In order to solve the problem of the stress-strain evolution of rock mass during the extraction of coal, the numerical integration of the system of solid mechanics equations was performed in a dynamic formulation using the finite difference method. The latter requires some explanation. Generally, the stress-strain evolution of rock mass undergoes quasi-static conditions, i.e. the kinetic energy is negligible, all elements of rock mass are in a quasi-static equilibrium, and strain rates are meagre. However, roof cavings are dynamic phenomena characterized by high strain rates. Thus, both tendencies, i.e. the quasi-static equilibrium between the cavings and the intensification of the deformation process during them, need to be properly described. An intrinsic property of a dynamic formulation is a description of deformation processes with strain rates that cannot be disregarded [22]. For low strain rates, a special technique of slow loading is applied. The gob is increased in a manner so as to maintain quasi-static conditions of rock-mass elements, i.e. every other face advancement step is made once per several thousands of integration time steps to maintain the quasi-static condition of the free surface in the gob area [19,23].

The system of equations includes the laws of mass (Eq. (1)) and momentum (Eq. (2)) conservation written in the Lagrangian form: (1)ρV=ρ0V0 (2)ρv˙i=σij,j+ρFi where ρ₀ and ρ is the reference and current value of density, respectively; V₀ and V the reference and current values of volume, respectively; v_i the components of velocity vector; σ_ij the components of Cauchy stress tensor; and F_i the components of mass force.

The system also includes the geometrical relations for strain rate and vorticity tensors (Eqs. (3) and (4)): (3)2ε˙ijT=vi,j+vj,i (4)2ω˙ij=vij−vj,i where ε_ij is the strain tensor.

It is evident in Fig. 2 showing a typical borehole log that the structure of rock mass is presented by many bedding planes of different rocks. The rocks, especially the rock mass, generally have considerable anisotropy of the physical and mechanical characteristics. There are several ways for simulating the state of stress and strain of rock mass under the action of different forces; one is to explicitly consider structural inhomogeneity and input the features of bedding into the model, supposing that the rock material between the bedding planes is isotropic. Following this way, a quasi-isotropic model of the rock mass was constructed, and the complexity of mechanical behaviour description of rocks was reduced, because only two constants in the equation of state, which could be readily obtained from the corresponding experiments, are needed for the description of elastic behaviour of isotropic materials. Also, at the depths of <1 km, where the majority of operating mines are located, the values of vertical stress are relatively lower than those obtained, e.g. under impact loading; thus a linear law relating the components of stress and strain rate tensors–equations of hypoelastic media can be used.

The volumetric and deviatoric parts of the stress tensor can be separated, and hence the hypoelastic media relations are written in the following forms (Eqs. (5) and (6) for volumetric and deviatoric parts, respectively): (5)P˙=−K(V˙V−θ˙P) (6)S˙ij+Sikω˙kj−Skjω˙ik=2μ[ε˙ijT−13(V˙V−θ˙P)δij−ε˙ijP] where P is the hydrostatic pressure; K the bulk modulus; S_ij the components of deviatoric stress tensor; μ the shear modulus; εijP the components of inelastic strain tensor; and θ and θ^P the volumetric elastic and inelastic strains, respectively.

The Jaumann corotational derivative is applied in Eq. (6) in order to subtract the rotation of a cell (particle or element of the medium) as a whole. Inclusion of the corotational derivative is dictated by the fact that the local rotation of element (cell) due to plastic deformations might be very strong in the band of localized inelastic deformation. Thus, to obtain the correct estimations of stress, one needs to subtract the rotation. The total strain consists of two parts: the elastic part (εijE) and the inelastic part (εijP), so that εijT=εijE+εijP.

It should be mentioned that inclusion into consideration of inelastic deformation of rock-mass elements is of great importance. When the coal seam is mined out, the virgin stress state is violated. This leads to considerable inhomogeneity and nonstationarity of the stress state of rock-mass elements, especially those near newly formed cavities, gateways, mining face, pillars, etc. In these regions of stress amplification, the rocks lose their stability and transit to the inelastic state and sometimes fracture, achieving a residual state [4]. Thus, a mathematical model should include the constitutive equations for inelastic strains and fracture of rocks. In this work, inelastic deformation of rock-mass elements is described by the modified Drucker-Prager model with a non-associated flow rule [24]. The inelastic strain rate tensor components are defined according to the Nikolaevskii plastic potential (Eq. (8)) from the theory of plasticity (Eq. (7)) [25–29]: (7)ε˙ijP=λ˙∂g(σij)σij (8)g(σij)=J2−ΛP(2Y+αP)+const where λ˙ is the multiplier used in theory of plasticity; g(σ_ij) the equation of plastic potential; J₂ the second invariant of deviatoric stress tensor; and Λ the dilatancy factor.

