Parameter Sensitivity Analysis for Long-Term Nuclide Migration in Granite Barriers Considering a 3D Discrete Fracture–Matrix System

Abstract

As a geological barrier for high-level radioactive waste (HLW) disposal in China, granite is crucial for blocking nuclide migration into the biosphere. However, the high uncertainty associated with the 3D geological system, such as the stochastic discrete fracture networks in granite, significantly impedes practical safety assessments of HLW disposal. This study proposes a Monte Carlo simulation (MCS)-based simulation framework for evaluating the long-term barrier performance of nuclide migration in fractured rocks. Statistical data on fracture geometric parameters, on-site hydrogeological conditions, and relevant migration parameters are obtained from a research site in Northwestern China. The simulation models consider the migration of three key nuclides, Cs-135, Se-79, and Zr-93, in fractured granite, with mechanisms including adsorption, advection, diffusion, dispersion, and decay considered as factors. Subsequently, sixty MCS realizations are performed to conduct a sensitivity analysis using the open-source software OpenGeoSys-5 (OGS-5). The results reveal the maximum and minimum values of the nuclide breakthrough time T_t (12,000 and 3600 years, respectively) and the maximum and minimum values of the nuclide breakthrough concentration C_max (4.26 × 10⁻⁴ mSv/a and 2.64 × 10⁻⁵ mSv/a, respectively). These significant differences underscore the significant effect of the uncertainty in the discrete fracture network model on long-term barrier performance. After the failure of the waste tank (1000 years), nuclides are estimated to reach the outlet boundary 6480 years later. The individual effective dose in the biosphere initially increases and then decreases, reaching a peak value of C_max = 4.26 × 10⁻⁴ mSv/a around 350,000 years, which is below the critical dose of 0.01 mSv/a. These sensitivity analysis results concerning nuclide migration in discrete fractured granite can enhance the simulation and prediction accuracy for risk evaluation of HLW disposal.

1. Introduction

Nuclear energy holds vast potential as a low-carbon energy source [1,2]. However, reprocessing spent fuel produces high-level radioactive waste (HLW), characterized by potent radioactivity, extreme toxicity, and a prolonged half-life. Any leakage of this waste could precipitate severe environmental repercussions and pose significant threats to human health. Currently, deep geological disposal, often designed based on multi-barrier systems comprising engineered and geological barriers, is considered a technically feasible and safe method for the long-term management of HLW [3,4]. These multi-barrier systems, illustrated in Figure 1, are aimed at preventing or delaying the movements of nuclides into the biosphere, reducing environmental impacts to an acceptable level, and achieving long-term safe isolation exceeding 10,000 years. Crystalline or granitic rock, known for its geological stability, low permeability, and desirable mechanical properties, is extensively recognized as a suitable host rock for HLW disposal repositories in countries such as Sweden, Switzerland, Finland, and China, among others. Nonetheless, granite as a fractured porous medium harbors fractures that can potentially serve as potential pathways for groundwater flow, allowing nuclide transport into the biosphere and ultimately compromising the service life of the repository. Thus, accurate characterization of the long-term nuclide migration properties in fractured rocks and precise prediction of their transport processes hold significance for assessing the long-term barrier performance and safety assessment of HLW disposal repositories.

Concerning the long-term barrier performance assessment of fractured geological barriers, extensive studies have been conducted on the flow and transport characteristics of nuclides in fractured rocks. These studies include field and laboratory tests [6,7], such as in situ drilling tracer and underground research laboratory (URL) nuclide migration tests [8,9,10,11], theoretical analyses [12,13,14], and numerical simulations [15,16,17,18]. In situ tracer tests using non-sorbing and sorbing tracers were conducted at the Äspö site in Sweden from 1995 to 2005 to understand nuclide migration and retention processes in fractured rocks [8,19,20]. Similarly, large-scale in situ tracer tests were performed at the Forsmark candidate area for a deep repository in Sweden to investigate the connectivity between water-bearing structures and their transport properties [9,21]. Soler et al. [10] and Ikonen et al. [22] performed long-term in situ diffusion experiments spanning two years at the Grimsel Test Site (GTS) in Switzerland. They found significant diffusion of the tracers into the granitic matrix. Tanaka et al. [11] carried out in situ tracer tests at the Mizunami URL in Japan to determine the transport properties of the rock mass and develop evaluation methods for crystalline rock. These in situ tracer tests provide crucial information on transport parameters (such as the length of the flow path, flow velocity, and longitudinal dispersivity) and retention parameters (retardation factors) for predicting long-term nuclide migration. However, identifying potential pathways in geological barriers and inferring long-term nuclide migration over thousands of years from in situ tracer experiments remain a challenge.

