Modeling Reservoir Temperature
Transients and Reservoir-Parameter
Estimation Constrained to the Model
Obinna O. Duru, SPE, and Roland N. Horne, SPE, Stanford University
Summary
Permanent downhole gauges (PDGs) provide a continuous source
of downhole pressure, temperature, and sometimes flow-rate data.
Until recently, the measured temperature data have been largely
ignored, although a close observation of the temperature measurements reveals a response to changes in flow rate and pressure. This
suggests that the temperature measurements may be a useful source
of reservoir information.
In this study, reservoir temperature-transient models were
developed for single- and multiphase-fluid flows, as functions of
formation parameters, fluid properties, and changes in flow rate
and pressure. The pressure fields in oil- and gas-bearing formations
are usually transient, and this gives rise to pressure/temperature
effects appearing as temperature change. The magnitudes of these
effects depend on the properties of the formation, flow geometry,
time, and other factors and result in a reservoir temperature distribution that is changing in both space and time. In this study, these
thermometric effects were modeled as convective, conductive, and
transient phenomena with consideration for time and space dependencies. This mechanistic model included the Joule-Thomson
effects resulting from fluid compressibility and viscous dissipation
in the reservoir during fluid flow.
Because of the nature of the models, the semianalytical solution technique known as operator splitting was used to solve them,
and the solutions were compared to synthetic and real temperature
data. In addition, by matching the models to different temperaturetransient histories obtained from PDGs, reservoir parameters such
as average porosity, near-well permeabilities, saturation, and some
thermal properties of the fluid and formation could be estimated.
A key target of this work was to show that temperature measurements, often ignored, can be used to estimate reservoir parameters,
as a complement to other more-conventional techniques.
Introduction
The behavior of pressure transients in reservoir and wellbore flow
has been studied extensively and has been applied in conventional
well-test analysis for reservoir description, parameter estimation
for formation characterization, and evaluation of well and field
performance. However, in most of these conventional pressuretransient analyses, the temperature distributions in the reservoir
and wellbore have been assumed isothermal. Although the flow
is generally nonisothermal, the temperature changes have been
considered to be relatively small and, hence, negligible for consideration in the analysis of flow behavior.
Most authors in petroleum reservoir engineering have studied
heat transfer mainly as a wellbore phenomenon. Ramey (1962)
developed a model for the prediction of wellbore-fluid temperature as a function of depth for injection wells and expanded this
model to give the rate of heat loss from the well to the formation,
assuming steady-state heat flow in the wellbore and unsteady radial
conduction in heat transfer to the Earth. Horne and Shinohara
Copyright © 2010 Society of Petroleum Engineers
This paper (SPE 115791) was accepted for presentation at the SPE Annual Technical
Conference and Exhibition, Denver, 21–24 September 2008. Original manuscript received
for review 3 March 2009. Revised manuscript received for review 11 March 2010. Paper
peer approved 22 June 2010.
December 2010 SPE Reservoir Evaluation & Engineering
(1979) presented single-phase heat-transmission equations for both
production and injection geothermal-well systems by modifying
Ramey’s model. Shiu and Beggs (1980) presented another modification of Ramey’s model to predict the wellbore-temperature
profile for a producing well, in which the temperature of fluid
entering the wellbore from the reservoir is known. Sagar et al.
(1991) developed a steady-state two-phase model for the wellboretemperature distribution accounting for Joule-Thomson effects
owing to heating/cooling caused by pressure changes within the
fluid during flow.
A number of authors have examined heat transfer in porous
media, during flow of the reservoir fluids. A good number of
these studies are in groundwater hydrology, with a few in petroleum reservoir engineering. Bravo and Jiang (2001) showed how
groundwater-temperature measurements can be used to constrain
parameter estimation in a groundwater-flow model. The order of
magnitude of the changes in the temperature measurements used in
their study is close to that seen in oil- and gas-flow systems (a few
degrees Fahrenheit). Woodbury and Smith (1988) and Rath et al.
(2006), also presented similar findings, in which thermal data could
be inverted with hydraulic data for reservoir-parameter estimation.
Again, in orders of magnitude, these temperature measurements
compare with those seen for oil and gas systems.
Valiullin et al. (2004) presented a simplified treatment of the
temperature distribution in the formation when the pressure field
in the reservoir changes. They showed that adiabatic and JouleThomson effects and effects caused by heat of phase transition
(gas liberation from oil) may be present during fluid flow in a
hydrocarbon-saturated porous medium. Ramazanov and Parshin
(2006) went on to develop an analytical model that described the
formation temperature distribution in a reservoir, while accounting
for phase transitions. They solved a steady-state convective thermal flow model with constant flow rate and extended it to cases
with phase changes. Ramazanov and Nagimov (2007) presented a
simple analytical model to estimate the temperature distribution in
a saturated porous formation with variable pressure but constant
flow rate. Their investigation showed that for a single-phase fluid
in a homogeneous reservoir, temperature/pressure effects such as
Joule-Thomson behavior can cause the temperature in the reservoir
to vary significantly when reservoir pressure is changing in time.
Models to couple the wellbore and reservoir systems in solving the heat-transfer problem have been presented by Dawkrajai
(2004), Izgec et al. (2007), and Yoshioka (2007). Izgec et al.
(2007) provided a transient-wellbore-temperature simulator coupled with a variable-Earth-temperature scheme for predicting
wellbore-temperature profiles in flowing and shut-in wells. Their
study looked at the mechanism of heat transfer in the wellbore and
the interaction with surrounding formation without consideration
for possible changes in the reservoir-fluid temperature before entry
into the wellbore. Dawkrajai (2004) and Yoshioka (2007) solved
the full energy-balance equation for the temperature distribution in
a reservoir, coupled to a wellbore model. Both approaches made
considerations for Joule-Thomson and frictional-heating effects
but assumed a constant flow rate and steady-state conditions in
arriving at the solution.
