Download - Temperature Reservoir
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MODELING OF RESERVOIR TEMPERATURE
TRANSIENTS, AND PARAMETER ESTIMATION
CONSTRAINED TO
A RESERVOIR TEMPERATURE MODEL
A THESIS SUBMITTED TO THE DEPARTMENT OF
ENERGY RESOURCES ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
By
Obinna Duru
JUNE 2008
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I certify that I have read this thesis and that in my opinion it is fully adequate, in
scope and in quality, as partial fulfilment of the degree of Master of Science in Energy
Resources Engineering.
Dated: June 2008
Principal advisor:Prof. Roland Horne
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This work is dedicated to the little children
suffering in war-torn African nations
...... there is light at the end of the tunnel
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Table of Contents
Table of Contents v
List of Figures vii
Abstract x
Acknowledgements xii
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Literature Review 4
3 Theory and Methodology 7
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Alternating Conditional Expectations (ACE) . . . . . . . . . . . . . . 8
3.2.1 Optimal Transformations[3]: . . . . . . . . . . . . . . . . . . . 8
3.3 Mechanistic Modeling of Fluid Temperature Transients . . . . . . . . 9
3.3.1 Distributed Reservoir Temperature Model . . . . . . . . . . . 9
3.3.2 Operator Splitting and Adaptive Time Stepping (OSATS) . . 15
3.3.3 Solution of the Thermal Convection-Diffusion Model by OSATS 17
3.4 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Wellbore Temperature Transient Model . . . . . . . . . . . . . . . . . 26
3.6 Coupling the Reservoir Model to the Wellbore Model . . . . . . . . . 28
4 Qualitative Evaluation and Sensitivity Analysis 29
4.1 Optimal Transformation for Estimation of Functional Relationship . . 29
4.2 Qualitative Evaluation of Model - Single-phase Case . . . . . . . . . . 31
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4.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Sampling the Distribution of the Parameter Space . . . . . . . . . . . 42
5 Results and Discussion 44
5.1 Synthetic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2.1 Single-phase System . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Two-phase System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Potential Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Scope for Further Study 60
6.1 Updating the Boundary Conditions . . . . . . . . . . . . . . . . . . . 60
6.2 Updating the Model to Treat Three-phase Systems (Gas-Oil-Water) . 60
6.3 Reservoir Models with Fluid Injection or Phase Transition . . . . . . 61
6.4 Cointerpretation of Pressure, Rate and Temperature . . . . . . . . . . 61
6.5 Heterogenous Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Nomenclature 63
References 65
Appendix 68
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List of Figures
4.1 Flow rate data (left), pressure data (right) from DAT.1 data set. . . . 30
4.2 Temperature data (left) from DAT.1 data set, optimal regression (right). 30
4.3 Qualitative comparison using DAT.1 (first representative transient). . 32
4.4 Qualitative comparison using DAT.1 (second representative transient). 32
4.5 Qualitative comparison using DAT.2. . . . . . . . . . . . . . . . . . . 33
4.6 Sensitivity to porosity (top) and fluid Joule-Thomson coefficient (bot-
tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.7 Sensitivity to thermal diffusivity length (top) and fluid thermal con-
ductivity (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8 Sensitivity to reservoir permeability (top) and fluid viscosity (bottom). 36
4.9 Sensitivity to gauge placement. . . . . . . . . . . . . . . . . . . . . . 37
4.10 Temperature estimation with b = 11m, over 0-350,000 data points
(left), actual data (right). . . . . . . . . . . . . . . . . . . . . . . . . 40
4.11 Temperature estimation with b = 8m, over 200,000-350,000 data points
(left), actual data (right). . . . . . . . . . . . . . . . . . . . . . . . . 40
4.12 Temperature estimation with b = 12m, over 300,000-400,000 data
points (left), actual data (right). . . . . . . . . . . . . . . . . . . . . . 41
4.13 Temperature estimation with b = 10m, over 100,000-200,000 data
points (left), actual data (right). . . . . . . . . . . . . . . . . . . . . . 41
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4.14 Two-dimensional marginal distribution of porosity and oil Joule-Thomson
coefficient from DAT.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 Single-phase: temperature match using rate data with random noise. 46
5.2 Single-phase: temperature match using rate data with random noise,
and true temperature with random noise. . . . . . . . . . . . . . . . 46
5.3 Single-phase: temperature match with random noise added to the
true temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.4 Single-phase: temperature match using rate data with random noise,
with different transient region. . . . . . . . . . . . . . . . . . . . . . . 47
5.5 Two-phase: temperature match using rate data with random noise,
and true temperature with random noise. . . . . . . . . . . . . . . . 48
5.6 Matching DAT.1 using OSATS ( = 0.196, = 1.16 107(K/Pa),b = 9.2m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.7 Matching DAT.1 using OSATS ( = 0.21, = 9.0 109(K/Pa),b = 6.12m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.8 Matching DAT.1 using OSATS ( = 0.235, = 4.48 108(K/Pa),b = 7.817m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.9 Matching DAT.1 using OSATS ( = 0.2, = 7.7 108(K/Pa),b = 5.2m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.10 Matching DAT.1 using OSATS ( = 0.228, = 1.09 109(K/Pa),b = 3.0m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.11 Matching DAT.1 using OSATS ( = 0.21, = 4.3 108(K/Pa),b = 5.16m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.12 Matching DAT.2 using OSATS ( = 0.182, = 5.95 108(K/Pa),b = 5.92m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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5.13 Matching DAT.1 using fully numerical solution ( = 0.286, = 1.53108(K/Pa), b = 2.0m). . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.14 Matching DAT.1 using fully numerical solution ( = 0.287, = 9.0 109(K/Pa), b = 2.6m,f = 0.11(W/m.K), s = 4.0(W/m.K)). . . . 55
5.15 Matching using two-phase model ( = 0.25, Sw = 0.3, b = 5.36m). . . 56
5.16 Three-dimensional spatial temperature distribution at a time instant
using DAT.1 data set. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.17 Two-dimensional spatial temperature distribution at a time instant
using DAT.1 data set. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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Abstract
Permanent downhole gauges (PDGs) provide a continuous source of downhole pres-
sure, temperature and sometimes rate data. Until recently, the measured temperature
data have been largely ignored. However, a close observation of the temperature mea-
surements reveals that the temperature responds to changes in flow rate and pressure,
which implies that the temperature data may be a source of reservoir information.
In this work, the Alternating Conditional Expectations (ACE) technique was ap-
plied to temperature and flow rate signals from PDGs to establish the existence of
a functional relationship between them. Then, performing energy, mass and mo-
mentum balances, reservoir temperature transient models were developed for single-
and multiphase fluids, as functions of formation parameters, fluid properties, and
changes in rate and pressure. The pressure field in oil and gas bearing formations
are usually nonstationary. This gives rise to pressure-temperature effects appearing
as temperature changes in the porous medium when the pressure field is nonstation-
ary. 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. Therefore, in this study, reservoir ther-
mometric effects were modeled as convective, conductive and transient phenomena
with consideration for time and space dependencies. This mechanistic model included
the Joule-Thomson effects due to fluid compressibility, and viscous dissipation in the
reservoir during fluid flow in accounting for the reservoir temperature dependence on
x
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changing pressure/flowrate fields.
Numerical solution schemes as well as the semianalytical scheme - Operator Split-
ting and Time Stepping (OSATS) were used to solve the models, and the solutions
closely reproduced the temperature profiles seen in real measured data. By matching
the models to different temperature transient histories obtained from PDGs, reservoir
parameters namely porosity and saturation and fluid Joule-Thomson coefficient could
be estimated. The significant contributions of this work include a method which:
Utilizes temperature data measured by PDGs.
Provides a way to estimate porosity and potentially saturation.
May provide a less expensive substitute for downhole flow rate measurement.
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Acknowledgements
I would like to thank my advisor Professor Roland Horne for his advice, guidance,
and encouragement through the course of this research.
Financial support from the Stanford Graduate Fellowship is greatly acknowledged.
The H.L. and Janet Bilhartz-ARCO Fellowship which was awarded to me was crucial
to the successful completion of this project.
Of course, I am grateful to my parents for their patience and love. My mother
- Lady Nelly Duru, whom I consider a lioness in character and in learning, and my
father - Sir Obinna Duru, a sheer genius. I would also like to thank my siblings for
their support.
I wish to thank the following: Uzoamaka (my best friend), my cousin Ijeoma
who refused to talk to me until I finished writing this report, Jerome (for being a
big brother), Antoine (for teaching me the French culture), Nilufer (for being a great
friend), Seun in UT Austin, and many others who in one way or the other have helped
shape my life.
Importantly, I can not thank Voke enough....although she was 7,814 miles away
from me, she offered to be my wake-up alarm every single morning and spurred me to
my responsibilities everyday. Her encouragements, support and concern were priceless
- she was there all the time, like my guardian angel.
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Chapter 1
Introduction
1.1 Background
Long term reservoir monitoring using permanent download monitoring devices has
been a continuous source of downhole data in the form of pressure, temperature and
sometimes flow rate. These tools provide access to data acquired continuously over
long periods of time which provide reservoir information at a much larger radius of
investigation than conventional wireline testing.
The behavior of pressure transients in reservoir and wellbore flow has been studied
extensively, and applied in conventional well test analysis for reservoir description,
parameter estimation for formation characterization and evaluation of well and field
performance. In recent times, with data convolution and deconvolution techniques as
well as data filtering and tuning, conventional pressure transient analytical methods
have also been applied to pressure data from permanent downhole gauges (PDGs),
increasing the usefulness of these data.
