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Wellbore heat-transfer modeling and applications
Article in Journal of Petroleum Science and Engineering · May 2012
DOI: 10.1016/j.petrol.2012.03.021
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Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
Contents lists available at SciVerse ScienceDirect
Journal of Petroleum Science and Engineering
journal homepage: www.elsevier.com/locate/petrol
Wellbore heat-transfer modeling and applications☆
A.R. Hasan a,⁎, C.S. Kabir b
a
b
Texas A&M University, United States
Hess Corporation, United States
a r t i c l e
i n f o
Article history:
Received 6 September 2011
Accepted 14 March 2012
Available online 26 March 2012
Keywords:
heat transfer
transient fluid and heat flows
wellbore-fluid temperature
borehole gauge placement
flow rate estimation
annular-pressure buildup
a b s t r a c t
Fluid temperature enters into a variety of petroleum production–operations calculations, including well drilling and completions, production facility design, controlling solid deposition, and analyzing pressuretransient test data. In the past, these diverse situations were tackled independently, using empirical correlations with limited generality. In this review paper, we discuss a unified approach for modeling heat transfer
in various situations that result in physically sound solutions. This modeling approach depends on many
common elements, such as temperature profiles surrounding the wellbore and any series of resistances for
the various elements in the wellbore. We show diverse field examples illustrating this unified modeling approach in solving many routine production–operations problems.
© 2012 Elsevier B.V. All rights reserved.
Contents
1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . .
Modeling approach . . . . . . . . . . . . . . . . . . .
2.1.
Formation/fluid heat exchange . . . . . . . . . .
2.1.1.
Wellbore resistances . . . . . . . . . .
2.1.2.
Energy balance for single conduit . . . .
2.1.3.
Heat flow through multiple conduits . . .
2.2.
Transient flow. . . . . . . . . . . . . . . . . .
3.
Applications . . . . . . . . . . . . . . . . . . . . . .
3.1.
Borehole gage placement in transient testing . . .
3.2.
Rate computation with WHP and temperature data
3.2.1.
Multirate gas-well test example . . . . .
3.2.2.
Multirate oil-well test . . . . . . . . . .
3.3.
Annular-pressure buildup (APB) . . . . . . . . .
3.3.1.
Field example. . . . . . . . . . . . . .
4.
Discussion . . . . . . . . . . . . . . . . . . . . . . .
5.
Summary . . . . . . . . . . . . . . . . . . . . . . .
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
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127
128
128
128
129
129
129
130
130
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135
135
1. Introduction
☆ J. of Petroleum Science & Engineering Invitational paper.
⁎ Corresponding author at: Department of Petroleum Engineering, Texas A&M
University, College Station, TX 77843, United States. Tel.: + 1 979/847 8564.
E-mail address: rashid.hasan@pe.tamu.edu (A.R. Hasan).
0920-4105/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.petrol.2012.03.021
Significant advances have occurred in wellbore-fluid temperature
modeling since the pioneering work of Ramey (1962). That seminal
work addressed single-phase fluid flowing through a single conduit
in a line-source well. The line-source well implies that the logarithmic
approximation of the exponential–integral function applies for the
128
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
heat diffusion problem at hand. Subsequently, Alves et al. (1992),
Hasan and Kabir (1994), and Sagar et al. (1991) improved Ramey's
model by allowing two-phase flow, changes in well deviation, and
variable thermal properties. For obtaining the temperature of circulating fluids, such as in cementing operations or mud circulations,
Davies et al. (1994) reported that making direct measurements is
the most common method.
In the 1970s, Raymond (1969), Arnold (1970) and others offered
a numerical solution for circulating fluid temperature. However, not
until the mid-1990s, when Hasan et al. (1996) and Kabir et al.
(1996a, 1996b) advanced their models, did simple analytical expressions for calculating circulating fluid temperatures become available.
Similarly, for gas-lift operations, until the late 1990s, the primary
method available for calculating fluid temperatures was the Shiu
and Beggs (1980) empirical correlation. Although gas-lift and mud
circulation are physically similar processes, the Shiu and Beggs correlation bears no resemblance to the analytical expressions presented by Hasan and Kabir (1996). We also note that until the late1990s, there was no model or correlation available for estimating
fluid temperature when production or injection occurred through
multiple strings. Most of the models and their applications were
compiled in a textbook by Hasan and Kabir (2002). As expected,
many advances have been made since 2002, and this paper attempts
to encapsulate some of them.
In essence, this review paper offers a unifying approach to
modeling wellbore heat-transfer processes for many practical situations. Our approach depends on the common physical elements in all
cases. For example, the temperature profile surrounding the wellbore controlling heat exchange between the formation and wellbore
is the same for all these cases because the effect of the wellbore as a
heat source/sink on the infinite-acting surrounding formation remains unchanged. Resistances to axial heat transfer from the wellbore fluid to the formation are very similar in all cases. For flow
though multiple strings, the same heat-transfer principle can be applied to all strings with appropriate heat-transfer coefficients representing each element and temperature difference governing heat
flow.
