Sampling stages Phreatic well fields

The reconnaissance stage consists of systematic sampling, using a fixed vertical sampling distance, of one available borehole (Figure 7.5). At this stage systematic sampling is preferred because the depths of reactive layers are not yet known. The disadvantage of systematic sampling is that the fixed distance might by chance correspond to a regular vertical sequence of reactive layering. Therefore, random sampling is advised for the second quantification stage. The reconnaissance sampling yields first indications on the position of the most reactive layers and a first indication of the average content of the reactive component (x¯) of the major reactive layer (Figure 7.5).

The second, quantification stage aims at improving the precision of the estimate of the average content of the reactive component and quantifying the uncertainty using a 95% confidence interval. As above, random sampling is recommended to obtain unbiased estimates of the average content of the reactive layer (Figure 7.5). For sample size n less than 30, the parametric 95% confidence interval is given by:

(7.1) where x¯ = average content of sample, s = standard deviation of sample, u = expected value of the population average, and t0,975,n-1= two-sided critical value for the Student’s t-distribution

with n-1degrees of freedom. The random sampling is preferably carried out on several boreholes to ensure that the estimates are spatially representative (Figure 7.5). However, a practical constraint of the presented sampling strategy is the restriction to the use of planned drilling programs (section 7.4).

The sampling results of the reconnaissance stage are used to determine the required sample size in the quantification stage. The sample size is adapted to the amount of variation, based on

Stage Phreatic well fields Deep-well recharge systems

Reconnaissance first estimate of average content first estimate of percentiles and coefficient of variation and coefficient of variation

Quantification estimates of average content estimates of percentiles

and quantified uncertainty and quantified uncertainty (confidence interval) (confidence interval) Table 7.1 - Specific information goals for the reconnaissance and quantification stages

the coefficient of variation (s/x¯), measured during the reconnaissance stage (Figure 7.5). For a normally distributed data set, the following equation describes the relation between the coefficient of variation and the relative precision as a percentage of estimated average content:

(7.2) where r is relative precision, n is sample size and t is critical value for the t-distribution for

n < 30. A relative precision of 50% means that the probability is less than 5% that the

population average u is outside the range x¯ - 0.5 x¯ < µ < x¯ + 0.5 x¯.

Following equation (7.2), a larger sample size is required for large s/x¯ to obtain similar precision of the estimate. Figure 7.6 shows the required sample size to obtain 30% and 50% relative precision for different values of s/x¯. Appendix IV discusses the use of the normal distribution for the determination of the sample size and shows that when using Figure 7.6 the required sample size is likely to be overestimated for n < 15.

As discussed in section 7.2, the sampling should provide data that are useful at the scale of model cells of the transport model (scale level (c) of Figure 7.1). Transport models for use at phreatic well fields need model layers of 1 to 5 m thickness in order to accurately predict the downward movement of the pollution fronts and to control the effects of numerical dispersion (Van Vught & van der Eijnden 1999, Uffink et al. 2001, Uffink 2001). Using this vertical dis- cretisation scale, the model results can also be compared with monitoring results from mixed samples of well screens of 2 m length (Chapter 6).

In the Netherlands, geochemically undisturbed samples are usually taken using thin wall tube samplers of about 0.5 m length. A decision was made to acquire average values for a 0.5 m depth interval of the aquifer by homogenizing the whole content of the sampling device or by mixing of a number of random samples from the core (Figure 7.5). The sample volumes

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systematic

sampling determinereactive layers

random

sampling mix samplefrom core analysis

Reconnaissance stage Q 1 2 3 lab Quantification stage s/x 0 5 10 15 20 25 30 35 de pt h (m ) determine sample size use in transport models specific reactive phases bulk analysis

grain size average content x

confidence interval β + 95% 12 3 – 1 1/2β x - x +1/2β - – –

Figure 7.5 - Overview of the sampling strategy at phreatic well fields with indication of the reconnaissance stage (systematic sampling of one borehole) and the quantification stage (random sampling from available boreholes), sample treatment, chemical and physical analysis and data use.

obtained have a similar support as the model cells, and no further correction for the support effect is necessary in the data analysis.

Deep-well recharge systems

The sampling stages at deep-well recharge systems correspond largely to the description for the phreatic well fields. However, the aim is to estimate the percentiles of the frequency dis- tribution in order to quantify the variability of the reactive properties with depth. Systematic sampling of the entire infiltration aquifer is recommended for the reconnaissance stage in order to obtain information over the complete depth range (Figure 7.7). The reconnaissance

sampling yields a first indication of the frequency distribution in the infiltration aquifer, including an initial estimate of the percentiles and the coefficient of variation s/x¯ .

This information is subsequently used to determine sample size for the quantification stage. The sampling in this stage aims at quantifying the precision of percentiles of the frequency distribution, using 95% confidence intervals. Non-parametric methods are preferred for the assessment of the confidence intervals, because parametric confidence intervals on percentiles depend strongly on the assumed frequency distribution (Helsel & Hirsch 1992). This effect is less for confidence intervals on the average contents at phreatic well fields as explained in Appendix IV.

In the example discussed in the next section, the 12.5, 37.5, 62.5 and 87.5 percentiles are used for illustration. The precision may be assessed using a non-parametric 95% confidence interval on the estimated percentile, using a normal approximation to the binomial distribution for sample size n > 20 (Helsel & Hirsch 1992). The confidence interval is calculated as:

(7.3) (7.4) where p is the percentile (0.75 for the 75 percentile) and z0.025and z0.975are the critical values

of the normal distribution. The data are ordered according to rank and Ruand Rlrepresent the ranks that correspond to the upper and lower limits of the confidence interval.

No parametric methods are available to determine the required sample size for obtaining an estimate with a pre-specified relative precision without assuming a distinct probability

80 70 60 50 40 30 20 10 0 sa m pl e si ze ( n) 0 0.4 0.8 1.2 1.6 2.0 2.4

coefficient of variation initial sample

precision (%) 30 50

Figure 7.6 - Required sample size to achieve 30% and 50% relative precision of the average content as a function of coefficient of variation s/x¯ .

distribution. Instead, bootstrapping may be used to get a general idea of the required sample size. In appendix IV, data from the Oostrum aquifer are used for bootstrapping (see next section for details about the Oostrum aquifer). The bootstrapping results indicated that a larger sample size is required to attain accurate estimates of a percentile than to attain accurate estimates of the average content. A suggestion for sample size is given in Appendix IV.

Transport models for use at deep-well recharge systems usually have model layers of 5-10 m thickness (for example Stuyfzand 1998). Therefore, mixing the contents of the 0.5 m thin wall tube samplers was considered reasonable to attain sample volumes of the largest possible support as input for the model cells.