Solutions for Groundwater Flow to a Pumping Well with an Exponentially Decreasing Variable-Rate in a Sloping Aquifer

During aquifer pumping tests in the field, the attenuation process of the pumping rate is generally monitored, and the aquifer may be slanted, especially in riverbank terraces, piedmont plain and other similar terrains. The combination of these two factors (attenuation process and aquifer slope) has not received enough academic attention, therefore, a two-dimensional groundwater flow model is developed for an exponentially decreasing variable rate pumping well in a sloping aquifer in this study. The new semi-analytical solution (based on a linearized free-surface boundary) is obtained using the Laplace transform and finite Fourier cosine transform. Meanwhile, to depict the linearized boundary precisely with the first-order free surface equation, a numerical model that considers the decline of the water table is also developed. Results show that under the effect of the variable rate, the time-drawdown curve presents a downward trend at intermediate times. Additionally, the error caused by the linearization of the free-surface equation on the analytical model is found to be acceptable except when the sloping angle is larger than π/6. Furthermore, since this study extends the work of Huang et al. (2014), who considered a constant pumping rate, the new solution was found better to interpret real pumping test data reported by Antonio and Pacheco (2002), and the laboratory experiment results. Overall, the new solution proves to be useful for parameter inversion accurately as the average pumping rate tended to underestimate hydraulic parameters when fitting short-term pumping data.