Rock Mechanics and Rock Engineering

, Volume 52, Issue 1, pp 133–147

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

• Katherine Sánchez
• Sergio Palma
• Raúl L. Castro
Original Paper

## Abstract

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.

## Keywords

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

## Abbreviations

$$\vec {u}$$

Velocity field of the fluid (m/s)

$$K$$

Hydraulic conductivity, (m2)

$$\nabla$$

$$\phi$$

Porosity of the medium

$$\rho$$

Fluid density (kg/m3)

$$k$$

Intrinsic permeability of the medium, (m/s)

$$\mu$$

Dynamic fluid viscosity (Pa s)

$$g$$

Gravitational constant (m/s2)

$${D_p}$$

Size grain (m)

$$Re$$

Reynolds number

$$\upsilon ~$$

Kinematic viscosity (m2/s)

$$\tau$$

Stress tensor (MPa)

$$\nabla P$$

Gradient of the fluid pressure (MPa/m)

$${u_{xp}}$$

Horizontal velocity of particles (m/s)

$${u_{yp}}$$

Vertical velocity of particles (m/s)

C

Constant along streamlines

$$\pi$$

Pi number

e

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)

Q

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

$${H_{{\text{IEZ}}}}$$

Height of the isolated extraction zone (m)

$${W_{{\text{IEZ}}}}$$

Width of the isolated extraction zone (m)

$$\overrightarrow {{v_F}}$$

Velocity of the dilation front (m/s)

$$\overrightarrow {{v_p}}$$

Particles velocity (m/s)

$${\text{d}}{y_f}~$$

Displacement of the dilation front velocity (m)

HIMZ

Height isolated movement zone (m)

WIMZ

Width isolated movement zone (m)

$$\beta$$

Convective velocity vector (m/s)

D

Diffusion coefficient (m2/s)

F

Arbitrary source term

$${c_{{\text{art}}}}$$

Artificial diffusion coefficient

$${\delta _{{\text{id}}}}$$

Tuning parameter of the artificial diffusion

$$h$$

Mesh element size (m)

## Notes

### Acknowledgements

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.

