Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 24

chemical flooding as a method to improve the recovery rates of oil fields and to get
the oil trapped in the pores of rocks. In secondary oil recovery water is injected to
move out the oil (water flooding). The injection of surfactants (surfactant flooding) to
reduce the interfacial tension between the oil and the aqueous phases, thus allowing
the recovery of oil trapped in small pores, has also been suggested (Lake 1996 ). As
before, this type of compositional flow can be modelled using a mass balance equa-
tion for each component, Darcy's law for the phase velocities, and an energy balance
equation. Under the assumption of thermal equilibrium among the components and
fluid phases, this latter equation can be written in terms of the common temperature.
In contrast to the exposition of previous section, an overall mass conservation or
continuity equation for the pressure of the acqueous phase must be added, where the
effects of the slight compressibility of the solid and coexisting fluid phases can be
incorporated.
The mass conservation equation of component k can be expressed in terms of the
total concentration of this component in the multiphase fluid per unit pore volume.
This equation is similar in form to Eq. ( 87 )

∂ (ˆˁ c k )

+∇· (ʳ kc ˁ v c k ) =∇· D k ʱ ·∇ c k ʱ −∇· c k ʱ J + I k , (103)

ʱ

∂

t

ʱ ʱ

except that now

I k = I k ʱ = ˆ S r k ʱ + ( 1 − ˆ) r kR + L k , (104)

ʱ

ʱ ʱ

where r k ʱ and r kR are the chemical reaction rates of component k in the fluid phase

ʱ and rock phase, respectively, and L k is the injection/production rate of the same

component per unit volume, and

1

m

c k = −

1 dž

c i S ʱ c k ʱ +dž c k , (105)

ʱ

i =

where c k is the overall concentration of component k , including the adsorbed phases,
m is the number of volume-occupying components,

c k is the adsorbed concentration,
and the term between parentheses represents the reduction of the pore volume due
to adsorption effects. In order to ensure mass conservation, the following constraints
must hold for each phase

dž

m m m

r k ʱ = r kR = 0 , D k ʱ ·∇ c k ʱ = 0 , (106)

k = 1 k = 1 k = 1

where the summations are taken over the total number of adsorbed components.