This paper presents three different examples illustrating coupled fluid
and heat flow modeling. First, we discuss issues with gage placement during transient data acquisition in a borehole. Second, flow rate estimation
with wellhead temperature (WHT) and wellhead pressure (WHP) is discussed. Finally, a case study shows the importance of modeling annularpressure build-up that leads to improved management of production
rates.
radius. We also developed an expression for heat diffusion, Q (per
unit well depth), from formation to the wellbore (or vice-versa)
that applies to all wellbores irrespective of their configuration:
Q≡−
2πke
ðT wb −T ei Þ:
TD
ð1Þ
In Eq. (1), Tei is the initial (undisturbed) formation temperature,
Twb is the wellbore/formation interface temperature, and TD is the dimensionless temperature function that depends on dimensionless
producing time tD,
h
pffiffiffiffiffii
−0:2t D
−t
þ 1:5−0:3719e D
T D ¼ ln e
tD :
ð2Þ
Heat diffusion from the wellbore fluid gradually raises the temperature of the surrounding formation. Therefore, as Eq. (2) shows, even for
steady production the heat diffusion from wellbore to the formation
changes with time, but the dependence decreases with increasing time.
2.1.1. Wellbore resistances
Radial heat transfer occurs between the wellbore fluid and the formation, overcoming resistances offered by the tubing wall, tubing–
casing annulus, casing wall, and cement, as shown in Fig. 1 for production through a single string. Because the resistances are in a series,
and therefore additive, an overall-heat-transfer coefficient (based on
tubing inside area) can be defined as
1
1
r lnðr to =r ti Þ
r ti
¼
þ ti
þþ
Ut
hti
kt
r to ðhc þ hr Þ
r lnðr co =r ci Þ r ti lnðr wb =r co Þ
þ ti
þ
:
kc
kcem
ð3Þ
This general expression can be easily modified by adding or deleting resistances as the particular situation demands. Heat transferred
to the wellbore/formation interface from the wellbore fluid is, therefore, given by
Q ¼ −2πr ti U t T f −T wb :
ð4Þ
Tci
T ti
Tf
T to
Tco
Twb
In his pioneering work, Ramey (1962) modeled the wellbore as a
line source and the surrounding earth as an infinite sink. He assumed
that heat diffusion in the vertical direction is negligible compared to
that in the horizontal plane, thereby rendering the governing differential equation to be second-order linear in time (production or
shut-in) and space. Ramey adapted the solution offered by Carslaw
and Jaeger (1959) for the system. We (Hasan and Kabir, 1994) improved upon that solution by considering the wellbore to have a finite
Tei
Q
Tubing
2.1. Formation/fluid heat exchange
Casing
Earth/
Formation
Cement
Modeling heat transport in any system requires an energy balance
for the wellbore fluid. Usually, the fluid element receives heat
through fluid convection and loses heat to the surroundings through
conduction. Heat loss (or gain) to the surroundings by the wellbore
fluid depends on (i) the formation temperature distribution in the
presence of a heat source/sink like the wellbore, (ii) temperature differences, and (iii) the resistances to heat transfer within the elements
of the wellbore.
Annulus
2. Modeling approach
dto
dco
dcemo
Fig. 1. Heat flow through a series of resistances.
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
Because this heat received by the fluid must equal that coming
from the formation, we equate Eq. (1) with Eq. (4) to obtain
Q ≡−LR wcp T f −T ei
ð5Þ
where
LR ≡
2π
cP w
rti U t ke
:
ke þ ðr ti U t T D Þ
ð6Þ
Note that Eq. (5), with an appropriate expression for LR, applies to
nearly all situations one can envision when modeling wellbore heat
transfer. As we showed, even for offshore production, when LR accounts for sea-water cooling, Eq. (5) represents heat loss from the
wellbore fluid to the surroundings.
2.1.2. Energy balance for single conduit
All that remains to obtain the governing equation for fluid temperature is to write an energy balance equation for fluid in a control volume.
Fig. 2 shows the schematic that forms the basis for such an energy
balance.
For the simple case of fluid being produced or injected through a
tubing, the energy-balance equation can be written in terms of the
fluid temperature as follows:
dT f
dp 1
Q g sinα
v dv
þ
¼ CJ
∓ þ
−
dz cp
w
Jg c
Jgc dz
dz
ð7Þ
where the negative sign for Q represents the case for fluid production
and the positive sign is for fluid injection. With reasonable assumptions,
Hasan et al. (2009) rendered Eq. (7) to be a first-order differential equation and used a segmented approach to obtain the following solution:
1−eðz−zj ÞLR
T f ¼ T ei þ
LR
"
#
g sinα
z−z L
g G sinα þ φ−
þ eð j Þ R T f j −T eij :
cp
ð8Þ
Eq. (8) uses the boundary condition that the fluid temperature at
the measured distance, zj (the “entrance” to the section), is known
and designated as Tfj. The values of the parameters, Tei, gG, ϕ, α, etc.,
to be used in Eq. (6) are those that apply to this section. Fig. 3
a)
shows the model's performance for an offshore well producing a
26°API oil at 2100 STB/D with a GOR of 3000 scf/STB.
2.1.3. Heat flow through multiple conduits
In practice, there are many situations when fluid flow occurs
through multiple conduits, causing heat exchange amongst the fluids
in conduits, as well as that with the surrounding formation. Some of
the practical examples include drilling (drilling fluid descending
through the drilling tube and ascending through the annulus), gaslift, production from different zones through casing as well as through
tubing, etc. Fig. 4, displaying the last example, shows how the basic
approach discussed for flow through a single conduit can be extended
to such cases.
