Heterogeneous Multi-Rate mass transfer models in OpenFOAM

We implement the Multi-Rate Mass Transfer (MRMT) model for mobile-immobile transport in porous media within the open-source finite volume library \textsc{OpenFOAM}\reg \citep{Foundation2014}. Unlike other codes available in the literature [Geiger, S., Dentz, M., Neuweiler, I., SPE Reservoir Characterisation and Simulation Conference and Exhibition (2011); Silva, O., Carrera, J., Dentz, M., Kumar, S., Alcolea, A., Willmann, M., Hydrology and Earth System Sciences 13, (2009)], we propose an implementation that can be applied to complex three-dimensional geometries and highly heterogeneous fields, where the parameters of the MRMT can arbitrarily vary in space. Furthermore, being built over the widely diffused OpenFOAM\reg library, it can be easily extended and included in other models, and run in parallel. We briefly describe the structure of the library that includes the formulation of the MRMT based on the works of [Haggerty, R., Gorelick, S.M., Water Resources Research 31, (1995)] and [F. Municchi and M. Icardi Phys. Rev. Research 2, 013041, (2020)]. The implementation is verified against benchmark solutions and tested on two- and three-dimensional random permeability fields. The role of various physical and numerical parameters, including the transfer rates, the heterogeneities, and the number of terms in the MRMT expansions is investigated. Finally, we illustrate the significant role played by heterogeneity in the mass transfer when permeability and porosity are represented using Gaussian random fields.

[1]  Allen F. Moench,et al.  Double‐Porosity Models for a Fissured Groundwater Reservoir With Fracture Skin , 1984 .

[2]  X. Sanchez‐Vila,et al.  On matrix diffusion: formulations, solution methods and qualitative effects , 1998 .

[3]  Ivan C. Christov,et al.  A two-fluid model for numerical simulation of shear-dominated suspension flows , 2018, International Journal of Multiphase Flow.

[4]  Brian Berkowitz,et al.  Theory of anomalous chemical transport in random fracture networks , 1998 .

[5]  Daniel M. Tartakovsky,et al.  Perspective on theories of non-Fickian transport in heterogeneous media , 2009 .

[6]  Philippe Gouze,et al.  Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion: NON-FICKIAN DISPERSION AND HETEROGENEOUS DIFFUSION , 2008 .

[7]  M. Icardi,et al.  Mechanisms of dispersion in a porous medium , 2017, Journal of Fluid Mechanics.

[8]  Brian Berkowitz,et al.  Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport , 2003 .

[9]  Matteo Icardi,et al.  Computational analysis of transport in three-dimensional heterogeneous materials , 2019, Comput. Vis. Sci..

[10]  W. Jury,et al.  LINEAR TRANSPORT MODELS FOR ADSORBING SOLUTES , 1993 .

[11]  Rodney O Fox,et al.  CFD simulation of shear-induced aggregation and breakage in turbulent Taylor-Couette flow. , 2005, Journal of colloid and interface science.

[12]  Ismael Herrera,et al.  Integrodifferential equations for systems of leaky aquifers and applications: 1. The nature of approximate theories , 1973 .

[13]  Anett Georgi,et al.  Colloidal activated carbon for in-situ groundwater remediation--Transport characteristics and adsorption of organic compounds in water-saturated sediment columns. , 2015, Journal of contaminant hydrology.

[14]  Stefan Radl,et al.  Spatial Filtering for Scale Bridging and Its Application to Transport in Dense Particle Beds , 2018 .

[15]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[16]  John M. Zachara,et al.  Scale‐dependent desorption of uranium from contaminated subsurface sediments , 2008 .

[17]  M. Bayani Cardenas,et al.  Mass Transfer Between Recirculation and Main Flow Zones: Is Physically Based Parameterization Possible? , 2019, Water Resources Research.

[18]  P. Domenico,et al.  Physical and chemical hydrogeology , 1990 .

[19]  E. A. Sudicky,et al.  Contaminant transport in fractured porous media: Analytical solution for a single fracture , 1981 .

[20]  Pierre M. Adler,et al.  Taylor dispersion in porous media: analysis by multiple scale expansions , 1995 .

[21]  Mikko Voutilainen,et al.  Solutions to and Validation of Matrix-Diffusion Models , 2011 .

[22]  Rajandrea Sethi,et al.  Recirculation zones induce non-Fickian transport in three-dimensional periodic porous media. , 2016, Physical review. E.

[23]  Stefan Radl,et al.  Momentum, heat and mass transfer simulations of bounded dense mono-dispersed gas-particle systems , 2018 .

[24]  Yongchen Song,et al.  Study of Selected Factors Affecting Hydrate-Based Carbon Dioxide Separation from Simulated Fuel Gas in Porous Media , 2013 .

[25]  M. Icardi,et al.  Macroscopic models for filtration and heterogeneous reactions in porous media , 2020, Advances in Water Resources.

[26]  Federico Municchi,et al.  Generalized multirate models for conjugate transfer in heterogeneous materials , 2019, Physical Review Research.

