Huipodi coal mine is located in the south of Fenxi County, northwest of Linfen, Shanxi Province. The average thickness of No.11 coal seam is 2.96 m, with a distance of 15.14–35.89 m from the Ordovician limestone, and an average of 25.4 m. The main aquifers affecting the mining of the No.11 coal seam in the research area are the K2 limestone aquifer of the Taiyuan Formation at the top and the Ordovician limestone aquifer group at the bottom. K2 limestone has an average thickness of 8.9 m, with moderate to weak water bearing capacity, mainly composed of static reserves, and is prone to drainage; The Ordovician limestone is a thick layer of limestone with strong water bearing capacity, and is the main aquifer that affects the safe mining of coal seams in this area. It is relatively close to the No.11 coal seam and is the key aquifer of this study.

We took the mining process of the coal with the risk of water inrush from floor in Huipodi Coal Mine, Xibeifen, Hongkong County, Shanxi Province as an example, the RFPA^2D-Flow (Li et al., 2009a; Li et al., 2009b; Li et al., 2011) software was used to simulate the damage evolution process and water inrush characteristics of the floor rock mass with hidden faults during mining. The model consists of three irregular hidden small faults located at 100 m, 175 m, and 250 m on the left boundary of the model, all of which are water conducting faults. From the perspective of fluid-structure interaction, the whole process of hidden fault activation, fracture evolution and coalescence, and water inrush caused by the damage of floor aquifuge in the coal mining process of the working face were analyzed.

The basic equations of seepage include the stress balance equation, constitutive equation, and seepage continuity equation. According to the static equilibrium condition of any rock element, the effective stress equilibrium differential equation of the research object can be established in a three-dimensional Cartesian coordinate system: ∂ σ i j ∂ x i j + f = 0   i , j = 1 , 2 , 3 (1)

Where f is the volumetric force, m·s^-2.

In practical engineering, the seepage of fluid in rock mass is non steady state seepage, and its continuous equation is expressed as follows: ∂ ρ n ∂ t + ▽ ⋅ ρ V = 0 (2)

Where f is the fluid density, kg/m³, n is the porosity, V is the seepage velocity.

When the seepage satisfies Darcy’s law and the coordinate axis direction is consistent with the main direction of the seepage coefficient tensor, there are: V x = − k x ρ g ∂ p ∂ x , V y = − k y ρ g ∂ p ∂ y , V z = − k z ρ g ∂ p ∂ z + ρ g (3)

Where k _x, k _y, and k _z are the permeability coefficients in the x, y, and z directions, g is the gravitational acceleration.

Assuming that the fluid is incompressible, the continuity equation for seepage is: 1 ρ g ∂ ∂ x k x ∂ p ∂ x + ∂ ∂ y k y ∂ p ∂ y + ∂ ∂ z k z ∂ p ∂ z + ∂ ∂ z k z = ∂ n ∂ t (4)

The solution to the coupled seepage stress equation can be obtained by combining the equilibrium equation, constitutive equation, and seepage continuity equation under the control of seepage boundary conditions.

Based on the constitutive relation of uniaxial tension and compression, the meso-fracture damage constitutive coupling equation of the element is introduced.

(1) unit damage constitutive relation under uniaxial tension

The permeability damage relationship equation of rock micro elements under uniaxial tension is basically the same as that under compression. When the element reaches the damage threshold of uniaxial tensile strength f _t: σ 3 ≤ − f t (5)

The damage variable D is expressed as: D = 0 ε t 0   < ε 1 − f t r E 0 ε ε t s   ≤ ε < ε t 0   1 ε < ε t s   (6)

Where f _tr is the residual uniaxial tensile strength. The change of permeability coefficient of the unit is as follows: λ = λ 0 e − β σ 3 − α p ε t 0   < ε ξ λ 0 e − β σ 3 − α p ε t s   ≤ ε < ε t 0   ξ ′ λ 0 e − β σ 3 − α p ε < ε t s   (7)

Where ξ ′ is the coefficient of increase of permeability when the element fails.

