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ISSN : 1738-1894(Print)
ISSN : 2288-5471(Online)

Journal of Nuclear Fuel Cycle and Waste Technology Vol.20 No.3 pp.269-278
DOI : https://doi.org/10.7733/jnfcwt.2022.022

Identification of Mechanical Parameters of Kyeongju Bentonite Based on Artificial Neural Network Technique

Minseop Kim¹

, Seungrae Lee²*

, Seok Yoon¹

, Min-Kyung Jeon²

¹Korea Atomic Energy Research Institute, 111, Daedeok-daero 989beon-gil, Yuseong-gu, Daejeon 34057, Republic of Korea
²Korea Advanced Institute of Science and Technology, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

^* Corresponding Author.
Seungrae Lee, Korea Advanced Institute of Science and Technology, E-mail: srlee@kaist.ac.kr, Tel: +82-42-350-3617

Received May 30, 2022 ; Revised June 17, 2022 ; Approved July 6, 2022

Abstract

The buffer is a critical barrier component in an engineered barrier system, and its purpose is to prevent potential radionuclides from leaking out from a damaged canister by filling the void in the repository. No experimental parameters exist that can describe the buffer expansion phenomenon when Kyeongju bentonite, which is a buffer candidate material available in Korea, is exposed to groundwater. As conventional experiments to determine these parameters are time consuming and complicated, simple swelling pressure tests, numerical modeling, and machine learning are used in this study to obtain the parameters required to establish a numerical model that can simulate swelling. Swelling tests conducted using Kyeongju bentonite are emulated using the COMSOL Multiphysics numerical analysis tool. Relationships between the swelling phenomenon and mechanical parameters are determined via an artificial neural network. Subsequently, by inputting the swelling tests results into the network, the values for the mechanical parameters of Kyeongju bentonite are obtained. Sensitivity analysis is performed to identify the influential parameters. Results of the numerical analysis based on the identified mechanical parameters are consistent with the experimental values.

Key Words : Bentonite , Sensitivity analysis , Artificial neural network , Parameter identification

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1. Introduction

In Korea, several hundred tons of high-level nuclear waste from nuclear power plants are generated every year, and the waste storage capacity is expected to reach a saturation point before 2038 [1]. Because of the high-level radiation and heat from such waste, there is an urgent need for research on permanent high-level nuclear waste disposal facilities. The deep geological disposal method, which is one of the various available disposal methods, disposes of such waste by burying it at a depth of 500 m below the ground using an engineered barrier system (EBS) and natural barrier system (NBS). An EBS is composed of canisters that contain the waste, with a buffer and backfill [2]. Among these components, the buffer fills the voids in the repository system to prevent the leakage of potential radionuclides from damaged canisters [3,4]. Compacted bentonite has been proposed as a candidate buffer material for high-level nuclear waste disposal systems. Because bentonite is composed of smectite, it expands when exposed to water, causing a high swelling pressure in a repository system. [5].

Various models have been proposed to describe the swelling behavior of bentonite. One of the most notable models, the Barcelona basic model (BBM), is a mechanical model that simulates the behavior of partially saturated ground and is widely applied to soils that undergo large expansion due to water [6,7]. Currently, there are no studies related to the properties of the BBM for the case of partially saturated Kyeongju bentonite of Korea.

Such hydro-mechanical model parameters should be characterized through experimental programs [8,9]. However, these experiments are very complicated and require long periods of time to produce an unsaturated state through different methods according to the suction range. Although there have been several studies in which geotechnical parameters were obtained through a back analysis instead of direct experiments [10-12], few studies have employed machine learning methods to identify geotechnical parameters.

In this study, instead of a direct test, the results of a relatively simple swelling pressure experiment [5] were compared to those of a numerical model to propose a method for determining some BBM parameters. The most influential factors were selected through a sensitivity analysis, and the values of these factors were obtained using an artificial neural network (ANN).

2. Methodology

2.1 Identification Process

The process of this study is illustrated in the flowchart shown in Fig. 1.

Fig. 1

Flowchart of study.

The procedure of the sensitivity analysis and parameter identification was as follows.

Step 1. Random sets of parameters were obtained using Latin hypercube sampling [13]. Virtual swelling pressure graphs were then derived by inputting the values of these sets into a numerical model that emulated a swelling test condition.
Step 2. The expansion parameters that influenced the swelling pressure were selected using a sensitivity analysis.
Step 3. The relationship between the swelling phenomenon and expansion parameters was determined using the ANN. The fitting parameters of these swelling pressure graphs were selected as input values, and the expansion parameters of the numerical model were selected as the output values of the ANN.
Step 4. The expansion factors of Kyeongju bentonite were determined by inputting the fitting parameters of the performed swelling test to the ANN.