Numerical modeling, coupled with in situ tracer tests and laboratory experiments, provides insights into the long-term performance of disposal repositories [7]. Several simulations have recently been performed to characterize the long-term barrier performance of HLW disposal repositories [13,17,23,24]. Ling et al. [23] employed GoldSim software (https://www.goldsim.com/) to simulate nuclide migration and assess the effectiveness of a geological barrier at the Beishan repository in China, focusing on the 1000 years post-closure of the repository. Duan et al. [24] used Monte Carlo simulation to model structural damage and estimate radiation levels over 1 million years following the closure of the disposal repository. Jia et al. [13] proposed a simulation framework to quantify uncertainty in nuclide migration associated with HLW disposal. They conducted statistical analysis and global sensitivity assessments by integrating the nuclide migration simulator FRACPIPE with the uncertainty quantification tool PSUADE. These studies simulated the interactions and dependencies between repository subsystems and nuclide transport through engineered and geological barriers, simplifying the barrier into a one-dimensional model for fractured media. Nonetheless, these studies did not consider stochastic discrete fracture networks, which govern the realistic behaviors of groundwater flow and nuclide migration within geological barriers, including their geometries, connectivity, and hydraulic properties.

The discrete fracture network (DFN) method is extensively used in modeling fractured rocks because of its ability to explicitly represent discrete fractures [25,26,27,28]. This approach has also been widely used to predict long-term nuclide migration in fractured rocks. Posiva, the Finnish nuclear waste disposal company, established a three-dimensional (3D) DFN model based on the in situ geometric parameters of fractures to calculate the nuclide migration paths in repositories using the ConnectFlow code [29]. Using the dfnWorks framework, Hyman et al. [30] investigated the evolution of the solute transport properties in a kilometer-scale 3D DFN for the SKB (Swedish Nuclear Fuel and Waste Management Company) site in Forsmark, Sweden. Wei et al. [17] combined the DFN model and probabilistic distribution to evaluate the risk of nuclear waste disposal in China and analyze areas with probabilities of contamination. While it accurately captures the geometric and topological characteristics of fracture networks, the DFN model may not be directly suitable for problems involving interactions between fracture networks and the surrounding rock matrix [31,32,33]. This constraint may be addressed using the discrete fracture–matrix (DFM) model. This model enables direct simulation of the solute exchange at the interface between fractures and the rock matrix, allowing for a more comprehensive analysis of solute transport in fracture–matrix systems, and has been widely employed to analyze solute migration in fractured rocks [34,35,36,37]. Ben Abdelghani et al. [38] used the HydroGeoSphere model to explore the mechanism by which conductivity and fracture network properties influence contaminant transport in unsaturated water flow within fractured rock characterized by orthogonal fracture networks. Ma et al. [18] proposed the unified pipe–network method to simulate and analyze nuclide migration in highly fractured rocks with complex fracture networks. Hu et al. [37] and Hyman et al. [39] developed an efficient numerical simulation workflow for investigating problems in fractured porous media using the discrete fracture–matrix approach. Using the multiscale fracture integrated equivalent porous medium method, which uniformly characterizes multiscale fractures (large-, medium-, and small-scale fractures) and the rock matrix, Ma et al. [40] simulated the fluid flow and solute transport in a multiscale fracture–matrix system. The aforementioned studies emphasize the potency of the DFM model in simulating complex heterogeneous fractured media. Although the DFM model has provided valuable insights into the fluid flow and solute transport in fractured rocks, a critical gap remains in our understanding of the long-term barrier performance of repositories, particularly when considering stochastic DFM systems. Numerical simulation based on DFM models offers a feasible approach for predicting nuclide migration in fractured granite. By incorporating the effects of matrix diffusion, this model can provide valuable insights into the long-term behavior of nuclides in geological barriers, thus improving the safety assessment of HLW disposal.

Motivated by the aforementioned unresolved concerns, this study aims to analyze the nuclide migration behavior and evaluate the long-term efficacy of a granite barrier, considering a stochastic discrete fracture–matrix system. This paper is organized as follows: A simulation framework is proposed for evaluating the long-term barrier performance of nuclide migration in a 3D discrete fracture–matrix system by using Monte Carlo simulation (MCS) in Section 2. Statistical data regarding the fracture geometric parameters, on-site hydrogeological conditions, and relevant migration parameters obtained from the research site in Northwestern China are presented. In Section 3, sixty 3D DFM realizations are performed using the open-source software OpenGeoSys-5 (OGS-5). These simulations investigate the migration of three key nuclides—namely Cs-135, Se-79, and Zr-93—within fractured granite. Several mechanisms are considered: adsorption, advection, diffusion, dispersion, and decay. Section 4 discusses the sensitivity analysis of the nuclide breakthrough time T_t and nuclide breakthrough concentration C_max. These analyses are intended to elucidate nuclide migration behaviors and predict the long-term performance of the granite barrier for HLW disposal.

2. Methodology

2.1. Monte Carlo Simulation

Figure 2 illustrates the MCS procedures for assessing the long-term performance of a granite barrier. First, the input parameters, including the geometric indicators of the research object (length, width, depth) and fractures (fracture orientation, length, and density), the key nuclides, physical considerations, and hydrogeological conditions, are obtained. Numerous 3D DFM models are then generated based on the geometric parameters of fractures by using Monte Carlo simulation. Appropriate theoretical analytical models relevant to the study area are chosen, including the rock matrix–fracture interaction, sorption phenomena, and decay processes. Moreover, OGS calculations for N cases of 3D DFM models are performed to determine the flow and concentration fields, breakthrough curves (BTCs), breakthrough time (T_t), and maximum breakthrough concentration (C_max). Finally, a statistical analysis of the output T_t and C_max is conducted to estimate the long-term geological barrier performance of the HLW repository.

2.2. The Three-Dimensional Discrete Fracture–Matrix System

Figure 3 presents the main components of and a conceptual model of nuclide migration in a hypothetical scenario of HLW disposal. The minimum distances from the disposal pit to the water-conducting fractures, water-conducting faults, and rock boundaries are 10, 100, and 1000 m, respectively, as previously reported by Chen et al. [41]. In the current study, the focus is on the geological barrier encompassing the water-conducting fractures (denoted in red in Figure 3). This barrier represents a typical discrete fracture–matrix system. In this system, the following assumptions were made:

(1)

The convection, molecular diffusion, adsorption, and decay mechanisms of the nuclides in the fractured rock are considered.

(2)

The scenario assumes the failure of the containers and buffer materials within a disposal pit 1000 years post-closure of the disposal facility. Thus, the disposal pit is directly connected to the water-conducting fractures, causing nuclide leakage into the fractured granite barrier under groundwater flows.

(3)

The water-conducting fractures in the geological barrier exhibit a stochastic distribution owing to the difficulty of identifying all the fracture geometries at the disposal site. In the following simulation, the water-conducting fracture zones are represented by a discrete fracture–matrix system with scale of 100 m.

(4)

The distances of the disposal pits to the water-conducting fractures varied because the disposal pits were aligned with the disposal facility. For simplicity, we consider the nearest disposal pit, 20 m high and 2 m wide, as the pollution source.

In recent decades, extensive field surveys have been conducted in various preselected regions in the Beishan area of China [4,42,43,44]. These studies have yielded geometric parameters of the fractures, including their orientation, trace length, and density, measured on site. With these parameters, a DFN model can be randomly generated using MCS. For more complex fracture systems, it is shown that power laws and fractal geometry provide widely applicable tools to describe fracture networks [45,46,47]. Specifically, power-law scaling has been applied to characterize trace length, spatial distribution, fracture aperture, fracture roughness, etc., that influence the connectivity of fractured rocks. In the current study, the DFN models are represented by circular disks within a cubic domain, and the geometric parameters of the fractures, such as fracture orientation, dip angle, and density, as reported by Xu et al. [44], are listed in

Table 1. (a) Parameters of fracture sets at the Beishan site in China [44]. (b) The mean fracture length/size at the Beishan site in China [44,48,49].

