In this paper, a multiscale modelling and simulation of destress blasting of rock mass is presented. The proposed and novel approach combines two separate 3D solutions: the first was obtained for the smallscale problem, face(s) blasting process, and the second for the global scale problem, seismic wave propagation within very large volumes of surrounding rock mass. Both the approaches were based on explicit dynamic modelling methodology using the central difference method. In the local case, the arbitrary Lagrangian–Eulerian (ALE) procedure with the Jones–Wilkins–Lee (JWL) equation defining an explosive material was used. For this purpose, a selected volume of a rock mass comprising a blasted mining face was modelled in detail. From the numerical simulation, pressure distribution over the modelled rock was obtained, which was the basis for an initial condition for the global 3D FE model. The peak particle velocity (ppv) distribution obtained from finite element analysis was compared with experimental outcomes. A reasonable agreement between both results was achieved; therefore, the adopted multiscale modelling approach confirmed its effectiveness and that it can be successfully implemented in further engineering practice.
The flattype copper ore body in Polish deep underground mines is basically excavated with drillandblast technology which seems to be relatively well suited to the hardness of the local rocks and to local mining/geological conditions. Besides the direct production potential, blasting works are also recognized as one of the most important active methods of seismic events prevention. This is achieved through strainrelease massive simultaneous blasting engaging ten to forty production faces near potentially unstable main roof strata and/or pillars. The strainrelease effect is in proportion to the rock mass ability for strain energy accumulation. A significant number of recorded seismic events may be directly explained by the blasting works’ inducing effects. However, at present time, these operations are conducted intuitively based on previous experience as well as on a trialanderror approach rather than upon an intentional and scientifically justified approach. In result, they are not as effective as they could be.
In the increasingly difficult geological and mining conditions in which extraction of copper ore deposits in Poland is conducted, ensuring an effective and safe mining becomes a key task and a significant challenge for mine operators. Recently, new geomechanical hazards have been identified under these conditions, particularly those related to induced seismicity, which are mostly attributable to high stress values, lower deformability, and higher strength of rocks surrounding a deep deposit. Therefore, more attention should be paid to the dual role played by multiplefaced blasting operations since it is a technology which provides sustainable high production and it is applied for rock burst hazards prevention.
The overall goal of the research is to develop and implement firing patterns for the multifaced blasting production process which may generate the effect of wave amplification due to the interference of postblasting seismic waves [
Numerical modelling of rock masses subjected to blast loading has long history. Back in 1986, Taylor and his coworkers [
Within the area of numerical methods of mechanics, problems that arise from multiscale character of investigated phenomena are usually addressed by the socalled “globallocal” approach. Traditionally, it is done by local refinement of the model in critical areas [
To override multiscaleinduced problems, the authors present a novel numerical submodelling approach which shall provide a better understanding of the blasting process for practitioners. This approach includes modelling from the face firing to the induced seismic wave propagation in remote areas of surrounding rock mass. While the overall idea of a hybrid, explicitexplicit or explicitimplicit, approach is not new [
The pressure distribution over the SRH, obtained from the smallscale model simulation, was used as an initial condition for the global 3D finite element (FE) model. Subsequently, the peak particle velocity distribution obtained from finite element analysis (FEA) using the global model was compared with experimental outcomes. A reasonable agreement between the results was observed; therefore, the adopted multiscale modelling method confirmed its effectiveness and that it can be successfully implemented in similar problems.
Two different numerical models were used to conduct FEA of rock mass located around the given roomandpillar panel in one of the KGHM mines:
Case no. 1: smallscale (local) problem—timedependent blasting process developing in the SRH fired at the chosen mining face of rock wall.
Case no. 2: global scale problem—seismic wave propagation within very large volume of surrounding rock mass, following the SRH blasting.
Generally, the proposed coupling procedure is based on the following consecutive phases:
The history of the detonation pressure within each side of the SRH was determined. Since the utilizing time step size should cover an extremely short period of time, the model cannot be too large and therefore also cannot represent the rock mass located in more remote locations. However, after detonation, the ground velocity can rapidly drop down and it may stabilize on the level of 5000.0–6000.0 m/s at a distance of few meters from the fired SRH. Thus, one may use the results of the smallscale problem solutions as the initial conditions for the globalscale problem.
Taking into account that the main roof strata in Polish copper mines are mostly composed of competent sediment rocks (dolomite/anhydrite/sandstone), it is justified to provide them with equations of state (EOS) adequate for the elastic continuous body. Afterwards, using any standard 3D FEM dynamic code, one may model very large volumes of rock mass surrounding the considered blasted face using small time step size (e.g., Δ
Numerical studies of local and global blasting process were carried out using the LSDyna commercial code [
The Arbitrary Lagrangian–Eulerian formulation (ALE) algorithm is adopted based on the previous paper [
The coupling between fluidlike and solid domains was one of the most challenging tasks during the modelling process. Unfortunately, parameters responsible for the coupling are not universal and have to be estimated for a specific problem [
Coupling penaltybased method scheme [
Generation of blast wave was modelled using the JWL equation of state. The equation describes behaviour of the detonation products has the following form [
All required parameters of an emulsion high explosive (HE) used in mining industry were defined based on the results of the socalled cylindrical test (Table
Material properties for HE with EOS [
Constant  Value  Unit 

