9. Application of VIP - Burgberg Test Case 141
9.2. Probabilistic Well Vulnerability Results - VIP Applied to Burgberg
The Fig. 9.2 shows the isopercentilesα= [0.1,0.5,0.9]for the four probabilistic well vulner-ability criteria, based on the critical values from Tab. 9.2. Quite intuitively one can observe, that the smaller the non-compliance probabilityα(i.e., the larger the corresponding reliabil-ity levelβ), the larger the required delineated area. The 50th-percentile line, for example, encloses the areaA50outside of which the respective vulnerability criterion is violated with less than50 %probability (see Eq. 4.2). The area between theA50- and theA10-line may be interpreted as the safety margin to achieve a reliability of90 %. Thus the reliability of the supply system based on the area A(β) can be plotted for each possible reliability level β against the (areal) costs, see Fig. 6.2.
The filled (red) dot within Fig. 9.2 marks the location S (1.2km upstream of the well field with grid coordinates xgrid = [239; 167]; or in easting and northing coordinates xs = [3589240; 5383470]m), where a spill event for the sake of discussion is assumed, in order to compare results and illustrate the VIP-based management concept.
Peak arrival time (VIP 1) W V C1,crit=τcrit 50 [d]
Maximum Concentration (VIP 2) W V C2,crit=ζcrit 1·10−8 [-]
Reaction Time (VIP 3) W V C3,crit=τreact 10 [d]
Well exposure Time (VIP 4) W V C4,crit =τexp 30 [d]
Table 9.2.: Critical levels of well vulnerability for the Burgberg test case.
144ApplicationofVIP-BurgbergTestCa
0.90 0.50 0.10
Hürben
Burgberg
Lone
3588000 3589000 3590000 3591000
538300053840000
0.90 0.50 0.10
Hürben
Burgberg
Lone
3588000 3589000 3590000 3591000
538300053840000
0.90 0.50 0.10
Hürben
Burgberg
Lone
3588000 3589000 3590000 3591000
538300053840005385000
0.90 0.50 0.10
Hürben
Burgberg
Lone
3588000 3589000 3590000 3591000
538300053840005385000
Figure 9.2.: Probabilistic vulnerability isopercentiles (VIP) for the four intrinsic well vulnerability criteria (VIP 1: top left, VIP 2: top right, VIP 3: bottom left, VIP 4:bottom right) fromn= 1000conditioned realizations.
9.2.1. VIP 1: Probability of Early Peak Arrival Time
The first VIP map (see Fig. 9.2, top left) represents the regulations for the German inner wellhead protection area, when using the critical travel timeW V C1,crit = τcrit = 50d(see Tab. 9.2). Here, the peak arrival time is used instead of bulk arrival time, because this is the more conservative and more meaningful definition of arrival time in the advective-dispersive context (see Section 8.3). The thick solid line (purple) shows the existing well-head protection zone within the Burgberg catchment. At the spill siteS (red dot) selected for illustration, the mean arrival time of peak concentration equals142days, with23 % prob-ability that peak arrival time is faster thanτcrit = 50d. Apparently, the peak arrival times may easily fall belowτcrit, while the mean arrival time is much larger thanτcrit = 50days.
This fact stems from the strong skewness of the arrival time distribution (see Fig. 9.3), which has often been found to be close to log-normal under uncertainty of hydraulic conductivity (e.g., Cvetkovic et al., 1992; Dagan and Nguyen, 1989; Nowak et al., 2008). The isopercentile lines of peak arrival time are shown for the non-compliance levelα= [0.1,0.5,0.9], indicated by the three colors in Fig. 9.2. A more detailed discussion on the first VIP map, how peak arrival time is used for risk-aware delineation is provided in Section 9.3.
9.2.2. VIP 2: Probability of Insufficient Plume Dilution
The second VIP map (see Fig. 9.2, top right) shows the probability map that a point-like instantaneous contaminant spill of unit mass (e.g.,1kg) causes a peak concentration in the well of more than W V C2,crit = ζcrit = 1·10−8 mkg2 (see Tab. 9.2). In case of a one-time spill event of a water-soluble compound at locationS, I assume a total contaminant mass ofms = 10,000kgto be infiltrated to the aquifer. The expected peak concentration for this contaminant in the well is then calculated by multiplyingms with the mean value of the second well vulnerability criterion. The expected concentration for a unit-mass spill in the test case at locationS is3.9×10−8[−](see Fig. 9.2, top right), which leads to an expected peak concentration in the above contaminant example of3.9×10−1 mg/l. Single realizations can be more or less diluted, thus leading to higher and lower possible peak concentrations as discussed in Fig. 8.5. The probability to not comply with the critical dilution factorζcrit
at positionS is92 %. This indicates strongly that storing or handling such an amount of contaminants at locationSshould be avoided.
9.2.3. VIP 3: Probability of Too Little Available Response Time
The third VIP map shows the probabilities that an early-alert and emergency reaction system with a necessary reaction time ofW V C3,crit =τreact = 10dwill fail. Such an early-warning system could be a network of monitoring wells, sampled at least everyτreact- days, including the time for sample analysis and installation of mitigation measures to reduce or even avoid the impact on the drinking water supply. The reaction time is defined as the time available until a certain threshold concentration ofccrit = 1·10−7mg/lis exceeded within the well for a unit mass release. The areal extensions of VIP contours for reaction time are smaller than
0 100 200 300 400 500 600 700 0
1 2 3 4
x 10
−3Peak arrival time [d]
PDF [−]
E(tpeak)
Figure 9.3.: Probability density distribution of peak arrival time for a spill event at loca-tionS.
those for the corresponding peak arrival time, because the first arrival of low concentrations ccritis much earlier than peak arrival (compare Fig. 4.2 and Fig. 8.5). The mean reaction time for a unit spill at locationS (in all realizations whenccrit is exceeded) is66days, which is by far larger than the critical valueτreact= 10days. The probability that the reaction time is smaller (larger) than the given critical duration is15 % (85 %). The conditional probability of having a sufficient available reaction time in critical cases (whereccritis exceeded) is not85 % but only22 %because realizations withc > ccritbehave differently than those withc≤ccrit.
