An Improved Method for Estimating in Situ Stress in an Elastic Rock Mass and Its Engineering Application

Open Geosci. 2016; 8:523–537

Research Article Open Access

Qitao Pei*, Xiuli Ding, Bo Lu, Yuting Zhang, Shuling Huang, and Zhihong Dong An improved method for estimating in situ stress in an elastic rock mass and its engineering application

DOI 10.1515/geo-2016-0047 Received Jan 20, 2016; accepted Jun 30,2016 1 Introduction

Abstract: The main contribution of this paper is to develop Knowledge of the in situ stress is a basic requirement a method to determine the in situ stress on an engineering for the design and construction of underground projects. scale by modifying the elasto-static thermal stress model Especially for those involving underground excavation (Sheorey’s model). The suggested method, firstly, intro- projects, an understanding of the initial state of stress, i.e., duces correction factors for the local tectonism to reflect that prior to any excavation or construction, is essential. the stress distribution difference caused by local tectonic Generally speaking, the in situ stress values in the movements. The correction factors can be determined by three mutually perpendicular directions of the Earth’s the least-squares approach based on laboratory tests and crust are unequal. Vertical stress can be obtained eas- local in situ stress measurements. Then, the rock elastic ily based on the overburden pressure without causing modulus is replaced by rock mass elastic modulus so as to much error in most instances [1]. However, the horizon- show the effect of rock discontinuities on the in situ stress. tal stresses can be affected significantly by plate tecton- Combining with elasticity theory, equations for estimating ics, major geological features, topography, etc., which are the major and minor horizontal stresses are obtained. It is difficult to estimate. Karl and Richart [2] performed many possible to reach satisfactory accuracy for stress estima- studies on the distribution characteristics of in situ stress tion. To show the feasibility of this method, it is applied to in sedimentary rocks. Li [3] proposed a method for estimat- two deep tunnels in China to determine the in situ stress. ing in situ stress in coal and soft rock masses. Brown and Field tests, including in situ stress measurements by con- Hoek [4] summarised the relationship between horizontal- ventional hydraulic fracturing (HF) and rock mass mod- to-vertical in situ stress ratio k with depth of cover by ulus measurements using a rigid borehole jack (RBJ), are analysing a large amount of measured data. Furthermore, carried out. It is shown that the stress field in the two deep González de Vallejo and Hijazo [5] plotted a large dataset tunnels is dominated by horizontal tectonic movements. of stress magnitudes versus depth on a global scale. She- The major and minor horizontal stresses are estimated, re- orey [6] presented an elasto-static thermal stress model of spectively. Finally, the results are compared with those de- the Earth to estimate crustal stress, but did not consider rived from the HF method. The calculated results in the two the main factors (e.g., local tectonic movements) affect- tunnels roughly coincide with the measured results with ing the state of stress. Other methods such as the geolog- an average of 15% allowable discrepancy. ical (tectonic structure analysis), or seismic (focal mech- anisms), can only determine the orientations of principal Keywords: in situ stress; field tests; Sheorey’s model; stress stresses rather than their magnitude. To reflect the influ- estimation; hydraulic fracturing ence 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 prob- *Corresponding Author: Qitao Pei: Key Lab. of Geotechnical Me- abilistic method and the empirical relationship between chanics and Engineering of Ministry of Water Resources, Yangtze River Scientific Research Institute, Wuhan 430010, P. R. China; the Tectonic Stress Index and k values. In addition, the in- Email: [email protected], Tel.: +86-27-82826540 crease of in situ stress due to local factors was expressed Xiuli Ding, Bo Lu, Yuting Zhang, Shuling Huang, Zhihong Dong: by the Stress Amplification Factor, which could provide an Key Lab. of Geotechnical Mechanics and Engineering of Ministry of estimate of structural stresses in rock masses for rock ex- Water Resources, Yangtze River Scientific Research Institute, Wuhan cavations [7, 8]. Although many scholars have undertaken 430010, P. R. China

© 2016 Q. Pei et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 524 Ë Q. Pei et al. much valuable research into stress estimation, the major 35 km and minor horizontal stresses on an engineering-scale re- Crust(solid) main hard to determine. Upper mantle

