Applied Science and Innovative Research 

ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) 

Vol. 8, No. 3, 2024 

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81 

Original Paper 

Research on the Influencing Factors of Pulsed Hydraulic 

Fracturing Effects in Hot Dry Rock Based on Numerical 

Simulation  

Qinqin Zou1, Liliang Gong2 & Ziwang Yu1* 

1 College of Construction Engineering, Jilin University, Changchun, 130026, China 

2 Jianghe Anlan Engineering consulting Co., Ltd, Zhengzhou, 450003, China 

* Corresponding author, Ziwang Yu, College of Construction Engineering, Jilin University, Changchun, 

130026, China 

E-mail: yuziw@jlu.edu.cn  

 

Received: June 15, 2024          Accepted: July 12, 2024          Online Published: July 17, 2024 

doi:10.22158/asir.v8n3p81            URL: http://doi.org/10.22158/asir.v8n3p81 

 

Abstract 

Geothermal energy, representing one of the five principal non-carbon-based renewable resources, is 

crucial in the global energy transition and in mitigating greenhouse gas emissions. Hot Dry Rock 

(HDR) reservoirs, rich in geothermal resources, predominantly utilize hydraulic fracturing, the most 

extensively researched and applied technology. However, fracturing these robust HDR formations 

poses significant challenges compared to traditional hydraulic fracturing in sedimentary rocks in the 

oil and gas industry. Cyclic pulse fracturing, an innovative mode of fracturing, has received limited 

academic attention, particularly regarding its permeability enhancement mechanism under varying 

thermal and pressure conditions, which remains largely unexplored. Building on prior laboratory 

experiments, this study employs the pore pressure cohesive zone model within ABAQUS software to 

elucidate the initiation and expansion of hydraulic fractures. Numerical simulations investigated the 

influence of critical parameters such as pulse pattern, temperature, geostress differential, and 

confining pressure on fracture propagation. The findings indicate that cyclic pulse hydraulic fracturing 

can significantly reduce the breakdown pressure in HDR, with reductions up to 37.68% under certain 

conditions. Notably, at constant amplitude, increased frequency exacerbates damage to hot dry rocks; 

conversely, at constant frequency, higher amplitudes result in greater damage. Additionally, pulse 

hydraulic fracturing improves the fracturing process, yielding a more intricate fracture network. In 

HDR pulse hydraulic fracturing, the temperature disparity between the fracturing fluid and the rock 

markedly impacts the breakdown pressure. Cold water interacting with a high-temperature geo-thermal 

reservoir triggers a thermal shock effect, diminishing initiation pressure and expediting fracture 



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development and expansion. The impact of geostress differential and confining pressure on pulse 

hydraulic fracturing is complex, influencing not only the breakdown pressure but also the direction and 

extent of fracture expansion. 

Keywords 

hydraulic fracturing, hot dry rock, numerical simulation, multistage cyclic pulses  

 

1. Introduction 

Geothermal resources encompass geothermal energy, fluids, and their beneficial components found on 

Earth that can be economically harnessed by humans. These resources are classified into three 

categories: shallow geothermal resources, hydrothermal resources, and dry-heat rock resources. The 

latter category comprises high-temperature rock bodies typically exceeding 180°C, located at depths of 

several kilometers, and possessing minimal or no internal subsurface fluids, which are dense and 

impermeable (Wu et al., 2012; Yang et al., 2023). Geothermal energy is utilized in a variety of 

applications including power generation, heating and cooling, spa treatments, and industrial processes. 

As one of the five principal non-carbon-based renewable energy sources, geothermal energy plays a 

crucial role in facilitating the global energy transition and mitigating greenhouse gas emissions. 

Enhanced Geothermal Systems (EGS) are pivotal in developing geothermal resources within dry, hot 

rock (HDR) formations. The fundamental process involves drilling and fracturing the reservoir to 

create a sizable and complex fracture network, thereby increasing the permeability of the thermal 

reservoir formations (Gong et al., 2023). Reservoir modification plays a critical role in various major 

subterranean energy projects, including EGS, oil shale, and shale gas developments, where hydraulic 

fracturing emerges as the most extensively researched and applied technology. Unlike traditional oil 

and gas fields, where sedimentary rocks are subjected to hydraulic fracturing, the technique faces 

significant challenges when applied to the harder, dry, and hot rock environments typical of HDR. 

