Radionuclide Decay Chains Research Articles

Recursive analytical solution for nonequilibrium multispecies transport of decaying contaminant simultaneously coupled in both the dissolved and sorbed phases

Multispecies transport analytical models that solve advection-dispersion equations (ADEs) are efficient tools for evaluating the transport of decaying contaminants and their sequential products. This study develops a novel semi-analytical model to simulate the multispecies transport of decaying contaminants, considering nonequilibrium sorption and decay in both dissolved and sorbed phases. First-order reversible kinetic sorption equations with decay processes are coupled to ADEs. Recursive analytical solutions, using the Laplace transform and generalized integral transform, are developed to address the mathematical complexity of the governing equations. The model's simulation results show excellent agreement with both numerical models and existing analytical solutions. Applied to a four-member radionuclide decay chain, the model reveals that including decay in the sorbed phase results in a lower concentration of the first member and avoids underestimating the radioactivity concentrations of daughter elements. These differences in dissolved radioactivity concentrations between models with and without sorbed phase decay may impact health risk assessments for radioactive waste disposal. Finally, this study provides a more sophisticated mathematical tool for analyzing multispecies transport in real field conditions where nonequilibrium sorption processes predominantly occur.

An Adjoint Sensitivity Model for Transient Sequentially Coupled Radionuclide Transport in Porous Media

AbstractWe present an efficient mathematical/numerical adjoint sensitivity analysis model for transient radionuclide transport through heterogeneous porous media. This work extends our previous research on radionuclide transport in a steady‐state regime. Both continuous and discrete adjoint approaches have been developed in this current research. The mathematical equations associated with the adjoint system and required for the continuous approach have been derived. The methodology for deriving the discrete adjoint solution is also presented.For the homogeneous case, the adjoint system for radionuclide decay chain is solved analytically. The four‐member decay chain (238Pu → 234U → 230Th → 226Ra) is considered for validation. The validity of the analytical adjoint sensitivity model has been shown using two illustrative examples associated with two performance measures. The analytical adjoint states are compared with calculated numerical results and show excellent agreements. The sensitivity coefficients are also computed numerically using the perturbation method. The numerical sensitivity coefficients successfully reproduce the analytically derived sensitivities based on adjoint states. The adjoint model has been coupled with a derivative‐based global sensitivity method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. An illustrative case study of spent fuel and high‐level waste (SF/HLW) repository involving 6 radionuclides and 48 parameters has been considered for identifying the most important parameters with respect to the maximum radionuclide dose at the interface geosphere/biosphere. Based on the results of sensitivity analysis, the uncertainty of the maximum radionuclide dose has been performed using Monte Carlo simulations.

Simulating Three-Dimensional Plume Migration of a Radionuclide Decay Chain through Groundwater

In this study, we present a semi-analytical model for simulating three-dimensional radioactivity transport of a radionuclide decay chain and assessing the radiological dose impact on the general public. The mechanisms and processes considered in the model include the one-dimensional advection, hydrodynamic dispersion in longitudinal and two lateral directions, linear equilibrium sorption, and first-order radioactive decay reactions. The semi-analytical model is derived for a semi-infinite domain, and the solutions in the Laplace domain for members of the decay chain are first generalized in a compact format. The concentrations in the original domain of each nuclide are independently evaluated with the help of the efficient and robust Laplace numerical inverse algorithms. The accuracy of the derived semi-analytical model is demonstrated by comparison of our developed model with an existing analytical model described in the literature. The results of the verification exercise indicate that the derived semi-analytical model is accurate and robust. The developed semi-analytical model is applied to an illustrative example that simulates the three-dimensional plume migration of a radionuclide decay chain on both the temporal and spatial scales. Moreover, the time histories of the radiological doses at different distances from the inflow source boundary are presented to understand the potential radiological impact on the general public. The developed model facilitates rapid assessment of the radiological impact posed by the presence of radionuclides in the environment because of leakage from a nuclear waste repository or accidental discharges from nuclear facilities.

