About the Symmetry of the Upscaled Equivalent Transmissivity Tensor

The main point of this paper is to propose a non-local three-dimensional hydraulic conductivity full tensor upscaling algorithm and code. The algorithm is capable of transforming very refined cell conductivity models into coarse block conductivity models for quick and accurate solution of the groundwater flow equation. Flow rate and hydraulic head gradient are the variables used to relate the outputs from the fine scale model to the outputs from the coarse scale model. The flows and gradients computed at the coarse scale blocks should match the average values of the corresponding quantities observed at the fine scale. The algorithm is geared towards its use in conjunction with finite-difference codes for the solution of the groundwater flow equation capable of handling full tensor hydraulic conductivities. The finite-difference formulation of the groundwater flow equation requires specifying the hydraulic conductivity at the block interface in order to compute the discharge crossing the interface. Most finite-difference codes take as input conductivities at the blocks and then perform some type of averaging to come up with the interblock conductivity. Typically the harmonic mean of the conductivities of neighboring blocks is used. We propose computing directly the interblock conductivity during the upscaling process thus avoiding the need to average already averaged (upscaled) values of adjacent blocks. This approach also circumvents the problem associated with averaging conductivity tensors when their principal directions are not aligned with the Cartesian axes. The proposed algorithm is successfully demonstrated in four synthetic examples with spatially isotropic, anisotropic and bimodal conductivities fields at the fine scale. The computer code is provided with explanation of the input parameters and output results.

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“…The tensor must be symmetric and positive definite (Bear, 1972;Farmer, 2002;Giudici and Vassena, 2007), that is,…”

Section: Upscaling Methodologymentioning

confidence: 99%

Three-dimensional hydraulic conductivity upscaling in groundwater modeling

Zhou

Gómez‐Hernández

2010

Computers & Geosciences

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“…In [6] the same track of the proof given by [4] for the continuous case is followed; gradients are substituted with finite differences approximations and integrals are substituted with sums over a block.…”

Section: The Computation Of the Symmetric Equivalent Conductivity Tensormentioning

confidence: 99%

“…In [6] we extend this theorem to a physically based conservative discrete model using the integrated finite differences method (IFD), proposed in the ground water literature by [7] and applied by some authors to model regional flow [5,12]. In [6] the focus is on 2D stationary flow in a confined aquifer, so that the physical parameter to be considered is the aquifer transmissivity; the proof of the symmetry of the upscaled transmissivity tensor given by [4] is followed and it is rigorously demonstrated that K is symmetric even in the discrete case for physically based conservative numerical models.…”

Section: Introductionmentioning

confidence: 99%

“…In [6] the focus is on 2D stationary flow in a confined aquifer, so that the physical parameter to be considered is the aquifer transmissivity; the proof of the symmetry of the upscaled transmissivity tensor given by [4] is followed and it is rigorously demonstrated that K is symmetric even in the discrete case for physically based conservative numerical models.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of this paper is to test the result obtained in [6] on some numerical examples: the equivalent K tensor is computed for two faces of a volume of glaciofluvial sediments belonging to Pleistocene sequences of the Ticino basin (Northern Italy) described in [15]. We show that IFD can be useful to model media which are the union of many zones, each of which characterized by a single hydrofacies: in fact this method permits the discretisation of the discontinuities between different zones in a more accurate way than standard finite differences techniques which cannot handle non-rectangular grid cells.…”

Section: Introductionmentioning

confidence: 99%

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Application of integrated finite differences to compute symmetrical upscaled equivalent conductivity tensor

Vassena¹,

Giudici²

2007

WIT Transactions on Engineering Sciences, Vol 56

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The equivalent conductivity tensor is computed with a method based on the solution of the balance equation at the fine scale. In particular the balance equation is solved on a block assigning Dirichlet boundary conditions that vary linearly with the space coordinates and the equivalent conductivity tensor is the linear tensor relating average flux and hydraulic gradient. Previous works prove that this method yields a symmetric equivalent conductivity tensor both for continuous domains and for discrete models based on integrated finite differences. Here the equivalent conductivity tensor is computed for two lateral faces of a volume of glacio-fluvial sediments and the results are compared with those obtained with a standard finite differences method on square grids with different spacings.

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Stochastic analysis of three-dimensional hydraulic conductivity upscaling in a heterogeneous tropical soil

Godoy

Zuquette

Gómez‐Hernández

2018

Computers and Geotechnics

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Hydraulic conductivity (K) heterogeneity is seldom considered in geotechnical practice for the impossibility of sampling the entire area of interest and for the difficulty of accounting for scale effects. Stochastic three-dimensional K upscaling can tackle these two problems, and a workflow is described with an application in a tropical soil. The application shows that K heterogeneity can be incorporated in the daily practice of the geotechnical modeler while discussing the aspects to consider when performing the upscaling so that the upscaled models reproduce the average fluxes at the fine scale.

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About the Symmetry of the Upscaled Equivalent Transmissivity Tensor

Cited by 5 publications

References 18 publications

Three-dimensional hydraulic conductivity upscaling in groundwater modeling

Three-dimensional hydraulic conductivity upscaling in groundwater modeling

Application of integrated finite differences to compute symmetrical upscaled equivalent conductivity tensor

Stochastic analysis of three-dimensional hydraulic conductivity upscaling in a heterogeneous tropical soil

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