700 km (solid,plastic) Field testing is a direct method used to obtain the 2900 km Moho discontinuity orientations and magnitude of in situ stresses. In recent Lower mantle decades, in situ stress testing equipment and relative (solid,plastic) 5150 km methods have made significant progress. In 2003, the In- Gutenberg 6371 km ternational Society for Rock Mechanics (ISRM) published discontinuity some suggested methods for determining rock stress [9– Outer core(liquid) 12]. These methods cover the basic principles of overcoring and hydraulic fracturing/hydraulic testing of pre- existing fracture methods. However, conventional meth- Inner core(solid) ods 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 Figure 1: Cross-section of the Earth (modified from Sheorey’s model in situ stress regions, or at great depth, core disking ren- (1994)). ders the over-coring method inapplicable, and reduces the success rate of the hydraulic fracturing method [13]. horizontal free surface [14]: Here, a method for estimating the major and minor horizontal stresses in an elastic rock mass at engineering- σzz = 훾H (1) scale, is presented. Unlike the general Sheorey’s model, the improved method takes into account not only the vari- ν σ = σ = 훾H (2) ation of elastic constants, density, and thermal expansion xx yy 1 − ν coefficient of the crust and mantle, but also the stress dis- Where, 훾 is the bulk unit weight of the material, ν is Pois- tribution difference caused by local tectonic movements. son’s ratio, and H is the depth from the free surface. The In addition, the rock mass elastic modulus, which can re- terms σxx and σyy are two horizontal stresses, and σzz sym- flect the effect of rock discontinuities onthe in situ stress, bolises the vertical stress at depth H. especially at shallow crustal depths, is adopted to replace However, a large number of field measurements show the rock elastic modulus. The improved method can be that the horizontal stresses commonly do not follow Equa- much better applied to local, shallow rock masses rather tion (2), and in many places are several times larger than than at a global, or regional, scale. Finally, the results of the vertical stress. So, assumption (2) is likely to be inap- the application of this method to the determination of in plicable in many cases. situ stress around two deep tunnels in China are presented. 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 2 Estimation method of in situ should be zero at the mantle-core interface (Gutenberg discontinuity) rather than at the crust-mantle interface (Mo- stress based on Sheorey’s model horovičić discontinuity). Supposing that the Earth’s crust is taken as a solid 2.1 Overview of methods spherical shell filled with an incompressible liquid, the equilibrium equation is [15]: So far, the formation mechanism of crustal stress is not yet dσ 2(σ − σ ) clear. As a result, the methods for its estimation always in- r − θ r − 훾 = 0 (3) volve some simplifying assumptions. The most common dr r practice is to assume lateral confinement (i.e., no horizon- Here σr denotes the radial (vertical) stress, and σθ is the tal displacement anywhere due to gravitational loading). tangential (horizontal) stress in polar coordinates (r, θ).

Based on the mentioned assumption, the following equa- The relationship between σr (σθ) with the radial displace- tions are often used to determine the initial state of in situ stress in a relatively uniform soil or rock mass beneath a Estimating in situ stress in an elastic rock mass and its application Ë 525

Table 1: Values of different parameters for isotropic rocks [6].

Location Slice No. R (103 km) E (GPa) 훾 (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

Boundaries between adjacent slices in crust

R12 R7 R6 R5 Boundaries R4 between adjacent R3 slices in mantle R2 R1 Liquid core

Figure 2: Simplified spherical shell model of the Earth (modified from Sheorey’s model (1994)). ment u can be described as follows [15]: Figure 3: Comparison of the in situ stress ratio k obtained by Sheo- rey’s model (1994) with the field data published by Hoek and Brown [︂ ]︂ E du u ⎫ (1980) as well as by González de Vallejo and Hijazo (2008). σr = (1 − ν) + 2v ⎪ (1 + v)(1 − 2v) dr r ⎬⎪ [︂ ]︂ (4) E du u ⎪ σθ = v + ⎪ vertical stress can be obtained [6]: (1 + v)(1 − 2v) dr r ⎭

