# Modified Fracture Mechanics Approach for Hydraulic Fracturing Stress Measurements

Authors: Guiyun Gao; Chenghu Wang; Hao Zhou; Pu Wang
Journal: Geofluids
 (2020)

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## Abstract

Hydraulic fracturing (HF) test has been widely used to determine in situ stress. The use of a conventional continuum method for this purpose has raised considerable controversies concerning field tests, particularly in the determination of the maximum horizontal principal stress under preexisting fractures. Fracture mechanics methods are very promising when considering preexisting cracks. However, most fracture mechanics methods do not include the effects of confinement on fracture parameters that depend on confining stress. In the present paper, we proposed a modified approach based on fracture mechanics for stress determination considering the relation between fracture toughness and confining stress based on the Rummel and Abou-Sayed methods. Then, we conducted true triaxial hydraulic fracturing tests under different stress ratios for granite and sandstone specimens to verify the proposed approach. The observed typical pressure-time curves indicate that in the conducted hydraulic fracturing tests, the steady fracture growth was attained. Moreover, we demonstrated that the stress ratios influence crack orientations. The horizontal maximum principal stresses determined using the modified Rummel method achieve the lowest relative error compared with other considered stress estimation approaches. This modified fracture mechanics method could be used as a potential alternative approach to obtain a considerably more precise estimation of the maximum horizontal stress in hydraulic fracturing stress determination.


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## Body

## 1. Introduction

In situ stress measurement can be performed to address a wide range of geomechanical problems, including faulting, earthquakes, and shale gas exploitation, as well as geothermal and geotechnical problems [1–9]. A hydraulic fracturing (HF) test (also referred to as a minifrac or microfrac test), in which a particular small volume of fluid is injected to create a hydraulic fracture, has been widely applied to in situ stress determination. The largest advantage of such test over other stress measurement approaches lies in its capability of estimating both stress magnitudes and orientations [10, 11]. In particular, a reliable estimate of minimum horizontal principal stress Shmin can be obtained by instantaneous shut-in pressure at which vertically induced hydraulic fractures in a vertical borehole arrest and start to close [12, 13]. However, concerning maximum horizontal principal stress SHmax, it can be evaluated indirectly based on elastic theory and Kirsch’s solution which is only valid for a homogenous defect-free continuum [12]. It requires an accurate determination of either fracture reopening pressure or the rock tensile strength suitable for in situ HF tests. The applicability of this continuum method to SHmax estimation has been discussed in many studies [14, 15]. The assumptions of Kirsch’s solution are not reasonable when hydraulic fractures or preexisting natural fractures exist [16]. Both borehole image logging and core analysis have revealed that the natural or mechanically induced fractures near a borehole may exist [6, 17]. When the length of fractures exceeds a certain value, the influence of a fracture cannot be neglected [18]. The natural fractures, induced fractures, or other discontinuities such as bedding planes influence stress determination in terms of preexisting fracture surface opening or even fracture initiation and propagation [19]. The generalized HF method, called hydraulic tests on preexisting fractures (HTPF) method, was introduced by Cornet and Valette [20] based on the exact stress balance between the fluid pressure and normal stress across the preexisting fracture. However, the HTPF method is time-consuming and requires a minimum of 14–15 tests to determine the stress state [12]. Diagnostic fracture injection test (DFIT), which was originally coined by Halliburton and has been widely used by industries [21], is a comprehensive small-volume fracture injection test. DFIT can be used to estimate the properties of rock formations including the leak-off coefficient, permeability, fracture closure pressure, and formation pressure [22]. However, DFIT estimates only the minimum principal stress and mainly focuses on leak-off behavior or preexisting fracture extension when considering preexisting natural fractures [21]. Here, we intend to determine maximum principal stress which cannot be accurately estimated by conventional methods for stress determination from hydraulic injection tests. Moreover, fracture initiation is the main concern for stress measurement that needs to be analyzed using the fracture mechanics method.Linear elastic fracture mechanics has been used for stress evaluation in HF tests and has been proven to be appropriate in many cases [23–26]. For in situ stress measurements, fracture toughness is an important parameter in this type of fracture mechanics approaches that can be determined through fracture tests using a cracked chevron-notched Brazilian disc or a semicircular bend specimen as suggested by ISRM [27, 28]. However, fracture toughness depends on the stress state and can be affected by confining stress that has barely been considered and incorporated in the analysis of stress determination using fracture mechanics approaches [29–31]. Fracture toughness obtained based on the in situ hydraulic fracture measurement can be one to two orders of magnitude larger than that acquired as a result of a laboratory core experiment. Laboratory rock fracture testing results have indicated that fracture toughness can increase by 10%–350% with increasing confining pressure [32–34]. Therefore, this effect needs to be considered when the principles of fracture mechanics are used for HF analysis for a more precise assessment of in situ stress.In the present study, we modify the existing fracture mechanics approaches for HF tests to incorporate the confining stress effect on fracture toughness. Then, we conduct the true triaxial HF tests under different stress ratios using the granite and red sandstone samples. The stresses applied to these cubic specimens are estimated using the proposed fracture mechanics method and conventional continuum approaches to verify the validity of the modified fracture mechanics method.