Numerical modeling of colloid-facilitated radionuclide decay chain transport in a coupled fracture–matrix system

A numerical model is developed to simulate the colloid-facilitated transport of radionuclides in a single fracture-host rock coupled system with planar geometry. Such a system may be encountered in case of leaching and transport of radionuclides from a high-level radioactive waste repository in deep geological formations. The presence of colloids in the system reduces the retardation effects on the radionuclide migration and can change the concentration distribution of radionuclides in the system significantly. The processes such as advection, dispersion, surface sorption, radioactive decay and diffusive loss to the host rock are considered for transport in the fracture, whereas dispersion, adsorption and radioactive decay are included for transport in the host rock. Since the radionuclides to be disposed into the high-level waste repository may include a decay chain of progeny, the performance assessment methodology must incorporate the ingrowth of progeny with time to carry out realistic estimates. The inclusion of decay chain buildup and transport is essential to avoid underestimation with respect to the environmental impact assessment for the repository. Hence, the numerical model based on implicit finite difference scheme is developed with the inclusion of decay chain transport. The model is verified by comparing with well-known analytical solutions. The application of the model is demonstrated by applying it to a five-member decay chain of 238U. With increase in the concentration of colloids, the concentrations of the radionuclides in the fracture water increase and also reach the steady state at an earlier time. The concentrations of radionuclides almost become steady throughout the distance of 200 m in the fracture as the colloid concentration reaches 5 kg m−3 or more. Sensitivity analysis is carried out with respect to various model parameters such as dispersion coefficient in fracture, dispersion coefficient in the porous host rock, porosity of the host rock and distribution coefficient of radionuclide with respect to the mobile colloids. The developed model can be a useful tool for the performance assessment of high-level radioactive waste repositories.

Analytical Solution for Multi-Species Contaminant Transport in Finite Media with Time-Varying Boundary Conditions

Most analytical solutions available for the equations governing the advective–dispersive transport of multiple solutes undergoing sequential first-order decay reactions have been developed for infinite or semi-infinite spatial domains and steady-state boundary conditions. In this study, we present an analytical solution for a finite domain and a time-varying boundary condition. The solution was found using the Classic Integral Transform Technique (CITT) in combination with a filter function having separable space and time dependencies, implementation of the superposition principle, and using an algebraic transformation that changes the advection–dispersion equation for each species into a diffusion equation. The analytical solution was evaluated using a test case from the literature involving a four radionuclide decay chain. Results show that convergence is slower for advection-dominated transport problems. In all cases, the converged results were identical to those obtained with the previous solution for a semi-infinite domain, except near the exit boundary where differences were expected. Among other applications, the new solution should be useful for benchmarking numerical solutions because of the adoption of a finite spatial domain.

The effects of preferential flow and soil texture on risk assessments of a NORM waste disposal site

This paper investigates the environmental fate of radionuclide decay chains (specially the 238U and 232Th series) being released from a conventional mining installation processing ore containing natural occurring radioactive materials (NORMs). Contaminated waste at the site is being disposed off in an industrial landfill on top of a base of earth material to ensure integrity of the deposit over relatively long geologic times (thousands of years). Brazilian regulations, like those of many other countries, require a performance assessment of the disposal facility using a leaching and off-site transport scenario. We used for this purpose the HYDRUS-1D software package to predict long-term radionuclide transport vertically through both the landfill and the underlying unsaturated zone, and then laterally in groundwater. We assumed that a downgradient well intercepting groundwater was the only source of water for a resident farmer, and that all contaminated water from the well was somehow used in the biosphere. The risk assessment was carried out for both a best-case scenario assuming equilibrium transport in a fine-textured (clay) subsurface, and a worst-case scenario involving preferential flow through a more coarse-textured subsurface. Results show that preferential flow and soil texture both can have a major effect on the results, depending upon the specific radionuclide involved.

Analytical modeling for colloid-facilitated transport of N-member radionuclides chains in the fractured rock

A previous analytical model for N-member radionuclide decay chains has been extended to include the effect of radionuclide sorption with groundwater colloids. Published distribution coefficients were employed in the nuclide decay chain to illustrate the present model. The colloid concentration was assumed constant in time and space owing to equilibrium between colloid generation and sedimentation by chemical and/or physical perturbations. Furthermore, the diffusion of colloids into the rock matrix was ignored because the diameter of colloid is relatively large and colloids and fracture surfaces are like-charged. The results indicated that colloids could facilitate the transport of radionuclides and the large adsorbability of nuclides with colloids enlarged the effect of acceleration by colloids. The influence of colloids on the radionuclide transport was expected to be crucial to the actinides with large adsorbability; however, the present results revealed that the low-adsorbing nuclides whose parent nuclides have large capability of sorption could be also facilitated significantly by colloids indirectly. Therefore, the role of colloids played in the transport of the radionuclides decay chain should be assessed carefully in the radioactive waste disposal. The analytical method presented herein is helpful to verify/validate further complex far-field models.