(a)
Set	Dip Direction (°)	Dip Angle (°)	k (Fisher Constant)	Number	Probability
1	79.2	60.3	19.05	52	0.347
2	330.9	24.9	23.32	30	0.200
3	286.3	75.8	33.96	38	0.254
4	250.5	73.6	15.86	30	0.200
(b)
References		Lei [48]	Xu et al. [44]	Li [49]
Mean fracture length/size		17.08–22.20	2.798–5.883	0.345–1.178

a. The fracture orientation (dip direction and dip angle) follows a Fisher distribution according to Xu et al. [44], whereas the fracture center location and density are assumed to have a uniform distribution. It should be noted that the statistical results on the mean fracture length/size in the Beishan area vary greatly [44,48,49], as summarized in Table 1b. To optimize subsequent simulations and achieve a balance between the computational power and calculation efficiency, this study used the data obtained by Lei et al. [48]. Based on their study, the fracture length/size follows a normal distribution, and the mean fracture length/size and standard deviation were assumed to be 20 m and 10 m, respectively. In addition, when constructing the DFN models, the stop condition for generating fractures in the domain was controlled by giving a total number of fractures to constrain the final number of fractures in the matrix instead of a density index, and to guarantee the generation of the four fractures sets, probability, which was calculated according to the ratio of the number of each set to the total number of the four sets, was used to generate the final DFN models.

To maintain generality and conduct a sensitivity analysis of the nuclide migration characteristics within the water-conducting fracture zones, 60 independent 3D DFM realizations with different geometries were established using the open-source tool dfnWorks [30] based on the parameters in Table 1a. These realization numbers are consistent with our previous work [50], which are enough to demonstrate the nuclide migration results. Furthermore, the fracture aperture was defined in relation to the fracture size with a power-law relationship of $b=5.0\times {10}^{-4}\sqrt{r}$, which is a widely used relationship, as reported in Bonnet et al. [45] and Hyman et al. [51]. Therefore, the fracture aperture varies among the DFNs, with larger fractures having wider hydraulic apertures than the smaller fractures. Figure 4 presents a realization of the 3D DFM model, together with the corresponding meshing result. The model depicts a cubic domain with a side length of 100 m; variations in the fracture color represent differences in the fracture apertures, with the color intensity reflecting the size of the aperture.

It should be noted that the relationship between the nuclide migration characteristics and the connectivity of the fracture network structures, such as the percolation probability and geological entropy, is an important issue for studying nuclide migration in fracture–matrix systems. However, for each DFM realization generated in this study, the fractures intersected with at least one other fracture, and the DFM consisted of at least one connected flow channel with fractures connecting the inlet and outlet surfaces. Equally, isolated fractures that did not intersect with any other fracture and would not contribute to the flow were removed after the DFM was generated. Therefore, all the final DFM models studied in this paper were percolation models.

2.3. Flow and Transport Equations

The governing equations in this study include fluid flow and mass transport in fractured and porous media [52,53]. For saturated rock fractures, the mass balance equation for laminar fluid flow can be expressed as

$$b{S}_f\frac{\partial P}{\partial t}+\nabla ·b{\mathit{u}}_f=0$$

(1)

where $S_f$ is the storage coefficient of the fractures (1/Pa), P is the water pressure, ${\rho }_w$ is the water density, $\mathrm{g}$ is the gravitational acceleration, H is the hydraulic head, and ${\mathit{u}}_f$ is Darcy’s velocity in the fractures (m/s), which can be calculated according to the cubic law

$${\mathit{u}}_f=-\frac{b^2}{12\mu }\left(\nabla P+{\rho }_w\mathrm{g}\right)$$

(2)

where μ is the dynamic viscosity.

For the fluid flow in the rock matrix, which is a porous medium, the mass balance equation can be expressed as

$$S_m\frac{\partial P}{\partial t}+\nabla ·{\mathit{u}}_m=0$$

(3)

where $S_m$ is the storage coefficient of the rock matrix (1/Pa), and ${\mathit{u}}_m$ is Darcy’s velocity in the rock matrix (m/s), which is given by

$${\mathit{u}}_f=-\frac{\mathit{K}}{\mu }\left(\nabla P+{\rho }_w\mathrm{g}\right)$$

(4)

where K is the permeability (tensor) of the porous medium (m²).

The parameters in Equations (2)–(4) are listed in

Table 2. (a) Parameters for Equations (2)–(4) in the 3D DFM simulations. (b) Parameter definition in Equation (5).

(a)
Symbol	Parameter	Value	Unit
$S_f$	Storage coefficient of fractures	1	1/Pa
$S_m$	Storage coefficient of rock matrix	1	1/Pa
g	Gravitational acceleration	9.8	kg/m3
${\rho }_w$	Water density	1000	kg/m3
μ	Dynamic viscosity	1 × 10−3	Pa s
(b)
Symbol	Parameter	Fractures (π = f)	Matrix (π = m)
Cπ	Concentration	Cf	Cm
θπ	Porosity	1	θ
τπ	Tortuosity	1	τ
uπ	Velocity	uf	um
$D_h^{\pi }$	Hydrodynamic dispersion coefficient	αLuf + $D^{*}$	τ$D^{*}$
Rπ	Retardation coefficient	1 + 2Kf/b	1 + ${\rho }_b$Km/θ
λ	Decay constant	ln2/t0.5

a. As for our saturated flow simulation, the storage coefficients of the fractures and the rock matrix are set to 1 (1/Pa).