## 2. Fracture Mechanics Approach for HF Stress Measurements

### 2.1. Conventional and Fracture Mechanics Approaches for HF Stress Measurements

The conventional theory underlying HF stress measurements is based on Kirsch’s solution in an elastic medium obtained considering stress around a circular hole [35, 36]. The principal stresses in a vertical borehole corresponding to the critical state under fluid pressure can be expressed as follows [35]:
(1)SHmax=T0+3Ps−Pb−P0,or
(2)SHmax=3Shmin−Pr−P0,(3)Shmin=Ps,where SHmax and Shmin are the maximum and minimum horizontal principal stress, respectively; T0 is the tensile strength of surrounding rock; P0 is the pore pressure; Pb is the breakdown pressure; Ps is the instantaneous shut-in pressure. Vertical stress can be calculated based on the overburden weights of rock at respective depths. Compared with equation (1), equation (2) is more commonly used in practice in the cases that do not require considering tensile strength measurements in cored rocks despite the difficulty in the interpretation of Pr [14].However, the conventional theory underlying HF ignores the possibility of preexisting fractures, as well as their initiation during HF, which may result in misleading in the HF process and the unreliability of equations (1) and (2) in SHmax calculation [23]. For example, it was discovered that natural fractures in core samples from the tests on the Devonian Shale corresponding to the Rome Basin of West Virginia violated the assumptions used in the conventional methods in the calculation of in situ stress [23]. The fracture mechanics approach for HF stress determination was therefore deemed neccesary.There were several attempts to interpret HF based on fracture mechanics, including the numerical and theoretical approaches [23, 25, 26, 37, 38], but very few of them attempted to estimate in situ stresses based on fracture mechanics [23, 25]. Abou-Sayed et al. [23] proposed an approach based on the fracture mechanics concepts to evaluate the in situ stress by conducting field and laboratory tests. To realize such a method, they assumed two symmetrical fractures emanating radially from a borehole in an infinite medium. In the case of the original crack perpendicular to Shmin, the maximum horizontal principal stress SHmax can be defined as follows [23]:
(4)SHmax=KIcG−Fπa1/2−FG−FPb+GG−FPs,where KIc is the fracture toughness, a is the half crack length, G and F are the functions of a/R [39], and R is the radius of a borehole.The other fracture mechanics approach to perform the HF stress measurement was introduced by Rummel [25] grounded on the basic principle of the superposition of stress intensity factors corresponding to each loading source acting on the symmetrical double cracks with half-length a:
(5)KISHmax,Shmin,p,pa=KISHmax+KIShmin+KIp+KIpa,where KI indicates the stress intensity factor for the crack propagation of mode I fracture, p is the fluid pressure acting on the wall of a hole, and pa=pax,0 characterizes the fluid pressure distribution along the crack from x=R,−R to x=R+a,−R−a. According to the general formulation of the stress intensity factor for a tension crack with half-length a in an infinite plate, as given by Paris and Sih [39], and considering Kirsch’s equation, we can derive the first two terms at the right of equation (5) as follows:
(6)KISHmax=−SHmaxRfb,KIShmin=−ShminRgb,where fb=−2b2−1/πb71/2, gb=πb1/21−2/πaarcsin1/b+2b2+1b2−1/πb71/2, and b=1+a/R. Assuming that no fluid penetrates into existing cracks, the third term in equation (5) can be expressed as follows:
(7)KIp=pRh0b,where h0b=1.3b−1/1+b3/2+7.8sinb−1/2/2b5/2−1.7. Then, the pressure acting within a crack yields stress intensity that can be defined as follows:
(8)KIpa=pRhab.Ifpax=p, for example, we obtain ha=πb1/21−2/πaarcsin1/b.Thereafter, the maximum horizontal principal stress introduced in equation (5) underlies the following relation at an unstable crack extension:
(9)SHmax=−KIc/R−Shming+h0+haPbf.In the fracture mechanics approaches proposed by Abou-Sayed et al. [23] and Rummel [25], a key parameter is fracture toughness KIc. However, the fracture toughness obtained based on the in situ hydraulic fracture measurement is larger than that acquired as a result of conducting a laboratory core experiment [40] and has a strong dependency on confining stress [30, 32, 41]. Therefore, if we apply equation (4) or (9) to analyze the maximum horizontal principal stress, we may incorrectly estimate the in situ stress or that derived in a laboratory tests.


### 2.2. Confining Effects and Modified Approach for HF Stress Measurements Based on Fracture Mechanics

Laboratory rock fracture tests revealed that fracture toughness increased with confining stress [32, 34]. Most researchers have indicated the presence of the positive relation between confining stress and fracture toughness. Based on experimental results, Winter [42] suggested formulating this relation as follows:
(10)KIc′=1+0.038Pconf×KIc0,where KIc′ is the fracture toughness under confining pressure (MPa m1/2), KIc0 is the fracture toughness at ambient pressure (MPa m1/2), and Pconf is confining pressure (MPa). This relation conforms to the results of Schmidt and Huddle [34] and Abou-Sayed [23], as demonstrated by Stoeckhert et al. [43]. Müller [41] proposed a similar relation by incorporating the pore pressure based on the three-point bending test under confining pressure. Vásárhelyi [31] assumed that the connection between the fracture toughness of gneiss and the hydrostatic pressure was linear and proposed a linear dependency similarly as in equation (10) by fitting the experimental data, which varied according to the direction of anisotropy. Funatsu et al. [29], however, formulated the relation between fracture toughness and confining pressure using the R curve method based on the square of confining pressure.However, the above equations have a critical drawback, i.e., the dimension problem, due to the simple mathematical fitting of the experimental data. Here, a semiempirical method can be utilized to analyze the relation between fracture toughness and confining pressure, combined with the experimental data reported in the related research works [29, 30, 32, 42–46]. The tensile strength of rock material needs to be considered when we explore the correlation between fracture toughness and confining pressure to retain dimension consistency. As a result of analyzing the fracture toughness variation depending on different values of confining pressure, we proposed a uniform form equation as follows:
(11)KIc′=1+CPconfT0×KIc0,whereT0 is tensile strength and C is the dimensionless coefficient that corresponds to the dependence of fracture toughness on confining stress together with T0 and KIc0. The relation between fracture toughness and confining pressure was concisely represented by fitting the experimental data and using the above equation [47]. The fitting results also revealed that coefficient C could be approximated to ~0.5, while the range of tensile strength for various rocks here was 0.6~25 MPa, as presented in our previous work [47]. Accordingly, equation (11) can be reduced as follows:
(12)KIc′=1+0.5PconfT0×KIc0.Once we obtain the fracture toughness under the confining stress state, we can use this modified fracture toughness equation to realize an approach of the fracture mechanics for HF stress measurement. The modified Abou-Sayed and Rummel methods forSHmax are defined similarly as in equations (4) and (9) except substituting KIc in these equations by KIc′ as follows:
(13)SHmax=KIc′G−Fπa1/2−FG−FPb+GG−FPs,(14)SHmax=−KIc′/R−Shming+h0+hapcf.