(2) Constitutive relationship of element damage during uniaxial compression of rocks

When subjected to uniaxial compression, the Mohr-Coulomb criterion is chosen as the failure criterion for the element, which is: F = σ 1 − σ 3 1 + ⁡ sin ⁡ ϕ 1 − ⁡ sin ⁡ ϕ ≤ f c (8)

Where φ is the internal friction angle of the rock, f _c is the uniaxial compressive strength of rocks.

When the shear stress value reaches the Mohr-Coulomb damage threshold, the damage variable D is given according to the following equation: D = 0   ε c 0   < ε   1 − f c r E 0 ε ε ≥ ε c 0   (9)

Where f _cr is the residual compressive strength under uniaxial compression, ε _c0 is the maximum compressive strain, ε is residual strain.

According to the experimental summary, damage will cause a rapid increase in the permeability coefficient of the specimen, and the change in the permeability coefficient of the unit is determined by the following equation: λ = λ 0 e − β σ 1 − α p   D = 0 ξ λ 0 e − β σ 1 − α p   D > 0 (10)

The study shows that the permeability coefficient of the rock specimen with a through crack after reaching the peak is about 100 times higher than that of the specimen with a non-through shear band. Therefore, when the tensile strain reaches the ultimate tensile strain, the unit will completely lose its bearing capacity and stiffness, and can be set as an air unit (i.e., D=1).

Considering the heterogeneity of rock material, a numerical calculation model for water inrush simulation was established based on the two-dimensional plane strain model and the geological conditions of Huipodi coal mine. The model size was 480 m × 280 m, divided into 480 × 280 A total of 134400 units. It was divided into 18 rock layers. The mechanical parameters of each rock layer material are shown in Table 1. Joints were set between the layers to represent the weak layer between the rock layers.

The calculation model is shown in Figure 1. The rock mass bears dead load and confined water pressure. The boundary conditions of the model were as follows: the left and right boundaries of the model were horizontally fixed and vertically free, the bottom boundary was horizontally unconstrained and vertically fixed. The 180 m high fixed head boundary was set to simulate the confined water pressure of Ordovician limestone. The distributed excavation method was used to simulate the mining of the working face. The open-off cut was located at 150 m to the right of the model. The excavation length of each step was 10 m, a total of 15 steps, and the cumulative excavation was 150 m.

The formation process of water inrush channels activated by hidden faults is shown in Figure 2 and Figure 3. According to the simulation results, it can be found that the floor rock layer was damaged below the working face due to the formation of a rapid pressure change zone between the advanced support pressure on one side of the coal pillar below the working face and the pressure relief zone on the goaf side. Due to the presence of hidden faults, the depth of damage to the floor near the fault was much greater than on the other side. Under the action of mining induced stress and confined water pressure, faults exhibit small-scale activation, which was limited to the stress disturbance range in the mining failure zone. The emergence of mining damage areas and fault activation areas has created favorable conditions for the formation of the next step of water inrush channels. When the working face advanced to a distance of 20–30 m from the open-off cut, regional damage begins to occur on the floor (as shown in Figures 2A, B), and cracks within the water conducting fault zone of the floor begin to develop. The first and second hidden faults exhibited a certain range of activated areas (as shown in Figure 3A). The second hidden fault has a relatively larger activation range due to its closer proximity to the goaf. Under the action of mining induced stress and pressurized water pressure (including hydraulic scouring and hydraulic wedging), cracks begin to develop in the water conducting fault zone of the floor, forming a certain range of activation zones within the fault zone, that was, hidden faults were in the activation stage.

