Hydrogeology Journal

, Volume 12, Issue 5, pp 497–510 | Cite as

Determining the equivalent permeability tensor for fractured rock masses using a stochastic REV approach: Method and application to the field data from Sellafield, UK

Paper

Abstract

A numerical procedure to determine the equivalent permeability tensor of a fractured rock is presented, using a stochastic REV (Representative Elementary Volume) concept that uses multiple realizations of stochastic DFN (Discrete Fracture Network) models. Ten square DFN models are generated using the Monte Carlo simulations of the fracture system based on the data obtained from a site characterization program at Sellafield, Cumbria, UK. Smaller models with varying sizes of from 0.25 m×0.25 m to 10 m×10 m are extracted from the generated DFN models and are used as two-dimensional geometrical models for calculation of equivalent permeability tensor. The DFN models are also rotated in 30º intervals to evaluate the tensor characteristics of calculated directional permeability. Results show that the variance of the calculated permeability values decreases significantly as the side lengths of the DFN models increase, which justifies the existence of a REV. The REV side length found in this analysis is about 5 m and 8 m with 20% and 10% acceptable variations, respectively. The calculated directional permeability values at the REV size have tensor characteristic that is confirmed by a close approximation of an ellipse in a polar plot of the reciprocal of square roots of the directional permeability.

Keywords

Permeability tensor Discrete Fracture Network (DFN) Representative Elementary Volume (REV) Equivalent continuum approach Fractured rocks 

Résumé

L’approche de milieu continu équivalent est une méthode efficace d’analyse de l’écoulement à grand échelle dans les roches fracturées. Il se pose des problèmes concernant l› échelle auquelle on peut déterminer les propriétés équivalentes ainsi que sur les possibilités d’appliquer les principes de la mécanique de milieu continu au l’écoulement dans les roches fracturées. Dans cet article on présente une technique pour déterminer le tenseur de perméabilité des roches fracturées en utilisant l ‹homogénéisation numérique et le procédé d’upscaling, basé sur le concept de Volume élémentaire Représentative (VER) et sur des réalisations multiples des simulations stochastiques sur des Systèmes de Fractures Discrets (SFD). Les paramètres clefs d’un système de fracture, comme la densité, les limites de la longueur de la trace et la constante de Fisher pour l’orientation des fractures, ont été obtenus d’après les caractéristiques du site de Sellanfield en Cumbria-Grande Bretagne. Par la méthode de Monte Carlo on a engendré dix modèles stochastiques de SFD sur des carrés de dimensions de 300 m×300 m. On a extrait de ces dix modèles un nombre de 120 modèles plus petites, dont les dimensions se rangent entre 0.25 m×0.25 m et 10 m×10 m, qui ont été utilisés comme des modèles géométriques pour l’analyse de l’écoulement avec un code des éléments distincts. Une série de dix SFD ont été tournés chaque 30» dans le sens des aiguilles d’une montre affin de calculer le tenseur de perméabilité. Il a résulté que la variance de la perméabilité calculée décroît à mesure que la longueur du modèle augmente, ce qui justifie l’existence d’un VER. On a trouvé que les échelles des cotés du VER sont de 5m pour une variation acceptable de 20%, caractérisée pat le coefficient de variation et respectivement de 8 m pour une variation acceptable de 10%. Le caractère tensoriel de la perméabilité directionnelle calculée est confirmé par une ellipse qui a résulté en représentant en coordonnées polaires les racines carrés des valeurs de la perméabilité directionnelle. Affin d’obtenir des distributions statistiques plus générales et représentatives. on a augmenté à 50 le nombre de réalisations multiples pour les échelles de 0.25 m, 0.5 m,1 m, 5 m et 10 m. On espère que les résultats de cet étude peuvent appuyer les futures analyses stochastiques à grande échelle des processus couplés thermo-hydro-mecaniques et de transport dans des massifs fracturés pour l’évaluation de la sécurité et de la performance du stockage souterraine de déchet nucléaires.

Resumen

La aproximación continua equivalente es un método efectivo para análisis de flujo en fluidos a gran escala en roca fracturada. Algunas dudas que han surgido son: En que escala pueden determinarse las propiedades equivalentes, y si es apropiado aplicar principios de mecánica continua a los problemas de flujo en rocas fracturadas. En el presente artículo se presenta un procedimiento para determinar el tensor de permeabilidad equivalente en rocas fracturadas, usando una homogenización numérica y una aproximación incremental basada en el concepto VER (Volumen Elemental Representativo) y de una realización múltiple de modelos estocásticos RFD (Red de Fracturas Discreta). Los parámetros críticos del sistema de fracturas para generaciones RFD, tales como densidad, límite de corte de la longitud de traza y constante de Fisher (para orientación de fracturas), se obtuvieron de la caracterización de un sitio en Sellafield, Cumbria, Reino Unido. Diez modelos estocásticos cuadrados RFD, con tamaño de 300 m×300 m, son generados usando simulaciones Montecarlo del sistema de fracturas. Además de los diez modelos, 120 modelos más pequeños, con tamaños variando desde 0.25 m×0.25 m hasta 10 m×10 m, son extractados y usados como modelos geométricos bidimensionales para el llamado elemento de código distinto (UDEC) para un análisis de flujo en fluidos. Una serie de diez modelos RFD son rotados en dirección de las manecillas del reloj con un intervalo de 30”, para evaluar las características del tensor de la permeabilidad direccional calculada. Los resultados muestran que la varianza de los valores de la permeabilidad calculada decrece significativamente, mientras la longitud lateral de los modelos cuadrados RFD se incrementa, lo cual justifica la existencia de un VER. La longitud lateral encontrada en este análisis del VER, es de alrededor de 5 m, con una variación aceptable del 20%, diseñada por un coeficiente de variación y una escala de 8 m, con una variación aceptable del 10% respectivamente. Los valores de permeabilidad direccional calculada correspondientes al tamaño del VER, tienen características de tensor, lo cual se confirma por una aproximación cercana a la gráfica de una elipse polar de raíces cuadradas recíprocas de los valores de permeabilidad direccional. En las escalas seleccionadas de 0.25 m, 0.5 m, 1 m, 5 m y 10 m, el número de realización múltiple fue extendido hasta cincuenta, con el fin de obtener una distribución estadística representativa y más general. Se espera que los resultados obtenidos en este estudio apoyen futuras aplicaciones a gran escala de análisis estocásticos continuos de procesos termo-hidro-mecánicos acoplados y procesos de transporte en masas fracturadas de roca, para evaluaciones de seguridad y comportamiento de depósitos subterráneos potenciales para desechos nucleares.

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Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Department of Land and Water Resources EngineeringRoyal Institute of Technology (KTH)Sweden
  2. 2.GeoforschungszentrumPotsdamGermany

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