Simulation Study on Rock Crack Expansion in CO₂ Directional Fracturing

Abstract

In underground construction projects, traversing hard rock layers demands concentrated CO₂ fracturing energy and precise directional crack expansion. Due to the discontinuity of the rock mass at the tip of prefabricated directional fractures in CO₂ fracturing, traditional simulations assuming continuous media are limited. It is challenging to set boundary conditions for high strain rate and large deformation processes. The dynamic expansion mechanism of the 3D fracture network in CO₂ directional fracturing is not yet fully understood. By treating CO₂ fracturing stress waves as hemispherical resonance waves and using a particle expansion loading method along with dynamic boundary condition processing, a 3D numerical model of CO₂ fracturing is constructed. This model analyzes the dynamic propagation mechanism of 3D spatial fractures network in CO₂ directional fracturing rock materials. The results show that in undirected fracturing, the fracture network relies on the weak structures near the rock borehole, whereas in directional fracturing, the crack propagation is guided, extending the fracture’s range. Additionally, the tip of the directional crack is vital for the re-expansion of the rock mass by high-pressure CO₂ gas, leading to the formation of a symmetrical, umbrella-shaped structure with evenly developed fractures. The findings also demonstrate that the discrete element method (DEM) effectively reproduces the dynamic fracture network expansion at each stage of fracturing, providing a basis for studying the CO₂ directional rock cracking mechanism.

1. Introduction

In today’s rapidly globalizing and urbanizing world, underground space construction is becoming increasingly important for providing additional space. Many countries and regions around the world have already reaped significant benefits from underground construction. For example, Russia’s Moscow Metro construction project [1], Ecuador’s Church of Our Lady of the Snows site preservation project [2], India’s Sri Lanka deep excavation project [3], Germany’s Frankfurt city center high-rise construction project [4], and Afghanistan’s underground canal construction project [5], etc. The rapid development of modern urban construction has intensified the competition for urban land resources, posing greater challenges for the development and utilization of underground space [6]. Excavation techniques such as foundation pits and tunnel excavation benefit from advancements in rock cracking technology [7]. Traditional non-blasting rock cracking methods, like static expansion agents and mechanical rock breaking, often face issues such as low efficiency, long duration, and high costs [8]. While blasting is widely used, it brings environmental problems like flying rocks, vibration, dust, and noise [9]. CO₂ fracturing technology offers a new solution, with lower vibration intensity compared to traditional blasting and a wider range of rock fragmentation [10]. It utilizes the energy from the phase transformation of industrial waste CO₂, reducing urban carbon emissions compared to gunpowder blasting. This method supports urban sustainable development and offers significant environmental benefits [11].

Rock cracking is a universal and regular natural phenomenon [12,13,14]. Several scholars have studied the mechanism of CO₂ fracturing in rocks through experiments. Zhang [15] used CT scanning to quantify fracture development in CO₂-fractured rocks. Li [16] conducted on-site experiments to study peak vibration velocity and rock cracking effects. Sampath [17] analyzed the interaction between heterogeneous rock materials and supercritical CO₂. Wang [18] researched the fracturing damage range and radial vibration characteristics of rock-like materials using a new indoor CO₂ fracturing experimental platform. However, these experimental studies still face some challenges. Complex industrial conditions, such as in situ stress, rock joint surfaces, and formation water content, present higher demands for understanding the CO₂ rock fracturing mechanism. In terms of laboratory experiments, Skoczylas [19] argued that scientific experiments should be repeatable, and mechanism studies should yield universal results. Tian [20] noted that the size effect of rocks can significantly impact their physical and mechanical properties. In construction projects, hard rock layers often require precise blasting energy guidance. Blasting in hard rock layers can cause significant deformation, damage, and instability in nearby rock layers [21]. Optimizing blasting parameters is challenging [22], and issues like overbreak [23] occur often. The directional CO₂ fracturing has proven effective for this. Therefore, numerical simulation research is an economical and efficient method to study the mechanism of directional CO₂ rock fracturing.

