experimental study and modeling of surge and swab pressures for yield-power-law drilling fluids

UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

EXPERIMENTAL STUDY AND MODELING OF SURGE AND SWAB

PRESSURES FOR YIELD-POWER-LAW DRILLING FLUIDS

A THESIS

SUBMITTED TO THE GRADUATE FACULTY

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE

By

FREDDY ESTEBAN CRESPO MOLINA

Norman, Oklahoma

2011

EXPERIMENTAL STUDY AND MODELING OF SURGE AND SWAB

PRESSURES FOR YIELD-POWER-LAW DRILLING FLUIDS

A THESIS APPROVED FOR THE

MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL

ENGINEERING

BY

Dr. Ramadan Ahmed, Chair

Dr. Subhash Shah

Dr. Samuel Osisanya

"Commit to the Lord whatever you do, and your plans will succeed”

Proverbs 16:3

To God my Lord,

You found me, saved me, and became the center of my life.

iv

ACKNOWLEDGEMENTS

I want to express my most sincere gratitude to my advisor Dr. Ramadan

Ahmed for giving me the opportunity to work on one of the most exciting topics

in drilling engineering. I am extremely thankful for his patience, support,

encouragement, inspiration and good humor during my graduate studies.

I also would like to thank my thesis committee members Dr. Subhash

Shah and Dr. Samuel Osisanya for their contributions and suggestions during the

review process of my work. Special thanks to Mr. Joe Flenniken, for his help

during the experimental study. Special thanks to the faculty and staff of the

Mewbourne School of Petroleum and Geological Engineering especially Mrs.

Shalli Young, Mrs. Sonya Grant and Ms. Summer Shije for their kindness and

support.

I want to thank my mom Janeth for her unconditional love and inspiration.

Thanks for being my partner and helping me make this dream possible. You are

the most amazing mom in the world. I want to express my gratitude to the love of

my life: my sister Alejandra for being a source of continuous encouragement and

laughter. Thank you for spending hours talking to me on the phone. I admire you

deeply for being an overcomer. Thanks to my dad Freddy Humberto for opening

v

his heart to me and giving us a second chance. I cannot wait for all the things God

has in store for our relationship. Thank to my step dad Octavio for being such a

blessing to my family.

Thanks to my girlfriend Catherine Bosma for driving me crazy and being

that spark that makes life more enjoyable. I will never forget the seeds you

planted in my life. You are such a breath of fresh air Amorsito.

Big thanks to my good friend Amin Mehrabian (MF). Thanks for his

Nivel-ness, his friendship and for giving me a huge insight into the world of

research. Thanks to my very special friends Paola, Andres Castano, Candace,

Pedrito, Amanda, and other friends from the Latin Dance Club and COLSA for

their friendship, support and all the fun.

Special thanks to my spiritual mom Beth, my American mom Linda and

my LifeGroup friends specially Nathan, Andrew, Ben and Amber for their

prayers, blessings and encouragement during this journey with the Lord.

Special thanks to my roommates “La Bleeds” for being my family here in

Oklahoma and for all the fun. Thanks for all the memories and for helping me

become a leader and a well rounded person.

vi

TABLE OF CONTENTS

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

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

ABSTRACT ......................................................................................................... xii

1. INTRODUCTION ......................................................................................... 1

1.1. OVERVIEW................................................................................................. 1

1.2. PROBLEM DESCRIPTION ........................................................................ 2

1.3. OBJECTIVES .............................................................................................. 3

1.4. OUTLINE..................................................................................................... 4

2. LITERATURE REVIEW ............................................................................. 5

2.1. PREVIOUS WORK ..................................................................................... 5

2.1.1. Field Studies .......................................................................................... 5

2.1.2. Laboratory Studies ................................................................................. 9

2.2. MODELING............................................................................................... 10

2.2.1. Steady-State Models ............................................................................ 10

2.2.2. Unsteady-State Models ........................................................................ 17

2.3. DYNAMIC EFFECTS ON SURGE AND SWAB PRESSURES ............. 17

2.4. ECCENTRICITY EFFECTS ON SURGE AND SWAB PRESSURES ... 18

3. SURGE AND SWAB PRESSURE MODELING ..................................... 20

3.1. THEORETICAL MODEL ......................................................................... 20

3.2. REGRESSION MODEL ............................................................................ 29

3.3. MODEL VALIDATION ............................................................................ 33

3.4. PARAMETRIC STUDY ............................................................................ 41

4. EXPERIMENTAL STUDY ........................................................................ 45

4.1. EXPERIMENTAL SETUP ........................................................................ 45

4.1.1. Vertical Test Section ........................................................................... 47

vii

4.1.2. Guiding Rod ........................................................................................ 48

4.1.3. Variable Speed Motor .......................................................................... 49

4.1.4. Pressure Transducer ............................................................................. 50

4.1.5. Data Acquisition System ..................................................................... 50

4.1.6. Fluid Mixing and Collection Tanks ..................................................... 51

4.2. SYSTEM CALIBRATION ........................................................................ 52

4.3. TEST PROCEDURE.................................................................................. 53

4.5. TEST MATERIALS .................................................................................. 55

4.4. RECORDED DATA PROCESSING ......................................................... 58

5. RESULTS AND DISSCUSSION ................................................................ 60

5.1. NEWTONIAN FLUIDS ............................................................................ 61

5.2. POWER-LAW FLUIDS ............................................................................ 63

5.3. YIELD-POWER-LAW FLUIDS ............................................................... 66

5.4. DISCUSSION ............................................................................................ 69

5.5. PRACTICAL IMPLICATIONS ................................................................ 73

6. CONCLUSIONS AND RECOMMENDATIONS .................................... 75

6.1. CONCLUSIONS ........................................................................................ 75

6.2. RECOMMENDATIONS ........................................................................... 76

REFERENCES .................................................................................................... 77

NOMENCLATURE ............................................................................................ 85

APPENDIX A ...................................................................................................... 90

APPENDIX B ...................................................................................................... 93

APPENDIX C ...................................................................................................... 95

APPENDIX D ...................................................................................................... 99

APPENDIX E .................................................................................................... 102

viii

LIST OF TABLES

Table 3.1: Rheological parameters of drilling fluids ............................................ 30

Table 3.2: Pipe-wellbore configurations ............................................................... 31

Table 4.1: Rheology of test fluids ......................................................................... 58

Table D.1: Fann model 35 (#1/5 spring) measurements for light mineral oil ...... 95

Table D.2: Fann model 35 (#1 spring) measurements for mineral oil .................. 95

Table D.3: Fann model 35 (#1 spring) measurements for 1.0% PAC .................. 96

Table D.4: Fann model 35 (#1 spring) measurements for 0.75% PAC ................ 96

Table D.5: Fann model 35 (#1 spring) measurements for 0.56% PAC ................ 96

Table D.6: Fann model 35 (#1 spring) measurements for mix 0.28% + 0.22%

Xanthan Gum ........................................................................................................ 97

Table D.7: Fann model 35 (#1 spring) measurements for 1.0% Xanthan Gum ... 97

Table D.8: Fann model 35 (#1 spring) measurements for 0.67% Xanthan Gum . 97

Table D.9: Fann model 35 (#1 spring) measurements for 0.44% Xanthan Gum . 98

Table E.1: Surge pressure gradient readings for mineral oil ................................ 99

Table E.2: Surge pressure gradient readings for light mineral oil ........................ 99

Table E.3: Surge pressure gradient readings for 1.0% PAC ................................. 99

Table E.4: Surge pressure gradient readings for 0.75% PAC ............................. 100

Table E.5: Surge pressure gradient readings for 0.56% PAC ............................. 100

Table E.6: Surge pressure gradient readings for mix 0.28% + 0.22% Xanthan

Gum..................................................................................................................... 100

Table E.7: Surge pressure gradient readings for 1.0% Xanthan Gum ................ 101

Table E.8: Surge pressure gradient readings for 0.67% Xanthan Gum .............. 101

ix

LIST OF FIGURES

Figure 2.1: Annulus pressure measured during swab tests (Wagner et al. 1993) ... 7

Figure 2.2: Typical pressure data while tripping-in (Rudolf and Suryanarayana,

1998) ....................................................................................................................... 9

Figure 2.3: Schematic of back extrusion experiment (Osorio and Steffe, 1991) .. 10

Figure 2.4: Clinging constant determination plot (Burkhardt, 1961) ................... 12

Figure 2.5: Dimensionless pressure gradient determination plot for diameter ratio

of 0.3 (Chukwu and Blick, 1989).......................................................................... 15

Figure 2.6: Annular and equivalent slot geometry................................................ 16

Figure 2.7: Effect of yield stress on surge pressures (Lal, 1983) ......................... 18

Figure 3.1: Representation of a concentric annulus as a slot ................................ 21

Figure 3.2: Velocity profile of yield-power-law fluid through a slot ................... 22

Figure 3.3: Characteristic curves to determine surge and swab pressure ............. 29

Figure 3.4: Comparison of predictions of numerical and regression models

(Newtonian fluids) ................................................................................................ 33

Figure 3.5: Comparison of predictions of numerical and regression models

(Power-Law Fluids) .............................................................................................. 34

Figure 3.7: Comparison of predictions of numerical and regression models (Yield-

Power-Law Fluids)................................................................................................ 35

Figure 3.8: Comparison between correlation and Newtonian model (Bourgoyne,

1986) Predictions .................................................................................................. 36

Figure 3.9: Comparison of Schuh’s solution with regression model for different

Power-Law Fluids ................................................................................................. 37

Figure 3.10: Comparison of correlation predictions and solution by Burkhardt

(1961) for different Bingham-Plastic Fluids ......................................................... 38

x

Figure 3.11: Comparison of the correlation predictions with the back extrusion

technique (Osorio and Steffe, 1991) for a specific Yield-Power-Law Fluid ........ 39

Figure 3.12: Comparison of different models for Power-Law Fluid .................... 40

Figure 3.13: Comparison of different models for Bingham Plastic Fluid ............ 41

Figure 3.15: Effect of fluid behavior index on surge pressures for Power-Law

Fluids at different speeds ...................................................................................... 42

Figure 3.16: Surge pressures vs. tripping speed for different yield stresses ......... 43

Figure 3.17: Surge pressure vs. diameter ratio for different tripping speeds ........ 44

Figure 4.1: Schematic view of experimental Setup .............................................. 46

Figure 4.2: Actual view of experimental setup ..................................................... 47

Figure 4.3: Vertical test section ............................................................................ 48

Figure 4.4: Variable speed motor.......................................................................... 49

Figure 4.5: Differential pressure transducers with pressure tapings ..................... 50

Figure 4.6: Personal computer .............................................................................. 51

Figure 4.7: Upward pipe speed as a function of voltage ....................................... 53

Figure 4.8: Downward pipe speed as a function of voltage .................................. 53

Figure 4.9: Fann 35 rotational viscometer ............................................................ 55

Figure 4.10: Rheology of PAC based test fluids................................................... 57

Figure 4.11: Rheology of Xanthan Gum based test fluids .................................... 57

Figure 4.12: Surge pressure measurement with established steady flow condition

............................................................................................................................... 59

