Wellbore flow-rate solution for a constant-head test in two-zone finite confined aquifers

Chi Shan Tsai, Hund-Der Yeh*

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Scopus citations


The solution describing the wellbore flow rate in a constant-head test integrated with an optimization approach is commonly used to analyze observed wellbore flow-rate data for estimating the hydrogeological parameters of low-permeability aquifers. To our knowledge, the wellbore flow-rate solution for the constant-head test in a two-zone finite-extent confined aquifer has never been reported so far in the literature. This article is first to develop a mathematical model for describing the head distribution in the two-zone aquifer. The Laplace domain solutions for the head distributions and wellbore flow rate in a two-zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large-time solution for the wellbore flow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L'Hospital's rule. The result shows that the large-time solution is identical to the steady-state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large-time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellbore flow rate.

Original languageEnglish
Pages (from-to)3216-3224
Number of pages9
JournalHydrological Processes
Issue number21
StatePublished - 15 Oct 2012


  • Analytical solution
  • Aquifer test
  • Bromwich integral
  • Composite aquifer
  • Laplace transform
  • Residue theorem
  • Thiem equation

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