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Upscaling of Stochastic Micro Model for Suspension Transport in Porous Media


Micro scale population balance equations of suspension transport in porous media with several particle capture mechanisms are derived, taking into account the particle capture by accessible pores, that were cut off the flux due to pore plugging. The main purpose of the article is to prove that the micro scale equations allow for exact upscaling (averaging) in case of filtration of mono dispersed suspensions. The averaged upper scale equations generalise the classical deep bed filtration model and its latter modifications.

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C :

Suspended particle concentration distribution by sizes, L−4

C t :

Concentration distribution for suspended particles trapped in cut-off accessible pores, L−4

c :

Total suspended particle concentration, L−3

f :

Fractional flow function, dimensionless

H :

Pore concentration distribution, L−3 or L−4

h :

Total pore concentration (density), L−2 or L−3

j :

Jamming ratio, dimensionless

j 0 :

Maximum value of jamming ratio for non-zero accessibility


Absolute permeability, L2

k a (σ):

Permeability of accessible part of the porous medium, L2

k na (σ):

Permeability of inaccessible porous medium, L2

k c (σ):

The total of pore accessible conductivities weighted with capture probability, L2

k 1 :

Conductivity of a single pore, L4

l :

Characteristic microscopic length, L

L :

Length of the core, L

P :

Pressure, M/LT

p(r s /r p ):

Overall capture probability, dimensionless

p a (r s /r p ):

Attachment capture probability, dimensionless

q(r p ):

Total flow rate in a single pore, L−3T−1

q a (r p , r s ):

Flow rate through accessible cross section of a single pore, L−3T−1

r :

Size of a particle or of a pore, L

s 1 :

Pore cross-section, L2

t :

Time, T

T :

Fast dimensionless time

U :

Total velocity of the flux, LT−1

U a :

Velocity of the accessible flux, LT−1

U na :

Velocity of the inaccessible flux, LT−1

v :

Concentration front velocity in 1d filtration flow, LT−1

x :

Coordinate, L

X :

Dimensionless coordinate

y :

Independent variable in system of differential-integral equations

α :

Critical porosity fraction, dimensionless

\({\varepsilon}\) :

Small parameter that equals to the ratio between the injected concentration and the critical porosity fraction


Filtration coefficient, L−1


Dimensionless filtration coefficient

μ :

Dynamic viscosity, ML−1T−1

η :

Collision efficiency

ν(r s /r p ):

Single pore flux reduction factor, dimensionless

σ :

Volumetric concentration of captured particles, L−3


Size distribution of the captured particle concentration, L−4

\({\underline \Sigma}\) :

Distribution of the captured particle concentration over the pore and particle radii, L−5

τ :

Slow dimensionless time

\({\phi (x, t)}\) :

Porosity, dimensionless

\({\phi _a (r_s, x, t)}\) :

Accessible porosity for a particle of the size r s , dimensionless

\({\phi _{na} (r_s, x, t)}\) :

Inaccessible porosity for a particle with size r s , dimensionless

χ(r s , r p ):

Accessible fraction of a single pore cross-section, dimensionless

a :


na :


s :

Suspended (solid) particle

p :


v :



Initial condition


Single pore (cross section, conductivity)


Lower percolation threshold corresponding to connectivity of accessible pores


Boundary condition


Upper percolation threshold corresponding to connectivity of inaccessible pores


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Bedrikovetsky, P. Upscaling of Stochastic Micro Model for Suspension Transport in Porous Media. Transp Porous Med 75, 335–369 (2008).

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  • Colloid
  • Suspension
  • Porous media
  • Transport
  • Averaging
  • Upscaling
  • Size distribution
  • Retention
  • Size exclusion
  • Straining
  • Accessibility
  • Stochastic model