Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 14, No. 3, 2025 295 Prediction of Mine Water Inflow Based on Growth Curve Model Shang Wang1, Qingge Zhang2, Shiyuan Tian1, * 1School of Resources and Environment, Henan Polytechnic University, Jiaozuo, Henan, China 2Faculty of Land Resources Engineering, Kunming University of Science and Technology, Kunming, Yunnan, China *Corresponding Author: Shiyuan Tian Abstract: During the construction and operation of a coal mine, water inrush incidents frequently occur. Minor water-related accidents can cause delays in mining operations, while major incidents can result in significant losses and casualties. This paper takes a coal mine in Shanxi as a case study, employing growth curve models and optimal weighted coefficient models to predict the mine's water inflow over the next five years. The predicted water inflow for the No. 4 coal seam, when mining in a single area, is 534.07 m3/h, with a maximum inflow of approximately 694.29 m3/h. Using the analytical method's prediction results as a reference, future scenarios are forecasted through the growth curve model approach. Analysis of the prediction accuracy reveals that the optimal coefficient combination model, established using three growth curve models, has the highest accuracy at 91.52%. The findings of this study provide valuable insights for future water management in coal mining operations. Keywords: Mine Water Inflow; Growth Curve Model; Optimal Weights. 1. Introduction During normal coal mining operations, inrush water can pose threats not only to the safety of the mining site and workers but also to the surrounding environment, leading to severe and irreversible ecological damage[1],[2]. Therefore, accurately predicting the water inflow in a coal mine during future extraction activities is crucial for providing a scientific basis to protect water resources and quality in mining areas[3]. The principle of the growth curve model method relies on certain mathematical growth curves. It begins by analyzing the data and general trends of water inflow, followed by the identification of an appropriate curve model. By substituting known data into the model to solve for parameters, the equation can be determined, and predictions can be made for the desired time frame. This method is particularly accurate for predicting future water inflow in mines that have been operational for many years. Previous studies, such as those by Peng Erlei[4], have successfully utilized this method, combining it with a composite model to predict water inflow in the Zhongtai Coal Mine. In this paper, the growth curve models, including the Logistic model, Gompertz model, and modified exponential model, are applied to predict the water inflow in the mining area. A comparison of the prediction accuracies of the four models is conducted, establishing a weighted coefficient model based on the three growth curves. 2. Hydrogeological Overview of the Mine 2.1. Hydrogeological Conditions Based on the analysis of mine data, the underground water system in the mining area can be divided into three distinct groundwater systems: the Cambrian-Ordovician carbonate rock karst fracture aquifer, the Carboniferous-Permian clastic rock fracture aquifer, and the Neogene loose clastic sediment pore aquifer. The primary source of groundwater replenishment in this mining area is precipitation. Rainwater moves to the subterranean layer through various conduits, forming groundwater, which typically flows from west to east. The distribution of the underground aquifers is summarized in Table 1. Table 1. Statistics of Aquifer Distribution in the Mining Area Main Aquifer Layer Main Lithology Water Yield Thickness (m) Water Level Elevation (m) Middle-Lower Ordovician Dolomitic Limestone Weak-Strong 0.60–16.50 1042.01–1065 Upper Carboniferous Taiyuan Group Fine to Medium-grained Sandstone Weak 13.52–47.61 1052.02–1065.14 Lower Permian Shanxi Group Fine to Coarse Sandstone and Gravel Weak 15.18–41.18 1014.12–1068.23 Neogene Lower-Middle Fine to Medium-grained Sandstone Weak- Medium 20.3–242.7 1072.93–1212.74 Neogene Upper Sand, Gravel, and Clay Weak- Medium 36.3–153.7 1065.2–1189.52 The main aquitards in the mining area are the clastic aquitard of the Middle Carboniferous and the clastic aquitard of the Middle and Lower Permian. The main lithology and thickness of the aquitards are shown in Table 2. 