Uranium migration prediction model based on particle swarm optimization and application of uranium migration prediction model in deep geological treatment

Abstract

The invention relates to a uranium migration prediction model based on particle swarm optimization and application of the uranium migration prediction model in uranium migration prediction in granite fractures. According to the method, by introducing a retention coefficient and an exponential decay item coupled with pH sensitivity, solute retention behaviors of a fissure domain and a matrix domain are dynamically corrected, and accurate description of adsorption / desorption, precipitation and re-release processes of U (VI) under an acid-alkali condition is realized. Parameter inversion is performed on the model by combining a particle swarm optimization (PSO) algorithm, so that the calculation efficiency and the prediction precision are remarkably improved. Experimental verification shows that the model has high stability and reliability (SSE is smaller than or equal to 0.026, RMSE is smaller than or equal to 0.045, and Error is smaller than or equal to 2.82%) under various pH, pressure and temperature conditions, and an efficient and interpretable analysis tool is provided for uranium migration risk assessment in the deep geological disposal environment.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of radionuclide migration prediction, and in particular to a uranium migration prediction model based on particle swarm optimization and application of the model to uranium migration prediction in deep granite fissures. BACKGROUND

[0002] Deep geological disposal is the main way of safe disposal of high-level waste, and the migration behavior of radionuclides in granite fissure systems is the key to evaluating long-term safety. The traditional dual-domain model has a contradiction between analytical efficiency and chemical interpretability, and it is difficult to simultaneously depict the pH-dependent adsorption / desorption process and the long tail decay of solutes. Existing researches mostly use numerical simulation or empirical correction, which has high computational complexity and limited generalization ability.

[0003] In recent years, with the rapid development of deep geological disposal technology for high-level radioactive waste, the migration behavior and kinetic characteristics of radionuclides in granite structures have become a research hotspot (Wang et al., 2022, 2025; Hu et al., 2024; Tang et al., 2024; Duan et al., 2025a, b). The granite single fissure system, due to its complex pore and permeability structure and significant multiscale heterogeneity, has become an ideal system for studying the interaction of groundwater and rock and the process of nuclide migration (Du et al., 2022; Jia et al., 2022; Ma et al., 2023; Duan et al., 2025). However, the traditional continuous medium model often faces a contradiction between analytical efficiency and physical-chemical interpretability when dealing with the coupling of fissure-matrix dual domains, the interaction of chemical reactions and environmental factors (such as pH), and it is difficult to simultaneously reveal the mass exchange between fissure and matrix domains, adsorption-desorption and precipitation processes (Schwartz, 2012; Trinchero and Iraola, 2020; Tian et al., 2025; Du et al., 2022).

[0004] To remedy this deficiency, researchers have been introducing different physical-chemical coupling mechanisms into the dual-domain or continuum framework (Chopra et al., 2015; Harvey, 2019; Yu et al., 2022; Tanbay and Durmayaz, 2023). Yang et al. (2020) implemented a preliminary characterization of multi-scale retention effects based on the classical dual-domain diffusion model by introducing a mass exchange rate constant between fractures and matrix into the one-dimensional analytical solution, but its assumption of linear adsorption and single kinetic process made it difficult to reflect the regulation of environmental factors on reaction rate. Subsequently, Wei et al. (2021) embedded a nonlinear Langmuir adsorption term in the model, revealing the modification of adsorption isotherms on the shape of migration curves under pH conditions. Although this study improved the accuracy of the model in describing the adsorption process under weak acid-weak base environments, it still did not include chemical reaction generation / dissolution processes. Trinchero and Iraola (2016) coupled the dual-domain framework with complex geological structures using three-dimensional finite element numerical simulation, achieving a comprehensive evaluation of large-scale heterogeneity on diffusion and retention, but the high computational overhead made it difficult to perform rapid parameter sensitivity analysis; Mahmoudzadeh and Crawford (2023) incorporated multi-component chemical reactions (complexation, precipitation, redox) into this framework, which improved the chemical complexity of the model, but its highly nonlinear characteristics significantly reduced the parameter identifiability and solution stability. Li et al. (2024) proposed a semi-empirical pH coupling scheme, which improved the prediction accuracy in neutral to weakly alkaline environments by calibrating pH-dependent adsorption / desorption rate constants, but failed to unify the tail decay characteristics into an analytical expression. Although the above studies have promoted the development of dual-domain models at different levels, there is still a lack of a model framework that can both retain the efficiency of analytical solutions and fully couple pH effects and tail decay kinetics (Jobmann, 2016; Noseck et al., 2021).

[0005] SUMMARY

[0006] As a major long-lived nuclide with high radioactivity and strong chemical activity, U(VI) exhibits strong pH-dependent surface adsorption behavior in the subsurface environment (Bachma and Merkel, 2011; et al., 2015), and are prone to form low-solubility precipitates or complexes under acidic or alkaline conditions (Gaskova and Boguslavsky, 2013; Szecsody et al., 2013), while the adsorption performance is also regulated by factors such as ionic strength, temperature, and organic matter (e.g., humic acid) (Wang et al., 2019), thereby significantly affecting its migration retardation characteristics and far-field diffusion behavior. Therefore, in the evaluation of deep geological disposal, taking U(VI) as the research object not only can deepen the theoretical understanding of its adsorption-desorption, diffusion, and convective transport mechanisms in different rock masses and barrier materials, but also can provide direct experimental and simulation guidance value for the design of high-level radioactive waste repositories, long-term safety assessment, and risk prediction (Alexander et al., 2015; Cao et al., 2019; Flámiková and 2020; Gong et al., 2024).

[0007] In order to balance the calculation efficiency and physical interpretability of the analytical solution, and fully reflect the coupling effect of chemical reaction and pH, the present invention innovatively proposes a uranium migration prediction model based on particle swarm optimization. The model of the present invention is based on a "double erfc function + pH coupling exponential decay" coupling model.

[0008] A uranium migration prediction model based on particle swarm optimization, characterized by comprising the following steps:

[0009] S1, constructing a double-domain model, dividing the fracture system of granite into fracture domain and matrix domain, and describing the convective-dispersion and diffusion retention behavior thereof, respectively;

[0010] S2, introducing a pH-coupled retention coefficient correction term, and modifying the retention factors of the fracture domain and the matrix domain by exponentiation;

[0011] S3, adding a pH-dependent exponential tail decay term to dynamically describe the long tail release behavior of the solute;

[0012] S4, using a particle swarm optimization (PSO) algorithm to globally invert the model parameters to optimize the fitting accuracy.

[0013] A uranium migration prediction method based on particle swarm optimization, characterized by comprising the following steps:

[0014] S1, constructing a double-domain model, dividing the fracture system of granite into fracture domain and matrix domain, and describing the convective-dispersion and diffusion retention behavior thereof, respectively;

[0015] S2, introducing a pH-coupled retention coefficient correction term, and modifying the retention factors of the fracture domain and the matrix domain by exponentiation;

[0016] S3, adding a pH-dependent exponential tail decay term to dynamically describe the long tail release behavior of solutes;

[0017] S4, using particle swarm optimization (PSO) algorithm to globally invert the model parameters and optimize the fitting accuracy.

