Correction: Renard, P.; Ababou, R. Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses).
Geosciences 2022, 12, 269

Philippe Renard, Rachid Ababou

► To cite this version:

Philippe Renard, Rachid Ababou. Correction: Renard, P.; Ababou, R. Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses). Geosciences 2022, 12, 269. Geosciences, 2023, 13 (8), pp.242. 10.3390/geosciences13080242. hal-04338736

HAL Id: hal-04338736

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Submitted on 12 Dec 2023

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Correction

# Correction: Renard, P.; Ababou, R. Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses). Geosciences 2022, 12, 269

Philippe Renard 1,2,* and Rachid Ababou 3

1 Centre for Hydrogeology and Geothermics, University of Neuchâtel, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland

2 Department of Geosciences, Sem Sælands vei, University of Oslo, 1 Geologibygningen, 0371 Oslo, Norway

3 Institut de Mécanique des Fluides de Toulouse, CNRS—University of Toulouse, 2 Allée Camille Soula, 31400 Toulouse, France; ababou@imft.fr

* Correspondence: philippe.renard@unine.ch

The authors would like to correct the published article [1].

At the end of Section 4.2.3 in the original publication, Equation (88) and the paragraph preceding it were incorrect. This text and equation were supposed to introduce the Vectorial Surface Flux (VSF) equivalent permeability of a 2D fracture network, but the paragraph and the equation incorrectly referred to it the Net Surface Flux (NSF) criterion.

A correction has, therefore, been made at the end of Section 4.2.3. K from Vectorial Surface Flux (VSF), which is located in the first paragraph below the title Equivalent K with the VSF Criterion for a 2D Fracture Network (Summary), as follows:

In the case of a 2D fracture network, and for constant head gradient conditions (GRD), the equivalent permeability resulting from the VSF criterion is of the form:

$$K_{VSF} = \frac{1}{2L} \mathbf{U} \cdot (\mathbf{S}_{DF} - \mathbf{S}_{DI}\mathbf{L}_{II}^{-1}\mathbf{L}_{IF}) \cdot \mathbf{X} \quad (88)$$

The rest of the text below Equation (88) remains unchanged, and the authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.

## Reference

1. Renard, P.; Ababou, R. Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses). Geosciences 2022, 12, 269. [CrossRef]

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**Citation:** Renard, P.; Ababou, R. Correction: Renard, P.; Ababou, R. Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses). Geosciences 2022, 12, 269. Geosciences 2023, 13, 242. https://doi.org/10.3390/geosciences13080242

Received: 17 July 2023
Accepted: 18 July 2023
Published: 11 August 2023

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**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).