Where E is rock elastic modulus. k = 0.25 + 7E(0.001 + 1/H) (5) To determine crustal stress, Sheorey treated the Earth Based on Equation (5), the k-depth relationships with as a solid spherical shell, and presented an elasto-static different rock modulus are shown in Figure 3. This figure thermal stress model (Sheorey’s model) by considering the also shows the minimum and maximum envelope lines ac- variation of elastic constants, density and thermal expan- cording to Hoek and Brown (1980) as well as González de sion coefficient of the crust and mantle. The simplified Vallejo and Hijazo (2008). It can be seen that the maximum spherical shell model of the Earth is shown in Figure 2, or minimum envelope lines are generally similar, with less which consists of a total of 12 slices. Some parameters of data scatter for depths greater than 1000 m. The curve ob- each slice are given in Table 1. Here β is linear thermal tained by Sheorey’s model is a good mean fit to the mea- expansion coefficient and R refers to the inner radius of i sured data. So, it is reasonable to apply the model to esti- spherical slice i. The calculations are then carried out by mate in situ stress. putting the parameters in Table 1 into Sheorey’s model, In most cases, Equation (6) provides a more general and the in situ stress ratio k of mean horizontal stress to (though not mathematically exact) form of the in situ stress 526 Ë Q. Pei et al. ratio k for the Earth’s crust [6]: that case, local stress variations are too small to be repre- sented on a global scale [17]. The distributions and orien- ν βEG (︂ 1000)︂ k = + 1 + (6) tations of stresses on continental and regional scales can 1 − ν (1 − ν)훾 H be found in the World Stress Map (WSM) [18]. However, the Where G is thermal gradient, and β is linear thermal ex- local stress distribution may vary considerably due to het- pansion coefficient. erogeneity and anisotropy of the rock mass as well as geological structures therein. Current research shows that tectonic stresses, caused by a pervasive force field imposed 2.2 Comments on Sheorey’s model by active tectonics or past tectonic events, are the main causes of stress in the lithosphere. Zang and Stephans- Although the in situ stress ratio k for Sheorey’s model has son [19] summarised the influence of tectonic stresses on a tendency to be consistent with the field data, some key the rock engineering and divided it into three levels: the problems still exist in Equation (6), which can be listed as first-order is global patterns of tectonic stress due tothe follows: relative displacement of plates or isostasy; the second- order is mountain scale tectonic stress, which can vary sig- 1. The elastic modulus applied to the model is only for nificantly over short distances; and the third-order is fault- rocks rather than rock masses. As a result, the cal- scale stress. Faults are one of the main tectonic structures culated results will neglect the effect of rock discon- which can affect stress magnitude and direction [20]. So, tinuities (e.g., joints or faults) on the in situ stress. different order tectonic stresses are scaled based on their In practice, the calculated results are proved to be coherent domain over the region in which a stress compo- significantly larger than the measured results inthe nent can be supposed to be uniform. shallow crust. So, it is necessary to determine the Generally speaking, the main factors affecting stress relationship between rock elastic modulus and rock distribution on an engineering scale may include geolog- mass elastic modulus. ical and structural anisotropies, sedimentary loads, relief 2. Based on Sheorey’s model, the major and minor hor- effects, glacial rebound, loads produced by submarine el- izontal stresses cannot be obtained, respectively. In evations (or the convexity of the oceanic lithosphere), rock most cases, we pay more attention to the major hor- composition, and geomechanical behaviour [5]. The mag- izontal stress and shear stress rather than the mean nitude of the in situ stress turns out to be non-uniform ow- horizontal stress. Sheorey was also aware of the is- ing to the local influence of discontinuities, faults, dikes, sue, and conducted further research thereof [16]; heterogeneities, intrusive bodies, and folds [21]. Litholog- however, his work still could not solve the problem ical heterogeneities and structural anisotropies may also perfectly. lead to stress concentrations. The process of erosion and 3. The distribution of horizontal stress is quite uneven denudation may cause high horizontal stresses, especially in the shallow crust due to the great differences in for deep-cut river valleys. The geological history and be- local tectonic movements. Although we can predict havioural evolution of rocks may exert a significant influ- the direction of the major horizontal stress caused ence on the long-term state of stress. In addition, as af- by geological phenomena (e.g., faults or folds), it is fected by the disturbance of excavations, the orientation still difficult to determine the influence of the tecton- and magnitude of in situ stress in the vicinity of the exca- ism on the horizontal stress. So, we should consider vation section vary considerably. the effect of local tectonic movements onthe in situ stress values in engineering applications. 3 Improved Sheorey’s method 2.3 The main factors affecting the distribution of in situ stress at For a uniform soil or rock mass, Equation (6) can be ex- engineering-scale pressed as: a k = + b (7) When it comes to the factors affecting the orientation and H magnitude of in situ stresses, there is the problem of scale. 1000βEG ν βEG Where a = (1−ν)훾 , b = 1−ν + (1−ν)훾 . Compared with the continental or global scale, engineer- In Equation (7), the relationship between the in situ ing projects rarely occur at more than 2000 m in depth. In stress ratio k and depth is inversely proportional. For a Estimating in situ stress in an elastic rock mass and its application Ë 527 certain depth, the value of k is mainly dependent on the obtain the structural characteristics of fractured rock. By two parameters a and b. Moreover, when the buried depth using such imaging with acoustic and optical televiewers, changes slightly in the shallow crust, the thermal expan- continuous and oriented 360∘ views of the borehole wall sion coefficient β, and thermal gradient G, can be treated are available, from which the character, inter-relationship, as constants. Then, the in situ stress ratio k is taken to con- and orientations of lithologic and structural planar fea- sist of a constant term and a hyperbolic term. In addition, tures can be easily defined [23, 24]. For a given depth range the two parameters a and b are only dependent on rock [h1, h2], the RMDI can be expressed as [25]: properties, which reflect the intensity of tectonic move- ∫︀ h2 f (z)dz ments. However, tectonic stress (including magnitude and RMDI = h1 (10) direction) often changes significantly in different parts of ∆h the study area. As a result, it is not easy to determine the Here, z is the depth. f (z) is an integrity index density func- parameters a and b accurately. tion of the rock mass which can be determined by Equa- To reflect the influence of local tectonic movements on tion (11): the in situ stress, we introduce the local tectonic correction {︃ factors ξ and η. The following equation provides a more 0 (Completely fragmented rock) f(z) = (11) common form of the in situ stress ratio k for the Earth’s δ (Not completely fragmented rock) crust: Where, δ is an impact factor for rock fragmentation which ν βE G (︂ 1000)︂ k = + rm 1 + × ξ + η (8) depends on filling type and rock mass size. For filling con- 1 − ν (1 − ν)훾 H ditions, taking clay filling as an example, δ is taken as zero.

