# A Dynamic Modeling Approach to Predict Water Inflow during Karst Tunnel Excavation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of the Study Area

^{−1}, which represents the rapid discharge of large conduits or karst caves with good connectivity, large flow, fast recession, and short duration; the decay coefficient of the second segment of the curve is α = 0.043 d

^{−1}, which represents the discharge of extensional fissures, fault fissures, and dissolved fissures with certain connectivity. In this case, the flow decay/decline speed is slow, and the duration is long. The integration calculation of each section of the decay curve can estimate the amount of water discharged for each subdomain. The calculation results showed that the discharge of karst conduits or caves accounts for 24.1% of the total discharge, and the discharge of general karst fissures, structural fissures, and pores accounts for 75.9% of the total discharge. It also suggested that the main storage spaces of karst groundwater in the ZSX basin are dissolution fissures, pores, and small karst conduits, and the large karst conduits are mostly water-conducting channels. The karst medium of the study area has apparent duality.

## 3. Modeling Approach

#### 3.1. CFP Model

^{−1}], ${h}_{m}$ is the head in the karst matrix cell [L], $W$ is the volumetric flux per unit volume [T

^{−1}], ${S}_{s}$ is the specific storage [L

^{−1}], and $t$ is time [T]. The conduit is composed of nodes and cylindrical conduits. The flow state in the conduit is divided into laminar and turbulent flow. The laminar flow state is described by the Hagen–Poiseuille Equation (2), and the turbulent flow state is depicted by the Darcy–Weisbach Equation (3). The switch between laminar flow and turbulent flow is judged by the Reynolds number, and the linear Equation (4) calculates the amount of water exchange between the karst conduit and the karst matrix.

^{3}T

^{−1}], $d$ is the diameter of the conduit [L], $g$ is the acceleration of gravity [LT

^{−2}], $\Delta {h}_{c}$ is the head loss of the conduit [L], $\nu $ is the kinematic viscosity of water [L

^{2}T

^{−1}], $\tau $ is the tortuosity of the conduit, $\Delta l$ is the length of the conduit [L], $v$ is the mean water flow velocity [LT

^{−1}], and ${k}_{c}$ is the mean roughness height of the conduit [L].

^{3}T

^{−1}] between the conduit and the matrix, ${\alpha}_{ex}$ is the water flow exchange coefficient [L

^{2}T

^{−1}], ${h}_{c}$ represents the water head in the conduit, and ${h}_{m}$ denotes the matrix water head. A negative value for ${Q}_{ex}$ means that the water flows from the matrix to the conduit, and a positive value for ${Q}_{ex}$ means that the water flows from the conduit to the matrix.

#### 3.2. Boundary Conditions

#### 3.3. Aquifer Hydraulic Properties

#### 3.4. Conduit Parameter

^{2}·d

^{−1}identified in their study was taken as the exchange coefficient of the three conduits. The average annual water temperature of 16 °C was set as the water temperature of the model, and the Reynolds number was set as the reference value of large-scale simulation [18]. The values of the conduit parameters are shown in Table 2.

#### 3.5. Model Representation

^{2}and the finite difference grid was divided into 236 columns and 112 rows with a computational cell size of 50 m × 50 m. The study area has abundant rainfall, with an average annual precipitation of 1734 mm and an annual average evaporation of 1286 mm during the hydrometeorological monitoring period. From April to September of every year is considered the wet season, while the dry season can last from October to March of the following year. Because there is no concentrated irrigation recharge and groundwater exploitation in the area, groundwater discharge is controlled by the Jinzhu River, Dongxi River, Toudaogou River, and Kongxian River. The only recharge of groundwater in the area is from precipitation, including two recharge ways of diffuse recharge to the matrix, and concentrated recharge to the conduit. The amount of diffuse recharge can be calculated by the rainfall infiltration coefficient, and the rainfall infiltration coefficient in karst areas is related to the surface morphology and rainfall characteristics [23]. Based on the distribution of karst depressions in the study area, four infiltration zones were determined, whose rainfall, underground river, and spring discharge were monitored. The infiltration coefficient of each zone was calculated according to the water balance principle (Figure 7).

^{2}was up to 0.94, and the linear regression equation was as follows:

#### 3.6. Dynamic Excavation

^{−3}m/d on the 18th day after one excavation [27].

