Page 481 - Petrophysics

RADIAL LAMINAR FLOW OF GAS 449

the continuity equation:

ia aP
--(prv) = 0- (7.95)
r ar at
and the equation of state for real gas:

MP
p= -- (7.96)
RT z
Assuming cg is approximately equal to Up, the diffusivity equation
describing the real gas flow in cylindrical porous and permeable rock is:

(7.97)

where pg is estimated at the average reservoir pressure p. This differential
equation has essentially the same form as the diffusivity Equation 7.54,
which was derived for incompressible fluids, except that the dependent
variable, p, has been replaced by p2. This similarity suggests that the
solutions to Equation 7.97 also will be of the same form as those for
Equation 7.54.
Real gas-flow equations differ from incompressible fluid flow equations
because the gas-flow rate, q, varies with pressure due to the compressi-
bility of gas. To make the wellbore flow rate a constant, let:

where q and q,, are expressed, respectively, in bbl/D and MSCF/D, and
the gas formation volume factor is defined by Equation 7.90 in bbl/SCF.
Thus:
z
q = (5.04TQ - (7.99)
P
the volumetric flow rate, in bbl/D, at any radius in the reservoir, according
to Darcy's law, where the permeability is expressed in mD, is:

(7.100)

If Equations 7.99 and 7.100 are substituted in Equation 7.65 and the
variables separated, one obtains:

(1.404 x pdp
1-(1-f)- -= (7.101)
r2] Tq,, PSZ
r,'