Comprehensive models for heat transport in a porous medium
from mass, energy, and momentum balance are presented in
Bear (1972), Bejan (2004), and Nield and Bejan (1998). Thermal
873
4
9000
2.5
×10
8900
2
8700
Flowrate, bbl/d
Pressure, psi
8800
8600
8500
8400
8300
1.5
1
0.5
8200
8100
(a)
0
200
400
600
800
0
1000
0
200
(b)
Time, hrs
400
600
800
1000
Time, hrs
Fig. 1—Pressure data (a) and flow-rate data (b).
diffusion, convection, effects resulting from the fluid compressibility, and viscous dissipation (mechanical power required to extrude
the fluid through the pore) are incorporated into the models, and
the final forms take the structure of convection/diffusion models
with source/sink terms.
Many of these previous attempts at developing an interpretation method for temperature profiles in wellbore/reservoir systems
have remained largely qualitative. While some have concentrated
on wellbore thermal exchanges because of conduction and convection, assuming that the produced fluid enters the wellbore at the
geothermal temperature, others have constrained their analyses to
convective effects only in steady-state formulations. A few have
considered the effects of heating or cooling of the produced fluid
before it enters the wellbore caused by factors such as the JouleThomson effect, adiabatic expansion, and viscous dissipation.
Unlike previous studies, this work presents models that consider
reservoir thermometric effects, modeled as convective, conductive,
and transient phenomena with consideration for time (transient)
and space dependencies, by considering the full underlying physical phenomena. The models and solutions presented here apply to
both steady- and unsteady-state heat transfer, and they represent
coupled reservoir and wellbore systems.
Demonstration of Functional Relationship
Using Optimal Transformations
To proceed with the study of the physics behind the observed
temperature response to changes in flow rate and pressure, there
is the need to establish the existence of a functional relationship
between the temperature, pressure, and flow-rate signals. The
technique chosen for this investigation was the iterative nonparametric regression tool, known as alternating conditional expectation (ACE), originally proposed by Breiman and Friedman (1985).
ACE allows for the estimation of optimal transformations that
may lead to the maximal multiple correlation between a response
variable (temperature in this case) and a set of predictor variables
(pressure, rate, and time), and these transformations are useful in
establishing the existence of a functional relationship between the
response variable and the predictor variables. ACE yields optimal
transformations of the variables, and the correlations between
these transformations have been shown to be optimal for regression between the variables. Here, using a field-data set obtained
from a permanent downhole monitoring tool in a well, the ACE
method was applied to the pressure, rate, and temperature data to
establish the existence or otherwise of a correlation and functional
form for their relationship.
Figs. 1 and 2 show the plots of rate, pressure, and temperature
data and the plot of the regression on the optimal transformations.
874
The optimal transformation functions (Fig. 2b) showed a correlation coefficient of 0.99. Essentially, this indicates the ability to
predict one of the three signals based on the values of the other
two, signifying that temperature is well correlated with flow rate
and pressure, that a functional relationship may exist between
them, and that this functional form can be extracted from any
representative data set.
Reservoir Temperature Model
The model for the flow of energy-carrying fluids through a porous
medium involves formulating the energy- and material-conservation
equations. Bejan (2004) and Bear (1972) have presented a comprehensive thermodynamic approach to obtaining a representative model for
temperature distribution in a porous medium. The model accounted
for spatial distribution and transient effects in the formation.
In a flowing well, the pressure and flow-rate measurements
recorded by permanent monitoring gauges are not constant. For
gauges placed close to the sandface flow area, these changes
reflect effects resulting from flow in the reservoir, and they cause
a temperature field to develop in the reservoir, driven primarily
by thermodynamic effects such as Joule-Thomson heating (or
cooling), adiabatic expansion, and heat of phase transitions. Other
effects such as viscous dissipation, equal to the mechanical power
needed to extrude the viscous fluid through the pore, and frictional
heating between the fluid and rock matrix during the fluid flow
are also factors that contribute to the evolution of a nonuniform
temperature field in the medium.
The Joule-Thomson effect is the change in the temperature of
a fluid caused by expansion or compression of the fluid in a flow
process involving no heat transfer or work (constant enthalpy),
and it results from a combination of the effects of fluid compressibility and viscous dissipation. The Joule-Thomson effect caused
by the expansion of oil in a reservoir or wellbore results in the
heating of the fluid because the value of the Joule-Thomson coefficient is negative for oil. The coefficient has a positive value for
real gases, and the consequent behavior for gas flow is a cooling
effect. Theoretically, the Joule-Thomson coefficient for ideal gases
is zero. Combined with other factors, upon expansion of the fluid
and subsequent flow of liquid oil and/or water out of the reservoir,
the wellbore and near-wellbore regions in the reservoir become
heated above the normal static reservoir temperature. By convection, diffusion, and further generation of heat energy owing to
these effects, a nonuniform temperature is created, which spreads
into the reservoir. Conversely, during no-flow conditions (shut-ins),
the regions already heated lose heat to the surrounding formation
through conduction and the result is a temperature decline at a rate
determined by the thermal diffusivity of the medium.
December 2010 SPE Reservoir Evaluation & Engineering
2
146.5
1.5
146
0.5
145
Σ Φi
Temperature, °F
1
145.5
144.5
0
-0.5
144
-1
143.5
-1.5
143
0
200
400
(a)
600
800
1000
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Φ0
(b)
Time, hrs
Fig. 2—Temperature data (a) and optimal regression with ACE (b).
