NUMERICAL EVALUATION OF EXPONENTIAL INTEGRAL - THEIS WELL FUNCTION APPROXIMATION

This paper reviews various approximation methods for evaluating one fo rm of the exponential integral known as the Theis well function W(u) a nd presents an algorithm that can be easily implemented to one's desir ed accuracy for 0 < u < infinity. The algorithm is based on a combinat ion of a fast-converging series representation for small u and an easy -implementing Gauss-Laguerre quadrature formula when 21 becomes large. The partition point for Ir varies with the desired program accuracy a nd efficiency. The error analyses suggested that a partition point in the interval of 1 less than or equal to u less than or equal to 4 can yield sufficiently accurate results for practical applications of grou ndwater problems with minimum computational effort. A comparison of th e results revealed that the popular Stehfest inverse Laplace transform technique generally yielded a relative error several orders of magnit ude greater than that obtained by the proposed algorithm. For large u, the approximation accuracy increased rapidly using the quadrature for mula while the results of the Laplace inversion technique showed numer ical oscillation between some positive and negative values. (C) 1998 E lsevier Science B.V.