296 Table 2. Statistics of Aquitards in the Mining Area The main aquiclude main rocks water- resisting property Thickness ( m ) Middle Carboniferous clastic rock aquiclude Mudstone, sandstone moderation 23.68~44.27 Clastic rock aquiclude in the middle and lower part of Permian mudstone good 0~414.88 2.2. Mine water filling factors The primary source of direct water inflow into the coal mine is the fractured water from the sandstone of the Shanxi Group. The secondary source of replenishment is the karst fractured water from the carbonate rocks of the Middle Ordovician. Additionally, due to the unique lithology of the roof strata of the No. 4 coal seam, loose aquifers may also serve as an indirect source of water inflow into the coal seam. The fractured sandstone aquifer of the Taiyuan Group typically exhibits underdeveloped fracture systems and does not represent a major source of inflow. 2.3. Characteristics of mine water inflow Real-time monitoring of the mine's water inflow has been conducted through a flow automatic observation station. The dynamic curve of the mine water inflow from 2016 to 2024 is illustrated in Figure 1. 2016/1 2017/1 2018/1 2019/1 2020/1 2021/1 2022/1 2023/1 2024/1 2025/1 100 200 300 400 500 600 In fl ow o f w at er /m 3 穐 -1 Time/s Figure 1. Mine water inflow dynamic curve It can be seen from Fig.1 that with the increase of time, the mine water inflow increases year by year, and it has been increasing steadily from 2016 to 2021, reaching the maximum water inflow in 2021, and then gradually tends to be stable. 3. Hydrogeological Overview of the Mine 3.1. Prediction of mine water inflow Based on the hydrogeological conditions of the coal mine, the direct water recharge aquifer of the No. 4 coal seam in the mining area can be generalized as homogeneous. Analysis of inrush data and conditions from the past few years indicates that under normal circumstances, the water consumption in the mining area remains relatively stable. The large well method is employed to transform the originally irregular and complex boundary into a "large well" for easier calculation. The prediction formula for water inflow from the overlying aquifer is as follows: 2 r 0 0 2 0.05692 lg lg H M M h Q K R r     （ ） (1) Where: Qr is the normal water inflow of the expected mine roof aquifer ; K is the permeability coefficient ; H is the height of water column ; m is the thickness of aquifer ; r0 is the reference radius ; R0 is the reference influence radius ; h is the distance from the water level to the bottom ; among them : 0 0 0 10.2062 1000 / R R r R H K r F      (2) Where: R is to calculate the influence radius ; F is the area taken. The prediction formula of water inflow in floor aquifer is as follows : 0 00.11375 / (log logfQ KMH R r  ） (3) Qf represents the expected normal water inflow rate from the mine’s floor aquifer. Under the current mining conditions, the height of the "three zones" for the No. 4 coal seam and the depth of disturbance to the floor involves both the Upper and Lower Shihezi Group sandstone aquifers. Based on the water withdrawal test results and monitoring data from hydrogeological boreholes that expose the relevant aquifers, the overlying aquifer of the No. 4 coal seam corresponds to 297 the Upper and Lower Shihezi Group aquifers. Data from the No. 2 intake shaft shows a permeability coefficient of 0.0014 m/d and a water level elevation of 1254.99 m. The floor aquifer of the No. 4 coal seam corresponds to the Shanxi Group aquifer, with data from boreholes 3407 and 27158 indicating a permeability coefficient of 0.0733 m/d and a water level elevation of 1068.02 m. The elevation of the bottom of the No. 4 coal seam is 684.80 m. Based on the borehole data from the water withdrawal tests, the thickness of the overlying aquifer for the No. 4 coal seam is found to range from 22.70 m to 162.56 m, while the thickness of the floor aquifer ranges from 1.45 m to 24.24 m. By substituting all the relevant data into the formula, the