J171768 DOI: 10.2118/171768-PA Date: 3-February-16
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Total Pages: 11
Nodal Analysis for Unconventional
Reservoirs—Principles and Application
Wentao Zhou, Raj Banerjee, and Eduardo Proano, Schlumberger
Summary
Nodal analysis is the standard technique used to evaluate the performance of integrated production systems. Two curves represent
the capacities of the inflow and of the outflow, and the intersection of the two curves gives the solution operating point. Limitations of traditional nodal analysis include:
• Results are offered only at a snapshot, not as a function of
time.
• Inflow-performance-relationship (IPR) models are limited,
with none available for shale gas wells.
• Analysis is performed on a well-by-well basis, with no
account of multiwell interference.
We propose a new nodal-analysis method that enables the study
of transient production systems, such as unconventional reservoirs, with IPR models generated from a high-speed semianalytical reservoir simulator and outflow curves generated from a
steady-state pipeline simulator. The use of analytical reservoir
simulation allows accurate, reliable modeling of the real inflow
system. The new approach studies the time-lapse behavior of the
system, with consideration of production history and neighboringwell interference.
This new method enables the study of transient deliverability
at the wellhead, where the measurement is usually available, and
shows the time-lapse relationship between wellhead pressure and
production rate. We provide examples of wellhead deliverability
and choke management and explain advantages of the method
with case studies involving tight and shale wells. The method is
also applied to design and optimize artificial lift in unconventional
wells and to study the method’s validity over time. In addition,
we discuss an example of operational well dynamics with timelapse nodal analysis.
Furthermore, this new method generates discussion about
some concepts that are often taken for granted—What should be
the definition of IPR in a transient production system? On the IPR
curve, is the zero-rate-pressure the reservoir pressure? Can IPR
curves at two different timesteps cross each other? Finding the
answers to these questions will help us better understand production systems.
The commonly used productivity-index (PI) method is
reviewed and compared with the new method. Results show that
one should not use the PI method when well operational conditions change.
Introduction
Nodal analysis has long been a key method used to evaluate the
performance of an integrated production system (Mach et al.
1979). Components in the system can include reservoir, completion, tubing string, subsurface safety valves, surface choke, flowline, and separator. Nodes are placed to segment the system; each
segment is defined by different equations or correlations. One can
select the solution node, for example, at the bottomhole location.
Pressure drops or gains from the starting point are added until the
solution node is reached, which gives the inflow capacity or
inflow-performance relationship (IPR). The same calculation
applies from the solution node to the endpoint, which gives the
outflow capacity or tubing-performance curve. The intersection of
the two curves gives the solution operating point.
Traditional nodal analysis is “static” and considers a snapshot
of the whole production life. It assumes a pseudosteady-state or
steady-state condition of the system. The IPR correlations used to
describe the inflow system are based on certain assumptions, and
are available for limited models: for example, vertical well, horizontal well, and fractured well. The analysis is performed on a
well-by-well basis and without considering the interference effect
of a neighboring well.
Transient Inflow-Performance Relationship (IPR)
in Literature and Current Practice
When it comes to production in tight or shale formulations, it is
widely accepted that pseudosteady-state or steady-state IPR curves
are not applicable, and a transient time-changing IPR is required.
Meng et al. (1982) developed a transient IPR for vertically
fractured wells that is based on a semianalytical model at constant-rate production, described by the following equation:
m½ pwf ðtÞ ¼ m½pi ðtÞ mwD ðtDxf ; Fcd Þ 1424 qg ðtÞ T
;
kh
ð1Þ
where mðpÞ is real-gas pseudopressure, pwf is flowing bottomhole
pressure, pi is initial pressure, mwD is the dimensionless pseudopressure-drop solution of a fractured vertical well, tDxf is dimensionless time, Fcd is dimensionless conductivity, qg is gas flow rate,
and T is reservoir temperature. With Eq. 1, one can compute the
IPR at any timestep t when given the input parameters. These IPRs
and the outflow curve at different wellhead pressures are shown in
Fig. 1. With this transient IPR, two things are noteworthy:
• pwf ¼ pi if qg ¼ 0, meaning that IPR always starts from
ð0; pi Þ.
• IPR at time t is predetermined and not affected by the production history before time t.
We will compare those findings with the results from the new
approach proposed by this paper in later sections.
Transient IPR curves for other models are found in the literature and in industry software. In general, if a dimensionless pressure solution is pD ðtÞ, then IPR at any time t is given by
pwf ðtÞ ¼ pi 141:2qlB
pD ðtÞ: . . . . . . . . . . . . . . . . . . ð2Þ
kh
Or denote the unit rate-pressure solution as
pu ðtÞ ¼
141:2lB
pD ðtÞ;
kh
then,
pwf ðtÞ ¼ pi qpu ðtÞ: . . . . . . . . . . . . . . . . . . . . . . . . . ð3Þ
C 2016 Society of Petroleum Engineers
Copyright V
This paper (SPE 171768) was accepted for presentation at the Abu Dhabi International
Petroleum Exhibition and Conference, Abu Dhabi, 10–13 November 2014, and revised for
publication. Original manuscript received for review 9 October 2014. Revised manuscript
received for review 16 March 2015. Paper peer approved 28 April 2015.
