## 1 Introduction  
Hydraulic cracks are prevalent in geotechnical engineering, and water pressure can cause the tensile stress at the crack tip to significantly increase compared to normal cracks, leading to easier crack initiation. Furthermore, under the action of hydraulic pressure and far-field external loads, hydraulic cracks will rapidly initiate, expand, and connect, eventually forming a crack network and increasing the difficulty of conducting geotechnical safety assessments. Crack networks are a considerable threat to the safety of hydraulic engineering (such as in the case of dams [1] and hydraulic tunnels [2]); however, their formation has a positive significance in terms of improving the efficiency of mining engineering (such as in the context of shale gas extraction [3], geothermal energy development [4], and CO2 geological storage [5]). Whether in the case of anti-cracking in hydraulic engineering or cracking in mining engineering, hydro-mechanical coupling stress-intensity factor (SIF) calculations for multiple cracks are the basis of the analyses. SIF calculations can be used to differentiate crack initiation mechanisms (including crack initiation load and initiation angle) under hydro-mechanical coupling conditions in geotechnical engineering and can also be used to determine whether a crack network will form in the engineering structure, thus providing a theoretical basis for conducting safety evaluations in the context of geotechnical engineering. Therefore, it is of great significance to study the calculation methods used for hydro-mechanical coupling SIFs for multi-cracks. Currently, the calculation method used in the case of multi-crack SIFs focuses mainly on mechanical loading. [PERSON] and [PERSON] [6] used the alternating method and the iterative method to solve multi-crack SIFs under load on the crack surface. [PERSON] and [PERSON] [7] used the dislocation theory and the superposition method to obtain multi-crack SIFs under far-field compressive stress. [PERSON] et al. [8] adopted the energy method to calculate multi-crack SIFs in symmetrical structures. However, these methods have limitations since they are applicable only to specific structural and loading situations. In order to overcome these problems, scholars have conducted further research. [PERSON] and [PERSON] [9] proposed the Schwartz-Neumann alternating method to determine the SIFs of co-linear cracks with arbitrary boundary conditions. [PERSON] et al. [10] and [PERSON] [11] used the Laplace transform method and the extended superposition method to obtain the SIFs for arbitrarily distributed multi-cracks, respectively. [PERSON] et al. [12] modified [PERSON]'s method based on the assumption of non-uniform terms and obtained the SIFs for multi-cracks with small spacings. Nevertheless, since these complex auxiliary functions (e.g., the Laplace transform function, the dislocation density function, and the alternating transform function) depend on the loading conditions, they are difficult to apply when solving cases involving complex hydro-mechanical coupling loading. The calculation methods used to solve multi-crack hydro-mechanical coupling SIFs rely mainly on numerical methods. [PERSON] et al. [13] developed a numerical model to analyze hydro-mechanical coupling SIFs and initiation lengths based on finite fracture mechanics (FFMs). [PERSON] et al. [14] adopted the second-order numerical manifold method and the singular boundary element method to calculate hydro-mechanical coupling SIFs under seepage conditions. [PERSON] et al. [15] established a new finite element model based on the equivalence theory and calculated hydro-mechanical coupling SIFs with wing cracks. [PERSON] et al. [16] used the virtual crack closure technique (VCCT) to calculate the hydraulic coupling stress-intensity factors for arbitrarily shaped cracks. [PERSON] et al. [17] adopted the level set method to describe holes and cracks in a numerical model and combined it with the extended finite element method (XFEM) to obtain hydro-mechanical coupling SIFs.