## 4. Results and Discussion

#### 4.1. Model Calibration

^{2}), respectively, which are employed to assess the goodness of fit of the model [31]. In the above equations, ${h}_{s}^{t}$ is the simulated value of hydraulic head at the time of $t$, ${h}_{o}^{t}$ is the observed value of hydraulic head at the time of $t$, $\overline{{h}_{o}}$ is the mean value of observed hydraulic heads, $\overline{{h}_{s}}$ is the mean value of simulated hydraulic heads, and T is the final simulation time. The R

^{2}ranges between 0 and 1, where one means that the simulated value is equal to the observed value and zero means that there is no correlation between the simulated and observed values.

^{2}was 0.991 and the corresponding NSE was 97.3%, respectively. Figure 11b displays the hydraulic head distribution of the study area before tunnel excavation, which served as the initial groundwater seepage field for water inflow prediction.

#### 4.2. Water Inflow Prediction

^{3}·d

^{−1}according to the CFP model prediction, and the MODFLOW model prediction had a greater mean absolute error value of 3984.88 m

^{3}·d

^{−1}. The water inflow curve in Section 2 has obvious rise and decay trends. During the wet season, the water in the matrix will be recharged by the karst conduit with the increase of conduit flow, and the matrix cells of the model will additionally be recharged [33,34]. Figure 12a shows that the prediction results of the MODFLOW model are smaller than the measured data because it only has matrix cells, while the CFP model prediction results are closer to the actual water inflow. In the dry season, karst conduits will “snatch” the water volume of the matrix due to the reduction of precipitation recharge, resulting in an additional drop of water volume from the matrix cell. Since there is no conduit cell in the MODFLOW model, the prediction results were too large to meet good agreement with the measured water inflow, even though it could reflect its downward trend, while the prediction results of the CFP model were closer to the observed value. The peak value of water inflow in this section was up to 15,897.14 m

^{3}·d

^{−1}, and the corresponding value of the CFP model prediction was 13782.35 m

^{3}·d

^{−1}, indicating a percentage error of 13.3%. The prediction value of the MODFLOW model was 10453.827 m

^{3}·d

^{−1}with a percentage error of 34.2%. Thus, the statistical results demonstrated that the predicted value of the CFP model was closer to the actual water inflow, reflecting the characteristics of the water inflow time series.

^{3}·d

^{−1}according to the CFP model prediction, with a larger value of 3162.99 m

^{3}·d

^{−1}by the MODFLOW model (Figure 12b). The tunnel path intersects the SHD1 conduit at DK241+012, 103 m above the tunnel. When the tunnel was excavated to this position, the peak field value of the water inflow was up to 10,492.00 m

^{3}·d

^{−1}in the entire section. The corresponding CFP model predicted value was 9921.97 m

^{3}·d

^{−1}with a percentage error of 5.4%, while the prediction result by the MODFLOW model was 6634.56 m

^{3}·d

^{−1}with a percentage error of 36.8%. When the tunnel was excavated underneath the karst conduit, the water inflow increased sharply. The conduit cell of the CFP model provided water volume to the matrix cell as an additional recharge and described the dynamic and sharp increase of water inflow in this section more accurately. Since there are no conduit cells in the MODFLOW model and the excavation time is at the end of the tunnel construction, the matrix cells within the tunnel area tend to be dry. The characteristics of “sudden rise and fall” of water inrush into the tunnel cannot be described by the MODFLOW model, thus, the prediction results were in poor agreement with the observed values.

## 5. Conclusions

^{2}= 0.991 and an NSE of 97.3%, which indicated that the CFP model successfully simulated the characteristics of karst groundwater flow.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Hydrogeology of the study area: (

**a**) recession curve of ZSX conduit flow; (

**b**) lithology and hydrogeology of the tunnel area.

**Figure 11.**Calibrated CFP model: (

**a**) comparison of simulated and observed hydraulic heads; (

**b**) calibrated groundwater seepage field of the study area.

Tunnel Section | Mileage | Section Length (m) | Excavation Days (d) |
---|---|---|---|

