# Research on Sustainable Development of Mining Goaf Management Based on Economic Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mined-Out Area Management Economic Model

#### 2.1. Postulated Conditions

- It is assumed that mining enterprises are completely rational and risk-neutral, that is, follow the traditional assumption of an “economic man” in economics.
- It is assumed that mining enterprises obtain profits through tailings sales and goaf filling and consider maximizing their own interests as the ultimate goal.
- It is assumed that the tailings content of ore extracted due to goaf filling is ignored. The main purpose of this assumption is to simplify the model and build an economic model of goaf filling management in a single period.
- It is assumed that the market demand for tailings x follows a random distribution on $(0,U)$; its probability density function is$f(x)$ and its cumulative distribution function is$F(x)$; its inverse function $F-1(x)$ is an increasing function, and $F(U)=1$.
- Based on the actual situation and simplification of tailings treatment, it is assumed that there is a cost of tailings treatment for mining enterprises, that is, unit tailings treatment cost.
- It is assumed that the selling price of tailing sand in the market is an exogenous parameter, and there is a maximum selling price of tailing sand according to the provisions of the national price management regulations on commodity prices.
- It is assumed that the additional ore extracted after the goaf filling will also generate income, and the mining enterprise will sell the ore at the market price; that is, the unit marginal income of the goaf filling is the ore sales price.

#### 2.2. Symbol Description and Constraint

- X: The actual demand quantity of tailing sand on the market, in tons.
- U: The maximum quantity of tailing sand required on the market, in tons.
- Q: The total amount of tailings produced by mining enterprises, in tons.
- q: The quantity of tailings sold by mining enterprises (decision variable), $0\le q\le Q$; the unit is tons, and the conversion factor between the quality and volume of tailings is ${\rho}_{1}$
- S: Total filling amount of goaf; the unit is ${\mathrm{m}}^{3}$.
- p: Sales price of tailing sand; the unit is CNY/ton.
- $\stackrel{\mathrm{-}}{\mathrm{p}}$: The upper limit of the sales price of tailings; the unit is CNY/ton.
- d: Tailings pre-treatment cost; the unit is CNY/ton.
- e: Ore sales price (unit marginal revenue of goaf filling); the unit is CNY/ton.
- c: The cost of purchasing ultra-high-water backfilling materials by mining enterprises; the unit is CNY/ton, and the conversion factor between the quality and volume of new inorganic solid water materials is ${\rho}_{2}$, ${\rho}_{2}<{\rho}_{1}$.
- M: Goaf ore production; unit is tons.
- m: The amount of ore mined due to gob filling, in tons, $m=\alpha M$, α is the rate of resource increasing.

#### 2.3. Model Building

#### Analysis of Optimal Decision for Mine Enterprise

- (2-1) When $c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}<\stackrel{\mathrm{-}}{\mathrm{p}}$, $p$ satisfies $c\frac{{\rho}_{2}}{{\rho}_{1}}+d<p<c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}$. By $c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}<\stackrel{\mathrm{-}}{\mathrm{per}}$, we gain: $0<Q<{F}^{-1}(1-\frac{d{\rho}_{1}}{\stackrel{\mathrm{-}}{\mathrm{p}}{\rho}_{1}-c{\rho}_{2}})$.
- (2-2) When $c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}\ge \stackrel{\mathrm{-}}{\mathrm{p}}$, we obtain: $c\frac{{\rho}_{2}}{{\rho}_{1}}+d<p<\stackrel{\mathrm{-}}{\mathrm{p}}$. By $c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}\ge \stackrel{\mathrm{-}}{\mathrm{p}}$, we gain: $Q\ge {F}^{-1}(1-\frac{d{\rho}_{1}}{\stackrel{\mathrm{-}}{\mathrm{p}}{\rho}_{1}-c{\rho}_{2}})$.

#### 2.4. Classification of Goaf Filling Schema

- Schema I: ultra-high-water filling materials mode;
- Schema II: ultra-high-water filling materials + mixed filling mode of partial tailings;
- Schema III: ultra-high-water filling materials + mixed filling mode of all tailings.