Most research on numerical simulation of CO₂ fracturing assumes continuous media. Zhu [24] suggested that a CO₂ fracturing numerical model based on the continuum assumption can better achieve macroscopic mechanical analysis. Arjomand [25] argued that finite element models can effectively simulate the rock fracturing process under multi-field coupling conditions during CO₂ fracturing. Sun [26] studied crack propagation under different initial stress conditions. Zhang [27] examined initiation pressure and the number and radius of vent holes in CO₂ fracturing. Peng [28] found that CO₂ fracturing speed is related to stress wave intensity, vent hole count, and expansion area. Treating CO₂ high-pressure fluid jets as stress waves in a semi-elastic space helps explain dynamic stress during CO₂ gas explosions [29]. However, CO₂ fracturing in rock mass involves discontinuity and large deformation [30]. Jiao [31] suggested that the DEM method is effective for simulating the CO₂ fracturing process, as it can analyze the forces between fractured rock particles and the fluid flow between them. Since the viscous resistance of CO₂ fracturing is very low (almost equal to zero), this approach is beneficial for a deeper analysis of the CO₂ fracturing damage process. Traditional numerical simulations are limited in modeling rock fractures in drilled rock masses and cannot fully address current research needs [32]. To better describe the evolution and development of internal rock fractures, the particle flow method has become essential [33]. Some studies have proven the effectiveness of the particle flow method. For example, Liu [34] used the CFD-DEM method to study the flow behavior of proppant during the CO₂ fracturing process. Zhang [35] investigated the fracturing mechanism of fluid in secondary fractures during CO₂ fracturing. Wang [36] studied the fluid pressure distribution on fracture surfaces in CO₂ fracturing.

As early as 2020, our team conducted discrete element method (DEM) numerical simulations on CO₂ fracturing in rock materials [37]. However, using 2D longitudinal section models has limitations. Firstly, 2D boreholes in these models resemble faults rather than complete drilling structures. Secondly, fracturing causes more severe damage along the direction of the blasting rod, creating a three-dimensional fracture network instead of a two-dimensional cross-section. Current research on dynamic fracture network expansion is insufficient. Therefore, we propose a numerical model for directional CO₂ fracturing based on discrete media. This model studies the distribution, deformation, and displacement changes in rock fracture networks during CO₂ directional fracturing, providing a basis for designing CO₂ directional fracturing parameters.

2. Materials and Methods

CO₂ fracturing can significantly deform rock masses. Traditional simulations, which assume continuous media, struggle with setting boundary conditions. Therefore, we used Particle Flow Code (PFC 5.0) 3D numerical simulation software, which assumes discrete media, for modeling and analysis. To compare the impact of directional fractures on the displacement field of CO₂-fractured rock, we created simulation samples both with and without directional fractures. Then, we studied the surface and internal displacements of these samples based on crack propagation laws.

2.1. Parameter Calibration

2.1.1. Samples for Laboratory Testing

To establish a numerical model, we prepared samples using a mixture of gypsum and water in a 7:3 ratio to ensure consistent mechanical strength [38]. Gypsum is selected for its ability to mimic the brittleness of rock materials and its resistance to air leakage. Although the strength of gypsum may be affected by curing time [39,40], for the sake of experimental repeatability and convenience, we conducted experiments using samples under the given experimental conditions only. After thoroughly mixing the materials according to the specified ratio, we poured them into a rectangular mold sized 70 mm × 70 mm × 140 mm. To achieve uniformity, we oscillated the mold and inserted a gas burst tube into the center. The mixture was allowed to set in the mold for 72 h, then removed from the mold and placed in a constant temperature and humidity chamber set at 19–24 °C and 20–30% relative humidity for 28 days. Finally, the surfaces of the samples were polished and repaired as shown in Figure 1.

