Upscaled models for time-varying solute transport: Transient spatial-Markov dynamics

Correlated velocity models (CVMs) have proven themselves to be effective tools for describing a wide range of solute transport behaviors in heterogeneous porous media. In particular, spatial Markov models (SMMs) are a class of CVMs where subsequent Lagrangian velocities along transport trajectories depend only on the current velocity, and not on past history. Such models provide a powerful tool for modeling transport in terms of a limited number of flow properties, such as the Eulerian point distribution of (flow) velocities, tortuosity, and the spatial scale of persistence of velocities. However, to date, all SMM modeling frameworks and applications have assumed that the underlying flow is steady-state. In this work, we extend SMMs to the case of timevarying flows. We propose, compare, and validate alternative numerical implementations, and we determine conditions for validity and efficiency based on standard physical quantities used to describe flow and transport at the Darcy scale. The models require additional information relative to a steady-state velocity SMM and we discuss the conditions under which this extra burden is warranted. We also provide clear, deterministic tests for the validity of the transient SMM, termed the slow variation'' and fast propagation'' criteria, which offer clear guidance on when transient, upscaled models are reasonable to employ. Our work forms the basis of a new framework allowing for the application of efficient upscaled models of transport to realistic transient flow conditions.

Automated summary

Correlated velocity models have proved useful in describing solute transport behaviors in porous media. This study extends the application of spatial Markov models to time-varying flows and proposes alternative numerical implementations. The validity and efficiency of the models are determined based on standard physical quantities, and clear tests are provided for the applicability of transient, upscaled models.