When production history is available, the current common
practice to obtain the transient IPR is to calculate the well productivity index (PI) with the traditional equation
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Pressure (psi)
2,000
1,500
Pwh = 1,000 psi
1,000
Pwh = 800 psi
Pwh = 600 psi
500
Pwh = 330 psi
G
F
t7 t6
0
0
100
200
E
D
t5
t4
300
C
B
t3
400
A
t2
500
600
t1
700
800
Fig. 1—Transient IPR (Meng et al. 1982).
qðtÞ
. . . . . . . . . . . . . . . . . . . . . . . . . ð4Þ
pi pwf ðtÞ
to capture the time-changing behavior of well productivity. Future
IPRs are then obtained by simple extrapolation. As an example, a
shale-oil production history is as shown in Fig. 2. The PI is then
fitted with the red line and extrapolated for a future transient PI,
as in Fig. 3. Another approach is to match the production history
by reservoir simulation or rate-transient analysis (RTA), and to
predict future production with the history-matched model and calculate a future PI with Eq. 4. Whether obtained by simple extrapolation, RTA, or simulation, the predicted IPRs are then used, for
example, for future design of well completion and artificial lift.
Note that, when discussing IPR, all methods described in this
section use the same traditional PI concept (i.e., Eq. 4) involving
initial reservoir pressure pi . For the sake of discussion, we name
this group of methods the PI Method, which we will compare
with the new method proposed in this paper.
2000
1800
pwf ¼ gðnÞ ðqÞ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð6Þ
where gðnÞ represents the inflow curve at the nth timestep. The
semianalytical reservoir model can be single well or multiwell.
For a multiwell model, all neighboring well production is taken
into account and has an impact on the IPR. The intersection of the
inflow curve and outflow curve gives the solution of rate and BHP
ðnÞ
at the current timestep pwf and rate qðnÞ , which concludes the
computation of this step:
(
(
ðnÞ
pwf
pwf ¼ hðnÞ ðqÞ
: . . . . . . . . . . . . . . . . . . . . ð7Þ
!
ðnÞ
pwf ¼ g ðqÞ
qðnÞ
Simulation then moves on to the next timestep. The whole process continues until it reaches the final timestep.
Time-lapse nodal analysis gives the solution at requested timesteps, which, when taken all together, show the evolution of
Qo
9000
1600
Oil-Production Cumulative (STB)
5000
6000
7000
8000
BHP
3000
4000
1400
1200
1000
800
Bottomhole Pressure (psi)
600
qo
200
1000
0
0
2000
Oil-Production Rate (STB/D)
where hðnÞ represents the outflow curve at the nth timestep. Inflow
performance, rather than with correlations as in traditional nodal
analysis, is obtained by running a semianalytical reservoir simulation (Busswell et al. 2006; Gilchrist et al. 2007; Thambynayagam
2011; Zhou et al. 2013). It has the flexibility of modeling the transient behavior of real reservoir and well configurations.
For instance, for a system (such as shown on the left in Fig. 5)
with 2 years of production history, the objective of inflow simulation is to get the relationship between the BHP and rate for the
current timestep. It is obtained by running simulation from the
start of production, with all the historical rates, rates solved from
previous timesteps, and an assumed rate for the current timestep.
One can try a multitude of rates with a corresponding group of
BHP responses. These rates and their BHP responses, represented
on a plot of rate vs. BHP, define the inflow-performance relationship (IPR) for the current timestep, as on the right in Fig. 5. In a
mathematical form, this is
10000 11000 12000
Time-Lapse Nodal-Analysis Principles
In numerical reservoir simulation, one can define outflow curves
for wells and run simulation with wellhead-pressure (WHP) control. Time-lapse nodal analysis mimics the numerical simulation
approach and conducts nodal analysis through timestepping, as
shown in Fig. 4.
400
Total Pages: 11
pwf ¼ hðnÞ ðqÞ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5Þ
Production Rate (Mscf/D)
PIðtÞ ¼
Page: 246
Each timestep is composed of two primary parts—inflow simulation and outflow simulation. The intersection of the inflow
curve with the outflow curve gives the operating point, which is
the solution of the rate and bottomhole pressure (BHP), when
given a wellhead pressure. The calculation then proceeds to the
next timestep. Outflow simulation, as usual, is performed with a
steady-state pipeline simulator with flow correlations. In this paper, outflow curves are simulated with Hagedorn and Brown
(1965) correlation for vertical flow, Beggs and Brill (1973) revised correlation for horizontal flow, and Moody (1947) for single-phase flow. For a given WHP, it gives a relationship between
production rate q and BHP pwf . This is expressed in a mathematical form,
t 1 = 20 days
t 2 = 35 days
t 3 = 50 days
t 4 = 100 days
t 5 = 150 days
t 6 = 250 days
t 7 = 300 days
2,500
Stage:
Oct 2011
Jan 2012
Jul 2012
Aug 2012
Oct 2012
Date
Bottomhole Pressure
Oil-Production Cumulative
Oil-Production Rate
Fig. 2—Production history of a shale oil well.
246
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Pl = q/(Pi-Pwf ), (STB/D/psi)
100
Start
10
1
1
Next timestep
0.1
Multiwell inflow
simulation
0.01
0.001
1
10
100
1,000
10,000
Outflow simulation
2
3
Time (days)
Calculate operating
point(s)
Fig. 3—PI vs. time.
production. For example, in Fig. 6, early-time and late-time IPR
curves obtained from the simulator, together with the outflow
curve throughout the time period, yield the production rate and
BHP as functions of time, as shown in Fig. 7. Another important
observation is that average reservoir pressure pavg , which is a
required input for traditional nodal analysis, is not required in
time-lapse nodal analysis. Instead, pavg is an output.