## 3. Results

#### 3.1. Selection of Mine Filling Mode

- (1)
- $p=200$, ${\rho}_{1}=20$, ${\rho}_{2}=5$, $U=\mathrm{20,000}$, $d=20$, $c=0:1:100$;
- (2)
- $p=200$, ${\rho}_{1}=20$, ${\rho}_{2}=5$, $U=\mathrm{20,000}$, $d=0:1:50$, $c=50$;
- (3)
- $p=50:1:300$, ${\rho}_{1}=20$, ${\rho}_{2}=5$, $U=\mathrm{20,000}$, $d=20$, $c=50$.

#### 3.2. Expected Income Analysis of Mining Enterprises

#### 3.3. Comparison and Analysis with Non-Filling Mode

- (1)
- When $\frac{dQ-M\alpha e+c{\rho}_{2}(S-Q/{\rho}_{1}){\rho}_{2}}{Q-{\int}_{0}^{Q}F(x)dx}<\frac{d}{1-F(Q)}$, so there is always ${\prod}_{g}^{A{O}^{\mathrm{*}}}>0$, so mining enterprises would like to fill the mined-out areas.
- (2)
- When $\frac{dQ-M\alpha e+c{\rho}_{2}(S-Q/{\rho}_{1}){\rho}_{2}}{Q-{\int}_{0}^{Q}F(x)dx}>\frac{d}{1-F(Q)}$, and when ${p}_{0}<p<\stackrel{\mathrm{-}}{\mathrm{p}}$, ${\prod}_{g}^{A{O}^{\mathrm{*}}}>0$, so mining enterprises would like to fill the mined-out areas.
- (3)
- When $\stackrel{\mathrm{-}}{\frac{dQ-M\alpha e+c{\rho}_{2}(S-Q/{\rho}_{1}){\rho}_{2}}{Q-{\int}_{0}^{Q}F(x)dx}+\frac{c{\rho}_{2}}{{\rho}_{1}}}>\stackrel{\mathrm{-}}{\mathrm{p}}$, no exiting ${\prod}_{g}^{A{O}^{\mathrm{*}}}>0$, so mining enterprises would not like to fill the mined-out areas.