[0018] In one possible embodiment, the pH-coupled retention coefficient correction term is implemented by the following formula:

[0019] R i,eff = R i exp(k i (pH-pH ref ))(i = 1, 2) (1)

[0020] wherein

[0021] R i,eff : represents the fracture / matrix retention factor after exponential correction coupled with pH;

[0022] R 1， R2: are the retention factors of the fracture domain and the matrix domain, respectively;

[0023] k1, k2: represent the coupling coefficients of pH on the retention factor;

[0024] pH: the pH value in the migration process, specified by experiment or model;

[0025] pH ref : reference pH value, "equivalent reference" used to normalize the parameter scale in the fitting process.

[0026] In one possible embodiment, the pH-dependent exponential tail decay term is implemented by the following formula:

[0027]

[0028] wherein:

[0029] K exp : reference decay rate;

[0030] k pH，exp : represents the coupling coefficient of pH on the exponential decay, linear adjustment of pH on the length of the tail, positive value makes the tail shorter (fast regression) at high pH, and the tail longer (slow regression) at low pH;

[0031] pH: the pH value in the migration process, specified by experiment or model;

[0032] pH ref : reference pH value;

[0033] λexp : represents the exponential decay rate constant (s -1 ), adjusts the steepness of the tail decay;

[0034] t: represents the actual time during the experiment;

[0035] τ: represents the delay time, the delay in the onset of the actual kinetic process from the start of the reaction or measurement.

[0036] In one possible embodiment, the parameter update formula of the particle swarm optimization (PSO) algorithm is:

[0037]

[0038] wherein:

[0039] the velocity of the i-th particle in the j-th dimension;

[0040] the position of the i-th particle in the j-th dimension;

[0041] the historical personal best position of the i-th particle (i.e. the personal best solution, Personal best, p best );

[0042] the best position found in the entire swarm (i.e. the global best solution, Global best, g best );

[0043] r1, r2: uniform distribution random numbers obeying [0, 1];

[0044] c1, c2: individual learning factor and social learning factor, commonly taking the value of 2;

[0045] ω: inertia weight, used to balance the global search and local search capabilities.

[0046] In one possible embodiment, the method is suitable for predicting the migration behavior of U(VI) in granite single fracture system, covering a pH range of 2.9 to 9.4.

[0047] In one possible embodiment, a drift correction term is further included to compensate for the systematic error in the experiment, and its expression is:

[0048] C drift = k lin · t + k quad · t 2 + k pH · (pH - pH ref )· t (5)

[0049] where k lin , k quad , and k pH are drift coefficients.

[0050] In one possible embodiment, the final expression of the model in step S4 is:

[0051]

[0052] where: t e = max(t - τ, 0)

[0053] L: represents the column length (m);

[0054] v: represents the flow rate (m / s);

[0055] t e : represents the effective time;

[0056] t: represents the actual time during the experiment (s);

[0057] τ: represents the delay time, the delay in the onset of the actual kinetic process from the start of the reaction or measurement;

[0058] A, B: represent the weight of the fast / slow zone in the overall erfc response;

[0059] D1, D2: represent the effective dispersion coefficients (m 2 / s) of the fast fracture and slow matrix domains;

[0060] R 1,eff , R 2,eff : represent the fracture / matrix retention factors after exponential correction by pH coupling;

[0061] R1, R2: are the retention factors of the fracture and matrix domains, respectively;

[0062] k1, k2: represent the coupling coefficients of pH on the retention factors;

[0063] pH: pH value during migration, specified by experiment or model;

[0064] pH ref : reference pH value

[0065] k exp : reference decay rate;

[0066] k pH : represents the linear drift coupling coefficient (concentration / pH-time);

[0067] k pH,exp: Represents the coupling coefficient of pH to exponential decay;

[0068] k lin k quad and k pH Drift coefficient;

[0069] γ exp : Represents the exponential decay rate constant (s) -1 Adjust the steepness of tail decline;

[0070] C0: Indicates the offset concentration during measurement.

[0071] An application of a particle swarm optimization-based uranium migration prediction model in uranium migration prediction in granite fractures, characterized by the following steps:

[0072] S1. Collect experimental data of granite single-fracture rock columns, including breakthrough curves under different pH conditions;

[0073] S2. Use the PSO algorithm to invert the model parameters and obtain the optimal parameter combination;

[0074] S3. Predict the migration behavior of U(VI) in the fracture-matrix system based on the optimized model;

[0075] S4. Verify the consistency between the model's prediction results and the experimental data, and evaluate the prediction accuracy.

[0076] A particle swarm optimization (PSO)-based uranium migration prediction model employs a framework of "dual ERFC functions + pH-coupled exponential decay." Two retention coefficients coupled to pH sensitivity are used to correct solute retention behavior in the fracture and matrix domains, respectively. An exponential tail decay term varying with pH is introduced to achieve a dynamic coupled description of U(VI) adsorption / desorption and precipitation processes. A series of column experiments on a single-fracture column of the Beishan Xinchang granite were conducted, and the PSO algorithm was used for parameter inversion, effectively avoiding the problems of local minima and excessive computation in high-dimensional nonlinear optimization. Results show that this model can accurately predict front arrival time, peak shape, and long-tail decay with only a few parameters, providing an efficient, interpretable, and widely applicable analytical tool for uranium migration risk assessment in deep geological remediation environments.

[0077] The beneficial effects of this invention:

[0078] 1. Accurate prediction of uranium migration behavior under complex chemical conditions was achieved through a pH coupling mechanism.

[0079] 2. The PSO algorithm significantly improves the efficiency of parameter inversion and avoids the problem of local minima.

[0080] 3. The model structure is simple, and the parameters are few, and is suitable for rapid evaluation in multiple scenes.

[0081] Other aspects and advantages of the present application will become apparent from the detailed description, taken in conjunction with the accompanying drawings, illustrating the principles of the present application. BRIEF DESCRIPTION OF DRAWINGS

[0082] Figure 1 : Schematic diagram of granite single fissure rock column experimental device.

[0083] Figure 2 : Particle swarm optimization (PSO) iteration flowchart.

[0084] Figure 3 : Comparison diagram of model prediction value and experimental data under normal temperature and pressure (pH = 2.9, 8.2, 9.4).

[0085] Figure 4 : Residual diagram of model prediction value and experimental value under three pH conditions (pH = 2.9, 8.2, 9.4).

[0086] Figure 5 : Comparison diagram of model prediction value and experimental value under pressure and temperature conditions ((a, b): group a, F = 20N, T = 28℃; (c, d): group b, F = 20N, T = 40℃; (e): group c, F = 50N, T = 23℃).

[0087] Figure 6 : Residual diagram of model prediction value and experimental value under pressure and temperature conditions ((a, b): group a, F = 20N, T = 28℃; (c, d): group b, F = 20N, T = 40℃; (e): group c, F = 50N, T = 23℃).

