Optimization of Key Parameters for Coal Seam L-CO₂ Phase Transition Blasting Based on Response Surface Methodology

Abstract

Liquid carbon dioxide (L-CO₂) phase transition blasting technology, known for its high efficiency, environmental friendliness, and controllable energy output, has been widely applied in mine safety fields such as coal roadway pressure relief and coal seam permeability enhancement. However, the synergistic control mechanism between L-CO₂ blasting loads and in situ stress conditions on coal seam fracturing and permeability enhancement remains unclear. This study systematically investigates the key process parameters of L-CO₂ phase transition blasting in deep coal seams using response surface methodology and numerical simulation. First, three commonly used L-CO₂ blasting tubes with the overpressure of 150 MPa, 210 MPa, and 270 MPa were selected, and the corresponding material parameters and state equations were established. A dynamic mechanical constitutive model for a typical low-permeability, high-gas coal seam was then developed. A numerical model of L-CO₂ phase transition blasting, considering fluid–solid coupling effects, was then constructed. Multiple experiments were designed based on response surface methodology to evaluate the effects of blasting pressure, in situ stress, and stress difference on L-CO₂ fracturing performance. The results indicate that the overpressures of the three simulated blasting loads were 156 MPa, 215 MPa, and 279 MPa, respectively, and the load model closely matches the actual phase blasting load. L-CO₂ blasting creates a plastic deformation zone and a pulverized zone around the borehole within 500 μs to 800 μs after detonation, with a tensile fracture zone appearing at 2000 μs. By analyzing radial and tangential stresses at different distances from the explosion center, the mechanical mechanisms of fracture formation in different blast zones were revealed. Under the in situ stress conditions of this study, the number of primary fractures generated by the explosion ranged from 0 to 12, the size of the pulverized zone varied from 1170 cm² to 2875 cm², and the total fracture length ranged from 44.4 cm to 1730.2 cm. In cases of unequal stress, the stresses display axial symmetry, and the differential stress drives the fractures to expand along the direction of the maximum principal stress. This caused the aspect ratio of the external ellipse of the explosion fracture zone to range between 1.00 and 1.72. The study establishes and validates a response model for the effects of blasting load, in situ stress, and stress difference on fracturing performance. A single-factor analysis reveals that the blasting load positively impacts fracture generation, while in situ stress and differential stress have negative effects. The three-factor interaction model shows that as the in situ stress and stress difference increase, their inhibitory effects become stronger, while the enhancement effect of the blasting load continues to grow. This research provides a theoretical basis for blasting design and fracture propagation prediction using L-CO₂ phase transition blasting in the coal seam under varying in situ stress conditions, offering valuable data support for optimizing the process of L-CO₂ phase transition fracturing technology.

1. Introduction

In recent years, as shallow coal resources are increasingly depleted, the depth of coal mining worldwide continues to increase. Studies show that the gas pressure and gas content in coal seams rise with increasing mining depth, accompanied by a continuous increase in the original stress of the coal seams. This significantly increases the likelihood of gas outbursts and coal and gas outbursts [1]. Consequently, coal mine gas has become one of the major safety hazards restricting deep coal mining [2]. Gas extraction at the White Haven coal mine in the UK began as early as the 18th century. In the United States, research into gas control started in the early 20th century. In 1969, the U.S. Mine Safety and Health Administration (MSHA) initiated gas control in mined-out areas, achieving positive results. After 1973, coal mine gas extraction technologies began to be widely adopted in the U.S. Since 2002, the “Safety Regulations for Coal Mines” issued by the National Mine Safety Administration of China have stipulated that for high-gas and outburst-prone mines, coal seam gas must be pre-drained before mining to ensure that gas pressure and gas content are reduced below the safety limits of 0.74 MPa and 8 m³/t, respectively. It is clear that gas extraction is a fundamental strategy for preventing and controlling coal mine gas disasters [3]. Since most high-gas and outburst-prone coal seams in China typically exhibit low porosity and low permeability, measures such as pressure relief and enhanced permeability or fracturing are necessary to induce a network of fractures in the coal seam that facilitates gas migration [4]. Among these, fracturing techniques primarily target single coal seams lacking protective layer mining conditions. These techniques have developed into water-based fracturing technologies represented by hydraulic fracturing and hydraulic punching, as well as non-water fracturing technologies represented by deep hole blasting and liquid nitrogen fracturing [5,6,7].

The liquid carbon dioxide (L-CO₂) phase change blasting technology originated from the Cardox tube technology invented in the United States in 1914 and has been applied in coal seam mining, step blasting, tunnel excavation, and other fields [8]. The L-CO₂ phase change blasting tube typically consists of a detonator, a cracking device, and a liquid storage tank. The detonator generates an electric spark that activates the activator loaded in the cracking device, causing the L-CO₂ in the storage tank to suddenly heat and pressurize, rapidly vaporizing and expanding to generate a powerful dynamic impact and high-pressure gas wedge effect on the surrounding rock [9]. Investigations have shown that in the 1940s, 25% of underground coal mines in the United States utilized L-CO₂ phase change blasting technology, with an annual blasting volume exceeding 2.8 million times [10,11,12]. Compared to other fracturing technologies, L-CO₂ phase change blasting offers advantages such as high operational efficiency, absence of toxic and harmful gasses, low blasting vibration response, adjustable and controllable blasting energy, fewer application restrictions, and high social acceptance, thus attracting widespread attention in fields like coal seam fracturing and transformation. Reports indicate that this technology has been widely applied in the comprehensive management of mine gas in China, including roadway pressure relief excavation, coal seam pressure relief and permeability enhancement, and top-coal safety fracturing [13]. Several scholars have demonstrated through field experiments that L-CO₂ phase change blasting can effectively increase coal permeability and enhance gas extraction efficiency. For instance, Wang et al. showed that the quantity of gas extracted from a single borehole controlled by CO₂ fracturing increased by about four times, and the outburst suppression effect was evident after comparing the gas desorption index of drilling cuttings within 20 m before and after blasting in the heading face [14].

