Equivalence of Finite Dimensional Input–Output Models of Solute Transport and Diffusion in Geosciences

Abstract

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 multi-rate mass transfer and multiple interacting continua 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 for obtaining equivalence. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute, and, eventually, on their interaction with soil microbes and minerals.

Inherent relevance of MRMT models to concentration variance and mixing-induced reactivity

“…has null entries. In [Rapaport et al, 2017], it is proved that for matrices of the form of Eq. 7the vector R −1 S A(2 : n + 1, 1) has non-null entries exactly when the pair (A, B) is controllable.…”

“…Babey et al [2015] and Rapaport et al [2017] have demonstrated that any mobile-immobile model identified as SINC in the previous section, i.e. whatever the connectivity of its immobile zones, is equivalent to a unique (up to the numbering of the immobile zones) MRMT model, proving the original model to be minimal.…”

“…The example of Appendix A1 does not show any visual symmetry or redundant structure although it is not minimal. Minimality of the system can however be tested by its controllability and observability properties [Andréa-Novel and de Lara, 2013] (here observability is equivalent to controllability because input and output are in the same mobile zone: see [Rapaport et al, 2017]). A system is said controllable if, for all couple of state vectors (C a , C b ), there exists a finite time T ≥ 0 and a input concentration…”

“…Relevance of MRMT models to reactive transport be sought here as a SINC model with less immobile zones (n < n) before applying the algorithm given in [Rapaport et al, 2017] to then obtain an equivalent MRMT model withn immobile zones.…”

“…Among them, the mobile-immobile Multi-Rate Mass Transfer models (MRMT) [Carrera et al, 1998;Haggerty and Gorelick, 1995] does not only propose efficient characterization and upscaling methodologies [Willmann et al, 2008;Babey et al, 2015;Rapaport et al, 2017] but also a natural bridge to equivalent concentration distributions, which relevance can be assessed to model reactive transport [Donado et al, 2009;Henri and Fernandez-Garcia, 2015;Sanchez-Vila et al, 2010;Soler-Sagarra et al, 2016]. Synthetic experiments have shown that MRMT models provide close approximations of bulk reactivity even in non-linear equilibrium and kinetically-controlled cases [Babey et al, 2016].…”