Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect
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...
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MDPI AG
2022-02-01
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| Series: | Water |
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| Online Access: | https://www.mdpi.com/2073-4441/14/4/624 |
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| author | Chaloemporn Ponprasit Yong Zhang Wei Wei |
| author_facet | Chaloemporn Ponprasit Yong Zhang Wei Wei |
| author_sort | Chaloemporn Ponprasit |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T20:50:36Z |
| format | Article |
| id | doaj.art-5fbbd66529464576b75c29a484ece2ea |
| institution | Directory Open Access Journal |
| issn | 2073-4441 |
| language | English |
| last_indexed | 2024-03-09T20:50:36Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Water |
| spelling | doaj.art-5fbbd66529464576b75c29a484ece2ea2023-11-23T22:34:50ZengMDPI AGWater2073-44412022-02-0114462410.3390/w14040624Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale EffectChaloemporn Ponprasit0Yong Zhang1Wei Wei2 Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USASchool of Environment, Nanjing Normal University, Nanjing 210023, ChinaBackward 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.https://www.mdpi.com/2073-4441/14/4/624backward modelgoverning equationscale effectbackward location probabilitybackward travel time probability |
| spellingShingle | Chaloemporn Ponprasit Yong Zhang Wei Wei Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect Water backward model governing equation scale effect backward location probability backward travel time probability |
| title | Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect |
| title_full | Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect |
| title_fullStr | Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect |
| title_full_unstemmed | Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect |
| title_short | Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect |
| title_sort | backward location and travel time probabilities for pollutants moving in three dimensional aquifers governing equations and scale effect |
| topic | backward model governing equation scale effect backward location probability backward travel time probability |
| url | https://www.mdpi.com/2073-4441/14/4/624 |
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