About equivalence of finite dimensional input-output models of solute transport and diffusion in geosciences
Résumé
Underground media are characterized by their high surface to volume ratio and by their slow solute movements that overall promote strong water-rock interactions. As a result, water quality strongly evolves with the degradation of anthropogenic contaminants and the dissolution of some minerals. Chemical reactivity is first determined by the residence time of solutes and the input/output behavior of the system, as most reactions are slow and kinetically controlled. Especially important are exchanges between high-flow zones where solutes are transported over long distances with marginal reactivity and low-flow zones in which transport is limited by slow diffusion but reactivity is high because of large residence time. The dominance of these characteristic features up to some meters to hundreds of meters have prompted the development of numerous simplified models starting from the ``double-porosity'' concept. In these models, solutes move quickly by advection in a first porosity with a small volume representing fast-flow channels and slowly by diffusion in a second large homogeneous porosity. Exchanges between the two porosities is diffusion-like. Such models have been widely extended to account not only for one diffusive-like zone but for many of them with different structures and connections to the advective zone. Such extensions are thought to model both the widely varying transfer times and the rich water-rock interactions. The two most famous ones are the Multi-Rate Mass Transfer model (MRMT) and Multiple INteracting Continua model (MINC). Theoretical grounds are however missing to identify classes of equivalent porosity structures, effective calibration capacity on accessible tracer test data, and influence of structure on conservative as well as chemically reactive transport. One can wonder if MRMT and MINC models are not too restrictive to represent real structures. In this work, we show that for a large class of finite dimensional input-output positive systems that represent networks of transport and diffusion of solute in geological media, there exist equivalent MRMT and MINC representations, which are quite popular in geosciences. Moreover, we provide explicit methods to construct these equivalent representations. The proofs show that controllability property is playing a crucial role. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute.