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Darcy’s model with optimized permeability distribution for the simulation of stokes flow and contaminant transport in karst aquifers

Modèle de Darcy avec une distribution optimisée de la conductivité hydraulique pour la simulation du débit de stokes et du transport de contaminants dans les aquifères karstiques

Modelo de Darcy con distribución de permeabilidad optimizada para la simulación del flujo de stokes y el transporte de contaminantes en acuíferos kársticos

岩溶含水层中斯托克斯流与污染物运移模拟的渗透率分布优化的达西模型

Modelo de Darcy com distribuição de permeabilidade otimizada para simulação de fluxo de stokes e transporte de contaminantes em aquíferos cársticos

Abstract

Simulation of fluid flow in karst aquifers is challenging because of the presence of porous and free-flow regions within a single aquifer system. One simple but less accurate approach to model such a system is to use Darcy’s model. This model has much lower computational overhead than many other more rigorous approaches. The results obtained from the Darcy model are accurate for the porous regions of the aquifer, but it produces inaccurate results inside the caves. An approach is proposed called the Darcy model with optimized permeability distribution (DMOPD) for modeling accurate pressure and velocity distributions within the aquifer while retaining almost the same computational cost as the Darcy model. This method comprises three main steps. First, at the first time-step, the pressure and velocity distribution of the entire aquifer is solved using the Brinkman model. Then each cave is divided into an odd number of zones, with the middle zone assigned a maximum permeability value. Subsequently, the permeability ratios of the other surrounding zones are estimated using a global optimization technique. The permeability ratio is the ratio of permeability in that zone to the maximum permeability within the cave. Finally, the Darcy model is run for the remaining time steps using the optimized values of permeability obtained in the second step. Example applications presented show that this method is able to approximate the Brinkman model very well and much faster when compared to the Brinkman model.

Résumé

La simulation des écoulements dans les aquifères karstiques constitue un défi à cause de la présence de régions poreuses et d’écoulements libres dans un même et unique système aquifère. Une approche simple mais moins précise pour modéliser un tel. système est d’utiliser le modèle de Darcy. Ce modèle nécessite un temps de calcul beaucoup moins important que beaucoup d’autres approches plus rigoureuses. Les résultats obtenus à l’aide d’un modèle de Darcy sont de bonne précision dans les régions poreuses de l’aquifère, mais il produit des résultats imprécis à l’intérieur des cavités. Une approche est proposée dénommée le modèle de Darcy avec une distribution optimizée de la conductivité hydraulique (DMOPD) pour simuler de manière plus précise les distribution de la charge hydraulique et les vitesses d’écoulement au sein de l’aquifère tout en conservant presque le même temps de calcul que le modèle de Darcy. Cette méthode comprend trois principales étapes. Tout d’abord, à la première étape, la distribution de la charge hydraulique et de la vitesse d’écoulement de l’aquifère entier est résolue en utilisant le modèle de Brinkman. Ensuite, chaque cavité est divisée en un nombre impair de zones, avec la zone médiane assignée une valeur de conductivité hydraulique maximale. Par la suite, les rapports de conductivité hydraulique des autres zones environnantes sont estimés à l’aide d’une technique d’optimisation globale. Le rapport de conductivité hydraulique est le rapport de la conductivité hydraulique dans cette zone à la conductivité hydraulique maximale à l’intérieur de la cavité. Finalement, le modèle de Darcy est exécuté pour les étapes de temps restantes en utilisant les valeurs optimisées de conductivité hydraulique obtenues dans la deuxième étape. Des exemples d’application présentés montrent que cette méthode est capable de bien approximer le modèle Brinkman et de manière beaucoup plus rapide en comparaison au modèle de Brinkman.

