1 Introduction

Sustainable management of groundwater resources has been paid attention to in many countries (Gu et al. 2016b; Blake et al. 2021; Stephens et al. 2021; Arnaud et al. 2022; Colyer et al. 2022). In China, approximately 8 billion tons of groundwater is being ruined by coal mining every year, but the utilization rate is only approximately 25% (Xie and Wang 2014). Rational utilization and protection of water resources in the mining process have become major long-standing concerns for the Chinese coal development industry. Faced with such a situation, Qian (2010) and Qian et al. (2007) proposed the concept of green coal mining and developed several areas of research on water-preserving mining technology in the western mining area. Accordingly, many scholars in China have conducted corresponding research. Gu (2015) and Gu et al. (2016a; b) took the lead in proposing the academic idea of Water Storage in Underground Reservoirs (WSIUR) and constructed a theoretical framework and technical system for the protection and utilization of water resources in underground reservoirs in coal mines in the western mining area. Underground reservoir technology uses rock mass void storage holes in goafs formed by coal mining and connects safe coal pillars with artificial dam bodies to form reservoir dam bodies. Moreover, mine water storage facilities and water intake facilities are being constructed, and coal mine underground reservoir projects are being developed based on the complete utilization of the natural purification effect of the rock masses in goafs on the mine water. At present, the Shendong mining area has successfully built 35 coal mine underground reservoirs, providing more than 95% of the industrial water in the mining area and effectively alleviating the conflict between coal mining and water resource protection in the western mining area (Gu et al. 2016a). The technical principle of underground coal mine reservoirs is shown in Fig. 1 (Gu et al. 2016a).

Fig. 1
figure 1

Schematic of underground reservoirs (Gu et al. 2016a; Yang 2019)

Full size image

Due to the development of underground reservoir technology, an increasing number of coal mines have adopted this method to solve groundwater-related problems. Another concern reviewed by many scholars relates to the stability of coal pillar dams used in the construction of underground reservoirs. The stability of coal pillar dams under the long-term water storage of underground reservoirs has been studied by Gu (2015) and Gu et al. (2016a; b). The stratification of sedimentary rock mass introduces spatial variability in hydraulic conductivity, primarily between individual horizontal rocks (Maier et al. 2022). Jing and Xing (2021) investigated crack propagation in jointed rocks and discussed its effect on rock macro fracture resistance. Wang et al. (2022) investigated the fluid flow processes considering the fracture roughness and environmental stress on rough granite fractures in an experimental triaxial cell. The relationship of the permeability of water with the fractures and stress conditions was verified. The shear strength of the joint sample declines with an increment in joint saturation (Hu et al. 2022). The anisotropic characteristics of coal and rock due to the existence of discontinuities have been investigated by Yang et al. (2015), and the property cannot be ignored in the study of stress-seepage coupling. To reveal the weakening effect of water on the mechanical properties of coal pillar dams in underground reservoirs, Tang et al. (2022) investigated the relationship of rock mechanical parameters such as uniaxial compressive strength and elastic modulus with water content and analyzed the rupture process of coal samples with different water contents based on cumulative acoustic emission counting. The experimental results show that the peak stress, modulus of elasticity, and strain softening modulus are linearly related to the moisture content. The post-peak stress and post-peak gradient decrease with increasing moisture content. Guo et al. (2021) conducted an experimental study on the permeability of the sandstone of the coal seam roof under different loading paths to address the seepage behavior of the coal seam surrounding the rock and the problem of sudden inflow of water. The influence of the surrounding pressure on the permeability was discussed, and it was revealed that the permeability decreases with increasing surrounding pressure, and increasing the number of this loading and unloading will weaken the effect. Wang and Li (2020) used laboratory simulation experiments to carry out stability-related experiments studying the connection between two dam bodies. They pointed out that water has a non-negligible softening effect on dam bodies, and the pore cracks in the joint should be reduced to ensure the safe and stable operation of dam bodies and provide an important basis for the optimal design of underground reservoir dam bodies. Li et al. (2019) focused on percolating pores and fractures and used water drive tests, magnetic resonance imaging, and grey-scale calculation methods to analyze the effects of each factor on the water intrusion degree (WID) of coal-rock reservoirs. According to the test, the WID first increased and then tended to be constant when increasing the fracture porosity. The waterlogging effect is different depending on the connectivity and permeability of the pores. Yao et al. (2019) studied the width of coal pillar dams considering the overburden pressure and water pressure in combination with laboratory mechanical tests. The researcher explored the influence of water action on the stability of coal-pillar dams and provided a useful reference for the design of coal-pillar dams in underground reservoirs. Wang et al. (2019) constructed a three-dimensional numerical model of the dynamic responses of coal pillar dams in multi-face mining, revealed the damage and failure law of coal pillar dams, and provided a scientific basis for the stability control of coal pillar dams. Zhong et al. (2020) used methods of theoretical analysis and numerical simulation to analyze the stress distribution and displacement characteristics of coal pillars with different widths and studied the stability of coal pillars in gob-cut roadways and their reasonable width, which improved roadway maintenance conditions. Gu et al. (2018) investigated the changes in static and dynamic mechanical properties of coal rocks under different soaking times by considering the influence of soaking time on the stability of coal rocks. They pointed out that the effect of water content on the mechanical properties of coal rock at the early stage of soaking precedes the effect of water-coal interaction, and the effect of water-coal interaction gradually increases with the increase of soaking time. Wei (2008) proposed a new method considering the plastic softening characteristics of coal materials and verified the rationality of the method through open-pit mining engineering examples, which can be well applied in practical engineering.