Let us focus on estimating the temperature profiles for both the tubing and the annular fluids. To accomplish this task, one needs to set up
an additional energy balance between the annular and tubing fluids.
The heat exchange between tubing and annular fluid is represented by the following expression
Q ta ¼ 2πr ti U t ðT a −T t Þ:
ð9Þ
Eq. (9) makes the final governing differential equation to be of
second order, which is given by
B′ d2 T t
″ dT t
′
T t þ T es þ g G zsinα þ D ¼ 0:
þB
LR dz2
dz
2.2. Transient flow
Petroleum production often involves shutting in, restarting, or
changing the production schedule. Such rate changes in a well causes
transients in mass, heat, and momentum fluxes because of the compressible nature of the fluids and the depth of modern wells. For example, when a producing well is shut-in at the wellhead, fluid
compressibility allows continuous fluid influx into the wellbore,
known as afterflow. The afterflow rate influences the heat lost by the
wellbore fluid, which in turn affects fluid properties and the pressure
Wellhead
z=0
Mudline
ad
z
Q
Tf
L
dz
z = Lj Tf = T fj
z
Tf
Q
z+dz
α
B o tt o
mhole
α
ð10Þ
Earlier, we (2002) showed the solution for tubing- and annular-fluid
temperatures for this case. A similar approach for developing fluid temperature models have been presented for gas-lift, mud circulation, and
production through two tubings by Hasan and Kabir (2002), among
others.
b)
We ll
he
129
j
z = Lj+1 Tf = T fj+1
z=L
Bottomhole
Fig. 2. Energy balance across (a) a uniform pipe angle and (b) variable angle and pipe diameter segments.
130
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
Temperature, o F
50
70
90
110
250
130
150
170
200
WHT, oF
0
3,000
Depth, ft
Model
6,000
150
100
Data
9,000
50
12,000
0
Data
Analytical
Numerical
0
15,000
18,000
10
15
20
25
Time, hr
30
35
40
45
Fig. 5. Modeling WHT during a multirate test in a gas well.
Fig. 3. Eq. (8) mimics a well's performance in an offshore setting.
profile in the wellbore. This coupled nature of fluid, momentum, and
heat flows complicates the modeling effort. Yet, proper modeling and
simulation of fluid flow and heat transfer in a well is necessary to analyze transient test data and to design/maintain flow lines and facilities.
Using the approach outlined above, Kabir et al. (1996a, 1996b) developed a transient wellbore/reservoir model by adding timedependent (or storage) terms that account for the transient and
coupled nature of these transport processes. The set of governing differential equations was discretized and solved numerically. Fig. 5
shows good agreement between their model estimated fluid temperature and data from a deep gas well in the Gulf of Mexico.
Kabir et al. (1996a, 1996b) noted that in addition to accounting for
energy accumulation (storage) of the fluid, an accurate transient
model must also account for the energy stored in the tubing/casing/
cement material in the wellbore. The wellbore thermal-storage effect
is associated with each transient period, which is depicted in Fig. 5.
Another important finding of this study was that the effect of mass
transient or afterflow on energy transport becomes negligible very
rapidly in most cases. Hasan et al. (2005) took advantage of that finding and decoupled heat transfer from the fluid and momentum transports by neglecting afterflow completely. This decoupling of heat flow
from the two other transport processes allowed for development of
the following governing equation:
!
dT f
wcp LR
wcp
∂T f
g sinθ
¼
T ei −T f þ
þ ϕ−
cp g c J
dt
mcp ð1 þ C T Þ
mcp ð1 þ C T Þ ∂z
ð11Þ
with the following analytical expression for flowing fluid temperature
as a function of depth and producing time
T f ¼ T ei þ
5
1−e−at ðz−LÞLR
ψ:
1−e
LR
ð12Þ
Fig. 5 shows that while the more rigorous numerical simulation
better represents the field data (Kabir et al., 1996a, 1996b), the
Ta
L2
T
u
b
i
n
g
Tt
Zone A
A
n
n
u
Casing
l
u
s
Q
Qta Formation/
Earth
L1
Zone B
Fig. 4. Energy balance for flow through two conduits.
analytical expression represented by Eq. (12) matches the data with
reasonable accuracy. Subsequently, Izgec et al. (2007b) improved
upon the model and showed the benefit of an analytical fluid temperature expression that can be easily combined with a reservoir model
for handling the coupled fluid and heat flow problems.
3. Applications
In this section we show applications of various heat-transfer
models associated with fluid flow in wellbores. Specifically, we discuss issues with borehole gage placement in off-bottom locations
during transient testing, rate calculations from WHP and WHT data,
and annular-pressure build-up threatening well integrity associated
with high-rate production.
3.1. Borehole gage placement in transient testing
The essence of this section owes largely to the study of Izgec et al.
(2007a, 2007b), with the underpinning work done by Kabir and
Hasan (1998) and Hasan et al. (1998). The production of both
single-phase oil and gas flow is considered in a deepwater setting.
Let us consider shut-in temperature and density profiles in a
10,000-ft well containing single-phase oil, as shown in Fig. 6. As the
figure indicates, the calculated temperature profile after 0.1 h of
shut-in is markedly different from those at late times when thermal
equilibrium is attained. The consequent changes in the density profiles in nonlinear fashion pose a challenge in pressure estimation anywhere in the wellbore as required in off-bottom gage situations. Of
course, exposure of the wellbore above the mudline, such as in a
deepwater setting adds further complications owing to the presence
of a large heat sink.