Figure 4.13: Surge pressure measurement without established steady flow

condition (Mineral Oil; 0.7 ft/s) ............................................................................ 59

xi

Figure 5.1: Friction Factor vs. Generalized Reynolds Number for experimental

data ........................................................................................................................ 61

Figure 5.2: Surge pressure gradient vs. trip speed with regular mineral oil. ........ 62

Figure 5.3: Surge pressure gradient vs. trip speed with light mineral oil ............. 63

Figure 5.4: Surge pressure gradient vs. trip speed with 1.0% PAC ...................... 64

Figure 5.5: Surge pressure gradient vs. trip speed with 0.75% PAC .................... 65

Figure 5.6: Surge pressure gradient vs. trip speed with 0.56% PAC .................... 65

Figure 5.7: Surge pressure gradient vs. trip speed for polymer mix ..................... 66

Figure 5.8: Surge pressure gradient vs. trip speed with 1.0% Xanthan Gum ....... 67

Figure 5.9: Yield Surge pressure gradient vs. trip speed with 0.67% Xanthan Gum

............................................................................................................................... 68

Figure 5.10: Surge pressure gradient vs. trip speed with 0.44% Xanthan Gum. .. 68

Figure 5.11: Surge Pressures at different annular eccentricities (1.0% PAC; 0.2

ft/s) ........................................................................................................................ 71

Figure 5.12: Effect of static time on surge pressure measurements (1.0% Xanthan

Gum; 0.05 ft/sec) .................................................................................................. 72

xii

ABSTRACT

Surge and swab pressures can be generated during different stages of well

construction operations by tripping-in, tripping-out or reciprocation of the

drillstring in the wellbore. This phenomenon is of economic importance for the oil

industry, especially in wells with narrow margin between pore and fracture

pressure gradients. Moreover, an accurate surge pressure model is very vital in

designing slim holes and casing operations with low annular clearance. Inaccurate

prediction of surge and swab pressures can lead to a number of costly drilling

problems such as lost circulation, formation fracture, fluid influx, kicks, and

blowouts.

Field measurements indicate that pressure surges strongly depend on

drillpipe tripping speeds, wellbore geometry, flow regime, fluid rheology, and

whether the pipe is open or closed. Although a large number of field and

modeling studies were conducted in the past to investigate surge and swab

pressures, experiments under controlled laboratory conditions have never been

reported. Most existing surge/swab models have been developed for Bingham

plastic and power-law fluids. However, these rheology models cannot adequately

describe the flow behavior of most of drilling fluids used in the field. The yield-

power-law (YPL) model best describes the rheology of most used drilling and

xiii

completions fluids. Despite its high accuracy in predicting the flow properties of

drilling fluid, surge and swab pressure models for YPL fluid are lacking.

This thesis presents a new steady-state theoretical model, which is casted

into a simplified dimensionless correlation to predict surge and swab pressures for

YPL fluids. An analytical solution for steady-state laminar flow in narrow slot is

developed to approximate and model the flow in a concentric annulus with inner

pipe axial movement. The analytical solution involves solving a system of non-

linear equations. Accurate predictions are presented as a family of curves, though

not in convenient forms. Thus, a numerical scheme has been developed to solve

the system of non-linear equations. After conducting an extensive parametric

study and applying regression techniques, a simplified dimensionless correlation

has been developed that does not require a cumbersome numerical procedure.

Correlation predictions have been validated by direct comparison with other

existing models and experimental measurements. A parametric study showing the

effects of rheological parameters on surge and swab pressures has been carried

out.

Experimental investigation of the effects of fluid properties and drilling

parameters on surge and swab pressures under laboratory conditions has also been

undertaken. Tests were performed in an experimental setup that has the capability

of varying the tripping speed and accurately measuring the surge or swab

xiv

pressure. The setup consists of fully transparent polycarbonate tubing and inner

steel pipe which moves axially using a speed controlled hoisting system. During

the experiments, several Newtonian and non-Newtonian fluids were tested.

The performances of both theoretical and regression models have been

rigorously tested by direct comparison with experimental data. In most cases, a

satisfactory agreement has been obtained between predictions and measurements.

Results confirm that trip speed, fluid rheology and annular clearance have a

significant effect on surge pressure.

1

1. INTRODUCTION

1.1. OVERVIEW

Wellbore hydraulics has received increased attention in the past few years

as deepwater drilling and new technologies such as slim hole and casing drilling

techniques have emerged in the industry. As thousands of wells are drilled every

year, challenges associated with downhole pressure management have become

more critical. Pressure variations in the wellbore may be caused by tripping-in or

tripping-out drillstring, or reciprocation of casing in the borehole. The pressure

change can increase (surge) or decrease (swab) the bottomhole pressure.

Accurate prediction of surge and swab pressures is crucial in terms of

estimating the maximum tripping speeds to keep the wellbore pressures within

specific limits (pore and fracture pressure). It also plays a major role in running

casings, particularly with narrow annular clearances.

Surge and swab pressures have been a constant area of research. As the oil

and gas industry is moving towards drilling more challenging and complex wells,

the ability to accurately predict pressure variations in the wellbore allows a better

optimization of wellbore hydraulics and can lead to more successful drilling

operations.

2

1.2. PROBLEM DESCRIPTION

Surge and swab pressures have been known to cause formation fracture,

lost circulation and well control problems. Often surge and swab pressures can be

generated due to viscous drag of the fluid in contact with the drillstring and

sudden pipe acceleration (inertial effects), both resulting from pipe movement and

fluid displacement when the drillstring moves along the wellbore.

High surge pressure can lead to lost circulation, either by sudden

fracturing the formation, or continuous fluid loss into the permeable formation.