[0088] Figure 7 : Comparison diagram of model prediction value and experimental value under room temperature and no external force conditions ((a): pH = 2.96; (b): pH = 6.54).

[0089] Figure 8 : Residual diagram of model prediction value and experimental value under room temperature and no external force conditions ((a): pH = 2.96; (b): pH = 6.54). DETAILED DESCRIPTION

[0090] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present application,

[0091] Obviously, the described embodiments are only part of the embodiments of the present application, not all the embodiments.

[0092] The components of the embodiments of the present application generally described and illustrated in the accompanying drawings can be arranged and designed in a variety of different configurations. Therefore, the following detailed description of the embodiments of the present application provided in the accompanying drawings is not intended to limit the scope of the application, but is merely representative of selected embodiments of the present application. Based upon the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative work are within the scope of the present application.

[0093] Hereinafter, the terms "include", "have", and their conjugates, used in various embodiments of the present application, merely indicate the presence of the features, numbers, steps, operations, elements, components, or combinations thereof, and should not be construed as excluding the possibility of the presence or addition of one or more other features, numbers, steps, operations, elements, components, or combinations thereof.

[0094] In addition, the terms "first", "second", "third", and the like are used only to distinguish descriptions, and should not be understood as indicating or implying relative importance.

[0095] Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which various embodiments of the present application belong. The terms (such as those defined in commonly used dictionaries) should be interpreted as having a meaning that is the same as the contextual meaning in the relevant technical field and should not be interpreted in an idealized or overly formal sense, unless clearly defined in various embodiments of the present application.

[0096] To balance the computational efficiency and physical interpretability of analytical solution and fully reflect the coupling effect of chemical reaction and pH, this study innovatively proposes a uranium migration prediction model based on particle swarm optimization, which is based on the "double erfc function + pH coupling exponential decay" framework. In the model, the solute retention behavior in the fracture domain and the matrix domain is modified by two pH-sensitive coupled retention coefficients, and an exponential tail decay term that adjusts with pH is added to realize the dynamic coupling description of the U(VI) adsorption / desorption and precipitation process. Combined with the series column experiment of the granite single fracture column in Beishan Xinchang, the particle swarm optimization (PSO) algorithm is used for parameter inversion, which effectively avoids the local minimum and large amount of calculation in high-dimensional nonlinear optimization. The results show that this model can accurately predict the front arrival time, peak shape and long tail decay under the premise of few parameterization, providing an efficient, interpretable and strong applicability analysis tool for uranium migration risk assessment in deep geological disposal environment.

[0097] A uranium migration prediction model based on particle swarm optimization, characterized by comprising the following steps:

[0098] S1, a dual-domain model is constructed, and the fracture system of the granite is divided into a fracture domain and a matrix domain, and the flow-dispersion and diffusion retention behaviors thereof are described respectively;

[0099] S2, a pH-coupled retention coefficient correction term is introduced, and the retention factors of the fracture domain and the matrix domain are corrected by exponentiation;

[0100] S3, a pH-dependent exponential tail decay term is added, and the long tail release behavior of the solute is dynamically described;

[0101] S4, a particle swarm optimization (PSO) algorithm is used to globally invert the model parameters, and the fitting precision is optimized.

[0102] A uranium migration prediction method based on particle swarm optimization, characterized by comprising the following steps:

[0103] S1, a dual-domain model is constructed, and the fracture system of the granite is divided into a fracture domain and a matrix domain, and the flow-dispersion and diffusion retention behaviors thereof are described respectively;

[0104] S2, a pH-coupled retention coefficient correction term is introduced, and the retention factors of the fracture domain and the matrix domain are corrected by exponentiation;

[0105] S3, a pH-dependent exponential tail decay term is added, and the long tail release behavior of the solute is dynamically described;

[0106] S4, a particle swarm optimization (PSO) algorithm is used to globally invert the model parameters, and the fitting precision is optimized.

[0107] In a possible embodiment, the pH-coupled retention coefficient correction term is realized by the following formula:

[0108] R i,eff =R i exp(k i (pH-pH ref )) (i=1,2) (1)

[0109] Wherein

[0110] R i,eff : indicates the corrected effective retention factor;

[0111] R1, R2: the retention factors of the fracture domain and the matrix domain, respectively;

[0112] k1, k2: indicate the coupling coefficients of pH on the retention factors;

[0113] pH: the pH value in the migration process, specified by experiment or model;

[0114] pH refReference pH value, "equivalent benchmark" used to normalize parameter scale during fitting.

[0115] In one possible embodiment, the pH-dependent exponential tail decay term is implemented by the following equation:

[0116]

[0117] where:

[0118] K exp : Benchmark decay rate;

[0119] k ph，exp : Coupling coefficient expressing the influence of pH on the exponential decay, linear adjustment of tail length by pH, positive values make the tail shorter (fast return) at high pH, longer tail (slow return) at low pH;

[0120] pH: pH value during the migration process, specified by experiment or model;

[0121] pH ref : Reference pH value;

[0122] l exp : Exponential decay rate constant (s -1 ), adjusts the steepness of the tail decay;

[0123] t: Actual time elapsed during the experiment;

[0124] r: Delay time, delay in the onset of the actual kinetic process from the start of the reaction or measurement.

[0125] In one possible embodiment, the parameter update formula for the Particle Swarm Optimization (PSO) algorithm is:

[0126]

[0127] where:

[0128] Velocity of the i-th particle in the j-th dimension;

[0129] Position of the i-th particle in the j-th dimension;

[0130] Historical personal best position of the i-th particle (i.e. personal best solution, Personal best, p best );

[0131] Best position found among all particles (i.e. global best solution, Global best, g best );

[0132] r1, r2: uniform distribution random numbers subject to [0, 1];

[0133] c1, c2: individual learning factor and social learning factor, usually taking the value of 2;

[0134] ω: inertia weight, used to balance the global search and local search capabilities.

[0135] In a possible embodiment, the method is suitable for predicting the migration behavior of U(VI) in granite single fracture system, covering a pH range of 2.9 to 9.4.

[0136] In a possible embodiment, a drift correction term is further included to compensate for system errors in the experiment, and its expression is:

[0137] C drift = k lin ·t + k quad ·t 2 + k pH ·(pH - pH ref )·t (5)

[0138] where k lin , k quad and k pH are drift coefficients.