Reservoir Temperature Model in 1D Cylindrical Coordinate
System. Single-Phase Formulation. To derive the energy equation for a homogeneous porous medium, the energy equations for
the solid and fluid parts are derived separately from the first law
of thermodynamics and are averaged over an elemental control
volume to obtain the general form of the model. The consideration
is for nonisothermal flow of a nonideal fluid in a porous medium.
The change in kinetic and potential energies of the flow will be
taken as negligible. An assumption of local thermal equilibrium
between the fluid and the porous matrix will also be made. Using
volumetric averaging to combine the heat transfer model in the
solid matrix with the model from the fluid heat transport gives
the general form
∂T
␳ c Q 1 ∂T
⎡⎣(1 − ␾ )(␳s cs ) + ␾ (␳ f c f ) ⎤⎦ + f f
∂t
2␲ h r ∂r
⎡⎣(1 − ␾ )␭s + ␾␭ f ⎤⎦ ∂ ∂T
∂p
=
+ ␤T ␾
r
∂t
∂r ∂r
Q 1 ∂p
+ v␳ g
+
−
2␲ h r ∂r 2␲ h r ∂r
r
␤TQ 1 ∂p
. . . . . . . . . . . . . (1)
The mass-balance equation takes the form
␳
∂␳ f
∂t
+
1 ∂(rv␳ f )
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
r ∂r
The flow is assumed to obey Darcy’s law and the equation for
Darcy flow, which is given by
v=
k ∂p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
␮ ∂r
Eqs. 2, 4, and 5 form the governing equations for 1D thermal
transport in a homogeneous porous medium. Other assumptions
made in deriving the equations include a homogeneous, isotropic
medium and thermal equilibrium between the solid matrix and the
fluid in the pores.
Multiphase Formulation. Assumptions made for the multiphase formulation are similar to those made in the single-phase
case, with the addition of negligible capillary effects. The thermal
model in a 1D radial coordinate system for the multiphase system
becomes
␳ f cfQ
and other
2␲ h␭e
␣t
p
r
T − T∞
dimensionless variables, rD = , TD =
, tD = 2 , ε D = ε o ,
T∞
rw
T∞
rw
p
p
␩ D = ␩ o , and pD = , and rearranging the formulation and
T∞
po
neglecting gravity effects, the equation becomes
∂T
⎡⎣(1− ␾ )(␳s cs ) + ␾ (␳ wc fw sw + ␳oc fo so ) ⎤⎦
∂t
∂T
+ ␳ wc fw sw vw + ␳oc fo so vo + ␳ gc fg sg vg
∂r
⎡(1 − ␾ ) ␭s + ␾ ␭w sw + ␭o so + ␭g sg ⎤ ∂ ∂T
⎦ r
=⎣
. . . . . . . . . . . . (6)
∂r ∂r
r
∂p
+ ␳ wc fw sw␩ w + ␳oc fo so␩o + ␳ gc fg sg␩ g ␾
∂t
∂p
+ ␳ wc fw sw vwεw + ␳oc fo so voεo + ␳ gc fg sg vgεg
∂r
∂TD Pe ∂TD 1 ∂
Pe ∂pD
∂T
∂p
.
+
=
rD D + ␩ D␾C D + εD
∂t D rD ∂rD rD ∂rD ∂rD
rD ∂rD
∂t D
. . . . . . . . . . . . . . . . . . . . . . . . (2)
On rearrangement, and keeping the formulation in dimensional
form, the equation reduces to
Defining a dimensionless Péclet number Pe =
The initial and boundary conditions are
TD ( rD , 0 ) = 0,
pD ( rD , 0 ) = 1,
TD ( rD → ∞, t D ) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
and pD ( rD → ∞,t D ) = 1
The form of Eq. 2 is the convection/diffusion equation with source/
sink terms. The second term on the right-hand side of Eq. 2 is the compressibility term, while the last term is the viscous-dissipation term.
December 2010 SPE Reservoir Evaluation & Engineering
(
(
(
(
)
)
)
)
∂T
∂T ␣ ∂ ∂T
∂p
∂p
+u
=
+ ␩ * + J * . . . . . . . . . . . . . . . . . (7)
r
∂t
∂r r ∂r ∂r
∂t
∂r
The mass-balance equations and the Darcy flow equations,
respectively, take the form
∂(␳ w sw ) 1 ∂(rvw ␳ w sw )
+
= 0,
∂t
r
∂r
∂(␳o so ) 1 ∂(rvo ␳o so )
␾
= 0,
+
r
∂r
∂t
∂(␳ g sg ) 1 ∂(rvg ␳ g sg )
␾
+
=0
r
∂t
∂r
␾
; . . . . . . . . . . . . . . . . . . . . . (8)
875
By the method of characteristics,
and
k ⎛ ∂p ⎞
k ⎛ ∂p ⎞
k ⎛ ∂p ⎞
vw = w ⎜ w ⎟ , vo = o ⎜ o ⎟ , vg = g ⎜ g ⎟ . . . . . . . (9)
␮ w ⎝ ∂r ⎠
␮o ⎝ ∂r ⎠
␮ g ⎝ ∂r ⎠
Eqs. 6 through 9 are the formulation for the temperature distribution in a reservoir during multiphase flow, with initial and boundary
conditions similar to those defined for the single-phase case.
