Page 318 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 253

At higher flow rates, in addition to the viscous force component represented by Darcy's
equation, there is also an inertial force acting due to convective accelerations of the
6
fluid particles in passing through the pore spaces . Under these circumstances the
appropriate flow equation is that of Forchheimer (1901), which is

dp µ u 2 (8.19)
dr = k u βρ+

In this equation the first term on the right hand side is the Darcy or viscous component
while the second is the non-Darcy component. In this latter term, β is the coefficient of
intertial resistance and, as the following dimensional analysis shows, has the
-1
dimension (length) .

2
dp ML 1 M 2 L
β
22 = [] ρ 3 u 2
dr TL L L T
β = L − 1

The non-Darcy component in equ. (8.19) is negligible at low flow velocities and is
generally omitted from liquid flow equations. For a given pressure drawdown, however,
the velocity of gas is at least an order of magnitude greater than for oil, due to the low
viscosity of the former, and the non-Darcy component is therefore always included in
equations describing the flow of a real gas through a porous medium.

Because of this it should be necessary to use the Forchheimer equation, rather than
that of Darcy, in deriving the basic radial differential equation for gas flow (refer
Chapter 5, sec. 2). Fortunately, even for gas, the non-Darcy component in equ. (8.19)
is significant only in the restricted region of high pressure drawdown, and flow velocity,
close to the wellbore.

Therefore, the non-Darcy flow is conventionally included in the flow equations as an
additional skin factor, that is, as a time independent perturbation affecting the solutions
of the basic differential equation in the same manner as the van Everdingen skin
(Chapter 4, sec. 7). Forchheimer's equation was originally derived for the flow of fluids
in pipes where at high velocity there is a distinct transition from laminar to turbulent
flow. In fluid flow in a porous medium, however, for most practical cases in reservoir
engineering, the macroscopic flow is always laminar according to the definitions of
classical fluid dynamics. What is referred to as the non-Darcy component does not
correspond with classical ideas of turbulent flow but, as stated earlier, is due to the
accelerations and decelerations of the fluid particles in passing through the pore
spaces. Nevertheless, Forchheimer's equation can be used to describe the additional
pressure drop due to this phenomenon, by integrating the second term on the right
hand side of equ. (8.19), as follows.

e r q 2
∆ p non Darcy = βρ dr
w r 2 π rh

or expressed as a drop in the real gas pseudo pressure, using equ. (8.8)