ECD en Hgh Temp Wells

UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

EVALUATION OF EQUIVALENT CIRCULATING DENSITY OF DRILLING

FLUIDS UNDER HIGH PRESSURE-HIGH TEMPERATURE CONDITIONS

A THESIS

SUBMITTED TO THE GRADUATE FACULTY

in partial fulfillment of the requirement for the

degree of

MASTER OF SCIENCE

(Petroleum Engineering)

By

Oluseyi Harris

Norman, Oklahoma

2004

EVALUATION OF EQUIVALENT CIRCULATING DENSITY OF

DRILLING FLUIDS UNDER HIGH PRESSURE-HIGH TEMPERATURE

CONDITIONS

A THESIS APPROVED FOR THE

MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL

ENGINEERING

BY

Chair: Dr. Samuel Osisanya

Dr. Subhash Shah

Member:

Member:

Dr. Djebbar Tiab

ACKNOWLEDGEMENTS

The author wishes to express his profound gratitude and appreciation

for Dr. Samuel Osisanya. His guidance, moral and financial support, and

encouragement were invaluable. The author would like to thank the members

of the thesis committee, Dr Samuel Osisanya, Dr Subhash Shah, and Dr.

Djebbar Tiab for their helpful comments and suggestions. Heartfelt thanks go

to Dr. Subhash Shah for his assistance in allowing use of WCTC facilities in

performing research for this thesis. The author wishes to extend special

thanks to colleagues whose assistance and encouragement was invaluable

during the course of this research work- Ricardo Michel-Villazon, Aristotelis

Pagoulatos, Kayode Aremu, Kola Ayeni.

The author wishes to thank his other half, Lola for always being there.

The author would also like to express immeasurable gratitude towards his

parents for their constant and unwavering support and faith. Last and most

importantly, thanks and praise are extended to God almighty who alone

makes all things possible.

Oluseyi Harris

Norman, Oklahoma July, 2004

iv

TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS ..............................................................................iv

LIST OF TABLES ........................................................................................viii

LIST OF FIGURES.........................................................................................ix

ABSTRACT....................................................................................................xi CHAPTER PAGE

1. FORMULATION OF THE PROBLEM .........................................................1 1.1. Introduction ..........................................................................................1

1.2. Literature Review .................................................................................3

1.3. Objectives and Scope of Work...........................................................13

1.4. Study Organization ...........................................................................14

2.FUNDAMENTAL CONCEPTS FOR ESTIMATION OF EQUIVALENT STATIC AND CIRCULATING DENSITY ....................................................15 2.1 Equivalent Static density ......................................................................15

2.2 Estimating Equivalent Static Density....................................................18

2.2.1 Compositional Models ...................................................................18

2.2.1.1 Volumetric Models for Mud Constituents ..................21

2.2.2 Empirical Models ...........................................................................23

2.3 Equivalent Circulating density ..............................................................23

2.4 Frictional Pressure Loss.......................................................................24

2.5 Fluid Rheology .....................................................................................26

2.5.1 Bingham Plastic Model ..................................................................27

2.5.2 Power Law Model ..........................................................................28

2.5.3 Herschel-Bulkley Model.................................................................30

2.5.4 Casson Model ...............................................................................31

v

2.5.5 Ellis Model .....................................................................................31

2.5.6 Carreau Model...............................................................................32

2.6 Temperature and Pressure Dependent Rheological Parameters.......33

2.6.1 Temperature/Pressure Dependent Plastic Viscosity....................33

2.6.2 Temperature Dependent Yield point..............................................35

2.7 Bingham Plastic Pressure Loss Equations......................................36

3.DRILLING FLUID TEMPERATURE PROFILE ESTIMATION ...................40 3.1 Heat Transfer in the Wellbore ..............................................................41

3.2 Analytical Method.................................................................................43

3.2.1 Assumptions of Analytical Model ..................................................43

3.2.2 Heat Balance in the DrillPipe........................................................44

3.2.3 Heat Balance in the Annulus ........................................................45

3.2.4 Heat Flow in the Formation and System Heat Balance ................46

3.3 Numerical Method................................................................................50

3.3.1 Equations Governing Heat transfer in the Wellbore and Formation

...............................................................................................................50

3.3.2 Discretizing Heat Flow Equations for Finite difference Analysis ....53

3.4 Summary.........................................................................................68

4.DEVELOPMENT AND VALIDATION OF THE DYNAMIC DENSITY SIMULATOR AND MODELLING OF DYNAMIC DENSITY .......................69 4.1 Program Lay-Out.............................................................................70

4.2 DDS Program Execution .................................................................71

4.2.1 General Well Parameters Form ...................................................71

4.2.2 Mud Properties Form ...................................................................77

4.2.3 Formation Properties Form ..........................................................77

4.2.4 Heat Transfer Coefficients Form..................................................77

4.2.5 Results and Results Form ...........................................................80

4.3 Equations used in DDSimulator Program........................................82

4.3.1 Fluid Properties ...........................................................................82

4.3.2 Temperature Profile Estimation ...................................................83

vi

4.3.3 Equivalent Hydrostatic Head and ECD ........................................84

4.4 Model Validation..............................................................................84

4.5 Dynamic Density Estimation............................................................91

Summary..................................................................................................107

5.SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS...................108 5.1 Summary.......................................................................................108

5.2 Conclusions...................................................................................109

5.3 Recommendations ........................................................................110

NOMENCLATURE ......................................................................................112

REFERENCES............................................................................................115

APPENDIX ..................................................................................................119 Code for DDSimulator Program ...............................................................119

vii

LIST OF TABLES TABLE PAGE

4.1 : Well and mud circulating properties for a gulf coast well…………….85

4.2 : Simulated Well Conditions………………………………………………92

4.3 : Results of Well Simulation………………………………………………92

4.4 : Well simulation results for parameters detailed in Table 4.2 with

gG = 0.015 oF/ft……………………………….……………..……….…..96


gG = 0.025 oF/ft………………………………………………….….……96


inlet fluid temperature = 80 oF……………..……………….….….…….97


circulation rate = 210 gal/min…………………………………..……...102

4.8 : Well Simulation Results for Parameters Detailed in Table 4.2

with Circulation Rate = 300 bbl/hr…………………………………….105

viii

LIST OF FIGURES FIGURE PAGE

1.1 : Schematic Diagram of Fluid in the Well bore

at the Start of Circulation………………………………………………..9

2.1 : Volumetric behavior of various liquids under varying conditions of

temperature and pressure.....………………………………………….17

2.2 : Shear-thinning in a typical non-Newtonian Fluid….....……………...29

2.3 : Flow curves for time-independent fluids……..……………………….29

3.1 : Schematic of Heat Balance for Fluid Circulating in a Wellbore…....42

3.2a : Solution grid for Finite Difference Analysis…..……………………....51

3.2b : Heat Flow at Formation Annulus Boundary…..……………………...51

3.3 : Finite Difference Grid…...………………………………………………54

3.4 : Heat Balance at Bottom-Hole…..……………………………………..60

4.1 : DDSimulator Program Flow Chart…...………………………………..72

4.2 : Title Form…..……………………………………………………………73

4.3 :DDSimulator Launch Command Button…..…………………………..74

4.4 : Well Parameters Form………..………………………………………..75

4.5 : Mud Properties Form………………………..………………………….76

4.6 : Formation Properties Form…………………………………………....78

4.7 : Heat Transfer Coefficients Form……..…………………..…..……….79

4.8 : Results Form………………………………………………..…………..80

4.9 : A Sample Temperature Profile Using Excel Graph Feature…..…...81

4.10 : Plot of Temperature Profile For Gulf Coast Well…………………….86

ix

4.11 : Well Temperature Profile While Circulating Field Salt Water………89

4.12 : Temperature Profile For Gulf Coast Well…………………………….90

4.13 : Temperature Profile in 17200ft well after 5hrs……………………….93

4.14 : Annular Pressure Profile in 17200ft well after 5hrs………………….94

4.15 : Equivalent Circulating Density in 17200ft well after 5hrs…………...94



4.18 : Equivalent Circulating Density in 17200ft well after 5hrs…………...98



4.21 : Equivalent Circulating Density in 17200ft well after 5hrs………….100

4.22 : Temperature Profile in 17200ft well after 5hrs………………………100

4.23 : Annular Pressure Profile in 17200ft well after 5hrs…………………101

4.24 : Equivalent Circulating Density in 17200ft well after 5hrs…………..101

4.25 : Temperature Profile in 17200ft well after 5hrs………………………103

4.26 : Annular Pressure Profile in 17200ft well after 5hrs…………………103

4.27 : Equivalent Circulating Density in 17200ft well after 5hrs………..…104

4.28 : Temperature Profile in 17200ft well after 5hrs………………..…..…105

4.29 : Annular Pressure Profile in 17200ft well after 5hrs…………..…..…106

4.30 : Equivalent Circulating Density in 17200ft well after 5hrs………..…106

x

ABSTRACT

The effects of the temperature and pressure conditions prevalent in

high temperature/high pressure wells on the equivalent circulating density of

drilling fluids in a circulating wellbore as well as the bottom-hole pressure are

studied. High temperature conditions cause the fluid in the wellbore to

expand, while high pressure conditions in deep wells cause fluid

compression. Failure to take these two opposing effects into account can lead

to errors in the estimation of bottom-hole pressure on the magnitude of

hundreds of psi. The rheological behavior of drilling fluids is also affected by

the temperature and pressure conditions.

A Bingham plastic model was used to simulate the temperature and

pressure dependent rheological behavior of the drilling fluids studied, with the

rheological parameters expressed as functions of temperature and pressure.

Analytical and numerical methods for estimating the temperature profile in a

circulating well-bore were studied. A simulator called DDSimulator was

developed using visual basic to simulate the wellbore during circulation. This

simulator can develop the temperature and pressure profiles of a wellbore

during circulation, and compute the bottom-hole pressure and equivalent

circulating density taking into account the temperature and pressure

conditions in the wellbore. The Crank-Nicolson numerical discretizing scheme

was employed in the DDSimulator for the evaluation of the temperature

profile.

xi

From the results of the DDSimulator, it was found that the geothermal

gradient has a great effect on the bottom-hole temperature and pressure, and

the equivalent circulating density that will occur in a circulating well-bore. It

was also found that the inlet pipe temperature did not have a significant effect

on the bottom-hole temperature and pressure. This is even more the case in

deep wells, and in areas with high geothermal gradient. The circulation rate

plays an important role in the bottom-hole temperature and pressure that will

occur in circulating well.

The major technical contribution of this work is the development of the

DDSimulator. The density and rheological properties of the drilling fluid in the

wellbore can be estimated in order to adequately evaluate the bottom-hole

pressure during fluid circulation. DDSimulator allows the evaluation of the

bottom-hole pressure taking into account the variation in the volumetric and

rheological properties of the drilling fluid under high temperature and high

pressure conditions in the wellbore. The effects of variation in the inlet fluid

temperature, circulation rate, and geothermal gradient are explored and

discussed in this work.

xii

Chapter 1

FORMULATION OF THE PROBLEM

1.1. Introduction

Drilling fluids are in general complex heterogeneous mixtures of

various types of base fluids and chemical additives that must remain stable

over a range of temperature and pressure conditions. The properties of these

complex mixtures, such as equivalent static density (ESD) and the rheological

properties of the fluid mixture determine pressure losses in the system while

drilling. It is often assumed that these properties and thus the equivalent

circulating density (ECD) are constant throughout the duration of drilling

activities. This assumption can prove to be quite wrong in cases where there

is large variation in the pressure/temperature conditions, such as in high

pressure-high temperature (HPHT) wells, and deep-water drilling, where low

temperature conditions are encountered very close to the sea bed.

In HPHT wells, as the total vertical depth increases, there is an

increase in the bottom-hole temperature, as well as the hydrostatic head of

the mud column. These two factors have opposing effects on equivalent

circulating density. The increased hydrostatic head causes increase in the

equivalent circulating density due to compression. The increase in

temperature on the other hand, causes a decrease in the equivalent

circulating density due to thermal expansion. It is most often assumed that

1

these two effects cancel each other out. This is not always the case,

especially in high temperature, high pressure wells.

Large variations in equivalent circulating density can also occur when

drilling in deep water environments where relatively cold temperatures are

encountered in the riser, near the ocean bed. In deepwater wells1, the seabed

temperature can be as low as 30 oF and hydrostatic pressures at the bottom

of the riser will be 2700 psi, with a mud density of 8.6 lb/gal and a water depth

6000-ft. The low temperature conditions can cause severe gelling of the

drilling fluid, especially in oil-base muds (OBM). Failure to take this effect into

account will result in underestimation of the equivalent circulating density of

the drilling fluid.

Errors in the estimation of equivalent circulating density have an

especially disastrous effect when drilling through formations with a small gap

between pore pressure, and the pressure at which the formation will fracture.

In such cases, the margin for error is very small and thus, the equivalent

circulating density must be estimated precisely. Disregarding pressure and

temperature effects in this case can lead to greater probability for the

occurrence of kicks, and blow-outs due to under-balanced pressure or fluid

loss to the formation (lost circulation and formation damage) due to

overbalance pressure.

Various experimental studies have also shown drilling fluid rheology to

be very pressure and temperature dependent 2,3. Rheological parameters

such as viscosity and yield stress affect frictional pressure losses during the

2

flow of drilling fluids. Failure to take into account the dependence of these

parameters on temperature-pressure conditions can result in obtaining

erroneous values for equivalent circulating density, which takes into account

the hydrostatic head of the drilling fluid as well as the pressure loss it

experiences during flow.

The focus of this research is to study the effect of pressure and

temperature on equivalent static density as well as equivalent circulating

density of drilling fluids.

1.2. Literature Review

Numerous publications4-10 have dealt with the behavior of equivalent

static density of drilling fluids in response to variations in pressure-

temperature conditions. Various models4-10 have been proposed in order to

characterize this relationship, with some models being empirical in nature,

and some being compositional. The compositional model4-5 characterizes the

volumetric behavior of drilling fluids based on the behavior of the individual

constituents of the drilling fluid. Hence, prior knowledge of the composition of

the drilling fluid is required for application of the compositional model.

In the compositional model, the density of any solids content in the

drilling fluid is taken to be independent of temperature and pressure. It is

assumed that any change in density is due to density changes in the liquid

phases. It is also assumed that there are no physical and/or chemical

interactions between the solid and liquid phases in the drilling fluid, or that the

3

solid phase is inert. Hoberock et al4 proposed the following compositional

model for equivalent static density of drilling fluids.

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

+++=

111,

2

1

2

1

1122

w

wvw

o

ovo

vccvssvwwvoo

ff

ffffTP

ρρ

ρρ

ρρρρρ (1.1)

where,

ρο1, ρw1 = density of oil and water at temperature T1 and

pressure P1, respectively

ρο2, ρw2 = density of oil and water at temperature T2 and

pressure P2, respectively

f vo, fvw, fvs, fvc = fractional volume of oil, water, solid weighting

material, and chemical additives, respectively

P1, P2 = pressure at reference and condition “2”

T 1, T2 = temperature at reference and condition “2”

Application of the compositional model requires some knowledge of

how the densities of each liquid phase in the mud, usually water and some

type of hydrocarbon, change with changes in temperature and pressure. The

static mud density at elevated pressure and temperature can be predicted

from knowledge of mud composition, density of constituents at ambient or

standard temperature and pressure, and density of liquid constituents at

elevated temperature and pressure.

Peters et al5 applied the Hoberock et al4 compositional model

successfully to model volumetric behavior of diesel-based and mineral oil-

4

based drilling fluids. In their study, they measured the density of the individual

liquid components of each drilling fluid at temperatures varying from 78-350

oF and pressures varying from 0-15,000 psi. Using this data in conjunction

with Hoberock et al’s compositional model, they were able to predict the

density of the drilling fluids at the elevated temperature-pressure conditions.

The model predictions yielded an error of <1% over the range of temperature

and pressure examined.

Sorelle et al6 proposed equations expressing the relationship between

water and hydrocarbon (diesel oil No. 2) densities, and temperature/pressure

for use with the compositional model with some success. Kutasov8 analyzed

pressure-density-temperature behavior of water and proposed a similar

equation, which was reported to yield very low error in predicting water

densities in the HTHP region.

Isambourg et al7 proposed a nine-parameter polynomial model to

describe the volumetric behavior of the liquid phases in drilling fluids, which is

applicable in the range of 14.5-20,000 psi and 60-400 oF. This model

characterizes the volumetric behavior of the liquid phases in the drilling fluid

with respect to temperature and pressure, and is applied in a similar

compositional model to that proposed by Hoberock et al4. The model also

assumes that all volumetric changes in the drilling fluid is due to the liquid

phase, and application of the model requires a very accurate measurement of

the reference mud density at surface conditions.

5

Kutasov8 proposed an empirical equation of state (EOS) model for

drilling fluids to express the pressure-density-temperature dependent

relationship. As is the case for the compositional model, mud density using

Kutasov’s empirical equation of state is evaluated relative to its density at

standard conditions (p= 14.7 psi, T = 60 oF). He applied the equation of state

with a temperature-depth relationship in order evaluate hydrostatic pressure

and equivalent static density as a function of depth.

Babu9 compared the accuracy of the two compositional models

proposed by Sorelle et al4 and Kutasov8 respectively, and the empirical model

proposed by Kutasov8 in predicting the mud weights for 12 different mud

systems. The test samples consisted of 3 water based muds (WBM), 5

OBM’s formulated using diesel oil No. 2, and 4 OBM’s formulated using

mineral oil. Babu9 found that the empirical model yielded more accurate

estimates for the pressure-density-temperature behavior of a majority of the

muds over the range of measured data more accurately than the

compositional model. He also concluded that the empirical model has more

practical application because unlike compositional models, it is not hindered

by the need to know the contents of the drilling fluid in question.

Drilling fluids contain complex mixtures of additives, which can vary

widely with the location of the well, and sometimes with different stages in the

same well. This was especially apparent in the behavior of the drilling fluids

prepared with diesel oil No. 2. Different oils available under the category of

diesel oil No. 2 that were used in the preparation of OBM’s can exhibit

6

different compressibility and thermal expansion characteristics, which were

reflected in the pressure-density-temperature dependent behavior of the fluids

prepared with them.

Research has also been reported on characterizing drilling fluid

rheology at high temperature/high pressure conditions. Rommetveit et al11

approached their analysis of shear stress/shear rate data at high temperature

and pressure by multiplying shear stress by a factor which depends on

pressure, temperature and shear rate. Coefficients of this multiplying factor

are fitted to shear stress/shear rate data directly without extracting rheological

parameters such as yield stress first. This eliminates the need to characterize

the behavior of each rheological parameter relative to pressure and

temperature changes. In essence, they obtain an empirical model in which

the effects of variation in all rheological parameters that describe fluid flow

behavior are lumped together.

Another approach to the analysis of temperature and pressure effects

on drilling fluid rheology is to consider the effect of temperature and pressure

changes on each rheological parameter that describes the behavior of the

fluid. The two most common models3 considered for such an analysis are the

Herschel-Bulkey/Power law model and the Casson model which is an

acceptable description of oil based mud rheology. Of these two models, the

Herschel-Bulkley model is the most robust, as it is a three parameter model

as opposed to the Casson model which is a two parameter model. In the

analysis performed by Alderman et al3 on shear stress/shear rate data, the

7

Herschel-Bulkley/Power and Casson models were considered. The behavior

of each rheological parameter in these models with respect to changes in

temperature and pressure was investigated. They studied a range of fluids

covering un-weighted and weighted bentonite water-based drilling fluids with

and without deflocculant additives.