Then, the following equation for the inelastic strain rate tensor components can be obtained: (9)ε˙ijP=(Sij+23Λ(Y−α3J1)δij)λ˙,θ˙P=ε˙iiP where α is the internal friction factor; and J₁ the first invariant of stress tensor.

Usage of the non-associated flow rule gives an independent rate of dilatancy from internal friction coefficient and describes more precisely the behaviour of rocks. A comprehensive review and a choice of dilatancy parameters of rocks can be found in [30]. In this work, the fact is ignored that the dilatancy factor of rocks strongly depends on confinement stress and accumulated inelastic strain. An interval was selected: (10)Λ=const=tan(φexp8)−tan(φexp4) where φ_exp is an internal friction angle in the Coulomb-Mohr criterion.

Eq. (10) is a satisfactory assumption for the purpose of this work and is consistent with the recommendations on the choice of dilatancy factor by Hoek and Brown [31]. The multiplier λ˙ is defined in the calculations when Eq. (11) is satisfied (Drucker-Prager conical yield surface). (11){f(σij,D)=−αP+J2−YY=Y0(1−D) where f(σ_ij,D) is the equation of yield surface; Y₀ and Y the reference and current values of “cohesion”, respectively; and D the damage measure.

When deriving the non-associated plastic flow rule, a procedure of stresses correction can be inserted into the available solver [25].

It is notable that both α and Y are the material constants associated with rock cohesion and internal friction angle of Coulomb-Mohr criterion within the Drucker-Prager model [24]. For simplicity, α is further referred to as the internal friction factor and Y the cohesion.

In order to describe the fracture process of rock-mass elements, the theory of continuum damage mechanics (CDM) is applied in this work. The basic ideas and equations of CDM were formulated by Kachanov and Rabotnov [32,33]. The damage process of material was treated as a progressive loss of continuity under applied stresses. Further, a correlation between the damage and fracture mechanics was revealed on the basis of thermodynamics, and it was shown elsewhere that these theories were equivalent [34,35]. The applicability of CDM was demonstrated in many works for different materials, media, and loading conditions, etc [29,35]. The following kinetic equation for the damage measure time derivative is used in this work: (12){dDdt=(σ−σ0)2(σ*)2t*σ=−αP+J2 where σ₀ and σ is the threshold and Drucker stress in the damage measure, respectively; and σ* and t* the model parameters. The model under development was earlier validated against the experiment by consideration of several simple model cases [36–38].

The parameters of the proposed model of damage accumulation are discussed below in detail. The damages start accumulating if σ > σ₀; otherwise, there is no damage accumulation and D = 0. It is evident that σ₀ is the threshold stress for damage initiation. The big question is: what is the threshold? By the trial and error method, we found the solution to this problem. All types of stress state are conditionally “located” between the uniaxial tension and uniaxial compression. Thus, for pressure-sensitive materials, the threshold is:

At negative pressure: (13)σ0=σt(13+α3)

And at positive pressure: (14)σ0=σc(13−α3) where σ_t is the uniaxial tensile strength (UTS); and σ_c the uniaxial compressive strength (UCS).

These two values of σ₀ are the Drucker stress values at uniaxial tension and at uniaxial compression, respectively [24]. If the values of σ_t and σ_c are substituted into Eqs. (13) and (14), then σ₀ at positive and negative pressure equals Y, respectively. In other words, the yield surface would, in this case, coincide with the surface of damage accumulation. However, our simulations showed that in the case of coincidence of yield and damage initiation surfaces, the rocks tend to behave much too viscous at chosen parameters, which contradicts with the observations in mines.