Computational cost of time-lapse nodal analysis, thanks to the
use of either analytical reservoir simulation or superposition, is
minimal. Examples shown in this paper take no more than a fraction of a second. This is the case for the PI method too.
Understanding Inflow-Performance Relationship
(IPR) From Time-Lapse Nodal Analysis:
Superposition Rule
An IPR derived from semianalytical reservoir simulation shows
distinct behavior, sometimes contradicting conventional nodalanalysis wisdom. To explain this behavior, we need to understand
how semianalytical reservoir simulation is performed, and most
importantly, to understand the superposition rule.
The analytical solution of the pressure-diffusion equation follows this superposition rule: One can obtain the variable-rate solution from the constant-rate solution.
Denote a unit-rate solution, or consider the pressure drop
because of a unit-rate production Dp pi pwf ðtÞ ¼ pu ðtÞ. Then,
the pressure of the varying rate qðtÞ is
ðt
DpðtÞ ¼ qðsÞ
dpu ðt sÞ
ds; . . . . . . . . . . . . . . . . . . . ð8Þ
dt
0
or for discrete rate changes,
Pressure (psia)
Solved
No
4
Last timestep
Yes
End
Fig. 4—Time-lapse nodal-analysis work flow.
pwf ðtÞ ¼ pi ½q1 pu ðtÞ þ
N
X
ðqj qj1 Þpu ðt tj1 Þ:
j¼2
ð9Þ
Compare Eq. 9 with Eq. 3; we see that the productivity-index
(PI) method takes no account of superposition.
If rate history is known, qi ¼ qðti Þ; i ¼ 1 N 1, the pressure response at the Nth timestep is
(
)
N1 X
qj pu ðtN tj1 Þ pu ðtN tj Þ
pwf ðtN Þ ¼ pi j¼1
qN pu ðtN tN1 Þ;
ð10Þ
which is essentially the IPR at the Nth timestep, with pwf ðtN Þ as a function of qN . From this equation,
nXN1 we can see that this IPR curve starts
o at
a pressure p0 ¼ pi q ½p ðt tj1 Þ pu ðtN tj Þ , and
j¼1 j u N
the slope is pu ðtN tN1 Þ, as shown in Fig. 8.
Besides the rate superposition just described, there is also pressure superposition: One can obtain the variable-bottomhole-
New timestep
BHP
2000
1000
WHP
400
300
200
Liquid Rate (STB/D)
Gas Rate (Mscf/D)
BHP
Rate
500
0
2009
2010
2011
2012
Rate
Date/Time
Fig. 5—Calculated inflow (left), inflow curve, and outflow curve at timestep n (right).
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10,000
5,000
Pressure (psia)
pwf
5,000
WHP
10
5
0
Liquid Rate (B/D)
Pwf (psia)
10,000
pavg
10,000
Liquid Volume (MMSTB)
Inflow curve
Operating points
Outflow curve
02/29/2012 14:02:30
03/01/2012 14:02:30
03/02/2012 14:02:30
03/03/2012 14:02:30
03/04/2012 14:02:30
03/05/2012 14:02:30
03/06/2012 14:02:30
03/13/2012 14:02:30
03/29/2012 14:02:30
04/28/2012 14:02:30
05/28/2012 14:02:30
06/27/2012 14:02:30
07/27/2012 14:02:30
08/26/2012 14:02:30
09/25/2012 14:02:30
10/25/2012 14:02:30
11/24/2012 14:02:30
12/24/2012 14:02:30
01/23/2012 14:02:30
02/22/2012 14:02:30
qo
30,000
Qo
25,000
Apr
2012
0
Jul
Oct
Jan 2013
Date/Time
Fig. 7—Time-lapse nodal-analysis results.
0
2e5
4e5
6e5
Liquid Rate (B/D)
BQN
is the average reservoir pressure,
/hct A
70:6Bl
4A
ln c
and the slope
is the well PI, according to Eq.
kh
e CA rw2
A-25 in Blasingame and Lee (1986). Therefore, for such systems,
time-lapse nodal analysis reduces to traditional nodal analysis.
where p0 ¼ pi 0:2339
Fig. 6—Inflow and outflow curves in time-lapse nodal analysis.
pressure (BHP) solution from the constant-BHP solution by superposition, with the following equation:
ðt dqu ðt sÞ
ds; . . . . . . . . . . . . . . ð11Þ
qðtÞ ¼ pi pðsÞ
dt
0
or for discrete BHP changes,
(
)
N1
X
qðtN Þ ¼
ðpi pj Þ qu ðtN tj1 Þ qu ðtN tj Þ
j¼1
ðpi pN Þqu ðtN tN1 Þ: . . . . . . . . . . . . . . ð12Þ
The effect of gas nonlinearities is handled by gas pseudopressure and gas pseudotime. For the sake of simplicity, this is not discussed in this paper.
The superposition rule holds true for tight formations and conventional reservoirs with high-to-medium permeability. It holds
for a transient, pseudosteady-state, or steady-state system. For a
well centered in a bounded, circular conventional reservoir, Eq.
10 becomes the following [according to Eq. A-42 in Blasingame
and Lee (1986)]:
BQN
70:6Bl
4A
ln c
;
qN
pwf ðtN Þ ¼ pi 0:2339
kh
e CA rw2
/hct A
ð13Þ
Discussion of Inflow-Performance-Relationship
(IPR) Curve
The new concept of time-lapse nodal analysis brings new ideas to
consider in how we think of the inflow-performance curve. Some
examples follow.