Several scholars have conducted relevant physical experiments to study the mechanism of L-CO₂ phase change fracturing technology in coal rock. Cao et al. placed L-CO₂ phase change blasting tubes within a core holder casing and subjected coal samples to dynamic impacts at different blasting pressures [15]. They performed field emission scanning electron microscope (FESEM) tests on the fragmented coal samples, revealing that CO₂ blasting generated a significant number of micron-scale tri-radial-wing fractures, greatly enhancing the connectivity of coal fractures. Sun et al. conducted CO₂ blasting fracturing experiments on coal-like material samples with 400 mm edge lengths under various stress conditions using a triaxial load platform [16]. Shang et al. performed CO₂ blasting fracturing experiments on coal samples at blasting pressures of 10 MPa to 25 MPa, indicating that continuously increasing blasting pressure reduces the fracture extension range while significantly increasing the crushed zone area, thus necessitating the identification of reasonable blasting pressures in engineering applications [17]. Kang et al. theoretically analyzed and divided the L-CO₂ phase change fracturing process into two coupled processes: blasting-induced coal fractures and blasting-induced damage for eliminating stress concentration [18]. The aforementioned studies analyzed the impact of blasting pressure and in situ stress on the effect and mechanism of L-CO₂ phase change fracturing from the perspectives of coal micro-fracture and macro-damage. However, due to experimental constraints, they did not consider the synergistic effects of in situ stress and blasting load. Jia et al. conducted numerical simulations using explicit dynamics analysis software ANSYS/LS-DYNA to study how coal seam physical parameters (in situ stress, gas pressure, and elastic modulus) and blasting parameters (blasting peak pressure and borehole diameter) affect fracturing outcomes, revealing that physical parameters have a minor influence on the scale of fracturing, while blasting parameters have a significant impact [19]. Yuan et al. constructed a FLAC numerical model to investigate the effects of energy release direction, coal rock mechanical properties, blasting parameters, in situ stress, and borehole parameters on phase change fracturing outcomes, concluding that coal rock compressive strength and cracking hole spacing are critical factors affecting fracturing effectiveness [20]. Zhang et al. accurately calibrated the dynamic constitutive model of coal rock (Holmquist–Johnson–Cook model, HJC model) through static and dynamic experiments, providing an in-depth analysis of the mechanisms of L-CO₂ blasting fracturing [21]. These studies systematically clarified the influence of various factors on the effectiveness of L-CO₂ phase change fracturing through numerical simulations, providing theoretical support for optimizing fracturing processes. However, the aforementioned numerical simulations overly simplified aspects such as the L-CO₂ phase change load form or coal rock mechanical constitutive relationships, and did not delve into the interactive effects of in situ stress parameters and blasting loads on fracturing outcomes.

In summary, there remains a lack of numerical simulation methods that can accurately model the entire process of L-CO₂ phase change blasting fracturing in deep coal seams under high in situ stress. Based on the limitations of the aforementioned research, this paper systematically conducts numerical simulations of L-CO₂ phase change blasting fracturing in deep coal seams. The research objective is to clarify the evolution patterns and key factors controlling the fracturing effects under the synergistic action of blasting load and in situ stress. Firstly, using LS-DYNA R11.1.0 dynamic simulation software, the material parameters and state equations of L-CO₂ phase change blasting tubes at blasting pressures of 150 MPa, 210 MPa, and 270 MPa were calibrated. Secondly, considering the fluid–solid coupling effects, a numerical model for L-CO₂ phase change blasting in a coal seam with a 10 m edge length was constructed. Multiple response surface experiments were designed based on the response surface method, considering blasting pressure, in situ stress, and stress difference. The effectiveness of L-CO₂ fracturing was scientifically evaluated based on five response indicators: the borehole radius after blasting, crushed zone area, total fracture length, and shape factor of the crushed zone’s secondary fracture area. This study can provide a reference for the process design of L-CO₂ blast fracturing technology.

2. Numerical Simulation Method

2.1. L-CO₂ Phase Change Blast Tool and Process

The physical diagram and structural composition of the L-CO₂ phase transition blasting tool, produced by Zhaodong Machinery Co., Ltd., Jingmen City (China) are shown in Figure 1. The blasting tool mainly includes a gas inlet valve, a chemical energizer, a pressure-resistant shell, a rupture disk, and a blast head with drilled holes. Before leaving the factory, the tool is filled with L-CO₂ using a filling device. After transportation to the working space, the chemical energizer is connected to the high-energy ignition system via a wire. The blasting tool is then pushed to the designated position in the drilled hole using a push rod, and the hole is appropriately sealed externally. Finally, the high-energy ignition system detonates the chemical energizer, triggering a rapid increase in temperature and pressure of the L-CO₂, resulting in a violent phase transition. When the gas pressure in the pressure-resistant chamber exceeds the rupture pressure of the rupture disk, high-pressure CO₂ is released through the blast head into the coal seam, fracturing the coal layer.

The pressure–temperature phase diagram of CO₂ is shown in Figure 2. With changes in state parameters, CO₂ exhibits four phases: solid, gas, liquid, and supercritical states. When the explosive device is not activated, the pressure of carbon dioxide is around 10 MPa, and the temperature is generally below 20 °C, as depicted in State 1 of the figure. Once the activator is triggered by the operator through the explosive device, carbon dioxide is rapidly heated, resulting in a swift increase in pressure within the sealed space. In this high-temperature and high-pressure state, CO₂ exists in a liquid and supercritical coexistence state, illustrated in State 2 of the figure. The physical property changes of CO₂ during this process are shown in process ①. Following the rupture of the high-pressure blasting disk, the CO₂ in the storage tank is released, causing a sudden drop in pressure and temperature, leading to a gas–liquid coexistence state, and ultimately transforming into a gaseous state, as shown in State 3. The physical property changes of CO₂ during this process are shown in process ②.

2.2. Material Model Selection for Numerical Simulation

To study the L-CO₂ phase transition blasting permeability enhancement mechanism at an engineering scale, a large-scale numerical simulation model under in situ stress conditions is established. LS-DYNA R11.1.0, powerful dynamic simulation software, is widely used for dynamic impact simulations of rock and concrete materials. This study employs LS-DYNA R11.1.0 to numerically simulate the phase transition blasting of liquid carbon dioxide at an engineering scale, elucidating the mechanisms of L-CO₂ phase transition fracturing and crack propagation under the combined effects of in situ stress and explosion load.