1 | DK234+506-DK237+400 | 2894 | 580 |

2 | DK237+400-DK240+060 | 2660 | 540 |

3 | DK240+060-DK241+848 | 1788 | 360 |

4 | DK241+848-DK243+535 | 1687 | 341 |

Parameters | ZSX Conduit | SHD1 Conduit | SHD2 Conduit | Unit | Explanation |
---|---|---|---|---|---|

DIAMETER | 1.5 | 1.0 | 1.0 | (m) | Conduit diameter |

TORTUOSITY | 1.1 | 1.1 | 1.1 | - | Conduit tortuosity |

RHEIGHT | 0.0001 | 0.0001 | 0.0001 | (m) | Conduit wall roughness |

LCRITREY | 2000 | 2000 | 2000 | - | Reynolds number |

TCRITREY | 4000 | 4000 | 4000 | - | Reynolds number |

K_EXCHANGE | 25 | 25 | 25 | (m^{2}·d^{−1}) | Exchange coefficient |

Precipitation (mm) | Duration (h) | Maximum Intensity (mm·h^{−1}) | Average Intensity (mm·h^{−1}) | Proportion of Concentrated Recharge | ||
---|---|---|---|---|---|---|

ZSX Conduit | SHD1 Conduit | SHD2 Conduit | ||||

114.2 | 42 | 21.6 | 2.719 | 0.571 | 0.536 | 0.554 |

64.4 | 36 | 10.8 | 1.79 | 0.440 | 0.422 | 0.432 |

64 | 63 | 12.6 | 1.02 | 0.128 | 0.137 | 0.184 |

355.2 | 142 | 23.2 | 2.501 | 0.518 | 0.476 | 0.503 |

216.8 | 58 | 13.2 | 3.576 | 0.677 | 0.636 | 0.652 |

34.8 | 16 | 15.6 | 2.175 | 0.473 | 0.442 | 0.492 |

Hydraulic Conductivity | Initial Value (m·d^{−1}) | Calibrated Value (m·d^{−1}) |
---|---|---|

HK_Zbd2 | 0.4 | 0.667 |

HK_Zbd1 | 0.13 | 0.125 |

HK_Zan2 | 0.1 | 0.413 |

HK_O42 | 0.2 | 0.137 |

HK_O21 | 0.3 | 0.3 |

HK_Zan1 | 0.4 | 0.16 |

HK_O22 | 0.3 | 0.37 |

VK_Zan2 | 0.01 | 0.011 |

HK_O41 | 0.1 | 0.177 |

HK_Cam1 | 0.3 | 0.284 |

VK_Pt2 | 0.01 | 0.011 |

HK_Pt2 | 0.1 | 0.457 |

HK_Pt1 | 0.25 | 0.5 |

HK_O32 | 0.4 | 0.403 |

HK_O12 | 0.1 | 0.15 |

HK_OS2 | 0.4 | 0.2 |

HK_Cam2 | 0.2 | 0.25 |

VK_Zbd2 | 0.04 | 0.04 |

HK_O31 | 0.2 | 0.2 |

HK_O11 | 0.1 | 0.141 |

VK_O22 | 0.01 | 0.037 |

HK_OS1 | 0.2 | 0.21 |

VK_OS2 | 0.04 | 0.093 |

VK_O42 | 0.02 | 0.02 |

VK_Zbd1 | 0.16 | 0.63 |

VK_O12 | 0.01 | 0.015 |

HK_Zan3 | 0.01 | 0.01 |

VK_Cam2 | 0.04 | 0.025 |

VK_O32 | 0.04 | 0.04 |

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**MDPI and ACS Style**

Bai, Y.; Wu, Z.; Huang, T.; Peng, D. A Dynamic Modeling Approach to Predict Water Inflow during Karst Tunnel Excavation. *Water* **2022**, *14*, 2380.
https://doi.org/10.3390/w14152380

**AMA Style**

Bai Y, Wu Z, Huang T, Peng D. A Dynamic Modeling Approach to Predict Water Inflow during Karst Tunnel Excavation. *Water*. 2022; 14(15):2380.
https://doi.org/10.3390/w14152380

**Chicago/Turabian Style**

Bai, Yang, Zheng Wu, Tao Huang, and Daoping Peng. 2022. "A Dynamic Modeling Approach to Predict Water Inflow during Karst Tunnel Excavation" *Water* 14, no. 15: 2380.
https://doi.org/10.3390/w14152380