MSc Petroleum Engineering

Project Report 2015/2016

Edsson Ricardo Martínez Villarreal

Impact of Sensor Placement on Transient Temperature Signal and

Temperature Transient Analysis in Advanced Wells.

Heriot-Watt University

School of Energy Geoscience Infrastructure and Society

Institute of Petroleum Engineering

Khafiz Muradov

Akindolu Dada

Ehsan Nikjoo

i

Declaration

I, Edsson Ricardo Martínez Villarreal, confirm that this work submitted for assessment

is my own and is expressed in my own words. Any uses made within it of the works of

other authors in any form (e.g. ideas, Eq.s, figures, text, tables, programs) are properly

acknowledged at the point of their use. A list of the references employed is included.

Signed

Date, 18th April 2016.

ii

Acknowledgements

I wish to express sincere thanks and appreciation to Heriot-Watt University for the

support of this work.

I want to express my gratitude for the financial support of the Secretaría Nacional de

Educación Superior, Ciencia y Tecnología (SENESCYT) in Ecuador for the auspice of

this program.

I am deeply grateful to Akindolu Dada, Ehsan Nikjo and Khafiz Muradov for his

guidance throughout the execution of this project.

iii

SUMMARY

Transient temperature signal can be measured at different points in the wellbore for test

interpretation and production monitoring.

Temperature transient analysis (TTA) uses the collected signal to model inflow

performance into the well, and thus, provide a technique to identify and quantify the

phases produced. This method is valuable for zonal monitoring and control in advanced

wells that are completed with flow control devices and/or permanent downhole gauges.

When transient data is gathered in a different point than the midpoint of producing

interval (MPP), heat transfer between fluid in the wellbore and the surrounding

formation can affect the collected data.

This paper investigates the optimal gauge distance that minimizes the signal attenuation

due to wellbore thermal effects and presents an insight into the variables involved in

wellbore heat transfer and its impact on the transient signal.

To achieve this objective, the wellbore heat transfer problem was simulated for a dry

gas well in different flow conditions to provide a transient signal of temperature for

various gauge locations in a 500m segment. The signal attenuation was analysed by

comparing the drawdown features calculated in each scenario. The variables included in

this analysis were; gauge location, wellbore diameter and inclination, thermal properties

of the formation and geothermal gradient.

Simple correlations are developed for designing gauge placement based on the

completion design production scenario.

iv

TABLE OF CONTENTS

Summary .......................................................................................................................... iii

List of Figures .................................................................................................................. vi

List of Tables .................................................................................................................. vii

List of abbreviations ...................................................................................................... viii

Nomenclature................................................................................................................... ix

1 Aim and objectives .................................................................................................. xii

2 Literature review ....................................................................................................... 1

2.1 Concepts and definitions in heat transfer. .......................................................... 1

2.1.1 Enthalpy (h). ............................................................................................... 1

2.1.2 Heat capacity at Constant Pressure (Cp) .................................................... 1

2.1.3 Heat capacity at Constant Volume (Cv) ..................................................... 1

2.1.4 Thermal conductivity (K). .......................................................................... 2

2.1.5 Thermal Diffusivity (α) .............................................................................. 2

2.1.6 Joule Thomson effect.................................................................................. 2

2.1.7 Density (ρ). ................................................................................................. 3

2.1.8 Thermal expansion. .................................................................................... 3

2.1.9 Geothermal gradient ................................................................................... 4

2.2 Mechanisms of heat transfer. ............................................................................. 4

2.2.1 Thermal Conduction. .................................................................................. 5

2.2.2 Heat Convection ......................................................................................... 5

2.2.3 Radiation. .................................................................................................... 6

2.3 Wellbore heat Transfer. ..................................................................................... 6

2.3.1 Formation Temperature Distribution. ......................................................... 7

2.3.2 Wellbore Fluid Energy Balance. ................................................................ 8

2.3.3 Wellbore Fluid Temperature. ................................................................... 10

2.4 Overall Heat transfer coefficient. ..................................................................... 11

2.5 Downhole gauge placement ............................................................................. 13

2.5.1 Intelligent well monitoring. ...................................................................... 13

2.5.2 Transient data acquisition ......................................................................... 14

2.6 Drawdown response derivative behaviour. ...................................................... 16

3 Methods used – workflow ....................................................................................... 18

3.1 Wellbore/reservoir simulation ......................................................................... 18

3.1.1 Model set up and description .................................................................... 19

v

3.1.2 Processing of temperature transients and sensitivity cases. ..................... 22

3.1.3 Case study – Sensitivity to wellbore diameter. ........................................ 22

4 Results ..................................................................................................................... 26

4.1 Wellbore segmentation length ......................................................................... 26

4.2 Sensitivity study for wellbore diameter. .......................................................... 27

4.3 Sensitivity study to flow rate. .......................................................................... 28

4.4 Sensitivity study for wellbore deviation .......................................................... 29

4.5 Sensitivity study to formation thermal properties. ........................................... 31

4.5.1 Sensitivity study to formation heat capacity. ........................................... 32

4.5.2 Sensitivity study to thermal conductivity ................................................. 33

4.5.3 Sensitivity study to formation density. ..................................................... 33

4.6 Sensitivity study to Geothermal Gradient. ....................................................... 34

5 Discussion ............................................................................................................... 35

5.1 Wellbore diameter. ........................................................................................... 36

5.2 Flow rate. ......................................................................................................... 37

5.3 Wellbore deviation ........................................................................................... 37

5.4 Formation thermal properties ........................................................................... 37

5.5 Geothermal gradient ........................................................................................ 38

6 Economic impact of the work ................................................................................. 39

7 Conclusions ............................................................................................................. 40

8 References ............................................................................................................... 42

9 Appendices .............................................................................................................. 47

9.1 Essential Simulator Equations. ........................................................................ 47

9.2 Pressure drop in a gas well. ............................................................................. 48

vi

LIST OF FIGURES

Figure 1. Schematic of wellbore heat problem (Ramey, 1962). ....................................... 8

Figure 2. General configuration of wellbore surroundings (from Hasan and Kabir,

1994). .............................................................................................................................. 11

Figure 3. General characteristics of the temperature semi-log plot (Kutun, Inanc, &

Satman, 2015). ................................................................................................................ 17

Figure 4. Transient mass, momentum and energy balance in a gas well (from Kabir,

C.S. et. al, 1996). ............................................................................................................ 18

Figure 5. Wellbore section model layout built in OLGA. .............................................. 19

Figure 6. Transient temperature signal and flow rate measured at MPP. OLGA mass

source. ............................................................................................................................. 21

Figure 7. Transient temperature signal response from the OLGA simulation model. Base

case. ................................................................................................................................ 23

Figure 8. Semi-log plot for TTS during the second drawdown period. Wellbore

segment: 100m above MPP. ........................................................................................... 24

Figure 9. Log-log diagnostic plot for TTS during the second drawdown period.

Wellbore segment: 100m above MPP. ........................................................................... 24

Figure 10. Accuracy of semi-log slope during the second drawdown period. ............... 25

Figure 11. Attenuation function for diameter sensitivity analysis in function of sensor

placement. ....................................................................................................................... 26

Figure 12. Accuracy of semi-log slope. Sensitivity to wellbore segmentation. ............. 26

Figure 13. Overall heat transfer coefficient. Wellbore diameter sensitivity analysis..... 28

Figure 14. Accuracy of semi-log slope. Sensitivity to flow rate signal.......................... 28

Figure 15. Attenuation function for flow rate sensitivity analysis. ................................ 29

Figure 16. Attenuation function for wellbore inclination. Sensitivity case a) y = 500 m.

........................................................................................................................................ 30

Figure 17. Attenuation function for wellbore inclination. Sensitivity case b) L = 500 m.

........................................................................................................................................ 30

Figure 18. Overall heat transfer coefficient. Deviated wellbore sensitivity case b) L =

500 m. ............................................................................................................................. 30

Figure 19. Attenuation function for formation heat capacity sensitivity analysis. ......... 32

Figure 20. Overall heat transfer coefficient. Heat capacity sensitivity analysis............. 32

Figure 21. Attenuation function for formation thermal conductivity sensitivity analysis.

........................................................................................................................................ 33

Figure 22. Overall heat transfer coefficient. Thermal conductivity sensitivity analysis. 33

Figure 23. Attenuation function for formation density sensitivity analysis. .................. 34

Figure 24. Overall heat transfer coefficient. Formation density sensitivity analysis. .... 34

Figure 25. Attenuation function for geothermal gradient sensitivity analysis. .............. 35

Figure 26. Overall heat transfer coefficient. Geothermal gradient sensitivity analysis. 35

vii

LIST OF TABLES

Table 1. Dry gas properties for the simulator ................................................................. 19

Table 2. Properties of pipe and formation walls for the wellbore section. ..................... 20

Table 3. Initial conditions for interpolation of fluid properties in OLGA. ..................... 20

Table 4. Values of sensitivity variables evaluated in the wellbore heat-transfer

simulation. ...................................................................................................................... 22

Table 4. Values of sensitivity variables for formation properties. ................................. 31

viii

LIST OF ABBREVIATIONS

API. Mass density.

