Approximation for the Theis Well Function Using Ramanujan’s Series and Bounds for the Exponential Integral
Publication: Journal of Hydrologic Engineering
Volume 29, Issue 2
Abstract
The solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, is provided in terms of the exponential integral, which is famously known as the well function. For the computation of this function in practical applications, it is important to develop not only accurate but also a simple approximation that requires evaluation of the fewest possible terms. To that end, introducing Ramanujan’s series expression, this work proposes an approximation to the exponential integral using Ramanujan’s series for the small argument () and an approximation based on the bound of the integral for the other range (. The evaluation of the proposed approximation results in the most accurate formula compared with existing studies, which possess the maximum percentage error of 0.05%. Further, the proposed formula is much simpler to apply because it contains just the product of exponential and logarithm functions. To further check the efficiency of the proposed approximation, we considered a practical example for evaluating the discrete pumping kernel, which shows the superiority of this approximation over the others. Finally, the authors hope that the proposed efficient approximation can be useful for groundwater and hydrogeological applications.
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Data Availability Statement
No data were used to support this study.
Acknowledgments
The authors dedicate this work to the mathematical genius Srinivasa Ramanujan. Also, we convey our sincere thanks to the reviewers and editorial board for their insightful comments, which helped improve the quality of the work.
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© 2024 American Society of Civil Engineers.
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Received: Apr 17, 2023
Accepted: Dec 6, 2023
Published online: Feb 15, 2024
Published in print: Apr 1, 2024
Discussion open until: Jul 15, 2024
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