With the development of computer technology, numerical simulation methods have been widely used in the field of rock seepage mechanics, and many numerical calculation methods have also been produced. This kind of method can comprehensively account for many factors, and its calculation results are intuitive. For the seepage-damage coupling mechanism of water inrush, it is generally necessary to introduce media fracture and damage judgment criteria into commercial programs such as FLAC and UDEC or numerical models based on theories of elastic–plastic mechanics, fracture mechanics, and damage mechanics. The permeability-damage evolution equation describing the media failure expansion zone has been embedded to study the seepage-damage coupling behavior in the process of hydraulic fracturing or water inrush (Wang and Park 2002; Wu et al. 2004; Zhang and Shen 2004; Mu et al. 2020). Li and Holt (2001) used a mesomechanical model to couple seepage pressure to study the hydraulic fracturing process. The main advantage of the models is their ability to describe the mechanical properties of microstructure microcracks, which can relate the macro mechanical behavior of rocks to the microstructural mechanisms of crack development. This is one of the important numerical methods for effectively analyzing the seepage and water inrush problems of fractured rock masses under the condition of stress-seepage coupling. Xu et al. (2006) studied the coal and gas outbursts using the RFPA code, in which the elastic modulus of the elements will gradually decrease considering the damaged elements. Mining-induced strata movement in an underground coal mine was investigated using the RFPA code by Xu et al. (2015), which shows a good understanding of the deformation and failure characteristics of rock strata. Han et al. (2022) proposed an engineering-scale fluid–solid coupling simulation method by using FLAC3D code. They revealed the damage mechanism of coal pillars under the action of water weakening and analyzed the influence of mining disturbance, water level height in the mining area, and coal pillar width on the stability of coal pillars. Shan and Lai (2018) carried out numerical simulations on the flow-solid coupling of coal rock body damage considering initial ground stress using COMSOL Multiphysics. They found that the largest flow velocities occurred at the main fractures, with larger hydraulic gradients and flow velocities at smaller fractures. External loading increases macroscopic fracture development, average porosity, and permeability in the coal rock body. Chen et al. (2017) carried out numerical simulation experiments of fluid–solid coupling using a discrete element code with two working conditions of coal seams near water-bearing seams and near faults. The characteristics of fracture development, water pressure, and seepage rate changes in coal seams under the effect of fluid–solid coupling were studied. Cheng et al. (2013) applied the coupled modeling method of stress field and seepage field in FLAC code to simulate the dynamic rupture process of coal rocks under different fault dip angles. The influence of fault dip angle on rock damage under the action of flow-solid coupling was verified. The results hold a specific guiding significance for deep coal rock disaster prevention and control. Maihemuti et al. (2016) established an equivalent continuous medium mathematical model for the coupling of seepage and stress fields. Rock deformation and damage under the coupling of seepage and stress in the reservoir slope of an open-cast coal mine hydropower station were tested, which provided a reasonable reference for the water–rock coupling problem in rock slopes. Li et al. (2013) realized the seepage-stress coupling relationship with the help of the secondary development of the FLAC3D code. Combining the specific engineering geological conditions of the edge water-filled and water-conducting collapse column, they carried out a numerical simulation of the whole process of working face advancement and an analysis of the related mechanical mechanism. Chu and Xu (2007) deduced the finite element formulation of the coupled equations based on the assumption that the solid skeleton and water are compressible, which has a certain significance for understanding the transient analysis of the seepage field in fractal media. Yang et al. (2011, 2007) studied the secondary effect of water pressure on the rock medium after considering the damage to the rock mass based on RFPA2D. They analyzed the problem of water inrush in underground coal mines, which is a good reflection of the seepage damage-stress coupling problem. Zheng and Zhu (2001) established a coupled analysis model for the seepage field and damage field of fractured rock masses and successfully applied the simulation of slope seepage damage and mine water inrush disaster mechanisms. Bruno et al. (2001) used the coupled flow of particles and fluids from fractures and mud injection in weakly consolidated granular media in simulations and established discrete particle models and pointed out that interparticle mechanics can be coupled with fluid flow dynamics and can be used to typically simulate particle deformation and fracture more realistically than continuum models. Fu et al. (2022) established a new coupling model involving fluid pressure and mechanical damage in FLAC3D and applied the model to explore seepage evolution, rock deformation, and lining stability in tunnel excavation. Gao et al. (2022) carried out a hollow cylinder hydraulic fracturing test simulation based on F-RFPA and verified the accuracy and reliability of the stress seepage coupling simulation of fractured rock masses. Wang and Xie (2022) studied the seepage stress coupling behavior of rock samples with multiple non-parallel fractures based on the RFPA2D-flow code and found that the fracture strength decreases with the increase of fracture density. Yang et al. (2022) established a hydro-mechanical coupling model of a rough single crack and conducted the shear seepage test. The results provide a reference for the water force coupling characteristics of rough fractures with different dip angles under three-dimensional compression shear stress. Song et al. (2022) studied the mechanical properties and the application of differential pressure cyclic loading in coal samples. The results provide practical guidance for the design, operation, and maintenance of underground reservoirs in coal mines. The researches show that research on the coupled seepage-stress mechanical properties of complex fractured rock masses needs to be combined with reliable numerical simulation methods. In the case of finely characterizing fracture field distributions, a stress-seepage coupled mechanical model can be established that comprehensively accounts for the complex distribution of fractures and the influence of geometric shapes. Ultimately, it is possible to obtain more scientific fracture development characteristics and mechanical response laws of coal pillars.

