Page 273 - Fundamentals of Enhanced Oil and Gas Recovery

Enhanced Gas Recovery Techniques From Coalbed Methane Reservoirs 261

n
Ω is the volume of REV, and @Ω is the surface enclosing Ω. ~ is a unit normal vector
perpendicular to @Ω:
These two equations should be written for each component. The unknowns are
,b w ,b g , S g , S w φ, j i ;~ g , and ~ w . In modeling fluid flow in CBM, it is usually
u
u
y i g , y i w
assumed that (1) the diffusion in cleats is negligible, compared to bulk flow, (2) gas is
not dissolvable in water phase, (3) no water vapor exists in gas phase, e.g., for a four-
5 0.
component y H 2 O g 5 y CH 4w 5 y N 2w 5 y CO 2w
Therefore, the equations for cleats will be simplified as follows:
@
ð ð ð ð

y i g g ~ g ~ndA 1 y i g g φS g dV 2 ~ q dV 2 ~ j i ~ndA 5 0 (8.29)
u
b
b
ig
@Ω Ω @t Ω @Ω
@
ð ð ð
b w ~ w ~ndAÞ1 ð b w φS w ÞdV 2 ~ q dV 5 0 (8.30)
u
ð
@Ω Ω @t Ω H 2 O w
N p
X
S l 5 1 (8.31)
l51
N c 21
X
5 1 (8.32)
y i g
i51
where N c is the number of components and N p is the number of phases.
Eq. (8.29) should be written for all the components except water. For matrix, it is
typically assumed that (1) there is no bulk flow, (2) fluid is in adsorbed state, and (3)
there is no water in matrix. The following equation represents molar balance in coal
matrix. It also should be written for all the components except water.
ð ð
@C i
dV 1 ~ j i ~ndA 5 0 (8.33)
Ω @t @Ω
where C i is the molar density of component i in the matrix.
Totally, there are 2N c 1 1 equations and 3N c 1 4 unknowns, e.g., for the four-
, b g , b w , ~ g , ~ w , S g , S w , φ,
u
u
component case, unknowns are y CH 4g , y N 2g , y CO 2g
.
C CH 4 ; C CO 2 ; C N 2 ; j CH 4 , j CO 2 ; and j N 4
A few constitutive equations are required to have the same number of equations
and variables.

8.5.2.1.2 Darcy’s Law
u
Using Darcy’s law, flow rate ~ l will be related to the phase pressure p l
Kk rl
~ l 52 rp l 2 ρ g (8.34)
u
μ l
l