As a consequence of thermal effects, the rate of pressure change at
any point in the wellbore does not necessarily reflect the same at the
sandface for single-phase oil. Therefore, one cannot readily use pressures at any point of the column, without accounting for the effects of
temperature, to estimate the formation parameters. For single-phase
oil experiencing drawdown, we demonstrate this point by plotting
the changes in semilog slope (m*), which is directly proportional to
the formation conductivity, as the virtual gage moves up in the wellbore. Fig. 7 shows the calculated error in semilog slope as a function
of depth. The percentage error in the slope increases with decreasing
depth until it reaches an equilibrium value. For this particular case,
identification of the radial flow regime was feasible on the derivative
plot up to the mudline.
In buildup simulations, difficulties arise for identification of the radial flow regime as pressure data is gathered at increasing distances
from the sandface. The reason for this kind of behavior is twofold.
After the valve at the mudline is completely closed, the mass flow
rate at the upper parts of the wellbore diminishes rapidly. By contrast,
the incoming flow at the sandface maintains a velocity profile up to a
certain depth, thereby creating a fully transient profile throughout
the wellbore. Once the mass flow rate ceases, the temperature of
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
131
20
75
50
10
Buildup
Drawdown
25
0
0
10,000
Error in m*, %
Error in m*, %
100
0
30,000
20,000
Depth, ft
Fig. 8. Error in semilog slope for drawdown and buildup as a function of depth.
Fig. 6. Wellbore temperature and density profiles during well shut-in.
the wellbore fluid decreases rapidly, particularly around the mudline,
thereby triggering rapid changes in fluid properties.
For a buildup test, continual changes in fluid properties combined
with afterflow may mask the radial flow regime, or lead to an erroneous
interpretation on a log–log plot. The presence of gas, such as in two- or
three-phase flow, complicates the situation further. Fig. 8 presents the
percentage error on the semilog slope for a buildup test conducted at
different locations in the wellbore. Also, the error on the slope from a
drawdown run, as shown in Fig. 7, is included for comparison.
Notice that the error curve for the buildup tests disappears after
reaching a certain depth. That is because radial flow cannot be identified beyond 3000 ft from the sandface, implying that no meaningful
information can be extracted. Fig. 9 shows the derivative plot generated at 3000 ft away from the sandface. Duration of the buildup test
would not change the signature on the log–log plot in this case because at the end of the run, the bottomhole pressure is only one
psia lower than the initial-reservoir pressure.
For the single-phase gas simulations, let us consider an offshore
wellbore model with the same water depth to investigate mudline
issues during gas production. Because the gas pressure–volume–
temperature (PVT) properties are much more sensitive to temperature
changes, both flow and shut-in periods exhibited trend reversals during
simulations at increasing distances from the sandface. Because gasses
intrinsically have much lower heat capacity and, therefore, lower enthalpy than a liquid, heat dissipation occurs much faster, leading to a
trend reversal of pressure. Fig. 10 compares the bottomhole and mudline pressures for flow and shut-in periods.
During drawdown, pressure at the mudline increases even though
the bottomhole pressure declines in accord with normal behavior.
Conversely, during the shut-in period, pressure at the mudline declines continuously. This trend reversal at the mudline is a direct consequence of temperature response, which eliminates the possibility of
extracting any formation parameters with conventional pressuretransient analysis. Similarly, this inverted pressure behavior precludes one from doing wellhead-to-bottomhole pressure conversion
or reverse simulation. Fig. 11 shows the derivative plot for the shutin period generated with pressure data collected at 3000 ft from the
sandface. Here, the radial flow is hard to discern.
Fig. 12 shows the changes in mudline pressure in relation to temperature. The similarity in pressure and temperature trends clearly
demonstrates the strong connection between the two responses, as
one may surmise intuitively from the real-gas law. One consequence
of this gas-well behavior is that significant heat loss not only occurs at
the seabed, but its effect is transmitted in the form of pressure thousands of feet below the mudline.
We observed that the trend reversal for drawdown pressure is a
strong function of the gas production rate and the duration of the production period. The initial pressure trend reversal takes place regardless
of the production rate. However, the duration of this reverse-trend period depends on the rate, and with continued production the trend
gradually begins to mimic that of the bottomhole pressure. Again, placing the gage close to the sandface is the only way to ensure data quality.
3.2. Rate computation with WHP and temperature data
Izgec et al. (2009) described two methods for computing flow
rates from WHT and WHP data. Here, we provide a brief description
of those methods and present two field examples. The dependence
of fluid temperature on flow rate can be used to estimate production
(or injection) rate when fluid temperature is known. For the usual
case of a wellbore surrounded by earth, when Eq. (13) applies, the
mass flow rate w is given by
w≡
ke
1
2πr to U t
ke
f ðT Þ: ð13Þ
¼
ke þ ðrti U t T D Þ LR
cp
ke þ ðrti U t T D Þ
2πr ti U t
cp
20
Δ p , Δ p ', psi
Error in m*, %
100
10
10
1
0
0
10,000
20,000
Depth, ft
Fig. 7. Error in semilog slope as a function of depth.
