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REAL GAS FLOW: GAS WELL TESTING 240
1 1 ∂Ζ 1
c = − ≈ (1.31)
p Ζ∂ p p
which again is highly pressure dependent and automatically violates the above
condition that cp << 1.
These problems, although severe, are not insurmountable. Nevertheless, it was not
until the mid sixties that reliable analytical solutions of equ. (5.1 ) were developed. Two
separate solution methods were published almost simultaneously in 1966; these are
2
- the Russell, Goodrich et. al., p formulation 1
2
- the Al-Hussainy, Ramey and Crawford, real gas pseudo pressure formulation .
Both techniques will be described in this chapter although the latter, for reasons
explained in the text, is preferred. To illustrate the difference in approach the radial
semi-steady state inflow equation, equivalent to equ. (6.12), will be derived in secs. 8.3
and 8.4, using both methods. Having thus established an analogy between liquid and
real gas flow equations, the constant terminal rate solution for gas is stated by
inference and its application described in detail in the remainder of the chapter.
Because of the great disparity between gas rates measured at the surface (Q) and in
the reservoir (q) it has become conventional to express gas flow equations using
surface rates, at standard conditions, with all parameters expressed in field units. This
practice will be adhered to in this chapter, using the following units
Q - Mscf/d µ - cp(= µ g)
(at 60°F and 14.7 psia) Z - dimensionless
t - hours p - psia
k - mD T - °R (460+°F)
h, r - ft
In all equations µ and Z are evaluated at some defined reservoir condition. The basic
derivations of the flow equations in sections 8.3 through 8.8 will still be performed in
Darcy units, with conversion to field units being made upon achieving the desired form
of equation.
8.3 THE RUSSELL, GOODRICH, et. al. SOLUTION TECHNIQUE
The authors approached the problem by making the initial assumption that it was
possible to linearize equ. (5.1) for real gas flow in precisely the same manner as for
liquid flow, described in Chapter 5, sec. 4. Admittedly, this approach should yield
inaccurate results. However, Russell and Goodrich also designed a numerical model of
a single well draining a radial volume element which itself was subdivided into finite
grid blocks as shown in fig. 8.1.