DTS Distributed temperature sensors

GLV Gas Lift Valve

JT Joule-Thomson

MPP Midpoint perforations of the producing interval.

HTHP High temperature - High Pressure.

TTS Transient temperature signal

TTA Temperature transient analysis

ix

NOMENCLATURE

A flow cross-sectional area.

b Surface geothermal temperature.

CJ Joule-Thompson coefficient.

Cp Specific heat at constant pressure of fluid, Btu/lb-°F.

d flow string diameter

D non-Darcy flow coefficient, D/Mscf.

E internal energy.

f Moody friction factor, dimensionless.

g acceleration due to gravity.

gT geothermal gradient.

H enthalpy.

hc convective heat transfer coefficient for annulus fluid, Btu/°F-hr-ft2.

hr radiative heat transfer coefficient for annulus fluid, Btu/°F-hr-ft2.

k conductivity, Btu/day-ft-°F.

L length of wellbore measured from perforations.

LR relaxation distance – wellbore heat-transfer model.

m mass of gas per unit length of tubing/casing/cement system.

P pressure

q fluid flow rate

Q heat flow rate from or to the wellbore, Btu/lb.

r radius

t producing time.

T temperature, °F.

TD dimensionless temperature

x

tD thermal diffusion time ( = ke t / rw2ρe Cpe)

U over-all heat-transfer coefficient, Btu/°F-hr-ft2.

v velocity.

W fluid mass rate.

Z gas-law deviation factor, dimensionless.

z depth below surface, ft.

β Coefficient for thermal expansion.

γg gas gravity

ε is the emissivity of the surface

η parameter combining the Joule-Thompson and kinetic energy effects.

θ pipe inclination angle from horizontal, degrees.

ρ Density.

σ Stefan-Boltzmann constant =1.713 x 10-9 Btu/ft2-hr-°R4.

ϕ porosity.

%t accuracy of the calculated slope respect to its value at sand face.

xi

Subscripts

bh bottom-hole.

c casing

cem cement

e earth – undisturbed formation.

f fluid.

G geothermal.

ins insulation.

l liquid.

m multiphase.

t tubing

wb wellbore-earth interface.

xii

1 AIM AND OBJECTIVES

The accuracy of transient temperature analysis depends on the quality of data available

for interpretation. The objective of this study is to analyse the effect of sensor placement

on the acquired transient temperature signal.

For this purpose, wellbore thermal effects were modelled in a simulator and its

influence on the transient signal was estimated by the attenuation obtained at different

positions downstream of the midpoint perforations.

This analysis will provide an initial understanding on how to account for thermal effects

at any given gauge location in order to restore the features of the temperature signal

expected at midpoint perforations.

The importance of this problem arises when data gathering takes place at a point other

than the midpoint of producing interval, either by restrictions on the wellbore or by

completion design, which can result in serious interpretation problems of the transient

signal.

1

2 LITERATURE REVIEW

2.1 CONCEPTS AND DEFINITIONS IN HEAT TRANSFER.

Before discussing the wellbore heat transfer problem, some concepts and definitions are

necessary.

2.1.1 Enthalpy (h).

Is the sum of thermal and flow energies in a given mass of the material based on a

reference state. The specific enthalpy, h, of a substance is defined as the enthalpy

content per unit mass of substance, and is equal to the internal energy per unit mass, u,

plus a flow term given by the product of pressure and specific volume.

ℎ = 𝑢 + 𝑃𝑉 = 𝑢 + 𝑃 𝜌⁄ (1)

2.1.2 Heat capacity at Constant Pressure (Cp)

Specific heat capacity or thermal capacity is the quantity of heat or change in enthalpy

required to increase a unit mass of a substance by one degree in temperature, while

maintaining constant pressure. This property characterise the variation of the fluid

temperature due to the absorption of heat.

𝐶𝑃 = (𝜕ℎ

𝜕𝑇)

𝑃 (2)

2.1.3 Heat capacity at Constant Volume (Cv)

Is the quantity of heat or change in internal energy required to increase the temperature

of a unit mass of the material by one degree of temperature, while maintaining constant

volume.

𝐶𝑃 = (𝜕𝑢

𝜕𝑇)

𝑉 (3)

2

Since the heat capacity of a phase is not a strong function of temperature except near the

critical temperature (Elshahawi, Osman, & Sengul, 1999), it is common to express

change in specific enthalpy and internal energy as:

𝑑ℎ = 𝐶𝑃 𝑑𝑇 (4)

𝑑𝑢 = 𝐶𝑉 𝑑𝑇 (5)

2.1.4 Thermal conductivity (K).

Thermal conductivity is the property that determines the quantity of heat transferred

through the material per unit time per unit thickness under a unit temperature gradient.

It is defined by Fourier’s Law presented in Eq. 11 (Elshahawi, Osman, & Sengul, 1999).

Heat transfer occurs at a lower rate across materials of low thermal conductivity than

across materials of high thermal conductivity.

2.1.5 Thermal Diffusivity (α)

Thermal diffusivity is defined as the ratio of the thermal conductivity to the volumetric

heat capacity. For a porous medium, this volumetric heat capacity is given by the

density-heat capacity product, which must apply to both the fluid and the rock in the

reservoir (Elshahawi, Osman, & Sengul, 1999).

𝛼 =𝑘

𝜌 𝐶𝑃 (6)

2.1.6 Joule Thomson effect.

The Joule-Thomson (JT) effect is described by the change in fluid temperature due to

the change in pressure for the condition of constant enthalpy, i.e. no heat is exchanged

with the environment. (Muradov, 2010).

3

𝐶𝐽 ≡ (𝜕𝑇

𝜕𝑃)

ℎ (7)

Petroleum liquids become hotter at as pressure decrease at downhole conditions

(Muradov, 2010). Eq. (7) yields:

𝐶𝐽 (𝑙) = −1

𝐶𝑝𝜌(1 − 𝑇 𝜌

𝑑

𝑑𝑇(

1

𝜌)

𝑃

) = −1

𝐶𝑃𝑙 𝜌𝑙

(1 − 𝛽𝑇) (8)

For gas flow, a pressure decrease at downhole will cause gases to cool (Muradov,

2010). For real gases Joule-Thomson coefficient can be expressed as:

𝐾𝐽𝑇 (𝑔) = −1

𝐶𝑝𝜌(1 −

1

𝑧

𝑑

𝑑𝑇(𝑧𝑇)𝑃) = −

1

𝐶𝑝𝜌(1 − 1 −

𝑇

𝑧

𝑑

𝑑𝑇(𝑧)𝑃) =

𝑇

𝐶𝑃 𝑔 𝜌𝑔 𝑧(

𝑑𝑧

𝑑𝑇)

𝑃 (9)

The average Joule-Thomson coefficient is the sum of the coefficients for each

individual phase, weighted by the phase volume factions and the phase velocities. Slip

effects allow the phase velocities to have significantly different values. (Muradov,

2010)

2.1.7 Density (ρ).

When density changes across the diameter of the tube are large, e.g. when the fluid is

near the critical point, the variable density can affect the transfer of momentum and

heat.

For flow in vertical tubes large density variations can also affect the heat transfer by

inducing natural convection.

2.1.8 Thermal expansion.

Thermal expansion is the tendency of matter to change in shape, area, and volume in

response to a change in temperature, through heat transfer. The volumetric thermal

expansion coefficient describes relative change in volume due to temperature change.

4

Two simple cases are isobaric change, where pressure is held constant, and adiabatic

change, where no heat is exchanged with the environment.

In the general case of a gas, liquid, or solid, the volumetric coefficient of thermal

expansion is given by:

𝛽 =1

𝑉(

𝜕𝑉

𝜕𝑇)

𝑝 (10)

In the case of a gas, the fact that the pressure is held constant is important, because the

volume of a gas will vary appreciably with pressure as well as temperature.

2.1.9 Geothermal gradient

The geothermal gradient reflects the amount of temperature rise as the well depth

increases. Normally about 1.1 to 1.8°F per 100 ft of true vertical depth of increase is

observed near the surface in most of the world.

An increase in the difference between the temperature of the fluid in the wellbore and

the surrounding formation increases the driving force for heat transfer. Heat loss

experienced by the fluid as it flows up the well causes its temperature to decrease.