30,000
0.1
1
10
Elapsed Time, hrs
Fig. 9. Log–log plot for a drawdown in an oil well.
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A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
15,700
19,530
15,650
19,510
Pressure, psia
19,520
Pressure, psia
Pressure, psia
Bottomhole
80
15,700
Mudline
70
60
15,650
50
Mudline Pressure
Temperature, o F
132
Mudline Temperature
15,600
15,600
19,500
0
10
20
30
40
40
0
50
10
30
40
50
Time, hrs
Time, hrs
Fig. 12. Pressure and temperature behavior at the mudline.
Fig. 10. Pressure at the mudline and bottomhole for a deepwater gas well.
For the submerged section of an offshore well, when Eq. (4) applies, the term in the bracket in Eq. (6) is replaced by 1, leading to
w¼
20
2πr ti U t
f ðT Þ:
cP
ð14Þ
In both Eqs. (13) and (14), the temperature function, f(T), is given by
T f −T ei −eðz−zj ÞLR T f j −T eij
:
f ðT Þ ¼ g G sinα þ φ− g sinα
1−eðz−zj ÞLR
c
ð15Þ
p
The following two examples illustrate the usefulness of the above
approach in estimating rate from temperature data.
3.2.1. Multirate gas-well test example
This example comes from a vertical gas well in a high-temperature
reservoir (Kabir et al., 1996a, 1996b), shown earlier in Fig. 5. Pressure
and temperature data were available both at the bottomhole and
wellhead from a multirate test. Fig. 13 shows that the transient
WHT was matched with the single-point method en route to the computing rate. Fig. 14 compares and contrasts the measured rates with
those obtained from the two computational methods. One important
difference between the entire-wellbore and single-point methods is
that the transient nature of rate is captured by the latter. We think
this is a significant development in that even many surface sensors
may not be able to capture the subtle rate variation during a transient
test. Put another way, the ever-changing rate, which is a direct reflection of temperature, suggests a dominating wellbore storage effect in
this 20,000-ft well with 0.35 md formation permeability.
Questions arise about the quality of rate solution. We investigated
the sensitivity of the estimated flow rate with the entire-wellbore
modeling approach. To illustrate the effect of temperature error on
rate calculations, we considered the recorded WHT for the second
rate period involving 10 MMscf/D. Fig. 15 shows that the method is
quite robust. Indeed, rate estimation is not very sensitive to WHT;
an error of 5 °F in WHT leads to a 7% error in the estimated rate.
3.2.2. Multirate oil-well test
This field example was discussed earlier by Izgec et al. (2007b). In
this deepwater setting, a combined cleanup and shut-in period of
about 70 h preceded the variable-rate test, lasting about 60 h. A
shut-in test followed. Because of the transient nature of the flow
problem, we invoked the single-point approach in estimating rate.
Fig. 16 depicts the temperature data measured at 9600 ft measured
depth (MD); the reservoir depth occurs at 26,500 ft MD. Also shown
in Fig. 16 is the quality of the temperature match, en route to computing the rate history. Fig. 17 compares and contrasts the computed rate
history with that measured at surface. Overall, the match appears
good with the exception of those occurring at the highest rates. Physical limitations of rate metering capability at surface required that the
rate in excess of 9000 STB/D be diverted to another vessel. Issues with
metering at the secondary facility precipitated the rate discrepancy.
Let us explore how the computed rate history impacts the
pressure-buildup analysis that followed the 60-hr flow period, discussed earlier by Izgec et al. (2007b). The maximum discrepancy in
rate is about 16%, occurring at approximately 25 h. Because the
point of discrepancy occurs at a distant past relative to the buildup
test, the error in the permeability-thickness product, kh, turns out to
be minimal; only 2% lower than that estimated previously. However,
had this rate discrepancy occurred just preceding the buildup test, the
magnitude of error in the kh estimation would have been directly
proportional to the rate itself. This example underscores the importance of validating rate history prior to any transient analysis. Matching data at both ends of the peak rates provides confidence in our
ability to compute rate from temperature measurements at any
point in the wellbore.
3.3. Annular-pressure buildup (APB)
Hasan et al. (2010) describe the essence of this section. The mass
of fluid trapped in an enclosed annulus may experience a significant
pressure increase when it receives heat from the producing fluids in
the tubing string and has limited or no space to expand. We may
240
210
10
Data
Calculated
o
Δ p 'n , psi
WHT, F
180
150
120
90
60
1
0.1
1
10
Elapsed Time, hrs
30
0
10
20
30
Time, hr
Fig. 11. Derivative diagnosis of gas-well build-up for a gage at 3000 ft from the
sandface.
Fig. 13. Matching WHT with a single-point approach.
40
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
20
160
o
Data
140
WHT, oF
Entire wellbore
Rate, MMscf/D
133
Single point
10
120
100
Data
Calculated
80
0
0
0
10
20
30
20
40
60
Time, hr
40
Time, hr
Fig. 16. Matching cell temperature at 9600 ft MD.
