In search of governing equations for scalar dissipations
Résumé
The idea of simplifying multicomponent reactive transport in heterogeneous porous media by first quantifying the potential mixing without reactions via scalar dissipation rates is continuing to produce promising results in quantitative hydrogeology. These metrics that originated in turbulence and combustion modeling are being applied to increasingly sophisticated flow and transport scenarios, more representative of the frozen turbulence of natural subsurface flow, typically through particle tracking or other lagrangian approaches. Alternatively instantaneous rates of scalar dissipation are obtained through eulerian modeling of mixing distributed over a given domain. The latter approach generally doesn't track the historical sequence of mixing that reacting solutes experience, and the former is generall posited without specification of the (eulerian) governing equation for the evolution of the scalar dissipations endured by reacting solutes. We demonstrate the construction of such governing equations by extending advective dispersive transports to include an exposure-time dimension along which the cumulative scalar dissipation accumulates at a velocity proportional to the local mixing. This approach is applied to several canonical base cases of transport with mixing for demonstration. The solution of such exposure-time extended mass conservation equations gives the evolution of passive solutes over space, time, and cumulative scalar dissipation, or mixing. This is exactly what is needed to close the loop between scalar dissipation metrics and reactive transport