Page 452 - Petrophysics 2E

420 PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Dividing Equation 7.7 by Equation 7.8 and solving for the variable
pressure, p, yields:

X
P = (P2 - Pl), + PI (7.9)

This equation indicates that the pressure behavior of a linear flow system
during steady-state flow is a straight line as a function of distance.

LINEAR FLOW OF GAS

Consider the same linear flow system of Figure 7.1, except that
the flowing fluid is now natural gas. Because the gas expands as the
pressure declines, however, the pressure gradient increases toward the
downstream end and, consequently, the flow rate q is not constant, but
is a function of p. Assuming that Boyle’s law is valid (gas deviation factor
z = 1) and a constant mass flow rate, i.e., pq is constant, one can write:

Plql = p2q2 = Pq = PCl (7.10)

where subscripts denote point of measurement,q is the mean flow rate
and p is the mean pressure. Combining this relationship with Darcy’s
law, i.e., Equation 7.5, gives:

92P2 kAdP
q=-=-- (7.11)
P Pg dx
where pg is the viscosity of gas in CP units. Separating variables and
integrating between p1 and p2, and 0 and L gives:

q2P2~Ldx (7.12)
= -- kA SP2 PdP
Ps p1
or:

p2q2 = - kA (2) PFP2 (7.13)
CLgL
The mean flowrate expression that follows can be derived by combining
Equations 7.10 and 7.13:
(pl
q = (E) - p2)(p1 + p2)
(7.14a)
2