However, in conventional pressure transient analysis, the temperature distribu-
tions in the reservoir and wellbore have been assumed isothermal. The temperature
changes associated with fluid flow had been considered to be relatively small and
1
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2hence negligible for any consideration in the analysis of flow behavior of most flu-
ids. An analysis of temperature measurements, at a fine scale using continuous data
from PDGs, has shown that the temperature of the fluids responds to changes in flow
conditions in the reservoir. Generally, the flow is not isothermal when the scale of
observation and resolution of the temperature data is refined. This study attempts to
identify the underlying physical phenomena responsible for this temperature transient
behavior and its possible application to reservoir characterization and evaluation of
well performance.
1.2 Statement of the Problem
Many previous attempts at developing interpretation method for temperature profiles
in wellbore-reservoir systems have remained largely qualitative. Most of the analyses
have concentrated on wellbore thermal exchanges due to conduction and convection,
assuming that the produced fluid enters the wellbore at the geothermal temperature
Maubeuge et al. [12]. Others have attempted the study of thermometric fields in
reservoirs and porous systems, but have constrained the 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 due to factors like the
Joule-Thomson effect, adiabatic expansion and viscous dissipation.
The pressure field in oil and gas bearing formations are usually nonstationary [5].
This gives rise to pressure-temperature effects appearing as temperature changes in
the porous medium when the pressure field is nonstationary. 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. Therefore, in this study, reservoir thermometric effects were modeled
as convective, conductive and transient phenomena with consideration for time and
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3space dependencies. This mechanistic model included the Joule-Thomson effects due
to fluid compressibility, and viscous dissipation in the reservoir during fluid flow in
accounting for the reservoir temperature dependence on changing pressure/flowrate
fields.
As a result of these investigations, it was found that in addition to establishing a
representative model for the temperature distribution in the reservoir, reservoir and
flow properties could be estimated in an inverse optimization problem.
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Chapter 2
Literature Review
Several authors have studied the thermodynamics of flow through porous media and
wellbore systems, especially in the context of heat convection and conduction. One
of the earliest works in this regard was by Ramey [9] who developed a model for the
prediction of wellbore fluid temperature as a function of depth for injection wells.
Ramey expanded this model to give the rate of heat loss from the well to the for-
mation, assuming steady-state flow in the wellbore and unsteady radial conduction
in heat transfer to the earth. Horne and Shinohara [7] presented single-phase heat
transmission equations for both production and injection geothermal well systems by
modifying Rameys model as a way of calculating the heat losses between wellhead
and reservoir in order to evaluate reservoir temperature. Shiu and Beggs [20] pre-
sented another modification of Rameys model to predict the wellbore temperature
profile for a producing well, where the temperature of fluid entering the wellbore from
the reservoir is known. These wellbore models considered heat transfer as strictly con-
vection and conduction phenomena with fluid entering the formation at a constant
temperature from the reservoir.
Izgec et al.[8] presented a model that applied to coupled wellbore and reservoir sys-
tems and provided a transient wellbore temperature simulator coupled with a variable-
earth-temperature scheme for predicting wellbore temperature profiles in flowing and
4
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5shut-in wells. Again, their study looked at the mechanism of heat transfer in the
wellbore and the interaction with surrounding formation without consideration for
for possible changes in the reservoir fluid temperature before entry into the wellbore.
Sagar et al. [18], developed a steady-state two-phase model for the wellbore tem-
perature distribution accounting for Joule-Thomson effects due to heating/cooling
caused by pressure changes within the fluid during flow. They considered Joule-
Thomson effects as a possible heat source/sink during fluid flow and applied the
model to estimate heat losses in gas flow.
Valiullin et al. [22] presented a treatment of the temperature distribution in the
formation when the pressure field in the reservoir changes, and showed that indeed,
adiabatic and Joule-Thomson effects as well as effects due to heat of phase transition
(gas liberation from oil) may be present during fluid flow in a hydrocarbon satu-
rated porous medium. They designed experiments to estimate the thermodynamic
coefficients, namely the Joule-Thomson coefficient and adiabatic coefficients.
Ramazanov and Parshin [16] went on to develop an analytical model that described
the formation temperature distribution in a reservoir, while accounting for phase tran-
sitions. They solved a steady-state convective thermal flow model with constant flow
rate and extended it to cases with phase changes. In 2007, Ramazanov and Nagimov
[15] presented a simple analytical model to estimate the temperature distribution in
a saturated porous formation at variable bottomhole pressure. Their investigation
showed that for a single-phase fluid in a homogenous reservoir, temperature-pressure
effects such as Joule-Thomson can cause the temperature in the reservoir to very
significantly when reservoir pressure is changing in time. Ramazanov and Nagimov
solved the convective thermal transport model with variable pressure but constant
flow rate.
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6Attempts to solve the full energy balance equation for the temperature distribu-
tion in a reservoir was made by Dawkrajai [4] and Yoshioka [23]. Both presented
equations for reservoir and wellbore heat flow and developed prediction models for
the temperature and pressure. In an inversion step, they showed a means for the
identification of water and gas entry into a well. Both approaches made considera-
tions for Joule-Thomson and frictional heating effects but assumed a constant flow
rate, and steady-state conditions in arriving at the solution to their models.
Bear [1], Bejan [2] and Neild and Bejan [13] presented a comprehensive model
for heat transport in a porous media from mass, energy and momentum balance.
Thermal diffusion and convection and effects due to the fluid compressibility, vis-
cous dissipation (mechanical power required to extrude the fluid through the pore)
were incorporated into the model and the final form presented took the form of
a convection-diffusion model with source/sink terms. These studies discussed the
possibility of the optimization of the fluid space configuration for minimal thermal
resistance in a porous medium heat exchanger.
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Chapter 3
Theory and Methodology
3.1 Introduction
In order to proceed with the study of the physics behind the observed temperature
response to changes in flow rate and pressure, as a reservoir temperature distribu-
tion problem, there is the need to establish the existence of a functional relationship
between the temperature, and pressure and flow rate. The technique chosen for this
investigation was the nonparametric regression tool, Alternating Conditional Expec-
tation (ACE) originally proposed by Breiman and Friedman [3]. ACE allows for the
estimation of optimal transformations that may lead to the maximal multiple corre-
lation between a response variable (temperature in this case) and a set of predictor
variables (pressure, rate and time). These transformations are useful in establish-
ing the existence of a functional relationship between the response variable and the
predictor variables.
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83.2 Alternating Conditional Expectations (ACE)
ACE is a nonparametric iterative approach at estimating optimal transformations
of a data set to obtain maximal correlations between observed variables in the data
set. The power lies in its ability to perform this regression without making a priori
assumptions of functional forms of the relationships between the variables. The op-
timal transformations are based solely on the data set and unlike Neural Networks,
ACE can facilitate physically based function identification. ACE can also incorporate
multiple and mixed variables, both continuous (e.g. permeability, porosity, pressure)
and categorical (e.g. rock types).
3.2.1 Optimal Transformations[3]:
Given a real response (dependent) variable Y , and a p-dimensional vector X =
X1, ...Xp of predictor (independent) variables, define a set of arbitrary transforma-
tions (Y ), 1(X1), ...p(Xp). Suppose that a regression of the transformed response
variable on the sum of the transformed predictors result in the error:
e2(, 1, ..., p) = E
{[(Y )
pi=1
i(Xi)]2
}(3.2.1)
Then the transformations (Y ), 1(X1), ..., p(Xp) are said to be optimal for the
regression if they satisfy that:
e2(, 1, ..., p) = min
,1,...,pe2(, 1, ..., p) (3.2.2)
The correlation coefficient between the transformed response variable and sum of
the transformed predictor variables (under the constraints, E[2(Y )] = 1 and E[2s(X)])
is given by:
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9(, s) = E[(Y )s(X)] (3.2.3)
where s(X) =p
i=1 i(Xi)
The optimal transformations (Y ), 1(X1), ..., p(Xp) fulfill the maximal correla-
tion condition because, as shown in Breiman and Friedman [3], they satisfy that:
(, s) = max,1,...,p
(, s) (3.2.4)
Breiman and Friedman [3] also showed that these optimal transformations for
correlation are also optimal for regression.
3.3 Mechanistic Modeling of Fluid Temperature
Transients
3.3.1 Distributed Reservoir Temperature Model
Theory of Thermometry in a Fluid-saturated Porous Media
The flow of energy-carrying fluids through a porous media has been studied for many
years. Bejan[2] and Bear[1] presented a comprehensive thermodynamic approach to
obtaining a representative model for temperature distribution in a porous media. The
model accounted for spatial distribution and well as transient effects in the formation.
Sharafutdinov[19], Filippov and Devyatkin[5] and Ramazanov and Nagimov[15] also
followed similar approaches in developing their models for temperature distribution
in a fluid saturated porous stratum.
In a flowing well, the pressure and flow rate measurements by permanent mon-
itoring gauges are not constant. For gauges placed close to the sandface flow area,
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these changes reflect the dynamics of the flow in the reservoir. These flow dynamics
cause a temperature field to evolve in the reservoir, driven by thermodynamic effects
such as the Joule-Thomson heating (or cooling), adiabatic expansion, and heat of
phase transitions. Other effects namely the viscous dissipation, equal to the mechan-
ical power needed to extrude the viscous fluid through the pore, as well as 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.
