# Inversion Analysis of Impervious Curtain Permeability Coefficient Using Calcium Leaching Model, Extreme Learning Machine, and Optimization Algorithms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calcium Leaching Model

#### 2.1. Controlling Equations

^{2+}in the pore solution; $D$ represents the diffusion coefficient; and ${R}_{C}$ means the rate of solid-phase calcium decomposition.

#### 2.2. The Solid-Phase Calcium Decomposition Model

^{2+}in different compounds, depending on the calcium content; $R$ means the gas constant; $T$ is the temperature; ${\tau}_{leach}$ denotes the characteristic time of calcium leaching; ${A}_{s}$ represents the chemical affinity that controls the force of chemical reactions; ${c}_{Ca}$ stands for the present calcium concentration in the pore fluid; $\kappa \left(\overline{s}\right)$ is the equilibrium constant; and ${c}_{Ca}^{eq}$ and ${s}_{Ca}^{eq}$ mean the calcium concentration in the pore fluid and solid skeleton at the equilibrium point, respectively.

#### 2.3. Pore Parameter Evolution Equation

#### 2.3.1. Kozeny–Carman Equation

#### 2.3.2. Porosity Variation Equation

^{3}in this paper; ${A}_{s}$ is the chemical potential; and $\eta $ denotes the coefficient affecting the micro-diffusion of the Ca

^{2+}in the pores. The values of different $\eta $ are indicated in the finite element analysis part.

#### 2.3.3. Diffusivity Evolution Equation

^{2+}which is taken as $4.5\times {10}^{-10}$ m/s; ${\phi}_{c}\left(x,0\right)$ means the initial capillary porosity; $H()$ is the Heaviside function; and ${\phi}_{c}\left(x,t\right)$ denotes the capillary porosity.

## 3. The Objective Function

## 4. Prediction Model of Permeability Coefficient

#### 4.1. Orthogonal Design Method

#### 4.2. Extreme Learning Machine

^{th}implied layer unit and ${\mathrm{v}}_{i}$ means the weight of the i

^{th}implied layer output unit. The output layer contains only one unit because of the integration between hydraulic head and leakage. One point to note is that there may be situations where the matrix cannot be inverted since the mapping function is initialized randomly. To solve this problem, the mapping function is chosen to be a Sigmoid function to ensure that the output matrix of the hidden layer achieves full row rank or full column rank.

#### 4.3. Predictive Modeling Procedure

## 5. Application Case

#### 5.1. Project Overview

#### 5.2. Finite Element Analysis

#### 5.2.1. Finite Element Model

#### 5.2.2. Boundary and Initial Conditions

^{3}and 0 mol/m

^{3}, respectively.

#### 5.2.3. Calculation Parameters

#### 5.3. Results of the Simulation

#### 5.3.1. The Simulated Parameters

^{−6}. In the SSA strategy, the number of sparrows is 50, the proportion of discoverers is 0.7, the proportion of followers is 0.1, the proportion of vigilants is 0.2, the maximum number of iterations is 200, and the safety threshold is 0.6. In the PSO strategy, the updated speed of the particle is the sum of its own speed inertia, self-cognition, and social cognition in the previous step. The initial population number is 20, the maximum number of iterations is 200, and the inertia weight is 0.8. The first and second learning factors are 1.5 and 1.5, respectively.

#### 5.3.2. Simulation Results of Hydraulic Head and Leakage

#### 5.3.3. Permeability Coefficient of the Impervious Curtain

^{−6}, two orders of magnitude higher than the initial permeability coefficient. The permeability coefficient on the downstream side of the curtain is 2.72 × 10

^{−7}, with an increase in one order of magnitude. Figure 8c presents the distribution of the curtain permeability coefficient at different elevations after the leaching time of 50 years. The permeability coefficient of the upstream side of the curtain at the elevation of 72 m is 1.45 × 10

^{−6}, while that of the downstream side is 3.45 × 10

^{−7}. The permeability coefficient at the altitude of 72 m is almost half an order of magnitude higher than the other two elevations. The higher the elevation, the more pronounced the increase in the permeability coefficient of the impervious curtain.