When the working face advanced to 50 m, due to the intensified destructive effect of mining activities on the initial stress field, the mining damaged zone of the floor increases, and downward open cracks appear on the floor. As well, the activated area of hidden faults gradually extends towards the working face. The cracks in the floor damaged zone within the goaf gradually connected with hidden faults, and confined water begins to flow out, as shown in Figure 2B and Figure 3B. After the working face advanced to 70 m, the third hidden fault also begins to locally activate. The fissures in the coal wall develop, and the confined water ascends, interacting with the fracture field of the second fault, further expanding the damage area of the floor strata, corresponding to Figure 2C and Figure 3C. During this process, the floor connected with the water conducting fissure of the second hidden fault, where the water inrush channel gradually formed and the seepage discharge increased, corresponding to Figure 2D and Figure 3D. After the working face advanced to 100 m (as shown in Figure 2E and Figure 3E), the activated area of hidden fault further expands, the aquifer rock mass between the mining damage area and the fault activation area continued to be damaged and destroyed, and the diversion fissure zone continued to expand and connect with each other. As the working face continued to advance, the fissure of the third hidden fault gradually connectd with the mining damage area of the floor. When the excavation reached to 110 m (as shown in Figure 2F and Figure 3F), the seepage passage penetrated. It can be found that the coalescence of water conducting fissures in the middle fault accelerates the expansion speed of the right fault fissure.

The variation character of seepage field in the numerical simulation of the water inrush process raised by activation of hidden faults as shown in Figure 4. From the simulated result it can be found that the mining activities have a significant impact on the permeability of the floor with hidden faults. During the process of advancing the working face, the action of mining stress and confined water made fissures of hidden faults expand rapidly, and the permeability has been greatly improved. At the same time, the confined water within the fault also has a scouring and expansion effect on the fault fissures, which accelerated the development speed of the fissures. New cracks were generated and interconnected, ultimately reduced the permeability and integrity of the aquifer rock mass of the floor. When the working face advanced to 70 m, the mining damage area connected with the conductive layer of the hidden fault, formed the first water inrush channel, and caused water inrush (Figure 4D corresponds to Figure 2D and Figure 3D). As the working face advanced to 110 m, the third water conducting zone of the fault was formed. During the entire advancing process of the working face, the seepage field (shown by the arrow in Figure 4) remained consistent with the fracture propagation direction of the hidden fault, and the seepage field evolved synchronously with the fracture field of the hidden fault. The variation range of the seepage field was the same as the propagation area of cracks (Figures 4A–F; Figures 2A–F; Figures 3A–F). There were spatial and temporal differences in the fracture penetration rules of hidden faults at different positions of the floor. The closer the distance to the goaf, the more likely it was to occur the water inrush raised by activation of hidden faults. With the increased of pressure relief effect in the goaf, the possibility of water inrush from hidden faults gradually increased.

The coal wall near the working face produced stress release due to excavation unloading, and the value of bearing pressure was small. After the excavation of the coal seam, the overall variation trend of the value of bearing pressure in the direction away from the mining area was: it first increases, and then started to decrease until the initial rock stress state. After the first weighting, as the working face advanced forward, due to the increase in excavation space, the increased in the bearing pressure concentration value caused by the advancing unit distance of the working face gradually decreases, and the variation tends to be flat, as shown in Figure 5. When the working face was advanced by about 95 m–100 m, the support stress of the coal wall was about 4 times the original stress, reached a maximum value of about 35 MPa, and then showed a gradually decreasing trend. As can be seen from Figure 6, the floor forms a stress concentration within a certain range near the open-off cut. As the working face continued to advance, the concentrated stress gradually increased, and the concentrated stress at the corresponding position of the floor below the open-off cut extends to a certain extent to the corresponding position above the goaf, that was, an elastic-plastic high stress zone was formed within 12 m around the open-off cut and the coal wall. The main stress of the protective layer in this range was the largest, and the coal rock undergoes compression shear failure, which could easily connect with the uplift fracture formed by the activation of hidden faults to form a fracture channel. Compared to other areas, due to the largest water head pressure difference between the coal seam and the goaf in this area, it was extremely prone to water inrush, causing accidents.

It can be found from Figures 5–7 that the presence of hidden faults has accelerated the destruction process of the water-resisting of floor. As the working face continued to advance, the permeability coefficient of the floor had undergone significant changes. Under the combined action of mining stress and confined water pressure, the confined water flows along the fracture channel from high pressure areas to low pressure areas. Between the initial stage of mining and the range of 50–60 m, the water head pressure gradient near the coal wall at the open-off cut and working face was larger, the stress on the coal wall was not released, its permeability was somewhat reduced, and the corresponding flow rate was also smaller. However, the floor stress in the goaf was released, and the elastic energy was released, resulting in floor heave and tensile cracks. The water head pressure decreased rapidly, and the corresponding flow rate also continuously increased. In particular, with the advancement of the working face, three hidden faults become activated, cracks expand and connect, and the formation of water inrush channels, the water flow increases rapidly, especially after the working face advances to 80 m, the flow distribution characteristics become more obvious.

Comprehensively analyzed the stress change state, damage zone characteristics, and fracture distribution of the floor with concealed faults, taking into account the change characteristics of mining stress, confined water pressure, seepage velocity, and water inrush quantity, the characteristics of water inrush were summarized as follows: When no faults were encountered in the working face, the failure characteristics were similar to those of the complete floor. When encountering hidden faults, the integrity of the floor decreased, and the damage zone of the floor was easily connected with the aquifer, and water inrush from the floor was easily induced. Due to the high abutment pressure near the coal wall, the head pressure increased, made the risk of water inrush greater. When there were faults behind the working face, seepage mainly occurred in the water inrush channels connected by the faults. Therefore, the existence of faults made the division of water inrush characteristics significantly different from that of the complete floor. Moreover, when there were multiple hidden small faults in the floor, it showed alternating changes between the water inrush growth area and the flow stable area with similar cycle characteristics.

The hidden faults in the coal seam floor are generally small geological structures with small planar distribution and limited vertical distance, which can be simplified as central oblique cracks under complex stress and osmotic pressure. Combined with the results of numerical simulation, the necessary condition of water inrush from fault is the initiation and propagation of the crack under the action of dynamic and static loads. Assuming that the length of the crack is 2a and the pore water pressure within the crack is p, and assuming that the pore water pressure exerts equal forces along each direction of the crack, the angle between the incident direction of the stress wave and the x-axis is α, the angle between the crack and the vertical principal stress σ ₁ is β, and the angle at which the polar coordinates are constructed with the crack tip as the origin is τ (As shown in Figure 9). When the influence of dynamic load effect was not considered, the static stress state on the crack surface can be expressed as (positive in tension and negative in compression in fracture mechanics): σ n = − σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P τ = − σ 1 − σ 3 2 sin ⁡ 2 α (11)

From Eq. 11, it can be found that there are both normal and shear stresses on the cracks, and the propagation and instability mode of the cracks is the I-II composite type. According to whether the normal stress of the fracture surface is tensile or compressive, the cracks can be divided into open cracks and closed cracks, thereby determining whether the expansion mode of rock mass cracks is compression-shear composite fracture or tension-shear composite fracture. In fracture mechanics, crack propagation and failure are divided into I-II type tension-shear propagation and II type compression-shear propagation based on whether the normal stress of the crack is a tensile or a compressive stress. According to the theory of fracture mechanics, in the linear elastic range, when cracks have the same propagation mode under multiple loads, the stress intensity factor at the crack tip can be calculated using the superposition method (Fan, 2006). Therefore, the dynamic and static stress intensity factors included can be used to obtain the equivalent stress intensity factors for cracks under combined dynamic and static loads using the linear elastic superposition principle, namely,: K E = K E J + K E D (12) where K _E is the stress intensity factor for hydrous cracks, K _EJ is the static stress intensity factor under static loads, K _ED is the dynamic stress intensity factor under dynamic loads, and the crack propagation mode generated by static load and dynamic load is the same.

The type II dynamic stress intensity factor generated by dynamic load can be calculated by the following equation (Fan, 2006) K I ′ t = τ 1 π a K I 2 exp − i ω t − δ 2 1 K I I ′ t = τ 1 π a K I I 2 exp − i ω t − δ 2 2 (13) where τ 1 = μ α 2 ψ 0 , α=ω/c_s, ω is the circular frequency of the stress wave, c_s is the SV wave velocity, α is the wave number (1/length), ψ ₀ is a constant (1/length²), μ is the Lame constant, δ 2 1 and δ 2 2 are the phase angles, K I 2 and K I I 2 are the dynamic stress intensity factors divided by their ω= Static value corresponding to 0.