## 2.1. Conventional and Fracture Mechanics Approaches for HF Stress Measurements

The conventional theory underlying HF stress measurements is based on Kirsch’s solution in an elastic medium obtained considering stress around a circular hole [35, 36]. The principal stresses in a vertical borehole corresponding to the critical state under fluid pressure can be expressed as follows [35]:
(1)SHmax=T0+3Ps−Pb−P0,or
(2)SHmax=3Shmin−Pr−P0,(3)Shmin=Ps,where SHmax and Shmin are the maximum and minimum horizontal principal stress, respectively; T0 is the tensile strength of surrounding rock; P0 is the pore pressure; Pb is the breakdown pressure; Ps is the instantaneous shut-in pressure. Vertical stress can be calculated based on the overburden weights of rock at respective depths. Compared with equation (1), equation (2) is more commonly used in practice in the cases that do not require considering tensile strength measurements in cored rocks despite the difficulty in the interpretation of Pr [14].However, the conventional theory underlying HF ignores the possibility of preexisting fractures, as well as their initiation during HF, which may result in misleading in the HF process and the unreliability of equations (1) and (2) in SHmax calculation [23]. For example, it was discovered that natural fractures in core samples from the tests on the Devonian Shale corresponding to the Rome Basin of West Virginia violated the assumptions used in the conventional methods in the calculation of in situ stress [23]. The fracture mechanics approach for HF stress determination was therefore deemed neccesary.There were several attempts to interpret HF based on fracture mechanics, including the numerical and theoretical approaches [23, 25, 26, 37, 38], but very few of them attempted to estimate in situ stresses based on fracture mechanics [23, 25]. Abou-Sayed et al. [23] proposed an approach based on the fracture mechanics concepts to evaluate the in situ stress by conducting field and laboratory tests. To realize such a method, they assumed two symmetrical fractures emanating radially from a borehole in an infinite medium. In the case of the original crack perpendicular to Shmin, the maximum horizontal principal stress SHmax can be defined as follows [23]:
(4)SHmax=KIcG−Fπa1/2−FG−FPb+GG−FPs,where KIc is the fracture toughness, a is the half crack length, G and F are the functions of a/R [39], and R is the radius of a borehole.The other fracture mechanics approach to perform the HF stress measurement was introduced by Rummel [25] grounded on the basic principle of the superposition of stress intensity factors corresponding to each loading source acting on the symmetrical double cracks with half-length a:
(5)KISHmax,Shmin,p,pa=KISHmax+KIShmin+KIp+KIpa,where KI indicates the stress intensity factor for the crack propagation of mode I fracture, p is the fluid pressure acting on the wall of a hole, and pa=pax,0 characterizes the fluid pressure distribution along the crack from x=R,−R to x=R+a,−R−a. According to the general formulation of the stress intensity factor for a tension crack with half-length a in an infinite plate, as given by Paris and Sih [39], and considering Kirsch’s equation, we can derive the first two terms at the right of equation (5) as follows:
(6)KISHmax=−SHmaxRfb,KIShmin=−ShminRgb,where fb=−2b2−1/πb71/2, gb=πb1/21−2/πaarcsin1/b+2b2+1b2−1/πb71/2, and b=1+a/R. Assuming that no fluid penetrates into existing cracks, the third term in equation (5) can be expressed as follows:
(7)KIp=pRh0b,where h0b=1.3b−1/1+b3/2+7.8sinb−1/2/2b5/2−1.7. Then, the pressure acting within a crack yields stress intensity that can be defined as follows:
(8)KIpa=pRhab.Ifpax=p, for example, we obtain ha=πb1/21−2/πaarcsin1/b.Thereafter, the maximum horizontal principal stress introduced in equation (5) underlies the following relation at an unstable crack extension:
(9)SHmax=−KIc/R−Shming+h0+haPbf.In the fracture mechanics approaches proposed by Abou-Sayed et al. [23] and Rummel [25], a key parameter is fracture toughness KIc. However, the fracture toughness obtained based on the in situ hydraulic fracture measurement is larger than that acquired as a result of conducting a laboratory core experiment [40] and has a strong dependency on confining stress [30, 32, 41]. Therefore, if we apply equation (4) or (9) to analyze the maximum horizontal principal stress, we may incorrectly estimate the in situ stress or that derived in a laboratory tests.


## 2.2. Confining Effects and Modified Approach for HF Stress Measurements Based on Fracture Mechanics

Laboratory rock fracture tests revealed that fracture toughness increased with confining stress [32, 34]. Most researchers have indicated the presence of the positive relation between confining stress and fracture toughness. Based on experimental results, Winter [42] suggested formulating this relation as follows:
(10)KIc′=1+0.038Pconf×KIc0,where KIc′ is the fracture toughness under confining pressure (MPa m1/2), KIc0 is the fracture toughness at ambient pressure (MPa m1/2), and Pconf is confining pressure (MPa). This relation conforms to the results of Schmidt and Huddle [34] and Abou-Sayed [23], as demonstrated by Stoeckhert et al. [43]. Müller [41] proposed a similar relation by incorporating the pore pressure based on the three-point bending test under confining pressure. Vásárhelyi [31] assumed that the connection between the fracture toughness of gneiss and the hydrostatic pressure was linear and proposed a linear dependency similarly as in equation (10) by fitting the experimental data, which varied according to the direction of anisotropy. Funatsu et al. [29], however, formulated the relation between fracture toughness and confining pressure using the R curve method based on the square of confining pressure.However, the above equations have a critical drawback, i.e., the dimension problem, due to the simple mathematical fitting of the experimental data. Here, a semiempirical method can be utilized to analyze the relation between fracture toughness and confining pressure, combined with the experimental data reported in the related research works [29, 30, 32, 42–46]. The tensile strength of rock material needs to be considered when we explore the correlation between fracture toughness and confining pressure to retain dimension consistency. As a result of analyzing the fracture toughness variation depending on different values of confining pressure, we proposed a uniform form equation as follows:
(11)KIc′=1+CPconfT0×KIc0,whereT0 is tensile strength and C is the dimensionless coefficient that corresponds to the dependence of fracture toughness on confining stress together with T0 and KIc0. The relation between fracture toughness and confining pressure was concisely represented by fitting the experimental data and using the above equation [47]. The fitting results also revealed that coefficient C could be approximated to ~0.5, while the range of tensile strength for various rocks here was 0.6~25 MPa, as presented in our previous work [47]. Accordingly, equation (11) can be reduced as follows:
(12)KIc′=1+0.5PconfT0×KIc0.Once we obtain the fracture toughness under the confining stress state, we can use this modified fracture toughness equation to realize an approach of the fracture mechanics for HF stress measurement. The modified Abou-Sayed and Rummel methods forSHmax are defined similarly as in equations (4) and (9) except substituting KIc in these equations by KIc′ as follows:
(13)SHmax=KIc′G−Fπa1/2−FG−FPb+GG−FPs,(14)SHmax=−KIc′/R−Shming+h0+hapcf.


