Well hydraulics with the Weber-Goldstein transforms

Two new integral transforms, ideally suited for solving boundary value problems in well hydraulics, are derived from one of the Goldstein identities which generalizes a corresponding Weber identity. The two transforms are, therefore, named the Weber-Goldstein transforms. Their properties are presented. For the first, second, and third type boundary conditions, the new transforms remove the radial portion of a Laplacian in the cylindrical coordinates. They are used to straightforwardly rederive known solutions to the problems of a fully penetrating flowing well and a fully penetrating pumped well. A novel solution for a fully penetrating flowing well with infinitesimal skin situated in a leaky aquifer is also found by means of one of the new transforms. This solution is validated by comparison to a numerical solution obtained via the finite-difference method and to a quasi-analytic solution obtained by numerical inversion of the corresponding solution in the Laplace domain. Based on the new solution, a flowing well test is proposed for estimating the hydraulic conductivity and specific storativity of the aquifer and the skin factor of the well. The test can also be used in a constant-head injection mode. A type-curve estimation procedure is developed and illustrated with an example. The effectiveness of the test in estimating the well skin factor and aquifer parameters depends on the availability of data on the sufficiently early well response

Hydraulics of a partially penetrating well: solution to a mixed-type boundary value problem via dual integral equations

New semi-analytic solutions are obtained for well response to the pumping test and slug test performed on a partially penetrating well. The solutions account not only for the wellbore storage, infinitesimal skin, and aquifer anisotropy, but also for the mixed-type boundary condition at the well face, which is novel. The solutions are obtained via the method of dual integral equations (DE). The new solutions are computationally robust and efficient, about one to two orders of magnitude faster than the corresponding finite difference solutions. Existing approximate solutions obtained with flux-flux discontinuous boundary conditions are compared to our DE solutions. The accuracy of the approximate solutions appears to be adequate for slender well screens. Our DE solution is computationally more efficient than the approximate solutions. Tn the range where the approximate solutions are less accurate the DE solution is about an order of magnitude faster. More important, the new solutions provide the correct distribution of the point flux (local velocity) along the well screen, unlike all existing solutions. This feature is essential in cases where vertical variations of hydraulic conductivity are sought (e.g. in flowmeter tests), and for tracer tests. (C) 1998 Elsevier Science B.V. All rights reserved