Bayesian reversal of the liquid level trajectory in a draining tank for pollution forensics

2026-05-27

AI Summary

This study addresses the challenge of estimating discharge volume from a leaking tank when the initial liquid level is unknown. The authors propose a novel approach that integrates a physics-based model with Bayesian inversion. Building upon Torricelli’s law, they formulate a drainage dynamics model and combine it with observed final liquid levels and drainage duration to infer the initial liquid height via Bayesian statistical inversion. To account for unrepresented physical processes in the model, an empirical discrepancy function is incorporated. This work represents the first application of Bayesian inversion coupled with a dynamic liquid-level model for pollution source reconstruction, enabling not only accurate recovery of the initial state but also quantification of associated uncertainties. Experimental validation using tank drainage tests confirms the method’s efficacy and reveals that inference uncertainty increases with longer drainage durations.

Abstract

Storage tanks for hazardous liquids are common in industry and agriculture. During a pollution incident, liquid may drain from a storage tank through a small hole, crack, or pipe. After containing the leak, estimating the discharged volume of liquid is essential for public safety, regulatory assessment, and remediation. When the original inventory of liquid is unknown, this constitutes an inverse problem. In this work, we present a framework for inferring the initial liquid level in a partially drained tank from the observed final liquid level after a pollution incident and an estimate of the drainage duration. Because the drainage dynamics, model parameters, and observations are uncertain, we employ Bayesian statistical inversion to combine prior physical knowledge with experimental liquid level time series data to predict the initial liquid level with quantified uncertainty. We use a physics-based model based on Torricelli's law to describe the tank-draining dynamics and augment it with an empirical discrepancy function to account for missing or imperfectly modeled physics. In our experiments with a tank draining of water, we found that our inferred initial liquid level was accurate, although uncertainty increased with drainage duration. Beyond its application to pollution forensics, this work may also serve as a hands-on classroom project illustrating dynamic modeling, model discrepancy, and Bayesian inference.

Problem

• pollution forensics
• inverse problem
• liquid level estimation
• tank drainage
• Bayesian inference

Innovation

• Bayesian inference
• inverse problem
• Torricelli's law
• model discrepancy
• pollution forensics