Simulating Anomalous Migration of Radionuclides in Variably Saturation Zone Based on Fractional Derivative Model

Abstract

The migration of radioactive waste in geological environments often exhibits anomalies, such as tailing and early arrival. Fractional derivative models (FADE) can provide a good description of these phenomena. However, developing models for solute transport in unsaturated media using fractional derivatives remains an unexplored area. This study developed a variably saturated fractional derivative model combined with different release scenarios, to capture the abnormal increase observed in monitoring wells at a field site. The model can comprehensively simulate the migration of nuclides in the unsaturated zone (impermeable layer)—saturated zone system. This study fully analyzed the penetration of pollutants through the unsaturated zone (retardation stage), and finally the rapid lateral and rapid diffusion of pollutants along the preferential flow channels in the saturated zone. Comparative simulations indicate that the spatial nonlocalities effect of fractured weathered rock affects solute transport much more than the temporal memory effect. Therefore, a spatial fractional derivative model was selected to simulate the super-diffusive behavior in the preferential flow pathways. The overall fitness of the proposed model is good (R² ≈ 1), but the modeling accuracy will be lower with the increased distance from the waste source. The spatial differences between simulated and observed concentrations reflect the model’s limitations in long-distance simulations. Although the model reproduced the overall temporal variation of solute migration, it does not explain all the variability and uncertainty of the specific sites. Based on the sensitivity analysis, the fractional derivative parameters of the unsaturated zone show higher sensitivity than those of the saturated zone. Finally, the advantages and limitations of the fractional derivative model in radionuclide contamination prediction and remediation are discussed. In conclusion, the proposed FADE model coupled with unsaturated and saturated flow conditions, has significant application prospects in simulating nuclide migration in complex geological and hydrological environments.

1. Introduction

With its vast reserves and high power generation efficiency, nuclear energy has become one of the most promising energy sources for the future, and a large amount of nuclear waste has been left in the environment [1]. How to safely dispose of it and prevent leakage into the environment has become an urgent issue. Currently, geological disposal is the internationally recognized method for handling nuclear waste, ensuring the permanent isolation of radioactive materials. Low-level waste (LLW) landfill sites are facilities specifically designed for disposing of low-activity radioactive waste. However, the radioactive decay period of nuclear waste is exceptionally long. In complex geological environments, disposal facilities are prone to gradual aging or damage due to natural disasters such as earthquakes, leading to the instability of the main structures of the disposal sites. This instability can result in the leakage of nuclear waste. Groundwater carries dissolved radionuclides into the biosphere, further threatening the safety of living organisms [2]. Studying the migration patterns in the unsaturated and saturated zones is essential to rapidly predict future diffusion characteristics in the event of radionuclide leakage and to facilitate emergency response measures.

Geological disposal involves encapsulating nuclear waste using engineered methods and natural vadose zone barriers to isolate it from the environment. During the site selection phase for landfill sites, countries prioritize studying radionuclide migration within the vadose zone. Radionuclides first leak into the unsaturated zone before reaching the saturated zone, making research on the retardation and degradation in the unsaturated zone is critically important [3]. Posal sites are typically located in regions where evaporation far exceeds precipitation, such as arid deserts and Gobi areas [4]. Relevant studies have investigated radionuclide migration behavior in unsaturated zones and dynamic environments through modeling, field monitoring, and experimental analyses while evaluating corresponding safety countermeasures. These studies reveal the environmental dispersion mechanisms of low-adsorption radionuclides and identify key pollution control challenges [5,6,7]. In the event of leakage from nuclear storage facilities into aquifers, radionuclides may persist in the ecosystem for extended periods. When radiation doses exceed permissible limits, the probability of human cancers and organ failure significantly increases. This necessitates parallel research on radionuclide transport behavior in saturated zones. The site selection and risk assessment of nuclear waste repositories typically rely heavily on numerical modeling and vadose zone characterization to predict radionuclide migration pathways and potential human exposure risks [8,9,10]. Practical challenges such as difficulties in deep subsurface sampling, soil structure disturbance, monitoring equipment vulnerability, and parameter uncertainties collectively constrain the accuracy of radionuclide migration modeling, while long-term monitoring faces both technical reliability and cost-effectiveness challenges.

Whether studying the unsaturated zone independently or treating the unsaturated and saturated zones as an integrated system, most research overlooks the highly heterogeneous nature of the medium. Moreover, the migration behavior of radionuclides is influenced by chemical adsorption-desorption, reaction rates, and radioactive decay, leading to anomalous diffusion that does not conform to Fick law. This solute transport process is defined as anomalous diffusion [11]. In anomalous transport, the relationship between the mean square displacement of contaminant concentration and time is nonlinear. Anomalous diffusion is categorized into sub-diffusion and super-diffusion. When the medium exhibits heterogeneity, spatial variability, and preferential flow channels, water, and its dissolved substances and colloids may bypass the majority of the soil pore structure [12]. This can lead to super-diffusion in radionuclide migration, causing radionuclide concentrations to exceed safety thresholds prematurely and significantly increasing the risk of radionuclide leakage [13,14].

Researchers have developed various models to describe solute super-diffusion phenomena and have made progress in modeling the transport of heavy metals and other pollutants, such as the stochastic splitting operator method of the traditional advection-diffusion model [15], the mobile-immobile model [16], the continuous-time random walk model [17], and the fractional advection-diffusion model [18]. The fractional derivative model can capture the historical dependence and global correlation of solute transport in complex media, offering advantages such as fewer parameters, clear physical meaning, and precise description [19,20,21]. However, most of the existing fractional derivative models are limited to the simulation of the saturated zone, and there is a lack of effective characterization of the anomalous migration mechanism of nuclides dominated by preferential flow in the unsaturated zone. This knowledge gap directly restricts the safety assessment of low-level radioactive waste landfills. Fractional derivative models primarily focus on simulating and predicting solute transport in the saturated zone, with limited research on the unsaturated zone. The shallow burial of low-level waste repositories and surface freeze-thaw cycles can readily alter the pore structure in unsaturated zones, potentially triggering preferential flow and accelerating radionuclide diffusion. The preferential flow channels are narrow and located underground, making it difficult to observe and repair the preferential flow paths in a timely manner. By the time the nuclide concentration in the observation well rises, it has caused a large-scale diffusion, significantly increasing the restoration cost. This paper introduces the fractional derivative model into the unsaturated zone to simulate the rapid diffusion behavior of nuclides affected by preferential flow after penetrating the unsaturated zone. This model is able to characterize the impact of preferential flow existing in the unsaturated—saturated zone on nuclide migration, reducing the deviation of prediction results. It enables the model to more accurately capture the abnormal diffusion behavior of nuclides, such as early arrival, aiming to reveal the abnormal migration mechanism and improve the predictability of model parameters. It can be further applied to the main body of the landfill pit where preferential flow is formed due to factors such as permafrost degradation, geological disasters, rainfall/irrigation infiltration, and human disturbance, causing material leakage.