Formation temperature is assumed to vary linearly with depth and thus, geothermal

temperature can be described as a linear function: TG = gT Z+b, where b is the surface

geothermal temperature.

2.2 MECHANISMS OF HEAT TRANSFER.

This section describes the mechanisms of heat transfer that allows to comprehend the

wellbore heat problem: conduction, convection and radiation.

5

2.2.1 Thermal Conduction.

In this process heat is transferred through non-flowing materials by molecular collisions

of particles and movement of electrons within a body from a region of high temperature

to another region of low temperature (Elshahawi, Osman, & Sengul, 1999). Heat

conduction is described by Fourier’s Law:

�̇�𝑧 = −𝐾𝜕𝑇

𝜕𝑧 (11)

For the wellbore problem considered, the thermal resistance of pipe or casing can often

be neglected because the thermal resistance of steel is much lower compared to the

cement and formation. Heat-transfer film coefficients of liquid water or condensing

steam also offer little resistance to heat flow. However, gas film coefficients and

thermal conductivity of insulating materials, e.g. the cement sheet between the casing

and hole, act as a resistance and thus reduce the over-all heat transfer coefficient

(Ramey, 1962).

2.2.2 Heat Convection

Convective heat transfer is the process in which energy is transferred by a flowing fluid.

For a production fluid inside the tubing the density of the hot fluid at the centre of the

pipe is less than that next to the tubing wall. This difference of density creates a

buoyancy force that acts against the viscous force and generates a circular motion of the

fluid inside the tubing, carrying energy with it (Hasan & Kabir, 1994).

The convective energy flux occurs in a direction that parallels that of the fluid and is

given by:

𝑑𝑄 = 𝑑𝐻 + 𝑑𝐸𝑝𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 + 𝑑𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐 = − [𝑑𝐻 + 𝑔 𝑑𝑧 + 𝑑 (𝑞2

𝜙2)] (12)

6

In the case of radial flow of an incompressible liquid, the potential and kinetic energy

changes are negligible (Elshahawi, Osman, & Sengul, 1999), and the total rate of

convective energy transfer is given by:

𝑑𝑄 = −𝜌𝑉𝑑ℎ = −𝑊 𝑑𝐻 = − 𝑊 𝐶𝑃 𝑑𝑇 (13)

2.2.3 Radiation.

Radiation is the transfer of heat by means of electromagnetic radiation. The rate of

radiation heat transfer from a heated surface per unit surface area is given by

𝑄 = 𝜎𝜀𝑇4 (14)

Emissivity (ε), is a dimensionless quantity equal to or less than one, which depends on

the nature of the surface. In his description of the wellbore heat problem, (Ramey, 1962)

pointed out that heat may be transferred from tubing to casing by radiation. However, in

most cases of oil production, the temperature difference across the annulus is usually

small, and only natural convection is considered as heat transfer mechanism.

Radiation is not normally considered to be an important heat transfer mechanism in

porous media (Elshahawi, Osman, & Sengul, 1999).

2.3 WELLBORE HEAT TRANSFER.

Two primary classes of models exist in the literature for quantitative temperature

analysis:

The first model was proposed by Ramey (1962) in which analytical expressions for the

wellbore temperature were obtained. This model consider the heat flow problem in

steady-state by neglecting heat conduction in the vertical direction, changes in the fluid

injection rate, horizontal temperature gradients, and any variations of either the heat

capacities or the densities of the formation materials or the injected fluids. Other

implicit assumptions were that the heat flows through various thermal resistances in the

7

immediate vicinity of the well-bore considerably faster than in the formation, and that

the fluid and rock within any small element of the formation are at the same temperature

(Elshahawi, Osman, & Sengul, 1999).

The second class includes McKinley’s model (1986) and other similar models which

rely on an energy balance for the produced fluids and neglect the formation properties

(Elshahawi, Osman, & Sengul, 1999), provides a review of these models.

2.3.1 Formation Temperature Distribution.

The transient unidimensional heat flow around the well is given by the following partial

differential equation:

1

𝑟

𝜕

𝜕𝑟(𝑘𝑖𝑟

𝜕 𝑇𝑖

𝜕𝑟) = 𝜌𝑖𝑐𝑝𝑖

𝜕 𝑇𝑖

𝜕𝑡 (15)

(Hasan & Kabir, 1994) developed a formation temperature distribution model, TD,

applicable for a finite wellbore inner boundary condition that allows easy calculation of

wellbore heat loss and flowing fluid temperature for steady state, two-phase flow.

Assuming radial symmetry around a wellbore the unsteady-state 3D heat diffusion can

be simplified as a 2D problem. Initially the formation temperature remains unalterable

with time and at the outer boundary the formation temperature does not change with

radial distance (Hasan & Kabir, 1994). The following algebraic approximation is

calculated:

𝑇𝐷 = 1.1281√𝑡𝐷(1 − 0.3√𝑡𝐷), 10−10 ≤ 𝑡𝐷 ≤ 1.5

𝑇𝐷 = (0.4063 + 0.5 𝑙𝑛𝑡𝐷) (1 +0.6

𝑡𝐷) , 𝑡𝐷 > 1.5

(16)

Where the dimensionless time is given by tD = αt/rw2, and the thermal diffusivity; α = ke

t / rw2ρe Cpe. The initial conditions are as follows:

8

t→0: lim Te → Tei.

r → ∞: lim ∂Te/∂r → 0

This expression is used to formulate a relation for heat transfer between the

wellbore/soil interface and the surrounding formation as follows:

𝑑𝑄

𝑑𝑧= −

2𝜋 × 𝑘𝑒

𝑊 × 𝑇𝐷

(𝑇𝑤𝑏 − 𝑇𝑒) (17)

Wellbore fluid temperature is governed by the rate of heat loss from the wellbore to the

surrounding formation, which in turn is a function of depth and production/injection

time (Hasan & Kabir, 1994).

2.3.2 Wellbore Fluid Energy Balance.

Ramey developed an approximate solution to the transient heat-conduction problem

involved in movement of fluids through a wellbore (Muradov, 2010). In this solution,

two main assumptions are considered: (1) heat flows radially away from the wellbore,

and (2) heat flow through various thermal resistances in the wellbore can be represented

by steady state solutions (Ramey, 1962).

Figure 1. Schematic of wellbore heat problem (Ramey, 1962).

The classical model developed by Ramey starts with the total energy equation, Eq.12:

9

𝑑𝐻 + 𝑑𝐸𝑝𝑜𝑡𝑒𝑛𝑐𝑖𝑎𝑙 + 𝑑𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐 = 𝑑𝑄 (18)

Where H is the fluid enthalpy, E – energy, and Q – heat transferred to the formation.

Heat flow rate of the fluid, and hence temperature profile in the wellbore, is

proportional to its mass rate, heat capacity and density. Observable changes in

temperature profile are used to locate injection or production zones, as well as the

relative flow allocation between zones (Muradov, 2010).

The application of Eq. 18 for a two-phase system leads to the following Eq. that

describes the heat loss experienced by the fluid as it flows up the well:

𝑑𝑇𝑓

𝑑𝑧=

1

𝐶𝑃

𝑑𝐻

𝑑𝑧+ 𝐶𝐽

𝑑𝑝

𝑑𝑧

𝑑𝑇𝑓

𝑑𝑧=

1

𝐶𝑃(

𝑑𝑄

𝑑𝑧− 𝑔 sin 𝜃 − 𝑣

𝑑𝑣

𝑑𝑧) + 𝐶𝐽

𝑑𝑝

𝑑𝑧

(19)

At steady state, the rate of heat flow through a wellbore per unit length of the well,

dQ/dz, can be expressed as a function of wellbore temperature, Twb:

𝑑𝑄

𝑑𝑧= −

2𝜋 𝑟𝑡𝑜 𝑈

𝑊(𝑇𝑓 − 𝑇𝑤𝑏) (20)

The overall heat transfer coefficient, U, depends on resistance to heat flow from the

fluid in the tubing to the surrounding formations. An expanded description presented in

the following sections.

Combining Eq. 17 for heat transfer at the formation/wellbore interface using the

formation temperature distribution TD, and Eq. 20 for radial heat transfer between the

fluid and the surrounding soil, we can derive the following expression for heat transfer

from the wellbore/soil interface to the soil (Hasan & Kabir, 1994).

𝑑𝑄

𝑑𝑧= −

2𝜋

𝑊(

𝑟𝑡𝑜 𝑈 𝑘𝑒

𝑘𝑒 + 𝑇𝐷 𝑟𝑡𝑜 𝑈) (𝑇𝑓 − 𝑇𝑒) (21)

10

2.3.3 Wellbore Fluid Temperature.

The expression for the fluid temperature variation with well depth, for a single-phase

liquid, is obtained from equations 19 and 21.