## 6. Discussion

## 7. Conclusions

- (1)
- Based on ELM and four optimization algorithms, the inversion values of leaching parameters by four algorithms show small differences. The results are all within the parameter range.
- (2)
- Four sets of leaching parameter results were used in the positive analysis. Among the four results, the curves fitted by PSO corresponding to the leaching parameters are in the best agreement with the measured values and show the highest prediction accuracy, which indicates that the inversion method is reliable and effective.
- (3)
- The values of permeability coefficients on the upper and upstream sides are greater than in other areas, showing that these areas are the most vulnerable parts of the grout curtain. Focusing on these vulnerable parts and strengthening safety management is necessary for the safety of water conservancy projects. The results illustrate the feasibility of the inversion analysis for obtaining the calcium leaching parameters under advection-diffusion-driven leaching processes.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kailai, Z.; Zhenzhong, S.; Lei, G. Advances in cement-based materials leaching test. Adv. Sci. Technol. Water Resour.
**2018**, 38, 86–94. (In Chinese) [Google Scholar] - Zongpan, L.; Hanchu, T. Design of curtain reinforcement and application of modified cement in Daheiting reservoir. Hydropower Eng. Design.
**2003**, 22, 11–13. (In Chinese) [Google Scholar] - Phung, Q.T.; Maes, N.; Jacques, D.; De Schutter, G.; Ye, G. Investigation of the changes in microstructure and transport properties of leached cement pastes accounting for mix composition. Cem. Concr. Res.
**2016**, 79, 217–234. [Google Scholar] [CrossRef] - Hiroshi, S.; Akira, D. Leaching tests on different mortars using accelerated electrochemical method. Cem. Concr. Res.
**2000**, 30, 1815–1825. [Google Scholar] - Gawin, D.; Pesavento, F.; Schrefler, B.A. Modeling of cementitious materials exposed to isothermal calcium leaching, considering process kinetics and advective water flow. Part 1: Theoretical model. Int. J. Solids Struct.
**2008**, 45, 6221–6240. [Google Scholar] [CrossRef] - Gawin, D.; Pesavento, F.; Schrefler, B.A. Modeling of cementitious materials exposed to isothermal calcium leaching, considering process kinetics and advective water flow. Part 2: Numerical solution. Int. J. Solids Struct.
**2008**, 45, 6241–6268. [Google Scholar] [CrossRef] - Nelio, H.; Juan, C.B.; Wagner, F.S. A three-parameter Kozeny–Carman generalized equation for fractal porous media. Chem. Eng. Sci.
**2010**, 65, 4432–4442. [Google Scholar] - Henderson, N.; Brettas, J.C.; Sacco, W.F. Applicability of the three-parameter Kozeny–Carman generalized equation to the description of viscous fingering in simulations of water flood in heterogeneous porous media. Adv. Eng. Softw.
**2015**, 85, 73–80. [Google Scholar] [CrossRef] - Jun, K.; Yoshihiro, K.; Jun, Y.; Tenma, N. Pore-scale modeling of flow in particle packs containing grain-coating and pore-filling hydrates: Verification of a Kozeny-Carman-based permeability reduction model. J. Nat. Gas Sci. Eng.
**2017**, 45, 537–551. [Google Scholar] - Lala, A.M.S. Modifications to the Kozeny-Carman model to enhance petrophysical relationships. Explor. Geophys.
**2017**, 49, 553–558. [Google Scholar] [CrossRef] - Gerard, B.; Le Bellego, C.; Bernard, O. Simplified modelling of calcium leaching of concrete in various environments. Mater. Struct.
**2002**, 35, 632–640. [Google Scholar] [CrossRef] - Phung, Q.T.; Maes, N.; Jacques, D.; Perko, J.; De Schutter, G.; Ye, G. Modelling the evolution of microstructure and transport properties of cement pastes under conditions of accelerated leaching. Constr. Build. Mater.
**2016**, 115, 179–192. [Google Scholar] [CrossRef] - Keshu, W.; Lin, L.; Wei, S. Solid–liquid equilibrium curve of calcium in 6mol/L ammonium nitrate solution. Cem. Concr. Res.
**2013**, 53, 44–50. [Google Scholar] - Keshu, W.; Lin, L.; Wei, S. Experimental and modelling research of the accelerated calcium leaching of cement paste in ammonium nitrate solution. Constr. Build. Mater.
**2013**, 40, 832–846. [Google Scholar] - Lambert, C.; Buzzi, O.; Giacomini, A. Influence of calcium leaching on the mechanical behavior of a rock-mortar interface: A DEM analysis. Comput. Geotech.
**2010**, 37, 258–266. [Google Scholar] [CrossRef] - Kailai, Z.; Zhenzhong, S.; Liqun, X.; Tan, J.C.; Yang, C. Durability control index of anti-seepage curtain considering the effect of advection-diffusion-driven leaching. J. Hydraul Eng.
**2020**, 51, 169–179. (In Chinese) [Google Scholar] - Zhou, C.B.; Liu, W.; Chen, Y.F.; Hu, R.; Wei, K. Inverse modeling of leakage through a rockfill dam foundation during its construction stage using transient flow model, neural network, and genetic algorithm. Eng. Geol.
**2015**, 187, 183–195. [Google Scholar] [CrossRef] - Ren, J.; Shen, Z.Z.; Yang, J.; Yu, C.-Z. Back analysis of the 3D seepage problem and its engineering applications. Environ. Earth Sci.
**2016**, 75, 113. [Google Scholar] [CrossRef] - Borazjani, S.; Hemmati, N.; Behr, A.; Genolet, L.; Mahani, H.; Zeinijahromi, A.; Bedrikovetsky, P. Determining water-oil relative permeability and capillary pressure from steady-state coreflood tests. J. Pet. Sci. Eng.
**2021**, 205, 108810. [Google Scholar] [CrossRef] - Janis, V.; Uldis, B.; Tomas, S.; Sennikovs, J.; Timuhins, A. Hydrogeological model of the Baltic Artesian Basin. Hydrogeol. J.
**2013**, 21, 845–862. [Google Scholar] - Chi, S.; Ni, S.; Liu, Z. Back Analysis of the Permeability Coefficient of a High Core Rockfill Dam Based on an RBF Neural Network Optimized Using the PSO Algorithm. Math. Probl. Eng.
**2015**, 118, 124042. [Google Scholar] - Bahrami, A.; Aghamir, F.; Bahrami, M.; Khodaverdiloo, H. Inverse modeling towards parameter estimation of the nonlinear soil hydraulic functions using developed multistep outflow procedure. Hydrol. J.
**2020**, 590, 125446. [Google Scholar] [CrossRef] - Tan, J.C.; Xu, L.Q.; Zhang, K.L.; Khodaverdiloo, H. A Biological Immune Mechanism-Based Quantum PSO Algorithm and Its Application in Back Analysis for Seepage Parameters. Math. Probl. Eng.
**2020**, 2020, 2191079. [Google Scholar] [CrossRef] - Larry, P.C.; Thomas, M.A.; John, A.A.; Yidana, S.M. Investigation of critical hydraulic gradient and its application to the design and construction of bentonite-grout curtain. Environ. Earth Sci.
**2019**, 78, 370. [Google Scholar] - Chen, Y.F.; Zhou, C.B.; Sheng, Y.Q. Formulation of strain-dependent hydraulic conductivity for a fractured rock mass. Int. J. Rock Mech. Min. Sci.
**2007**, 44, 981–996. [Google Scholar] [CrossRef] - Chen, Y.F.; Hu, S.H.; Zhou, C.; Jing, L. Micromechanical modeling of anisotropic damage-induced permeability variation in crystalline rocks. Rock Mech. Rock. Eng.
**2014**, 47, 1775–1791. [Google Scholar] [CrossRef] - Lingireddy, S. Aquifer parameter estimation using genetic algorithms and neural networks. Civil Eng. Env. Syst.
**1998**, 15, 125–144. [Google Scholar] [CrossRef] - Karpouzos, D.K.; Delay, F.; Katsifarakis, K.L.; de Marsily, G. A multi-population genetic algorithm to solve the inverse problem in hydrogeology. Water Resour. Res.
**2001**, 37, 2291–2302. [Google Scholar] [CrossRef] - Garcia, L.A.; Shigidi, A. Using neural networks for parameter estimation in ground water. Hydrol. J.
**2006**, 318, 215–231. [Google Scholar] [CrossRef] - Chang, Y.C.; Yeh, H.D.; Huang, Y.C. Determination of the parameter pattern and values for a one-dimensional multi-zone unconfifined aquifer. Hydrogeol. J.
**2008**, 16, 205–214. [Google Scholar] [CrossRef] - Dietrich, C.R.; Newsam, G.N. Suffificient conditions for identifying transmissivity in a confifined aquifer. Inverse Probl.
**1990**, 6, 21–28. [Google Scholar] [CrossRef] - Ulm, F.; Torrenti, J.; Adenot, F. Chemoporoplasticity of calcium leaching in concrete. J. Eng. Mech.
**1999**, 125, 1200–1211. [Google Scholar] [CrossRef] - Kuhl, D.; Falko, B.; Meschke, G. Coupled chemo-mechanical deterioration of cementitious materials. Part I: Modeling. Int. J. Solids Struct.
**2004**, 41, 15–40. [Google Scholar] [CrossRef] - Kuhl, D.; Falko, B.; Meschke, G. Coupled chemo-mechanical deterioration of cementitious materials Part II: Numerical methods and simulations. Int. J. Solids Struct.
**2004**, 41, 41–67. [Google Scholar] [CrossRef] - Van Eijk, R.J.; Brouwers, H.J.H. Study of the realation between hydrated portland cement composition and leaching resistance. Cem. Concr. Res.
**1998**, 28, 815–828. [Google Scholar] [CrossRef] - Gong, W.Y.; Cai, Z.H.; Jiang, L.X. Enhancing the performance of differential evolution using orthogonal design method. Appl. Math. Comput.
**2008**, 206, 56–69. [Google Scholar] [CrossRef] - Li, X.; Zhenzhong, S. Permeability coefficient of complex earth rock dam based on elm-ga inversion model and its application. J. Water Resour. Power
**2021**, 39, 86–90. (In Chinese) [Google Scholar] - Zhanping, S.; Annan, J.; Zongbin, J. Back Analysis of Geomechanical Parameters Using Hybrid Algorithm Based on Difference Evolution and Extreme Learning Machine. Math. Probl. Eng.
**2015**, 2015, 821534. [Google Scholar] - Youliang, C.; Yang, C.; Gang, X. Analysis of dam deformation prediction model based on improved bat algorithm and optimized limit learning machine. J. Surv. Mapp. Bull.
**2021**, 9, 68–73. (In Chinese) [Google Scholar]