2. Methodology Development

The medium without dead ends has a strong water conductivity and serves as the main channel for nuclide transportation. This paper refers to relevant studies and mainly considers: (1) In the actual site, the medium is a heterogeneous anisotropic medium with unobstructed and no dead ends; (2) The migration of nuclides in the medium mainly includes convection and diffusion; (3) There is an adsorption effect of nuclides on the medium surface [22,23,24]. Due to the fact that the research scale of this study is only several decades, the decay of radioactive nuclides is ignored. And this site is a landfill with a concentrated stacking place for pollutants, so the source and sink terms are considered in the equation.

2.1. Fractional Derivative Equation for Saturated Zone

Early studies [25,26,27] were mainly based on the classical advection-diffusion model of Fick’s law. This model is derived based on the hydrodynamic dispersion theory, mass conservation, and the principle of continuity. The advection-diffusion equation based on the Fick law is expressed as Equation (1).

$$R{\displaystyle \frac{{𝜕C}_{(x,t)}}{𝜕t}}=D_x{\displaystyle \frac{{𝜕}^2{C}_{(x,t)}}{𝜕x^2}}-V_x{\displaystyle \frac{𝜕C_{\left(x,t\right)}}{𝜕x}}+f(x,t)$$

(1)

where, $C(x,t)$ is the particle concentration at the space point x and time t; $D_x$ is the dispersion coefficient in the x direction; $V_x$ is the average advection velocity in the $x$ direction of the medium; and $f(x,t)$ is the source and sink term at the space point $x$ and time $t$.

Solute transport exhibits heavy-tail characteristics and early arrival [28,29]. Such behavior exhibits pronounced space nonlocality and long-range correlation [30]. Therefore, it is essential to consider the time dependence and spatial hysteresis in the analysis. During the migration of radionuclides in the environment, they are subject to strong equilibrium adsorption, making it imperative to account for both time dependence and spatial nonlocality. To address these aspects, the time-fractional derivative, which captures time dependence, and the space-fractional derivative, which characterizes spatial nonlocality, are introduced. Given the strong adsorption effect of the medium on radionuclides, a surface retardation factor will also be incorporated into the equation to account for this phenomenon. The one-dimensional and two-dimensional FADE models describing the radionuclide transport process in media are provided by Equations (2) and (3), respectively. The one-dimensional FADE model is applied to simulate solute transport in the unsaturated zone, while the two-dimensional FADE model is used for simulating solute transport in the saturated zone.

$${\displaystyle \frac{{𝜕}^{\alpha }{C}_{(x,t)}}{𝜕t^{\alpha }}}={\displaystyle \frac{D_x}{R_d}}{\displaystyle \frac{{𝜕}^{\beta }{C}_{(x,t)}}{𝜕x^{\beta }}}-{\displaystyle \frac{V_x}{R_d}}{\displaystyle \frac{𝜕C_{\left(x,t\right)}}{𝜕x}}+f(x,t)$$

(2)

$${\displaystyle \frac{{𝜕}^{\alpha }{C}_{(x,y,t)}}{𝜕t^{\alpha }}}={\displaystyle \frac{D_x}{R_d}}{\displaystyle \frac{{𝜕}^{\beta }{C}_{(x,y,t)}}{𝜕x^{\beta }}}-{\displaystyle \frac{V_x}{R_d}}{\displaystyle \frac{𝜕C_{\left(x,y,t\right)}}{𝜕x}}+{\displaystyle \frac{D_y}{R_d}}{\displaystyle \frac{{𝜕}^{\beta }{C}_{(x,y,t)}}{𝜕y^{\beta }}}-{\displaystyle \frac{V_y}{R_d}}{\displaystyle \frac{𝜕C_{(x,y,t)}}{𝜕y}}+f(x,y,t)$$

(3)

where $C(x,y,t)$ is the particle concentration at the space point $(x,y)$ and time t; $R_d$ is the retardation factor; $V_y$ represents the average flow velocity in the y-direction of the medium; $D_y$ denotes the dispersion coefficient in the y-direction of the medium; $f(x,y,t)$ signifies the source and sink term at the space point $(x,y)$ and time t; $\alpha (0<\alpha \le 1)$ is the time-fractional derivative, and $\beta (1<\beta \le 2)$ is the space-fractional derivative.

The initial and boundary conditions are set as shown in Equation (4):

$$\begin{gathered} C_{\left(x,y,0\right)}=\Psi \left(x\right) \\ {C}_{\left(\mathrm{0,0},t\right)}=0,C_{\left(x,0,t\right)}={\phi }_1\left(t\right),C_{\left(0,y,t\right)}={\phi }_2\left(t\right) \end{gathered}$$

(4)

The two-dimensional space-time fractional equation is solved using a stable difference method developed by Lu et al. [31]. The fractional derivative α is capable of capturing the sub-diffusion phenomena in the transport of radionuclides. The smaller the value of α, the stronger the sub-diffusion behavior of radionuclides, resulting in a longer retention time within the medium. This leads to a pronounced tailing phenomenon in the solute breakthrough curve. The fractional derivative β is capable of capturing the super-diffusion behavior in the transport of radionuclides. The larger the value of β, the faster the transmission speed of radionuclides in the channels, leading to an increase in the peak value of the breakthrough curve. This indicates that when α = 1, the equation simplifies to the s-FADE model, and when β = 2, it simplifies to the t-FADE model. When α = 1 and β = 2, the equation simplifies to the classical advection-diffusion equation. Due to the large area of the study region, it can be considered an open boundary. Therefore, the effects of boundary-induced rebound, aggregation, and other influences on solute particles are not taken into account [32].

2.2. Unsaturated-Fractional Derivative Model Development

Compared to the saturated zone, the transport of pollutants in the unsaturated zone is additionally influenced by the soil moisture content [27]. Therefore, to comprehensively simulate the actual conditions of pollutant transport, the volumetric soil moisture content is incorporated into the fractional derivative advection-diffusion model to simulate the migration of radionuclides in the vadose zone. For the unsaturated zone, only one-dimensional vertical infiltration is considered, leading to a revision of Equation (2) as follows:

$${\displaystyle \frac{{𝜕}^{\alpha }{C}_{(x,t)}}{𝜕t^{\alpha }}}={\displaystyle \frac{D_{un}\left(\theta \right)}{R_d}}{\displaystyle \frac{{𝜕}^{\beta }{C}_{(x,t)}}{𝜕x^{\beta }}}-{\displaystyle \frac{V_x\left(\theta \right)}{R_d}}{\displaystyle \frac{𝜕C_{\left(x,t\right)}}{𝜕x}}+f(x,t)$$

(5)

where, $D_{un}\left(\theta \right)$ is the solute hydrodynamic dispersion coefficient of unsaturated soil (cm/min), $V\left(\theta \right)$ is the seepage velocity in the unsaturated zone.