2.1.2. Samples for Numerical Simulation

To conduct true triaxial compression numerical simulations using PFC 3D and accurately reflect the brittleness of rock materials, we utilized a parallel bonding model for particle bonding in our modeling approach. To establish the shale numerical model, the parallel bond (PB) model and the smooth joint (SJ) model were used to simulate the particle contact in the shale matrix and bedding plane [41]. By applying the M-C criterion to analyze the failure of the particle flow, we fit the material strength envelope using Equation (1) to determine the critical strength of the specimen under varying pressure conditions.

$$\tau =c+\sigma \tan \phi$$

(1)

where τ is the shear stress, MPa, c is the cohesive force, MPa, σ is the normal stress, MPa, φ is the internal friction angle of the material, °.

When choosing initial parameters, we assumed that the numerical simulation model reflects the critical strength of the actual sample. Following Li’s recommendation [42], the discrete element model initiates relative sliding of particles after bonding failure, with the particle friction coefficient μ set at 0.5. Other parameters are adjusted based on their relationships until the numerical simulation model matches the macroscopic mechanical properties of the real sample.

Table 1. Numerical model parameters.

Particle Parameter	Values	Bonding Parameter	Values
Minimum particle radius (mm)	0.5	knContact	0.8 × 106
Maximum particle radius (mm)	1.5	ksContact	0.6 × 106
Particle density (kg/m3)	1500	pbkncontact	3.0 × 1010
Contact damping	0.7	pbkscontact	4.0 × 1010
Friction coefficient	0.5	tenPbond	1.0 × 106
		cohPbond	1.1 × 106
		fapbond	30
		gapbond	1.0 × 10−6
		fricContract	0.2

details the consistent numerical model parameters for specimens with and without prefabricated cracks. Uniaxial compression tests were conducted using standard methods, and Figure 2 shows the stress–strain curve.

As shown in Figure 2, due to the characteristics of the numerical model’s loading mechanism, the curve of the numerical model is nearly linear in the initial loading stage, while the actual uniaxial compression test curve is slightly lower and shows an “S” shape before failure. In the modeling process, the particles have no gaps and are in a balanced state, whereas in the initial stage of the uniaxial compression test, there is a compaction phase due to natural voids and other factors. As the load continues to increase, the sample begins to enter the plastic stage, and the difference between the numerical model results and the actual loading process gradually decreases. After reaching the peak strength, the sample begins to fail, and both curves show a downward trend, eventually overlapping. Therefore, the numerical model can effectively reflect the mechanical properties of the actual sample and can be used for CO₂ dynamic load testing.

2.2. Model Establishment

We created a simulated specimen measuring 70 mm × 70 mm × 140 mm, with a rigid cluster of dimensions 10 mm × 35 mm positioned centrally as the blasting tube. For the model without prefabricated cracks, we used a sample composed of 97,314 randomly generated particles ranging in radius from 0.5 mm to 1.5 mm, illustrated in Figure 3a. In addition to the above specifications, the sample was modeled with prefabricated cracks by introducing a symmetrical vertical crack with dimensions of 0.4 mm thickness, 10 mm length, and extending 35 mm above and below the blasting tube, as per experimental parameters. This sample included a total of 97,313 randomly generated particles with radii ranging from 0.5 mm to 1.5 mm, depicted in Figure 3b. The model’s boundary conditions included walls on the left and right sides, servo loading on the top and bottom, and structural constraints as shown in Figure 3c.

2.3. CO₂ Fracturing Loading Settings

2.3.1. CO₂ Dynamics Loading

During CO₂ fracturing, when the distance between the rock mass and the blasting hole is minimal, the explosive stress propagates outward in spherical waves (radius r₀) from the perforation point. To balance the loading area, two hemispherical surfaces with a radius of 5 mm are positioned 5 mm away from the end of the blasting tube. Spherical particles with a radius of 0.5 mm are selected for force loading, as shown in Figure 4.

If the initial pressure of CO₂ fracturing is P₀, then the pressure p(t) of CO₂ fracturing varies uniformly over time and follows a simple harmonic law.

$$p(t)=P_0{e}^{j\omega t}$$

(2)

where P₀ is the initial fracturing pressure, Pa.

By integrating the contact areas between each small ball particle and the particles, the total force exerted by all loaded steel balls approximates the hemispherical surface area S = 2πR². When the gas explosion pressure is 1.0 MPa, the loading waveform is depicted in Figure 5.