2.2 Materials and Equipment of Swelling Pressure Test

The bentonite used in the swelling pressure tests was Kyeongju bentonite collected from Kyeongju, Korea, which is a natural smectite. It was fractioned to obtain bentonite powder with a particle size of < 2 μm [14]. The bentonite powder, which had a water content of 13%, was compacted by applying isotropic pressures, and the dry densities of the blocks were 1.59 and 1.73 g·cm⁻³. The compacted bentonite samples were 50 mm in diameter and 20 mm in height.

A photo and schematic view of the swelling pressure test is presented in Figs. 2 and 3. The temperature of the oedometer was adjusted to 30℃, and a vertical load of 0.5 kN was applied. After that, distilled water was injected at a pressure of 1 MPa through the bottom surface of the sample to hydrate it. When the measurement of swelling pressure steadied, the sample was assumed to be saturated [15].

Fig. 2

Photograph of sample placed in oedometer [4].

Fig. 3

Geometry and boundary conditions of numerical model of swelling pressure.

2.3 Numerical Model

2.3.1 Two-phase Flow Model

COMSOL Multiphysics 5.6, which is based on a finite element technique (FEM) [16], was used to simulate the swelling behavior of the bentonite. For a hydraulic–mechanical coupled analysis, two mass balance equations and a momentum balance equation were used. Eq. 1 gives the mass balance equation for a porous media with two phases (gas, and liquid) as follows:

\frac{\partial}{\partial t} \sum_{i} ϕ S_{i} ρ_{i} X_{i}^{x} + \nabla \cdot (J_{A}_{i}^{x} + J_{D}_{i}^{x}) = Q_{m}

(1)

where ϕ is the porosity, S_i is the i state phase saturation, ρ_i is the density of the i state phase, and $X_{i}^{x}$ is the mass fraction. The superscripts denote air (x = a) and water (x = w) components, whereas the subscripts refers gaseous (i = g) and liquid phases (i = l). The total mass flux of component x consists of advective part $J_{A}_{i}^{x}$ and diffusive part $J_{D}_{i}^{x}$ . Darcy’s law [17] is used to express adjective fluxes $J_{A}_{i}^{x}$ , whereas Fick’s rule [18] is used to represent diffusive mass fluxes $J_{D}_{β}^{x}$ in the medium, and each flux term can be expressed as Eqs. 2 and 3.

J_{A}_{i}^{x} = \sum_{i} X_{i}^{x} (- K (\nabla P_{i} - ρ_{i} g))

(2)

J_{D}_{i}^{x} = \sum_{i} (- ϕ S_{i} ρ_{i} D_{i}^{x} \nabla X_{i}^{x})

(3)

In these equations, K is the hydraulic conductivity, P is the pressure, g is the gravitational acceleration vector, and $D_{i}^{x}$ is the diffusion coefficient of component x in phase.

2.3.2 BBM

The BBM [6,9] was used to interpret the mechanical performance of the partially saturated bentonite. The BBM is an elastoplastic model for unsaturated soils that is an extension of the modified cam clay model (MCCM) [19]. To account for the influence of saturation, suction is introduced as an independent variable, and the plastic yield surface is analyzed in a three-dimensional (3D) stress space (p, q, s). This is how the surface equation is written:

q^{2} - M^{2} (p + p_{s}) (p_{0} - p) = 0

(4)

where p is the mean stress, q is the deviatoric stress, M is the slope of the critical line, p_s is the soil strength in extension, and p₀ is the pre-consolidation pressure. The same interpretation as the MCCM can be achieved in the case of a saturated state (i.e. suction is equal to zero), whereas the elastic slope of the compressibility curve versus the net mean stress, κ_i (s), and the plastic slope of the compressibility curve, λ(s), are dependent on the degree of saturation (Fig. 4). To represent this phenomenon, the following equations are proposed:

λ (s) = λ (0) [(1 - r) exp (- γ s) + r], and

(5)

κ_{i} (s) = κ_{i o} (1 + α_{i} s)

(6)

Fig. 4

Compression curves for saturated and unsaturated soil (modified based on [6]).

where r and γ are parameters characterizing the changes in the soil stiffness with suction, and κ_io and α are parameters for the elastic curve slope.