The nuclide transport through the discrete fracture–matrix system, as shown in Figure 3 and Figure 4, can be calculated using the transport equation with source terms, given by

$$\frac{\partial {\theta }_{\pi }{C}_{\pi }}{\partial t}=\frac{1}{R_{\pi }}\left[\left.\nabla \cdot {\theta }_{\pi }{D}_h^{\pi }\left(\nabla C_{\pi }\right)-\nabla \cdot ({\mathit{u}}_{\pi }{C}_{\pi }\right)\right]-\lambda {\theta }_{\pi }{C}_{\pi }$$

(5)

Definitions of the parameters in Equation (5) are listed in Table 2b, and the corresponding parameters for the simulations can be found in Table 4 in Section 2.4.1.

Notably, the hydrodynamic dispersion coefficient $D_h^{\pi }$ considers the effects of effective molecular diffusion, $D_m^{\pi }={\tau }_{\pi }{D}^{*}$, and mechanical dispersion, $D_d^{\pi }={\alpha }_L^{\pi }{u}_{\pi }$. In addition, θ is the granite rock’s porosity, $\tau$ represents the granite rock’s tortuosity, ${\rho }_b$ is the granite rock’s density, $D^{*}$ is the molecular diffusion coefficient in water, α_L is the longitudinal dispersivity, and $t_{0.5}$ is the half-life of the nuclides.$K_f$ and $K_m$ denote the distribution coefficients of the fracture and the rock matrix, respectively.

In the current study, the open-source finite element method software platform OGS was employed to solve the flow and transport equations. Nuclide exchange between the fractures and the matrix was achieved by coupling the conforming vertices at the fracture–matrix interface. Detailed verification of the numerical method is provided in our previous study [37].

2.4. Simulation Settings

2.4.1. Boundary and Hydrogeological Conditions

Figure 4b and Figure 5 present the boundary conditions for the 3D DFM model. As shown in Figure 4b, the boundary conditions in the 3D DFM model consist of fixed water pressure at the inlet and outlet faces with $P{|}_{S_{in}}=p_1$ and $P{|}_{S_{out}}=p_2$, and no flow occurs at the other four faces, with $\mathit{n}·\nabla P{|}_{S_{im}}=0$, where S_in, S_out, and S_im denote the inlet and outlet face and the four impermeable faces, respectively; $\mathit{n}$ represents the normal unit vector at the boundary surface. The pollution source is positioned at the center of the upstream or inlet face (S_in) for a clear observation of the nuclide migration in the geological barrier. Subsequently, the nuclides migrate in the fracture network channel under the hydraulic gradients between the upstream and downstream water pressures. The evolution of the nuclide release rate downstream over time can then be calculated. Thus, the long-term performance of the geological barrier at a scale of 100 m can be evaluated and analyzed.

China is currently in the exploratory phase of studying the characteristics of spent fuel waste. Considering the similarities between spent fuel waste in China and Japan, this study mainly referred to the H12 report of the Japan Atomic Energy Commission when selecting key nuclides and relevant parameters [54]. According to the simulation results in the H12 report, the release rate of nuclides entering the biosphere is primarily attributed to the nuclides Cs−135, Se−79, and Zr−93 owing to their small distribution coefficient, high solubility, and strong migration ability. Thus, these three nuclides were selected as the key nuclides in our study. Figure 6 illustrates the release rates of the three key nuclides, Cs−135, Se−79, and Zr−93, from the engineering barrier system (EBS) using data from the H12 report, which lists the source concentrations at the inlet boundary. As shown in the graph, the release rates of Cs−135 and Zr−93 initially rise and then fall over time, whereas that of Se−79 gradually decreases over time.

In addition, the migration characteristic parameters of Cs−135, Se−79, and Zr−93 are summarized in

Table 3. Parameters of key nuclides Cs−135, Se−79, and Zr−93 [54].