Obtaining the load form for L-CO₂ phase transition fracturing is critical for the reliability of the numerical simulation. The instantaneous heating and vaporization of L-CO₂ produce high-pressure gaseous CO₂, which applies explosive impact loads to the engineering rock mass. This process represents a typical physical explosion process without chemical reactions, relying primarily on gas impact generated by phase transition expansion to fracture the rock mass. This study selects the *MAT_HIGH_EXPLOSIVE_BURN material model built into LS-DYNA R11.1.0, paired with the *EOS_LINEAR_POLYNOMIAL equation of state, specifically designed to describe the physical processes of phase transition blasting. Through extensive trial calculations, model parameters that can produce different L-CO₂ phase transition explosive loads are obtained. The LINEAR_POLYNOMIAL equation of state is shown in Equation (1):

$$P=C_0+C_1\mu +C_2{\mu }^2+C_3{\mu }^3+\left(C_4+C_5\mu +C_6{\mu }^2\right)E$$

(1)

The state equation is applicable for describing the pressure state of the gas as well as the compression and expansion processes. Therefore, this study selects this equation of state, which effectively simulates the adiabatic expansion in L-CO₂ phase transition blasting. In the equations, P represents the pressure of the detonation products, in MPa; E is the unit initial internal energy, in J; and C₀, C₁, C₂, C₃, C₄, C₅, and C₆ are the coefficients of the polynomial. μ = ρ/ρ₀ and ρ/ρ₀ is the ratio of current density of reference density. Since the L-CO₂ phase transition blasting process does not involve chemical reactions, it is assumed that the gas follows the Gamma law, which models an ideal gas with a constant adiabatic index γ. The relationship between pressure P, density ρ, and specific internal energy E under the Gamma law is given by P = (γ − 1)ρE. By simplifying the LINEAR_POLYNOMIAL, a gas state equation consistent with the Gamma law can be derived. This simplification primarily employs the following equations, Equations (2)–(4):

$$C_0=C_1=C_2=C_3=C_6=0$$

(2)

$$C_4=C_5=\gamma -1$$

(3)

$$\gamma ={\displaystyle \frac{C_p}{C_v}}$$

(4)

where C_p is the specific heat capacity at constant pressure, in J/(kg·K); and C_v is the specific heat capacity at constant volume, in J/(kg·K).

The validity of the mechanical model and parameters for coal rock materials is crucial for accurately reflecting the fracture distribution patterns during L-CO₂ phase transition blasting in coal seams. Unlike quasi-static loads, L-CO₂ phase transition fracturing, as a dynamic impact fracturing method, exhibits entirely different mechanical response characteristics under dynamic loads, primarily manifesting as significant strain rate effects and strengthening effects. Therefore, the constitutive equations for coal rock under quasi-static conditions cannot capture its impact dynamic behavior. The RHT constitutive model, modified from the HJC model by Riedel, Hiermaier, and Thoma, effectively simulates the dynamic response of rock and concrete materials, as well as the tensile–shear damage process. It has been widely applied in fields like blasting engineering, with ample data available for reference [22]. The failure surface equation of the RHT model is given by Equation (5):

$${\sigma }_{eq}^{*}\left(p,\theta ,\dot{\epsilon }\right)=Y_{TXC}^{*}\left(p\right)R_3\left(\theta \right)F_{RATE}\left(\dot{\epsilon }\right)$$

(5)

where ${\sigma }_{eq}^{*}={\sigma }_{eq}/f_c$ is the normalized equivalent stress. $f_c$ is the uniaxial compressive strength of the coal rock, in MPa. $\theta$ is the similarity angle (Lode angle), in degrees. $\dot{\epsilon }$ is the equivalent strain rate. $Y_{TXC}^{*}\left(p\right)$ represents the compressive meridian. $F_{RATE}\left(\dot{\epsilon }\right)$ is the strain rate enhancement factor, and $R_3\left(\theta \right)$ is a function of the Lode angle and the ratio of the deviatoric stress at the Lode angle on the tension–compression meridian (Q₂). Additionally, the RHT model effectively considers the confining pressure effect, strain hardening, and damage softening of rock materials through the Mie–Gruneisen equation of state, making it particularly suitable for simulating the damage and fracture of coal under high-strain-rate conditions, while also accounting for the impacts of plastic volume changes and compaction effects in rock [23]. Consequently, the RHT model is employed to simulate the damage and fracture patterns in coal rock, with the fracture distribution represented by the damage variable D in the RHT constitutive model, defined in Equation (6):

$$D={\displaystyle \sum \left(\Delta {\epsilon }_p/{\epsilon }_p^f\right)}$$

(6)

where

$$\Delta {\epsilon }_p={\epsilon }_p^f-{\epsilon }_p$$

(7)

$${\epsilon }_p^f=\left\{\begin{array}{c}{D}_1\left[p^{*}-\left(1-D\right)p_t^{*}\right]\to p^{*}\ge \left(1-D\right)p_t^{*}+{\left({\epsilon }_p^m/D_1\right)}^{{\displaystyle \frac{1}{D_2}}}\\ {\epsilon }_p^m\to \left(1-D\right)p_t^{*}+{\left({\epsilon }_p^m/D_1\right)}^{{\displaystyle \frac{1}{D_2}}}>p^{*}\end{array}\right.$$

(8)

where ${\epsilon }_p$ is the plastic strain; ${\epsilon }_p^f$ is the plastic strain at failure; $\Delta {\epsilon }_p$ is the difference between the failure plastic strain and the current plastic strain; $D_1$ and $D_2$ are damage parameters in the RHT model; $p^{*}$ is the normalized pressure of compressive strength, with $p^{*}=p/f_c$, where $p$ is the current applied pressure in MPa; $p_t^{*}$ is the failure cutoff pressure; and ${\epsilon }_p^m$ is the intermediate value of the plastic strain.

2.3. Model Parameter Assignment and Simulation Plan

2.3.1. Parameter Assignment for Numerical Model

The L-CO₂ blasting device can generate different explosion pressures by controlling the filling amount and the rupture pressure of the pressure–shearing plate. This study selects three common explosion pressures on the market, 150 MPa, 210 MPa, and 270 MPa, for a process parameter optimization analysis. Based on surveys conducted by our research team with multiple manufacturers and suppliers of L-CO₂ blasting devices, these three explosion pressure levels are the most common on the market and can accommodate the majority of L-CO₂ blasting pipe models. Through extensive trial calculations, EOS state parameters that match the model dimensions are determined, as shown in

Table 1. Parameter value table for *EOS_LINEAR_POLYNOMIAL equation of state.

