Time-evolving dispersion (TED) model: towards a more realistic representation of Darcy-scale mixing in porous media
Authors: Tajima Satoshi , Tokunaga Tomochika , Liu Jiaqi , Dentz Marco 
Published: February 2023

The advection-dispersion equation has been a key tool for modelling Darcy-scale solute transport. In the equation, local-scale mixing is expressed by the local dispersion coefficient, which considers mixing by molecular diffusion and pore-scale variability in flow velocity [1]. Mixing is enhanced by velocity variability on the Darcy scale, represented by spatial heterogeneity in hydraulic conductivity (K) [1].&#160;&#160;&#160;&#160; Observations show that the dispersion coefficient increases with the spatial scale [e.g., 2]. This effect is expressed by the macrodispersion coefficient increasing proportionally to the correlation length of heterogeneity in K [2]. Conventional upscaled advection-dispersion models assume constant macrodispersion coefficients [2], yet they typically overestimate the dispersive effect and can cause problems such as &#8220;back dispersion&#8221;, an unrealistic spreading of the solute in the direction opposite to the flow when groundwater flows towards a high concentration zone [3]. Explicitly representing local scale medium heterogeneity mitigates the overestimation of dispersion, however, this downscaled approach (hereinafter called the &#8220;local dispersion model&#8221;) requires the full representation of heterogeneity in K with a high spatial resolution. This comes at a high computational cost in numerical simulation whereas the detailed representation of the full K variability on the field scale is typically not feasible.&#160;&#160;&#160;&#160; Dentz et al. [4] showed that the effective dispersion coefficient (D) evolves temporally and asymptotically reaches a macrodispersion coefficient. They derived explicit analytical expressions for an idealized setting with an isotropic velocity spectrum in a steady state and constant local dispersion. From this finding, we hypothesise that accounting for the temporal evolution in D can mitigate the overestimation of dispersive processes in the macrodispersion model. Prospecting further applications to complex settings, this study aims to find an empirical formulation of the time-evolving dispersion (TED) coefficient. In this model, D is set to be identical to the molecular diffusion coefficient at the initial stage, and then it increases exponentially over time and asymptotically reaches a value identical to the macrodispersion coefficient after sufficient time has elapsed. We implemented the TED model by modifying the MODFLOW [5] source code and compared the simulation results with those from the macrodispersion and local dispersion models.&#160;&#160;&#160;&#160; The results from the TED model showed concentration distributions similar to the local dispersion model and less dispersive than those from the macrodispersion model, especially at early times in the simulations. The temporal evolution of D assumed in the TED model well matched that calculated from the spatial variance of the concentration distributions obtained by the local dispersion model. Therefore, the TED model can be an alternative to the conventional models for modelling Darcy-scale solute transport.&#160;References[1] Dentz, M., Hidalgo, J. J., & Lester, D. (2022). Transport in Porous Media.[2] Gelhar, L. W. & Axness, C. L. (1983). Water Resources Research, 19(1), 161&#8211;180.[3] Konikow, L. F. (2011). Ground Water, 49(2), 144-159.[4] Dentz, M, Kinzelbach, H., Attinger, S., & Kinzelbach, W. (2000). Water Resources Research, 36(12), 3591&#8211;3604.[5] Langevin, C., Hughes, J., Banta, E., Provost, A., Niswonger, R., & Panday, S. (2017). MODFLOW 6.