2.2.1. Seepage Velocity in the Unsaturated Zone $V\left(\theta \right)$

The unsaturated soil seepage equation incorporating the space-fractional derivative is used to calculate the seepage velocity in the unsaturated zone, as shown in Equation (6):

$${\displaystyle \frac{𝜕}{𝜕y}}[D(\theta ){\displaystyle \frac{{𝜕}^{(\beta -1)}\theta }{𝜕y^{(\beta -1)}}}]={\displaystyle \frac{𝜕\theta }{𝜕t}}$$

(6)

where, $\theta$ represents the volumetric soil moisture content (cm³/cm³), and $D\left(\theta \right)$ is the water hydraulic diffusion coefficient of the unsaturated soil.

When using a power-law form of the diffusion coefficient $D\left(\theta \right)=D_0{\theta }^n$ [30] in the equation, Equation (6) can be expressed as Equation (7):

$${\displaystyle \frac{𝜕}{𝜕y}}[D_0{\theta }^n{\displaystyle \frac{{𝜕}^{(\beta -1)}\theta }{𝜕y^{(\beta -1)}}}]={\displaystyle \frac{𝜕\theta }{𝜕t}}$$

(7)

where $n$ is an empirical value, typically ranging from 4 to 8. In this study, the intermediate value of 6 is selected; $D_0$ is the initial hydraulic diffusion coefficient of the unsaturated soil.

The seepage velocity in the unsaturated zone is given by Equation (8):

$$V(\theta )=-D_0{\theta }^n{\displaystyle \frac{{𝜕}^{(\mathsf{\beta }-1)}\mathsf{\theta }}{𝜕{\mathrm{x}}^{(\mathsf{\beta }-1)}}}$$

(8)

2.2.2. Solute Hydrodynamic Dispersion Coefficient $D_{un}\left(\theta \right)$

During the initial stage of soil infiltration, solute transport is primarily governed by mechanical dispersion. As the soil approaches saturation, the soil matrix potential tends to zero, and under the influence of factors such as gas resistance, the water flow stabilizes, molecular diffusion playing a dominant role. Numerous studies have demonstrated that although the contributions of molecular diffusion and mechanical dispersion to solute transport in the unsaturated zone vary at different stages, neglecting either of them would result in significant discrepancies between measured and calculated values [33,34,35,36]. Therefore, the commonly used hydrodynamic dispersion coefficient is applied for calculations [37,38], as shown in Equation (9):

$$\begin{gathered} D_m=D_{w}a{e}^{b\theta } \\ {D}_h=\lambda {\nu }^n \\ {D}_{un}\left(\theta \right)=D_m+D_h=D_{w}a{e}^{b\theta }+\lambda {V\left(\theta \right)}^n \end{gathered}$$

(9)

where, $D_m$ is the molecular diffusion coefficient (cm/min), $D_h$ is the mechanical dispersion coefficient (cm²/min), and $D_w$ is the diffusion coefficient of solutes in free water (cm/min). The parameter $\mathit{\lambda }$ stands for dispersity (cm). Within the soil suction range of 0.3–15 atm, b = 10, and a is an empirical coefficient ranging from 0.005 to 0.001. In one-dimensional scenarios, $D_h$ is proportional to the first power of velocity $V$.

The initial and boundary conditions are set as shown in Equation (10):

$$C_{\left(x,0\right)}=\Psi \left(x\right),C_{\left(0,t\right)}=0$$

(10)

where, $x$ and t represent the boundaries in the $x$ and $t$ directions.

The space-fractional seepage model for unsaturated soil infiltration velocity is solved using the finite difference method developed by Wang et al. [39] for one-dimensional case. The calculation flowchart is shown in Figure 1.

3. Methodology Validations

To verify that the fractional derivative model for the unsaturated zone is applicable to solute transport in the unsaturated zone, experimental data on solute transport in the unsaturated zone were collected from published papers. The verification data are all from Diao [40] and Tran et al. [41]. Selecting microplastics and arsenic as test data for simulating the accuracy of solute transport in the unsaturated zone. On the one hand, it is because there are many and detailed studies on both. As a representative organic matter, the adsorption and migration behavior of microplastics in the environment is a current research hotspot. Microplastics, as carriers for comigration, prolong the environmental retention time and diffusion range of pollutants and are dynamically transferred between the water-soil-atmosphere-biosphere. The mobility of arsenic highly depends on its form and environmental conditions. Reductive environments, low pH, low iron content, and environments rich in organic matter usually promote the activation and migration of arsenic. At the same time, many studies have also been conducted on the adsorption and migration characteristics of arsenic. On the other hand, it is because other research data are not easy to obtain.

Diao [40] reported data on the migration of microplastics in well-sorted porous media. Tran et al. [41] presented data on the migration of arsenic (AS) in screened soils. Both approaches relied on sand column experiments to obtain experimental data, where the flow rate was maintained stable under the action of a turbulence pump. Therefore, the validation of the unsaturated zone model assumes steady-state flow conditions. In both datasets, the fractional derivatives (α and β) and the retardation factors require calibration, while the other parameters in the model were calibrated based on the experimental datasets. The fitting results (R² > 0.9, NSE >0.95) indicate that the assumed simulated curves agreed well with the observations.

In the simulation of microplastic migration, 1 um pristine and aged polystyrene microplastics and 30–40 mesh quartz sand are selected as porous media for the experiment. The setting conditions of the six groups of experiments are that the ionic strength is equal to zero (IS = 0), and the concentrations of humic acid (HA) are 0 mg/L, 5 mg/L, and 15 mg/L, respectively [40]. The fitting parameters of the space-time fractional derivatives model are presented in

Table 1. Fractional derivatives parameters for the transport behavior of microplastics in unsaturated porous media.

	Paraments	HA = 0 mg/L		HA = 5 mg/L		HA = 15 mg/L
Types of Microplastics		α	β	α	β	α	β
Pristine microplastics		1	1.83	1	1.81	1	1.89
Aged microplastics		1	1.88	1	1.89	1	1.96

. As shown in Table 1 under identical environmental conditions, the β values of pristine microplastics are consistently lower than those of aged microplastics. For the same type of plastic, the β value increases with rising humic acid concentration. In other words, when the IS = 0, increasing humic acid concentration promotes the aging of microplastics. This results in a weaker spatial nonlocality of microplastic transport in unsaturated porous media, meaning the concentration at the observation point is less influenced by all points in space (rather than just adjacent points). Consequently, the peak concentration increases, as illustrated in Figure 2. β effectively characterizes the influences on the migration of microplastics in the unsaturated zone.