Blast Load = 150 MPa
Characteristic parameters	C0	C3/Bar	C6/Bar	Em/J	V
Value	0	0.2950	0.2950	2.60 × 103	1.0
Blast load = 210 MPa
Characteristic parameters	C0	C3/Bar	C6/Bar	Em/J	V
Value	0	0.2950	0.2950	4.80 × 103	1.0
Blast load = 270 MPa
Characteristic parameters	C0	C3/Bar	C6/Bar	Em/J	V
Value	0	0.2950	0.2950	3.70 × 103	1.0

, along with the material model parameters for L-CO₂ listed in

Table 2. Parameter value table for *MAT_HIGH_EXPLOSIVE_BURN energetic material.

Characteristic Parameters	ρ0/(kg·m−3)	d/(m·s−1)	Pcj/MPa
Value	915	4000	270

.

The parameters of the RHT constitutive model are complex and require the clarification of the 34 undetermined parameters in the model based on the elastic limit surface equation, failure surface equation, linear strengthening segment equation, and damage softening phase equation of the RHT model. Previous research indicates that the undetermined parameters mainly fall into four categories: fixed parameters, experimental test parameters, numerical calculation parameters, and parameters that cannot be directly determined [24]. This study references the RHT constitutive parameters obtained from the #2 coal seam of the Baijiao Coal Mine in Yibin, Sichuan Province, as shown in

Table 3. RHT constitutive parameters for the #2 coal seam of the Baijiao Coal Mine in Yibin, Sichuan Province.

Parameter Category	Parameter	Parameter Value	Parameter	Parameter Value
Fixed parameters	Failure compression strain rate, ${\dot{\epsilon }}_{\mathrm{c}}$/s−1	3.0 × 1025	Failure tensile strain rate, ${\dot{\epsilon }}_t$/s−1	3.0 × 1025
	Reference compression strain rate, ${\dot{\epsilon }}_0^c$/s−1	3.0 × 10−5	Failure tensile strain rate, ${\dot{\epsilon }}_0^t$/s−1	3.0 × 10−6
	Damage factor, D2	1.0	Factor related to the Rothe angle, B	0.0105
	Tensile yield surface parameters, $G_t^{*}$	0.7	EOS polynomial parameter, T2/GPa	0.0
Experimental testing parameters	UCS, fc/MPa	21.1	Poisson’s ratio, v	0.27
	Elastic modulus, E/GPa	8.38	Longitudinal wave velocity, c0/(m·s−1)	1525
	Density, ρ0/(kg·m−3)	1397	Initial porosity, α0	11.8
Calculation parameters	Elastic shear modulus, G/GPa	E/(2 (1 + v)), 3.30	Pore collapse pressure, pcl/MPa	2fc/3, 14.1
	Compression strain rate index, βc	4/(20 + 3fc), 48.02 × 10−3	Tensile strain rate index, βt	2/(20 + fc), 48.66 × 10−3
	Hugoniot polynomial parameter, A1/GPa	ρ0·c02, 0.454	Hugoniot polynomial parameter, A2/GPa	A1 (2s − 1), 0.464
	Hugoniot polynomial parameter, A3/GPa	A1(3s2 − 4s + 1), 0.011	EOS polynomial parameter, T1	A1, 0.454
	EOS polynomial parameter, B0	2s − 1, 2	EOS polynomial parameter, B1	2s − 1, 2
Orthogonal experiment-determined parameters	Residual surface parameters, Af	1.6	Residual surface parameters, Nf	0.61
	Relative shear strength, $f_s^{*}$	0.18	Relative tensile strength, $f_t^{*}$	0.1
	Rhode angle correlation factor, Q0	0.61	Compression yield surface parameters, $G_c^{*}$	0.35
	Shear modulus reduction factor, XI	0.45	Damage factor, D1	0.035
	Minimum-damage residual strain, ${\epsilon }_p^m$	0.005	Residual surface parameters, Af	1.25
	Residual surface parameters, Nf	0.45	Pore compaction pressure, $P_{comp}$/GPa	0.55
	Porosity index, n	2.0	Related material parameter, s	1.50

[25]. The gas content of the coal seam in the Baijiao Coal Mine ranges from 12.10 to 18.86 m³/t, and the coal seam permeability coefficient is between 0 and 0.29 m²/(MPa²·d), classifying it as a low-permeability coal seam group. Therefore, applying L-CO₂ phase transition blasting for permeability enhancement in the #2 coal seam of this mine is of practical significance. Additionally, the detailed RHT constitutive parameters of this coal seam ensure the reliability of the research results.

2.3.2. Numerical Simulation Model and Plan

Figure 3 presents the numerical model for simulating L-CO₂ phase transition blasting of coal. This model can apply confining pressures in the x and y directions and simulate the phase transition fracturing process of L-CO₂ within the borehole. Given that the axial length of the wellbore is much greater than its diameter, the model is simplified to a quasi-two-dimensional model aligned with the vertical axis of the wellbore. The model consists of a coal layer measuring 10,000 mm × 10,000 mm × 10 mm. A borehole with a diameter of φ100 mm is prefabricated at the center, referencing the actual drilling size of φ94 mm commonly used in coalbed methane drilling. According to the equivalence principle, the borehole is filled with an L-CO₂ material model of φ80 mm × 10 mm, with the L-CO₂ model coupled with the coal layer through an air coupling domain of φ2000 mm × 10 mm using a Euler–Lagrange method (ALE) for fluid–solid coupling simulation. In this study, the coal material model employs the MAT_RHT constitutive model constructed in Section 2.3.1; air uses the MAT_Null material model combined with the *EOS_LINEAR_POLYNOMIAL equation of state; L-CO₂ employs the *MAT_HIGH_EXPLOSIVE_BURN material model alongside the *EOS_LINEAR_POLYNOMIAL equation of state. All contact types in the model are defined as automatic surface-to-surface contact. The outer boundary of the coal layer is defined as a non-reflective boundary to simulate a coal layer extending into infinite space. The in situ stress is applied to the model boundary using the dynain method commonly used in LS-DYNA R11.1.0. First, stress loads are applied in the x and y directions, and the model reaches a stress equilibrium state after 10,000 μs, generating a dynain file for this stress state. This dynain file is then reintroduced along with the air coupling domain and the L-CO₂ model, inheriting the stress field from the dynain file to conduct the numerical simulation of L-CO₂ phase transition blasting under in situ stress loading conditions.

