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

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100 CHAPTER 4. MODELING TRANSIENT GROUNDWATER . . . block tensor, the underlying random function used to generate the fine scale realizations has no interference on the upscaling; however, for the third step, the application of EnKF to non-multiGaussian parameter fields is more difficult, some researchers propose moving on to particle filtering (Arulampalam et al., 2002), some others have worked on variants of the EnKF to handle the non-multiGaussianity (Zhou et al., 2011); the non-stationarity is not an issue, since the EnKF deals, by construction, with non-stationary states. 4.5 Conclusion The “missing scale” issue brought out by Tran (1996) is still, today, much overlooked. Data, particularly conductivity data, are collected at smaller support volumes and in larger quantities than years ago, yet, when constructing a numerical model based on these data, the discrepancy between the scale at which data are collected and the scale of the numerical model is most often disregarded. We have presented an approach to rigorously account for fine-scale conductivity measurements on coarse-scale conditional inverse modeling. The resulting model is composed of an ensemble of realizations of conductivity tensors at a scale (much) coarser than the scale at which conductivities were measured. The ensemble of final realizations is conditioned to both conductivity and piezometric head measurements. The latter conditioning is achieved by using the ensemble Kalman filter on realizations of conductivity tensors. To handle the tensor parameters, we propose to work with the invariants of the tensors, instead of their representations on a specific reference system, this approach allows the ensemble Kalman filter to perform a tensor updating which produces realizations that are conditioned to the transient piezometric head measurements.
Bibliography Aanonsen, S. L., Naevdal, G., Oliver, D. S., Reynolds, A. C., Valles, B., 2009. Ensemble kalman filter in reservoir engineering - a review. SPE Journal 14 (3), 393–412. Arulampalam, M. S., Maskell, S., Gordon, N., Clapp, T., 2002. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on signal processing 50 (2), 174–188. Bear, J., 1972. Dynamics of fluids in porous media. American Elsevier Pub. Co., New York. Behrens, R., MacLeod, M., Tran, T., Alimi, A., 1998. Incorporating seismic attribute maps in 3D reservoir models. SPE Reservoir Evaluation & Engineering 1 (2), 122–126. Burgers, G., van Leeuwen, P., Evensen, G., 1998. Analysis scheme in the ensemble Kalman filter. Monthly Weather Review 126, 1719–1724. Capilla, J., Llopis-Albert, C., 2009. Gradual conditioning of non-Gaussian transmissivity fields to flow and mass transport data: 1. Theory. Journal of Hydrology 371 (1-4), 66–74. Capilla, J. E., Rodrigo, J., Gómez-Hernández, J. J., 1999. Simulation of nongaussian transmissivity fields honoring piezometric data and integrating soft and secondary information. Math. Geology 31 (7), 907–927. Carrera, J., Alcolea, A., Medina, A., Hidalgo, J., Slooten, L., 2005. Inverse problem in hydrogeology. Hydrogeology Journal 13 (1), 206–222. Chen, Y., Zhang, D., 2006. Data assimilation for transient flow in geologic formations via ensemble Kalman filter. Advances in Water Resources 29 (8), 1107–1122. Deutsch, C. V., Journel, A. G., 1998. GSLIB, Geostatistical Software Library and User’s Guide, 2nd Edition. Oxford University Press, New York. 101
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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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38 BIBLIOGRAPHY Berkowitz, B., Sche
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40 BIBLIOGRAPHY Gómez-Hernández,
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42 BIBLIOGRAPHY Salamon, P., Fernà
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44 CHAPTER 3. TRANSPORT UPSCALING U
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150 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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