Pressure at Zero Rate Is the Buildup Pressure, Not Necessarily
the Reservoir Pressure. According to the conventional IPR concept, pressure at zero rate is the average reservoir pressure. But
from the preceding analysis, we can see this pressure is actually
the buildup pressure for a shut-in (rate equals zero). In high-permeability conventional reservoirs, it may build up to the actual
reservoir pressure; then, the meaning of pressure at zero rate converges to the meaning in the conventional concept. For unconventional shale reservoirs, the pressure may not be able to reach
reservoir pressure within the duration of the shut-in.
One Can Define Productivity Index (PI) With Buildup
Pressure. According to Eq. 10 and as shown in Fig. 8, the slope
of the curve is pu ðtN tN1 Þ. If the timestep size, Dt ¼ tN tN1 ,
is constant, the slope will be identical at all times, making the
well PI constant. For a conventional reservoir system, as
described in Eq. 13, this corresponds to a constant-productivity
well with reducing reservoir pressure. For shale wells, if we propose a new way of defining the well PI,
PI ¼
pi –
{Σ
N–1
j =1 qj [pu
}
(tN – tj–1) – pu (tN – tj )]
pu (tN – tN–1)
pwf
0
q
Fig. 8—IPR from superposition.
248
qðtÞ
; . . . . . . . . . . . . . . . . . . . . . . . ð14Þ
pbu ðtÞ pwf ðtÞ
where pbu is the buildup pressure at q ¼ 0; this corresponds to a
constant-productivity well with reducing buildup pressure. This
new definition, compared with the traditional definition with
respect to initial reservoir pressure or average reservoir pressure,
is more practical and suitable to model shale wells, because
buildup pressure is measurable.
IPR Depends On Production History. The IPR of a well is generally considered a predefined attribute independent of outflow
curves and production history. For example, in Fig. 9, a set of
IPRs is predefined at three times and used for analysis for any
outflow curves (e.g., tubing-performance curves TPC1 and
TPC2), which then gives the two production predictions, as
shown on the right. From time-lapse nodal analysis, however,
IPR at one timestep is the result of superposition and is related to
production history before this timestep, which, in turn, is
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TPC1
q
TPC2
pwf
IPR@t3
0
IPR@t2
q@TPC2
IPR@t1
q@TPC1
q
t
Fig. 9—IPR curves independent of outflow curves.
12,000
12,000
Op. points
Op. points
10,000
10,000
TPC–1
TPC–2
IPR, t = 30 days
IPR, t = 30 days
8,000
IPR, t = 60 days
IPR, t = 90 days
6,000
IPR, t = 120 days
IPR, t = 150 days
4,000
IPR, t = 180 days
BHP (psi)
BHP (psi)
8,000
IPR, t = 60 days
IPR, t = 90 days
6,000
IPR, t = 120 days
IPR, t = 150 days
IPR, t = 180 days
4,000
IPR, t = 210 days
IPR, t = 210 days
2,000
IPR, t = 240 days
IPR, t = 240 days
2,000
IPR, t = 270 days
IPR, t = 270 days
0
IPR, t = 300 days
0
500
1,000 1,500
Rate (B/D)
IPR, t = 300 days
0
2,000 2,500
0
500
1,000 1,500
Rate (B/D)
2,000 2,500
Fig. 10—IPR curves dependent on outflow curves: IPR curves for TPC1 (left) and TPC2 (right).
dependent on the outflow curve. As a result, IPR curves cannot
be predefined and can only be determined during the analysis.
For example, in Fig. 10, IPR curves for TPC1 (high, constant
pwf ) are higher than those for TPC2 (low, constant pwf ). In practical application, it means that
• For new well, IPR calculation for timestep n needs to take
into account the superposition effect of the production rate
calculated from timestep 1 to n 1.
• For wells with long history, similar to the technique used in
well-test analysis, the rate history occurred a long time
before the analyzed period could be simplified and the rate
variations that happened immediately before should be taken
more accurately (Bourdet 2002).
4,500
4,500
4,000
4,000
3,500
3,500
Op. points
3,000
TPC
2,500
IPR, t = 10 days
IPR, t = 20 days
2,000
IPR, t = 30 days
IPR, t = 40 days
(dt = 10 days)
IPR, t = 50 days
1,500
1,000
IPR, t = 60 days
500
BHP (psi)
BHP (psi)
IPR Is a Function of Timestep. Well productivity is normally
considered a well property associated with time (e.g., 05/26/2014)
but not with timesteps (e.g., delta time, 1 day, 1 month, and 1
year). For example, one always asks this kind of question: What
is my well productivity today? From time-lapse nodal analysis,
however, we can see that IPR is strongly related to the timesteps
used. With a constant unit rate, pressure drop always increases as
time increases [i.e., pu ðDt1 Þ < pu ðDt2 Þ if Dt1 < Dt2 ]; therefore,
we will have low productivity at large timesteps, regardless of the
time itself. Results shown on the left plot in Fig. 11 are from the
nodal analysis with a constant pwf ¼ 1; 500 psi at equal timesteps,
t ¼ 10; 20; 30; 40; 50; 60 days. Because the timestep remains
the same (10 days), all IPRs share the same slope. The right plot
in Fig. 11 shows results of the same nodal analysis, but one timestep (from the time interval 30 days to 40 days) is divided into
five substeps of 2 days each, t ¼ 32; 34; 36; 38; 40 days. The
IPRs at these five substeps show a distinctly different slope. Both
analyses give the same result, as shown on the left plot of Fig. 12.