In exploring hydraulic fracturing methods, Arno Zang introduced the concept of Fatigue Hydraulic 

Fracturing (FHF) in 2013, proposing various fluid injection schemes, such as cyclic, progressive, and 

fatigue hydraulic fracturing. This approach, differing from traditional methods with steadily increasing 

injection pressure, tends to create broader fracture zones, showcasing that varying injection schemes 

significantly impact hydraulic fracture development (Zang et al., 2013). In 2015, Reza Safari's 

evaluation of the geomechanical aspects of pulse fracturing through numerical simulations revealed 

that pulsed loading could lead to a ductile/brittle transition in shale rock's fracture network (Safari et al., 

2015). Luke P. Frash furthered this research by hydraulically augmenting granite samples with 

mechanical impulse hydraulic fracturing (MIHF) as a re-stimulation technique to enhance well 

injection capacity (Frash et al., 2015). In 2017, Chen et al. employed a three-axis loaded impulse 

hydraulic fracturing test system to explore the effects of varying pulse frequency and fluid viscosity on 

coal rock, revealing that pulse hydraulic fracturing promotes mutant crack expansion and reaming 

effects (Jiang-zhan et al., 2017). Zimmermann et al. conducted tests in the Äspö hard rock laboratory 

using new cyclic and pulse injection schemes in 2019, followed by an optimization study 



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(Zimmermann et al., 2018). In 2020, Kang et al. at MIT showed that cyclic fracturing could lower 

in-situ fracture pressure in sandstone samples (Kang et al., 2020). In 2022, Chang et al. investigated 

black shale's hydraulic fracturing from the Sichuan Basin's Lower Silurian Longmaxi Formation, 

employing constant-flow, cyclic, shut-in interval, and low-frequency pulse booster injections. They 

found cyclic fracturing more likely to induce fatigue damage, creating subcritical microfractures that 

expand into macrofractures, with cyclic water injection reducing rock fracture pressure by 

approximately 24%. Although shut-in well intermittent water injection also reduced pressure, it was 

less effective in enhancing fracture complexity. Pulse fracturing yielded the most intricate fracture 

patterns among the tested schemes, with rapid pressure changes promoting multiple fractures around 

the borehole (Chang et al., 2022). Despite these advancements, cyclic pulse fracturing's permeability 

enhancement mechanisms under various temperatures and pressures remain largely unexplored. 

In the current scenario, significant opportunities exist to enhance the permeability optimization of 

hydraulic fracturing technology for Enhanced Geothermal Systems (EGS) reservoir modification. The 

hydraulic fracturing process, characterized by varying fluid injection patterns and fluid seepage rates 

within the rock, generates diverse fracture pressures. By deviating from the conventional constant-flow 

injection scheme and integrating monitoring arrays within the fracture monitoring site, along with the 

deployment of efficient pulse and fatigue fracturing techniques, an improved injection strategy can be 

devised for permeability enhancement. However, these methods are currently in the stages of 

theoretical exploration and preliminary testing, with engineering construction predominantly reliant on 

engineers' experience and field feedback, indicating an urgent need for technological advancements. 

This paper investigates the impact of varying the pulsing pattern, temperature, geostress differences, 

and confining pressures on the expansion of fractures in dry hot rocks. By comparing fracture pressure, 

length, and the number of fracture units under diverse conditions, it unveils the pattern of enhanced 

permeability in dry hot rocks post-hydraulic fracturing. This study offers solutions to complex and 

practical engineering challenges, facilitating the achievement of optimal fracturing outcomes and 

guiding engineering practices effectively. 

 

2. Theoretical Background 

2.1 Theoretical Foundations 

The two-dimensional elastic stresses around a circular cavity within a homogeneous linear elastic solid, 

as described by Kirsch's well-known solution (1898), can be accurately determined. While the stress 

state in laboratory triaxial hydraulic fracturing tests inherently involves three dimensions, it can be 

effectively reduced to a two-dimensional stress state on a planar plate perpendicular to the horizontal 

drilling axis, as illustrated in Figure 1. 



