atw 2018-05v6


atw Vol. 63 (2018) | Issue 5 ı May

The fulfilment of the admissibility

conditions, however, is not the

endpoint of engineering endeavor.

Supplementary benefits may be

achiev able within reasonable efforts.

It is therefore important to explore

repo sitory configurations beyond

their strict admissibility.

Since every disposal project needs

to fulfil particular national regulations,

it is not possible to address all

situations at once. In the following,

two situations shall be considered as

examples to illustrate the interdependence

between spatial and thermal

dimensioning: (1) A facility

planned for argillaceous rock in

northern Switzerland by the Swiss

National Cooperative for the Disposal

of Radioactive Waste (Nagra), with

baseline configuration C a (Table 1).

This project is currently in the stage

of site-selection under the supervision

of the Swiss Federal Office of Energy.

(2) A facility targeted for crystalline

rock in southern Finland by the

Finnish expert organisation for

nuclear waste management (Posiva

Oy), with baseline configuration C b

(Table 1). This project has been approved

by the Finnish Radiation and

Nuclear Safety Authority.

The thermal dimensioning of a

repository for high-level waste and

spent fuel requires the mutual

comparison of a large number of

engineering configurations comprising

large amounts of individual

heat sources. Therefore, the evolution

of temperature at decision-critical

points P i must be evaluated in reasonable

CPU time. For the purpose of this

study, a computational model based

on the numerical integration of

analytically calculated Green’s functions

for all relevant heat sources has

been implemented [Carslaw and

Jaeger 2011; Myers et al. 2015; Heierli

2016]. The method remains insensitive

to length scales, which greatly

facilitates parameter studies.

The purpose of the Green’s function

model is to predict the temperature

increase u(t,P) within a few

degrees Kelvin at location P and time t

in a homogeneous (but not necessarily

isotropic) volume of the circumambient

rock to a repository of

arbitrary configuration. The model is

not designed to compute the temperature

in engineered components such

as tunnel lining, inside the backfill or

waste containers, nor in remote rock


The positions of heat sources in

the repository are modelled in 3

dimensions. The (real) heat sources

are supplemented by (imaginary)

images sources to account for isotherm

conditions on a given boundary

surface, e.g. the earth surface in the

present work. The image sources

are sited on the mirror image of

the repository with respect to the

boundary surface. The instant power

release of each image source is equal

and opposite in sign to the power

release of the corresponding heat

source. The heat sources and the

image sources are modelled in

dependence to their distance to P.

Sources sited remotely from the point

of observation are modelled as single

point sources, whereas nearby sources

are modelled as a mesh of points sited

on the surface of the waste canister,

to account for container geometry.

By virtue of the superposition principle,

contributions from the repository

are treated separately from

those originating from geothermal

heat. Aspects of secondary importance

are omitted in the model: It is

assumed that the radioactive decay of

the activated waste represents the

dominating source of heat in the

repository. Other sources or sinks,

such as enthalpy of reactions taking

place underground, initial heat content

of waste canisters, ventilation

and construction of caverns are

neglected. Heat is conservatively assumed

to be transported by conduction.

Thermal properties are assumed

invariant in time. Waste disposal is

assumed to take place instantly, rather

than in a period of one or two decades.

The geometry is assumed to remain

stable over the time range considered.

Further implementation details are

given in Heierli [2016].

The main advantage of the Green’s

function method is that the temperature

at an arbitrary point in

the underground can be computed independently

of the temperature of any

other points at previous times,

resulting in a fast algorithm allowing

for the calculation of a large variety of

configurations. The CPU time for the

simulation of the configurations considered

in this study, calculated over

100’000 y in full 3D for P, range

between 30 s and 1 minute per

Point P Location Significance of P

con figuration. In previous studies, the

Green’s function method has been

compared with results from sophisticated

numerical models and found to

produce unbiased results within 2 °C

standard deviance or 2 % of maximum

value, provided that P is chosen in the

appropriate domain of application

(see above) and that the thermal

properties of the host rock formation

are homogeneous in space [Myers et

al. 2015; Heierli 2016].


The evolution of temperature at

decision points P 1 and P 2 (Figure 1a,

Table 2) has been calculated for the

baseline configuration C a in argillaceous

rock, as well as for a number

of parameter variations thereof. The

results for u(t,P 1 ; C a ) and for u(t,P 2 ;

C a ) are presented in Figures 2 and 3

respectively. The same procedure

was applied for decision point P 3 in

con figuration C b for crystalline rock

(Figure 1b, Table 2). Corresponding

results for u(t,P 3 ; C b ) are presented

in Figure 4. The y-axis in the graphs

designates the change in temperature

at P i . In order to obtain the resulting

temperature, the local undisturbed

temperature has to be added (this is

approximately 38 °C in 650 m depth

in Figure 2 and 37 °C in 630 m depth

in Figure 3). In panels a to f, one and

only one engineering parameter has

been changed with respect to the

baseline configuration. Panel g shows

the temperature for both the baseline

configuration and alternative configurations

with three parameters

changed simultaneously, leading to a

lower and shorter temperature peak.

The uppermost curves in every panel

are identical and represent the temperature

evolution in the baseline

configuration. The dashed curves

represent asymptotic limits. The

arrows indicate the path of the temperature

peak under variation of one

parameter, e.g. cooling time in panel

a. The arrows points towards increasing

technical effort.


In the baseline configuration C a for

argillaceous rock, the temperature

P 1 P 0 + (x = 0, y = 1.50, z = 0 ) Retrievabilty, degradation of backfill material.

P 2 P 0 + (0, 0. 20.0) Pore water pressure

P 3 P 0 + (0, 0.88, 0) Retrievability, degradation of bentonite.

| | Tab. 1.

Location of decision points P i relative to the midpoint P 0 of a central batch. The x-axis is taken parallel

to the drift axis, the y-axis perpendicular to x in the plane of the repository and the z-axis is taken

perpendicular to the repository plane.


Decommissioning and Waste Management

Scope for Thermal Dimensioning of Disposal Facilities for High-level Radioactive Waste and Spent Fuel ı Joachim Heierli, Helmut Hirsch, Bruno Baltes

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