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Hydrogeology Journal

, Volume 26, Issue 6, pp 1903–1916 | Cite as

Upscaling permeability for three-dimensional fractured porous rocks with the multiple boundary method

  • Tao ChenEmail author
  • Christoph ClauserEmail author
  • Gabriele Marquart
  • Karen Willbrand
  • Thomas Hiller
Paper

Abstract

Upscaling permeability of grid blocks is crucial for groundwater models. A novel upscaling method for three-dimensional fractured porous rocks is presented. The objective of the study was to compare this method with the commonly used Oda upscaling method and the volume averaging method. First, the multiple boundary method and its computational framework were defined for three-dimensional stochastic fracture networks. Then, the different upscaling methods were compared for a set of rotated fractures, for tortuous fractures, and for two discrete fracture networks. The results computed by the multiple boundary method are comparable with those of the other two methods and fit best the analytical solution for a set of rotated fractures. The errors in flow rate of the equivalent fracture model decrease when using the multiple boundary method. Furthermore, the errors of the equivalent fracture models increase from well-connected fracture networks to poorly connected ones. Finally, the diagonal components of the equivalent permeability tensors tend to follow a normal or log-normal distribution for the well-connected fracture network model with infinite fracture size. By contrast, they exhibit a power-law distribution for the poorly connected fracture network with multiple scale fractures. The study demonstrates the accuracy and the flexibility of the multiple boundary upscaling concept. This makes it attractive for being incorporated into any existing flow-based upscaling procedures, which helps in reducing the uncertainty of groundwater models.

Keywords

Fractured rocks Hydraulic properties Numerical modeling Groundwater flow Geostatistics 

Augmentation scalaire de la conductivité hydraulique pour des roches poreuses fracturées en trois dimensions à l’aide de la méthode à limites multiples

Résumé

L’augmentation scalaire de la conductivité hydraulique de blocs maillés est. cruciale pour les modèles numériques d’écoulement souterrain. Une nouvelle méthode d’augmentation scalaire est. proposée pour des roches poreuses fracturées en trois dimensions. L’objectif de l’étude était de comparer cette méthode avec la méthode d’augmentation scalaire Oda communément utilisée et la méthode de la moyenne des volumes. Premièrement, la méthode des limites multiples et son cadre de calcul ont été définis pour des réseaux de fractures stochastiques tridimensionnels. Ensuite, les différentes méthodes d’augmentation scalaire ont été comparées pour un jeu de fractures pivotées, pour des fractures tortueuses et pour deux réseaux de fractures discrètes. Les résultats calculés par la méthode des limites multiples sont comparables avec ceux des deux autres méthodes et coïncident bien avec la solution analytique pour un jeu de fractures pivotées. Les erreurs sur le débit du modèle de fractures équivalentes sont plus faibles avec la méthode des limites multiples. En outre, les erreurs des modèles de fractures équivalentes augmentent lorsque l’on passe de réseaux de fractures bien connectées à des réseaux mal connectés. Enfin, les composantes diagonales des tenseurs de la conductivité hydraulique équivalente présentent une distribution de type normal ou log-normal pour le modèle de réseaux de fractures bien connectées avec des fractures de taille infinie. En revanche, ils présentent une distribution en loi puissance pour les réseaux de fractures mal connectés avec des fractures d’échelles multiples. L’étude démontre la précision et la flexibilité du concept d’augmentation scalaire par multiples limites. Cela le rend attractif pour l’incorporer dans toutes les procédures existantes d’augmentation scalaire basées sur les écoulements, afin de réduire l’incertitude des modèles numériques d’écoulement souterrain.

Escalado de la permeabilidad para rocas porosas fracturadas en tres dimensiones con el método de límites múltiples

Resumen

El proceso de escalado de las cuadrículas es crucial para los modelos de agua subterránea. Se presenta un nuevo método de escalado para rocas fracturadas porosas en tres dimensiones. El objetivo del estudio fue comparar este método con el método comúnmente utilizado de escalado de Oda y con el método promedio del volumen. En primer lugar, se definió el método de límites múltiples y su marco computacional para las redes de fracturas estocásticas tridimensionales. Luego, se compararon los diferentes métodos de escalado para un conjunto de fracturas rotadas, para fracturas tortuosas y para dos redes discretas de fracturas. Los resultados calculados por el método de límites múltiples son comparables con los de los otros dos métodos y se ajustan mejor a la solución analítica para un conjunto de fracturas rotadas. Los errores en el índice de flujo del modelo de fractura equivalente disminuyen cuando se usa el método de límite múltiple. Además, los errores de los modelos de fractura equivalentes aumentan desde redes de fracturas bien conectadas a redes de fracturas pobremente conectadas. Finalmente, los componentes diagonales de los tensores de permeabilidad equivalentes tienden a seguir una distribución normal o log-normal para el modelo de red de fractura bien conectado con un tamaño de fractura infinito. Por el contrario, exhiben una distribución de la ley de potencia para la red de fracturas mal conectadas con fracturas de múltiples escalas. El estudio demuestra la precisión y la flexibilidad del concepto de escalado por límites múltiples. Esto lo hace atractivo para ser incorporado en cualquier procedimiento de escalado basado en flujo existente, lo que ayuda a reducir la incertidumbre de los modelos de agua subterránea.

基于多边界法的三维裂隙介质渗透率尺度提升

摘要

网格块的渗透率尺度提升对地下水渗流模型至关重要。本文提出一种新的三维裂隙介质尺度提升方法,即多边界尺度提升方法,并与常用的Oda方法和体积平均法进行了比较 。首先,推导了三维裂隙介质的多尺度提升方法并介绍在随机裂隙网络中的计算步骤。然后,考虑不同的裂隙介质,将上述尺度提升方法的计算结果进行了对比,包括一系列旋转裂隙、曲折裂隙和两个离散裂隙网络。结果表明,使用多边界方法得到的结果与其他两种方法总体相近。对于旋转裂隙,使用多边界法得到的等效渗透率与解析解拟合最好;且在多边界方法基础上生成的等效裂隙模型具有较小流量误差。此外,当裂隙网络连通性由好变差,等效裂隙模型的计算误差会增加。最后,在连通性好的裂隙网络,等效渗透率张量的主对角元素呈正态或对数正态分布;在连通性差的裂隙网络,呈幂律分布。本研究表明了多边界尺度提升概念的精确性和灵活性,这将使其能够嵌入到任意基于流动计算的裂隙介质尺度提升步骤中,以减小地下水渗流模型的不确定性。

Aumento de escala de permeabilidade para rochas fraturadas tridimensionais com o método de fronteira múltipla

Resumo

Aumento de escala de permeabilidade de blocos em malha é crucial para os modelos hidrogeológicos. Um novo método de aumento de escala para rochas fraturadas tridimensionais é apresentado. O objetivo desse estudo foi comparar esse método com o método usual Oda de aumento de escala e o método da média do volume. Primeiro, o método de fronteira múltipla e sua estrutura computacional foram definidos para redes de fraturas tridimensionais estocásticas. Então, os diferentes métodos de mudança de escala foram comparados para um conjunto de fraturas rotacionadas, para fraturas tortuosas e para duas redes de fraturas discretas. Os resultados calculados pelo método de fronteira múltipla são comparáveis ​​aos dos outros dois métodos e melhoram a solução analítica para um conjunto de fraturas rotacionadas. Os erros na taxa de fluxo do modelo de fratura equivalente diminuem ao usar o método de fronteira múltipla. Além disso, os erros dos modelos de fratura equivalente aumentam de redes de fraturas bem conectadas para redes mal conectadas. Finalmente, as componentes diagonais dos tensores de permeabilidade equivalentes tendem a seguir uma distribuição normal ou log-normal para o modelo de rede de fraturas bem conectadas com tamanho de fratura infinita. Em contrapartida, eles exibem uma distribuição pela lei da potência para a rede de fraturas mal conectadas com fraturas de escala múltipla. O estudo demonstra a precisão e a flexibilidade do conceito de aumento de escala em fronteira múltipla. Isso torna atrativo para ser incorporado em qualquer procedimento existente de aumento de escala baseado em fluxo, o que ajuda a reduzir a incerteza dos modelos hidrogeológicos.

Notes

Acknowledgements

The fracture network data are attached as electronic supplementary material (ESM) and the upscaling codes are available upon request from the first author. The authors appreciate the people at Golder Associates for supplying the academic license for FracMan software and their kind advice. The authors are very grateful to the two anonymous reviewers, the associate editor Sylke Hilberg, and the technical editorial advisor Sue Duncan, for their valuable comments and corrections, as well as the editor Martin Appold for his critical suggestions, which improved the quality of the paper.

Funding Information

This study was supported by the China Scholarship Council-RWTH Aachen University Joint PhD Program (201304190043).

Supplementary material

10040_2018_1744_MOESM1_ESM.pdf (50 kb)
ESM 1 (PDF 50 kb)

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Applied Geophysics and Geothermal Energy, E.ON Energy Research CenterRWTH Aachen UniversityAachenGermany
  2. 2.College of Earth Science and EngineeringShandong University of Science and TechnologyQingdaoChina

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