𝑑𝑇𝑓

𝑑𝑧=

𝑇𝑒 − 𝑇𝑓

𝐿𝑅−

𝑔 sin 𝜃

𝐶𝑃+ 𝐶𝐽

𝑑𝑝

𝑑𝑧−

𝑣𝑑𝑣

𝐶𝑃

𝐿𝑅 =𝐶𝑃 𝑊

2𝜋(

𝑘𝑒 + 𝑟𝑡𝑜 𝑈 𝑇𝐷

𝑟𝑡𝑜 𝑈 𝑘𝑒)

(22)

Where LR is the inverse relaxation distance proportional to the mass flow-rate and heat

transfer coefficient between the wellbore and the surroundings. Eq. 22 represents a

system where steady-state fluid flow occurs in the well bore while unsteady-state heat

transfer takes place in the formation as described by (Ramey, 1962).

This expression can be been written at bottom-hole conditions in order to obtain a

relation for fluid temperature as a function of well depth and producing time:

𝑇𝑓 = 𝑇𝑒 + 𝐿𝑅[1 − 𝑒(𝑍𝑏ℎ−𝑍) 𝐿𝑅⁄ ] (−𝑔 sin 𝜃

𝐶𝑃+ 𝜂 + 𝑔𝑇 sin 𝜃) + 𝑒(𝑍𝑏ℎ−𝑍) 𝐿𝑅⁄ (𝑇𝑓(𝑏ℎ) − 𝑇𝑒(𝑏ℎ))

𝜂 = 𝐶𝑓

𝑑𝑃

𝑑𝑧−

𝑣𝑑𝑣

𝐶𝑃

(23)

The value of the parameter η does not vary with well depth and depend on several

variables, such as the mass rate, gas liquid ratio, and wellhead pressure (Hasan & Kabir,

1994).

Formation temperature is assumed to vary linearly with depth. When different geologic

formations with differing geothermal gradients are encountered at various depths, the

computation may be divided into different zones with constant geothermal gradients

(Hasan & Kabir, 1994).

Eq. 23 is used to estimate wellbore fluid temperature. This approach considers wellbore

heat transfer by conduction, convection, and radiation, and estimates the formation

11

temperature by the use of the transient solution model, TD, for unsteady-state heat

diffusion in the reservoir under the assumption that physical and thermal properties of

earth and wellbore fluids do not vary with temperature and that heat will transfer

radially in the formation.

(Mendes, Coelho, Guigon, Cunha, & Landau, 2005), presents an analytical technique

for the solution of transient heat conduction in the wellbore by using a set of normalized

variables to substitute the physical parameters of the problem. In this approach, the heat

conduction equation is evaluated through the separation of variables method, using

Bessel functions.

2.4 OVERALL HEAT TRANSFER COEFFICIENT.

Heat transferred from the wellbore fluid to the soil overcomes resistances imposed by

the tubing wall, tubing insulation, tubing/casing annulus, casing wall, and the cement

depending on wellbore completion (Figure 2). These resistances are in series, and

except for the annulus, the only mechanism of transfer involved is conductive heat

transfer (Hasan & Kabir, 1994).

Figure 2. General configuration of wellbore surroundings (from Hasan and Kabir, 1994).

12

As described in the previous section, the radial heat flow through a wellbore can be

expressed as in terms of the overall heat transfer coefficient that considers the net

resistance to flow from the fluid inside the tubing to the surrounding formation. The

overall coefficient can be expressed as follows:

1

𝑈𝑡𝑜=

𝑟𝑡𝑜

𝑟𝑡𝑖ℎ𝑡𝑜+

𝑟𝑡𝑜 ln(𝑟𝑡𝑜 𝑟𝑡𝑖⁄ )

𝑘1+

𝑟𝑡𝑜 ln(𝑟𝑖𝑛𝑠 𝑟𝑜⁄ )

𝑘𝑖𝑛𝑠+

𝑟𝑡𝑜

𝑟𝑖𝑛𝑠(ℎ𝑐 + ℎ𝑟)+

𝑟𝑡𝑜 ln(𝑟𝑐𝑜 𝑟𝑐𝑖⁄ )

𝑘𝑐𝑎𝑠+

𝑟𝑡𝑜 ln(𝑟𝑤𝑏 𝑟𝑐𝑜⁄ )

𝑘𝑐𝑒𝑚 (24)

Eq. (24) is general and includes resistances owing to insulation and heat loss through

radiation. The resistance to heat transfer offered by the annulus is difficult to estimate.

The local heat transfer coefficient will be negligible if the annulus is under vacuum, but

heat may be transferred from tubing to casing by radiation. In the case of a steam

injection or geothermal well, the annulus may be filled with gas (either natural gas or

air), in which case, mechanisms for heat transfer may include both radiation and natural

convection

The radiative heat transfer coefficient, hr, can be estimated from the following

expression:

ℎ𝑟 =𝜎(𝑇𝑖𝑛𝑠

∗2 + 𝑇𝑐𝑖∗2)(𝑇𝑖𝑛𝑠

∗ + 𝑇𝑒∗)

1𝜀𝑖𝑛𝑠

+𝑟𝑖𝑛𝑠𝑟𝑐𝑖

(1

𝜀𝑐𝑖− 1)

(25)

The asterisks denote absolute temperatures.

(Ramey, 1962), further described that the effect of scale and wax deposition in tubing

and casing can be included by addition of terms similar to those for heat transfer

through fluid films.

It should be noted that some of the elements presented in Figure 2 offer negligible

resistance to heat flow. In general, resistances to heat flow through the tubing or casing

metal may be neglected due to its high conductivity. In most cases, tubulars are not

13

insulated and temperatures are too low (less than 400°F) for radiation to be significant.

(Hasan & Kabir, 1994), presented a simplified expression for the overall heat transfer

coefficient, considering also that for a typical oil well, tubing insulation is absent:

𝑈𝑡𝑜 = [1

ℎ𝑐+

𝑟𝑡𝑜 ln(𝑟𝑤𝑏 𝑟𝑐𝑜⁄ )

𝑘𝑐𝑒𝑚]

−1

(26)

When the wellbore is exposed to seawater or air, Eq. (24) is rewritten in terms of the

wellbore fluid temperature, instead of the wellbore formation interface temperature, and

the ambient temperature.

The heat-transfer coefficient outside the wellbore, owing to water or air convection,

may be estimated using any standard correlation.

2.5 DOWNHOLE GAUGE PLACEMENT

2.5.1 Intelligent well monitoring.

Advanced wells are completed with permanent downhole gauges and flow control

devices to monitor and control of the near-wellbore region. There are different types of

sensors in the oil industry including wellbore internal sensors and casing internal

sensors. Some of this sensors are used for wellbore measurement and some for imaging

the distribution of reservoir attributes away from the well. The use of distributed

temperature sensors (DTS) is now a common practice for zonal monitoring through

real-time measurement of pressure and temperature profiles.

Transient data from the well is essential for reservoir management because it allows to

recognise types and amounts of fluid entering along the wellbore, and thus, to identify

water or gas influx that guides the action of downhole control devices (Yoshioka K. ,

Zhu, Hill, Dawkrajai, & Lake, 2005).

14

Transient data is combined with a physical wellbore model that describes transient fluid

flow in the wellbore and reservoir. These models include mass and energy balances of

fluid flow and account for Joule-Thomson effects, and convective and conductive heat

transfer described in section 2.1.

The use of pressure sensors to collect transient data has experienced a continuous

improvement in technology related to gauge resolution, data-sampling frequency,

transmission capacity and reliability. Current fibre optic measurements can provide a

near-continuous profile of distributed temperature with resolution less than 0.1°C, over

a distance of several kilometres, with a spatial resolution of one meter, and with a

measurement time of typically a few minutes (Yoshioka K. , Zhu, Hill, Dawkrajai, &

Lake, 2005).

Single point data measurement is usually carried out by high precision quartz crystal P

and T gauges. Recent advances in sensor reliability has been don through Micro Electro-

Mechanical Systems and Silicon on Insulator technology for HPHT environments (Silva

Junior, Muradov, & Davies, 2012).

Fiber optic sensors technology have been applied to permanent downhole monitoring

systems for its reliability and also due to the potential of measuring a variety of physical

quantities in the same cable. This type of sensors can be applied in complex Distributed

Temperature Sensing (DTS) systems providing a spatial component for a continuous

wellbore temperature profile in real-time. (Silva Junior, Muradov, & Davies, 2012).

2.5.2 Transient data acquisition

In many occasions the design of the completion and/or physical restrictions, such as

plugs formed by solids deposition, prevent the gauge to be set at the midpoint producing

interval. Test data can be interpreted by gathering transient temperature signal (TTS) at

15

various points in the wellbore, however, when data collection takes place at a point

other than the midpoint perforation (MPP), wellbore thermal effects affect the TTS and

can lead to misdiagnosis of the reservoir model.

