Page 113 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 95

Integration of Eq. (3.34) at the core inlet using boundary condition
(Eq. (3.39)) allows calculating the strained concentration at the inlet:
x 5 0:σ s 5 0: (3.40)

Before attempting to solve the above system of equations, first the fol-
lowing dimensionless parameters are introduced to produce dimensionless
relationships:

Ð t
x Uy ðÞdy c σ s σ a pk 0
X 5 ;T 5 0 ;C 5 ;S s 5 ;S a 5 ;Λ5λL;P 5 ;
L φL Δσ φΔσ φΔσ UμL
(3.41)
where L is the core length, φ is the rock porosity, Δσ 5 σ cr (U 0 ) σ cr (U 1 )
is the total detached concentration when the fluid velocity is increased
from U 0 to U 1 , and μ is the fluid viscosity.
Substituting these parameters into the system of Eqs. (3.33 3.36)
yields, for the mass balance:
@ @C
ð C 1 S s 1 S a Þ 1 α 5 0: (3.42)
@T @X
The kinetics of particle straining is now given by:

@S s
5 ΛαC: (3.43)
@T
The attached concentration is similarly expressed as:
S a 5 S cr UðÞ: (3.44)

And the dimensionless modified Darcy’s law becomes:
1 @P
1 52 : (3.45)
1 1 βφΔσS s @X

In dimensionless coordinates, the initial and boundary conditions
become, respectively:

T 5 0:C 5 1; S s 5 0; S a 5 S cr U 1 Þ; (3.46)
ð
X 5 0:C 5 0; S s 5 0: (3.47)
In the following section, the solution for the dimensionless
Eqs. (3.42 3.45) subject to initial and boundary conditions (Eqs. (3.46
and 3.47)) is presented.