# Comparison of Various Growth Curve Models in Characterizing and Predicting Water Table Change after Intensive Mine Dewatering Is Discontinued in an East Central European Karstic Area

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mining Water Extraction, a Central-Eastern European Example

^{3}min

^{−1}by the beginning of the 1960s and even 800 m

^{3}min

^{−1}for some years [30]. The long-term dewatering had its effect not only on the mining sites themselves, but over a wider area covering of several tens of kilometer. The amount abstracted exceeded the natural recharge, causing ever more extensive cones of depression in the karst water level of the formation and a decrease in groundwater level [31]. According to pers. comm. of officials, 20% of the abstracted water ended up in the drinking water supply, the remaining portion reached the surface water bodies (rivers and lakes).

^{3}min

^{−1}. This then decreased, reaching a level of 60 m

^{3}min

^{−1}at the beginning of the 2000s, due to rising water prices and, as a result, more economical water consumption [30]. From the beginning of the 1990s, the amount of natural recharge started to exceed the artificial dewatering, so the recovery of the aquifer began and still continues. Since then, the system has been trying to return to its near-natural state performed in the 1950s, which led to economic and technical-engineering problems as well [32]. Specifically, industrial and residential and public buildings were previously built-in areas where the karst water level was low, while mining was being actively carried out; however, without the artificial dewatering, these could not have been built there. Therefore, the determination of the spatiotemporal distribution of the rise in karst water levels is essential in the future planning of any industrial real estate investment.

#### 1.2. Aims of the Study

## 2. Materials and Methods

#### 2.1. Hydrogeological Description of the Study Area

#### 2.2. Dataset Description

#### 2.3. Applied Methodology

#### 2.4. Cluster and Discriminant Analysis

#### 2.5. Trend Estimation

#### 2.5.1. Growth Curves

#### 2.5.2. Model Fitting

^{2}value, in the present case called “correlation index” determined by Kenney and Keeping [64] (Equation (1)), which is not similar to the frequently used Pearson or Spearman correlation coefficient. It takes into account the spread of the modelled values by standardizing the squared error of the prediction (SSE) with the sum of squares total (SST) (see Equation (1)). In other words, the smallest the SSE, the closer r

^{2}will get to 1. The best fit was found with changing each parameter iteratively to maximize the r

^{2}value.

^{2}was calculated based on this function (Equation (1)):

#### 2.6. Deterministic Model

#### 2.7. Variography and Interpolation

_{e}= a × √3 have to be determined; a

_{e}is then used for further geostatistical modeling. A theoretical model can be given by calculating the empirical semivariogram using the algorithm of Matheron [67] (Equation (2))

_{0}) and the reduced sill (c), and the range (a), which is the distance within which the samples have an influence on each other and beyond which they are uncorrelated [68,69]. If the semivariogram does not have a rising part and the points of the empirical semivariogram align parallel to the abscissa, a nugget-type variogram is obtained. In this case, the sampling frequency is insufficient to estimate the sampling range using variography [50,70].

#### 2.8. Used Software

## 3. Results

#### 3.1. Cluster Analysis

#### 3.2. Trend Estimation

#### 3.2.1. Determining the Best-Fitting Model

^{2}> 0.9). However, it could be established that there is not a single function among those examined that best describes the recovery process in the Transdanubian Range. In most cases, the Richards (29.91% of cases) model proved to be the most accurate. However, the accuracy of the fitting—that is, the value of r

^{2}—showed very little difference between the functions in several cases. Because the value of r

^{2}can be similarly high in the case of different functions, the question of which functions fit best was addressed, taking into consideration not only the case with the largest r

^{2}value, but all of the cases in which ${r}^{2}\ge {r}_{max}^{2}-0.005$ (Table 1). However, even despite this, the Richards model proved to be the functions describing the recovery process the best. It should be noted that the difference between the goodness of fits (r

^{2}values) obtained for the different functions was quite small.

#### 3.2.2. Limitations of Trend Estimation

^{2}and the water level values predicted for January 2030 would change if a longer period was missing from the beginning of the recovery time series. The aim was to rank the functions in the case of 1–5 years of missing data compared to the case of time series with no missing data (Table 3). Although the best-fitting function does indeed change in many cases for the five randomly selected wells, there was no remarkable change in the estimated values. In the case of 5 years of missing data, the largest change in the predicted water level occurred at the Bakonybél-2a well, where the deviation is about 2.5% compared to the prediction based on the whole recovery time series. It is also noticeable that the time series of Bakonybél-2a are characterized by significant, long-term water level fluctuations (Figure 7).

#### 3.3. Variography and Predicted Karst Water Level Maps

## 4. Discussion

^{2}). This means that it was not only able to estimate the trend of the measured data but also its spatial variability among the wells.

## 5. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Map of the study area, the Transdanubian Range, located in Hungary, Europe (upper left inset maps). The red dots indicate the karst water monitoring wells. The names of the mining centers and the names of geographical units are also indicated in the map.

**Figure 2.**Number of karst water monitoring well providing monthly water level data over the period studied. The focus (recovery) period of the study is marked by a red double arrow.