## References

1. Alabi OO (2011) Validity of Darcy’s law in laminar regime. Electron J Geotech Eng 16:27–40Google Scholar
2. Arora KR (2009) Soil mechanics and foundation engineering (geotechnical 7th engineering) edition. Standard Publishers Distributors, DelhiGoogle Scholar
3. Bard Y (1974) Nonlinear parameter estimation, vol 1209. Academic Press, New YorkGoogle Scholar
4. Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics, vol 14. Cambridge University Press, Cambridge
5. Bear J (1988) Dynamics of fluids in porous media. Dover Publications Inc, New YorkGoogle Scholar
6. Bear J, Bachmat Y (1990) Introduction to modeling phenomena of transport in porous media. Kluwer, Dordrecht
7. Bear J, Cheng AHD (2010) Modelling groundwater flow and contaminant transport, vol 23. Springer, Amsterdam
8. Bergmark JE (1975) The calculation of drift spacing and ring burden for sublevel caving. LKAB memo # RU 76-16Google Scholar
9. Brinkman HC (1947) A calculation of the viscosity and the sedimentation constant for solutions of large chain molecule staking into account the hampered flow of the solvent through these molecules. Physica 13:447–448
10. Buckingham E (1914) On physically similar systems: illustrations of the use of dimensional equations. Phys Rev 4:345–376
11. Butcher R, Joughin W, Stacey TR (2000) Methods of combating mudrushes on diamond and base metal mines. Safety in Mines Research Advisory Committee (SIMRAC), JohannesburgGoogle Scholar
12. Butcher R, Stacey TR, Joughin WC (2005) Mud rushes and methods of combating them. J S Afr Inst Min Metall 105:817–824Google Scholar
13. Call and Nicholas(1998) IOZ wet muck study. PT Freeport internal report (Unpublished) Google Scholar
14. Caram H, Hong DC (1991) Random-walk approach to granular flows. Phys Rev Lett 67:828–831
15. Castro R, Trueman R, Halim A (2007) A study of isolated draw zones in block caving mines by means of a large 3D physical model. Int J Rock Mech Min Sci 44:860–870
16. Castro R, Basaure K, Palma S, Vallejos J (2017) Geotechnical characterization of ore related to mudrushes in block caving mining. J S Afr Inst Min Metall 117:275–284
17. Chen G (1997) Stochastic modeling of rock fragment flow under gravity. Int J Rock Mech Min Sci 34:323–331
18. Dullien FA (1992) Porous media: fluid transport and pore structure, 2nd edn. Academic Press, San DiegoGoogle Scholar
19. Durlofsky L, Brady JF (1987) Analysis of the Brinkman equation as a model for flow in porous media. Phys Fluids 30:3329–3341
20. Faghri A, Zhang Y (2006) Transport phenomena in multiphase systems, 1st edn. Academic Press, San DiegoGoogle Scholar
21. Fitts CR (2012) Groundwater science, 2nd edn. Academic Press, San DiegoGoogle Scholar
22. Garcés D, Castro R, Valencia ME, Armijo F (2016) Assessment of early mud entry risk for long term cave mining applications. 1st International Congress on Underground Mining U-Mining, Santiago, pp 428–439Google Scholar
23. Gavin H (2011) The Levenberg–Marquardt method for nonlinear least squares curve-fitting problems. Course lectures:experimental systems. Department of Civil and Environmental Engineering, Duke University, Durham. http://people.duke.edu/~hpgavin/ce281/lm.pdf Accessed 31 May 2017
24. Ghidaoui MS, Kolyshkin AA (1999) Some global properties of flow in a heterogeneous isotropic porous medium. Mech Res Commun 26:161–166
25. Gibbings JC (2011) Dimensional analysis. Springer, London
26. Hancock W, Weatherley D, Chitombo G (2012) Modeling the gravity flow of rock using the discrete element method. In: Proceedings of the sixth international conference and exhibition on mass mining, Canadian Institute of Mining, Metallurgy and Petroleum, 6972, OntarioGoogle Scholar
27. Hekmat A, Castro R, Navia I, Sánchez LK, Palma S (2016) Mud inflow risk assessment in block caving operation based on AHP comprehensive method. In: Proceedings of risk and resilience mining solution, VancouverGoogle Scholar
28. Holder A, Rogers AJ, Bartlett PJ, Keyter GJ (2013) Review of mud rush mitigation on Kimberley’s old scraper drift block caves. J S Afr Inst Min Metall 113:529–537Google Scholar
29. Islam MF, Lye LM (2009) Combined use of dimensional analysis and modern experimental design methodologies in hydrodynamics experiments. Ocean Eng 36:237–247
30. Jakubec J, Clayton R, Guest A (2012) Mud rush risk evaluation. In: Proceedings of the sixth international conference and exhibition on mass mining, Canadian Institute of Mining, Metallurgy and Petroleum, 6860, OntarioGoogle Scholar
31. Janelid I, Kvapil R (1966) Sublevel caving. Int J Rock Mech Min Sci Geomech Abstr 3:129–132
32. Kasenow M (2002) Determination of hydraulic conductivity from grain size analysis. Water Resources Publication, LLC, DenverGoogle Scholar
33. Kuchta ME (2002) A revised form of the Bergmark-Roos equation for describing the gravity flow of broken rock. Miner Resour Eng 11:349–360
34. Kvapil R (1965) Gravity flow of granular materials in hoppers and bins in mines—II. Coarse material. Int J Rock Mech Min Sci Geomech Abstr 2:277–292
35. Lara N (2014) Análisis histórico de las variables operacionales asociadas al ingreso de agua-barro en el sector Reserva Norte, División El Teniente, Codelco. Dissertation, Universidad de ChileGoogle Scholar
36. McCarthy PL, Harvey S (1998) Inrushes and subsidence, vol 3. Australasian Institute of Mining and Metallurgy, QueenslandGoogle Scholar
37. McNearny RL, Abel JF (1993) Large-scale two-dimensional block caving model tests. Int J Rock Mech Min Sci Geomech Abstr 30:93–109
