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Semi-empirical equations for the systematic decrease in permeability with depth in porous and fractured media

Equations semi-empiriques de la diminution systématique de la perméabilité dans des milieux poreux et fracturés

Ecuaciones semi-empíricas para la disminución sistemática en la permeabilidad con la profundidad en medios porosos y fracturados

孔隙和裂隙介质渗透性随深度衰减趋势的半经验公式

Equações semi-empíricas para a diminuição sistemática da permeabilidade com a profundidade em meios porosos e fracturados

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Abstract

Permeability loss with depth is a general trend in geological media and plays an essential role in subsurface fluid flow and solute transport. In the near surface zone where groundwater movement is active, the decrease in permeability with depth is dominated by the mechanical compaction of deformable media caused by the increase in lithostatic stress with depth. Instead of using empirical equations from statistical analysis, by considering the well-defined relationships among permeability, porosity, fracture aperture and effective stress under lithostatic conditions, new semi-empirical equations for the systematic depth-dependent permeability are derived, as well as the equations for the depth-dependent porosity in a porous medium and the depth-dependent fracture aperture in a fractured medium. The existing empirical equations can be included in the new equations as special cases under some simplification. These new semi-empirical equations perform better than previous equations to interpret the depth-dependent permeability of the Pierre Shale (with a maximum depth of approximately 4,500 m) and the granite at Stripa, Sweden (with a maximum depth of about 2,500 m).

Résumé

La perte de perméabilité avec la profondeur est une tendance générale dans le milieu géologique et joue un rôle essentiel sur le flux de subsurface et le transport de soluté. Dans la zone d’écoulement de nappe proche de la surface, la diminution de perméabilité avec la profondeur est contrôlée par la compaction mécanique d’un milieu déformable causée par l’accroissement de la pression de couverture avec la profondeur. Au lieu d’utiliser des équations empiriques tirées de l’analyse statistique, en considérant les relations bien définies entre perméabilité, porosité, ouverture des fractures et pression, on a établi de nouvelles équations semi-empiriques corrélant, avec la profondeur, perméabilité, porosité d’un milieu poreux, ouverture de la fracturation d’un milieu fracturé. Les équations empiriques existantes peuvent être comprises dans les nouvelles équations en tant que cas particuliers à condition de quelque peu simplifier. Ces nouvelles équations semi-empiriques rendent, mieux que les équations précédentes, compte de la perméabilité en fonction de la profondeur à Pierre Shale (à une profondeur maximale d’environ 4,500 m) et dans le granite de Stripa, Suède (à une profondeur maximale d’environ 2,500 m).

Resumen

La pérdida de la permeabilidad con la profundidad es una tendencia general en los medios geológicos y juega un rol esencial en el flujo de fluidos subsuperficiales y transporte de solutos. En la zona cercana a la superficie donde el movimiento del agua subterránea es activo, la disminución de la permeabilidad con la profundidad está dominada por la compactación mecánica del medio deformable provocado por el incremento de la presión litostática con la profundidad. En lugar de usar ecuaciones empíricas del análisis estadístico se desarrollaron nuevas ecuaciones semiempíricas para dependencia sistemática de la profundidad considerando las relaciones definidas de los pozos entre permeabilidad, porosidad, apertura de las fracturas y presión efectiva bajo condiciones litostáticas, así como también las ecuaciones para la dependencia con la profundidad de la porosidad de un medio poroso y dependencia de la apertura de la fractura con la profundidad de un medio fracturado. Las ecuaciones empíricas existentes pueden ser incluidas en las nuevas ecuaciones como casos especiales bajo alguna simplificación. Estas nuevas ecuaciones semiempíricas son más adecuadas que las ecuaciones previas en la interpretación de la dependencia de la permeabilidad con la profundidad de Pierre Shale (con una profundidad máxima de aproximadamente de 4,500 m) y el granito en Stripa, Suecia (con una profundidad máxima de alrededor de 2,500 m).

摘 要

渗透性随埋深减小是地质介质的普遍规律, 对地下流体运动和溶质运移具有重要影响. 在地下水很活跃表层区, 渗透性随深度衰减的趋势主要取决于地层压力增加引起的介质压缩变形. 与基于统计分析得到经验公式不同, 本文考虑渗透率, 孔隙度, 裂隙张开度和有效应力在自重应力条件下的明确相互关系, 推导了半经验半理论式的渗透率-埋深趋势方程, 并且得到了孔隙介质的孔隙度和裂隙介质的隙宽随埋深变化的公式. 现有的经验公式可以作为本文新公式的某些简化特例. 在解释Pierre页岩(最大深度达4,500 m)和瑞典Stripa场地花岗岩 (最大深度达2,500 m) 渗透率-埋深特征方面, 新的半经验公式比过去的公式更有效.

Resumo

A perda de permeabilidade com a profundidade é uma tendência geral nos meios geológicos e desempenha um papel essencial no escoamento de fluidos e no transporte de soluto na subsuperfície. Na zona próxima da superfície, onde o movimento das águas subterrâneas está activo, a diminuição da permeabilidade com a profundidade é dominada pela compactação mecânica do meio deformável causada pelo aumento da pressão litostática com a profundidade. Em vez de se utilizarem equações empíricas provenientes da análise estatística, considerando as bem definidas relações entre permeabilidade, porosidade, abertura das fracturas e pressão efectiva sob condições litostáticas, derivam-se novas equações semi-empíricas para a sistemática permeabilidade dependente da profundidade, assim como equações para a porosidade dependente da profundidade num meio poroso e para as aberturas das fracturas dependentes da profundidade num meio fracturado. Fazendo algumas simplificações, as equações empíricas existentes podem ser incluídas nas novas equações, como casos especiais. Estas novas equações semi-empíricas resultam melhor do que as anteriores equações na interpretação da permeabilidade dependente da profundidade dos Xistos de Pierre (com uma profundidade máxima de aproximadamente 4,500 m) e do granito de Stripa, Suécia (com uma profundidade máxima de cerca de 2,500 m).

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Acknowledgements

This research is financially supported by the Yalongjiang River Joint Fund established by the National Natural Science Foundation of China and the Ertan Hydropower Development Company (Grant No. 50639090). The research is also partially supported by the Program of Outstanding Overseas Chinese Young Scholars established by the National Natural Science Foundation of China (Grant No. 40528003). We appreciate valuable comments from Dr. D. Hart and two anonymous reviewers as well as constructive suggestions from the editors, Dr. J. Bahr and Dr. J. Jiao.

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  1. School of Water Resources & Environment, China University of Geosciences, Xueyuan Load 29, Haidian District, Beijing, 100083, China

    Xiao-Wei Jiang, Xu-Sheng Wang & Li Wan

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  1. Xiao-Wei Jiang
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Correspondence to Xu-Sheng Wang.

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Jiang, XW., Wang, XS. & Wan, L. Semi-empirical equations for the systematic decrease in permeability with depth in porous and fractured media. Hydrogeol J 18, 839–850 (2010). https://doi.org/10.1007/s10040-010-0575-3

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