The PKN model is based on the following assumptions: the rock is an elastic and brittle material, the height of the fracturing crack is constant, the cross section of the crack is elliptical with the maximum crack width in the middle of the crack, and the fracture toughness has no effect on the geometric deformation of the fracture (Wang et al., 2017). The fracture model of PKN is shown in Fig. 1.

Under the assumption of neglecting the compressibility of the fracturing fluid, Nordgren (1972) derived the fluid continuity equation for the fracturing fluid in the crack, as shown in Eq. (1). 1∂q∂x+qt+∂q∂t=0 In Eq. (1), q(x,t) represents the volumetric flow rate of fluid through the cross section of the crack. qt(x,t) represents the volumetric flow rate of fluid lost per unit length of the crack. A(x,t) represents the cross-sectional area of the crack. The calculation formulas for the crack length L, local fracture width w, and pore pressure pw in the absence of fluid loss are as follows (Nordgren, 1972): 2L=0.68GQ31-υμh415t45,3w=2.51-υμQ2Gh15t15,4pw=2.5G4μQ21-υ4h615t15. In Eqs. (2)–(4), G represents the shear modulus of the rock, ν represents Poisson's ratio of the rock, h represents the length of the crack, Q represents the fluid injection rate, and μ represents the viscosity of the fracturing fluid.

The basic principle of in situ stress measurement based on hydraulic fracturing involves placing drill rods and packers into a borehole using a drilling rig to measure their positions. Fluid is injected into the packers through a loading control system, isolating a test section within the borehole, and the fluid is further injected into the test section until fracturing occurs.

As shown in Fig. 2, the first highest-pressure value is recorded as the fracturing pressure Pb. Then the pressure drops rapidly to a state of fluid seepage into the fracture and remains constant. At this point, the pump is turned off to stop loading, and the pressure in the fracturing section decreases rapidly, causing the fracture to close quickly. When the fracture is in the near-closed state, the rate of pressure decrease slows down, and the pressure value at this time is recorded as the instantaneous closure pressure Ps. After releasing the pressure, reloading causes the fracture to reopen, and the pressure value at this time is recorded as the reopening pressure Pr (Wang et al., 2017).

According to the elastic theory and the PKN mechanical model, as shown in Fig. 3, the fracturing pressure of the rock in the fracturing section is 5Pb=3σh-σH+T. Among them, σH and σh are the maximum and minimum horizontal principal stresses, respectively, and T is the tensile strength of the rock. The fractures induced by hydraulic fracturing are vertical fractures and perpendicular to the direction of the minimum horizontal principal stress. Equation (5) indicates that the fracturing pressure of rocks is independent of the size of the borehole and the elastic modulus of the rock and is mainly determined by the tensile strength of the rock and the magnitude of the in situ stress around the borehole.

The underground measurement components mainly consist of small-diameter packers and underground connecting installation rods. The function of the packers is to seal a section of intact rock wall in the borehole, forming a sealed space. The connecting installation rods are used in a multi-segment connection manner to connect the packers to the surface control components at the borehole entrance.

The surface control components consist of a high-pressure fluid control system and a controllable flow high-pressure oil pump. The high-pressure fluid control system is a key component for fluid extraction, transmission, and control during hydraulic fracturing in situ stress testing. The inlet of the high-pressure fluid control system is connected to the outlet of the high-pressure water pump, and the two outlet ports of the high-pressure fluid control system are, respectively, connected to the seat sealing pressure circuit and the fracturing pressure circuit of the connecting installation rod, achieving dual-circuit control of high-pressure fluid for hydraulic fracturing.

The high-pressure fluid control system includes a high-pressure oil pump, an oil tank, a pressurized ball valve, a relief ball valve, a needle valve, a pressure gauge, a pressure sensor, a PC data acquisition and processing unit, and multiple three-way pipe fittings. It can sequentially perform high-pressure permeability self-checking and sensor calibration tests, a seat sealing function test, a high-pressure output function test of the fracturing segment, and a pressure relief function test. This system realizes the integration and implementation of various high-pressure fluid control functions during the hydraulic fracturing in situ stress measurement process. Figure 9 shows the schematic diagram of the high-pressure fluid control system connection.

As shown in Fig. 9, the high-pressure oil pump pumps the liquid from the oil tank to the three-way junctions. The three-way junctions are connected to the channels of the packer and the fracturing segment, respectively. In the packer channel, there are sequentially connected needle valves and two three-way junctions. The third channel of the two three-way junctions is connected to the pressure gauge and the pressure sensor, respectively. In the fracturing segment channel, there are sequentially connected pressurized ball valves and three three-way junctions. The third channels of the three-way junctions are connected to the pressure gauge, pressure sensor, and relief ball valve, respectively. The relief ball valve outputs to the oil tank via a pipeline. Both the sensors output to the PC data acquisition and processing unit.