The drilling fluid that has entered into the fractured formation causes a drop in the

fluid level, resulting in a reduced wellbore hydrostatic pressure. This reduction in

mud hydrostatic pressure allows formation fluids to enter the wellbore, which

may lead to a kick and eventually a blowout.

Pressure reduction due to swabbing can lead to the flow of formation fluid

into the wellbore and generate a kick. Excessive swab pressures are a major

source of blowouts. Also, pressure changes caused by alternating between surge

and swab pressures due to reciprocation, such as those made on connections may

cause hole sloughing, or other unstable hole conditions, including solids fill on

bottom.

The yield-power-law (YPL) model best fits the rheological properties of

most of drilling fluids and aqueous clay slurries (Fordham et al., 1991; Hemphil et

3

al., 1993; Merlo et al., 1995; Maglione and Ferrario, 1996; Kelessidis et al., 2005;

Kelessidis et al., 2007). However, no general analytical solution for annular flow

of yield-power-law to calculate surge and swab pressures has been reported in the

literature. The YPL rheology model involves three parameters to describe flow

behavior of drilling fluids. However, this makes the mathematical modeling of the

surge and swab flows more complex. Also, the lack of experimental studies under

controlled laboratory conditions is a limiting factor in understanding surge and

swab phenomena. Therefore, a continued research effort is required to develop

more accurate models and better understand surge and swab pressures.

1.3. OBJECTIVES

The principal objective of this study is to improve our understanding of surge and

swab pressures and investigate the effects of fluid rheology, diameter ratio, and

pipe velocity. The main objectives of this research are:

1. To develop a new model that allows accurate steady-state calculations of

surge and swab pressures for yield-power-law fluids in concentric annulus.

2. To carry out a regression study in order to develop a simplified dimensionless

correlation to predict swab and surge pressures in a more convenient way.

3. To develop a new test setup that has the capability to vary the trip speed and

accurately measure the surge or swab pressures.

4

4. To validate the newly developed models, and other existing models, by direct

comparison with experimental results.

1.4. OUTLINE

A general overview and extensive literature review of surge and swab

pressures, experimental studies, and theoretical models have been presented

(Chapter 2). In order to develop a model that allows better prediction of surge

and swab pressures, the steady flow of YPL fluid in concentric annuli is

represented by a narrow slot (Chapter 3). The solution is presented numerically

and as a family of curves. After performing parametric study, regression

techniques were applied to develop a simplified regression model (dimensionless

correlation). Comparison with experimental results and previously published

analytical solutions validates the predictions of the correlation for Newtonian and

Non-Newtonian fluids. Moreover, experimental investigations have been

conducted to study the effects of different fluid properties and drilling parameters

on surge and swab pressures (Chapter 4). Experiments were carried out using a

newly developed test setup that has the capability to accurately measure surge or

swab pressures. Experimental results were analyzed and compared with

predictions of the new and existing models to rigorously test their performance

(Chapter 5). Conclusions and recommendations for further studies have also

been presented (Chapter 6).

5

2. LITERATURE REVIEW

2.1. PREVIOUS WORK

2.1.1. Field Studies

A number of studies (Moore 1965; Clark and Fontenot 1974; Lal 1983;

Clark 1956) were undertaken to investigate the effects of fluid properties and

drilling parameters on surge and swab pressures. Generally, it was found that

pressure surges depend strongly on drillpipe tripping speed, wellbore geometry,

flow regime, fluid rheology, and whether the pipe is open or closed. Early studies

of surge and swab (Cannon, 1934; Horn, 1950; Goins et al., 1951) were carried

out to investigate drilling problems associated with pressure variations in the

wellbore. These studies demonstrated that lost circulation, formation fracture,

fluid influx, kicks, and blowouts are connected to excessive surge and swab

pressures due to high tripping speeds. Based on the outcomes of these studies, the

following observations can be made:

• Surge and swab pressures increase with tripping speed;

• Swab pressures can potentially cause blowouts;

• Tripping-out of the wellbore may be a contributing factor to blowouts;

6

• Surge pressure can be the main cause of lost circulation;

• Surge pressure with closed-end pipe are higher than those with open-end.

Very limited field measurements that show detailed surge and swab

pressure tests are available. Burkhardt (1961) presents surge pressure data for a

100 ft study well, which was specially instrumented to measure bottomhole

pressures. His data is very instructive and provides a good test reference for

analytical models, but do not represent a real well situation since the well

dimensions and smaller than a regular well. Much more useful data was gathered

by Clark and Fontenot (1974), who conducted surge tests on two wells. The first

was an 18,500 ft well in Mississippi. The second well was a 15,270 ft well in

Utah. Clark and Fontenot provide very complete information on drillstring

velocity and drilling fluid properties throughout the tests, and full information on

pressure measurements. They found that control of pipe speed while tripping is

necessary to minimize downhole pressure surges.

Wagner et al. (1993) presents actual surge and swab field data during

tripping and circulating operations which include both surface and downhole

measurements. A series of three field tests were performed in each of the two

study wells in the Gulf of Mexico. The first was a deep onshore exploration well

in Mississippi that was drilled to a depth of 19,600 ft. The second was a slightly

deviated development well in shallow waters of offshore Louisiana that was

drilled to 15,384 ft depth. Results show that pipe velocity is proportional to the

7

surge/swab pressures (Fig. 2.1). Also, it is noted that the pressure is increasing

with time due to the increase in hydrostatic pressure as a function of depth.

Figure 2.1: Annulus pressure measured during swab tests (Wagner et al. 1993)

White and Zamora (1997) gathered surge and swab pressure data from a

12,710 ft well in the Gulf of Mexico. Although, measurements were limited by

technical constraints, the effect of tripping speeds and acceleration is observed.