In order to estimate equivalent circulating density, it is important to take

into account the effects of temperature and pressure on fluid rheology. Two

methods are proposed to accomplish this by Rommetveit et al11. They

propose a stationary or static method and a dynamic method. In both

methods, the contributions of hydrostatic and frictional pressure losses in high

pressure/high temperature wells to the equivalent circulating density were

considered. The variation in temperature vertically along the well bore is

taken into account for both models, and drilling fluid properties are allowed to

vary relative to temperature.

The dynamic method however, also takes into account transient

changes in temperature i.e. change in temperature over time. This effect is

especially important in the case where circulation has been stopped for a

significant amount of time. The drilling fluid temperature will begin to

approach the temperature of the formation. Once circulation commences

again as shown in Fig. 1.1, the lower part of the annulus will be cooled by

cold fluid from the drill string and the upper part of the annulus will be warmed

by hotter fluid coming from the bottom-hole. During this transient period, fluid

density and rheological characteristics can change rapidly due to rapid

8

changes in temperature. Research on this effect is still at a very early stage

and will not be taken into account during this study.

Cooler fluid from drill pipe cooling down the annulus

Warmer fluid from bottom-hole warming up the upper annulus

Drill Pipe

Figure 1.1- Schematic Diagram of Fluid in the Well bore at the Start of Circulation

9

Alderman et al3 performed rheological experiments on water based

drilling fluids over a range of temperatures up to 260 oF and pressures up to

14,500 psi, using both weighted and unweighted drilling fluids. Rheograms

were obtained for the water based drilling fluids, holding temperature constant

and varying pressure, and vice versa. It was found that the Herschel-Bulkley

model yielded the best fit to the experimental data. Other models that were

investigated are the Bingham plastic model, and the Casson model which

some authors argue is the best model for characterizing oil-based drilling fluid

rheology.

For the Herschel-Bulkley model, it was found that the fluid viscosity at

high shear rates increased with pressure to an extent, which increases with

the fluid density, and decreases with temperature in a similar manner to pure

water. Alderman et al3 found the yield stress to vary little with pressure-

temperature conditions. The yield stress remained essentially constant with

respect to temperature until a characteristic threshold temperature is attained.

This threshold temperature was found to depend on mud composition. Once

this threshold is reached, the yield stress increases exponentially with 1/T.

Alderman et al3 also found that the power law exponent increased with

temperature, and decreased with pressure. This lead them to conclude that

the Casson model will become increasingly inaccurate at these two extremes,

that is, at high temperature and low pressure.

The estimation of ECD under high temperature conditions requires

knowledge of the temperatures to which the drilling fluid will be subjected to

10

downhole. As the fluid is circulated in the wellbore, heat from the formation

flows into the wellbore causing the wellbore fluid temperature to rise. This

process is more pronounced in deep, hot wells where the temperature

difference between the formation and the well-bore fluid is greater. The

process is very dynamic at early times, that is, at the commencement of

circulation, with great changes in fluid temperature occurring over small

intervals of time.

There are two major methods for estimating the down-hole

temperature of drilling fluid. The first is the analytical method. This method

assumes constant fluid properties. Ramey13 solved the equations governing

heat transfer in a well bore for the case of hot-fluid injection for enhanced oil

recovery. His solution permits the estimation of the fluid, tubing and casing

temperature as a function of depth. He assumed that heat transfer in the well

bore is steady state, while heat transfer in the formation is unsteady radial

conduction.

Holmes and Swift14 solved the heat transfer equations analytically for

the case of flow in the drillpipe and annulus. They assumed the heat transfer

in the wellbore to be steady state. However, they used a steady-state

approximation to the transient heat transfer in the formation. They justified

this assumption by asserting that the heat transfer from the formation is

negligible in comparison to the heat transfer between the drill pipe and

annular sections due to the low thermal conductivity of the formation.

11

Arnold15,16 also solved the heat transfer equations analytically for both

the hot-fluid injection case and the fluid circulation case. However, in

circulation case, he did not assume steady state heat transfer in the

formation. He represented the transient nature of heat flow from the formation

with a dimensionless time function that is independent of depth16. Kabir et al17

also solved a similar set of equations, but for the case of flow down the

annulus and up the drill pipe. They also assumed transient heat flow in the

formation, and evaluated a number of dimensionless time functions.

The second method of estimating fluid temperature during circulation

involves allowing the fluid properties such as heat capacity, viscosity, and

density to vary with the temperature conditions. This method involves solving

the governing heat transfer equations numerically using a finite difference

scheme. Marshal et al18 created a model to estimate the transient and steady-

state temperatures in a well bore during drilling, production and shut-in using

a finite difference approach.

Romero and Touboul19 created a numerical simulator for designing and

evaluating down-hole circulating temperatures during drilling and cementing

operations in deep-water wells. Zhongming and Novotny20 developed a finite

difference model to predict the well bore and formation transient temperature

behavior during drilling fluid circulation for wells with multiple temperature

gradients and well bore deviations.

12

1.3. Objectives and Scope of Work

The main objective of this work is to evaluate changes in drilling fluid

density with variations in temperature and pressure conditions and

characterize how these changes in equivalent circulating density are affected

by the composition of the drilling fluid. Specifically, the objectives of this work

are;

1. Evaluate changes in static density of drilling fluids with changes in the

temperature-pressure conditions

2. Evaluate changes in the rheological behavior of drilling fluids with

changes in the temperature-pressure conditions and ascertain the

degree of the resultant effect on the dynamic ECD.

3. Evaluate different methods of estimating the circulating fluid

temperature gradient in the well bore and the effects on frictional

pressure loss and hence on ECD.

The objectives of this work are accomplished with the development of a

Dynamic Density Simulator. This simulator was developed in the visual basic

language and will allow the estimation of the equivalent circulating density

under high pressure and high temperature conditions.

13

1.4. Study Organization

The fundamental concepts involved with hydrostatic pressure and

frictional pressure loss are discussed in Chapter Two. The most commonly

used rheological models for characterizing drilling fluid flow in conjunction

with frictional pressure loss calculation methods are also discussed. The

equations that express viscosity as a function of temperature and pressure

will be discussed here. Chapter Three discusses the heat transfer equations

in the well bore and the analytical and numerical methods for estimating the

temperature profile in a circulating well. Chapter Four covers the development

of the ECD simulator for high-pressure/high-temperature wells. The chapter

covers the modeling procedure, model validation, and discusses the results.

Chapter five covers the summary of the results, conclusions, and

recommendations.

14

Chapter 2

FUNDAMENTAL CONCEPTS FOR ESTIMATION OF EQUIVALENT STATIC AND CIRCULATING DENSITY

In today’s drilling industry, deeper and hotter wells are increasingly

being drilled. In order to maintain proper well control, prevent lost circulation,

and accurately analyze fracture gradient data, it is of paramount importance

to accurately predict the density of the drilling fluids used in drilling these

wells, under high temperature-high pressure conditions. Drilling fluids in

general become compressed under high pressure, and expand with

temperature. Hence, their down-hole densities are often quite different from

their surface densities, which are usually measured during drilling operations.

The fundamental concepts of equivalent static density and equivalent

circulating density are reviewed in this chapter.

2.1 Equivalent Static density

The equivalent static density of a drilling fluid is an expression of the

hydrostatic pressure exerted by the fluid. Hydrostatic pressure in turn can be

defined as the pressure exerted at any point by a vertical column of fluid. The

hydrostatic pressure is a function of the density of the fluid, and the height of

the fluid column. Hydrostatic pressure is expressed in field units as follows.

15

P = 0.052ρh (2.1)

Where,

P = pressure, psi

ρ = fluid density, lbm/gal (ppg)

h = height of fluid column, ft

This simple equation assumes the fluid in question to be incompressible. If

the temperature and pressure in the mud is low, the use of constant surface

mud density in conjunction with the above equation will yield a reasonable

approximation of the bottom-hole density.

Equivalent static density however, must take into account the effects of

temperature and pressure conditions present in the well. Excluding these

factors in the estimation of bottom-hole pressure in the case of deep, hot

wells, can yield figures that are in error by hundreds of psi. Figure 2.1 shows

the effects of temperature and pressure on the density of various base liquids

that can be used in drilling fluids. As expected, these figures show that

density increases with increasing pressure, but decreases as temperature

increases. However, as depth increases, temperature effects tend to

dominate pressure effects, so that the net result is decreasing mud density

with increasing depth.

16

oF

Figure 2.1- Volumetric Behavior of Various liquids

Under varying conditions of Temperature and Pressure4

17

2.2 Estimating Equivalent Static Density

There are two main methods of characterizing the variation in

equivalent static density of drilling fluids in response to changes in

temperature and pressure conditions; empirically based models, and

compositional models. The empirical model provides explicit empirically

derived equations for estimating mud density at various temperature-pressure

conditions. The compositional model however takes into account the

volumetric behavior of each of the individual mud constituents in response to

variations in temperature and pressure.

2.2.1 Compositional Models

The compositional model proposed by Hoberock et al4 is derived as

follows. In order to account for the variation in compressibility across the

different constituents present in a drilling fluid, it is necessary to perform a

material balance on the drilling fluid as a whole. In the model, it is assumed

that all solids present in the drilling fluid are incompressible. Consider a

drilling fluid that consists of oil and water phases, solid weighting material,

and chemical additives. The volume and weight of the drilling fluid at some

reference temperature (p1, T1) would be expressed as follows

18

V1 = Vo + Vw + Vs (2.2)

W = ρo1Vo + ρw1Vw + ρsVs (2.3)

Where,

Vo, Vw,Vs = volume of oil, water, and solids

W = weight

ρo1, ρw1 = density of oil and water phases at reference

conditions (p1, T1)

ρs = density of solid content

Ideal mixing is assumed in Eqs. 2.2 and 2.3. Once the drilling fluid is

subjected to a new set of temperature-pressure conditions (p2, T2), the

volume of the fluid will change due to the compressibility of the liquid phases.

The new drilling fluid volume is thus expressed as

V2 = Vo + Vw + Vs + ∆Vo + ∆Vw (2.4)

From the law of conservation of mass, the change in volume of the liquid

phases can be expressed as follows.

oo

ooo VVV −⎟⎟

⎠

⎞⎜⎜⎝

⎛=∆

2

1

ρρ (2.5)

ww

www VVV −⎟⎟

⎠

⎞⎜⎜⎝

⎛=∆

2

1

ρρ (2.6)

From Eqs. 2.3 and 2.4, the new mud density at T2 and p2 will be as follows.

( )wocswo

sswwoom VVVVV

VVVTp∆+∆+++

++=

ρρρρ 1122, (2.7)

19

where the subscript m refers to the drilling mud. Substituting Eqs. 2.5 and 2.6

into Eq. 2.7 and dividing by the original total volume at pressure p1 and

temperature T1, the following equation is obtained.

( )

1112

1

12

1

1111

11

22 ,

VV

VV

VV

VV

VV

VV

VV

VV

Tpcsw

w

wo

o

o

cc

ss

ww

oo

m

+++

+++=

ρρ

ρρ

ρρρρρ (2.8)

Consider the volume fraction fx of each component to be

VVf x

x = (2.9)

where Vx = volume of component x

fx = volume fraction of component x

V = total volume

Taking into account that

fo + fw + fs = 1 (2.10)

then Eq. 2.8 can be expressed as

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

++=

111,

2

1

2

1

1122

w

ww

o

oo

sswwoom

ff

fffTp

ρρ

ρρ

ρρρρ (2.11)

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

=111

,

2

1

2

1

122

w

ww

o

oo

mm

ffTp

ρρ

ρρ

ρρ (2.12)

where ρm1 is the mud density at temperature T1 and pressure p1. From the

above equation, it is evident that the mud density at elevated temperatures

and pressures can be predicted based on knowledge of the mud constituents

and the volumetric behavior of each constituent relative to variations in

20

temperature and pressure. Various authors have proposed equations

expressing the volumetric behavior of water, and oil phases that may be

present in drilling fluids.

2.2.1.1 Volumetric Models for Mud Constituents

Sorelle et al6 proposed the following empirical expressions for the

volumetric behavior of the water phase, and diesel oil No. 2.

ρo = Ao + A1(T) + A2(p-po) (2.13)

ρw = Bo + B1(T) + B2(p-po) (2.14)

where

Ao = 7.24032 Bo = 8.63186

A1 = -2.84383 * 10-3 B1 = -3.31977 * 10-3

A2 = 2.75660 * 10-5 B2 = 2.37170 * 10-5

The equation for the water phase was obtained by curve fitting data from

tables of physical constants, while that of the diesel oil No. 2 was obtained by

curve fitting data from experiments.

Kutasov 21 analyzed the pressure-density-temperature behavior of

water, and proposed the following similar empirical equation.

(2.15) ( ) ( ) ( )[ ]20

70

40

6 10*0123.510*2139.210*0997.33619.8 TTTTppw e −−−−− −−−

=ρ

where po and To represent standard temperature (59 oF) and pressure (14.7

psia).

21

Isambourg et al7 proposed a nine parameter model to express the

volumetric behavior of liquids in response to variations in temperature and

pressure. The model is expressed as follows.

Vr(p, T) = k00 + k01T + k10P + k02T2 + k20P2+k11pT

+ k12pT2 + k21p2T + k22p2T2 (2.16)

where Vr(p, T) is the volumetric ratio at pressure p, and temperature T.

( )( )oo TpVolume

TpVolumeratioVolumetric,,

= (2.17)

Eq. 2.16 is valid in the range of 14.5 to 20,000 psia, and 68 to 392 oF and can

be used to estimate fluid density at elevated temperatures and pressures in

the following manner.

( ) ( ) ( )( )TpV

TpVTpTpr

oorooff ,

,,, ρρ = (2.18)

The above equation is a simplification of the compositional model proposed

by Hoberock et al4. Isambourg et al7 also proposed the following equation to

express variations in the density of solid weighting material with changes in

temperature-pressure conditions.

( ) ( )( )[ ] ([ )]osos

ooss ppbTTa

TpTp−+−+

=1*1,, ρρ (2.19)

where

as = 0.8*10-4 oC-1, thermal expansivity of barite

bs = -1.0*10-5 bar-1, compressibility of barite

22

2.2.2 Empirical Models

Kutasov8 proposed the following empirical three-parameter equation of

state to describe the volumetric behavior of drilling fluids.

(2.20) ( ) ( ) ( )[ ]2ooo TTTTpp

mom e −±−−−= γβαρρ

where,

ρmo = mud density at standard pressure and temperature

(14.5 psia, 59oF)

α, β, γ = empirical constants

Kutasov’s model applies to both water-based and oil-based drilling fluids, and

treats the particular drilling fluid as a continuous phase. Hence, knowledge of

the volumetric behavior of each of the constituents of the drilling fluid is not

required.

2.3 Equivalent Circulating density

The equivalent circulating density of a drilling fluid can be defined as

the sum of the hydrostatic head of the fluid column, and the pressure loss in

the annulus due to fluid flow. It is expressed as density term at the point of

interest.

23

( frictionchydrostatiecd PPh

∆+∆=052.01ρ ) (2.21)

where,

ρecd = equivalent circulating density (lb/gal)

∆Phydrostatic = Hydrostatic head of fluid column (psi)

∆Pfriction = Pressure drop due to friction in the annulus (psi)

As stated before, the hydrostatic pressure of the drilling fluid is affected by the

temperature-pressure conditions present in the well-bore, and the depth of

the well-bore. The frictional pressure loss term in the above equation however

is affected by the well-bore and drill string geometry, fluid rheology, and the

pump rate or fluid flow rate.

2.4 Frictional Pressure Loss

The frictional pressure loss is the loss in pressure during fluid flow due

to contact between the fluid and the walls of the flow conduit. When fluid

moves past the solid interface, a boundary layer is formed adjacent to the wall

of the flow conduit. The viscous property of the fluid creates a variation in the

flow velocity normal to the solid interface, ranging from zero at the pipe wall

with a no-slip assumption and maximum velocity at the edge of the boundary

layer. This variation in fluid velocity represents a loss in momentum and a

resistance to flow. The associated pressure loss is directly proportional to the

24

length of the flow conduit, the fluid density, the square of the fluid velocity,

and inversely related to the conduit diameter.

LD

vfp ∆=∆22 ρ (2.22)

where

∆p = frictional pressure loss

f = Fanning friction factor

ρ = fluid density

∆L = conduit length

V = fluid velocity

D = pipe diameter

In the case of non-circular flow conduits, the diameter parameter is replaced

by the equivalent diameter.

w

fe P

AD 4= (2.23)

where

De = equivalent diameter

Af = cross-sectional area

Pw = wetted perimeter

The variable “f” in equation 2.22 is known as the Fanning friction factor. The

friction factor can be defined as the ratio between the force exerted on the

walls of a flow conduit as a result of fluid movement, and the product of the

characteristic area of the flow conduit and the kinetic energy per unit volume

of the fluid.

25

2.5 Fluid Rheology

Rheology can be defined as the science and study of the deformation

and flow of matter, in this case drilling fluids. It is also the characteristic of the

particular fluid in reference to its flow behavior. Rheological models seek to

characterize this flow behavior by developing relationships between applied

shear stress, and the shear rate of the fluid. Based on the nature of this

relationship, fluids in general can be classified as Newtonian, non-Newtonian,

and visco-elastic fluids.

Newtonian Fluids- Newtonian fluids are fluids in which the ratio between

applied shear stress, and the rate of shear is constant with respect to time

and shear history. The relationship characterizing Newtonian fluids is

expressed mathematically as follows:

γµτ &= (2.24)

where,

τ = shear stress

γ& = shear rate

µ = viscosity

Examples of Newtonian fluids are water, light hydrocarbons, and all gases.

Non-Newtonian fluids- Non-Newtonian fluids are fluids whose viscosity varies

with time and shear history. This class of fluids can be further subdivided into

time-dependent and time-independent fluids. Time-dependent fluids are

26

fluids, in which the viscosity varies with time at a constant shear rate, while

time-independent fluids are fluids whose viscosity is constant over time at a

constant shear rate. Most drilling fluids are non-Newtonian fluids.

Visco-elastic Fluids- These are fluids which exhibit both viscous and elastic

behavior. When subjected to stress, they deform and flow like true fluids, but

once the stress is removed, they regain some of their original state like solids.

Examples of visco-elastic fluids include flour dough, and polymer melts.

The following are the rheological models that characterize the various

types of non-Newtonian fluids.

2.5.1 Bingham Plastic Model

The Bingham plastic model is a two-parameter time-independent

rheological model that accounts for the stress required to initiate fluid flow in

viscous fluids. This initial stress is referred to as the yield stress. Once the

yield stress is overcome, the fluid is represented as a Newtonian fluid, which

is shown by the linear relationship between the applied stress and the rate of

shear. The constitutive equation for the Bingham plastic model is given as

follows.

γµττ &po += (2.25) oττ >

where

τo = yield stress µp = plastic viscosity

27

Although the Bingham plastic model does account for the yield stress, it can

be inadequate for characterizing some types of drilling fluids, as it does not

account for their shear thinning property.

2.5.2 Power Law Model

The power law model is also a time-independent two parameter

rheological model like the Bingham plastic model. However, where the

Bingham plastic model expresses a linear relationship between shear stress

and shear strain, the power law model uses a non-linear relationship which

can better characterize the shear-thinning characteristics of most common

drilling fluids. The following is the constitutive equation for the power law

model.

(2.26) nkγτ &=

where

k = consistency index

n = flow behavior index

k and n are constants characteristic of a particular fluid. k is a measure of the

consistency of the fluid, the higher the value of k the more viscous the fluid; n

is a measure of the degree of non-Newtonian behavior of the fluid. In cases

where the flow behavior index is equal to 1, the power law model describes

the behavior of a Newtonian fluid. In situations where the flow behavior index

is between 0 and 1, the fluid is referred to as pseudoplastic and shear-

28

thinning. Shear-thinning refers to the reduction in viscosity with the shear rate.

