Backward probabilities have been used for decades to track hydrologic targets such as pollutants in water, but the convenient deviation and scale effect of backward probabilities remain unknown. This study derived backward probabilities for groundwater pollutants and evaluated their scale effect in heterogeneous aquifers. Three particle-moving methods, including the backward-in-time discrete random-walk (DRW), the backward-in-time continuous time random-walk (CTRW), and the particle mass balance, were proposed to derive the governing equation of backward location and travel time probabilities of contaminants. The resultant governing equations verified Kolmogorov’s backward equation and extended it to transient flow fields and aquifers with spatially varying porosity values. An improved backward-in-time random walk particle tracking technique was then applied to approximate the backward probabilities. Next, the scale effect of backward probabilities of contamination was analyzed quantitatively. Numerical results showed that the backward probabilities were sensitive to the vertical location and length of screened intervals in a three-dimensional heterogeneous alluvial aquifer, whereas the variation in borehole diameters did not influence the backward probabilities. The scale effect of backward probabilities was due to different flow paths reaching individual intervals under strong influences of subsurface hydrodynamics and heterogeneity distributions, even when the well screen was as short as ~2 m and surrounded by highly permeable sediments. Further analysis indicated that if the scale effect was ignored, significant errors may appear in applications of backward probabilities of groundwater contamination. This study, therefore, provides convenient methods to build backward probability models and sheds light on applications relying on backward probabilities with a scale effect.

Groundwater vulnerability assessment helps subsurface water resources management by providing scientific information for decision-makers. Rigorous, quantitative assessment of groundwater vulnerability usually requires process-based approaches such as groundwater flow and transport modeling, which have seldom been used for large aquifer-aquitard systems due to limited data and high model uncertainty. To quantify the vulnerability of regional-scale aquifer-aquitard systems in the East Gulf Coastal Plain of Alabama, a three-dimensional (3D) steady-state groundwater flow model was developed using MODFLOW, after applying detailed hydrogeologic information to characterize seven main aquifers bounded by aquitards. The velocity field calibrated by observed groundwater heads was then applied to calculate groundwater age and residence time for this 3D aquifer-aquitard system via backward/forward particle tracking. Radioactive isotope data (14C and 36Cl) were used to calibrate the backward particle tracking model. Results showed that shallow groundwater (<300 ft below the groundwater table) in southern Alabama is mainly the Anthropocene age (25–75 years) and hence susceptible to surface contamination, while the deep aquifer-aquitard systems (700 ft or deeper below the groundwater table) contain “fossil” waters and may be safe from modern contamination if there is no artificial recharge/discharge. Variable horizontal and vertical vulnerability maps for southern Alabama aquifer-aquitard systems reflect hydrologic conditions and intermediate-scale aquifer-aquitard architectures in the regional-scale models. These large-scale flow/transport models with coarse resolutions reasonably characterize the broad distribution and vertical fluctuation of groundwater ages, probably due to aquifer-aquitard structures being captured reliably in the geology model. Parameter sensitivity analysis, vadose zone percolation time, wavelet analysis, and a preliminary extension to transient flow were also discussed to support the aquifer vulnerability assessment indexed by groundwater ages for southern Alabama.

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