Colloid-Radionuclide Transport Within Fractured Rock: Potential Impact of Nonlinear Sorption Kinetics

A numerical model was developed to analyze radioniclide transport within saturated fractured rock that accounts for the effect of nonlinear kinetic sorption of radionuclides on groundwater colloids. The interactions between radionuclides and colloids are assumed to be nonlinear and kinetic, while sorption of radionuclides on fracture surfaces and in rock matrix is described by a sorption distribution coefficient. Colloids may move with a velocity that is higher than the mean groundwater velocity. However, as there are insufficient data with which to assign a priori colloid velocity, we use a theoretical model based on hydrodynamic chromatography to evaluate the colloid velocity within a single fracture.Calculation results show that external surface forces acting on colloids could alter both the mobility of colloids and the host population of radionuclides in groundwater. The results also indicate that colloid-facilitated transport occurs depending on colloid concentration. Moreover, a simple two-member radionuclide decay chain is assumed and incorporated into the kinetic model.

Development of a solution method for the differential equations arising in the biosphere module of the BNFL's suite of codes MONDRIAN

British Nuclear Fuels plc owns and operates the near-surface Drigg disposal facility for low level radioactive waste. The long-term performance of the site is modelled by a suite of computer codes called MONDRIAN. One of the modules of MONDRIAN deals with the transport of radionuclides through the environment, and this paper reports on the current status of this module (BIOS). We derive the basic set of working equations from first principles and show clearly how the approximate nature of the final equations is arrived at. This is done by an averaging process leading to compartments, in and out of which radionuclides, solids and water can flow. The equations allow radioactive decay chains and an arbitrary number of compartments. There is also the facility to deal with changes in the rate coefficients, thereby simulating different environmental states. It is also possible to include the creation of new compartments arising as a consequence of climatic variations. In addition to developing a new differential equation solver which is now incorporated in the BIOS module of MONDRIAN, we have demonstrated the relative efficiency of this in comparison with a previously employed differential equation solver and have compared the benefits with an alternative approach that restricted the solution to the case which required all the retardation factors to be equal. The comparison is based upon a 31 compartment biosphere model with an eight member radionuclide decay chain. Verification against the probabilistic assessment code MASCOT is also reported to further increase confidence.

Two-proton radioactivity as a genuine three-body decay: The 19Mg probe

The chapter includes a history of the discovery and characterization of radioactivity. It follows with a description of the properties of atomic constituents, and the relation between mass and energy. This is followed with a treatment on the properties of the nucleus, nuclear forces, binding energy, nuclear models, and the relativistic properties of nuclear radiation. Natural and artificially produced radionuclides are discussed including radionuclides of cosmogenic origin and natural radionuclide decay chains. Nuclear reactions are discussed including reaction types, energy of reactions (Q value), and reaction cross section. A treatment of alpha decay, beta decay including negatron emission, positron emission, electron capture (EC), double beta (ββ) decay, and the interactions of alpha and beta radiation with matter. Also discussed are internal conversion and Auger electron emissions and a detailed treatment of neutron sources, interaction of neutrons with matter, neutron attenuation and cross section, and neutron decay. The wave-particle dual nature of matter is discussed and a treatment of electromagnetic radiation or photons including the mechanisms of photon interaction with matter. Cherenkov radiation, its origin, and properties are discussed. The origins, properties and applications of synchrotron radiation are also discussed. The chapter continues with a treatment of nuclear recoil and the calculations of recoil energy following alpha, beta, gamma, X-ray, and neutrino emission in radionuclide decay. Cosmic radiation is discussed including the origins, properties, classification, and showers of the cosmic radiation. A treatment of radiation dose, stopping power, and linear energy transfer is included. The principles of radionuclide decay, ingrowth, and equilibrium are included. There is also a discussion of radioactivity units and the correlation of radioactivity and radionuclide mass.