Where Erm stands for rock mass elastic modulus. The two Otherwise, the value of δ is dependent on rock mass size. parameters ξ and η are local tectonic correction factors Based on the published literature [25], as the rock mass which will be determined by the least-squares approach changes from intact, massive, bedded, fractured, to non- based on local stress measurements and laboratory tests. compact states, the value of δ changes from 1.0, 0.8, 0.5,

For a certain project, the parameters β, G, 훾, ν and Erm in 0.2, to 0.1, respectively. Equation (8) can be easily determined based on laboratory In addition, studies [26, 27] show that horizontal stress tests and empirical formulas. Then, the in situ stress ra- mainly consists of two parts: one is the horizontal com- tio k in Equation (8) can be taken as the function of ξ and ponent of gravitational stress, and the other is the hor- η. Furthermore, the least-squares fitting analysis is carried izontal tectonic stress. When the stress field is mainly out to determine the initial values of ξ and η based on local caused by gravity, horizontal stresses can be estimated by stress measurements. If some stress values deviate consid- Equation (2). However, as the horizontal tectonic move- erably, they should be neglected so as to obtain the opti- ment predominates, the following equations provide the mum tectonic correction factors. It is noted that if the study basic relationship for elastic rock behaviour, with the con- area suffers the same tectonic movements, the parameters straints that one of the principal stresses is vertical and ξ and η can be treated as constants. varies linearly with depth [3]: When the rock mass elastic modulus is unknown, we ⎫ σV = 훾H⎪ can indirectly determine it based on laboratory tests and ⎬⎪ rock quality information. Guo, et al. [22] provided the fol- σH = λσV + σT (12) ⎪ lowing relationship between the factors Erm/Ei and RMDI σh = λ(σV + σT )⎭ through analysing many measurements: Where σH and σh are the major and minor horizontal Erm = 0.73 × RMDI − 0.079 (9) stresses, respectively, σT is the horizontal tectonic stress, E i and λ is the lateral pressure coefficient due to gravitational

Where Ei is the intact rock elastic modulus, and RMDI is loading with λ = ν/(1 − ν). a rock mass integrity index, which can be determined by As mentioned before, the ratio k = (σH + σh)/(2σV ). using Equations (10) and (11). Then, Equation (8) can be rewritten as: The indicator RMDI can be defined as a ratio of in- (︂ )︂ σ + σ ν βErm G 1000 tact rock to entire rock mass in certain range. It can reflect H h = + 1 + × ξ + η (13) 2σV 1 − ν (1 − ν)훾 H (quantitatively) the effect of discontinuities (e.g., joints or faults) on rock integrity based on drilling images. Drilling In Equation (12), both σH and σh have the common images, from the use of a borehole camera, is often used to part in σT . Then, we can get the relationship between σH 528 Ë Q. Pei et al.

