(F21) Part of the modelling exercise was to examine if an analytical model could be used to estimate, at least approximately, the rise of tritium from contaminated groundwaters up to the soil surface. A strong incentive for this approach was that an analytical solution, even if restrictive regarding the conditions of application, would be of interest for a rapid estimation of tritium dispersion and would not suffer from some of the errors that may be associated with numerical solutions.
(F22) An analytical solution of tritium transport through a soil column can only be obtained when the transport equation is linear with constant coefficients over space and time. In a very strict sense, this would require the soil column to maintain constant uniform properties from top to bottom over the whole duration of the time interval considered. In the current problem,
this condition is not fulfilled, but the space varying and time-fluctuating coefficients can nevertheless be approximated by their averages calculated over a certain interval of space and time. With this approximation, the transport of tritium can be described by a simple transport equation with constant coefficients (3), and the associated Dirichlet boundary condition (3bis):
x C
C is the concentration of tritium in liquid phase;
t is the time;
x is the distance from the boundary;
D is the dispersion coefficient; and
v is the pore water velocity and Ȝ is the radioactive decay constant.
For a semi-infinite medium with origin at the boundary, the transient solution of the system (3) and (3bis) can be taken from Bear (1972), or, if it is considered that the radioactive decay can be neglected, from the simpler formula of Lapidus and Admunson (1952):
¸¹
which z is the depth above ground level taken positive downwards, and w is the pore water velocity also taken to be positive when oriented downwards.
(F23) The main problem is to evaluate the average value of the pore water velocity, wm, and the average value of the dispersion coefficient, Dm, for the period of interest. The average velocity wm is proportional to the infiltration flux, which is generally positive but may be negative when capillary rise dominates. The average dispersion coefficient Dm, mostly related to mechanical dispersion, is proportional to the water flux regardless of its direction.
Taeschner (ZSR) (this work), as described in his soil model (Annex II–B), assumes that the water flux so defined, although actually varying through the soil column, can be approximated for the entire soil column by the mean value of precipitation and infiltration fluxes (absolute values).
F5. INFLUENCE OF ELEMENTAL PROCESSES ON TRITIUM TRANSPORT (F24) The results to be intercompared are presented and discussed in Section F6 below. They were obtained by taking into account all three main processes that contribute to the transport of tritium in the soil, namely convection, molecular diffusion and mechanical dispersion. One important first task initially assigned to the modellers was to assess the relative importance of the three different transport processes. The participants in the exercise were asked to calculate the profiles of tritium concentration at a given date, when taking into account all three processes together or only selected ones, switching off the others (see Scenario 2.4 description in Annex II–F). This exercise was limited to the case of a bare soil, the specific configuration of soil and watertable given in the scenario description and the set of evaporation data provided to the modellers.
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(F25) The results demonstrating the influence of the different processes are illustrated in Figure F1 which gives the tritium profiles calculated for the end of July for a bare soil. The profiles obtained when assuming only convection and molecular diffusion processes are operating differ considerably from those obtained when assuming mechanical dispersion also operates. The concentration at a 10 cm-depth is at least six orders of magnitude lower in the first case than in the second one. This indicates that the mechanical dispersion due to the irregularities of the flow pattern must be considered, at least for the conditions of the proposed scenario.
TABLE F1. COMPARISON OF MODELS USED FOR SCENARIO 2
Institution (Modeller)
Modelling Approach for Transient Moisture Flow and Tritium Transport ANDRA
(C. Meurville) Approach: 3-D finite-volume code (PORFLOW developed by ACRI).
Solution Method: Preconditioned conjugate gradient for pressure; Alternate Direction Implicit method (ADI) for transport
Discretisation: Vertical 1-D grid: 196 layers 0.005 m, 2 boundary layers 0.0025 m. Time increment: modified by user to ensure stability (360 s for Scenario 2.4 and 36 s for Scenario 2.5)
Consultant
(Y. Belot) Approach: 1-D finite-difference code (TRIMOVE developed by Y. Belot).
Solution Method: Implicit method for moisture flow and tritium transport
Discretisation: vertical 1-D grid: 101 layers 0.01 m. Time increment to ensure stability (30 s) CEA
(G. Guinois) Approach: 2-D finite-volume code (METIS developed by Ecole des Mines).
Solution method: Conjugate gradient method
Discretisation: 100 layers of 0.01 m. Time increment automatically calculated FZK
(W. Raskob) Approach: 1-D compartment model (developed by W. Raskob)
Solution method: Explicit method for moisture flow and tritium transport
Discretisation: 10 layers of 0.1 m and 1 surface layer of 0.01 m. Time increment 360 s JAERI
(H. Yamazawa) Approach: 1-D finite-difference code with tritium routines (SOLVEG developed by H.
Yamazawa)
Solution method: semi-implicit for dispersion; explicit for transport (low numerical dispersion scheme)
Discretisation: 200 layers of 0.005 m. Time increment: 2 s NIPNE
(D. Galeriu)
Approach: 1-D finite-difference code Solution method: Implicit method.
Discretisation: 40 layers of 0.025 m. Time increment 3.6 s to 324 s STUDSVIK
(O. Edlund)
Approach: 1-D finite-difference code
Solution method: Implicit method for moisture flow and tritium transport
Discretisation: vertical 1-D grid: 101 layers of 0.01 m. Time increment to ensure stability (30 s)
VNIIEF
(A. Golubev) Approach: 1-D finite-difference code (developed by S. Mavrin) Solution method: implicit method
Discretisation: 50 layers of 0.02 m. Automatic time increment 1 s to 86 s ZSR
(M. Täschner)
Approach: Analytical solution of the mass transport equation for an homogeneous semi-infinite medium with prescribed concentration at the boundary (Bear 1972).
Solution method: The pore water velocity is proportional to the infiltration flux.
The dispersion coefficient is proportional to the mean value of precipitation and infiltration fluxes.
(F26) These very preliminary calculation tests can only be considered to be indicative. They were not sufficient to study the interdependency of the different elemental processes that operate within the soil, and the contribution of each of them to the movement of tritium and its evaporation at soil surface. This was studied more deeply by one of the TWG participants (Yamazawa, 2000). In this parallel work, H. Yamazawa (JAERI) carried out simulations in which the different factors contributing to tritium movement were varied in order to get an
insight into their respective influence. It appears that the balance of precipitation over evaporation is the most important factor in the movement of tritiated water from a sub-surface source. Tritium has a tendency to move downward when the balance is positive and upward when the balance is negative. But in the case of the downward movement some tritium lags behind, while in the case of upward movement some tritium is transported faster. This behaviour is the result of mechanical dispersivity, which plays an important role in determining the distribution of tritium concentration through the soil profile. Molecular diffusion is generally negligible in comparison to mechanical dispersion.