The limiting viscosity is known as the viscosity at infinite shear, ∞µ (Fig. 2.2).

∞µ

µο

µ

γ

Figure 2.2- Shear-thinning in a typical non-Newtonian Fluid

Shear Rate

Dilatant

Newtonian

Pseudo-plastic

Bingham Plastic

Yield Pseudo-plastic

Shear Stress

Figure 2.3- Flow curves for time-independent fluids

29

When the flow behavior index is greater than 1 the fluid is referred to

as dilatant and shear thickening. This is shown in Fig. 2.3. The dimensions of

k are dependent on the value of n; hence k is not a material property but an

empirical constant. In general, the shear-thinning behavior is more desirable

in drilling fluids; hence drilling fluids tend to be pseudo-plastic fluids. Shear-

thinning behavior is desirable in drilling fluids because it allows the fluid to

carry cuttings even while it is at rest due to the fluid thickness, and at the

same time lowers pumping costs because the fluid becomes thinner and

easier to pump as it is sheared.

2.5.3 Herschel-Bulkley Model

The Herschel-Bulkley model is a time-independent three parameter

rheological model that accounts for both the yield stress, and the non-linear

relationship between shear stress and shear rate exhibited by most drilling

fluids. The constitutive equation is given below.

no kγττ &+= oττ > (2.27)

where



τo = yield stress

The Herschel-Bulkley model is also used widely in the oil industry to

characterize drilling fluids as well as fracturing fluids.

30

The above three rheological models are the most commonly used in

the oil industry for the characterization of drilling fluids. There are, however,

various other rheological models that can and have been used. The following

are a few of these models.

2.5.4 Casson Model

The Casson model is a two-parameter model originally developed in

order to characterize the rheology of pigment-oil suspensions. The

constitutive equation is given as follows.

γττ &ko += oττ ≥ (2.28)

where

τo = yield stress

k = Casson model constant, similar to the consistency index for

Power-law model

The Casson model is also commonly used to characterize the rheology of

blood.

2.5.5 Ellis Model

The Ellis model is a three-parameter model which accounts for a

Newtonian region at low shear rates, while still expressing a power law type

dependence at higher shear rates. These are the initial flat plateau and

31

successive straight line segments of Fig. 2.2. The constitutive equation is

given as follows.

γ

ττ

µγτµ α

&&

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+

== −1

21

1

oa (2.29)

where

µa = apparent viscosity µo = low shear rate viscosity

τ1/2 = shear stress @ µa = µ0/2 α = Ellis model parameter

From eq. 2.29, a plot of ⎟⎟⎠

⎞⎜⎜⎝

⎛−1ln

a

o

µµ versus

21

lnττ will yield a slope of (α-1).

Hence, if µo and τ1/2 are known, α can be estimated. µo refers to the viscosity

at very low shear rates, i.e. as τ tends to zero.

2.5.6 Carreau Model

The Carreau model is a four-parameter model developed in order to

account for the entire flow curve shown in Fig. 2.2, i.e. the two Newtonian-like

flow regions at very high shear rates and very low shear rates characterized

by flat plateaus and the power-law region in the middle. The constitutive

equation for the Carreau model is as follows.

32

( )[ ]( )2

121

−

∞

∞ +=−− n

o

γλµµµµ

& (2.30)

where

oµ = low shear rate viscosity

∞µ = viscosity at infinite shear

λ = time constant

n = exponential constant

The λ parameter is a time constant calculated from the point on the flow curve

where the flow behavior transitions from the lower Newtonian region to the

power law region.

2.6 Temperature and Pressure Dependent Rheological

Parameters

In order to estimate the flow behavior of drilling fluids under high

temperature-high pressure conditions, the following variation on the Bingham

plastic model proposed by Politte22 will be applied.

2.6.1 Temperature/Pressure Dependent Plastic Viscosity

Politte22 analyzed rheological data for diesel based drilling fluid and

found the plastic viscosity tracked the behavior of the base oil. Hence, the

plastic viscosity of the oil-based drilling fluid is normalized with the viscosity of

the base oil. The plastic viscosity will be normalized with the viscosity of the

33

base fluid at reference conditions. The steps of this method are detailed as

follows:

1. Measure the plastic viscosity of the drilling fluid at reference conditions

(PV0).

2. Calculate the base oil viscosity at the reference conditions (µo) and at

the temperature and pressure conditions of interest (µT,P).

3. Calculate the plastic viscosity at the conditions of interest using the

following equation.

o

PToPT PVPV

µµ ,

, = (2.31)

Politte22 concluded that this procedure will be valid regardless of the type of

base oil used. He obtained the following equations from the analysis of diesel

viscosity data.

( ) ⎟⎠⎞⎜

⎝⎛ +++++

= ρρµ

111111

110GFPETPDTBACTPP (2.32)

1000 ≤ P ≤ 15000

75 ≤ T ≤ 300

(2.33) 222

22222 TFTEPDPCPTBA +++++=ρ

A1 = -23.1888 A2 = 0.8807

B1 = -0.00148 B2 = 1.5235*10-9

C1 = -0.9501 C2 = 1.2806*10-6

D1 = -1.9776*10-8 D2 = 1.0719*10-10

E1 = 3.3416*10-5 E2 = -0.00036

F1 = 14.6767 F2 = -5.1670*10-8

34

G1 = 10.9973

Where

µ = viscosity (cp)

ρ = density (lb/gal)

T = temperature (oF)

P = pressure (psi)

Further analysis with other oils by Politte led to the conclusion that Eqs. 2.32

and 2.33 are applicable for estimating the downhole plastic viscosity

regardless of the type of base oil used.

2.6.2 Temperature Dependent Yield point

Politte22 concluded from his analysis of rheological data for emulsions

that the yield point is not a strong function of pressure, and becomes

progressively less as temperature increases. The effects of temperature on

the yield point are, however, hard to predict, as there are chemical as well as

particle effects that have to be considered. Politte22 advises that in situations

where it may be important to know the precise behavior of the drilling fluid,

the yield point should be measured on a viscometer capable of such

measurements. If the equipment is not available, he provides the following

steps based on an empirical equation obtained from the analysis of diesel oil

based drilling fluid.

35

1. Measure the yield value at the reference conditions (YVTo).

2. Calculate the yield value of the drilling fluid at the temperature of

interest (YVT) using the following equation.

23

133

23

133

−−

−−

++++

=oo

yoy TCTBATCTBAττ (2.34)

90 T ≤ ≤ 300

A3 = -0.186

B3 = 145.054

C3 = -3410.322

Where

τy = yield point (lbf/100ft2)

T = temperature (oF)

Since yield value is dependent on the chemical attractions between the

particles present in the drilling fluid, Eq. 2.34 cannot be used to estimate the

yield value of drilling fluids that have base fluids of significantly different

chemistry from No. 2 diesel oil.

2.7 Bingham Plastic Pressure Loss Equations

In order to evaluate frictional pressure loss, it is first necessary to determine if

the flow is laminar or turbulent. The apparent viscosity of the fluid is first

calculated using the following equations.

36

Apparent Newtonian viscosity in pipes-

v

dypa

τµµ

66.6+= (2.35)

Apparent Newtonian viscosity in the annulus-

vdey

pa

τµµ

5+= (2.36)

where

µa = apparent viscosity (cp) d = pipe Diameter (in)

µp = plastic viscosity (cp) de = equivalent annular

τy = yield point (lbf/100ft2) diameter (in)

v = average fluid velocity

The apparent viscosity is then used in place of the Newtonian viscosity in

order to calculate the Reynolds number according to the following equation.

a

dvNµρ928

Re = (2.37)

where

ρ = fluid density (lb/gal)

d = pipe diameter or equivalent annular diameter (in)

v = average fluid velocity (ft/s)

aµ = apparent viscosity (cp)

A Reynolds number greater than 2100 indicates turbulent flow. Depending on

the flow regime, the frictional pressure drop can be calculated using the

following equations.

37

Laminar Flow

Pipe

Ldd

vP yp

f ∆⎟⎟⎠

⎞⎜⎜⎝

⎛+=∆

2251500 2

τµ (2.38)

Annulus

( ) ( ) Ldddd

vP yp

f ∆⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

−=∆

122

12 2001000τµ

(2.39)

Turbulent Flow

( ) 395.0log41Re10 −= fN

f (Colebrook Equation) (2.40)

where p

dvNµρ928

Re = (2.41)

∆PF = Frictional pressure drop (psi)

∆L = Flow conduit length (ft)

ρ = fluid density (lb/gal)

d = pipe diameter or equivalent annular diameter (in)

v = average fluid velocity (ft/s)

pµ = plastic viscosity (cp)

d1 = inner annular wall (in)

d2 = outer annular wall (in)

If the flow regime is turbulent, once the friction factor has been obtained, the

frictional pressure drop can be found with Eq. 2.22.

38

2.8 Summary

For the purposes of this study, the compositional model for

characterizing the volumetric behavior of drilling fluids as expressed in Eq.

2.12 will be applied in conjunction with Eqs. 2.14 and 2.33 to express the

behavior of the major fluid constituents. In order to characterize the flow

behavior of the drilling fluid under high temperature-high pressure conditions,

the Bingham plastic model with temperature/pressure dependent model

parameters will be applied. The Bingham plastic model was chosen because

it is the most commonly used rheological model on the oil field and models

the behavior of a wide variety of fluids.

39

Chapter 3

DRILLING FLUID TEMPERATURE PROFILE ESTIMATION

As fluid flows in the wellbore, it absorbs heat from the formation,

causing a rise in its temperature. This rise in temperature in turn can lead to

changes in the fluid’s volumetric and rheological behavior, and thus the

frictional pressure drop. This effect is more pronounced in deep high

temperature wells and fluids with temperature sensitive rheological

properties11. Estimation of fluid temperature in the drill pipe and the annulus is

thus necessary in order to calculate the frictional pressure drop for a number

of well construction operations.

Fluid temperature within the wellbore will vary with depth and time with

this variation being especially pronounced at early times when the

temperature within the wellbore has not stabilized appreciably. The

temperature profile within the wellbore can be estimated analytically, or

numerically. This chapter gives a description of the heat transfer processes

that take place in the wellbore and the methods for estimating the

temperature profile.

40

3.1 Heat Transfer in the Wellbore

Figure 3.1 shows a schematic of drilling fluid circulating in the wellbore

and the associated heat transfer process over a differential element of length

∆z. The figure shows heat flow from the formation into the annular section

through convection (qfa). This rate of heat flow by convection into the annulus

is much greater than the rate of heat conduction in the formation. This is due

to the relatively low heat conductivity of the formation. This fact will be

important when modeling the heat transfer process in the wellbore. The fluid

within the drillpipe receives heat from the annulus via convection on the pipe

surface on the inside and outside of the drill pipe, and conduction through the

drillpipe itself (qap). There is heat flow in and out of the differential elements

within the drillpipe and annulus due to the bulk flow of fluid (qp(z), qp(z+Dz),

qa(z+Dz), qa(z) respectively).

Two methods have emerged for estimating the temperature profile in

the wellbore during circulation. They are the analytical method and the

numerical method. The analytical method entails solving the equations

governing heat transfer in the wellbore analytically, that is, assuming constant

fluid and formation properties. This method is best applied to systems of

simple geometry as in the case of a single casing string and inner drill pipe.

The numerical method uses a finite difference scheme to represent the

wellbore/formation system. Systems of great complexity can be better

handled using this method, and it has the added advantage of allowing

variable fluid and formation properties.

41

z

z +

qfa

qap

qa(z+∆

Annulus

Formation

qa(z)

qa(z+∆

qp(z)

qa(z+∆

Ta

Tp

qa(z)

Drill Pipe

Figure 3.1- Schematic of Heat Balance for Fluid Circulating in a Wellbore

42

3.2 Analytical Method

The temperature of the fluid within the drillpipe and the annulus is

described by two coupled ordinary differential equations. The temperature in

the formation is determined by the geothermal gradient coupled with the

transient formation heat conduction function, f(tD)13. The function accounts for

the un-steady state heat conduction in the formation. In order to solve these

equations, boundary conditions are required. The boundary conditions

applied are as follows.

Boundary Conditions

• The inlet fluid temperature coming into the drillpipe at the surface.

• The fluid temperature in the drillpipe and the annulus are equal at the

bottom-hole.

3.2.1 Assumptions of Analytical Model

The assumptions used in deriving and solving the equations governing

heat transfer within the wellbore are stated below16.

• The analytical method assumes constant fluid properties.

• Heat generated by viscous forces, friction, and changes in potential

energy are negligible.

• The formation is radially symmetric and infinite with respect to heat flow.

• Heat flow within the wellbore is rapid compared to heat flow within the

formation. Hence, heat flow within and across the wellbore conduits is

43

assumed to be steady-state, and heat flow within the formation is

assumed to be transient.

3.2.2 Heat Balance in the DrillPipe

Heat enters the differential element in the drillpipe from two sources;

bulk fluid flow qp(z), and from convection and conduction through the drillpipe

wall, qap. Heat leaves the differential element through bulk fluid flow qp(z+∆z).

The heat balance of the differential element in the drillpipe yields the following

equation

qp(z) + qap = qp(z+∆z) (3.1)

where,

qp(z) =zpflTmc (3.2)

qap = ( ) ( )( )dzzTzTUr papp −π2 (3.3)

qp(z+z)= zzpflTmc

∆+ (3.4)

to yield

( ) ( )( )zzpflpappzpfl TmcdzzTzTUrTmc

∆+=−+ π2 (3.5)

rearranging,

( ) ( )( 02 =−− zTzTUrz

Tmc papp

pfl π )

δδ

(3.6)

where,

m = mass flow rate of drilling fluid (lb/hr)

cfl = fluid Heat Capacity (Btu/lb-oF)

44

Tp = temperature of drillpipe fluid as a function of depth (oF)

Ta = temperature of annular fluid as a function of depth (oF)

rp = radius of drillpipe (ft)

Up = equivalent heat transfer coefficient across pipe wall

(Btu/hr-ft2-oF)

z = depth (ft)

3.2.3 Heat Balance in the Annulus

Heat enters the differential element in the annulus from the formation

by convection (qfa), and through bulk fluid flow (qa(z+∆z)). Heat leaves the

differential element through convection and conduction through the pipe wall

(q(ap)) and through bulk fluid flow (qa(z)). This process yields the following

equation.

qfa + qa(z+Dz) = qap + qa(z) (3.7)

where

qfa = ( ) ( )( dzzTzTUr aiaa )−π2 (3.8)

qa(z+Dz) = zzaflTmc

∆+ (3.9)

qap = ( ) ( )( )dzzTzTUr papp +π2 (3.10)

qa(z) = zaflTmc (3.11)

to yield

( ) ( )( )zzaflaiaa TmcdzzTzTUr

∆++−π2

( ) ( )( )zaflpapp TmcdzzTzTUr ++= π2 (3.12)

45

rearranging,

( ) ( )( ) ( ) ( )( ) 022 =++−−zTmcdzzTzTUrdzzTzTUr a

flpappaiaa δδππ (3.13)

where

ra = radius of annulus (ft)

Ua = equivalent heat transfer coefficient across formation/annulus

interface (Btu/hr-ft2-oF)

Ti = temperature at interface between Formation and Annulus (oF)

3.2.4 Heat Flow in the Formation and System Heat Balance

The heat flow from the formation is given by the following equation.

( ) ( ) ( )( dzzTzTtfkq iFD

Ff −= )π2 (3.14)

where

kF = formation thermal conductivity (Btu/ft-oF-hour)

TF = temperature of Formation according to the undisturbed

geothermal gradient (oF)

f(tD) = dimensionless time function

The dimensionless time function16 is given as

( )( )

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛−

=

tr

trEi

tfa

a

D

α

α

4exp

421

2

2

(3.15)

FF

F

ck

ρα = (3.16)

46

where

ρF = formation density (lb/gal)

cF = formation heat capacity (Btu/lb-oF)

Ei = exponential Integral function

It can be observed that the heat flow from the formation should be equal to

the heat flow into the annulus by convection. Thus, qf = qfa. The temperature

of the interface between the formation and the annulus can thus be eliminated

as follows.

( ) ( ) ( )( ) ( ) ( )( )dzzTzTUrdzzTzTtfk

aiaaiFD

F −=− ππ 22

rearranging,

( ) ( )

( ) ( )aaD

F

aaFD

F

i

Urtfk

TaUrTtfk

Tππ

ππ

22

22

+⎟⎠⎞⎜

⎝⎛

+⎟⎠⎞⎜

⎝⎛

= (3.17)

From Eq. 3.2,

pp

a TdzdT

T += β (3.18)

dzdT

dzTd

dzdT ppa += 2

2

β (3.19)

where,

pp

fl

Urmcπ

β2

= (3.20)

Inserting Eqs. 3.17, 3.18, and 3.19 into Eq. 3.13 will yield an equation that is

in terms of Tp alone. The equation is given below.

47

02

2

=+−− Fppp TT

dzdT

dzTd

βσβ (3.21)

where,

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

Faa

DaaFfl kUr

tfUrkmcπ

σ2

(3.22)

The formation temperature is based on the geothermal gradient. Therefore,

TF = TFs + gGz (3.23)

Where,

TFs = surface formation temperature (oF)

gG = geothermal gradient (oF/ft)

Eq. 3.21 then becomes,

zgTTdzdT

dzTd

GFsppp −−=−− βσβ 2

2

(3.24)

The solution of the above ordinary differential equation is given below.

(3.25) ( ) GFsGzz

p gTzgeCeCzT βγγ −+++= 2121

where

αβ

αβββγ

242

1++

= (3.26)

αβ

αβββγ

242

2+−

= (3.27)

From Eq. 3.25,

( )

Gzzp geCeC

dzzdT

++= 212211

γγ γγ (3.28)

48

By inserting Eqs. 3.25 and 3.28 into Eq. 3.18, we obtain the following

equation for the temperature in the annulus.

( ) ( ) ( ) FsGzz

a TzgeCeCzT +++++= 2211 11 21 βγβγ γγ (3.29)

In order to obtain the constants C1 and C2, the following boundary conditions

are applied16.

@ z = 0 Tp(z) = Tps

@ z = L Tp(z) = Ta(z)

where,

Tps = fluid temperature at drillpipe inlet or at the surface (oF)

L = the total vertical depth of the well (ft)

By applying the boundary conditions, the following expressions are obtained

for the constants C1, and C2.

12

21 12

2

γγγ

γγ

γ

LLdiff

LG

eeTeg

C−

−= (3.30)

12

12 12

1

γγγγγ

γ

LLdiff

LG

eeTeg

C−

+−= (3.31)

where,

Tdiff = (TFs – Tps – βgG) (3.32)

49

3.3 Numerical Method

The numerical method involves solving the equations governing heat

flow in the wellbore and formation, using finite difference technique. Heat

transfer in the wellbore is assumed to be steady-state, while heat transfer

between the formation and annulus is treated as unsteady-state heat flow.

The solution grid used is shown in Fig. 3.2.

3.3.1 Equations Governing Heat transfer in the Wellbore and Formation

The equation of conservation of energy for a control volume inside the

drill pipe20 is given as

( ) ( ) ( )[ ] ( ) ( )t

tzTcr

ztzT

mctzTtzTtzUr pflp

pflpapp ∂

∂+

∂

∂=−

,,,,,2 2ρππ (3.33)

where

rp = radius of pipe (ft)

Up = heat Transfer coefficient across boundary layer on outer pipe

surface, pipe wall, and boundary layer on inner pipe surface.