According to the contact stiffness values in Table 1 and the initial pressure for carbon dioxide phase change explosion in Equation (2), the model is loaded by varying the particle expansion radius. The range of variation for the particle expansion radius is given in Equation (3).

$$Dr={\displaystyle \frac{2\pi r0}{Kr}}p$$

(3)

where D_r represents the particle expansion; K_r is the indirect contact stiffness of particles.

2.3.2. Dynamic Boundary Conditions

To simulate rock layers using spheres in contact with each other and considering the constitutive characteristics, it is necessary for the boundaries to absorb incident wave energy to simulate the infinite continuous medium outside the weak layer. Typically, boundary forces are applied to boundary particles to achieve this objective. The relationship between boundary forces and particle motion is depicted in Equation (4).

$$F=-2R\rho C_p{v}_b$$

(4)

where R is the particle radius, m. v_b is the particle velocity, m/s.

At the model boundary, boundary particle contact forces are specified to simulate the transmitting boundary. Considering the diffusion effect during the blasting process, the boundary conditions are adjusted [43,44], as shown in Equation (5).

$$F=\left\{\begin{array}{c}-\epsilon \cdot 2R\rho C_{pr}{v}_{br}\\ -\eta \cdot 2R\rho C_{p\theta }{v}_{b\theta }\end{array}\right.$$

(5)

where ε and η are the dispersion effect correction coefficients for longitudinal and transverse waves, respectively, taken as 0.35, C_pr and C_pθ represent the longitudinal and transverse wave velocities, m/s, v_br and v_bθ represent the radial and tangential velocities of particles, m/s.

3. Results and Discussion

3.1. Model Validation

After modeling and debugging, the final simulation results were compared with the actual samples failure, as shown in

Table 2. Comparison of damaged forms.

Model	Destructive Forms
	Real	Numerical
non-prefabricated cracks
prefabricated cracks

.

According to Table 2, there is a high similarity between the numerical and real samples in the distribution of crack spaces. The crack morphology and maximum crack length are also similar, which verifies the feasibility of the numerical model.

3.2. Characteristics of Crack Network

The transverse load on the upper and lower sections of the numerical model specimen is a constant 1.0 MPa. According to the loading method described in Section 2, the numerical model underwent 16 ms of gas explosion loading to achieve the final distribution of crack networks, as shown in

Table 3. Impact of directional cracks on fracture networks.

Model	Frontal Section	Lateral Section
non-prefabricated cracks
prefabricated cracks

. From Table 3, a CO₂ load with an initial pressure of 1.0 MPa was applied at the hemispherical positions of the blasting load. The loading parameters were set as specified, with a loading time of 16 ms. Evidently, multiple main cracks were generated near the blasting hole, accompanied by numerous network-like secondary crack structures at the ends of the main cracks, indicating well-developed fractures.

According to Table 3, directional cracks significantly promote the generation and radial extension of crack networks during fracturing. This indicates that directional cracks enhance the internal failure of rock mass, increasing the number of cracks from 184,514 to 186,173, with an overall increase of 1659 cracks due to prefabricated cracks. The development of frontal cracks reflects the expansion around the blasting borehole, and a distinct directional crack network forms along the depth of the cracks on the side.

It is evident that directional cracks have the capability to alter the direction of crack development to a certain extent and enhance the fracturing effect in the following ways:

(1)

Direction Influence: In samples without directional cracks, crack formation around the blasting borehole is random and primarily depends on the weak structure of the material. Directional cracks cause the crack direction to deviate, mainly expanding along the prefabricated cracks.

(2)

Development Promotion: Directional cracks promote the development of larger failure areas with a greater extension range, resulting in a more concentrated crack network.

(3)

Increased Microcracks: The sample without directional cracks generated 184,514 microcracks, whereas the sample with directional cracks generated 186,173 microcracks. This shows that prefabricated cracks significantly impact crack propagation, resulting in a deeper internal crack network in the oriented crack specimen.