**Figure 5.**Comparison of measured and simulated values at piezometer ${P}_{4}^{9}$: (

**a**) hydraulic head; (

**b**) hydraulic head relative error.

**Figure 6.**Comparison of measured and simulated values at the dam foundation: (

**a**) leakage; (

**b**) leakage relative error.

**Figure 7.**Impervious curtain porosity evolution in the leaching process: (

**a**) porosity distribution; (

**b**) porosity evolution at 63 m elevation in a century; (

**c**) porosity evolution of three positions.

**Figure 8.**Impervious curtain permeability coefficient evolution in the leaching process; (

**a**) permeability coefficient distribution; (

**b**) permeability coefficient evolution at 63 m elevation in a century; (

**c**) permeability coefficient evolution of three positions.

Test Number | Permeation Parameters | Adaptability Value | ||
---|---|---|---|---|

${\mathit{\Omega}}_{0}$ | $\mathit{n}$ | ${\mathit{k}}_{\mathit{r}}$ | ||

1 | 5000 | 500 | 1 × 10^{−7} | 0.004774 |

2 | 5000 | 800 | 4.6 × 10^{−7} | 0.244917 |

3 | 5000 | 1100 | 8.2 × 10^{−7} | 0.537412 |

4 | 5000 | 1400 | 2.8 × 10^{−7} | 1.086679 |

5 | 5000 | 1700 | 6.4 × 10^{−7} | 1.409979 |

6 | 8000 | 500 | 8.2 × 10^{−7} | 0.01401 |

7 | 8000 | 800 | 2.8 × 10^{−7} | 0.371828 |

8 | 8000 | 1100 | 6.4 × 10^{−7} | 1.055184 |

9 | 8000 | 1400 | 1 × 10^{−7} | 1.258973 |

10 | 8000 | 1700 | 4.6 × 10^{−7} | 1.458692 |

11 | 11,000 | 500 | 6.4 × 10^{−7} | 0.019933 |

12 | 11,000 | 800 | 1 × 10^{−7} | 0.640415 |

13 | 11,000 | 1100 | 4.6 × 10^{−7} | 1.239977 |

14 | 11,000 | 1400 | 8.2 × 10^{−7} | 1.636077 |

15 | 11,000 | 1700 | 2.8 × 10^{−7} | 2.748317 |

16 | 14,000 | 500 | 4.6 × 10^{−7} | 0.035461 |

17 | 14,000 | 800 | 8.2 × 10^{−7} | 0.696786 |

18 | 14,000 | 1100 | 2.8 × 10^{−7} | 1.298395 |

19 | 14,000 | 1400 | 6.4 × 10^{−7} | 2.484507 |

20 | 14,000 | 1700 | 1 × 10^{−7} | 2.892282 |

21 | 17,000 | 500 | 2.8 × 10^{−7} | 0.038281 |

22 | 17,000 | 800 | 6.4 × 10^{−7} | 0.702525 |

23 | 17,000 | 1100 | 1 × 10^{−7} | 1.879253 |

24 | 17,000 | 1400 | 4.6 × 10^{−7} | 2.515877 |

25 | 17,000 | 1700 | 8.2 × 10^{−7} | 4.068743 |

Material | Parameter | Notation | Value |
---|---|---|---|

Rock | Bulk density | ${\gamma}_{r}$ | 25.40 kN/m^{3} |

Rock | Initial porosity | ${\phi}_{f}$ | 0.10 |

Rock | Initial diffusivity | ${D}_{r0}$ | 1.47 × 10^{−11} m^{2}/s |

Concrete | Bulk density | ${\gamma}_{c}$ | 23.51 kN/m^{3} |

Concrete | CH content | ${C}_{c\_CH}$ | 3027 mol/m^{3} |

Concrete | CSH content | ${C}_{c\_CSH}$ | 6054 mol/m^{3} |

Concrete | Initial porosity | ${\phi}_{\mathrm{c}0}$ | 0.10 |

Concrete | Initial diffusivity | ${D}_{\mathrm{c}0}$ | 7.10 × 10^{−12} m^{2}/s |

Impervious curtain | CH content | ${C}_{CH}$ | 3027 mol/m^{3} |

Impervious curtain | CSH content | ${C}_{CSH}$ | 6054 mol/m^{3} |

Impervious curtain | Initial porosity | ${\phi}_{0}$ | 0.15 |

Impervious curtain | Initial diffusivity | ${D}_{0}$ | 9.87 × 10^{−12} m^{2}/s |

Impervious curtain | Intact/leached bulk density | ${\rho}_{0}/{\rho}_{L}$ | 30.6/145.8 |

$\mathbf{Skeleton}$ $\mathbf{Compound}$ | $\mathbf{C}{\mathbf{a}}^{2+}$ $\left(\mathbf{mol}/{\mathbf{m}}^{3}\right)$ | $\mathbf{d}{\mathit{s}}_{\mathit{C}\mathit{a}}/\mathbf{d}{\mathit{c}}_{\mathit{C}\mathit{a}}$ | $\mathbf{Diffusivity}$ $\left({\mathbf{m}}^{2}/\mathbf{s}\right)$ | ${\mathit{\tau}}_{\mathit{leach}}$ $\left(\mathbf{s}\right)$ | $\frac{1}{\mathit{\eta}}$ $\left(\mathbf{mol}/\left(\mathbf{J}\mathit{\xb7}\mathbf{s}\right)\right)$ |
---|---|---|---|---|---|

CH | 19~22 | 2142 | 1.47 × 10^{−9} | 1.17 × 10^{4} | 3.45 × 10^{−8} |

C-S-H | 2~19 | 203 | 1.62 × 10^{−9} | 5.88 × 10^{2} | 7.00 × 10^{−9} |

C-S-H | 0~2 | 1910 | 1.83 × 10^{−9} | 6.52 × 10^{3} | 6.20 × 10^{−8} |

Inversion Parameter | ${\Omega}_{0}$ | $\mathit{n}$ | ${\mathit{k}}_{\mathit{r}}$ $\left(\mathit{m}/\mathit{s}\right)$ | |
---|---|---|---|---|

Optimization Algorithm | ||||

GA | 9334 | 1029 | 5.36 × 10^{−7} | |

SA | 13,583 | 1036 | 1.53 × 10^{−7} | |

SSA | 12,756 | 1225 | 1.48 × 10^{−7} | |

PSO | 12,253 | 1594 | 8.83 × 10^{−8} | |

Range of parameter | 5000~17,000 | 500~1700 | 1.0 × 10^{−8}~1.0 × 10^{−6} |

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**MDPI and ACS Style**

Shu, Y.; Shen, Z.; Xu, L.; Zhang, K.; Yang, C.
Inversion Analysis of Impervious Curtain Permeability Coefficient Using Calcium Leaching Model, Extreme Learning Machine, and Optimization Algorithms. *Appl. Sci.* **2022**, *12*, 3272.
https://doi.org/10.3390/app12073272

**AMA Style**

Shu Y, Shen Z, Xu L, Zhang K, Yang C.
Inversion Analysis of Impervious Curtain Permeability Coefficient Using Calcium Leaching Model, Extreme Learning Machine, and Optimization Algorithms. *Applied Sciences*. 2022; 12(7):3272.
https://doi.org/10.3390/app12073272

**Chicago/Turabian Style**

Shu, Yongkang, Zhenzhong Shen, Liqun Xu, Kailai Zhang, and Chao Yang.
2022. "Inversion Analysis of Impervious Curtain Permeability Coefficient Using Calcium Leaching Model, Extreme Learning Machine, and Optimization Algorithms" *Applied Sciences* 12, no. 7: 3272.
https://doi.org/10.3390/app12073272