An Analytical Solution of Gibson’s Model for Estimating the Pore Water Pressures in Accreting Deposition of Slurried Material Under One-Dimensional Self-Weight Consolidation. Part I: Pervious Base


Every year, mines produce a large amount of tailings that must properly be managed. For most cases, the tailings are sent by pipes and disposed of in tailings dams. Another more and more common practice is to send a part of the tailings with or without cement binder to underground to fill mine voids (stopes). In civil engineering, a similar practice is the deposition of dredged sludge pumped and confined in a containment structure (dam for most cases). In these cases, an important challenge is to determine the pore water pressure (PWP) associated with the self-weight consolidation of the slurried backfill disposed layer by layer. A solution to the self-weight consolidation of accreting (increase in thickness) deposition of slurried material was first proposed by Gibson in late 1950s. Recently, it has been revisited and applied to mine stopes backfilled with slurried materials. The solution can only be evaluated numerically. In this paper, a truly analytical solution is proposed that is based on the Gibson approach and can be readily used to manually calculate the variation and evolution of the PWP within accreting deposition. The effect of the filling rate, fill height and backfill properties on the variation and evolution of the PWP is analyzed. The solution is proposed for accreting deposition of slurried backfill placed on pervious base. An analytical solution for estimating the PWP within slurried backfill disposed on impervious base is presented in a companion paper.

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The authors would like to acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC 402318), Institut de recherche Robert-Sauvé en santé et en sécurité du travail (IRSST 2013-0029), Fonds de recherche du Québec—Nature et Technologies (FRQNT 2015-MI-191676, 2017-MI-202116), and industrial partners of the Research Institute on Mines and the Environment (RIME UQAT-Polytechnique;

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Correspondence to Jian Zheng.


Appendix 1: Sample Calculations of PWP Distribution with Microsoft Excel®

Excel® of Microsoft is used to illustrate the application of the proposed analytical solution to estimate the PWP distribution and evolution in the backfill. The following geometry and material parameters are considered:

  • H = 8 m (pour height)

  • γ = 20 kN/m3

  • c v  = 0.1 m2/h

  • m = 0.2 m/h

Figure 12 shows the detailed calculation considering h 0 = 0.5 and n ranging from −55 to 55. Beyond this range, the value of f(y) becomes zero. In this illustrative calculation, the (excess) PWP at several points is calculated with z varying from 0.001 to 8 m. It can be seen that the calculation is simple and easy to do with Excel®.

Fig. 12

Sample calculations of the (excess) PWP using the proposed solution (Eq. 12) with Excel®

Appendix 2: Sensitivity Analyses of h 0 and n in the Application of the Proposed Analytical Solution (Eq. 12)

The proposed analytical solution (Eq. 12) contains two parameters h 0 and n. To obtain accurate and reliable results, h 0 should be small enough while the range of n should be large enough. This requires a sensitivity analysis of these two parameters.

To illustrate this sensitivity analysis, the calculation example shown in “Appendix 1” is taken again here. Figure 13 shows the variation of the PWP at z = 6 m obtained by applying Eq. 12 with different h 0 and n. One sees that the results can be considered stable and accurate enough when h 0 reduces to 0.5 and n ranges from −55 to 55.

Fig. 13

Sensitivity analysis of h 0 and n in application of the proposed analytical solution (Eq. 12) to ensure stable results

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Zheng, J., Li, L., Mbonimpa, M. et al. An Analytical Solution of Gibson’s Model for Estimating the Pore Water Pressures in Accreting Deposition of Slurried Material Under One-Dimensional Self-Weight Consolidation. Part I: Pervious Base. Indian Geotech J 48, 72–83 (2018).

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  • Tailings dam
  • Backfilled stope
  • Dredged sludge
  • Self-weight consolidation
  • Gibson’s solution
  • Analytical solution
  • Pore water pressure (PWP)
  • Pervious base