Page 236 - Fundamentals of Reservoir Engineering

OILWELL TESTING 174

in which t D is the dimensionless flowing time prior to closure and is therefore a constant
while ∆t D is the dimensionless closed in time corresponding to the pressure p ws, the
latter two being variables which can be determined by interpretation of the pressure
chart retrieved after the survey.

For small values of ∆t, p ws is a linear function of In (t+∆t)/∆t, which can be verified by
adding and subtracting ½ In (t D +∆t D ) to the right hand side of equ. (7.32) and
evaluating p D (∆t D) for small ∆t using equ. (7.23). Thus,
2kh 4t
∆
π
(p − p ) = p (t + ∆ t ) − 1 2 In D ± 1 2 ln (t + ∆ t )
ws
i
D
D
D
D
D
qµ γ
which can alternatively be expressed as
2kh t +∆ t 4t +∆ t D )
π
( D
(p − p ) = 1 2 ln + p (t + ∆ t ) − 1 2 ln (7.35)
D
i
D
D
ws
qµ t ∆ γ
in which dimensionless time has been replaced by real time in the ratio t+∆t/∆t. Again,
for small values of the closed-in time ∆t
ln (t +∆ t ) ≈ ln (t )
D
D
D
and
p D (t +∆ t ) ≈ p (t )
D
D
D
D
and equ. (7.35) can be reduced to
2kh t +∆ t 4t
π
(p − p ) = 1 2 ln + p (t ) − 1 2 ln D (7.36)
D
ws
D
i
qµ t ∆ γ
Since the dimensionless flowing time t D is a constant then so too are the last two terms
on the right-hand side of equ. (7.36) and therefore, for small values of ∆t a plot of the
observed values of p ws versus In (t+∆t)/∆t should be linear with slope m = q / 4 kh ,
µ
π
from which the value of the permeability can be determined. This particular
4
presentation of pressure buildup data is known as a Horner plot and is illustrated in
fig. 7.9.
Equation (7.36) is the equation describing the early linear buildup and due to the
manner of derivation is only valid for small values of ∆t. Nevertheless, having obtained
such a straight line it is perfectly valid to extrapolate the line to large values of ∆t in
which case equ. (7.36) can be replaced by

2kh t +∆ t 4t
π
t
(p − p ws(LIN) ) = ½ ln + p D () − ½ln D (7.37)
D
i
qµ t ∆ γ
in which p ws, the actual pressure in equ. (7.36), is now replaced by p ws(LIN ) which is
simply the pressure for any value of ∆t on the extrapolated linear trend and while the
latter may be hypothetical it is, as will be shown, mathematically very useful. The
equation can be used in two ways. Firstly, drawing a straight line through the early