However, as shown on the right in Fig. 12, the IPR curve at t ¼ 40
days with the 10-day step is steeper, or appears as lower
Op. points
TPC
IPR, t = 10 days
3,000
IPR, t = 20 days
2,500
IPR, t = 30 days
2,000
IPR, t = 32 days
1,500
IPR, t = 36 days
IPR, t = 34 days
IPR, t = 38 days
IPR, t = 40 days
(dt = 2 days)
IPR, t = 50 days
1,000
500
IPR, t = 60 days
0
0
1,000
2,000 3,000
Rate (B/D)
4,000
0
0
1,000 2,000 3,000 4,000 5,000
Rate (B/D)
Fig. 11—Results of time-lapse nodal analysis with timestep of 10 days (left), with substeps of 2 days between 30 and 40 days
(right).
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4,500
2,500
4,000
3,500
3,000
1,500
BHP (psi)
Rate (B/D)
2,000
1,000
2,500
Op. points
TPC
IPR, t = 10 days
(dt = 10 days)
IPR, t = 10 days
(dt = 2 days)
2,000
1,500
500
1,000
0
0
10
20
30
40
Time (days)
Rate(dt = 10 days)
50
60
500
70
0
0
Rate(with substeps)
1,000
2,000
3,000
Rate (B/D)
4,000
Fig. 12—Production rate (left), and IPR comparison (right).
100
10
1
0
200
400
600
800
1,000
1,200
1,800
1,600
1,400
1,200
1,000
800
600
400
200
0
1,200
1,000
800
600
400
200
0
BHP (psi)
1,000
Rate (B/D)
Rate (STB/D)
10,000
TLNA-Rate
PI-Rate
TLNA-Pwf
PI-Pwf
0
50 100 150 200 250 300 350
Time (days)
Time (days)
Fig. 14—PI method and time-lapse nodal analysis give the
same result.
Fig. 13—Rate profile at constant BHP.
productivity, than that with the 2-day step, and the two curves
intersect at the operating point.
The dependence of the IPR on timestep is important because it
means we cannot use the same set of IPRs for long-term production prediction with a monthly timestep and for operational well
dynamics when a timestep is on the order of minutes and hours.
This is elaborated in a later section.
Comparison Between Time-Lapse Nodal-Analysis
and the Productivity-Index (PI) Methods
We use a synthetic example under a variety of scenarios to compare the PI-analysis method with the time-lapse nodal-analysis
method. The synthetic model is a shale oil well in a reservoir with
initial reservoir pressure of 4,200 psi. The well-production profile
at constant pwf ¼ 2,000 psi is as shown in Fig. 13. For the PI
method, one can simply calculate the transient PI at each month by
Eq. 4: PI ¼ qðtÞ=ð4; 200 2; 000Þ. However, in the time-lapse
1,400
2,500
2,000
1,000
800
1,500
600
1,000
400
500
200
0
0
BHP (psi)
Rate (B/D)
1,200
TLNA-Rate
PI-Rate
TLNA-Pwf
PI-Pwf
0
50 100 150 200 250 300 350
Time (days)
Fig. 15—PI method and time-lapse nodal analysis give similar
result.
250
nodal-analysis method, PI at timestep N is calculated by pressuresuperposition (Eq. 12), considering the history pj ; j ¼ 1 N 1.
Flat Tubing-Performance Curve. Under constant bottomhole
pressure (BHP), the two methods give the same result (Fig. 14).
Upward Tubing-Performance Curve. Even for a slightly
upward-trending tubing-performance curve, the two methods give
similar results (Fig. 15).
The PI method gives the same or similar results as time-lapse
nodal analysis for two previous cases, because the condition under
which the system is evaluated, constant BHP [flat tubing-performance curve (TPC)] and slightly upward-trending TPC, is within
the assumption of the PI method, that is, stable production condition. When production condition changes, however, as in the next
two cases, the PI method fails to capture the right behavior.
Effect of Shut-in. Suppose the well produced for 3 months and
then was shut down for 3 months and opened again afterward
under the same pwf before the shut-in. Production after the shut-in
would increase because of the charging of the formation during
shut-in. The time-lapse nodal analysis correctly captures this
behavior, compared with the result from numerical reservoir simulation in red squares (Fig. 16). In contrast, the PI method fails to
account for the shut-in, giving exactly the same production trend
despite the shut-in. Also, examine pwf during the shut-in, from 90
days to 180 days; the PI method moves pwf back to the initial
pressure of 4,200 psi, but the time-lapse analysis correctly models
the pressure buildup.
Two Tubing-Performance Curves. When the operational parameters of the well change—for example, when the choke is
opened wider or artificial lift is installed and the BHP is lowered
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1,800
1,600
1,400
1,200
1,000
800
600
400
200
0
0
50
100
150
Time (days)
Stage:
200
Page: 251
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0
250
Total Pages: 11
TLNA-Rate
BHP (psi)
Rate (B/D)
J171768 DOI: 10.2118/171768-PA Date: 3-February-16
PI-Rate
Rate from Res. Sim.
TLNA-Pwf
PI-Pwf
Fig. 16—PI method fails to account for the shut-in.
1,400
2,500
2,000
1,000
800
1,500
600
1,000
400
500
200
0
BHP (psi)
Rate (B/D)
1,200
0
TLNA-Rate
PI-Rate
TLNA-Pwf
PI-Pwf
0
50 100 150 200 250 300 350
Time (days)
Fig. 17—PI method fails to capture the rate increase caused by
lower BHP.
from 2,000 psi to 1,100 psi, we see from Fig. 17 that the timelapse nodal-analysis method captures the rate increase correctly,
and the PI method underestimates the increase.
