Page 203 - Applied Petroleum Geomechanics

198 Applied Petroleum Geomechanics

There are several other methods used to determine the closure pressure
from the pressure decline following shut-in. The G-time function
technique uses a plot of pressure as a function of pressure versus G-time
(Nelson et al., 2007). The G-time function was constructed to account
for the temporal and spatial variations of leak-off observed in the rock as the
created hydraulic fracture propagates and recedes. It is a dimensionless time
function relating shut-in time (t) to total pumping time (t p ) (at an assumed
constant rate). The dimensionless pumping time used in the G-function is
defined as follows (Barree et al., 2009):
Dt D ¼ðt t p Þ=t p (6.21)
where t is the elapsed total time from the start of fracture initiation; t p is the
total pumping time (elapsed time from fracture initiation to shut-in) in
consistent time units.
The G-function can be expressed in the following:
4
GðDt D Þ¼ ½ gðDt D Þ g 0 Š (6.22)
p
where g 0 is the dimensionless loss-volume function at shut-in (t ¼ t p or
Dt D ¼ 0). It is the computed value of g at shut-in (Eq. (6.23) or (6.24)).
For the assumption of low leak-off, or high efficiency where the open
fracture area after shut-in varies approximately linearly with time in low-
permeability formations, the intermediate function g(Dt D ) is computed in
the following equation (Barree et al., 2009):

4 1:5 1:5
gðDt D Þ¼ ð1 þ Dt D Þ Dt D (6.23)
3
For high leak-off, or low-efficiency fluids where the fracture surface
area varies with the square root of time after shut-in, the intermediate
function g(Dt D ) is as follows:
0:5
gðDt D Þ¼ð1 þ Dt D Þsin 1 ð1 þ Dt D Þ 0:5 þ Dt D (6.24)

On a plot of pressure and its derivatives versus G-function calculated
from Eqs. (6.22) to (6.23) (Barree et al., 2009), as shown in Fig. 6.6, the
fracture closure pressure can be easily determined.
There are two ways to record the pressures when performing the LOT or
FPIT (formation pressure integrity test). One is to record the pump pressure
from the surface; the other is to measure it through downhole pressure
gauges or tools, such as the pressure while drilling (PWD or APWD).