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

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86 CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . Table 4.1: Definition of Cases depending on the the different sets of conditioning data. Conditioning Data Case A Case B Case C Case D Hydraulic conductivities(K) No Yes No Yes Dynamic piezometric heads(h) No No Yes Yes to-row interblock tensors (K b,r ). All interblock tensors are transformed into their corresponding triplet of invariants prior to starting the EnKF algorithm. For illustration purposes, Figure 4.3 shows the resulting triplets for the reference field. This figure will be used later as the reference upscaled field to analyze the performance of the proposed method. On the right side of Figure 4.4, the simulated piezometric heads at the end of the 60th time step are displayed side by side with the simulated piezometric heads at the fine scale. The reproduction of the fine scale spatial distribution by the coarse scale simulation is, as can be seen, very good; the average absolute discrepancy between the heads at the coarse scale and heads at the fine scale (on the block centers) is only 0.087 m. 4.3.3 Case Studies Four cases, considering different types of conditioning information, are analyzed to study the performance of the proposed approach (see Table 4.1). They will show that the coupling of the EnKF with upscaling can be used to construct aquifer models that are conditional to conductivity and piezometric head data, when there is an important discrepancy between the scale at which the data are collected and the scale at which the flow model is built. The cases will serve also to carry out a standard worth-of-data exercise in which we analyze the trade-off between conductivity data and piezometric head data regarding aquifer characterization. Case A is unconditional, 200 realizations are generated according to the spatial correlation model given by Equation (4.11) at the fine scale. Upscaling is performed in each realization and the flow model is run. No Kalman filtering is performed. Case B is conditional to logconductivity measurements, 200 realizations of logconductivity conditional to the 100 logconductivity measurements of Figure 4.4B are generated at the fine scale. Upscaling is performed in each realization and the flow model is run. No Kalman filtering is performed. Cases A and B act as base cases to be used for comparison when the piezometric head data are assimilated through the EnKF.
CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . 87 �� Figure 4.3: Upscaled values for the interblock tensor components: ln(Kmax), ln(Kmin) and rotation angle for the maximum component measured from the x-axis θ (in degrees), for both the interblocks between columns and the interblocks between rows. Upscaling method used: Laplacian with a skin of 10 m ��
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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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Page 70 and 71: 38 BIBLIOGRAPHY Berkowitz, B., Sche
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Page 74 and 75: 42 BIBLIOGRAPHY Salamon, P., Fernà
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Page 102 and 103: 70 BIBLIOGRAPHY Dagan, G., 1994. Up
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Page 106 and 107: 74 BIBLIOGRAPHY Zhang, Y., 2004. Up
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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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