Rate Transient Analysis
1-4: TRADITIONAL
DECLINE ANALYSIS
1. Traditional (Arps) Decline Curves
23-32: RADIAL TYPE
CURVES
23. Calculations for Oil
(Agarwal-Gardner Type Curves)
3. Exponential Decline
EXPONENTIAL DECLINE:
•
•
•
•
23-24: RADIAL FLOW MODEL: TYPE CURVE
ANALYSIS
Decline rate is constant.
All radial flow type curves are based on the same
reservoir model:
Log flow rate vs. time is a straight line.
Flow rate vs. cumulative production is a straight line.
Well in centre of cylindrical homogeneous reservoir.
Decline rate is not constant (D=Kqb).
•
•
•
•
Straight line plots are NOT practical and b is determined
by nonlinear curve fit.
•
The shapes are different because of
different plotting formats.
•
Each format represents a different “look” at the data
and emphasizes different aspects.
Provides minimum EUR (Expected Ultimate Recovery).
HYPERBOLIC DECLINE:
•
•
b value
0
0.1-0.4
0.4-0.5
0.5
0.5-1.0
2. Decline Rate Definitions
Reservoir Drive Mechanism
Single phase liquid (oil above bubble point)
Single phase gas at high pressure
Solution gas drive
Single phase gas
Effective edge water drive
Commingled layered reservoirs
Information content of all type curves
(Figures 25-32) is the same.
rwa
26. Blasingame: Integral-Derivative
1. qDd vs. tDd (Figure 25).
2. Rate integral (qDdi) vs. tDd (has the same shape
as qDd).
3. Rate integral-derivative (qDdid) vs. tDd (Figure 26).
Decline rate is directly proportional to flow rate (b=1).
Log flow rate vs. cumulative production is a straight line.
•
Boundary-dominated flow only.
Constant operating conditions.
In general: qDd
qD bDpss , tDd
2
t
b Dpss DA
• bDpss is a constant for a particular well / reservoir
Developed using empirical relationships.
configuration.
Quick and simple to determine EUR.
EUR depends on operating conditions.
Does NOT use pressure data.
b depends on drive mechanism.
27. Agarwal-Gardner: Rate (Normalized)
5-10: FETKOVICH
ANALYSIS
5. Analytical: Constant Flowing Pressure
6. Analytical: Constant Flowing Pressure
27-28: AGARWAL-GARDNER
•
Notes:
1. Pressure derivative is defined as pDd
d ( pD )
d (ln t DA )
2. Inverse of pressure derivative is usually too noisy
and inverse of pressure integral-derivative is
used instead.
boundary-dominated stems.
• qDd and tDd definitions are convenient for
production data analysis.
•
•
28. Agarwal-Gardner: Integral-Derivative
• qD and tDA definitions are similar to well testing.
• Normalized rate (q/ p or q/ pp) is plotted.
• Three sets of type curves:
1. qD vs. tDA (Figure 27).
2. Inverse of pressure derivative (1 / pDd) vs. tDA
(not shown).
3. Inverse of pressure integral-derivative (1 / pDid)
vs. tDA (Figure 28).
• qD and tD definitions are similar to well test.
• Convenient for transient flow.
• Results in single transient stem but multiple
7. Empirical: Arps Depletion Stems
re
Skin factor represented by rwa.
• qDd and tDd definitions are similar to Fetkovich.
• Normalized rate (q/ p or q/ pp) is plotted.
• Three sets of type curves:
SUMMARY:
•
•
•
•
•
•
•
No flow outer boundary.
25-26: BLASINGAME
25. Blasingame: Rate (Normalized)
4. Harmonic Decline
HARMONIC DECLINE:
•
•
24. Calculations for Gas
(Agarwal-Gardner Type Curves)
29-30: NORMALIZED PRESSURE
INTEGRAL (NPI)
29. NPI: Pressure (Normalized)
Convenient for boundary-dominated flow.
Results in single boundary-dominated stem but
multiple transient stems.
30. NPI: Integral-Derivative
• pD and tDA definitions are similar to well testing.
• Normalized Pressure ( p/q or pp /q) is plotted
8. Empirical: Arps-Fetkovich Depletion Stems
rather than normalized rate (q/ p or q/ pp).
•
Three sets of type curves:
1. pD vs. tDA (Figure 29).
2. Pressure integral (pDi) vs. tDA (has the same
shape as pD).