2.4. Numerical Simulation Response Surface Design

The response surface analysis method is advantageous due to fewer experimental groups, high prediction accuracy, and robust model validation capabilities, making it suitable for optimizing process parameters. Through the response surface method, not only can the relationship between the response target and a design variable be obtained, but also the interaction effects of different design variables on the response target can be analyzed. This study employs the Box–Behnken response surface method to conduct response surface design. This research selects the in situ stress, stress differential, and explosion load as the dominant factors affecting the blasting effectiveness of a single coal seam. Therefore, based on the in situ stress conditions of common kilometer-deep coal layers in China and the actual conditions of L-CO₂ phase transition blasting technology, a three-factor, three-level response surface design scheme is developed, as shown in

Table 4. Range table of values for the three factors in the response surface design.

Factor	Variable	Level 1	Level 2	Level 3
σy/MPa	A1	5	10	15
σx − σy/MPa	A2	0	5	10
Pmax/MPa	A3	150	210	270

. The selection of parameters is based on the following considerations. Since coal seams in China are typically found at depths of less than 1000 m, where horizontal stress predominates, this study sets σ_y to be smaller than σ_x. Based on the common relationship between vertical stress and burial depth, typical vertical stresses for depths of 200 m, 400 m, and 600 m are chosen, with values of 5 MPa, 10 MPa, and 15 MPa, respectively. To reflect the influence of stress difference on fracture propagation, three levels of differential stress—0 MPa, 5 MPa, and 10 MPa—are selected. The L-CO₂ phase transition blasting load is set to 150 MPa, 210 MPa, and 270 MPa, as previously calibrated. A total of 13 groups of numerical calculation model parameters are ultimately obtained, as shown in

Table 5. Three factors and three-level response surface design table.

No.	A1/MPa	A2/MPa	A3/MPa	No.	A1/MPa	A2/MPa	A3/MPa
1	5	5	150	9	5	0	210
2	5	5	270	10	15	0	210
3	15	5	150	11	5	10	210
4	15	5	270	12	15	10	210
5	10	0	150	13	10	5	210
6	10	0	270	14	10	5	210
7	10	10	150	15	10	5	210
8	10	10	270	--	--	--	--

.

Using the response surface analysis method requires first clarifying the response target. To comprehensively elucidate the effects of the vertical stress, stress differential, and explosion pressure on coal fracturing effects, this study extracts five fracturing parameters as response targets. To illustrate the determination method of response targets, for example, the damage cloud diagram is extracted after 3000 μs following L-CO₂ phase transition blasting for the simulation group numbered 13, as shown in Figure 4. Based on the partition theory of borehole blasting, the coal layer sequentially forms a smash district, fracture zone, and vibration zone from the borehole to the far field after the blasting. The rapid pressure increase caused by the L-CO₂ phase transition leads to significant expansion at the borehole boundary, reflecting the power exerted during the rapid-pressure-rise phase of the blasting. Thus, the radius of the borehole after plastic deformation is defined as R_well, cm. In the near-borehole area, the dynamic compressive stress generated by the explosion load in the coal seam exceeds the dynamic compression strength of the coal, leading to compression–shear failure and forming a compression–shear smash district with shear cracks, as shown in the enlarged view in Figure 4. The area of the region with a damage degree of 1 was extracted and measured using ImageJ (FIJI win 64), denoted as A_crush, cm². The lengths of the major and minor axis of the outer ellipse enclosing the compression–shear smash district were extracted, and their ratio was defined as the shape factor of the smash district, ξ_crush. This parameter reflects the extent to which differential stress controls the fracture morphology. The fracture zone is the main area for gas permeability enhancement in the coal seam. The mechanism by which explosive loads generate fractures in the coal layer is that the circumferential tensile stress component of the stress wave exceeds the dynamic tensile strength of the coal mass, resulting in multiple radial fractures. The total length of all radial fractures is defined as the total length of fractures, L_frac, cm. Similarly to the shape factor of the smash district, the shape factor of the fracture zone is defined as ξ_frac.

3. Results

3.1. In Situ Stress Analysis

In situ stress is a key formation parameter affecting the effectiveness of blasting and fracturing. In this study, the response surface method is used to design the initial conditions for numerical simulations, encompassing a total of nine in situ stress states. To intuitively illustrate the influence mechanism of in situ stress on blasting and fracturing effects, a polar coordinate system is established with the borehole center as the origin. Radial and tangential stress contour maps induced by in situ stress are drawn, as shown in Figure 5. The numerical unit in the color bar is 1 × 10⁵ MPa, with negative values indicating compressive stress.

First, it is observed that the stress contour maps for the states (5 MPa, 5 MPa), (10 MPa, 10 MPa), and (15 MPa, 15 MPa). The results indicate that under conditions of equal in situ stress in the x and y directions, both radial and tangential stresses exhibit a center-symmetric distribution. Specifically, radial stress increases with distance from the origin, while being lowest at the borehole boundary. Tangential stress shows stress concentration near the borehole, decreasing as the distance from the origin increases. The tangential stress concentrations for the three groups are 7.313 MPa, 13.220 MPa, and 17.990 MPa, respectively.

As in situ stress increases, both radial and tangential stresses significantly rise, with the tangential stress concentration around the borehole becoming more pronounced. When the two directional stresses are unequal, radial and tangential stresses symmetrically distribute along the x and y axes, with the radial compressive stress in the x direction being greater than that in the y direction. Conversely, the tangential stress in the x direction is significantly lower than that in the y direction. Additionally, the stress concentration area around the borehole displays a flattened distribution with the y direction as the long axis. This evidence suggests that the stress difference increases the radial stress component in the direction of maximum stress while reducing the tangential stress component, alleviating tangential stress concentration around the borehole, which is more conducive to the expansion of explosive fractures.