water inflow can be calculated, and the results are presented in Table 3. Table 3. Calculation of Water Inflow for No. 4 Coal Seam Mining Area Roof and Floor Strata K H M F r0 R R0 Q Total Area 1 Roof Strata 0.0014 570.19 122.32 15.09 2191.64 217.74 2409.38 241.24 500.87 Floor Strata 0.0733 383.22 13.91 1058.92 3250.56 259.63 Area 2 Roof Strata 0.0014 570.19 70.80 10.17 1799.22 217.74 2016.96 121.63 365.49 Floor Strata 0.0733 383.22 15.34 1058.92 2858.14 243.86 Area 3 Roof Strata 0.0014 570.19 60.32 12.28 1977.08 217.74 2194.82 114.42 263.65 Floor Strata 0.0733 383.22 8.70 1058.92 3036.00 149.23 Area 4 Roof Strata 0.0014 570.19 59.19 16.71 2306.28 217.74 2524.02 130.16 406.26 Floor Strata 0.0733 383.22 14.18 1058.92 3365.20 276.10 Area 5 Roof Strata 0.0014 570.19 98.53 15.43 2216.19 217.74 2433.93 200.98 416.30 Floor Strata 0.0733 383.22 11.43 1058.92 3275.11 215.32 Area 6 Roof Strata 0.0014 570.19 64.10 15.26 2203.95 217.74 2421.69 134.36 470.40 Floor Strata 0.0733 383.22 17.46 1058.92 3262.87 336.04 Area 7 Roof Strata 0.0014 570.19 62.64 13.01 2034.99 217.74 2252.73 121.86 475.55 Floor Strata 0.0733 383.22 20.14 1058.92 3093.91 353.69 Average Value 414.07 The calculation results indicate that during the mining of Coal Seam 4, the water inflow in the mining area ranges from 263.65 to 500.87 m3/h, with an average inflow of 414.07 m3/h. If the water inflow in the roadway is calculated to be 120 m3/h, then the total water inflow for Coal Seam 4, when mining is conducted in only one mining area, would amount to 534.07 m3/h. Based on empirical data, the maximum water inflow is generally 1.3 times the normal inflow. Therefore, the maximum water inflow for Coal Seam 4 is approximately 694.29 m3/h. 3.2. Prediction of Mine Water Inflow Using Different Curve Models 3.2.1. Different Growth Curve Models (1) Logistic Growth Curve Model The Logistic growth curve model, also known as the self- limiting equation, was introduced in the 19th century primarily to address issues related to population growth in ecology. This model is also applicable for predicting water inflow in mines. The prediction formula is as follows: 2/dQ dt Q Q   , 0＞ , 0＞ (4) Variable Substitution: 2 / /Q dQ dt      (5) 1 / /dQ dt      (6) To find the general solution to the non-homogeneous linear equation. 1 - 1dt dt e dt t t t t t tQ Ce e dt Ce e e dt Ce e e                          (7) We need to determine C. Setting Q=Q0, t=t0, we can derive the expression for C. 0 0 01 0 1t t tQ Ce e e        (8) 00 0 tQ C e Q      (9) Substituting the expression for C yields the following result: 0 0 01 0 t tQ e Q Q Q         （ ） (10) Thus, 0 10 0 0 / / 1 / 1 / 1) 1 t t t tQ Q e Q e                      （ ）（ ） （ (11) Let 1 /K   , 1A  , 0 1 0( / 1 tB Q e   ） then the above equation can be expressed as: 1 1 11 A t K Q B e  (12) Equation (12) represents the water inflow growth curve model derived using the Logistic growth curve model, where K1 denotes the maximum value of the water inflow Q. (2) Modified Exponential Curve Model The modified exponential curve model, also known as the simple extrapolation method, involves fitting a curve based on exponential functions to the past measured data of the 298 object being predicted. This establishes a model that can describe the development of the object and allows for predictions by substituting in new states. The condition for fitting the modified exponential curve model is that the development process of the target to be calculated approximately follows an exponential function curve and does not exhibit sudden changes. This model can clearly be used for predicting water inflow in mines. Letting 1 1/Q Q , 11/M M , 1 /A B K , 1AB e in Equation (12) be represented as follows: 1 1 1 1 1 tA B Q K K       (13) 1 tQ K AB  (14) Equation (14) provides the model for mine water inflow Q1 concerning tt derived from the modified exponential curve model theory. (3) Gompertz Curve Model By letting YQ e , 2 KK e , AA e2  , BB 2 in Equation (14), we can derive the formula: 2 3 2 2 tBQ K A (15) This represents the mine water inflow Q3 concerning time tt obtained using the Gompertz curve model theory. Using the known data on mine water inflow, we can solve the aforementioned three curve models and predict mine water inflow based on their respective solutions. 