Then, the experiments on the determination of tensile and compressive strengths of rocks were looked through (the experimental values are related to laboratory scale specimens, Table 1). Fig. 3a illustrates the yield surface for the case where the experimental cohesion (Y_exp) and experimental internal friction factor (α_exp) are taken as the basic strength characteristics of the Drucker-Prager model (grey line in Fig. 3a). The uniaxial tensile strength (σ_t,DP) was calculated from cohesion and internal friction factor (blue dashed line in Fig. 3a) by using the well-known formula [24]. At the same time, the value of tensile strength (red dashed line in Fig. 3a) was experimentally obtained. It can be seen that σ_t,DP > σ_t,exp, i.e. the calculated value σ_t,DP is an overestimated value of tensile strength (see the comparison of these two values for all rocks in Table 1). Overestimation of tensile strength by Drucker-Prager criterion is also mentioned in [39]. Other physical–mechanical properties of rocks necessary for numerical simulation are listed in Table 2 [40].

Thus, the fracture criterion has to be updated. If the elastic stress path No. 1 occurs (dark yellow dashed and dotted lines in the Fig. 3a), no damage is accumulated; but if the pressure becomes less than −σ_t,exp/3, the cell is fractured (we immediately ascribe D = 1 in this case, tensile cut-off).

If we perform a backward calculation and take experimental values of σ_t,exp and σ_c,exp as basic strength characteristics of the Drucker-Prager model and calculate cohesion (Y_DP) and internal friction factor (α_DP), two inequations can be obtained: Y_DP < Y_exp and α_DP > α_exp. In our opinion, such a misfit between the calculated parameters and experimental data is accounted for by different mechanisms of deformation of rocks at uniaxial and multiaxial loading, which is revealed by the corresponding experiments. In other words, UCS and UTS are useful in determining the strength envelope in the region of negative pressure and thus allow the entire description of rock behavior to be completed.

The new surface (purple line in Fig. 3a) is built in terms of Y_DP and α_DP. This surface was proposed in the 2D Haigh-Westergaard stress space to be the surface of damage accumulation onset. Thus, the stress difference (σ–σ₀) in the numerator of Eq. (12) equals −αDPP+J2−YDP. If the elastic stress path No. 2 is taken (green dash-dot line in Fig. 3a), damages start accumulating when the new surface is reached. The damage accumulation onset is schematically marked with a circle. After that point, σ–σ₀ > 0. If D = 1, then the cell is fractured and the strength of rocks turns to residual.

Degradation of rock properties in the regions of compressive stress demands a detailed investigation, however this discussion is omitted herein since the fracturing of rock mass elements usually occurs in the regions where tensile stress prevails. In other words, in the framework of the model only shear plastic strains can accumulate rather than damages if the yield criterion is satisfied in the regions of positive pressure P > 0 (Fig. 3b). Such an approach is applicable since the crack opening is impossible at high compressive stress values.

In fact, a piece-wise linear approximation of yield strength envelope of rocks with three segments was proposed. If compared to the review of experiments on rock yield strength envelopes reported by Hoek and Martin, then it might be a compulsory assumption [41]. A typical parabolic yield strength envelope is approximated by a piecewise linear function with several segments, which derivation is practically easier for the determination of inelastic strain tensor components.

The critical stress σ* in the denominator of Eq. (12) is equal to initial values of σ_t,exp and σ_c,exp for tension and compression, respectively.

The rate of damage accumulation is controlled by the parameter t*, which is the only adjustable parameter of the proposed model. It has the physical meaning of the characteristic time of a fracture incubation process, while in our model it is a fitting constant. The value of t* was selected on the basis of the field observations data available from different mines of the Kondomsky deposit, e.g., if the average value of the first roof caving, in particular, geotechnical conditions according to field data is equal to the step of the first main roof caving (L_fc), then t* is selected to make the step of the first in the calculations close to L_fc. Table 3 summarizes the field data from 4 mines of the southern Kuzbass. These values were used for the selection of t* in simulation. Thus, t* is an empirically derived parameter based on the analysis of the rock response in the Kuzbass coal mines. It might be taken as the field time of first caving expressed in terms of arbitrary units (numerical integration time).

The material after its fracture at this point no longer resists tensile stresses (all stress tensor components are nullified) but resists shear and compressive stresses with local strength τ = α_expP.

