Page 99 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 81

and Born Repulsive potentials (Derjaguin and Landau, 1941; Elimelech
et al., 2013; Gregory, 1981; Hogg et al., 1966; Israelachvili, 2011; Verwey
et al., 1999).

V 5 V LVW 1 V EDL 1 V BR ; (3.9)
where V is the total energy, V LVW is the London-Van der Waals potential,
V EDL is the Electrical Double Layer potential, and V BR is the Born
Repulsive potential.
The London-Van der Waals potential is the attraction between two
closely separated surfaces and will mostly be attractive for the interaction
between particles and grains in natural porous media. The force arises
from the spontaneous electrical and magnetic polarizations, giving a
fluctuating electromagnetic field within the medium. The classical
approach to evaluating the London-Van der Waals interaction between
two macroscopic bodies is obtained by pairwise summation of all the rele-
vant intermolecular interactions. Retarded London-Van der Waals energy
for sphere-plate interaction is given by the following formula (Gregory
(1981)):

A 132 r s 5:32h λ w
V LVW 52 1 2 ln 1 1 ; (3.10)
6h λ w 5:32h
where A 132 is the Hamaker constant, λ w is the characteristic wavelength
of interaction, and h is the surface-to-surface separation distance.
The expression for the Hamaker constant has the form (Israelachvili,
2011):
3 ε 1 2 ε 3 ε 2 2 ε 3
A 132 5 k B T
4 ε 1 1 ε 3 ε 2 1 ε 3
(3.11)
2 2 2 2
n 2 n n 2 n
3hv e 1 3 2 3
1 p ffiffiffi n o ;
1=2
8 2 n 1n 2 n 1 n 2 n 1n 2 1=2 1 n 1 n 2
2
2
2

2
1 3 2 3 1 3 2 3
where k B 5 1.381 3 10 223 J/K is Boltzmann constant, ε 1 , ε 2 ,and ε 3 are
static dielectric constants of the particle, grain, and fluid, respectively, n 1 , n 2 ,
and n 3 are the refractive indices of particle, grain, and fluid, respectively, and
15 21
v e 5 3.0 3 10 s is the constant value of absorption frequency.
The Born repulsive potential is a short-range repulsion that originates
from the strong repulsive forces between the atoms as their electron shells
interpenetrate each other. An expression for the Born repulsion potential