#### 3.4. Application

^{3}. According to the actual production experience of similar mines, the design requires the newly generated gob to be filled within 1 to 2 months after mining, and the total production of tailing sand is 730,000 t. According to the situation in Linyi and surrounding areas, the demand for tailing sand in the market is about 2,000,000 t, and the selling price of tailing sand is CNY 50 /ton. The price ceiling is CNY 65/ton, the tailing sand reservation treatment cost is CNY 20 /ton, the cost of the new high-water filling material of Shandong Gufengyuan Technology Co., Ltd. (Zaozhuang City, China) is CNY 900/ton, and the resource increase rate is set at 0.15. Please see Table 2.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wei, Z.X.; Zhao, M.; Xie, F.; Hong, C.S.; Hong, D.J.; Hui, D.; Yang, D. Detection of former goaf and analysis of deformation characteristics of overburden in Dameidong coal mine. Front. Earth Sci.
**2023**, 11, 1111745. [Google Scholar] [CrossRef] - Blinova, F.; Ponomarenko, T.; Knysh, V. Analyzing the Concept of Corporate Sustainability in the Context of Sustainable Business Development in the Mining Sector with Elements of Circular Economy. Sustainability
**2022**, 14, 8163. [Google Scholar] [CrossRef] - Xu, Z.C.; Xu, W.; Zhu, Z.H.; Zhao, J.Y. Research on monitoring and stability evaluation of ground subsidence in gypsum mine goaf. Front. Environ. Sci.
**2023**, 10, 1097874. [Google Scholar] [CrossRef] - Luthra, S.; Mangla, S.K.; Sarkis, J.; Tseng, M.L. Resources melioration and the circular economy: Sustainability potentials for mineral, mining and extraction sector in emerging economies. Resour. Policy
**2022**, 77, 102652. [Google Scholar] [CrossRef] - Smoliński, A.; Malashkevych, D.; Petlovanyi, M.; Rysbekov, K.; Lozynskyi, V.; Sai, K. Research into Impact of Leaving Waste Rocks in the Mined-Out Space on the Geomechanical State of the Rock Mass Surrounding the Longwall Face. Energies
**2022**, 15, 9522. [Google Scholar] [CrossRef] - He, R.X.; Liu, H.; Ren, F.Y.; Li, G.H.; Zhang, J.; Zhou, Y.J. Comprehensive Evaluation and Decision for Goaf Based on Fuzzy Theory in Underground Metal Mine. Adv. Civ. Eng.
**2022**, 2022, 3104961. [Google Scholar] [CrossRef] - Yang, L. Study on Safety Hazard Investigation and Treatment Scheme in a Marble Mine Goaf. Acad. J. Arch. Geotech. Eng.
**2022**, 4, 24–30. [Google Scholar] - Han, C.P.; Du, C.; Zu, F.J.; Gao, J.H. Factors Influencing the Stability of a Slope Containing a Coal Seam in a Goaf. Appl. Sci.
**2022**, 12, 11699. [Google Scholar] [CrossRef] - Ding, Z.; Liu, P.; Cui, P.; Hong, C.Y. Strength Development and Environmental Assessment of Full Tailings Filling Materials with Various Water-to-Binder Ratios. Metals
**2023**, 13, 122. [Google Scholar] [CrossRef] - Yang, P.; Liu, L.; Sou, Y.L.; Qu, H.S.; Xie, G.; Zhang, C.X.; Deng, S.C.; Lv, Y. Investigating the synergistic effects of magnesia-coal slag based solid waste cementitious materials and its basic characteristics as a backfill material. Sci. Total Environ.
**2023**, 880, 163209. [Google Scholar] [CrossRef] - Li, T.; Chen, G.B.; Qin, Z.C.; Li, Q.H.; Cao, B.; Liu, Y.L. The gob-side entry retaining with the high-water filling material in Xin’an Coal Mine. Geomech. Eng.
**2020**, 22, 541. [Google Scholar] - Behera, S.K.; Ghosh, C.N.; Mishra, K.; Mishra, D.P.; Prashant, S.; Mandal, P.K.; Buragohain, J.; Sethi, M.K. Utilization of lead–zinc mill tailings and slag as paste backfill materials. Environ. Earth Sci.
**2020**, 79, 1–8. [Google Scholar] [CrossRef] - Xie, S.R.; Wang, E.; Chen, D.D.; Li, H.; Jiang, Z.S.; Yang, H.Z. Stability analysis and control technology of gob-side entry retaining with double roadways by filling with high-water material in gently inclined coal seam. Int. J. Coal Sci. Tech.