## 3. True Triaxial HF Tests

### 3.1. Test Apparatus and Specimen Processing

The true triaxial testing apparatus designed by the former Institute of Crustal Dynamics, China Earthquake Administration, was used to address true triaxial confinement. This apparatus comprises three main parts: loading frames, servo-hydraulic system, and control system as well as transducers and software (Figure1). The loading frames have stiffness higher than 1 GN/m and can accommodate specimens with a dimension of 700×500×500mm3. The servo-hydraulic system and the loading frames are designed to produce output forces up to 1000 and 400 kN in the vertical and two horizontal directions, respectively, which can yield a vertical stress up to 100 MPa and horizontal stresses up to 40 MPa on a 100×100m2 cross-sectional area. The servo control system controls the independent loading and maintenance of three principal stresses. A hand pump with a maximum pulse output of 60 MPa was used for hydraulic loading so that the loading rate was monitored to maintain consistency as much as possible. Hydraulic pressure was monitored and recorded using a high-precision sensor and subacquisition system.Figure 1
(a) True triaxial HF testing system and (b) schematic drawing of a specimen under triaxial loadings and hydraulic pressure.
(a)
(b)Two kinds of rock specimens, granite and red sandstone, were selected for hydraulic fracturing tests. The granite samples were collected in Lushan city, Jiangxi province, China, while the red sandstone samples were obtained in Yunnan province. The rock samples we used were from quarries near the surface beneath the weathering layer. The lithology of the granite samples is two-mica granite of the Xingzi formation (HtX). Moreover, the lithology of sandstone samples is red sandstone of the Shuanglongtan formation (є2s), the Middle Cambrian system. The physical and mechanical properties of these rock samples are summarized in Table1. Fracture toughness was evaluated through the ISRM-suggested semicircular bend test [28].Table 1
Physical and mechanical properties of the rock samples.
Lithology
Density(g/cm3)
Elastic modulus(GPa)
Poisson’s ratio
Tensile strength(MPa)
Fracture toughness(MPa m1/2)
Granite
2.65
50.0
0.18
10.87
1.07
Red sandstone
2.50
28.8
0.18
5.3
0.78The granite and red sandstone specimens were processed into100mm×100mm×100mm cubes. A through hole with a diameter of 10 mm was drilled in each granite specimen. Concerning the red sandstone specimen, the diameter of the through borehole was 15 mm with a pair of symmetrical cracks in the direction of one principal stress (Figure 2). The length of these prefabricated cracks was approximately 2.5 mm. The open-hole HF experiments were conducted by injecting fluid directly into the boreholes of cubic specimens (Figure 1(b)).Figure 2
Typical specimens used in true triaxial HF tests. (a) Granite specimen with a through hole. (b) Through-hole red sandstone specimen with the diametrically opposed cracks.
(a)
(b)


### 3.2. Experiment Design

Based on the horizontal stress ratio range in mainland China [48], the experiment program was designed, as shown in Table 2. The direction of wellbores was along the direction of F3 to simulate vertical stress in a vertical borehole, while horizontal stresses were perpendicular to a borehole (Figure 1(b)). Concerning the specimens with prefabricated cracks, the maximum horizontal principal stress was parallel to their direction. The horizontal stress ratios kHmax=SHmax/Sv and khmin=Shmin/Sv were in the range of 1.5~3.0 and 0.5~1.5, respectively. The ratio between the minimum and maximum horizontal stresses was from 0.5 to 0.75, representing the typical value of tectonic stress in China.Table 2
Experiment design.
Sample no.
S
Hmax
(MPa)
S
hmin
(MPa)
S
v
(MPa)
k
Hmax
k
hmin
S
hmin
/
S
Hmax
HF-JG-2
9.83
4.86
6.81
1.44
0.71
0.49
HF-JG-5
14.53
7.33
4.86
2.99
1.51
0.50
HF-HS-1
9.76
6.11
4.93
1.98
1.24
0.63
HF-HS-2
9.95
6.22
6.20
1.61
1.00
0.63
HF-HS-3
9.90
7.47
4.99
1.98
1.50
0.75
HF-HS-4
9.84
4.94
6.16
1.99
0.80
0.50
HF-HS-5
9.91
5.98
4.98
1.66
1.20
0.60


### 3.3. Experiment Procedure

In the conducted experiments, preloads were almost simultaneously applied in three directions at a low loading rate (0.015 MPa/s, corresponding to ~30% of the following normal loading rate of 0.05 MPa/s) to a value of 5 kN (Figure3). In the loading stage (under stress boundary conditions), stress σ3, corresponding to F3, was first applied in the horizontal direction, parallel to the borehole, and then, the maximum and minimum horizontal stresses were simultaneously applied in F1 and F2 directions, respectively, perpendicular to the borehole. Finally, the hydraulic stress was applied to pressurize the test interval at the constant flow rate using a hand pump to simulate the HF process for stress measurements. As soon as fracture breakdown (or reopening) pressure was reached and pressure stabilized, we stopped the pump and recorded the closure of fractures. Then, we depressurized the hole and waited before executing the next injection cycle. Finally, at least five injection cycles were implemented. Cyclic hydraulic pressure with respect to time was recorded for stress determination.Figure 3
Typical loads applied to a cubic specimen.


