A prediction method for long-term surface subsidence considering the mining-induced stratum creep effect and its application

Abstract

Considering the rock creep effect can effectively improve the prediction accuracy of surface subsidence in goafs, which is particularly important for determining construction timing and ensuring the safe operation of surface structures. On basis of the time-dependent deformation characteristics of the overlying strata in mining areas, a nonlinear viscoelastic‒plastic (NVEP) model for accurately describing the three-stage creep behavior of rocks was established. The three-dimensional creep constitutive equations of this model were derived, and a secondary development of the creep model was implemented on the FLAC^3D numerical simulation platform. A creep parameter inversion method for overlying strata in goaf areas was proposed by combining creep simulation analysis with a genetic algorithm. A specific project was selected as a case study, and the creep model and its related parameters were subsequently used to predict the long-term surface settlement behavior, providing a scientific basis for determining the appropriate construction timing for the surface railway in the goaf area of the region. The numerical simulation results indicate that the surface subsidence exhibits an exponential decay trend, which can be divided into three distinct stages: an initial rapid settlement phase (0–2 years), a transitional phase (2–3 years), and a long-term stabilization phase (beyond 4 years). On the basis of the railway construction specifications and the results of the numerical simulation analysis, the feasibility of constructing a railway on the goaf surface is assessed. The findings indicate that railway construction above the goaf in this area should be postponed for at least five years after the cessation of mining activities.

Introduction

As a major coal producer and consumer, China sources more than 80% of its coal from underground mining^1,2. The underground caving methods used for coal extraction result in the formation of goaf areas after the coal has been mined, where the overlying strata gradually collapse^3,4,5. This subsidence propagates to the surface, ultimately forming funnel-shaped settlement zones that extend from the goaf centers to the periphery^6,7. Mining-induced surface subsidence not only damages surface infrastructure and building deformation but also lowers groundwater tables and contaminates soil and water^8,9, which significantly limits sustainable mining development and remains an active frontier in mining engineering research.

Mining-induced surface subsidence involves a multidimensional dynamic problem that includes coal seam extraction, overburden strata movement, and surface deformation. Throughout this process, the geometric configuration of the rock strata has nonlinear time-dependent characteristics resulting from the coupling of multiple physical fields, including stress field redistribution and displacement field transmission. To elucidate surface deformation mechanisms, extensive studies have focused on the temporal evolution of goaf subsidence. Sun et al.¹⁰ proposed a new theoretical method to predict the strata movement boundary and surface subsidence caused by inclined coal seam mining, which considers the influence of key strata, rock quality and coal seam dip angle. Cheng et al.¹¹ established an improved Knothe time function model on the basis of the inversion of creep curves and surface subsidence curves. The empirical model can predict the dynamic process of surface subsidence affected by coal mining well. Ding et al.¹² integrated the Boltzmann function with a transformed normal distribution time function to establish a novel B-normal dynamic prediction model, which can accurately predict mining-induced surface displacement. Wang et al.¹³ developed novel time-function models for mining subsidence prediction via the probability integral method and with consideration of surface deformation features. Zhou et al.¹⁴ proposed a Morgan–Mercer–Flodin time function for predicting progressive surface subsidence caused by coal mining on the basis of model assumptions and formula derivations. Researchers have reported significant findings on the evolution of surface subsidence in goaf areas through time-function modeling. However, these methods rely on empirical or theoretical assumptions and disregard rock mass constitutive relationships and geological conditions, which limits insights into subsidence mechanisms and long-term deformation patterns.

The surface subsidence in mine-out areas is caused by the cumulative movement and deformation of the rock strata. Owing to the inability of time function models to account for the effect of rock strata properties on subsidence deformation, their predictive reliability significantly decreases under complex geological conditions. As a result, several researchers have extensively investigated surface subsidence caused by overburden displacement through numerical and physical simulations. Xu et al.¹⁵ investigated the movement laws of the overlying strata at the working face ends and assessed their effects on surface deformation via field monitoring as well as physical and numerical modeling. Chai et al.¹⁶ analyzed ground surface movement and deformation in deep and extra-thick coal seam mining under ultra-thick conglomerate at the Qianqiu Coal Mine via physical modeling. Liu et al.¹⁷ analyzed the surface subsidence of coal mines using a novel bonded block numerical method. Zhao and Konietzky¹⁸ proposed a numerical modeling strategy to predict uplift during the flooding process of an excavated coal mining area, which considers three phases: the virgin state before mining, the stage immediately after mine closure, and the flooding phase. Cheng et al.¹⁹ analyzed the whole strata movement process of the footwall induced by underground mining with the UDEC method. Fathi Salmi et al.²⁰ analyzed the effects of underground mining on the stability of slopes via numerical modelling. Numerical simulations provide important technical support for analyzing the characteristics of strata movement and surface subsidence after coal mining. However, the majority of studies focus on analyzing the effect of strata fracture and collapse on surface subsidence after coal extraction and rely on elastoplastic theory, with minimal consideration given to the long-term effects of strata creep on surface subsidence.

Although initial subsidence due to rock stratum collapse during mining has been investigated extensively, the time-dependent settlement caused by strata creep after mining ends critically affects the safety of surface structures. Conventional prediction methods for surface subsidence, which often overlook the viscoelastic‒plastic creep behavior of rock masses, fail to accurately represent this prolonged deformation. As a result, design margins for critical infrastructure may be underestimated, potentially compromising long-term stability and safety. To address these limitations, first, this study established a nonlinear viscoelastic-plastic creep model for describing the three creep stages by adding threshold elements and a viscoplastic model to the Burgers model. Second, the creep equation of the model was derived, and the secondary development of the model was implemented on the FLAC^3D platform. Third, the prediction and analysis of the surface subsidence in a mine was selected as a case study, the creep model parameters were inverted with field monitoring data, and the long-term surface subsidence pattern of the region was investigated through numerical simulation. Last, on the basis of numerical simulation results and relevant standards^21,22, the optimal timing for railway construction in mining-affected areas was determined. These findings provide theoretical foundations and establish practical engineering benchmarks for the feasibility analysis of railway construction in mining areas.