Initial density, 
1130.0  kg/m^{3} 
Detonation velocity  4805.0  m/s 
Chapman–Jouget pressure  7400.0  MPa 

252000.0  MPa 

15570.0  MPa 

6.08  — 

2.05  — 

0.25  — 
Detonation energy per unit volume, 
3700.0  MPa 
In the local approach, the rock mass was considered as isotropic, with the properties of one type of rock, e.g., dolomite. From the available material models in LSDyna, the JohnsonHolmquist II (JH2) was used for predicting behaviour of the dolomite [
The normalized intact strength of the JH2 is given by
The accumulated damage resulted from the fracture of material is given by
In Table
Material properties for JH2 material for dolomite.
Constant  Value  Unit 

Density, 
2840.0  kg/m^{3} 
Young’s modulus, 
77320  MPa 
Bulk modulus, 
50287  MPa 
Shear modulus, 
31084  MPa 
Poisson’s ratio, 
0.24  — 
Hugoniot elastic limit, HEL  2500  MPa 
HEL pressure, 
1440  MPa 
Maximum tensile strength, 
30.23  MPa 
Intact strength coefficient, 
0.73  — 
Fractured strength coefficient, 
0.56  — 
Strain rate coefficient, 
0.026  — 
Intact strength exponent, 
0.57  — 
Fractured strength exponent, 
0.57  — 
Bulk factor, 
1  — 
Damage coefficient, 
0.001  — 
Damage coefficient, 
1.17  — 
Pressure coefficient 2, 
−78000  MPa 
Pressure coefficient 3, 
9400000  MPa 
Maximum normalized fracture strength, 
0  — 
For the numerical model development in the local scale, a selected mining face of rock mass was modelled in detail. The cubicallike geometry of the rock mass had the following dimensions: width
Model of the mining face (submodel) with initialboundary conditions.
Based on the developed model, pressure histories were measured at the mining face sides, representing the finite elements (segments) of the mining face in the global model. Therefore, the pressure histories from the local approach were the initial loading condition in the global modelling.
The proposed approach has been studied based on one of the sections in the Lubin mine where copper ore is exploited using the roomandpillar technology. The copper ore deposit in the analysed area is located at the depth of 742.0 m below the ground and is horizontally flat. It is covered immediately by very thick and stiff main roof strata consisting of 20.0 m layer of dolomite overlaid by 137.0 m thick strong anhydrite plate and 233.0 m thick competent sandstone stratum (Figure
Geological layers in the considered area.
Geological data of the considered area.
Type of rock  Thickness (m)  Compressive strength (MPa)  Tensile strength (MPa)  Shear strength (MPa)  Young’s modulus (MPa)  Poisson’s ratio 