Solution
Operator-Splitting Method. Eqs. 2 and 7 are convection/
diffusion-type equations with forcing terms. Analytical solutions
for convection/diffusion equations exist, but with the forcing
terms, in the form that appears in Eqs. 2 and 7, the equations
become more complicated and not easily amenable to analytical solutions. The operator-splitting method (Valocchi and
Malmstead 1992; Khan and Liu 1995; Kacur and Frolkovic
2002; Remešiková 2004) is a semianalytic method that allows
the model to be broken into two different parts. Then, at each
timestep, the nonlinear transport part and the nonlinear diffusion
part are solved separately. Holden et al. (2000) showed the theoretical basis for this technique. An advantage of this process is
that appropriate and different solution techniques may be applied
to the two parts, and where numerical solutions are used, different
timesteps can be chosen for each part.
In this work, the operator-splitting approach was used to solve the
thermal model given by Eqs. 2 and 7. The method adopted was
1. Decouple the model into two parts: the convection-transport
part and the diffusion part.
2. At each timestep, first solve the hyperbolic convection-transport
part, accounting for variable flow rate, and heat generation because
of viscous-dissipation, frictional, and Joule-Thomson effects.
3. Solve the diffusion part at the same timestep, adaptively modifying the timestep to ensure stability if the solution is numerical.
4. Continue until the last timestep.
Solution of the Thermal Models by Operator Splitting. The
assumptions made in solving the single-phase reservoir thermal
model were constant fluid Joule-Thomson coefficient, adiabatic
expansion coefficient, and thermal conductivities (these parameters
are assumed to be weak functions of temperature) and constant
fluid viscosity and formation porosity.
Solution of the Convective-Transport Part for Single-Phase
Flow. The convection equation with its initial condition becomes
∂TD Pe ∂TD
∂p
Pe ∂pD
, . . . . . . . . . . . . . . (10)
+
= ␩ D␾C D + εD
∂t D rD ∂rD
∂t D
rD ∂rD
TD (t D = 0) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)
Using the method of characteristics yields
rD2 = rD21 + 2 Pe ⋅ t D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)
and
TD = TD (rD , 0) + ε D [ pD − pD (rD , 0)]
−
␩# + εD
ln R
tD
∫
0
, . . . . . . . . . . . . . . . . . . . (13)
∂pD ( rD , t D )
d␶ D
∂␶ D
where ␩ = ␩ D␾C . The integral can be evaluated numerically, or
analytically, using the classical solutions of pressure-transient
problems. In the work, both approaches were used and gave
similar results.
Solution of the Convective-Transport Part for Multiphase
Flow. The convection equation with its initial condition is
#
∂T
∂T
∂p
∂p
+C
= ␩* + J *
∂t
∂r
∂t
∂r . . . . . . . . . . . . . . . . . . . . . . . . . (14)
T (t = 0) = To (r )
876
⎡
⎛
⌳o ⌳g ⎞ ⎤
−
⎥
⎢ ␳ wc fw sw ⎜ 1 −
⌳
⌳T ⎟⎠ ⎥
⎝
Q
T
r 2 = r12 + T ⎢
( t − t1 ), . . . . . . . . . (15)
␲h ⎢
⌳g ⎥
⌳
⎥
⎢ + ␳oc fo so o + ␳ gc fg sg
⌳T
⌳T ⎥⎦
⎢⎣
⌳ w , ⌳ o , ⌳ g , and ⌳T
are
the
water
mobility,
where
oil mobility, gas mobility, and total mobility, respectively, and
⎛ ␳ c s ␩ + ␳oc fo so␩o ⎞
⎛ ␳ c s v ε + ␳oc fo so voεo ⎞
␩ * = ⎜ w fw w w
␾ , J * = ⎜ w fw w w w
⎟
⎟⎠ .
cm
cm
⎝
⎠
⎝
The final form of the solution closely follows the solution for
the single-phase case:
T (r , t ) = T0 (r ) + ε * [ p(r , 0) − p(r , t )]
␩ * + ε * ∂p ( r , t )
d␶
ln R ∫0 ∂␶
t
−
, . . . . . . . . . . . . . . . . . . (16)
where
ε* =
␳ wc fw swε w␭w + ␳oc fo soε o␭o
. . . . . . . . . . . . . . . . . . . . . . (17)
␳ wc fw sw␭w + ␳oc fo so␭o
Again, the integral can be evaluated numerically or analytically, at
each timestep, as in the single-phase case.
Solution of the Diffusion Part. The form of the diffusion
problem is similar for both single and multiphase formulations and
takes the same general solution.
∂TD 1 ∂TD ∂ 2TD
=
+
∂t D rD ∂rD
∂rD2
0 < r < ∞, . . . . . . . . . . . . . . . . . . (18)
with initial and boundary conditions:
TD (t D = 0) = F (rD )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)
TD (rD = 0) = finite
Özişik (1993), using the method of integral identity described
by Masters (1955), showed that the solution to Eq. 18 and Eq. 19
is of the form
TD (rD , t D ) =
1
2t D
⎡ '
⎛ rD2 + rD' 2 ⎞
⎛ rD rD' ⎞ ⎤ '
'
r
exp
−
(
)
I
F
r
⎢
D
D
o
∫
⎜⎝
⎜⎝ 2t ⎟⎠ ⎥ drD ε ,
4 t D ⎟⎠
D
⎦
rD' ⎣
. . . . . . . . . . . . . . . . . . . . . . . (20)
bD
where 0 < bD < ∞ is equivalent to the dimensionless thermal-diffusivity length and takes the form b = t D . The solution of the
diffusion part for the multiphase problem is also similar to Eq. 20,
with the appropriate terms.
Therefore, in the operator-splitting approach, at each timestep,
Eqs. 13 and 16 are evaluated for the solution of the convective part
at that step. Then, the solution forms the initial condition F (rD )
[or F (r )] in the solution of the diffusion problem, Eqs. 18 and 19.