The author uses the COMSOL Multiphysics code to study different types of seepage and seepage field-stress field coupling problems in coal pillar dams. A coupled seepage-stress mechanical model considering anisotropy was established and discussed. The evolution law of the stress field, seepage field, and damage field under anisotropic conditions was investigated. This has great theoretical significance and practical value for revealing the seepage mechanism of coal pillar dam bodies, the stability analysis of dam bodies, and the proposal of the support design scheme.

2 Anisotropic constitutive relation

The jointed rock mass consists of many discontinuities, which make it discontinuous and anisotropic. Considering coal as a special kind of rock, coal rock can be regarded as a continuous medium, which should take into account the anisotropy of rock strata. To establish an anisotropic mechanical relationship, the following assumptions need to be introduced (Wang et al. 2013, 2018; Yang et al. 2015): (1) the anisotropy of coal and rock mass is mainly caused by the Class IV and Class V structural planes, which are usually randomly distributed in parallel sets. The IV or V-class structure is a general classification for structures that have worse connectivity than that of I, II, or III-class structures; (2) the coal-rock mass containing the structural plane is regarded as a uniform anisotropic elastic–plastic body; (3) the permeability and deformation parameter tensors of coal and rock mass on the REV scale have characterization significance.

2.1 Stress

According to linear elasticity theory, the constitutive relation of the rock mass can be determined by Hooke's law, as expressed by Eq. (1),

$$\varepsilon_{ij} = \left[ S \right]\sigma_{ij}$$
(1)

where [S] is the compliance matrix, εij is the strain, and σij is the stress.

The problem of deep underground coal mining can be regarded as a plane strain problem for the section parallel to the coal pillar dam because the direction of the coal pillar and the mined-out area for water storage are generally long. Referring to the reports published by Wang et al. (2013), Yang et al. (2015), and Wang et al. (2018), it can be seen that for the plane strain problem, the elastic tensor of the anisotropic model in Eq. (1) is defined as follows:

$$\left[ {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {\varepsilon_{22} } \\ {\varepsilon_{12} } \\ \end{array} } \right] = \left[ S \right]\left[ \sigma \right] = \left[ {\begin{array}{*{20}c} {\frac{1}{{E_{1} }} - \frac{{\nu_{31}^{2} }}{{E_{3} }}} & { - \left( {\frac{{\nu_{12} }}{{E_{1} }} + \frac{{\nu_{31}^{2} }}{{E_{3} }}} \right)} & 0 \\ { - \left( {\frac{{\nu_{12} }}{{E_{1} }} + \frac{{\nu_{31}^{2} }}{{E_{3} }}} \right)} & {\frac{1}{{E_{1} }} - \frac{{\nu_{31}^{2} }}{{E_{3} }}} & 0 \\ 0 & 0 & {\frac{{2\left( {1 + v_{12} } \right)}}{{E_{1} }}} \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{12} } \\ \end{array} } \right]$$
(2)

where [S] is the flexibility matrix under the plane strain problem and its inverse matrix, and [E] is the stiffness matrix of the material. Therefore, the constitutive relation can be expressed in the form of stress–strain as (Wang et al. 2018),

$$\left[ {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{12} } \\ \end{array} } \right] = \left[ E \right]\left[ \varepsilon \right] = \left[ {\begin{array}{*{20}c} {\frac{{E_{1}^{2} \left( {E_{3} - E_{1} v_{31}^{2} } \right)}}{\Delta }} & {\frac{{ - E_{1}^{2} \left( {E_{3} v_{21} + E_{1} v_{31}^{2} } \right)}}{\Delta }} & 0 \\ {\frac{{ - E_{1}^{2} \left( {E_{3} v_{21} + E_{1} v_{31}^{2} } \right)}}{\Delta }} & {\frac{{E_{1}^{2} \left( {E_{1} v_{31}^{2} - E_{3} } \right)}}{\Delta }} & 0 \\ 0 & 0 & {\frac{{E_{1} }}{{2\left( {1 + v_{12} } \right)}}} \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {\varepsilon_{22} } \\ {\varepsilon_{12} } \\ \end{array} } \right]$$
(3)

where Δ is represented as in Eq. (4),

$$\Delta = - E_{1} E_{3} + E_{1} E_{3} v_{21}^{2} + E_{1}^{2} v_{31}^{2} + E_{1}^{2} v_{31}^{2} + 2E_{1}^{2} v_{21} v_{31}^{2}$$
(4)

The elastic parameters E1 and E3 and Poisson's ratio v31 and v12 are essentially the values representing different research directions of the rock mass. However, in traditional rock mechanics tests (including laboratory tests and field tests), it is difficult to directly reflect the elastic properties of rock masses. The Hoek–Brown criterion (Hoek 2002; Zhang and Zhu 2007; Saroglou and Tsiambaos 2008; Li et al. 2011) can be used to calculate the basic mechanical properties of rock masses by introducing the Geological Strength Index (GSI). This criterion cannot be used to obtain anisotropic characteristics, so the analysis method of the anisotropy of jointed rock masses needs to account for damage theory. The author regards the primary joint damage in the coal pillar as a macroscopic damage field. In elastic damage mechanics, the elastic modulus of the connecting material may decrease. Therefore, the elastic modulus of the damaged coal pillar is defined as follows (Lemaitre and Desmorat 2005; Xu et al. 2015):

$$E= E_{0} \left( {1 - D} \right)$$
(5)

where D is the damage variable of the rock mass, and E and E0 are the elastic moduli of the jointed rock mass and rock matrix, respectively. Based on geometric damage theory (Kawamoto et al. 2010), the damage tensor of the rock mass can be obtained as shown in Eq. (6),

$$D_{ij} =\frac{l}{V}\mathop \sum \limits_{k = 1}^{N} \alpha^{\left( k \right)} \left( {n^{\left( k \right)} \otimes n^{\left( k \right)} } \right)\left( {i, \, j = 1,2,3} \right)$$
(6)

where N is the number of joints in the joint sample, l is the minimum joint spacing in the joints, V is the rock mass volume, n(k) is the normal vector of the kth joint, and a(k) is the trace length of the kth joint (in the two-dimensional case).

