Hydrologists often attempt to estimate formation properties from aquifer tests for which only the hydraulic responses in a pumped well are available. Borehole storage, turbulent head losses, and borehole skin, however, can mask the hydraulic behavior of the formation inferred from the water level in the pumped well. Also, in highly permeable formations or in formations at significant depth below land surface, where there is a long column of water in the well casing, oscillatory water levels may arise during the onset of pumping to further mask formation responses in the pumped well. Usually borehole phenomena are confined to the early stages of pumping or recovery, and late-time hydraulic data can be used to estimate formation properties. In many instances, however, early-time hydraulic data provide valuable information about the formation, especially if there are interferences in the late-time data. A mathematical model and its Laplace transform solution that account for inertial influences and turbulent head losses during pumping is developed for the coupled response between the pumped borehole and the formation. The formation is assumed to be homogeneous, isotropic, of infinite areal extent, and uniform thickness, with leakage from an overlying aquifer, and the screened or open interval of the pumped well is assumed to fully penetrate the pumped aquifer. Other mathematical models of aquifer flow can also be coupled with the equations describing turbulent head losses and the inertial effects on the water column in the pumped well. The mathematical model developed in this paper is sufficiently general to consider both underdamped conditions for which oscillations arise, and overdamped conditions for which there are no oscillations. Through numerical inversion of the Laplace transform solution, type curves from the mathematical model are developed to estimate formation properties through comparison with the measured hydraulic response in the pumped well. The mathematical model is applied to estimate formation properties from a singlewell test conducted near Waialua, Oahu, Hawaii. At this site, both the drawdown and recovery showed oscillatory water levels in the pumped well, and a step-drawdown test showed that approximately 86% of the drawdown is attributed to turbulent head losses. Analyses at this site using late-time drawdown data were confounded by the noise present in the measured water levels due primarily to nearby irrigation wells and ocean tides. By analyzing the early-time oscillatory recovery data at the Waialua site, upper and lower bounds were placed on the transmissivity, T, storage coefficient, S, and the leakance of the confining unit, K′/B′. The upper and lower bounds on T differ by a factor of 2. Upper and lower bounds on S and K′/B′ are much larger, because drawdown stabilized relatively quickly after the onset of pumping.
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Estimating formation properties from early-time oscillatory water levels in a pumped well