Approximation of the exponential integral (well function) using sampling methods

Abstract

• This study proposes a new method for exponential integral approximation. • The methods are based on probabilistic sampling. • Three different sampling methods have been used; LHS, OA and MCS. • The results of sampling methods are good and accurate compared to other methods. Exponential integral (also known as well function) is often used in hydrogeology to solve Theis and Hantush equations. Many methods have been developed to approximate the exponential integral. Most of these methods are based on numerical approximations and are valid for a certain range of the argument value. This paper presents a new approach to approximate the exponential integral. The new approach is based on sampling methods. Three different sampling methods; Latin Hypercube Sampling (LHS), Orthogonal Array (OA), and Orthogonal Array-based Latin Hypercube (OA-LH) have been used to approximate the function. Different argument values, covering a wide range, have been used. The results of sampling methods were compared with results obtained by Mathematica software, which was used as a benchmark. All three sampling methods converge to the result obtained by Mathematica, at different rates. It was found that the orthogonal array (OA) method has the fastest convergence rate compared with LHS and OA-LH. The root mean square error RMSE of OA was in the order of 1E−08. This method can be used with any argument value, and can be used to solve other integrals in hydrogeology such as the leaky aquifer integral.