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Hydrogeology Journal

, Volume 27, Issue 1, pp 55–60 | Cite as

Review: Revisiting the Theis solution derivation to enhance understanding and application

  • Luke FloresEmail author
  • Ryan T. Bailey
Paper
  • 106 Downloads

Abstract

The Theis solution is perhaps the most influential and frequently used analytical model in groundwater hydrology. Its publication in 1935 led to immediate and continued use for simulating hydraulic head drawdown, in both confined and unconfined aquifers, as a tool in aquifer parameter estimation. For educational purposes, the Theis solution and the related Jacob’s approximation often serve as the backbone for teaching pumping-well theory, including topics such as boundary conditions in aquifers, image well theory, linear superposition, and pumping-induced streamflow depletion. Clearly, a thorough understanding of the Theis solution is critical for groundwater engineers and hydrologists. However, the solution often is presented as a “black box”, neglecting the actual origins of its derivation and accompanying physical context. This can lead to misconceptions about the model and its inherent limitations. In this paper, a physically based detailed derivation of the Theis solution is presented, along with a method of calculating drawdown from a pumping well without resorting to the final Theis equation. Examples of both constant-rate pumping and variable-rate pumping are presented and compared to results using the original Theis solution. In particular, variable pumping rates are accounted for by direct numerical integration of an earlier form in the original Theis derivation, removing the need for linear superposition of solutions in time. In this way, it is hoped the paper will provide a method of calculation that ties the model user to the physical meaning of the solution, including its assumptions.

Keywords

Groundwater hydraulics Analytical solutions Education Theis Foundations (pedagogy) 

Article de synthèse: Revisiter la dérivation de la solution de Theis pour améliorer sa compréhension et son application

Résumé

La solution de Theis est sans doute le modèle analytique le plus influent et le plus couramment utilisé en hydrogéologie. Sa publication en 1935 conduisit à son utilisation immédiate et continue depuis lors pour simuler les rabattements, aussi bien dans les aquifères captifs que libres, ainsi qu’en tant qu’outil pour l’estimation des paramètres des aquifères. Dans un objectif pédagogique, la solution de Theis et l’approximation de Jacob qui lui est associée servent souvent de base pour enseigner la théorie des puits de pompage, en intégrant aussi des sujets tels que les conditions aux limites dans les aquifères, la théorie du puits image, la superposition linéaire et la diminution du débit des cours d’eau induite par les pompages. Clairement, une compréhension approfondie de la solution de Theis est capitale pour les hydrogéologues. Cependant, la solution est souvent présentée comme une “boîte noire”, en négligeant les vraies origines de sa dérivation et le contexte physique qui l’accompagne. Cela peut conduire à des concepts erronés à propos du modèle et de ses limitations propres. Dans cet article, une dérivation de la solution de Theis basée sur des concepts physiques est présentée ainsi qu’une méthode de calcul du rabattement induit par un puits de pompage sans avoir recours à l’équation de Theis finale. Des exemples de pompages tant à débit constant et à débit variable sont présentés et comparés aux résultats obtenus avec la solution de Theis d’origine. Les pompages à débit variable sont notamment représentés au moyen d’une intégration numérique directe d’une forme précédente de la dérivation originale de Theis, ce qui supprime le besoin de superposition linéaire des solutions avec le temps. Il est ainsi espéré que cet article fournira une méthode de calcul qui relie l’utilisateur du modèle à la signification physique de la solution, en intégrant ses hypothèses.

Revisión: Reconsideración de la derivación de la solución Theis para mejorar su comprensión y aplicación

