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Revisiting the Domenico plume formula through saddle-point asymptotics
Kentaro Miyamoto1, Makoto Yasojima1, Hiroaki Takemori1
1Shimadzu Techno-Research, Inc., 1 Nishinokyo Shimoaicho, Nakagyo-ku, Kyoto 604-8436, Japan.
Abstract:
The Domenico plume formula is widely used for screening contaminant migration, yet its relation to the reactive advection-dispersion equation has remained unclear. Starting from the Green's function solution for a finite rectangular source patch, we derive a finite-time saddle-point expansion and show that the standard Domenico expression is the far-field leading-order term of a large-distance expansion. The same analysis yields explicit next-to-leading-order (NLO) and next-next-to-leading-order (NNLO) corrections. The NLO term captures transient departures near the advancing front, whereas the NNLO term captures the residual steady-state discrepancy caused by transverse dispersion. The resulting hierarchy explains how the Domenico formula emerges from the Green's function solution and identifies the regime in which it remains accurate. Environmental Implication By identifying when the Domenico formula is reliable and when it is not, this framework supports more defensible screening of contaminant migration from finite sources. The resulting validity map helps practitioners judge when a compact plume formula is adequate for estimating threshold exceedance distances and when direct numerical evaluation is needed to avoid non-conservative site decisions.
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