start 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). Stage 1: Selecting the in situ stress measured data 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 Stage 2: Determining parameters increase the number of in situ stress measurements. describing rock mass properties It is noted that if at a location the in situ stresses are measured in competent parts of the same formation, then Stage 3: Preliminary determination of the the corresponding stresses in an adjacent location can be parameters ξ and η by least square approach estimated by the relationship: ⎫ σH1 λ1σV1 + 2β1Erm1Gξ(H1 + 1000) + 2η(1 − ν1) Stage 4: Estimating the horizontal = ⎪ stresses by the improved method σH2 λ2σV2 + 2β2Erm2Gξ(H2 + 1000) + 2η(1 − ν2)⎪ ⎬⎪ σ = 훾H Recalculating V 2 Stage 5: Comparing the calculated ξ and η ⎪ σh1 λ1σV1 + 2λ1β1Erm1Gξ(H1 + 1000) + 2ην1 ⎪ results with the measured ones = ⎭⎪ σh2 λ2σV2 + 2λ2β2Erm2Gξ(H2 + 1000) + 2ην2 (17) No α ≤ 0.2 Yes Where the suffix 1 denotes the hard (or soft) rock inwhich stress measurements have been carried out: suffix 2 refers end to the hard (or soft) rock at an adjacent location where the in situ stresses are to be estimated. Figure 4: Flow chart describing the improved method for estimating in situ stress proposed in this study. 4 Engineering application – a case and σh by removing σT . study σh ⎫ σH = − (1 − λ) × σV ⎬ λ (14) To show the feasibility of this improved method, it is σV = 훾H⎭ applied to Meihuashan Tunnel to determine the in situ Based on Equations (13) and (14), we can obtain: stresses around it. ⎫ σH = λσV + 2βErm Gξ(H + 1000) + 2η(1 − ν)⎪ ⎬⎪ σV = 훾H (15) 4.1 Project overview ⎪ σh = λσV + 2λβErm Gξ(H + 1000) + 2ην⎭ Meihuashan Tunnel is located in Longyan City, Fujian According to Equation (15), the in situ stress at an ar- Province, southeast of China. It is 15,780 m in length with bitrary depth can be estimated. Figure 4 illustrates the a maximum depth of 688.21 m. The location of the tunnel flowchart of the improved method. is shown in Figure 5(a). To verify the reliability of this method in estimation According to tectonics, the tunnel lies in the depres- of in situ stress, it is necessary to compare the calculated sion belt of southwest of Fujian. Figure 5(b) shows the geo- results with field measurements. logical cross-section along the longitudinal axis of the tun- In stage 5, we define nel. There are seven faults exposed along the tunnel. In n addition, NE-NEE and NNE trending faults are well devel- ∑︀ ⃒σ 2 − σ 2⃒ ⃒ ci mi ⃒ oped, and control the distribution of strata and intrusive α = i=1 , i = 1, 2, 3, ···, n (16) n bodies in this area. The rocks around the tunnel are sand- ∑︀ 2 σci i=1 stone, siltstone of the Permian Longtan formation, early Jurassic Yanshanian granite, and Devonian quartz sand- Where σ and σ are the calculated and measured ci mi stone. At the test site, the rock is mainly moderate weath- stresses at point i, respectively. According to Equation (16), ered granite with a medium to coarse-grained structure. if α ≤ 0.2, the process is terminated and we will have Estimating in situ stress in an elastic rock mass and its application Ë 529

(a)

(b)

Figure 5: Location map and geological cross-section along Meihuashan Tunnel: (a) Location map (satellite imagery from Google Earth Ver. 7.1.5.1557. (October 11, 2014). Longyan City, China. 25∘ 10’ 50.91” N, 116∘ 49’ 46.78” E, Eye alt. 64,080 m. Digital Globe 2016), (b) Geologi- cal cross-section and test site [31].

4.2 Field tests stress. In addition, by neglecting the effect of excavation on the in situ stress in the vertical borehole (Figure 6(b)),

Within the project area, the in situ stress and rock mass the stresses were such that σH > σV > σh. That is, the elastic modulus measurements were carried out. The im- stress field at the test site is dominated by horizontal tec- plementation process of the field tests is described as fol- tonic movements. lows.

4.2.2 Rock mass elastic modulus test 4.2.1 In situ stress test Rock mass elastic modulus tests in the vertical borehole In situ stress testing by the hydraulic fracturing method were carried out using a rigid borehole jack (type of BJ- was carried out. The test site in Figure 5(b) is located in the 91A) [28]. The rigid borehole jack was developed by In- tunnel area, and two testing boreholes have been placed. stitute of Rock and Soil Mechanics, Chinese Academy of One is the horizontal with a maximum depth of 33.9 m. The Sciences, and has been widely used in many projects. Fig- other is the vertical with its depth ranging from 3.9 m to ure 7 shows a typical pressure versus deformation history 41.9 m. The in situ stress measurements in the horizontal recorded at a depth of 19.2 m. and vertical borehole are shown in Tables 2 and 3, respec- Based on the measured data, the rock mass elastic tively. Also, Figure 6 shows the variation of the in situ stress modulus at the measurement points can be obtained by measurements with borehole depth. Equation (18). Based on Table 2 and Figure 6(a), we can conclude ∆P E = KBDT(υ, ψ) (18) that the maximum principal stress is horizontal. When rm ∆D the horizontal borehole depth exceeds 21.9 m, the mini- Where Erm is the measured rock mass elastic modulus, K is mum horizontal stress approximately equals the vertical the influence coefficient under three-dimensional test con- 530 Ë Q. Pei et al.

Table 2: In situ stress measurements in horizontal borehole.