Rock creep constitutive model of rock strata

Establishment of the nonlinear creep model

Previous studies have indicated that rock typically exhibits three stages of creep under high-stress conditions—decaying creep, steady-state creep, and accelerated creep— whereas rocks that have been damaged or fractured by mining activities are more susceptible to creep deformation^{23,24,25,26,27}. Currently, classical creep models, such as the Burgers model, play significant roles in analyzing the time-dependent deformation behavior of geotechnical materials in underground engineering^28,29,30. However, these classical models are not fully applicable to the analysis of surface subsidence induced by the time-dependent deformation of roof strata in coal seams for the following reasons. First, the significant differences in lithology and stress levels of the roof rock result in various creep behaviors among different rock layers. Second, the behaviors of roof rock collapse and continuous compaction after coal seam mining are related to the nonlinear characteristics of rock accelerated creep. Third, during coal mining, the stress field in roof strata continuously adjusts as the rock mass deforms, resulting in transient elastic, viscoelastic, and viscoplastic deformation behavior. On the basis of the time-dependent mechanical behavior of rock strata, a creep model that can describe the nonlinear viscoelastic‒plastic deformation characteristics of rocks is particularly critical for analyzing the long-term subsidence law of the surface behind roof collapse in goafs. In this study, a novel creep model (NVEP model) is established by connecting the nonlinear viscoplastic model in series with the Burgers model, the yield failure of which is governed by the Mohr‒Coulomb criterion. Moreover, the model characterizes the time-dependent deformation of roof strata under different stress levels by incorporating a stress-dependent creep threshold component. The nonlinear creep model is shown in Fig. 1.

Fig. 1

Nonlinear creep model of roof rock strata.

This model reveals that the rock mass exhibits only transient elastic deformation at low stress levels (i.e. σ < σ_k), below the creep threshold. At this stage, rock strata deformation is time-independent; consequently, the creep model can be described by Hooke’s law as follows:

$${\sigma _{\text{e}}}={E_{\text{M}}}{\varepsilon _{\text{e}}}$$

(1)

When stress levels exceed the creep threshold but remain below the long-term strength of the rock (i.e., σ_k ≤ σ < σ_s), the rock mass undergoes viscoelastic deformation. At this stage, the creep model can be described as follows:

$$\left\{ \begin{gathered} {\sigma _{\text{e}}}={E_{\text{M}}}{\varepsilon _{\text{e}}} \hfill \\ {\sigma _{{\text{ve-1}}}}={E_{\text{K}}}{\varepsilon _{{\text{ve-1}}}}+{\eta _{\text{K}}}{{\dot {\varepsilon }}_{{\text{ve-1}}}} \hfill \\ {\sigma _{{\text{ve-2}}}}={\eta _{\text{M}}}{{\dot {\varepsilon }}_{{\text{ve-2}}}} \hfill \\ {\sigma _{{\text{ve-1}}}}={\sigma _{{\text{ve-2}}}}={\sigma _{\text{e}}}=\sigma \hfill \\ \varepsilon ={\varepsilon _{\text{e}}}+{\varepsilon _{{\text{ve-1}}}}+{\varepsilon _{{\text{ve-2}}}} \hfill \\ \end{gathered} \right.$$

(2)

According to Eq. (2), the constitutive equation for the creep model is derived as follows:

$$\sigma +\left( {\frac{{{\eta _K}}}{{{E_K}}}+\frac{{{\eta _M}}}{{{E_K}}}+\frac{{{\eta _M}}}{{{E_M}}}} \right)\dot {\sigma }+\frac{{{\eta _K}{\mkern 1mu} {\eta _M}}}{{{E_K}{\mkern 1mu} {E_M}}}{\mkern 1mu} \ddot {\sigma }={\eta _M}{\mkern 1mu} \dot {\varepsilon }+\frac{{{\eta _K}{\mkern 1mu} {\eta _M}}}{{{E_K}}}{\mkern 1mu} \ddot {\varepsilon }$$

(3)

When the stress level remains constant at σ₀, we obtain the following:

$$\ddot {\sigma }=\dot {\sigma }=0$$

(4)

The creep equation is obtained by substituting Eq. (4) into Eq. (3):

$$\varepsilon =\frac{{{\sigma _0}}}{{{E_{\text{M}}}}}+\frac{{{\sigma _0}}}{{{\eta _{\text{M}}}}}t+\frac{{{\sigma _0}}}{{{E_{\text{K}}}}}\left( {1 - {e^{ - \frac{{{E_{\text{K}}}}}{{{\eta _{\text{K}}}}}t}}} \right)$$

(5)

When stress levels exceed the long-term strength of the rock (i.e., σ ≥ σ_s), the rock mass exhibits significant nonlinear viscoelastic‒plastic deformation. Previous studies have indicated that this nonlinear deformation during the accelerated creep stage results from a stress-induced reduction in the viscosity coefficient. Therefore, we replace the viscous component in the Saint-Venant body with a nonlinear component to characterize the mechanical response at this stage, which is expressed as follows:

$$\eta (t)={\eta _{\text{c}}}{e^{ - at}}$$

(6)

where a (a > 0) represents the creep index, which is affected by stress levels and lithology, and controls the rheological rate of the rock during the accelerated creep stage. Consequently, the rheological model consists of the following system of equations:

$$\left\{ \begin{gathered} {\sigma _{\text{e}}}={E_{\text{M}}}{\varepsilon _{\text{e}}} \hfill \\ {\sigma _{{\text{ve-1}}}}={E_{\text{K}}}{\varepsilon _{{\text{ve-1}}}}+{\eta _{\text{K}}}{{\dot {\varepsilon }}_{{\text{ve-1}}}} \hfill \\ {\sigma _{{\text{ve-2}}}}={\eta _{\text{M}}}{{\dot {\varepsilon }}_{{\text{ve-2}}}} \hfill \\ {\sigma _{{\text{vp}}}}={\eta _{\text{c}}}{e^{ - at}}{{\dot {\varepsilon }}_{{\text{vp}}}} \hfill \\ {\sigma _{{\text{ve-1}}}}={\sigma _{{\text{ve-2}}}}={\sigma _{\text{e}}}={\sigma _{{\text{vp}}}}+{\sigma _{\text{s}}}=\sigma \hfill \\ \varepsilon ={\varepsilon _{\text{e}}}+{\varepsilon _{{\text{ve-1}}}}+{\varepsilon _{{\text{ve-2}}}}+{\varepsilon _{{\text{vp}}}} \hfill \\ \end{gathered} \right.$$

