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UNCERTAINTY, VARIABILITY AND SENSITIVITY ANALYSIS APPLIED TO THERAGENA MODEL OF RADON GENERATION, ENTRY AND ACCUMULATION

INDOORS

Ll. Font, C. Baixeras and C. Domingo

Grup de Física de les Radiacions. Edifici Cc. Universitat Autònoma de Barcelona.E-08193 Bellaterra. Spain

The application of a radon model is useful to understand the processes that drive the radon gasbehaviour from its sources to its accumulation indoors. Since in a given inhabited house the detailedknowledge of the values of all the parameters that affect indoor radon levels is not available, theresponse of the model has to be explored in a reference site in which all the parameters are supposedto be known. We call this site the reference configuration. In this paper we report on the procedurefollowed to carry out uncertainty, sensitivity and variability analysis of the model response for areference configuration that corresponds to a single family multi-zone house. We have obtained fromthe uncertainty analysis that, assuming a normal distribution of all the input parameters with a 10%relative standard deviation (RSD), the model outputs present a RSD in the range [17-22]%. Thesensitivity analysis reflects, in general, a good behaviour of the model, in the sense that its responsedescribes a realistic behaviour of the system. The variability analysis has shown that the model isapplicable to a wide range of situations, and that the most relevant parameters for the referenceconfiguration are: the soil gas-permeability (obtained from the mean soil grain diameter), theventilation rate of the rooms, the air-exchange rate between the basement and room 2, the soil-indoorpressure difference, the open area and the concrete radium content.Keywords : Radon, modelling, uncertainty analysis, sensitivity analysis, variability analysis.

INTRODUCTIONThe application of a radon model is useful to understand the processes that drive the radon gasbehaviour from its sources to its accumulation indoors. When applying a given radon model, it isnecessary to assign values to the parameters of the model. These values can be either the result of ameasurement, taken from the literature, or obtained from a fit in a known situation. Once the valueshave been assigned, the model can be used to make predictions, to characterise the influence of agiven parameter, or to simulate radon levels in different sites. It is important to study the responseof the model under different conditions in order to determine the uncertainty of the modelpredictions, to identify the parameters which variation most affect the response of the model, and toestablish the applicability of the model to different sites. We have applied the RAGENA (RadonGeneration, ENtry and Accumulation indoors) model to a reference configuration that correspondsto a single family multi-zone house (Font 1997). In this paper we report on the procedure followedto carry out uncertainty, sensitivity and variability analysis of the model response for a givenreference configuration.

APPLICATION OF THE RAGENA MODEL TO THE REFERENCE CONFIGURATIONThe RAGENA model has been designed to be adapted to real inhabited houses and to take intoaccount all the radon sources, and most of the processes and parameters affecting indoor radonlevels from a dynamic point of view (Font 1997, Font et al. 1999a). It has been successfully appliedto a Mediterranean climate inhabited house (Font et al. 1999b) and to a Swedish house (Font et al.1999c). Since in a given inhabited house the detailed knowledge of the values of all the parameters

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that affect indoor radon levels is not available, the response of the model has to be explored in areference site in which all the parameters are supposed to be known. We call this site the referenceconfiguration, which corresponds to a single family multi-zone house with features realistic enoughto be representative of a real inhabited house. Fig 1 shows a diagram of the reference configuration.The house has a single basement room, a ground floor with two rooms (1 and 2) and a first floorwith two rooms (3 and 4). A 1 mm-width crack along all wall joints in direct contact with soil ispresent. Each room has a window. In addition, room 1 has the entrance door. The rooms areinterconnected in the following way: rooms 1,2 and 3,4 are connected through a door; rooms 2,3and 2, basement are connected through steps and a trap door. The effective volume is the same forall the rooms as we have not considered any furniture in the house and steps do not have asignificant volume. Room 4 has water supply available. We do not consider any heating, ventilatingand air-conditioned (HVAC) system or any gas supply. There are two types of building materials.The building shell and the floors are made from concrete. The thin walls are made from bricks. Allbrick and concrete surfaces are covered with a 0.05 m layer of plaster and paint. The soilunderneath the house is assumed to be thick enough to neglect the contribution of the bedrock to thesoil radon concentration and that of the water table. The values of the parameters that characterisethe building, site and radon processes for the studied reference configuration are given in Font(1997). The model has been applied to the reference configuration under both static and dynamicconditions. The model outputs are: the radon concentration in the soil underneath the house, in thedifferent building material walls, in each room; and the entry rate from each source. These outputsallow to characterise the relative importance of each source and/or process (Font 1997).