(Btu/hr-ft2-oF)

Ta = temperature in the annulus (oF)

Tp = temperature in the drillpipe (oF)

m = mass Flow rate of drilling fluid (lb/hr)

cfl = heat capacity of drilling fluid (Btu/lb-oF)

50

r

z

Annuulus DrillPipe

Formation

Figure 3.2a- Solution grid for Finite Difference Analysis

q1 q2

Heat Flow by Convection

Heat Flow by Conduction

Control Volume in

the Annulus

Control Volume in

the

Figure 3.2b- Heat Flow at Formation Annulus Boundary

51

The equation for conservation of energy for a control volume inside the

annulus20 is given as follows:

( ) ( ) ( )[ ] ( ) ( ) ( )[ ]tzTtzTtzUrtzTtzrTtzUr pappaafaa ,,,2,,,,2 −−− ππ

( ) ( ) ( )z

tzTcmt

tzTcrr afl

oa

flpa ∂∂

−∂

∂−=

,,22ρπ (3.34)

where

ra = annular radius (ft)

Ua = heat transfer coefficient across annulus/formation interface

(Btu/hr-ft2-oF)

TF = formation Temperature (oF)

The temperature in the formation is given by the following equation20.

( ) ( )t

trzTr

trzTrrr

FF

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂

∂∂ ,,1,,1

α (3.35)

where

α = formation transmissivity (kF/ρcF)

kF = formation conductivity

ρ = formation density

cF = formation heat transfer coeeficient

Note that heat flow in the formation is assumed to occur radially only. At the

boundary of the formation, heat exchange between the formation and annulus

is governed by Eq. 3.36. The equation is derived by performing a heat

balance about a sufficiently small control volume within the formation adjacent

to the annulus.

52

@ r = ra

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡

∂∂

+−−r

trzTkrtzTtrzTtzUr aFFaaaFaa

,,2,,,,2 ππ

( )t

trzTcrr aFFa ∂

∂∆=

,,2 ρπ (3.36)

The first term on the left hand side of Eq. 3.36 represents the rate at which

heat is leaving the formation boundary by convection (q1 in Fig. 3.2b). The

second term on the left hand side of the equation represents the rate at which

heat enters the control volume by conduction (q2 in Fig. 3.2b). The right-hand

side of Eq. 3.36 represents the rate at which heat accumulates in or is lost

from the control volume at the formation boundary, leading to changes in

temperature.

3.3.2 Discretizing Heat Flow Equations for Finite difference Analysis

The solution of the equations governing heat transfer in the wellbore

and formation will be propagated across the formation and well bore using the

finite difference grid shown in Fig. 3.3. The solution will be advanced starting

at the outer boundary of the formation in the r-direction until the temperature

field is mapped for the entire formation cross-section at a particular time-step.

The temperature in the wellbore and formation are expressed as follows.

Tp(z, t) = Tp(i∆z, n∆t) = ( )nipT

Ta(z, t) = Ta(i∆z, n∆t) = ( )niaT

TF(z, r, t) = Ta(i∆z, j∆r, n∆t) = ( )njiFT ,

53

r

j = 0 j = 1 j = 2 i = 0

i = 1

i = 2

i = 3

z

FormatioWellbore

Figure 3.3- Finite Difference Grid

54

where,

i = depth coordinate j = radial coordinate n = time coordinate

Two discretization schemes were considered for the equations describing

heat flow in the wellbore (3.33 & 3.34). They were first discretized using

explicit finite differences as follows.

Eq. 3.33 can be expressed as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )t

TTcr

zTT

cmTTTT

Urnip

nip

flp

nip

nip

fl

onip

nip

nia

nian

ipp ∆

++

∆

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−

++

−−−

12111

222 ρππ

for i = 1,2,3,…,I-1,I n = 1,2,3,…N-1,N (3.37)

Parameters bearing coordinate n are known while parameters bearing

coordinate n+1, i.e. at the next time step, are not known. Equation 3.37 is

rearranged with the known parameters on right-hand side and the unknown

parameters on the left-hand side as follows.

( ) ( ) ( ) ( )nip

flp

ppfl

o

nia

flp

pnia

flp

pnip T

crz

UrzcmtT

crtU

Tcr

tUT

1211

−−+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

∆

⎟⎠⎞

⎜⎝⎛ ∆−∆

+∆

+∆

=ρπ

π

ρρ

( ) nip

flp

ppfl

o

Tcrz

Urzcmt

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

∆

⎟⎟⎠

⎞⎜⎜⎝

⎛∆+∆

−+2

1ρπ

π

(3.38)

Equation 3.38 in essence, expresses the temperature at the particular

coordinate of interest as a weighted average of temperatures at spatial

coordinates located nearby in the pipe and annulus, at the previous time-step.

55

Stability Criterion- The coefficients of the known pipe and annulus

temperatures (i.e. in the present time-step) in Eq. 3.38 must be positive in

order to arrive at a stable solution of the equation. A negative coefficient

would not make physical sense, because it would be saying that the hotter

the temperature is at a coordinate near or at the particular coordinate of

interest for the present time-step, the colder the temperature will be at the

particular coordinate of interest at the next or future time-step. Therefore, the

coefficients must all be greater than or equal to zero. In fact, it is usually

desirable to avoid zero coefficients. Application of this rule to Eq. 3.38, results

in the following constraints.

⎟⎠⎞

⎜⎝⎛ ∆+

∆≤∆

ppfl

oflp

Urzcm

crzt

π

ρπ 2

(3.39)

(3.40) ppfl

oUrzcm π∆≥

The equation describing heat flow in the annulus (Eq. 3.34) is discretized as

follows.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−

+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−

+ −−−−

222

222 1111

nip

nip

nia

nian

ipp

nia

nia

niF

niFn

iaa

TTTTUr

TTTTUr ππ

( ) ( ) ( ) ( ) ( )

tTT

crrzTT

cmnia

nia

flpa

nia

nia

fl

o

∆+

−+∆−

−=+

−1

221 ρπ

for i = 1,2,3,…,I-1,I n = 1,2,3,…N-1,N (3.41)

Equation 3.41 is rearranged, with known parameters on the right hand side of

the equation and unknown parameters on the left hand side as follows.

56

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )nip

flpa

ppnip

flpa

ppniF

flpa

aaniF

flpa

aania T

crrtUr

Tcrr

tUrT

crrtUrT

crrtUrT 2212222122

11 −

∆+

−

∆+

−∆

+−

∆=

−−+− ρρρρ

( ) ( )nia

flpa

ppaafl

o

Tcrrz

UrzUrzcmt1221 −

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−∆

⎟⎠⎞

⎜⎝⎛ ∆+∆+∆

−+ρπ

ππ

( ) ( )nia

flpa

ppaafl

o

Tcrrz

UrzUrzcmt

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−∆

⎟⎠⎞

⎜⎝⎛ ∆−∆−∆

+ 22ρπ

ππ (3.42)

Stability Criterion- Application of the rule of positive coefficients to Eq. 3.42

results in the following constraints.

( )

⎟⎠⎞

⎜⎝⎛ ∆+∆+

−∆≤∆

ppaafl

oflpa

UrzUrzcm

crrzt

ππ

ρπ 22

(3.43)

(3.44) ppaafl

oUrzUrzcm ππ ∆+∆≥

Boundary conditions in the Wellbore- The following boundary conditions are

applied to the wellbore.

@ i = 0 (pipe inlet) ( ) psnip TT = for all n.

@ i = imax (bottom-hole) ( ) ( )nia

nip TT = for all n.

where

Tps = temperature of fluid at pipe inlet

The second boundary condition states that at the bottom of the well, the pipe

and annulus fluid temperatures are equal. Note that Eq. 3.38 is solved from

57

the surface to the bottom in the direction of fluid flow, while Eq. 3.42 is solved

from bottom-hole to surface, also in the direction of fluid flow.

The explicit method detailed above was tested and found to be too

slow, that is, it took a longer time to converge on an answer. This was largely

due to the constraints placed on the time step. An alternative discretizing

scheme, the Crank-Nicolson method26, was then used. This scheme yields an

efficient, easy to use finite difference scheme which gives more accurate

solutions without constraints on the time step used.

Using the Crank-Nicolson scheme, Eq. 3.33 is discretized as follows.

( ) ( ) ( ) ( ) ( ) ( ) ( ) nip

nia

nipp

nip

nia

nipp TTUrTTUr −−+− +++ θπθπ 122 111

( ) ( )( )

( ) ( ) ( ) ( )t

TTcr

zTT

zTT

cmnip

nip

flp

nip

nip

nip

nip

fl

o

∆

−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∆

−−+

∆

−=

+

−+

+

−

+

+

1211

11

11

21

2ρπθθ (3.45)

Eq. 3.45 is rearranged as

( ) ( ) ( ) ( ) 11

112

11 2

22

+

+

+++

− ∆+⎟

⎟⎠

⎞⎜⎜⎝

⎛+

∆+

∆− n

ipfl

o

nip

nipp

flpnip

fl

o

Tz

cmTUr

tcr

Tz

cmθθπ

ρπθ

( )( ) ( ) ( ) ( ) ( )( )nip

fl

o

nip

nipp

flpnip

fl

o

Tz

cmTUr

tcr

Tz

cm1

2

11

2121

2 +−−

∆−⎟

⎟⎠

⎞⎜⎜⎝

⎛−+

∆+−

∆= θθπ

ρπθ

( ) ( )( ) ( ) ( ) 11212 +++−+ nia

nipp

nia

nipp TUrTUr θπθπ (3.46)

Note that in Eq. 3.46, all the terms on the left hand side are unknowns while

all the terms on the right-hand side are known except the last one which is the

temperature in the annulus at the depth of interest during the current time

step. This problem is solved by taking an initial guess of the temperature

profile in the annulus. This guess is the temperature profile at the previous

58

time-step. Hence, the last term on the right-hand side of Eq. 3.46 will be a

known term.

The equation describing heat flow in the annulus (Eq. 3.34) is

discretized using the Crank-Nicolson method as follows.

( ) ( ) ( ) ( ) ( ) ( ) ( ) nia

niF

niaa

nia

niF

niaa TTUrTTUr −−+− +++

0,11

0,1 122 θπθπ (3.47)

( ) ( ) ( ) ( ) ( ) ( ) ( ) nip

nia

nipp

nip

nia

nipp TTUrTTUr −−−−− +++ θπθπ 122 111

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∆−

−+∆−

−∆

−−= −+

+−

++

+

zTT

zTT

cmt

TTcrr

nia

nia

nia

nia

fl

onia

nia

flpa 21

211

11

11

122 θθρπ

Equation 3.47 is rearranged as

( ) ( ) ( ) ( ) ( ) ( ) 11

11122

11 2

222

++

++++− ∆

−⎟⎟⎠

⎞⎜⎜⎝

⎛++

∆

−+

∆nia

fl

o

nia

niaa

nipp

flpania

fl

o

Tz

cmTUrUr

tcrr

Tz

cmθθπθπ

ρπθ

( )( ) ( ) ( ) ( ) ( ) ( ) ( nia

niaa

nipp

flpania

fl

o

TUrUrt

crrT

zcm

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−

∆

−+−

∆−= − θπθπ

ρπθ 12121

21

22

1 )

( )( ) ( ) ( )( ) ( ) ( )( )niF

niaa

nip

nipp

nia

fl

o

TUrTUrTz

cm0,

11 12121

2θπθπθ −−−+−

∆− +

( ) ( ) ( ) ( ) 10,

111 22 ++++ ++ niF

niaa

nip

nipp TUrTUr θπθπ (3.48)

Note that all the terms on the left hand side of Eq. 3.48 are unknowns while

all the terms on the right hand side are known with the exception of the last

term. The sixth term is already known because the temperature profile in the

drill pipe at any given time-step is evaluated before the temperature profile in

the annulus. The problem of the last term is solved by making an initial guess

59

of the temperature profile in the immediate adjacent formation. The initial

guess is taken to be the temperature profile at the previous time step.

Boundary Conditions in the Wellbore - A heat balance is performed at the

bottom of the wellbore taking into account the bottom-hole boundary

condition. Figure 3.4 shows a diagram of the heat balance.

( )1max−ipT

( ) maxiT

imax

`

Figure 3.4- Heat Balance at Bottom-Hole

60

Note,

Timax = ( )maxipT = ( ) maxiaT

Since @ i = imax (Bottom-hole), ( )maxipT = ( ) maxiaT

Performing a heat balance about the bottom-hole volume element in the

wellbore yields the following equation.

( ) ( ) ( ) ( ) tTczrTTzUrTcmTcm flaiiFaaifl

o

ipfl

o

∂∂∆

=−∆

+−− 22

2 2maxmaxmax1max

ρππ (3.49)

Where T = Tp, Eq. 3.49 is discretized as follows.

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1max

10max,max1max

1max

11max

1 ++

−

++

−−∆+−−+− n

ipniFaa

nip

nipfl

onip

nipfl

oTTzUrTTcmTTcm θπθθ

( ) ( ) ( ) ( ) ( )tTT

czrTTzUrnip

nip

flanip

niFaa ∆

−∆=−−∆+

+

max1

max2max0max, 2

1 ρπθπ (3.50)

Eq. 3.50 is rearranged as

( ) ( ) 1max

21

1max 2++

− ⎟⎟⎠

⎞⎜⎜⎝

⎛

∆

∆−−+ n

ipfla

fl

onipfl

oT

tzcr

cmTcmρπ

θθ

( )( ) ( ) ( ) ( ) 10max,max

2

1max 211 +

−∆−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∆

∆−−−+−= n

iFaanip

flafl

onipfl

oTzUrT

tzcr

cmTcm θπρπ

θθ

( )( ) ( ) ( )( )niaaa

niaaa

niFaa TzUrTzUrTzUr max

1max0max, 11 θπθπθπ −∆−+∆+−∆− + (3.51)

Equation 3.46 in conjunction with Eq. 3.51, thus form a tridiagonal system of

equations. The system of equations is solved easily using the Thomas

algorithm26.

Where T = Ta, Eq. 3.49 is discretized as follows.

61

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1max

10max,max1max

1max

11max

1 ++

−

++

−−∆+−−+− n

ianiFaa

nia

nipfl

onia

nipfl

oTTzUrTTcmTTcm θπθθ

( ) ( ) ( ) ( ) ( )t

TTczrTTzUr

nia

nia

flania

niFaa ∆

−∆=−−∆+

+max

1max2

max0max, 21 ρπθπ (3.52)

Eq. 3.52 is rearranged as follows

( ) ( ) ( )( )nipfl

onipfl

oniafl

o

aafla TcmTcmTcmzUr

tzcr

1max1

1max1

max

2

12 −

+

−

+ −+=⎟⎟⎠

⎞⎜⎜⎝

⎛+∆+

∆

∆θθθθπ

ρπ

( ) ( )( )niFaa

niFaa TzUrTzUr 0max,

10max, 1 θπθπ −∆+∆+ +

( ) ( ) ( )nia

flaaafl

oT

tzcr

zUrcm max

2

211 ⎟

⎟⎠

⎞⎜⎜⎝

⎛

∆

∆+−∆−−−+

ρπθπθ (3.53)

Equation 3.47 is solved in conjunction with Eq. 3.53 to yield the temperature

profile in the annulus.

Heat Flow Equations in the Formation

Equation 3.35 governs heat flow in the formation and is discretized

using the Taylor series approximation of derivatives in conjunction with the

Crank-Nicolson method of finite differences. The Crank-Nicolson method

results in an implicit set of linear algebraic equations which must be solved

simultaneously. However, this method has the advantage of being

unconditionally stable. There is no constraint set on the size of the time-step

to be used. Equation 3.35 is thus discretized as follows.

62

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

∆

+−+

∆

+−=

∆

− +−+

+++

−+

21,,

11,

2

11,

1,

11,,

1, 22

2 r

TTT

r

TTTt

TT njiF

njiF

njiF

njiF

njiF

njiF

njiF

njiF α

( ) ( ) ( ) ( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∆

−

∆+

∆

−

∆+ −+

+−

++

rTT

rjrTT

rj

njiF

njiF

njiF

njiF

21

21

21,1,

11,

11,α (3.54)

Rearranging Eq. 3.54 so that unknowns are on the left-hand side,

( ) ( )( ) ( ) nji

njiF

rrnjiFr

nijiF

rr Taj

aTaTaj

a,

11,

1,

1, 24124 φααααα =⎟

⎠⎞⎜

⎝⎛ −−+++⎟

⎠⎞⎜

⎝⎛ − +

+++

− (3.55)

where,

( ) ( )( ) ( )njiF

rrnjiFr

nijiF

rrnji Ta

jaTaTa

ja

1,,,, 24124 +− ⎟⎠⎞⎜

⎝⎛ ++−+⎟

⎠⎞⎜

⎝⎛ +−= αααααφ

( )2rtar ∆

∆=

Formation Boundary Conditions- At the formation and annulus interface, the

boundary condition can be expressed as follows.

aaFaF

F TUTUr

Tk =+∂

∂− (3.56)

Equation 3.47 can be expressed as follows.

( ) ( ) ( ) ( )n

iaaniFa

niF

niF

F TUTUrTT

k =+∆

−− −

0,1,1,

2 (3.57)

Rearranging Eq. 3.57 for time-steps n and n+1, we obtain the following.

( ) ( ) ( ) ( )niF

F

niF

niaa

niF T

krTTUT 1,0,1,

2+

∆−=− (3.58)

( ) ( ) ( ) ( ) 11,

10,

111,

2 ++++− +

∆−= n

iFF

niF

niaa

niF T

krTTUT (3.59)

63

Note that there in Eqs. 3.49 and 3.50, there is a node j = -1. This is an

imaginary node located outside of the formation in the annulus. Near the

boundary, the equation describing internal heat flow in the formation (Eq.

3.35) is replaced with the following equation26.

2

2

2rT

tT FF

∂∂

=∂

∂ α (3.60)

This is done to avoid the apparent singularity which occurs at the node j = 0.

Equation 3.60 can be expressed as

( ) ( )t

TT njiF

njiF

∆

−+,

1,

( ) ( ) ( )( )

( ) ( ) ( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

∆

+−+

∆

+−= +−

++

++−

21,,

11,

2

11,

1,

11, 22

22

rTTT

rTTT n

jiFn

jiFn

jiFn

jiFn

jiFn

jiFα (3.61)

rearranging at node j = 0, we obtain

( ) ( )( ) ( ) ni

niFr

niFr

niFr TaTaTa 0,

11,

10,

11, 21 φααα =−++− +++

− (3.62)

where

( ) ( )( ) ( )niFr

niFr

niiFr

ni TaTaTa 1,0,,0, 21 ααφ +−+= − (3.63)

The temperatures at the imaginary node j = -1 are eliminated from Eqs. 3.62

and 3.63 by inserting Eqs. 3.58 and 3.59 to obtain

( ) ( ) ni

niFr

niFa

Frr TaTU

kraa 0,

11,

10, 2221 φααα =−⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆++ ++ (3.64)

64

where

( ) ( )niaa

Fr

niaa

Fr

ni TU

kraTU

kra ∆

+∆

= + 22 10, ααφ

( ) ( )niFr

niFa

Frr TaTU

kraa 1,0, 2221 ααα +⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∆+−+ (3.65)

Equations 3.64 and 3.65 express the temperature in the formation at the

formation-annulus interface. Note that in Eq. 3.65, the first term on the right

hand side contains the annulus temperature for the future time step. This

does not create a problem because at the time when the temperature in the

formation is evaluated, the temperature in the wellbore for the future time-step

has already been evaluated. Hence, the temperature in the annulus for the

future time step is already known. At the outer boundary of the formation, a

sufficient radius of consideration is chosen at which the temperature

disturbance caused by the flow of fluid in the wellbore is no longer felt in the

formation. This length is generally taken to be 10 ft from the wellbore14. At this

outer boundary, the temperature conforms to the undisturbed geothermal

gradient.

@ j = J for all n (3.66) ( ) GGFSn

jiF TzigTT =∆+= **,

Equations 3.55, 3.64, and 3.66 thus form a complete set of linear algebraic

equations which describe the temperature at every node in the formation.