(7)

The constitutive equation derived from Eq. (7) is then expressed as follows:

$$\frac{{\dot {\sigma }}}{{{\mkern 1mu} {\eta _{\text{K}}}}} - \frac{{{E_{\text{K}}}\dot {\varepsilon }}}{{{\eta _{\text{K}}}}}+\frac{{{E_{\text{K}}}\dot {\sigma }}}{{{\eta _{\text{K}}}{E_{\text{M}}}}}+\frac{{{E_{\text{K}}}\sigma }}{{{\eta _{\text{K}}}{\eta _{\text{M}}}}}+\frac{{\sigma - {\sigma _{\text{s}}}}}{{{\eta _{\text{K}}}{\eta _{\text{c}}}{\mkern 1mu} {\mkern 1mu} }}{E_{\text{K}}}{e^{at}}={\mkern 1mu} \ddot {\varepsilon } - \frac{{\ddot {\sigma }}}{{{E_{\text{M}}}}} - \frac{{\dot {\sigma }}}{{{\eta _{\text{M}}}}}+\left( {2a{\sigma _{\text{s}}} - a\sigma - \dot {\sigma }} \right)\frac{{{e^{at}}}}{{{\eta _{\text{c}}}}}$$

(8)

When the stress level remains constant at σ₀, we obtain the creep equation as follows:

$$\varepsilon =\frac{{{\sigma _0}}}{{{E_{\text{M}}}}}+\frac{{{\sigma _0}}}{{{\eta _{\text{M}}}}}t+\frac{{{\sigma _0}}}{{{E_{\text{K}}}}}\left( {1 - {e^{ - \frac{{{E_{\text{K}}}}}{{{\eta _{\text{K}}}}}t}}} \right)+\frac{{{\sigma _0} - {\sigma _{\text{s}}}}}{{a{\eta _{\text{c}}}}}{e^{at}}$$

(9)

The one-dimensional creep equations for the novel nonlinear viscoelastic‒plastic creep model are expressed as follows:

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(10)

In three-dimensional stress states, the spherical stress tensor controls volumetric deformation, whereas the deviatoric stress tensor controls shape distortion. Previous have indicated that these stress components control the creep process^31,32. Consequently, the three-dimensional constitutive relations for the modified Burgers creep model can be expressed as follows:

$$\left\{ \begin{gathered} {\sigma _{ij}}={\sigma _{\text{m}}}{\delta _{ij}}+{s_{ij}} \hfill \\ {\varepsilon _{ij}}={\varepsilon _{\text{m}}}{\delta _{ij}}+{e_{ij}} \hfill \\ \end{gathered} \right.$$

(11)

where σ_m represents spherical stress tensor (σ_m = σ_ii/3), s_ij denotes the deviatoric stress tensor, ε_m indicates the spherical strain tensor, and e_ij represents the deviatoric strain tensor.

The 3D Hooke’s law in the elastic state is expressed as follows:

$$\left\{ \begin{gathered} {\varepsilon _{\text{m}}}=\frac{{{\sigma _{\text{m}}}}}{{3K}} \hfill \\ {e_{ij}}=\frac{{{s_{ij}}}}{{2G}} \hfill \\ \end{gathered} \right.$$

(12)

where K represents the bulk modulus (K = E/2(1–2µ)) and G denotes the shear modulus (G = E/2(1 + µ)).

According to Eq. (11), the 3D nonlinear creep constitutive equations are expressed as follows:

$$\left\{ \begin{gathered} \varepsilon _{{ij}}^{{\text{e}}}=\frac{{{\sigma _{\text{m}}}}}{{3K}}{\delta _{ij}}+\frac{{{s_{ij}}}}{{2{G_{\text{M}}}}} \hfill \\ \varepsilon _{{ij}}^{{{\text{ve}}}}=\frac{{{s_{ij}}}}{{2{G_{\text{K}}}}}\left( {1 - {e^{ - \frac{{{G_{\text{K}}}}}{{{\eta _{\text{K}}}}}}}} \right)+\frac{{{s_{ij}}}}{{2{\eta _{\text{M}}}}}t \hfill \\ \varepsilon _{{ij}}^{{{\text{vp}}}}=\frac{1}{{2a{\eta _{\text{c}}}}}\left\langle {\phi \left( {\frac{F}{{{F_0}}}} \right)} \right\rangle \frac{{\partial Q}}{{\partial {\sigma _{ij}}}}{e^{at}} \hfill \\ \end{gathered} \right.$$

(13)

where F denotes the yield function with F₀ as its initial value; Q represents the plastic potential function, which is assumed to be identical to that in the Mohr‒Coulomb model; and ϕ indicates a power-law function whose exponent is typically 1 for rock materials.

Under conventional triaxial creep conditions (σ₂ = σ₃), the constitutive equations simplify from Eq. (12) to:

$$\left\{ \begin{gathered} {\sigma _{\text{m}}}=\frac{1}{3}({\sigma _1}+2{\sigma _3}) \hfill \\ {s_{ij}}=\frac{2}{3}({\sigma _1} - {\sigma _3}) \hfill \\ \end{gathered} \right.$$

(14)

By substituting Eqs. (13) and (14) into Eq. (11), we obtain the 3D creep constitutive equations for the proposed nonlinear viscoelastic‒plastic model:

$${\varepsilon _{11}}=\left\{ {\begin{array}{*{20}{l}} {\frac{{{\sigma _1} - {\sigma _3}}}{{3{G_{\text{M}}}}}+\frac{{{\sigma _1}+2{\sigma _3}}}{{9K}}}&{({\sigma _{11}} \leqslant {\sigma _{\text{k}}})} \\ {\frac{{{\sigma _1} - {\sigma _3}}}{{3{G_{\text{M}}}}}+\frac{{{\sigma _1}+2{\sigma _3}}}{{9K}}+\frac{{{\sigma _1} - {\sigma _3}}}{{3{\eta _{\text{M}}}}}+\frac{{{\sigma _1} - {\sigma _3}}}{{2{G_{\text{K}}}}}\left[ {1 - \exp ( - \frac{{{G_{\text{K}}}}}{{{\eta _{\text{K}}}}}t)} \right]}&{({\sigma _{\text{k}}}<{\sigma _{11}},F<0)} \\ {\frac{{{\sigma _1} - {\sigma _3}}}{{3{G_{\text{M}}}}}+\frac{{{\sigma _1}+2{\sigma _3}}}{{9K}}+\frac{{{\sigma _1} - {\sigma _3}}}{{3{\eta _{\text{M}}}}}+\frac{{{\sigma _1} - {\sigma _3}}}{{2{G_{\text{K}}}}}\left[ {1 - \exp ( - \frac{{{G_{\text{K}}}}}{{{\eta _{\text{K}}}}}t)} \right]}&{} \\ {+\frac{{{\sigma _1} - {\sigma _3}{N_\varphi } - 2C\sqrt {{N_\varphi }} }}{{2a{\eta _c}}}{e^{at}}}&{({\sigma _{\text{k}}}<{\sigma _{11}},F \geqslant 0)} \end{array}} \right.$$

(15)

To confirm the advantages of the proposed creep model over the classical model in characterizing the accelerated creep stage of rock, the creep test data from Reference³³ were used to fit the creep models. The creep model parameters obtained from the fitting are listed in Table 1. The comparison between the theoretical curves of creep models and the experimental data is shown in Fig. 2. In the initial stage of creep, the theoretical predictions of all three models exhibit reasonable agreement with the sandstone test results, indicating that each model can adequately capture the early-stage creep behavior of rock. However, as creep duration increases, discrepancies among the models and between the models and experimental observations become progressively more pronounced. Notably, the theoretical curve of the NVEP model demonstrates a consistently closer fit to the experimental data across the entire creep process, particularly during the later stages. Additionally, to investigate the influence of the creep index on creep behavior, theoretical creep curves under different creep index values (a = 1, 3, 5, 7) were analyzed. According to the parameters listed in Table 1; Fig. 3 presents the theoretical curves of the NVEP model for different creep indices. The results show that the creep index has a significant effect on the creep behavior. As the creep index increases, accelerated creep occurs earlier, and the nonlinear deformation characteristics during this stage become increasingly pronounced. It is crucial for accurately predicting the long-term deformation and stability of rock strata in engineering applications.

Table 1 Creep parameters of different models.

Fig. 2

Comparison between the theoretical curves of creep models and the experimental data³³.

Fig. 3

Effect of the creep index on the theoretical curves of NVEP model.

Numerical implementation of the model

Numerical implementation of the theoretical model requires the derivation of 3D creep equations in the form of the finite difference method (FDM) and their integration into FLAC^3D. The deviatoric behavior in the modified creep model is expressed as follows.

Specifically, the total strain rate is expressed as follows:

$${\dot {e}_{ij}}=\dot {e}_{{ij}}^{{\text{e}}}+\dot {e}_{{ij}}^{{{\text{ve}}}}+\dot {e}_{{ij}}^{{{\text{vp}}}}$$

(16)

The elastic element is expressed as follows:

$${s_{ij}}=2{G_{\text{M}}}e_{{ij}}^{{\text{e}}}$$

(17)

The viscoelastic element is expressed as follows:

$$\left\{ \begin{gathered} {s_{ij}}=2{\eta _{\text{K}}}\dot {e}_{{ij}}^{{{\text{ve1}}}}+2{G_{\text{K}}}e_{{ij}}^{{{\text{ve-1}}}} \hfill \\ {s_{ij}}=2{\eta _{\text{K}}}\dot {e}_{{ij}}^{{{\text{ve-}}2}} \hfill \\ \dot {e}_{{ij}}^{{{\text{ve}}}}=\dot {e}_{{ij}}^{{{\text{ve}}1}}+\dot {e}_{{ij}}^{{{\text{ve-}}2}} \hfill \\ \end{gathered} \right.$$

(18)

The viscoplastic element is expressed as follows:

$$\left\{ \begin{gathered} \dot {\varepsilon }_{{ij}}^{{{\text{vp}}}}=\frac{F}{{2{\eta _c}}}\frac{{\partial Q}}{{\partial {\sigma _{ij}}}}{e^{at}} - \frac{1}{3}e_{{{\text{vol}}}}^{{{\text{vp}}}}{\sigma _{ij}} \hfill \\ e_{{{\text{vol}}}}^{{{\text{vp}}}}=\frac{F}{{2{\eta _c}}}(\frac{{\partial Q}}{{\partial {\sigma _{11}}}}+\frac{{\partial Q}}{{\partial {\sigma _{22}}}}+\frac{{\partial Q}}{{\partial {\sigma _{33}}}}) \hfill \\ \end{gathered} \right.$$

(19)

The total deviatoric strain increment is derived from Eq. (16):

$$\Delta {e_{ij}}=\Delta e_{{ij}}^{{\text{e}}}+\Delta e_{{ij}}^{{{\text{ve}}}}+\Delta e_{{ij}}^{{{\text{vp}}}}$$

(20)

In the incremental formulation, we use the following expression:

$${\bar {S}_{ij}}=\frac{{S_{{ij}}^{{\text{N}}}+S_{{ij}}^{{\text{O}}}}}{2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Delta {S_{ij}}=S_{{ij}}^{{\text{N}}} - S_{{ij}}^{{\text{O}}}$$

(21)

where the superscripts N and O represent new and old values, respectively, in the timestep procedure.