Measuring point No. Borehole depth/m σ1 (MPa) σ3 (MPa) σV (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/m3, the below is same.

35 30 25 20 15

Pressure (MPa) 10 5 (a) 0 0 100 200 300 400 Deformation (0.001 mm)

Figure 7: A typical pressure versus deformation history recorded at a depth of 19.2 m in vertical borehole.

parameters ∆P and ∆D refer to the incremental pressure and deformation, respectively and T (ν, ψ) is a coefficient related to bearing plate arc ψ 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 (b) stress at the point of measurement is inconsistent with that Figure 6: Variation of the measured in situ stresses with horizontal of the measured rock mass elastic modulus. To obtain the and vertical boreholes depth: (a) Horizontal borehole, (b) Vertical rock mass elastic modulus at the measuring points for in borehole. situ stress, we should calculate the rock mass elastic modulus at the required depth based on the measured data. ditions with K = 0.93, B is the pressure modified coeffi- As the depth of the adjacent measuring points changes by cient with B = 0.98, and D is the borehole diameter. The no more than 3 m, a two-dimensional linear interpolation Estimating in situ stress in an elastic rock mass and its application Ë 531

Table 3: In situ stress measurements in vertical borehole.

Measuring point Borehole σH (MPa) σh (MPa) σV (MPa) Azimuth σH No. depth/m 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.

Table 4: The rock mass elastic modulus in vertical borehole.

Measuring Measured results Calculated results

point No. Borehole depth (m) Erm (GPa) Borehole depth (m) Ecm (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 61.11 532 Ë Q. Pei et al.

Table 5: Coeflcient of the thermal expansion β of some rocks.

Rock type β (10−6 /∘C) Granite [29] 6–9 Limestone [29, 30] 3.7–10.3 Sandstone [29] 5–12 Marble [29] 3–15 Conglomerate [30] 9.1 Dolomite [30] 8.1 Breccia [30] 4.1–9.1 Schist [29] 6–12

(a) bilised zone. So, the depth of the stress disturbance zone affected by the excavation of the tunnel is about 11.9 mac- cording 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 borehole depth. In addition, the depth of the stress disturbance zone, as affected by the excavation of the tunnel, isabout (b) 17.9 m according to the vertical borehole. When the depth is less than 17.9 m, it is actually a secondary stress field, Figure 8: Characteristics of stress field zoning along horizontal which cannot reflect the initial stress state at the test site. and vertical boreholes depth: (a) Horizontal borehole, (b) Vertical Therefore, the measurements from points 1 to 7 should be borehole. removed to avoid their undue influence of the accuracy of the in situ stress determination. method can be used to obtain Ecm where desired. There- after, the rock mass elastic modulus Ecm at the measuring points for in situ stresses is obtained (Table 4). 4.3.2 Determination of the parameters

To quantify the rock mechanical properties of granite, 4.3 In situ stress determination based on uniaxial, and triaxial, compression tests are carried out. the improved method Based on the test data, v = 0.25 and r = 0.026 MPa/m. The coefficient of linear thermal expansion β is not widely 4.3.1 Selection of the measured results reported, but Table 5 provides the values for some rocks available from published literature [29, 30]. From Table 5, Due to the excavation of the tunnel, the stress field in the we can take the thermal expansion coefficient β for gran- region of the horizontal borehole depth can be divided into ite as 6.0 × 10−6/∘C. For crustal rocks, the thermal gradient three parts, as shown in Figure 8(a). When the borehole G is taken as 0.024∘C [16]. In addition, through collecting depth is less than 5.9 m, the in situ stress measurements be- a large number of in situ stress measurements under the come smaller, and we can take that region as the stress re- same tectonism, we used the least-squares approach to de- lief zone. As the borehole depth ranges from 5.9 m to 11.9 m, termine the two constants ξ and η in Equation (8), and the the in situ stress measurements appear to be larger, and we distribution of error can thus be controlled. The best ap- can take it as stress concentration zone. When the bore- proximate solution for this over-determined problem was hole depth exceeds 11.9 m, the in situ stress measurements found when ξ = 0.85 and η = 0.11 [31]. change slightly, and we take the region as a stress sta- Estimating in situ stress in an elastic rock mass and its application Ë 533

Table 6: Comparison of measured in situ stresses with those calculated using the improved method in Meihuashan Tunnel.

Measuring Borehole Ecm Measured results Calculated results Mean horizontal stress point depth (GPa) (MPa) (MPa) (MPa)

No. (m) σH σV σh σH σh 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

Figure 9: The relationship between σH +σh − ν and βErm G (︀1 + 1000 )︀ 2σV 1−ν (1−ν)훾 H for Meihuashan Tunnel. Figure 10: Comparison of the mean measured horizontal stress calculated by Sheorey’s model and estimated using the improved 4.3.3 Estimation of in situ stress method.