[0139] In a possible embodiment, the final expression of the model in step S4 is:

[0140]

[0141] where: t e = max(t - τ, 0)

[0142] L: represents the column length (m);

[0143] v: represents the flow rate (m / s);

[0144] t e : represents the effective time;

[0145] t: represents the actual time during the experiment (s);

[0146] τ: represents the delay time, the actual kinetic process is delayed from the start of the reaction or measurement;

[0147] A, B: represent the weight of the fast / slow zone in the overall erfc response;

[0148] D1, D2: represent the effective dispersion coefficients of the fast fracture and slow matrix domain (m 2 / s);

[0149] R 1,eff , R 2,eff : represents the fissure / matrix retention factor after the pH coupling exponential correction;

[0150] R1, R2: reference retention factor obtained by global fitting;

[0151] k1, k2: represents the coupling coefficient of pH on the retention factor;

[0152] pH: pH value in the migration process, specified by experiment or model;

[0153] pH ref : reference pH value

[0154] k exp : reference decay rate;

[0155] k pH : represents the linear drift coupling coefficient (concentration / pH·time);

[0156] k pH,exp : represents the coupling coefficient of pH on exponential decay;

[0157] k lin , k quad and k pH : drift coefficient;

[0158] γ exp : represents the steepness of the exponential decay rate constant (s -1 ) adjustment of the tail fading;

[0159] C0: represents the offset concentration at the time of measurement.

[0160] A uranium migration prediction model using particle swarm optimization is applied to the prediction of uranium migration in granite fissures, characterized by the following steps:

[0161] S1, collecting experimental data of granite single fissure rock column, including breakthrough curves under different pH conditions;

[0162] S2, using PSO algorithm to inverse the model parameters, and obtaining the optimal parameter combination;

[0163] S3, based on the optimized model, predicting the migration behavior of U(VI) in the fissure-matrix system;

[0164] S4, verifying the consistency of the model prediction results with the experimental data, and evaluating the prediction accuracy.

[0165] The method used in the present application will be described by way of example using specific embodiment 1:

[0166] Step 1: Preparation of granite samples and fracture formation

[0167] The granite samples used in the experiment were taken from the original rock mass at a depth of 280 m in the Xinchang underground of the Beishan preselected area of the Chinese high-level waste geological disposal repository. First, the mineral composition of the rock samples was quantitatively analyzed using an X-ray diffractometer (model: Philips MPD) after being brought back to the laboratory. The results showed that the quartz content in the granite was 25.6%, plagioclase accounted for 63.1%, and biotite accounted for 11.9%.

[0168] After preliminary cutting and polishing, the rock mass was processed into cylindrical samples with a diameter of 50 mm and a length of 100 mm. To create a single fracture in the rock column, a Brazilian split experiment (Singh et al., 2015) was used to apply a horizontal splitting force to the cylindrical sample, causing it to produce a through fracture along the predetermined direction.

[0169] Step 2: Solution preparation

[0170] The solution used in all experiments was based on natural groundwater. After chemical analysis, the main cations in the surface water sample were K + , Na + , Ca 2+ , and Mg 2+ , with concentrations of 7.10, 455.10, 95.95, and 18.05 mg / L, respectively. The main anions were F - , HCO3 - , Cl - , and SO4 2- , with concentrations of 3.05, 121.29, 651.04, and 670.40 mg / L, respectively (Zhou et al., 2023; Wu et al., 2024).

[0171] The preparation method of the uranium stock solution is as follows: Take the analytical pure uranium nitrate (UO2(NO3)2·10H2O), weigh an appropriate amount, and dissolve it in deionized water to prepare a high-concentration uranium standard solution. The working solution required for the experiment is prepared by diluting a certain volume of uranium stock solution into groundwater, so that the initial uranium concentration reaches 100 mg / L. Then, use trace amounts of NaOH or HCl solution to adjust the pH of the uranium working solution to the predetermined value, and place it under the monitoring of temperature, pH meter, etc. until the pH is stable. The column experiment can be used. In addition to the uranium reagent, all other chemical reagents are of analytical purity and are strictly prepared and stored according to the standard operating procedures.

[0172] Step 3: Column migration experiment

[0173] Column migration experiment was conducted in a prepared single-fracture granite column. The experimental device Figure 1 ) mainly consists of three parts, including the liquid supply system, the fractured column and the automatic collection system.

[0174] Before the experiment, the fracture was pre-saturated with background groundwater for 24 h to ensure uniform ion environment and stable pH inside the fracture. The column experiment was divided into dynamic adsorption and elution stages.

[0175] Dynamic adsorption stage: The uranium working solution with the target pH was input into the fractured column at a constant concentration, and the pump-controlled flow rate was maintained at 2.25 mL / h (equivalent to a fracture equivalent linear velocity of 1.4 × 10 -4 m / s). The input continued until the uranium concentration in the outlet liquid no longer changed significantly and reached a steady-state rising platform. During this period, the effluent was collected at intervals, and the uranium concentration was determined using a UV-visible spectrophotometer with appropriate color reagents. The concentration at each time was expressed as a relative concentration C / C0, where C0 was the input flow concentration (100 mg / L).

[0176] Elution stage: After the dynamic adsorption experiment, the input flow was switched to a solution containing only background groundwater, and elution was carried out at the same flow rate. The effluent was continuously collected and the uranium content was determined until the uranium concentration in the sample decreased to the detection limit or was essentially zero.

[0177] Based on the data of the relative concentration of the effluent collected during the experiment varying with time, the breakthrough curves (BTC) of uranium under different pH conditions were plotted. On this basis, the particle swarm optimization algorithm (PSO) was used to fit and analyze the model parameters, and a mathematical model reflecting the influence of pH changes on uranium migration behavior was constructed.

[0178] Step 4: Two-domain model (corresponding to S1, the granite fracture system is divided into fracture domain and matrix domain, and the convection-dispersion and diffusion retention behaviors are described respectively)

[0179] To comprehensively depict the complex transport process of nuclides in the granite fracture-matrix system, which is dominated by both the fracture region convection-mechanical dispersion and the matrix region molecular diffusion retention and subsequent release, the invention is based on the classic dual-domain model. The model divides the porous medium into two parts: one part is the fracture domain with high porosity and strong connectivity, in which fluid convection is dominated by the average velocity v and accompanied by mechanical dispersion; the other part is the matrix domain with low porosity and weak connectivity, in which solute mainly realizes retention and release by molecular diffusion. The fracture domain and the matrix domain are coupled through a first-order reversible mass transfer term, and the mass transfer rate is characterized by the parameter α (unit: s -1 ) reflecting the interface non-equilibrium exchange strength:

[0180]

[0181] Where C f and C m are the concentrations in the fracture and matrix domains, respectively.

[0182] Under the assumption of one-dimensional parallel fractures, step or constant inlet concentration, and flow rate, the fracture-matrix mass conservation equation can be expressed as:

[0183]

[0184] Where:

[0185] C f : nuclide concentration in the fracture domain;

[0186] C m : nuclide concentration in the matrix domain;

[0187] R 1， R2: the retention factors of the fracture domain and the matrix domain, respectively;

[0188] v: the average flow rate in the fracture;

[0189] D f , D m : total dispersion / diffusion coefficients of the fracture and the matrix, respectively;

[0190] λ: nuclide decay constant;

[0191] Γ: first-order mass transfer rate between the fracture and the matrix;

[0192] t: actual time during the experiment.

