Small Time Asymptotics of Solute Transport at High Peclet Number and the Effect on Velocity Correlation

Abstract:

In order to predict transport of solutes, upscaling techniques are often applied. After the amount of time it takes the solute to sample all of the velocities in the system, the upscaling process is well understood and fairly simple to implement. But in highly heterogeneous velocity fields, this amount of time may be prohibitively long. When there is a need to predict transport at earlier times, the upscaling process is more difficult because the solute tends to stay on or near its initial streamline, inducing a correlation between its average velocity over fixed distances (or times), which must be accounted for. A Spatial Markov model was developed in 2008 that does just that^[1,2]. It accounts for the velocity correlation by treating the transport process as a Markov Chain. This model has been successfully applied to predict solute transport in a large variety of complicated flow fields and is becoming increasing popular. It almost seems as though it works for every situation, but so far no rigorous study has gone into determining its limitations. 

So we have decided to take a step back and ask: when is this model valid? We understand the asymptotic behavior in the limit as t→ ∞, but what about in the limit as 1/t→ ∞ (or t→ 0)? Are the assumptions of the Spatial Markov model valid over all length (and time) scales? It turns out that the answer is no. At very early times, the transport process is diffusion dominated, leading to non-monotonic correlation between solute particles' average velocity over consecutive space and time steps. The assumptions of the Spatial Markov model only hold after this early diffusive regime ends and the correlation function peaks. We find the location of the peak in the correlation function for transport in simple stratified flows and show the effect of using the Spatial Markov model over length scales on either side of the peak.