Fig. 14. Comparing measured and computed rates with both methods.
write an expression (Oudeman and Baccareza, 1995; Oudeman and
Kerem, 2004) describing the three components contributing to the
annular pressure increase by the following:
Δp ¼
αl
1
1
ΔT−
ΔV a þ
ΔV l
κT Va
κT Vl
κT
ð16Þ
where κT is the coefficient of isothermal compressibility, αl is the coefficient of thermal expansion, Vl is the volume of annular liquid, and
Va is the annular volume. The first term implies liquid expansion, the
second term accounts for volume change in the annulus owing to tubular buckling, and the third term includes liquid influx (ΔVl positive)
or efflux (ΔVl negative) in the annulus. Because the first term is by far
the most dominant in a sealed annulus, accounting for well over 80%
of pressure increase in most cases, our modeling approaches center
around this term. In the following, we describe the development of
two methods for estimating APB. In both methods, we adapt the
first term in the Oudeman–Bacarreza model and integrate it over
the entire wellbore.
To account for the changes in volume of trapped fluid as a function
depth, one needs to use the following expression:
∂P ¼ −
ð∂V=∂T Þp
∂T ¼ ðα=κ Þ∂T
ð∂V=∂p
ÞT
2
∑ M=ρ ð∂ρ=∂T Þp ∂T−fΔV gbv
¼
:
∑ M=ρ2 ð∂ρ=∂P ÞT
ð17Þ
3.3.1. Field example
High rate single-phase oil production occurs from a 12,000 ft well.
First, we modeled this well behavior with the semisteady-state
approach for pressure increase in 2002 and 2005. Pressure build-up
was observed in the 7-in production casing owing to the heating of
annular fluid by the producing fluid in the tubing string. As Fig. 18
shows, the rise in annulus pressure is directly related to the
increase in flow rate. With the increased rate, the available energy
for heat transfer increases proportionately, leading to the increased
annular pressure. Fig. 19 makes this point amply clear. Because the
fluid cannot expand in the annular confined space, its low
compressibility manifests in terms of increased pressure with
increased temperature, resulting in the APB behavior.
Fig. 20 shows the match of the semisteady-state model response
with the WHT in the tubing. Note that the early-time mismatch is a
result of the model starting production in a cold well when in actuality the well had been in production for some time. However, this mismatch did not impede the model's ability to match the annular
pressure rise, as the pressure match testifies.
In 2005, some annular liquid was bled off to relieve pressure. As a
consequence, the higher producing rate was restored while the annulus pressure decreased, as Fig. 21 exhibits. This bleed-off volume
was not reported, however. When we used a bleed-off volume of
79 gal, the model was able to reproduce the annular pressure decline
quite accurately. Fig. 22 displays both this pressure and temperature
match.
Although the semisteady-state model provided very encouraging
results, we used the transient model to compare and contrast with
those obtained earlier for the well. As Fig. 23 shows, the issue with
higher WHT prediction was improved significantly with the transient
formulation. The quality of APB prediction is comparable, as Fig. 24
demonstrates. Note that the late-time pressures reflect the bleeding of
79 gal of annular fluid.
9
14,000
6
12,000
Rate, STB/D
%Error in Rate
16,000
3
0
-3
10,000
8,000
6,000
4,000
-6
2,000
-9
Data
Calculated
0
-6
-4
-2
0
o
2
4
WHT Error, F
Fig. 15. How WHT error manifests in rate calculation uncertainty.
6
0
20
40
Time, hr
Fig. 17. Computed oil rate compares favorably with measured values.
60
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A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
Annulus Pressure, psig
Annulus Pressure, psig
1,200
1,000
800
600
400
200
1,600
1,200
Annulus
Bleedoff
800
400
0
0
5,000
10,000
15,000
20,000
25,000
0
5,000
10,000
Oil Rate, STB/D
15,000
20,000
25,000
Oil Rate, STB/D
Fig. 18. Increasing production rate causes increased heat transfer, leading to APB.
Fig. 21. Annulus pressure bleedoff leads to restoration of high-producing rates in 2006.
1,600
215
1,400
210
205
1,200
200
1,000
195
800
190
o ×
600
Data
185
Semisteady-State
400
0
1,000
2,000
3,000
4,000
Tubinghead Temp., o F
With the advent of drilling in deeper horizons both onshore and offshore, heat-transfer modeling has understandably gained importance
over the last decade. In particular, increasing water depths have presented numerous field development challenges from drilling to testing
to flow assurance. Of course, pipe metallurgy and corrosion mitigation,
among others, become integral parts of well and facility design. This
paper merely attempted to focus on a small subset of heat-flow problems related to transient testing and production monitoring. Let us reiterate the specific elements addressed here.
Placement of permanent downhole sensors is often dictated by completion hardware and other logistical issues. As a consequence, a pressure gage is often placed hundreds of feet away from the point of fluid
Annulus Pressure, psig
4. Discussion
180
5,000
Producing Time, hr
Fig. 22. Matching tubinghead temperature and annular-pressure build-up in 2006.
1,200
1,000
215
Annulus Pressure, psig
Annulus Pressure, psig
1,400
800
600
400
200
0
170
180
190
200
210
220
o
Tubinghead Temperature, F
210
205
200
195
190
Semisteady-State
185
Transient
180
0
1,000
Fig. 19. Increase in annulus pressure is directly related to increased tubinghead temperature.
1,400
2,000
3,000
4,000
5,000
Producing Time, hr
Fig. 23. Comparing tubinghead temperature predictions in 2006.
