The diffusing-velocity random walk: a spatial-Markov formulation of heterogeneous advection and diffusion

Abstract

Abstract

The chemical continuous time random walk framework for upscaling transport limitations in fluid–solid reactions

“…Future work will also aim to generalize the approach to higher-order reactions involving multiple transported components, multiple simultaneous reactions, and heterogeneity (spatial and temporal variability) in the solid-phase reactant distribution along the interface. The PTRW method used here is based on that employed in [85] to simulate conservative transport in a number of stratified flows, where it was validated against theoretical predictions for dispersion, concentration distributions, breakthrough curves (first passage times across a plane), and Lagrangian velocity distributions. The conservative transport algorithm consists in discretizing the Langevin equation ( 1) for a set of Lagrangian particles, or trajectories, with prescribed initial conditions.…”

Sharp Transition to Strongly Anomalous Transport in Unsaturated Porous Media

“…We thus approximate the flow rate decay along the depth as $$${q}_{\mathrm{d}}(z\vert \ell )\approx {q}_{\mathrm{0}}{e}^{-z/{a}_{\mathrm{m}}}H(\ell -z),$$$ where q 0 = q d (0| ℓ ) is the flow rate at the contact with the backbone and H is the Heaviside step function. Since this flow rate profile is monotonically decreasing, the associated PDF for a given q 0 and ℓ can be computed as (Aquino & Le Borgne, 2021) $$${p}_{\mathrm{Q}}^{d}(q\vert \ell ,{q}_{\mathrm{0}})={\left(\ell {\left.\frac{d{q}_{\mathrm{d}}(z\vert \ell )}{dz}\right\vert }_{z={z}_{\mathrm{q}}(q)}\right)}^{-1},$$$ where z q ( q ) is the point at which the flow has a given value q , that is, q d [ z q ( q )| ℓ ] = q . Thus, inverting Equation for depth as a function of flow rate, computing dq d ( z | ℓ )/ dz , and substituting, Equation becomes $$${p}_{\mathrm{Q}}^{d}(q\vert \ell ,{q}_{\mathrm{0}})=\frac{{a}_{\mathrm{m}}}{\ell q}H({q}_{0}-q)H(q-{q}_{0}{e}^{-\ell /{a}_{\mathrm{m}}}),$$$ for the dead‐end flow rate PDF …”

Using Complex Probability Amplitudes to Simulate Solute Transport in Composite Porous Media

“…6b). The flux-weighted velocity PDF of the composite domain was used to compute the velocity correlations because it provides a reliable analog for the Lagrangian velocities (trajectories) (Dentz et al, 2016;Aquino & Le Borgne, 2021). S-Lagrangian velocity correlation (Le Borgne et al, 2008) was computed throughout the composite domain, evaluated at a distance of one model cell-size (i.e.…”