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 a full-range approximation to the exponential integral using Ramanujan's series for the small argument (u \leq 1) and an approximation based on the bound of the integral for the other range (u \in (1,100]). The evaluation of the proposed approximation results in the most accurate formulae compared to the existing studies, which possess the maximum percentage error of 0.05\%. Further, the proposed formula is much simpler to apply as it contains just the product of exponential and logarithm functions. To further check the efficiency of the proposed approximation, we consider 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.

翻译：描述水平承压含水层中非稳定地下水流动的水头下降控制方程的解，以指数积分形式给出，即著名的井函数。在实际应用中计算该函数时，不仅要开发精确的近似方法，还需确保其简单性，以便于评估尽可能少的项。为此，引入Ramanujan级数表达式，本文提出指数积分的全范围近似：对于小参数（u ≤ 1）采用Ramanujan级数，对于另一范围（u ∈ (1,100]）采用基于积分界限的近似。与现有研究相比，所提出的近似公式在评估时得到最精确的结果，最大百分比误差为0.05%。此外，该公式仅包含指数函数和对数函数的乘积，应用更为简便。为进一步验证所提近似的效率，我们考虑一个评估离散抽水核函数的实际示例，结果表明该近似优于其他方法。最终，作者希望所提出的高效近似可用于地下水和水文地质应用。