Steady state calibration

Most of the uncertainties in predicting real aquifer behaviours are due to lack of adequate data for assessing the hydraulic parameters of the aquifer and its spatial variability (SEPA, 2006). In order for a groundwater model to be used in any type of predictive mole, it must be demonstrated that the model can successfully simulate observed aquifer behaviour; where significant differences exist, it is an indication that the initial parameter estimates are inadequate and that more reliable estimates must be obtained by a formal calibration of the model. The best way to know whether or not these parameters accurately reflect the true behaviour of the aquifer is to compare the computed heads with the observed heads. If this is not satisfactory, then the only way to obtain representative aquifer parameters is by calibrating the model, i.e. during which aquifer parameters are varied until the simulated heads are acceptably close to those observed. Calibration is the process wherein certain parameters of the model such as recharge and hydraulic conductivity are altered in a systematic fashion and the model is repeatedly run until the computed solution matches field-observed values within an acceptable level of accuracy.

Calibration begins by choosing the calibration targets and determining the ranges for all potential parameters that can be adjusted during calibration. The model calibration process adjusts model parameters from their initial values until the calibration goal which

involves the minimization of an objective function is achieved. The objective function is the sum of squares of the residuals, i.e.

)

and N is the total number of target wells.

GMS software provides a number of criteria for testing the adequacy of the calibration.

These can be quantitative and semi quantitative.

Quantitative calibration criteria test the runs statistic value to ensure that the residual are random and the correlation between ordered weighted residuals and normal order statistics to ensure that they are normally distributed. The non-parametric runs test can be described as follows (Adeloye and Montaseri, 2002):

Let the objective be to test whether the data sample (in this case the residuals) Y_i, i

=1,….,n is random based on the runs of the data with respect to the median of the observation. The procedure is therefore as follows:

1. Determine the median of the observation. To do this, sort the sample in increasing

2. Examine each data item in turn to see whether or not it exceeds the median. If a data item exceeds the median, then this is a success case (replaced by letter S) but if it does not exceed the median, it is a failure case (denoted by letter F). Cases that are exactly equal to the median are excluded.

3. Count the number successes and denote this by n₁; similarly denote the number failures by n2. In general, n = n1 + n2 except where some of the values are omitted as explained in step 2 above.

4. Determine the total number of runs in the data. A run is a continuous sequence of S‟s until it is interrupted by an F and vice versa. Let the total number of runs be denoted by R.

5. Compute the test statistic

  standard normal distribution. Hence obtain critical values of the standard normal distribution for the chosen significance level  and denote these by z_/2

7. Compare the z obtained in step 5 (see equation (5.3)) with the critical values z_/2. Reject Ho if z < z_/2 or z > z_/2. In general the critical z vales are tabulated in standard statistical textbooks but for z_/2= 1.96, 1.65 and 1.28 for the 5%, 10%, and 20% significance level respectively. Since the run statistic (-0.0887) in Table 5.9 is higher than -1.96, then we do not have any statistical evidence to reject the null hypothesis. The residuals can therefore be considered to be random at the 5% level

To test for normality of the residual, it is necessary to test the statistical significance of the correlation between the residuals and the normal order statistics. Consider the null hypothesis

H_o: R = 0, against the alternative hypothesis H1: R ≠ 0,

where R is the correlation coefficient. The appropriate test statistic for these hypotheses is (Montgomery and Runger, 2003, page 402):

)

which has the t distribution with n-2 degrees of freedom if Ho is true. Therefore, the null hypothesis will be rejected if the calculated t_o < -t α/2,n-2 or t_o > t α/2,n-2.

The sample size for the example in Table 5.9 is 61; hence the corresponding critical value of the t-statistic at the 5% level, t 0.025, 59 = 1.96. Also from the results in Table 5.9, the estimated correlation coefficient R = 0.985. Thus,

85 hypothesis at the 5% level, in other words we can accept the alternative hypothesis that the residuals are normally distributed at the 5% level.

Semi quantitative calibration criteria involve ensuring that:

1. Parameters adjusted during calibration should be consistent with field measured values.

2. Groundwater flow direction in the key area of the site should be matched by the model.

3. Important hydrological features such as groundwater structures, shapes and divides should be replicated by the model.

The GMS software provides a number of automated calibration tools as well as a trial and error method to iteratively adjust model parameters until the model computed values match the field observed values to an acceptable level of agreement. Model calibration can be done either manually or by using automated methods. The common practice is to use both methods.

In many cases, calibration can be achieved much more rapidly with an inverse model. The GMS software contains an interface to three inverse models similar to the use of equation 5.1: MODFLOW 200 PES process, PEST and UCODE. An inverse model is an internal process which is MODFLOW 200 PES process or an external utility (PEST and UCODE) that automates the parameter estimation process (EMRL, 2004). It systematically adjusts a user-defined set of input parameters until the difference between the computed and observed values of heads is minimized. MODFLOW 200 PES process was used in this study because layer 2 encompasses a diverse mix of deposits such as conglomerate, limestone, mudstone and siltstone. It also has locality variation in lithology. Therefore, PES process calibration would be the best to represent smaller K zones variations for this complex and heterogamous layer. This involved identifying polygonal zones of hydraulic conductivity, making the zones as parameters, and assigning a starting value for each zone. The PES Process will then adjust the K values assigned to the zones as it attempts to minimize the residual error between computed versus observed heads and flows. Bahremand and De Smedt (2008) used a model-independent parameter estimator, PEST, in their study titled Distributed Hydrological Modelling and Sensitivity

Analysis in Torysa Watershed, Slovakia. The results of this study demonstrated that the use of combining a GIS-based hydrological model with PEST can produce calibrated parameters that are physically sensible.

Using 42 control points (see Table 4.4 of Chapter 4), the hydraulic conductivity (K) distribution for layer 2 was derived based on field pumping test transmissivity values obtained from the exploration well drilling data using an average saturated thickness of 150 m ( MWR, 1997e). This layer is much more complex as it encompasses a diverse mix of deposits such as conglomerate, limestone, mudstone and siltstone. Therefore, the K data were subdivided into smaller zones and adjusted as illustrated in Figure 5.9a by initial trial and error calibration method. In doing this, the mentioned K control points were used as guide to establish the reasonable initial starting K distribution zones compatible with the regional generated contours heads from the exploration wells in 1997 (see Figure 5.7) and the existing observation wells in these zones. Then automatic steady state calibration for these polygonal zones was carried out to determine more accurate estimates of the hydraulic parameters. Figure 5.9 (a- and b) shows the initial hydraulic conductivity values (0.3 – 65 m/day) and the final calibrated K values (0.55 – 554 m/day) for layer 2 (alluvium) respectively. This large difference in K is reflecting the heterogeneity deposits of this layer (mudstone to conglomerate).

On the other hand, the available data of the layer 1 and the homogeneity in this layer suggest that parameter zonation for K is not essential (MWR, 1997e). Therefore, an average K value of 4 m/day has been adopted for the aeolianite in the model which was acceptable value after the calibration.

Sixty one observation wells were selected to carry out the calibration process; 21 and 40 observation wells were used for layer 1 and layer 2 respectively. In order to evaluate the

calibration performance, the comparison between the observed and simulated heads is reported in Tables 5.7 and 5.8 for layer 1 and layer 2 respectively. The differences between the observed and simulated heads are maximum two metres with exception of well WAB236 showing 3.1 m and 3.2 m for layer 1 and layer 2 respectively. This close difference indicates that there is good overall agreement between observed and simulated water levels throughout the model domain.

Furthermore, the software automation calibration statistics reported a value of 0.985 for the correlation between weighted residuals and normal order statistics, which is greater than 0.963 (the critical value for the correlation at the 5% significance level). This means that one can accept the hypothesis that the weighted residuals are independent and normally distributed at the 5% significance level. The calibration summary statistics are presented in Table 5.9.