The FDM formulas from Eqs. (19) to (21) are expressed as follows:

$$\Delta e_{{ij}}^{{\text{e}}}=\frac{{S_{{ij}}^{{\text{N}}} - S_{{ij}}^{{\text{O}}}}}{{2{G_{\text{M}}}}}$$

(22)

$$\left\{ \begin{gathered} \Delta e_{{ij}}^{{{\text{ve}}}}=\Delta e_{{ij}}^{{{\text{ve-1}}}}+\Delta e_{{ij}}^{{{\text{ve-2}}}} \hfill \\ \Delta e_{{ij}}^{{{\text{ve-1}}}}=\frac{1}{{4{\eta _{\text{M}}}}}(S_{{ij}}^{{\text{O}}}+S_{{ij}}^{{\text{N}}})\Delta t \hfill \\ \Delta e_{{ij}}^{{{\text{ve-2,N}}}}=\frac{1}{A}\left[ {\frac{{\Delta t}}{{4{\eta _{\text{K}}}}}(S_{{ij}}^{{\text{O}}}+S_{{ij}}^{{\text{N}}})+B\Delta e_{{ij}}^{{{\text{ve-2,O}}}}} \right] \hfill \\ \end{gathered} \right.$$

(23)

$$\Delta e_{{ij}}^{{{\text{vp}}}}=\frac{F}{{2{\eta _c}}}\frac{{\partial Q}}{{\partial {\sigma _{ij}}}}{e^{at}}\Delta t$$

(24)

where $A=1+\frac{{{G_{\text{K}}}\Delta t}}{{2{\eta _{\text{K}}}}}$ and $B=1 - \frac{{{G_{\text{K}}}\Delta t}}{{2{\eta _{\text{K}}}}}$.

The new deviatoric stress tensor is expressed as follows:

$$S_{{ij}}^{{\text{N}}}=\left\{ \begin{gathered} 2{G_{\text{M}}}\Delta {e_{ij}}+S_{{ij}}^{{\text{O}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({S_{ij}} \leqslant {\sigma _{\text{k}}},F \leqslant 0) \hfill \\ \frac{1}{a}\left[ {\Delta {e_{ij}}+bS_{{ij}}^{{\text{O}}} - (\frac{B}{A} - 1)\Delta e_{{ij}}^{{{\text{ve-2,O}}}}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ({S_{ij}}>{\sigma _{\text{k}}},F \leqslant 0) \hfill \\ \frac{1}{a}\left[ {\Delta {e_{ij}}+bS_{{ij}}^{{\text{O}}} - (\frac{B}{A} - 1)\Delta e_{{ij}}^{{{\text{ve-2,O}}}} - \Delta e_{{ij}}^{{{\text{vp}}}}} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (F>0) \hfill \\ \end{gathered} \right.$$

(25)

where $a=\frac{1}{{2{G_{\text{M}}}}}+\frac{{{\text{\varvec{\Delta}}}t}}{4}\left( {\frac{1}{{{\eta _{\text{M}}}}}+\frac{1}{{A{\eta _{\text{K}}}}}} \right)$, $b=\frac{1}{{2{G_{\text{M}}}}} - \frac{{{\text{\varvec{\Delta}}}t}}{4}\left( {\frac{1}{{{\eta _{\text{M}}}}}+\frac{1}{{A{\eta _{\text{K}}}}}} \right)$

The FDM formula for spherical stress is expressed as follows:

$$\sigma _{0}^{{\text{N}}}=\sigma _{0}^{{\text{O}}}+K(\Delta {e_{{\text{vol}}}} - \Delta e_{{{\text{vol}}}}^{{\text{p}}})$$

(26)

During constitutive model implementation in FLAC^3D, the new trial stress components are computed via Eqs. (25) and (26). The principal stresses are subsequently determined, enabling evaluation of the yield function F. If F < 0, the trial stress is updated to be valid. If F > 0, the stresses are corrected on the basis of the increase in viscoplastic strain. The stress components projected back to the principal axes are expressed as follows:

$$\left\{ \begin{gathered} \sigma _{1}^{{\text{N}}}=\hat {\sigma }_{1}^{{\text{N}}} - \left[ {{\alpha _1}\Delta \varepsilon _{1}^{{\text{p}}}+{\alpha _2}(\Delta \varepsilon _{2}^{{\text{p}}}+\Delta \varepsilon _{3}^{{\text{p}}})} \right] \hfill \\ \sigma _{2}^{{\text{N}}}=\hat {\sigma }_{2}^{{\text{N}}} - \left[ {{\alpha _1}\Delta \varepsilon _{2}^{{\text{p}}}+{\alpha _2}(\Delta \varepsilon _{1}^{{\text{p}}}+\Delta \varepsilon _{3}^{{\text{p}}})} \right] \hfill \\ \sigma _{3}^{{\text{N}}}=\hat {\sigma }_{3}^{{\text{N}}} - \left[ {{\alpha _1}\Delta \varepsilon _{3}^{{\text{p}}}+{\alpha _2}(\Delta \varepsilon _{1}^{{\text{p}}}+\Delta \varepsilon _{2}^{{\text{p}}})} \right] \hfill \\ \end{gathered} \right.$$

(27)

where ${\alpha _1}=K+\frac{2}{{3a}}$ and${\alpha _2}=K - \frac{1}{{3a}}$.

For shear yielding:

$$\left\{ \begin{gathered} \sigma _{1}^{{\text{N}}}=\hat {\sigma }_{1}^{{\text{N}}} - {\lambda ^{\text{s}}}\left( {{\alpha _1} - {\alpha _2}{N_\psi }} \right) \hfill \\ \sigma _{2}^{{\text{N}}}=\hat {\sigma }_{2}^{{\text{N}}} - {\lambda ^{\text{s}}}{\alpha _2}\left( {1 - {N_\psi }} \right) \hfill \\ \sigma _{3}^{{\text{N}}}=\hat {\sigma }_{3}^{{\text{N}}} - {\lambda ^{\text{s}}}\left( {{\alpha _2} - {\alpha _1}{N_\psi }} \right) \hfill \\ {\lambda ^{\text{s}}}=\frac{F}{{2{\eta _c}{e^{ - at}}}}\Delta t \hfill \\ \end{gathered} \right.$$

(28)

For tensile yielding:

$$\left\{ \begin{gathered} \sigma _{1}^{{\text{N}}}=\hat {\sigma }_{1}^{{\text{N}}} - {\lambda ^{\text{s}}}\left( {{\alpha _1} - {\alpha _2}{N_\psi }} \right) \hfill \\ \sigma _{2}^{{\text{N}}}=\hat {\sigma }_{2}^{{\text{N}}} - {\lambda ^{\text{s}}}{\alpha _2}\left( {1 - {N_\psi }} \right) \hfill \\ \sigma _{3}^{{\text{N}}}=\hat {\sigma }_{3}^{{\text{N}}} - {\lambda ^{\text{s}}}\left( {{\alpha _2} - {\alpha _1}{N_\psi }} \right) \hfill \\ {\lambda ^{\text{t}}}=\frac{F}{{2{\eta _c}{e^{ - at}}}}\Delta t \hfill \\ \end{gathered} \right.$$

(29)

On the basis of the stress increment formulation of the aforementioned viscoelastic‒plastic model, the computational code was programmed in VC++. Through the UDM (user-defined model) interface of FLAC^3D, the code was compiled into a “.dll” file, enabling secondary development of the constitutive model within FLAC^3D.

