fault: implications for the magnitude of shear stress at depth. J. Geophys. Res. 1980, 85(B11), 6157–6173.10.1029/JB085iB11p06157Search in Google Scholar

[21] Stephansson O. Rock stress in the Fennoscandian shield [J]. Comprehensive rock engineering – rock testing and site characterization, 1993, 3, 445–459.10.1016/B978-0-08-042066-0.50024-0Search in Google Scholar

[22] Guo Q., Ge X. R., Che A. L., Research on relationship of rock mass integrity index and rock mass elastic modulus. Chin. J. Rock Mech. Eng., 2011, 30(S2), 3914–3919 (In Chinese with English abstract)Search in Google Scholar

[23] Williams J H, Johnson C D. Acoustic and optical borehole–wall imaging for fractured–rock aquifer studies [J]. Journal of Applied Geophysics, 2004, 55(1): 151–159.10.1016/j.jappgeo.2003.06.009Search in Google Scholar

[24] Wang C. Y., Ge X. R., Bai S. W. The digital panoramic borehole camera system and its application [J]. Rock and soil Mechanics, 2001, 22(4): 522–525. (In Chinese with English abstract)Search in Google Scholar

[25] Wang C. Y., Hu P. L., Sun W. C., Method for evaluating rock mass integrity based on borehole camera technology. Rock Soil Mech., 2010, 31(04), 1326–1330 (In Chinese with English abstract)Search in Google Scholar

[26] CAI Meifeng. Principle and techniques of in–situ stress measurement [M]. Beijing: Science Press, 2000.Search in Google Scholar

[27] Hu B., Feng X.T., Huang X. H., et al. Regression analysis of initial geo–stress field for left bank high slope region at Longtan hydropower station[J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 22(11): 4055–4064. (In Chinese with English abstract)Search in Google Scholar

[28] Luo C. W., Li H. B., Ma P., Application of borehole jack in deformation modulus measurement of rock mass in the foundation of Baihetan hydropower station. Journal of Yangtze River Scientific Research Institute, 2012, 29(8), 76–82 (In Chinese with English abstract)Search in Google Scholar

[29] Haas C. J., Static stress–strain relationships. Physical properties of rocks and minerals (Touloukian, YS, Judd, WR & Roy, RF, eds.), Hemisphere Publishing Corporation, New York, 1989, 123–176Search in Google Scholar

[30] Schäfer K., In–situ strain measurements in Libya//Tectonic Stresses in the Alpine–Mediterranean Region. Springer Vienna, 1980, 49–6110.1007/978-3-7091-8588-9_7Search in Google Scholar

[31] Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Rockburst and Large Deformation in Meihuashan Tunnel of GL–4 Tenders of Ganlong Railway. Wuhan, Report No. IRSM– 2012–122, 2012Search in Google Scholar

[32] Yangtze River Scientific Research Institute. The in–situ stress and borehole deformation tests of Qinling Tunnel of Hanjiang to Weihe River Project in Shaanxi Province. Wuhan, Report No. 2011099YT, 2011Search in Google Scholar

**Received:**2016-1-20

**Accepted:**2016-6-30

**Published Online:**2016-10-15

**Published in Print:**2016-10-1

© 2016 Q. Pei *et al*., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.