They can be expressed in matrix form as follows.

(3.67) [ ] Ψ=Ω T

This is done at each depth and at each time-step n as follows:

65

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=Ω

100000

00000001

KK

MM

MM

K

CBA

CBACBA

( )( )( )

( ) ⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

+

+

+

+

1,

12,

11,

10,

nJiF

niF

niF

niF

T

TTT

T

M

M

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=Ψ

−

G

nJi

ni

ni

ni

T1,

2,

1,

0,

φ

φφφ

M

where

⎟⎠⎞⎜

⎝⎛ −= 24

rr aj

aA αα ( )raB α+= 1 ⎟⎠⎞⎜

⎝⎛ −−= 24

rr aj

aC αα

Equation 3.56 is a tridiagonal system of linear algebraic equations, and can

thus be solved using the simple but efficient Thomas Algorithm.

Numerical Procedure-

The following is a summary of the steps taken in the numerical solution.

1. The initial conditions of the system are specified (time t = 0). The initial

temperature conditions in the formation conform to the geothermal

gradient. The initial temperature conditions in the wellbore also

66

conform to the formation geothermal gradient. This condition is chosen

because it is found that after sufficient time during a trip, the

temperature of the fluid in the wellbore is equal to temperature of the

formation.

2. The temperature profile in the drill pipe is evaluated first using Eqs.

3.46 and 3.51. It is first necessary to guess the temperature profile in

the annulus at the current time step in order to evaluate the drill-pipe.

The initial guess is taken to be the temperature profile in the annulus at

the previous time step.

3. Based on the newly evaluated drill-pipe temperature, the annular

temperature profile is evaluated using Eqs. 3.48 and 3.53. Note that it

is necessary to guess the temperature profile in the immediate

adjacent formation at the current time-step. The guess chosen is the

temperature profile in the previous time step.

4. The temperature profile in the formation is then evaluated at the

current time step based on the newly evaluated annulus profile. The

results of the procedure are then compared with the initial guesses. If

the error is insignificant, the next time step is evaluated. If there is

significant error, the whole procedure is repeated with the current

temperature profiles in the annulus and formation being used as the

guesses. This procedure is repeated until the calculations are

completed for the total circulation time.

67

3.4 Summary

The analytical and numerical methods for estimating the temperature

profile within the wellbore and formation have been described. It is assumed

that heat flow in the wellbore occurs rapidly in comparison to heat flow in the

formation. Heat flow in the wellbore occurs by bulk fluid flow, convection

across surface films on the pipe conduit and on the outer wall of the annulus,

and conduction through the pipe wall. Heat flow in the formation occurs by

conduction in the radial direction only. For the numerical method, the

equations governing heat flow in the wellbore and formation are solved using

the Crank-Nicolson method which results in a system of linear algebraic

equations. The numerical method is used in the development of the Dynamic

density Simulator.

68

Chapter 4

DEVELOPMENT AND VALIDATION OF THE DYNAMIC DENSITY SIMULATOR AND MODELLING OF DYNAMIC DENSITY Drilling operations are being conducted to deeper depths as the need

to supplement dwindling hydrocarbon resources forces exploration into more

unconventional environments. Deeper wells mean the drilling fluid will

encounter higher temperatures and pressures. As discussed in previous

chapters, it is not expected that the volumetric and rheological properties of

the fluid remain constant under these conditions. It is thus necessary to

predict these downhole conditions and their impact on the drilling fluid

behavior. This allows for precise drilling fluid selection and preparation, and

accurate estimation of the maximum allowable pump pressure.

The following chapter contains a description, development, and

validation of the Dynamic Density Simulator (DDS) and analysis of the results

of equivalent circulating density estimation under high-temperature/high-

pressure conditions. The DDS program is a predictive tool that will allow the

drilling engineer to predict the down-hole temperature/pressure conditions

that will be encountered and the resultant change in drilling fluid rheological

behavior. The simulator was written using Visual Basic for applications

automated through Microsoft Excel. The user interface is integrated with

Excel and initiated with a command button that is integrated into the main

69

worksheet menu. This format was chosen because it allows ease of use and

accessibility. It also allows manipulation of generated results with Excel

utilities such as worksheets and graphs. The following is a detailed

description and explanation of the program layout.

4.1 Program Lay-Out

The program interface is executed with a series of user forms, which

will accept data pertaining to the well bore, drilling fluid, and formation

parameters and return the temperature profiles in the wellbore and formation,

pressure losses in the wellbore and the ECD of the circulating fluid. The user

can navigate between forms and input data at leisure using the “back” and

“next” buttons. Once all the parameter values have been entered into the

program, the results are displayed on a “results” form. The following is the

sequence of forms used in the program.

1. “frmStart”- This is the starting form and the form that is displayed

when the program is initiated. On this form, the option is given to initiate a

new well profile.

2. “frmWellProps”- This form allows input of the well bore parameters. These

include the total vertical depth of the well, drill string dimensions such as

inner and outer drill pipe diameters, drill bit dimensions, and information

such as the inlet pipe temperature and the circulation rate.

3. “frmMudProps”- This form accepts the mud parameters such as

rheological data, volumetric and constituents data, and thermal data.

70

4. “frmFormationProps”- This form allows input of the thermal properties of

the surrounding formation.

5. “frmHeatTransfer”- This form allows input of the heat transfer coefficients.

6. “frmResults” - This form displays the frictional pressure losses and ECD

results.

Figure 4.1 shows a step-by-step sequence of data entry, computation and

results display.

4.2 DDS Program Execution

The first form that is displayed once the program is executed is the

program title page. From this form, a new well profile can be initiated. Figure

4.2 shows a screen capture of the title form. The form is initiated by clicking

on a command button that is integrated into Excel’s set of main menu

commands as shown in the screen capture in Fig. 4.3. The following

sequence of forms will be described as follows.

4.2.1 General Well Parameters Form

On this form, the dimensions and configuration of the drill-string are

specified. These include the dimensions of the drill-pipe, heavy-weight drill-

pipe and drill collars. The bit size, circulation rate, inlet pipe temperature and

total vertical depth are also specified. A screen capture of this form is shown

in Fig. 4.4.

71

START (Initiate Well

Profile)

INPUT WELL PARAMETERS

INPUT MUD PARAMETERS

INPUT FORMATION PARAMETERS

Drill String Configuration/Geometry

Bit Configuration

TVD, Circulation Rate, Inlet Pipe Temperature

EVALUATE WELLBORE TEMPERATURE PROFILE

CALCULATE BOTTOM-HOLE PRESSURE & ECD

DISPLAY

INPUT HEAT TRANSFER COEFFICIENTS

STOP

Reference Conditions

Thermal Properties

Density and Constituents

Rheological Properties

Figure 4.1- DDSimulator Program Flow

Chart

72

Figure 4.2- Title Form

73

DDSimulator Command

Button

Figure 4.3- DDSimulator Launch Command Button

74

Figure 4.4- Well Parameters Form

75

Figure 4.5- Mud Properties Form

76

4.2.2 Mud Properties Form

The properties of the drilling fluid are entered on this form. These

include the rheological properties such as plastic viscosity and yield strength,

the density and constituents, and the thermal properties such as thermal

conductivity and heat capacity. The density and rheological parameters are

obtained at certain reference temperature and pressure. These reference

conditions are also entered. Figure 4.5 shows a screen capture of the form.

4.2.3 Formation Properties Form

The properties of the formation are entered on this form. These

properties include the density, geothermal gradient, surface temperature, heat

conductivity, specific heat capacity, and maximum radius of interest. The

maximum radius of interest refers to the minimum radius at which the

formation no longer sees the temperature disturbance as a result of

introducing the drilling fluid into the well. This radius is usually about 10-ft

from the well-bore. Figure 4.6 shows a screen capture of the formation

properties form.

4.2.4 Heat Transfer Coefficients Form

The overall heat transfer coefficients for heat transfer across the

annulus-formation interface, and across the drill-pipe wall are entered on this

form. Figure 4.7 shows a screen capture of the heat transfer coefficients form.

77

Figure 4.6- Formation Properties Form

78

Figure 4.7- Heat Transfer Coefficients Form

79

4.2.5 Results and Results Form

Once all the system parameters have been specified, the simulator can

then evaluate the temperature profile inside the drill-pipe and annulus and the

resultant variation in density and rheological parameters. The frictional

pressure drop in the annulus and drill-pipe, the bottom-hole pressure and the

ECD are then determined. The results are presented on the result form and

temperature profile is presented graphically using the Excel graph feature.

The result form is shown in Fig. 4.8 and a sample temperature profile is

shown in Fig. 4.9.

Figure 4.8- Results Form

80

Figure 4.9- A Sample Temperature Profile Using Excel Graph Feature

81

4.3 Equations used in DDSimulator Program

The DDS program uses the numerical method to evaluate the

temperature profile in the wellbore and near formation environment. Taking

the temperature profile into account, the simulator computes equivalent

hydrostatic head and frictional pressure loss in the wellbore during circulation.

The equations used in the simulator are as follows:

4.3.1 Fluid Properties

The density of the fluid in the wellbore is computed using the

compositional method according to the following equation.

( )⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+

=111

,

2

1

2

1

122

w

ww

o

oo

mm

ffTp

ρρ

ρρ

ρρ (2.12)

The volumetric behavior of the oil component is calculated as follows:

(2.33) 222

22222 TFTEPDPCPTBAo +++++=ρ

where

A2 = 0.8807 B2 = 1.5235*10-9

C2 = 1.2806*10-6 D2 = 1.0719*10-10

E2 = -0.00036 F2 = -5.1670*10-8

The volumetric behavior of the water component is calculated as follows:

ρw = Bo + B1(T) + B2(p-po) (2.14)

where

Bo = 8.63186 B1 = -3.31977 * 10-3 B2 = 2.37170 * 10-5

82

The density of the fluid in the annulus is computed for 200 discrete lengths of

the wellbore, that is, the total vertical depth is divided into 200 discrete

lengths. The temperature dependent plastic viscosity of the oil component of

the fluid in the well bore is calculated according to the following equation.

( ) ⎟⎠⎞⎜

⎝⎛ +++++

= ρρµ

111111

110GFPETPDTBACTPP (2.32)

1000 ≤ P ≤ 15000

75 ≤ T ≤ 300

where

A1 = -23.1888 B1 = -0.00148 C1 = -0.9501

D1 = -1.9776*10-8 E1 = 3.3416*10-5 F1 = 14.6767

G1 = 10.9973

The steps used in the DDSimulator to calculate the plastic viscosity and yield

value of the drilling fluid are detailed in Sections 2.6.1 and 2.6.2 of Chapter 2.

The apparent viscosity of the drilling fluid is calculated according to Eqs. 2.36

and 2.36. The apparent viscosity is calculated for 200 discrete lengths of the

drill pipe.

4.3.2 Temperature Profile Estimation

The temperature profiles in the drill pipe and annulus are evaluated

implicitly using the Crank-Nicolson method. This discretizing scheme was

chosen because it is an efficient, easy to use scheme that allows for accurate

solutions without constraints on the time step used. The equations and

solutions steps used are detailed in Chapter 3.

83

4.3.3 Equivalent Hydrostatic Head and ECD

The hydrostatic head for each of the discrete sections of the wellbore

for which the density is known is computed according to Eq. 2.1. In order to

compute the frictional pressure loss, the flow regime must first be known.

Thus, the Reynolds number is first computed according to Eq. 2.37. If the flow

regime is laminar, the frictional pressure drop is computed using Eq. 2.38 or

2.39. If the flow regime is turbulent, the friction factor is computed according

to Eq. 2.40 and 2.41. The frictional pressure loss is then computed according

to Eq. 2.22. The equivalent circulating density is then calculated according to

Eq. 2.21.

4.4 Model Validation

As stated previously, the numerical method is applied in the

DDSimulator for temperature profile estimation in the wellbore. This method is

chosen because it can model more complex geometries than the analytical

method. In order to validate the temperature estimation capability of the

simulator, fluid circulation in a Gulf-Coast well was modeled. The well

parameters are as detailed in Table 4.1. The numerical results from the

DDSimulator were compared with the results obtained using the analytical

method. Fig. 4.10 shows a plot of the temperature profile in the pipe and

annulus obtained with the numerical and analytical methods.

84

Well Geometry Well Depth, ft 15000 Drill Stem OD, in. 6-5/8 Drill-Bit Size, in. 8-3/8 Circulation Rate, bbl/hour 300 Mud Properties Inlet Temperature, oF 75 Plastic Viscosity, cp 20.9 Yield Strength, lbf/100 ft2 35.3 Thermal conductivity, Btu/ft-oF-hour 1 Specific Heat, Btu/lb-oF 0.4 Density, lb/gal 10 Formation Properties Thermal conductivity, Btu/ft-oF-hour 1.3 Specific Heat, Btu/lb-oF 0.2 Density, lb/cu ft 165 Surface Earth Temperature, oF 59 Geothermal Gradient, oF/ft 0.0127

Table 4.1 WELL AND MUD CIRCULATING PROPERTIES FOR A GULF COAST WELL14

85

0

2000

4000

6000

8000

10000

12000

14000

16000

0 50 100 150 200 250 300

Temperature (oF)

Dep

th (f

t)

Pipe Numerical Annulus NumericalGeothermal Gradient Pipe AnalyticalAnnulus Analytical

Figure 4.10- Temperature Profile For

Gulf Coast Well

86

Figure 4.10 shows the good agreement for the temperature profile

between the numerical model and the analytical method. The maximum

deviation between the two methods was less than 2%. The predicted flowing

bottom-hole temperature in the well matched the observed flowing bottom-

hole temperature of 186 oF. Figure 4.10 also shows that the maximum

temperature in the well-bore may not occur at the bottom of the hole. As seen

in the figure, the maximum temperature in the well-bore for this particular

case occurs in the annulus several feet above the total vertical depth (TVD).

This agrees with observations made by several authors13-17.

The shape of the temperature profile occurs as a result of the heat flow

equilibrium attained by the fluid as it flows down the drill-pipe and up the

annulus. As the fluid flows down the drill-pipe, it gains heat from the annular

fluid thereby increasing in temperature until it reaches the bottom of the hole.

Once the fluid enters the annulus, it starts to lose heat to the relatively cooler

drill pipe. However, for a certain length in the annulus, the formation is still

hotter and some heat is lost to the annular fluid. The annular fluid thus

increases in temperature, until the heat lost to the drill-pipe is greater than the

heat gained from the formation or the annular fluid temperature is actually

higher than the formation temperature, whichever occurs first. This process

results in the highest well-bore temperature occurring in the annulus some

length above the TVD, and the unique shape of the temperature profile.

The numerical model was also used to simulate reported field data.

The following data was obtained from a well in Matagorda County, Texas23.

87

The well had been drilled and cased with 5-1/2-in., 17 lbf/ft casing, and 2-1/2-

in. tubing was set without a packer at 8650-ft. Tests were conducted on the

well measuring bottom-hole circulating temperature while circulating field salt

water at 84 and 252 gal/min. The initial undisturbed static bottom-hole

temperature was 250 oF, with a temperature gradient of 2.03 oF/100 ft. The

well was circulated at 84 gal/min for 2 hours and 40 minutes. The bottom-hole

temperature of the fluid dropped to 213 oF. The well was then logged and

circulated again at 252 gal/min. The bottom-hole temperature at the beginning

of the second circulation period was 224 oF. The well was circulated for 56

minutes, at the end of which the bottom-hole temperature dropped further to

196 oF. These conditions were simulated using the numerical method, and

the results are shown in Figs. 4.11 and 4.12.

The bottom-hole temperature at the end of first circulation period was

estimated to be 230 oF. This value has an 8% difference relative to the actual

measured value of 213 oF. The estimated bottom-hole temperature at the end

of the second circulation period was 199 oF. This is a 1.5 % difference from

the actual measured value of 196 oF.

88

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 50 100 150 200 250 300

Depth (ft)

Tem

pera

ture

(o F)

Pipe Numerical Annulus Numerical Geothermal Gradient

Circulation Rate- 84 gal/min Circulation Time- 2 hrs 40 min Fluid- Field Salt Water

Temperature (oF)

Figure 4.11- Well Temperature Profile While Circulating Field Salt Water

89

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 50 100 150 200 250 300

Depth (ft)

Tem

pera

ture

(o F)


Circulation Rate- 252 gal/min Circulation Time- 56 min Fluid- Field Salt Water

Temperature

Figure 4.12- Temperature Profile For Gulf Coast Well

90

4.5 Dynamic Density Estimation

The effects of temperature and pressure on the equivalent circulating

density in a high temperature-high pressure well were simulated using the

DDSimulator. The properties of the first well that was simulated are detailed

in Table 4.2. The temperature profile for the well is shown in Fig. 4.13. The

temperature profile indicates that the temperature in the well-bore is higher

than the formation temperature for a large portion of the hole. As fluid moves

from the bottom of the hole upwards in the annulus, it looses heat to the pipe,

and for about 2000 ft up the annulus. It also gains heat from the formation.

Beyond this point, the annular temperature is higher than the formation

temperature. Thus, heat moves from the annulus into the formation as well as

into the drill-pipe. The rate of heat transfer across the pipe wall is very high

due to the high heat conductivity of steel and the high flow rate. Hence, the

temperature profiles in the annulus and drill pipe are very close.

The results of the ECD calculations are detailed in Table 4.3. The ECD

based on constant fluid properties (i.e. independent of the

temperature/pressure conditions) was evaluated for comparison. The bottom-

hole pressure taking into account the temperature-pressure dependence of

the fluid properties was 218 psi lower than the bottom-hole pressure obtained

using constant fluid properties. This is due to the volumetric behavior of the

drilling fluid. The decrease in the density of the fluid due to temperature is

more pronounced than the increase in density due to the pressure. Hence,

91

the bottom-hole pressure is less than one would expect if the density of the

drilling fluid remained constant.

General well Properties Well Depth (L) 17200 ft Outer Drill Pipe Radius (rp) 0.208333 ft Annulus Radius (ra) 0.354167 ft Circulation Rate 400 bbl/hr Circulation Time (hr) 5 hr

Inlet Mud Temperature (Tps) 120 oF Mud Properties Viscosity (µfl) (@ reference conditions) 50.82 lb/(ft-hour) Yield Value (@ reference conditions) 10 lbf/100ft2

Thermal Conductivity (kfl) 1 Btu/(ft-oF-hour) Specific Heat (cfl) 0.4 Btu/(lb-oF) Density (ρfl) (@ reference conditions) 16.8 lb/gal Oil Fraction 0.594 Water Fraction 0.066 Formation Properties

Thermal Conductivity (kF) 0.3 Btu/(ft-oF-hour) Specific Heat (cF) 0.21 Btu/(lb-oF) Density (ρF) 165 lb/ft3 Surface Earth Temperature (TFs) 70 oF

Geothermal Gradient (gG) 0.020 oF/ft

Temperature/Pressure

Dependent Constant Property Difference

Bottom Hole Pressure (psi) 22020 22238 -218

ECD (ppg) 24.6 24.9 -0.3

Table 4.3 Results of Well Simulation

Table 4.2 Simulated Well Conditions

92

Figure 4.14 shows a plot of the bottom-hole pressure versus depth. It

can be observed that there is a steady increase in pressure as the depth

increases with the final bottom-hole pressure obtained with constant fluid

density being higher than that obtained with a temperature/pressure

dependent fluid density. Figure 4.15 shows the temperature/pressure

dependent behavior of the fluid density. As depth increases the equivalent

circulating density continues to decrease as a result of the greater effect of

fluid expansion due to temperature, as opposed to compression due to the

increased down-hole pressure.

