Page 316 - Fundamentals of Gas Shale Reservoirs

296 PERFORMANCE ANALYSIS OF UNCONVENTIONAL SHALE RESERVOIRS

where
2453 qB c 1 1
p t (13.63)
k 32 / t t t p p t 0 p t (13.70)
sp p w D w D
16
/
/
where G t D 1 t D 32 t 32 1 (13.71)
D
3
t
p p () p ( t ) 0 (13.64)
ws wf tt t
t D p (13.72)
13.6.1.2 Radial Flow Regime t p t p
We rewrite Equation 13.69 as follows:
k
p m log tm log t log . 3230 .869 s
r
r
cr 2 p m G t D (13.73)
t w
(13.65)
where
where
C t
p p pt() (13.66) m L p (13.74)
i w c 2
f
.
162 6 qB
m r (13.67) x
kh c 1 2 h f (13.75)
rw
k k k (13.68) f 2 E r f
r xy f
x f t
13.6.2 Mini-Frac Test h f q p (13.76)
f
In the absence of mini‐DST, we can rely on mini‐frac r f 2 hC L
pressure–time data to determine matrix permeability. We can
also analyze the mini‐frac pressure falloff, after injection has where
ceased, for the closure pressure p and leakoff coefficient C .
c
L
Closure pressure is the pressure at the inflection point on the x x thin fat fracture
,
f
pressure falloff curve. This pressure is equal also to minimum h f h thin long fracture
,
horizontal stress σ for mini‐frac in vertical wells. In this f f
h
case, the formation breakdown pressure is a function of σ r f r p ,eenny shaped fracture
f
h
and σ , given by Equation 13.81. For horizontal wells, the
H
formation breakdown pressure, given by Equations 13.82 After calculating C from Equation 13.74, we use the fol-
L
and 13.83, depends on the direction of the well with respect lowing leakoff equation to calculate the formation matrix
to the principal stresses in the horizontal plane. permeability.
Determination of the in situ matrix permeability is a most
crucial parameter for use in reservoir engineering analysis kc t (13.77)
.
and forecasting. In the absence of a viable pressure or rate C L 0 00118 p c ; C in ft / min
L
transient test, we can estimate matrix permeability from the
leakoff coefficient appearing in Equation 13.74. This p p p (13.78)
equation is the slope of fracture‐closure pressure falloff in c c
the mini‐frac test (Economides and Nolte, 1987). Specifically, Thus,
the analysis requires plotting Δp(Δt ) versus G(Δt ) to yield
D
D
a straight line. From the slope, we can calculate formation C 2
permeability using Equation 13.79. k m L 2 (13.79)
.
The mini‐frac equations, developed by Nolte in the publi- c 0 00118 p c
t
cation by Economides and Nolte (1987), are
Finally, we estimate the average fracture width during
C t fluid injection from the following equation:
p L p G t D (13.69)
c 2 f w f c f p hf p c (13.80)