(1) TypeⅠ-Ⅱ tension-shear composite propagation

When the pore water pressure in the fracture is greater than the normal stress generated by the remote stress on the fracture surface, the normal stress on the fracture surface is tensile stress, and the fracture is in an open state. The propagation problem of an open fracture belongs to the I-II tension-shear composite type. When the normal stress at the hydrous crack surface of the rock mass is the tensile stress, the crack will expand under tensile stress. The approximate fracture criteria commonly used in engineering (Huang et al., 2000; Wang et al., 2004) are used for analysis, and the criterion for instability propagation can be expressed as K I C = K I + K I I + K I d + K I I d (14) where K _IC is the stress intensity factor for I-II type tension-shear propagation; K _I and K _II are Type I and Type II stress intensity factors caused by static loads (crustal stress, pore water pressure, etc.), as shown in Eq. 14; K _Id and K _IId are Type I and Type II stress intensity factors caused by dynamic loads K I = σ π a K I I = τ π a (15)

Taking Eq. 11 into Eq. 15, the static stress intensity factor at the crack tip under the static load can be obtained K I = σ π a = − π a σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P K I I = τ π a = − π a σ 1 − σ 3 2 sin ⁡ 2 α − f σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P (16)

According to Eq. 3, the maximum values of K _Id and K _IId can be obtained after ignoring time and phase, and they are substituted with Eq. 13 and (16) into Eq. 14 to obtain the stress intensity factor K _IC for crack failure of tension-shear propagation under the dynamic load K I C = − π a σ 1 − σ 3 2 sin ⁡ 2 α − f σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P − π a σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P + τ 1 π a K I 2 + τ 1 π a K I I 2 (17)

At this time, the corresponding water pressure p is the critical water pressure for the tension-shear propagation mode of the crack, recorded as P _lj P l j = σ 1 + σ 3 − σ 1 − σ 3 cos ⁡ 2 α 2 1 + f + σ 1 − σ 3 sin ⁡ 2 α 2 1 + f + f σ 1 + σ 3 − σ 1 − σ 3 cos ⁡ 2 α 2 1 + f + K I C 1 + f π a − τ 1 K I 2 + τ 1 K I I 2 1 + f (18)

(2) Type II compression-shear propagation

When the normal stress at the hydrous crack surface of the rock mass is compressive stress, the crack will close under the compressive stress, forming a pure type II compression-shear propagation mode (Wang et al., 2010). According to the maximum hoop stress theory (Li and Yang, 2006; Li et al., 2015), a polar coordinate system is established with the crack tip as the polar coordinate origin. The hoop stress can be expressed as σ θ = 1 2 2 π r cos θ 2 3 K I I + K I I ′ sin ⁡ θ − K I + K I ′ 1 + ⁡ cos ⁡ θ (19)

The angle between the section where the maximum hoop stress σ _θmax is located and the original crack line can be used to determine the initial crack angle of crack propagation θ ₀ . At this time, the stress intensity factor for crack initiation can be expressed as K e = σ θ r , θ 0 2 π r (20)

When the influence of crack thickness and tip curvature radius is ignored, the initial crack angle of crack compression-shear propagation is always arctan 2 2 (Huang et al., 2000). Simultaneous Eqs 19, 20, when the normal stress at the hydrous crack surface of the rock mass is compressive stress, a pure type II compression-shear propagation mode is formed, and the stress intensity factor for type II compression-shear propagation of the crack can be obtained K I I C = 2 3 K I I + K I I ′ (21)

Combined with the results of numerical simulation, it can be concluded that the water inrush from the floor of the hidden fault is caused by dual actions: ① the dynamic and static load actions caused by the collapse of overlying strata lead to the activation of the water-bearing hidden fault; ② under the continuous action of high pressure water pressure, the crack of rock mass weakens and changes the effective stress between cracks of rock mass, which causes the crack to expand and intersect, the permeability has been greatly improved, and the confined water in the fault has the effect of scouring and dilatancy on the cracks, which accelerates the growth of cracks. The closer to the goaf, the more likely it is for hidden fault activation and water inrush to occur.

The critical water pressure for the compression-shear failure mode of cracks at different depths is much smaller than the critical water pressure for the tension-shear failure mode of cracks, and the critical water pressure for the compression-shear failure of cracks is easier to meet than the critical water pressure required for the tension-shear failure. Therefore, in natural construction force environment such as in-situ stress and high water head pressure, hydraulic fracturing of rock mass under the disturbance of engineering measures such as excavation is the compression-shear failure mode (Wang et al., 2004; Chen et al., 2007; Wang and Yao, 2022). When the normal stress at the hydrous crack surface of the rock mass is compressive stress, the cracks undergo closure and compaction, and the cracks appear to be uniformly relieved, while transmitting corresponding shear stress and normal stress. The shear stress can cause crack surface growth. Due to the crack closure and compaction, the frictional force will be generated along the crack surface under the action of normal stress to resist the propagation and sliding of the crack surface. According to this, the effective shear stress on the crack of the floor rock mass causes the compressive-shear failure and water inrush in the fractured rock mass, and the effective shear stress can be expressed as τ ′ = 0 τ < f σ + c τ − f σ + c τ > f σ + c (22) where f is the coefficient of friction, c is the cohesion of the crack surface.

According to Eq. 22, adjust the K _II in Eq. 15 to: K I I = τ ′ π a (23)

The calculation method is the same as that for the stress intensity factor for tension-shear propagation failure. According to Eq. 18, the maximum value of K I I ′ after ignoring the time and phase can be obtained. By substituting it with Eqs 22, 23 into Eq. 21, the stress intensity factor K _IIC for compression-shear propagation failure of the crack under dynamic loads can be obtained K I I C = 2 π a 3 σ 1 − σ 3 2 sin ⁡ 2 α − f σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − P + c + τ 1 + τ 1 K I I 2 (24)

At this point, the corresponding water pressure p is the critical water pressure for the compression-shear propagation mode of the crack under the superimposed dynamic and static loads, recorded as P _yj P y j = σ 1 + σ 3 2 − σ 1 − σ 3 2 cos ⁡ 2 α − 1 f ⋅ σ 1 − σ 3 2 sin ⁡ 2 α + τ 1 K I I 2 − 3 K I I C 2 π a − c (25)

The process of hydraulic fracturing of hydrous cracked rock mass under dynamic loads requires three stages, namely, the splitting potential period of cracked rock mass, the crack propagation period of rock mass, and the crack coalescence and expansion period of rock mass. During the splitting potential period of cracked rock mass, the water pressure P _s at the edge of the crack ≥ the water pressure P _c at the tip of the crack, and the cracked rock mass undergoes propagation and splitting. With the increase of time, the water pressure P _t at the crack tip gradually develops into karst water pressure, and the dynamic load disturbance causes damage to the rock mass, making the critical water pressure P _c of the cracked rock mass continuously decrease. When P _t=P _s>P _c, the accumulation of fracture energy ends and enters the rock mass crack propagation period (Guo and Qiao, 2012; Guo J. Q. et al., 2018).

The excavation disturbance of the working face has changed the effective stress between rock cracks, and seriously destroys the static stability of the water-resisting layer of the floor. It not only increases the depth of floor plate failure and the height of the pressurized water bearing uplift zone, but also greatly promotes the expansion of water bearing cracks and the activation of floor plate faults, leading to the activation of hidden fault tip cracks and the occurrence of propagation and penetration, resulting in a significant improvement in permeability, until the formation of water-inrush channel induced water-inrush disaster.

The excavation disturbance of the working face changes the effective stress between the cracks in the rock mass, leading to the activation of the cracks at the tip of the hidden fault and the occurrence of propagation and coalescence until the formation of water inrush channels, inducing water inrush disasters. Therefore, the rock mass between the floor with hidden faults and the water-bearing hidden faults is divided into two parts, namely, the mining damage and destruction zone L _c, and the distance between the crack tip and the mining damage zone of the floor is the complete rock layer zone L _s (Chen et al., 2007; Yang and Zhang, 2016), as shown in Figure 8. The mining of the working face has formed a mining damage and destruction zone L _c, whose size can be determined through monitoring measurements (Zhang et al., 2013; Zhang et al., 2022). Whether the hydraulic fracturing zone can reach the thickness of the complete rock layer zone during mining requires theoretical analysis to determine. It is beneficial to take reinforcement measures in advance to ensure safe mining of the working face.

(2) Complete rock layer zone L _s

Under the action of high water pressure, the compressive bearing capacity of rock mass under certain geometric dimensions is limited. When the ultimate compressive bearing capacity of rock mass is exceeded, the rock mass will rupture and damage along its weakest point. According to Eq. 25, the splitting failure of cracked rock mass is affected by σ ₁,σ ₃ and dynamic stress intensity factors K I I 2 , while σ₁ and σ₃ are mainly affected by overburden pressure and lateral pressure coefficient. Since the coal seam depth is basically unchanged during the advancing process of the working face, the overburden pressure is basically unchanged. Therefore, the change of critical water pressure mainly depends on the change of lateral pressure coefficient and K I I 2 . The decrease in the lateral pressure coefficient will lead to the decrease in the critical water pressure for hydraulic fracturing, while the excavation disturbance of the working face will cause the unloading of the mining damage zone rock mass of the floor. The lateral pressure coefficient in the disturbed area can be expressed as λ x = λ 1 − exp L s 1.1 R − 1.7 (26) where L _s is the distance from the edge of the crack zone of the hidden fault to the innermost edge of the excavation disturbance damage zone of the working face (i.e., the thickness of the complete rock layer zone), and R is the equivalent radius.

During the advancing process of the working face, the distance from the edge of the initial crack zone of the hidden fault to the innermost edge of the mining damage zone of the working face is L _s (complete rock layer zone). If the critical water pressure for hydraulic fracturing here is less than the pressurized water pressure, fracturing failure occurs. At this point, L _s is the thickness of the hydraulic fracturing zone. According to the relationship between the lateral pressure coefficient and the critical water pressure, when the lateral pressure coefficient is 1, L _s can be expressed as L s 1 = 11 R 17 ⋅ ln ⁡ λ − ⁡ ln λ − 1 (27)

Take the value of λ _x as 1, and the critical water pressure for the propagation failure of hydrous cracks in the rock mass within the section obtained from Eq. 25 is: P l j = σ 1 − 1 f τ 1 K I I 2 − 3 K I I C π a − c (28)

Take σ ₁= λσ ₃= λγH, γ is the weight of the overburden stratum. Based on the analysis of P _s>P _lj and P _s<P _lj, the calculation equation for the thickness L _s of the complete rock layer zone under the combined action of dynamic and static loads are derived, as shown in Eq. 29 (P _s>P _lj) and Eq. 30 (P _s<P _lj) L s = 11 R 17 ⋅ ln ⁡ λ − ⁡ ln λ − 2 f P s π a − 3 K I I C − 2 c π a + 2 τ 1 K I I 2 π a γ H π a f − f ⁡ cos ⁡ 2 α − ⁡ sin ⁡ 2 α − f + f ⁡ cos ⁡ 2 α + ⁡ sin ⁡ 2 α f − f ⁡ cos ⁡ 2 α − ⁡ sin ⁡ 2 α (29) L s = 11 R 17 ⋅ ln ⁡ λ − ⁡ ln λ − 2 f P s π a − 3 K I I C − 2 c π a + 2 τ 1 K I I 2 π a γ H π a f − f ⁡ cos ⁡ 2 α + ⁡ sin ⁡ 2 α − f + f ⁡ cos ⁡ 2 α − ⁡ sin ⁡ 2 α f − f ⁡ cos ⁡ 2 α + ⁡ sin ⁡ 2 α (30)

Note: In the calculation process of L _s, in order to unify the units, this item 2 f P s π a − 3 K I I C − 2 c π a + 2 τ 1 K I I 2 π a needs to be multiplied by 1000.

