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Modified Formulations of Particle Deposition and Removal Kinetics in Saturated Porous Media

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Abstract

Important revisions on the particulate deposition and removal rate equations are offered for flow of particulate suspensions through saturated porous media by distinguishing between the factors effecting the mechanisms of various particle deposition and removal processes, and their rate coefficients and parameters. Formulation considers particle transport by convection and dispersion toward open and partially jammed pores, flux of particles above a critical minimum in suspension, unblocked particles available for removal at the pore surface, shear-force below critical for deposition of suspended particles, shear-force above critical for entrainment of unblocked surface particles, active ion concentration below critical for spontaneous particle release from pore surface, critical pore-throat-to-particle diameter ratio for pore-throat jamming, deformation and pore-sealing effects of elastic particles, internal filter cake formation by particle accumulation behind pore throats, normal-force above critical for dislocation of deposits from pore throats, and thermal-, concentration-, stress-, and mobility-shock phenomena. Inherent limitations of previous outstanding rate equations are alleviated by means of theoretical reasoning and/or experimental observations.

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Abbreviations

a :

Half-axis of ellipsoidal particle (m)

A :

Surface area of bulk porous media normal to flow direction (m\(^{2}\))

\(A_{0}\) :

Cross-sectional area of a single pore throat (m\(^{2}\))

\(A_{1}\), \(A_{2}\), and \(A_{3}\) :

Empirically determined fitting parameters

\(A_\mathrm{s}\) :

Specific pore-surface area of the hydraulic flow paths (m\(^{2}\))

\(A_\mathrm{t}\) :

Total cross-sectional area of pore throats available (m\(^{2}\))

\(A_\mathrm{pt}\) :

Areas of the plugged pore throats (m\(^{2}\))

\(A_\mathrm{nt}\) :

Areas of nonplugged pore throats (m\(^{2}\))

b :

Half-axis of ellipsoidal particle (m)

\(B_{1}, B_{2}\), and \(B_{3}\) :

Empirically determined fitting parameters

c :

Fractal coefficient

C :

Fractal coefficient

\(C_\mathrm{cr}\) :

Critical active ion concentration of the flowing solution, normality

\(C_\mathrm{s}\) :

Effective salinity of the cation of the flowing solution causing particle detachment, normality

\(C_{2}\) and \(C_{3}\) :

Empirical parameters

d :

Fractal dimension

D :

Fractal dimension

\(D_{\mathrm{b}}\) :

Bulk coefficient of particle dispersion (m\(^{2}\)/s)

\(D_\mathrm{p}\) :

Mean diameters of the particles (m)

\(D_\mathrm{t}\) :

Mean diameters of the pore throats (m)

\(D_\phi \) :

Mean pore size (m)

E :

Activation energy (J/kmol)

\(f_\mathrm{i}({x_\mathrm{i}})\) :

Functions of properly selected variables \(x_\mathrm{i}\)

\(f_\sigma \left( {J_\sigma -J_{\sigma _\mathrm{cr} } } \right) \) :

Function of \(\left( {J_\sigma -J_{\sigma _\mathrm{cr} } } \right) \)

\(f_\tau \left( {F_\mathrm{cr} -F} \right) \) :

Function of \(\left( {F_\mathrm{cr} -F} \right) \)

\(f_F \left( {F-F_\mathrm{cr} } \right) \) :

Function of \(\left( {F-F_\mathrm{cr} } \right) \)

\(f_C \left( {C_\mathrm{cr}-C_\mathrm{s}}\right) \) :

Function of \(\left( {C_\mathrm{cr}-C_\mathrm{s}}\right) \)

F :

Tangential-force or shear-force acting over the pore-surface deposits (N)

\(F_\mathrm{cr}\) :

Critical shear-force required for entrainment of the surface deposits (N)

G :

Ratio of the fluid viscous forces to the particles elastic forces, dimensionless

H :

Hamaker coefficient (J)

j :

Dispersive mass flux of the particles of the suspension of particles flowing through porous media (kg/m\(^{2}\)/s)

\(J_\sigma \) :

Volumetric flux of the migrating particles (m\(^{3}\)/m\(^{2}\)/s)

\(J_{\sigma _\mathrm{cr}}\) :

Critical minimum volumetric flux (m\(^{3}\)/m\(^{2}\)/s)

\(J_w\) :

Mass flux of the migrating particles of suspension flowing through porous media (kg/m\(^{2}\)/s)

k :

Rate coefficient

\(k_{0}\) :

Coefficient

\(k_1 ,k_{10} ,k_{11}\) :

Pore-surface particle deposition rate coefficients

\(k_2 , k_{20} ,k_{21}\) :

Pore-surface particle removal rate coefficients

\(k_3 , k_{30} ,k_{31}\) :

Pore-throat particle deposition rate coefficients

\(k_{4},k_{40} ,k_{41} ,k_{42} ,k_{43}\) :

Pore-throat unplugging rate coefficients

\(k'\) :

Empirically determined parameter

\(k_\mathrm{nt}\) :

Rate coefficient of open pore-throat particle deposition by jamming

\(k_\mathrm{pt}\) :

Rate coefficient of pore-space filling behind jammed or plugged pore throats

\(k_{\mathrm{vo}}\) :

Empirical parameter

\(k_\tau \) :

Empirical parameter

K :

Permeability of porous media (m\(^{2}\))

\(K_\mathrm{pt}\) :

Permeability of the pore-throat clogging deposits (m\(^{2}\))

l :

Distance between the particles (m)

L :

Porous core sample length (m)

m :

Mass of particles retained per unit bulk volume of porous media (kg/m\(^{3}\))

\(\dot{m}\) :

Net rate of particle loss by flowing suspension by deposition in porous media by various processes (kg/m\(^{3}\)/s)

\(m_{11}, m_{12,} m_{13, }m_{21} ,m_{22} ,m_{23} ,m_{24}\) :

Empirical exponents

\(m_{31}, m_{32,}m_{41}, m_{42}, m_{43}, m_{44}\) :

Empirical parameters

M :

Number of factors

\(M_\mathrm{cr}\) :

Critical mobility number

\(n'\) :

Empirically determined parameter

\(n_\mathrm{t}\) :

Total number of pore throats

\(n_{\mathrm{pt}}\) :

Number of plugged pore throats

\(n_\mathrm{nt}\) :

Number of nonplugged pore throats

N :

Number of variables

\(N_\mathrm{D}\) :

Ellipsoidal particle deformation number, dimensionless

p :

Fluid pressure (Pa)

q :

Volumetric flow rate of the flowing fluid through porous media (m\(^{3}\)/s)

\(q_\mathrm{nt}\) :

Volumetric flow rate of the flowing fluid through the unplugged pore throats (m\(^{3}\)/s)

\(q_\mathrm{pt}\) :

Volumetric flow rate of the flowing fluid through the plugged pore throats (m\(^{3}\)/s)

R :

Mass fraction of the particles of a suspension retained by a sieve after filtration

\(R_\mathrm{g}\) :

Universal gas constant [8.314 kJ/(kmol K)]

\({Re}_\mathrm{cr}\) :

Critical Reynolds number

\({Re}_\mathrm{p}\) :

Reynolds number for particles migrating in suspension of particles flowing through porous media, dimensionless

u :

Volumetric fluid flux (m\(^{3}\)/m\(^{2}\)/s)

v :

Interstitial fluid velocity through porous media (m/s)

\(v_\mathrm{p}\) :

Particle velocity (m/s)

x :

Distance in the Cartesian coordinate (m)

\(x_\mathrm{i}: i= 1,2,\ldots ,N\) :

Properly selected variables

t :

Time (s)

T :

Temperature (K)

v :

Interstitial fluid velocity through porous media (m/s)

\(v_\mathrm{cr}\) :

Critical interstitial fluid shear velocity (m/s)

w :

Mass fraction of particles in the flowing suspension (kg particles/kg suspension)

\(W_\phi \) :

Width of slits (m)

\(\alpha _\mathrm{b}\) :

Longitudinal dispersivity (m)

\(\beta \) :

Pore-throat-to-particle diameter ratio

\(\beta _\mathrm{cr}\) :

Critical pore-throat-to-particle diameter ratio causing pore-throat blocking

\(\beta _\infty \) :

Empirically determined fitting parameter

\(\rho \) :

Density of the flowing suspension of particles (kg/m\(^{3}\))

\(\rho _\mathrm{p}\) :

Particle material mass density (kg/m\(^{3}\))

\(\phi \) :

Instantaneous porosity

\(\phi _o\) :

Initial porosity of porous media

\(\varepsilon \) :

Volume occupied by the deposited particles per unit bulk volume

\(\varepsilon _{\max }\) :

Maximum capacity of the pore-throat particle deposition when all pore throats are plugged by particles

\(\varepsilon _o\) :

Average volume of a single pore-throat clogging deposit per unit bulk volume of porous media

\(\lambda \) :

Empirical decay constant

\(\gamma _\mathrm{s}\) :

Specific weight of particles (N/m\(^{3}\))

\(\dot{\gamma }\) :

Fluid shear rate (1/s)

\(\mu \) :

Fluid viscosity (Pa s)

\(\sigma \) :

Volume fraction of particles in the flowing suspension (m\(^{3}\)/m\(^{3}\))

\(\sigma _\mathrm{cr}\) :

Critical minimum volumetric concentration of particles in the flowing suspension (m\(^{3}\)/m\(^{3}\))

\(\sigma _\mathrm{in} \hbox { and }\sigma _\mathrm{out}\) :

Inlet and outlet fluid particle volume concentrations (m\(^{3}\)/m\(^{3}\))

\(\tau \) :

Shear-stress applied over the surface of deposited particles (Pa)

\(\tau _\mathrm{cr}\) :

Critical shear-stress (Pa)

\(\tau _\phi \) :

Tortuosity of porous media, dimensionless

\(\eta \) :

Fraction of unblocked deposited particles available over the pore surface

\(\eta _\mathrm{s}\) :

Particle shear modulus (Pa)

\(\omega \) :

Ellipsoidal particle aspect ratio

o:

Initial or coefficient

cr:

Critical value

i:

Index

t, pt, and nt:

All, plugged, and nonplugged pore throats

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  1. Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma, 100 East Boyd, SEC Room 1210, Norman, OK, 73019-1003, USA

    Faruk Civan

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Civan, F. Modified Formulations of Particle Deposition and Removal Kinetics in Saturated Porous Media. Transp Porous Med 111, 381–410 (2016). https://doi.org/10.1007/s11242-015-0600-z

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