Thermal effects manifest themselves in two ways: a non-isothermal environment and

the local thermal cell created by the gauge carrier. The farther the gauge is from the

MPP, the larger the distortion (Kabir & Hasan, Does Gauge Placement Matter in

Downhole Transient-Data Acquisition, 1998).

Numerous studies have been published to assess the impact of sensor placement on

transient pressure and temperature signals.

(Kabir & Hasan, Does Gauge Placement Matter in Downhole Transient-Data

Acquisition, 1998), presented an example of a synthetic case for a gas well where test

data was collected 1,200 ft above the MPP in a 9,000 ft well. They observed that

thermal diffusion dominates over pressure diffusion in a high transmissivity reservoir,

causing a declining pressure during build up tests and an increasing pressure during

drawdown tests. This analysis suggested that even if the data were collected at the MPP,

build-up would not have shown the expected increasing trend because pressure diffuses

very rapidly in this type of system.

A study from (Sidorova, Shako, Pimenov, & Theuveny, 2015) about the impact of

vertical location respect to the MPP, and horizontal location respect to the well axis, of

the gauge, confirm that geothermal gradient and heat exchange between wellbore fluid

and reservoir has the greater influence in production well test TTS.

(Izgec, Cribbs, Pace, Zhu, & Kabir, 2007), performed a detailed uncertainty analysis

with experimental design to study the placement of permanent downhole pressure

sensors. This investigation corroborated that gauge location is the key factor in

16

collecting analysable data, especially in gas wells in which rapid heat loss induces far

larger error than in an oil well.

Placing a gauge close to the MPP allows to achieve an isothermal or near isothermal

wellbore condition. When the gauge is placed at the MPP only the Joule-Thomson

heating or cooling effect can introduce temperature change in the fluid, therefore,

transient data can reflect the information of an isothermal reservoir removing

uncertainty in fluid gradient beneath the tool.

When gauges are installed at increasing distance away from the MPP, heat transfer can

occur between the hot reservoir fluid and the cold formation. In this scenario, gauge

placement optimisation can be done using experimental design in which simple

correlations allow to check the validity of subsurface data in specific wellbore/reservoir

environments.

2.6 DRAWDOWN RESPONSE DERIVATIVE BEHAVIOUR.

Transient, sand-face temperature changes originated by the adiabatic

expansion/compression of the fluids in the reservoir provide extensive information for

supporting well production diagnostics and optimisation.

The temperature behaviour derived from the general theory on wellbore heat

transmission contains four time periods that describe the qualitative aspects of sand-face

and wellbore temperature response: early-time period, transition period, intermediate-

time period, and a late-time period. Figure 3 shows the various flow regimes in a semi-

log plot of temperature versus time, as presented in (Kutun, Inanc, & Satman, 2015).

17

Figure 3. General characteristics of the temperature semi-log plot (Kutun, Inanc, & Satman,

2015).

In Fig. 4, ΔT corresponds to the difference between the wellhead temperature, Ttop, and

the surface temperature, Tsurf. It can be written in terms of the Ramey number:

NRa = 2πkL/(w Cp):

∆𝑇 =𝑇𝑡𝑜𝑝 − 𝑇𝑠𝑢𝑟𝑓

𝑁𝑅𝑎𝑇𝐷 [1 − 𝑒

−(𝑁𝑅𝑎

𝑇𝐷⁄ )

] (27)

At early times when wellbore convection dominates, a log-log plot of dT/dlnt versus t

gives a unit slope straight line due to thermal storage. The intermediate-time behaviour

is observed when heat transfer between wellbore fluid and the surrounding formations

becomes significant and is recognized by -1/2 slope straight line on a log-log plot of

dT/dlnt versus t. At late times the wellbore heat transmission solution converges to the

cylindrical-source solution transferring heat at constant temperature (Kutun, Inanc, &

Satman, 2015).

The application of permanent pressure gauges and temperature sensors is to estimate

permeability and skin factor of the reservoir. Transient temperature signal at the sand

face makes possible to identify any behind pipe channelling, gas entry, and other

18

expected and unexpected behaviours that are continuously in develop (Nadri Pari,

Kabir, Mahdia Motahhari, & Turaj Behrouz, 2009). (Kabir, Hasan, Jordan, & Wang,

1996) addressed the aspects of transient testing in High temperature - High Pressure

reservoirs (HTHP) by proper modelling of the transient nature of mass, momentum and

energy for an accurate translation of the wellhead measurements of P and T into bottom-

hole pressure.

3 METHODS USED – WORKFLOW

3.1 WELLBORE/RESERVOIR SIMULATION

The wellbore was modelled in OLGA Dynamic Multiphase Flow Simulator. The

simulation basics apply the mass, momentum and energy conservation equations for the

fluid in the wellbore, while the reservoir flow is modelled as a mass source to specify

the flow rate and temperature input data.

Figure 3 shows a scheme for a control volume of unit length within the wellbore for the

basis of the balances. The governing equations are discussed in Appendix 8.1.

Figure 4. Transient mass, momentum and energy balance in a gas well (from Kabir, C.S. et. al,

1996).

19

3.1.1 Model set up and description

A vertical wellbore segment of 500 m length was modelled in OLGA to simulate the

heat transfer problem presented in figure 4. Fluid properties were calculated from PVT

data obtained from a dry gas reservoir.

Mass, momentum and energy balances, along with the gas PVT relation, are used to

generate the constitutive equations for pressure and temperature calculations. Table 1

presents the properties of the fluid modelled in the simulator.

Table 1. Dry gas properties for the simulator

Specific gravity 0.605

H2S [mole fraction] 0.0001

CO2 [mole fraction] 0

N2 [mole fraction] 0

OLGA applies one global time-step for the time integration based on the limitation that

a fluid particle should not spend less than one time-step on passing through any

numerical section length of a pipe. The spatial integration was performed on a defined

grid discretized in 10 segments of equal length.

Figure 5. Wellbore section model layout built in OLGA.

20

To initiate the flow computation, the program assigns the geothermal temperature

profile to the wellbore fluid and estimates the pressure along the wellbore for the no-

flow initial condition. The fluid temperature is calculated from the overall heat transfer

coefficient, U. The basic thermal model of the simulator calculates the inner wall heat

transfer coefficient considering the wellbore and formation material properties, and the

geothermal temperature profile (ambient conditions). The properties of the pipe and

formation walls are presented in table 2 and initial conditions used for model

initialization are shown in table 3.

Table 2. Properties of pipe and formation walls for the wellbore section.

Material Formation Steel

Heat Capacity [J/Kg-C] 1256 500

Conductivity [W/m-C] 1.59 500

Density [Kg/m3] 2243 7850

Thickness 15 0.0106

Table 3. Initial conditions for interpolation of fluid properties in OLGA.

Initial condition Value Units

Te reservoir 322 [°K]

Out node temperature 298.15 [°K]

P reservoir 140 [bara]

Out node Pressure 110 [bara]

The boundary conditions of temperature and flow rate were given as time series to

model the transient signal at the MPP, as shown in figure 6.

21

Figure 6. Transient temperature signal and flow rate measured at MPP. OLGA mass source.

The analytical solution for transient temperature around a cased and cemented wellbore

assume that heat is transferred radially from the bore-face to the formation through a

multi-layered cylinder of different materials in perfect thermal contact, with

homogeneous and time independent physical properties (Mendes, Coelho, Guigon,

Cunha, & Landau, 2005).

In this application conductive heat transfer occurs through the tubular into the formation

while convective heat transport take place between the produced gas in the wellbore and

the surrounding formation. The main assumptions considered in the simulation model

are detailed following:

- Heat flux is radial through the various layers that compose the wellbore completion

until a radius of thermal influence inside the rock.

- Each cylindrical layer has homogeneous and time independent thermal properties to

represent the materials that compose the system.

- There is no heat generation inside the system.

- The cylindrical layers are in perfect thermal contact. This means that heat flux is

continuous at the surface contacts.

0

5,000

10,000

15,000

20,000

38

40

42

44

46

48

50

52

0 20 40 60 80 100 120

Tem

per

atu

re [

C]

Time [h]

Temperature transient signal and flow rate

T source [C] Flow rate [Mscf/d]

Flo

w r

ate

[Msc

f/d

]

22

3.1.2 Processing of temperature transients and sensitivity cases.

Simultaneous solution of the mass, momentum and energy balance equations, Appendix

8.1, results in pressure and temperature data calculated as a function of well depth and

time. To evaluate the temperature changes along the wellbore in a single-phase dry-gas

production system, we studied the following cases:

Table 4. Values of sensitivity variables evaluated in the wellbore heat-transfer simulation.

Parameter Base case Case 1 Case 2 Case 3

Internal wellbore diameter, [in] 10 7 4.5 2.75

Wellbore inclination, [degree] 0 30 60 -

Formation heat capacity, [J/Kg-°K] 1256 2616 920 -

Formation conductivity, [W/m-°C] 1.59 2.82 4.22 -

Formation density, [Kg/m3] 2243 2654 3000 -

Geothermal gradient, [°K/m] 0.048 0.1 0.024 -

The model cases were simulated by varying only one sensitivity variable while keeping

constant the remaining parameters established in the base case. This procedure allowed

to obtain a simulated TTS that responds to a specific sensitivity variable in different

position along the pipe length.

To summarize, the simulation model allowed calculating fluid temperature by

considering thermal effects in the wellbore, as described in section 2.3. This

temperature is obtained as a TTS at different depths, due to the initial conditions

imposed to the model, and were compared by calculating the drawdown derivative

response.

A case study is presented below to provide an overview of the analysis.

3.1.3 Case study – Sensitivity to wellbore diameter.

The TTS calculated from the simulation model, considering the base case presented in

table 4, is shown in figure 7 below.

23

Figure 7. Transient temperature signal response from the OLGA simulation model. Base case.

Figure 7 shows how wellbore thermal effects produces a decrease in the TTS as gas

flows up from the MPP. An explanation to this trend is detailed:

As described in previous sections, geothermal temperature increases with well depth.

Thus, for upward flow, formation temperature will be cooler and gas will lose heat due

to its higher temperature compared to the surroundings.

Pressure drop created at the producing interval and frictional effects also induces

cooling due to the pressure dependence of the Joule-Thomson coefficient. Therefore,

upward gas flow is cooled by both the temperature change in the formation and pressure

drop in the wellbore.

In order to compare the main features of the transient signal, the diagnostic derivative

plot was obtained for each one of the four-drawdown periods registered in the data.

Figures 8 and 9 present an example of the semi-log and derivative plots, respectively.

30.0

32.0

34.0

36.0

38.0

40.0

42.0

44.0

46.0

48.0

50.0

0 20 40 60 80 100 120

Tem

per

atu

re [

C]

Time [h]

Transient temperature signal

50 m 100 m 150 m 200 m250 m 300 m 400 m 500 m

24

Figure 8. Semi-log plot for TTS during the second drawdown period. Wellbore segment: 100m

above MPP.

Figure 9. Log-log diagnostic plot for TTS during the second drawdown period. Wellbore

segment: 100m above MPP.

The parameter used to estimate the TTS attenuation with well depth was the slope

calculated in the semi-log plot, based on the drawdown derivative response described in

section 2.6. It is considered that fluid heat loss induces an increasing error in semi-log

slope as the transient signal is obtained downstream.

40.0

40.5

41.0

41.5

42.0

42.5

43.0

43.5

44.0

0.1 1.0 10.0 100.0

Tem

per

atu

re [

C]

Time [h]

Semi-log plot: Drawdown 2. L = 100m (above MPP)

10 in 7 in 4.5 in

0.10

1.00

0.1 1.0 10.0 100.0Tem

p. d

eriv

ativ

e: δ

T/δ

(Ln

t)

Time [h]

Log-log plot: Drawdown 2. L = 100m (above MPP).

10 in 7 in 4.5 in

25

Figure 10 shows the results calculated for the sensitivity analysis to wellbore diameter

in all pipe segments.

Figure 10. Accuracy of semi-log slope during the second drawdown period.

The final approach to estimate an attenuation relationship with gauge placement was to

fit a model using regression analysis in order to draw conclusions about how changes in

the sensitivity variables are associated with changes in the temperature response.

The value of R2 presented in figure 11 is a statistical measure of how close the data are

to the fitted regression line. In general, the higher the R2 value, the better the model fits

the data.

93

%

86

%

78

%

70

%

62

%

55

%

49

%

43

%

38

%

95

%

89

%

82

%

76

%

70

%

64

%

58

%

53

%

48

%

96

%

92

%

87

%

83

%

78

%

73

%

69

%

64

%

60

%

98

%

94

%

91

%

88

%

84

%

81

%

78

%

74

%

71

%

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP[M]

Accuracy of semi-log slope: Drawdown 2

10 in 7 in 4.5 in 2.75 in

26

Figure 11. Attenuation function for diameter sensitivity analysis in function of sensor

placement.

4 RESULTS

4.1 WELLBORE SEGMENTATION LENGTH

The number of segments in which the wellbore is divided to simulate the transient

signal was obtained from its influence on the calculated semi-log slope.

Figure 12. Accuracy of semi-log slope. Sensitivity to wellbore segmentation.

y = 6E-07x2 - 0.0018x + 1.0967 R² = 0.9991

y = -0.0012x + 1.0605 R² = 0.9986

y = -0.0009x + 1.0523 R² = 0.9992

y = -0.0007x + 1.0396 R² = 0.9991

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

0 100 200 300 400 500 600 700

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

10 in 7 in 4.5 in 2.75 in

93

%

86

%

78

%

70

%

62

%

55

%

49

%

43

%

38

%

93

%

85

%

77

%

69

%

61

%

54

%

48

%

42

%

37

%

93

%

85

%

77

%

69

%

61

%

54

%

48

%

42

%

36

%

20%30%40%50%60%70%80%90%

100%110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Accuracy of semi-log slope: Drawdown 2

10 segments 20 segments 40 segments

27

Figure 12 shows that an increase in the number of wellbore segments produces a similar

behaviour for the drawdown response, i.e. the accuracy of the semi-log slope is not

affected by an increase in the resolution of the special integration. Besides, it was

observed that an increase in the number of segments above 20, resulted in a noisy

transient signal during simulation.

4.2 SENSITIVITY STUDY FOR WELLBORE DIAMETER.

The effect of the pipe diameter in TTS was analyzed by calculating the temperature

profile in four cases: 10in, 7 in, 4 ½ in, 2 ¾ in. Pressure boundaries were kept constant

during the simulation while the flow rate and temperature transient signals were an

input for the model.

Figure 7 presented the TTS calculated for different points along a wellbore segment of

10in diameter. Similar TTS were obtained for the range of diameters.

Figures 10 and 11 presented the accuracy of semi-log slope and attenuation functions

for the sensitivity analysis, respectively.

Wellbore diameter[in] Fitted regression line.

10 in %𝑡 = −6.56 × 10−4 𝐿 + 1.0396

7 in %𝑡 = −9.06 × 10−4 𝐿 + 1.0523

4 ½ in %𝑡 = −1.18 × 10−3 𝐿 + 1.0605

2 ¾ in %𝑡 = −6.2 × 10−7𝐿2 − 1.76 × 10−3 𝐿 + 1.0967

Where: L is the length of wellbore measured from perforations and %t is the accuracy of

the calculated slope respect to its value at sand face.

The profile of the overall heat transfer coefficient calculated for the different wellbore

diameters is presented in figure 13.

28

Figure 13. Overall heat transfer coefficient. Wellbore diameter sensitivity analysis

4.3 SENSITIVITY STUDY TO FLOW RATE.

Because flow rate is part of the boundary condition for transient signal, its effect was

evaluated by changing the values of the data series in two scenarios: increasing flow

rate to twice its value, and reducing flow by half.

Figure 14. Accuracy of semi-log slope. Sensitivity to flow rate signal.

0

50

100

150

200

250

300

350

400

450

500

20 30 40 50 60 70 80 90 100 110

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

10 in 7 in 4.5 in 2.75 in

93

%

86

%

78

%

70

%

62

%

55

%

49

%

43

%

38

%

88

%

74

%

62

%

51

%

41

%

33

%

27

%

21

%

17

%

97

%

92

%

88

%

83

%

79

%

74

%

69

%

65

%

61

%

0%10%20%30%40%50%60%70%80%90%

100%110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Accuracy of semi-log slope: Drawdown 2

Base case Half flow Double flow

29

Figure 15. Attenuation function for flow rate sensitivity analysis.

4.4 SENSITIVITY STUDY FOR WELLBORE DEVIATION

Deviations of the well trajectory may cause variations in temperature and pressure

profiles when compared with a vertical wellbore.

The sensitivity analysis for wellbore deviation was conducted considering the angle to

its axis (θ), with the following values: 0° (vertical well), 30° and 60°.

The TTS was calculated allowing for the variation of the geothermal gradient according

to the true vertical depth obtained for the deviated wellbore in two scenarios:

a) True vertical depth: y = 500 m.

b) Length of the wellbore segment: L = 500 m.

The results of these models are presented in figures 16 and 17. It can be noted that the

accuracy of the semi-log plot slope value is slightly lower when y = 500 m because the

length of the wellbore increases with increasing deviation, and thus increases fluid

residence time leading to a cooler temperature up in the wellbore.

