Synthetic Case Study

For the synthetic pumping test we generate a reference log-conductivity field log(K)^{r e f}from a selection-Gaussian distribution with parameters£^SG_{r e f} =(µ^{r e f}_r˜ ,µ^{r e f}_∫ ,æ^{r e f}_r˜ ,ß^Ω_r˜,∞^{r e f},A^{r e f}). The parameters values are listed in Table 5. The spatial structure of the generated log-hydraulic conductivity field is in agree-ment with the two-layer system of hydraulic conductivity distributions recognized at the field site (see Figure 6). We assume an effective aquifer storativityS₀of 8.6£10^°3m^°1which is typical for confined systems.

For the SEnKF, see Section 3.3.2.3, the prior ensemble for the log-conductivity field is generated from a selection-Gaussian random field log(K) with parameters£^SG_K =(µ^k_r˜,µ^k_∫,æ^k_r˜,ß^Ω_r˜,∞^k,A^k). The param-eters (∞^k,A^k) are chosen such that the prior marginal distributions be bimodal, with modes close to those of the spatial histogram of the reference log conductivity fieldlog(K)^{r e f} (Figure 7). For the EnKF, see Section 3.1, the prior ensemble for the log-conductivity field is generated from a Gaussian random field log(K) defined by'_n(log(K);µ_ri_n,æ²_rß^Ω_r) whereµ_r=°7 andæ_r =p

1.6 are chosen such that the marginal prior distribution covers the spatial histogram of the reference log-conductivity field (Figure 7). The spatial correlation (n£n)-matrixß^Ω_· is defined by the second order exponential spatial

correla-Table 2: Parameter values for reference and prior model

Ref µ^{r e f}_r˜ () µ^{r e f}_∫ () æ^{r e f}_r˜ () ∞^{r e f}() A^{r e f}() -7 0 0.6 0.99 [(°1,°0.1][[0.1,+1)]ⁿ Prior SEnKF µ^k_r˜() µ^k_∫() æ^k_r˜() ∞^k() A^k()

-7 0 1.6 0.95 [[°3,°0.2][[0.3,3.3]]ⁿ

tion functionΩ(ø;dx,dy,dz)=exp[°(ø²_x/d²_x+ø²_y/d²_y+ø²_z/d_z²)]. This spatial correlation function is used to generate the reference field and for both prior models with parameters valuesdx=8m,dy=4m and d_z=1m. While the prior for the SEnKF has a larger marginal variance, it has a comparable spread

Log-conductivity (m/s)

Density

Log-conductivity (m/s)

Density

Figure 7: Prior marginal distribution for the SEnKF (left) and the EnKF (right) compared to the spatial histogram of the reference log-conductivity field

to the prior for the EnKF (Figure 7). The prior distribution for the effective storativityS0is defined by '₁(S0;8.6£10^°3,1£10^°6). During the data assimilation, covariance localization, see Section 3.3.1.1, is used in both methods.

4.2.1 Results

Figures 8 and 9 show the predicted drawdown recorded at the nine observation wells defined in the model for the SEnKF and EnKF, respectively. The dark gray area represents the 70% prediction interval, the light gray area represents the 90% prediction interval. The red lines are the true unperturbed synthetic draw-down, and the red crosses represent drawdown perturbed by observation error with standard deviation æ^t_d=0.05. The value ofæ^t_dis representative of the expected accuracy of modern sensors for hydraulic head.

4 ENSEMBLE METHODS APPLIED TO GROUNDWATER FLOW 29

Figure 8: Predicted drawdown from the SEnKF at the observation wells for pumping test 1b compared to the true (red line) and measured drawdowns (red crosses). The dark gray area represents the 70%

prediction interval, the light gray area represents the 90% prediction interval.

The SEnKF and the EnKF appear to predict convincingly the true unperturbed drawdown. The spread of the predicted drawndown is larger for the SEnKF than for the EnKF, and it seems more commensurate with the observed drawdown for the former than the latter. This suggests that the EnKF might be under-estimating the uncertainty despite the good predictions of the true unperturbed drawdowns.

Figure 10 displays the 3D reference log-conductivity field together with the 3D predicted log-conductivity field from the SEnKF and the EnKF. Recall that the predicted spatial variables provided by the SEnKF and the EnKF will be smoother than the truth. The SEnKF prediction represents to some extent the two-layer structure observed in the reference field while the EnKF prediction exhibits the aforementioned regres-sion towards the mean.

Figure 11 shows the predicted log-conductivity along well 2 for the SEnKF and the EnKF. The SEnKF appears to better predict the log-conductivity along this well as it is able to identify the aforementioned layered structure of the reference field, at least at this measurement well, while the EnKF does not.

These observations are consistent with the analysis of Figure 10. Recall again that spatial predictions are smoother than the truth.

Figure 12 shows the coverage probability of the reference log-conductivity field for the SEnKF and the EnKF prediction intervals. Visual inspection of the curves indicates that the SEnKF slightly overestimates the prediction intervals while the EnKF clearly underestimates them. Conservatively overestimating the prediction intervals appear preferable to the opposite.

Figure 9: Predicted drawdown from the EnKF at the observation wells for pumping test 1b compared to the true (red line) and measured drawdowns (red crosses). The dark gray area represents the 70%

prediction interval, the light gray area represents the 90% prediction interval.

Table 3: RMSE and SSIM and coverage comparing the predictions of the different methods to the true log diffusivity field

SEnKF EnKF

Initial M AE 1.26 1.15

M AE 1.05 1.10

Posterior SSI M 0.0704 0.0464

To further quantify the performance of the data assimilation experiments, we estimate the mean absolute error (MAE) between the predicted log-conductivity fields and the reference field. We want the MAE to be as small as possible. We also compare the structural similarity index (SSIM) (e.g. Zhou et al.

(2004)), a metric used in image analysis to measure the degradation of the structural information of an image. We want the SSIM to be as large as possible. The results are detailed in Table 3, both metrics are favorable to the SEnKF. In particular the SEnKF offers a 17% relative reduction in MAE while the EnKF only gives 4%.

4 ENSEMBLE METHODS APPLIED TO GROUNDWATER FLOW 31

Figure 10: Reference log-conductivity field (left), mean log-conductivity prediction from the SEnKF (cen-ter) and mean log-conductivity prediction from the EnKF (right)

4.2.2 Closing remarks

The synthetic case study demonstrate that the SEnKF provides more accurate predictions of the reference log-conductivity field than the EnKF. Moreover, the prediction intervals for the drawdowns provided by the SEnKF appear as more realistic than the ones provided by the EnKF. To further test the performance of the SEnKF for real applications, where information about the true aquifer properties is limited, we apply the presented methodology to data collected from real pumping tests.