Resumen

La simulación del flujo de fluidos en los acuíferos kársticos es un desafío debido a la presencia de regiones porosas y de flujo libre dentro de un solo sistema acuífero. Un enfoque simple pero menos preciso para modelar tal sistema es utilizar el modelo de Darcy. Este modelo tiene una sobrecarga computacional mucho menor que muchos otros enfoques más rigurosos. Los resultados obtenidos del modelo Darcy son exactos para las regiones porosas del acuífero, pero produce resultados inexactos dentro de las cavernas. Se propone un enfoque llamado modelo Darcy con distribución de permeabilidad optimizada (DMOPD) para modelar distribuciones precisas de presión y velocidad dentro del acuífero mientras se mantiene casi el mismo costo computacional que el modelo Darcy. Este método comprende tres pasos principales. Primero, en el primer paso, la distribución de presión y velocidad de todo el acuífero se resuelve usando el modelo Brinkman. Luego, cada caverna se divide en un número impar de zonas, y a la zona media se le asigna un valor máximo de permeabilidad. Posteriormente, se estiman los índices de permeabilidad de las demás zonas circundantes mediante una técnica de optimización global. El índice de permeabilidad es la relación entre la permeabilidad en esa zona y la permeabilidad máxima dentro de la caverna. Finalmente, el modelo Darcy se ejecuta para los pasos de tiempo restantes utilizando los valores optimizados de permeabilidad obtenidos en el segundo paso. Las aplicaciones de ejemplo presentadas muestran que este método es capaz de aproximarse muy bien al modelo de Brinkman y mucho más rápido cuando se compara con dicho modelo.

摘要

由于单一含水层系统中存在多孔的自由流区域,岩溶含水层中流体流动的模拟具有挑战性。对这种系统进行建模的一种简单但不太准确的方法是使用Darcy模型。与许多其他更严格的方法相比,该模型的计算量要低得多。从Darcy模型获得的结果对于含水层的多孔介质区是准确的,但在洞穴内产生的结果不准确。因此提出了一种方法,称为优化渗透率分布的达西模型(DMOPD),用于准确模拟含水层内压力和流速分布,同时保留与达西模型几乎相同的计算成本。该方法包括三个主要步骤。首先,在第一个时间步长,使用Brinkman模型求解整个含水层的压力和流速分布。然后,将每个洞穴划分为奇数个区域,并为中间区域分配最大渗透率值。随后,使用全局优化技术估算其他周围区域的渗透率比。渗透率比是该区域渗透率与洞穴内最大渗透率之比。最后,使用在第二步中获得的渗透率优化值,在剩余的时间步长中运行Darcy模型。本文阐述的应用案例表明,该方法能够很好地逼近Brinkman模型,而且计算快于Brinkman模型。

Resumo

A simulação do fluxo de fluidos em aquíferos cársticos é desafiadora devido à presença de regiões porosas e de fluxo livre dentro de um único sistema aquífero. Uma abordagem simples, porém menos precisa, para modelar esse sistema utiliza o modelo de Darcy. Esse modelo gera menos sobrecarga computacional do que muitas outras abordagens mais rigorosas. Os resultados obtidos no modelo de Darcy são precisos para as regiões porosas do aquífero, mas produz resultados imprecisos no interior das cavernas. É proposta uma abordagem denominada modelo de Darcy com distribuição de permeabilidade otimizada (MDDPO) para modelar distribuições precisas de pressão e velocidade no aquífero, mantendo-se quase o mesmo custo computacional do modelo de Darcy. Este método compreende três etapas principais. Na primeira etapa a distribuição de pressão e velocidade de todo o aquífero é resolvida usando o modelo de Brinkman. Em seguida, cada caverna é dividida em um número ímpar de zonas, atribuindo um valor máximo de permeabilidade para a zona intermediária. Posteriormente, as taxas de permeabilidade das outras zonas circundantes são estimadas usando uma técnica de otimização global. A taxa de permeabilidade é a variação da permeabilidade de uma zona até a permeabilidade máxima dentro da caverna. Finalmente, o modelo de Darcy é executado para as etapas de tempo restantes usando os valores otimizados de permeabilidade obtidos na segunda etapa. Exemplos de aplicações apresentadas mostram que esse método é capaz de aproximar muito bem o modelo de Brinkman e muito mais rapidamente quando comparado ao modelo de Brinkman original.

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Acknowledgements

The authors express their sincere gratitude towards the editor and the anonymous reviewers for their comments and suggestions.

Funding

The authors would like to acknowledge the College of Petroleum Engineering and Geosciences, King Fahd University of Petroleum and Minerals for providing funding for this research.

Author information

Correspondence to Abeeb A. Awotunde.

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Jamal, M.S., Awotunde, A.A. Darcy’s model with optimized permeability distribution for the simulation of stokes flow and contaminant transport in karst aquifers. Hydrogeol J (2020). https://doi.org/10.1007/s10040-020-02124-y

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Keywords

  • Numerical modeling
  • Karst
  • Brinkman’s equation
  • Contaminant transport
  • Darcy’s model