Page 97 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 79

3.2.1 Torque balance of forces acting on particle
Attached particles present in porous media are adhered either on the
grains comprising the rock matrix or an internal filter cake formed by
other attached particles. Particle detachment depends on the mechanical
equilibrium of forces acting on the particle. Drag, electrostatic, lift, and
gravitational are the most important forces acting on the attached parti-
cles. Drag and lift forces act to detach particles from the surface; whereas,
electrostatic and gravitational forces act to maintain the particle attached
to the surface, considering the particles sitting on grains in the porous
media (Muecke, 1979; Sarkar and Sharma, 1990).
Detachment of a particle attached to the rock can occur through one
of three primary mechanisms: horizontal translation along the rock sur-
face, translation vertically away from the surface, or rotation around the
rock asperity of neighboring particles. These equilibria can be evaluated
by summing either the acting forces tangential or normal to the surface,
or summing the torques generated by each acting force. By Newton’s sec-
ond law of motion, the balance among these forces is such that any of
these sums equals zero will give a mechanical equilibrium condition (i.e.,
the threshold between attachment and detachment). Analysis of these
forces for fine particle detachment in porous media has shown that parti-
cle rotation, or rolling, is significantly more likely than translation
(Sharma et al., 1992). This simplifies the quantitative prediction of
detachment by reducing the three equations outlined above to simply the
torque balance.
The torque balance of detaching and attaching forces acting on the
particle is expressed as Eq. (3.1) (Bradford et al., 2013). The normal
forces are responsible for the deformation of the particle in contact with
the grain surface. If the particle rotates around a point of contact with the
surface, the lever arm of the normal forces l n is assumed to be equal to
the radius of the contact area deformation by the normal force. Thus, l n
can be calculated from Hertz’s theory as (Derjaguin et al., 1975;
Schechter, 1992):

3
l 5 F e r s 4 ; (3.4)
n ; K 1 2 ν 2 1 2 ν 2
4K 1 2
3 1
E 1 E 2
where K is the composite Young’s modulus, ν is the Poisson’s ratio, E is
the Young’s modulus, and the subscripts 1 and 2 refer to the particle and
the grain, respectively. Once the normal lever arm is calculated, the drag