To further analyze the impact of directional cracks on the fracturing process, the numbers of crack curves and the bond fracture curves (Figure 6) were plotted.

According to Figure 6, during the initial stage of crack development (0~6 ms), the spherical particles in the sample experience uniform stress. As the load surpasses the maximum load-bearing capacity of the parallel bonding bonds, microcracks form, leading to the gradual emergence and relatively stable increase in strain localization bands on the sample. By the time the loading period exceeds 12 ms, the sample exhibits developed cracks and through cracks, with a minor increase in the number of stable cracks. It becomes evident that after 12 ms, the rate of bond fracture tends towards stabilization, indicating a cessation of sample failure.

The initial count of bonded bonds was 393,400, and initially, the bonding rate remained relatively stable. Subsequently, from 2 ms onwards, the number of failures began to increment gradually, reaching a stable state by 8 ms. However, following 12 ms, due to the establishment of through cracks, further extension of crack tips ceased, resulting in a final reduction in bonded bonds to 272,000.

3.3. Surface Displacement

It is important to note a key difference between the experiment and simulation processes: during testing, gas gradually fills the crack tip, accumulating energy until it reaches a steady state sufficient to initiate crack propagation again—a process known as crack tip re-splitting. In contrast, in simulations, force is directly applied to the particles comprising the sample, simulating the initial expansion process when gas between cracks achieves a steady state and adequate energy for expansion. As a result, the simulation replicates the entire process from crack initiation to complete failure. Based on experimental results under 1.0 MPa gas explosion pressure and a 16 ms loading time, DEM software was employed to simulate and analyze surface crack development on the sample. Figure 7 illustrates the study of surface crack evolution using the 1.0 MPa gas explosion model as an example.

Based on Figure 7, the fracturing process shows symmetrical behavior on the top surface of the specimen, expanding outward from the crack tip. Initially, a few spherical particles near the crack tip detach from the surface due to loading, causing local peeling of particles. CO₂ splitting results in an elliptical displacement pattern along the crack tip, with the maximum surface particle displacement recorded at −2.21 mm (Figure 7a). As loading continues, the displacement caused by gas splitting at the crack tip increasingly dominates. The displacement distribution shifts from an elliptical pattern to an umbrella-shaped push-type failure, reaching a maximum displacement of −3.51 mm (Figure 7b,c). In the advanced stage of fracturing, visible damage occurs at the prefabricated crack tip due to gas explosion. The crack opens at a specific angle, leading to substantial particle detachment from the sample surface and forming a pronounced through-crack structure. The sample experiences splitting failure, with a maximum displacement of −4.153 mm (Figure 7h). Additionally, initial cracks, primary pores, and other factors play a significant role in directing crack distribution. The crack tip is crucial for the re-expansion of high-pressure carbon dioxide gas that fractures the rock mass. The crack tip itself exhibits a symmetrical umbrella-shaped structure with symmetrical growth.

3.4. Vertical Section Displacement of the Blast Hole

The interior of the drilling and fracturing process involves the injection of CO₂ gas, resulting in cracks forming along the drilling direction in a similar distribution pattern. To analyze the development of cylindrical cracks vertically, a 3D model captures the central position profile of the gas explosion hole. Figure 8 displays the displacement cloud map of the explosion process.

According to Figure 8, when carbon dioxide splits, the gas explosion hole near the rock wall experiences a hemispherical harmonic wave load. This creates symmetrical elliptical cracks above and below the borehole. The initial damage starts at the crack tip, with a maximum displacement of just 1.43 mm (Figure 8a,b). After 6 ms, the weak structures around the borehole, such as prefabricated cracks, first reach compressive strength, creating an air release channel. This leads to the formation of a push-type failure zone extending from the borehole perforation to its edge. Later, local damage in the weak layer acts as the starting point of failure, extending deeper into the rock mass, with a maximum displacement of 2.51 mm (Figure 8c,d). Then, the blasting zone centered on the hole gradually forms. The rock near the hole faces compressive stress, while the edges of the sample face tensile stress, causing overall displacement. When the load exceeds the rock’s tensile or shear strength, a main crack forms, leading to layer cracking (Figure 8e–h).