Creep parameter inversion method at the engineering scale

The development of the creep model in a numerical calculation platform is the premise for achieving the transformation from a theoretical model to an engineering application, and the creep parameters must be determined at the engineering scale to obtain reliable simulation results. In long-term stability analyses of underground engineering, the identification of creep parameters on the basis of creep models and in-situ rock mass deformation monitoring data is considered relatively reliable. With the advancement of computer technology, information theory, system theory, genetic algorithms, and neural networks have become the common methods for parameter inversion analysis. Genetic algorithms (GAs) are widely used extensively in multi-parameter inversion because of their advantage of adaptive heuristic critical random global search optimization³⁴. When the creep model is used to predict and analyze long-term surface settlement, the model parameters are identified via genetic algorithms. The identification process for the creep parameters of each rock stratum is presented as follows:

(1) Under actual geological engineering conditions, a three-dimensional numerical calculation model of working face mining is established. Static calculations of coal seam mining are performed to obtain surface settlement data without considering the time effect. These data are then compared with the monitoring data to modify the rock mass parameters used in the static calculation.

(2) According to the static analysis, the creep model parameters are initially set on the basis of the rock lithology, and creep simulations are conducted to determine the parameter range. The parameter ranges {x₁, x₂, and x₃} represent the Kelvin shear modulus, Kelvin viscosity, and Maxwell viscosity of the rock layers, respectively.

(3) On the basis of the initial parameters, a random method is used to generate 24 initial genetic chromosomes, A₁ to A₂₄, within the ranges [x₁/10, 10x₁], [x₂/10, 10x₂], and [x₃/10, 10x₃], to complete the initial population setup of the creep model parameters.

(4) The pre-developed creep model for computational execution is invoked, and the relative error between the same-generation simulation results and the on-site monitoring results is calculated via Eq. (30).

$$f({A_i})=\sqrt {\frac{{\sum\limits_{{i=1}}^{5} {{{({x_i} - {x_{i,real}})}^2}} }}{5}}$$

(30)

where f(A_i) denotes the error value at monitoring point i, x_i represents the numerical simulation result at that time step, and x_{i, real} denotes the field-measured displacement at the corresponding time step.

(5) For the sample set ${P_s}=\left\{ {{A_1},{A_2}, \cdots ,{A_{24}}} \right\}$, where each individual A_i has an error value f(A_i), the selection probability P(A_i) is determined via Eq. (31). The options with lower error values are easier to choose.

$$P(Ai)=\frac{{f({A_i})}}{{\sum\limits_{{k=1}}^{{24}} {f({A_i})} }}$$

(31)

(6) Twenty-four offspring are selected as the new parameter population for simulation calculation by calling the gene mutation function. Steps (4) and (5) are repeated until the error is less than 0.01 to obtain the optimal solution and creep parameters.

A workflow diagram of the aforementioned methodology is presented in Fig. 4.

Fig. 4

Flowchart for inverse analysis of creep parameters in rock layers.

Engineering implementation

Project overview

A railway serving a coal mine will be constructed along the northern perimeter of the industrial park with a northeast–southwest alignment. According to the coal mine data, the mining of longwall faces 5–20109, 5–20110, 5–20111, 5–20112, and 5–20113 could impact railway construction, as shown in Fig. 5. To evaluate the effect of the mining of the above working face on the construction of the railway line and to provide guidance for the follow-up railway construction schedule, analysis of the long-term surface subsidence law of the goaf on the basis of the actual geological conditions is critical. Therefore, on the basis of the established nonlinear creep model, the surface settlement pattern and its influence range after mining near the railway line are analyzed, and the railway construction schedule is evaluated in accordance with railway construction specifications.

Fig. 5

Overview of pre-laid railway.

Creep parameters inversion

Currently, the working face near the railway line in Fig. 5 is still in the mining scheme demonstration phase, so no surface subsidence monitoring data is available for direct calibration of the numerical model parameters. To ensure the reliability of the long-term subsidence predictions, this study leverages surface subsidence data from a geologically similar working face in the same region, applying the method outlined in Sect. 2.3 to identify the strata creep parameters. The numerical simulation model shown in Fig. 6 was established on the basis of the borehole data from the area around the 5–20101 working face. The model is 500 m in length, 200 m in width, and 150 m in height. The model boundary conditions include a fixed base, normal stress constrained on all lateral boundaries, and a free surface at the top. The gravitational acceleration in the downward vertical direction was 10 m/s². The physical and mechanical properties of the distinct rock and soil strata are detailed in Table 2, and the interface properties were a normal stiffness of 10 GPa, a tangential stiffness of 10 GPa, a cohesion of 0.1 MPa, and a friction angle of 30°.

Fig. 6

Numerical calculation model of 5–20101 working face.

Table 2 Physical and mechanical parameters of different rock strata.Physical and mechanical parameters of different rock strata.

In addition, the tectonic stress is not significant in the shallow buried 5^-2 coal seam. Consequently, the in-situ stress field is dominated by gravitational forces, with the vertical stress (σ_v) and horizontal stress (σ_h) calculated as follows:

$$\left\{ \begin{gathered} \sigma _{v} = \gamma H \hfill \\ \sigma _{h} = K_{0} \sigma _{v} \hfill \\ \end{gathered} \right.$$

(32)

where γ represents the unit weight of rock or soil, H represents the burial depth, and K₀ represents the coefficient of lateral pressure.