As described before, in situ stress at eleven measuring the comparison of measured in situ stresses by HF with the points are estimated based on Equation (15). To begin with, finally calculated values by the improved method inMei- the relationship between σH +σh − ν and βErm G (︀1 + 1000 )︀ 2σV 1−ν (1−ν)훾 H huashan Tunnel. This comparison shows the acceptability can be preliminarily obtained based on the local stress of the results of the improved method with regard to the es- measurements and Equation (13). Then, the values of ξ timation of the in situ stresses. and η are considered equal to 0.85 and 0.11, respectively, Furthermore, the comparison of the mean measured as initial values. The first trial results show that α = 0.275 horizontal stress and that calculated by Sheorey’s model, which was greater than 0.2. Therefore, ξ and η should be as well as those found using the improved method, is modified by removing some measured data which deviate shown in Figure 10. It can be seen that the in situ stresses considerably from the mean. When the new values of ξ estimated by the improved method are closer to the mea- and η are equal to 0.72 and 0.08, α is decreased to 0.149 sured results than those found by Sheorey’s model. The (Fig. 9). Then, the precision requirement for estimating the in situ stress in the project area is satisfied. Table 6 shows 534 Ë Q. Pei et al.

Figure 11: Geological cross-section along Qinling Tunnel and its location [32]. calculated results using Sheorey’s model differ signifi- ite based on borehole #6. The rock mass is relatively in- cantly from the measured results on the whole. tact. 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 verti- 5 Case study cal 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 sta- As an example of the use of the improved method, an- ble, ranging from 12.8 MPa to 13.0 MPa. The in situ stress other case study to determine the in situ stresses for the and rock mass elastic modulus measurements in the verti- Qinling water-conveyance tunnel project in China was per- cal borehole #6 are shown in Table 7. Similarly, the rock is formed (Fig. 11). The tunnel is 81.779 km long, and its max- mainly moderate weathered gneiss as evinced by arising imum overburden depth is 2004 m. It lies in the west of from borehole #3. The rock mass is intact and hard. The Qinling Mountain, Shaanxi Province, northwest of China. overburden depth at the test site is 1000 m. As the hori- The topography is controlled mainly by tectonics, and the zontal depth exceeds 11.0 m, the minimum principal stress faults are well-developed. The rocks around the tunnel are remained almost stable with an average value of 14.0 MPa. formed in the Cretaceous, Carboniferous, and Devonian The in situ stress and rock mass elastic modulus measured systems, and are predominantly: granite, quartz schist, in the vertical borehole (#3) are shown in Table 8. gneiss, granodiorite, sandstone, phyllite, and micaceous For the purpose of evaluating the applicability of the schist. improved method, the measured data in borehole #6 are In situ stress testing was undertaken by the hydraulic used to determine the local tectonic correction factors ξ fracturing method, and rock mass elastic modulus testing and η, and the measurements in borehole #3 are mainly was carried out using a rigid borehole jack [32]. Boreholes used for validation purposes. For borehole #6, we can get #3 and #6 were arranged in the tunnel site (Fig. 11). At ν = 0.25, r = 0.027 MPa/m, and β = 7.3 × 10−6 /∘C the test site, the rock is mainly moderate weathered gran- from laboratory tests [32]. The thermal gradient G is taken Estimating in situ stress in an elastic rock mass and its application Ë 535

Table 7: In situ stress measurements in vertical borehole #6.

Measuring Borehole Erm (GPa) σH (MPa) σh (MPa) σV (MPa) Azimuth σH 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.

Table 8: In situ stress measurements in vertical borehole #3.

Measuring Borehole Erm (GPa) σH (MPa) σh (MPa) σV (MPa) Azimuth σH 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.

Equation (19): ⎫ σH = λσV + 1.28βErm G(H + 1000) + 0.36(1 − ν)⎪ ⎬⎪ σV = 훾H (19) ⎪ σh = λσV + 1.28λErm Gξ(H + 1000) + 0.36ν⎭ In borehole #3: v = 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 substi- tuted into Equation (19), and the resulting maximum and minimum horizontal stresses are listed in Table 9. The parameter α 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 Figure 12: Comparison of the mean measured horizontal stress stresses estimated by the improved method are closer to calculated by Sheorey’s model and estimated using the improved those measured than those estimated by the use of Sheo- method. rey’s model. Therefore, the improved method in this paper is practical with regard to the estimation of the in situ stress as 0.024 ∘C/m. In addition, the measured in situ stresses state. are analysed by the least-squares procedure. The best ap- proximate solution for this over-determined problem was found when ξ = 0.64 and η = 0.18. The parameter α was 6 Conclusions 0.146, which satisfied the precision requirements for stress determination. According to tectonics, the tunnel belongs An improved method for estimating the magnitude of hor- to the same tectonic background; the stress state for the izontal stresses on an engineering scale based on Sheo- elastic rock in the tunnel can then be estimated by use of rey’s model is presented. It considers not only the variation of elastic constants, density, and thermal expansion 536 Ë Q. Pei et al.