3. Pressure integral-derivative (pDid) vs. tDA (Figure
30).
Replot on Log-Log Scale
31-32: TRANSIENT-DOMINATED DATA
31. Rate (Normalized)
10. Fetkovich/Cumulative Type Curves
•
9. Fetkovich Type Curves
• qD and tD definitions are similar to well testing.
• Normalized rate (q/ p or q/ pp) is plotted.
• Three sets of type curves:
SUMMARY:
•
•
•
Combines transient with boundary-dominated flow.
•
•
•
•
•
•
Constant operating conditions.
Transient: Analytical, constant pressure solution.
1. qD vs. tD (Figure 31).
2. Inverse of pressure integral (1 / pDi) vs. tD (not
shown).
3. Inverse of pressure integral-derivative (1 / pDid)
vs. tD (Figure 32).
Boundary-Dominated: Empirical, identical to traditional
(Arps).
Used to estimate EUR, skin and permeability.
EUR depends on operating conditions.
Does NOT use pressure data.
Cumulative curves are smoother than rate curves.
Combined cumulative and rate type curves give more
unique match (Figure 10).
33-40: FRACTURE
TYPE CURVES
33. Rate
11. Comparison of
qD and 1/pD
32. Integral-Derivative
Similar to Figures 27 & 28 but uses tD instead of tDA.
This format is useful when most of the data are in
TRANSIENT flow.
11-14: MODERN DECLINE
ANALYSIS: BASIC
CONCEPTS
34. Integral-Derivative
33-37: FINITE CONDUCTIVITY FRACTURE
•
Fracture with finite conductivity results in bilinear flow
(quarter slope).
Material Balance Time (tc) effectively converts constant
pressure solution to the corresponding constant rate
solution.
•
Dimensionless Fracture Conductivity is defined as:
•
Exponential curve plotted using Material Balance Time
becomes harmonic.
•
•
Fracture with infinite conductivity results in linear flow
(half slope).
Material Balance Time is rigorous during
boundary-dominated flow.
•
For FCD>50, the fracture is assumed to have infinite
conductivity.
12. Equivalence of
qD and 1/pD
11-12: MATERIAL BALANCE TIME
•
Actual Rate Decline
FCD
Constant Rate
Q
q
tc
Q
1
q
t
0
qdt
35. Elliptical Flow: Integral-Derivative
Q
Actual Time (t)
kf w
kxf
36. Elliptical Flow: Integral-Derivative
37. Elliptical Flow: Integral-Derivative
Material Balance Time
13. Concept of Rate Integral
(t c) = Q /q
14. Derivative and Integral-Derivative
13-14: TYPE CURVE INTERPRETATION AIDS
Rate (Normalized)
•
Combines rate with flowing pressure.
Integral (Normalized Rate)
•
Smoothes noisy data but
attenuates the reservoir signal.
Derivative (Normalized Rate)
•
Amplifies reservoir signal but
amplifies noise as well.
Integral-Derivative (Normalized Rate)
•
15-18: GAS FLOW
CONSIDERATIONS
15. Darcy’s Law
38-40: INFINITE CONDUCTIVITY FRACTURE
Smoothes the scatter
of the derivative.
38. Blasingame: Rate and Integral-Derivative
39. NPI: Pressure and Integral-Derivative
40. Wattenbarger: Rate
16. Pseudo-Pressure (pp)
15-16: PSEUDO-PRESSURE
Gas properties vary with pressure:
•
•
Z-factor (Pseudo-Pressure, Figures 15 & 16)
•
Compressibility (Pseudo-Time, Figures 17 & 18)
•
Pseudo-pressure corrects for changing viscosity and
Z-factor with pressure.
•
In all equations for liquid, replace pressure (p) with
pseudo-pressure (pp).
Viscosity (Pseudo-Pressure & Pseudo-Time, Figures
15, 16 & 18)
Note: For gas,
41-43: HORIZONTAL WELL TYPE CURVES
17-18: PSEUDO-TIME
17. Gas Compressibility Variation
•
•
Compressibility represents energy in reservoir.
•
Ignoring compressibility variation can result in
significant error in original gas-in-place (G) calculation.
•
Pseudo-time(ta) corrects for changing viscosity and
compressibility with pressure.
•
Pseudo-time calculation is ITERATIVE because it
depends on μg and ct at average reservoir pressure,
and average reservoir pressure depends on G (usually
known).