From the perspective of rock mass damage characterization, rock mass presents anisotropy. Its essence is that the material exhibits anisotropy during damage; that is, there is a damage tensor. Sidoroff (1981) first introduced the application of damage theory in rock mechanics. According to the energy equivalence principle, the flexibility matrix of a jointed rock sample can be expressed as follows:

$$S_{ij}^{\prime} = (1 - D_{i} )^{ - 1} S_{{i}_{j}} (1 - D_{j} )^{ - 1}$$
(7)

where Sij is the flexibility matrix of the rock matrix, and Di and Dj are the principal damage values in the i and j directions, respectively. For the plane strain problem, the constitutive relation (Wang et al. 2013, 2018; Yang et al. 2015) with the same coordinate and main damage direction can be expressed as Eq. (8),

$$\left[ {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{E_{0} \left( {v_{0} - 1} \right)(D_{1} - 1)^{2} }}{{2v_{0}^{2} + v_{0} - 1}}} & {\frac{{ - E_{0} v_{0} \left( {D_{1} - 1} \right)\left( {D_{2} - 1} \right)}}{{2v_{0}^{2} + v_{0} - 1}}} & 0 \\ {\frac{{ - E_{0} v_{0} \left( {D_{1} - 1} \right)\left( {D_{2} - 1} \right)}}{{2v_{0}^{2} + v_{0} - 1}}} & {\frac{{E_{0} \left( {v_{0} - 1} \right)(D_{2} - 1)^{2} }}{{2v_{0}^{2} + v_{0} - 1}}} & 0 \\ 0 & 0 & {\frac{{E_{0} \left( {D_{1} - 1} \right)\left( {D_{2} - 1} \right)}}{{2\left( {1 + v_{0} } \right)}}} \\ \end{array} } \right]\,\left[ {\begin{array}{*{20}c} {\varepsilon_{11} } \\ {\varepsilon_{22} } \\ {\varepsilon_{12} } \\ \end{array} } \right]$$
(8)

where E0 is the elastic modulus of the rock matrix, v0 is the Poisson's ratio of the rock, which can be obtained directly from laboratory tests, and the principal damage values Di and Dj in Eq. (7) can be calculated based on geometric damage theory. Therefore, all elements in this matrix can be obtained by the aforementioned method. Considering the coal pillar as a special rock mass, the anisotropic constitutive relation of the coal pillar dam body can be determined.

2.2 Seepage

The permeability coefficient of the coal pillar is the quantitative form of its permeability and is also the basis for solving the equivalent continuous medium seepage field. Based on the properties of the fractured rock mass and the actual engineering design, a coal pillar with jointed fractures can be regarded as an anisotropic continuum. According to the theory of Biot (Zeitoun and Wakshal 2013), the relationship between the fissure water pressure p and the permeability coefficient tensor Kij can be expressed as

$$K_{ij} \nabla^{2} p = \frac{1}{Q}\frac{\partial p}{{\partial t}} - \alpha \frac{{\partial \varepsilon_{v} }}{\partial t} = 0$$
(9)

where \(\varepsilon_{v}\) is the volume strain and Q is the Biot coefficient. The expression of the layered coal seam mainly includes the permeability coefficient along the bedding and vertical bedding and the permeability coefficient in the principal direction of the performance. For two-dimensional problems, the four components Kij of the permeability coefficient tensor determine the hydraulic conductivity tensor, which can be written in matrix form (Wang et al. 2013),

$$K_{ij} = \left[ {\begin{array}{*{20}c} {K_{11} } & {K_{12} } \\ {K_{21} } & {K_{22} } \\ \end{array} } \right]$$
(10)
$$\left\{ {\begin{array}{*{20}c} {K_{11} = \frac{{K_{11}^{^{\prime}} + K_{22}^{^{\prime}} }}{2} + \frac{{K_{11}^{^{\prime}} - K_{22}^{^{\prime}} }}{2}\cos 2\theta } \\ {K_{12} = K_{21} = - \frac{{K_{11}^{^{\prime}} - K_{22}^{^{\prime}} }}{2}\sin 2\theta } \\ {K_{22} = \frac{{K_{11}^{^{\prime}} + K_{22}^{^{\prime}} }}{2} - \frac{{K_{11}^{^{\prime}} - K_{22}^{^{\prime}} }}{2}\cos 2\theta } \\ \end{array} } \right.$$
(11)

where \({K_{{i_j}}}\) and \(K_{ij}^{^{\prime}}\) are the hydraulic conductivities in the principal material direction, and coordinate direction, respectively, and θ is the angle between the coordinate direction and the principal material direction.

2.3 Coordinate system transformation

For plane problems, there is a certain angle between the selected coordinate system and the principal material direction. Therefore, when establishing the mechanical calculation relationship, this angle needs to be considered. Suppose θ is the angle between the selected coordinate system and the principal material direction, as shown in Fig. 2 (Bruno et al. 2001); 1 and 2 are the material coordinate directions. For the anisotropic constitutive relation of coal rock, θ is the angle between the coordinate system and the principal direction of the bedded coal specimen.

Fig. 2
figure 2

The relationship between the reference frame and material master direction

Full size image

According to the physical meaning of the void fluid pressure, the void fluid pressure component does not require coordinate transformation. According to the coordinate transformation of stress in elastic theory, the constitutive in the non-material principal direction is listed by Eq. (12),

$$\left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\tau_{xy} } \\ \end{array} } \right\} = \left[ {T_{\sigma } } \right]^{ - 1} \left[ Q \right]\left[ {T_{\varepsilon } } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\gamma_{xy} } \\ \end{array} } \right\} - \alpha_{ij} \left\{ {\begin{array}{*{20}c} P \\ P \\ 0 \\ \end{array} } \right\}\delta_{ij}$$
(12)

In Eq. (12), [Tσ]−1 and [Tε] can be expressed as Eqs. (13) and (14),

$$\left[ {T_{\sigma } } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {\cos^{2} \theta } & {\sin^{2} \theta } & { - 2\sin \theta \cos \theta } \\ {\sin^{2} \theta } & {\cos^{2} \theta } & {2\sin \theta \cos \theta } \\ {\sin \theta \cos \theta } & { - \sin \theta \cos \theta } & {\cos^{2} \theta - \sin^{2} \theta } \\ \end{array} } \right]$$
(13)
$$\left[ {T_{\varepsilon } } \right] = \left[ {\begin{array}{*{20}c} {\cos^{2} \theta } & {\sin^{2} \theta } & {\sin \theta \cos \theta } \\ {\sin^{2} \theta } & {\cos^{2} \theta } & { - \sin \theta \cos \theta } \\ { - 2\sin \theta \cos \theta } & {2\sin \theta \cos \theta } & {\cos^{2} \theta - \sin^{2} \theta } \\ \end{array} } \right]$$
(14)

3 Anisotropic characteristics of layered coal pillar

The proposed anisotropic constitutive relation could consider the influence of structure planes on mechanical behavior. Compared with the conventional isotropic constitutive relations, the proposed model could thus better study the anisotropic properties of the bedded coal pillars. The application example is the coal pillar dam of the Daliuta 5–2 coal seam in the underground reservoir in Shenmu city. This coal seam is buried at a depth of 34.40–40.10 m, and the floor elevation is + 1000 ~  + 1030 m. The coal seam does not contain gangue and has a simple structure. The roof of the coal seam is mainly composed of mudstone and siltstone; the bottom plate is mainly composed of siltstone. The coal seam is generally very thick, and the entire bifurcation area can be mined. Its thickness changes slowly and with obvious regularity; the structure is simple, the coal is single, and it is a stable coal seam. The author focuses on the stress and seepage anisotropy characteristics of the coal pillar dam of the 5–2 coal seam in the underground reservoir.

3.1 Mechanical parameters

Gu (2015) pointed out that coal pillar dam bodies are reconstructed based on existing safe coal pillars or waterproof coal pillars. The traditional safety coal pillar accounts for the mine pressure only and performs the function of isolation and load bearing. However, as a permanent hydraulic structure, the coal pillar dam of underground reservoirs in coal mines accounts for not only the effect of mine pressure but also issues such as water pressure, anti-seepage, and reliability. Research has shown that the basic conditions for the stability of coal pillar dam bodies are as follows: After plastic deformation occurs on both sides of the coal pillar, there is an elastic core with a certain width in the center of the coal pillar. Its width should not be less than twice the height of the coal pillar and should also meet the anti-seepage requirements for water storage. The study determined that the width of the coal pillar dam body of the underground reservoir of the coal mine in the Shendong mining area is 20–30 m. The author selects the underground reservoir No. 4 of the Daliuta Coal Mine as the project prototype. Its overlying rock stratification and physical and mechanical parameters are shown in Table 1. The coal pillar dam is 20–30 m wide and 3–5 m high. Since the sampling process directly leads to damage in the layered coal, the author did not directly test the anisotropic mechanical index of the coal. By consulting a large number of reports (Wang 2013; Shi et al. 2015; Meng et al. 2016; Yang 2016; Li and Li 2017; Pang et al. 2019; Zhi et al. 2022; Li 2021), the mechanical indices of the 5–2 coal seam and its surrounding rock can be obtained (listed in Table 1). The elastic modulus of coal is 6.26 GPa, and Poisson's ratio is 0.22. According to the principle of the difference between the strength in the principal direction of the material and the strength in the principal vertical direction, the author obtained preliminary values for compressive strength, tensile strength, and shear strength (listed in Table 2) and then evaluated the damaged area. Based on the research by Zhang (2006), the anisotropic elastic mechanic indices of the coal pillar rock samples are listed in Table 2. The parameters are obtained according to Table 1 based on numerical or experimental reports.

Table 1 Physical and mechanical parameters of rocks in the simulation
Full size table
Table 2 Mechanical parameters of the bedding coal and rocks
Full size table

3.2 Numerical model

According to the actual size of the coal pillar dam body, a two-dimensional mechanical model with a length of 80 m and a height of 50 m is established, as shown in Fig. 3. The coal pillar model is 30 m wide and 5 m high, and the gob model is 7 m wide and 5 m high.

