Page 354 - Reservoir Formation Damage

334 Reservoir Formation Damage

and the conservation of the ion exchange sites by

«„ = (13-17)
jt=i

where N is the total number of chemically reacting species. ^ ;(^a) z and
Sf(E a] represent the cations attached to the active exchange sites.
Sears and Langmuir (1982) report that ion exchange and adsorption
reactions in soils typically require a time of seconds to days to attain
equilibrium. Therefore, Jennings and Kirkner (1984) describe these
reactions by rate equations for full kinetic modeling. Applying their
approach to Eq. 13-16 according to Chang and Civan (1997) yields the
following kinetic expression for the rates of consumption of Sj (E a ) and
production of Sf(E a] , respectively: J

as,
iy „ _ --T
' dt
(13-18)

where (|) is the porosity and kj and k' r j denote the rate coefficients for
the forward and reverse reactions, respectively. If /" is the reaction rate
th
for the exchange of the i cation present in aqueous solution with the
j th cation attached to the a th site on the mineral surface, and I r is the
rate of the reactions of the /"'cation of the aqueous solution other than
adsorption, the transport equation for the /""cation present in aqueous
solution is given by (Lichtner, 1985):

M

o=l j=l r=l (13-19)

where e a denotes the volume fraction of the aqueous phase in the bulk
of porous formation and c i denotes the concentration of the /"" cation in the
aqueous solution, expressed in moles per unit volume of the aqueous phase.
The balance of the /""cation adsorbed on the a' h site of the mineral
surface is given by (Litchner, 1985):

a? (13-20)