3.5. Radial Vertical Section Displacement

The vertical section along the drilling direction shows how cracks develop in front of the blasting rock during gas explosion. Figure 9 displays the displacement cloud map.

According to Figure 9, the drilling structure initially causes a circular damage layer around the drilling hole, with a maximum displacement of 1.47 mm (Figure 9a,b). As carbon dioxide gas splits the rock, deep cracks gradually form and extend deeper into the rock mass (Figure 9c,d). Subsequently, the cracks continue to develop until the changes become less significant. At the rear of the specimen, the maximum displacement was 1.41 mm, while at the front, particles detached from the specimen structure due to penetrating cracks, resulting in a significant displacement of 4.71 mm.

From the simulations, it is evident that DEM effectively replicates the extent of gas explosion stages under high strain rates and large deformations. Moreover, the presence of discontinuous structures does not impact the simulation outcomes.

4. Limitations and Future Work

The fracture network created by CO₂ fracturing is more complex. Li [45] demonstrated using a 2D DEM model that CO₂ fracturing results in more complex fracture patterns than hydraulic fracturing. By examining the 3D fracture network formed during CO₂ fracturing, we found that it creates a more intricate spatial fracture network compared to other explosive methods. This is because CO₂ fluid can more easily penetrate the tips of fractures than other high-energy gases. McCraw [46] suggested that by adjusting parameters like particle cohesion and bonding models in discrete element models, the dynamic response of particles can be accurately reproduced. Tatomir [47] also noted that discrete element software can control stress-induced damage at a microscale. Thus, 3D discrete element models have proven effective in reproducing internal fracture networks in rocks.

Directional CO₂ fracturing can effectively control the direction and extent of fracture development. Gheibi [48] argued that existing faults and defects within rocks can promote fracture extension. By analyzing displacement and fracture network maps, it can be seen that directional fracturing guides the direction of damage, expands the maximum impact range of cracks, and alters the fracture network’s shape.

Practical impact of this study: The DEM model effectively replicates the microscopic fracturing process of rocks, successfully reproducing complex fracture networks under CO₂ fracturing. This could support research in critical areas such as underground space protection, mine gas extraction, and underground carbon storage. Additionally, CO₂ directional fracturing demonstrates the importance of induced cracks in fracture formation and expansion, offering valuable insights for excavating hard rock.

There are still limitations in current research: the DEM numerical model contains a large number of particles, making it challenging to simulate field-scale industrial experiments and proportionate fracturing processes. While the particle radius expansion method partially reflects the stress wave propagation during fracturing, research on crack propagation mechanisms under multi-field coupling conditions remains insufficient. Future research could improve by analyzing and calculating fluid parameters, integrating them into fracturing simulation loading processes to more accurately depict the CO₂ fracturing mechanism.

5. Conclusions

To address the discontinuous failure in rock materials during CO₂ fracturing and explore the dynamic expansion of three-dimensional fractures in CO₂ directional fracturing, numerical simulations were conducted using the DEM software. The study focused on the morphology of failures, distribution of fracture networks, and mechanisms of fracture propagation in rocks. The main conclusions are as follows:

(1)

In samples without directional cracks, cracks around the blasting borehole form randomly, largely influenced by the weak structure of the rock material. Introducing directional cracks causes cracks to propagate along the prefabricated crack direction. In the simulated conditions, the directional crack model showed a reduction in bonding keys to 2.72 × 10⁵, indicating an increase of 1659 cracks compared to the non-directional crack model.

(2)

As fracturing progresses, the tip of the directional crack initially shows an elliptical displacement pattern on the specimen’s surface. This pattern gradually transitions into an umbrella-shaped distribution of progressive failure, symmetrically developing. The maximum displacement reaches −4.153 mm.

(3)

Based on the simulation results, using the discrete element method effectively replicates the dynamic expansion of fracture networks throughout each stage of the CO₂ fracturing process. This method serves as a foundation for studying the mechanism behind CO₂-induced rock fracturing.