3.2.2. Curve Model Calculation (1) Solution of the Modified Exponential Curve Model The mine water inflow Q changes with time t. Let the water inflow sequence be Qt,t=t1,t2、t3....tn. To solve for parameters A, B, and K, we can use the three-segment method, yielding K=487.8, A=−451.19, and B=0.79. This leads to the modified exponential curve equation: 1 487.8 451.19 0.79tQ    (16) The solution process for the Logistic growth curve model is similar to that of the modified exponential model, where we solve for parameters K1, A1, and B1. We find K1=492.31, A1=0.36, and B1=3.11. By substituting the observed water inflow data from recent years into the above equation, we obtain: 2 0.36 492.31 1 3.11 t Q e  (17) The solution process for the Gompertz growth curve model follows the same method as that of the modified exponential model. We solve for parameters K2, A2, and B2, finding K2=491.7, A2=0.2033, and B2=0.7533. Substituting the observed water inflow data from recent years into the above equation yields: 0.7533 3 491.7 0.2033 t Q   (18) Using the modified exponential model for prediction and analyzing the results, the results are shown in Table 4. Table 4. Analysis of Predicted Values using Modified Exponential Model Model Particular year Actual value (m3ꞏh-1) Predicted value (m3ꞏh-1) Absolute error (m3ꞏh-1) Relative error (%) Precision (%) Modified Exponential Model 2016 145.11 131.36 13.75 0.094755703 0.905244297 2017 223.58 206.21 17.37 0.077690312 0.922309688 2018 226.30 265.35 39.05 0.172558551 0.852835877 2019 233.99 312.06 78.07 0.333646737 0.749823752 2020 338.98 348.97 9.99 0.029470765 0.971372897 2021 455.92 378.12 77.8 0.170643973 0.829356027 2022 403.67 401.15 2.52 0.006242723 0.993757277 2023 437.67 419.34 18.33 0.041880869 0.958119131 2024 404.58 433.72 29.14 0.07202531 0.932813797 Logistic Growth Model 2016 145.11 155.31 10.2 0.070291503 0.934324899 2017 223.58 195.84 27.74 0.124071921 0.875928079 2018 226.30 239.43 13.13 0.058020327 0.945161425 2019 233.99 283.45 49.46 0.211376555 0.825507144 2020 338.98 325.15 13.83 0.040798867 0.959201133 2021 455.92 362.35 93.57 0.205233374 0.794766626 2022 403.67 393.78 9.89 0.024500211 0.975499789 2023 437.67 419.14 18.53 0.042337834 0.957662166 2024 404.58 438.86 34.28 0.084729843 0.921888529 Gompertz Growth Model 2016 145.11 148.09 2.98 0.020536145 0.979877102 2017 223.58 199.11 24.47 0.109446283 0.890553717 2018 226.30 248.86 22.56 0.099690676 0.909346621 2019 233.99 294.38 60.39 0.258087952 0.794856988 2020 338.98 334.11 4.87 0.014366629 0.985633371 2021 455.92 367.52 88.4 0.193893666 0.806106334 2022 403.67 394.88 8.79 0.021775212 0.978224788 2023 437.67 416.83 20.84 0.047615784 0.952384216 2024 404.58 434.17 29.59 0.073137575 0.931846972 299 Model predicts the normal water inflow and the maximum water inflow (calculated as 1.3 times the normal inflow) for the coal mine over the next five years, as presented in Table 5. Table 5. Prediction Results of Modified Exponential Model for 2025-2029 Model Particular year Normal water inflow (m3ꞏh-1) Maximum water inflow (m3ꞏh-1) Modified Exponential Model 2025 445.08 578.604 2026 454.05 590.265 2027 461.14 599.482 2028 466.74 606.762 2029 471.16 612.508 Logistic Growth Model 2025 453.75 589.875 2026 464.76 604.188 2027 472.76 614.588 2028 478.50 622.05 2029 482.60 627.38 Gompertz Growth Model 2025 447.71 582.023 2026 458.18 595.634 2027 466.23 606.099 2028 472.39 614.107 2029 477.08 620.204 3.3. Optimal Weighted Coefficient Model In the previous sections, three different curve models were used to predict the water inflow of the coal mine, each yielding different results. There are noticeable discrepancies when compared to actual data. To minimize these errors, methods can be employed to reduce discrepancies. Given the existence of three models, the optimal weighted coefficient method from economics can be utilized to construct a combination model. This involves assigning a weight to each model and combining the three models into a new model, expressed as: Q=mQ1+nQ2+pQ3, where m,n,p are all less than 1 and satisfy the equation m+n+p=1. Next, we