Nuclide	Parameters of the Nuclides			Parameters of the Granite Mass
	Half-Life t0.5 (Year)	Decay Coefficient λ (year−1)	Dose Coefficient (Sv/Bq)	Diffusion Coefficient D* (m2/s)	Effective Diffusion Coefficient Dd (m2/s)	Distribution Coefficient K (m3/kg)
Cs−135	2.30 × 106	9.56 × 10−15	2.69 × 10−14	1.5 × 10−9	3 × 10−12	0.05
Se−79	6.50 × 104	3.38 × 10−13	8.9 × 10−14			0.01
Zr−93	1.53 × 106	1.43 × 10−14	5.5 × 10−15			0.1

. These parameters include their half-lives, effective diffusion coefficients in granite, distribution coefficients, and dose coefficients. The main pathway by which nuclides could enter the human body from the disposal repository is ingestion; with this factor considered, the ingestion dose coefficient from the International Commission on Radiological Protection was used as the dose coefficient.

Table 4. Hydrogeological conditions in the 3D DFM simulations [18,41,55].

Symbol	Parameter	Value	Unit
J	Hydraulic gradient	0.1%~1%	-
K	Permeability coefficient of granite	3.9 × 10−21~6.6 × 10−17	m2
θ	Porosity of granite	0.58	-
${\rho }_b$	Density of granite	2670	kg/m3
τ	Tortuosity of granite	0.1	-

lists the hydrogeological parameters, including the hydraulic gradient and physical characteristics, of granite rock [41,55]. The site investigations and permeability tests in the Beishan area, China, suggested that the hydraulic conductivity of granite rock in the section of interest (i.e., 400–600 m below the ground surface) is extremely low, ranging between 3.9 × 10⁻¹⁴ and 6.6 × 10⁻¹⁰ m/s (corresponding permeability, 3.9 × 10⁻²¹ and 6.6 × 10⁻¹⁷ m²). Consequently, a permeability value within this range was chosen in this study, and the permeability coefficient of granite was set to 8.3 × 10⁻²⁰ m². Furthermore, to consider potential adverse factors affecting nuclide migration in the geological barrier, the hydraulic gradient was set to 1%.

2.4.2. Safety Assessment Considerations

In the present study, we propose two indicators for evaluating the long-term performance of the granite barrier [55]: (1) Time span: Safety and geological stability assessments from several countries, including Sweden, Finland, Japan, France, the United States, and Germany, suggest that the radiological risk significantly decreases over 1 million years, the maximum duration considered in these studies. Given this finding, we selected a simulation period of 1 million years and a reference time of 10 million years for our analysis of the peak release rates. (2) Safety metrics: Safety assessment indicators include the release rate and the effective dose. The release rate quantifies the rate of nuclide release in Bq/a during the same period. The effective dose represents the cumulative radiation dose (in mSv/a) received by an individual during a given period.

Table 5. Dose limits used by different countries in disposal safety assessments [55].

Country	Type of Geological Barrier	Safety Indicator	Limit Value (mSv/a)
Sweden	Granite	Dose	0.014
Finland			0.1
Japan			0.01
Canada			0.3
Switzerland	Claystone	Dose	0.1
France			0.25
Belgium			0.1
United States	Tuff	Dose	0.15 (10,000 years) 1.0 (1,000,000 years)
Germany	Salt rock		0.01

lists the dose limits used by different countries in disposal safety assessments. The dose limit varies from 0.01 mSv/a to 0.25 mSv/a. In the current study, the minimum value (0.01 mSv/a) was selected as the dose limit, and the maximum effective dose, C_max, was considered for evaluating nuclide decay. The time at which the nuclide concentration reaches C_max was defined as the decay time t_{c, max}. Therefore, based on the 100,000-year upper limit in safety assessments conducted by different countries, barrier failure or success was determined given the following conditions:

(1)

When t_{c, max} ≤ 100,0000 years, if C_max ≤ 0.01 mSv/a, the barrier was deemed effective; otherwise, if C_max > 0.01 mSv/a, the barrier was deemed failed.

(2)

When t_{c, max} > 100,0000 years, the barrier exceeded the upper limit and was deemed effective. Owing to the high toxicity and radioactivity of nuclides, they pose a significant threat to human life and health once they enter the biosphere. Accordingly, we defined the arrival time of the nuclides in the biosphere, T_t, as the moment when the effective dose is equal to 1.0 × 10⁻¹⁰ mSv/a. Thus, T_t was used to reflect the nuclide migration speed in the geological barrier.