The PI method yields a good approximation when well operational conditions remain unchanged. If there are changes in, for
example, choke positions, lifting mechanisms, or shut-ins, the PI
method will greatly underestimate the production after the
change, and one should not use that method.
Application of Time-Lapse Nodal Analysis
The following examples show the application of time-lapse nodal
analysis. Although the examples are either tight or shale wells,
the principles apply to studies of any type of wells when the traditional nodal analysis falls short of expectations.
Wellhead Deliverability. Well deliverability is usually referenced at bottomhole conditions in terms of the productivity index
(PI) (STB/D/psi). However, bottomhole deliverability is difficult
to use operationally if downhole-pressure measurement is not
available in relevant time. Because wellhead pressure (WHP) is
readily available most of the time, wellhead deliverability will
shed light on well performance and guide well operations. For
example, Fig. 18 is the crossplot of wellhead pressure vs. gas rate
for a tight gas well. The color scale represents time—blue refers
to time back in history and red refers to the most-recent time. The
dots are aligned in a set of straight lines corresponding to different
choke positions. What is the wellhead deliverability today? If it is
Curve A, the well has much more production potential than if it is
Curve B.
To predict wellhead-deliverability curves, we can history
match the production with rate-transient analysis and predict into
the future at different wellhead pressures with time-lapse nodal
analysis, as shown in Fig. 19. Keeping the same WHP of 1,689
psi would continue the rate-declining trend (brown curve in the
NA sensitivity plot with History Curve 1
Wellhead Pressure (psia)
4,000
2,000
Wellhead Pressure
Time
A
0
03/08/2009
B
1,000
2,000
Production Rate_1;qg (Mscf/D)
03/22/2010
3,000
Fig. 18—WHP vs. rate, and wellhead-deliverability curve.
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Stage:
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Calc_BHP_Go
Calculated reservoir pressure
Wellhead pressure
Production Rate_1:qg
Press (Line style)
Pwf (Line style)
qg (Line style)
Wellhead pressure (Line style)
Wellhead pressure: 1,000 psia (Color)
Wellhead pressure: 1,250 psia (Color)
Wellhead pressure: 1,689 psia (Color)
Wellhead pressure: 250 psia (Color)
Wellhead pressure: 750 psia (Color)
4,000
Pressure (psia)
Total Pages: 11
2,000
Gas Rate (Mscf/D)
4,000
2,000
Sep
Oct
Nov
Dec
Jan 2010
2009
Feb
Mar
Apr
May
Jun
Date/Time
Fig. 19—Production history matching (before February 2010) and prediction with time-lapse nodal analysis with different WHPs.
bottom pane of the figure); lowering WHP to 250 psi would boost
the rate from 1 MMscf/D to 4 MMscf/D before it falls back again
because of depletion. One can plot together WHP vs. predicted
rate at different timesteps, as in Fig. 20, which shows the wellhead deliverability. One can use these curves in well operations—
for example, in choke management.
Shale-Oil Well-Production Performance. A 4,600-ft-long shale
oil well at a depth of 6,500 ft has 28 hydraulic fractures and a tubing size of 2.875 in. For simplicity, fractures are equidistant and
of equal length. We analyzed the sensitivity of WHP with the
time-lapse nodal-analysis method. Fig. 21 shows the inflow-performance relationship (IPR) and tubing-performance curves
(TPCs) at eight timesteps for two WHPs: 100 and 500 psi. The
upper TPC is for WHP ¼ 500 psi; the lower one is for WHP ¼ 100
psi. The IPR at the first timestep (the outermost one) is the same
for both pressures. Starting from the second timestep (because the
history from previous timesteps is different), the IPR curves differ
for these two cases. In Fig. 22, the result, plotted as a function of
time, shows that cumulative production at low WHP is approximately twice that of the cumulative production at high WHP. Production at WHP ¼ 500 psi would reach the economic limit of 20
B/D in 7 months; for WHP ¼ 100 psi, the limit would be reached
NA sensitivity plot with History Curve 1
Wellhead pressure
Wellhead Pressure (psia)
4,000
Operation Points (Line style)
Wellhead pressure: 1,000 psia (Color)
Wellhead pressure: 1,250 psia (Color)
Wellhead pressure: 1,689 psia (Color)
Wellhead pressure: 250 psia (Color)
Wellhead pressure: 750 psia (Color)
* After the end of the
production history
(January 30, 2010)
2,000
11 days*
Pavg = 2,190 psia
0
91 days
Pavg = 1,443 psia
123 days
Pavg = 1,276 psia
61 days
Pavg = 1,617 psia
30 days
Pavg = 1,940 psia
08/21/2009
2,000
Production Rate_1;qg (Mscf/D)
Time
06/01/2010
4,000
Fig. 20—Wellhead-deliverability curves.
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Inflow curve
Pressure (psia)
Inflow Wellhead Pressure: 100 psia
Inflow Wellhead Pressure: 500 psia
Operating points
Outflow Wellhead Pressure: 100 psia
Outflow Wellhead Pressure: 500 psia
3,000
12/04/2013 00:00:00
12/24/2013 00:00:00
07/23/2014 00:00:00
03/24/2014 00:00:00
3,000
Pres
Pwf
qo
Qo
Wellhead Pressure
Wellhead Pressure: 100 psia
Wellhead Pressure: 500 psia
2,000
1,000
07/22/2014 00:00:00
03/19/2015 00:00:00
07/11/2016 00:00:00
0
Liquid Volume (MMSTB)
2,000
1,000
0.04
Qo@pwh = 100 psi
Liquid Rate (B/D)
Pwf (psia)
02/26/2019 00:00:00
102
0.02
Qo@pwh = 500 psi
qlim : 20 B/D
101
0.00
2014
2015
2016
2017
Date/Time
2018
2019
Fig. 22—Time-lapse nodal-analysis results.