38. Melo F, Vivanco F, Fuentes C, Apablaza V (2007) On drawbody shapes: from Bergmark–Roos to kinematic models. Int J Rock Mech Min Sci 44:77–86
39. Melo F, Vivanco F, Fuentes C, Apablaza V (2008) Kinematic model for quasi-static granular displacements in block caving: dilatancy effects on drawbody shapes. Int J Rock Mech Min Sci 45:248–259
40. Melo F, Vivanco F, Fuentes C (2009) Calculated isolated extracted and movement zones compared to scaled models for block caving. Int J Rock Mech Min Sci 46:731–737
41. Mullins WW (1972) Stochastic theory of particle flow under gravity. J Appl Phys 43:665–678
42. Navia IM (2014) Análisis del ingreso de agua-barro al sector Diablo Regimiento, División El Teniente. Dissertation, Universidad de ChileGoogle Scholar
43. Nedderman RM (2005) Statics and kinematics of granular materials, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
44. Nedderman RM, Tüzün U (1979) A kinematic model for the flow of granular materials. Powder Tech 22:243–253
45. Nguyen HD (1995) Probabilistic modeling of moisture flow in layered vadose zone: applications to waste site performance assessment. Int J Eng Sci 33:1345–1355
46. Nield DA, Bejan A (2006) Convection in porous media, 3rd edn. Springer, New YorkGoogle Scholar
47. Palmstrom A, Stille H (2007) Ground behaviour and rock engineering tools for underground excavations. Tunn Undergr Space Tech 22:363–376
48. Peters DC (1984) Physical modeling of the draw behavior of broken rock in caving. Colo Sch Mines Q 79:1Google Scholar
49. Power GR (2004) Modelling granular flow in caving mines: large scale physical modelling and full scale experiments. Dissertation, The University of QueenslandGoogle Scholar
50. Rustan A (2000) Gravity flow of broken rock: What is known and unknown. In: Proceedings of the third international conference and exhibition on mass mining, The Australasian Institute of Mining and Metallurgy, Brisbane, pp 557–568Google Scholar
51. Samadani A, Pradhan A, Kudrolli A (1999) Size segregation of granular matter in silo discharges. Phys Rev E 60:7203.
52. Samosir E, Basuni J, Widijanto E, Syaifullah T (2008) The management of wet muck at PT Freeport Indonesia’s deep ore zone mine. In: Proceedings of the fifth international conference and exhibition on mass mining, LuleåUniversity of Technology, Luleå, pp 323–332Google Scholar
53. Schlegel F (2014) Understanding stabilization methods. Comsol blog: technical content. COMSOL, Inc., Burlington. https://www.comsol.com/blogs/understanding-stabilization-methods. Accessed 14 Aug 2018
54. Shamey R, Zhao X (2014) Modelling, simulation and control of the dyeing process. Woodhead Publishing, New YorkGoogle Scholar
55. Shi Z, Wang X (2007) Comparison of Darcy’s law. the Brinkman equation, the modified NS equation and the pure diffusion equation in PEM fuel cell modeling. In: Proceedings of the COMSOL conference 2007, BostonGoogle Scholar
56. Szirtes T (2007) Applied dimensional analysis and modeling, 2nd edn. Butterworth-Heinemann, OxfordGoogle Scholar
57. Tabatabaian M (2014) COMSOL for engineers. Mercury learning and information, DullesGoogle Scholar
58. Todd DK (1980) Groundwater hydrology, 2nd edn. Wiley, New YorkGoogle Scholar
59. Trueman R, Castro R, Halim A (2008) Study of multiple draw-zone interaction in block caving mines by means of a large 3D physical model. Int J Rock Mech Min Sci 45:1044–1051
60. Tuller M, Or D (2002) Unsaturated hydraulic conductivity of structured porous media. Vadose Zone J 1:14–37
61. Valencia M, Basaure K, Castro R, Vallejo J (2014) Towards an understanding of mud rush behaviour in block-panel caving mines. 3erCongreso Internacional en Block Caving, Santiago, pp 363–371Google Scholar
62. Vallejos J, Basaure K, Palma S, Castro R (2017) Methodology for a risk evaluation of mud rushes in block caving mining. J S Afr Inst Min Metall 117:491–497
63. Van Golf-Racht TD (1982) Fundamentals of fractured reservoir engineering, vol 12. Elsevier, AmsterdamGoogle Scholar
64. Vivanco F, Melo F (2013) The effect of rock decompaction on the interaction of movement zones in underground mining. Int J Rock Mech Min Sci 60:381–388
65. Vivanco F, Watt T, Melo F (2011) The 3D shape of the loosening zone above multiple draw points in block caving through plasticity model with a dilation front. Int J Rock Mech Min Sci 48:406–411
66. Vukovic M, Soro A (1992) Determination of hydraulic conductivity of porous media from grain-size distribution. Water Resources Publications, LittletonGoogle Scholar
67. Vutukuri VS, Singh RN (1995) Mine inundation-case histories. Mine Water Environ 14:107–130Google Scholar
68. White FM (2008) Mecánica de fluidos, 6th edn. McGraw-Hill, New YorkGoogle Scholar
69. Wicaksono D, Silalahi K, Sryanto I, Soebari L, Ekaputra A, De Jong G (2012) Potential hazard map for the wet muck flow prevention at the deep ore zone (DOZ) block cave mine, Papua, Indonesia. In: Proceeding TPT XXI PERHAPS, pp 87–95Google Scholar
70. Widijanto E, Sunyoto WS, Wilson A, Yudanto W, Soebari L (2012) Lessons learned in wet muck management in Erstberg East Skarn system of PT Freeport Indonesia. In: Proceedings of the fifth international conference and exhibition on mass mining, Canadian Institute of Mining, Metallurgy and Petroleum, 6780, OntarioGoogle Scholar

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## Authors and Affiliations

• Katherine Sánchez
• 1
• 2
• Sergio Palma
• 1
• 3
• 4
• Raúl L. Castro
• 1
• 2
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