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Figure 1. Conversion from 3-D to 2-D 

 

Therefore, we can derive the following equation based on Kirsch's theory: 

 

𝜎𝑟 =
(𝜎𝐻 + 𝜎ℎ)(1 −

𝑟𝑤
2

𝑟2)

2
−

(𝜎𝐻 − 𝜎ℎ) (1 − 4
𝑟𝑤

2

𝑟2 + 3
𝑟𝑤

4

𝑟4) cos 2𝜃

2
+ 𝑃𝑤

𝑟𝑤
2

𝑟2
 

(1) 

 

𝜎𝜃 =
(𝜎𝐻 + 𝜎ℎ)(1 +

𝑟𝑤
2

𝑟2)

2
−

(𝜎𝐻 − 𝜎ℎ) (1 + 3
𝑟𝑤

4

𝑟4) cos 2𝜃

2
− 𝑃𝑤

𝑟𝑤
2

𝑟2
 

(2) 

When the absolute value of 𝜎𝜃 is greater than the tensile strength 𝑇, tensile fracture of the hole wall 

occurs. Therefore, when 𝜃 = 0 or 𝜋 and 𝑟 = 𝑟𝑤, the breakdown pressure is: 

 𝑃𝑏 = 3𝜎ℎ − 𝜎𝐻 − 𝜎𝜃 = 3𝜎ℎ − 𝜎𝐻 − 𝑇 (3) 

 
휀𝑟 =

1

𝐸
(𝜎𝑟 − 𝑣𝜎𝜃) 

휀𝜃 =
1

𝐸
(𝜎𝜃 − 𝑣𝜎𝑟) 

(4) 

Therefore, we can program MATLAB to calculate the stresses and strains of the single-hole model 

based on the above equations. 

Where 𝜎𝑟 is the radial stress; 𝜎𝜃 is the tangential stress; 휀𝑟 is the radial strain; 휀𝜃 tangential strain; 

𝜎𝐻 is the maximum horizontal principal stress; 𝜎ℎ is the minimum horizontal principal stress; 𝑟𝑤 is 

the radius of the hole; 𝑟 is the distance from the hole center to the measurement point; 𝑃𝑤 is the 

injection pressure in the hole; 𝑃𝑏 is the breakdown pressure; 𝐸 is Young's modulus; 𝑣 is Poisson's 

ratio; 𝜎𝑟𝑖 is radial induced stress; 𝜎𝜃𝑖 is tangential induced stress; 휀𝑟𝑖 is the radial induced strain; 

휀𝜃𝑖 is the tangential induced strain; 𝜃 is the angle between the line between the measurement point 

and the center of the hole and the direction of the maximum principal stress. 

2.2 Flow Equations 

In hydraulic fracturing, intra-fracture fluid pressure serves as the primary energy source for fracture 

expansion. Assuming the fracturing fluid is incompressible and neglecting the impacts of fluid inertia 

and fracture wall roughness, flow within the viscous domain is categorized into tangential flow along 

the viscous unit and normal flow perpendicular to the viscous unit, as illustrated in Figure 2. 



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Figure 2. The Flow Patterns of Pore Fluid in Cohesive Elements (Ma, 2023) 

 

Tangential flow equations: 

Specifically, the flow and pressure gradients in the tangential direction are consistent with the cubic 

seepage equation, which can be expressed as (Ma et al., 2023): 

 
𝑞 = −

𝑤3

12𝜇
∆𝑝𝑓 (5) 

Where 𝑞 denotes the volume flow rate through the fracture cross-section, 𝑤 denotes the fracture 

width that varies continuously with time, 𝜇 denotes the viscosity of the fracturing fluid, and ∆𝑝𝑓 

denotes the fluid pressure gradient along the viscous zone. 

Normal flow equation: 

Fluid in the fracture flows mainly in the tangential direction, and a small amount of fluid filters out 

through the normal flow of the fracture to the surface layer above and below the viscous unit. The 

normal flow of fracturing fluid in a viscous unit is as follows: 

 𝑞𝑡 = 𝑐𝑡(𝑝𝑓 − 𝑝𝑡) 

𝑞𝑏 = 𝑐𝑏(𝑝𝑓 − 𝑝𝑏) 
(6) 

Where 𝑐𝑡, 𝑐𝑏 denote the filtration loss coefficients of the fracturing fluid in the upper and lower walls 

of the fracture, respectively; 𝑝𝑓 denotes the fluid pressure of the fracturing fluid in the fracture; 𝑝𝑡 , 𝑝𝑏 

denote the pore pressures in the upper and lower walls of the fracture, respectively. 