## 3.1. Test Apparatus and Specimen Processing

The true triaxial testing apparatus designed by the former Institute of Crustal Dynamics, China Earthquake Administration, was used to address true triaxial confinement. This apparatus comprises three main parts: loading frames, servo-hydraulic system, and control system as well as transducers and software (Figure1). The loading frames have stiffness higher than 1 GN/m and can accommodate specimens with a dimension of 700×500×500mm3. The servo-hydraulic system and the loading frames are designed to produce output forces up to 1000 and 400 kN in the vertical and two horizontal directions, respectively, which can yield a vertical stress up to 100 MPa and horizontal stresses up to 40 MPa on a 100×100m2 cross-sectional area. The servo control system controls the independent loading and maintenance of three principal stresses. A hand pump with a maximum pulse output of 60 MPa was used for hydraulic loading so that the loading rate was monitored to maintain consistency as much as possible. Hydraulic pressure was monitored and recorded using a high-precision sensor and subacquisition system.Figure 1
(a) True triaxial HF testing system and (b) schematic drawing of a specimen under triaxial loadings and hydraulic pressure.
(a)
(b)Two kinds of rock specimens, granite and red sandstone, were selected for hydraulic fracturing tests. The granite samples were collected in Lushan city, Jiangxi province, China, while the red sandstone samples were obtained in Yunnan province. The rock samples we used were from quarries near the surface beneath the weathering layer. The lithology of the granite samples is two-mica granite of the Xingzi formation (HtX). Moreover, the lithology of sandstone samples is red sandstone of the Shuanglongtan formation (є2s), the Middle Cambrian system. The physical and mechanical properties of these rock samples are summarized in Table1. Fracture toughness was evaluated through the ISRM-suggested semicircular bend test [28].Table 1
Physical and mechanical properties of the rock samples.
Lithology
Density(g/cm3)
Elastic modulus(GPa)
Poisson’s ratio
Tensile strength(MPa)
Fracture toughness(MPa m1/2)
Granite
2.65
50.0
0.18
10.87
1.07
Red sandstone
2.50
28.8
0.18
5.3
0.78The granite and red sandstone specimens were processed into100mm×100mm×100mm cubes. A through hole with a diameter of 10 mm was drilled in each granite specimen. Concerning the red sandstone specimen, the diameter of the through borehole was 15 mm with a pair of symmetrical cracks in the direction of one principal stress (Figure 2). The length of these prefabricated cracks was approximately 2.5 mm. The open-hole HF experiments were conducted by injecting fluid directly into the boreholes of cubic specimens (Figure 1(b)).Figure 2
Typical specimens used in true triaxial HF tests. (a) Granite specimen with a through hole. (b) Through-hole red sandstone specimen with the diametrically opposed cracks.
(a)
(b)


## 3.2. Experiment Design

Based on the horizontal stress ratio range in mainland China [48], the experiment program was designed, as shown in Table 2. The direction of wellbores was along the direction of F3 to simulate vertical stress in a vertical borehole, while horizontal stresses were perpendicular to a borehole (Figure 1(b)). Concerning the specimens with prefabricated cracks, the maximum horizontal principal stress was parallel to their direction. The horizontal stress ratios kHmax=SHmax/Sv and khmin=Shmin/Sv were in the range of 1.5~3.0 and 0.5~1.5, respectively. The ratio between the minimum and maximum horizontal stresses was from 0.5 to 0.75, representing the typical value of tectonic stress in China.Table 2
Experiment design.
Sample no.
S
Hmax
(MPa)
S
hmin
(MPa)
S
v
(MPa)
k
Hmax
k
hmin
S
hmin
/
S
Hmax
HF-JG-2
9.83
4.86
6.81
1.44
0.71
0.49
HF-JG-5
14.53
7.33
4.86
2.99
1.51
0.50
HF-HS-1
9.76
6.11
4.93
1.98
1.24
0.63
HF-HS-2
9.95
6.22
6.20
1.61
1.00
0.63
HF-HS-3
9.90
7.47
4.99
1.98
1.50
0.75
HF-HS-4
9.84
4.94
6.16
1.99
0.80
0.50
HF-HS-5
9.91
5.98
4.98
1.66
1.20
0.60


## 3.3. Experiment Procedure

In the conducted experiments, preloads were almost simultaneously applied in three directions at a low loading rate (0.015 MPa/s, corresponding to ~30% of the following normal loading rate of 0.05 MPa/s) to a value of 5 kN (Figure3). In the loading stage (under stress boundary conditions), stress σ3, corresponding to F3, was first applied in the horizontal direction, parallel to the borehole, and then, the maximum and minimum horizontal stresses were simultaneously applied in F1 and F2 directions, respectively, perpendicular to the borehole. Finally, the hydraulic stress was applied to pressurize the test interval at the constant flow rate using a hand pump to simulate the HF process for stress measurements. As soon as fracture breakdown (or reopening) pressure was reached and pressure stabilized, we stopped the pump and recorded the closure of fractures. Then, we depressurized the hole and waited before executing the next injection cycle. Finally, at least five injection cycles were implemented. Cyclic hydraulic pressure with respect to time was recorded for stress determination.Figure 3
Typical loads applied to a cubic specimen.


## 4. Results and Discussion

### 4.1. HF Test Results of Granites

To confirm the validity of the proposed HF testing system under triaxial loading, we first conducted two HF tests using granite samples. Figure4 represents the hydraulic pressure changes with the time of the considered granite specimens in true triaxial and hydraulic loading. Breakdown pressure Pb of two granite specimens HF-JG-2 and HF-JG-5 is 14.42 MPa and 18.73 MPa, respectively. The fracture reopening pressure Pr and instantaneous shut-in pressure Ps are also clearly observed in each pressure cycle and determined using the interactive software developed base Visual Basic [49]. The calculated HF parameters and stresses based on equations (2) and (3), respectively, are summarized in Table 3.Figure 4
Typical experimental pressure-time curves corresponding to the HF tests of granite specimens.Table 3
Summary of the experimental stress measurement data and the results based on equations (2) and (3).
No.
Rock type
HF parameters (MPa)
Calculated stress (MPa)
Applied stress (MPa)
Relative error (%)
P
b
P
r
P
s
T
S
Hmax
S
hmin
S
Hmax
S
hmin
e
Hmax
e
hmin
HF-JG-2
Granite
14.42
8.40
5.16
6.02
7.08
5.16
9.83
4.86
27.98
6.24
HF-JG-5
Granite
18.73
8.77
7.63
9.96
14.12
7.63
14.53
7.33
2.82
4.11
HF-HS-1
Sandstone
13.01
7.75
6.16
5.31
10.73
6.16
9.76
6.11
9.89
0.79
HF-HS-3
Sandstone
14.56
9.79
7.41
4.77
12.44
7.41
9.95
6.22
25.63
0.80
HF-HS-4
Sandstone
10.61
7.69
5.34
2.92
8.33
5.34
9.90
7.47
15.35
8.11
HF-HS-5
Sandstone
9.70
7.83
5.72
1.87
9.33
5.72
9.84
4.94
5.88
4.28
Notes:Pb—breakdown pressure; Pr—fracture reopening pressure; Ps—instantaneous shut-in pressure; T—tensile strength; Shmin—minimum horizontal principal stress; SHmax—maximum horizontal principal stress. Here, eHmax=SHmax1−SHmax/SHmax and ehmin=Shmin1–Shmin/Shmin.The maximum and minimum horizontal principal stresses applied on specimen HF-JG-2 are 9.83 and 4.86 MPa, respectively, while the corresponding calculated stresses from the HF tests are 7.08 and 5.16 MPa, respectively (Table3). The relative error associated with the determined maximum and minimum horizontal principal stresses is 27.98% and 6.24%, respectively. The calculated maximum and minimum horizontal principal stress estimates for specimen HF-JG-5 are 14.12 and 7.63 MPa, respectively, and the corresponding errors with respect to the applied maximum and minimum horizontal principal stress are 2.82% and 4.11%, respectively. The tensile strength values calculated based on the HF data are 6.02 MPa and 9.96 MPa, which are smaller than the tensile test results (namely, ~10.87 MPa).For granite samples, we located and identified the induced fractures using the ink dyeing method. Figure5 represents the fracture morphology after applying hydraulic fracturing to specimens HF-JG-2 and HF-JG-5. The length values of the cracks induced by the HF of these specimens were approximately 20 mm. However, the crack direction of HF-JG-2 was close to the maximum horizontal principal stress, while the orientation of the other specimen HF-JG-5 deviated by 45° from the maximum horizontal principal stress.Figure 5
Fracture morphology after the HF tests of specimens: (a) HF-JG-2 and (b) HF-JG-5.
(a)
(b)Considering the simulated critical injection pressure versus the fracture length as shown by Stoeckhert [46], the critical injection pressure generally increases with an increase in confining stress. In the conducted tests, the first injection cycle indicated the stable crack propagation, and the crack arrested finally. However, crack reopening occurred in the subsequent injection processes without further magnification of the crack length or rapidly reaching the boundary of the rock cube. Therefore, the crack lengths in the conducted tests are finite and deviated from the maximum principal stress orientation due to the confining stress effect.