3. Simulation Results

3.1. Flow Field

Figure 7 depicts the flow field distribution within the DFNs and the entire fracture–matrix system. Under a 1% hydraulic gradient, groundwater flows through the channels formed in the fracture network. The heterogeneity of the fracture networks leads to a significantly uneven distribution of the flow field, with a maximum flow velocity of 7.5 × 10⁻⁶ m/s and an average flow velocity of 7.46 × 10⁻⁷ m/s. The velocity within the rock mass approaches zero, which is attributed to the extremely low permeability of the granite rock (8.3 × 10⁻²⁰ m²).

3.2. Evolution of the Concentration Field

To elucidate the evolutionary process of nuclide migration within the geological barrier, we selected a typical case from the 60 realizations. To clarify the detailed migration results inside the fractured rocks, the nuclide concentration distributions within both the DFNs and the rock matrix were separately examined, and the concentration distributions in the rock matrix were plotted according to a series of slices at X = 0, 25, 50, 75, and 100 m, Y = 50 m, and Z= 50 m, respectively. Figure 8 and Figure 9 present the migration results on the total nuclide release rates at different moments within the DFNs and the rock matrix (single waste package), respectively. As time progresses, the nuclides migrate within the fracture channels of the geological barrier in response to groundwater advection, molecular diffusion, and other occurrences. Simultaneously, these nuclides infiltrate into the surrounding granite matrix through matrix diffusion. This process widens the distribution range, ultimately reaching the outflow boundary of the geological barrier. However, the successive decay mechanism of Se−73, Cs−137, and Zr−93 prompts a decrease in the nuclide concentrations within the fracture channels after 500,000 years, further narrowing their distribution range.

3.3. Breakthrough Curves

Figure 10 shows the concentration breakthrough curves (BTCs) at the outflow boundary of the geological barrier for the key nuclides Cs−135, Se−79, and Zr−93, calculated from all 60 random realizations. The solid black line represents the cumulative concentration of the three nuclides. The BTCs exhibit variability due to the inherent uncertainty associated with the randomly distributed fracture network structures. Analyzing the average outflow BTCs from all the scenarios, we can conclude that the nuclides reach the boundary at approximately 7000 years (10⁻⁴ Bq/a) within 10 million years following the failure of the waste canister. Subsequently, the total nuclide release rate initially increases and then decreases, reaching a peak value of C_max = 51.6 Bq/a at around 350,000 years. At 1 million years, the total nuclide release rate is 36.2 Bq/a. An analysis of the outflow concentrations of the three nuclides indicates that Se-73 is the earliest to exhibit a change in concentration. The total release rate of the nuclides in the initial stages is mainly attributed to Se−73 with its relatively strong migration ability (small retardation coefficient) and faster efflux rate. Owing to its markedly shorter half-life (65,000 years) than that of Cs−137 (2.3 million years), Se−73 begins to decay during migration over time, prompting a gradual decrease in its concentration. Meanwhile, the Cs−137 and Zr−93 concentrations gradually increase. As the migration time reaches the half-life of Cs−137 and Zr−93, the total release rates of these nuclides gradually decrease. When the Cs−137 concentration rises, its effect on the total release rate of the nuclides is predominant given its significantly higher source concentration compared with the other two nuclides (Figure 6).

4. Discussion

4.1. Sensitivity Analysis of Breakthrough Time

In evaluating the long-term performance of the geological barrier in this study, the individual effective dose is analyzed, and the breakthrough time (T_t) and maximum breakthrough concentration (C_t) of the nuclides are evaluated. A total of 60,000 waste packages are estimated to be generated given the nuclear power development level of China in 2020 [55]. Under the extreme condition that all 60,000 waste packages break and release nuclides after 1000 years, the curve of the total individual effective dose over time can be calculated, as illustrated in Figure 11.

Figure 12 illustrates the nuclide breakthrough time (T_t), together with a statistical histogram and cumulative probability density curve. As shown in Figure 12a, the maximum and minimum values of T_t are 12,000 and 3600 years, respectively, with a significant difference of 3.3 times. This significant difference emphasizes the noticeable effect of the uncertainty in the 3D DFM model on the breakthrough time results. As shown in Figure 12b, the breakthrough time of the nuclides obtained from the uncertainty analysis of the 3D DFM models theoretically follows a normal distribution, with a mean and standard deviation of 6486 and 2412 years, respectively.