Data 1 to 4 respectively represent the experimental data of groups 1_1 to 1_4 of Tran (V = 0.036 cm/min) [41]. The soil samples are taken from a site in South Korea. The organic matter content is 3.53%, the Fe content is 13.5 g/kg, the coefficient of uniformity is 20, 30% of the soil particles have a particle size less than 0.2 mm, and the PH is 4.9. The sole distinction among groups 1–4 pertains to the bulk density of the medium, which was varied as 1.16, 1.26, 1.36, and 1.46 g/cm³. The original study divided the delay factor into mobile and immobile water zones, whereas the fractional derivatives model only considered the overall delay factor. According to the calibration results of the fractional derivatives model presented in

Table 2. Parameters of AS transport behavior in unsaturated soil.

Data	α	β	Rd	Bulk Density
1	1	1.9	5	1.16
2	1	1.9	6	1.26
3	1	1.88	6	1.36
4	1	1.7	6	1.46

, the value of R_d falls within a reasonable range compared to the data from the original study. As illustrated in Figure 3 the simulation demonstrates a favorable fitting performance. The Pearson correlation coefficient for bulk density exceeds 0.05, leading the original study to conclude that bulk density has no significant influence on the retardation factor. However, analysis based on the fractional derivative model reveals that β exhibits a negative correlation with bulk density. Specifically, as bulk density increases, β decreases, indicating stronger spatial nonlocality. The differences in soil bulk density were attributed to varying degrees of compaction during the experiments. This could be due to the disruption of the original pore structure during the compaction process, further enhancing the soil’s heterogeneity and resulting in a gradual decrease in the parameter β. The spatial fractional parameter also effectively characterizes the influences on the migration of AS in the unsaturated zone.

4. Field Application

4.1. Natural Conditions of the Study Area

This study focuses on near-surface disposal for low-level radioactive waste. The landfill structure, from top to bottom, consists of the following layers: covering soil, multiple impermeable layers, a layer of slightly radioactive waste, and a final impermeable layer, with the bottommost layer comprising natural media. The leakage point is estimated to be located approximately 800 m from the J1 observation well, about 2.6 km from the J3 observation well, and roughly 2.9 km from the J4 observation well.

The average groundwater depth in this area is 10 m. There are rock weathering fissures running 8 m below the surface, which can easily form preferential pathways for the migration of site contaminants, leading to super-diffusion phenomena and reducing the long-term operational safety of the landfill pit. Therefore, modeling studies have been conducted to investigate the influence of weathering fractures on contaminant migration. Typical observation well data were selected as the basis for simulation and prediction, with the wells arranged from left to right as J1, J3, and J4, as shown in Figure 4.

4.2. Parameter Configuration

4.2.1. Analysis of Measured Data

A simulation study was conducted for the 30-year operational period of the landfill from 1993 to 2022. Since the observation well concentration data is only available on a quarterly basis from 2007 to 2021, the evaluation of the goodness-of-fit was solely based on the data within this timeframe. The background value is determined based on the measured data from 2007 to 2021.

Since the lateral migration velocity of soil water in the vadose zone is extremely slow, the site contaminant transport model is simplified as follows: precipitation infiltrates and dissolves the pollutants buried in the site, penetrates the unsaturated zone to reach the aquifer, and then migrates and diffuses through weathered fissures or pores in the rock layer to the J1, J3, and J4 monitoring wells. The longitudinal dispersion coefficient is 22.68 m²/d. Gelhar et al. [42] indicate that within an experimental range of 10⁻¹ to 10⁵, the longitudinal dispersity ranges from 10⁻² to 10⁴ m, with no significant difference in dispersity between fractured media and porous media, by analyzed dispersion data from 59 large regions worldwide. Therefore, in this scenario, despite the differences in migration pathways and the unknown proportion of fractures and pores in the medium, the longitudinal dispersion coefficient based on dispersion test data provided by the geological survey department is still reasonable.

As shown in Figure 5, the rise in nuclide concentration in the J1 monitoring well occurs later compared to the J3 and J4 monitoring wells, with its early-stage concentration remaining relatively stable. The concentration trends in the J3 and J4 monitoring wells exhibit similarities Therefore, it is assumed that the initial release moment of concentration occurred before 2014 during the period from 2007 to 2021, and the model was calibrated accordingly. Based on this, it is inferred that the J3 and J4 monitoring wells may be located on the same spatially connected groundwater flow path. Additionally, the concentration in the J4 monitoring well shows a declining trend in the later stage.

4.2.2. Release Scenario Determination

The migration of nuclides in the unsaturated zone is primarily influenced by water movement, concentration gradients, and the adsorption of nuclides by geological media. In the landfill environment, the release of nuclides mainly occurs under two types: dynamic leaching and static release. Dynamic leaching primarily refers to the release of nuclides caused by water movement, while static release mainly involves the release of contaminants driven by concentration gradients. Radionuclides persist in the soil environment for extended periods, making it difficult to accurately determine their release patterns. Due to the substantial accumulation of pollutants, the initial release concentration can be maintained over a short period. Under the influence of precipitation, there is no noticeable trend in the variation of pollutants within the landfill pit.

Zhou [43] conducted extensive soil column experiments simulating precipitation to investigate the release characteristics of radionuclides. The study revealed that the release of nuclides under dynamic leaching conditions can be divided into three stages: rapid dissolution, abrupt decline in dissolution, and saturation. The nuclides are initially eluted by flowing water and rapidly enter the aqueous phase, leading to a fast release of uranium during this stage. Since the total amount of uranium waste is fixed, the dissolution of nuclides sharply decreases as the concentration of uranyl on the surface diminishes. This process gradually transitions to the diffusion of inner-layer nuclides into the aqueous phase until the release of nuclide concentrations stabilizes entirely by diffusion. After prolonged depletion, the nuclide concentration gradually decreases to match the environmental background level. Yang [44] conducted research on the measured concentrations of leachate from large-scale heavy metal tailings and found that over several decades, the release concentration of tailings did not exhibit a long-term stable declining trend. Instead, the release concentration increased during low-intensity precipitation. Conversely, during high-intensity precipitation, the rapid infiltration rate of water prevented the timely and rapid release of pollutants into the aqueous phase, resulting in a decrease in concentration. However, a study on a landfill in Shanghai [45] found that the concentration of landfill leachate increased under the influence of precipitation. Under the influence of precipitation, the release patterns of pollutants from large-scale landfills do not yield a definitive conclusion.