Joule-Thomson effect is the change in the temperature of a fluid due to expan-
sion or compression of the fluid in a flow process involving no heat transfer or work
(constant enthalpy). This change is due to a combination of the effects of fluid com-
pressibility and viscous dissipation. The Joule-Thomson effect due to the expansion
of oil in a reservoir or wellbore results in the heating of the fluid because of the value
of the Joule-Thomson coefficient of oil - it is negative for oil. The coefficient has a pos-
itive value for real gases and the consequent cooling effect is more prominent in gases.
Theoretically, the Joule-Thomson coefficient for ideal gases is zero implying that the
temperature of ideal gases would not change due to a pressure change if the system
is held at constant enthalpy. Combined with other factors, on expansion of the fluid
and subsequently flow of liquid oil and/or water out of the reservoir, the wellbore and
near wellbore areas in the reservoir become heated above the normal static reservoir
temperature. By convection, diffusion and further generation of heat enerygy due 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 diffusion and result in a temperature decline
at a rate determined by the thermal diffusivity of the medium.
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11
Reservoir Temperature Model in One-dimensional Cylindrical Coordinate
System
Single-phase Formulation
To derive the energy equation for a homogenous porous medium, the energy equations
for the solid and fluid parts are derived separately from the first law of thermody-
namics, and averaged over an elementary control volume to obtain the general form
of the model. The consideration is for a nonisothermal flow of a nonideal fluid in a
porous media. The change in kinetic and potential energies of the flow will be taken
as negligible.
For the solid part, assuming no internal heat generation per unit volume of the
solid material, the energy conservation equation associated with flow in an elastic
porous medium becomes:
scsT
t= ks
rrT
r(3.3.1)
where (, c, k)s are properties of the solid matrix.
The energy conservation equation at any point in space, occupied by the fluid is
given by:
fcpf
(T
t+ up
T
r
)= kf
rrT
r+ T
p
t+ Tv
p
r vp
r+ vg (3.3.2)
where (, c, k)f are the fluid properties and T is the temperature for both the fluid
and solid. An assumption of local thermal equilibrium has been made between the
fluid and the porous matrix. Bejan[2] notes that this assumption is valid only for
small-pore media such as geothermal and oil reservoirs. Bejan[2] also showed that
Eqn. (3.3.1) and Eqn. (3.3.2) can be combined by volumetric averaging to give the
final form of the model as:
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12
((1 )(scs) + (fcf )) Tt
+ fcfvT
r=
((1 )s + f )r
rrT
r+ T
p
t+ Tv
p
r vp
r+ vg (3.3.3)
On rearrangement and assuming negligible gravity effects, this becomes:
T
t+ u
T
r=
r
rrT
r+ C
p
t+ u
p
r(3.3.4)
The form of Eqn. (3.3.4) is the convection-diffusion type partial differential equa-
tion with source/sink terms. The second term on the right hand side of Eqn. (3.3.4)
is the compressibility term, while the last term is the viscous dissipation term.
The mass balance equation takes the form:
ft
+1
r
(rvf )
r= 0 (3.3.5)
The flow is assumed to obey Darcys law, and the equation for Darcy flow given
by
v =k
p
r(3.3.6)
completes the formulation.
Where:
u = Cv
v = superficial velocity, ms
C = volumetric heat capacity ratio,cfcm
Cf = volumetric heat capacity of fluid,J
m3K
cm = (1 )(scs) + (fcf ) , volumetric heat capacity of fluid saturated rock, Jm3K = porosity
= density
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= TCf
, adiabatic expansion coefficient, KPa
= T1Cf
, Joule-Thomson coefficient, KPa
= 1( T
)p , thermal expansion coefficient,1K
= mcm
, thermal diffusivity, m2
s
= thermal conductivity, WmK
m = f + (1 )s, thermal conductivity of fluid saturated rock
Eqns (3.3.4), (3.3.5) and (3.3.6) form the governing equations for one-dimensional
thermal transport in a homogenous porous medium. The assumptions made in de-
riving the equations were:
The medium is homogenous, such that the solid and fluid permeating the poresare evenly distributed throughout the porous medium.
The medium is isotropic such that permeability, k and thermal conductivity do not depend on the direction of the experiment.
At any point in the porous medium, the solid matrix is in thermal equilibriumwith the fluid in the pores.
Darcys law applies.
Two-phase Formulation
The assumptions made for the two-phase formulation are similar to those made in the
single phase case, with the addition of negligible capillary effects. The thermal model
in a one-dimensional radial coordinate system for the two-phase system becomes:
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14
((1 )(scs) + (wcfwsw + ocfoso)) Tt
+ (wcfwswvw + ocfosovo)T
r=
((1 )s + (wsw + oso))r
rrT
r+ (wcfwsww + ocfosoo)
p
t
+ (wcfwswvww + ocfosovoo)p
r
(3.3.7)
On rearrangement and with negligible gravity and capillary effects, this becomes:
T
t+ u
T
r=
r
rrT
r+
p
t+ J
p
r(3.3.8)
The mass balance equation takes the form:
(wsw)
t+
1
r
(rvwwsw)
r= 0 (3.3.9)
(oso)
t+
1
r
(rvooso)
r= 0 (3.3.10)
The Darcy flow equation becomes:
vw =kww.(pwr
) (3.3.11)
vo =koo.(por
) (3.3.12)
Where:
C = volumetric heat capacity ratio,(wcfwsw+ocfoso)
cm
Cf = volumetric heat capacity of fluid,J
m3K
cm = (1 )(scs) + ((wcfwsw) + (ocfoso)) , volumetric heat capacity of fluid sat-urated rock, J
m3K
=(wcfwsww+ocfosoo
cm
)
-
15
J =(wcfwswvww+ocfosovoo
cm
) = T
Cf, adiabatic expansion coefficient, K
Pa
= T1Cf
, Joule-Thomson coefficient, KPa
= 1( T
)p , thermal expansion coefficient,1K
= mcm
, thermal diffusivity, m2
s
= thermal conductivity, WmK
m = f + (1 )s, thermal conductivity of fluid saturated rocksubscripts:
w = water
o = oil
f = fluid
Eqns 3.3.8 through 3.3.12 are the equations defining the formulation for the tem-
perature distribution in a reservoir during two-phase flow. Eqn. 3.3.8 is a convection-
diffusion equation. Analytical solutions for convection-diffusion equations have been
the subject of research and numerical solutions have serious issues with stability due
in part to the nature of the model - a combination of a hyperbolic convective transport
and parabolic diffusion transport models. In this work, a semianalytical technique
used in ground water transport was used to solve this problem. The technique is
known as Operator Splitting and Time Stepping (OSATS), and was developed for the
solution of contaminant transport in ground water hydrology.[10][17]
3.3.2 Operator Splitting and Adaptive Time Stepping (OS-
ATS)
Kacur[10] and Remesikova[17] described methods for solving the convection-diffusion
type problem using the operator splitting approach. The operator splitting method
-
16
breaks the model into two different parts, the transport part and the diffusion part.
Then, at each time step, the nonlinear transport part and the nonlinear diffusion
part are solved separately. The semianalytical nature comes from the fact that the
solution is obtained in time sequences, with the solution at a time step depending on
the solution at all previous time steps. Holden et al.[6] showed the theoretical basis
for this technique.
Kacur[10] showed a precise way of solving the following type of problems:
F (u)
t+ v(x)
u
x x
(D(x, t))u
x= f(x, u) (3.3.13)
with the boundary and initial conditions:
u(0, t) = co(t) (3.3.14)
u(L, t) = 0, for x (0, L), t > 0 (3.3.15)
u(x, 0) = uo(x) (3.3.16)
First, the transport part is solved, which presents a hyperbolic problem of the
form:
F (u)
t+ v(x)
u
x= 0, t (tj1, tj) (3.3.17)
with boundary conditions of the form of Eqns. 3.3.14 and 3.3.15, and initial
condition of the form:
U(x, tj1) = uj1 (3.3.18)
-
17
The solution obtained is denoted as u1/2j := U(x, tj). Then the diffusion transport
part is solved, which has the form:
F (u)
t x
(D(x, t))u
x= f(x, u), t (tj1, tj) (3.3.19)
With the same boundary condition but with initial condition given by the solution
of the convective part -
U(x, tj1) = u1/2j (3.3.20)
Finally, the solution at that time step is put as:
uj := U(x, tj) (3.3.21)
Which is the solution of Eqn. 3.3.19. This is continued until the solution at the
last time step is obtained, which becomes the final solution of the model.
3.3.3 Solution of the Thermal Convection-Diffusion Model
by OSATS
The OSATS approach was used to solve the thermal model Eqns. 3.3.4 and 3.3.8.
The methodology adopted was:
Decouple the model into two parts: the convection transport part and thediffusion part.
At each time step, first solve the hyperbolic convection transport part, account-ing for variable flow rate, as well as heat generation due to viscous dissipation,
frictional and Joule-Thomson effects.
Then solve the diffusion part at the same time step, adaptively modifying thetime step to ensure stability if solution is numerical.
Continue until the last time step.
