Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard...

Aquitards significantly affect groundwater flow in multi-aquifer systems through adjacent aquifer leakage. Despite this, studies focusing on their heterogeneity and the non-conventional diffusion patterns of their flow are still limited. In this study, a factional derivative approach was first extended to explore the time-dependent behavior of flow transport in the aquitard. Two analytical solutions were derived for specific discharges in independent aquitards under different boundary conditions. The findings revealed that aquitard flow exhibits obvious anomalous diffusion behaviors, characterized by slower decay and heavy-tailed specific discharge data. The fractional derivative model provided a more accurate representation of this behavior than traditional models, as evidenced by its superior agreement with experimental data. Moreover, a transient model for pumping tests in a leaky aquifer system was developed, incorporating the memory effect of anomalous flow and vertical heterogeneity in aquitards. Relevant semi-analytical solutions were derived to explore the impacts of memory factor β and decay exponent of aquitard hydraulic conductivity (K) on the leakage aquifer system. Theoretical results demonstrated that stronger memory effect reduces drawdowns in the aquitard and confined aquifer during mid-to-late times. A larger dimensionless decay exponent (A_d) decreases aquitard drawdown and increases aquifer drawdown at late times. Sensitivity analysis showed aquitard drawdown experiences two peaks in sensitivity to β and A_d at early- or mid-times, affected by memory effect and decay exponent of aquitard K, signifying maximal impact at these specific intervals. This study provides a practical model to effectively manage groundwater resources by accurately reflecting aquitard memory and heterogeneity effects.