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.

翻译：在工业与农业领域，储存危险液体的储罐十分常见。污染事故发生时，液体可能通过小孔、裂缝或管道从储罐中泄漏。在控制泄漏后，估算泄漏液体体积对于公共安全、监管评估及环境修复至关重要。当泄漏前储罐的初始存量未知时，这构成了一个反问题。本研究提出一个框架，通过污染事故后观测到的最终液位及泄漏持续时间的估算值，推断部分排放储罐的初始液位。由于排放动力学、模型参数及观测数据均存在不确定性，我们采用贝叶斯统计反演方法，将先验物理知识与实验液位时间序列数据相结合，以量化不确定性的方式预测初始液位。基于托里拆利定律构建描述储罐排放动力学的物理模型，并辅以经验性差异函数补偿缺失或不完善的物理建模。在水箱排放实验中，我们发现推断的初始液位具有较高准确性，但不确定性随排放持续时间增加而增大。除应用于污染溯源外，本工作还可作为涉及动态建模、模型差异及贝叶斯推断的实践性课堂项目。