The final solution obtained in Eq. 20 is taken as the temperature
of the system at that timestep.
Wellbore Temperature-Transient Model and Coupling With
Reservoir Model. Downhole monitoring tools are usually located
some feet (usually more than 30 m) above the perforation/production zone. The tool-placement constraint is one that is imposed by
the design of the completions, although the optimal location for
pressure- and temperature-data management would be a position
as close to the perforations as possible, to give measurements that
are comparable with their sandface values. This disparity in location calls for a coupling of the reservoir temperature model to a
wellbore model to account for heat loss that may occur between
December 2010 SPE Reservoir Evaluation & Engineering
146.5
146.5
146
146
145.5
Temperature, °F
Temperature, °F
145.5
φ=0.05
φ=0.1
φ=0.2
φ=0.3
φ=0.4
φ=0.5
φ=0.6
145
144.5
145
k=50 md
k=100 md
k=200 md
k=400 md
k=800 md
k=1600 md
k=3200 md
144.5
144
143.5
144
0
5
10
(a)
15
20
25
143
30
0
5
10
(b)
Time, hrs
15
20
25
30
Time, hrs
Fig. 3—Sensitivity to porosity (a) and permeability (b).
the fluid in the wellbore and the surrounding formation when the
fluid flows from the sandface to the gauge location.
The wellbore model used in this work was obtained by modifying the solution proposed by Izgec et al. (2007).
T f (r , t ) = Tei
1 − e − aLR t
+
LR
where LR =
⎧
⎡
( g sin ␪ ) ⎤ ⎫
( z − L ) LR
⎤⎦ ⎢ gG sin ␪ + ⌿ −
⎪⎪ ⎡⎣1 − e
⎥ ⎪, . . . . . . (21)
c p Jgc ⎥⎦ ⎪⎬
⎢⎣
⎨
⎪
⎪ ⎡
− L e ( z − L ) LR T fin − Tein ⎤⎦
⎭⎪
⎩⎪ ⎣ R
(
)
2␲ ⎛ rtoU to ke ⎞
,
wc p ⎜⎝ ke + rtoU toTD ⎟⎠
␣t
dp
TD = ln ⎡⎣e −0.2 tD + 1.5 − 0.3719e − tD ⎤⎦ t D , t D = 2 , ⌿ = ε , ␪ is
rw
dz
(
)
wellbore-inclination angle, T fin is fluid temperature entering the sandface from formation (solution to the reservoir temperature model),
Tei is geothermal temperature of the formation at gauge location,
and Tein is geothermal temperature of the formation at bottomhole.
In principle, the closer the gauge is to the perforations, the
more able the overall model would be to resolve effects caused
by the reservoir flow. If the gauge is moved farther away from the
perforations, then the model begins to lose the ability to capture
effects caused by the reservoir flow mechanism because most of
the temperature change will be caused by the wellbore heat loss.
The two models are coupled through the temperature of the fluid at
the bottomhole location. The bottomhole temperature is estimated
using the reservoir model and is used as an input into the wellbore
model to estimate the fluid temperature at the gauge location. The
sensitivity of the overall model to the distance between the gauge
and the perforations will be considered in a sensitivity analysis.
Results and Discussion
Sensitivity Analysis. The thermal model contains many parameters, some of whose values are uncertain. In the sensitivity analysis
presented in Figs. 3 through 5, some of the important parameters
were varied up to orders of magnitude that are similar to their
actual variability in reality, and the model solution was checked
for its sensitivity to these values. The parameters tested were the
permeability and porosity of the formation, the Joule-Thomson and
adiabatic-expansion coefficients of the fluids, thermal-diffusivity
length, and distance of the permanent downhole gauge from the
perforations. Some other parameters in the model can be obtained
easily from measurements and laboratory testing. These parameters include the fluid viscosities, gauge distance from perforations,
densities, and thermal conductivities of the rock and fluid. Others
146
150
145
Temperature, °F
Temperature, °F
145.8
145.6
145.4
ε =4e−12 K/Pa
ε =4e−11 K/Pa
ε =4e−10 K/Pa
ε =4e−09 K/Pa
ε =4e−08 K/Pa
ε =6e−08 K/Pa
ε =8e−08 K/Pa
145.2
145
144.8
(a)
140
135
130
125
120
115
0
5
10
15
20
Time, hrs
25
μ=1e−09 K/Pa
μ=3e−09 K/Pa
μ=5e−09 K/Pa
μ=8e−09 K/Pa
μ=1e−08 K/Pa
μ=2e−08 K/Pa
μ=3e−08 K/Pa
30
(b)
0
5
10
15
20
25
30
Time, hrs
Fig. 4—Sensitivity to Joule-Thomson coefficient (a) and adiabatic-expansion coefficient (b).
December 2010 SPE Reservoir Evaluation & Engineering
877
146
146
145.5
145
Temperature, °F
Temperature, °F
145
144
143
142
β=6 m
140
0
20
40
(a)
144
143.5
h=5 m
h=10 m
h=20 m
h=40 m
h=80 m
h=160 m
h=320 m
143
β=9 m
β=12 m
β=15 m
β=20 m
141
144.5
142.5
60
80
100
142
120
0
5
10
(b)
Time, hrs
15
20
25
Time, hrs
Fig. 5—Sensitivity to thermal-diffusivity length (a) and gauge distance from perforation (b).
the perforations is large and, hence, the parameters estimated in
the inverse problem come with a large confidence interval. On the
other hand, using DTS data, or PDG data, in which the gauge is
close to the perforation allows the model to resolve effects caused
by the reservoir flow without being masked excessively by the heat
loss in the wellbore. In this work, we found that a gauge situated
up to 80 m above the perforation would still be able to capture
effects caused by the reservoir flow, without excessive masking
from the wellbore heat loss. However, this number depends on the
perforations and the types of fluid flowing in the system and may
not be generally optimal.