[0193] Under the assumption of one-dimensional parallel fractures and constant inlet gradient, the analytical solution can be derived as a superposition of two complementary error functions (erfc) (Huang et al., 2021; Ma et al., 2022):

[0194]

[0195] where:

[0196] L: represents the column length (m);

[0197] v: represents the flow velocity (m / s);

[0198] D1, D2: represent the effective dispersion coefficients of the fast (fracture) and slow (matrix) domains (m 2 / s);

[0199] R1, R2: are the retention factors of the fracture and matrix domains, respectively;

[0200] A, B represent the weights of the fast / slow zones in the overall erfc response.

[0201] Step 5: pH coupling and empirical correction (corresponding to S2, the introduction of a pH-coupled retention factor correction term, by exponentializing the retention factors of the fracture and matrix domains, S3, the addition of a pH-dependent exponential tailing term, dynamically describing the long-tailing release behavior of solutes)

[0202] In deep geological environments or laboratory column experiments, the solution pH not only affects the charge state and coordination chemistry of nuclides, but also significantly adjusts the migration characteristics of fracture-matrix systems by changing the mineral surface electrical properties and interfacial mass transfer resistance (Glover, 2018; Deng and Spycher, 2019; Townsend et al., 2019). Therefore, the present invention directly couples pH as part of the intrinsic mechanism of the model based on the two-domain analytical solution, and supplements it with empirical correction terms to account for non-ideal behavior under complex chemical conditions.

[0203] First, the retention factor (adsorption strength) is exponentially modified by pH coupling:

[0204] R i,eff i i exp(k i (pH-pH ref )) (i=1,2) (4)

[0205] where

[0206] R i,eff : represents the fracture / matrix retention factor after exponential modification by pH coupling;

[0207] R1, R2: Retention factor of fissure and matrix domain, respectively;

[0208] k1, k2: Coupling coefficient of pH to retention factor;

[0209] pH: pH value during transport, specified by experiment or model;

[0210] pH ref : Reference pH value, "equivalent benchmark" used to normalize parameter scale during fitting.

[0211] The pH-coupled retention factor can continuously and smoothly describe the sensitivity of pH to adsorption equilibrium, avoiding recalibrating a completely new R for each pH point, and also retaining the physical "multiplicativity" in the model - the acid-base effect is proportionally superimposed on the benchmark retention factor to map the multiplicative effect of pH change on adsorption equilibrium (distribution coefficient K d ) directly to the retention factor R. Then superimpose an exponential decay term at the tail:

[0212]

[0213] Where:

[0214] K exp : Benchmark decay rate;

[0215] k pH，exp : Coupling coefficient of pH to exponential decay, linear adjustment of tail length by pH, positive value makes the tail shorter (fast regression) at high pH, and longer (slow regression) at low pH;

[0216] pH: pH value during transport, specified by experiment or model;

[0217] pH ref : Reference pH value;

[0218] λ exp : Exponential decay rate constant (s -1 ), adjust the steepness of the tail decay;

[0219] t: Actual time during the experiment;

[0220] τ: Delay time, the delay in the start of the actual kinetic process from the start of the reaction or measurement.

[0221] Exponential decay characterizes the rate of concentration "regression platform" in the tail of the breakthrough curve (after C / C0→1), i.e. the combined effect of matrix diffusion release and adsorption-desorption kinetics.

[0222] Simultaneously introduce multi-order drift and linear pH drift terms to capture weak baseline flow rate / detection drift;

[0223] Linear drift:

[0224] C lin (t) = k lin t e (6)

[0225] where:

[0226] C lin represents the concentration contribution due to linear background change, with units consistent with concentration;

[0227] k lin represents the linear drift coefficient (concentration / time);

[0228] t e represents the experimental run time (s).

[0229] Quadratic drift is used for more complex system drift trends;

[0230]

[0231] where:

[0232] C quad : represents the amount of concentration drift that grows with the square of time, with units consistent with concentration;

[0233] k quad : represents the quadratic drift coefficient (concentration / time);

[0234] t e : represents the experimental run time (s).

[0235] pH-coupled linear drift is used to correct for secondary drift caused by the interaction of pH with time.

[0236] C pH (t) = k pH (pH - pH ref )t e (8)

[0237] where:

[0238] C pH : represents the concentration contribution due to linear background drift caused by the pH level deviating from the reference value, with units consistent with concentration; k pH : represents the linear drift coupling coefficient, (concentration / pH-time);

[0239] pH: the pH value during the migration process, specified by experiment or model;

[0240] pH refReference pH value, "equivalent benchmark" used in the fitting process to normalize the scale of the parameters;

[0241] t e : denotes experimental run time (s).

[0242] Finally, a time lag τ and a constant offset C0 are added to correct for injection delay and baseline drift, respectively.

[0243] With the above corrections, the expression for the normalized concentration C = C / C0 is:

[0244]

[0245] where: t e = max(t-τ,0)

[0246] L: denotes column length (m);

[0247] v: denotes flow rate (m / s);

[0248] t e : denotes effective time;

[0249] t: denotes actual time during the experiment (s);

[0250] τ: denotes delay time, the delay in the onset of the actual kinetic process from the start of the reaction or measurement;

[0251] A, B: denote the weight of the fast / slow zone in the overall erfc response;

[0252] D1, D2: denote the effective dispersion coefficients (m2 / s) of the fast fissure and slow matrix domains; 2

[0253] R 1,eff , R 2,eff : denote the fissure / matrix retention factors after exponential correction with pH coupling;

[0254] R1, R2: are the retention factors for the fissure and matrix domains, respectively;

[0255] k1, k2: denote the pH coupling coefficients for the retention factors;

[0256] pH: pH value during migration, specified by experiment or model;

[0257] k exp : benchmark decay rate;

[0258] k pH : denotes linear drift coupling coefficient (concentration / pH-time)

[0259] ​k pH,exp : represents the coupling coefficient of pH on exponential decay;

[0260] γ exp : represents the steepness of tail decay adjusted by exponential decay rate constant (s -1 ).

[0261] C0: represents the offset concentration at the time of measurement.

[0262] This method not only retains the physical mechanism of the dual-domain model, but also achieves high-precision fitting of the shape and amplitude of the breakthrough curve under different pH conditions through pH coupling and polynomial modification.

[0263] 2.6. Particle swarm optimization (PSO) (corresponding to S4, global inversion of model parameters using particle swarm optimization (PSO) algorithm to optimize fitting precision)

[0264] To overcome the challenges of high-dimensional nonlinearity, multi-modal search space, and dependence on gradient information in the solution process in parameter inversion based on the dual-domain model, and to avoid falling into local minimum, the particle swarm optimization (PSO) algorithm is introduced in the initial inversion stage. This method was proposed by Kennedy and Eberhart in 1995, and is based on the simulation of group cooperative behavior. It has the advantages of simple structure, few parameters, and fast convergence speed. In PSO, a particle represents a candidate solution, and searches for the optimal solution in the solution space through position and velocity. Under the guidance of individual and group optimization, it iterates continuously and gradually approaches the global optimal solution. Figure 2 The overall process of PSO algorithm in the parameter inversion process of this study is given.