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0062x2 - 0.162x + 1.1667 R² = 0.9996

y = -0.0448x + 1.0534 R² = 0.9991

0%10%20%30%40%50%60%70%80%90%

100%110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

Base case Half flow Double flow

30

Figure 16. Attenuation function for wellbore inclination. Sensitivity case a) y = 500 m.

Figure 17. Attenuation function for wellbore inclination. Sensitivity case b) L = 500 m.

Figure 18. Overall heat transfer coefficient. Deviated wellbore sensitivity case b) L = 500 m.

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0018x2 - 0.0921x + 1.101 R² = 0.9992

y = 0.0022x2 - 0.1011x + 1.1102 R² = 0.9992

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

0 deg 30 deg 60 deg

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0017x2 - 0.0903x + 1.0991 R² = 0.9992

y = 0.0017x2 - 0.0907x + 1.0993 R² = 0.9992

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

0 deg 30 deg 60 deg

0

100

200

300

400

500

20 30 40 50 60 70 80 90

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

30 deg 0 deg 60 deg

31

The regression models calculated from the sensitivity analysis is presented below:

Dev. angle a) y = 500 m b) L = 500 m

θ = 0° %𝑡 = −1.55 × 10−3 𝐿2 + 0.088 𝐿 + 1.1 %𝑡 = −1.55 × 10−3 𝐿2 + 0.088 𝐿 + 1.1

θ = 30° %𝑡 = −1.77 × 10−3 𝐿2 + 0.092 𝐿 + 1.1 %𝑡 = −1.68 × 10−3 𝐿2 + 0.090 𝐿 + 1.1

θ = 60° %𝑡 = −2.24 × 10−3 𝐿2 + 0.101 𝐿 + 1.11 %𝑡 = −1.71 × 10−3 𝐿2 + 0.091 𝐿 + 1.1

The parameters of these equations are the same as defined in section 4.1.

4.5 SENSITIVITY STUDY TO FORMATION THERMAL PROPERTIES.

In order to define the heat transfer properties in thermal computations, physical

properties of the materials associated with pipe wall, pipeline coating, insulation, soil

and also shape must be specified.

The simulation of the wellbore environment was built by using a simple model with

constant thermal properties in OLGA. The equations solved for heat transfer

computations are that of heat transfer in solid medium.

Three sensitivity cases where analysed to study the effect of formation thermal

properties in wellbore heat transfer: specific heat capacity, thermal conductivity and

rock density.

Table 5. Values of sensitivity variables for formation properties.

Formation

properties

Formation Heat

Capacity [J/Kg-C]

Thermal conductivity

[W/m-C]

Density

[kg/m3]

Base case 1256 1.59 2243

Case 1 2616.75 2.82 2654

Case 2 920 4.22 2710

32

4.5.1 Sensitivity study to formation heat capacity.

Heat capacity sensitivity cases considered a range of values between the base case and

the value presented in (Kabir & Hasan, Does Gauge Placement Matter in Downhole

Transient-Data Acquisition, 1998).

Figure 19. Attenuation function for formation heat capacity sensitivity analysis.

Figure 20. Overall heat transfer coefficient. Heat capacity sensitivity analysis.

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0013x2 - 0.0846x + 1.0945 R² = 0.999

y = 0.0016x2 - 0.0885x + 1.0974 R² = 0.9991

0%

20%

40%

60%

80%

100%

120%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

Cp = 1256 [J/Kg-K] Cp = 2616.75 [J/Kg-K] Cp 920 [J/Kg-C]

0

100

200

300

400

500

20 30 40 50 60 70 80 90

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

Cp = 2616 [J/Kg-K] Cp = 920 [J/Kg-K] Cp = 1256 [J/Kg-K]

33

4.5.2 Sensitivity study to thermal conductivity

The sensitivity values of thermal conductivity consider the measurements of thermal

properties of North Sea rocks.

Figure 21. Attenuation function for formation thermal conductivity sensitivity analysis.

Figure 22. Overall heat transfer coefficient. Thermal conductivity sensitivity analysis.

4.5.3 Sensitivity study to formation density.

The variations in formation density considered the densities of the formation rocks:

sandstone (quartz) and carbonates (dolomite).

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0029x2 - 0.1177x + 1.1311 R² = 0.9986

y = 0.004x2 - 0.1375x + 1.1543 R² = 0.9981

10%

30%

50%

70%

90%

110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

K = 1.59 [W/m-C] K = 2.82 [W/m-C] K = 4.22 [W/m-C]

0

100

200

300

400

500

20 40 60 80 100 120 140 160 180

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

K = 1.59 [W/m-C] K = 2.82 [W/m-C] K = 4.22 [W/m-C]

34

Figure 23. Attenuation function for formation density sensitivity analysis.

Figure 24. Overall heat transfer coefficient. Formation density sensitivity analysis.

4.6 SENSITIVITY STUDY TO GEOTHERMAL GRADIENT.

The values considered for the sensitivity case considered the double and half of the one

provided in the wellbore simulation model.

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0015x2 - 0.0868x + 1.096 R² = 0.9991

y = 0.0015x2 - 0.0868x + 1.096 R² = 0.9991

0%

20%

40%

60%

80%

100%

120%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

ρ = 2243 [Kg/m3] ρ = 2654 [Kg/m3] ρ = 3000 [Kg/m3]

0

100

200

300

400

500

20 30 40 50 60 70 80 90

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

ρ = 2243 [Kg/m3] ρ = 2654 [Kg/m3] ρ = 3000 [Kg/m3]

35

Figure 25. Attenuation function for geothermal gradient sensitivity analysis.

Figure 26. Overall heat transfer coefficient. Geothermal gradient sensitivity analysis.

5 DISCUSSION

The model developed was used to study the variation in estimated TTS for various

conditions of production and formation properties in a gas well.

To simplify the analysis, the semi-log slope was selected as the dependent variable to

evaluate solutions for the sensitivity analysis influencing wellbore heat transfer. It

should be noted that the entire range of variables used in this analysis should not be

y = 0.0016x2 - 0.0878x + 1.0967 R² = 0.9991

y = 0.0016x2 - 0.0901x + 1.0989 R² = 0.9992

y = 0.0017x2 - 0.0904x + 1.0992 R² = 0.9992

20%

30%

40%

50%

60%

70%

80%

90%

100%

110%

50 100 150 200 250 300 350 400 450 500

% A

CC

UR

AC

Y

VERTICAL DISTANCE FROM MPP [M]

Attenuation function: Drawdown 2

GT = 0.048 [K/m] GT = 0.1 [K/m] GT = 0.024 [K/m]

0

100

200

300

400

500

20 30 40 50 60 70 80 90

Pip

elin

e le

ngt

h [

m]

U [W/m2-C]

Overall heat transfer coefficient

GT = 0.1 [K/m] GT = 0.048 [K/m] GT = 0.024 [K/m]

36

encapsulated because unreal wellbore models are generated that do not provide any

solution.

Because gas PVT properties are more sensitive to changes in temperature, both

drawdown and build up periods exhibited trend reversals during simulations at

increasing distances from the MPP. This thermal storage implies the distortion period

associated with a temperature transient. The sharp rise and decline in simulated build up

and drawdown temperature shows the importance of accounting for the heat capacity of

the tubular and cement sheath.

5.1 WELLBORE DIAMETER.

The initial trend observed for a reduction in the wellbore diameter from 10in to 7in and

4 ½ in respectively, is a decrease in temperature of the transient signal.

The straight-line equations presented in figure 11 shows that the effects of wellbore

diameter could not be represented in a single tendency. However, it is clear that a

reduction in wellbore diameter produces an increase in the fluid temperature because of

the increase in its velocity, which in turn reduces the time to transfer heat to the

surroundings.

The profile of the overall heat transfer coefficient calculated for the different wellbore

diameters shows an augmented value due to an increase in gas velocity.

It can be observed that for a given gauge position, a reduction in wellbore diameter

reduces the attenuation of TTS respect to the original source. This effect arises from the

decrease in time that the fluid in the wellbore has to transfer heat with the surroundings.

37

5.2 FLOW RATE.

Despite semi-log slope was evaluated in a period of constant flow rate, attenuation

functions for flow rate could not match in a regression model.

The overall heat transfer coefficient increases with an increase in flow rate as expected.

This behaviour was attributed to the increased gas velocity at high production rates, as

similar effect can be observed when the diameter of the wellbore is reduced.

5.3 WELLBORE DEVIATION

The results of wellbore deviation simulation were presented in figures 16 and 17. It can

be noted that the accuracy of the semi-log plot slope value is similar in both models

(y=500m and L=500m) and that his influence in the wellbore heat transfer can be

described by a single correlation that only depends on the sensor location.