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 50 100 150 200 250 300 350 400 450

Temperature (oF)

Dep

th (f

t)


Circulation Rate- 400 gal/minCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft

Figure 4.13- Temperature Profile in 17200-ft well after 5 hrs

93

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 5000 10000 15000 20000 25000

Dep

th (f

t)

Annular Pressure (psi)

Variable Fluid Properties Constant Fluid Properties

Circulation Rate- 400 gal/minCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft


Figure 4.14- Annular Pressure Profile in 17200-ft well after 5 hrs

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

24 24.5 25 25.5 26 26.5 27 27.5 28 28.5

Dep

th (f

t)

Equivalent Circulating Density (ppg)

Temperature/Pressure DependentConstant Fluid Properties

Circulation Rate- 400 gal/min Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft


Figure 4.15- Equivalent Circulating Density in 17200-ft well after 5 hrs

94

The well parameters detailed in Table 4.2 were simulated using water

based drilling fluid. The results are shown in Table 4.4, and in Figs. 4.16 to

4.18. The results showed a similar trend to that obtained with oil-based drilling

fluid with the ECD obtained taking pressure and temperature conditions into

account being lower than the ECD calculated assuming constant fluid

properties.

The effects of the temperature gradient of the formation were also

studied. The same well properties as detailed in Table 4.2 were used. The

results are displayed in Tables 4.5 and 4.6, and in Figs. 4.19 to 4.24. As seen

in Tables 4.5 and 4.6, increase in the geothermal gradient results in a larger

difference between, the bottom-hole pressure estimated taking into account

the temperature/pressure conditions, and the bottom-hole pressure estimated

assuming constant fluid density and viscosity with the constant property

bottom-hole pressure being higher. A geothermal gradient of 1.5 oF/100-ft

results in a difference of 110 psi while a geothermal gradient of 2.5 oF/100-ft

results in a difference of 325 psi. These results show the higher fluid

expansion that occurs with a higher geothermal gradient. This trend is also

displayed in Figs. 4.21 and 4.24. Failure to take this effect into account during

drill operations could lead to the occurrence of a kick and possibly a blow-out.

95

Table 4.4 Well Simulation Results for Parameters Detailed in Table 4.2 with Water-Based mud




ECD (ppg) 20.8 21.0 -0.2




ECD (ppg) 24.7 24.9 -0.2

Table 4.5 Well Simulation Results for Parameters Detailed in Table 4.2 with gG = 0.015 oF/ft

96




ECD (ppg) 24.5 24.9 -0.4

Table 4.6 Well Simulation Results for Parameters Detailed in Table 4.2 with gG = 0.025 oF/ft

0

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14000

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0 50 100 150 200 250 300 350 400 450

Temperature (oF)

Dep

th (f

t)


Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- WBM Geothermal Grad- 0.02 oF/ft

Figure 4.16- Temperature Profile in 17200-ft well

after 5 hrs

97

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20.5 21 21.5 22 22.5 23 23.5 24 24.5D

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99

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101

The effect of varying the inlet temperature of the drilling fluid was also

studied. The well parameters detailed in Table 4.2 were simulated with an

inlet temperature into the drill pipe of 80 oF. Although the return temperature

coming up the annulus was reduced to 87 oF, the bottom-hole temperature

and pressure after 5 hours of circulation did not change significantly from the

values obtained with an inlet temperature of 120 oF. The results are shown in

Table 4.7 and in Figs. 4.25 to 4.27. The ECD profile in the well during

circulation also did not change appreciably from the case of 120 oF pipe inlet

temperature. This trend indicates that the inlet temperature over a certain

range does not play an important role in the overall wellbore heat transfer

mechanism compared to the geothermal gradient.




ECD (ppg) 24.6 24.9 -0. 3

Table 4.7 Well Simulation Results for Parameters Detailed in Table 4.2 with Inlet Fluid Temperature = 80 oF

102

0

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12000

14000

16000

18000

20000

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Temperature (oF)

Dep

th (f

t)


Circulation Rate- 400 bbl/hrCirculation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ftInlet Temp = 80 oF

Figure 4.25- Temperature Profile in 17200-ft well

after 5 hrs

0

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0 5000 20000 25000

Dep

th (f

t)

10000 15000



Circulation Rate- 400 bbl/hr Circulation time- 5 hrs Fluid- OBM Geothermal Grad- 0.02 oF/ft Inlet Temp = 80 oF


Figure 4.26- Annular Pressure Profile in 17200-ft

well after 5 hrs

103

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24 24.5 25 25.5 26 26.5 27 27.5

Equivalent Circulating Density (ppg)28 28.5

Dep

th (f

t)





The effects of the circulation rate on the bottom-hole pressure were

also studied. The well parameters in Table 4.2 were simulated with a

circulation rate of 300 bbl/hr. The results are shown in Table 4.8 and Figs.

4.28 to 4.30. The difference in the bottom-hole pressure estimated with

constant fluid properties and temperature-pressure dependent properties rose

to 297 psi. This could be due to the fact that the bottom-hole temperature

increases slightly at the lower rate and will thus result in increased expansion

of the drilling fluid and a greater reduction in the fluid density. The circulation

rate thus plays a great role in the temperature and pressure profiles that will

occur in a circulating well.

104

Table 4.8 Well Simulation Results for Parameters Detailed in Table 4.2 with Circulation Rate = 300 bbl/hr




ECD (ppg) 21.0 21.3 -0.3

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105

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106

Summary

A drilling hydraulic simulator called DDSimulator was developed using

Microsoft Visual Basic for Applications automated through Excel. This format

was chosen for ease of use and accessibility and access to Excel’s powerful

dynamic graphing capability. The simulator allows estimation of the

temperature profile in the wellbore during circulation and estimation of the

frictional pressure drop under high-temperature/high-pressure conditions. The

complete code for the DDS program is documented in the Appendix A.

High-temperature/high-pressure well conditions were simulated. It was

found that the bottom-pressure in the well is lower for the oil based mud that

was simulated when the temperature-pressure conditions prevalent in the

well-bore during circulation are taken into account. This indicates that the

effect of the fluid expansion due to temperature was more pronounced than

the effect of compression as a result of the increased pressure down-hole.

Temperature thus plays a more pronounced role in this particular case. This

is further confirmed by the further drop in bottom-hole pressure with

increasing geothermal gradient. The inlet temperature of the drilling fluid into

the drill-pipe was not found to have a significant effect on the bottom-hole

pressure even though it had an effect on the return temperature out of the

annulus. The circulation rate was found to play an important role in the

temperature profile that develops in a well during circulation.

107

Chapter 5

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

5.1 Summary

The objective of this study was to explore the effects of the

temperature and pressure conditions prevalent in high temperature/high

pressure wells on the equivalent circulating density of the drilling fluid and on

the bottom-hole pressure. The high temperature conditions cause expansion

of the drilling fluid while the high pressure conditions at deeper depths cause

compression. In the industry, these two counter effects were thought to

cancel each other resulting in constant surface fluid density throughout the

length of the well-bore during circulation. However, industry experience has

shown that predicted bottom-hole pressures assuming constant surface fluid

properties are often in error by hundreds of psi.

The above objective was achieved with the development of a simulator

called DDSimulator. This is a hydraulic simulator that computes the bottom-

hole pressure and ECD throughout the length of a circulating well-bore taking

into account the temperature and pressure conditions in the well-bore. The

program can compute the temperature and pressure profile in a circulating

well. The effects of temperature and pressure on the density and viscosity of

drilling fluids was studied in chapter 2, along with frictional pressure loss

108

estimation. Analytical and numerical methods for estimating the temperature

profile in a circulating well-bore were studied in chapter 3. The Crank-

Nicolson numerical discretizing scheme was employed in the DDSimulator for

the evaluation of the temperature profile in a circulating well. Taking the

temperature profile into account the program then estimates the frictional

pressure loss, equivalent circulating density, and bottom-hole pressure using

methods detailed in Chapter 2.

In the case of the oil based drilling fluid that was simulated, it was

found that the bottom-hole pressure estimated taking into account

temperature and pressure conditions, is lower than if the fluid properties are

taken to be independent of temperature and pressure. This indicates that the

temperature effect of fluid expansion is more pronounced than the

compression effect due to pressure. Thus, a reduction in the fluid density

occurs. It is also important to note that the increased temperature results in

lower fluid viscosity and thus lower frictional pressure drop.

5.2 Conclusions

Based on the simulations that were performed, the following

conclusions were drawn.

1. Temperature and pressure effects play an important role in the bottom-

hole pressure that will occur in deep hot wells.

2. Higher geothermal gradients lead to lower bottom-hole pressure.

109

3. The inlet pipe temperature does not have a significant effect on the

bottom-hole pressure.

4. Higher circulation rates result in lower bottom-hole temperature and

higher bottom-hole pressure.

5. The objectives of the study were achieved by developing the Dynamic

Density Simulator. The simulator allows evaluation of the bottom-hole

pressure and equivalent circulating density taking into account the

temperature and pressure conditions in the well-bore.

5.3 Recommendations

It is recommended that a similar set of circulating wellbore simulations as

detailed in this study be carried out before the commencement of drilling

operations where it is known that high pressure and temperature conditions

may be encountered. This will minize the occurrence of common drilling

problems such as

• Premature intake of formation fluid into the wellbore (kick).

• Formation damage

• Unnecessary trips

thereby reducing the total drilling cost.

The following areas have been identified for further improvement.

• Simulations should be carried out with more varieties of drilling fluids,

including synthetic oil based drilling fluids, and drilling fluids with

110

chemical additives such as surfactants, flocculants, and fluid loss

reducers. Some experimental work may be required, as data of the

volumetric behavior as well as rheological behavior of drilling fluid

components with respect to temperature and pressure are not

abundant in the literature.

• Simulations should be carried out under deep water conditions where

cold temperatures and multiple temperature gradients are

encountered.

111

NOMENCLATURE

Ao, A1, A2 = Empirically determined parameter in Sorrelle et al6 model

Af = Cross-sectional area

as = 0.8*10-4 oC-1, thermal expansivity of barite

Bo, B1, B2 = Empirically determined parameter in Sorrelle et al6 model

bs = -1.0*10-5 bar-1, compressibility of barite

D = Pipe diameter

De = Equivalent diameter

f = Friction factor

fx = volume fraction of component x

f vo, fvw, fvs, fvc = Fractional volume of oil, water, solid weighting material, and

chemical additives, respectively

h = height of fluid column, ft


L = Conduit length


P = pressure, psi

Pw = Wetted perimeter

P1, P2 = Pressure at reference and condition “2”

T 1, T2 = Temperature at reference and condition “2”

V = Total volume

Vo, Vw,Vs,Vc = Volume of oil, water, solids and chemical additives

112

Vx = volume of component x

W = Weight

YV = Yield value (lbf/100ft2)

∆Phydrostatic = Hydrostatic head of fluid column (psi)

∆Pfriction = Pressure drop due to friction in the drill string and annulus

(psi)

∆p = Frictional pressure loss

α = Ellis model parameter

γ& = shear rate

λ = time constant

µa = apparent viscosity

µ = viscosity

µo = low shear rate viscosity

µp = plastic viscosity

∞µ = viscosity at infinite shear

ρ = fluid density, lbm/gal (ppg)

ρο1, ρw1 = Density of oil and water at temperature T1 and pressure P1,

respectively

ρο2, ρw2 = Density of oil and water at temperature T2 and pressure P2,

respectively

ρo1, ρw1 = Density of oil and water phases at reference conditions (p1,

T1)

113

ρs, ρc = Density of solids content and chemical additives

ρecd = equivalent circulating density (lb/gal)

τ = shear stress

τo = yield stress

τ1/2 = shear stress @ µa = µ0/2

A1 to G1 = Empirical parameters in temperature/pressure dependent

diesel

viscosity equation

A2 to F2 = Empirical parameters in temperature/pressure dependent

diesel

density equation

A3,B3, C3 = Empirical parameters in temperature/pressure yield value

equation for oil based drilling fluid

114

REFERENCES

1. Davison, J.M., Clary, S., Saasen, A., Allouche, M., Bodin, D., Nguyen,

V.A.: “ Rheology of Various Drilling Fluid Systems Under Deepwater

Drilling Conditions and the Importance of Accurate Predictions of

Downhole Fluid Hydraulics”, SPE 56632, Houston, Oct 3-6, 1999.

2. Houwen, O.H., Geehan, T.: “ Rheology of Oil-Base Muds”, SPE 15416,

New Orleans, LA, 5-8 Oct, 1986.

3. Alderman, N.J., Gavignet, A., Guillot, D., Maitland, G.C.: “High-

Temperature, High-Pressure Rheology of Water-Base Muds”, SPE

18035, Houston, TX, 2-5 Oct, 1988.

4. Hoberock, L.L., Thomas, D.C., Nickens, H.V.: “Here’s How

Compressibility and Temperature Affect Bottom-Hole Mud Pressure”,

OGJ, Mar 22, 1982, p. 159.

5. Peters, E.J., Chenevert, M.E. and Zhang, C.: “A Model for Predicting

the Density of Oil-Based Muds at High Pressures and Temperatures”,

SPEDE (June 1990) 141-148; Trans., AIME, 289.

6. Sorelle, R.R., Jardiolin, R.A., Buckley, P., Barios, J.R.: “Mathematical

Field Model Predicts Downhole Density Changes in Static Drilling

Fluids”, SPE 11118, New Orleans, Sept 26-29, 1982.

7. Isambourg, P., Anfinsen, B.T., Marken, C.: “Volumetric Behavior of

Drilling Muds at High Pressure and High Temperature”, SPE 36830,

Milan, Italy, Oct 22-24, 1996.

115

8. Kutasov, I., and Sweetman, M.: “Method Predicts Equivalent Mud

Density”, OGJ, Sept 24, 2001, p. 57.

9. Babu D. R.: “Effects of P-ρ-T Behavior of Muds on Static Pressure

During Deep Well Drilling”, SPE 27419, SPEDC, June 1996, pp. 91-97.

10. McMordie Jr., W.C., Bland, R.G. and Hauser, J.M.: “Effect of

Temperature and Pressure on the Density of Drilling Fluids”, SPE

11114, New Orleans, Sept. 26-29, 1982.

11. Rommetveit, R., Bjorkevoll, K.S.: “Temperature and Pressure Effects

on Drilling Fluid Rheology and ECD in Very Deep Wells”, SPE 39282,

Bahrain, 23-25 Nov, 1997.

12. Baranthol, C., Alfenore, J., Cotterill, M.D., Poux-Guillaume, G.:

“Determination of Hydrostatic Pressure and Dynamic ECD by

Computer Models and Field Measurements on the Directional HPHT

Well 22130C-13”, SPE 29430, Amsterdam, 28 Feb-2 Mar, 1995.

13. Ramey, H.J., Jr: “Wellbore Heat Transimission,” JPT(April 1962) 427-

35

14. Holmes, C.S., Swift, S.C.: “Calculation of Circulating Mud

Temperatures,” JPT(May 1970) 670-74

15. Arnold, F.C.: “Temperature Profile During Heated Liquid Injection,” Int.

Comm. Heat Mass Transfer, Vol. 16, pp. 763-72.

16. Arnold, F.C.: “Temperature Variation in a Circulating Wellbore Fluid,”

Journal of Energy Resources, Vol. 112, pp. 79-83.

116

17. Kabir, C.S., Hasan, A.R., Kouba, G.E., Ameen, M.M.: “Determining

Circulating Fluid Temperature in Drilling, Workover, and Well-Control

Operations,” SPE 24581, Washington, DC, Oct 4-7, 1992.

18. Marshal, T.R., Lie, O.H.: “A Thermal Transient Model of Circulating

Wells: 1. Model Development,” SPE 24290, Stavanger, Norway, May

25-227, 1992.

19. Romero, J. and Touboul, E.: “Temperature Prediction for Deepwater

Wells: A Field Validated Methodology,” SPE 49056, New Orleans,

Sept. 27-30, 1998.

20. Chen, Z., Novotny, J.: “Accurate Prediction Wellbore Transient

Temperature Profile Under Mulitple Temperature Gradients: Finite

Difference Approach and Case History,” SPE 84583, Denver, Oct 5-8,

2003.

21. Kutasov, I.M.: “Water FV Factors at Higher Pressure and

Temperatures,” Oil & Gas J. (Mar, 20, 1989) 102-104.

22. Politte, M.D.: “Invert Oil Mud Rheology as a Function of Temperature,”

SPE13458, New Orleans, Mar 6-8, 1985.

23. Tragasser, A.F., Crawford, P.B., Horace, R.: “A Method for Calculating

Circulating Temperatures,” Journal of Petroleum Technology, Vol. 19,

pp. 1507-1512, 1967.

24. Raymond, L.R.: “Temperature Distribution in a Circulating Drilling

Fluid,” Journal of Petroleum Technology, Vol. 21, pp. 333-341, 1969.

117

25. Muneer, T., Kubie, J., and Grassie, T.:Heat Transfer-A Problem

Solving Approach, Taylor & Francis Group, New York and London,

2003; pg 231.

26. Ozisik, M.N.:Finite Difference Methods in Heat Transfer, CRC Press,

Boca Raton, AnnArbor, London, and Tokyo, 1994; pg 99-137.

27. Kraus, A.D., Aziz, A., and Welty, J.:Extended Surface Heat Transfer,

John Wiley & Sons, 2001; pg 165-172, 181-190.

28. Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids, Clarendon

Press, Oxford, 1986; pg 220-224.

29. Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G.: Fundamentals of

Momentum Heat and Mass Transfer, John Wiley & Sons, 2001.

30. Dusinbere, G.M.:Heat Transfer Calculations by Finite Differences,

International Textbook Company, Scranton, Pennsylvania, 1961; pg 8-

23, 103-106.