[0265] Let the search space be n-dimensional, and there are N particles in the population. The position and velocity of the i-th particle are represented as:

[0266] x i = (x i1 , x i2 ,..., x in ) (10)

[0267] v i = (v i1 , v i2 ,..., v in ) (11)

[0268] In each iteration, the velocity and position of the particle are updated according to the following formula:

[0269]

[0270] Where:

[0271] Velocity of the i-th particle in the j-th dimension;

[0272] Position of the i-th particle in the j-th dimension;

[0273] Personal best position of the i-th particle in history (i.e. Personal best, p best );

[0274] Global best position found in the swarm (i.e. Global best, g best );

[0275] r1, r2: Uniformly distributed random numbers in [0, 1];

[0276] c1, c2: Personal and social learning factors, usually set to 2;

[0277] ω: Inertia weight, used to balance global and local search capabilities.

[0278] Inertia weight ω has important influence on convergence speed and global search capability, usually a linear decreasing strategy is adopted:

[0279]

[0280] Where:

[0281] ω max , ω min : Initial and minimum inertia weight, respectively;

[0282] T max : Maximum number of iterations;

[0283] t: Current iteration number.

[0284] PSO algorithm iterates until the stopping condition is met (reached the maximum number of iterations or error threshold), finally outputs the optimal solution machine corresponding to the objective function value.

[0285] The particle swarm optimization (PSO) optimization strategy is adopted to calibrate the experimental breakthrough curve under three calibration conditions of pH 2.9, 8.2 and 9.4. The optimal parameter set obtained is shown in Table 1. On this basis, the subsequent model fitting and verification work all follow this parameter set.

[0286] Table 1 Key parameters obtained by calibration set fitting

[0287]

[0288]

[0289] Dual-zone retention and diffusion: The ratio of diffusion / dispersion coefficients D1 / D2 ≈ 1 × 10-5 between the fracture and matrix zones was measured by the dual-domain model fitting of the Beishan exploration pit field tracer interference test. 4 (Zhang et al.,2025). This order is highly consistent with the fast-slow channel contrast observed in this study in granite fracture-matrix, further confirming the typical dual-channel transport behavior in the system under study. In addition, early multi-tracer forced gradient experiments in crystalline granite also reported the same order of magnitude of the dual-domain coefficient ratio (Reimus et al., 2002).

[0290] Effect of pH coupling on R1, R2: A number of experimental evidence and numerical simulations show that the regulatory effect of pH on adsorption-desorption kinetics in the fracture channel is significantly stronger than that in the matrix channel: basic research has revealed that the adsorption-desorption process itself is highly sensitive to pH (Goldberg et al., 2007; Chorover and Brusseau, 2008), and the fracture surface, due to its fractal heterogeneity and enhanced surface activity, further amplifies this pH dependence (Chugunov and Fomin, 2020; Lawrence et al., 2024), which is also consistent with the positioning of key chemical reaction domains in porous media (Zachara et al., 2016). The fracture zone k1 = 0.9335 is nearly twice as high as the matrix domain k2 = 0.5641, consistent with the conclusion that solute adsorption / desorption in the fracture channel is more sensitive to pH.

[0291] Exponential decay rate coupled with pH: The decay rate is jointly controlled by adsorption-desorption kinetics and matrix diffusion release. k pH，exp = 1.0821 indicates that the tail decay rate increases with increasing pH, and at high pH, there is little adsorption or retention, so the "exponential decay" behaves faster (i.e., "decay rate increases"). This trend is consistent with the known pH response: U(VI) is widely stable in the form of UO2 2+ ion in the aqueous phase, but it is easily adsorbed to clay, organic matter or iron hydroxide surface at low pH; while in neutral to alkaline environment, U(VI) more forms calcium-uranium carbonate complexes (CaUO2(CO3)3 2+ , Ca2UO2(CO3)3, etc.), which have weak adsorption ability to mineral surfaces, leading to increased U(VI) mobility (Ilton et al., 2012; Szecsody et al., 2013; Du et al., 2017; Skierszkan et al., 2020).

[0292] Concentration offset: According to the performance evaluation of spectrophotometer, the baseline offset measured in the low concentration range of 0-0.05 mg / L is about C0=-0.20, which is highly consistent with the system offset of-0.1 to-0.3 usually occurring in spectrophotometer at low concentration, which needs to be corrected by zero point correction (Adeeyinwo et al., 2013; Akash & Rehman, 2020).

[0293] Drift term: Cirpka et al. (2011) found that the effective transverse dispersion coefficient does not increase with the increase of transport distance through the flux-related second moment analysis of steady-state solute cross-transport mixing in two-dimensional heterogeneous porous media, indicating that the high-order drift term can be ignored.

[0294] The method used in the present application will be exemplarily described below by taking specific embodiment 2 as an example:

[0295] 1. Model construction

[0296] Dual-domain division: fissure domain (advection-dispersion dominated) and matrix domain (diffusion dominated).

[0297] pH coupling correction: the retention factor is corrected by an exponential function to reflect the influence of pH on adsorption / desorption.

[0298] Tail attenuation: a pH-dependent exponential attenuation term is introduced to describe the long-term release behavior of solute.

[0299] 2. Parameter inversion

[0300] The PSO algorithm is used to optimize the model parameters, including the retention factor, diffusion coefficient, pH coupling coefficient, etc., to ensure the global optimal solution.

[0301] 3. Experimental verification

[0302] Figure 3 The experimental data and the model prediction values are compared, corresponding to the conditions of pH 2.9, 8.2 and 9.4. Each figure is labeled with the goodness-of-fit indicators (SSE, RMSE, R 2 ).

[0303] As Figure 3 shown in FIG. a, under the condition of pH 2.90 (strong acid), the model accurately captures the adsorption kinetics of U(VI) in the form of UO2 2+ in the aqueous phase, with extremely high goodness-of-fit (SSE=0.006, RMSE=0.020, R 2 =0.997), and the scatter points almost completely fall within the ±1σ confidence band of the curve, reflecting the high reliability of the model in simple adsorption process.

[0304] In the pH 8.25 (near neutral and alkaline) environment, considering the CaUO2(CO3)3 2- complexation mechanism described by Skierszkan et al. (2020), the model still showed excellent performance: SSE = 0.001, RMSE = 0.012, R 2 = 0.997. This result indicates that the model structure can fully integrate the calcium-carbonate uranium complexation mechanism and accurately reproduce the experimental data Figure 3 (b), verifying its excellent adaptability to complex complex systems in neutral and alkaline conditions.

[0305] Even in the pH 9.40 (alkaline) interval, in the presence of UO2(OH)2, UO2(CO3)2 2- / UO2(CO3)3 4- and other multiple species, the model still maintains a high level of fitting quality Figure 3 (c), showing good stability and generalization ability. The residual points are symmetrically distributed along the zero line Figure 4 (c), without systematic deviation, proving that the model can still provide reliable predictions under complex chemical equilibrium conditions.

[0306] In the Beishan Xinchang granite single fracture rock column experiment, the model showed high precision and accuracy (R 2 ≥ 0.986) in the pH 2.9-9.4 range, verifying its reliability and stability.