250
1,600
1,000
200
800
600
150
400
Data
200
Semisteady-State Model
0
100
0
400
800
1,200
Time, hr
Fig. 20. Matching tubinghead temperature and APB in early life.
Annulus Pressure, psig
o
Tubinghead Temp., F
Annulus Pressure, psi
1,200
1,400
1,200
1,000
Data
Semisteady-State
Transient
800
600
400
0
1000
2000
3000
4000
Producing Time, hr
Fig. 24. Comparing APB predictions in 2006.
5000
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
entry. Questions arise whether pressure data so collected are free of
wellbore thermal effects. For a given gage location, flow rate plays the
most dominant role in distorting the pressure response from a
transient-pressure analysis standpoint. In other words, the larger the
rate, the larger the thermally induced distortion. Consequences of potentially inaccurate pressure data can be significant, from both
pressure-transient and rate-transient analyses' viewpoints.
The two methods presented for estimating flow rate from temperature data have potential applications in a large spectrum of situations. For instance, rate allocation has a large uncertainty bar in
most business settings; either method can work well after calibration
is attained with dedicated well measurements. Many well completions prevent the placement of permanent sensors close to perforations. In this setting, the single-point method offers an opportunity
to capture the rate because off-bottom measurements will afford significant heat transfer between the wellbore fluid and the formation,
leading to significant temperature perturbation. While such offbottom measurements pose a challenge in estimating formation parameters with pressure-transient analysis (Izgec et al., 2007b), they
offer a better opportunity for estimating rate. Exploration-well testing is another area where the single-point method appears particularly appealing; the oil-well test is a case in point. In all cases, we view
these methods to be complementary to conventional metering.
Increased drilling in deeper waters makes the prediction and management of APB imperative. The two analytical models used in this paper provide methods for estimating APB. Real-time monitoring of pressure and
temperature in various annuli and the use of these models makes dayto-day management of APB much more viable. In fact, a calibrated
model provides clues about the maximum allowable rate commensurate
with the system's mechanical integrity. Moreover, the model can guide an
engineer on the bleed volume needed to improve the operable range of
production rate; Fig. 22 illustrates such an example. The field example
shows that both models allow easy application to managing APB en
route to preserving well integrity. In so doing, aspects of flow assurance
and real-time production management are intrinsically addressed.
5. Summary
1. A unified approach to modeling wellbore heat transfer under diverse situations has been presented, and elements common to
these situations have been identified. Analytical fluid temperature
expressions for both the single and multiple flow conduits are
available. In addition, expressions for transient flowing fluid
temperature are available for the single conduit case. These transient expressions allow coupling with fluid flow models for
handling many different types of flow problems, both steady- and
unsteady-state, encountered in the production of hydrocarbons.
2. Analyzing field data reveals that thermal effects may severely impact the outcome of gage response. Thermally-induced distortion
contributes to a larger error in semilog slope for a buildup test
than in a drawdown test. Quality of the estimated reservoir parameters suffers as the gage moves away from the perforations. The
low-heat capacity of gas and the consequent rapid heat loss induces far larger error in the semilog slope in a gas well than in
an oil well, everything else being equal.
3. Rate computation is feasible with two methods, predominantly
from temperature measurements. In the entire-wellbore approach,
both WHT and WHP are matched en route to estimating the flow
rate. In contrast, the single-point method works off of the temperature measured at any point in the wellbore. Field data corroborates the estimated rates for both gas and oil wells.
4. Wellhead fluid temperature in the tubing string is directly correlated with APB; the model allows for establishing the magnitude
of the rate reduction needed to alleviate APB. Field examples illustrate applications of the methodology used, both in terms of diagnosis and mitigation.
135
Nomenclature
a
lumped parameter defined by Eq. (A-3)
cp
specific heat capacity of fluid, Btu/lbm-°F
CT
thermal storage parameter (=m′E′/mE), dimensionless
DG
perforation-to-gage distance, ft
G
gravitational acceleration, ft/s 2
gc
conversion factor, 32.17 lbm-ft/lbf/s 2
gG
geothermal temperature gradient, °F/ft
Em*
error in semilog slope, percent
gG
geothermal gradient, °F/ft
h
formation thickness, ft
k
permeability, md
ka
thermal conductivity of annular fluid, Btu/h-ft-°F
kc
thermal conductivity of casing material, Btu/h-ft-°F
kcem
thermal conductivity of cement, Btu/h-ft-°F
ke
thermal conductivity of earth, Btu/h-ft-°F
LR
relaxation parameter defined by Eq. (2)
M
mass of fluid per ft of wellbore, lbm/ft
m′
mass of wellbore system per unit depth, lbm/ft
m*
semilog slope (=162.6 Bμ/kh), psi/log-cycle
pn
normalized pseudopressure, psia
pwf
flowing bottomhole pressure, psig
pwh
flowing wellhead pressure, psig
qo
oil flow rate, STB/D
Q
heat flow rate per unit length of wellbore, Btu/h-ft
Qta
heat flow between tubing and annulus fluid per unit length
of wellbore, Btu/h-ft
rci
casing inside radius, ft
rco
casing outside radius, ft
rti
tubing inside radius, ft
rto
tubing outside radius, ft
rw
wellbore radius, ft
s
mechanical skin, dimensionless
t
producing time, h
tD
dimensionless time, h
Ta
annulus fluid temperature, °F
TD
dimensionless temperature
T or Tf
fluid temperature, °F
Te
earth or formation temperature, °F
Tei
undisturbed earth or formation temperature, °F
Tt
tubing fluid temperature, °F
Twb
wellbore/earth interface temperature, °F
Ut
overall-heat-transfer coefficient, Btu/h-ft 2-°F
V
fluid velocity, ft/s
Va
annular volume, ft 3
Vl
annular volume occupied by liquid, ft 3
ΔVa
change in annular volume due to buckling, ft 3
ΔVl
volume of liquid influx/efflux in the annulus, ft 3
W
mass rate, lbm/h
Z
variable well depth from surface, ft
γo
oil gravity, °API
γg
gas gravity (air = 1), dimensionless
α
well inclination from horizontal, degree
μ
fluid viscosity, cp
ρe
earth density, lbm/ft 3
ϕ
lumped parameter used in Eq. 8, °F/ft
ψ
gGsinθ + ϕ − (gsinθ/cpJgc)
References
Alves, I.N., Alhanati, F.J.S., Shoham, O., 1992. A unified model for predicting flowing
tempertaure distribution in wellbores and pipelines. SPEPE 7 (6), 363–367.
