Rock Mechanics and Rock Engineering

, Volume 52, Issue 1, pp 133–147 | Cite as

Numerical Modelling of Water Flow Through Granular Material for Isolated and Simultaneous Extractions in Block Caving

  • Katherine Sánchez
  • Sergio PalmaEmail author
  • Raúl L. Castro
Original Paper


In this work, we numerically solve the Brinkman–Darcy equation coupled to the granular kinematic model using the finite elements method in 2D, to describe the entry of water into draw points in Block Caving mining. We perform a total of 990 numerical simulations incorporating the relative change of local rock density (\(\Delta \rho /{\rho _p}\)), particle size (\({D_p}\)), extraction area (\(S\)) and the separation between draw points (\(L\)). We propose two mathematical models using scale arguments for estimating the velocity of the water in the draw point as a function of two and three dimensionless numbers (for isolated and simultaneous extractions, respectively). The relative error in the estimation of the results using the mathematical model for the set of numerical experiments ranges from 0.83 to 6.09%, where the greatest deviations correspond to \({D_p}\) = 6 mm. The proposed models allow estimating the water velocity at the draw point, which in turn helps to predict the time and place where there is a greater probability of a mud rush occurrence. The results can be applied in the design and optimisation of extraction sequences when the water present in the subsoil of a mine is a relevant factor to consider.


Block caving Mud rush Porous medium Brinkman–Darcy equation Granular materials Kinematic model 


\(\vec {u}\)

Velocity field of the fluid (m/s)


Hydraulic conductivity, (m2)


Gradient of the hydraulic head (m−1)


Porosity of the medium


Fluid density (kg/m3)


Intrinsic permeability of the medium, (m/s)


Dynamic fluid viscosity (Pa s)


Gravitational constant (m/s2)


Size grain (m)


Reynolds number

\(\upsilon ~\)

Kinematic viscosity (m2/s)


Stress tensor (MPa)

\(\nabla P\)

Gradient of the fluid pressure (MPa/m)


Horizontal velocity of particles (m/s)


Vertical velocity of particles (m/s)


Constant along streamlines


Pi number


Euler’s number

\({\rho _p}\)

Density of the medium (particles) inside the IMZ (kg/m3)

\(\Delta \rho\)

Local density change introduced by the rock motion (kg/m3)

\({\rho _0}\)

Initial density or outside of the IMZ (kg/m3)


Extraction rate (e.g. in 2D m2/day or in 3D m3/day)


Height of the isolated extraction zone (m)


Width of the isolated extraction zone (m)

\(\overrightarrow {{v_F}}\)

Velocity of the dilation front (m/s)

\(\overrightarrow {{v_p}}\)

Particles velocity (m/s)


Displacement of the dilation front velocity (m)


Height isolated movement zone (m)


Width isolated movement zone (m)


Convective velocity vector (m/s)


Diffusion coefficient (m2/s)


Arbitrary source term


Artificial diffusion coefficient

\({\delta _{{\text{id}}}}\)

Tuning parameter of the artificial diffusion


Mesh element size (m)



This study was financially supported by the Grant “PiensaCobre” under the auspices of the Corporación Nacional del Cobre de Chile (CODELCO) and the Centre for Mathematical Modelling (CMM) of the University of Chile. The authors are greatly grateful to the Block Caving Laboratory and Advanced Mining Technology Center (AMTC) of the University of Chile. The authors thank H. Rivera and J.P. Le Roux and Xavier Emery for providing much-appreciated comments, which helped to improve the clarity of this manuscript.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Advanced Mining Technology CenterUniversity of ChileSantiagoChile
  2. 2.Block Caving Laboratory, Department of Mining EngineeringUniversity of ChileSantiagoChile
  3. 3.School of Civil EngineeringThe University of QueenslandBrisbaneAustralia
  4. 4.Aix-Marseille Université, CNRS, IUSTIMarseilleFrance

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