**2022**, 9, 52. [Google Scholar] [CrossRef] - Li, Y.F.; Hu, N.Y.; Ye, Y.C.; Wu, M.L. Improved catastrophe progression method of underground goaf risk evaluation. J. Int. Fuzzy Syst.
**2023**, 44, 1–16. [Google Scholar] [CrossRef] - Ma, T.X.; Lin, Y.; Zhou, X.B.; Zhang, M.Z. Grading Evaluation of Goaf Stability Based on Entropy and Normal Cloud Model. Adv. Civ. Eng.
**2022**, 2022, 9600909. [Google Scholar] [CrossRef] - Wang, F.F.; Zou, P.; Meng, Z.H.; Li, A.B.; Liu, Z.Y.; Hu, W.J.; Ma, Z. Study on Stability of Goaf Pillars in Daqiao Phosphate Mine: Theoretical Calculation and Field Investigation. Geotech. Geol. Eng.
**2019**, 37, 1483–1492. [Google Scholar] [CrossRef] - Liu, R. The Mechanical Model on Surface Subsidence Caused by Backfilling Mined-Out Area on Thick Loose Loess Layer: Taking a Sedimentary Bauxite Mine as an Example. Geotech. Geol. Eng.
**2022**, 41, 1083–1099. [Google Scholar] [CrossRef] - Jia, H.W.; Yan, B.X.; Guan, K.; Liu, H.L.; Wu, Q.Z.; Yin, Y.T.; Liu, H.X. Stability Analysis of Shallow Goaf Based on Field Monitoring and Numerical Simulation: A Case Study at an Open-Pit Iron Mine, China. Front. Earth Sci.
**2022**, 10, 897779. [Google Scholar] [CrossRef] - Belibi, T.A.E.; Yin, S.H.; Wang, L.M. Investigation on Mechanical Characteristics and Microstructure of Cemented Whole Tailings Backfill. Minerals
**2021**, 11, 592. [Google Scholar] [CrossRef] - Nabassé, J.F.; Koupouli, T.; Belem, P.; Rivard, H.E. Direct shear tests on cemented paste backfill–rock wall and cemented paste backfill–backfill interfaces. J. Rock Mech. Geotech. Eng.
**2016**, 8, 472–479. [Google Scholar] - Nasir, O.; Fall, M. Shear behaviour of cemented pastefill-rock interfaces. Eng. Geol.
**2008**, 101, 146–153. [Google Scholar] [CrossRef] - Uskov, V.A.; Kondrat’ev, S.A.; Neverov, S.A. Economic Expediency of Copper Ore Mining with Waste Rock Fill of Secondary Stopes in the West Wing of the Oktyabrsky Deposit. J. Min. Sci.
**2017**, 53, 1090–1095. [Google Scholar] [CrossRef] - Richard, G.; Ziggah, Y.Y.; Gyimah, M.D. A novel risk-adjusted discount rate methodology for mine economic evaluation—A case study. Int. J. Min. Miner. Eng.
**2022**, 13, 162–184. [Google Scholar] - Valentín, P.; Luis, S.A.; Joaquín, S.; David, A.L.P. Profitability Index Maximization in an Inventory Model with a Price- and Stock-Dependent Demand Rate in a Power-Form. Mathematics
**2021**, 9, 1157. [Google Scholar] - Askar, S.S.; AI-Khedhairi, A. Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game. Mathematics
**2020**, 8, 1458. [Google Scholar] [CrossRef] - Lee, S.; Choi, D.H. Dynamic pricing and energy management for profit maximization in multiple smart electric vehicles charging stations: A privacy-preserving deep reinforcement learning approach. Appl. Energy
**2021**, 304, 117754. [Google Scholar] [CrossRef] - Xiao, X.Q.; Zhu, Z.Q.; Fu, Z.T.; Mu, W.S.; Zhang, X.S. Carbon Footprint Constrained Profit Maximization of Table Grapes Cold Chain. Agronomy
**2018**, 8, 125. [Google Scholar] [CrossRef] - Li, X.M.; Miu, L.X.; Xu, Q.Q. Review on the research progress of newsboy model. Stat. Decis. Mak.
**2008**, 17, 11–14. [Google Scholar] - Guo, Y.; Yan, R.; Wang, H. Maximization of container slot booking profits for carriers in the liner shipping industry. J. Shipp. Trade
**2021**, 6, 1–10. [Google Scholar] [CrossRef] - Yu, H.B. The impact of demand uncertainty on minimizing cost and maximizing profit on the newsboy problem. Syst. Eng. Theory Pract.
**2014**, 34, 1756–1768. [Google Scholar] - Ahiad, M.; Tal, A. The newsvendor problem with a non-stationary demand process and exact accounting of holding costs. IFAC–PapersOnLine
**2022**, 55, 2761–2766. [Google Scholar] - Prasad, D.T.P.; Jha, J.K. A two-period newsvendor model for prepositioning with a post-disaster replenishment using Bayesian demand update. Soc. Econ. Plan. Sci.
**2021**, 78, 101080. [Google Scholar]