-
18
Solution of the Single-phase Formulation by OSATS
The following assumptions were made in solving the single-phase reservoir thermal
model:
Constant fluid Joule-Thomson, adiabatic expansion coefficient, and thermalconductivity i.e. these parameters are assumed to be weak functions of temper-
ature
Constant fluid viscosity and formation porosity
Negligible gravity effects
A. Solution of the convective transport part:
The convection equation with its initial condition becomes:
T
t+ u
T
r= C
p
t+ u
p
r(3.3.22)
T (t = 0) = To(r) (3.3.23)
Using the method of characteristics, the solution yields:
dr
dt= u(r, t) = Cv(r, t) (3.3.24)
Such that for constant rate,
r2 = r21 +qC
pih(t t1) (3.3.25)
Another form of Eqn. 3.3.25 starts by solving for the pressure terms independently
from the material balance equation, and using the Darcy equation to replace velocity
in Eqn. 3.3.24. If the total compressibility of the porous medium is considered
negligible, then the pressure equation reduces to:
-
19
1
r
rrp
r= 0 (3.3.26)
With boundary conditions:
p(r = re) = pe (3.3.27)
p(r = rw) = (t) (3.3.28)
where re = external radius and rw = wellbore radius.
This yields:
p(r, t) = pe +pe (t)ln(rerw
) . ln( rre
)(3.3.29)
And:
p
r= [
pe (t)ln(rerw
) ].1r
(3.3.30)
Applying this to Eqn. 3.3.24 by replacing the velocity with the Darcy law equiv-
alent gives the form:
r2 = r21 2(pet s(t)) (3.3.31)
where:
= kC lnR
,
R = ln rerw,
s(t) =t0
()d
The parameter (t) which is the sandface pressure (bottomhole pressure) in the well
bore is readily obtainable from solutions of classical pressure transient problems for
different reservoir models.
-
20
Alternatively, where the total compressibility of the system is not negligible, the
solution of the pressure equation comes in the form:
p =q
4pikhEi
(r2wpict
4kt
)(3.3.32)
Continuing the method of characteristics solution for the temperature yields,
dT
dt=
t0
dp
dtdt+ ( )
t0
p
d (3.3.33)
with = C.
This becomes:
T (r, t) = T0(r) [p(r, 0) p(r, t)] lnR
t0
() ln
(r
re
)d (3.3.34)
Ramazanov and Nagimov [15] showed that by using the average time theorem,
the integral on the left hand side of Eqn. 3.3.35 can be closely approximated when
an optimal average time is used. Therefore, combining with Eqn. 3.3.31, and apply-
ing the average time theorem, the final approximate solution for wellbore sandface
temperature becomes:
T (rw, t) = T0(r1) [p(r, 0) (t)] lnR
[(t) (0)] ln(
r21 2(pez s(z)re
)(3.3.35)
where:
r1 =r2w 2(pet s(t) (3.3.36)
[0 z t]
z = average time.
-
21
An optimal choice for z must be used, and Ramazanov and Nagimov [15] suggested
t2< z < t.
B. Solution of the diffusion part:
The form of the diffusion problem is:
1
T
t=
1
r
T
r+2T
r2(3.3.37)
0 < r
-
22
Solution of the Two-phase Formulation by OSATS
A. Solution of the convective transport part:
The convection equation with its initial condition is:
T
t+ C
T
r=
p
t+ J
p
r(3.3.41)
T (t = 0) = To(r) (3.3.42)
The solution follows closely the approach used for single phase formulation.
By method of Characteristics,dr
dt= C (3.3.43)
For constant rate, let
qT = qo + qw (3.3.44)
qo =oTqT (3.3.45)
and
qw = qT qo = qT(
1 oT
)(3.3.46)
where:
o = oil mobility(koo
)w = water mobility
(kww
)T = o + w
The solution to Eqn. 3.3.43 becomes
r2 = r21 +qTpih
{wcfwsw(1 o
T) + ocfoso
oT
}(t t1) (3.3.47)
-
23
Again, we obtain another form of the solution Eqn. 3.3.47 by solving the pressure
equation, and using the Darcy equation to replace velocity in Eqn. 3.3.43. Assuming
total compressibility of the porous medium is negligible, then the pressure equation
reduces to:
1
r
rrp
r= 0 (3.3.48)
with boundary conditions
p(r = re) = pe (3.3.49)
p(r = rw) = (t) (3.3.50)
where re = external radius and rw = wellbore radius.
Therefore,
p(r, t) = pe +pe (t)ln(rerw
) . ln( rre
)(3.3.51)
and
p
r= [
pe (t)ln(rerw
) ].1r
(3.3.52)
Applying this to Eqn. 3.3.43 by replacing the velocity with the Darcy law equiv-
alent and solving gives
r2 = r21 2(pet s(t)) (3.3.53)
where
={wcfwsww + ocfosoo}
cm lnR
R = lnrerw
-
24
s(t) =
t0
()d
The parameter (t) which is the sandface pressure (bottomhole pressure) in the
well bore is readily obtainable from solutions of classical pressure transient problems
for multiphase flow, and for different reservoir models.
The temperature model therefore becomes,
dT
dt=
t0
dp
dtdt+ ( )
t0
p
d (3.3.54)
where
= wcfwswww+ocfosooowcfwsww+ocfosoo
This yields the solution
T (r, t) = T0(r) [p(r, 0) p(r, t)] lnR
t0
() ln
(r
re
)d (3.3.55)
Using the average time theorem, the integral on the left hand side of Eqn. 3.3.55
can be closely approximated when an optimal average time is used. Therefore, the
final approximate solution for wellbore sand face temperature becomes:
T (rw, t) = T0(r1)[p(r, 0)(t)] lnR
[(t) (0)] ln(
r21 2(pez s(z)re
)(3.3.56)
with
r1 =r2w + 2(pet s(t))
0 < z < t, z = average time. An optimal choice for z must be used.
-
25
B. Solution of the diffusion part:
The form of the diffusion problem is
1
T
t=
1
r
T
r+2T
r2(3.3.57)
0 < r
-
26
was to use Operator Splitting and Time Stepping (OSATS), in which case convec-
tive transport part was solved analytically (since it is easier to solve this way) and
numerical discretization methods were used the diffusion transport part.
The analytical solution of the convective transport problem has been shown in
preceding sections. In discretizing the diffusion part, the finite difference scheme was
adopted, and the coordinate system was taken to be cartesian. This gave:
1
T
t=
1
r
T
r+2T
r2(3.4.1)
Which became (in two-dimensional cartesian coordinate system) :
1
T
t=2T
x2+2T
y2(3.4.2)
1
T n+1i,j T n+1i,j =
1
x2T ni+1,j T ni1,j +
1
y2T ni,j+1 T ni,j1 (3.4.3)
subject to the same boundary and initial conditions.
Therefore, in the OSATS routine, at each time step, the convective problem was
solved analytically, while the diffusion problem was numerically approximated in the
central difference formulation shown in Eqn. 3.4.3 and solved in an explicit scheme.
3.5 Wellbore Temperature Transient Model
Permanent downhole monitoring tools are usually located some hundreds of feet (200
- 300 ft) above the perforation/production zone. The tool placement constraint is one
that is imposed by the design of the completions, and the optimal location for pressure
and temperature data management would be a position as close to the perforation as
possible, to give measurements that are comparable with their sandface values. This
calls for a coupling of the reservoir temperature model to a wellbore model to account
-
27
for heat loss that may occur between 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 from the work of Izgec et al.
[8]. The solution to the model is a modification of Rameys model [9] to account for
heat transfer at shut-in times when flow rate is zero and heat transfer in the wellbore
is only by conduction into the formation.
Izgec et al. [8] showed that the distribution of temperature in a wellbore can be
obtained, as a fucntion of depth, by
Tf (r, t) = Tei+1 eaLRt
LR
[1 e(zL)LR](gG sin + (g sin )
cpJgc
)+e(zL)LR (Tfbh Tebh)
(3.5.1)
where
LR is called the relaxation parameter, defined by:
LR =2piwcp
[rtoUtoke
ke+rtoUtoTD
]
with
TD = ln [e0.2tD + (1.5 0.3719etD)]tD
tD =tr2w
= wellbore inclination angel
= dpdz
Tfbh = fluid temperature entering the sandface from formation (solution to the
reservoir temperature model)
Tei = geothermal temperature of formation at gauge location
-
28
Tebh = geothermal temperature of formation at bottomhole
3.6 Coupling the Reservoir Model to the Wellbore
Model
The issue of gauge placement, usually at some distance away from the perforation calls
for the coupling of the reservoir model to a wellbore model to account for heat loss
during flow between the perforation and the gauge location. In principle, the closer
the gauge to the perforation, the closer the overall model would be to the reservoir
model and the better the reservoir model can be used for further reservoir/formation
analysis.
The two models are coupled through the temperature of the fluid at the bottomhole.
The bottomhole temperature is estimated using the distributed reservoir model, and
used as an input into the wellbore model to estimate the fluid temperature at the
gauge location. The sensitivity of this overall model to the distance between the
gauge and the perforation will be revisited in a sensitivity analysis test to determine
the viability of using the model in reservoir studies, since the further away the gauge
is from the perforation, the less sensitive the solution will be to the reservoir model,
and more sensitive to the wellbore model.
-
Chapter 4
Qualitative Evaluation and
Sensitivity Analysis
4.1 Optimal Transformation for Estimation of Func-
tional Relationship
In order to show that a functional relationship may exist between temperature as a
response variable, and rate and pressure as the predictor variables, the nonparamet-
ric regression method known as Alternating Conditional Expectations (ACE) was
applied to a field data set. ACE yields optimal transformations of the variables,
and the correlations between these transformations have been shown to be optimal
for regression between the variables. These transformations are also useful in estab-
lishing the existence of a functional relationship between the response and predictor
variables. Using a field data set obtained from 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
29
-
30
relationship. Figures 4.1 and 4.2 show the plots of rate, pressure and temperature
data, and the plot of the regression on the optimal transformations.
