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90 CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . were not used for conditioning. Figure 4.7 shows the ensemble mean and variance of the piezometric heads at the 60th time step, while Figure 4.8 shows the ensemble average absolute bias. Figure 4.9 shows the ensemble mean and variance of ln(Kmax) for interblocks between rows, and Figure 4.10 shows the ensemble average absolute bias. Finally, Table 4.2 shows the metric performance measurements for ln(Kmax) between rows and for piezometric heads at the 30th, 60th and 90th time steps. 4.4.1 The EnKF Coupled with Upscaling The EnKF has the objective of updating conductivity realizations so that the solution of the flow equation on the updated fields will match the measured piezometric heads. Analyzing cases C and D in Figure 4.5, we can observe how the updated fields, when piezometric head is assimilated by the EnKF, produce piezometric head predictions that reproduce the measured values very well, particularly when compared with case A, which corresponds to the case in which no conditioning data are considered. Notice also that piezometric head data are assimilated only for the first 66.7 days (the period in which the heads are almost perfectly reproduced in the EnKF updated fields) while the rest of the simulation period serves as validation. Additional validation of the EnKF generated realizations, is given in Figure 4.6 that shows two of the piezometers not used for conditioning; we can also observe the improvement in piezometric head reproduction for cases C and D as compared to case A. Furthermore, the analysis of Figure 4.7 shows how, for cases C and D, the average spatial distribution, at the end of time step 60, follows closely the reference piezometric head distribution, while the ensemble variance is reduced to very small values everywhere. The ensemble average head bias is also noticeably reduced when conditioning to heads, not just at the conditioning locations (as expected) but also elsewhere. A final analysis to show how conditioning to the heads improves the overall reproduction of the head spatial distribution is by looking at the metrics displayed in Table 4.2. Comparing cases B and C, it is interesting to notice the increasing impact of the conditioning to piezometric heads as time passes; at time step 30, the initial effect of just conditioning to hydraulic conductivity measurements (which occurs from time step 0 ) is still larger than just conditioning to the heads measured during the first 30 time steps, but at time step 60, this effect is clearly reversed, and it is maintained to time step 90 even though the heads between steps 60 and 90 are not used for conditioning. As expected, conditioning to both piezometric heads and hydraulic conductivities gives the best results in terms of smallest bias and smallest spread.
CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . 91 Piezometric Head [m] Piezometric Head [m] Piezometric Head [m] Piezometric Head [m] 1 0 -1 -2 -3 -4 -5 -6 -7 1 0 -1 -2 -3 -4 -5 -6 -7 1 0 -1 -2 -3 -4 -5 -6 -7 1 0 -1 -2 -3 -4 -5 -6 -7 Case A (W1) 100 200 300 400 500 Time [day] Case B (W1) 100 200 300 400 500 Time [day] Case C (W1) 100 200 300 400 500 Time [day] Case D (W1) 100 200 300 400 500 Time [day] Piezometric Head [m] Piezometric Head [m] Piezometric Head [m] Piezometric Head [m] 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Case A (W9) 100 200 300 400 500 Time [day] Case B (W9) 100 200 300 400 500 Time [day] Case C (W9) 100 200 300 400 500 Time [day] Case D (W9) 100 200 300 400 500 Time [day] Figure 4.5: Piezometric head time series in the reference field and simulated ones for all cases at wells W1 (left column) and W9 (right column). The piezometric heads measured at these wells during the first 67.7 days were used as conditioning data for cases B and D.
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Upscaling and Inverse Modeling of G
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c⃝ Copyright by Liangping Li 2011
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Abstract The need to reduce the com
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improved as more data is assimilate
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Resumen La necesidad de reducir el
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Por último, en el tercer bloque, e
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Resum La necessitat de reduir el co
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flux i el transport (conductivitat
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Acknowledgements I want to thank my
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xx CONTENTS 3 Transport Upscaling U
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xxii CONTENTS
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xxiv LIST OF FIGURES 2.7 Flow compa
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xxvi LIST OF FIGURES 4.4 Reference
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xxviii LIST OF FIGURES
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1.1 Motivation and Objectives 1 Int
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CHAPTER 1. INTRODUCTION 3 Chapter 2
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6 CHAPTER 2. A COMPARATIVE STUDY OF
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8 CHAPTER 2. A COMPARATIVE STUDY OF
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38 BIBLIOGRAPHY Berkowitz, B., Sche
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Page 102 and 103: 70 BIBLIOGRAPHY Dagan, G., 1994. Up
Page 104 and 105: 72 BIBLIOGRAPHY Lawrence, A. E., Sa
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Page 170 and 171: 138 BIBLIOGRAPHY Journel, A., 1974.
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140 BIBLIOGRAPHY Wen, X., Deutsch,
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142 CHAPTER 6. GROUNDWATER FLOW INV
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170 BIBLIOGRAPHY Deutsch, C. V., Jo
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172 BIBLIOGRAPHY Lee, S., Carle, S.
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174 BIBLIOGRAPHY Yeh, W., 1986. Rev
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176 CHAPTER 7. CONCLUSIONS travels
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178 CHAPTER 7. CONCLUSIONS predicti
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180 APPENDIX A. FLOWXYZ3D - A THREE
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182 APPENDIX A. FLOWXYZ3D - A THREE
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