Page 213 - Reservoir Formation Damage

194 Reservoir Formation Damage

dm p/dt + k em p = [k rk eALm pg /q]exp(-k et) (10-49)

The solution of Eq. 10-49, subject to the initial condition m p=m* p
(previously deposited particles), is obtained by the integration factor
method as:

(10-50)

in which

C u=k rk eALm p/q (10-51)

Then, the area occupied by the remaining particles is given by
Eq. 10-29 as:

A (10-52)
P =

and the area open for flow is given by Eq. 10-28 as:
(10-53)

Eliminating A fo between Eqs. 10-48 and 53, substituting Eqs. 10-47 and
52 for A* and A p, and then applying Eq. 10-10 for A f and A* f yields
the following diagnostic equation:

(K/K*)' = (10-54)

in which

(10-55)

and

C 8 = C n/m* = k rk eALm pJ(qm* p) (10-56)

Normally, m pg =m* p. Wojtanowicz et al. (1987) simplified Eq. 10-54 by
substituting C 7 = 0 when the mass of the particles initially available on
the pore surface is small compared to the mass of particles deposited later
(i.e., mJsO).
If only pore sweeping occurs, then k r « k e. Thus, substitute k r = 0
in Eq. 10-56 to obtain C 8 =0 and Eq. 10-54 becomes: