Page 306 - Fundamentals of Gas Shale Reservoirs

286 PERFORMANCE ANALYSIS OF UNCONVENTIONAL SHALE RESERVOIRS

q 1 13.3.2 Flow Rate Transient Analysis (RTA) and its
N() i (13.7) Relation to Rate Decline Analysis
D 1 b
i
Arps introduced Equation 13.1 to petroleum engineers and
economists based on empirical observations of declining
13.3.1.3 Harmonic Decline (b = 1)
well flow rates in petroleum reservoirs (Arps, 1944; Garb
and Smith, 1987). The most common decline rate equation is
qt() q 1 D t 1 (13.8) the hyperbolic equation with 0 < b < 1. Nevertheless, as
i i
pointed out earlier, in unconventional oil reservoirs, we com-
q monly observe the decline rates b to start as four, decline to
Nt() i ln 1 Dt (13.9)
i
D i two, and eventually approach zero. After a short period of
production, when bD t  1, Equation 13.1 takes the follow-
i
N() (13.10) ing form:
13.3.1.4 Beyond Hyperbolic (b ≥ 1) qt() qbD b / 1 t b / 1 (13.18)
i i
qt() q 1 bDt b / 1 (13.11) Thus,
i i

q 1 1 bD i 1 b / 1 b /
Nt() i 1 e Dt i 1 bDt 1 b 1 (13.12) t (13.19)
b 1 D i i qt() q i
N() (13.13)
13.3.2.1 Bilinear Flow Regime For b = 4.0, we obtain
the classical bilinear flow equation—a signature of a vertical
13.3.1.5 Nominal and Effective Decline Rate Terminology hydraulic fracture, which is observed in field about a‐third of
The initial decline rate D , known as the nominal decline
i the time (Lacayo and Lee, 2014; Patzek et al., 2013).
rate, will not remain constant unless the decline is
exponential. The definition of decline rate D(t) at time t is 14
/
1 4D i 4 t (13.20)
d ln q D qt() q i
Dt() i (13.14)
dt 1 bD t
i
13.3.2.2 Linear Flow Regime For b = 2.0, we obtain the
In economic analysis of projects, engineers use the effec- classical linear flow toward a vertical hydraulic fracture,
tive decline rate, defined below: which is observed in the majority of stimulated wells.

/
q q qt qt 1 2D i 12 t (13.21)
d 1 2 1 2 (13.15) qt() q
q qt i
1 1
Multiplying by Δp, and using formation volume factor to
The following equation relates the effective decline rate d convert surface flow rate to bottom‐hole flow rate, we obtain:
to the nominal decline rate D(t):

d 1 e D (13.16) p p 2 D i / 12 t (13.22)
()
qt B qB i
i
Finally, the following equation relates the cumulative
production N(t), initial rate q , and current flow rate q(t): Now, we compare the above equation with the RTA
i
equation for multistage hydraulic fractured well, given
below:
b 1
q q
Nt() i i 1 (13.17)
b ( ) 1 D qt() 12 /
2
.
i p 4 064 ( 2 / ) 1 t 141.2 s face
qB k hny c k h hf
We used Equations 13.6 and 13.12 separately to derive f ,eff hf hf t fm f ,eff
Equation 13.17. (13.23)