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

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108 CHAPTER 5. JOINTLY MAPPING HYDRAULIC . . . or land surface models in general only the model states are updated, whereas in petroleum engineering and hydrogeology both system parameters and state variables are commonly addressed (Naevdal et al., 2005). The EnKF has been successfully applied to assimilate dynamic piezometric head data to improve model predictions (e.g., Chen and Zhang, 2006; Hendricks Franssen and Kinzelbach, 2008; Sun et al., 2009; Li et al., 2011c; Zhou et al., 2011a). With regard to assimilating concentration data, Huang et al. (2008) conducted a synthetic experiment and calibrated hydraulic conductivity fields by assimilating piezometric head and concentration data. In their experiment, flow was at steady-state. Liu et al. (2008) estimated multiple parameters (i.e., hydraulic conductivity, dispersivities, mobile/immobile porosities) by assimilating piezometric head and concentration data in the steady-state flow model for the MADE site. It is worth to note that they used constant values as the prior estimates for the mentioned parameters, and perturbed the parameters, by assimilating observation data via EnKF, to yield heterogeneous fields. Our aim is to quantify the uncertainty of parameters and states starting with heterogeneous fields by conditioning on the direct measurements. Schöniger et al. (2011) assimilated normal-score transformed concentration data to calibrate hydraulic conductivities. They concluded that the improvement by the normal-score transformation (as compared with the classical EnKF, which uses untransformed data) is limited because after univariate normal transformation of the state variable, the concentration distribution is far from multi-Gaussian. In comparison with the effort devoted to characterize the spatial variability of hydraulic conductivity by conditioning state information, less attention has been paid to identifying the spatial variability of porosity, probably due to its relatively small spatial variability ranging from 0.1 to 0.55 in unconsolidated granular aquifers (Freeze and Cherry, 1979). Additionally, various authors (e.g., Hassan, 2001; Riva et al., 2008; Hu et al., 2009; Jiang et al., 2010) have demonstrated (both in synthetic examples and real aquifers) the significance of accounting for the heterogeneity of porosity on predictions of solute movement. We will demonstrate the capability of the EnKF to jointly map the hydraulic conductivity and porosity fields by assimilating dynamic piezometric head and multiple concentration data. Few studies have considered the conditioning with help of both multiple concentration data and dynamic piezometric head data to characterize unknown parameters. Also, to the best of our knowledge, this is the first work proposing the joint estimation of spatially distributed hydraulic conductivity and porosity fields in hydrogeology. The remaining of this paper is organized as follows. We first summarize in section 5.2 the mathematical framework of the EnKF and discuss the jointly mapping of hydraulic conductivity and porosity by assimilating multiple concentration data. In section 5.3, a synthetic example is used to demonstrate
CHAPTER 5. JOINTLY MAPPING HYDRAULIC . . . 109 the effectiveness of the EnKF. The paper ends with summary and conclusions in section 5.4. 5.2 Data Assimilation with the EnKF First, the flow and transport equations (i.e, the transfer functions) will be presented, and then the algorithm of EnKF is introduced with emphasis on the assimilation of concentration data. 5.2.1 Flow and Transport Equations The well known flow equation of an incompressible or slightly compressible fluid in saturated porous media can be expressed by combining Darcy’s Law and the continuity equation (Bear, 1972; Freeze and Cherry, 1979): ∇· ( K∇h ) = S ∂h ∂t + W (5.1) where K is the hydraulic conductivity [LT −1 ] (which, without loss of generality, will be considered as a scalar at the characterization scale), h is the piezometric head [L ]; W represents sources or sinks [L 3 T −1 ]; S is the specific storage coefficient [L −1 ]; t is the time [T ]; ∇· = (∂/∂x + ∂/∂y + ∂/∂z) is the divergence operator of a vector field, and ∇ = (∂/∂x, ∂/∂y, ∂/∂z) T is the gradient operator of a scalar field. Solute transport with linear equilibrium adsorption is governed by the following differential equation (Bear, 1972; Freeze and Cherry, 1979): ϕR ∂c ∂t = ∇(ϕD · ∇c) − ∇qc (5.2) where c is solute concentration of solute in the water phase [ML −3 ]; ϕ is the porosity [dimensionless]; D is the local hydrodynamic dispersion coefficient tensor [L 2 T −1 ], with eigenvalues associated with principal axes parallel and perpendicular to the direction of flow, defined as DI = αL|q| + Dm, DII = αT |q| + Dm, DIII = αT |q| + Dm (αL and αT are respectively the longitudinal and transverse pore-scale dispersivity; Dm is the molecular diffusion coefficient set to zero in this study, and q is the Darcy velocity given by q = −K∇h [LT −1 ] ); R is retardation factor expressed by R = 1 + ρbKd/ϕ (ρb is the bulk density of soil; Kd is the distribution coefficient). 5.2.2 Ensemble Kalman Filter Extensive descriptions of EnKF and its algorithm can be found in Burgers et al. (1998) and Evensen (2003). Here, we mainly focus on the use of EnKF
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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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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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Page 102 and 103: 70 BIBLIOGRAPHY Dagan, G., 1994. Up
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158 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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