Figure 4.1: Flow rate data (left), pressure data (right) from DAT.1 data set.
Figure 4.2: Temperature data (left) from DAT.1 data set, optimal regression (right).
The optimal transformation functions showed a correlation coefficient of 0.99.
This means that temperature is well correlated with flow rate and pressure, that a
functional relationship may exist between temperature, and rate and pressure and
this functional form can be extracted from any representative data set.
-
31
4.2 Qualitative Evaluation of Model - Single-phase
Case
Having seen, from ACE analysis, that a functional form may exist between temper-
ature as a dependent variable, and rate and pressure as the independent predictor
variables, the mechanistic models developed for their relationship were tested quali-
tatively for a check of reproducibility of trends seen in the data used. Using arbitrary
but typical and physically meaningful values of the model parameters, the following
results were generated for qualitative evaluation of the model and the solution strat-
egy adopted here. The model were also checked for reproducibility of transient trends
seen in the measured data.
The input data sets used were flow rate information from two real fields, and here
called DAT.1 and DAT.2 data sets. These were obtained from permanent downhole
gauges (PDGs). The data sets consist of PDG measurements of flow rate, pressure and
temperature with time for different wells in different fields. Using the representative
flow rate data as input, and thermal model developed, the temperature profile was
simulated for each representative data input set.
Figures 4.3, 4.4, and 4.5 show that the model and the solution qualitatively cap-
tured the changes, effects and trends in the data, in acceptable details. The overall
shapes are reproduced by the formulation/solution using arbitrary model parameters.
This forms a motivation for performing sensitivity analysis on the model parameters,
for using the model in parameter estimation in an inverse problem and for subsequent
uncertainty analysis.
-
32
Figure 4.3: Qualitative comparison using DAT.1 (first representative transient).
Figure 4.4: Qualitative comparison using DAT.1 (second representative transient).
-
33
Figure 4.5: Qualitative comparison using DAT.2.
4.3 Sensitivity Analysis
Many variables/parameters are present in the model formulation and uncertainties
in their values present a challenge in further processing and utilization of the formu-
lation and solution methodology presented in this work, hence the need to test the
sensitivity of the formulation and solution to different values of these parameters.
The following parameters were tested for sensitivity of the solution to their values:
the porosity of the formation, Joule-Thomson coefficient of the fluids, reservoir thick-
ness, fluid viscosity, thermal conductivity of rock and fluid, permeability, thermal
diffusivity length, distance of permanent downhole gauge from the perforation and
the geothermal gradient.
-
34
Figure 4.6: Sensitivity to porosity (top) and fluid Joule-Thomson coefficient (bottom).
-
35
Figure 4.7: Sensitivity to thermal diffusivity length (top) and fluid thermal conductivity(bottom).
-
36
Figure 4.8: Sensitivity to reservoir permeability (top) and fluid viscosity (bottom).
-
37
Figure 4.9: Sensitivity to gauge placement.
It is clear from Figures 4.6 to 4.9 that the temperature formulation and solution is
sensitive to most of the model parameters. The parameters with the most prominent
sensitivity (> 50% in temperature estimation for
-
38
estimation, permeability of the medium was specified, reducing the final parameter
space to formation porosity, formation thermal diffusivity length and fluid Joule-
Thomson coefficient. Sensitivity to the gauge distance presents a problem in using the
formulation presented in this work for possible reservoir parameter estimation. The
temperature of the fluid recorded at the gauge location is a combined effect of thermal
transient processes in the reservoir which delivers fluid of changing temperature (with
constant/changing rate) to the wellbore from the reservoir, and the heat loss to the
external formation during flow up the wellbore to the gauge location. This makes the
magnitude of heat loss in the wellbore very important, and calls for a careful use of
the formulation developed here if the gauge distance is very large. This is because,
at a very large distance away from the formation, the magnitude of the heat loss in
the wellbore will mask the effect of the reservoir thermal transients and hence lead to
poor reservoir parameter estimation. By repetitive trials, the optimal distance of the
gauge from the perforation to ensure obtaining representative reservoir parameters is
< 100m (300ft). This however is the average distance currently used in many field
applications.
Diffusivity length issues - single-phase
The challenge presented by the optimal selection of the diffusivity length parameter,
b, became apparent. Since this parameter depends on the thermal diffusivity and the
length of shut-in time (diffusion-dominated heat transfer period), different shut-in
regimes required different optimal diffusivity length because of differing shut-in time
durations.
As a test for this, the flow rate from DAT.1 data set (800 hrs, 466000 data points)
was used as input to the model to predict the temperature over the entire duration
of the measurement. Uniform thermal diffusivity length was assumed over several
-
39
transients and revealed a complete loss of the diffusion behavior later in time. Using
a uniform but different diffusivity length, b = 11m, over the data region 0 - 350,000
(data point counter on x-axis)(Figure 4.10), b = 8m over the data region 200,000 -
400,000,(Figure 4.11) and b = 12m over 300,000 - 400,000 (Figure 4.12). These plots
are shown only for qualitative reasons since model parameters were specified arbi-
trarily and were generated to see the behavior of the different shut-in regions with
different diffusivity length scale, as well as to reveal the effects later in time.
-
40
Figure 4.10: Temperature estimation with b = 11m, over 0-350,000 data points (left), actualdata (right).
Figure 4.11: Temperature estimation with b = 8m, over 200,000-350,000 data points (left),actual data (right).
-
41
Figure 4.12: Temperature estimation with b = 12m, over 300,000-400,000 data points (left),actual data (right).
Figure 4.13: Temperature estimation with b = 10m, over 100,000-200,000 data points (left),actual data (right).
The results show that while the model has the ability to predict the temperature
profile in the reservoir, the accuracy of that prediction depends on the diffusivity
-
42
length that characterizes the behavior of the profile at shut-in periods. No one dif-
fusivity length value will characterize the entire model over a long period of time
with recurring transients of different durations. Therefore, the optimal choice of this
length scale must be made and would not be one uniform value over several tran-
sient periods or over data taken for a relatively long time. Figure 4.13 shows that an
optimal selection should be found over each representative transient, separately and
independent of previous or subsequent shut-ins.
4.4 Sampling the Distribution of the Parameter
Space
The nature of the distribution of the parameter space is not known explicitly since the
model is nonlinear and the solution is semianalytic. Monte Carlo simulations was per-
formed to generate the one-dimensional and two-dimensional marginal distributions
of the parameter space. The distributions were then sampled for identification of the
optimal search space for parameter estimation, as well as to check for multimodalities
in the distribution of model parameters, using the method of volumetric probabilities
introduced by Tarantola [21]. The two-dimensional marginal distribution for the ra-
dial system, with porosity and oil Joule-Thomson coefficient as parameters is shown
in Figure 4.14.
-
43
Figure 4.14: Two-dimensional marginal distribution of porosity and oil Joule-Thomsoncoefficient from DAT.1.
Figure 4.14 shows that the distribution of the parameter space, in two-dimensional
marginal distribution sense is unimodal. The plots also show that the optimal pa-
rameter space for both porosity and oil Joule-Thomson coefficient as captured by the
model is within the feasible range and the values are physically realistic.
-
Chapter 5
Results and Discussion
The model solution, unique to the boundary condition chosen in the formulation, was
matched to the temperature data using the flow rate as input. In the case of field
data collected over long periods in time, because of thermal diffusivity length issues,
representative transient regions were selected, to ensure that a constant diffusivity
length could be used. In the optimization routine, the parameters that were perturbed
to establish the match were porosity , oil Joule-Thomson coefficient , fluid thermal
conductivity f (in some instances as a check) and the optimal diffusivity length
b. The values of these parameters obtained at the optimal match were taken to be
estimates of the optimal values of the parameters.
5.1 Synthetic Data
As a check on the procedure, synthetic data were generated using the model devel-
oped in this work. The synthetic data were generated in three forms:
normally distributed random noise (with mean 20% of the range of the flow
44
-
45
rate from DAT.1 data set) was added to the flow rate data, and this was used
as input to generate a temperature data set using the single-phase model
again, normally distributed random noise was added to the temperature dataobtained above to create a second noisy temperature data set
using a different transient region in DAT.1 data set, the step above was repeatedto generate a third temperature data set with appropriate noise added to it,
using the single-phase model
the two-phase model was also used to generate a temperature data set
Each of the four temperature data sets above was used as true measurement in
an inversion step to attempt to reestimate the model parameters used in generating
the data sets. This was done to test the robustness of the formulation and solution
strategy, and to examine the possibility of using them in an inverse model for param-
eter estimation.
-
46
Figure 5.1: Single-phase: temperature match using rate data with random noise.
Figure 5.2: Single-phase: temperature match using rate data with random noise, and truetemperature with random noise.
-
47
Figure 5.3: Single-phase: temperature match with random noise added to the true tem-perature.
Figure 5.4: Single-phase: temperature match using rate data with random noise, withdifferent transient region.
-
48
Figure 5.5: Two-phase: temperature match using rate data with random noise, and truetemperature with random noise.
For Figure 5.1, the data were generated using = 0.3, = 4.5 108(K/pa)and b = 7.0m. The optimal parameter values after the match were = 0.227, =
3.5 108(K/pa) and b = 7.3m.