[0307] In summary, this model can accurately reproduce the migration behavior of U(VI) in single fracture media from strong acid to alkaline conditions with low error and R 2 values as high as 0.986-0.997. It successfully integrates adsorption, complexation, and hydrolysis equilibrium mechanisms, not only verifying the consistency of theory and experiment, but also providing a reliable and generalizable tool for future uranium migration prediction under multiple species and conditions.

[0308] To quantify the improvement effect of different components on the overall prediction accuracy of the model, the invention constructs four nested sub-models (A-D), and statistics the SSE, RMSE and R 2 changes of each (Table 2).

[0309] Table 2 Performance comparison of each nested model (A-D) in SSE, RMSE and R 2

[0310]

[0311] ​Model A (basic dual-domain-dual erfc framework): This sub-model only contains two types of retention factors and diffusion coefficients in fractures and matrix (R1, R2, D1, D2) and two coupling coefficients (k1, k2), and the fitting accuracy is low (R 2 = 0.521), which highlights that it is difficult to simultaneously consider the peak and long tail response characteristics by using a dual-domain model.

[0312] Model B (introducing pH-coupled adsorption mechanism): On the basis of sub-model A, an empirical decay parameter (k exp , λ exp ) is added, and the initial concentration C0 and the retention time constant τ are added to capture the adjustment effect of pH on the adsorption-desorption kinetics of solute, and the SSE is reduced from 2.272 to 1.622 (28.6% reduction), and R 2 is increased to 0.658.

[0313] Model C (linear and quadratic drift terms are added): On the basis of sub-model B, background drift correction terms k lin and k quad are further added to compensate for the system drift during the injection and measurement of the solution. This modification reduces the SSE to 0.936 again (42.2% lower than model B), and R 2 reaches 0.797, significantly optimizing the fitting tail and background noise matching.

[0314] Model D (comprehensive introduction of pH-dependent decay): Finally, on the basis of model C, pH-coupled exponential decay terms (k pH，exp , k pH ) are introduced, so that the model can adapt to the tail decay rate under different pH conditions. The SSE is sharply reduced to 0.026 (98.8% reduction compared with model A), and the RMSE is only 0.026, and R 2 reaches 0.994, indicating that the model has almost perfectly matched the experimental data in terms of peak position, transport speed, and long tail decay characteristics.

[0315] As can be seen, the step-by-step introduction of each component enhances the model's ability to describe physical and chemical processes at different levels—from basic dual-domain coupling to pH modulation, to drift correction and exponential decay, achieving geometric-level improvement in fitting accuracy.

[0316] The following will use specific embodiment 3 to exemplarily illustrate the method used in the present application:

[0317] In order to further verify the robustness of the model when the physical boundary condition changes, the present application takes the uranium migration under pressurized and heated conditions as an example, and constructs three different pressure and temperature combination tests:

[0318] Group a (F=20N, T=28°C): pH=2.53, 8.21

[0319] Group b (F=20N, T=40°C): pH=2.54, 10.26

[0320] Group c (F=50N, T=23°C): pH=2.90

[0321] After the same prediction-calibration procedure, the prediction ability indicators are as follows:

[0322] Group a:

[0323] pH=2.53: SSE=0.002, RMSE=0.014, R 2 =0.998, Error=1.25%( Figure 5 a, Figure 6 a)

[0324] pH=8.21: SSE=0.004, RMSE=0.018, R 2 =0.995, Error=2.11%( Figure 5 b, Figure 6 b)

[0325] Group b:

[0326] pH=2.54: SSE=0.005, RMSE=0.022, R 2 =0.997, Error=0.91%( Figure 5 c, Figure 6 c)

[0327] pH=10.26: SSE=0.001, RMSE=0.007, R 2 =0.998, Error=0.97%( Figure 5 d, Figure 6 d)

[0328] Group c:

[0329] pH=2.90: SSE=0.003, RMSE=0.017, R 2 =0.996, Error=1.35%( Figure 5 e, Figure 6 e)

[0330] From the above, the model not only performs outstandingly under normal chemical conditions, but also can stably cope with external pressure and temperature changes, providing a powerful tool for accurate simulation of radionuclide migration processes in granite single fissures.

[0331] The method used in the present application will be exemplarily illustrated below with specific Example 4:

[0332] To systematically evaluate the generalization ability of the "double erfc + pH coupled exponential decay" coupled model, a set of new experimental data not used for calibration was selected, covering independent tests under different pH, pressure (F) and temperature (T) conditions. The test procedure and results are summarized as follows:

[0333] (1) Verification of pH sensitivity at room temperature without external force

[0334] Under room temperature (23°C) and external pressure-free environment, only the pH of the uranium solution was adjusted to test the response ability of the model to chemical boundary changes. The test points included pH = 2.96 and pH = 6.54. After the original model was predicted, combined with multivariate second-order polynomial calibration, the model predicted value and experimental value were compared as shown in Figure 7

[0335] Under the above conditions, the calibrated model showed a very high goodness of fit, with random residual distribution and no obvious systematic bias Figure 8 ), indicating that the model can accurately capture the dual influence of pH changes on the tail decay and peak position of uranium migration.

[0336] (2) Comprehensive applicability test under pressure and temperature conditions

[0337] To further verify the robustness of the model when the physical boundary conditions change abruptly, three different pressure and temperature combination tests were constructed: group a (F = 20N, T = 28°C): pH = 2.53, 8.21; group b (F = 20N, T = 40°C): pH = 2.54, 10.26; group c (F = 50N, T = 23°C): pH = 2.90. The comparison of model predicted value and experimental value and residual plot of each group are as follows: Figures 5-6 .

[0338] ​Overall, the model showed good predictive ability under all test conditions, with no systematic bias in the residual distribution, fully demonstrating the stability and reliability of the model under the condition of multiple coupling of external physical field (pressure, temperature) and chemical field (pH). The model coupled with the "fast channel" and "slow matrix" erfc function can analyze the main peak and long tail decay process, and its physical interpretability is better than that of single speed field or simple exponential decay model. The two pH-sensitive coupling coefficients act on the fracture domain and matrix domain respectively, accurately reflecting the combined effect of pH on adsorption-desorption and matrix diffusion, so that the model maintains high fitting degree under low pH (high adsorption) and high pH (low adsorption) conditions. In view of the experimental noise and instrument drift, this study introduces a second-order polynomial calibration to fine-tune the predicted value, thereby greatly improving the fitting accuracy (average RMSE reduced by about 20%), and the calibrated residual has no significant systematic trend, ensuring the objectivity of the results.

[0339] In summary, the "double erfc + pH coupling exponential decay" coupling model not only has excellent performance under normal chemical conditions, but also can stably cope with external pressure and temperature changes, providing a powerful tool for accurate simulation of radionuclide migration in granite single fracture.

