Research Article

Linear Boltzmann Equation for Solute Dispersion in Heterogeneous Media Under Non-Ergodic Conditions

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Abstract

We study solute dispersion under non-ergodic conditions using a Boltzmann equation for the evolution of the joint distribution of the position and speed of solute particles in steady spatially heterogeneous flow fields. We show that the Boltzmann equation is equivalent to a time-domain random walk in which particle speeds follow a stationary spatial Markov process. The analysis of particle velocity series for flow in heterogeneous flow fields indicates that velocity transitions can be characterized by Gaussian copulas. This transport framework allows to systematically study the impact of non-ergodic source conditions on solute dispersion across scales. In systems with strong medium heterogeneities and velocity contrasts, the dispersion behavior depends critically on the velocity distribution in the source zone. That is, it depends on whether the solute can initially sample the full flow variability (ergodic conditions), or whether it can sample only a part of the flow spectrum (non-ergodic conditions). We study the evolution of solute dispersion for particle injections in high, intermediate and low velocity regions. We find that non-ergodic initial conditions have a significant impact on dispersion at early and intermediate times with different scaling exponents than expected for ergodic conditions. They can give rise to distinctly bimodal particle distributions, and are imprinted in the peak behaviors of breakthrough curves. These results shed new light on the interpretation of dispersion data and the modeling and prediction of dispersion in heterogeneous media from the pore to the regional scales.

Keywords: Dispersion, Linear Boltzmann Equation, Heterogeneous Media, Ornstein-Uhlenbeck Process, Copulas, Markov Processes, First Passage Times

How to Cite:

Dentz, M. & Massoudieh, A., (2025) “Linear Boltzmann Equation for Solute Dispersion in Heterogeneous Media Under Non-Ergodic Conditions”, ARC Geophysical Research 1(1): 4. doi: https://doi.org/10.5149/ARC-GR.1401