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Mathematical models for transport process between compartments The material balances between compartments are shown in Figure 3 which are considered in the mathematical models. fijNi (Bq y-1) Compartment j · Inventory of nuclide N: Nj (Bq) Compartment i · Inventory of nuclide N: Ni (Bq) · Loss of Nuclide N due to decay: λNNi (Bq y-1) · Increase of nuclide N due to ingrowth from parent nuclide M: λMNi (Bq y-1) fjiNj (Bq y-1) Source term: Si(t) (Bq y-1) Figure 3 Schematic diagram of inter-compartmental transfer The change in quantity of nuclide N (Bq) existing in compartment i is expressed as:
DOI: 10.7717/peerj.8002/fig-4
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This is the donor-controlled, first-order linear ordinary differential equation for transport in compartment models. The following definitions are used: Ni: quantity of radionuclide N in compartment i (Bq); Nj: quantity of radionuclide N in compartment j (Bq); Mi: quantity of radionuclide M in compartment i (Bq), M is a parent nuclide of N in a decay chain; Si(t): quantity of external source term of radionuclide N into compartment i at time t (Bq year−1); λN: decay constant for radionuclide N (y−1); λM: decay constant for radionuclide M (y−1); fji: transfer coefficient of nuclide N from compartment j(≠i) to compartment i (y−1); fij: transfer coefficient of nuclide N from compartment i(≠j) to compartment j (y−1). The equations used to calculate transfer rates between compartments, unless otherwise notified, based on various sources of literature accounting for the availability of the needed data[1,2,[] Smith G.S.: Biosphere modeling and dose assessment for Yucca mountain, Final Report,TR- 107190 (EPRI, U.S.A 1996). -[] Karlsson S.: Models for dose assessments: Model adapted to the SFR-area, Sweden,TR-01-04 (SKB, Sweden 2001). ,[] Eagan M.J.: Analysis of critical issues in biosphere assessment modeling and site investigation, SSI Report 2003 (SSI, Sweden 2003). ]. The main formulas describing transport processes between compartments in solid and liquid phases are as follows. l Infiltration (and other downward losses) from surface soil The transfer rate coefficient in soil by infiltration and recharge finf, y-1, is given by:
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where dd: annual infiltration/recharge rate (m/y) Rdsed: retardation coefficient for the soil compartment from which infiltration/recharge is coming θsedw: water-filled porosity of the soil compartment from which infiltration/recharge is coming dsed: depth of the soil compartment from which infiltration/recharge is coming(m) Rdsed is calculated using the following equation:
DOI: 10.5194/hess-2016-613-rc1
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where θsed: total porosity of the soil or sediment compartment θgsed: grain density of the soil or sediment compartment (kg/m3) θsedw: water-filled porosity of the soil or sediment compartment Kdsed: sorption coefficient of the soil or sediment compartment (m3/kg) l Erosion The rate coefficient for the transfer of radionuclides from soil to sinks (i.e. out of the system) by erosion fero, y-1, is given by:
DOI: 10.5772/25111
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Esed: erosion rate for the soil compartment (m y-1) dsed: depth of soil compartment (m) l Ingestion of water The rate coefficients for the transfer of nuclides from well to animal and individual by ingestion are fingwa, y-1 and fingwi, y-1 respectively which are given by:
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INGfwa1: consumption rate of water by the animal (m3/y) Vwell_water: volume of groundwater of well in 1 year (m3)
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INGfw: consumption rate of water by the individual (m3/y) Vwell_water: volume of groundwater of well in 1 year (m3) l Ingestion of soil The rate coefficients for the transfer of nuclides from soil to animal and individual by ingestion are fingsa, y-1 and fingsi, y-1 respectively which are given by:
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INGssa1: consumption rate of soil by the animal (kg/y) Vsurface_soil: volume of surface soil (m3) ρsed: density of the soil or sediment compartment (kg/m3)
DOI: 10.17816/0321-4443-105664-60257
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INGsed: consumption rate of soil by the individual (kg/y) l Ingestion of plant The rate coefficients for the transfer of nuclides from plant to animal by ingestion are fingpa, y-1 which are given by:
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INGfodd1: consumption rate of fodder by the animal (kg/y) Ypast1: yield of pasture in 1 year (kg m-2) Apast: area of pasture (m2) l Uptake from soil
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The rate coefficient for the transfer of radionuclides from soil to plant by uptake fupt, y-1, is given by: Spast: soil contamination on pasture [kg(dry weight soil)/kg(fresh weight of pasture)] Ypast: yield of pasture (kg m-2y-1) Apast: area of pasture (m2) ρsed: density of the soil or sediment compartment (kg/m3)
DOI: 10.17816/0321-4443-105664-60257
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Mathematical models for exposure modes Exposure pathways for exposed individuals are classified into three modes: external exposure, internal exposure by ingestion and internal exposure by inhalation. The annual individual dose for each exposure mode is calculated by these formulas for exposure pathways based on the nuclide concentration in each compartment. The main formulas for exposure modes are as follows. l External exposure The individual dose Dext (Sv/y) from external exposure is given by: Dext=DCFext Ocomp Cext (11) where DCFext: the dose conversion factor for external exposure [(Sv/h)/(Bq/m3)] Ocomp: the individual occupancy (h/y) Cext: the nuclide concentration in water, soil or sediment (Bq/m3) l Internal exposure by ingestion The individual dose Ding (Sv/y) from internal exposure by ingestion is given by: Ding=DCFing ING Cing (12) where DCFing: the dose conversion factor for ingestion(Sv/Bq) ING: the individual consumption rate of food (kg/y) or (m3/y) Cing: the nuclide concentration in the food (Bq/kg) or (Bq/m3) l Internal exposure by inhalation The individual dose Dinh (Sv/y) from internal exposure by inhalation is given by: Dinh=DCFinhBRcomp OcompCinh (13) where DCFinh: the dose conversion factor for inhalation (Sv/Bq) BRcomp: the human breathing rate (m3/h) Ocomp: the individual (h/y) Cinh: the nuclide concentration in the air (Bq/m3) Data for biosphere assessment In the early stage of biosphere assessment, some of the data required for biosphere assessment models are not available in China so far. But data from other countries or IAEA can be used to some extent. The three types of data are as follows:
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Data for compartments The size, porosity, density, concentration and other characteristic parameters of compartments in biosphere model have been determined by choosing the proper and conservative values consistent with the assumption of fast and homogeneous mixing of contaminants by physical, chemical and biological processes.
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Data for transport process between compartments The parameters used in calculation of transport process between compartments in the biosphere model are important such as infiltration, erosion, ingestion etc.
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Data for exposure pathways Data used in the assessment of exposure pathways are collected mainly from other countries and IAEA. However, for information specific to China, such as population and food consumption and some other factors, parameters can be obtained from field investigation or literature and statistical data published in China. Result and analysis Based on the biosphere conceptual model, mathematical model, data and the nuclide release rates from the geosphere, the results of mathematical simulation for individual doses in Beishan, China are discussed which is implemented using Amber software. The critical group is local adults live in grazing. In order to understand the behavior of the biosphere model, especially the timescale over which the model responds to changes in the release from the geosphere, the calculations assume that releases of each nuclide of 1 Bq/y are maintained until steady-state biosphere doses to individual are obtained. The results of calculation are shown in Figure 4. Figure 4 Response of the biosphere assessment model to steady and unit flux input(1 Bq/y) of nuclides The Figure 4 shows that the timescale for the distribution of nuclides in the biosphere is less than ten thousand years and is thus shorter than typical timescale for variations in geosphere release which is similar with the results from other countries such as Japan, Korea, Switzerland, etc. Furthermore, the code is run to steady state for unit input of each nuclide and the results are used to derive "biosphere dose conversion factors (BDCFs)" for conversion of release rates from the geosphere to individual doses. The results of doses comparison indicate the sequence of dose from different exposure pathways is as follow: Drinking water > Ingestion of animal products > Inhalation of dust > External irradiation > Consumption of soil. Conclusions On the basis of BIOMASS methodology, biosphere conditions and FEPs analysis, the biosphere model for Beishan, China (BIOMOBEIC) for performance assessment on high level radioactive waste(HLW) disposal repository has been constructed. Furthermore, Based on the biosphere conceptual model, mathematical model and data preparation, the results of mathematical simulation for individual doses in Beishan, China are discussed which is implemented using Amber software. Finally, the biosphere dose conversion factors (BDCFs) are obtained which are critical factors for conversion of release rates from the geosphere to individual doses in biosphere assessment and performance assessment. Prospect The biosphere model for Beishan, China is constructed and biosphere dose conversion factors are obtained in this study, but many issues still need to be further studied such as sensitivity and uncertainty analysis, 14C/3H/222Rn transfer processes, climate and environment change, human intrusion, comparing with other codes, etc. The study on modelling in biosphere for performance assessment on HLW disposal repository is very helpful to conduct the project of geological disposal of HLW which is ongoing in China. On the basis of this study, the first safety assessment of Chinese deep geological disposal on HLW will can be performed. Acknowledge This work is supported by the project of the technical cooperation between China and France in the field of high level radioactive waste disposal (STC No. CNNC_BGRI_2011_01). The author would like to thank all collaborators from and support of the Laboratory for Transfer Modelling in the Environment (LMTE) of CEA Cadarache center and IAEA for providing the Amber software as the mathematical simulation tool. References
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