The precipitation in the study area has the rainy and hot seasons coinciding, with frequent night rain and strong winds. The annual average temperature ranges from −2.4 °C to 1.4 °C, and the annual average precipitation is 426.8 mm, with the precipitation mainly concentrated in summer. The monitoring data shows that the precipitation in this area is concentrated from May to September. Affected by the precipitation, the average groundwater level in the third quarter rises by 1 m. This study assumes that the release pattern of landfill materials under rainfall follows a decay release model, where the rise in groundwater levels during the rainy season results in pollutants being alternately released from the saturated and unsaturated zones.

For the simulation study of this landfill, two release scenarios were established: nuclide leakage when the groundwater level is stable; and nuclide leakage when the groundwater level fluctuates. In the first scenario, the nuclide is released either uniformly or in a decrease manner, whereas in the second scenario, only the attenuated release of the nuclide is considered. Uniform release assumes a constant concentration of pollutant release under prolonged precipitation leaching conditions; decrease release implies that, under extended precipitation leaching, the pollutant is gradually depleted, and its concentration decreases according to a specific reduction pattern. The first release scenarios only include the leaching release of radionuclides, which then penetrate the unsaturated zone to reach the saturated zone. The second release scenario occurs during the rainy season when the groundwater level rises and the bottom of the disposal site is situated within the saturated zone. In this case, pollutant release includes both leaching release and static release. Assuming the pollutant is located 5 m below the ground and the landfill material has a thickness of 5 m. According to monitoring data, the groundwater level fluctuates by approximately 1 m between the rainy season and the dry season.

The initial conditions and boundary conditions for the release under the stable groundwater level situation are shown in Equations (11) and (12).

Uniform release:

$$C_{\left(1,t:t\_N\right)}=90,C_{(0,t)}=0$$

(11)

Decrease release:

$$C_{\left(1,t\_start:t\_N\right)}=C,C_{(0,t)}=0$$

(12)

The initial and boundary conditions for release under groundwater level fluctuation scenarios are shown in Equations (13) and (14):

Unsaturated zone:

$$C_{\left(2,t\_start:t\_N\right)}=C,C_{(0,t)}=0$$

(13)

Saturated zone:

$$C_{\left(2,tt\_starti:tr\_endi\right)}=90,C_{(0,t)}=0$$

(14)

4.3. Simulated Results

4.3.1. Radionuclide Migration in the Unsaturated Zone

Figure 6 illustrates the temporal dynamics of pollutant concentration in the unsaturated zone under three typical release scenarios (D₀ = 22; n = 4; β = 1.98; R_d = 2). D₀ is measured by the indoor soil experiment. The unsaturated zone is composed of the surface cover soil, the impermeable layer, slightly radioactive waste, and the bottom natural soil. The study regards it as a whole. The fractional derivative β = 1.98, and it can be obtained that the locality of the unsaturated zone is relatively weak. There are cracks or large pores in the multi-layer soil structure, and the structure is damaged. The possible reason is that the soil body undergoes freeze-thaw cycles, and the original structure is damaged. The appearance time of the solute peak is advanced, and the peak value is increased, which aggravates the pollution degree. R_d = 2 means that the adsorption capacity of the surface soil medium for nuclides is weak. It can be inferred that the descending stage of the solute breakthrough curve is relatively fast, shortening the residence time of the solute in the unsaturated zone.

Under uniform release conditions (Figure 6), the concentration of the contaminant in the unsaturated zone rapidly increased to a peak value (approximately 100 µg/L) during the initial observation period and remained relatively stable over subsequent time intervals. This suggests that the persistent external influx leads to continuous accumulation of the contaminant in the unsaturated zone, with no significant natural attenuation or migration-dilution phenomena observed. Prolonged exposure to high concentrations of contaminants due to leaks increases the risk of failure in landfill anti-seepage engineering, posing significant hazards. In the scenario of decrease release (Figure 6), the concentration of contaminants initially rises rapidly before gradually declining. The initial concentration reaches approximately 80 µg/L, followed by a decline until the pollutant concentration approaches zero. Given the finite total amount of pollutants in the landfill, the release rate of the pollution source decreases over time, enabling gradual natural purification without human intervention. However, the duration of natural purification is prolonged, exerting a long-term impact on the environment. Under the third release scenario (Figure 6), fluctuations in groundwater levels significantly alter the pollutant concentration at the observation points. The concentration curve exhibits periodic oscillations, with the fluctuation cycle aligning with the frequency of water level changes. During the rainy season, as the water level rises, the pollutants at the bottom come into contact with the aquifer, causing the observation points to be influenced by dual releases from both the unsaturated and saturated zones. During the dry season, as the water level declines, the observation points are solely influenced by the transport of pollutants in the unsaturated zone. The concentration at the observation points shows a significant increase in the initial phase, followed by a notable decrease in pollutant concentration in the later stage. In the initial phase, pollutants released from sources located in the unsaturated zone must traverse this zone to reach the observation points, resulting in a heavy tail effect. Consequently, they do not immediately impact the concentration at the observation points. During the rainy season, the groundwater level rises, initiating the release of pollutants from sources in the saturated zone. Conversely, in the dry season, as the groundwater level declines, the release of pollutants from sources in the saturated zone ceases. In the mid-term, the concentration released from pollution sources in the saturated zone decreases, leading to a reduction in pollutant concentration at the observation points after mixing during the rainy season. In the later stage, the pollutant concentration at the observation points during the rainy season is influenced by the dual effect of reduced release concentrations from both the unsaturated and saturated zone pollution sources.

Under the decrease release mode, although the release intensity of the pollution source shows a decreasing characteristic over time, and the natural attenuation process can effectively reduce the migration flux of pollutants, due to the retardation effect of the unsaturated zone medium on pollutants, the residence time of pollutants in the vadose zone is prolonged. This effect is reflected in the actual observation as a downward trend in pollutant concentration. Although the prevention and control pressure is less than that of the stable release, it is also worthy of attention. When the groundwater level fluctuates, the pollutant release time is advanced. Groundwater will carry a substantial mass of nuclide pollutants to the shallow part in advance, causing them to migrate to the shallow part in advance, significantly expanding the pollution range in a short period of time and significantly increasing the urgency of pollutant treatment. Seasonal fluctuations bring more uncertainties. Also, as decrease release, in the later stage of pollution, the concentration at the observation point still remains at a high level, and there is still a risk of exceeding the limit of the environmental quality standard for a long time in the later stage.

