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Time-Dependent Stability Analyses of Side-Exposed Backfill Considering Creep of Surrounding Rock Mass

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Abstract

The stability of side-exposed backfill is essential to ensure a successful mining operation. Until now, it has been analyzed without considering the creep of rock mass. In practice, stope excavation and backfilling are always scheduled with different time during which fill mechanical properties can evolve and rocks exhibit more or less creep deformation. In this study, time-dependent stability and minimum required cohesion (cmin) of side-exposed backfill associated with the creep of surrounding rock mass are, for the first time, analyzed through numerical modeling with FLAC3D. A distinction is made between the cohesion at failure and cmin. Results show that the empty time of primary stope does not significantly affect the stability and cmin. When mine depth is small and the rock exhibits little creep, it deserves to wait longer time before adjacent extraction for the backfill to gain more strength. When the mine depth is large or/and the rock exhibits heavy creep, the instability of side-exposed backfill can be dictated by crushing failure. A stronger backfill means also a harder backfill, which absorbs larger compressive stress and is more prone to be crushed. In this condition, a softer backfill can be better through the use of lower binder content or/and with a shorter curing time. The adjacent secondary stope should be filled as soon as possible to avoid failure of side-exposed backfill. More simulations were done on the effects of stope geometry and mechanical properties of backfill and rock mass on the stability and cmin of side-exposed backfill.

Highlights

  • The empty time of primary stope does not significantly affect the stability of side-exposed backfill.

  • When the mine depth is small and the rock exhibits little creep, it deserves to wait longer time before adjacent extraction for the backfill to gain more strength.

  • When the mine depth is large and the rock exhibits heavy creep, a softer backfill can be better through the use of lower binder content or with a shorter curing time to prevent crushing failure.

  • The extracted secondary stope should be filled as soon as possible to avoid failure of side-exposed backfill.

  • Optimization of sizes for the primary and secondary stopes is needed to minimize the overall backfilling cost.

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Abbreviations

B :

Stope width (m)

B D :

Width of the numerical model (m)

c :

Backfill cohesion (kPa)

c min :

Minimum required cohesion (kPa)

c R :

Rock mass cohesion (MPa)

D :

Mine depth (m)

E :

Backfill Young’s modulus (MPa)

E K :

Young’s modulus of the Kelvin–Voigt element (Pa or GPa)

E M :

Young’s modulus of the Maxwell element (Pa or GPa)

E R :

Rock mass Young’s modulus (Pa or GPa)

h :

Height (m)

H :

Stope height (m)

H D :

Height of the numerical model (m)

G :

Backfill shear modulus (MPa)

G K :

Shear modulus of the Kelvin–Voigt element (Pa or GPa)

G M :

Shear modulus of the Maxwell element (Pa or GPa)

k n :

Normal stiffness of fill–rock interface (Pa/m)

k s :

Shear stiffness of fill–rock interface (Pa/m)

K :

Backfill bulk modulus (MPa)

K r :

Lateral earth pressure coefficient

K 0 :

At-rest earth pressure coefficient

K R :

Rock mass bulk modulus (Pa or GPa)

lfob :

Lower limit of the unbalanced force ratio

L b :

Strike length of the primary stope (m)

L s :

Strike length of the secondary stope (m)

L D :

Length of the numerical model (m)

t :

Time (s, h, or day)

t c :

Curing time (day)

t d :

Delayed time before filling the primary stope (day)

t e :

Exposure time before filling the secondary stope (day)

t f :

Time period of 28 days for backfill parameters evolution (day)

t i :

Initial timestep (s)

t m :

Maximum timestep (s)

T :

Backfill tensile strength (kPa)

T R :

Rock mass tensile strength (kPa)

ufob :

Upper limit of the unbalanced force ratio

V :

Values of E, c and T

V f :

Final values of E, c and T after a curing time of tf

σ yy :

Horizontal stress (kPa)

σ zz :

Vertical stress (kPa)

ϕ :

Backfill internal friction angle (°)

ϕ R :

Rock mass internal friction angle (°)

γ :

Backfill unit weight (kN/m3)

γ R :

Rock mass unit weight (kN/m3)

ν :

Backfill Poisson’s ratio

ν R :

Rock mass Poisson’s ratio

ψ :

Backfill dilation angle (°)

ψ R :

Rock mass dilation angle (°)

σ m :

Spherical stress tensor

ε ij :

Strain tensor

S ij :

Deviatoric stress tensor

η K :

Viscosity coefficient of the Kelvin–Voigt element (Pa s)

η M :

Viscosity coefficient of the Maxwell element (Pa s)

Δz min :

Smallest size of backfill element (m)

References

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Acknowledgements

The authors acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC 402318), China Scholarship Council (201706420059), and industrial partners of the Research Institute on Mines and the Environment (RIME UQAT-Polytechnique; http://rime-irme.ca/). The authors thank the anonymous reviewers for their constructive comments that are helpful to improve the quality of the paper.

