# Dynamic Parameter Calibration of an Analytical Model to Predict Transient Groundwater Inflow into a Tunnel

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## Abstract

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## 1. Introduction

_{s}(specific storage) combinations for a high-K zone and simulated the daily flow rate at the intersection of a tunnel and that high-K zone. The sensitivities of the time-varying drainage process to the K-Ss combinations were compared, and the results showed that the discharge process was very sensitive to the hydraulic parameters; the maximum flow rate increased with the value of K as well as S

_{s}. By analyzing the linear relationship between the parameters and the decay exponent of flow rate, it was found that the variability of the K value had a more significant effect on the flow rate [3]. However, research is still lacking on the automatic calibration of the hydrogeological parameters by some optimal algorithm to achieve a good fitting of calculation versus observation of groundwater inflow rate into a tunnel.

## 2. Perrochet Model and Dynamic Parameter Calibration Process

^{3}/T) of the tunnel excavated sectors with time t in a subvertical multilayered heterogeneous stratum (the exposed rock mass of the tunnel is divided into N sectors, as shown in Figure 1) [5]:

_{i}is the moment when the ith sector of the tunnel is excavated (T); the subscript i indicates the ith sector of the tunnel, a positive integer of 1 ≤ i ≤ N; L is the length of each sector (L); x is the length of the tunnel excavation sector along the axis (L), 0 ≤ x ≤ L; v is the tunnel drilling speed (L/T); K is the hydraulic conductivity (L/T); S is the specific storage coefficient (1/L); s is the water level drop depth (L); r is the radius of the tunnel (L); and H(L

_{i}− x) is the Heaviside step-function.

_{i}, L

_{i}, s

_{i}, v

_{i}, r

_{i}are set as known variables, their values have been specified right after the tunnel division. The measurement of groundwater inflow rate in the tunnel is denoted as Q

^{obs}, which are M dimension time series data. In general, M is much larger than N; in other words, the quantity of water inflow measurements is far more than the number of tunnel sectors. Q

_{i}

^{obs}is referred to as the time series data of the inflow observation in the ith sector, and m

_{i}is the number of measured inflows in the ith sector.

- Initial values assignment: The initial values of K
_{i}^{in}and S_{i}^{in}are assigned to the parameters of hydraulic conductivity and specific storage for sector i. - Water inflow calculation: This step is denoted as “Perrochet’s Equation” in the flow chart in Figure 2. A MATLAB code script for inflow calculation has been written according to Equation (1). After the initial values of K
_{i}^{in}and S_{i}^{in}are assigned, the code runs, then the inflow calculation is obtained for the ith sector, which is referred to as Q_{i}^{cal}. It is also a time series including m_{i}values, exactly the same dimension as its paired observation Q_{i}^{obs}. - Parameter calibration: Once the Q
_{i}^{obs}and Q_{i}^{cal}are available, the trust region reflection algorithm will be applied to calibrate and optimize K_{i}and S_{i}. The number of iterations in the optimization depends on the stopping condition of the solver, namely, Tolerance, referred to as Tol. When the tolerance is lower than the threshold ε (ε = 1 × 10^{−6}), the stop criterion is met, then the iteration of the solver will be terminated. In this step, the proposed optimal algorithm will run repeatedly until the stop criterion is met; at that time, the optimized hydraulic conductivity K_{i}^{op}and specific storage coefficient S_{i}^{op}are determined as well. - Results output: The optimized K
_{i}^{op}and S_{i}^{op}will be stored in a database to form parameter sets, while the Coefficient of Determination is calculated as:$$CoefficientofDetermination({R}^{2})=1-\frac{SSE}{SST}$$_{i}^{obs}and calculations Q_{i}^{cal}, and SST stands for the sum of squares. This coefficient is often used to evaluate the fitting effect of optimization. The time-dependent water inflow curves Q_{i}^{obs}-t and Q_{i}^{cal}-t will be plotted as well, to visually display the fitting effect between calculation and measurement. - Next calculating sector: As the calibration goes to the (i + 1)th sector, the entire process will be conducted again, as shown in the flowchart in Figure 2. The optimized parameters of the ith sector will be used for water inflow calculation in step (2), and the parameters of the (i + 1)th sector will be calibrated in step (3).

^{−10}–10

^{1}m/d, while it is 10

^{−10}–10

^{−1}m

^{−1}for S.

**H**is a symmetric matrix of second derivatives, D is a diagonal scaling matrix, ∆ is the trust-region radius > 0, and ‖‖ is the second norm [25].

