Abstract
Two-dimensional (2D) and 3D numerical simulations of the dispersive Henry problem show that heterogeneity affects seawater intrusion differently in 2D and 3D. When the variance of a multi-Gaussian isotropic hydraulic conductivity field increases, the penetration of the saltwater wedge decreases in 2D while it increases in 3D. This is due to the combined influence of advective and dispersive processes which are affected differently by heterogeneity and problem dimensionality. First, the equivalent hydraulic conductivity controls the mean head gradient and therefore the position of the wedge. For an isotropic medium, increasing the variance increases the equivalent conductivity in 3D but not in 2D. Second, the macrodispersion controls the rotation of the saltwater wedge by affecting the magnitude of the density contrasts along the saltwater wedge. An increased dispersion due to heterogeneity leads to a decreasing density contrast and therefore a smaller penetration of the wedge. The relative magnitude of these two opposite effects depends on the degree of heterogeneity, anisotropy of the medium, and dimension. Investigating these effects in 3D is very heavy numerically; as an alternative, one can simulate 2D heterogeneous media that approximate the behaviour of the 3D ones, provided that their statistical distribution is rescaled.
Résumé
Des simulations bi et tri dimensionnelles du problème de Henry relatif à la dispersion montrent que l’hétérogénéité affecte l’intrusion marine différemment en 2D et 3D. Quand la variance d’un champ de conductivité hydraulique isotrope multi-gaussien s’accoît, la pénétration du biseau salé diminue en 2D, tandis qu’elle augmente en 3D. Ceci est dû à l’influence combinée de processus advectifs et dispersifs, affectés de façon différente par l’hétérogénéité et par la dimensionnalité. Tout d’abord, la conductivité hydraulique équivalente contrôle le gradient de charge moyen et par suite la position du biseau. Dans un milieu isotrope, l’augmentation de la variance augmente la conductivité équivalente en 3D, mais pas en 2D. Deuxièmement, la macrodispersion contrôle la déviation du biseau salé en affectant l’amplitude des contrastes de densité le long du biseau. Une augmentation de la dispersion liée à l’hétérogénéité conduit à une diminution du contraste de densité et par suite à une moindre avancée du biseau. La résultante de ces deux effets antagonistes dépend des degrés d’hétérogénéité, d’anisotropie et des dimensions du milieu. La simulation numérique de ces effets en 3D est très complexe; on peut lui substituer une simulation en 2D dans un milieu hétérogène reproduisant approximativement le comportement d’un milieu 3D, à condition que leurs distributions statistiques soient équivalentes.
Resumen
Simulaciones numéricas bidimensionales (2D) y tridimensionales (3D) del problema dispersivo de Henry muestran que la heterogeneidad afecta la intrusión de agua de mar de manera diferente en 2D y 3D. Cuando la varianza de un campo isotrópico multi- Gausiano de conductividad hidráulica se incrementa, la penetración de la cuña de agua salada decrece en 2D mientras se incrementa 3D. Esto es debido a la influencia combinada de procesos dispersivos y advectivos que se ven afectados de diferente forma por la heterogeneidad y el problema de la dimensionalidad. Primero, la conductividad hidráulica equivalente controla el gradiente hídrico medio y por lo tanto la posición de la cuña. Para un medio isotrópico, al incrementar la varianza se incrementa la conductividad equivalente en 3D pero no en 2D. Segundo, la macrodispersión controla la rotación de la cuña de agua salada afectando la magnitud de los contrastes de densidad a lo largo de la cuña de agua salada. Un incremento en la dispersión debido a la heterogeneidad conduce a un decrecimiento del contraste de la densidad y por lo tanto una menor penetración de la cuña. La magnitud relativa de estos dos efectos opuestos depende del grado de heterogeneidad, la anisotropía del medio, y la dimensión. Investigar estos efectos en 3D es numéricamente muy pesado, como una alternativa, uno puede simular un medio heterogéneo en 2D que se aproxima al comportamiento del 3D, con la condición que su distribución estadísticas sea reescalada.
摘要
弥散亨利问题的二维 (2D) 和三维 (3D) 数值模拟表明非均质性对海水入侵的影响在2D和3D条件下有所不同. 当多高斯各向同性渗透系数场的变异增加时, 盐水楔的楔入在2D条件下减弱, 在3D条件下则增强. 这是由于对流和弥散过程的综合影响, 而这两个过程又受非均质性和问题维数不同的影响. 首先, 等效渗透系数控制平均水力坡度, 因此控制着盐水楔的位置. 对于各向同性介质, 如增加变异, 则3D条件下的等效渗透系数增加, 而2D条件下则不变. 其次, 宏观弥散通过影响沿盐水楔密度差异的数量级控制着盐水楔的转动. 由于非均质性引起的弥散增加使密度差异降低, 盐水楔的楔入减弱. 这两个相反影响的相对幅度取决于非均质程度, 介质的各向异性和维数. 应用数值模拟在3D条件下查明这些影响型工作量很大, 作为替代方案, 在其统计分布重新标定的情况下, 可以通过模拟2D非均质介质来近似三维.
Resumo
Simulações numéricas bidimensionais (2D) e 3D do problema dispersivo de Henry mostram que a heterogeneidade afecta a intrusão marinha de forma diferente em 2D e 3D. Quando a variância multi–Gaussiana da condutividade hidráulica isotrópica aumenta, a penetração da cunha de água salina diminui em 2D, enquanto aumenta em 3D. Isto é devido à influência combinada dos processos advectivos e dispersivos, os quais são afectados de maneira diferente pela heterogeneidade e dimensão do problema. Primeiro, a condutividade hidráulica equivalente controla o gradiente médio e, por conseguinte, a posição da cunha. Para um meio isotrópico, incrementando a variância incrementa–se a condutividade equivalente em 3D, mas não em 2D. Em segundo lugar, a macro–dispersão controla a rotação da cunha de água salgada, afectando a magnitude dos contrastes de densidade ao longo da cunha salina. Uma maior dispersão devido à heterogeneidade leva a um decréscimo dos contrastes de densidade e, portanto, a uma menor penetração da cunha. A importância relativa desses dois efeitos opostos depende do grau de heterogeneidade, da anisotropia do meio, e da dimensão. Investigar esses efeitos em 3D é, do ponto de vista numérico, muito pesado, logo, como alternativa, pode simular-se um meio heterogéneo 2D que aproxime o comportamento obtido em 3D, desde que a sua distribuição estatística seja redimensionada.
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Acknowledgements
This work has been funded by the Swiss National Science Foundation under Grants: 207020-110017 and PP002-106557. The authors thank G. de Marsily, R. Ababou, J. Carrera and P. Perrochet for providing valuable suggestions on earlier version of the manuscript as well as Christian Langevin and the anonymous reviewers for their critical but constructive review comments. The authors are also grateful to R. Bouhlila and F. Cornaton for useful discussions about the study.
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Kerrou, J., Renard, P. A numerical analysis of dimensionality and heterogeneity effects on advective dispersive seawater intrusion processes. Hydrogeol J 18, 55–72 (2010). https://doi.org/10.1007/s10040-009-0533-0
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DOI: https://doi.org/10.1007/s10040-009-0533-0