4.3.2. The Fitting Result of Nuclide Migration in the Observation Well

Analyzing the model fitting parameters listed as

Table 3. Parameters of the Saturated Zone Model.

Well		J1	J3	J4
Paraments	Release scenarios	Stable groundwater level: Uniform release
Vx|Vy/(m· d−1)		0.65|0.002	7|0.06	10|0.08
Dx |Dy/(m2·d−1)		22.68|0.22	22.68|0.22	22.68|0.22
Rd		6	6	8
α		1	1	1
βx|βy		1.9|2	1.4|2	1.4|2
Paraments	Release scenarios	Stable groundwater level: Decrease release
Vx|Vy/(m · d−1)		1|0.002	7|0.06	10|0.08
Dx |Dy/(m2·d−1)		22.68|0.22	22.68|0.22	22.68|0.22
Rd		7	6	8
α		1	1	1
βx|βy		1.9|2	1.4|2	1.4|2
Paraments	Release scenarios	Fluctuating groundwater levels
Vx|Vy/(m · d−1)		1|0.002	9|0.06	10| 0.08
Dx |Dy/(m2 · d−1)		22.68|0.22	22.68|0.22	22.68|0.22
Rd		7	6	8
α		1	1	1
βx|βy		1.9|2	1.4|2	1.4|2

, the fitting parameters for the J3 and J4 observation wells are similar, with the maximum lateral groundwater flow velocity reaching 11 m/d and the minimum at 7 m/d, significantly higher than the flow rate calibration results from the J1 observation well. The pollutant response in the J3 and J4 observation wells is relatively rapid. Therefore, it can be deduced that the J3 and J4 observation wells are situated along the same solute transport channel with rapid flow, while the J1 observation well lies independently on a relatively slow pathway.

The time-fractional derivative α for all three observation wells is equal to 1. The FADE model simplifies to a s-FADE model, which aligns with findings from previous studies [18,46]. The smaller the value of α, the more pronounced the heavy tail of the solute in the medium, resulting in slower diffusion and higher pollutant concentrations as diffusion stabilizes. All three observation wells are located on preferential groundwater flow channels, experiencing minimal influence from temporal memory effects, which leads to the occurrence of super-dispersion phenomena. However, compared to the preferential flow channel where J3 and J4 observation wells are located, the groundwater flow velocity in the preferential flow channel of J1 is slower. The spatial fractional derivative β represents the heterogeneity of the porous medium; the larger the value of β, the weaker the heterogeneity of the porous medium and, consequently, the less the retardation effect on pollutant transport. The β value for the J1 observation well is higher than that for the J3 and J4 observation wells. Consequently, the primary reason for the significantly lower concentration in the J1 observation well compared to the others is the complexity of the groundwater flow field. Although the retarding effect of the medium on pollutants is relatively weak, the slower groundwater flow rate leads to a slower response of pollutants in the J1 observation well. The migration paths reaching the J3 and J4 observation wells are significantly longer than that of the J1 observation wells. The longer the migration path, the greater the complexity of the porous medium along the way. Although geological exploration indicates no significant fracture development in the area, small-scale anisotropy of fractures can still influence macroscopic solute migration [17].

Analysis of the fitted curves, as shown in Figure 7, reveals that for observation well J1, under the different pollutant release scenarios, both simulated and predicted concentrations exhibit no decline. The rate of pollutant concentration increase remains rapid, indicating a high potential for further escalation. Under the uniform release scenario of pollutant concentrations at observation wells J3 and J4, the rate of increase in simulated concentrations tends to stabilize in the later stages. Influenced by groundwater levels, the pollutant concentrations at J3 and J4 wells have already peaked. Under the attenuation release scenario, due to the continuous decline in pollutant leakage concentrations from the source, the initial rate of increase in pollutant concentrations observed in the monitoring wells is the slowest compared to the other two scenarios.

Under the fluctuating state of the groundwater level, the early rise in groundwater levels caused the premature release of pollutants, expanding the contamination scope. The early concentrations of pollutants in the three observation wells showed a small increase. However, due to subsequent attenuation, the concentrations in the observation wells declined accordingly. Observation wells J3 and J4, located along the preferential flow pathways for pollutant migration, exhibited rapid responses to pollutant leakage from the source, with initial concentrations rising sharply and stabilizing quickly in the later stages. When pollutants penetrate the unsaturated zone, they show seasonal fluctuations, but there are no obvious fluctuations in the three observation wells. This is mainly because the concentration at the observation point reflects the comprehensive effect of pollutant release and transport, rather than real-time fluctuations. That is, nuclide transport has time nonlocal. The nuclide particle transport process is related not only to the current state but also to the state at previous times. The rapid dilution in the rainy season and the enrichment in the dry season are averaged over a longer time scale, which buffers the changes in nuclide concentration at the observation point. Even if the surface concentration fluctuates sharply, the deep groundwater may show a stable concentration due to the mixing effect.

The observational data indicate that the pollutant concentrations in observation wells J3 and J4 are significantly higher than those in observation wells J1. In the long term, the medium along the diffusion pathway to the observation well J1 exhibits stronger adsorption of radionuclides, with a larger ratio of retardation zones to mobile zones. Since pollutants in the retardation zones can only enter the mobile zones through diffusion, this leads to a tailing phenomenon in the migration of radionuclides [47].

Based on the predictive analysis from 2021 to 2030 (with the measured data up to 2021), the nuclide concentrations in the observation wells J1, J3, and J4 have exceeded the limit. Although the nuclide concentration in the J1 observation well is currently much lower than that in the J3 and J4 observation wells, and the leakage and pollution situation is relatively mild, it shows an increasing trend under the three analysis conditions and has not yet reached the peak, with a strong degree of uncertainty. Specifically, in the stable release scenario, the nuclide concentration in the J1 observation well rises rapidly; in the decrease release, regardless of whether the groundwater level fluctuates, the upward trend of the nuclide concentration slows down in the later stage. In the next 9 years, the nuclide concentration in the J1 observation well is still an object that needs to be focused on for observation. Without artificial intervention, the nuclides near the J1 observation well will exist in the environment for a longer time. The greatest adverse impact it has on the ecological environment is still unknown. The trends of future nuclide concentration changes in the J3 and J4 observation wells are the same. Under the stable release condition, the nuclide concentration has reached the peak and will remain stable in the next 9 years (J3 ≈ 60 µg/L, J4 ≈ 75 µg/L). When the pollution source concentration is decreased, the nuclide concentrations have reached the peak (J3 ≈ 45 µg/L, J4 ≈ 55 µg/L), and a significant downward trend has emerged. The fluctuation of the groundwater level has slowed down the rate of the decrease in the pollutant concentration. When carrying out engineering treatment in the J1 observation well, continuous investment of human and material resources is still required to observe the concentration in the well. The nuclide leakage channels connected to the J3 and J4 observation wells can choose the natural attenuation method to slowly reduce the nuclide concentration. Or, a permeable reactive barrier can be established on the pollutant transportation channel to further prevent nuclide leakage. Avoid further accelerating the nuclide release due to construction disturbances to the geological structure.

