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

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88 CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . 320 North Head field (reference) .0 .0 East 320 0.0 -2.0 -4.0 -6.0 -8.0 320 North Head field (reference) .0 .0 East 320 Figure 4.4: Reference piezometric head at the 60th time step. Left, as obtained at the fine scale; right, as obtained at the coarse scale Case C is conditional to piezometric heads. The same 200 coarse realizations from Case A serve as the initial ensemble of realizations to be used by the EnKF to assimilate the piezometric head measurements from locations W1 to W9 for the first 60 time steps (66.7 days). Case D is conditional to both logconductivity and piezometric heads. The same 200 coarse realizations from Case B serve as the initial ensemble of realizations to be used by the EnKF to assimilate the piezometric head measurements from locations W1 to W9 for the first 60 time steps (66.7 days). In Cases C and D we use the measured heads obtained at the fine scale in the reference realization as if they were measurements obtained at the coarse scale. There is an error in this assimilation that we incorporate into the measurement error covariance matrix. Specifically we here assumed a diagonal error covariance matrix, with all the diagonal terms equal to 0.0025 m 2 ; this value is approximately equal to the average dispersion variance of the fine scale piezometric heads within the coarse scale blocks. 4.3.4 Performance Measurements Since this is a synthetic experiment, the “true” aquifer response, evaluated at the fine scale, is known. We also know the upscaled conductivity tensors for the reference aquifer, which we will use to evaluate the performance of the updated conductivity tensors produced by the EnKF. The following criteria will be used to analyze the performance of the proposed method and the worth of data: 1. Ensemble mean map. (It should capture the main patterns of variability of the reference map.) 0.0 -2.0 -4.0 -6.0 -8.0
CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . 89 2. Ensemble variance map. (It gives an estimate of the precision of the maps.) 3. Ensemble average absolute bias map, ϵX, made up by: ϵXi = 1 Ne Ne ∑ r=1 |Xi,r − Xi,ref |, (4.13) where Xi is the parameter being analyzed, at location i, Xi,r represents its value for realization r, Xi,ref is the reference value at location i, and Ne is the number of realizations of the ensemble (200, in this case). (It gives an estimate of the accuracy of the maps.) 4. Average absolute bias AAB(X) = 1 Nb Nb ∑ i=1 ϵXi , (4.14) where Nb is the number of interblocks when X is coarse logconductivity tensor component, or the number of blocks when X is piezometric head. (It gives a global measure of accuracy.) 5. Square root of the average ensemble spread AESP (X) = [ 1 Nb Nb ∑ i=1 σ 2 Xi ] 1/2 , (4.15) where σ2 is the ensemble variance at location i. (It gives a global Xi measure of precision.) 6. Comparison of the time evolution of the piezometric heads at the conditioning piezometers W1 to W9, and at the control piezometers W10 to W13. (It evaluates the capability of the EnKF to update the forecasted piezometric heads using the measured values.) 4.4 Discussion Ensembles of coarse realizations for the four cases have been generated according to the conditions described earlier. Figure 4.5 shows the evolution of the piezometric heads in piezometers W1 and W9 for the 500 days of simulation; the first 60 steps (66.7 days) were used for conditioning in cases C and D. Similarly, Figure 4.6 shows piezometers W10 and W13; these piezometers
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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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10 CHAPTER 2. A COMPARATIVE STUDY O
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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
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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
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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 168 and 169: 136 BIBLIOGRAPHY Chen, Y., Zhang, D
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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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