In Figure 5.2, the data were generated using = 0.3, = 4.5 108(K/pa) andb = 7.0m and the match occurred at = 0.231, = 3.27 108(K/pa) and b = 7.2m.In Figure 5.4, the data were generated with = 0.3, = 4.5 108(K/pa) andb = 8.0m and the match occurred at = 0.259, = 1.51 108(K/pa) and b = 8.32mFinally, in the two-phase case shown in Figure 5.5, the data were generated with
= 0.3, Sw = 0.45 and b = 2.7m and the match occurred at = 0.145, Sw = 0.493
and b = 3.0m.
The plots show good matches between the true data and simulated results for
the tests on both models (single- and two-phase models). Also, the values of the
-
49
model parameters obtained after the matching were close to the original values of
the parameters. This shows that the formulation with the solution method used was
able to closely reestimate the parameters used in generating a synthetic data set and
so lends support to the attempt to use the model formulation in an inverse step for
reservoir parameter estimation by matching the model to real field temperature data.
5.2 Field Data
5.2.1 Single-phase System
Two different sets of field data were used to test the model, as well as obtain model
parameters at optimal match of the model to the data. The data sets have been named
DAT.1 and DAT.2. DAT.1 set is an 800-hr long measurement, each measurement
taken every 6 secs. DAT.2 is 24-hr long with each measurement taken every second.
Again, representative transient regions were selected, to ensure a constant diffusivity
length and the parameters for used in the inverse problem were porosity , oil Joule-
Thomson coefficient , fluid thermal conductivity f (in some instances as a check)
and the thermal diffusivity length b.
The following plots were obtained using the semianalytical solution technique (Op-
erator Splitting and Adaptive Time Stepping [OSATS]) discussed in Chapter 3.0, and
the values of the model parameters at optimal match are shown in the caption of each
figure.
-
50
Figure 5.6: Matching DAT.1 using OSATS ( = 0.196, = 1.16107(K/Pa), b = 9.2m).
Figure 5.7: Matching DAT.1 using OSATS ( = 0.21, = 9.0 109(K/Pa), b = 6.12m).
-
51
Figure 5.8: Matching DAT.1 using OSATS ( = 0.235, = 4.48 108(K/Pa), b =7.817m).
Figure 5.9: Matching DAT.1 using OSATS ( = 0.2, = 7.7 108(K/Pa), b = 5.2m).
-
52
Figure 5.10: Matching DAT.1 using OSATS ( = 0.228, = 1.09109(K/Pa), b = 3.0m).
Figure 5.11: Matching DAT.1 using OSATS ( = 0.21, = 4.3108(K/Pa), b = 5.16m).
-
53
Figure 5.12: Matching DAT.2 using OSATS ( = 0.182, = 5.95 108(K/Pa), b =5.92m).
Figures 5.6 to 5.11 show good matches between the single-phase model and DAT.1
data set. Figure 5.12 also show the match using the single-phase model on DAT.2
data set. Both direct search (Genetic algorithms) and gradient based (Levenberg
Marquardt) optimization techniques were used in the inverse problem to run each
case, and the optimal parameters at the matches were always approximately equal
for each optimization algorithm used. These values of the model parameters (also
reservoir and fluid properties) obtained at optimal match are physically meaningful,
and the porosity values are well within the range of porosity values for carbonate
reservoirs (for DAT.1 data set), the Joule-Thomson coefficient values also satisfy the
range of values of the coefficient that have been obtained for different types of crude
oil. The thermal diffusivity length correlation to the duration of shut-in for each
transient is also very obvious. The longer the shut-in (when input flow rate is zero),
the longer the optimal thermal diffusivity length estimated from the inverse problem.
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54
A fully numerical solution of the the model formulation was also attempted
for comparison with results from the semianalytical solution. It must be noted
that numerical solution to convection-diffusion problems, especially where there are
source/sink terms, are known to be very unstable. A way out was to use Opera-
tor Splitting and Time Stepping (OSATS), in which case numerical discretization
methods were used to solve each of the decoupled convection and diffusion transport
parts.
Figure 5.13: Matching DAT.1 using fully numerical solution ( = 0.286, = 1.53 108(K/Pa), b = 2.0m).
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55
Figure 5.14: Matching DAT.1 using fully numerical solution ( = 0.287, = 9.0 109(K/Pa), b = 2.6m,f = 0.11(W/m.K), s = 4.0(W/m.K)).
Figures 5.13 and 5.14 show that the numerical scheme could also match the data.
The model parameters at optimal match are comparable to the values obtained using
the more stable semianalytic solution scheme.
5.3 Two-phase System
The two-phase formulation has the fluid saturations included among the model pa-
rameters. The saturation variables in the formulation are static (not modeled to
change with time). However, since permanent monitoring devices take measurements
over long periods of time, saturation changes with time can be modeled as a time-lapse
problem, in which effective values of saturations are estimated at each time.
Saturation data to fully test this formulation was not readily available at the time
of writing this report. Therefore, data from the single-phase system (DAT.1) was
used to test the formulation. The intent of the inversion was to check if the inversion
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56
process would drive the water saturation to the specified critical value when data
acquired from single-phase oil flow is used. Therefore, as in the single-phase case, the
model developed here, unique to the boundary condition chosen in the formulation,
was matched to the temperature data using the flow rate as input. Representative
transient regions were selected, to ensure a constant diffusivity length and the param-
eters for estimation were porosity , water saturation Sw, and the optimal thermal
diffusivity length b. The critical water saturation value used was Swc = 0.2.
Figure 5.15: Matching using two-phase model ( = 0.25, Sw = 0.3, b = 5.36m).
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57
Figure 5.15 shows the plots of the match of the two-phase model to data single-
phase system. Since the data set used was that from a single-phase oil flow, the
inversion optimization step was expected to drive the water saturation to its critical
value at optimal match. The initial value of water saturation was set at 0.6. Then a
gradient-based search was used to obtain the match of the model to the data. The
estimated value of water saturation at optimal match (according to tolerance set on
the optimization routine) was Sw = 0.28. The algorithm drove the water saturation
towards the critical water saturation specified, at each iteration. Using direct search
(Genetic algorithm), similar results were also obtained.
5.4 Potential Applications
The models developed in this study have been shown to have the potential to help
characterize a reservoir using only rate and temperature data from any permanent
downhole monitoring source. Until recently, temperature data measured have been
largely ignored. Specifically, this study provides a method for the estimation of poros-
ity in homogenous fields, as well as potential for estimation of saturation distribution
in a reservoir using temperature data only. Being rate-dependent, the temperature
data could also offer a way to estimate sandface flowrate by inverting the temperature
model to estimate flowrate.
Spatial distribution of porosity and saturation fields
(heterogeneous field preliminary study)
The formulations and their solutions allow for local point-to-point estimation of tem-
perature in a discretized spatial grid. Each local estimation is dependent on the local
values of the model parameters, hence allowing for the estimation of local (grid) val-
ues of model parameters, each depending on the value of the temperature at the grid.
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58
This forms the basis for the estimation of heterogeneous parameter fields. Results
can serve as secondary conditioning data for porosity and saturation fields in reservoir
modeling. Examples of such estimations are shown in Figures 5.16 and 5.17.
Figure 5.16: Three-dimensional spatial temperature distribution at a time instant usingDAT.1 data set.
Data from interference well testing or horizontal wells could be used in check-
ing/validating these characterization of temperature fields in space.
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59
Figure 5.17: Two-dimensional spatial temperature distribution at a time instant usingDAT.1 data set.
-
Chapter 6
Scope for Further Study
6.1 Updating the Boundary Conditions
Only one boundary condition has been specified in the solution obtained in this
problem (bounded for convective, infinite radial for diffusion). Different reservoir
configurations will require different boundary condition (e.g. where the reservoir is
bounded by an aquifer), and will require the specification of the appropriate boundary
conditions in the solution.
6.2 Updating the Model to Treat Three-phase Sys-
tems (Gas-Oil-Water)
The formulations presented in this study considered single-phase oil, and two-phase
oil-water systems. In a single-phase gas system, some conditions and assumptions
made must be relaxed to accommodate specific physical conditions that come with
gas systems. For example, gas densities are not constant and are specifically affected
60
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61
by changes in temperature. Similarly, gas thermal conductivity is a strong function
of temperature as well as gas viscosities. Most of these conditions were relaxed in
building the single-phase oil model. Similar considerations must also be made in
developing the three-phase gas-oil-water model.
6.3 Reservoir Models with Fluid Injection or Phase
Transition
The formulation in this study assumed no fluid injection and considered temperature
changes due strictly to fluid expansion and viscous dissipation. However, the idea
behind the model can be extended to also account for cases where there is fluid
injection into the reservoir. Similarly, phase transitions (e.g. liberation of free gas in
the reservoir) can also be considered for a more complete picture of thermal effects
in the reservoir.
6.4 Cointerpretation of Pressure, Rate and Tem-
perature
This model presents an additional source of information in transient analysis - tem-
perature. Using ACE, it was shown that a functional relation exists between temper-
ature, rate and pressure. This presents the possibility of cointerpreting rate, pressure
and temperature.
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62
6.5 Heterogenous Fields
As shown earlier, if the appropriate data is available (e.g. temperature distribution
in space from a horizontal well, or temperature measured at different wells producing
from the same reservoir - interference testing), the spatial distribution of the model
parameters could be estimated and used to characterize the heterogenous fields of
porosity and saturation.