The geological data shows that the average thickness (M) of the Kuzbass working coal seams is within the range of 2–3 m. In this work, several coal seams with the working heights H_m > 1 m were taken as case studies. The burial depths of all working seams of the Kondomsky deposit vary along the exploration lines, according to [21]. In order to estimate the steps of main roof bed weightings for different seams, for the model, the burial depth of the seams was selected under consideration from the field data range. Generally, the lengths of longwall panels vary from 1 to 4 km. It is not, however, necessary to simulate the opening of such a length; moreover, this demands large computing power. Previous calculations and literature data show that it is enough to simulate at best about 200 m of the face advancement in order to understand the features of damage accumulation, abutment pressure amplification, and evolution of other parameters of the state of stress and strain. It was also found out that the lateral and vertical dimensions of the model have to be at least 2 or 3 times bigger (about 400 and 600 m, respectively) than those of the gob (about 200 m), which would allow decreasing the boundary effects. Fig. 4 shows a 2D roof and floor structural model and a loading and mining scheme for Seam 9a (as an example) of the Kondomsky deposit. The locations of longwall face retreat onset and finish are shown by white quads, with the distance between them being 200 m.

The following boundary conditions were applied to the model: (15){vy=0,xi∈B1vx=0,xi∈B2,B4σijnj=σoverburden ,xi∈B3σijnj=0,xi∈B5 where x_i is the Cartesian coordinates; σ_overburden the weight of the overburden strata implicitly accounted in the model; n_j the components of a normal vector; and δ_ij the Kronecker delta.

The hydraulic support line is modelled by a pressure zone (2 MPa) moving towards the longwall face retreat occupying 5 cells behind the face (3.5 m). The hydraulic support pressure value is generally chosen close to the lithostatic pressure in order to support the rocks near the face and not to produce an overload of the surrounding rocks.

At the time marked as the “End time of the gravitational state of stress formation” in Fig. 5c, the distribution of stress tensor components is the result of gravity forces. The total number of the elements (cells) is 900 and 510 in x and y direction, respectively; the mesh step is 70 cm, so the lateral and vertical dimensions of the computational domain are 630 and 357 m, respectively.

The numerical simulation of Seam 3 as an example shows that the virgin state of the rock mass is considerably disturbed when the longwall face retreat starts. The pressure patterns combined with patterns of accumulated damages for two consecutive time points of face retreat are shown in Fig. 5a and b. The red rectangles in Fig. 5a and b indicate the volume of extracted coal. The pressure range to negative values was specially re-scaled only in order to identify the regions of negative pressure, as potential regions of cracking (black and grey regions in Fig. 5a and b). These patterns are typical for all seams at the onset of face advancement. The patterns show that the arch-shaped local regions of the rock mass, under and above the gob, experience negative pressure from the very beginning of extraction. The larger the extracted volume, the larger the regions of negative pressure.

The dependence of pressure on time in one of the immediate roof points was also recorded (Fig. 5c). After the onset of face retreat, the virgin state of stress due to gravity forces, which was formed in the point close to the mining chamber, is violated and changes the sign several meters after the face advancement. When the pressure becomes negative, indicating tensile loading of rocks, the fracture criterion might be met as it occurs in the monitored point (Fig. 5c). After the failure, the rocks experience a residual strength and cave. Then, obviously, P = 0. Before the caving of rocks, it is observed that some periodic pressure jumps caused by the advancing face (Fig. 5d).

If we look at the next stages of face advancement, it is seen that the propagation of localized bands of plastic deformation violates the stress state to a more significant extent. The growing gob volume gradually increases the hanging roof length and shifts the zone of negative pressure towards the surface. Thus, the fracture process gradually entrains the roof and forms an arch-shaped fracture-rich zone (Fig. 6). One can see that the bands of localized inelastic deformation, which contour the fracture-rich zone, follow the lines of the maximum tangential stress. Due to the arch effect, there is a transient contour, where the stress state changes its type and sign (Fig. 7). The Lode parameter characterizing the type of stress state was calculated. Its pattern illustrates that the band of localized plastic deformation butts against the zone of compressive stresses (μ_σ = 2(S₂ − S₃)/(S₁ − S₃) − 1, μ_σ → −1, where S₁, S₂ and S₃ are the principal deviatoric stresses). The transient contour marked in Fig. 7 becomes an obstacle for band propagation towards the surface and leads to deviation from the initial path along the direction of the maximum pair tangential stress, which appears to be a more preferable path.

When all conditions listed in Section 3.1 for damage measure are satisfied, cohesion begins degrading in some point of the rock mass. Stress relaxation gives rise to energy redistribution. The amplification of stresses and hence the formation of a local critical state occurs in some neighbouring regions causing further rock failures (Fig. 8a–c). Together with the build-up of gob, this process helps in sustaining the damage accumulation. As a result, a large number of the bands of localized accumulated damage divide the main roof and overburden the strata into a hierarchically organized system of blocks of different scales (Fig. 8d).