4.2. Sensitivity Analysis of Breakthrough Concentration

Figure 13 presents the maximum breakthrough concentration (C_max) of the nuclides alongside a statistical histogram and cumulative probability distribution density curve. Figure 13a shows that the maximum and minimum values of C_max are 4.26 × 10⁻⁴ and 2.64 × 10⁻⁵ mSv/a, respectively, with the former being 16.1 times greater than the latter, indicating a significant difference. As shown in Figure 13b, C_max in all the realizations theoretically follows a normal distribution, with a mean and standard deviation of 7.84 × 10⁻⁵ and 6.04 × 10⁻⁵ mSv/a, respectively, indicating that the uncertainty of the DFM models significantly affects the maximum individual effective dose. This effect further influences the long-term performance of the geological barrier. These findings, combined with Figure 11, suggest that the individual effective dose reaches its peak state (C_max = 4.26 × 10⁻⁴ mSv/a) at around 350,000 years, which falls below the dose limit of 0.01 mSv/a. This preliminary assessment indicates that the geological barrier at a scale of 100 m exhibits satisfactory long-term performance.

5. Conclusions

This study introduces a methodology for assessing the long-term barrier performance of low-permeability fractured rock masses using MCS and a 3D DFM model. The research focuses on a granite geological barrier designated for a proposed high-level radioactive waste (HLW) disposal repository in Northwestern China. It investigates the mechanism by which various parameters related to key nuclides—Cs−135, Se−79, and Zr−93—affect the long-term performance of the geological barrier.

The results indicate that the nuclide breakthrough time (T_t) ranges from 3600 to 12,000 years, with the maximum value being 3.3 times greater than the minimum. Similarly, the maximum and minimum values of the maximum nuclide breakthrough concentration (C_max) are 4.26 × 10⁻⁴ mSv/a and 2.64 × 10⁻⁵ mSv/a, respectively, with the former being 16.1 times higher than the latter. These significant differences underscore the significant effect of the uncertainty in 3D DFM models on the long-term performance of geological barriers. After the failure of the waste tank (1000 years), nuclides are predicted to reach the outlet boundary approximately 6480 years later. Subsequently, the individual effective dose in the biosphere initially increases and then decreases. It peaks at C_max = 4.26 × 10⁻⁴ mSv/a around 350,000 years, which remains below the critical dose of 0.01 mSv/a. These findings suggest the modeling workflow considering a 3D discrete fracture–matrix system can provide a reference for the design of the potential geological disposal of high-level radioactive waste. Furthermore, the sensitivity analysis results concerning nuclide migration in discrete fractured granite can enhance the simulation and prediction accuracy for risk evaluation of HLW disposal.

In this study, we present an uncertainty analysis of field-data-based randomly distributed fracture network structures on long-term nuclide migration in granite barriers considering a 3D DFM model, which is crucial for assessing and managing the risks associated with nuclear waste disposal. The uncertainty analysis is conducted in two steps. First, a large number of (60) Monte Carlo simulations are conducted to provide statistically representative results on the nuclide migration processes using randomly generalized multiple parameter distributions. Secondly, based on these results, the variability in the breakthrough curves, times, and concentrations caused by the inherent uncertainty associated with the randomly distributed fracture network structures is systematically investigated. By quantifying and addressing the uncertainty of fracture network structures, the limitations of the models and results can be better understood, and the results can provide more reliable and robust assessments of the long-term performance of granite barriers for nuclear waste repositories. Furthermore, uncertainty and parameter sensitivity analysis for solute or nuclide migration in fractured rock considering a 3D discrete fracture–matrix system can also offer valuable insights into mineral crystallization/dissolution, groundwater contamination remediation, oil/gas and geothermal energy resources for exploitation. For instance, parameters associated with fractured rock aquifers such as the fracture aperture, connectivity, hydraulic conductivity, hydraulic gradient, diffusion coefficient, porosity, dispersity, and so on are key factors influencing the flow and concentration field in a fracture–matrix system; uncertainty and parameter sensitivity analysis can provide insights into the factors that most strongly influence the fluid flow and solute transport, allowing for the identification of critical parameters and potential pathways for contaminant migration. It may also provide some theoretical guidance for future underground energy developments and environmental assessments in fracture reservoirs.

In the future, a better representation of the 3D DFM models that incorporates a more realistic geometric structure can provide valuable insights into the behavior of fractured rocks under different conditions. Meanwhile, increasingly available field data can be obtained from ongoing repository construction work, or that under planning, of underground research laboratories (URLs), and therefore a more comprehensive and convergent understanding of the long-term performance of HLW repositories for post-closure safety analyses can be obtained based on the simulation methods. Thus, more systematic uncertainty and parameter sensitivity analysis algorithms are necessary to optimize the design and manage the risks associated with long-term nuclide migration in geological repositories, ensuring the safe and sustainable disposal of nuclear waste.