Page 493 - Petrophysics

TURBULENT FLOW OF GAS 46 I

Using the same units, the Reynolds number of a porous rock is equal to:

Pkqm
Repr = (7.118)
6.33 x 10lOApg
where the permeability k is expressed in mD and the viscosity in cP. The
porous media Reynolds number, Rep,, is unitless.
Figure 7.10 shows three regions [40]:

(a) for Re,, < 0.08, the curve is a straight line of slope equal to - 1;
(laminar flow)
(b) a transition region for 0.08 < Rep, < 8; and
(c) a horizontal line for Re,, > 8. (turbulent flow)

It is important to note that the unit-slope line and the horizontal line
intercept at Re,, = 1. The existence of a straight line for small Reynolds
numbers indicates that the pressure drop (p1 - p2) for a given porous
medium is directly proportional to the flow rate (q,), and that the laminar
flow regime is dominant. Darcy’s law is applicable during this portion
of the curve only because the magnitude of the group of terms (ppv2)
in Equation 7.108 is too small to be detected in experimental data [42].
As the flow rate increases and Repr becomes larger, the turbulent flow
regime becomes increasingly dominant. The horizontal portion of the
fpr-vs.-Repr curve corresponds to the so-called non-Darcy flow, or fully
turbulent flow regime. Katz and Lee and Firoozabadi and Katz suggested
abandoning the concept of Darcy and non-Darcy flow [42, 441. They
recommended the use of viscous Darcy flow” to describe the flow regime
observed at low flow rates, and for high velocity flow to use “quadratic
Darcy flow.” Viscous Darcy flow theoretically occurs only when the flow
rate is infinitely small [44].

EXAMPLE

Solve the previous example using the friction factor plot for porous
and permeable rock (Figure 7.10).

SOLUTION

The variables have to be converted to the units used to derive
Equations 7.117 and 7.118, which were used to generate the log-log plot
of fpr-vs.-Rer. Thus, from the previous example,
p = 3 x 107ft-l.
A = 3.5/30.482 = 3.767 xlOU3 ft2.
p = (4.64 x 10-3)(30.483/453.6) = 0.29 lb/ft3.