Page 48 - Hybrid Enhanced Oil Recovery Using Smart Waterflooding

40 Hybrid Enhanced Oil Recovery using Smart Waterflooding

In a multiphase and multicomponent system, the At the equilibrium, the change in Gibbs free energy is
Gibbs free energy varies by species of the system. The zero and the equilibrium constant can be calculated
partial molar Gibbs energy of the species can be defined directly with the change in the standard Gibbs free
using the theory of ideal solutions as shown in Eq. (3.3) energy following Eq. (3.9).
and it is known as a chemical potential. o
DG
ln K eq ¼ (3.9)
vG o RT
m i ¼ ¼ m þ RT ln a i (3.3)
i
vn i T; p The equilibrium constant is highly sensitive to
the temperature and less sensitive to the pressure
where i indicates the species, m i is the chemical potential
o
of species i, n i is the number of moles of species, m is a (Appelo & Postma, 1999). The variations of equilib-
i
constant, which is the chemical potential of species i at a rium constant with temperature are usually calculated
standard condition, R is the ideal gas constant, and a i is with van’t Hoff equation as shown in Eq. (3.10)
the activity of the species i. The standard condition (Appelo & Postma, 1999).
refers the pure state of the species i at the temperature DH 1 1
of the solution and one atmospheric pressure. log K eq;T 1 log K eq;T 2 ¼ (3.10)
2:303R T 1 T 2
Following the definition, the equilibrium of reactions
in the multiphase and multicomponent system ideally where T 1 and T 2 are the temperatures, and K eq;T 1 and
can be determined from the tabulated values of K eq;T 2 are the equilibrium constants at temperatures
standard chemical potential. T 1 and T 2 .
Acitivity
Equilibrium constant
The fundamental description of geochemical equilib- In the calculation of the chemical potential and equilib-
rium follows the law of mass action. The law of mass rium constant, the acitivity of species is necessary to
action can be applicable to any type of reactions: be defined. For an aqueous solution, the acitivity of
mineral dissolution, the complex formation between ion can be thought as the effective concentration of
species, the dissolution of gases in water, etc. For the ion and defined as in Eq. (3.11).
generalized reaction corresponding to Eq. (3.4),an (3.11)
a i ¼ g i m i
equilibrium constant determines the distribution of
where g i is the activity coefficient of species i and m i is
species in the equilibrium state and it is a function of
the molality of species i.
activity of species and stoichiometric coefficient
In an infinite diluted solution, the activity coefficient
following Eq. (3.5).
of species approaches unity and the acitivity of a solute
aA þ bB4cC þ dD (3.4) becomes its concentration following the definition of
c d acitivity. Otherwise, Debye-Hückel theory calculates
½CŠ ½DŠ
K eq ¼ (3.5) the activity coefficient. In electrolyte solutions, the
a b
½AŠ ½BŠ
theory allows an activity coefficient for a single ion to
where K eq is the equilibrium constant and [i] is the be determined on the basis of the effect of ionic interac-
activity of species i. tions. In an aqueous solution, negative ions become
The equilibrium constant also can be derived from surrounded by the cloud of positive ions, and vice versa.
the expressions of chemical potential and Gibbs free The real system would have lower Gibbs free energy
energy. The difference between the Gibbs free energy than a hypothetical system where the ions are
of the products and that of the reactants, as shown in completely distributed randomly. In the real system,
Eq. (3.6), can be rewritten in terms of chemical poten- the electrostatic interaction generates an activity
tial of Eq. (3.3). It is rearranged with Eq. (3.7) and coefficient to be less than unity. In an ideal solution
equivalent to Eq. (3.8). or infinite dilution system, an activity coefficient would
be close to the unity. When the ions are point charges
DG ¼ G products G reactants (3.6)
and the interactions are entirely electrostatic, the
!
c
a a d distribution of ions around any particular ion follows
o
o
o
o
DG ¼ cm þ dm am bm þ RT ln C D (3.7) the Boltzmann distribution. Then, the activity coeffi-
A
D
C
B
a b
a a
A B
cient is given by original Debye-Hückel model as
o
DG ¼ DG þ RT ln K eq (3.8) defined in Eq. (3.12).
o
where DG is the change in the standard Gibbs free energy log g i ¼ Az 2 p ffiffi I (3.12)
o
of the reaction and m is the standard chemical potential. i