4.4. Sensitivity Analysis

In complex mathematical models or engineering systems, variations in input variables can exert different degrees of influence on model outputs. To identify which input variables significantly impact the output results, sensitivity analysis methods are commonly employed. Sensitivity analysis not only aids in optimizing models but also provides valuable information for decision-making. This study employs the Morris sensitivity analysis method Morris [48] to evaluate the impact of various input variables on the model outputs. Compared to other sensitivity analysis methods, such as Variance Decomposition or Sobol Indices, the Morris method offers advantages, including high computational efficiency and suitability for high-dimensional problems. Morris’s sensitivity analysis primarily employs two indices to measure the influence of input variables: the first-order sensitivity index (S) and the total sensitivity index (T).

The first-order sensitivity index measures the direct influence of a single input variable on the model output, quantifying the impact of changes in that input variable on the output while keeping all other input variables fixed. The larger the first-order sensitivity index, the greater the direct influence of that variable on the output. Its calculation is represented by Equation (15), as shown below:

$$S_i={\displaystyle \frac{Var\left(E\right[Y\left|X_i\right])}{Var\left(Y\right)}}$$

(15)

where, $S_i$ is the first-order sensitivity index, $X_i$ represents the variable, and $Y$ denotes the model output variable.

The total sensitivity index measures the combined influence of an input variable and its interactions with other input variables on the model output. The closer the total sensitivity index is to 1, the greater the impact of the interactions between that variable and others on the output results. If the total sensitivity index is relatively small, it indicates that the influence of the variable is primarily due to its direct effects, with weaker interactions with other variables. The calculation of $T_i$ is expressed by Equation (16), as shown below:

$$T_i=1-{\displaystyle \frac{Var\left(E\right[Y\left|X_{~i}\right])}{Var\left(Y\right)}}$$

(16)

where, $X_{~i}$ represents the set of all variables excluding the i-th variable.

Since the time-fractional derivative (α = 1) under the three fitting scenarios, the model simplifies to a spatial fractional derivative model. The unsaturated zone groundwater flow velocity is derived from soil moisture content and the dispersion coefficient is determined based on indoor tests conducted on soil samples collected from the site. Therefore, to accurately simulate and predict the nuclide concentration in the observation well and estimate the contamination source location, sensitivity analysis is required for certain parameters in the unsaturated zone ($\beta , R$) and the saturated zone $(\beta \_x, \beta \_y, e\_x, e\_y, R_d)$. The parameter analysis range is either the range of the calibration results or ±10% of the calibrated values.

The analysis results are shown in Figure 8.

For sensitivity analysis targeting NSE, the first-order sensitivity indices (S) (1.5–0.82) are significantly higher than those for R², while the total sensitivity indices (T) remain at a high level (0.79–1.1). This suggests that the direct influence of parameters on capture the variability in observed data is more pronounced, while there remain strong synergistic effects among parameters. The spatial fractional derivative β holds an absolutely dominant position in the model’s fitting performance. β directly influences the core mechanisms of the model and exhibits complete synergistic effects with other parameters.

For sensitivity analysis targeting R², it is observed that the first-order sensitivity indices (S) are relatively low (0.01–0.014), while the total sensitivity indices (T) are generally high (0.799–1.0). This indicates strong interactions among the parameters, where variations in individual parameters have limited direct influence on the model’s explanatory power, but changes in parameter combinations exert significant effects.

The model exhibits significant nonlinear characteristics. The total sensitivity indices of all parameters are significantly higher than the first-order indices, indicating that the system exhibits strong nonlinear responses, high-order interactions among parameters, and the need to consider combined effects during parameter optimization rather than adjusting parameters individually. The sensitivity indices for RMSE are overall higher. From the trend of parameter variations, the sensitivity of saturated zone parameters is lower than that of unsaturated zone parameters. This implies that the concentration of contamination from the source, as it penetrates the unsaturated zone and reaches the saturated zone, has a stronger impact on the model’s fitting performance.

In the case of limited observation resources, priority should be given to strengthening the monitoring of the unsaturated zone, such as the on-site determination of soil moisture content, medium structure, adsorption coefficient, etc. Especially at the junction between the bottom of the landfill pit and the saturated zone. It is suggested to add monitoring points at the key interfaces of the unsaturated zone between the pollution source and the saturated zone. For example, tensiometers should be set at the bottom of the vadose zone to monitor the water movement front. Further concentration monitoring equipment should be set at the bottom to further calibrate the unsaturated zone breakthrough curve in the model, which will greatly improve the simulation and prediction accuracy of the model.

5. Discussion

5.1. Model Fitting Performance

The performance of the binning calculation model employed by Berghuijs [49]. The binning data in Figure 8 indicate the average observed concentrations within each bin interval (each interval represents 7% of the measurement time).

As illustrated in Figure 9, under the uniform release scenario, the model is able to account for the temporal variations in the majority of the observed data from wells J1, J3, and J4. However, it fails to fully explain the variability and uncertainty of concentrations at all time points for each observation well (The R² of the binned data is greater than that of the overall data; the NSE of the binned data is lower than that of the overall data). Similarly, under the attenuation release and groundwater level fluctuation scenarios, the model performs well in simulating the overall trends of the observed data but lacks strong explanatory power for concentrations at individual time points. Overall, under the three release scenarios, the R² values are all close to 1, indicating a good fit. Whether due to a constant pollution source input leading to a steady-state pollutant migration process, a pollution source release rate that decays over time, or irregular changes in the release scenario caused by water level fluctuations, these assumptions are based on actual scenarios and possess a certain degree of rationality. Furthermore, by flexibly adjusting parameters, a satisfactory fitting result has been achieved in all cases. However, due to the extensive temporal and spatial span of the simulation, many parameters are fitted values rather than actual measurements, which introduces some degree of fluctuation error. Nevertheless, these errors remain within a reasonable range.

The NSE under the attenuation release scenario is slightly lower than that of the other two release scenarios. In the long term, it is observed that pollutants are released in a decaying manner. Under fluctuating groundwater levels, R² and NSE are slightly higher than under the other two scenarios. Therefore, the impact of fluctuating groundwater levels on pollutant transport cannot be ignored.