Glacial deposits  336.0  —  —  —  75.0  0.30 
Motley sandstone  221.0  11.51  0.13  2.07  5000.0  0.15 
Clayey sandstone  12.0  1.73  0.002  0.21  3375.0  0.18 
Main anhydrite  137.0  19.53  0.24  3.12  1365.0  0.25 
Dolomite II  11.0  63.96  0.89  7.64  14575.0  0.26 
Dolomite I  9.0  149.60  2.15  18.44  24050.0  0.25 
Copper ore  3.0  116.4  6.90  19.80  8200.0  0.23 
Grey sandstone  10.0  25.10  1.10  4.80  4700.0  0.16 
Quartz sandstone  290.0  17.90  0.90  3.40  2550.0  0.13 
Except strength and deformation data, the rock mass must be characterized by dynamic parameters between which the structural damping coefficient and maximum frequency value within the ground response range are the most important.
As Preece [
As a global model for the problem, the multiplate overburden model was accepted with the following simplifying assumptions:
Overburden strata consist of several homogeneous rock plates reflecting the real lithology in the area
Technological and remnant pillars work effectively within postcritical phase (elasticplastic with strain softening behaviour)
The value of carried loads depends on pillar size and actual extraction ratio
The FE model was developed based on the geometry of the selected mining section in the Lubin mine (Figure
Selected section in Lubin mine considered in global FEA.
Model of blast in global scale with initialboundary conditions.
From the local modelling simulation, the damage of the rock in the mining face was obtained (Figure
Damage index over rock area after the blast.
Pressure distribution over rock area for selected time points (MPa). (a)
Pressure histories transfer from the segments of the mining face modelled in the local and global approach.
Pressure histories corresponding to the segments of the mining face modelled in the local approach.
The main objective of the paper was to compare the results obtained from numerical simulations of rock mass dynamic response of groupfaces blasting with the actual underground measurements from the seismic station. Example of obtained results in the form of excited vertical velocities distribution is presented in Figure
Vertical velocity propagation wave (mm/s) after 0.2 sec within the dolomite I layer.
It is worth noticing that the precision of the nonelectrical detonators used is not high enough since the monitored multiface blasting effects are visibly divided into five isolated period high rock vibrations, which lasted for more than four seconds in total. This kind of blasting cannot be considered as an instant and simultaneous process as one theoretically could assume. Taking this into account, instead of similarity in the velocity distribution, the authors have focused on similarity among ppv. In Figure
Measured and numerically assessed effects of multiplemining faces blasting.
The paper confirms the general ability of computer modelling for rock motion assessment due to dynamic phenomena such as multiple blasting of production faces. As shown above, a reasonable coupling of two different solutions based on a small and global interpretation of the multiface production blasting may produce very encouraging results which may be instantly applied in the prevention of the occurrence of seismic events in underground mines. However, the numerical model parameters (degree of damping, dominant frequency of the motion process, etc.) must be chosen wisely since some incorrect values may give unreasonable output data. The nonelectrical detonators, which somehow generate unpredictable delay times, are also very hard to introduce in the model. Due to the above shortcomings, in the authors’ opinion, a quantitative analysis of the results would be unjustified. On the other hand, the FEA results match well with the measured field data as far as the ppv is considered (Table
Comparison of ppv values obtained from FEA and experimental test.
Test  ppv value  Unit 

Experiment  0.515  mm/s 
FEA  0.516  mm/s 
Error  0.2  % 
Since ppv is widely used in the mining industry as one of the parameters to predict damage caused by blasting [
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This paper has been prepared through the Horizon 2020 EU funded project on “Sustainable Intelligent Mining Systems (SIMS)” (Grant Agreement No. 730302). The research was also supported by the Interdisciplinary Centre for Mathematical and Computational Modeling (ICM), University of Warsaw (Grant Agreement No. GB7319).