Page 46 - Reservoir Formation Damage

30 Reservoir Formation Damage

through the solid matrix according to Pick's second law over a short
distance near the surface of the solid exposed to aqueous solution, be-
cause the coefficient of water diffusion in solid is small. Thus, the water
absorption in the solid can be predicted by the one-dimensional transient-
state diffusion equation:

2 2
dc/dt - Dd c/dz ,0 < z < oo, t > 0, (2-2)

subject to the initial and boundary conditions given, respectively, by:
c = c 0 ,0<z<°°,f = 0 (2-3)

5 = -Ddc/dz = k( Cl - c), z=0, f >0 (2-4)

c = c Q,z-*°°,t>0 (2-5)

where c 0 and c are the initial and instantaneous water concentrations,
respectively, in the solid, Cj is the water concentration of the aqueous
solution, z is the distance from the pore surface, t is the actual contact
time, k is the film mass transfer coefficient, and D is the diffusivity co-
efficient in the solid matrix. Eq. 2-4 expresses that the water diffusion
into clay is hindered by the stagnant fluid film over the clay surface.
Thus, similar to Civan (1997), the analytical solution of Eqs. 2-2 through
2-5 can be used to express the cumulative amount of water diffusing into
the solid surface as given by Crank (1956):

2
S EE - f i-D^- dt = (h Dt}erfc
J dz dz h \ /
(2-6)

and the rate of water absorption is given by differentation of Eq. 2-6 as:

(2-7)

where h = kID.
Civan et al. (1989) have resorted to a simplified approach by assum-
ing that the film mass transfer coefficient k in Eq. 2-4 is sufficiently large