9.2.4. VIP 4: Probability of Excessive Exposure Time
The fourth VIP map shows the probability of exposure times being larger than the given critical exposure timeW V C4,crit =τexp = 30d(see Tab. 8.1). Exposure time is defined as the duration a well is contaminated with solute concentration above a critical threshold concen-trationccrit = 1·10−7mg/l, and thus unavailable for the water supply without treatment.
The mean well down-time for location S equals289days. The probability of critical well contamination for more thanτexp = 30dis92 %. 99 %of all realizations exceedingccritlead to well down-times of τexp ≥ 30d. In fact, contamination at almost any possible location within the catchment will lead to a well exposure duration larger thanτexp = 30d, for the considered concentration threshold level. Therefore, the map for VIP 4 and VIP 2 are prac-tically identical (see Section 8.2). In order to avoid the risk of long well down-times, the protection area has to be enlarged up to the level, where either the risk acceptance level is fulfilled or the economic viability is over-strained.
9.2.5. The Impact of Considering Specific Vulnerability
Next, a brief example is presented on how the information in the VIP maps can be modified for contaminant-specific properties (here: linear sorption) and how this affects the VIP maps evaluated via Eq. (5.43). Fig. 9.4 shows the two ensemble-averaged breakthrough curves at locationS, depending on the retardation factorRd. The base case (dark gray line) is calcu-lated without any attenuation potential besides the hydromechanical dispersion behavior of the aquifer. These intrinsic conditions are valid for conservative tracers or bacteriological and virological transport problems.
The other breakthrough curve (medium gray) shows a retarded concentration profile at the well withRd= 3.5due to linear sorption processes to the minerals. This value only serves to illustrate the flexibility of the vulnerability concept to account for fate and degradation processes due to the post-processing procedure. The retardation level is contaminant and site-specific and needs adaption to each catchment. The peak arrival time is shifted to later times, thus the isopercentile outlines within the first VIP map (peak arrival time) would be-come smaller than in the intrinsic case. Also, the intensity of the peak concentration in the well is reduced to much lower levels, causing smaller VIP contours for the second well vul-nerability criterion. The corresponding change of the third VIP map (threshold arrival time) is more difficult to explain. On the one hand, the time of exceeding a given threshold level would be postponed to later times (compare first VIP map, Fig. 9.2, top left), thus leading to increased reaction times. On the other hand, all concentrations are attenuated more strongly (compare second VIP map, Fig. 9.2, top right), such that the critical concentration would be exceeded in fewer realizations. In this specific example, both effects would roughly cancel out, and the third VIP map would almost not change. The same would be true for the fourth VIP map (exposure duration), as exposure time shrinks with increasing attenuation, while it increases with the slower overall speed of transport. Tab. 9.3 summarizes the correspond-ing probability values of non-compliance obtained when assumcorrespond-ing R = 3.5 for a spill at locationS.
9.2.6. Accuracy of Breakthrough Curve Reconstruction from Small Particle Numbers
To assess the accuracy of breakthrough curve reconstruction from small particle numbers via the inverse Gaussian distribution (Eq. 5.41), two different solute breakthrough curves are demonstrated at the well (see Fig. 9.5). Again, a contamination event at locationSserves
VIP 1 VIP 2 VIP 3 VIP 4
P1,S[%] ¯t1[d] P2,S[%] ¯cr[−] P3,S[%] ¯t3[d] P4,S[%] ¯t4[d]
intrinsic 23.2 142 91.8 3.9e−8 22.1 66.4 8.3 289.1
with sorption 2.7 499 33.9 1.1e−8 2.7 125.6 0.3 259.0
Table 9.3.: Ensemble-averaged well vulnerability criteria and the corresponding VIP value at locationSfor intrinsic transport conditions and with linear sorption (Rd= 3.5).
10 0
010
110
210
310
40.5
1 1.5 2
2.5 x 10
−8Time [days]
relative concentration [−]
intrinsic BTC retarded BTC
Figure 9.4.: Mean solute breakthrough curves fromn= 1000realizations within the well for a contamination spill at locationS. Dark gray line indicates intrinsic conditions and medium gray line is the breakthrough curve including sorption effects with Rd= 3.5. Please note the logarithmic scale of the time axis.
for illustration. The solid black line is based on a reference simulation withnp = 15,000,000 particles in a single realization, using the calibrated parameter set (e.g., Lang and Justiz, 2009). This number of particles follows a guideline of Kinzelbach (1988). However, all presented VIP calculations are based on calculations with a smaller number of particles np = 500,000in the backward transport modeling case. The reconstructed breakthrough curve based on the inverse Gaussian distribution (IGD, Eq. 5.41) and the smaller particle number is shown as the blue line in Fig. 9.5. The values of the four well vulnerability crite-ria computed with these two methods differ by△tpeak= 2 %,△cpeak = 26 %,△treact= 10 % and △texp = 7 %. The high deviation in peak concentration△cpeak stems only from the high fluctuations in particle arrival even in the reference simulation, and would be reduced to below 10 %, if applying a suitable noise filter to the reference BTC. These error levels fall far below the statistical fluctuation of the well vulnerability criteria between the real-izations. Thus, the reduced model seems to be acceptable for the vulnerability concept, reflected against the fact that it yields a computational speed up by a factor of30.