can use this approach to construct a new water inflow model that minimizes errors and yields more accurate predictions. Let:    n 1i itiiti2t2t11t FKFK FKFKF  (19) The prediction error can be expressed as:      n 1i n 1i itiititttt EKFKYFYE (20) The sum of squared prediction errors can be expressed as:        N N EKEX 1t 1t n 1i 2 iti 2 t ）（ (21)                    NN NN EEEE EEEE Z 1t ntnt 1t t1nt 1t ntt1 1t t1t1    (22) Thus:   1 1t 1t 1t nt t 1 t 1N n 2 2 t i it 1 i t 1 t 1 i 1 nt 1t nt nt t 1 t 1i . K E . . N N N T N N K E E E E X E K K K ZK E E E E K                                                     （ ） (23) Let  TR 11,1  . Since the sum of the weighting coefficients must equal 1, we have R=1. Typically, the matrix Z is invertible, so the optimal weighted coefficient vector exists and is unique. Thus: RZR RZ K Ti 1 1    (24) Using knowledge from linear algebra, this can be substituted to show that: RZR RZ K T *i   (25) Since  imin JJ  holds, it can be understood that the sum of squared prediction errors of the new combined model is less than that of any individual model. Where: Yt: Actual mine water inflow;Fit: Predicted water 300 inflow of the i-th model at time t;Ft: Predicted water inflow of the combined model;Eit: Error of the i-th model at time t;Et: Predicted error of the combined model;Ki: Weight coefficient of the i-th model; Let 1,2......t N ; 1,2......i n ;    n 1i i 1K . The calculation method is as follows: (1) Calculate the adjoint matrix Z*: * * ij n nZ z （ ） (26) Where zij∗ is the cofactor of zij. Calculate: * 1 n i ij i D m   (27) Solve: 1 2 1 n i i i D D D D D       (28) Solve: i i d K d  (29) By substituting the predicted results from the three models, we can obtain the most suitable weight coefficients, ensuring that the resulting error is less than that of any individual model. 15455 13778 14254 13778 14055 13819 14254 13819 13989 Z          Calculate the adjoint matrix Z∗ of the invertible matrix Z:            27386741171810339941788 17181033125598294648924 994178846489245228984 *Z Thus, we have: 198196969941788464892452289841 D 34389786171810331255982946489242 D 54509562273867411718103399417883 D 108719044545095623438978619819696 D 18.01  D D , 32.02  D D , 5.0 3  D D Then, we can establish the new model equation as: 321t 5.032.018.0 QQQQ  (30) According to formula (30), the results can be obtained as shown in Table 6: Table 6. Analysis of Predicted Values using the Combined Model Particular year Actual value (m3ꞏh-1) Predicted value (m3ꞏh-1) Absolute error (m3ꞏh-1) Relative error (%) Precision (%) 2016 145.11 147.39 2.280 0.015712218 0.984530837 2017 223.58 199.34 24.24 0.108417569 0.891582431 2018 226.30 248.81 22.51 0.099469730 0.909529360 2019 233.99 294.06 60.07 0.256720373 0.795721962 2020 338.98 333.92 5.060 0.014927134 0.985072866 2021 455.92 367.78 88.14 0.193323390 0.806676610 2022 403.67 395.66 8.010 0.019842941 0.980157059 2023 437.67 418.02 19.65 0.04489684 0.955103160 2024 404.58 435.59 31.01 0.076647387 0.928809201 Using this model, we predict the normal water inflow and maximum water inflow (calculated as 1.3 times the normal inflow) for the coal mine for the years 2025, 2026, 2027, 2028, and 2029, as presented in Table 7. Table 7. Prediction Results of Combined Model for Mine Water Inflow for 2025-2029 Particular year Normal water inflow (m3ꞏh-1) Maximum water inflow (m3ꞏh-1) 2025 449.17 583.921 2026 459.54 597.402 2027 467.40 607.620 2028 473.33 615.329 2029 477.78 621.114 4. Model Comparison Four different models were used to calculate the normal water inflow for the coal mine in the years 2025, 2026, 2027, 2028, and 2029. The prediction results from these four models will be compared and analyzed. The process is summarized in 301 Table 8. Table 8. Comparison of Results from Four Models Model Absolute Error Sum (m6ꞏh-2) Relative Error Sum (%) Standard Deviation (m3ꞏh-1) Relative Standard Deviation (%) Average Relative Error (%) Average Accuracy (%) Modified Exponential Model 15454.7528 0.1931 41.44 0.15 0.111 90.17 Logistic Growth Model 14055.1413 0.1217 39.52 0.12 0.0957 91.00 Gompertz Growth Model 13988.9817 0.1348 39.43 0.12 0.0932 91.43 Optimal Weighted Coefficient Model 13914.0469 0.1337 