118

APPENDIX Code for DDSimulator Program

119

Option Explicit 'This is a temperature profile object in which the methods to compute the pipe and annular 'temperature profiles are contained Dim Tax As WellProfile 'The following list conatins the depth coordinates at which temperature will be computed Dim Depth(200) As Double Private Sub ConstPropPressureDrop_Click() 'This sub computes the frictional pressure drop in the pipe and annulus with constant 'fluid properties Dim PlasticP As Double Dim PlasticA As Double Dim YieldP As Double Dim YieldA As Double Dim vP As Double Dim vA As Double Dim muP As Double Dim muA As Double Dim ReP As Double Dim ReA As Double Dim DPP As Double Dim DPA As Double Dim DppTotal As Double Dim DpaTotal As Double Dim i As Integer Dim imax As Integer imax = Tax.iTotal DppTotal = 0 DpaTotal = 0 'Compute Frictional Pressure Drop For i = 0 To imax If i > 0 Then 'compute plastic viscosity in cp PlasticP = Tax.RefMudPlasticViscosity PlasticA = Tax.RefMudPlasticViscosity 'Compute yield point in lbf/100ft^2 YieldP = Tax.RefMudYieldValue

120

YieldA = Tax.RefMudYieldValue 'Compute velocity in pipe and annulus vP = 4 * Tax.mRate / (7.48 * 3.142 * ((2 * Tax.rpi) ^ 2) * Tax.RefMudDensity) vA = 4 * Tax.mRate / (7.48 * 3.142 * (Tax.de ^ 2) * Tax.RefMudDensity) 'Compute apparent viscosity in pipe and annulus muP = PlasticP + (6.66 * YieldP * 2 * Tax.rp / vP) muA = PlasticA + (5 * YieldA * Tax.de / vA) 'Compute Reynold's # in pipe and annulus ReP = Tax.ReynoldsNum(Tax.RefMudDensity, vP, (2 * Tax.rp), muP) ReA = Tax.ReynoldsNum(Tax.RefMudDensity, vA, Tax.de, muA) 'Compute frictional pressure drop DPP = Tax.PressureDrop(ReP, (2 * Tax.rp), 0.01, Tax.DeltaZ, Tax.RefMudDensity, vP) DPA = Tax.PressureDrop(ReA, Tax.de, 0.03, Tax.DeltaZ, Tax.RefMudDensity, vA) DppTotal = DppTotal + DPP DpaTotal = DpaTotal + DPA End If Worksheets("AnnulusTemperature").Cells((i + 2), 12).Value = DppTotal Worksheets("AnnulusTemperature").Cells((i + 2), 13).Value = DpaTotal Next i End Sub Public Sub LoadProperties_Click() Set Tax = New WellProfile Tax.AnalyticalConstants End Sub Private Sub ComputeTempProf_Click() Dim i As Integer 'The "Interval" refers to the distance between depths at which temperature will be computed

121

Dim Interval As Double Interval = Tax.TVD / 200 For i = 0 To 200 Depth(i) = i * Interval Tax.AnalyticalComputeTemp Depth(i) Cells(i + 16, 8).Value = Tax.TPipe Cells(i + 16, 9).Value = Tax.TAnnulus If i < 300 Then End If Next i End Sub Private Sub NumericalTemperature_Click() 'Note that terms bearing an "N" at the end signify data at the time step that 'is currently being evaluated. Dim FormTemperature() As Double Dim PipeTemperature() As Double Dim AnnTemperature() As Double Dim FormTemperatureN() As Double Dim PipeTemperatureN() As Double Dim AnnTemperatureN() As Double 'These matrices store the pressure profile with the pipe and annulus Dim PipePressure() As Double Dim AnnPressure() As Double Dim PipePressureN() As Double Dim AnnPressureN() As Double 'Dim Uaaa() As Double 'Dim Uppp() As Double 'mud density in the pipe and annulus Dim rhoPipe() As Double Dim rhoAnnulus() As Double 'i - (depth), j - (radius), n - (time)

122

Dim i As Integer Dim j As Integer Dim n As Integer Dim imax As Integer Dim jMax As Integer Dim nMax As Integer 'This array is used to store the old pipe temperature values Dim store1() As Double Dim store2() As Double 'heat transfer coefficient for inner pipe surface Dim hi As Double 'error between guess and solution Dim err As Double Dim check1 As Double Dim check2 As Double Dim check3 As Double 'Terms used to evaluate the frictional pressure loss Dim PlasticP As Double Dim PlasticA As Double Dim YieldP As Double Dim YieldA As Double Dim vP As Double Dim vA As Double Dim muP As Double Dim muA As Double Dim ReP As Double Dim ReA As Double Dim DPP As Double Dim DPA As Double Dim DppTotal As Double Dim DpaTotal As Double imax = Tax.iTotal jMax = Tax.jTotal nMax = Tax.nTotal ReDim FormTemperature(imax, jMax) ReDim PipeTemperature(imax) ReDim AnnTemperature(imax) ReDim PipePressure(imax)

123

ReDim AnnPressure(imax) ReDim FormTemperatureN(imax, jMax) ReDim PipeTemperatureN(imax) ReDim AnnTemperatureN(imax) ReDim PipePressureN(imax) ReDim AnnPressureN(imax) ReDim rhoPipe(imax) ReDim rhoAnnulus(imax) 'ReDim Uaaa(iMax) 'ReDim Uppp(iMax) ReDim store1(imax) ReDim store2(imax) 'Worksheets.Add.Name = "PipeTemperature" 'Worksheets.Add.Name = "AnnulusTemperature" 'Worksheets.Add.Name = "FormationTemperature" 'Set initial conditions in the formation and annulus Tax.InitializeGrid FormTemperature(), PipeTemperature(), AnnTemperature(), PipePressure(), AnnPressure(), rhoPipe(), rhoAnnulus(), imax, jMax 'The following code computes temperature in the formation and wellbore For n = 1 To nMax 'Set initial guess for Ta(i,n+1), Ua(i,n+1), Tp(i,n+1), Up(i,n+1), Pp(i,n+1), and Pa(i,n+1) For i = 0 To imax AnnTemperatureN(i) = AnnTemperature(i) FormTemperatureN(i, 0) = FormTemperature(i, 0) 'PipePressureN(i) = PipePressure(i) AnnPressureN(i) = AnnPressure(i) 'rhoPipeN(i) = rhoPipe(i) 'rhoAnnulusN(i) = rhoAnnulus(i) Next i Do 'Evaluate the pipe Tax.EvaluatePipe FormTemperature(), FormTemperatureN(), PipeTemperature() _ , PipeTemperatureN(), PipePressure(), PipePressureN(), AnnTemperature() _

124

, AnnTemperatureN(), AnnPressure(), AnnPressureN(), imax 'Re-evaluate Pp(i,n+1) with the newly obtained Tp(i,n+1) For i = 0 To imax If i = 0 Then PipePressureN(i) = 14.7 Else rhoPipe(i - 1) = Tax.MudDensity(PipeTemperatureN(i - 1), PipePressureN(i - 1)) PipePressureN(i) = PipePressureN(i - 1) + 0.052 * rhoPipe(i - 1) * Tax.DeltaZ End If Next i 'Evaluate the Annulus Tax.EvaluateAnnulus FormTemperature(), FormTemperatureN(), PipeTemperature() _ , PipeTemperatureN(), AnnTemperature(), AnnTemperatureN(), PipePressure() _ , PipePressureN(), AnnPressure(), AnnPressureN(), imax, store1() 'Re-evaluate Pa(i,n+1) with the newly obtained Ta(i,n+1) For i = 0 To imax If i = 0 Then AnnPressureN(i) = 14.7 Else rhoAnnulus(i - 1) = Tax.MudDensity(AnnTemperatureN(i - 1), AnnPressureN(i - 1)) AnnPressureN(i) = AnnPressureN(i - 1) + 0.052 * rhoAnnulus(i - 1) * Tax.DeltaZ End If Next i 'Evaluate the formation Tax.EvaluateFormation FormTemperature(), FormTemperatureN(), AnnTemperature() _ , AnnTemperatureN(), AnnPressure(), AnnPressureN(), imax, jMax, store2() err = 0 'check for convergence

125

For i = 0 To imax check1 = ((AnnTemperatureN(i) - store1(i)) ^ 2) ^ (1 / 2) check2 = ((FormTemperatureN(i, 0) - store2(i)) ^ 2) ^ (1 / 2) check3 = ((PipeTemperatureN(imax) - AnnTemperatureN(imax)) ^ 2) ^ (1 / 2) If check1 > err Then err = check1 End If If check2 > err Then err = check2 End If If check3 > err Then 'err = check3 End If Next i Loop Until err <= 0.05 'update the temperature in the formation and wellbore For i = 0 To imax For j = 0 To jMax FormTemperature(i, j) = FormTemperatureN(i, j) Next j AnnTemperature(i) = AnnTemperatureN(i) PipeTemperature(i) = PipeTemperatureN(i) PipePressure(i) = PipePressureN(i) AnnPressure(i) = AnnPressureN(i) Next i 'Display results in formation For i = 0 To imax For j = 0 To jMax Worksheets("FormationTemperature").Cells((i + 2), (j + 2)).Value = FormTemperatureN(i, j) Next j Next i Next n 'Display results in wellbore For i = 0 To imax Worksheets("AnnulusTemperature").Cells((i + 2), 3).Value = AnnTemperatureN(i) Worksheets("AnnulusTemperature").Cells((i + 2), 2).Value = PipeTemperatureN(i)

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Next i DppTotal = 0 DpaTotal = 0 'Compute Frictional Pressure Drop For i = 0 To imax If i > 0 Then 'compute plastic viscosity in cp PlasticP = Tax.MudPlasticViscosity(PipeTemperatureN(i), PipePressureN(i)) PlasticA = Tax.MudPlasticViscosity(AnnTemperatureN(i), AnnPressureN(i)) 'Compute yield point in lbf/100ft^2 YieldP = Tax.MudYieldValue(PipeTemperatureN(i), PipePressureN(i)) YieldA = Tax.MudYieldValue(AnnTemperatureN(i), AnnPressureN(i)) 'Compute velocity in pipe and annulus vP = 4 * Tax.mRate / (7.48 * 3.142 * ((2 * Tax.rpi) ^ 2) * rhoPipe(i - 1)) vA = 4 * Tax.mRate / (7.48 * 3.142 * (Tax.de ^ 2) * rhoAnnulus(i - 1)) 'Compute apparent viscosity in pipe and annulus muP = PlasticP + (6.66 * YieldP * 2 * Tax.rp / vP) muA = PlasticA + (5 * YieldA * Tax.de / vA) 'Compute Reynold's # in pipe and annulus ReP = Tax.ReynoldsNum(rhoPipe(i - 1), vP, (2 * Tax.rp), muP) ReA = Tax.ReynoldsNum(rhoAnnulus(i - 1), vA, Tax.de, muA) 'Compute frictional pressure drop DPP = Tax.PressureDrop(ReP, (2 * Tax.rp), 0.01, Tax.DeltaZ, rhoPipe(i - 1), vP) DPA = Tax.PressureDrop(ReA, Tax.de, 0.03, Tax.DeltaZ, rhoAnnulus(i - 1), vA) DppTotal = DppTotal + DPP DpaTotal = DpaTotal + DPA End If Worksheets("AnnulusTemperature").Cells((i + 2), 4).Value = rhoPipe(i) Worksheets("AnnulusTemperature").Cells((i + 2), 5).Value = rhoAnnulus(i) Worksheets("AnnulusTemperature").Cells((i + 2), 6).Value = PipePressureN(i)

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Worksheets("AnnulusTemperature").Cells((i + 2), 7).Value = AnnPressureN(i) Worksheets("AnnulusTemperature").Cells((i + 2), 8).Value = DppTotal Worksheets("AnnulusTemperature").Cells((i + 2), 9).Value = DpaTotal Next i End Sub The following Code details the methods and characteristics of a well-bore profile object. Option Explicit 'Formation Properties Public kF As Double Public cF As Double Public FormationDensity As Double Public alpha As Double 'The maximum formation radius that will be considered (ft) 'a.k.a r-infinity Public rMax As Double 'Undisturbed formation temperature at the maximum depth considered Public Tmax As Double 'Mud Properties (densities in lb/gal) Public RefMudDensity As Double Public RefMudPlasticViscosity As Double Public RefMudYieldValue As Double Public OilFraction As Double Public WaterFraction As Double Public mRate As Double Public cfl As Double Public kfl As Double 'Reference mud conditions Public RefTemp As Double Public RefPress As Double 'annular radius Public ra As Double

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'outer pipe radius Public rp As Double 'inner pipe radius Public rpi As Double 'heat conduction coefficient of pipewall Public kp As Double 'heat conduction coefficient of cement Public kcement As Double 'equivalent diameter of the annulus Public de As Double 'Heat Transfer coefficients across pipewall and across annulus/formation interface 'in Btu/(hour-ft2-oF) Public Ua As Double Public Up As Double Public time As Double Public beta As Double Public gG As Double Public TVD As Double Public Tdiff As Double Public TFs As Double Public Tps As Double 'Analytical parameters Public DTime As Double Public DTimeFunc As Double Public sigma As Double Public gammaOne As Double Public gammaTwo As Double Public COne As Double Public CTwo As Double 'Temperature in the annulus and pipe (oF) terms used in analytical analysis Public TAnnulus As Double Public TPipe As Double 'time, depth and radius intervals Public DeltaT As Double Public DeltaZ As Double Public DeltaR As Double 'terms used in numerical analysis- see class initialize for definitions Public ar As Double

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Public az As Double 'Total number of depth, radial, and time steps Public iTotal As Double Public jTotal As Double Public nTotal As Double Private Sub Class_Initialize() mRate = Range("B20").Value cfl = Range("B11").Value kfl = Range("B10").Value kF = Range("B14").Value cF = Range("B15").Value FormationDensity = Range("B16").Value ra = Range("B5").Value rp = Range("B4").Value rpi = Range("B34").Value kp = Range("B25").Value Ua = Range("B24").Value Up = Range("B23").Value rMax = Range("B22").Value alpha = Range("B21").Value time = Val(Application.Worksheets("Sheet1").CirculationTime.Value) gG = Range("B18").Value TVD = Range("B3").Value TFs = Range("B17").Value Tps = Range("B7").Value kcement = 0.025 RefTemp = Range("B27").Value RefPress = Range("B28").Value RefMudDensity = Range("B29").Value RefMudPlasticViscosity = Range("B30").Value RefMudYieldValue = Range("B31").Value OilFraction = Range("B32").Value WaterFraction = Range("B33").Value de = 2 * (ra - rp) 'The following constants are declared for convenience beta = mRate * cfl / (2 * Pie * rpi * Up) Tdiff = TFs - Tps - beta * gG

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'Tdiff = Tps - TFs + beta * gG Tmax = TFs + gG * TVD 'numerical intervals DeltaT = 0.05 DeltaZ = TVD / 200 DeltaR = rMax / 100 'terms used in numerical analysis ar = DeltaT / (DeltaR ^ 2) az = DeltaT / (DeltaZ ^ 2) 'compute iTotal If ((TVD / DeltaZ) - Int(TVD / DeltaZ)) < 0.5 Then iTotal = Int(TVD / DeltaZ) Else nTotal = Int(TVD / DeltaZ) + 1 End If 'compute jTotal If ((rMax / DeltaR) - Int(rMax / DeltaR)) < 0.5 Then jTotal = Int(rMax / DeltaR) Else jTotal = Int(rMax / DeltaR) + 1 End If 'compute nTotal If ((time / DeltaT) - Int(time / DeltaT)) < 0.5 Then nTotal = Int(time / DeltaT) Else nTotal = Int(time / DeltaT) + 1 End If End Sub Private Function ExpInt(X As Double) As Double Rem C Rem DOUBLE PRECISION FUNCTION EXPIN(X) Rem C Rem C Rem C EXPIN is the function which computes Rem C the exponential integral. Rem C Rem C E1(x)=SIGMA(x---->infinity)(exp(-t)/t)dt Rem C Rem C

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Rem IMPLICIT REAL*8(A-H,O-Z) Rem COMMON AS,BS,S,ISTEP Rem C IF (X.LE.0) WRITE(10,10) X Dim Sign As Double Dim XX As Double Dim A1 As Double Dim A As Double Dim B As Double If (X <= 0#) Then Sign = -1# Else Sign = 1# Rem GoTo 2 End If XX = Abs(X) Rem C 10 FORMAT(//,2X,'X must be a positive number X= ',D19.8) If ((XX >= 0#) And (XX <= 1#)) Then GoTo 1 End If Rem C Rem C 1<=X<infinity. 5.1.56 Abramowitz & Stegun. Rem C A1 = XX ^ 4 A = A1 + 8.5733287401 * XX ^ 3 A = A + 18.059016973 * XX ^ 2 + 8.6347608925 * XX A = A + 0.2677737343 B = A1 + 9.5733223454 * XX ^ 3 B = B + 25.6329561486 * XX ^ 2 + 21.0996530827 * XX B = B + 3.9584969228 ExpInt = A / B / XX / Exp(XX) GoTo 2 Rem Return Rem C Rem C 0<=X<=1 5.1.53 Abramowitz & Stegun. Rem C 1 A = -0.57721566 + 0.99999193 * XX - 0.24991055 * XX ^ 2 A = A + 0.05519968 * XX ^ 3 - 0.00976004 * XX ^ 4 A = A + 0.00107857 * XX ^ 5

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ExpInt = A - Log(XX) / Log(Exp(1#)) Rem Return Rem End Rem C 2 ExpInt = Sign * ExpInt End Function Public Function Pie() As Double 'This function computes Pi Pie = Application.WorksheetFunction.Pi End Function Public Function Absolute(X As Double) As Double 'This function returns the absolute value Absolute = Application.WorksheetFunction.Abs(X) End Function Public Function NatLg(X As Double) As Double NatLg = Application.WorksheetFunction.Ln(X) End Function '---------------------------TRIDIAGONAL ALGORITHM Public Sub ThomasAlgorithm(A() As Double, B() As Double, C() As Double, D() As Double, X() As Double, n As Integer) Dim i As Integer For i = 1 To n B(i) = B(i) - A(i) * C(i - 1) / B(i - 1) D(i) = D(i) - A(i) * D(i - 1) / B(i - 1) Next i ' Back Substitution X(n) = D(n) / B(n) For i = n - 1 To 0 Step -1 X(i) = (D(i) - C(i) * X(i + 1)) / B(i) Next i End Sub

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Public Sub AnalyticalConstants() 'The following code will calculate the constants "sigma", "gammaOne", '"gammaTwo", "COne", and "CTwo" 'mRate = mass flow rate, cfl = fluid heat capacity, kF = formation 'conductivity, ra = annular radius, Ua = annular heat transfer coeff, 'aplha = k/(rho*cF), time = length of time of fluid circulation, 'beta = m*cfl/(2*Pi*rp*Up), gG = formatin temp grad. , TVD = Total 'vertical depth, Tdiff = (Tfs - Tps - beta*gG) 'Calculate dimensionless time "tD" 'DTime = (ra ^ 2) / (4 * alpha * time) DTime = (-1 * ra ^ 2) / (4 * alpha * time) 'Calculate dimensionless time function "f(tD)" DTimeFunc = 0.5 * ExpInt(DTime) / Exp(DTime) sigma = mRate * cfl * ((kF + ra * Ua * DTimeFunc) / (2 * Pie * ra * Ua * kF)) 'Compute "gammaOne" and "gammaTwo" gammaOne = (beta + ((beta ^ 2) + 4 * sigma * beta) ^ (1 / 2)) / (2 * sigma * beta) gammaTwo = (beta - ((beta ^ 2) + 4 * sigma * beta) ^ (1 / 2)) / (2 * sigma * beta) 'Compute "COne" and "CTwo" COne = (gG - (Exp(gammaTwo * TVD) * gammaTwo * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) CTwo = (-1 * gG + (Exp(gammaOne * TVD) * gammaOne * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) 'COne = (gG + (Exp(gammaTwo * TVD) * gammaTwo * Tdiff)) / (gammaTwo * Exp(gammaTwo * TVD) - gammaOne * Exp(gammaOne * TVD)) 'CTwo = (gG + (Exp(gammaOne * TVD) * gammaOne * Tdiff)) / (-gammaTwo * Exp(gammaTwo * TVD) + gammaOne * Exp(gammaOne * TVD)) End Sub Public Sub AnalyticalComputeTemp(z As Double) 'Computes temperature in the drill pipe and annulus at a particular depth TAnnulus = AnalyticalAnnularTemp(z) TPipe = AnalyticalPipeTemp(z)

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End Sub Private Function AnalyticalAnnularTemp(Depth As Double) As Double 'Computes the temperature in the annulus at a particular depth AnalyticalAnnularTemp = (1 + beta * gammaOne) * COne * Exp(gammaOne * Depth) + (1 + beta * gammaTwo) * CTwo * Exp(gammaTwo * Depth) + gG * Depth + TFs End Function Private Function AnalyticalPipeTemp(Depth As Double) As Double 'Computes the temperature inside the drill pipe at a particular depth AnalyticalPipeTemp = COne * Exp(gammaOne * Depth) + CTwo * Exp(gammaTwo * Depth) + gG * Depth + TFs - beta * gG End Function Public Function ReynoldsNum(rho As Double, v As Double, D As Double, mu As Double) As Double 'Computes the Reynold's number 'density (rho) is in lb/gal, velocity (v) is in ft/hr, equivalent diameter (D) is in ft 'apparent viscosity (mu) is in lb/ft-hr ReynoldsNum = (7.48 * rho * v * D) / mu End Function Private Function NusseltNum(Re As Double, Pr As Double, D As Double, L As Double) As Double 'This function computes the Nusselt number 'Re & Pr - dimensionless 'D - ft 'L - ft If L = 0 Then L = DeltaZ / 2 End If If Re < 2300 And L / D < (8 / (Re / Pr)) Then 'Seider and Tate(1936) correlation - Laminar NusseltNum = 1.86 * (Re * Pr * (D / L)) ^ (1 / 3) ElseIf Re < 2300 And L / D > (8 / (Re / Pr)) Then