We can also mention that the effect of an increase in fluid residence time is expected in

the case which y = 500 m, thus leading to a cooler temperature up in the wellbore.

(Hasan & Kabir, Aspects of Wellbore Heat Transfer During Two-Phase Flow, 1994).

An initial insight suggest that the turbulent regime developed in the wellbore is

governing the heat transfer process along the pipe. In this case, and depending on the

variation of the geothermal gradient according to the true vertical depth, the effect of

wellbore deviation is that of increasing fluid residence time (y = 500m), but does not

affect the heat flow per unit length in the wellbore.

5.4 FORMATION THERMAL PROPERTIES

Because gases have lower heat capacity, and thus, lower enthalpy than a liquid, heat

dissipation occurs faster, leading to a thermal storage distortion.

38

As it was expected, the overall heat transfer coefficient is not affected neither by the

specific heat capacity nor the formation density. The greater effect was observed for

thermal conductivity of the formation, which is part of the resistances to heat transfer

encountered in the system.

Figure 22 shows that an increase in thermal conductivity reduces the resistance to heat

flow, and thus, the overall heat transfer coefficient is affected positively. This increased

coefficient facilitates the thermal interaction between the fluid in the wellbore and the

surrounding formation promoting a larger attenuation in the transient signal when the

gas flows upward the wellbore.

5.5 GEOTHERMAL GRADIENT

The difference between the temperature of produced fluids and the geothermal gradient

provides the driving force for heat transfer. It was observed that a larger difference of

temperatures promotes a larger heat flow resulting in a larger attenuation of the transient

signal.

The sensitivity analysis showed that convective heat transfer is the most important

effect in signal attenuation. (Hasan & Kabir, Aspects of Wellbore Heat Transfer During

Two-Phase Flow, 1994) pointed this out. In his research, they demonstrated that the

omission of convective heat transfer in the annulus of a wellbore estimates too high

temperature for the fluid.

In the sensitivity cases analysed, the variables of density and heat capacity of the

formation, wellbore deviation and geothermal gradient does not affect the overall heat

transfer coefficient and thus, does not affect the heat flow in the wellbore. This means

that the effect of these variables is reflected in the absolute value of temperature but not

in the derivative signature of the transient signal.

39

On the other hand, a major attenuation of the features in the transient signal is obtained

for a change in the overall heat transfer coefficient. This study proposes polynomial

equations to correct the value of the transient signal to obtain the value at MPP,

however, the scope of its application is based on the range of the sensitivity variables

described previously.

6 ECONOMIC IMPACT OF THE WORK

The improvement in the technology of intelligent wells leads to several, and even

unexpected benefits to an oil company. Despite this fact, advanced completions has

some challenges regarding their reliability, cost effectiveness and implementation

phases.

There are different types of sensors, being the most common P, T and flow rate sensors

that are used to control well performance of the production zone. This application,

among others, makes sensor an essential component of an intelligent completion.

From the economic point of view cost of sensors depend on well depth. This fact arises

from the components (hardware) required to operate in a more severe environment,

which in turn makes a smart completion three or four times more expensive than a

conventional completion (Nadri Pari, Kabir, Mahdia Motahhari, & Turaj Behrouz,

2009).

During the development of this work, it has been mentioned that placement of gauges in

a location different that the MPP leads to an inaccurate signal and inability to carry out

reliable analysis.

(Nadri Pari, Kabir, Mahdia Motahhari, & Turaj Behrouz, 2009) provides an extensive

discussion about the smart well benefits considering further economic risk analysis.

The economic impact of this work arises from the reduced cost of installation and

operation for gauges placed above the producing interval. The accurate description of

40

thermal effects in the wellbore allows correcting any TTS, measured at a different depth

from the MPP, to provide reliable analysis of the inflow performance of a well.

This reduction of costs could make the application of smart completions more

adaptable, reducing further costs associated with well intervention in onshore and

offshore fields, and thus increasing the Net Present Value of a project.

7 CONCLUSIONS

This study provides a short description of wellbore thermal effects associated with

gauge placement. Gases have lower heat capacity, and thus, lower enthalpy than a

liquid, then heat dissipation occurs faster, leading to a thermal storage distortion in a

temperature transient period.

Sensitivity analysis showed that a major attenuation of the features in the transient

signal is obtained for a change in the overall heat transfer coefficient, principally by the

influence of convective heat transfer. These effects arise for changes in gas velocity,

that influences the residence time of the fluid in the wellbore, and the resistances to heat

flow present in the system, principally the formation conductivity.

The influence of flow rate, wellbore diameter and formation thermal conductivity (k),

could not be represented in a single tendency to associate its thermal effects with sensor

location. However, an accuracy over 90% was observed for the semi-log slope

calculated 50m above the MPP for all cases.

Density and heat capacity of the formation, wellbore deviation and geothermal gradient

does not affect the overall heat transfer coefficient and thus, does not affect the heat

flow per unit length in the wellbore. This means that the effect of these variables is

reflected in the absolute value of temperature but not in the derivative signature of the

41

transient signal. Therefore, the signal attenuation can be expressed as a single tendency

dependent on the gauge location.

Further work by means of experimental design would improve the correlations

calculated in this study to estimate wellbore thermal effects. These correlations can be

applied to real production data in order to analyse its general use.

42

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Mendes, R., Coelho, L., Guigon, J., Cunha, G., & Landau, L. (2005). Analytical

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47

9 APPENDICES

9.1 ESSENTIAL SIMULATOR EQUATIONS.

The simplified form of the material, momentum and energy balance Eq.s developed for

a gas reservoir at HTHP.

Material balance.

𝜕𝜌

𝜕𝑡+

𝜕

𝜕𝑧(𝜌𝑣) = 0 (A1)

Momentum Balance.

𝜕𝑣

𝜕𝑡+ 𝑣

𝜕𝑣

𝜕𝑧= −

1

𝜌

𝜕𝑝

𝜕𝑧− 𝑔 −

2𝑓𝑣2

𝑑𝑡 (A2)

Energy Balance.

�̃� = 𝐴 𝜌 𝑑𝐻

𝑑𝑡+

𝑑(𝑚 𝑐𝑝𝑇𝑓)

𝑑𝑡− 𝐴 𝜌 𝑅

𝑑(𝑍 𝑇𝑓)

𝑑𝑡+ 𝐴(𝐻 − 𝑍 𝑅 𝑇𝑓)

𝑑𝜌

𝑑𝑡+

𝑑

𝑑𝑧[𝑤 (𝐻 +

1

2𝑣2 + 𝑔𝑧)] (A3)

Simultaneous solution of the three conservation Eq.s and the sand face flow Eq. is

needed to obtain the values of the desired variables, pressure, temperature and flow rate

as functions of time and depth.

Analytical solution of these Eq.s is unlikely because of the complexity of the problem,

consequently, a numerical approach is adapted (Kabir, Hasan, Jordan, & Wang, 1996).

48

9.2 PRESSURE DROP IN A GAS WELL.

Information on variation of Tf with well depth and time is critical to calculate pressure

gradient in the wellbore using the energy balance Eq.:

−𝑑𝑃

𝑑𝑧=

𝑓𝑣2𝜌

2𝑔𝑐𝑑+

𝑔 sin 𝜃

𝑔𝑐𝜌 +

𝜌𝑣

𝑔𝑐

𝑑𝑣

𝑑𝑧 (A4)

For a gas well, fluid density may be expressed in terms of pressure, temperature and gas

gravity as:

𝜌 =29𝛾𝑔𝑝

𝑍𝑅𝑇 (A5)

Combining Eq. A4 and A5, neglecting acceleration, and using field units, the expression

for pressure drop in a dry-gas well of Cullender–Smith is obtained:

∫ ((𝑃 𝑇𝑓𝑍⁄ )𝑑𝑝

𝑓 𝑞2

𝑑5(𝐿 𝐷⁄ )+ (𝑃 𝑇𝑓𝑍⁄ )

2)

𝑝𝑤𝑓

𝑝𝑡𝑓

= 𝛾𝑔 𝑍 (A6)

Cullender–Smith expression can lead to significant errors in high velocity systems, or

flow occurring in low-pressure situations, because it neglects the kinetic-energy term. A

better approach comes from the use of the energy balance Eq..

Computation starts by evaluating Tf at a specific gauge position with Eq. 23. Input

parameters include static-reservoir temperature, geothermal gradient and flow rate,

among others. Knowing Tf and P, Z-factor is evaluated to calculate the average value of

the pressure gradient, dP/dz, at the midpoint of the first discretized segment in the

wellbore (Hasan & Kabir, Analytic Wellbore Temperature Model For Transient Gas-

Well Testing, 2003).

1

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(Muradov & Davies, 2012)