UNCERTAINTY ANALYSISIf a model is used to predict indoor radon levels, it is necessary to determine the uncertaintyassociated to any result given by the model to make possible the comparison between predicted andmeasured values. An uncertainty analysis accounts for the fact that the values of the inputparameters within a system are never precisely defined and are best described by a probabilitydistribution. The aim of such analysis is to estimate, in a given site, which is the uncertainty of themodel predictions.

The assignment of a given probability distribution or uncertainty to each input parameter can bevery complicated and depends on the knowledge available in a given specific site. Some parametersmay be measured experimentally with a given accuracy, while others may be estimated fromindirect measurements, and some others may be taken from the literature available. It is, therefore,of great importance to evaluate accurately the different sources of uncertainty for each inputparameter.

Procedure and resultsWe have assumed that all the RAGENA model input parameters are described by a normaldistribution around their reference configuration value, with a relative standard deviation (RSD) of10%. Fifty simulations have been performed under steady-state conditions to obtain the distributionof model outputs. Descriptive statistics of the indoor radon concentration values obtained with themodel in this analysis is shown in table 1, where it can be seen that this uncertainty of 10% of allthe input parameters originates an uncertainty on the model outputs in the range [17-22] %

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SENSITIVITY ANALYSISThe objective of a sensitivity analysis is to determine, in a given site, which are the parameterswhich fluctuations have a higher effect on the response of the model. To perform such an analysis,the value of a single parameter is subject to small and/or sudden fluctuations around its mean valueto explore how sensitive are the model outputs to these fluctuations. If the response of the model isvery sensitive to small changes of the value of a given parameter, then this parameter is critical andmust be measured with a good accuracy.

Procedure and resultsThis analysis have been performed by stressing the model with three types of sudden singleparameter time-variations: i) step functions, where the studied parameter suddenly changes its valueto a new one at which it is kept; ii) pulses, where the parameter value experiments periodicinstantaneous raises and descends; and iii) sinusoidal waves, where the parameter value varies intime following a sinusoidal wave of a given frequency. For each type of sudden variation of thevalues, we have chosen the parameters most likely to follow the given variation. We have observedin all cases a good behaviour of the model in the sense that the model predictions can be imputed tothe physical system rather than to any computing problem. As an example, we present in Figure 2the response of the indoor radon concentrations obtained with the model to a step change of the soilwater saturation fraction in which, at the instant t=150 h, the water saturation fraction is multipliedby a factor of 2 and kept at this new value for the rest of the simulation. It can be seen that theincrease of the water saturation fraction from 0.35 to 0.70 results in a decrease of indoor radonlevels. This behaviour is consequence of the reduction of the soil effective diffusion coefficient andof the soil gas-permeability and consequently, of the radon entry from the soil. The time needed toreach the new steady state is different for each room. We characterise the dependence of the systemto the sudden fluctuation of the parameter value by the longest time needed to reach the 95% of theradon concentration corresponding to the new steady-state value (denoted by "response time") aswell as by the percentage of variation of the new steady-state value with respect to the old one. Inthis example, the response time has been 25 h, and the percentage of variation has ranged from –2.7% in room 4, to –36.7% in the basement, which is the room most affected by the change of theparameter, as it might be expected.

VARIABILITY ANALYSISThe aim of a variability analysis is to explore the capability of the model to be applied to verydifferent sites. In a given site, the values of a parameter may vary within a given range. However,the range of variation of the parameter value across different sites can be much wider. This is thecase of some relevant radon-related parameters as, for instance, the soil gas-permeability and thesoil effective diffusion coefficient, which can span respectively up to 9 and 5 orders of magnitude(Nazaroff et al. 1988). Obviously, in a specific site, the range of variation will be much smaller.Consequently, for a variability analysis, it is very important to take into account the completepossible range of variation of the input parameters of the model when considering different sites.An important feature of the variability analysis is then that, in contrast to the uncertainty andvariability analysis, it is not site-specific: it allows to determine the most relevant parameters from ageneric point of view. This most relevant parameters are those which value must be measured atevery specific site where the model is applied.

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Procedure and resultsWe have investigated the response of the model to the variation of a single parameter within itsrange of variation, obtained from the literature, keeping the other parameters constant at thereference configuration value. The minimum (Cmin) and maximum (Cmax) indoor radonconcentrations obtained with the model in each room when changing the parameter value have beenused to calculate the Variability Index (VI) of the studied parameter, defined as (Schell et al., 1996):

VI ==== 1 −−−− minC

maxC.