2.3 Elastic-plastic Damage Model for Viscous Units 

In this paper, a quadratic correlation function of the nominal stress ratio is utilized to determine if a 

crack will open. Complete damage is assumed when the sum of the squares of the critical stress ratios 

is equal to one. This criterion can be expressed as (Li et al., 2017): 

 
{

〈𝑡𝑛〉

𝑡𝑛
0

}

2

+ {
𝑡𝑠

𝑡𝑠
0

}
2

+ {
𝑡𝑡

𝑡𝑡
0}

2

= 1 (7) 

Where 𝑡𝑛
0 is the normal critical stress (rock tensile strength); 𝑡𝑠

0 and 𝑡𝑡
0 are the critical shear stresses; 

and the symbol  indicates that the pure compressive deformation or stress state does not initiate 

damage. 

When the stress state of the unit satisfies the above equation, the unit starts to incur damage, and as the 

separation displacement continues to increase, both the unit stiffness and the load carrying capacity 

continue to decrease, and eventually fail completely. By introducing the damage variable D to describe 



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the unit damage evolution process, the actual stress state of the unit is as follows (Camanho et al., 

2003). 

 
𝑡𝑛 = {

(1 − 𝐷)𝑡�̅�，𝑡�̅� ≥ 0

𝑡�̅�，                𝑡�̅� < 0 
 

𝑡𝑠 = (1 − 𝐷)𝑡�̅� 

𝑡𝑡 = (1 − 𝐷)𝑡�̅� 

(8) 

Where 𝑡�̅�, 𝑡�̅�, and 𝑡�̅� are the stress components estimated by the traction separation criterion before 

the damage; 𝑡𝑛, 𝑡𝑠, and 𝑡𝑡 are the actual stresses in the normal direction and the two shear directions, 

respectively; and D is the damage variable. 

Figure 3 illustrates the intrinsic curves depicting the relationship between effective traction and 

effective displacement in the viscous unit, where 𝐸 is the unit stiffness with respect to the geometric 

parameters; 𝛿𝑚
0  is the effective displacement at the time of the initial damage; and 𝛿𝑚

𝑓
 is the effective 

displacement at the time of complete separation. 

 

 

Figure 3. Ontological Relationship Curve of the Cohesion Unit 

 

3. Finite Element Modeling of Dry Hot Rock Fracturing 

Experiments, while valuable, are often time-consuming, labor-intensive, and costly. In contrast, 

numerical simulations offer a more time-efficient, cost-effective, and manageable approach, especially 

under complex conditions that challenge experimental setups. The finite element method (FEM) has 

emerged as a particularly effective simulation technique for hydraulic fracturing, as evidenced by its 

extensive application in the literature (Bao et al., 2014; Chen, 2012; Shauer & Duarte, 2019). Figure 4 

illustrates the finite element model, constructed based on the dimensions of the rock samples used in 

experiments. This model centrally places the injection holes and employs pulse pattern, temperature, 

geostress differential, and confining pressure as variable parameters to simulate the fracturing process. 



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Figure 4. Hydraulic Fracturing Cohesive Element Model 

 

The seepage-stress unit within ABAQUS represents the predominant choice for hydraulic fracturing 

simulations currently. However, this unit lacks a temperature feature. In response, this study leverages 

breakdown pressure data of rocks at various temperatures (referenced in Table 1 from existing literature 

(Cheng et al., 2021)) to calibrate the model's damage intensity. Such calibration facilitates the 

integration of temperature considerations into the hydraulic fracturing simulation, enabling a more 

nuanced understanding of the temperature's impact on rock behavior. 

 

Table 1. Fracturing Test Parameters and Breakdown Pressures (Cheng, 2021) 

Test 

No 

Injection Flow Rate 

(ml/min) 

Temperature 

(℃) 

Confining stresses 

(MPa) 

Breakdown pressure 

(MPa) 

1 10 30 4/12 30.28 

2 10 60 4/12 29.49 

3 10 90 4/12 28.32 

4 10 120 4/12 25.24 

5 10 150 4/12 24.68 

 

3.1 Model Setup and Validation 

In the simulations presented in this study, the Abaqus software's pore pressure cohesive unit is 

employed to model the initiation and extension of hydraulic cracks, as well as the tangential flow of 

fluid within these cracks and the normal filtration at their surfaces. The model's damage behavior 

follows the traction-separation criterion, while the fracture criterion is based on the quadratic nominal 

stress criterion, facilitating the simulation of hydraulic crack propagation. Additionally, the 

determination of hydraulic crack length and the quantification of damaged units are conducted through 

post-processing. The essential parameters utilized in the model are detailed in Table 2. 

