Transport in Porous Media

, Volume 95, Issue 3, pp 669–696 | Cite as

Multiscale Modeling of Colloid and Fluid Dynamics in Porous Media Including an Evolving Microstructure

  • Nadja Ray
  • Tycho van Noorden
  • Florian Frank
  • Peter Knabner
Article

Abstract

We consider colloidal dynamics and single-phase fluid flow within a saturated porous medium in two space dimensions. A new approach in modeling pore clogging and porosity changes on the macroscopic scale is presented. Starting from the pore scale, transport of colloids is modeled by the Nernst–Planck equations. Here, interaction with the porous matrix due to (non-)DLVO forces is included as an additional transport mechanism. Fluid flow is described by incompressible Stokes equations with interaction energy as forcing term. Attachment and detachment processes are modeled by a surface reaction rate. The evolution of the underlying microstructure is captured by a level set function. The crucial point in completing this model is to set up appropriate boundary conditions on the evolving solid–liquid interface. Their derivation is based on mass conservation. As a result of an averaging procedure by periodic homogenization in a level set framework, on the macroscale we obtain Darcy’s law and a modified averaged convection–diffusion equation with effective coefficients due to the evolving microstructure. These equations are supplemented by microscopic cell problems. Time- and space-dependent averaged coefficient functions explicitly contain information of the underlying geometry and also information of the interaction potential. The theoretical results are complemented by numerical computations of the averaged coefficients and simulations of a heterogeneous multiscale scenario. Here, we consider a radially symmetric setting, i.e., in particular we assume a locally periodic geometry consisting of circular grains. We focus on the interplay between attachment and detachment reaction, colloidal interaction forces, and the evolving microstructure. Our model contributes to the understanding of the effects and processes leading to porosity changes and pore clogging from a theoretical point of view.

Keywords

Periodic homogenization Evolving microstructure Level set function Colloidal transport Pore scale modeling Particle-surface interaction Pore clogging Porosity changes Multiscale coefficients Mixed finite elements 

List of Symbols

\({\cdot^{n,T}}\)

Physical quantity at t = tn on triangle \({T\in\mathcal T_H}\)

\({\cdot_0}\)

Physical quantity of order \({\epsilon^0}\)

\({\cdot_\epsilon}\)

Physical quantity on pore scale

\({\bar{\cdot}}\)

y-averaged physical quantity

A

Weighted porosity (-)

c

Concentration of colloidal particles (kg/m3)

D

Diffusivity of colloidal particles in the fluid (m2/s)

\({\bar{D}}\)

Effective diffusion tensor (m2/s)

δij

Kronecker delta

ej

jth unit vector (−)

\({\epsilon}\)

Scale parameter, ratio of pore size to domain size (−)

η

Kinematic viscosity of the fluid (m2/s)

f

Surface reaction rate (kg/m2/s)

F

Effective production/consumption (kg/m3/s)

Γ

Solid–liquid interface (m2)

Γ0

Solid–liquid interface within unit cell (m2)

\({\Gamma_\epsilon}\)

Interior boundary of \({\Omega_\epsilon ({m^2})}\)

\({\Gamma^{ij}_{\epsilon,s}}\)

Solid–liquid interface within scaled unit cell (m2)

h

Characteristic mesh size of the fine scale grid (m)

H

Characteristic mesh size of the coarse scale grid (m)

j

Mass flux (kg/m2/s)

kT

Boltzmann constant times absolute temperature (kg m2/s2)

K

Effective permeability tensor (m2)

\({\hat{K}}\)

Effective tensor (m2)

L

Level set function (−)

mcol

Molecular mass of colloidal particle (kg)

n

Time level index

ν

Exterior unit normal (−)

Ω0

Global domain (m3)

Ω0

Exterior boundary (m2)

Ωε

Periodic, perforated domain (m3)

p

Pressure (kg/m/s2)

\({\mathbb{P}_k}\)

Space of polynomials of order at most k

Φ

Total (interaction) energy (kg m2/s2)

R

Grain radius (m)

\({\mathbb{R}\mathbb{T}_0}\)

Raviart–Thomas approximation space of lowest order

ρ

Density of the solid phase including attached colloidal particles (kg/m3)

ρl

Density of the fluid (kg/m3)

t

Time (s)

tn

Time level after n steps (s)

tend

End time (s)

T

Triangle

\({\mathcal{T}}\)

Set of triangles or triangulation (\({\mathcal{T}_H}\) coarse-, \({\mathcal{T}_H}\) fine scale grid)

τ

Unit tangent (−)

τn

Time step size from tn-1 to tn (s)

u

Transformed concentration of colloidal particles (kg/m3)

v

Fluid velocity (m/s)

V

Effective transport velocity (m/s)

vν

Normal velocity of the solid-liquid interface (m/s)

x

Global space variable (m)

y

Microscopic space variable (m)

Y

Unit cell (m3)

\({Y^{ij}_\epsilon}\)

Scaled and shifted unit cell (m3)

Yl,0

Liquid phase within unit cell (m3)

|Yl,0|/|Y|

Porosity of the porous medium (−)

\({Y^{ij}_{{\rm l},\epsilon}}\)

Liquid phase within scaled and shifted unit cell (m3)

Ys

Solid phase within unit cell (m3)

\({Y^{ij}_{{\rm s},\epsilon}}\)

Solid phase within scaled and shifted unit cell (m3)

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Nadja Ray
    • 1
  • Tycho van Noorden
    • 1
  • Florian Frank
    • 1
  • Peter Knabner
    • 1
  1. 1.Department of MathematicsFriedrich-Alexander University of Erlangen-NurembergErlangenGermany

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