The ranges of variation of the model parameters and their corresponding VI for each room are givenin table 2. The VI ranges from 0 to 1; a value close to 1 means that the parameter has a large impacton the indoor radon concentration when varying within the range chosen, while a value close to 0means a low impact. Table 2 allows seeing which parameters are more relevant for the referenceconfiguration and in which room their variation is more important as well. The determination of themost relevant parameters for each room depends on the value of the VI considered as high. Takingas a “first order in importance” those parameters with the VI value higher than 0.800, we see that:

i) The mean soil grain diameter has a large impact on all the rooms of the house, mainly in thebasement and in the ground floor. This is consequence of its range of variation and itsinfluence on the soil permeability.

ii) The concrete radium content and emanation coefficient have also a large influence, speciallyin the first floor rooms, where the influence of the soil parameters is diminished.

iii) The soil-indoor pressure difference and the fraction of the open area, that is, the soil-houseinterface parameters affect basically the basement and the ground-floor rooms.

iv) Ventilation rates, as the main responsible for radon removal, are very important parameters,affecting specially the room considered.

v) The most important inter-zone flow is that between the basement and room 2, which affectsvery much room 2 radon concentration.

A closer view to table 2 shows that in general, the importance of the soil parameters decreases withthe height of the floor level, while the importance of the concrete parameters increases, as it mightbe expected. The contribution of the brick building material is low compared to the concretecontribution. Considering as “second order in importance” those parameters with VI in the range[0.400 - 0.800], we obtain the soil water saturation fraction, radium content, and maximumemanation coefficient, the air-exchange between rooms, and the surface to volume ratio of the room(S/V).

CONCLUSIONSIt is convenient to perform uncertainty and sensibility analysis when a radon model is applied to agiven site. The capability of a radon model to be applicable to different sites can be explored with avariability analysis. We have obtained from the uncertainty analysis that, assuming a normal

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distribution of all the input parameters with a 10% relative standard deviation (RSD), the modeloutputs present a RSD in the range [17-22]%. The sensitivity analysis reflects, in general, a goodbehaviour of the model, in the sense that its response describes a realistic behaviour of the system.The variability analysis has shown that the model is applicable to a wide range of situations, and theimportance of each input parameter on indoor radon concentration has been quantified by thevariability index. From this analysis, we have found that the most relevant parameters for thereference configuration are: the soil permeability (obtained from the mean soil grain diameter), theventilation rate of the rooms, the air-exchange rate between the basement and room 2, the soil-indoor pressure difference, the open area and the concrete radium content.

REFERENCES

[1] Font Ll. Radon generation, entry and accumulation indoors. PhD Thesis. Universitat Autònoma deBarcelona. Barcelona, 1997, 138 pp.

[2] Font Ll, Baixeras C, Domingo C, Fernández F. A first step towards an integrated approach formodelling indoor radon levels. In: Radon in the Living Environment, 19-23 April, Athens, Greece,1999a.

[3] Font Ll, Baixeras C, Domingo C, Fernández F. Experimental and theoretical study of radon levels andentry mechanisms in a Mediterranean climate house. Radiat. Meas. “in press” 1999b.

[4] Font Ll, Baixeras C, Jönsson G, Enge W, Ghose R,. Application of a radon model to explain indoorradon levels in a Swedish house. Radiat. Meas. “in press” 1999c.

[5] Nazaroff W.W, Nero A.V., Radon and its decay products in indoor air. John Wiley and sons, NewYork, 1988, pp 504.

[6] Schell W. R, Linkow I, Myttenaere C, Morel B, A dynamic model for evaluating radionuclidedistribution in forests from nuclear accidents. Health Phys. 1996; 70:318-335

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Table 1: Descriptive statistics of the indoor radon concentrations (Bq·m-3) obtained when all theinput parameters are given by a normal distribution of 10% Relative Standard Deviation(RSD) around the reference configuration. SD: standard deviation.

Basement Room 1 Room 2 Room 3 Room 4

Mean 370 60 69 48 46

SD 63 10 12 9 10

RSD (%) 17 17 17 18 22

Minimum 260 42 47 34 30

Maximum 578 89 98 67 80

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Table 2: The range of variation and the Variability Index in each room corresponding to eachparameter around the reference configuration. BM: Building materials.