Resumen

La solución de Theis es quizás el modelo analítico más influyente y utilizado con mayor frecuencia en la hidrología del agua subterránea. Su publicación en 1935 condujo a un uso inmediato y continuo para simular la depresión de la carga hidráulica, en acuíferos confinados y no confinados, como una herramienta para la estimación de parámetros del acuífero. Para fines educativos, la solución de Theis y la relacionada aproximación de Jacob a menudo sirven como la columna vertebral para enseñar la teoría de los pozos de bombeo, incluyendo temas tales como condiciones de contorno en acuíferos, teoría del pozo imagen, superposición lineal y agotamiento de flujo inducido por bombeo. Claramente, una comprensión profunda de la solución Theis es crítica para los ingenieros e hidrólogos de aguas subterráneas. Sin embargo, la solución a menudo se presenta como una “caja negra”, descuidando los orígenes reales de su derivación y el contexto físico que la acompaña. Esto puede conducir a conceptos erróneos sobre el modelo y sus limitaciones inherentes. En este trabajo, se presenta una derivación detallada basada en la física de la solución Theis, junto con un método para calcular la extracción desde un pozo de bombeo sin recurrir a la ecuación final de Theis. Se presentan ejemplos de bombeo a velocidad constante y de caudal variable y se comparan con los resultados que utilizan la solución Theis original. En particular, los caudales variables de bombeo se explican por la integración numérica directa de una forma anterior en la derivación original de Theis, eliminando la necesidad de una superposición lineal de soluciones en el tiempo. De esta forma, se espera que el trabajo proporcione un método de cálculo que vincule al usuario del modelo con el significado físico de la solución, incluidos sus supuestos.

综述:重温Theis解派生方法以增进对其了解和应用

摘要

Theis解可能是水文地质学中最具影响力和最常用的解析模型。1935年初次提出立即并且持续得到使用,在含水层参数估算中作为一项工具用来模拟承压和非承压含水层中的水头降深。针对教育目的,Theis解及相关的Jacob近似法常常作为抽水理论的教学支柱,这些抽水理论包括含水层中的边界条件、图像井理论、线性叠加、抽水引起的河流枯竭等诸方面。显然,彻底了解Theis解对于地下水工程师和水文地质工作者来说至关重要。然而,该解常常表示为“黑匣子”,忽略了其派生出的方法和相伴的物理背景。这会导致模型的错误想法及其固有的局限。本文中,详细论述了基于物理的Theis解派生方法,以及不依靠最终Theis解计算抽水井降深的方法。展示了恒定速度抽水和变化速度抽水的例子,并与采用经典Theis解得到的结果进行了对比。特别是,通过经典Theis解派生方法中较早形式的直接数值积分对变化的抽水速度进行了说明,及时除去了Theis解线性叠加的需求。这样,本文希望能提供一种计算方法, 把模型使用者和该解的物理意义、包括其假设捆绑在一起。

Revisão: Revisitando a derivação da solução de Theis para elevar o seu entendimento e aplicação

Resumo

A solução Theis é talvez o modelo analítico mais influente e frequentemente usado na hidrologia de águas subterrâneas. Sua publicação em 1935 levou à utilização imediata e continuada para simular o rebaixamento da carga hidráulica, em aquíferos confinados e não confinados, como uma ferramenta na estimativa de parâmetros do aquífero. Para fins educacionais, a solução Theis e a aproximação de Jacob frequentemente servem como a espinha dorsal para o ensino da teoria do poço de bombeamento, incluindo tópicos como condições de contorno em aquíferos, teoria do poço de imagem, superposição linear e depleção do fluxo de escoamento. Claramente, um entendimento completo da solução Theis é crítico para engenheiros e hidrólogos de águas subterrâneas. No entanto, a solução é frequentemente apresentada como uma “caixa preta”, negligenciando as origens reais de sua derivação e o contexto físico que a acompanha. Isso pode levar a equívocos sobre o modelo e suas limitações inerentes. Neste artigo, é apresentada uma derivação detalhada e fisicamente detalhada da solução de Theis, juntamente com um método de calcular o rebaixamento de um poço de bombeamento sem recorrer à equação final de Theis. Exemplos de bombeamento de taxa constante e bombeamento de taxa variável são apresentados e comparados aos resultados usando a solução original de Theis. Em particular, as taxas de bombeamento variáveis são explicadas pela integração numérica direta de uma forma anterior na derivação original de Theis, eliminando a necessidade de superposição linear de soluções no tempo. Desta forma, espera-se que o artigo forneça um método de cálculo que vincule o usuário do modelo ao significado físico da solução, incluindo suas suposições.

Notes

Acknowledgements

The authors wish to acknowledge Jean-Michel Lemieux and two anonymous reviewers for their suggestions which improved the final manuscript.

Supplementary material

10040_2018_1843_MOESM1_ESM.xls (28 kb)
ESM 1 (XLS 28 kb)
10040_2018_1843_MOESM2_ESM.py (1 kb)
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Copyright information

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

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringColorado State UniversityFort CollinsUSA

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