Page 49 - Hybrid Enhanced Oil Recovery Using Smart Waterflooding

CHAPTER 3 Modeling of Low-Salinity and Smart Waterflood 41

where A is the temperature-dependent constant, z i is with ionic strength. Above the ionic strength of
the charge of the ion, and I is the ionic strength of the 0.5 molal concentration, it predicts the coefficient to
solution. The ionic strength takes account of electro- increase. The B-dot model is known as to calculate the

þ
static effectiveness of polyvalent ions. It is defined as a accurate activities of Na and Cl up to several molal
function of molality of ions and charge of ion as shown concentrations of ionic strength. It can reasonably esti-
in Eq. (3.13). mate the activities of other species when ionic strength
is up to 0.3e1 molal concentrations (Bethke, 1996).
1 X 2
I ¼ m i z i (3.13)
2
Basic Reactions of Geochemistry
The original Debye-Hückel model is reasonable up
to ionic strength with 1 10 3 molal concentration. Aqueous reactions
In aqueous solutions, ions can be attached to other ions
At the higher ionic strength, the original Debye-
or complexes and form other complexes. Because these
Hückel becomes less accurate to predict the activity
reactions in aqueous solutions are incorporated in one
coefficient. The extended Debye-Hückel model of
phase, they are homogeneous reactions. The formation
Eq. (3.14) is applicable to the diluted electrolyte
of aqueous complexes is relatively fast reaction and
solution systems, which have ionic strength with less
follows the equilibria. The aqueous equilibrium
than 0.1.
reaction between the components in an aqueous phase
p ffiffi
Az 2 i I follows the law of mass action. For an example of the
log g i ¼ p ffiffi (3.14)
1 þ B _ a I aqueous reaction corresponding to Eq. (3.17), the dis-
tribution of the species is obtained by the equilibrium
where B is the temperature-dependent constant and _ a i is
constant of Eq. (3.18). The equilibrium constant is
the empirical ion-size parameter measuring the effective
diameter of the hydrated ion. also termed as a stability constant.
In addition, the Davies equation of Eq. (3.15) also Ca 2þ þ SO 2 4CaSO 4 (3.17)
calculates the activity coefficient. It is a variant of the 4
extended Debye-Hückel model. It is capable to be ½CaSO 4 Š ¼ 10 2:5
K eq ¼ (3.18)
used up to ionic strength up with 0.5 molal ½Ca 2þ Š SO 2
4
concentration.
p ffiffi I
log g i ¼ Az 2 i p ffiffi 0:3I (3.15) Mineral reactions
The mineral reactions of dissolution and precipitation
1 þ I
are heterogeneous reactions because the species
Another modified version of the extended Debye- involved in these reactions are in the different phases,
Hückel model is the B-dot model as defined in i.e., solid and aqueous phases. In the reaction of
Eq. (3.16). It is parameterized from 0 to 300 C for the calcium carbonate corresponding to Eq. (3.19), the

solutions of up to 3 molal ionic strength (Bethke, 1996).
law of mass action describes the equilibrium state of
p ffiffi
Az 2 i I mineral reactions. In the expression of equilibrium
log g i ¼ p ffiffi þ _ BI (3.16) constant of mineral reactions, the activity of a pure solid
1 þ B _ a I
is unity and the equilibrium constant is termed as the
where _ B is the temperature-dependent constant. solubility product of Eq. (3.20).
The previous Debye-Hückel, Davies, and B-dot
models depend on ionic strength and charge of ions. CaCO 3 4Ca 2þ þ CO 2 (3.19)
3
Debye-Hückel and B-dot models, additionally, are

K sp ¼ Ca 2þ CO 2 ¼ 10 8:48 at 25 C (3.20)
affected by the ion size and temperature-dependent 3
constants. The activity coefficient calculated by where the K sp is the solubility product.
Debye-Hückel model becomes unity when ionic Generally, the mineral reactions are slow kinetic
strength decreases to zero. In the high ionic strength reactions, and the achievement of equilibrium of
condition, the real activity coefficient increases, but mineral reactions requires relatively longer time than
calculated coefficient calculated by Debye-Hückel that of aqueous reactions. A saturation index is defined
model still decreases. Therefore, above the 0.1 molal to determine whether the mineral reactions are under
concentration of ionic strength, Debye-Hückel has less equilibrium or not. The saturation index is defined as
accuracy and other models are recommended to be the logarithm of a saturation state of Eq. (3.21). The
used. The Davies model gives more reasonable activity saturation state of Eq. (3.22) is defined as the ratio of
coefficient in the range of 0.3e0.5 molal concentrations ion activity product (IAP) to solubility product.