The partially saturated medium’s loading–collapse (LC) curve can be represented as the following equation:

p_{0} = p_{c} {(\frac{p_{0^{*}}}{p_{c}})}^{\frac{λ (0) - κ (s)}{λ (s) - κ (s)}}

(7)

where p₀^* is the pre-consolidation pressure in a saturated condition, and p_c is the reference pressure. A decrease in suction (an increase in the degree of saturation) shows the swelling phenomena when net mean stress p is constant. Specific volume changes with suction changes, $d v_{s}^{e}$ , and the elastic slope of the compressibility curve against the suction, κ_s, can be found as follows:

d v_{s}^{e} = - κ_{s} \frac{d s}{(s + p_{a t m})}

(8)

κ_{s} (p) = κ_{s 0} (1 + α_{s p} ln p / p_{c})

(9)

where p_atm is the atmospheric pressure, and κ_s₀ and α_sp are parameters for the elastic curve slope with the saturation change.

2.3.3 Material Properties

Based on the findings of a previous study [20], the intrinsic hydraulic conductivity of Kyeongju bentonite can be expressed as Eq. 10.

l o g (K) = - 3.806 γ_{d} + 0.008 T - 6.705

(10)

In the above equation, γ_d is the dry density (t/m³), and T is the temperature (℃).

For the water retention curve and relative permeability k_r, respectively, a van Genuchten model and an exponential model were utilized.

S_{e} = {\begin{matrix} \frac{S - S_{r}}{S_{s} - S_{r}} = {[\frac{1}{1 + {| α H_{p} |}^{n}}]}^{m} f o r H_{p} < 0 \\ 1 f o r H_{p} \geq 0 \end{matrix}

(11)

k_{r} = {\begin{array}{l} S_{e}^{L} & f o r H_{p} < 0 \\ 1 & f o r H_{p} \geq 0 \end{array}

(12)

The pressure head is H_p, and the current, maximum, and residual liquid saturation degrees are S, S_s, and S_r, respectively. Furthermore, m is the shape parameter for van Genuchten’s retention curve, represented as 1−(1/n). The input data for the rock used in the numerical model, as well as the hydraulic property employed in the numerical model, are summarized in Table 1 [2].

Table 1

Hydraulic property used in numerical model

3. Computer Design Modeling

3.1 Data Acquisition

In this study, the identification process was performed using the relationship between the BBM parameters for Kyeongju bentonite and the change in swelling pressure over time. In order to obtain such a relational expression through machine learning, it was necessary to secure a large amount of data. First, 500 random sets of κ_io, α_i, κ_s₀, and α_sp (Eqs. 6 and 7) were obtained through Latin hypercube sampling [13]. By inputting the values of this random set into the constructed numerical model, 500 virtual swelling pressure graphs were derived, as shown in Fig. 5(a).

Fig. 5

(a) Virtual swelling pressure growth and (b) fitting of experimental swelling pressure data.

To quantify the swelling pattern, fitting parameters a and b were derived by fitting the swelling graphs to the following equation:

P = \frac{t}{a \cdot t + b^{'}}

(13)

where P is the swelling pressure, and t is the time elapsed in the test. Fig. 5(b) shows the fitting curve of experimental swelling pressure data and derived fitting parameters a and b.

3.2 Sensitivity Analysis

A sensitivity analysis is a study that provides information on the impacts of input variables on target objectives [21]. This study used a variance-based sensitivity analysis (VBSA), which is also known as Sobol’s method. Sobol’s method, which operates within a probabilistic framework, decomposes the variance of a system’s or model’s output into fractions that can be attributed to inputs or sets of inputs [22]. The target objective, g(x), can be decomposed into terms of increasing dimensionality based on the input parameter, x [23], and this form, which is known as the ANOVA form, can be expressed as follows:

g (x) = g_{0} + \sum_{i} g_{i} (x_{i}) + \sum_{i} g_{i j} (x_{i}, x_{j}) + \dots + g_{1 \dots s} (x_{1}, \dots, x_{s})

(14)

where $\int_{0}^{1} g_{i_{1} \dots i_{t}} (x_{i_{1}}, \dots, x_{i_{t}}) d x_{k} = 0, k = i_{1}, \dots, i_{t}$

The term g_i(x_i) denotes the main effect of the factors, which refers to each individual input variable’s contribution to the uncertainty of the target objective. The total variance and partial variances are defined as follows:

D = \int g^{2} (x) d x - g_{0}^{2} and D_{i_{1} \dots i_{s}} = \int g_{i_{1} \dots i_{s}}^{2} d x_{i_{1}} \dots d x_{i_{s}}

(15)

The total and partial variances (Eq. 16) can be used to define the global sensitivity indices, and their sum is one (Eq. 17).