  1. (1)

    Stress boundary conditions The left and right end of the model limit the displacement in the x-direction, and the bottom end is fixed. The top has a load of 2.33 MPa to be equivalent to the upper 100 m of the rock formation. In addition, the model is subject to gravity.

  2. (2)

    Seepage boundary conditions The upper and lower boundaries of the model are waterproof boundaries, the left and right boundaries have no water pressure, and the wall of the goaf is attached with a water pressure of P0 = 1 MPa.

  3. (3)

    Calculation scheme The stress field, seepage field, and coal pillar damage are simulated under real conditions of the coal pillar dam.

Fig. 3
figure 3

Coal pillar model and boundary stress conditions

Full size image

3.3 Results and analysis

For the above model, the COMSOL Multiphysics code (COMSOL 2005) was used to calculate and solve the seepage stress coupling considering Darcy's law and solid mechanics, and the results are plotted. To further study the anisotropic characteristics of the coal pillar dam body, an isotropic model was established for calculation and comparison. In Fig. 4a–d show the computing cloud diagrams of the model, including the seepage pressure gradient map, the maximum principal stress distribution map, the anisotropic coal pillar damage map and the isotropic coal pillar damage map.

Fig. 4
figure 4

Simulation results of coal pillar dam a is the water pressure, b is the stress field, c is the anisotropic damage area, and d is the isotropic damage area

Full size image

Figure 4a shows that the seepage field has obvious symmetry. In the figure, the vertices at both ends of the coal pillar dam are in the gob; the water pressure is the largest, with a value of 0.94 MPa. The water flow velocity vector is mainly along the principal direction of seepage, and the water flow is primarily concentrated around the reservoir and on the side close to the coal pillar. From Fig. 4b, it can be seen that the symmetry of the maximum principal stress is obvious. It is mainly distributed at the sides, roof, and floor of the gob, and the maximum principal stress is the largest at the four ends of the gob, with a value of 12.7 MPa. In the figure, the stress value at the apex of the coal pillar dam is large and gradually decreases in a triangular shape. Figure 4c depicts the damage to the anisotropic coal pillar dam. The damaged area is mainly distributed in the goaf and shows obvious symmetry. Butterfly-shaped damage occurs in the roof and floor of the goaf, but the damage to the floor is smaller than that of the roof. The side gang damage is mainly triangular, and the roof in this area is at risk of falling. The damage to the coal pillar dam body is less, and the damage occurs when the two ends of the roof and floor of the coal pillar begin to deepen and penetrate the interior, and the failure mode is mainly cracking. In Figs. 1 and 2, it can be seen that failure is related to the coupling of the seepage field and stress field. Compared with Fig. 4d, the isotropic coal pillar damage is different, and the coal pillar failure position occurs on both sides of the coal pillar and extends to the interior in a fan shape. Its damage range is larger, accounting for 14.29%, and its damage mode is large area coverage. The roof and floor of the gob damage range are the same size and have the same risk of the roof falling. Compared with Fig. 4c, the damage of the goaf is smaller, and the damage shape is mainly in strips.

4 Analysis and discussion

It has been found that bedded rocks show different failures when the angles between the loading direction and dips of structure planes are different. The mechanical behavior of bedded coal rock at different dip angles is obviously different. For the bedded coal rocks, the principal angle coincides with the direction of the bedding planes in the coal seam. The influence of the anisotropic characteristics of the coal pillar dam on the seepage field, stress field, and evolution of the damage zone of the surrounding rock is studied. Different tensor principal direction angles are selected as 0°, 15°, 30°, 45°, 90°, and 135° values, respectively. The model is solved and calculated, and in turn, the numerical simulation results of the seepage field, stress field, and damage zone are analyzed. The coal pillar dam model is further studied, and different burial depths d are used for calculation. The change in the failure mode is observed, and the results of the displacement change and energy evolution are analyzed.

4.1 Anisotropic seepage

Figure 5 shows a graph of the seepage pressure gradient under the condition of different tensor principal direction angle values, in which the direction of the arrow represents the direction of the fluid velocity, and its magnitude represents the magnitude of the fluid velocity. The influence of the principal direction of penetration on the fluid flow field is very obvious. When the principal direction of seepage changes, the pressure gradient cloud map of the seepage field also changes. In the case of different angles, the water pressure distribution at both ends of the coal pillar dam has altering characteristics with the changing angle. When the angle is 0° or 90°, it can be seen that the water pressure is symmetrically distributed relative to the center of the two gobs, while other angles are asymmetrical. The principal directions of permeation at the angles of 45° and 135° are opposite to the pressure gradient distribution. The value of the principal direction angle has little effect on the seepage pressure value of the coal pillar dam body, and the maximum value coincides with the two ends of the roof and the floor of the coal pillar dam body at the side edge of the goaf, with a magnitude of 0.94 MPa; the minimum seepage pressure on the dam body is 0.78 MPa, which is located in the center of the dam body.

Fig. 5
figure 5

The pressure gradient value under different principal directions

Full size image

The 1–1’ (y = 22 m) intercept of the model is further studied. In Fig. 6, (a) and (b) show the fluid velocity curves in the X direction and the Y direction on the 1–1’ section. As seen from the figure, the magnitudes of the velocities at different θ values are quite different, but the changes are roughly the same. The penetration velocity at each angle is the minimum value in the center of the coal pillar, but the velocity at each angle is asymmetric, showing obvious anisotropy characteristics.