Variability indexParameter Range Basement 1 2 3 4Mean soil grain diameter (m) [10-6 - 10-3] 0.994 0.992 0.989 0.946 0.872Soil grain density (kg·m-3) [2650 - 2750] 0.016 0.014 0.010 0.004 0.002Soil water saturation fraction [0.01 - 0.99] 0.496 0.436 0.335 0.091 0.038Soil porosity [0.2 - 0.6] 0.261 0.235 0.189 0.057 0.024Soil radium content (Bq·kg-1) [10 - 150] 0.698 0.653 0.563 0.214 0.098Soil maximum emanationcoeff.

[0.02 - 0.7] 0.677 0.627 0.526 0.182 0.081

Max. migration distance (m) [2 - 15] 0.000 0.000 0.000 0.000 0.000Coeff. of Rn solubility in water [0.180 - 0.525] 0.074 0.067 0.050 0.015 0.007Concrete width (m) [0.1 - 0.4] 0.486 0.444 0.334 0.102 0.258Concrete porosity [0.12 - 0.27] 0.032 0.029 0.037 0.050 0.052Concrete density (kg·m-3) [1930 - 2260] 0.075 0.072 0.088 0.117 0.123Concrete Ra content (Bq·kg-1) [10 - 100] 0.590 0.570 0.660 0.790 0.814Concrete emanation coefficient [0.01 - 0.4] 0.700 0.680 0.766 0.883 0.904Concrete eff. diff. coeff. (m2·s-

1)[0.0076-2.1·]10-6

0.503 0.484 0.568 0.694 0.719

BM covering layer width (m) [0.01 - 0.1] 0.071 0.069 0.085 0.114 0.121BM covering factor [0.15 - 0.98] 0.126 0.121 0.151 0.204 0.213Brick width (m) [0.10 - 0.25] 0.001 0.007 0.008 0.015 0.016Brick porosity [0.24 - 0.26] 0.000 0.000 0.000 0.000 0.000Brick density (kg·m-3) [1950 - 2030] 0.000 0.000 0.000 0.002 0.000Brick radium content (Bq·kg-1) [20 - 200] 0.006 0.037 0.034 0.068 0.076Brick emanation coefficient [0.02 - 0.1] 0.003 0.021 0.019 0.037 0.041Brick eff. diff. coeff. (m2·s-1) [0.84·- 3.4·]10-7 0.000 0.000 0.000 0.000 0.000Soil - indoor pressure diff.(Pa)

[-5 - 15] 0.973 0.909 0.781 0.295 0.134

Fraction of open area [0.00001 - 0.1] 0.994 0.993 0.990 0.415 0.220Rooms 1 and 2 vent. rates (h-1) [0.1 - 1] 0.405 0.837 0.794 0.508 0.305Rooms 3 and 4 vent. rates (h-1) [0.1 - 1] 0.039 0.071 0.191 0.779 0.820Air-exchange basement-2 (h-1) [0.1 - 1] 0.703 0.251 0.999 0.085 0.078Air-exchange 2-3 (h-1) [0.1 - 1] 0.003 0.128 0.412 0.302 0.074Air-exchange 1-2 (h-1) [0.1 - 1] 0.007 0.379 0.473 0.086 0.045Air-exchange 3-4 (h-1) [0.1 - 1] 0.000 0.000 0.003 0.025 0.038Outdoor Rn concentr. (Bq·m-3) [0 - 10] 0.026 0.159 0.137 0.190 0.198Water use-rate (m3·h-1) [0.017 - 0.064] 0.000 0.000 0.000 0.000 0.000Water transfer efficiency [0.1 - 0.98] 0.000 0.000 0.000 0.000 0.000Water Rn concentr. (Bq·m-3) [1 - 1000]·103 0.000 0.000 0.000 0.000 0.000Basement S/V ratio (m-1) [1.10 - 3.75] 0.766 - - - -

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12

3 4

BasementSOIL

OUTDOORS

Figure 1: Diagram of the reference configuration.

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0 100 200 300 4000

100

200

300

400

Time (h)

Rado

n co

ncen

tratio

n (B

q/m

3)

B Basement C Room 1 D Room 2 E Room 3 F Room 4

0.0

0.2

0.4

0.6

0.8

1.0

Wat

er sa

tura

tion

fract

ion

G Water saturation fraction

Figure 2: Response of the indoor radon concentrations to a sudden change of the soil watersaturation fraction

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