### 4.2. Results of the HF Tests for Sandstones with Preexisting Cracks

Concerning the other considered specimens (red sandstones), the pressure-over-time curves are represented in Figure6. The breakdown pressure, instantaneous shut-in pressure, and fracture reopening pressure in each test are clearly visible with acceptable reproducibility in each cycle. The typical characteristic parameters of these tests were interpreted, as listed in Table 3. In the case of applying the conventional stress measurement of HF and equation (2), the stresses corresponding to each cubic specimen were determined (see Table 3). The stresses obtained using equations (2) and (3) are close to the applied ones (theoretical values). The relative errors of Shmin are less than 10%, while the errors of SHmax are slightly higher, falling in the range of 5.88%–25.63%. These results indicate that the conventional theory of HF based on the strength criterion can be used to approximately assess the stress values, particularly concerning the minimum horizontal principal stress.Figure 6
Pressure vs. time in the HF tests in the case of red sandstones.As we noted above, the maximum horizontal principal stresses are inaccurately estimated using the conventional stress measurement theory due to the presence of preexisting fractures, and hydraulic fractures may initiate and propagate along these preexisting cracks. Therefore, the fracture mechanics methods must be used in HF analysis in this case. We employed the modified approaches for HF stress determination to analyze the results of these experiments. Here, fracture toughness was revised by incorporating the confining stress effect. The modified Rummel and Abou-Sayed methods were both utilized to calculate the maximum principal stress. The estimated stress results are represented in Table4. The obtained results indicate that SHmax values obtained using the modified Rummel method are in the range of 10.1–10.96 MPa which are all lower than those determined through the modified Abou-Sayed method, thereby being closer to applied stresses (as shown in Table 4). The relative errors of SHmax obtained by the modified Rummel method are in the range of 2.0%–11.4%, while they are approximately in the range of 16.0%–86.5% in the case of the modified Abou-Sayed method. This finding indicates that the Rummel method based on the superposition of stress intensity factors corresponding to each loading source is more accurate than the Abou-Sayed method.Table 4
Summary of the experimental stress measurement data and results based on the modified fracture mechanics methods.
No.
HF parameters (MPa)
Calculated stress (MPa)
Error (%)
P
b
P
r
P
s
T
S
hmin
S
Hmax
R
S
Hmax
A
e
Hmax
R
e
hmin
A
HF-HS-1
13.01
7.75
6.16
5.31
6.16
10.28
11.45
5.32
17.29
HF-HS-3
14.56
9.79
7.41
4.77
7.41
10.10
13.97
2.05%
41.05%
HF-HS-4
10.61
7.69
5.34
2.92
5.34
10.96
11.49
11.40%
16.75%
HF-HS-5
9.70
7.83
5.72
1.87
5.72
10.91
18.49
10.07%
86.54%
Notes:SHmaxR is the maximum horizontal principal stress determined using the modified Rummel method; SHmaxA is the maximum horizontal principal stress obtained by the modified Abou-Sayed method; eHmaxR=SHmaxR−SHmax/SHmax, and eHmaxA=SHmaxA−SHmax/SHmax.The fracture morphology of the red sandstone specimens after the application of HF is represented in Figure7. The maximum horizontal principal stresses loaded on these specimens were the same; however, the minimum horizontal principal stresses differed. Specimen HF-HS-4 has the lowest horizontal stress ratio Shmin/SHmax, namely, 0.5, indicating a larger stress difference. We observed that the fracture of this specimen was nearly parallel to the maximum horizontal stress. Concerning specimen HF-HS-3, however, the stress ratio is the highest one which is close to 1. Therefore, the crack path of HF-HS-3 deviated considerably from the direction of the maximum horizontal principal stress, as shown in Figure 7. The fracture morphology of other specimens was similar to that of HF-HS-3 and HF-HS-4 except the differences in the crack length and deviation angles. A higher stress difference leads to a lower deviation of crack orientation from maximum horizontal principal stress.Figure 7
Fracture morphology of the red sandstone specimens after hydraulic fracturing.
(a)
HF-HS-1
(b)
HF-HS-3
(c)
HF-HS-4
(d)
HF-HS-5