 



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Table 2. Basic Parameters of the Model 

Parameter name 
parameter 

value 
Parameter name 

parameter 

value 

Model size (m) 0.3 × 0.3 
Fracturing fluid viscosity 

(Pa·s) 
0.001 

Mesh basic dimensions (m) 0.003 Injection length (m) 0.15 

Reservoir permeability（mD） 0.034 Cohesive cell thickness (m) 0.001 

Reservoir porosity (%) 3.22 
Viscosity regularization 

factor 
0.0001 

Specific gravity of fracturing fluid 

(N/m3) 
9800 In situ stress (MPa) 4/8/12 

Reservoir modulus of elasticity 

(GPa) 
39.99 Initial Pore Ratio 0.033 

Reservoir Poisson's ratio 0.28 
Fracturing fluid flow 

(ml/min) 
10 

Damage intensity (MPa) 8.5/20/20 Reservoir temperature (℃) 150 

Unit damage displacement (m) 0.001 
Fracturing fluid temperature 

(℃) 
20 

Coefficient of leaching 1 × 10−14 Injection time (s) 1 

 

To ascertain the reliability of our cohesive unit model, we conducted a series of hydraulic fracturing 

simulations at varying temperatures. The outcomes of these simulations were juxtaposed with 

laboratory experimental results to confirm the model's validity. Parameters included an injection flow 

rate of 10 ml/min, a maximum confining pressure of 12 MPa, and a minimum confining pressure of 4 

MPa. 

Figure 5 illustrates the comparison between the experimental and simulated fracture pressures across 

varying temperatures, revealing a maximum error of 3.65%. This paper also includes a validation study 

from Ren et al. on cyclic pulse hydraulic fracturing (Ren et al., 2023), presented in Figure 6, which 

compares experimental and simulated breakdown pressures under different fracturing modes; here, the 

discrepancy is reported at 5.47%. Furthermore, similar numerical simulation models have been 

corroborated by previous studies (Xue et al., 2023; Zhang et al., 2021), which consistently align the 

simulated values closely with experimental outcomes. These findings affirm that the numerical model 

is robust and the errors are within acceptable bounds. 



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Figure 5. Comparison of Experimental Breakdown Pressure and Simulated Breakdown Pressure 

at Different Temperatures 

 

 

Figure 6. Comparison of Experimental Breakdown Pressure and Simulated Breakdown Pressure 

under Different Fracturing Modes 

 

3.2 Numerical Simulation of Pulsed Hydraulic Fracturing 

Pulse hydraulic fracturing is an emerging technique that offers the advantages of lower fracture 

initiation pressures and enhanced fracture expansion post-compression, as detailed in references (Gao 

et al., 2022; Wei et al., 2023). This study investigates four distinct fracturing schemes, depicted in 

Figure 7. The initial scheme is the conventional injection method, where the injection rate uniformly 

increases to 10 ml/min within 0-0.2 seconds and then stabilizes. The subsequent schemes—second, 

third, and fourth—are multi-stage cyclic pulse injection methods. These stages are set to 20%, 40%, 

60%, and 80% of the conventional test rate. Initially, the amplitude size of the pulse intervals is set at 

10% and 20% of the interval's maximum value, with pulse frequencies of 3 Hz and 5 Hz. The pressure 

increments during the test proceed as follows: the first stage increases to 20%, the second stage from 20% 

to 40%, the third stage from 40% to 60%, and the fourth stage from 60% to 80%. 



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Figure 7. Four Different Injection Schemes Applied in Hydraulic Fracturing Simulations 

 

A total of thirty-two simulation tests, detailed in Table 3, were conducted to examine the hydraulic 

fracturing characteristics of dry hot rocks, focusing on the influences of pulse pattern, temperature, 

geostress difference, and confining pressure. Tests 1 to 20 evaluated the effect of pulse pattern on 

hydraulic fracturing under varying temperature conditions, maintaining a maximum principal stress of 

10 MPa and a minimum of 6 MPa, across four fracturing modes: continuous flow at 10 ml/min, Pulse1 

(low-frequency, low-amplitude), Pulse2 (high-frequency, low-amplitude), and Pulse3 (high-frequency, 

high-amplitude). Tests 17 to 28 investigated the impact of geostress difference, maintaining constant 

temperature and fracturing mode, with three stress ratios examined: 10 MPa/6 MPa, 10 MPa/8 MPa, 

and 10 MPa/10 MPa. Lastly, tests 25 to 32 assessed the role of confining pressure under uniform 

temperature and fracturing conditions, exploring two pressure settings: 10 MPa and 6 MPa. 