[0340] Based on the classical two-domain analytical solution, a "double erfc function + pH coupling exponential decay" coupling model with physical and chemical interpretability and high computational efficiency is proposed and verified. The model realizes the synchronous characterization of U(VI) adsorption-desorption, precipitation / re-release and convection-diffusion coupling process in granite single fracture system by embedding pH-dependent retention coefficient and tail exponential decay into double-channel erfc analytical solution. Model parameters are globally inverted based on particle swarm optimization (PSO) algorithm, which realizes the balance of convergence and optimality in high-dimensional and multi-peak target space, and provides a stable parameter basis for nuclide migration modeling under multi-factor control.

[0341] Through step-by-step comparative analysis of nested sub-models, the gain contribution of each key mechanism to fitting performance is clearly explained:

[0342] 1. The pH coupling adsorption mechanism significantly optimizes the front arrival time and peak shape fitting, revealing the differential regulation of acid-base conditions on fracture and matrix adsorption equilibrium;

[0343] 2. The drift correction term effectively compensates for the system error in the injection and measurement process, thereby significantly improving the fitting accuracy of the long tail decay;

[0344] 3. The introduction of exponential tail decay accurately describes the regression behavior under the joint control of matrix diffusion release and chemical reaction, enhancing the analytical ability of the model in long-term process.

[0345] In independent validation experiments including different pressure (20 N, 50 N) and temperature (23 ℃–40 ℃) combinations, the model also exhibits excellent prediction ability (SSE≤ 0.026, RMSE≤0.045, Error≤2.82%), demonstrating its superior generalization performance under the conditions of multi-physical field and multi-chemical field coupling. Compared with the traditional full numerical simulation method, the model can realize rapid inversion and sensitivity analysis with only a small number of parameters, providing a practical tool for deep geological disposal risk assessment and field engineering applications.

[0346] In the description of the present specification, the description referring to the terms "one embodiment", "some embodiments", "an example", "a specific example", or "some examples" and the like means that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present application. In the present specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner.

[0347] It should be understood that, although the present application has been specifically disclosed by preferred embodiments and optional features, modifications, improvements and changes to the disclosed embodiments of the present application can be made by those skilled in the art, and such modifications, improvements and changes are considered to be within the scope of the present application. The materials, methods and examples provided herein are representative and exemplary of the preferred embodiments, and are not intended to be limiting to the scope of the present application. The above is a specific description of the preferred embodiments of the present application, but the present application is not limited to the described embodiments, and those skilled in the art can make various equivalent modifications or replacements without departing from the spirit of the present application, and these equivalent modifications or replacements are all included in the scope defined by the claims of the present application.

Claims

1. A uranium transport prediction model based on particle swarm optimization, characterized in that, The model is constructed by the following steps: S1, a two-domain model is constructed, which divides the granite fracture system into fracture domain and matrix domain, and describes the convection-dispersion and diffusion retention behavior of each domain respectively; S2, a pH-coupled retention coefficient correction term is introduced, which modifies the retention factors of fracture domain and matrix domain by exponential correction; S3, a pH-dependent exponential tail decay term is added to dynamically describe the long tail release behavior of solute; S4, the particle swarm optimization (PSO) algorithm is used to globally invert the model parameters to optimize the fitting accuracy.

2. The model of claim 1, wherein, The pH-coupled retention coefficient correction term is realized by the following formula: R i,eff = R i exp(k i (pH-pH ref )) (i=1,2) (1) Wherein R i,eff : represents the modified effective retention factor; R1, R2: reference retention factors obtained by global fitting; k1, k2: pH coupling coefficients of retention factors; pH: pH value in the migration process, specified by experiment or model; pH ref : Reference pH value, "equivalent benchmark" used in the fitting process to normalize the scale of the parameters.

3. The model of claim 1, wherein, The pH-dependent exponential tail decay term is realized by the following formula: Wherein: K exp : reference decay rate; k pH,,exp : represents the coupling coefficient of the pH on the exponential decay, the linear adjustment of the pH on the tail length, positive value makes the tail shorter at high pH (fast return), longer tail at low pH (slow return); pH: pH value in the migration process, specified by experiment or model; pH ref : reference pH value; λ exp : represents the exponential decay rate constant (s -1 ), regulates how steep the tail-off is; t: actual time during the experiment; τ: delay time, the actual kinetic process is delayed from the start of the reaction or measurement.

4. The model of claim 1, wherein, The parameter update formula of the particle swarm optimization (PSO) algorithm is: Wherein: v ij (t) : velocity of the i-th particle in the j-th dimension; x ij (t) : position of the i-th particle in the j-th dimension; p ij (t) : the individual best position in the history of the i-th particle (i.e. the personal best, p best ); g j (t) : the best position found among all particles (i.e. global best, g best ); r1, r2: uniform distribution random numbers obeying [0, 1]; c1, c2: individual learning factor and social learning factor, commonly taking the value of 2; w: inertia weight, used to balance the global search and local search ability.

5. The model of claim 1, wherein, The model is applicable to the prediction of U(VI) migration behavior in granite single fracture system, covering the pH range of 2.9 to 9.

4.

6. The model of claim 1, wherein, It also includes a drift correction term to compensate for systematic errors in the experiment, and its expression is: C drift = k lin ·t + k quad ·t 2 +k pH ·(pH-pH ref )·t (5) where k lin , k quad , and k pH are drift coefficients.

7. The model of claim 1, wherein, The model also introduces a time lag term t and a constant offset C0, which are used to correct the injection delay and overall baseline drift, respectively.

8. The model of claim 1, the final expression of which is: wherein: t e = max(t - T, 0) L: represents the column length (m); v: represents the flow rate (m / s); t e : indicates the valid time; t: actual time during the experiment (s); τ: delay time, the actual kinetic process is delayed from the start of the reaction or measurement; A, B: weight of fast / slow zone in the overall erfc response; D1, D2: Represent the effective dispersion coefficients (m) of the fast fracture and slow matrix domains. 2 / s); R 1,eff , R 2,eff : represents the fracture / matrix retention factor after exponential correction coupled with pH. R1, R2: retention factors of fracture domain and matrix domain, respectively; k1, k2: pH coupling coefficients of retention factors; pH: pH value in the migration process, specified by experiment or model; pH ref : reference pH value k exp : reference decay rate; k pH : represents a linear drift coupling coefficient (concentration / pH-time); k pH,exp : represents the coupling coefficient of pH on the exponential decay; k lin , k quad , and k pH : drift coefficient; gamma exp : indicates the exponential decay rate constant (s -1 ) that modulates the steepness of the tail decay; C0: offset concentration at the time of measurement.

9. Use of the model according to any one of claims 1 to 8 for the prediction of uranium migration in fractures of granitic rocks, characterized in that, The following steps are included: S1, collect experimental data of granite single fracture rock column, including breakthrough curves under different pH conditions; S2, use the PSO algorithm to invert the model parameters to obtain the optimal parameter combination; S3, predict the migration behavior of U(VI) in the fracture-matrix system based on the optimized model; S4, verify the consistency of the model prediction results with the experimental data and evaluate the prediction accuracy.

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