The patterns for successive time points (Fig. 8a–d) indicate that the emerging localization bands are oriented both in the direction of maximum tangential stresses and in the sub-vertical and sub-horizontal manner. The latter bands are the regions of bed separation, while the former bands divide the layers into several blocks of different scales. A concurrent action of these two mechanisms results in failures. The localization bands oriented along the maximum tangential stresses tend to move towards the face zone of abutment pressure amplification. They form an “onion-ring” structure of localized damage. It is evident that when the rock mass is in the virgin state of stress, one of the principal stress axes, oriented perpendicular to the direction of vertical stress, is directed horizontally in the absence of tectonic stresses. The simulation data shows that trajectories of the maximum principal stress undergo a horizontal-to-sub-vertical rotation when the extraction of coal starts. Fig. 8a–d also illustrate the process of maximum principal stress axis rotation. An increased gob, in turn, causes the increase of rocks volume where the maximum principal stress axis rotation takes place. This appears to be the key mechanism responsible for roof cavings.

Abutment pressure is one of the geotechnical parameters having significant value in mining management. An extensive application of this parameter is accounted for by a large number of relatively simple ways to measure it, using the electric resistance, strain gauges, etc. The abutment pressure caused by the advancing face can be estimated by numerical simulation.

After analyzing the numerical simulation data, the maximum level of abutment pressure is observed about 1.5–2.5 m deep into the panel in front of the face prior to the first caving of the main roof. After the first caving, the drop of abutment pressure occurs, and there appears to be a plateau of maximum abutment pressure with small oscillations in the area of about 6–8 m deep into the panel in front of the face.

The abutment pressure diagrams for successive times of face retreat t_i (Fig. 9) show that the relative amplification (vertical stress concentration factor) lies in the range of 1.5–2.5. Following the first main roof caving, a decrease in the abutment pressure concentration is observed in front of the face (t₅ in Fig. 9), which implies a completed effective energy dissipation. Further, the concentration factor was relatively unchanged with small oscillations, i.e. it has a periodic character due to periodic roof cavings. It is notable that the abutment pressure influence extends to a distance as long as ≥150 m after the first caving of the main roof. The calculations also show that 1.5–2.5 m from the face deep into the panel coal deforms plastically (Fig. 10).

Qualitatively close results for abutment pressure diagrams and coal yielding in front of the face were obtained by [3,7,8].

Figs. 11–14 illustrate the estimations of the first and periodic cavings of the roofs of four different seams. We combined the structural roof and floor models of the corresponding seams with the patterns of accumulated damage due to longwall mining. This has been done in order to get a deeper insight into the features of roof deformation and to better estimate the steps of cavings. Table 3 summarizes the data of numerical modelling and provides a comparison with the field data.

Fig. 15 illustrates the dependence of steps of the first roof caving on the main roof thickness. In the considered range of thicknesses, it is linear with a positive slope. This dependence demonstrates a strong influence of the features of bedding and especially the main roof thickness on the steps of caving. The simulations show that the estimated main roof thicknesses are in the range of the field data.

In our opinion, the case of Polosuhinskaya mine is the most interesting one, where the step of first roof caving is rather high in comparison with the other seams. The roof of Seam 29a is composed of a relatively thick sandstone layer. This is the case of a poorly self-cavable roof. The simulations show that the first detachment crack in the main roof occurs at a distance of about 30 m from the immediate roof. The thickness of the immediate roof layer is also very large in comparison with other seams. Seam 29a belongs to the Lower Balahonskaya formation when the periods of sedimentation were long. Obviously, a failure of such a large thickness of the main roof demands bigger spans of the hanging roof, which was proved by our simulations.

Generally, the simulation results give slightly overestimated values of the first step of caving, which are approximately 10%. We believe that such an agreement of the model results with the field data is quite good. A small misfit between the numerical estimations and reality is due to a general complexity of the problem, caused by a large number of input parameters: physical-mechanical properties, stratigraphy, etc.

At the same time, the agreement between the numerical estimations of steps of periodic caving is much worse (Fig. 16). Values of the periodic steps are significantly underestimated due to: (1) After the first main roof caving, the face retreat rate generally increases by up to 30–40% of its initial rate. An increase in the retreat rate is likely to cause caving step growth. The change of the face retreat rate is not included into account; (2) After the first caving of the rocks, the gob properties recover with increasing distance behind the face. The recovery of the gob properties has not been included for consideration.