# Double-Borehole Superimposed Effect of a New Non-Explosive Directional Rock-Breaking Method
## 2 Methodology
### The Theoretical Model of Double-Borehole Directional Rock-Breaking by IESF  
#### 2.2.1 Spatial Distribution Pattern of High-Pressure Gas Pressure Generated by IESF  
Following rock fragmentation by IESF, the resulting two-dimensional fracture planes exhibit significantly greater length than width, with the fracture width progressively decreasing from the near-surface to deeper regions. Consequently, the fracture cross-section can be approximated as symmetrical triangular profiles, as shown in Figure 3.  
Then, the volume of the cavity inside the borehole and fracture is  
\[V=\pi r^{2}L+l(t)dL \tag{1}\]  
where \(V\) is the volume of the cavity inside the borehole and fracture, m\({}^{3}\). \(r\) is the radius of the borehole, m. \(L\) is the length of the charge section inside the borehole, m. \(l(t)\) is the fracture depth as a function of time \(t\), m. \(d\) is the width of the fracture in the borehole wall, m.  
Figure 2: Directional fracture initiation, propagation, and through process by IESF: (**a**) charging of IESF; (**b**) reaction of coal-based solid waste expansive agent; (**c**) fracture initiation; (**d**) fracture propagation; (**e**) fracture coalescence.  
The volume of gas generated by IESF can be viewed as a linear function of the mass of the expansive agent, so there is  
\[k=\frac{V_{0}}{m} \tag{2}\]  
where \(k\) is the IESF combustion coefficient per unit time, m\({}^{3}\cdot\)s/kg. \(V_{0}\) is the volume of gas generated after the reaction at room temperature and pressure, m\({}^{3}\). \(m\) is the mass of expansive agents involved in the reaction per unit time, kg/s.  
The volume of gas generated at room temperature and pressure can be expressed by the ideal gas equation. For the high-temperature and high-pressure gases generated by IESF in the borehole, the van der Waals equation [37] is used to express the relationship between the gas pressure and volume, and the pressure in the pore can be expressed as  
\[\frac{(p_{g}+\frac{an^{2}}{V^{2}})(V-nb)}{p_{0}V_{0}}=\frac{nRT}{nRT_{0}} \tag{3}\]  
where \(p_{g}\) is the gas pressure in the borehole, MPa. \(a\) is the parameter for measuring intermolecular gravity. \(b\) is the sum of the volumes contained in 1 mole of the molecule itself. \(n\) is the number of moles. \(p_{0}\) is the standard atmospheric pressure, MPa. \(T\) is the post-reaction temperature in the borehole, \({}^{\circ}\)C. \(T_{0}\) is the temperature corresponding to the combustion coefficient \(k\), \({}^{\circ}\)C.  
Substituting Equations (1) and (2) into Equation (3) can be obtained by  
\[p_{g}(t)=\frac{Tp_{0}kmt}{T_{0}[\pi r^{2}L+l(t)dL-nb]}-\frac{an^{2}}{V^{2}} \tag{4}\]  
Due to the influences of fracture surface roughness, surrounding rock permeability, and gas flow velocity, a pressure gradient inevitably exists in the gas pressure distribution within fractures. However, fracture surfaces in different rock types exhibit significant variations in roughness and permeability. The results of fracture propagation are influenced by the action of the gas within the crack as a whole, rather than the individual action of one point within the crack. Therefore, the calculation of the distribution of gas pressure is simplified considering the above situations. For computational simplification, this gradient  
Figure 3: Mechanical model of double-borehole crack propagation under in situ stress.  
can be approximated as uniform attenuation. Referencing [PERSON] et al.'s research on gas flow characteristics within fractures [38], the gas pressure distribution is expressed as  
\[p(x,t)=p_{g}(t)[1-\frac{x-r}{l(t)}] \tag{5}\]  
where \(p(x,t)\) is the gas pressure distribution in the fracture, MPa. \(p_{g}(t)\) is the gas pressure in the borehole at the moment \(t\), MPa. \(r\) is the radius of the borehole, m.  
Substituting Equation (4) into Equation (5) can be obtained by  
\[p(x,t)=\bigg{[}\frac{Tp_{0}kmt}{T_{0}[\pi r^{2}L+l(t)dL-nb]}-\frac{an^{2}}{V^ {2}}\bigg{]}[1-\frac{x-r}{l(t)}] \tag{6}\]