The Arnoldi reduction technique for efficient direct solution of radionuclide decay chain transport in dual-porosity media

An efficient technique is presented for the numerical solution of multi-species radionuclide decay chain transport problems in dual-porosity media. The method is based on the Arnoldi modal reduction technique that uses orthogonal matrix transformations to reduce the discretized transport equations. The reduced equation system is much smaller than the original one. This new system can be solved by a standard Crank–Nicolson scheme with very little computational effort and then the original solutions at desired time steps are obtained by using a matrix–vector multiplication. In this paper, we also develop two new alternative methods for choosing a common starting vector for all the transport species. The new methods can be used for most popular cases of first or second type boundary conditions and ensure the convergence of Arnoldi method. In addition, a new method for calculating mass exchange between the matrix block and fracture is presented. This method calculates the leakage terms directly using reduced space data for both slab and spherical matrix block and is highly efficient compared to the traditional iterative methods. The technique is verified through the comparison with analytical solutions. The efficiency and accuracy of the Arnoldi method are demonstrated by applying to the case study of three-species decay chain transport in a heterogeneous dual-porosity aquifer system. The proposed technique shows an impressive 98% saving in computing time and 75% saving in storage space for the case study problem. The Arnoldi reduction method (ARM) affords an efficient means of solving large problems particularly when time durations are long or many species are involved.

A general algorithm for radioactive decay with branching and loss from a medium.

Many areas in the field of health physics require evaluation of the change of radionuclide quantity in a medium with time. A general solution to first-order compartmental models is presented in this paper for application to systems consisting of one physical medium that contains any number of radionuclide decay chain members. The general analytical solution to the problem is first described mathematically and then extended to four applications: 1) evaluation of the quantity of radionuclides as a function of time, 2) evaluation of the time integral of the quantity during a time period, 3) evaluation of the amount in a medium as a function of time following deposition at a constant rate, and 4) evaluation of the time integral of the amount in a medium after deposition at a constant rate for a time. The solution can be applied to any system involving constant physical transfers from the medium and radioactive chain decay with branching in the medium. The general solution is presented for quantities expressed in units of atoms and activity. Unlike many earlier mathematical solutions, this solution includes chain decay with branching explicitly in the equations.

One‐dimensional analytical solutions for the migration of a three‐member radionuclide decay chain in a multilayered geologic medium

Analytical solutions are presented for the one‐dimensional convective‐dispersive and nondispersive equilibrium transport equations of a three‐member radionuclide decay chain in a multilayered porous media. The solution for the nondispersive case is exact. The solution for the general case, although analytic in the first layer, takes a semianalytical form in the subsequent layers by virtue of its numerical integration requirements. The solutions are based on the Laplace transformation technique. Two types of boundary conditions are considered at the source, i.e., a continuous and a band release mode. The accuracy of the solutions was satisfactorily tested on a selected number of problems for which experimental and analytical solutions were available. The practical use of the solutions in a two‐dimensional domain is illustrated by a scenario of radionuclide migration from a high‐level waste repository located in a saturated multilayered aquifer.

Diffusional transport of radionuclide chains in subseabeds

The diffusion-controlled migration of radionuclides released from a leaky canister located in a subseabed sedimentary layer, bounded at the top by the ocean and at the bottom by an impermeable basalt zone, is analyzed to determine the transport rate to the ocean floor. The unsteady-state diffusion equations for radionuclide decay chains have been solved by integral transform techniques to obtain analytical solutions for the concentration distribution, flux at the ocean-sediment interface and discharge rate to the ocean floor for each chain member. By means of a parametric study the effects of the major parameters on the release rate of radionuclides to the ocean floor are elucidated.

A two-dimensional finite-element solution for the simultaneous transport of water and multi solutes through a nonhomogeneous aquifer under transient saturated-unsaturated flow conditions

A two-dimensional finite-element model for the simultaneous solution of the transport of water and radionuclides in a nonhomogeneous anisotropic aquifer under saturated and unsaturated flow conditions is presented. The adsorption mechanism is described by a linear equilibrium isotherm. The model allows for a variety of boundary conditions; e.g. infiltration, drainage or evaporation. The solution to these equations is obtained through a Galerkin-type finite-element method using linear quadrilateral and triangular isoparametric elements, respectively. The model was tested by comparing predictive values with analytical solutions and experimental results for transient flow. Use of the model is illustrated by the analysis of the movement of hypothetical three-membered radionuclide decay chains leaching from the tailings pond of a typical uranium mill.