Their results also showed a higher pressure surge at the bottom of the drillstring

than at the top.

Other studies (Rudolf and Suryanarayana, 1997; Rudolf and

Suryanarayana, 1998) showed swab measurements recorded on every stand of the

6400

6500

6600

6700

0 100 200 300

An

nu

lus

Pre

ssu

re (

psi

)

Time (s)

Swab Pressure

8

drillpipe while tripping in a 15,000 ft well. Results confirm that surge pressure

peaks appear every time as the drillstring begins to trip (Fig. 2.2). The sudden

increase in tripping speed results in acceleration that generates pressure surge.

Also, they have shown that pipe elasticity, fluid compression and expansion,

bottomhole temperature, wellbore expansion and contraction, and the drillstring

oscillations, appear to all contribute to the pressure surge. Recent studies (Bing

and Kaiji, 1996; Thorsrud et al., 2000; Robello et al., 2003; Mitchell, 2004;

Rommetveit et al., 2005) presented extensive surge and swab pressure

measurements and modeling results. Results and observations of the studies are

consistent with many of previous investigations.

9

Figure 2.2: Typical pressure data while tripping-in (Rudolf and Suryanarayana, 1998)

2.1.2. Laboratory Studies

Laboratory data on surge and swab pressures is lacking. One technique

that has similar flow conditions as the current problem is known as back

extrusion. The procedure consists of forcing a cylindrical plunger down into a

fluid trapped in a cylinder forcing the fluid to flow upwards through a concentric

annular space (Fig. 2.3). This procedure is widely used to obtain rheology

parameters of thick food products at low speeds. Osorio and Steffe (1991)

presented a semi-empirical surge pressure model for a specific yield-power-law

fluid based on back extrusion measurements.

8000

8500

9000

9500

10000

0 2 4 6 8 10 12 14

Me

asu

red

Pre

ssu

re (

psi

)

Time (s)

10

Figure 2.3: Schematic of back extrusion experiment (Osorio and Steffe, 1991)

2.2. MODELING

2.2.1. Steady-State Models

Accurate surge/swab model predictions have been a constant area of

research. In the past, Cardwell (1953) and Ormsby (1954) attempted to explain

the physical causes, nature and magnitude of surge and swab pressures. Both

studies presented quantitative prediction techniques to determine these pressures

for Newtonian fluids in laminar and turbulent flow regimes. Only the pressure

losses arising from the viscous drag of the moving fluid with stationary pipe wall

was taken into consideration. Another study (Clark, 1955) introduced the case of a

moving inner pipe through a concentric annulus with Bingham plastic fluid.

r

Vp

ΔL

V(r)

Cylindrical

Plunger

Test Fluid

11

Pressure variations caused by sudden changes in pipe speed in addition to those

arising from viscous drag were included in the analysis. Idealized graphs and

equations for predicting surge and swab pressures in laminar and turbulent flow

regimes were presented. Burkhardt (1961) presented a semi-empirical method

describing quantitatively and theoretically pressure surges for a Bingham Plastic

fluid. The drilling fluid velocity resulting from the tripping is related to the trip

velocity:

(2.1)

where and are the effective fluid velocity and trip velocity, respectively.

is the proportionally constant known as clinging factor, which depends upon

the ratio of the pipe to hole diameter according to the curves presented in Fig. 2.4.

Burkhardt developed models that are used to predict the viscous drag surge

pressure. Model predictions compare favorably with actual pressure surge

measurements. In addition, Results showed that pressure surges are usually high

when running closed-end casing or drillpipe in the hole.

12

Figure 2.4: Clinging constant determination plot (Burkhardt, 1961)

Later, a numerical model (Schuh, 1964) was developed to compute surge

and swab pressure. Schuh’s solutions were patterned after studies presented by

Burkhardt (1961) and Clark (1955).

Another study (Fontenot and Clark, 1974) presented a comprehensive

technique for determining surge/swab pressure for both Bingham-plastic and

power-law fluids. Models presented in previous studies (Melrose et al., 1958;

Dodge and Metzner, 1959; Burkhardt, 1961; Schuh, 1964) were implemented in a

computer program to investigate the effects of different parameters including mud

properties, closed and open-ended pipe, well geometries, tool joints, drillpipe

rubbers, and bit nozzles. Model predictions showed good agreement with field

measurements.

13

Surge and swab pressure modeling have been also carried out by hydraulic

analysis of annular flow with axial motion of the inner pipe (Lin and Hsu, 1980;

Chukwu and Blick, 1989; Malik and Shenoy, 1991; Haige and Xisheng, 1996;

Filip and David, 2003) for different pipe/borehole configurations and fluid

rheology models. Lin and Hsu (1980) presented a numerical procedure for the

case of power-law fluid in concentric annulus. However, the procedure is too

complex for ready use in drilling applications. Some minor shortcomings to this

approach were indentified (Macsporran, 1982) and subsequently corrected (Lin

and Hsu, 1982).

Another study (Chukwu and Blick, 1989) applied the Couette flow with

pressure gradient to establish a relationship between inner pipe speed and pressure

variation in the wellbore resulting from the pipe movement. They related the

dimensionless flow rate and surge/swab pressure gradient resulting from tripping

and presented their solutions as a family of curves for different diameter ratios

(i.e. ratio of pipe diameter to hole/casing diameter). In order to find a specific

solution, the dimensionless annular flow rate generated by fluid displacement by

the inner pipe motion given by:

(2.2)

14

Having the value of the dimensionless annular flow rate, the dimensionless

pressure gradient is obtained from a type curve as shown in Fig. 2.5. The

surge/swab pressure value is obtained from the expression:

(

)

(

) (2.3)

where is the hole inside diameter, is the consistency index, is the fluid

behavior index, and is the dimensionless pressure gradient.

