Page 141 - Formation Damage during Improved Oil Recovery Fundamentals and Applications

120 Thomas Russell et al.

determined for each sample which allowed calculating the accumulated
concentration in the outlet over time. Figs. 3.17 3.19 present the results
of three corefloods.
The model contains four parameters: the drift delay factor α, the fil-
tration coefficient λ, the concentration of released particles Δσ, and the
formation damage coefficient β. The parameters are determined by the
least squares method, i.e., by minimization of the total deviation between
the modeling and laboratory curves for accumulated particle concentra-
tion and impedance. The reflective trust region algorithm (Coleman and
Li, 1996) was applied to solve the optimization problem using the soft-
ware MATLAB (Mathworks, 2010).
The initial value for the drift delay factor α is determined from the
stabilization time for impedance, which is equal to 1/α. Accumulated
particle breakthrough concentration is calculated from the exact solution:
ð T
c acc 5αΔσ C 1;yÞdy
ð
0
0; T ,1
8
>
Δσ αΛ
>
>
>
> 2a exp 2bT 21ÞÞ21Þ; 1,T ,1=α; a5 ; b5
> ð ð ð
< φΛ ð 12αÞ
5 ;
! ! !
12α
>
>
>
> 21 ; T .1=α
> 2a exp 2b
>
: α
(3.112)
where a and b are fitting parameters.
Fitting the curve for experimental accumulative particle breakthrough
concentration with exponential formula (Eq. (3.112)) enables the calcula-
tion of the coefficients a and b. The initial filtration coefficient Λ from
Eq. (3.112) for known α is determined from the expression for b. The
initial value for the concentration of released particles Δσ is determined
for the known filtration coefficient Λ from the expression for a. The ini-
tial value for the formation damage coefficient β is determined by fitting
the impedance data by the analytical formula (Eq. (3.111)). Afterward,
the initial values are used in a minimization algorithm for model adjust-
ment. Table 3.7 presents the tuned values of the four parameters. The
2
coefficient of determination R in all cases exceeds 0.95, demonstrating a
high agreement between the experimental data and the theoretical model.
The obtained values of the model coefficients α, λ, Δσ, and β in