3.2. L-CO₂ Phase Transition Blasting Load Analysis

To verify the effectiveness of the explosive load, pressure–time (p-t) curves are extracted from the air domain elements closest to the L-CO₂ phase transition fracturing pipe, with explosive pressures of 150 MPa, 210 MPa, and 270 MPa, as shown in Figure 6. The results indicate that the p-t curves for the three explosive loads are quite similar, exhibiting a rapid pressure rise followed by a slow pressure decline in a pulsed manner, which aligns well with previous physical experimental results [26]. The curves reach peak pressures within 50 μs, with peak pressures of 156 MPa, 215 MPa, and 279 MPa, respectively. As the peak pressure increases, the time required to reach it gradually decreases to 50 ms, 45 ms, and 40 ms. Therefore, the pressure rise rate of the explosive load also significantly increases, with rise rates for the three loads of 3.120 GPa/s, 4.778 GPa/s, and 6.975 GPa/s, respectively. This consistency in peak pressure and loading rates with previous experimental results is notable, and all values are significantly lower than the peak pressures and loading rates of conventional explosives, thus accurately reflecting the L-CO₂ phase transition blasting load parameters and greatly enhancing the reliability of the numerical simulation results [17].

3.3. Analysis of Explosive Fracture Propagation Characteristics

The damage parameter D of the RHT constitutive model effectively reflects the distribution pattern of fractures generated by explosive loading in coal rock. Fifteen groups of numerical simulations are executed based on the response surface design, yielding damage cloud maps of the coal seam after 3000 μs. Among these, #13, #14, and #15 are repeated tests; thus, only results from 13 groups of numerical simulations are presented, as shown in Figure 7. According to research by Wang et al., a damage parameter greater than 0.2 is defined as effective damage, so the color bar range for the damage cloud maps is set from 0.2 to 1.0 [27]. The results show that as the explosive load increases, both the area of the smash zone and the range of the fracture zone significantly expand. With rising in situ stress and stress differences, the extent of the fracture zone notably decreases. In cases 3 and 12, no significant radial fractures are observed. In comparison, while the area of the smash zone also decreases, the impact is less pronounced than in the fracture zone. As the stress difference increases, both the smash zone and fracture zone tend to extend toward the direction of maximum in situ stress (in this study, the x direction), with the degree of bias gradually increasing.

3.4. Analysis of Explosive Fracture Propagation Process

To clarify the operational process of L-CO₂ phase transition expansion fracturing in the coal seam, the evolution of damage cloud maps over 2000 μs from 13 groups of numerical simulations is summarized in

Table 6. Evolution table of explosive fractures.

No.	200 μs	500 μs	800 μs	1100 μs	1400 μs	1700 μs	2000 μs
1
2
3
4
5
6
7
8
9
10
11
12
13

. From 0 to 500 μs, L-CO₂ is activated and undergoes phase transition expansion, generating an instantaneous shock load significantly greater than the compressive strength of the coal body and the radial compressive stress component from in situ stress, resulting in crush–shear failure in the coal body. During this phase, a smash zone primarily forms in the coal seam, and significant hole enlargement effects occur. From 500 μs to 800 μs, the plastic deformation around the borehole and the smash zone are largely established. Between 800 μs and 1100 μs, due to the damping effect of the coal seam, the peak stress of the explosive shock load decays, causing the radial compressive stress generated by the explosive gasses in the reservoir to fall below the sum of the coal body’s compressive strength and the radial compressive stress component from in situ stress. However, due to the Poisson effect, the tangential tensile stress generated by the dynamic stress wave still exceeds the sum of the coal body’s tensile strength and the tangential compressive stress component from in situ stress, resulting in the initiation and rapid propagation of the main explosive fracture. Between 1100 μs and 1400 μs, radial fractures continue to propagate, gradually ceasing. By 2000 μs, the L-CO₂ shock fracturing zone is essentially formed, and the tangential tensile stress decreases to below the coal body’s tensile strength, failing to cause damage or fracture in the coal body, which instead leads to elastic vibrations. Consequently, from 2000 μs to 3000 μs, the range of the fracture zone does not change significantly and is not displayed in the table.

3.5. Dynamic Mechanical Response of Rock Reservoirs

The variation in radial and circumferential stress over time reflects the overall stress state of the material unit. To quantitatively analyze the stress evolution results, curves depicting the radial and circumferential stress changes with time at different positions in the coal seam, ranging from close to far from the origin in the x direction, are plotted in Figure 8. Following the L-CO₂ explosion, shock waves propagate into the coal seam from near to far, and material units at varying distances experience pressure pulses sequentially. Both the radial and circumferential stresses in the coal body exhibit compressive stress characteristics. During the propagation of the shock wave, the damping and energy-absorbing effects of the coal rock medium gradually reduce the amplitude of the shock wave, transitioning the propagation speed from supersonic to sonic, equating to the longitudinal wave speed of the material. Under this mechanism, the peak intensity of the radial shock stress in the near-well area significantly exceeds the dynamic compressive strength of the coal seam; for instance, at a distance of 60 cm from the borehole, the peak stress reaches 89.72 MPa, which is substantially greater than the compressive strength of the coal body, leading to crush–shear failure and the formation of a smash zone. As the shock wave propagates and attenuates, the peak value of the stress wave gradually falls below the dynamic compressive strength of the coal seam while still exceeding its dynamic tensile strength; for example, at a distance of 140 cm, the circumferential peak stress is −4.53 MPa, with the negative sign indicating tensile stress. At this moment, the coal body experiences dynamic tensile failure, resulting in the formation of a fracture zone. As the stress wave continues to propagate, the stress eventually decays below the dynamic tensile strength of the coal body, causing the explosive fractures to stop propagating.

4. Analysis of Response Model Results

4.1. Model Validity Testing

As discussed above, this study selects the borehole enlargement radius R_well, total area of the smash zone A_crush, shape factor of the smash zone ξ_crush, total length of fractures L_frac, and shape factor of the fracture zone ξ_frac as response targets. These response targets are defined as F₁, F₂, F₃, F₄, and F₅, respectively, to quantify the interactive effects of vertical in situ stress E₁, stress difference E₂, and explosive load E₃ on the plastic deformation around the borehole, the smash zone, and the fracture zone. A statistical analysis of the three response variables and five response targets yields the results shown in

Table 7. Response factors and response variables under different simulation conditions.

No.	E1/MPa	E2/MPa	E3/MPa	F1/mm	F2	F3/mm2	F4	F5/mm
1	5	5	150	1.17 × 103	1.00	2.64 × 103	1.02	12.47
2	5	5	270	8.55 × 103	1.03	1.80 × 103	1.04	11.63
3	15	5	150	1.09 × 103	1.57	2.29 × 103	1.16	12.09
4	15	5	270	4.44 × 103	1.10	1.17 × 103	1.10	11.28
5	10	0	150	1.08 × 103	1.06	1.82 × 103	1.09	10.68
6	10	0	270	8.19 × 103	1.21	1.38 × 103	1.09	9.81
7	10	10	150	1.71 × 103	1.15	2.65 × 103	1.28	13.40
8	10	10	270	7.24 × 103	1.42	1.89 × 103	1.01	12.71
9	5	0	210	1.40 × 103	1.00	1.70 × 103	1.00	10.37
10	15	0	210	5.94 × 103	1.72	1.29 × 103	1.14	9.81
11	5	10	210	1.73 × 103	1.00	2.87 × 103	1.01	13.57
12	15	10	210	1.10 × 103	1.33	2.38 × 103	1.06	12.86
13	10	5	210	9.63 × 103	1.20	1.81 × 103	1.05	11.84
14	10	5	210	9.66 × 103	1.20	1.82 × 103	1.04	11.73
15	10	5	210	9.61 × 103	1.20	1.80 × 103	1.07	11.96

.

To systematically quantify the control mechanisms of the selected response indicators F₁ to F₅ under the influence of three response variables E₁ to E₃ and their interactions, multiple regression fitting was performed for each response indicator, with the regression models summarized in

Table 8. Regression models of F₁–F_5.

Response Variable	Regression Models
F1	$F_1=7.597-0.081\cdot E_1-0.050\cdot E_2+0.025\cdot E_3$
F2	$F_2=1406.519-78.870\cdot E_1-46.932\cdot E_2+7.481\cdot E_3$
F3	$F_3=0.521+0.042\cdot E_1+0.033\cdot E_2+0.003\cdot E_3$
F4	$\begin{array}{l}{F}_4=2238.920+143.779\cdot E_1-43.684\cdot E_2-16.977\cdot E_3-7.302\cdot E_1{E}_2+0.006\cdot E_1{E}_3+0.148\cdot E_2{E}_3\\ -9.615\cdot E_1^2+2.722\cdot E_2^2+0.049\cdot E_3^2\end{array}$
F5	$F_5=1.047-0.00045\cdot E_1+0.042\cdot E_2-0.00002\cdot E_3$

.

Significance testing and a correlation analysis are primary methods for assessing the reliability of regression models. According to the ANOVA table shown in

Table 9. Analysis of variance.

Type	Response Variable	Model	E1	E2	E3
F-value	F1	252.40	50.65	19.62	686.92
	F2	35.64	40.36	14.29	52.29
	F3	2.94	3.99	6.53	0.06
	F4	15.85	63.39	30.79	25.18
	F5	4.49	0.00	13.41	0.04
p-values	F1	<0.0001	<0.0001	0.0010	<0.0001
	F2	<0.0001	<0.0001	0.0030	<0.0001
	F3	0.0802	0.0808	0.0339	0.8172
	F4	0.0036	0.0005	0.0026	0.0040
	F5	0.0274	0.9696	0.0037	0.8408

, the regression equations for response indicators F₁, F₂, and F₄ have p-values less than 0.05, with F₁ and F₂ having p-values below 0.0001, indicating highly significant regression results. Analyzing the F-values reveals that the L-CO₂ phase transition explosive load has the greatest impact on the borehole diameter and the area of the smash zone, followed by in situ stress, while the influence of stress difference on both the borehole diameter and smash zone area is minimal. The regression model for borehole diameter has an F-value of 686.92, significantly higher than the F-values for in situ stress and stress difference, confirming that the plastic deformation in the near-borehole region is primarily controlled by explosive load. In contrast, for the regression model of total fracture length, the F-value ranking is in situ stress > stress difference > explosive load, indicating that explosive fractures in the far field are mainly governed by in situ stress conditions. Furthermore, the p-values corresponding to F₃ and F₅ are greater than 0.05, indicating non-significant regression results. A comprehensive analysis of F-values and p-values suggests that the shape factors of the smash zone and fracture zone are primarily controlled by stress difference, which is consistent with previous studies. The borehole diameter following the L-CO₂ phase transition explosion is mainly influenced by explosive load, aligning with the control mechanism of the smash zone area.

An error analysis was conducted for the response models constructed from the selected five response indicators, as shown in

Table 10. Error analysis.

Statistical Items	F1	F2	F3	F4	F5
R2	0.986	0.907	0.688	0.966	0.550
R2A	0.982	0.881	0.454	0.905	0.428
R2P	0.972	0.818	0.552	0.858	0.037
CV/%	45.634	18.600	5.612	14.994	5.296
PA	1.365	8.985	4.949	15.407	13.461

. The results exhibit high consistency with the ANOVA findings, with R² values for F₁, F₂, and F₄ all exceeding 0.9, indicating low discrepancies between model predictions and actual values; both R²_A and R²_P values are high and close, suggesting that the response models can adequately predict the borehole diameter, smash zone area, and total fracture length. Additionally, the values for C_V and P_A also meet the required standards, collectively indicating a high level of reliability and accuracy in the experimental results.

4.2. Univariate Analysis

Based on the results of the ANOVA and error analysis, the impact relationships of response variables E₁ to E₃ on response targets F₁ to F₅ were determined, along with the significance of the models. The data indicate that the regression results for F₃ and F₅ are not significant. Therefore, we first analyze the univariate effects of E₁ to E₃ on F₁, F₂, and F₄. To intuitively present the influence trend of a single factor on the response variables, the other two response variables were set to their middle values, with vertical in situ stress at 10 MPa, stress difference at 5 MPa, and explosive load at 210 MPa. As shown in Figure 9a,b, the borehole radius and smash zone area show a consistent response to the three response variables, decreasing linearly with increasing vertical in situ stress and stress difference, while increasing linearly with increasing explosive load. Notably, the borehole radius is influenced far less by in situ stress compared to explosive load, indicating that the primary controlling factor for plastic deformation around the borehole is the explosive load. The smash zone area is mainly controlled by the same factors, as the radial compressive stress in the near-borehole region increases with distance from the borehole center, while the change in circumferential compressive stress is relatively minor. During the propagation of explosive load within the coal body, attenuation occurs, with the most rapid decay happening in the near-borehole area. Both factors act together in this region, resulting in the formation of a compressive–shear smash zone.