An analytical solution of the steady-state laminar flow of power-law fluid

in annulus resulting from the fluid displacement and axial motion of the inner

pipe was presented by Malik and Shenoy (1991). However, the solution was

limited to the calculation of the volumetric flow rate, and no discussion was

presented on its application to obtain surge or swab pressures.

Later, Haige and Xisheng (1996) presented a model that predicts pressure

surge in directional wells. The model considered the axial flow of Robertson-Stiff

fluid in concentric annuli. The equations were solved numerically and solutions

were presented as a family of curves. More recently, this approach has been

adopted (Filip and David, 2003) to include the effect of the inner cylinder

movement on the pressure gradient. The predictions of the model have shown a

satisfactory agreement with previous data (Malik and Shenoy, 1991) for power-

law fluids.

15

Figure 2.5: Dimensionless pressure gradient determination plot for diameter ratio of 0.3

(Chukwu and Blick, 1989)

Representation of the annulus as a slot (Fig. 2.6) is a commonly used

technique to simplify the mathematical analysis of the annular flow. The slot

model (i.e. approximate model) is valid for diameter ratios greater than 0.3

(Guillot and Dennis, 1988; Chukwu and Blick, 1989; Guillot, 1990; Bourgoyne et

al., 1991; Kelessidis et al., 2007; Crespo et al, 2010). Newtonian slot flow

16

between two parallel plates, one moving at a constant velocity while the other is

fixed, was carried out by Schlichting (1955). Their solution is simply the

superposition of the solution of two problems: flow between two parallel walls,

one of which is moving with no pressure gradient, and flow between two fixed

parallel walls because of a pressure gradient. For non-Newtonian fluids such a

simple superposition is not possible, as the flow coupling occurs due to the

apparent viscosity function. A complete solution for this problem using Ellis fluid

flow was presented by Wadhwa (1966). Later on, Flumerfelt et al. (1969)

presented both tabular and graphical solutions for the steady-state laminar flow of

power-law fluid and developed dimensionless correlations for general use.

Figure 2.6: Annular and equivalent slot geometry

rh

rh rp rp

L

H

17

2.2.2. Unsteady-State Models

Most field studies indicated that acceleration exacerbates surge and swab

pressures. During the last couple of decades, unsteady-state (transient) models

have been developed (Lal, 1983; Bing et al., 1995; Yuan and Chukwu, 1996) to

determine bottomhole pressure fluctuations due to pipe acceleration while

tripping which is a real-life situation in drilling operations. Model results are in

agreement with field studies showing that pipe acceleration can generate pressure

peaks.

2.3. DYNAMIC EFFECTS ON SURGE AND SWAB PRESSURES

In addition to the transient flow behavior, a number of studies (Lubinski et

al., 1977; Lal, 1983; Mitchell, 1988; Bing and Kaiji, 1996) included previously

neglected dynamic effects such as fluid inertia, fluid and wellbore

compressibility, and axial elasticity of moving pipe.

Lal (1983) presented a parametric study showing the effects of yield stress

on surge and swab pressures. Calculations indicate that when the fluid has high

yield stress values, the magnitude of the generated surge/swab pressure increases

(Fig. 2.7).

18

Figure 2.7: Effect of yield stress on surge pressures (Lal, 1983)

The model developed by Mitchell (1988) is one of the most accepted by

the drilling industry. The model is based on a steady-state approach for power-law

fluids. It has been extensively validated against field data (Wagner et al., 1993;

Robello et al., 2003; Rommetveit et al., 2005). The model has also been enhanced

to include the effects of temperature-dependent fluid rheology, fluid circulation,

acceleration, well deviation and eccentricity (Robello et al., 2003). However, a

detailed formulation of the model has not been published in the literature.

2.4. ECCENTRICITY EFFECTS ON SURGE AND SWAB PRESSURES

Eccentricity can have a significant effect in surge and swab pressures. The

surge pressures can be as much as 50 percent less than a concentric calculation

when the inner pipe lies to one side of the hole. For power-law fluids, Yang and

0

500

1000

1500

0 5 10 15 20 25 30

Pe

ssu

re (

psi

)

Time (s)

YP=15 lbf/100 ft²

YP=30 lbf/100 ft²

19

Chukwu (1995) applied their analytical technique to determine the surge or swab

pressure at specified steady velocity in an eccentric wellbore. The solutions of the

equations are presented in both dimensionless form and as a family of curves for

different eccentricity ratios and power-law fluid index values. A numerical study

(Hussain and Sharif, 1997) indicated the reduction of surge pressure with the

increase in eccentricity. For a partially blocked eccentric annulus with cuttings

bed, the surge pressure decreases with the increase in the bed thickness.

A simplified model using eccentricity geometry for Casson model fluids

was presented by Sun et al (2010). Numerical solution was applied to calculate

surge and swab pressures in horizontal wells.

20

3. SURGE AND SWAB PRESSURE MODELING

The phenomenon of annular fluid flow due to axial motion of the inner

pipe is modeled to predict surge and swab pressures using the narrow slot

approximation technique. This approach is used to simplify the analysis of the

surge and swab pressure for YPL fluid under steady laminar flow condition. The

model is valid for diameter ratios greater than 0.4. Model solutions require

iterative numerical procedures. Therefore, a simple dimensionless regression

model (correlation) has been developed using numerically obtained results.