Arnold, F.C., 1970. Temperature variation in a circulating wellbore fluid. J. Energy
Resour. Technol. 112 (79), 670–675. doi:10.1115/1.2905726.
136
A.R. Hasan, C.S. Kabir / Journal of Petroleum Science and Engineering 86–87 (2012) 127–136
Carslaw, H.S., Jaeger, J.C., 1959. Conduction of heat in solids, Second Edition. Oxford Science Publications, London.
Davies, S.N., et al., 1994. Field studies of circulating temperature under cementing conditions. SPEDC 9 (1), 12–16.
Hasan, A.R., Kabir, C.S., 1994. Aspects of heat transfer during two-phase flow in wellbores. SPEPF 9 (3), 211–216.
Hasan, A.R., Kabir, C.S., 1996. A mechanistic model for computing fluid temperature
profiles in gas-lift wells. SPEPF 11 (3), 179–185.
Hasan, A.R., Kabir, C.S., 2002. Fluid flow and heat transfer in wellbores. Society of Petroleum Engineers, Richardson, Texas.
Hasan, A.R., Kabir, C.S., Ameen, M., 1996. A fluid circulating temperature model for
workover operations. SPEJ 1 (2), 133–144.
Hasan, A.R., Kabir, C.S., Wang, X., 1998. Wellbore two-phase flow and heat transfer during transient testing. SPEJ 3 (2), 174–180.
Hasan, A.R., Kabir, C.S., Lin, D., 2005. Analytic wellbore temperature model for transient
gas-well testing. SPEREE 8 (3), 240–247.
Hasan, A.R., Kabir, C.S., Wang, X., 2009. A robust steady-state model for flowing-fluid
temperature in complex wells. SPEPO 24 (2), 269–276.
Hasan, A.R., Izgec, B., Kabir, C.S., 2010. Sustaining production by managing annularpressure buildup. SPEPO 25 (2), 195–203.
Izgec, B., Kabir, C.S., Zhu, D., Hasan, A.R., 2007a. Transient fluid and heat flow modeling
in coupled wellbore/ reservoir systems. SPEREE 10 (3), 294–301.
Izgec, B., Cribbs, M.E., Pace, S., Zhu, D., Kabir, C.S., 2007b. Placement of permanent
downhole pressure sensors in reservoir surveillance. SPEPO 24 (1), 87–95.
View publication stats
Izgec, B., Hasan, A.R., Kabir, C.S., 2009. Flow rate estimation from wellhead pressure and
temperature data. SPEPO 25 (1), 31–39.
Kabir, C.S., Hasan, A.R., 1998. Does gauge placement matter in downhole transient-data
acquisition? SPEREE 1 (1), 64–68.
Kabir, C.S., Hasan, A.R., Jordan, D.L., Wang, X., 1996a. A wellbore/reservoir simulator for
testing gas wells in high-temperature reservoirs. SPEFE 11 (2), 128–134.
Kabir, C.S., Hasan, A.R., Kouba, G.E., Ameen, M., 1996b. Determining circulating fluid
temperature in drilling, workover, and well control operations. SPEDC 11 (2),
74–79.
Oudeman, P., Baccareza, L.J., 1995. Field trial results of annular pressure behavior in a
high-pressure/high-temperature well. SPEDC 10 (2), 84–88.
Oudeman, P., Kerem, M., 2004. Transient behavior of annular pressure buildup in HP/
HT wells. Paper SPE 88735 presented at the 11th Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, UAE, 10–13 October.
Ramey Jr., H.J., 1962. Wellbore heat transmission. JPT 14 (4), 427–435 Trans., AIME,
225.
Raymond, L.R., 1969. Temperature distribution in a circulating drilling fluid. JPT 21 (3),
333–341.
Sagar, R.K., Doty, D.R., Schmidt, Z., 1991. Predicting temperature profiles in a flowing
well. SPEPE 6 (6), 441–448.
Shiu, K.S., Beggs, H.D., 1980. Predicting temperatures in flowing oil wells. J. Energy
Resour. Technol. 102 (2). doi:10.1115/1.3227845.