### 4.3. Discussion

Although equation (2) is still frequently used in field tests, as it does not require performing tensile strength measurements in cored rocks, a number of studies have argued about its reliability, especially in the case where no fracture reopening is clearly observed. The approach based on considering the tensile strength to calculate SHmax is formulated in equation (1). It is deemed valid from the theoretical point of view, as long as a reliable tensile strength estimate can be obtained. The values of SHmax estimated based on the tensile strength combined with the conventional formula provided in equation (1) are represented in Figure 8 (purple triangles). The relative errors of SHmax are in the range of 8.9%–31.1%. However, if preexisting cracks appear, these conventional equations based on the tensile strength are not applicable despite their relative conformity with applied stresses.Figure 8
Estimated maximum horizontal principal stresses derived using the considered methods. Equations (1) and (2) correspond to the conventional methods for stress determination, while the modified Rummel and Abou-Sayed methods are the fracture mechanics approaches, expressed in equations (14) and (13), respectively.We compared the results obtained using four considered methods: two conventional and two fracture mechanics approaches (see Figure8). We observed that the results obtained using the modified Rummel method (red columns) are closest to the applied stress values. The maximum relative error of the modified Rummel method is approximately 11.4%, while the errors associated with the other three methods are all larger than 20%. The errors of the estimated are induced not only by the interpretation of pressure with respect to a time curve but also by the measurement of the preexisting crack length and its propagation. The modified Rummel method based on fracture mechanics and the incorporation of the confining effect achieve the considerably more precise estimation of horizontal maximum principal stress.The key parameter that may influence stress determination using the fracture mechanics method in laboratory and field tests is the fracture length. For granite samples, we located and identified hydraulic fractures using the ink dyeing method and micrometer. For sandstone samples, the crack propagation path could be clearly observed by the traces of injection fluid. Although these methods may seem rough, they could help us locate and identify the near-surface fractures. In our future work, we plan to upgrade our true triaxial testing apparatus by incorporating an acoustic emission (AE) system to detect and identify induced fractures and microcracks during HF tests. Moreover, the high-energy computed tomography (CT) scan can be used to observe the fracture morphology of postfracturing rock samples.


## 4.1. HF Test Results of Granites

To confirm the validity of the proposed HF testing system under triaxial loading, we first conducted two HF tests using granite samples. Figure4 represents the hydraulic pressure changes with the time of the considered granite specimens in true triaxial and hydraulic loading. Breakdown pressure Pb of two granite specimens HF-JG-2 and HF-JG-5 is 14.42 MPa and 18.73 MPa, respectively. The fracture reopening pressure Pr and instantaneous shut-in pressure Ps are also clearly observed in each pressure cycle and determined using the interactive software developed base Visual Basic [49]. The calculated HF parameters and stresses based on equations (2) and (3), respectively, are summarized in Table 3.Figure 4
Typical experimental pressure-time curves corresponding to the HF tests of granite specimens.Table 3
Summary of the experimental stress measurement data and the results based on equations (2) and (3).
No.
Rock type
HF parameters (MPa)
Calculated stress (MPa)
Applied stress (MPa)
Relative error (%)
P
b
P
r
P
s
T
S
Hmax
S
hmin
S
Hmax
S
hmin
e
Hmax
e
hmin
HF-JG-2
Granite
14.42
8.40
5.16
6.02
7.08
5.16
9.83
4.86
27.98
6.24
HF-JG-5
Granite
18.73
8.77
7.63
9.96
14.12
7.63
14.53
7.33
2.82
4.11
HF-HS-1
Sandstone
13.01
7.75
6.16
5.31
10.73
6.16
9.76
6.11
9.89
0.79
HF-HS-3
Sandstone
14.56
9.79
7.41
4.77
12.44
7.41
9.95
6.22
25.63
0.80
HF-HS-4
Sandstone
10.61
7.69
5.34
2.92
8.33
5.34
9.90
7.47
15.35
8.11
HF-HS-5
Sandstone
9.70
7.83
5.72
1.87
9.33
5.72
9.84
4.94
5.88
4.28
Notes:Pb—breakdown pressure; Pr—fracture reopening pressure; Ps—instantaneous shut-in pressure; T—tensile strength; Shmin—minimum horizontal principal stress; SHmax—maximum horizontal principal stress. Here, eHmax=SHmax1−SHmax/SHmax and ehmin=Shmin1–Shmin/Shmin.The maximum and minimum horizontal principal stresses applied on specimen HF-JG-2 are 9.83 and 4.86 MPa, respectively, while the corresponding calculated stresses from the HF tests are 7.08 and 5.16 MPa, respectively (Table3). The relative error associated with the determined maximum and minimum horizontal principal stresses is 27.98% and 6.24%, respectively. The calculated maximum and minimum horizontal principal stress estimates for specimen HF-JG-5 are 14.12 and 7.63 MPa, respectively, and the corresponding errors with respect to the applied maximum and minimum horizontal principal stress are 2.82% and 4.11%, respectively. The tensile strength values calculated based on the HF data are 6.02 MPa and 9.96 MPa, which are smaller than the tensile test results (namely, ~10.87 MPa).For granite samples, we located and identified the induced fractures using the ink dyeing method. Figure5 represents the fracture morphology after applying hydraulic fracturing to specimens HF-JG-2 and HF-JG-5. The length values of the cracks induced by the HF of these specimens were approximately 20 mm. However, the crack direction of HF-JG-2 was close to the maximum horizontal principal stress, while the orientation of the other specimen HF-JG-5 deviated by 45° from the maximum horizontal principal stress.Figure 5
Fracture morphology after the HF tests of specimens: (a) HF-JG-2 and (b) HF-JG-5.
(a)
(b)Considering the simulated critical injection pressure versus the fracture length as shown by Stoeckhert [46], the critical injection pressure generally increases with an increase in confining stress. In the conducted tests, the first injection cycle indicated the stable crack propagation, and the crack arrested finally. However, crack reopening occurred in the subsequent injection processes without further magnification of the crack length or rapidly reaching the boundary of the rock cube. Therefore, the crack lengths in the conducted tests are finite and deviated from the maximum principal stress orientation due to the confining stress effect.