-
Nomenclature
b = thermal diffusivity length, m C = volumetric heat capacity ratio,(wcfwsw+ocfoso)
cm
Cf = volumetric heat capacity of fluid,J
m3K
cm = (1 )(scs) + ((wcfwsw) + (ocfoso)) , volumetric heat capacity of fluid sat-urated rock, J
m3K
p = pressure, Pa q = flowrate m3
s
r = radius
T = temperature, K
Tfbh = fluid temperature entering the sandface from formation (solution to the reser-
voir temperature model)
Tei = geothermal temperature of formation at gauge location
Tebh = geothermal temperature of formation at bottomhole
tD = dimensionless timetr2w
Uto = overall heat transfer coefficient,J
(secKm2)
u = Cv
v = superficial velocity, ms
Greek
= porosity
= density
= TCf
, adiabatic expansion coefficient, KPa
= T1Cf
, Joule-Thomson coefficient, KPa
= 1( T
)p , thermal expansion coefficient,1K
= mcm
, thermal diffusivity, m2
s
= thermal conductivity, WmK
m = f + (1 )s, thermal conductivity of fluid saturated rock
63
-
64
= optimal transformation of the dependent variable = optimal transformation of the independent variables = wellbore inclination angel
= dpdz
subscripts:
w = water
o = oil
f = fluid
-
References
[1] J. Bear, The Dynamics of Fluids in Porous Media, Dover Publications, Inc, 1972.
[2] A. Bejan, Convective heat transfer, Wiley, 2004.
[3] L. Breiman and J.H. Friedman, Estimating optimal transformation for multiple
regression and correlation, Journal of American Statistical Association 80 (1985),
no. 391, 580598.
[4] P. Dawkrajai, A.R. Analis, K. Yoshioka, D. Zhu, A.D Hill, and L.W Lake, A
comprehensive statistically-based method to interpret real-time flowing measure-
ments, DOE Report (2004).
[5] A.I. Filippov and E.M. Devyatkin, Barothermal effect in a gas-bearing stratum,
High Temperature 39 (2001), no. 2, 255263.
[6] H. Holden, K.H. Larlsen, and K.A. Lie, Operator splitting methods for degen-
erate convection-diffusion equations, II: Numerical examples with emphasis on
reservoir simulation and sedimentation, Comput. Geosci. 4 (2000), 287323.
[7] R.N. Horne and K. Shinohara, Wellbore heat loss in production and injection
wells, J. Pet. Tech (1979), 116118.
[8] B. Izgec, C.S. Kabir, D. Zhu, and A.R. Hasan, Transient fluid and heat flow
modeling in coupled wellbore/reservoir systems, SPE 102070 presented at the
SPE Annual Technical conference, San Antonio, Texas (Sept. 2006).
65
-
66
[9] H.J Ramey Jr., Wellbore heat transmission, JPT 435 (1962).
[10] J. Kacur and P. Frolkovic, Semi-analytical solutions for contaminant transport
with nonlinear soption in one dimension, University of Heidelberg, SFB 359 24
(2002), 120.
[11] J.I. Masters, Some applications of the p-function, Journal of Chem. Physics 23
(1955), 18651874.
[12] F. Maubeuge, M. Didek, M.B. Beardsell, E. Arquis, O. Bertrand, and J.P. Cal-
tagirone, Mother:a model for interpreting thermometrics, SPE 28588 presented
at the SPE Annual Technical Conference and Exhibition (Sept. 1994).
[13] D.A. Neild and A. Bejan, Convection in porous media, Springer, 1999.
[14] M.N. Ozisik, Heat conduction, Wiley-Intersciences, 1993.
[15] A. Sh. Ramazanov and V.M. Nagimov, Analytical model for the calculation of
temperature distribution in the oil reservoir during unsteady fluid inflow, Oil and
Gas Business Journal (2007).
[16] A. Sh. Ramazanov and A.V. Parshin, Temperature distribution in oil and water
saturated reservoir with account of oil degassing, Oil and Gas Business Journal
(2006).
[17] M. Remesikova, Solution of convection-diffusion problems with nonequilibrium
adsorption, Journal of Comp. and Applied Maths 169 (2004), 101116.
[18] R.K. Sagar, D.R. Dotty, and Z. Schmidt, Predicting temperature profiles in a
flowing well, SPE 19702 presented at SPE Annual Technical Conference and
Exhibition, San Antonio, TX (1989).
-
67
[19] R.F. Sharafutdinov, Multi-front phase transitions during nonisothermal filtration
of live paraffin-base crude, Journal of Applied Mechanics and Techincal Papers
42 (2001), no. 2, 284289.
[20] K.C. Shiu and H.D. Beggs, Predicting temperatures in flowing oil wells, J. Energy
Resources Tech (1989), 111.
[21] A. Tarantola, The Inverse Problem Theory and Model Parameter Estimation,
SIAM, 2005.
[22] R.A. Valiullin, R.F. Sharafutdinov, and A.Sh. Ramazanov, A research into ther-
mal fields in fluid-saturated porous media, Elsevier 148 (2004), 7277.
[23] K. Yoshioka, Detection of water or gas entry into horizontal wells by using per-
manent downhole monitoring systems, Ph.D. thesis, Texas A and M University,
Department of Petroleum Engineering, 2007.
-
Appendix
Matlab program files
Main calling file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Main Calling file. Main file for the optimization routine%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
clc
% x is the vector of parameters to be estinated. It comes as the vector
% [phi e bb lambdaf lambdas] where phi = porosity, e = JT coeff, lambda = thermal
% conductivity, bb = thermal diffusivity length. The vector x can be any
% length depending on number of parameters specified for estimation
display Using Tsimulator......
%%%Initial Guess
phid0 = log(0.300/0.2); ed0 = log((9.0*10^-8)/(8.0*10^-8)); bd0 = log(150/100);
lamdf0 = log(0.15/0.12); lamds0 = log(3/2); %re = log(100/100); h = log(30/30); nn = log(90/90);
x0 = [phid0 ed0 bd0 lamdf0 lamds0];% re h nn];
lb = [-1.08 -5.0 -2.350 -2.49 -1.386];% -3.3 -1.79 -1.5];
ub = [.9163 4.22 1.9163 1.200 0.9100 ];%3.30 1.500 2.05];
TypX = [phid0 ed0 bd0 lamdf0 lamds0];% re h nn];
%%%Optimization routine
nvars = 6;
[x, fval] = ga(@ParEstm,nvars,[],[],[],[],lb,ub); % Using Genetic algorithm, a direct search method
%%%Other algorithms to use
68
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69
% options = saoptimset(TolFun,1.00e-6);
% [x, fval] = simulannealbnd(@ParEstm,x0,lb,ub,options); % Using simulated
% annealing
% options = optimset(LevenbergMarquardt,on,LargeScale, off,MaxFunEvals,2000,TypicalX,TypX,Jacobian,off,TolFun,1e-15,TolX,1e-15,TolCon,1e-15);
% [x,fval,exitflag,output,lambda,grad,hessian] = fmincon(@ParEstm,x0,[],[],[],[],lb,ub,[],options); % Using LevenbergMarquardt
% options = optimset(LevenbergMarquardt,off,LargeScale, on,MaxFunEvals,2000,Jacobian,off,TypicalX,TypX,TolFun,1e-10,TolX,1e-10);
% %
% [x,resnorm,residual,exitflag,output,lambda,jacobian] = lsqnonlin(@ParEstm,x0,lb,ub,options);
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%ParEstimation program
function [F] = ParEstm(x)
x = real(x);
% load data
load B5.dat
time = B5(1:length(B5),1)*3600; %Time count (s),
press = B5(1:length(B5),2)*6894; %Pressure data, pa
temp = B5(1:length(B5),3) + 273.15; %Temperature data, K
rate = B5(1:length(B5),4)*0.00000184; %rate (m^3/sec)
P=press;
q=rate;
t=time;
Te=345.1; % Initial reservoir temperature, K
[ObjFunc] = ObjFun(P,q,t,temp,x,Te);
F = sqrt(sum(ObjFunc.^2));
[fid] = fopen(ResidualTrack.txt,at+);
fprintf(fid, %10.6f\n,ObjFunc);
