Examining the large-time wellbore flux of constant head test

Abstract

The function 1/[p ln (p/lambda)] comes from the wellbore flux solution in the Laplace domain for a constant head aquifer test when the Laplace variable p is small. The resulting inverse Laplace transform v(lambda t), representing the large-time wellbore flux, grows exponentially with time, which does not agree with the physical behavior of the wellbore flux. Based on this result, Chen and Stone (1993) asserted that the well-known relationship of small p-large t may fail to yield the correct large-time asymptotic solution. Yeh and Wang (2007) pointed out that the large-time wellbore flux is not n(lambda t) if the inversion of 1/[p ln (p/lambda)] is subject to the constraint Re p > lambda. Chen (2009) subsequently questioned the necessity of imposing this constraint on the Laplace transform in the inversion of the large-time wellbore flux. Motived by Chen's comment, we reexamine the inversion of 1/[p ln (p/lambda)] and demonstrate that this contradictory issue originates from a spurious pole introduced when applying the small p-large t correspondence to the Laplace domain solution. We explain why this occurs and why the actual wellbore flux at large times is proportional to the function N(lambda t), known as Ramanujan's integral. The function N(lambda t) does decay at large time, which agrees with the steady state wellbore flux of the constant head test.