0
200
400
Liquid Rate (B/D)
600
Fig. 21—Sensitivity analysis of wellhead pressure for a shale
oil well.
in 2 years. The bottomhole pressure (BHP) of the scenario with
WHP ¼ 500 psi is approximately 3,000 psi, and it is approximately 2,500 psi when the WHP ¼ 100 psi.
Artificial-Lift Design and Optimization for Shale Well. Next,
we will apply time-lapse nodal analysis and model the production
of a shale oil well under artificial lift and optimize the performance of the pump.
The shale oil well is the one used in Fig. 13, and the figure
shows the well production under constant pwf ¼ 2,000 psi. The
well has been under natural flow for 100 days, and we want to
investigate artificial-lift methods for the future. If the same natural
flow is maintained, production follows the same trend, and both
methods give the same result (Fig. 23). If an electrical submersible
pump (ESP) is installed, in this case a REDA DN1750 with 322
stages, and we set the frequency at 60 Hz, the production jumps to
1,200 B/D and falls back to 800 B/D in 100 days, as in Fig. 24. On
average, the PI method underestimates the rate by 200 B/D.
One should optimize the production rate through a pump so
that it falls within the recommended operating range to avoid
downthrust and upthrust. Variable-speed drives (VSDs) can vary
the speed of pumps and enable pumps to operate across a wider
range, improve performance, and optimize productivity. So, next,
we investigate the effect of different frequencies on the resulting
production profile.
As in Fig. 24, if the frequency is set at 45 Hz, the production
jumps to 1,100 B/D and falls back to 600 B/D in 100 days. On average, the PI method underestimates the rate by 160 B/D. If the
frequency is set at 30 Hz, the rate jumps to 800 B/D and falls back
to 500 B/D in 100 days. On average, the PI method underestimates
the rate by 85 B/D. In general, the effect of an ESP on production
Rate (B/D)
500
201
1,200
1,000
800
600
400
0
1,000
101 151
Time (days)
1,400
200
1,500
51
1,600
2,500
2,000
1
Operational Well Dynamics. The preceding examples show the
application of time-lapse nodal analysis on long-term production
behavior, on the order of months to years. The method also has
application in short-term operational well dynamics, those with a
time scale of hours to days. Timesteps can be as small as seconds.
As previously discussed, the timestep is a key parameter that determines the slope of the IPR curves, and the use of a wrong IPR
would give wrong results. Full study of well dynamics would
require the link of transient inflow simulation with transient outflow simulation. In this paper, we explain only the importance of
having the right transient inflow model by linking with a steadystate outflow simulation. A full study is the topic of another paper.
As an example, for the shale well in Fig. 23, under natural flow
after 100 days, the choke is opened at 8/64, 16/64, 32/64, and 64/
64 in. sequentially, each for 1 hour in a 4-hour cycle. The well
behavior is modeled with both methods, as shown in Fig. 26.
Notice the IPRs from time-lapse nodal analysis and those from the
PI method: Well productivity from time-lapse analysis is 12.5 B/D/
3,000
BHP (psi)
1,600
1,400
1,200
1,000
800
600
400
200
0
and the impact of a different frequency are underestimated by the
PI method.
The production rate at different frequency values, modeled
with both methods, is overlayed with the pump VSD curves in
Fig. 25. The rate predicted by the PI method is totally outside the
recommended operating range, whereas the rate profile from
time-lapse nodal analysis falls into the efficient range for 45 and
30 Hz. This means that the use of the PI method would result in a
false optimization of the pump.
Rate (STB/D)
0
TLNA-Rate
PI-Rate
Rate history
TLNA-Pwf
PI-Pwf
Pwf history
0
251
Fig. 23—Well follows the same trend with natural flow.
1
50
100
150
Time (days)
200
Hist
NF
TLNA-30 Hz
PI-60 Hz
TLNA-60 Hz
TLNA-45 Hz
TLNA-45 Hz
PI-30 Hz
250
Fig. 24—Production profile at 60 Hz (red), 45 Hz (green), and
30 Hz (blue), from time-lapse nodal analysis (solid line) and PI
method (dashed line).
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Reda - DN1750 Stages = 322
Reda - DN1750 Stages = 322
@90 Hz
@85 Hz
@80 Hz
@75 Hz
Head (ft)
Head (ft)
@90 Hz
18,000
16,000
14,000
12,000
10,000
@70 Hz
@65 Hz
@60 Hz
8,000
6,000
@55 Hz
@50 Hz
@45 Hz
@40 Hz
@35 Hz
@30 Hz
4,000
2,000
500
Total Pages: 11
18,000
16,000
14,000
12,000
10,000
8,000
6,000
@85 Hz
@80 Hz
@75 Hz
@70 Hz
@65 Hz
@60 Hz
@55 Hz
@50 Hz
@45 Hz
@40 Hz
@35 Hz
@30 Hz
4,000
2,000
1,000 1,500 2,000 2,500
Flow Rate (B/D)
3,000 3,500
500
1,000 1,500 2,000 2,500
Flow Rate (B/D)
3,000 3,500
Fig. 25—Pump VSD curves overlayed with the production profile at different frequency, from PI method (left) and time-lapse nodal
analysis (right).