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NusseltNum = 6.49 ElseIf ((2300 <= Re) And (Re <= 10000)) Then 'Hausen(1943) correlation - Transition NusseltNum = 0.116 * ((Re ^ (2 / 3)) - 125) * (Pr ^ (1 / 3)) * (1 + (D / L) ^ (2 / 3)) ElseIf Re > 10000 Then 'Seider & Tate(1936) correlation - Turbulent NusseltNum = 0.027 * (Re ^ 0.8) * (Pr ^ (1 / 3)) End If End Function Public Function FrictionFactor(Re, Roughness, Diameter) As Double 'The friction factor is computed using the Swamme and Jain correlation If Re < 2300 Then FrictionFactor = 64 / Re Else FrictionFactor = 1.325 / (NatLg((Roughness / Diameter) / 3.7) + (5.74 / (Re ^ 0.9))) ^ 2 End If End Function Public Function PressureDrop(Re As Double, Diameter As Double, Roughness As Double _ , L As Double, rho As Double, v As Double) As Double 'Re- Reynold's # 'D - diameter in question (ft) 'e - Pipe roughness (ft) Dim F As Double If Re < 2100 Then F = 64 / Re Else F = 1.325 / (NatLg((Roughness / Diameter) / 3.7) + (5.74 / (Re ^ 0.9))) ^ 2 'f = (0.79 * NatLg(Re) - 1.64) ^ (-2) End If PressureDrop = F * rho * (L / Diameter) * (v ^ 2) * 7.48 / (2 * 32.174 * (3600 ^ 2) * (12 ^ 2))

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End Function Private Function WaterDensity(T As Double, P As Double) As Double 'This function computes the water density at the given temperature and pressure 'The correlation used in this function was obtained from Sorelle (1982) 'This function gives water density in lb/gal 'P - psi, T - oF WaterDensity = 8.63186 + (-3.31977 * 10 ^ -3) * T + (2.3717 * 10 ^ -5) * P End Function Private Function OilDensity(T As Double, P As Double) As Double 'This function computes the oil density at the given temperature and pressure 'The correlation used in this function was obtained from Politte (1985) 'This function gives density in (g/cm^3) 'P - psi, T - oF OilDensity = 0.8807 + 1.5235 * (10 ^ -9) * P * T + 1.2806 * (10 ^ -6) * P _ + 1.0719 * (10 ^ -10) * (P ^ 2) + (-0.00036) * T _ + (-5.167 * 10 ^ -8) * T ^ 2 End Function Private Function OilViscosity(T As Double, P As Double) As Double 'This function computes the oil phase viscosity at the given temperature and pressure in "cp" 'P - psi, T - oF Dim A As Double Dim B As Double Dim C As Double Dim D As Double Dim E As Double Dim F As Double Dim G As Double A = -23.1888 B = -0.00148 C = -0.9501 D = -1.9776 * 10 ^ -8 E = 3.3416 * 10 ^ -5 F = 14.6767 G = 10.9973

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OilViscosity = P * ((T * P) ^ C) * 10 ^ (A + B * T + D * T * P + E * P + F * OilDensity(T, P) + G / OilDensity(T, P)) End Function Public Function MudDensity(T As Double, P As Double) As Double 'This function Computes the mud density in lb/gal at the given temperature 'and pressure using the compositional model 'P - psi, T - oF MudDensity = RefMudDensity / (1 + OilFraction * ((OilDensity(RefTemp, RefPress) / (OilDensity(T, P))) - 1) + WaterFraction * ((WaterDensity(RefTemp, RefPress) / WaterDensity(T, P)) - 1)) End Function Public Function MudPlasticViscosity(T As Double, P As Double) As Double 'This function computes the plastic viscosity of the mud at the given temperature and pressure 'in centipoise (cp) 'P - psi, T - oF MudPlasticViscosity = RefMudPlasticViscosity * OilViscosity(T, P) / OilViscosity(RefTemp, RefPress) End Function Public Function MudYieldValue(T As Double, P As Double) As Double 'This function computes the yield value at the given temperature and pressure 'in lbf/100ft^2 'P - psi, T - oF Dim A As Double Dim B As Double Dim C As Double A = -0.186 B = 145.054 C = -3410.322 If T >= 90 Then MudYieldValue = RefMudYieldValue * (A + B * (T ^ -1) + C * (T ^ -2)) / (A + B * (RefTemp ^ -1) + C * (RefTemp ^ -2)) Else MudYieldValue = RefMudYieldValue

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End If End Function Public Function ConvectiveHeatTransCoeff(T As Double, P As Double, L As Double _ , D1 As Double, D2 As Double) As Double 'This function computes the heat transfer coefficient 'T - oF, P - psi, L - ft, D - ft Dim rho As Double Dim mu As Double Dim velocity As Double Dim Re As Double Dim Pr As Double Dim Nu As Double 'Compute fluid density in lb/gal 'rho = MudDensity(T, P) rho = RefMudDensity 'Compute fluid viscosity in lb/ft-hr 'mu = 2.42 * MudPlasticViscosity(T, P) mu = 2.42 * RefMudPlasticViscosity 'Compute fluid velocity in ft/hr velocity = 4 * mRate / (7.48 * 3.142 * (D1 ^ 2) * rho) 'Compute Reynold's # Re = ReynoldsNum(rho, velocity, D1, mu) 'Compute Prandtl # Pr = mu * cfl / kfl 'Compute Nusselt # Nu = NusseltNum(Re, Pr, D1, L) 'Compute heat transfer coefficient ConvectiveHeatTransCoeff = Nu * kfl / D2 End Function Public Sub InitializeGrid(FormTemp() As Double, PipeTemp() As Double, AnnTemp() As Double, PipePress() As Double, Annpress() As Double, rhoP() As Double, rhoA() As Double, imax As Integer, jMax As Integer)

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Dim i As Integer Dim j As Integer Dim hi As Double 'Set initial conditions in the formation and annulus i.e. n = 0 PipePress(0) = 14.7 Annpress(0) = 14.7 For i = 0 To imax For j = 0 To jMax FormTemp(i, j) = TFs + gG * i * DeltaZ Next j PipeTemp(i) = TFs + gG * i * DeltaZ AnnTemp(i) = TFs + gG * i * DeltaZ 'Specific well initial conditions 'PipeTemp(i) = 134.7 + 0.0047 * i * DeltaZ 'AnnTemp(i) = 134.7 + 0.0047 * i * DeltaZ 'Set the initial hydrostatic pressure profile in the wellbore If i > 0 Then rhoP(i - 1) = MudDensity(PipeTemp(i - 1), PipePress(i - 1)) rhoA(i - 1) = MudDensity(AnnTemp(i - 1), Annpress(i - 1)) PipePress(i) = PipePress(i - 1) + 0.052 * rhoP(i - 1) * DeltaZ Annpress(i) = Annpress(i - 1) + 0.052 * rhoA(i - 1) * DeltaZ End If Next i End Sub Public Sub EvaluateFormation(FormTemp() As Double, FormTempN() As Double _ , AnnTemp() As Double, AnnTempN() As Double, Annpress() As Double, AnnpressN() As Double _ , imax As Integer, jMax As Integer, store() As Double) 'This sub evaluates temperature in the formation and at the formation boundary Dim A() As Double Dim B() As Double Dim C() As Double

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Dim D() As Double Dim X() As Double Dim E() As Double Dim F As Double Dim G() As Double Dim H As Double Dim K() As Double Dim L As Double Dim Q As Double Dim Uaa As Double Dim i As Integer Dim j As Integer ReDim A(jMax) ReDim B(jMax) ReDim C(jMax) ReDim D(jMax) ReDim X(jMax) ReDim E(jMax) ReDim G(jMax) ReDim K(jMax) 'Store the old guess for Tf(i,0,n+1) For i = 0 To imax store(i) = FormTempN(i, 0) Next i F = 1 + alpha * ar L = 1 - alpha * ar H = 1 + 2 * alpha * ar Q = 1 - 2 * alpha * ar For i = 0 To imax Uaa = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((100 - i) * DeltaZ), de _ , de) 'This is for a specific cased hole with cement Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _

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+ (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'Uaa = Ua 'construct matrix equation for formation at particular depth coordinate i For j = 0 To jMax If j = 0 Then A(j) = 0 B(j) = H + alpha * ar * 2 * DeltaR * Uaa / kF C(j) = -2 * alpha * ar D(j) = (alpha * ar * 2 * DeltaR * Uaa / kF) * AnnTemp(i) _ + (Q - alpha * ar * 2 * DeltaR * Uaa / kF) * FormTemp(i, j) _ + (2 * alpha * ar) * FormTemp(i, (j + 1)) _ + (alpha * ar * 2 * DeltaR * Uaa / kF) * AnnTempN(i) ElseIf j = jMax Then 'A(j) = 0 'B(j) = 1 'C(j) = 0 'D(j) = TFs + (gG * i * DeltaZ) E(j) = ar * alpha * ((1 / (4 * j)) - 1 / 2) G(j) = ar * alpha * (-(1 / (4 * j)) - 1 / 2) K(j) = ar * alpha * (-(1 / (4 * j)) + 1 / 2) A(j) = E(j) B(j) = F C(j) = 0 D(j) = K(j) * FormTemp(i, (j - 1)) + L * FormTemp(i, j) _ + (-G(j)) * (TFs + (gG * i * DeltaZ)) - G(j) * (TFs + (gG * i * DeltaZ)) Else E(j) = ar * alpha * ((1 / (4 * j)) - 1 / 2) G(j) = ar * alpha * (-(1 / (4 * j)) - 1 / 2) K(j) = ar * alpha * (-(1 / (4 * j)) + 1 / 2) A(j) = E(j) B(j) = F C(j) = G(j) D(j) = K(j) * FormTemp(i, (j - 1)) + L * FormTemp(i, j) _ + (-G(j)) * FormTemp(i, (j + 1)) End If

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Next j ThomasAlgorithm A, B, C, D, X, jMax 'update the temperature in the formation grid For j = 0 To jMax FormTempN(i, j) = X(j) Next j Next i End Sub Public Sub EvaluateAnnulus(FormTemp() As Double, FormTempN() As Double _ , PipeTemp() As Double, PipeTempN() As Double, AnnTemp() As Double, AnnTempN() As Double _ , PipePress() As Double, PipePressN() As Double, Annpress() As Double _ , AnnpressN() As Double, imax As Integer, store() As Double) Dim A() As Double Dim B() As Double Dim C() As Double Dim D() As Double Dim X() As Double Dim E As Double Dim F As Double Dim G As Double Dim H As Double Dim L As Double Dim M As Double Dim i As Integer Dim UaaN As Double Dim UppN As Double Dim hoN As Double Dim hiN As Double Dim Uaa As Double Dim Upp As Double Dim ho As Double Dim hi As Double 'These parameters will be used to compute the bottom-hole

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'annular temperature Dim Re As Double Dim mi As Double Dim fa As Double Dim so As Double Dim theta As Double ReDim A(imax) ReDim B(imax) ReDim C(imax) ReDim D(imax) ReDim X(imax) 'Store the old guess for Ta(i,n+1) For i = 0 To imax store(i) = AnnTempN(i) Next i E = mRate * cfl / (2 * DeltaZ) 'The following computes the bottom-hole annular temperature theta = 2 / 3 UaaN = ConvectiveHeatTransCoeff(AnnTempN(imax), AnnpressN(imax), ((200 - imax) * DeltaZ) _ , de, de) 'This is for a specific cased hole UaaN = ((1 / UaaN) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'UaaN = Ua Re = mRate * cfl mi = Pie * ra * UaaN * DeltaZ fa = 7.48 * MudDensity(AnnTemp(imax), Annpress(imax)) * Pie * (ra ^ 2) * DeltaZ * cfl _

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/ (2 * DeltaT) 'fa = 7.48 * RefMudDensity * Pie * (ra ^ 2) * DeltaZ * cfl / (2 * DeltaT) so = Re * theta * PipeTempN(imax - 1) + Re * (1 - theta) * PipeTemp(imax - 1) _ + mi * theta * FormTempN(imax, 0) + mi * (1 - theta) * FormTemp(imax, 0) _ + (-Re * (1 - theta) - mi * (1 - theta) + fa) * AnnTemp(imax) AnnTempN(imax) = so / (Re * theta + mi * theta + fa) For i = 0 To (imax - 1) 'These are the heat transfer coefficients at the current time-step UaaN = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((200 - i) * DeltaZ) _ , de, de) 'This is for a specific cased hole UaaN = ((1 / UaaN) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) hoN = ConvectiveHeatTransCoeff(AnnTempN(i), AnnpressN(i), ((100 - i) * DeltaZ) _ , de, de) hiN = ConvectiveHeatTransCoeff(PipeTempN(i), PipePressN(i), (i * DeltaZ) _ , (2 * rpi), (2 * rpi)) UppN = ((1 / hiN) + (Pie * (rpi ^ 2) / (hoN * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * DeltaZ)) ^ (-1) 'These are the heat transfer coefficients at the previous time-step Uaa = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((200 - i) * DeltaZ), de, 2 * ra) 'This is for a specific cased hole Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1)

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ho = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((100 - i) * DeltaZ), de, 2 * rp) hi = ConvectiveHeatTransCoeff(PipeTemp(i), PipePress(i), (i * DeltaZ), 2 * rpi _ , 2 * rpi) Upp = ((1 / hi) + (Pie * (rpi ^ 2) / (ho * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * DeltaZ)) ^ (-1) 'Uaa = Ua 'UaaN = Ua 'Upp = Up 'UppN = Up F = 2 * Pie * ra * UaaN G = 2 * Pie * rp * UppN H = 7.48 * MudDensity(AnnTempN(i), AnnpressN(i)) * Pie _ * ((ra ^ 2) - (rp ^ 2)) * cfl / (DeltaT) 'H = 7.48 * RefMudDensity * Pie * ((ra ^ 2) - (rp ^ 2)) * cfl / (DeltaT) L = 2 * Pie * ra * Uaa M = 2 * Pie * rp * Upp Select Case i Case 0 A(i) = 0 B(i) = F * theta + G * theta + H C(i) = -E * theta D(i) = -E * (1 - theta) * (AnnTemp(i) - 2) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) - E * theta * (AnnTempN(i) - 1)

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Case (imax - 1) A(i) = E * theta B(i) = F * theta + G * theta + H C(i) = 0 D(i) = -E * (1 - theta) * AnnTemp(i - 1) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) + E * theta * AnnTempN(imax) Case Else A(i) = E * theta B(i) = F * theta + G * theta + H C(i) = -E * theta D(i) = -E * (1 - theta) * AnnTemp(i - 1) _ + (H - L * (1 - theta) - M * (1 - theta)) * AnnTemp(i) _ + E * (1 - theta) * AnnTemp(i + 1) + L * (1 - theta) * FormTemp(i, 0) _ + M * (1 - theta) * PipeTemp(i) + F * theta * FormTempN(i, 0) _ + G * theta * PipeTempN(i) End Select Next i ThomasAlgorithm A, B, C, D, X, (imax - 1) 'Update annulus temperature in finite difference grid For i = 0 To (imax - 1) AnnTempN(i) = X(i) Next i End Sub Public Sub EvaluatePipe(FormTemp() As Double, FormTempN() As Double, PipeTemp() As Double _ , PipeTempN() As Double, PipePress() As Double, PipePressN() As Double _ , AnnTemp() As Double, AnnTempN() As Double, Annpress() As Double, AnnpressN() As Double _ , imax As Integer) Dim A() As Double Dim B() As Double Dim C() As Double Dim D() As Double Dim X() As Double

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Dim E As Double Dim F As Double Dim G As Double Dim H As Double Dim L As Double Dim M As Double Dim mi As Double 'This parameter is used to weight the numerical solution ' at the bottom of the hole. A value of half gives the 'Crank-Nicholoson scheme Dim theta As Double Dim i As Integer Dim Uaa As Double Dim Upp As Double Dim ho As Double Dim hi As Double ReDim A(imax) ReDim B(imax) ReDim C(imax) ReDim D(imax) ReDim X(imax) theta = 2 / 3 E = mRate * cfl / (2 * DeltaZ) 'Store the old guess for Tp(i,n+1) 'For i = 0 To iMax ' store(i) = PipeTemp(i, n) 'Next i 'This sets the inflow pipe temperature PipeTempN(0) = Tps For i = 1 To imax ho = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i), (i * DeltaZ), de, de)

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hi = ConvectiveHeatTransCoeff(PipeTemp(i), PipePress(i), (i * DeltaZ), 2 * rpi _ , 2 * rpi) Upp = ((1 / hi) + (Pie * (rpi ^ 2) / (ho * Pie * rp ^ 2)) _ + Pie * (rpi ^ 2) * NatLg(rp / rpi) / (2 * Pie * kp * i * DeltaZ)) ^ (-1) 'Upp = Up F = 7.48 * MudDensity(PipeTempN(i), PipePressN(i)) * Pie * (rp ^ 2) * cfl _ / (DeltaT) 'F = 7.48 * RefMudDensity * Pie * (rp ^ 2) * cfl / (DeltaT) G = 2 * Pie * rp * Upp H = 2 * Pie * rp * Upp Select Case i Case 1 A(i - 1) = 0 B(i - 1) = F + G * theta C(i - 1) = E * theta D(i - 1) = E * (1 - theta) * PipeTemp(i - 1) + (F - H * (1 - theta)) * PipeTemp(i) _ - E * (1 - theta) * PipeTemp(i + 1) + H * (1 - theta) * AnnTemp(i) _ + G * theta * AnnTempN(i) + E * theta * PipeTempN(i - 1) Case imax Uaa = ConvectiveHeatTransCoeff(AnnTemp(i), Annpress(i) _ , ((200 - i) * DeltaZ), de, 2 * ra) 'This is for a specific cased hole Uaa = ((1 / Uaa) _ + (Pie * (4.892 ^ 2) * NatLg(5.5 / 4.892) / (2 * Pie * kp * DeltaZ)) _ + (Pie * (4.892 ^ 2) * NatLg(6.108 / 5.5) / (2 * Pie * kcement * DeltaZ))) ^ (-1) 'Uaa = Ua L = mRate * cfl

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M = 7.48 * MudDensity(PipeTempN(i), PipePressN(i)) * Pie * (ra ^ 2) * DeltaZ _ * cfl / 2 * DeltaT 'M = 7.48 * RefMudDensity * Pie * (ra ^ 2) * DeltaZ * cfl / 2 * DeltaT mi = Pie * ra * Uaa * DeltaZ A(i - 1) = theta * L B(i - 1) = -theta * L - M C(i - 1) = 0 D(i - 1) = -(1 - theta) * L * PipeTemp(i - 1) + ((1 - theta) * L - M) _ * PipeTemp(i) - mi * theta * FormTempN(i, 0) _ - mi * (1 - theta) * FormTemp(i, 0) + mi * theta * AnnTempN(i) _ + mi * (1 - theta) * AnnTemp(i) Case Else A(i - 1) = -E * theta B(i - 1) = F + G * theta C(i - 1) = E * theta D(i - 1) = E * (1 - theta) * PipeTemp(i - 1) _ + (F - H * (1 - theta)) * PipeTemp(i) _ - E * (1 - theta) * PipeTemp(i + 1) _ + H * (1 - theta) * AnnTemp(i) + G * theta * AnnTempN(i) End Select Next i ThomasAlgorithm A, B, C, D, X, (imax - 1) 'Update pipe temperature in finite difference grid For i = 0 To (imax - 1) PipeTempN(i + 1) = X(i) Next i End Sub

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ECD en Hgh Temp Wells

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