After the three-dimensional model of numerical calculation is established, coal seam mining is conducted in timesteps of 4 m, and the surface settlement is analyzed. The vertical displacement contours of the overlying strata of the mine goaf are shown in Fig. 7. The results reveals that the maximum surface subsidence is 1521 mm after coal mining, which is almost consistent with the field-measured displacement of 1411 mm. Consequently, the results indicate that the mechanical parameters are suitable for static analysis. The creep constitutive model was activated after static analysis. The initial creep parameters for sandstone, mudstone, and coal were calibrated on the basis of previous studies^35,36,37. The inversion of the creep parameters of the strata was determined by the workflow diagram in Fig. 4. During this process, error analysis was conducted by comparing field-monitored and simulation-monitored data. The monitored points are shown in Fig. 8. Based on the surface subsidence monitoring data from the three measurement points, the optimal creep parameters were ultimately determined by the GAs, and the results are listed in Table 3.

To evaluate the prediction accuracy of the proposed model, numerical simulation analyses were conducted using the developed creep model and the identified model parameters. The creep duration was set to 123 days. The comparison of surface subsidence between field-monitored and simulation data is shown in Fig. 9. It can be observed that the time-dependent surface subsidence patterns derived from the numerical simulations align well with field monitoring data, with a coefficient of determination (R²) exceeding 0.95 at all stage. As a result, the proposed creep model, along with the creep parameters obtained through GA-FDM coupling, demonstrates sound rationality in predicting goaf-related subsidence.

Fig. 7

Vertical displacement contour map of the overlying strata in the goaf after mining of the 5–20101 working face.

Fig. 8

Layout diagram of the surface monitoring points above the 5–20101 working face.

Table 3 Inversion parameters of the creep mechanics for different rock strata.

Fig. 9

Chart of the comparison between the surface subsidence monitoring data and the numerical inversion results.

Numerical model for subsidence analysis

A numerical model was developed on the basis of geological data and the location of longwall faces 5–20109 to 5–20113. Given that consistent computational methods were applied to all the longwall face models, the surface subsidence was analyzed using a numerical model of longwall face 5–20109 as a representative case. FLAC^3D software was used to construct the numerical model for subsidence analysis as shown in Fig. 10. On the basis of geological data, the model incorporates eight strata: diluvial soil, fine sand, loess, medium sandstone, mudstone, fine sandstone, siltstone, and coal seam. The model is 300 m in length, 300 m in width, and 150 m in height, comprising 139,400 grid points and 110,952 zones. The boundary conditions include a free surface at the top, normal displacement constraints on the lateral boundaries, and full fixation at the base. Interfaces between rock layers were assigned a normal/shear stiffness of 30 GPa, a cohesion of 0.02 MPa, and a friction angle of 25°.

Fig. 10

Numerical simulation model for surface subsidence over the goaf of the 5–20109 working face.

During each simulation cycle, the longwall face advanced by 4 m, with 0.6 MPa of support pressure applied to the surrounding rock to simulate hydraulic shield resistance. After the mining of the working face is completed, the creep analysis mode in FLAC^3D opened, the developed viscoelastic-plastic mechanical model is called, and its creep parameters are set according to Table 3 for the creep calculation. The creep analysis terminated after 15 years of surface subsidence were simulated. Additionally, to monitor the variation law of surface subsidence after the mining of the working face, six measuring points (A₁ to A₆) are arranged on the surface of the working face model along its mining direction. The horizontal distances between these measuring points and the center of the mining area are 50 m, 100 m, 150 m, 200 m, 250 m and 300 m, as shown in Fig. 11.

Fig. 11

Layout plan of surface subsidence monitoring lines and points.

Analysis of the simulation results

Spatiotemporal evolution patterns of surface subsidence

The time-dependent surface subsidence curves along survey line A after the extraction of longwall face 5–20109 are shown in Fig. 12. These curves exhibit an S-type subsidence profile above the coal seam goaf. Notably, surface subsidence reaches a maximum value (1200 mm) in the zone extending more than 75 m behind the longwall termination line. The subsidence value decreases rapidly from 1200 mm to 500 mm between 75 m behind the line and 25 m in front of the line. The subsidence value decreases to 0 within the range of more than 125 m in front of the line. Within four years after-mining, surface subsidence within 125 m of the goaf periphery, particularly within 100 m, persistently increases at a decelerating rate. Within four years after-mining, subsidence stabilized as the overburden reached dynamic equilibrium.

Historical surface subsidence at different monitoring points along line A is shown in Fig. 13. The results indicate that subsidence along line A follows as exponential decay pattern for the post-mining years. According to the simulation results and corresponding fitting curves, the variation in the subsidence along line A is negligible four years after-mining.

Fig. 12

Surface subsidence variation curves of monitoring points along survey line A.

Fig. 13

Subsidence history curves of different monitoring points along survey line A.

Spatiotemporal evolution patterns of surface horizontal displacement

The time-dependent curves of the surface horizontal displacement along line A after the extraction of longwall face 5–20109 are shown in Fig. 14, which reveals that the surface horizontal displacement above the goaf generally exhibits a double-trough distribution pattern. Notably, the first trough occurs above the goaf, whose maximum displacement, ranging from 150 to 310 mm, is 75 m behind the termination line. The second trough occurs on the outer side of the goaf, where the maximum displacement is 150 m in front of the termination line. From the maximum displacement point of the second tough extending 150 m in front of the goaf, the displacement value gradually decreases to 0 mm. Moreover, the horizontal displacement fluctuates significantly near the double troughs within four years after-mining, whereas the fluctuation amplitude is less than 1.0 mm within four years after-mining, indicating that the deformation of the overlayers has no effect on railway construction.

The time-dependent curves of the surface horizontal displacement along line A after the extraction of longwall face 5–20109 are shown in Fig. 14, which reveals that the horizontal displacement above the goaf exhibits a double-trough distribution pattern. Notably, the first trough occurs within the goaf area, where the maximum displacement (ranging from 150 to 310 mm) is located 75 m behind the termination line. The second trough develops outside the goaf boundary, with its peak displacement 150 m in front of the termination line. From this second peak which extends 150 m beyond the goaf, the displacement values gradually decrease to 0 mm. Moreover, significant displacement fluctuations occur near both troughs within four years after-mining, whereas the amplitudes decrease below 1.0 mm thereafter, indicating that overburden deformation has a negligible effect on railway construction.

The monitored horizontal displacement at points along Line A are shown in Fig. 15. The results indicate rapid displacement changes during the third year, a gradual decrease from years 3 to 4, and near-zero values after year 4 at positions less than 125 m from the goaf edge. Moreover, the horizontal displacement amplitudes are negligible beyond 125 m from the goaf boundary.

Fig. 14

Curves of the variation in horizontal displacement for monitoring points along survey line A.

Fig. 15

Horizontal displacement history curves of various monitoring points along survey line A.

Feasibility assessment for railway construction

The railway construction over the goaf needs to comply with the relevant provisions in the Code for Coal Pillar Retention and Underwater, Building, Railway and Main Shaft Coal Mining (hereinafter referred to as the Code)²¹ and the Maintenance Rules for Conventional Railway Lines (hereinafter referred to as the Rules)²². Given that the passenger and freight traffic volume of this coal mine railway is less than 10 MT but more than 5 MT, the proposed railway is a class III railway according to the classification of railway protection grades in the Code. For class III railways, the main parameters of track deformation after track laying are clearly specified in the rules and code. The critical parameters for track deformation, including track gauge, alignment, cross level, and longitudinal level, are shown in Fig. 16. The Code specifies allowable deformations of 5.74 mm for the track gauge, 8.61 mm for the cross level, and 40 mm for the alignment, whereas the Rules defines a track gauge tolerance of + 6/−2 mm, with uniform 4 mm limits for the cross level, longitudinal level, and alignment. On the basis of the numerical simulation results, the linear difference method is used to calculate the static geometric irregularity deviation values of the borehole locations around the railway line shown in Fig. 4, and the potential deformation of the railway track for different post-mining years is analyzed (see Fig. 17).

Fig. 16

Schematic of the main deformation parameters for railway tracks.

It should be pointed out that due to the lack of on-site monitoring data, the feasibility analysis results of railway construction are based solely on numerical simulation studies, which means that the evaluation of the construction waiting period can only be conducted on an annual basis. As shown in Fig. 17, surface subsidence evolves through three consecutive phases: an initial rapid settlement period (0–2 years), a transitional phase (2–3 years), and a long-term stable period (beyond 4 years). During the rapid settlement stage, all the parameters decrease significantly: the longitudinal level decreases by 39.5%, from 24.12 mm to 14.6 mm, and the alignment deviation decreases by 28.3%, from 17.64 mm to 12.64 mm. In the transitional phase, the subsidence rate subsequently decreases markedly. The longitudinal level deviation further decreases by 37.3%, from 14.6 mm to 9.16 mm, whereas the track gauge deviation decreases by 60.5%, from 3.8 mm to 1.5 mm. During the stable period, all the deviations approach zero with annual changes of less than 1 mm, indicating that the surface deformation has effectively stabilized. In addition, according to the data in Fig. 17, after five years of mining, the deviations in the gauge, level, height, and axial direction are 0.4 mm, 0.6 mm, 0.8 mm, and 0.64 mm respectively, which meet the requirements for track deformation specified in the specifications and regulations.

For practical engineering, the continuous and slow surface subsidence will not only affect the structural safety and operational stability of the railway but also force enterprises to set a long “waiting period” to wait for the stabilization of surface subsidence before starting railway construction, which directly conflicts with enterprise production and operational efficiency. To address this contradiction, targeted control measures for long-term surface subsidence must be formulated based on the specific geological conditions of the mining area—especially the occurrence conditions of coal seams (e.g., seam thickness, burial depth, dip angle, and lithology of the overlying strata). Combined with the characteristics of thick coal seams and shallow burial depth in the mine, two technical schemes are recommended: (1) By filling the goaf with backfill materials, the collapse and deformation of the overlying strata can be inhibited at the source, thereby effectively controlling the transmission of mining-induced deformation to the surface and reducing the magnitude and duration of long-term surface subsidence; (2) By controlling the single-mining thickness of the coal seam, the degree of disturbance to the overlying strata is reduced, avoiding large-scale and long-term surface subsidence caused by excessive mining intensity. Whereas the implementation of the final plan requires establishing a quantitative indicator system across three dimensions—technical effectiveness, economic viability, and engineering feasibility—and applying a weighted scoring method for comprehensive evaluation.

Fig. 17

Static geometric irregularity deviation values of railway tracks constructed for different post-mining years.

Conclusions

In this study, a nonlinear viscoelastoplastic creep model was established, and corresponding creep functions were derived on the basis of the time-dependent deformation characteristics of roof strata. The novel creep model was used for simulation through a user-defined model in FLAC^3D. A method for inverse analysis of the creep parameters of the rock in goaf areas is proposed, combining creep simulation with genetic algorithms. A specific engineering project is selected as a background, and the temporal–spatial evolution law of surface movement and deformation in the goaf area is analyzed. The feasibility recommendations for constructing railways around goaf areas are provided in accordance with relevant railway construction specifications. The main conclusion are as follows:

(1) A novel nonlinear viscoelastoplastic creep model that can characterize deformation features across all stages of rock creep curves was developed. The 1D and 3D creep equations under constant stress were derived. A critical parameter sensitivity analysis indicates that with increasing creep exponent a, the nonlinear characteristics of the rock accelerated creep stage become more distinct, whereas the time required to reach this stage is significantly reduced. These findings validate the ability of the model to assess the complex nonlinear behavior of rocks during the accelerated creep stage.

(2) The novel viscoelastoplastic creep model was embedded into FLAC^3D via a user-defined module. A method for the inverse analysis of creep parameters in goaf areas was proposed, combining creep simulation with genetic algorithms. In a case study conducted at a specific coal mine, the creep parameters of the goaf rock mass were successfully calibrated. The simulation results strongly correlated with the in-situ surface subsidence monitoring data, confirming the applicability of both the novel creep model and the inversion method.

(3) The time-dependent surface displacement was analyzed, and the results indicate that subsidence at monitoring points exhibits an exponential decay pattern in the post-mining years, whereas horizontal displacements above the goaf exhibit a double-trough distribution. Surface subsidence progresses through three postmining phases: an initial rapid settlement period (0–2 years), a transitional phase (2–3 years), and a long-term stable period (beyond 4 years). According to the relevant railway construction specifications, the deviations in the gauge, level, height, and axial direction are all meet the requirements for track deformation after five years mining. To minimize the waiting period prior to railway construction, it is advisable to implement filling mining or limited-thickness mining methods.