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

Recommendations

Info

58 CHAPTER 3. TRANSPORT UPSCALING USING MULTI- . . . the main reasons for these results are the use of a full hydraulic conductivity tensor to represent the interblock property and the use of a skin to approximate the “real” boundary conditions around the interblock, in contrast with the simple-Laplacian approach which seeks a diagonal hydraulic conductivity tensor with boundary conditions directly at the block sides. (3) The significance of the skin size is evident as it was already pointed out by Zhou et al. (2010) (17% relative bias using interblock Laplacian with a skin size of 3 m, down to 9% relative bias using interblock Laplacian with a skin of 10 m in the x and y directions and 5 m in the z direction, see Figures 3.3C and 3.3D). The high variance of hydraulic conductivity, as is the case in this example with σ 2 lnk =4.0, can result in local flows departing significantly from the aver- age flow direction (along the x axis in this case), in which case the use of a full tensor and the skin size is more important. (4) For a mild isotropic heterogeneous field, the Landau-Lifshitz-Matheron conjecture (a close expression that gives the upscaled conductivity as a p-norm of the fine scale conductivities within the block, in which p only depends on the dimensionality of the problem) performs well (Desbarats, 1992). However, when the global variance increases, the conjecture loses its accuracy and it is better to resort to the numerical flow experiments as is the case here, i.e., the Laplacian-with-skin method (38% relative bias using conjecture to 9% relative bias using interblock Laplacian-with-skin of 10 m along rows, 10 m along columns and 5 m along layers), see Figure 3.3D and 3.3E). In short, the best reproduction of the fine scale flows is given by the interblock-centered Laplacian-with-skin approach. This scheme is retained for the subsequent transport upscaling. 3.3.3 Transport upscaling results We examined two transport upscaling approaches using the same set of upscaled hydraulic conductivities obtained in section 3.3.2; in the first one, we only model advection using the velocities from the coarse scale flow simulation, and in the second one, we include the multi-rate term in the transport equation at the coarse scale and perform transport upscaling to determine enhanced macrodispersion coefficients, upscaled effective porosities and the parameters of the multi-rate transfer model. The multi-rate model estimates the mass transfer parameters as described in section 3.2.3. It should be noted that we do not make the comparison with an intermediate model including only enhanced macrodispersion coefficients, since it has already been shown (e.g., Zinn and Harvey, 2003; Fernàndez-Garcia and Gómez-Hernández, 2007; Frippiat and Holeyman, 2008; Willmann et al., 2008; Fernàndez-Garcia et al., 2009) that upscaled macrodispersion coefficients are not sufficient to reproduce the transport behavior for highly heterogeneous media.
CHAPTER 3. TRANSPORT UPSCALING USING MULTI- . . . 59 As mentioned previously, the synthetic studies of Fernàndez-Garcia et al. (2009) have shown that the double-rate mass transfer model is better than the singe-rate model in 2D mass transport upscaling. Herein, we only consider the double-rate mass transfer model to represent the mass transfer process, i.e., in each coarse block, the solute transport is assumed to happen in three zones: transport in the mobile zone is mainly by advection, while transport in the other two immobile zones is by diffusion-like processes. With regard to the double-rate mass transfer model, the mass transfer rate density function f(α) and the memory function g(t) are: f(α) = β1 β δ(α − α1) + β2 δ(α − α2) β β1 g(t) = α1 β e−α1 β2 t + α2 β e−α2t (3.20) Accordingly, the parameters being estimated, are collected as a vector in Θ = [α1, α2, β1, β2, Al]. Notice that the parameters are spatially variable since they are estimated for each upscaled block independently. We compare the effectiveness of the transport upscaling by analyzing the breakthrough curves at different control planes in one specific realization and by looking at the ensemble results. For the ensemble results we will look at the early (5 th percentile of the BTC), median (50 th percentile) and late (95 th percentile) travel times. Results using the advective-only model and the double-rate transport model are shown (see Figure 3.4 for the reproduction of BTCs in one realization and Figure 3.5 for the ensemble behavior of early, median and late travel times). From these results, we see that: (1) In contrast to the advective-only model, the double-rate mass transfer upscale model displays a higher accuracy to reproduce the fine scale breakthrough curves, in particular, the late travel times. Therefore, it is important to include the fictitious mass transfer process for solute transport predictions after upscaling. (2) In agreement with the study of Fernàndez-Garcia and Gómez-Hernández (2007), it is shown that the advective-only model even when using a sophisticated hydraulic conductivity upscaling (interblock Laplacian-with-skin here) can result in overestimating the early travel times and underestimating the late travel times in very heterogeneous media. (3) The small deviations in the reproduction of the BTCs by the mass transfer model may be due to fact that the upscaled mass transfer parameters are derived from a one-dimensional analytical solution of the double-rate transport model (see Equation (3.9)).
Page 1:
Upscaling and Inverse Modeling of G
Page 4 and 5:
c⃝ Copyright by Liangping Li 2011
Page 7 and 8:
Abstract The need to reduce the com
Page 9 and 10:
improved as more data is assimilate
Page 11 and 12:
Resumen La necesidad de reducir el
Page 13 and 14:
Por último, en el tercer bloque, e
Page 15 and 16:
Resum La necessitat de reduir el co
Page 17 and 18:
flux i el transport (conductivitat
Page 19:
Acknowledgements I want to thank my
Page 22 and 23:
xx CONTENTS 3 Transport Upscaling U
Page 24 and 25:
xxii CONTENTS
Page 26 and 27:
xxiv LIST OF FIGURES 2.7 Flow compa
Page 28 and 29:
xxvi LIST OF FIGURES 4.4 Reference
Page 30 and 31:
xxviii LIST OF FIGURES
Page 33 and 34:
1.1 Motivation and Objectives 1 Int
Page 35:
CHAPTER 1. INTRODUCTION 3 Chapter 2
Page 38 and 39:
6 CHAPTER 2. A COMPARATIVE STUDY OF
Page 40 and 41: 8 CHAPTER 2. A COMPARATIVE STUDY OF
Page 42 and 43: 10 CHAPTER 2. A COMPARATIVE STUDY O
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
Page 76 and 77: 44 CHAPTER 3. TRANSPORT UPSCALING U
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
Page 108 and 109: 76 CHAPTER 4. MODELING TRANSIENT GR
Page 132 and 133: 100 CHAPTER 4. MODELING TRANSIENT G
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 140 and 141:
108 CHAPTER 5. JOINTLY MAPPING HYDR
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
136 BIBLIOGRAPHY Chen, Y., Zhang, D
Page 170 and 171:
138 BIBLIOGRAPHY Journel, A., 1974.
Page 172 and 173:
140 BIBLIOGRAPHY Wen, X., Deutsch,
Page 174 and 175:
142 CHAPTER 6. GROUNDWATER FLOW INV
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
170 BIBLIOGRAPHY Deutsch, C. V., Jo
Page 204 and 205:
172 BIBLIOGRAPHY Lee, S., Carle, S.
Page 206 and 207:
174 BIBLIOGRAPHY Yeh, W., 1986. Rev
Page 208 and 209:
176 CHAPTER 7. CONCLUSIONS travels
Page 210 and 211:
178 CHAPTER 7. CONCLUSIONS predicti
Page 212 and 213:
180 APPENDIX A. FLOWXYZ3D - A THREE
Page 214:
182 APPENDIX A. FLOWXYZ3D - A THREE
show all

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

Create successful ePaper yourself

Delete template?

Save as template?