Fig. 6
figure 6

Fluid velocity on the 1–1 'horizontal section directly above the coal pillar roof. a The speed in X direction, b the velocity in Y direction

Full size image

4.2 Anisotropic stress

Figure 7 shows the shear stress cloud diagram in the material coordinate system when the principal direction angle of the material is different. When the principal direction is different, the shear stress in the material coordinate system also changes: When θ = 0° and θ = 90°, the shear stress distribution is the same, but when θ = 90°, the maximum shear stress value of the coal pillar is 4.69 MPa. The maximum and minimum values are located at the top and bottom of the coal pillar, which are diagonally distributed and correspond to the value of θ in the principal direction. The shear stress distribution of θ = 15° and θ = 30° rotates with the value of θ in the principal direction. There is a difference in the distribution of shear stress between θ = 45° and θ = 135°, which does not correspond to θ but develops vertically on the roof and floor of the coal pillar. The maximum shear stresses of the six different working conditions are 3.66, 2.74, 1.72, 1.87, 4.69, and 1.11 MPa. When θ = 90°, the shear stresses on the roof and floor of the coal pillar dam body are the largest; that is, the dam body is the most dangerous at this time.

Fig. 7
figure 7

The shear stress filed under different principal directions

Full size image

4.3 Anisotropic damage zone

To define the damage zone, the Hoffman anisotropic strength criterion was used to illustrate the effect of the joint plane angle on the damage zone. This criterion was proposed by Hoffman (1967) which is a more reasonable criterion for anisotropic materials (Wang et al. 2013; Yang et al. 2015). The strength criterion for judging the failure of anisotropic materials is achieved mainly through the combination of a series of stress-strength indices. The equation is as follows:

$$\frac{{\sigma_{1}^{2} }}{{X_{t} X_{c} }} - \frac{{\sigma_{1} \sigma_{2} }}{{X_{t} X_{c} }} + \frac{{\sigma_{2}^{2} }}{{Y_{t} Y_{c} }} + \frac{{X_{c} - X_{t} }}{{X_{t} X_{c} }}\sigma_{1} + \frac{{Y_{c} - Y_{t} }}{{Y_{t} Y_{c} }}\sigma_{2}^{{}} + \frac{{\tau_{12}^{2} }}{{S^{2} }} = 1$$
(9)

where σ1 is the stress along the joint direction, σ2 is the stress perpendicular to the joint direction, τ12 is the shear stress in the joint direction, Xt and Yt are the tensile strength indices parallel to the direction of the layered rock joints, and perpendicular to the direction of the layered rock joints, Xc and Yc are the compressive strength indices parallel to the direction of the layered rock joints and perpendicular to the direction of the layered rock joints, and S is the shear strength index in the rock bedding direction, which is calculated based on the Mohr–Coulomb theory:

$$S = c + \sigma_{2} \tan \phi$$
(10)

where c is the cohesion along the principal direction of the material and ϕ is the friction angle along the principal direction of the material.

The damaged areas at different angles of the joint surface are shown in Fig. 8. Overall, the shape and distribution of the damaged area undergo obvious changes with the direction angle. When θ = 0°, the top of the rock mass is prone to cracking due to the low tensile strength of the coal pillar. The damaged area is mainly concentrated on the roof and floor of the goaf and extends to both ends of the coal pillar. When the angle increases to 15°, the shape and distribution of the damaged area tend to change. It can be seen that the damage areas increase significantly, and the distribution position rotates with the angle. Among them, the coal pillar damage is the most serious when θ = 45° and θ = 135°, and the damage areas of the two angles are symmetrically distributed. Figure 9 shows a comparison between the numerical results and the research results declared by Yang (2019) and Zhang et al. (2022). The failure area of the numerical result when θ = 0° (Fig. 9a) has a good agreement with the simulation results given by Yang (2019) (Fig. 9b). It also shows a similar pattern with the in-situ survey observed in the coal pillars (Zhang et al. 2022), as shown in Fig. 9c.

Fig. 8
figure 8

The damaged zone under different principal elastic direction

Full size image
Fig. 9
figure 9

Comparison of damaged zones of the coal pillars, a is the proposed numerical result, b is the damage zone by Yang (2019), and c is the in-situ survey by Zhang et al. (2022).

Full size image

Figure 10 shows a comparison of the damage of the coal pillar dam with different burial depths in the principal direction θ = 0°. The burial depth of the coal pillar was increased by 10 m to explore when the coal pillar would fail. The burial depths of the simulated coal pillar dam were 110, 120, 130, 140, 150, and 160 m, and with the increasing burial depth of the coal pillar dam, the damaged area also increased. When the burial depth increased to 110 m, triangular failure occurred at the vertices of both ends of the coal pillar. By comparing the damage situation at 100 m, it was found that the damage is connected by both ends of the top and bottom plates. At 120 m, the damage at both ends of the coal pillar is triangular and deepens inside, and the damage range also increases. At 130 m, the failure of the coal pillar is rectangular, and the failure in the coal pillar is serrated. At 140 m, the internal damage of the coal pillar extends to the outer wall, and the damaged area reaches 40% at this time, and the damage is more serious. At 150 m, both ends of the coal pillar are destroyed. At this time, the damaged area of the coal pillar accounts for 60%, and it can no longer be used as a dam body. At 160 m, the coal pillar is penetrated and damaged. The damage to the goaf is less affected by the burial depth of the coal pillar. The increase in the burial depth does not change the damage position, and the damage only increases slightly.