**Figure 3.**Flowchart of the steps taken in the study. The input and output information is indicated by boxes with a dashed outline.

**Figure 4.**Visualization of the first two canonical discriminating functions indicating the groups obtained in different colors.

**Figure 5.**Grouping of the karst water wells. The groups are indicated with different colors. The base map represents the annual average water level of the Transdanubian Range in 1990 (modified after Alföldi and Kapolyi [30]). The dendrogram of the hierarchical cluster analysis, showing the three groups, is shown on the upper left side.

**Figure 6.**Curves fitted on the monthly average karst water levels of well Bakonybél-2/a and Csákberény-86/a, where the brown line is the measured water level, the 10 colored curves are the fitted trend models described above (Section 2.5.1), and the blue horizontal line is the K value (maximum water level).

**Figure 7.**Forecast up to January 2030 in the Bakonybél-2a well with the best-fitting function in the case of 1–5 years of missing data from the beginning of the recovery time series.

**Figure 8.**Empirical semivariograms (blue dots) and fitted theoretical semivariogram (continuous black line) of the: trend estimation results (

**A**) and the deterministic MODFLOW derived results (

**B**) from water levels for January 2030. The parameters of the Gaussian semivariogram model fitted with trend estimation were C

_{0}= 1; C

_{0}+ C = 1758; a

_{e}= 325 km; r

^{2}= 0.9; RSS = 337,350, while for the deterministic model: C

_{0}= 1; C

_{0}+ C = 1090; a

_{e}= 286 km; r

^{2}= 0.88; RSS = 192,194. The bin width was 5.1 km, and 10 uniform bins were used; the dotted horizontal line represents the variance.

**Figure 11.**Difference map of water levels measured and those estimated using trend analysis for the beginning of 2016.

**Figure 12.**Difference map of water levels measured and estimated by MODFLOW for the beginning of 2016.

**Figure 13.**The difference map of water levels based on the values estimated by trend analysis and MODFLOW modelling for January 2030.

**Figure 14.**Z-scored time series of Group 1 (blue line), Group 2 (green line), and Group 3 (red line), 1995–2015.

**Table 1.**Number of best fits in the case of the different models for the 107 wells, based on the highest r

^{2}value (top row) and based on the cases in which the values of ${r}^{2}\ge {r}_{max}^{2}-0.005$ (bottom row).

Bertalanffy | Törnquist1 | Törnquist2 | Logistic | Delayed Log | ||

r^{2} = max. | % | 11.21% | 0.00% | 1.87% | 1.87% | 17.76% |

r^{2} ≥ (max.—0.005) | % | 32.71% | 3.74% | 6.54% | 20.56% | 33.64% |

Squared log | Gompertz | 63 percent | Johnson | Richards | ||

r^{2} = max. | % | 0.00% | 4.67% | 28.04% | 4.67% | 29.91% |

r^{2} ≥ (max.—0.005) | % | 22.43% | 31.78% | 48.60% | 17.76% | 52.34% |

**Table 2.**Number and rate of change of the best fitting function in the case of 1–5 years of missing data.

Missing Year(s) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Number of changed time series | 13 | 26 | 28 | 35 | 38 |

Percent of changed time series | 24.53% | 49.06% | 52.83% | 66.04% | 77.55% |

**Table 3.**Changes in r

^{2}values and water levels predicted for January 2030 in case of 1–5 years of data missing from the beginning of the time series, in the case of 5 wells representing the different parts of the whole study area.

Well | Number of Missing Year(s) | ||||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | ||

Bakonybél-2a | Best fitting function | Bertalanffy | Squared log | Richards | |||

max. r^{2} | 0.85 | 0.82 | 0.80 | 0.78 | 0.76 | 0.74 | |

January 2030 (m asl.) | 225.5 | 228.9 | 230.6 | 231.1 | 231.1 | 231.1 | |

deviation in forecast (%) | 0.00% | 1.52% | 2.27% | 2.47% | 2.51% | 2.49% | |

Csákberény-86a | Best fitting function | “63 percent” | Gompertz | Richards | |||

max. r^{2} | 0.99 | 0.98 | 0.98 | 0.98 | 0.98 | 0.97 | |

January 2030 (m asl.) | 156.1 | 155.6 | 155.7 | 155.4 | 157.1 | 157.9 | |

forecast deviation (%) | 0.00% | 0.32% | 0.31% | 0.50% | 0.62% | 1.11% | |

Duka-1 | Best fitting function | Delayed log | |||||

max. r^{2} | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.98 | |

January 2030 (m asl.) | 171.5 | 171.0 | 171.5 | 171.3 | 170.6 | 170.7 | |

forecast deviation (%) | 0.00% | 0.29% | 0.04% | 0.12% | 0.50% | 0.45% | |

Epöl-5 | Best fitting function | “63 percent” | |||||

max. r^{2} | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.98 | |

January 2030 (m asl.) | 134.8 | 134.0 | 134.7 | 134.0 | 134.0 | 134.6 | |

forecast deviation (%) | 0.00% | 0.57% | 0.07% | 0.61% | 0.62% | 0.17% | |

Tata-Pokol | Best fitting function | “63 percent” | Gompertz | ||||

max. r^{2} | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 | 0.97 | |

January 2030 (m asl.) | 146.8 | 147.4 | 147.4 | 147.0 | 147.0 | 149.9 | |

forecast deviation (%) | 0.00% | 0.38% | 0.43% | 0.13% | 0.14% | 2.08% |

**Table 4.**Changes in r

^{2}values and water levels predicted for January 2030 in the case of modifying the threshold value by max. +/− 10%, in the case of 5 wells representing the different parts of the whole study area.