18. Pseudo-Time (ta)
41. Blasingame: Integral-Derivative
42. Blasingame: Integral-Derivative
43. Blasingame: Integral-Derivative
Gas compressibility is strong function of pressure
(especially at LOW PRESSURES).
19-22: FLOWING
MATERIAL BALANCE
19. Oil: Flowing Material Balance
20. Gas: Determination of
bpss
44-45: WATER-DRIVE
TYPE CURVES
44. Blasingame: Rate
45. Agarwal-Gardner: Rate
Infinite Aquifer
Oil
Reservoir
Mobility ratio (M) represents the strength of the aquifer.
M
Gas
k aq μ res
kres μ aq
Copyright
•
Note: bpss is the inverse of productivity index and is
constant during boundary-dominated flow.
• M = 0 is equivalent to Radial Type Curves (Figures
25-32).
21. Gas: Flowing Material Balance
22. Procedure to Calculate Gas-In-Place
SUMMARY:
•
•
•
•
•
•
•
•
•
Uses flowing data. No shut-in required.
Applicable to oil and gas.
Determines hydrocarbon-in-place, N or G.
Oil (N): Direct calculation.
Gas (G): Iterative calculation because of pseudo-time.
Simple yet powerful.
Data readily available (wellhead pressure can be
converted to bottomhole pressure).
Supplements static material balance.
Ideal for low permeability reservoirs.
a
A
b
b
b
B
B
B
B
c
c
c
D
D
D
F
G
G
G
h
k
k
k
Dpss
pss
gi
o
oi
g
t
t
e
i
CD
p
pa
aq
f
semi-major axis of ellipse
area
hyperbolic decline exponent or
semi-minor axis of ellipse
dimensionless parameter
inverse of productivity index
formation volume factor
initial gas formation volume factor
oil formation volume factor
initial oil formation volume factor
gas compressibility
total compressibility
total compressibility at average reservoir pressure
nominal decline rate
effective decline rate
initial nominal decline rate
dimensionless fracture conductivity
original gas-in-place
gas cumulative production
pseudo-cumulative production
net pay
permeability
aquifer permeability
fracture permeability
k
k
k
K
L
M
N
N
p
p
p
p
p
p
p
p
p
p
p
p
p
q
q
q
h
res
v
p
O
D
Dd
Di
Did
i
p
p
pi
pwf
wf
D
Dd
horizontal permeability
reservoir permeability
vertical permeability
constant
horizontal well length
mobility ratio
original oil-in-place
oil cumulative production
pressure
average reservoir pressure
reference pressure
dimensionless pressure
dimensionless pressure derivative
dimensionless pressure integral
dimensionless pressure integral-derivative
initial reservoir pressure
pseudo-pressure
pseudo-pressure at average reservoir pressure
initial pseudo-pressure
pseudo-pressure at well flowing pressure
well flowing pressure
flow rate
dimensionless rate
dimensionless rate
q
q
q
Q
Q
r
r
r
r
s
S
S
t
t
t
t
t
t
t
t
t
T
w
x
Ddi
Ddid
i
Dd
e
eD
w
wa
gi
oi
a
c
ca
D
DA
Dd
Dxf
Dye
e
dimensionless rate integral
xf
dimensionless rate integral-derivative
ye
initial flow rate
yw
cumulative production
Z
Z
dimensionless cumulative production
exterior radius of reservoir
Zi
dimensionless exterior radius of reservoir
α
wellbore radius
φ
apparent wellbore radius
μ
skin
μaq
initial gas saturation
μg
initial oil saturation
μg
flow time
pseudo-time
μo
material balance time
μres
material balance pseudo-time
dimensionless time
Oil field units;
dimensionless time
dimensionless time
dimensionless time
dimensionless time
reservoir temperature
fracture width
reservoir length
2008 Fekete Associates Inc. Printed in Canada
Note: Pseudo-time in build-up testing is evaluated at well
flowing pressure NOT at average reservoir pressure.
fracture half length
reservoir width
well location in y-direction
gas deviation factor
gas deviation factor at
average reservoir pressure
initial gas deviation factor
constant
porosity
viscosity
aquifer fluid viscosity
gas viscosity
gas viscosity at average
reservoir pressure
oil viscosity
reservoir fluid viscosity
q (MMSCFD); t (days)
g
All analyses described can be performed using Fekete’s Rate Transient Analysis software