Fig. 10
figure 10

Damage area under different buried depths

Full size image

4.4 Anisotropic displacement

Figure 11 shows the displacement change of the coal pillar dam under seepage stress coupling conditions when the principal direction angle is different. Combined with Fig. 3, it can be seen that the model has displacement only in the Y direction; with the change in the principal direction value, the displacement direction of the coal pillar and the displacement size also change. By comparing the displacements of θ = 0° and θ = 90°, it is not difficult to see that the changes are the same and have obvious self-symmetry. When θ = 15° and θ = 30°, the displacements at both ends of the coal pillar have obvious deviations. At this time, the displacement cloud map of the coal pillar is no longer symmetrical, showing the characteristics of anisotropy. However, the displacement changes of the coal pillar dam body with θ = 45° and θ = 135° are symmetrical with each other, which is caused by the complementary rotation angle of the coal pillar.

Fig. 11
figure 11

The displacement nephogram of different directions of θ

Full size image

Furthermore, the displacement on horizontal section line 1–1' (y = 22 m) just above the roof of the coal pillar was analyzed. According to Fig. 12, the displacement of the coal pillar dam under different principal directions is different, with obvious anisotropic characteristics. When θ = 90°, the displacement of the coal pillar dam is the largest, and its peak value reaches 0.0296 m, and the displacement is the smallest when θ = 45° and θ = 135°. The displacement of the coal pillar dam body on both sides of the goaf is the most obvious. This is because the seepage flow of the water here is relatively large, the coal rock softens under the immersion of water, and its plastic deformation is obvious.

Fig. 12
figure 12

Displacement on the 1–1' horizontal section above the coal pillar roof

Full size image

4.5 Anisotropic energy evolution

Figure 13 is the cloud diagram of the energy evolution process of the coal pillar dam body at different principal direction angles. As the principal direction θ increases, the position of the maximum strain energy of the coal pillar dam also changes. Among them, the strain energy changes at θ = 0°, 45°, 90°, and 135° show symmetry, and the energy evolution processes of the coal pillar dam in other principal directions show obvious anisotropy. The positions of the maximum strain energy and the minimum strain energy are also significantly different at different principal direction angles. When θ = 0° and θ = 90°, the minimum strain energy is at the top of the gob and the coal pillar but at the top and bottom of the gob when θ = 45° and θ = 135°. The change in the principal direction has a significant effect on the strain energy of the coal pillar dam.

Fig. 13
figure 13

Different distribution of strain energy density of varied principal directions (kJ m−3)

Full size image

As shown in Fig. 14, further research is conducted on the model 1–1' (y = 22 m) section, and its energy evolution process is analyzed. By comparing the strain energy in different principal directions and positions, we can see that the changes in strain energy at both ends of the coal pillar dam are the most intense, and the peak value of strain energy density when the principal direction θ = 30° is the highest and reaches 5.56 kJ m−3. The seepage velocity on both sides of the goaf is the largest, and the strain energy density is also the largest because the immersion effect of water on the coal reduces its strength. The average strain energy densities in different principal directions show anisotropic characteristics; the maximum occurs when θ is 90°, and the minimum occurs when θ is 135°. The anisotropic behavior of the seepage-stress coupling mechanical model was introduced to study the deformation and damage zones of coal pillars of underground reservoirs. The influence of different depths of the coal seam was discussed. The results could give a good reference for the supporting design, future excavation, and stability of coal pillars. In addition, the proposed numerical model can predict the anisotropic damage areas of other coal dams by considering different anisotropic properties.

Fig. 14
figure 14

Strain energy density on the 1–1' horizontal section directly above the coal pillar roof

Full size image

5 Conclusion

According to the geometric damage theory and the discrete medium seepage method, the stress and seepage characteristics of the fracture surrounding rock of the coal-pillar dam are studied, and the stress-seepage coupling model of the coal-pillar dam is established, considering anisotropy. Taking the coal pillar dam of the underground reservoir No. 4 in the Daliuta Coal Mine in China as an example, the distribution law of its seepage field and stress field is analyzed. On this basis, combined with the Hoffman anisotropic strength criterion, the evolution law of the coal pillar damage zone is discussed. The main conclusions are as follows:

  1. (1)

    Based on the COMSOL Multiphysics analysis method, the anisotropic constitutive relation and strength criterion of coal and rock was established. The principal direction of the mechanical properties of coal and rock has a significant effect on the distribution of the surrounding rock stress field, seepage field, and the size of the damage zone, showing obvious anisotropy characteristics. This effect is the most significant when θ = 45° or θ = 135°.

  2. (2)

    The stress-seepage anisotropy mechanical properties analysis of the coal pillar dam was carried out. The damage and failure mode of the coal pillar dam is affected by the principal direction angle, and the tensile stress and shear stress play a major role in the failure mechanism. The damage of the coal pillar dam is mainly inward along the edge and corner, and the damage of the coal pillar is obvious when θ = 45° or θ = 135°.

  3. (3)

    The failure modes of different coal bedding angles were analyzed, and the damage at θ = 0° is consistent with the actual situation. With increasing burial depth, the damaged area of the coal pillar gradually expands. When the burial depth reaches 150 m, the coal pillar will be destroyed. At this time, the displacement damage area accounts for approximately 60% of the coal pillar. The change in the coal pillar displacement with the principal direction shows obvious anisotropy characteristics, and this effect is more significant when θ = 15° and θ = 30°. The change in the principal direction has an obvious influence on the change in the strain energy of the coal pillar dam.