Analytic expressions for the moving infinite line source model

TL;DR

This paper introduces new analytical expressions and first-order approximations for the moving infinite line source model, simplifying calculations and maintaining high accuracy for groundwater heat transfer analysis.

Contribution

It establishes a link between special functions and the model, and proposes integral-free, convergent power series expressions with low error for practical use.

Findings

New exact analytical expressions with no recursive evaluations

First-order approximations with less than 1% error using few terms

Enhanced understanding of convergence and validity domains

Abstract

Groundwater flow can have a significant impact on the thermal response of ground heat exchangers. The moving infinite line source model is thus widely used in practice as it considers both conductive and advective heat transfert processes. Solution of this model involves a relatively heavy numerical quadrature. Contrarily to the infinite line source model, there is currently no known first-order approximation that could be useful for many practical applications. In this paper, known analytical expressions of the Hantush well function and generalized incomplete gamma function are first revisited. A clear link between these functions and the moving infinite line source model is then established. Then, two new exact and integral-free analytical expressions are proposed, along with two new first-order approximations. The new analytical expressions proposed take the form of convergent power series involving no recursive evaluations. It is shown that relative errors less than 1% can be obtained with only a few summands. The convergence properties of the series, their accuracy and the validity domain of the first-order approximations are also presented and discussed.