## 4.2. Results of the HF Tests for Sandstones with Preexisting Cracks

Concerning the other considered specimens (red sandstones), the pressure-over-time curves are represented in Figure6. The breakdown pressure, instantaneous shut-in pressure, and fracture reopening pressure in each test are clearly visible with acceptable reproducibility in each cycle. The typical characteristic parameters of these tests were interpreted, as listed in Table 3. In the case of applying the conventional stress measurement of HF and equation (2), the stresses corresponding to each cubic specimen were determined (see Table 3). The stresses obtained using equations (2) and (3) are close to the applied ones (theoretical values). The relative errors of Shmin are less than 10%, while the errors of SHmax are slightly higher, falling in the range of 5.88%–25.63%. These results indicate that the conventional theory of HF based on the strength criterion can be used to approximately assess the stress values, particularly concerning the minimum horizontal principal stress.Figure 6
Pressure vs. time in the HF tests in the case of red sandstones.As we noted above, the maximum horizontal principal stresses are inaccurately estimated using the conventional stress measurement theory due to the presence of preexisting fractures, and hydraulic fractures may initiate and propagate along these preexisting cracks. Therefore, the fracture mechanics methods must be used in HF analysis in this case. We employed the modified approaches for HF stress determination to analyze the results of these experiments. Here, fracture toughness was revised by incorporating the confining stress effect. The modified Rummel and Abou-Sayed methods were both utilized to calculate the maximum principal stress. The estimated stress results are represented in Table4. The obtained results indicate that SHmax values obtained using the modified Rummel method are in the range of 10.1–10.96 MPa which are all lower than those determined through the modified Abou-Sayed method, thereby being closer to applied stresses (as shown in Table 4). The relative errors of SHmax obtained by the modified Rummel method are in the range of 2.0%–11.4%, while they are approximately in the range of 16.0%–86.5% in the case of the modified Abou-Sayed method. This finding indicates that the Rummel method based on the superposition of stress intensity factors corresponding to each loading source is more accurate than the Abou-Sayed method.Table 4
Summary of the experimental stress measurement data and results based on the modified fracture mechanics methods.
No.
HF parameters (MPa)
Calculated stress (MPa)
Error (%)
P
b
P
r
P
s
T
S
hmin
S
Hmax
R
S
Hmax
A
e
Hmax
R
e
hmin
A
HF-HS-1
13.01
7.75
6.16
5.31
6.16
10.28
11.45
5.32
17.29
HF-HS-3
14.56
9.79
7.41
4.77
7.41
10.10
13.97
2.05%
41.05%
HF-HS-4
10.61
7.69
5.34
2.92
5.34
10.96
11.49
11.40%
16.75%
HF-HS-5
9.70
7.83
5.72
1.87
5.72
10.91
18.49
10.07%
86.54%
Notes:SHmaxR is the maximum horizontal principal stress determined using the modified Rummel method; SHmaxA is the maximum horizontal principal stress obtained by the modified Abou-Sayed method; eHmaxR=SHmaxR−SHmax/SHmax, and eHmaxA=SHmaxA−SHmax/SHmax.The fracture morphology of the red sandstone specimens after the application of HF is represented in Figure7. The maximum horizontal principal stresses loaded on these specimens were the same; however, the minimum horizontal principal stresses differed. Specimen HF-HS-4 has the lowest horizontal stress ratio Shmin/SHmax, namely, 0.5, indicating a larger stress difference. We observed that the fracture of this specimen was nearly parallel to the maximum horizontal stress. Concerning specimen HF-HS-3, however, the stress ratio is the highest one which is close to 1. Therefore, the crack path of HF-HS-3 deviated considerably from the direction of the maximum horizontal principal stress, as shown in Figure 7. The fracture morphology of other specimens was similar to that of HF-HS-3 and HF-HS-4 except the differences in the crack length and deviation angles. A higher stress difference leads to a lower deviation of crack orientation from maximum horizontal principal stress.Figure 7
Fracture morphology of the red sandstone specimens after hydraulic fracturing.
(a)
HF-HS-1
(b)
HF-HS-3
(c)
HF-HS-4
(d)
HF-HS-5


## 4.3. Discussion

Although equation (2) is still frequently used in field tests, as it does not require performing tensile strength measurements in cored rocks, a number of studies have argued about its reliability, especially in the case where no fracture reopening is clearly observed. The approach based on considering the tensile strength to calculate SHmax is formulated in equation (1). It is deemed valid from the theoretical point of view, as long as a reliable tensile strength estimate can be obtained. The values of SHmax estimated based on the tensile strength combined with the conventional formula provided in equation (1) are represented in Figure 8 (purple triangles). The relative errors of SHmax are in the range of 8.9%–31.1%. However, if preexisting cracks appear, these conventional equations based on the tensile strength are not applicable despite their relative conformity with applied stresses.Figure 8
Estimated maximum horizontal principal stresses derived using the considered methods. Equations (1) and (2) correspond to the conventional methods for stress determination, while the modified Rummel and Abou-Sayed methods are the fracture mechanics approaches, expressed in equations (14) and (13), respectively.We compared the results obtained using four considered methods: two conventional and two fracture mechanics approaches (see Figure8). We observed that the results obtained using the modified Rummel method (red columns) are closest to the applied stress values. The maximum relative error of the modified Rummel method is approximately 11.4%, while the errors associated with the other three methods are all larger than 20%. The errors of the estimated are induced not only by the interpretation of pressure with respect to a time curve but also by the measurement of the preexisting crack length and its propagation. The modified Rummel method based on fracture mechanics and the incorporation of the confining effect achieve the considerably more precise estimation of horizontal maximum principal stress.The key parameter that may influence stress determination using the fracture mechanics method in laboratory and field tests is the fracture length. For granite samples, we located and identified hydraulic fractures using the ink dyeing method and micrometer. For sandstone samples, the crack propagation path could be clearly observed by the traces of injection fluid. Although these methods may seem rough, they could help us locate and identify the near-surface fractures. In our future work, we plan to upgrade our true triaxial testing apparatus by incorporating an acoustic emission (AE) system to detect and identify induced fractures and microcracks during HF tests. Moreover, the high-energy computed tomography (CT) scan can be used to observe the fracture morphology of postfracturing rock samples.


## 5. Conclusions

The determination of in situ stress using the HF tests based on the conventional method has raised considerable controversies with regard to field tests, especially in the case of preexisting fractures. To address this issue, in the present paper, we proposed a modified fracture mechanics approach for stress determination considering the confinement effect in a deep hole. The relation between fracture toughness and confining pressure was incorporated by including dimension transformation. Then, we developed the modified Rummel and Abou-Sayed methods based on fracture mechanics.Moreover, we conducted the true triaxial HF tests under different stress ratios for the granite and sandstone specimens to verify the applicability of the proposed approaches. We analyzed the typical pressure-time curves that indicated the steady fracture growth in these HF tests. Stresses were determined through the proposed fracture mechanics approaches and the methods based on conventional continuum mechanics. In the case without preexisting cracks, the conventional methods are deemed acceptable. In other cases with the preexisting fractures included, the modified fracture mechanics methods are confirmed to be reasonable and applicable. Analyzing the stress determination results obtained through different methods, we observed that the stresses determined using the modified Rummel method exhibit the lowest relative error and are consistent with applied stresses. The modification of the Rummel fracture mechanics method for stress determination allows obtaining the considerably precise estimation of maximum horizontal stress in HF stress measurement.



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