Table 9: Comparison of measured in situ stresses with those calculated using the improved method in vertical borehole #3.

Measuring Borehole Ecm Measured results Calculated results Mean horizontal stress point depth (GPa) (MPa) (MPa) (MPa)

No. (m) σH σV σh σH σh Measured Sheorey’s In the model 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 coefficient of the crust and mantle, but also the stress dis- accuracy of the equation remains a topic worthy of future tribution difference caused by local tectonic movements. research. So, the general Sheorey’s model, applicable at global or regional scale could be, with this modification, much bet- Acknowledgement: The authors gratefully acknowledge ter applied to local, shallow rock masses, where local tec- the financial support of the National Science Founda- tonic movements can be recognised and measured. Be- tion of China (Nos. 51539002, 51609018, 51379022) and the sides, the effect of rock discontinuities on the in situ stress National key research and development project of China is taken into account by replacing the rock elastic modulus (Nos. 2016YFC0401802, 2016YFC0401804). 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 References does not obey the principal of Sheorey’s model. On an engineering scale, the improved method can also be used to [1] Hoek E., Brown E. T., Underground Excavations in Rock. The In- estimate the in situ stress state for elastic rock masses in in- stitution of Mining and Metallurgy, London, 1980, 93–101 traplate regions with hard and more intact rock. The major [2] Karl T., Richart F. E., Stresses in rock about cavities. Geotech- and minor horizontal stresses can be determined. In addi- nique, 1952, 3(2), 57–90 tion, the proposed method can also be used to determine [3] Li W. P., A preliminary estimation method of geostresses in coal and soft rock masses. Chin. J. Rock Mech. Eng., 2000, 19(2), in situ stress in some places where field tests are hard to 234–237 (In Chinese with English abstract) carry out. [4] Brown E. T., Hoek E., Technical note trends in relationships be- The suggested method is applied to two deep tunnels tween measured in–situ stress and depth. Int. J. Rock Mech. in China and the results show that the calculated in situ Min. Sci. and Geomech. Abstr., 1978, 15(4), 211–215 stresses match those measured. Also, it is shown if the lo- [5] Hijazo, T., González de Vallejo, L.I. In-situ stress amplification cal tectonic correction factors are adequate, the stress state due to geological factors in tunnels: The case of Pajares tunnels, Spain[J]. Engineering geology, 2012, 137–138: 13–20. will be calculated with the average accuracy of 15% in an [6] Sheorey P. R., A Theory for In–Situ Stress in Isotropic and Trans- unknown location. Moreover, it is possible to achieve an versely Isotropic Rock. Int. J. Rock Mech. Min. Sci. and Geomech. estimated in situ stress with the desired error by changing Abstr., 1994, 31(1), 23–34 the provisions in the method. [7] González de Vallejo, L.I., Hijazo, T. A new method of estimat- Although we have tried to decrease the error as much ing the ratio between in situ rock stresses and tectonics based on empirical and probabilistic analyses[J]. Engineering geology, as possible, the difference between the measured in situ 2008, 101(3): 185–194 stress and the calculated results remained. On one hand, [8] González de Vallejo, L.I. Hijazo,T. Galera,J.M. Entezari, Z.A. the rock mass has all the characteristics of inhomogene- (2014). In–situ stress amplification in tunnels from Spain, Iran ity and discontinuity, while the improved method used and Chile estimated by TSI and SAF indices. In: Rock Engineer- for in situ stress determination is based on the theory of ing and Rock Mechanics: Structures in and on Rock Masses, elasticity and cannot reflect the non-linear mechanical be- 2014, 83: 523–528. [9] Hudson J. A., Cornet F. H., Christiansson R., ISRM suggested haviour of the rock mass. On the other hand, major geolog- methods for rock stress estimation – part 1: strategy for rock ical features will also influence the in situ stress measure- stress estimation. Int. J. Rock Mech. Min. Sci., 2003, 40(7), 991– ments. So, it is only a preliminary attempt to estimate the 998 in situ stress by using Equation (15) and how to improve the [10] Sjöberg J., Christiansson R., Hudson J. A., ISRM suggested methods for rock stress estimation – Part 2: overcoring meth- Estimating in situ stress in an elastic rock mass and its application Ë 537

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