Page 226 - Reservoir Formation Damage

206 Reservoir Formation Damage

u np=uK np/K (10-118)

Considering that the physical properties of the particles and the carrier
liquid are constant, the volumetric balance of particles in porous media
is given by:

9/9?((|>(j + e) + 9/9jt(a«) = 0 (10-119)

Substituting Eq. 10-100 into Eq. 10-119, and then rearranging, an
alternative convenient form of Eq. 10-119 can be obtained as:

($ 0 - e) 9a/9? + 3/9x(ou) + (1 - a) de/9r = 0 (10-120)

Following Gruesbeck and Collins (1982), Eq. 10-120 can be simplified
for cases where e and a are small compared to (|) 0 and unity, respectively,
and for constant injection rate, as:

(|) 0 — + u — + — = 0 (10-121)

The initial particle contents of the flowing solution and porous media
are assumed zero:

a = a 0 = 0, £ = 8 0, 0 < x < L, t = 0 (10-122)

where L is the length of porous medium. The particle content of the
suspension of particles injected into the porous media is prescribed as:

a = a in,x = Q,t>Q (10-123)

Alternatively, the pressures of the inlet and outlet ends of the porous
media instead of the flow rate can be specified. Then, the volumetric flux
can be estimated by the Darcy law:

(10-124)

Substituting Eq. 10-124 into the volumetric equation of continuity

(10-125)