Page 346 - Fundamentals of Reservoir Engineering

REAL GAS FLOW: GAS WELL TESTING 281

The theoretical equation describing the pseudo pressure drop during the second flow
period can be derived from the basic test equation (8.39) as

kh
′
′ −
=
−
(m(p ) m(p )) Q (m (t + ∆ t D + t ) m ( t ∆ D + t ))
D
1
D
D
D
D
i
wf
1422T 1 max max (8.58)
′ +
+ Qm (t ) Q S′ 2
2
D
2
D
where t′ is the time measured from the start of the second flow period at rate Q 2,
(fig. 8.14). This equation is analysed for transient conditions during the second flow
period, that is, for small values of t´. In this case the expression
′
′
∆
−
Q (m(t +∆ t D max + t ) m( t D max + t ))
D
D
D
D
D
1
1
in equ. (8.58) can be regarded as being constant. If both t 1 and ∆t max are short so that
both the m D functions can be evaluated under transient conditions the above statement
is quite correct and, in fact, the difference between the m D functions is both small and
constant. For a very long initial flow period, corresponding to a routine well survey
rather than an initial test, the difference between the m D functions can only be regarded
as constant on the grounds that t is small, which is always the case since the wellbore
pressure response at rate Q 2 is only being analysed during the brief, initial, transient
flow period. Therefore, equ. (8.58) implies that a plot of m(p wf) versus log t will be
linear, for transient flow, with slope
1637Q T
m = 2
kh
which leads to a re-determination of k. The skin factor can be evaluated by expressing
equ. (8.58) as
kh kh (m(p ) m(p )) Q m (t ) Q S′ (8.59)
′
−
i
wf
1422T (m(p ) m(p )) = 1422T i − ws + 2 D D ′ + 2 2
in which p′ is the hypothetical static pressure that would be obtained had the buildup
ws
been continued for a time ∆t max + t′ . The value of p′ will therefore increase as t´
ws
increases. Equation (8.59) can then be solved to give S′ as
2
(m(p ) − m(p ) k
′
−
−
S′ = S DQ = 1.151 ws 1 hr wf 1 hr − log 2 + 3.23 (8.60)
+
2
2
( c) r
m φµ iw
mp
in which both ( wf ) and ( ) are evaluated for t′ = 1 hour. The latter can be
mp′
ws
obtained by extrapolation of the final buildup trend for one hour after the buildup has
ceased. However, this correction is seldom applied and usually ( ) 1hr is set equal
mp′
ws
−
to ( ws ) , evaluated for the final closed in pressure.
mp
The following exercise illustrates the method of buildup analysis for a well test in a new
reservoir in which p i is the initial reservoir pressure.