Based on the above analysis, when there is high pressure water in the floor of water-bearing hidden faults, coal mining should be carefully carried out, construction scheme should be adjusted, or effective measures should be taken to ensure construction safety and prevent water inrush disasters. At this time, the minimum safety thickness in the floor of the working face is L = L c + L s (31)

The equation for calculating the minimum safe thickness of fractured rock mass was obtained by substituting the mining damage zone of the floor L _c (which can be obtained through actual measurement) and the hydraulic fracturing zone L _s into Eq. 31. L _s is calculated by choosing Eq. 29 when P _s>P _lj and Eq. 30 when P _s<P _lj.

From the above analysis, it can be concluded that the effect of dynamic loads has a great promoting effect on the pressure reduction and propagation of hydrous cracks. Dynamic load affects the fracture propagation mode by increasing pore water pressure, and the effect of dynamic loads will lead to the increase in pore pressure within the fracture, reducing the effective stress on the fracture surface; In addition, within the certain range of fracture dip angles, the effect of dynamic compressive stress will directly change the stress intensity factor at the crack tip, making the stress intensity factor at the crack tip increase. When the stress intensity factor reaches a critical value, it will lead to fracture propagation and coalescence failure of the rock mass.

Combined with theory and numerical simulation analysis, it can be concluded that when there are hidden faults in the floor, not only the failure depth of the floor and the height of the pressurized water bearing uplift zone increase rapidly under the combined action of dynamic and static loads, it also promotes the expansion of water-bearing cracks and the activation of floor hidden faults, and aggravates the formation of floor water-conducting channels, the risk of failure and instability of water-bearing hidden faults is greater than that of static loading, and it is easier to cause water inrush from floor.

From Equations (29), (30), and (31), it can be found that the minimum safe thickness of the outburst prevention layer of the floor with hidden faults in the working face is related to physical quantities such as the Ordovician limestone water pressure, the distribution characteristic parameters of hidden faults, and the mechanical parameters of the rock mass of the outburst prevention layer. According to Equation (31), the variation trend of the minimum safe thickness of the outburst prevention rock layer with various influencing factors can be analyzed, as shown in Figure 10. When the influence of a parameter was not discussed, the parameter was taken as a fixed value in the calculation of the minimum safe thickness, assuming that the pore water pressure P _s=2 MPa, and the dip angle of the hidden fault fissure β=30°, measured lateral pressure coefficient n=0.6, γ=26.50 kN/m³, type II compression-shear propagation stress intensity factor of the floor rock layer K _IIC=4.68 MN/m^3/2, H=45 m, the hydrous crack’s length 2a is equal to the initial fracture zone thickness of 3.4m, the friction coefficient f is 0.92, the cohesion force of crack surface c=1.4 MPa, R=5.65 m, and the mining damage zone thickness of the floor L _c=4.5 m (Discuss respectively that p _s , a, and H are the mining depth of the working face and the crack inclination angle is α).

By analyzing the variation trend of each curve in Figure 10, it can be found that with the increases of p _s, the minimum safe thickness of the outburst prevention layer approximately linearly increases, and with the pressure of water increases, the stress acting on the crack tip increases, making it easier for the crack to expand, resulting in an increase in the thickness of the outburst prevention layer. With the increases of the inclination angle of the crack, the thickness of the outburst prevention layer first increases and then decreases. When the dip angle α varies in the interval of 10°–60°, L increases from 4.74 m to 7.3 m and then decreases to 3.72 m. With the increases of crack length a, L continues to increase at a certain rate, and L continues to increase with the increase of crack length a. The crack length increases, so that the crack extension length was reduced, making it easier to penetrate the floor and form the water inrush channel. With the increases of mining depth H, the thickness of the outburst prevention layer continues to increase, the geostress increases, and the abutment pressure increase during coal mining, which makes the working face prone to water inrush disasters. Therefore, a larger anti-collision layer was needed to ensure the safety of the working face.