Although the slippery boundary condition (BC) has been validated to enhance fracture permeability (k), the coupling effects of heterogeneous slippery BC and inertia on k remain less understood. We used computational fluid dynamics to investigate the competing roles of slippery BC and inertial forces in controlling k evolution with increasing pressure gradient by designing six cases with different slip length scenarios for a two-dimensional natural fracture. Our results suggest that pronounced inertial effects were directly related to and demonstrated by the growth of recirculation zone (RZ); this caused flow regimes transitioning from Darcy to non-Darcy and significantly reduced k, with an identical tailing slope for six cases, regardless of the variability in slip lengths. Moreover, the slippery BC dominantly determine the magnitude of k with orders depending on the slip length. Lastly, our study reveals that the specific k evolution path for the case with a varying slip length was significantly different from other cases with a homogeneous one, thus encouraging more efforts in determining the slip length for natural fractures via experiments.

Karst groundwater systems are characterized by highly heterogeneous flow networks, resulting in complex responses of groundwater levels to precipitation variability. The Jinan Spring Basin, one of the most representative karst spring systems in northern China, has experienced substantial changes in spring discharge due to variations in precipitation, groundwater exploitation, and hydrogeological conditions. However, the temporal scales at which precipitation signals control groundwater-level fluctuations and the mechanisms governing their transmission within the karst aquifer remain poorly understood. In this study, daily precipitation data from 30 meteorological stations and groundwater-level records at Baotu Spring during 2016–2018 were analyzed using global wavelet spectrum (GWS) and wavelet transform coherence (WTC) approaches. The dominant precipitation cycles, scale-dependent precipitation–groundwater relationships, and phase-derived groundwater response lags were quantified to reveal the hydrological response characteristics of the Jinan karst system. Three prominent precipitation periods were identified at 17.37, 29.22, and 330.57 days. Groundwater responses presented clear temporal-scale dependence, with short-period signals showing rapid but localized responses, while intermediate and long-period signals demonstrated stronger and more persistent coherence. The percentage of significant coherence area (PASC) increased from 23.50% at the 0–17.37 day scale to 64.28% at the 29.22–330.57 day scale, indicating that accumulated precipitation rather than individual rainfall events exerts the dominant control on groundwater-level variations. The spatial distribution of response lags revealed that rapid responses (17.37 days) mainly occurred in the southern recharge areas, reflecting preferential recharge through well-developed karst conduits. Intermediate responses (29.22 days) showed a progressive increase in lag time from south to north, indicating the influence of regional groundwater flow and aquifer storage. Long-period responses (330.57 days) were locally enhanced near major faults, suggesting structural controls on deeper groundwater circulation. This study reveals that precipitation signals in the Jinan Spring Basin are transmitted through multiple groundwater circulation pathways with distinct temporal characteristics. The identified multi-scale response patterns provide new insights into the internal structure and hydrological functioning of karst aquifers and offer scientific support for sustainable management of spring water resources.