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Authors and Affiliations

  1. Department of Civil, Geological and Mining Engineering, Research Institute on Mines and Environment (RIME UQAT-Polytechnique), École Polytechnique de Montréal, C.P. 6079 succursale Centre-ville, Montréal, QC, H3C 3A7, Canada

    Ruofan Wang & Li Li

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  1. Ruofan Wang
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  2. Li Li
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Correspondence to Ruofan Wang.

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Appendices

Appendix A: The CVISC Model in FLAC3D and its Applicability Against Experimental Results

Figure 

Fig. 20
figure 20

Schematics of the a components, b strain–time curve under a stress state below MC criterion and c strain–time curve under a stress state above MC criterion of the CVISC model

Full size image

20 shows the components (Fig. 20a) and the strain–time curve (Fig. 20b, c) of the CVISC model. The constitutive equations of the Burgers models (a part of the CVISC model for creep behavior) are defined as (Jaeger 1969):

$${\varepsilon }_{ij}=\frac{{\sigma }_{m}}{3{K}_{\mathrm{R}}}+\frac{{S}_{ij}}{{2G}_{\mathrm{M}}}+\frac{{S}_{ij}\bullet t}{2{\eta }_{\mathrm{M}}}+\frac{{S}_{ij}}{{2G}_{\mathrm{K}}}\bullet \left[1-\mathrm{exp}\left(-\frac{{G}_{\mathrm{K}}\bullet t}{{\eta }_{\mathrm{K}}}\right)\right],$$
(4)

where εij is the strain tensor; Sij is the deviatoric stress tensor; σm is the spherical stress tensor.

When a rock is submitted to a stress state below MC yield criterion, the mechanical behavior is governed by the Burgers model as shown in Fig. 20b. The instantaneous strain is captured by the Maxwell’s spring element while the primary creep stage is captured by the Kelvin–Voigt element. The linear viscoelastic strain with time at a constant strain rate for the secondary creep stage of the rock mass is described by the Maxwell’s dashpot element. The tertiary creep stage cannot be described by the CVISC model. However, instantaneous plastic strain following the MC elasto-plastic model occurs if the stress state exceeds the MC yield criterion as shown in Fig. 20c. Therefore, the CVISC model is able to capture the creep behavior of rocks in the primary and secondary stages, but it cannot reflect the delayed failure and creep strain in the tertiary creep stage. Table

Table 4 Viscoelastic parameters for some typical rocks reported in the literature
Full size table

4 presents viscoelastic parameters for modeling some typical rocks with the CVISC model reported in the literature. In the table, EM and EK are Young’s modulus of elastic springs in Kelvin–Voigt and Maxwell elements of the CVISC model that can be used in one-dimensional condition of Eq. (4).

The applicability and capability of the CVISC model in FLAC3D can be verified by comparing with experimental results. Mansouri and Ajalloeian (2018) conducted conventional uniaxial compression creep tests on the rock salt specimens that have a diameter of 58 mm and a height of 116 mm. The tests were performed at room temperature using a hydraulic press to maintain the constant load while the axial creep strain was measured with dial-gauges for 14 days. Figure 

Fig. 21
figure 21

a The physical model of conventional uniaxial compression creep tests conducted by Mansouri and Ajalloeian (2018); b the numerical model built with FLAC3D

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21a shows the physical model of Mansouri and Ajalloeian (2018) creep tests. The numerical model of the creep tests is built with FLAC3D as illustrated in Fig. 21b. Vertical displacements are not allowed at the bottom of the numerical model while a uniform compressive stress is applied on the top to represent the constant load. The CVISC model is applied for the numerical model and the axial strain is calculated based on the average vertical displacement at the top recorded during the creep calculation.