[27]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[28]  Philippe Gouze,et al.  Effective non-local reaction kinetics for transport in physically and chemically heterogeneous media. , 2011, Journal of contaminant hydrology.

[29]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[30]  M. Dentz,et al.  Modeling non‐Fickian transport in geological formations as a continuous time random walk , 2006 .

[31]  Sebastian Geiger,et al.  A Novel Multi-rate Dual-porosity Model for Improved Simulation of Fractured and Multi-porosity Reservoirs , 2011 .

[32]  Sean Andrew McKenna,et al.  On the late‐time behavior of tracer test breakthrough curves , 2000 .

[33]  Jesús Carrera,et al.  A general real-time formulation for multi-rate mass transfer problems , 2009 .

[34]  Brian Berkowitz,et al.  Exploring the nature of non-Fickian transport in laboratory experiments , 2009 .

[35]  Schalk Cloete,et al.  Development and verification of anisotropic drag closures for filtered Two Fluid Models , 2018, Chemical Engineering Science.

[36]  Qinjun Kang,et al.  Upscaling pore‐scale reactive transport equations using a multiscale continuum formulation , 2007 .

[37]  Michel Quintard,et al.  A dual-porosity theory for solute transport in biofilm-coated porous media , 2013 .

[38]  Emil O. Frind,et al.  Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures , 1982 .

[39]  M. Dentz,et al.  Transport behavior of a passive solute in continuous time random walks and multirate mass transfer , 2003 .

[40]  Stefan Radl,et al.  Consistent closures for Euler-Lagrange models of bi-disperse gas-particle suspensions derived from particle-resolved direct numerical simulations , 2017 .

[41]  Sumit Mukhopadhyay,et al.  Analytical solutions of tracer transport in fractured rock associated with precipitation-dissolution reactions , 2011 .

[42]  Peter Englezos,et al.  Capture of carbon dioxide from flue or fuel gas mixtures by clathrate crystallization in a silica gel column , 2010 .

[43]  David A. Benson,et al.  A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations , 2009 .

[44]  Brian Berkowitz,et al.  Numerical simulation of non‐Fickian transport in geological formations with multiple‐scale heterogeneities , 2002 .

[45]  Auli Niemi,et al.  Geological Storage of CO2 in Deep Saline Formations , 2017 .

[46]  M. Darwish,et al.  The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM® and Matlab , 2015 .

[47]  Vladimir Cvetkovic,et al.  Modeling of Solute Transport in a 3D Rough-Walled Fracture–Matrix System , 2017, Transport in Porous Media.

[48]  Schalk Cloete,et al.  Heat transfer to a gas from densely packed beds of cylindrical particles , 2017 .

[49]  Philipp Neumann,et al.  Semantic interoperability and characterization of data provenance in computational molecular engineering , 2019, Journal of Chemical & Engineering Data.

[50]  R. Yates,et al.  Integrodifferential equations for systems of leaky aquifers and applications 3. A numerical method of unlimited applicability , 1977 .

[51]  Seyyed Mohammad Mousavi,et al.  Modeling and CFD simulation of flow behavior and dispersivity through randomly packed bed reactors , 2008 .

[52]  Brian Berkowitz,et al.  Non‐Fickian transport and multiple‐rate mass transfer in porous media , 2008 .

[53]  Rajandrea Sethi,et al.  Fine and ultrafine particle deposition in packed-bed catalytic reactors , 2019, Chemical Engineering Science.

[54]  Sebastian Geiger,et al.  A non-local two-phase flow model for immiscible displacement in highly heterogeneous porous media and its parametrization , 2013 .

[55]  Daniele Marchisio,et al.  Validation of a novel open-source work-flow for the simulation of packed-bed reactors , 2015 .

[56]  Turid Rustad,et al.  Superchilling of food: A review , 2011 .

[57]  Igor M. Sokolov,et al.  An improved scheme for a Robin boundary condition in discrete-time random walk algorithms , 2018, J. Comput. Phys..

[58]  Wei-xian Zhang,et al.  Iron nanoparticles for environmental clean-up: recent developments and future outlook. , 2013, Environmental science. Processes & impacts.

[59]  Nan Zhang,et al.  CFD analysis of fluid flow and particle-to-fluid heat transfer in packed bed with radial layered configuration , 2019, Chemical Engineering Science.

[60]  Jesús Carrera,et al.  Multicomponent reactive transport in multicontinuum media , 2009 .

[61]  Ivars Neretnieks,et al.  Diffusion in the rock matrix: An important factor in radionuclide retardation? , 1980 .

[62]  Rajandrea Sethi,et al.  Microscale colloidal transport simulations for groundwater remediation , 2016 .

[63]  C. Poinssot,et al.  Radionuclide behaviour in the natural environment: an overview , 2012 .

[64]  S. Gorelick,et al.  Multiple‐Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore‐Scale Heterogeneity , 1995 .

[65]  Rae Mackay,et al.  Contaminant transport in heterogeneous porous media: a case study. 1. Site characterisation and deterministic modelling , 1996 .