S_{i_{1} \dots i_{s}} = \frac{D_{i_{1} \dots i_{s}}}{D}

(16)

\sum_{s = 1}^{n} \sum_{i \dots i s}^{n} S_{i_{1} \dots i_{s}} = 1

(17)

The global sensitivity indices are commonly used to rank parameters, screen out unnecessary variables, and eliminate high-order members.

3.3 ANN

An ANN is a machine learning technique that can simulate the processing methods of neurons in the human brain in order to determine nonlinear and complex relationships between variables in a system [24,25]. ANNs can handle incomplete or imperfect data better than other statistical methods, allowing them to be used as global approximators even in the absence of knowledge [26]. Therefore, ANNs are emerging as powerful modeling tools.

Fig. 6 depicts the general concept of an ANN model. This structure, known as a multilayer perceptron model, is made up of an input layer, hidden layers, and an output layer. Each neuron is connected to all the neurons in the next layer, and the number of hidden layers and neurons is determined by trial and error based on the model’s complexity. The system’s input values (x_i) are propagated to subsequent layers via transfer functions by multiplying the value by an interconnection weight (w_ij), obtaining the sum of the products, and adding a bias (θ_j) (Eq. 18). The gradient descent method is used to iteratively adjust the weight and bias until the error is minimized based on deviations from the estimated output and the goal value. [10]. In this study, an ANN model with a single hidden layer was created using the Bayesian regularization algorithm [27,28].

y_{i} = f (\sum w_{i j} x_{i} + θ_{j})

(18)

Fig. 6

General concept of ANN model (modified based on [25]).

4. Results and Discussion

4.1 Sensitivity Analysis

The sensitivity analysis results of each expansion parameters κ_io, α_i, κ_s₀, and α_sp (Eqs. 6 and 7) on swelling pressure growth fitting parameters a and b (Eq. 13) are shown in Fig. 7. The higher the global sensitivity index, the higher the influence of parameters, and overall, the tendency was found to be consistent regardless of dry density. Among the four expansion parameters, κ_io and κ_s₀ influenced the value of b, and α_sp affected the value of a. On the other hand, α_i had little influence on the increase in the swelling pressure. According to Eq. 13, it can be seen that κ_io and κ_s₀ describe how rapidly the expansion occurs during the initial stages of saturation, whereas α_sp determines the final swelling pressure. Because α_i had little influence, it was excluded from the parameter identification. In addition, κ_io, which represented the elastic slope for a specific volume–mean stress variation (with zero suction), was excluded from the identification process because it had previously been determined by tests; instead, the experimental value was entered into the model [5]. As a result, the expansion factors, κ_s₀ and α_sp, were obtained using the ANN model.

Fig. 7

Sensitivity analysis results for swelling pressure with dry densities of (a) 1.59 g·cm⁻³ and (b) 1.73 g·cm⁻³.

4.2 Identification of Parameters Using ANN

The identification results for κ_s₀ and α_sp are listed in Table 2, and these values were compared with those of MX80 [29], as a representative bentonite. In the case of κ_s₀, its value increased with the dry density, but the value of α_sp did not show significant changes. The change in the swelling pressure over time was derived by inputting the value obtained through the ANN into the numerical model, and the results were compared with the experimental swelling data (Fig. 8). As a result, it was found that the two swelling pressure change graphs were almost identical.

Fig. 8

Comparison of numerical model and experimental swelling data.

Table 2

Identification results for κ_s₀ and α_sp

5. Conclusions

This study used a simple swelling pressure test, a numerical model, and an ANN to determine the BBM parameters for Kyeongju bentonite, which is found in Korea. A sensitivity analysis was performed to evaluate the influences of the four expansion parameters (κ_io, α_i, κ_s₀, and α_sp) on the swelling pressure. The fitting parameters, a and b, were selected as input values, and the expansion factors, κ_s₀ and α_sp, selected after the sensitivity analysis were selected as output values to construct the ANN model. The expansion factors of Kyeongju bentonite were determined by inputting the experimental data values to this ANN model, which represented the relationship between the swelling pressure graph and the expansion parameters. It was found that the results of the numerical analysis using the identified parameters and the experimental results had similar trends. This method could be applied to identify BBM parameters for other types of bentonite, as well as the properties of other ground materials.

Acknowledgements

This work was supported by the Nuclear Research and Development Program of the National Research Foundation of Korea (NRF) (2021M2E3A2041351) and the National Research Foundation of Korea Basic Research Project (2020R1A2B5B01002067) funded by the Korea government (Ministry of Science and ICT, MSIT).

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Tables

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