## 3. Case Studies

#### 3.1. Modane Exploratory Tunnel, France

^{2}

_{P}is 0.9555 by the trial-and-error method in the Reference [5], while the R

^{2}

_{10}is 0.9780 by DPC in this paper (k = 10, all sectors). Because R

^{2}

_{P}< R

^{2}

_{10}, the fitting effect of DPC is a little better. Figure 4b displays the scatter plots of the measured and calculated inflow.

^{2}are 0.9569, 0.9604, and 0.9591 for k = 1, 2, and 3, respectively. Therefore, the result R

^{2}

_{P}< R

^{2}

_{1}< R

^{2}

_{3}< R

^{2}

_{2}indicates that the fitting effect of DPCs are still slightly better than the trial-and-error method. It also demonstrates that k affects the DPC results; different k will lead to different parameters optimization and different fitting effects as well. Generally, a larger k achieves better fitting; for example, the biggest R

^{2}is 0.9780 when k = 10. However, a larger k also results in more computation and is more time-consuming.

_{ij}and S

_{ij}, where the subscript i represents the ith sector as aforementioned, while the superscript j denotes the jth times optimization for this sector. j should be a positive integer and meets 1 ≤ j ≤ k.

^{−3.26}–10

^{0.99}m/d in the sectors with relatively developed fractures, such as in sectors 2, 3, 4, 6, 8, and 9, as shown in Figure 6. On the other hand, the range is 10

^{−10.00}–10

^{−9.76}m/d in the sectors with weak fracture development, as in sections 1, 5, 7, and 10.

^{−10.00}–10

^{−1.00}m

^{−1}. In most sectors, the optimization of S is as stable as K. However, when sectors 2–4 and 3–5 were calibrated, S

_{3}of the 3rd sector changed dramatically, as shown in Figure 6c,d; this may be due to the sudden decrease of the inflow rate in the 5th sector after a steep rise in the 4th sector, seen in Figure 5c,d. Similarly, in Figure 6g,h, the S of the 7th and 8th sectors have increased obviously, in accordance with the significant inflow increase of the 9th sector, as shown in Figure 5g,h.

^{2}is a little smaller, and the fitting effect of sectors 6 and 7 is not good. Even if the fitting of the 6th sector was greatly improved compared to the 5th sector, it was still not a satisfactory fitting. The water inflow in the 6th sector rose sharply, in order to make a better fit for the subsequent decline sector; the K value in the 6th sector underwent a small decrease after optimization. However, even if the K and S were very small in the 7th sector, its dropping trend still could not be fitted well, as seen in Figure 6e–h. In summary, the abrupt change of inflow brings great difficulty to parameter optimization.

#### 3.2. Xiema Tunnel in Chongqing, China

_{2}s) sandstone with mudstone, Xintiangou Formation (J

_{2}x) sandstone with mudstone, Lower Ziliujing Formation (J

_{1}zl) sandstone with mudstone, Lower Zhenzhuchong Formation (J

_{1}z) sandstone with mudstone, Triassic Upper Xujiahe Formation (T

_{3}xj) sandstone shale, Middle Leikoupo Formation (T

_{2}l) dolomite limestone, Lower Jialingjiang Formation (T

_{1}j) limestone dolomite, and Feixianguan Formation (T

_{1}f) limestone with mudstone. The geological cross-section along the tunnel is shown in Figure 7a, and Figure 7b presents the curve of measured inflow versus location. According to the above analysis of the geological conditions of Xiema Tunnel and the geological cross-section, the rock formation in the entrance section of Xiema Tunnel is nearly vertical, which satisfies the prerequisite of using the Perrochet Equation to calculate the water inflow.

^{2}

_{X}is as high as 0.9284. Here, we calibrated K and S by using the DPC method; with the same tunnel division scheme, the whole 20 sectors were optimized simultaneously, and R

^{2}

_{20}is 0.9910 larger than R

^{2}

_{X}. The optimization results are shown in Figure 7c.

^{−4.08}–10

^{−2.50}m/d, and S is 10

^{−4.52}–10

^{−1.00}m

^{−1}in sectors where sandstone is interbedded with mudstone and limestone with mudstone; Type 2: in sectors where sandstone and shale mainly occur, K is 10

^{−5.16}–10

^{−4.11}m/d, and S is 10

^{−1.31}–10

^{−1.00}m

^{−1}; Type 3: in the carbonate rock strata, K is 10

^{−8.99}–10

^{−1.20}m/d, and S is 10

^{−9.31}–10

^{−1.00}m

^{−1}. Thus, the overall permeability of the rock mass through the tunnel is sequenced as: Type 2 < Type 1 < Type 3; the storability is: Type 1 < Type 2 < Type 3.

^{−1}in sectors with relative high storability, while lgS is −10.0 m

^{−1}in lower sectors, and lgS is −2.1 m

^{−1}for others. Likewise, when S is fixed, lgK is −1.0 m/d in sectors with greater permeability, while lgK is −10.0 m/d in sectors with lower permeability, and lgK is −3.4 m/d for the rest.

