Abstract
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.
Highlights
Introduction iationsKolmogorov derived his well-known transport equations describing the probability density of random-walk particles in both jump and diffusion processes in 1931 [1,2]: Licensee MDPI, Basel, Switzerland. h i ∂G ∂ ∂2 ∗ ∗ ∗ T= − ∑ ∗ ( Ai G ) + ∑ B (B ) G ∂t
We show for the first time that, by tracking backward in time, the widely used random walk and mass balance theories can conveniently lead to backward probability models
The calculated backward location probability (BLP) of contaminants in groundwater collected at different depths along the 2.5 m long screen of the monitoring well have similar main characteristics (Figure 3), except for the following subtle differences
Summary
The mass-balance method is proposed here for three-dimensional expansion. It is a reasonable assumption that each particle moves randomly in multiple dimensional spaces. Assuming that the particle generally moves from cell i to cell i + 1 under ambient conditions, the particle number flux from cell i + 1 to cell i per unit area and per unit time in the backward-in-time process is [34]:. Ω where the parameter φi represents the difference in probabilities when particles jump forward and backward along X-axis, so φi > 0; ω is the area of cell normal to X-axis [L2 ]; Ris the number of jumps per unit time for each particle [T−1 ]; and ∆x is the cell length [L].
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