Multi-Rate Mass Transfer (MRMT) models for general diffusive porosity structures

Abstract

International audienceWe determine the relevance of Multi-Rate Mass Tansfer (MRMT) models to general diffusive porosity structures. To this end, we introduce Structured INteracting Continua models (SINC) as the combination of a finite number of diffusion-dominated interconnected immobile zones exchanging with an advection-dominated mobile domain. It directly extends Multiple INteracting Continua framework [34] by introducing a structure in the immobile domain, coming for example from the dead-ends of fracture clusters or poorly-connected dissolution patterns. We demonstrate that, whatever their structure, SINC models can be made equivalent in terms of concentration in the mobile zone to a unique Multi-Rate Mass Transfer (MRMT) model [21]. We develop effective shape-free numerical methods to identify its few dominant rates, that comply with any distribution of rates and porosities. We show that differences in terms of macrodispersion are not larger than 50% for approximate MRMT models with only one rate (double porosity models), and drop down to less than 0.1% for five rates MRMT models. Low-dimensional MRMT models accurately approach transport in structured diffusive porosities at intermediate and long times and only miss early responses

A Domain Decomposition Approach to Finite-Epsilon Homogenization of Scalar Transport in Porous Media

“…Besides the fact that, contrary to MRMT models, we have derived a direct link (the closure problems) between the microscale geometry and the effective parameters, there are also differences in the macroscale form compared to MRMT models. To better understand this, we consider the generic form from [4] that we straightforwardly extend to the mobile-mobile cases by allowing each phase to flow. With our formalism, this reads…”

“…Further, boundary conditions, nonlinearities, source terms, topological effects, or transient phenomena can generate or amplify nonequilibrium effects-large gradients at the microscale-leading to non-Fickian transport. Several approaches [16,36] are used to describe these phenomena, including stochastic [23], nonlocal [31], higher-order, multirate mass transfer (MRMT) [27,4,53] and multicontinua models (e.g., mobile-mobile, mobile-immobile, dual-porosity, or dualpermeability). For MRMT and multicontinua models, the central point is the introduction of characteristic times that describe exchanges between different subdomains at the microscale by coupling equations representing each phase separately, as is the case of our approach.…”

“…One of the limitations of the VAT approach, which we overcome in this paper, is that there exists no general theory that deals with any number of subdomains, as is the case for empirical MRMT models [4]. Further, the nonequilibrium approach with the two-equation model is mostly used when two distinct physical phases (when the microscale is the pore-scale) or regions (when the microscale is the Darcy-scale in heterogeneous systems) can be identified [44], which is limiting.…”

Quantifying the impact of immobile water regions on the fate of nitroaromatic compounds in dual-porosity media

“…Mathematically, this exchange can be described either by a first-order transfer coefficient (van Genuchten and Wierenga, 1976;Haggerty and Gorelick, 1995;Babey et al, 2015;Van Genuchten and Wagenet, 1989) or a parameter which is based on the molecular diffusion coefficient of the solute of interest (van Genuchten et al, 1984;Tang et al, 1981;Sudicky and Frind, 1982;Neretnieks, 1980;Małoszewski and Zuber, 1985;Małoszewski and Zuber, 1990). The advantage of the second group of models is that a smaller number of fitting parameters is required.…”

On the validity of effective formulations for transport through heterogeneous porous media

“…MRMT is taken as a typical anomalous transport framework. Its advantage lies in providing local concentrations, which can be straightforwardly used to evaluate concentration variance, mixing and mixing-induced reactivity (Babey et al, 2014;Carrera et al, 1998;Haggerty and Gorelick, 1995), as well as the apparent reduction in the rate of kinetic reactions . The question is whether its validity as a representation of transport through HPM can be extended to reproduce the effects of the evolution of mixing rates resulting from the stretching and folding associated with complex velocity structures (de Anna et al, 2014b;Jimenez-Martinez et al, 2015;Le Borgne et al, 2015).…”