Characterization of Hydraulic Parameters via Sensitivity Maps for Frequency-Based Oscillatory Pumping in 2D, Confined, Weakly Heterogeneous Aquifers

Multi-frequency oscillatory pumping, in which the groundwater is extracted during a half period, and then reinjected, has recently been used to characterize aquifer heterogeneity. After the initial transition time, a steady periodic head can be observed at the observation well with constant amplitude and phase shift. However, the efficacy of utilizing multiple frequencies to enhance parameter estimation in hydraulic tomography is debatable. Most studies suggest that using multiple frequencies in joint inversion can gradually improve aquifer imaging. However, some researchers argue that additional frequencies add little to the resolution of hydraulic properties. This raises the question: how much non-redundant information do multiple frequencies provide? In this study, we derive closed-form analytical sensitivity maps (i.e., Fréchet kernels) of oscillatory pumping in 2D, unbounded, confined weakly heterogeneous aquifers by the sensitivity equation. It is found that the sensitivity maps of amplitude or phase shift are very similar to the sensitivity of drawdown to hydraulic parameters in constant-rate pumping. Fréchet kernels indicate that multi-frequency information, much like multi-time information, can be instrumental for hydrogeological parameter inversion. According to the Fréchet kernels and their derivatives, we propose a method to select the optimal observation frequency, that is, the distance between the observation well and the pumping well is converted to frequency based on the hydraulic background values. Finally, we compare the effectiveness of single-frequency and multi-frequency observations in inverse transmissivity and storativity modeling using the iterative ensemble smoother. The results show that multi-frequency oscillatory pumping can better characterize aquifer heterogeneity than a single frequency.