From Figure 9c, it can be observed that greater in situ stress correlates with a smaller total fracture length, showing an approximately negative exponential relationship, indicating that increased in situ stress significantly suppresses the propagation of explosive fractures. As stress difference increases, the total fracture length continues to decrease, though the influence is the smallest among the three response variables. Conversely, with increasing explosive load, the total fracture length significantly increases, although its impact is lower than that of in situ stress on total fracture length. These results demonstrate that the formation of explosive fractures is mainly controlled by in situ stress. This is due to the significant attenuation of shock waves in the far field, while the stress field generated by in situ stress within the coal body remains relatively stable. When the shock wave attenuates to a certain degree, in situ stress becomes dominant in the propagation of explosive fractures. Figure 9d indicates that the shape factor of the fracture zone is primarily controlled by stress difference, while the influences of in situ stress and explosive load are negligible. As the stress difference increases from 0 to 10 MPa, the shape of the fracture zone changes from circular to elliptical, with the ratio of the ellipse’s long to short axes increasing alongside the stress difference.

4.3. Response Surface Model

From the previous analysis, it is evident that the total fracture length F₄ is influenced not only by the three individual factors E₁ to E₃ but also by the interactions among these factors. The interaction surface plots for the three factors on F₄ were generated, with irrelevant factors set to their middle values, as shown in Figure 10. Figure 10 illustrates that the interaction surfaces of in situ stress–explosive load and stress difference–explosive load are quite similar; both decrease with increasing in situ stress or stress difference and increase with rising L-CO₂ phase transition explosive load. In contrast to the explosive load’s influence, the impacts of in situ stress and stress difference on the construction of the borehole fracture network are monotonically negative and distinct, with their interactions significantly reducing the network’s development. As seen in Figure 10a, with increasing in situ stress and stress difference, F₄ significantly decreases, and at lower in situ stress levels, the influence of in situ stress on F₄ is notably greater than that of stress difference. However, as in situ stress increases, the influence of stress difference on F₄ gradually intensifies. As in situ stress rises from 5 MPa to 15 MPa, the range of explosive load’s effect on F₄ shifts from 1058–1580 to 214–746, with the change range slightly increasing from 522 to 532. When stress difference increases from 0 to 10 MPa, the explosive load’s effect range on F₄ reduces from 1249–1695 to 607–1224, with the change range rising from 446 to 617. This is primarily due to the significant suppression of fracture network expansion under high-stress conditions, which may even hinder successful formation. Therefore, after increasing the explosive load, the formation and expansion of the fracture network demonstrate certain expansion and enhancement effects.

5. Discussions

To validate the reliability of this study, the results were compared with previous research. Ma et al. conducted experimental studies on liquid CO₂ phase transition blasting fracturing of coal rock model specimens [28]. The results showed that with increasing blasting pressure, both the number and complexity of cracks increased significantly. Similarly, the experimental results of Wang et al. also demonstrated that after gas blasting, distinct zones, broken zones, and crack zones appeared sequentially around the borehole [29]. These results are highly consistent with the numerical results obtained in this study. Moreover, physical experiments are limited by sample size and cannot determine the extent of fracture propagation. Therefore, this study compensates for the limitations of experimental research through numerical simulation methods. Compared with the single-factor analysis of Yuan et al., this study provides a method for investigating the interactions between various factors in blasting fracturing [20].

However, there are still some limitations in this study. First, due to limitations in computational resources, only a quasi-2D numerical model of the coal seam was established. As a result, the transmission of stress waves in the direction perpendicular to the plane was inevitably neglected during the blasting fracturing process [30]. In future studies, a 3D coal seam model will need to be developed. Second, this study did not consider the control effect of coal rank on blasting fracturing. In fact, the mechanical properties of different coal types significantly influence the propagation of explosive fractures. For example, Shang et al. studied the fracture propagation patterns in coals of varying hardness [31]. The results indicated that hard coals, with a high elastic modulus, tend to generate longer primary fractures under low blasting energy input, whereas soft coals exhibit better fracturing effects under high blasting energy input. Therefore, future research should calibrate the RHT constitutive parameters of coal samples from different coal ranks to evaluate the adaptability of L-CO₂ blasting fracturing for various reservoirs.

6. Conclusions

(1)

A coupled fluid–solid model that accurately reflects the L-CO₂ phase transition blasting fracturing process in coal seams was established. The state equation parameters for three typical L-CO₂ blasting tubes, with explosive loads of 150 MPa, 210 MPa, and 270 MPa, were calibrated with an error of no more than 4%. The RHT constitutive model, which accounts for the dynamic mechanical properties of the coal, was appropriately selected. A quasi-2D numerical model for L-CO₂ phase transition blasting fracturing of a 10 m × 10 m coal seam was constructed.

(2)

In situ stress was applied to the model. The results show that as in situ stress increases, both radial and circumferential stresses rise. When the stresses in both directions are unequal, the radial compressive stress in the direction of maximum in situ stress exceeds that in the direction of minimum in situ stress, while the circumferential stress is less than that in the direction of minimum in situ stress. Increased in situ stress significantly inhibits the propagation of explosive fractures, while stress difference facilitates the extension of explosive fractures along the direction of maximum principal stress.

(3)

The time-varying process of L-CO₂ phase transition blasting fracturing was analyzed. The results indicate that between 500 μs and 800 μs after an explosion, the plastic deformation around the borehole and the smash zone are essentially formed. By 2000 μs, the fissure zone due to L-CO₂ impact is largely established. With increasing explosive load, both the smash zone area and the fissure zone range significantly expand. The number of primary fractures generated by the explosion ranged from 0 to 12, the size of the pulverized zone varied from 1170 cm² to 2875 cm², and the total fracture length ranged from 44.4 cm to 1730.2 cm. As in situ stress rises, the range of the fissure zone will significantly decrease. As stress difference rises, the aspect ratio of the external ellipse of the explosion fracture zone will range between 1.00 and 1.72.

(4)

Using the total fracture length (F₄) as a response indicator, a response model was established to assess the effects of explosive load, in situ stress, and stress difference on explosive fracturing effectiveness. The results indicate interactions among the factors affecting explosive fracturing effectiveness. The suppressive effects of in situ stress and stress difference on fracturing effectiveness increase with their numerical values, while the enhancing effect of explosive load on fracturing effectiveness shows an increasing trend with a steeper slope as its value increases. This study provides data support for the optimization of L-CO₂ phase transition explosive fracturing processes.