 

 

 

 

 

 



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Table 3. Hydraulic Fracturing Simulation Results 

Test 

No 

water 

injection 

mode 

Temperature 

(℃) 

Maximum/minimum principal 

stress（MPa） 

Breakdown pressure 

(MPa) 

1 10ml/min 

30 

10/6 

33.16 

2 Pulse 1 30.17 

3 Pulse 2 28.60 

4 Pulse 3 24.19 

5 10ml/min 

60 

32.43 

6 Pulse 1 28.62 

7 Pulse 2 26.46 

8 Pulse 3 22.66 

9 10ml/min 

90 

29.57 

10 Pulse 1 24.95 

11 Pulse 2 24.08 

12 Pulse 3 19.80 

13 10ml/min 

120 

28.03 

14 Pulse 1 23.13 

15 Pulse 2 22.33 

16 Pulse 3 18.30 

17 10ml/min 

150 

27.55 

18 Pulse 1 21.94 

19 Pulse 2 19.58 

20 Pulse 3 17.17 

21 10ml/min 

150 10/8 

30.78 

22 Pulse 1 24.36 

23 Pulse 2 23.56 

24 Pulse 3 19.11 

25 10ml/min 

150 10/10 

33.55 

26 Pulse 1 28.69 

27 Pulse 2 26.38 

28 Pulse 3 25.18 

29 10ml/min 

150 6/6 

25.03 

30 Pulse 1 22.11 

31 Pulse 2 20.12 

32 Pulse 3 17.29 

 



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4. Simulation Results and Analysis 

4.1 Effect of Pulse Pattern 

The four distinct fracturing schemes employed in this study are illustrated in Figure 7. The first scheme 

is a conventional injection method, while the subsequent schemes—second, third, and fourth—are 

multi-stage cyclic pulse injection schemes. These methods were selected based on prior experimental 

evidence indicating that cyclic injection reduces the magnitude of seismic events. Although the 

improvement in hydraulic performance is less pronounced than with continuous injection, pulse 

pressurization effectively enhances permeability (Zhuang et al., 2020). 

Figure 8a presents simulation results from tests 1 through 20, demonstrating that the cyclic pulse water 

injection method significantly reduces the breakdown pressure in hydraulic fracturing of dry hot rock. 

A comparative analysis between pulse1 and pulse2 shows that increasing the pulse frequency from 3 

Hz to 5 Hz further reduces the breakdown pressure. Similarly, an increase in pulse amplitude from 10% 

to 20% (pulse2 to pulse3) results in additional reductions. At a rock temperature of 150°C, modifying 

the pulse pattern decreases the breakdown pressure from 27.55 MPa to 17.17 MPa, a 37.68% reduction. 

Furthermore, Figures 8b, c, and d indicate that high-frequency and high-amplitude pulse injections 

enhance the expansion of the hydraulic fracture network. Specifically, at 150°C, the number of fracture 

rupture units increased from 76 to 121, and the fracture length grew from 234.73 mm to 367.84 

mm—both metrics showing more than a 50% increase due to changes in pulse pattern. 

 

 

a. Effect of pulse pattern on fracture breakdown 

pressure 

b. Number of fracture rupture units and fracture 

lengths in pulse1 mode at 150℃ 



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c. Number of fracture rupture units and fracture 

length in pulse2 mode at 150℃ 

d. Number of fracture rupture units and fracture 

length in pulse3 mode at 150℃ 

   

Pulse1 Pulse2 Pulse3 

e. Hydraulic fracture under geostress of 6/10MPa at 150℃ 

Figure 8. Effect of Pulse Pattern on Crack Extension 

 

4.2 Effect of Temperature 

Fracturing construction in dry-heat rock formations diverges significantly from conventional hydraulic 

fracturing due to the unique conditions of high temperature and high pressure in dry-heat environments. 

Specifically, injecting low-temperature fracturing fluid into a high-temperature formation induces 

significant thermal stress, which influences the expansion of fractures. Additionally, the impact of this 

low-temperature fluid varies depending on the rock's mechanical properties, leading to distinct fracture 

expansion behaviors under thermal stress (Zhang et al., 2023; Zhang et al., 2019). 