Results for Synthetic Data. Synthetic data were generated using
the thermal model developed, and some flow-rate and pressure history. The model used in estimating the pressure from the flow rate
was the infinite-acting radial-flow model. Then, 5% random noise
was added to the generated data to create a second data set. The
synthetic data (true and noisy) are needed to check the robustness
of the model in parameter estimation. In essence, the data were
used as measurements in the inverse problem to re-estimate some
of the model parameters. Two tests were carried out on each data
set. In the first test, only one parameter was estimated (porosity).
In the second test, the number of variables was increased to five
to check how much uncertainty would be added to the estimations
by an increase in the number of unknowns.
147.5
147.5
147
147
Temperature, °F
Temperature, °F
that cannot be measured with a high degree of certainty enter the
parameter space for the inverse problem. Figs. 3 through 5 show
that the model solutions are sensitive to the model parameters
tested. Because the parameters were varied within the actual range
of magnitude of possible changes seen in real cases, it is clear that
these parameters can affect the shape and magnitude of the solution
and, in effect, the temperature distribution in the reservoir. Without
consideration for effects of noise (for the moment), it is clear that
permeability, porosity, adiabatic-expansion coefficient, and thermal-diffusivity length are parameters that affect the solution most
prominently. The Joule-Thomson coefficient is prominent only at
the points where there is a sudden change in the flow-rate behavior
for single-phase flows. It is also prominent in gas flows (shown in
the two-phase example case).
The effect of the gauge distance from the perforation zone is
also prominent and important, as shown in Fig. 6b. At larger distance, the temperature drop is larger and the drop results mainly
from the wellbore heat losses. At lower distance, the temperature
drop mainly is caused by effects from the reservoir. Therefore, in
order to obtain reservoir parameters from the solution, it is best
to have the gauge situated as close to the reservoir as possible.
Where this is not possible, the use of distributed-temperature-survey (DTS) data obtained at the sandface would be an appropriate
alternative. This is because the model loses the ability to resolve
effects caused by the reservoir flow when the gauge distance from
146.5
146
145.5
146.5
146
145.5
Model after match
Data with noise
Data
Model after match
Data
145
(a)
0
100
200
300
400
500
600
Time, hrs
700
800
900
145
1000
(b)
0
100
200
300
400
500
600
700
800
900
1000
Time, hrs
Fig. 6—Synthetic data (true: a, noisy: b). Matching using only one variable (porosity).
878
December 2010 SPE Reservoir Evaluation & Engineering
152
152
151
151
150
149
Temperature, °F
Temperature, °F
150
149
148
147
148
147
146
146
145
Data
Model after match
145
Data
Model with noise
Model after match
144
144
143
0
(a)
200
400
600
800
0
1000
Time, hrs
200
400
600
Time, hrs
(b)
800
1000
Fig. 7—Synthetic data (true: a, noisy: b). Matching using five variables (permeability, porosity, thermal diffusivity length, JouleThomson coefficient, and thermal conductivity of the reservoir rock).
In the data set shown in Fig. 6, the input porosity used in generating the data was ␾ = 0.25. The estimation using the pure data
(without noise) yielded ␾opt = 0.25, at the minimum. For the noisy
data, the minimum was achieved at ␾opt = 0.249. The matching was
relatively easy using only one variable. In Fig. 7, the number of
variables for estimation was increased to five. These were permeability k, porosity ␾, thermal-diffusivity length b, Joule-Thomsom
coefficient ε, and thermal-conductivity value of λs for the reservoir
rocks. The input parameters used in generating the data and the
results of the match are tabulated in Table 1.
The results show that in the case of using the data without
noise, the matching yielded model parameters that were sufficiently close to the values used in generating the synthetic
data. The slight variations from the true values can be attributed
to the inherent uncertainties when more than one parameter is
used in an inverse problem. This generally defines a confidence
interval around the estimations. With 5% added noise, the
parameter estimation also yields results that are close to the
true values of the parameters, with the slight variations caused
by the noise, as well as the fact that the inverse problem is
estimating more than one parameter. The underlying conclusion
from these observations is that the model and the solution are
able to resolve the effects caused by these parameters that are
being estimated. The solution does not fail when applied to an
inverse problem.
Before considering real field data, an important reflection at
this point would be the issue of noise in the temperature measurements. Generally, as will be seen with the field data, the
temperature variation is usually in the region of 1 through 10°F,
for producing wells without injection of hotter/colder fluids. The
temperature gauges, as evidenced in the number of different data
studied so far, have generally measured data with minimal noise,
enough for the effects caused by changing flow conditions to be
seen. As with any other problem, larger amounts of noise in a
data set obscure the data and reduce the ability of any solution to
resolve effects caused by the model parameters. This is patently
so with temperature measurements. However, unless the data are
extremely noisy (and, as such, useless), the temperature solution
is able to resolve effects caused by the model parameters, even
a temperature-change range of 1 through 10°F. This is because
most of the variations in the temperature are caused by changes
in the flow conditions such as changes in the fluid pressure and/or
flow rate, and changes in compositions as may be captured in the
gas/oil ratios (GORs).
Results for Field Data. The results of applying the model solution
to field data are presented here for data from two fields. In one
field, the flow was assumed to be near single-phase (low water cut
and low free gas) and was solved using the single-phase model
presented in this work. In the second field, the flow was two-phase
oil and gas and was solved using the two-phase-model solution.