fclose(fid);
[fid] = fopen(ParTrack.txt,at+);
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70
phi = 0.2*exp( x(1)); %porosity
e = (8.0*10^-8)* exp(x(2)); % % Joule thomson coefficient, K/Pa
bb = 100. * exp(x(4));
lamdf = 0.12*exp(x(5));
lamds = 2.0*exp(x(6));
% re = 100.*exp(x(7));
% h = 30.*exp(x(8));
% nn = 90.*exp(x(9));
fprintf(fid, %10.6f %10.12f %10.7f %10.6f %10.7f\n,phi, e, bb, lamdf, lamds);% re, h, nn);
fclose(fid);
Resnorm = sqrt(sum(ObjFunc.^2));
[fid] = fopen(ResNormTrack.txt,at+);
fprintf(fid, %10.6f\n,Resnorm);
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ObjFunc] = ObjFun(P,q,t,temp,x,Te)
%cfct= 2.*exp(x(3));
[Tmodel, init, fnl] = Tsim(P,q,t,x);
Tdata = temp(init:fnl);
Tcalc = (Te + Tmodel);
ObjFunc = Tdata - Tcalc;
% fopen(figure(1));
% plot((Te+Tmodel*x(3)),*r);
% hold on;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Tmodel, init, fnl] = Tsim(P,q,t,x)
%% Parameters for Estimation
% phi = x(1); %porosity
% e = x(2); % % Joule thomson coefficient, K/Pa
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71
% %lambdaf = x(3);
% cfct= x(3); % correction for wellbore heat loss
% bb = x(4);
phi = 0.2*exp( x(1)); %porosity
e = (8.0*10^-8)* exp(x(2)); % % Joule thomson coefficient, K/Pa
cfct= 2.0 * exp(x(3)); % correction for wellbore heat loss
bb = 100. * exp(x(4)); % Thermal diffusivity length, m
lambdaf = 0.12*exp(x(5)); %oil thermal conductivity, W/m.K
lambdas = 2.0*exp(x(6)); %formation rock thermal conductivity, W/m.K
%% fluid property
cf=1880; %oil spec heat, J/Kg.K
n=1.*10^-9; %adiabatic expansion coeff, K/Pa
beta= 0.001206; %thermal expansion coeff, 1/K
mu= 0.005; %current is from STARS at 157 F, 0.005; %viscosity, Pa.s
rhof=800; %oil density, Kg/m^3
%% Reservoir skeleton property/geometry
k = .7*10^-12; %permeability (2D), m^2
cs = 1000; %specific heat of solid, J/Kg.K
rhos=2300; %rock density, Kg/m^3
re=1000; % reservoir external radius, m
h=40; % reservoir height, m
D=3500; % well depth, m
zz=3400; % position of PDG above production zone, m
gG=0.0364; % geothermal gradient, K/m
g=9.81; % acceleration due to gravity, m/sec^2
cT=3.0; %
alphas=1.03*10^-6; % earth thermal diffusivity, m^2/sec
%% Wellbore property
rti=0.03; % tube inner radius, m
rto=0.04445; % tube outer radius, m
rci=0.08433; % casing inner radius, m
rco=0.0889; %casing outer radius, m
rwb=0.1; % wellbore radius, m
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72
%% combined property
ct= 1.16*10^-8; % formally *10^-8 % total compressibility, 1/Pa
cres=((1-phi)*(rhos*cs))+(phi*rhof*cf); % saturated reservoir volumetric heat capacity, J/K.m^3
%lambdar=((1-phi)*(lambdas*cs))+(phi*lambdaf*cf); %
lambdar=((1-phi)*(lambdas))+(phi*lambdaf);% saturated reservoir thermal conductivity, W/m.K
c=(rhof*cf)/cres;% vol heat capacity fraction, dimless
Ti=335.377; % Initial reservoir temperature, K
Pe = 3.2*10^7; % Initial reservoir pressure, pa
alpham=lambdar/cres; % thermal diffusivity of fluid saturated rock, m^2/sec
%%
R=re/rwb;
nu_x=n*phi*c;
big_phi = k*c/(mu*log(R));
T1 = 0.0;
q = q(7750:9000);
t = t(7750:9000);
init = 7750;
fnl = 9000;
diff = (fnl - init)+1;
% z = zeros(1,diff);
% phi_t = zeros(1,diff);
% M = zeros(1,diff);
% TT = zeros(1,diff);
% Ttt2 = zeros(1,diff);
% Ttt12 = zeros(1,diff);
% TT1 = zeros(1,diff);
% T = zeros(1,diff);
z(1) = (4/5)*t(1);
for i=2:diff
M(i) = (((q(i)-q(i-1))*mu/(4*pi*k*h))*(expint(rwb^2*phi*mu*ct/(4*k*(t(i)-t(i-1))))));
phi_t(i)=Pe + M(i);
tD(i) = k*t(i)/(phi*mu*ct*rwb^2);
tD = tD(i);
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73
if (tD < 10)
St(i) = (4/3)*((k/(pi*phi*mu*ct*rwb^2))^0.5)*(t(i)-t(i-1))^(3/2);
z(i) = (4/5)*t(i);
Sz(i) = (4/3)*((k/(pi*phi*mu*ct*rwb^2))^0.5)*(z(i)-z(i-1))^(3/2);
T(i) = T1 + e*(Pe - phi_t(i))-(((e+nu_x)/(log(R)))*(phi_t(i)-Pe)*log((sqrt(rwb^2 + 2*big_phi*St(i) - 2*big_phi*Sz(i)))/re));
else
St(i) = (t(i)-t(i-1))*(0.5*log(k*(t(i)-t(i-1))/(phi*mu*ct*rwb^2))-0.095465);
z(i) = (4/5)*t(i);
Sz(i) = (z(i)-z(i-1))*(0.5*log(k*(z(i)-z(i-1))/(phi*mu*ct*rwb^2))-0.095465);
T(i)=T1 + e*(Pe - phi_t(i))-(((e+nu_x)/(log(R)))*(phi_t(i)-Pe)*log((sqrt(rwb^2 + 2*big_phi*St(i) - 2*big_phi*Sz(i)))/re));
end
phi_tt = phi_t(i);
Stt = St(i);
Szz = Sz(i);
%end
%% Diffusion Part
nn=50;
aaa=0;
dx=(bb-aaa)/nn;
xs=(0.01:dx:bb);
jj=1;
kk=length (xs);
for jj=1:kk
Tr(jj)= T1+e*(Pe - phi_tt)-(((e+nu_x)/(log(R)))*(phi_tt-Pe)*log((sqrt(xs(jj)^2 + 2*big_phi*Stt - 2*big_phi*Szz))/re));
aa(jj)=exp(-(rwb^2+xs(jj)^2)/(4*t(i)*alpham));
bsl(jj)=besseli(0,((rwb*xs(jj))/(2*alpham*t(i))));
Tt(jj)= (1/(2*alpham*t(i)))*xs(jj)*aa(jj)*Tr(jj)*bsl(jj);
% aa(jj)=exp(-(rwb^2+xs(jj)^2)/(4*(t(i)-t(i-1))*alpham));
% bsl(jj)=besseli(0,((rwb*xs(jj))/(2*alpham*(t(i)-t(i-1)))),1);
% Tt(jj)= (1/(2*alpham*(t(i)-t(i-1))))*xs(jj)*aa(jj)*Tr(jj)*bsl(jj);
Tr1(jj) = T1+e*(Pe - phi_tt)-(((e+nu_x)/(log(R)))*(phi_tt-Pe)*log((sqrt(rwb^2 + 2*big_phi*Stt)/re)));
% Tt1(jj) = (1/(2*alpham*(t(i)-t(i-1))))*xs(jj)*aa(jj)*Tr1(jj)*bsl(jj);
Tt1(jj) = (1/(2*alpham*t(i)))*xs(jj)*aa(jj)*Tr1(jj)*bsl(jj);
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74
end
Ttt(i)=(0.5)*(Tt(1)+2*sum(Tt(2:kk-1))+Tt(kk))*dx;
Ttt1(i) = (0.5)*(Tt1(1)+2*sum(Tt1(2:kk-1))+Tt1(kk))*dx;
T1 = Ttt(i);
TT(i) = Ttt(i-1) + Ttt(i);
end
Tres = TT + Te;
wellbore();
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [] = wellbore()
%% Wellbore model
% sundry parameters
e = -e;
J = 1;
ke=2.423; % thermal conductivity of earth, watt/meter-K
kan=0.6629; % thermal conductivity of anulus material, watt/meter-K
kcem=6.96; % cement thermal conductivity, watt/meter-K
qc(1) = q(1);
tc(1) = t(1);
w(1) = qc(1)*rhof;
m=pi*zz*(rti^2)*rhof; % mass of fluid in bore volume between prod zone and PDG, kg
a(1) = w(1)*cf/(m*cf*(1+cT));
U=(rci*(log(rwb/rco)/kcem))^-1; % J/(sec-K-m^2)
TD(1) = -log(rco/(2*sqrt(alphas*tc(1))))-0.290;
Lr(1) = (2*pi/w(1)*cf)*(rto*U*lambdas/(lambdas+rto*U*TD(1)));
Tei = 294.26 + gG*3450;
Tein = 294.26 + gG*3500;
Fc(1) = -e*(rhof*g + qc(1)*mu/(k*pi*rci^2));
Zhi(1) = gG + Fc(1) - g/(J*cf);
Tf(1) = Tei + ((1-exp(-a(1)*tc(1)))/Lr(1))*(1-exp((zz-L)*Lr(1)))*Zhi(1) + exp((zz-L)*Lr(1))*(Lr(1)*(Tein - Tres(1)));
for jj = 2:diff
qc(jj) = q(jj);
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75
tc(jj) = t(jj);
w(jj) = qc(jj)*rhof;
m=pi*zz*(rti^2)*rhof; % mass of fluid in bore volume between prod zone and PDG, kg
a(jj) = w(jj)*cf/(m*cf*(1+cT));
U=(rci*(log(rwb/rco)/kcem))^-1; % J/(sec-K-m^2)
TD(jj) = -log(rco/(2*sqrt(alphas*tc(jj))))-0.290;
www = w(jj);
Lr(jj) = (2*pi/w(jj)*cf)*(rto*U*lambdas/(lambdas+rto*U*TD(jj)));
Tei = 294.26 + gG*3450;
Tein = 294.26 + gG*3500;
Fc(jj) = -e*(rhof*g + qc(jj)*mu/(k*pi*rci^2));
Zhi(jj) = gG + Fc(jj) - g/(J*cf);
Tf(jj) = Tei + ((1-exp(-a(jj)*Lr(jj)*tc(jj)))/Lr(jj))*(1-exp((zz-L)*Lr(jj)))*Zhi(jj) + exp((zz-L)*Lr(jj))*((Tein - Tres(jj)));
end
Tmodel = Tf;
%plot(Tf,r)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%