^{2}

_{K}by the first way is greater than R

^{2}

_{S}by the second way, indicating its better fitting effect over the second way, especially in the sectors where water inflow drops sharply as seen in Figure 9. Moreover, the fitting effect of the second way is significantly reduced from sectors 14 to 16, as shown in Figure 9b,c. In Figure 9c, when calibrating sectors 15–17, R

^{2}

_{S}is rather small by the second way, indicating a poor fitting.

## 4. Conclusions

- The proposed Dynamic Parameter Calibration (DPC) scheme, which can automatically and quickly optimize hydrogeological parameters by computer, has great advantages over the empirical trial-and-error method. The coefficient of determination R
^{2}by DPC is always greater than that by trial-and-error, which means a better fitting between observation and calculation. - The number of sectors being calibrated in each step can be set arbitrarily in this DPC scheme. This number has a great influence on the fitting effect, specifically, the more sectors being calibrated, the better the fitting. However, more sectors to be calibrated simultaneously means more parameters to be optimized, and this will lead to massive computation and time consumption. Moreover, the scenario with too many sectors to be optimized simultaneously deviates from the reality of dynamic construction and continuous advancement of tunnel engineering.
- Considering the uncertainty of parameters, the proposed calibration scheme can also conduct multiple optimizations in the same sector. The first case study of the Modane exploration tunnel shows that the hydraulic conductivity K of the same sector is relatively stable after multiple optimizations, and the change is insignificant. The optimization results of the specific storage coefficient S are relatively stable in most sectors, except in the sectors where the inflow changes dramatically.
- For the case where the tunnel passes through heterogeneous strata with different lithologies, the proposed DPC scheme is also applicable. The example of Xiema tunnel demonstrates that K and S will change abruptly near the rock boundary and fault influence zone, reflecting the changes of permeability and storability. The parameter sensitivity analysis shows that K is more sensitive to water inflow than S.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Three-dimensional (3D) schematic of a tunnel progressively drilled through a subvertical multilayered system: different shades of gray represent different permeability and storability strata (modified after Perrochet et al. [5]).

**Figure 3.**Three-dimensional schematic of the dynamic calibration process when k = 3. Red sectors indicate the parameters being calibrated; Blue sectors indicate that the parameters are fixed or have been calibrated.

**Figure 4.**Comparison between the calculated and observed transient groundwater inflow of the Modane tunnel: (

**a**,

**c**) time series curves; (

**b**,

**d**) scatter diagrams. (Water inflow measurements are from Perrochet et al. [5]).

**Figure 5.**Comparison between calculated versus measured inflow rate in the process of dynamic parameter calibration (k = 3, R

^{2}is the coefficient of determination of the optimized sectors).

**Figure 7.**(

**a**) Transverse geological cross-section of Xiema tunnel; (

**b**) measured groundwater inflow into tunnel during excavation: the dark gray and white color of the tunnel entrance represents the length of the different sectors, totaling 20 sectors; and (

**c**) optimized values of K and S for each sector (Plots (

**a**,

**b**) were modified after Long [26]).

**Figure 8.**Probability density, cumulative distribution, and other statistical characteristics of the optimized parameters (k = 3); (

**a**) lgK; (

**b**) lgS.

**Figure 9.**The comparison between the analytical calculation of the water inflow by DPC against the measured values. (R

^{2}

_{K}and R

^{2}

_{S}are the determination coefficients of two calibration processes, respectively. The shadow on the left side of the dashed line indicates the sectors with a fixed parameter, whereas the right side are the sectors with a parameter to be calibrated).

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## Share and Cite

**MDPI and ACS Style**

Zhu, R.; Xia, Q.; Zhang, Q.; Cao, C.; Zhang, X.; Mao, B.
Dynamic Parameter Calibration of an Analytical Model to Predict Transient Groundwater Inflow into a Tunnel. *Water* **2023**, *15*, 2702.
https://doi.org/10.3390/w15152702

**AMA Style**

Zhu R, Xia Q, Zhang Q, Cao C, Zhang X, Mao B.
Dynamic Parameter Calibration of an Analytical Model to Predict Transient Groundwater Inflow into a Tunnel. *Water*. 2023; 15(15):2702.
https://doi.org/10.3390/w15152702

**Chicago/Turabian Style**

Zhu, Rui, Qiang Xia, Qiang Zhang, Cong Cao, Xiaoyu Zhang, and Bangyan Mao.
2023. "Dynamic Parameter Calibration of an Analytical Model to Predict Transient Groundwater Inflow into a Tunnel" *Water* 15, no. 15: 2702.
https://doi.org/10.3390/w15152702