Six parameters were estimated in the single-phase case. These variables were permeability k, porosity ␾, thermal diffusivity length
b, Joule-Thomson coefficient ε, thermal-conductivity value of λs
for the reservoir rocks, and viscosity µ. In the two-phase oil-andgas case, the number of parameters increased to eight, and these
included the gas saturation Sg and the gas Joule-Thomson coefficient εg, in addition to the parameters estimated in the single-phase
care. These estimations were compared with average values of
the parameters obtained from the two fields. Because we added a
temperature model to the already-multivariable pressure equations,
the number of model parameters increased significantly and so also
did the uncertainties in the estimations and the computational cost
of solving the inverse problem.
Single-Phase Oil System. The data for the single-phase oil flow
were from an 800-hour history of temperature, pressure, and flow
rate, as shown in Figs. 8 and 9. The original data sets were presented earlier, in Figs. 1 and 2. Because of the large number of data
points in the data, the data set was divided into four different parts
that were matched independently while accounting for the histories
from previous segments. The histories were carried over through
the values of the initial pressure, flow rate, and temperatures in
TABLE 1—ESTIMATED PARAMETERS FROM MATCHING THE PURE DATA
AND NOISY DATA
k (md)
φ
b (m)
(K/Pa)
s
Parameter for data generation
100
0.25
5.0
1.0 10
8
Match: pure data
100
0.249
5.16
9.6 10
9
Match: noisy data
110
0.286
4.99
1.28 10
December 2010 SPE Reservoir Evaluation & Engineering
8
[W/(m·k)]
0.3
0.32
0.29
879
146.4
146.2
146.2
146
146
145.9
145.9
Temperature, °F
Temperature, °F
146.4
145.6
145.4
145.2
145
144.8
145.6
145.4
145.2
145
144.8
Data
Model after match
144.6
Data
Model after match
144.6
144.4
144.4
0
10
20
30
0
40
Time, hrs
(a)
50
100
Time, hrs
(b)
150
200
Fig. 8—Result of match to data for the single-phase oil flow (first half of field data).
each data segment. In Fig. 10, the entire field data were matched,
after sampling the data at every 20th data point. The tolerance set
for the objective-function minimization was low enough to have
meaningful parameter estimates that represented what is obtainable
in reality, but high enough to allow for realistically meaningful
length of time for the simulation.
Table 2 shows the results of the match of Figs. 8 through 10.
For clarity in representing the results, the data segment in Fig.
8a will be simply referred to as “8a” and that in Fig. 8b will be
referred to as “8b.” Similarly, Figs. 9 and Fig. 10 are called “9”
and “10.” Also reported are the percentage effect because of flow in
the wellbore (i.e., by how much the reservoir temperature variation
is masked by the flow in the wellbore up to the gauge location).
This parameter would be unnecessary if the gauges are close to
the perforations.
Table 2 shows results (within the objective-function tolerance)
that are close to the average values of the parameters obtained for
the field. The search space in each inverse problem was open (−∞
to +∞), and the convergence to these values at the optimal match
shows promise in the ability of the formulation and the solution
to resolve the parameters in the model. The band of uncertainty,
as seen from the range of the values, is obvious and has been discussed previously. It is clear why the full-field match has a longer
diffusivity length—this is because of the longer length of time
(800 hours) in the data. However, depending on the length of time
of a transient (one shut-in and one flowing period), the expected
diffusivity length would be different for each transient, and this
would affect the overall results if the full data of a multitransient
data set with multiple shut-ins are used. To properly capture this
effect and reduce its consequence on the results of the match, it is
better to divide the temperature data set into segments containing
fewer transients and shut-ins. Also interesting is the observation
that only approximately 25% of the effects measured by the PDG
are caused by the reservoir flow, on average. The other 75% is
caused by wellbore flow.
Two-Phase Oil-and-Gas System. The data for the two-phase
case are shown in Fig. 11a. The flow rate reported is the oil-flow
rate (reduced by a factor of 10 for scaling). The Joule-Thomson
coefficient for oil is negative, while the coefficient for gas is positive. This causes a heating of the fluid during flow of oil. As gas
starts to come out of solution and flow with the oil, the cooling
because of the gas expansion counteracts the heating caused by
oil expansion. If this continues, the temperature profile may show
an eventual drop in temperature even with high oil-flow rates
because of the cooling caused by more gas in the system. This
can be seen in Fig. 11a. Although the temperature dropped slightly
owing to the gradually decreasing flow rates (hence, smaller pressure changes) and fairly constant GOR, the temperature dropped
significantly more after 1,500 hours. The GOR can also be seen to
have increased significantly after 1,500 hours, from 1,000 to 2,000
145.5
146.5
146
Temperature, °F
Temperature, °F
145.5
145
144.5
144
145
144.5
Data
Model after match
Data
Model after match
143.5
144
143
0
(a)
20
40
60
Time, hrs
80
100
120
0
(b)
10
20
30
Time, hrs
40
60
80
Fig. 9—Results of match to data for the single-phase oil flow (second half of field data).
880
December 2010 SPE Reservoir Evaluation & Engineering
TABLE 2—RESULTS FOR THE SINGLE-PHASE FIELD CASE
k (md)
φ
b
(m)
Reported
average from
field
319
0.14-0.19
NA
NA
8a
142
0.18
2.3
1.28 10
8b
470
0.194
7.05
9a
200
0.15
9b
380
10
300
µ (cp)
Fwbf
NA
0.9
NA
8
0.1
2
30
2.36 10
8
0.5
2
23
2.93
2.94 10
7
0.1
0.7
20
0.23
8.0
3.9 10
8
0.3
8
20
0.40
24.
