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

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16 CHAPTER 2. A COMPARATIVE STUDY OF THREE- . . . The main shortcoming of this approach is that the assumption of a diagonal tensor is not well-founded for a heterogeneous aquifer. In other words, the heterogeneity within the block may induce an overall flux that is not parallel to the macroscopic head gradient, a behavior that cannot be captured with a diagonal tensor. This method has been widely used to calculate block conductivities in petroleum engineering and hydrogeology (e.g., Warren and Price, 1961; Bouwer, 1969; Journel et al., 1986; Desbarats, 1987, 1988; Deutsch, 1989; Begg et al., 1989; Bachu and Cuthiell, 1990). More recently Sánchez-Vila et al. (1996) utilized this approach to study the scale effects in transmissivity; Jourde et al. (2002) used it to calculate block equivalent conductivities for fault zones; and Flodin et al. (2004) used this method to illustrate the impact of boundary conditions on upscaling. It has also been employed by Fernàndez-Garcia and Gómez-Hernández (2007) and Fernàndez-Garcia et al. (2009) to evaluate the impact of hydraulic conductivity upscaling on solute transport. Some reasons favoring this approach are that it is not empirical but phenomenological, i.e., it is based on the solution of the groundwater flow equation, and it yields a tensor representation of the block conductivity, which would be exact for the case of perfectly layered media, with the layers parallel to the coordinate axes. 2.3.3 Laplacian-with-skin To overcome the shortcomings of the simple-Laplacian approach, the Laplacianwith-skin approach was presented by Gómez-Hernández (1991). In this approach, the block conductivity is represented by a generic tensor (not necessarily diagonal) and the local flow problem is solved over an area that includes the block plus a skin surrounding it (see Figure 2.6). The skin is designed to reduce the impact of the arbitrary boundary conditions used in the solution of the local flow problems letting the conductivity values surrounding the block to take some control on the flow patterns within the block. For a 3D block, the overall algorithm is summarized as follows: (1) the block to upscale plus the skin is extracted from the domain; (2) flow is solved at the fine scale within the block-plus-skin region for a series of boundary conditions; (3) for each boundary condition the spatially-averaged specific discharge (q) and gradient (J) are calculated as, ⟨qi⟩ = 1 ∫ qi(x)dx (2.4) V (x) ⟨Ji⟩ = 1 V (x) ∫ V (x) V (x) ∂h(x) dx (2.5) ∂xi where i refers to the three components of the vectors (i.e., qx, qy and qz; Jx,Jy and Jz); and (4) the tensor components of K b are determined by solving
CHAPTER 2. A COMPARATIVE STUDY OF THREE- . . . 17 the following overdetermined system of linear equations by a standard least squares procedure (Press et al., 1988). ⎡ ⎢ ⎣ ⟨Jx⟩1 ⟨Jy⟩1 ⟨Jz⟩1 0 0 0 0 ⟨Jx⟩1 0 ⟨Jy⟩1 ⟨Jz⟩1 0 0 0 ⟨Jx⟩1 0 ⟨Jy⟩1 ⟨Jz⟩1 ⟨Jx⟩2 ⟨Jy⟩2 ⟨Jz⟩2 0 0 0 0 ⟨Jx⟩2 0 ⟨Jy⟩2 ⟨Jz⟩2 0 0 0 ⟨Jx⟩2 0 ⟨Jy⟩2 ⟨Jz⟩2 · · · · · · · · · · · · · · · · · · ⟨Jx⟩n ⟨Jy⟩n ⟨Jz⟩n 0 0 0 0 ⟨Jx⟩n 0 ⟨Jy⟩n ⟨Jz⟩n 0 0 0 ⟨Jx⟩n 0 ⟨Jy⟩n ⟨Jz⟩n ⎤ ⎥ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ · ⎢ ⎥ ⎢ ⎥ ⎣ ⎥ ⎦ K b xx K b xy K b xz K b yy K b yz K b zz ⎡ ⎤ ⟨qx⟩1 ⎢ ⟨qy⟩1 ⎥ ⎤ ⎢ ⎥ ⎢ ⟨qz⟩1 ⎥ ⎢ ⎥ ⎥ ⎢ ⟨qx⟩2 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⟨qy⟩2 ⎥ ⎥ = − ⎢ ⎥ ⎢ ⎥ ⎢ ⟨qz⟩2 ⎥ ⎦ ⎢ · · · ⎥ ⎢ ⟨qx⟩n ⎥ ⎣ ⟨qy⟩n ⎦ ⟨qz⟩n (2.6) where 1, . . . , n refers to the different boundary conditions; K b xx · · · K b zz are the components of the upscaled equivalent conductivity tensor K b . In principle, in 3D, two sets of boundary conditions are sufficient to determine K b . However, from a practical point of view, the number of boundary conditions should be greater than two (n > 2) to better approximate all possible flow scenarios. Every three rows in Equation (2.6) are the result of enforcing Darcy’s law on the average values in equations (2.4) and (2.5) for a given boundary condition: ⟨q⟩ = −K b ⟨J⟩ (2.7) The block conductivity tensor must be symmetric and positive definite. Symmetry is easily enforced by making K b xy = K b yx, K b xz = K b zx and K b yz = K b zy. Positive definiteness is checked a posteriori. In case the resulting tensor is non-positive definite, the calculation is repeated either with more boundary conditions or with a larger skin size (Wen et al., 2003; Li et al., 2011). We note that the critical point in this approach is the selection of the set of n alternative boundary conditions. In general, this set of boundary conditions is chosen so as to induce flow in several directions (for instance, the prescribed head boundary conditions in Figure 2.6 induce flow at 0 ◦ , 45 ◦ , 90 ◦ and 135 ◦ angles with respect to the x-direction). For the boundary conditions, we have chosen to prescribe linearly varying heads along the sides of the blocks, other authors (Durlofsky, 1991) have proposed the use of periodic boundary conditions. Flodin et al. (2004) showed that the resulting block conductivities do not depend significantly on whether the boundary conditions are linearly varying or periodic.
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Page 7 and 8: Abstract The need to reduce the com
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Page 11 and 12: Resumen La necesidad de reducir el
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66 CHAPTER 3. TRANSPORT UPSCALING U
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68 CHAPTER 3. TRANSPORT UPSCALING U
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70 BIBLIOGRAPHY Dagan, G., 1994. Up
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72 BIBLIOGRAPHY Lawrence, A. E., Sa
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74 BIBLIOGRAPHY Zhang, Y., 2004. Up
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76 CHAPTER 4. MODELING TRANSIENT GR
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100 CHAPTER 4. MODELING TRANSIENT G
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102 BIBLIOGRAPHY Durlofsky, L. J.,
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104 BIBLIOGRAPHY Tran, T., 1996. Th
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106 CHAPTER 5. JOINTLY MAPPING HYDR
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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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