Well | Percentage of Original K Value | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

90% | 92% | 94% | 96% | 98% | 100% | 102% | 104% | 106% | 108% | 110% | ||

Bakonybél-2a | Best fitting function | Bertalanffy | ||||||||||

max. r^{2} | 0.84 | 0.84 | 0.84 | 0.84 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | 0.85 | |

January 2030 (m asl.) | 220.6 | 221.7 | 222.7 | 223.7 | 224.6 | 225.5 | 226.3 | 227.1 | 227.9 | 228.6 | 229.3 | |

forecast dev. (%) | 2.15% | 1.68% | 1.23% | 0.80% | 0.39% | 0.00% | 0.37% | 0.73% | 1.07% | 1.40% | 1.71% | |

Csákberény-86a | Best fitting function | “63 percent’ | ||||||||||

max. r^{2} | 0.98 | 0.98 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | |

January 2030 (m asl.) | 152.7 | 153.4 | 154.2 | 154.9 | 155.5 | 156.1 | 156.7 | 157.3 | 157.8 | 158.3 | 158.8 | |

forecast dev. (%) | 2.23% | 1.73% | 1.27% | 0.82% | 0.40% | 0.00% | 0.38% | 0.74% | 1.08% | 1.41% | 1.72% | |

Duka-1 | Best fitting function | “63 percent” | Bertalanffy | Delayed lg | ||||||||

max. r^{2} | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | |

January 2030 (m asl.) | 168.3 | 169.9 | 171.2 | 169.6 | 170.6 | 171.5 | 172.3 | 173.1 | 173.8 | 174.5 | 175.1 | |

forecast dev. (%) | 1.86% | 0.94% | 0.15% | 1.08% | 0.52% | 0.00% | 0.48% | 0.93% | 1.36% | 1.75% | 2.12% | |

Epöl-5 | Best fitting function | “63 percent” | Bertalanffy | |||||||||

max. r^{2} | 0.9895 | 0.9899 | 0.9902 | 0.9905 | 0.9907 | 0.9909 | 0.9911 | 0.9912 | 0.9914 | 0.9915 | 0.9916 | |

January 2030 (m asl.) | 133.35 | 133.69 | 134.00 | 134.29 | 134.56 | 134.82 | 135.06 | 135.28 | 135.49 | 135.68 | 135.78 | |

forecast dev. (%) | 1.09% | 0.84% | 0.61% | 0.39% | 0.19% | 0.00% | 0.18% | 0.34% | 0.50% | 0.64% | 0.72% | |

Tata-Pokol | Best fitting function | “63 percent” | ||||||||||

max. r^{2} | 0.9841 | 0.9846 | 0.9851 | 0.9855 | 0.9858 | 0.9861 | 0.9863 | 0.9866 | 0.9868 | 0.9869 | 0.9871 | |

January 2030 (m asl.) | 144.56 | 145.07 | 145.55 | 146.00 | 146.42 | 146.81 | 147.17 | 147.52 | 147.84 | 148.14 | 148.43 | |

forecast dev. (%) | 1.53% | 1.18% | 0.86% | 0.55% | 0.27% | 0.00% | 0.25% | 0.48% | 0.70% | 0.91% | 1.10% |

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## Share and Cite

**MDPI and ACS Style**

Modrovits, K.; Csepregi, A.; Kovácsné Székely, I.; Hatvani, I.G.; Kovács, J.
Comparison of Various Growth Curve Models in Characterizing and Predicting Water Table Change after Intensive Mine Dewatering Is Discontinued in an East Central European Karstic Area. *Water* **2021**, *13*, 1047.
https://doi.org/10.3390/w13081047

**AMA Style**

Modrovits K, Csepregi A, Kovácsné Székely I, Hatvani IG, Kovács J.
Comparison of Various Growth Curve Models in Characterizing and Predicting Water Table Change after Intensive Mine Dewatering Is Discontinued in an East Central European Karstic Area. *Water*. 2021; 13(8):1047.
https://doi.org/10.3390/w13081047

**Chicago/Turabian Style**

Modrovits, Kamilla, András Csepregi, Ilona Kovácsné Székely, István Gábor Hatvani, and József Kovács.
2021. "Comparison of Various Growth Curve Models in Characterizing and Predicting Water Table Change after Intensive Mine Dewatering Is Discontinued in an East Central European Karstic Area" *Water* 13, no. 8: 1047.
https://doi.org/10.3390/w13081047