Page 50 - Hybrid Enhanced Oil Recovery Using Smart Waterflooding

42 Hybrid Enhanced Oil Recovery using Smart Waterflooding

Eq. (3.27). The equilibrium constant of ion exchange
SI ¼ log U (3.21)
is termed with an exchange constant.
IAP
þ
U ¼ (3.22) A-X þ B 4B-X þ A þ (3.26)
K sp
þ
½A-XŠ½B Š
where SI indicates the saturation index, U is the satura- K A=B ¼ (3.27)
þ
tion state, and IAP is the ion activity product. The IAP is ½A Š½B-XŠ
defined as the product of the activities of species in the where X represents the clay surface, A and B are cations,
water, and the IAP of calcium carbonate is described in A-X and B-X are the adhered cations on the clay surface,
Eq. (3.23). and K A=B is the exchange constant.
Because the activity coefficients of A-X and B-X are

IAP ¼ Ca 2þ CO 2 (3.23)
3 applicable to the none of Debye-Hückel, Davies, and
The saturation index is a useful tool to state the B-dot models, a selectivity coefficient is normally
saturation condition of mineral reaction. When the used rather than the exchange constant. In the defini-
saturation index is equal to zero, there is an equilibrium tion of selectivity coefficient, the activities of A-X and
between the mineral and the solution. For the satura- B-X are replaced by equivalent fractions of A-X and
tion index is less than zero, there is a subsaturation state B-X. The equivalent fractions are defined as the
resulting in mineral dissolution. For the saturation fractions of cation exchange capacity (CEC) occupied
index is higher than zero, there is a supersaturation state by the particular ion. Introducing the concepts of
resulting in mineral precipitation. selectivity coefficient, equivalent fraction, and activity
The dissolution or precipitation of minerals is the of ions, the equilibrium constant is rewritten as in
result of multiple processes: (1) the transport of solutes Eq. (3.28). The equivalent fraction is also defined as
between solutions and the mineral surface, (2) the in Eq. (3.29).
adsorption and desorption of solutes at the surface,
þ
þ
zðA-XÞmðB ÞgðB Þ
0
(3) the hydration and dehydration of ions, and (4) K A=B ¼ þ þ (3.28)
zðB-XÞmðA ÞgðA Þ
surface migration. Some processes are faster than the
meq i-X
other processes. The slowest process dominates the zði-XÞ¼ meq i-X ¼ P (3.29)
overall rate of the mineral reaction. The slow mineral CEC meq i-X
A;B;.
reactions are considered as kinetically controlled reac-
0
tions. There are a number of theories to describe the where K A=B indicates the selectivity coefficient, z(A-X)
reaction kinetics. One of the successful theories is the and z(B-X) are the equivalent fractions of A-X and
general rate law of transition state theory (TST) B-X, meq i-X is the milliequivalent of the exchangeable
proposed by Eyring (1935). The reaction rate of the species i, and CEC is the cation exchange capacity.
mineral reaction is described in Eq. (3.24). There are three common conventions (Gaines-
Thomas, Vanselow, and Gapon) to describe the selec-

E a U tivity coefficient of ion exchange. For the homovalent
r ¼ b Ak a exp a H þ 1 (3.24)
RT K sp exchange reaction, there is no impact what convention
where r is the reaction rate, A is the reactive surface area is applied. For the heterovalent exchange reaction, the
b
type of convention results in the different results of
of the mineral, k a is the reaction rate constant, and E a is
ion exchange. For the exchange between sodium and
the activation energy of the reaction. The reaction rate
calcium cations, the Gaines-Thomas convention
depends on the pH. In addition, the reaction rate
describes the ion-exchange reaction and selectivity
constant is sensitive to the temperature following
coefficient as Eqs. (3.30) and (3.31).
Eq. (3.25).
1 1

þ
E a 1 1 Na þ Ca-X 2 4Na-X þ Ca 2þ (3.30)
k a ¼ k 0 exp (3.25) 2 2
R T T 0

2þ 0:5
zðNa-XÞ Ca
where k 0 is the reaction rate constant at reference K Ca=Na ¼ 0:5 (3.31)
0
þ
temperature and T 0 is the reference temperature. zðCa-X 2 Þ ½Na Š
In the descriptions of Eqs. (3.30) and (3.31), Vanse-
low convention uses the molar fraction instead of the
Ion exchange
Clay mineral exhibits an ion-exchange behavior with equivalent fraction. The Gapon convention describes
cations. The equilibrium of ion exchange between the activities of adhered ions with a fraction of the
two cations corresponding to Eq. (3.26) follows number of exchange sites as shown in Eq. (3.32).