Page 167 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

Formation Damage by Fines Migration: Mathematical and Laboratory Modeling, Field Cases 145

To integrate the right-hand side of Eq. (3.159), all terms must be
expressed in the variable T. As such, the equation for the parametric
curves (Eq. (3.158)) is solved to yield:

X 2 X 0 5 α T 2 T 0 Þ: (3.160)
ð
As all of these curves originate on the line X 5 T,so X 0 5 T 0 ,
yielding:
X 5 αT 1 T 0 1 2 αÞ: (3.161)
ð
Eq. (3.161) can be used in combination with Eq. (3.159) in the ODE
(Eq. (3.157)) to solve directly for the suspended concentration.
Behind the particle front, the PDE (Eq. (3.156)) is reduced to:

dC 1 @S a
52 ΛC 2 : (3.162)
dX α @T
which can be solved similarly using the expression for the change in the
attached concentration (Eq. (3.159)).
The final solution for the suspended concentration in all regions is
presented in Table 3.10. The solutions for the strained concentration and
impedance are derived using the solution for the suspended concentration
and by integrating Eqs. (3.95) and (3.98), respectively. The complete ana-
lytical expressions for these terms are given in Table 3.10.

3.6.3 Analysis of laboratory data and tuning of the model
coefficients
Applications of this model to practical problems follows a similar work-
flow to that outlined for the models given in previous sections. First and
foremost, laboratory testing should be completed, and the model should
be used to fit the model parameters. An example of this is presented
alongside a brief methodology to aid further work.
In typical experiments on fines migration, two experimental datasets
are used to treat the data; the pressure drop and the outlet particle con-
centration. For the case of model tuning, as before, these are transformed
for the convenience and accuracy of fitting.
Under the condition of a constant injection rate, the pressure drop
can be represented as the impedance:

Δp
J 5 : (3.163)
Δp 0