5,000
8/64
32/64
16/64
4,500
4,000
3,500
64/64
The huge difference in the IPRs and the resulting pressure and
rate profile have a big impact in multiphase-flow regimes and
dynamic well behavior. This topic will be further investigated
with both the transient inflow model and transient outflow model
in a separate paper.
BHP (psi)
3,000
2,500
2,000
IPR from timelapse nodal
analysis
1,500
1,000
IPR from PI method
500
0
0
500
1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000
Rate (B/D)
Fig. 26—IPR difference between time-lapse nodal analysis and
PI method.
psi and only 0.285 B/D/psi according to the PI method, a difference
of 44 times.
For the resulting rate and pressure profile, as shown in Fig. 27,
the PI method predicts a BHP fluctuation of approximately 1,800
psi, but the time-lapse method predicts that the BHP fluctuation
will be less than 200 psi. The rate from the PI method fluctuates
in a range of 500 B/D, whereas the time-lapse method predicts a
fluctuation range of 1,500 B/D in this 4-hour cycle.
2,000
1,800
Conclusions
The time-lapse nodal-analysis advantageous characteristics are
• Has application in both unconventional and conventional
resources.
• Captures the transient behavior of the reservoir.
• Models inflow performance relationship (IPR) with semianalytical reservoir simulation or superposition; no IPR correlations
are required.
• Evaluates production performance as a function of time, not as
a static snapshot.
• Captures wellhead deliverability as a function of time.
• Is suitable for study of operational well dynamics and artificiallift design and optimization.
In contrast, the traditional PI-method characteristics are
• Can only be used when well-operational conditions remain
unchanged.
• Underestimates production, and one should not use it for artificial-lift design and optimization for wells in tight/shale
formations.
• Overestimates bottomhole-pressure fluctuation and underestimates rate fluctuation.
3,500
3,000
1,600
1,200
2,500
2,000
1,000
800
600
1,500
1,000
400
200
500
BHP (psi)
Rate (B/D)
1,400
TLNA-Rate
PI-Rate
Rate
TLNA-Pwf
PI-Pwf
Pwf
0
100.00 100.05 100.10 100.15 100.20 100.25 100.30 100.35 100.40
Time (days)
Fig. 27—PI method overestimates pwf fluctuation and underestimates rate fluctuation.
254
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J171768 DOI: 10.2118/171768-PA Date: 3-February-16
Nomenclature
A ¼ reservoir area, ft2
B ¼ volume factor
ct ¼ total compressbility, 1/psi
CA ¼ Dietz shape factor
Fcd ¼ fracture conductivity, dimensionless
h ¼ pay-zone thickness, ft
k ¼ formation permeability, md
mwD ¼ pseudopressure drop of a fractured vertical well,
dimensionless
pavg ¼ average reservoir pressure, psi
pbu ¼ buildup pressure, psi
pi ¼ initial reservoir pressure, psi
pu ¼ unit rate pressure, psi/(B/D)
pwf ¼ flowing bottomhole pressure, psi
pwh ¼ wellhead pressure, psi
q ¼ production rate, B/D or Mscf/D
qo ¼ oil-production rate, B/D
qu ¼ unit drawdown rate, (B/D)/psi
Q ¼ cumulative production
Qo ¼ cumulative oil production, bbl
rw ¼ wellbore radius, ft
t ¼ time, days
tDxf ¼ dimensionless time
T ¼ temperature
c ¼ 0.577216, Euler’s constant
l ¼ viscosity, cp
/ ¼ porosity, fraction
Acknowledgments
The authors would like to thank Schlumberger for supporting this
work and giving permission to publish this paper.
References
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Mach, J., Proano, E., and Brown, K. E. 1979. A Nodal Approach for
Applying Systems Analysis to the Flowing and Artificial Lift Oil or
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Conversion Factors
bbl 1.589 873
cp 1.0*
ft 3.048*
in. 2.54*
psi 6.894 757
E01 ¼ m3
E03 ¼ Pas
E01 ¼ m
Eþ00 ¼ cm
Eþ00 ¼ kPa
*Conversion factor is exact.
Wentao Zhou is Production Product Champion with Schlumberger, based in Houston. In his current role, he works on software product development, product marketing, project
consulting, and technical training. Zhou’s products cover
areas such as analytical reservoir simulation, pressure- and
rate-transient analysis, unconventional-resources production,
artificial lift, reservoir modeling and simulation, and field-development planning. He joined Schlumberger in 2006. Zhou holds
an MSc degree in petroleum engineering from Stanford University and an MSc degree in fluid mechanics from Peking University, China.
Raj Banerjee is Advisor with Schlumberger. Based in Houston,
he is currently responsible for the development and deployment of digital-oilfield-software solutions globally. Banerjee is
well-published and has authored or coauthored more than a
dozen patents. He holds a PhD degree in applied mathematics from Imperial College London. Banerjee also holds an MS
degree in petroleum engineering from the University of Tulsa
and a BS degree in petroleum engineering from Indian School
of Mines.
Eduardo A. Proano is Advisor, Reservoir Discipline Career Manager, for Schlumberger. He has been with the company for
more than 27 years. Proano’s expertise includes production
optimization, well intervention, online production surveillance
and diagnosis, rapid field/asset wide evaluations for production enhancement, asset-management applications for well
productivity, reservoir management, and field-development
plan fast-track solutions. He holds BS and MS degrees in petroleum engineering and a Masters of Engineering Management
degree from the University of Tulsa.
February 2016 SPE Journal
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