5.8 10
7
0.8
10
20
data and reasonably accepted values obtained from other fields that
share the same characteristics as the subject field.
Although the model and solution closely followed the field
data for this case, as can be seen in Fig. 11b, the uncertainty in
the parameters estimated may be very large. This would be a
candidate for further research, especially on how to measure and
report these measurable thermal properties of multiphase systems
and how to minimize the uncertainties in the estimations of the
other parameters that cannot be measured.
146.5
146
145.5
Temperature, °F
s
[W/(m·k)]
(K/Pa)
145
144.5
144
143.5
Data
Model after match
143
0
200
400
600
Time, hrs
800
1000
Fig. 10—Results of match to data for the single-phase oil flow,
using full-field data.
scf/STB, even with continued moderate drop in the oil flow rate.
This is the counteracting cooling effect caused by more gas expansion. Fig. 11b is a plot of the match of the data to the model. The
results are presented in Table 3. It is important to state here that
the uncertainties in the values of the parameters used in this match
are even higher because more model parameters are needed, and
many of them come with the accompanying uncertainties in their
measurements. As a result, in the inverse problem, we attempted
to match eight parameters, while fixing the rest using average field
Conclusions
The models developed in this study have been shown to have the
potential to help characterize a reservoir by use of temperaturetransient measurements from any downhole monitoring source.
This shows one way of extracting the information inherent in the
temperature signals, which have not been fully used in the past.
The parameter estimation shown here can be used to complement
with the parameter-estimation step in pressure-transient analysis.
Specifically, this study provides a method for the estimation of
porosity and permeability and other thermal properties of the reservoir and reservoir fluids. The approach is also potentially a tool
for tracking the saturation in a reservoir. Being rate dependent,
the temperature data also offer a way to infer sandface flow rate
by inverting the reservoir temperature model. Finally, it is clear
that temperature provides an additional source of information in
transient analysis, and this presents the potential for cointerpreting
rate, pressure, and temperature histories.
Nomenclature
b = thermal-diffusivity length, m
c = pecific heat capacity, J kg ⋅ K
Time, hrs
500
1000
1500
2000
2500
3000
3500
4000
155.5
3500
155
3000
154.5
2500
154
2000
153.5
1500
1000
153
Temperature
152.5
152
500
GOR
Oil flowrate
0
500
1000
1500
2000
2500
3000
3500
4000
156
4500
4000
0
4500
155.5
155
Temperature, °F
0
GOR=scf/stb, flowrate=10*bbl/d
Temperature, °F
156
154.5
154
153.5
153
152.5
152
0
Time, hrs
(a)
(b)
Data
Model after match
1000
2000
3000
Time, hrs
4000
5000
Fig. 11—Data (a) and result of match (b) for the two-phase oil-and-gas case.
December 2010 SPE Reservoir Evaluation & Engineering
881
TABLE 3—RESULTS FOR THE TWO-PHASE OIL-AND-GAS FLOW
k
(md)
φ
b
(m)
Reported average from
field
400
0.23
NA
NA
11b
30
0.212
0.6
8.4 10
cm = (1− ␾ )(␳s cs ) + ␾ ⎡⎣(␳ wc fw sw ) + (␳oc fo so ) ⎤⎦, volumetric heat
capacity of fluid-saturated rock, J 3
m ⋅K
C = volumetric-heat-capacity ratio,
(␳ wc fw sw + ␳oc fo so + ␳ gc fg sg )
cm
k = permeability
p = pressure, Pa
3
Q, q = flow rate, m s
r = radius, m
S = saturation
t = time, seconds, hours
T = temperature, K, °F
U to = overall-heat-transfer coefficient, J
(K ⋅ m 2 ⋅ s)
u = Cv = superficial velocity, m s
␾ = porosity
␳ = density
␪ = wellbore-inclination angle
␤T
K
, adiabatic-expansion coefficient,
␩ =
Cf
Pa
␤T − 1
K
, Joule-Thomson coefficient,
ε =
Cf
Pa
1
1 ⎛ ∂␳ ⎞
⎜ ⎟ , thermal-expansion coefficient,
␳ ⎝ ∂T ⎠ p
K
␭
m2
␣ = m , thermal diffusivity,
cm
s
W
␭ = thermal conductivity,
(m ⋅ K )
␤ =
␭m = ␾␭ f + (1 − ␾ )␭s , thermal conductivity of fluid-saturated
rock
Subscripts
D = dimensionless
f = fluid
o = oil
w = water
Acknowledgments
We would like to acknowledge the support of the Stanford Graduate Fellowship and the Stanford University Research Institute on
Innovations in Well Testing for making this research possible.
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Obinna Onyinye Duru is a PhD candidate in the department of
energy resources engineering, at Stanford University. He holds
December 2010 SPE Reservoir Evaluation & Engineering
a BSc degree in chemical engineering from the University
of Lagos, and an MS degree in petroleum engineering from
Stanford University. Duru is also a research assistant with the
Stanford University Research Consortium on Innovations in Well
Testing. Roland N. Horne is the Thomas Davies Barrow Professor
of Earth Sciences at Stanford University, and was the Chairman
of Petroleum Engineering from 1995 to 2006. He holds BE, PhD,
and DSc degrees from the University of Auckland, New Zealand,
all in engineering science. Horne has been an SPE Distinguished
Lecturer, and has been awarded the SPE Distinguished
Achievement Award for Petroleum Engineering Faculty, the
Lester C. Uren Award, and the John Franklin Carl Award. He is
a member of the U.S. National Academy of Engineering and is
also an SPE Honorary Member.
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