39.32 0.12 0.0922 91.52 0 1 2 3 4 5 6 7 8 9 10 100 150 200 250 300 350 400 450 500 In fl ow o f w at er /m 3 穐 -1 Time/a Actual water inflow Amended index model Logistic growth model Gompertz growth model Hybrid model Figure 2. Trend Graph of Predictions from the Four Models Logistic growth model Amended index model Gompertz growth model Hybrid model 0.900 0.902 0.904 0.906 0.908 0.910 0.912 0.914 0.916 M ea n ac cu ra cy /% Mean accuracy Model Figure 3. Comparison of Prediction Accuracy Among the Four Models By analyzing the various indicators of the prediction results from the four models, it is observed that the optimal weighted coefficient model has a higher average accuracy than the other three models, achieving an average accuracy of 91.52%. Among them, the modified exponential model has the lowest accuracy at 90.17%. Additionally, the trends of the prediction results from the four models, as shown in Figure 2, indicate that the combined model yields the best prediction results, significantly reducing errors and improving prediction accuracy. From the calculation results of the aforementioned methods, the analytical method predicts the normal water inflow at 534.07 m3/h and the maximum value at 694.29 m3/h. Based on nine years of measured data, the highest water inflow 302 during this period was 531 m3/h in June and July of 2021. The predictions from the other methods also appear slightly higher, possibly due to the lack of information regarding the mining face area for the next five years when applying the analytical method. This could lead to an overestimation of water inflow. Nevertheless, the primary purpose of the analytical method is to verify whether the errors in other prediction methods are excessively large. The results from the other methods differ significantly, as each method simulates the actual hydrogeological conditions to varying degrees, resulting in different parameter estimates. As shown in Figure 3, the Gompertz model within the conventional three growth curve models has the highest prediction accuracy at 91.43%. The combined model, i.e., the optimal coefficient method, has higher prediction accuracy than the Gompertz model while being simpler to calculate. Therefore, the optimal weighted coefficient model is selected as the final prediction result, with the predicted water inflow from 2025 to 2029 being 449.17 m3/h, 459.54 m3/h, 467.40 m3/h, 473.33 m3/h, and 477.78 m3/h, respectively. 5. Conclusions This study focuses on a coal mine in Shanxi, using monthly water inflow data from 2016 to 2024 along with other information to explore the hydrogeological conditions of the coal mine in depth. The average water inflow of the coal at No. 4 was taken as a reference for using the analytical method. Additionally, the gray system prediction method and growth curve model methods were employed to scientifically predict the future water inflow of the mining area. The prediction accuracies of both the gray system prediction method and growth curve model methods were computed, and the growth curve model method with the highest accuracy was ultimately selected as the final prediction result. The main conclusions are as follows: (1) The large well method predicted the average water inflow of the No. 4 coal seam during normal mining at 417.07 m3/h and estimated that the water inflow in the coal mine would reach 534.07 m3/h when only one mining area is in operation, with a maximum water inflow of about 694.29 m3/h. (2) The water inflow predictions were made using the modified exponential model, Logistic model, Gompertz model, and a combined model with higher accuracy. The prediction accuracy of the modified exponential model was 90.17%, while the Logistic model achieved an accuracy of 91%. The Gompertz model had a prediction accuracy of 91.43%, and the combined model (optimal coefficient method) reached 91.52%. Thus, the combined model is determined to be the most accurate for this coal mine. (3) According to the predictions from the combined model, the water inflow from 2025 to 2029 will be 449.17 m3/h, 459.54 m3/h, 467.40 m3/h, 473.33 m3/h, and 477.78 m3/h, respectively. 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