Upscaling and Inverse Modeling of Groundwater Flow and Mass ...

Recommendations

Info

162 CHAPTER 6. GROUNDWATER FLOW INVERSE MODELING . . . Correct prior Wrong prior 240 North (B) (A) 240 .0 .0 (D) East 300 240 North .0 .0 East 300 Constant Head (h=0) No Flow 0 No Flow 300 3.0 (C) 240 2.0 1.0 0.0 -1.0 -2.0 North 3 Q=-4.0 m /d 3 Q=-90.0 m /d 3 Q=-11.0 m /d 3 Q=-150.0 m /d 3 Q=-15. 5 m /d Ensemble mean Ensemble variance -3.0 .0 .0 East 300 3.0 (E) 240 2.0 1.0 0.0 -1.0 -2.0 North -3.0 .0 .0 East 300 Figure 6.10: (A) flow configuration and spatial distribution of sampled head data. (B) the ensemble mean of logconductivity with correct prior model. (C) the ensemble variance of logconductivity with correct prior model.(D) the ensemble mean of logconductivity with wrong prior model. (E) the ensemble variance of logconductivity with wrong prior model. 3.0 2.0 1.0 0.0 3.0 2.0 1.0 0.0
CHAPTER 6. GROUNDWATER FLOW INVERSE MODELING . . . 163 6.4.2 Effect of localization It is well known that the standard EnKF has the problem of filter inbreeding, which leads to a final ensemble of realizations too similar among them resulting in the underestimation of uncertainty of flow and transport predictions (Hendricks Franssen and Kinzelbach, 2009; Devegowda et al., 2010; Zhou et al., 2011a). The ensemble size we have used is large, 1000 realizations, yet, when analyzing the piezometric head covariances, and cross-covariances with logconductivity we observe the same type of spurious correlation values that, eventually, lead to filter inbreeding; therefore, we have decided to apply localization techniques and compare the resulting ensembles with and without localization. In general, localization should increase the ensemble spread and, at the same time, reducing the RMSE, that is, there is a gain in accuracy at the cost of precision, which also helps to avoid filter inbreeding. Analyzing Figures 6.5 and 6.6 we observe that the ensemble mean hydraulic conductivity maps produced with the localized NS-EnKF are smoother than the same mean maps without localization, and capture better the main channels of the reference field. All artifacts in the non-localized realizations are removed after localization; the local ensemble variances in the localized ensembles increase, particularly significant values appear in the northwestern corner of the parallel flow exercise an area without conditioning data and too close to a prescribed head boundary where the sensitivity of the piezometric head to a change in logconductivity is close to zero. These results may indicate that even in this case with a large number of realizations, the ensemble variance estimate obtained from a non-localized NS-EnKF may be too optimistic. The effect of localization is also quite clear when analyzing the RMSE and ES of the calibrated conductivity fields (see Figure 6.7). On one hand, localization induces a reduction of the RMSE in all cases, and an increase in the ES. In all cases, these changes result in values of RMSE and ES that are very similar in value at the end of the updating steps, which is indicative, according to Chen and Zhang (2006), of reduced filter inbreeding. Taking the ratio ES/RMSE as an index of filter inbreeding, this is worse for the radial flow scenarios without localization. In previous synthetic studies, Chen and Zhang (2006) and Hendricks Franssen and Kinzelbach (2008) have found that 200 realizations are enough to avoid filter inbreeding without localization for multiGaussian fields. Here we observe that filter inbreeding can be serious when EnKF is used without localization in a highly channelized aquifer. Regarding sand connectivity it is not clear whether localization improves its characterization. The spread of the connectivity functions is much larger for the localized ensemble than for the non localized one, and, at least for parallel flow, the average of the connectivity functions does not get closer to the real one after localization; the small spread exhibited by the non localized
Page 1:
Upscaling and Inverse Modeling of G
Page 4 and 5:
c⃝ Copyright by Liangping Li 2011
Page 7 and 8:
Abstract The need to reduce the com
Page 9 and 10:
improved as more data is assimilate
Page 11 and 12:
Resumen La necesidad de reducir el
Page 13 and 14:
Por último, en el tercer bloque, e
Page 15 and 16:
Resum La necessitat de reduir el co
Page 17 and 18:
flux i el transport (conductivitat
Page 19:
Acknowledgements I want to thank my
Page 22 and 23:
xx CONTENTS 3 Transport Upscaling U
Page 24 and 25:
xxii CONTENTS
Page 26 and 27:
xxiv LIST OF FIGURES 2.7 Flow compa
Page 28 and 29:
xxvi LIST OF FIGURES 4.4 Reference
Page 30 and 31:
xxviii LIST OF FIGURES
Page 33 and 34:
1.1 Motivation and Objectives 1 Int
Page 35:
CHAPTER 1. INTRODUCTION 3 Chapter 2
Page 38 and 39:
6 CHAPTER 2. A COMPARATIVE STUDY OF
Page 40 and 41:
8 CHAPTER 2. A COMPARATIVE STUDY OF
Page 42 and 43:
10 CHAPTER 2. A COMPARATIVE STUDY O
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
38 BIBLIOGRAPHY Berkowitz, B., Sche
Page 72 and 73:
40 BIBLIOGRAPHY Gómez-Hernández,
Page 74 and 75:
42 BIBLIOGRAPHY Salamon, P., Fernà
Page 76 and 77:
44 CHAPTER 3. TRANSPORT UPSCALING U
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
70 BIBLIOGRAPHY Dagan, G., 1994. Up
Page 104 and 105:
72 BIBLIOGRAPHY Lawrence, A. E., Sa
Page 106 and 107:
74 BIBLIOGRAPHY Zhang, Y., 2004. Up
Page 108 and 109:
76 CHAPTER 4. MODELING TRANSIENT GR
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
100 CHAPTER 4. MODELING TRANSIENT G
Page 134 and 135:
102 BIBLIOGRAPHY Durlofsky, L. J.,
Page 136 and 137:
104 BIBLIOGRAPHY Tran, T., 1996. Th
Page 138 and 139:
106 CHAPTER 5. JOINTLY MAPPING HYDR
Page 140 and 141:
Page 142 and 143:
Page 144 and 145: 112 CHAPTER 5. JOINTLY MAPPING HYDR
Page 168 and 169: 136 BIBLIOGRAPHY Chen, Y., Zhang, D
Page 170 and 171: 138 BIBLIOGRAPHY Journel, A., 1974.
Page 172 and 173: 140 BIBLIOGRAPHY Wen, X., Deutsch,
Page 174 and 175: 142 CHAPTER 6. GROUNDWATER FLOW INV
Page 202 and 203: 170 BIBLIOGRAPHY Deutsch, C. V., Jo
Page 204 and 205: 172 BIBLIOGRAPHY Lee, S., Carle, S.
Page 206 and 207: 174 BIBLIOGRAPHY Yeh, W., 1986. Rev
Page 208 and 209: 176 CHAPTER 7. CONCLUSIONS travels
Page 210 and 211: 178 CHAPTER 7. CONCLUSIONS predicti
Page 212 and 213: 180 APPENDIX A. FLOWXYZ3D - A THREE
Page 214: 182 APPENDIX A. FLOWXYZ3D - A THREE
show all

Upscaling and Inverse Modeling of Groundwater Flow and Mass ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?