This study investigates the impact of temperature on the expansion of the fracture network in pulsed 

hydraulic fracturing within dry hot rock, analyzing fracture pressure, the number of fracture units, and 

fracture lengths. The model's rock temperatures vary from 30 ℃ to 150 ℃, while the fracturing fluid, 

maintained at room temperature, creates an increasing temperature differential with the rock as the rock 

temperature rises. Figure 9a demonstrates that with rising rock temperatures, observed in Tests 1 to 20, 

the fracture pressure in dry hot rock declines. Furthermore, Figure 9b reveals that as the temperature 

increases in the pulse1 fracturing mode, the number of fracture units rises from 52 to 76, and fracture 

lengths extend from 163.97 mm to 234.73 mm. Across the entire fracture network in dry hot rock, 

higher rock temperatures correlate with longer total fracture lengths when the fracturing fluid 



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temperature remains constant. The findings indicate that a larger temperature difference between the 

fracturing fluid and the rock enhances fracture network expansion. Additionally, rock temperature 

significantly affects the breakdown pressure of granite specimens, with the breakdown pressure 

decreasing as rock temperature increases under specific injection flow rates, stress conditions, and 

fracturing modes. 

 

 

a. Effect of dry hot rock temperature on fracture 

pressure 

b. Effect of dry hot rock temperature on the 

number of fracture units and fracture lengths 

   

30℃ 60℃ 90℃ 

  

120℃ 150℃ 

c. Hydraulic cracking in pulse1 pulse mode at different temperatures 

Figure 9. Effect of Dry Hot Rock Temperature on Crack Extension 

 

 



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4.3 Effect of Geostress Differential 

The impact of geostress differential on pulsed hydraulic fracturing-induced fractures is both complex 

and significant. Geostress differential arise from variations in formation stresses across different 

directions. These differentials influence the initiation, propagation, and orientation of fractures during 

hydraulic fracturing. Notably, fractures tend to propagate towards the direction of the highest principal 

stress, potentially causing deviations in fracture orientation when significant geostress differences exist. 

Additionally, geostress differential elevate the fracture pressure required, thereby increasing the volume 

and cost of the necessary fracturing fluid, and complicating the operation. Moreover, large geostress 

differential can hasten fracture closure post-operation, diminishing fracture conductivity, and 

destabilize fractures, making them more prone to premature closure or transformation into other forms. 

Consequently, understanding the role of geostress differential in hydraulic fracturing demands a 

thorough assessment of geological conditions, fracturing parameters, and formation stresses. 

Figure 10a illustrates the simulation results from Experiments 17 to 28, plotting geostress differential 

against breakdown pressure on the x-axis and y-axis, respectively. The scatter plots, categorized under 

four fracturing modes, are presented and analyzed in Figure 10a. It is evident that geostress differential 

significantly influences fracture extension, with breakdown pressure inversely related to the increase in 

stress difference (Li et al., 2015). At 150°C in pulse3 mode, the breakdown pressure for dry hot rock is 

25.18 MPa at a geostress differential of 0 MPa and decreases to 17.17 MPa at a 4 MPa difference, a 

reduction of 31.81%. 

Figure 10b depicts the expansion of pulse fracturing cracks under varying geostress differential, 

showing that larger stress differences correlate with more uniform crack morphology, clearer 

directional tendencies, and a broader main crack extension. At a 4 MPa geostress differential, the 

maximum geostress direction predominantly influences the fracture extension; at 2 MPa, the fracture 

morphology around the injection point becomes more complex; and at 0 MPa, fracture morphology 

exhibits significant randomness. This study confirms that the primary fracture in hydraulic fracturing 

consistently extends in the direction of the maximum principal stress, favoring the formation of a single 

flat fracture under high stress differences and a network of fractures under lower stress differences. 

 

 



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a. The effect of geostress differential on crack breakdown pressure 

   

6/10MPa 8/10MPa 10/10MPa 

b. Hydraulic cracking in pulse3 pulse mode with different geostress differential at 150°C 

Figure 10. Effect of Geostress Differential on Crack Extension 

 

4.4 Effect of Confining Pressure 

Figure 11a illustrates the simulation results from Experiments 25 to 32, plotting confining pressure 

against fracture pressure on the y-axis in a bar chart for four fracturing modes. The chart clearly 

demonstrates that confining pressure significantly impacts crack propagation, evidenced by the increase 

in fracture pressure as confining pressure rises. Figure 11b depicts at 150°C under the pulse3 mode, a 

confining pressure of 6 MPa results in two prominent primary cracks along the horizontal direction and 

an emerging secondary crack vertically; at 10 MPa, a crack develops in both the horizontal and vertical 

directions. 