The Nalin River coal mine in Wushen Banner, Ordos, Inner Mongolia, belongs to the Nalin River mining area in the Dongsheng coalfield. The stratigraphic division of this area falls within the Ordos subdivision of the North China Stratigraphic Region (Tian et al., 2011). The region, represented by the Dongsheng–Wushen Banner, exhibits a relatively complete development of Mesozoic and Cenozoic strata, referred to as the “Wushen Banner subregion”. The Nalin River exploration area is located in the southeastern part of the Wushen Banner subregion, as shown in Fig. 10.

The Dongsheng coalfield is located on the northeastern edge of the basin, while the no. 2 well of the Nalin River mining area is closer to the central zone of the basin. In the entire Ordos Basin, both in terms of basin genesis and coal accumulation patterns, the Upper Triassic Yanchang Formation (T3y) serves as the sedimentary basement for Jurassic coal-bearing basins and coal-bearing strata (Tian et al., 2011). Therefore, it is necessary to employ a small-diameter hydraulic fracturing in situ stress measurement system for deep-coal-mining operations in order to conduct stress testing of the geological stress in coal seams. This will be beneficial for analyzing the characteristics and stability of the geological stress field in the coal mining engineering area.

Selection of the test section. Based on the depth position of the complete core recorded and verified, as well as the positions required by the engineering design, consideration should be given to placing the packer at a location with smooth borehole walls and consistent aperture.

During the measurement process, each section usually undergoes three to five cycles in order to obtain reasonable stress parameters and accurately determine the fracture and extension status of the rock.

The stress values obtained from the three boreholes arranged in the engineering area are relatively consistent and increase with depth. The maximum and minimum horizontal principal stress values and vertical stress values were analyzed and statistically analyzed, and the average values for each measurement interval were calculated. The statistical results are shown in Table 3.

Overall, within the testing depth range, the relative magnitudes of the three principal stresses can be described as follows: SH>SV>Sh. This indicates that in the shallow crust of the region, horizontal stresses play a dominant role, and the present-day stress conditions are relatively intense. Based on the general statistical analysis in Table 3, it can be determined that the range of in situ stress values in the tested area should be as follows: the range of maximum horizontal principal stress values is 13.20–22.99 MPa, the range of minimum horizontal principal stress values is 6.05–11.79 MPa, and the range of vertical principal stress values is 13.87–14.25 MPa.

The relationship between the three principal stresses in this region is SH>SV>Sh, indicating a stress state favorable for slip on strike-slip faults. This result is consistent with the findings of Fan et al. (2003) using focal mechanism analysis, and Fan et al. (2003) also obtained corresponding conclusions that SH>SV>Sh, which suggest that seismic events in this region are primarily generated by slip on strike-slip faults. By conducting linear data fitting on the stress data, the distribution patterns of maximum, minimum, and vertical principal stresses with depth can be obtained (Fig. 11) and represented, respectively, as SH=-0.28+0.035×Z, Sh=-0.29+0.020×Z, SV=0.023×Z (with correlation coefficients of 0.99, 0.98, and 1). It can be observed that both the maximum and minimum horizontal principal stresses increase with depth. The calculated stress gradients for the maximum and minimum stresses are 0.035 and 0.020, respectively, indicating that the stress state in this region is conducive to slip fault activity. The relatively small stress gradient for vertical stress is due to the presence of predominantly loose soil in the upper covering layer at the measurement point, reflecting an overall lower average density characteristic. Due to the excellent linear correlation of the data, these fitting formulas can be used to predict and estimate the stress state in other parts of the mining area.

The data were plotted based on the depth distribution characteristics, resulting in a graph illustrating the lateral pressure coefficient, as shown in Fig. 12. It can be observed that the lateral pressure coefficient exhibits a relatively scattered distribution without displaying a common distribution pattern. Therefore, the arithmetic average of the lateral pressure coefficient was calculated for all the data. The average ratio of the maximum horizontal principal stress to the vertical principal stress is 1.81, while the average ratio of the minimum horizontal principal stress to the vertical stress is 0.82. The average ratio of the maximum horizontal principal stress to the vertical principal stress at this measurement point is higher than the value of 1.5 obtained at the Wushenqi 3–1 coal mine stress measurement point. This is because the measurement point includes more shallow data, which typically have higher measurement coefficient values. Overall, at a depth of 550 m, the stress level at this measurement point is comparable to that of the Wushenqi 3–1 coal mine stress measurement point.