Page 293 - Reservoir Formation Damage

Cake Filtration: Mechanism, Parameters and Modeling 273

Eliminating q between Eqs. 12-29 and 12-35 yields another expression as:

A 6 + D
c B
t = A ln C
5 5 A 5 0 + D C (12-36)
.B C

in which, usually, 8 0 =0at? = 0 (i.e., no initial filter cake).
Eq. 12-36 is different from Eq. 7-96 of Collins (1961) because Collins
did not consider the filter cake erosion. Therefore, Collins' equation
applies for static filtration. To obtain Collins' result, k e = 0 or 5 = 0 must
be substituted in Eq. 12-11. Thus, eliminating q between Eqs. 12-29 and
12-11, and then integrating, yields the following equation for the filter
cake thickness (Civan, 1998a):

(l/2)8 (12-37)

which results in Eqs. 7-96 of Collins (1961) by invoking Eqs. 12-18 for
=0, Eqs. 12-30, 12-31, and 12-12 and expressing the mass of sus-
$ f
pended particles per unit volume of the carrier fluid in terms of the
volume fraction, o p , of the particles in the slurry according to:

(12-38)

Civan (1998a) derived the expressions for the filtrate flow rate and the
cumulative filtrate volume by integrating Eq. 12-33 for 5 = 0 and apply-
ing Eq. 12-27, respectively, as:

(12-39)

and

(12-40)

Eq. 12-39 expresses that the filtrate rate declines by time due to static
filter cake build-up. Donaldson and Chernoglazov (1987) used an empirical
decay function: