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\begin{document}

\title{Stochastic dynamics of two-dimensional particle motion in Darcy-scale
heterogeneous porous media}



\author[1]{Aronne Dell'Oca}%
\author[2]{Marco Dentz}%
\affil[1]{Politecnico di Milano}%
\affil[2]{IDAEA-CSIC}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\begin{abstract}
We study the upscaling and prediction of dispersion in two-dimensional\\
heterogeneous porous media with focus on transverse dispersion. To this
end, we\\
study the stochastic dynamics of the motion of advective particles that
move\\
along the streamlines of the heterogeneous flow field. While
longitudinal\\
dispersion may evolve super-linearly with time, transverse dispersion
is\\
characterized by ultraslow diffusion, that is, the transverse
displacement variance grows asymptotically with the logarithm of time.
This remarkable behavior is linked to the solenoidal\\
character of the flow field, which needs to be accounted for in
stochastic\\
models for the two-dimensional particle motion. We analyze particle
velocities\\
and orientations through equidistant sampling along the particle
trajectories\\
obtained from direct numerical simulations. This sampling strategy
respects the flow structure, which is organized on a characteristic
length scale. Perturbation theory shows that the longitudinal particle
motion is determined by the variability of travel times, while the
transverse motion is governed by the fluctuations of the space
increments. The latter turns out to be strongly anti-correlated with a
correlation structure that leads to ultraslow diffusion. Based on this
analysis, we derive a\\
stochastic model that combines a correlated Gaussian noise for the
transverse\\
motion with a spatial Markov model for the particle speeds. The model
results\\
are contrasted with detailed numerical simulations in two-dimensional\\
heterogeneous porous media of different heterogeneity variance.%
\end{abstract}%



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