**Figure 2.**The enterprise’s optimal sales volume of tailings with $p$ should be listed as: (

**a**) the change trend in the optimal sales volume of enterprise tail sand with $p$ $(Q=\mathrm{10,000})$; (

**b**) the change trend in the optimal sales volume of enterprise tail sand with $p$ $(Q=\mathrm{19,500})$.

**Figure 4.**The enterprise’s optimal sales volume of tailings with $c$, $d$, and $p$(mode II): (

**a**) correlation between the optimal sales volume of enterprise tail sand and c; (

**b**) correlation between the optimal sales volume of enterprise tail sand and d; (

**c**) the correlation between the optimal sales volume of enterprise tail sand p.

**Figure 5.**The change trend of expected income of mining enterprise with $c$, $d$, $p$, and $Q$(mode I): (

**a**) correlation between expected return of mining enterprises and c; (

**b**) correlation between expected return of mining enterprises and d; (

**c**) correlation between expected return of mining enterprises and p; (

**d**) correlation between expected return of mining enterprises and Q.

**Figure 6.**The change trend of expected income of mining enterprise with $c$, $d$, $p$, and $Q$(mode II): (

**a**) correlation between expected return of mining enterprises and c; (

**b**) correlation between expected return of mining enterprises and d; (

**c**) correlation between expected return of mining enterprises and p; (

**d**) correlation between expected return of mining enterprises and Q.

**Figure 7.**The change trend of expected income of mining enterprise with $c$ and $Q$(mode III): (

**a**) correlation between expected return of mining enterprises and c; (

**b**) correlation between expected return of mining enterprises and Q.

The Total Amount of Tailings | The Sales Price of Tailings | Optimal Decision of Mining Enterprises |
---|---|---|

$0Q{F}^{-1}(1-\frac{d{\rho}_{1}}{\stackrel{\mathrm{-}}{\mathrm{p}}{\rho}_{1}-c{\rho}_{2}})$ | $c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}<p<\stackrel{\mathrm{-}}{\mathrm{p}}$ | Q |

$0<Q<{F}^{-1}(1-\frac{d{\rho}_{1}}{\stackrel{\mathrm{-}}{\mathrm{p}}{\rho}_{1}-c{\rho}_{2}})$ | $c\frac{{\rho}_{2}}{{\rho}_{1}}+d<p<c\frac{{\rho}_{2}}{{\rho}_{1}}+\frac{d}{1-F(Q)}$ | ^{1}${F}^{-1}(1-\frac{d{\rho}_{1}}{p{\rho}_{1}-c{\rho}_{2}})$ |

$Q\ge {F}^{-1}(1-\frac{d{\rho}_{1}}{\stackrel{\mathrm{-}}{\mathrm{p}}{\rho}_{1}-c{\rho}_{2}})$ | $c\frac{{\rho}_{2}}{{\rho}_{1}}+d<p<\stackrel{\mathrm{-}}{\mathrm{p}}$ | ${F}^{-1}(1-\frac{d{\rho}_{1}}{p{\rho}_{1}-c{\rho}_{2}})$ |

$Q\ge 0$ | $c\frac{{\rho}_{2}}{{\rho}_{1}}<p<c\frac{{\rho}_{2}}{{\rho}_{1}}+d$ | $0$ |

Symbol | Parameter | Symbol | Parameter |
---|---|---|---|

$U$ | 2,000,000 | $e$ | 1000 |

$Q$ | 730,000 | $c$ | 900 |

$S$ | 150,000 | $M$ | 18,250,000 |

$p$ | 50 | $\alpha $ | 0.15 |

$\overline{p}$ | 65 | ${\rho}_{1}$ | 200 |

$d$ | 20 | ${\rho}_{2}$ | 5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pang, C.; Shi, Y.; Liu, Y.
Research on Sustainable Development of Mining Goaf Management Based on Economic Models. *Sustainability* **2023**, *15*, 14772.
https://doi.org/10.3390/su152014772

**AMA Style**

Pang C, Shi Y, Liu Y.
Research on Sustainable Development of Mining Goaf Management Based on Economic Models. *Sustainability*. 2023; 15(20):14772.
https://doi.org/10.3390/su152014772

**Chicago/Turabian Style**

Pang, Chuming, Yongkui Shi, and Yang Liu.
2023. "Research on Sustainable Development of Mining Goaf Management Based on Economic Models" *Sustainability* 15, no. 20: 14772.
https://doi.org/10.3390/su152014772