Analysis of the fitting effect of concentration at different time points in observation wells, the NSE exhibits a clear increasing trend from monitoring well J1 to J4, indicating improved model performance with increasing simulation distance. However, this contrasts with the opposite trend to R². The possible reasons include: (1) Pollutant concentrations at J1 to J4 tend to stabilize in later stages (low variance), resulting in smaller denominators for NSE, where minor errors still yield higher NSE values. (2) Systematic overestimation in the late-stage concentration fitting at J3 and J4 reduces R². Additionally, spatial variations in aquifer parameters and cumulative geological heterogeneity lead to the discretization of solute transport characteristics, enhancing the randomness of particle movement. Currently, the lack of recent observational data makes it difficult to assess late-stage concentration changes in monitoring wells, further complicating model calibration.

While the overall fitting performance of the model is satisfactory under the three scenarios (R² ≈ 1), the spatial variation in NSE reveals the model’s limitations across different scales and mechanisms. In the near-source region (J1), the spatial variability of natural factors is minimal due to its proximity to the pollution source, resulting in higher reliability of the model’s simulation and prediction. In contrast, the far-source regions (J3, J4) exhibit significant medium heterogeneity due to their distance from the pollution source, leading to compromised fitting performance influenced by parameter uncertainties. In actual practice, the specific release scenario to be employed can be selected on a case-by-case basis, depending on the available data and the interpretability of the simulation.

5.2. Advantages and Disadvantages of the Model

This study employs the FADE model, treating the unsaturated-saturated zone as an integrated system, to comprehensively simulate the process of contaminants from the landfill pit penetrating the unsaturated zone after the failure of the impermeable barrier, and subsequently diffusing with groundwater flow to the observation wells in the saturated zone. This approach further improves upon traditional FADE models that focus solely on the saturated zone, thereby significantly broadening the model’s applicability.

The fitting results with observational data demonstrate that the FADE model excels in simulating super-diffusion phenomena under real-world conditions. In natural environments, super-diffusive pathways in groundwater significantly accelerate the migration of radionuclides. In practical applications, the majority of solute transport models in groundwater are based on the ADE model [50]. The rapid migration of radionuclides accelerates the spread of groundwater contamination, and traditional methods that fail to account for super-diffusion phenomena tend to underestimate the risk of pollution.

The fitting results of the three release scenarios exhibit significant differences in detail. However, it is noteworthy that the fitting results for the three observation wells are satisfactory across all three scenarios, this further confirms the broad applicability of the s-FADE model in simulating solute transport, making it suitable for studies involving super-diffusion. In practical applications, the choice of which release scenario to use can depend on the available data and the interpretability of the model. The advantage of the uniform release scenario lies in its ability to conservatively determine the maximum extent of contamination and the peak concentration of pollutants, making it particularly suitable for large-scale landfills where the concentration of pollutant leakage from the source remains constant. The advantage of the decay release scenario is its ability to reflect the progressive reduction in pollutant leakage concentration typical of most landfill pits, as well as to predict the time required for pollutants to reach their peak concentration at observation points over the long term. For pollutant leakage influenced by groundwater level fluctuations, particularly in aquifers primarily recharged by atmospheric precipitation, this scenario aligns more closely with real-world conditions, thereby enhancing the credibility and persuasiveness of the simulation results. To obtain more meaningful simulation results, it is essential to quantify the leakage concentration of radionuclides from the landfill.

The application of fractional derivative models to simulate solute release characteristics from landfills also has some limitations. Firstly, fractional derivative models lack the capability to set release concentrations as functions of both space and time variables. Secondly, the migration of radionuclides in groundwater involves complex physical, chemical, and biological processes, influenced not only by the properties of the pollutants but also by the characteristics of the medium. However, this study primarily focuses on the impact of hydrodynamic factors on the migration of radionuclides. Given the pronounced heterogeneity of soil, it is essential to consider the transient variations in boundary conditions and model parameters. The use of a uniform fractional derivative value in large-scale simulations may lead to deviations in physical significance, thus necessitating the development of variable fractional derivative models.

Although this study demonstrates that the FADE model can be simplified to the s-FADE model in the presence of solute fast transport channels, there is no quantitative description of these channels. These channels can only be qualitatively interpreted based on the observed solute transport data. Furthermore, no studies have provided a precise physical interpretation for α and β, nor have any established relationships between these parameters and the physicochemical properties of the medium or solutes based on extensive experimental data. The adjustment of parameters largely relies on empirical approaches, which hinders the broader application and scalability of this model.

6. Conclusions

For a long time, accurately understanding the migration and transformation patterns of radionuclides in soil has been crucial for the safe disposal of nuclear waste, particularly when predicting the anomalous migration behavior of radionuclides at various scales. Due to the unique characteristics of low-level radioactive waste disposal facilities featuring thick impermeable layers, contaminants initially penetrate through the unsaturated zone before entering the saturated zone, where rapid diffusion occurs via aquifer transport. Neglecting either hydrological domain would introduce significant prediction deviations. This study proposes a unified system integrating both unsaturated and saturated zones, developing a spatiotemporal fractional advection-dispersion model (FADE) to systematically simulate and quantitatively analyze the migration and diffusion dynamics of radionuclides in geological media at nuclear waste repositories. This model can, at the very least, simulate and predict the anomalous diffusion phenomena of radionuclides in the landfill pits discussed in this paper.

When fitting the results of observation well J1, the model reflects the combined influence of spatial heterogeneity and historical memory effects on solute transport in preferential groundwater flow paths. This further highlights the important role of small-scale fracture anisotropy in controlling macroscopic solute migration. When fitting the results of observation wells J3 and J4, the model confirms they share the same solute transport pathway. Compared to well J1, the faster groundwater flow velocity and longer migration paths in the J3 and J4 areas lead to more pronounced pollutant concentration responses. The smaller spatial fractional derivative β values indicate stronger medium heterogeneity.

Parameter sensitivity analysis reveals that the parameters in the unsaturated zone exhibit significantly higher sensitivity than those in the saturated zone. This underscores the necessity of including the unsaturated zone in groundwater solute transport simulations. Moreover, landfill pits are often isolated from the saturated zone through engineering measures. Ignoring the unsaturated zone would lead to overestimated simulation predictions, necessitating increased investment in protective engineering construction or repairs, thereby escalating financial burdens.

Overall, the FADE model provides a robust tool for describing the transport of radionuclides in complex geological media. However, further improvements and refinements are still needed in parameter settings and the coupling of multiple processes to enhance its accuracy and applicability. Future research should focus on refining parameter estimation methods, incorporating additional influencing factors, and validating the model across diverse geological settings to ensure its reliability in predicting radionuclide migration under various scenarios.