The analysis reveals that confining pressure's influence on fracture pressure and crack propagation in 

pulse hydraulic fracturing is a sophisticated physical process. An increase in confining pressure 

intensifies the stress environment surrounding the rock, consequently elevating the requisite pressure 

for fracturing. In this dynamic, the confining pressure in the direction of minimum principal stress 

critically facilitates the initiation and expansion of cracks. Moreover, a rise in confining pressure boosts 

the rock's elastic modulus, influencing crack propagation. The rock's compressive strength also 

escalates with confining pressure, necessitating higher pressures for fracturing. Additionally, confining 

pressure modifies the crack propagation mode, affecting crack length, width, and orientation relative to 



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the principal stress direction. Typically, higher confining pressures produce straighter and longer cracks, 

whereas lower pressures lead to more intricate crack pathways. Therefore, confining pressure 

profoundly affects fracture pressure and crack development during the hydraulic fracturing process. 

 

 

a. Effect of confining pressure on fracture breakdown pressure 

  

6/6MPa 10/10MPa 

b. Hydraulic fracture under different confining pressure pulse3 injection mode at 150℃ 

Figure 11. Effect of Confining Pressure on Crack Extension 

 

5. Conclusion 

Rock fracturing and crack propagation are crucial processes in hydraulic fracturing. In this paper, we 

investigate the pulsed hydraulic fracturing of hot dry rock by varying the pulsing pattern, temperature, 

geostress differential, and confining pressure. Based on the analysis of simulation results, the following 

conclusions can be drawn: 

(1) The employment of cohesive elements in hydraulic fracturing simulations offers a detailed and 

reliable methodology for analyzing and predicting rock fracturing behaviors. This cohesive element 

model utilizes a second-order stress criterion to assess the initiation of fractures. It incorporates a 

damage variable, D, to delineate the damage evolution within units, taking into account the rock's 

microstructure and crack development. Consequently, the model accurately depicts the initiation, 

expansion, and failure processes of cracks at a microscopic scale, enhancing the realism of rock 

fracture simulations. 



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(2) Compared to traditional hydraulic fracturing, pulsed hydraulic fracturing offers several distinct 

advantages. This technique can significantly reduce the breakdown pressure in hot dry rocks, achieving 

reductions up to 37.68% under specific conditions. Notably, at constant amplitude, increased frequency 

exacerbates damage to hot dry rocks; conversely, at constant frequency, higher amplitudes result in 

greater damage. Pulsed hydraulic fracturing not only improves the efficiency of fracturing but also 

fosters the development of a more intricate network of fractures. Furthermore, this method has the 

potential to lessen environmental impacts. 

(3) In the pulse hydraulic fracturing of hot dry rocks, the temperature differential between the 

fracturing fluid and the rocks markedly influences the breakdown pressure. As the rock temperature in 

the model escalates from 30°C to 150°C, the breakdown pressure decreases by approximately 30%. 

Notably, the introduction of cold water into a high-temperature geothermal reservoir triggers a thermal 

shock effect. This effect significantly accelerates the initiation and expansion of fractures within the 

reservoir, thereby enhancing the effectiveness of fracturing in the hot dry rock reservoir. 

(4) The impact of geostress differential and confining pressure on pulsed hydraulic fracturing in hot dry 

rocks is multifaceted, influencing not only the breakdown pressure and the morphology of the fracture 

face, but also the direction of fracture propagation and the aperture size. Studies have shown that an 

increase in geostress differential leads to a gradual decrease in breakdown pressure. In high-frequency, 

high-amplitude pulsed fracturing scenarios, fracture length can extend by up to 50%. Conversely, an 

increase in confining pressure results in higher breakdown pressure and more complex fracture 

propagation paths. 

 

Acknowledgments 

This work was supported by the National Natural Science Foundation of China (NSFC) [grant numbers 

42172274, 42141013, 42002258]; the Scientific Research Project of Education Department of Jilin 

Province (JJKH20241289KJ). 

 

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