Figure 

Fig. 22
figure 22

Comparisons between the experimental results of Mansouri and Ajalloeian (2018) with the analytical and numerical results of the CVISC model in FLAC3D

Full size image

22 illustrates the creep strain–time curve of Mansouri and Ajalloeian (2018) test under an applied stress of 15 MPa. The viscoelastic parameters of the CVISC model were first obtained through calibration by comparing the analytical solution (Eq. (4)) with experimental results. The calibrated viscoelastic parameters are KR, GK, ηK, GM, and ηM, as shown in Table

Table 5 Viscoelastic parameters for rock salt calibrated by comparing Eq. (4) with experimental results of Mansouri and Ajalloeian (2018)
Full size table

5. The parameters of cR = 8.5 MPa, ϕR = 35°, ψR = 0°, TR = 1.5 MPa were assumed based on the reported UCS which do not affect the numerical results of creep strain–time curve. These parameters along with other parameters of ti = 30 s, tm = 1000 s, lfob = 1 × 10−5, ufob = 3 × 10−5, and 100 steps for latency based on sensitivity analyses were then introduced in the numerical model. Comparisons between the experimental results of Mansouri and Ajalloeian (2018) with the analytical and numerical results of the CVISC model are shown in Fig. 22. The good agreements verify the applicability and capability of the CVISC model in FLAC3D.

Appendix B: Verification of the Applicability of the Numerical Model to Describing the Time-Dependent Closure of Underground Openings

The time-dependent closure of stope shown in Fig. 6 is a typical closure profile of underground openings (Malan 1999; Barla et al. 2010; Qi and Fourie 2019). The applicability of the numerical model (FLAC3D along with the CVISC creep model) can be further validated by comparing with the in-situ measurement of creep deformation around a tunnel. The Frejus tunnel between France and Italy is 12.78 km long and has a typical cross section as shown in Fig. 

Fig. 23
figure 23

adopted from Sulem et al. 1987)

a Cross-section and b measured time-dependent closure along line M′N′ in a section of the Frejus tunnel during the interruption of face advance (

Full size image

23a (Sulem et al. 1987). It was excavated in schist under a depth of 600–1200 m. In the construction of Frejus tunnel, the face advance was stopped for 22 days during which the walls’ time-dependent convergence along line M′N′ was monitored at a section that is 29 m behind the face. The measured results were reported by Sulem et al. (1987) and is shown in Fig. 23b. Due to that the face advance is interrupted, the closure shown in the Fig. 23b is only attributed to the creep behavior of rocks. The measured results were reproduced using FLAC3D to verify the applicability of the numerical model. Since the actual stress conditions and rock properties are not accurately known, some parameters were assumed and the numerical reproduction was performed by calibrating the viscoelastic parameters.

Figure 

Fig. 24
figure 24

Plane strain numerical model of the Frejus tunnel built with FLAC3D

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24 shows the plane strain numerical model of Frejus tunnel built with FLAC3D. The numerical model has a height of 411 m and a length of 414 m. Gravity is along the negative direction of the y-axis. The average depth of 900 m is considered and the lateral earth pressure coefficient Kr is applied as 1. The CVISC model is applied for the rock mass. The assumed parameters for rock mass include γR = 27 kN/m3, ER = 10 GPa, νR = 0.25, cR = 2 MPa, ϕR = 40°, ψR = 0°, TR = 150 kPa. One should note that elastic modulus and strength parameters mainly affect the instantaneous deformation which is not the objective here. Creep modeling parameters involve tm = 1000 s, ti = 30 s, latency of 50 steps, lfob = 1 × 10−6, and ufob = 3 × 10−6. In the numerical model, displacement along the third direction (z-axis) is restricted. Lateral displacements are prohibited along the left and right boundaries while for the bottom, all displacements are restricted. After excavating the tunnel, the model was solved to the equilibrium state and the instantaneous deformation was reset to 0. The time-dependent closure along line M′N′ was then calculated for 22 days. Based on the calibration, the applied viscoelastic parameters are GK = 5.62 GPa, ηK = 1.07 × 1015 Pa s, ηM = 9.75 × 1015 Pa s.

Figure 

Fig. 25
figure 25

adopted from Sulem et al. 1987) and numerical results of walls time-dependent closure in the Frejus tunnel

Comparisons between the in-situ measurements (

Full size image

25 shows the comparisons between the numerical results and in-situ measurements of walls time-dependent closure in the Frejus tunnel. A good agreement is obtained between the in-situ measurement and numerical results. The validity and applicability of FLAC3D along with the CVISC creep model are verified.

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Wang, R., Li, L. Time-Dependent Stability Analyses of Side-Exposed Backfill Considering Creep of Surrounding Rock Mass. Rock Mech Rock Eng 55, 2255–2279 (2022). https://doi.org/10.1007/s00603-022-02776-0

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