Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics - Selected Topics of Computational and Experimental Fluid Mechanics - page 23

To formally close this system of equations, we need some supplementary con-
straints. One class of supplementary constraints consists of thermodynamic relation-
ships for the phase densities, saturations, and concentrations as functions of the phase
pressure and fluid mixture composition (Allen 1985 ), which can be written as

c k ʱ = c k ʱ (ˁ ʱ , c 1 , c 2 ,..., c n ) , (98)

ˁ ʱ = ˁ ʱ ( p

ʱ ,

T , c 1 ʱ , c 2 ʱ ,..., c n ʱ ) , (99)

S

ʱ =

S

ʱ (

p

ʱ ,

c 1 , c 2 ,..., c n ) . (100)

The actual form of these constraints may imply simultaneous sets of nonlinear alge-
braic equations giving the phase densities, concentrations, and saturations implicitly.
In particular, this occurs when we use fugacity functions for the components in the
fluid phases in conjunction with an equation of state to solve for local thermodynamic
equilibria. In compositional flows, mass transfer between adjacent fluid phases is
characterized by the variation of mass distribution of each species in these phases.
In general, the coexisting fluid phases are assumed to be in equilibrium, which is
a reasonable physical condition because the interchange of mass between adjacent
phases occurs much faster than the fluid flow in the porous medium. Therefore, the
distribution of component k into two adjacent phases, say , is subject to the

condition of stable thermodynamic equilibrium, which results from minimizing the
Gibbs free energy of the compositional system. This condition can be expressed by
demanding that the fugacities of component k in the two phases be equal:

ʱ and ʲ

F k ʱ (

p ʱ , T , c 1 ʱ , c 2 ʱ ,..., c n ʱ ) = F k ʲ ( p ʲ , T , c 1 ʲ , c 2 ʲ ,..., c n ʲ ). (101)

The other class of supplementary relations includes constitutive relations for the
capillary pressures and relative permeabilities as functions of the phase saturations.
Equation ( 87 ), Darcy's law for the phase velocities v

, and Eq. ( 96 ) provide n + P + 1

ʱ

differential equations for n + ( n + 3 ) P + 1 independent variables for S , p , v , c k ,

ʱ ʱ ʱ

c k ʱ P remaining relations needed to solve the problem are given

by the saturation constraint ( 6 ), the

, and T .The ( n + 2 )

( P − 1 ) capillary pressures ( 21 ), the n constraints

( 79 ), the n ( P − 1 ) equal-fugacity constraints ( 101 ), and the P constraints

n

c k ʱ = 1 , (102)

k = 1

for the mass fraction balance.

5 Chemical Flooding Compositional Flow in Porous Media

In the previous section we have described the equations governing the flow of compo-
sitional fluids in a porous medium in the absence of sources and/or sinks of chemical
components by external means. It is a common practice in the oil industry to use