3.1. THEORETICAL MODEL

The annular flow is induced by the axial motion of the drillpipe as it

displaces the fluid trapped in the wellbore. The concentric annulus is represented

by an equivalent narrow slot (Fig. 3.1) where the top plate represents the drillpipe

that moves with a constant velocity and the lower plate represents a stationary

casing or hole. The following assumptions are made in the formulation of the

theoretical model:

Incompressible fluid (density of the fluid is constant);

Axial flow under steady state and isothermal conditions;

21

Fully developed laminar flow of YPL fluid;

A moving plate traveling at a constant velocity Vp; and

No slip conditions at the walls.

Figure 3.1: Representation of a concentric annulus as a slot

The annular velocity profile (Fig. 3.2) in the wellbore during tripping

operations is expected to have three distinct flow regions: i) outer sheared region

(Region I) within the boundary limits ii) plug zone (Region II) within

the boundary limits and, iii) inner sheared region (Region III) within

the boundary limits . The fluid in the plug zone (Region II) and outer

sheared regions (Region I and Region III) flows opposite to the direction in which

the upper plate moves. Some part of the fluid in the inner sheared region (i.e.

Region III which is close to the moving wall or drillpipe) moves in the same

direction of upper plate.

YPL Fluid

YPL Fluid

p

Velocity Profile

22

Figure 3.2: Velocity profile of yield-power-law fluid through a slot

In order to develop a hydraulic model, momentum balance of each layer is

first considered. For the sheared regions, applying the momentum balance the

shear stress distributions are expressed as:

Region I:

(3.1)

Region III:

(3.2)

where is the shear stress at the stationary wall. For the yield-power-law fluid,

the local shear stresses in Regions I and III are related to the local shear rates

using the constitutive equation as:

(

)

(3.3)

Region III

Region II

Region I

y2

y1

y2 – y1

H

Vp

23

and

(

)

(3.4)

respectively, where k and n are fluid consistency and behavior index. The

dimensionless velocity profiles in Region I and Region III are defined as:

(3.5)

Similarly, normalized coordinates of any point are expressed as:

(3.6)

where H and W are the slot clearance and width, respectively. The dimensionless

velocity distributions are obtained by combining Eqns. (3.1) through Eqn. (3.4),

and applying the boundary conditions in Region I (

) and Region III (

).

Dimensionless velocity profiles in Region I and Region III are:

For Region I:

[( ) ( )

] ; (3.7)

For Region III:

24

[( ) ( )

] ; (3.8)

where is the dimensionless pressure defined as:

(

) (

) (

)

(3.9)

The exponent is a function of the fluid behavior index:

(3.10)

Geometric analysis shows that the dimensionless plug thickness is simply the

difference between the dimensionless boundary limits. Hence:

(3.11)

where are the dimensionless boundary limits defined as:

(3.12)

Applying momentum balance in Region II, the dimensionless plug zone thickness

can be obtained from the following expression:

(3.13)

In the plug zone (Region II), the velocity distribution is uniform (i.e. plug velocity

is constant) and expressed in dimensionless form as:

25

( ) ; (3.14)

The velocity gradient is negative in the Region I and positive in the Region III. At

the edges of the plug zone ( and ), the local velocity Eqns. (3.7) and

(3.8) should give the same value. Thus:

( ) ( )

(3.15)

The total dimensionless flow rate is the sum of the flow rate in each region.

Hence:

∫ (∫

∫

∫

)

(3.16)

By substituting Eqns. (3.7), (3.8) and (3.14) into Eq. (3.16), the dimensionless

fluid flow rate is expressed as:

*(

)

+ [ ( ) ] [ ]

(

) ( )

( ) (3.17)

where the dimensionless fluid flow rate is:

(3.18)

26

To represent the wellbore, slot geometry parameters and are expressed in

terms of annular geometric dimensions and as follows:

( )

(3.19)

( ) (3.20)

For a closed-pipe case, the actual fluid flow rate in the annulus is equal to the rate

at which fluid is being displaced by running the drillpipe into the wellbore. This

means that circulation loss and wellbore ballooning effects are negligible. Hence,

the flow is expressed as:

( ) (3.21)

Subsequently, substituting Eqns. (3.19), (3.20) and (3.21) into Eqn. (3.18), the

dimensionless flow rate is calculated as:

( )

(3.22)

Combining Eqn. (3.17) and (3.22), we obtain:

( )

*(

)

+ [ ( ) ] [ ]

(

) ( )

( ) (3.23)

27

A simplified graphic solution procedure has been developed to obtain

solutions for the analytical model. The procedure requires the mud rheology,

wellbore geometry and pipe velocity as input parameters. To apply this method

for a specific case (fluid rheology/wellbore geometry combination), first Eqn.

(3.15) is expressed in this form:

(

)

( )

(

)(

)(

) (3.24)

Grouping the constant parameters:

(3.25)

(

) ( ) (

)

(3.26)

Eqn. (3.24) can be expressed as:

(

)

( )

(

) (3.27)

The following procedure is followed to determine the surge and swab pressures:

The parameters and are calculated for the specific annular geometry and

fluid rheology combinations.

The value of is obtained by solving Eqn. (3.27) by iteration or using the

modified Newton-Raphson technique for all combinations of and .

28

Substituting the obtained values of into Eq. (3.17), the dimensionless

fluid flow rates are determined for different values of .

A characteristic curve, which is a plot of versus is prepared for

different values of (Fig. 3.3).

From Eqn. (3.22), the dimensionless fluid flow rate is obtained.

Using the value of from the previous step as an input parameter, and from

the plot of , the corresponding value of at a given can be

obtained.

Finally, the following equation is used to calculate the surge or swab

pressure as:

( ) (3.28)

The procedure involving graphic methods, yields exact solutions for the

slot model, though not in convenient forms. It is also time consuming. Hence,

instead of this method, solutions were obtained using a direct numerical

technique. To get the numerical solution, we developed a computer code that

solves a system of four equations Eqns. (3.9), (3.13), (3.15), and (3.23). Since

some of these equations are non-linear and solutions cannot be obtained using the

conventional numerical methods, the computer code varies the pressure gradient

until the system of equations is fully satisfied. Using the code, extensive

29

numerical solutions were obtained varying pipe velocity, fluid properties, and

wellbore geometry.

Figure 3.3: Characteristic curves to determine surge and swab pressure

3.2. REGRESSION MODEL

A systematic regression analysis was carried out using the numerical

results to develop a simple regression model. A wide range of diameter

ratios ( ), various Newtonian and non-Newtonian fluids (Table 3.1),

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.020

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)

Dimensionless Flow Rate

Vp = 1.0 ft/sec

Vp = 2.0 ft/sec

Vp = 3.0 ft/sec

Vp = 4.0 ft/sec

Vp = 5.0 ft/sec

30

and wellbore/pipe configurations (Table 3.2) were considered. Tripping speeds of

up to 3.0 ft/s were considered.

Table 3.1: Rheological parameters of drilling fluids

Fluid Type Test Fluid , lbf/100 ft2 , lbfsn /100 ft2

Newtonian N1 0.0 0.05 1.00

Newtonian N2 0.0 0.16 1.00

Newtonian N3 0.0 0.43 1.00

Power-Law A1 0.0 4.38 0.38

Power-Law A2 0.0 1.74 0.53

Power-Law A3 0.0 1.74 0.56

Power-Law A4 0.0 1.93 0.52

Power-Law B1 0.0 1.37 0.36

Power-Law B2 0.0 0.52 0.61

Power-Law B3 0.0 0.73 0.59

Power-Law B4 0.0 0.78 0.59

Power-Law P1 0.0 4.72 0.57

Power-Law P2 0.0 1.44 0.67

Power-Law P3 0.0 0.35 0.80

Power-Law P4 0.0 1.62 0.50

Bingham Plastic E1 34.0 0.20 1.00






Bingham Plastic F1 61.8 0.04 1.00



Yield-Power-Law C1 18.8 8.00 0.35

31

Fluid Type Test Fluid , lbf/100 ft2 , lbfsn /100 ft2




Yield-Power-Law D1 13.0 2.84 0.43




Table 3.2: Pipe-wellbore configurations

dh (Casing ID) dp (Drillpipe OD) (dp/dh)

9.00 5.00 0.56

9.00 4.50 0.50

7.00 5.50 0.79

7.00 5.00 0.71

7.00 3.50 0.50

7.00 4.00 0.57

7.00 4.50 0.64

5.00 2.88 0.58

4.50 3.50 0.78

The regression model predicts surge or swab pressures conveniently for

laminar flow of yield-power-law fluids without requiring iterative procedure.

Predictions can be made for Newtonian ( ), power-law ( ),

and Bingham plastic ( ) fluids as well. The surge pressure is obtained using

the friction factor as:

(3.29)

32

where and are fluid density and wellbore length. and are trip speed and

annular clearance (i. e. hydraulic radius for flow between two parallel plates),

respectively. The relationship between the friction factor and generalized

Reynolds number is established methodically to resemble the pipe flow equation.

Hence, the friction factor is used. The expression for generalized

Reynolds number is given as:

( ) (3.30)

where is the modified Reynolds number. The yield stress factor is a

dimensionless parameter, which is greater than one for any fluid with yield stress.

This parameter is defined as:

[(

) (

)

] (3.31)

The modified Reynolds is expressed as:

( ) (

) (3.32)

where and are geometric parameters that vary with the diameter ratio

( ⁄ ). The geometric parameters are determined as:

33

(3.33)

(3.34)

3.3. MODEL VALIDATION

The prediction regression model is first compared with numerical

solutions to confirm its validity. Figures 3.4 to 3.7 compare predictions of the

regression model with numerical results for different fluids. Results show

excellent agreement between the model and numerical solutions. The maximum

discrepancy is approximately 10%.

Figure 3.4: Comparison of predictions of numerical and regression models (Newtonian

fluids)

0.01

0.10

1.00

10.00

0.01 0.10 1.00 10.00

Fri

ctio

n F

act

or

f (N

um

eri

cal

So

luti

on

)

Friction Factor f (Regression Model)

Fluid N1

Fluid N2

Fluid N3

10%

-10%

34

Figure 3.5: Comparison of predictions of numerical and regression models (Power-Law Fluids)

Figure 3.6: Comparison of predictions of numerical and regression models (Bingham Plastic

Fluids)

0.00

0.01

0.10

1.00

10.00

0.00 0.01 0.10 1.00 10.00

Fri

ctio

n F

act

or

(Nu

me

rica

l S

olu

tio

n)


Fluid A1

Fluid A2

Fluid A3

Fluid A4

Fluid B1

Fluid B2

Fluid B3

Fluid B4

Fluid P1

Fluid P2

Fluid P3

10%

-10%

0.00

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00 100.00

Fri

ctio

n F

act

or

f (

Nu

me

rica

l So

luti

on

)


Fluid E1

Fluid E2

Fluid E3

Fluid E4

Fluid E5

Fluid E6

Fluid F1

Fluid F2

10%

-10%

35

Figure 3.7: Comparison of predictions of numerical and regression models (Yield-Power-

Law Fluids)

In order to further evaluate the performance of the regression model, its

predictions have been also compared with predictions of existing models.

Newtonian surge pressure predictions have been compared with the analytical slot

flow (Fig. 3.8) solution for Newtonian fluids (Bourgoyne, 1986). Excellent

agreement is observed for all tested cases.

0.00

0.01

0.10

1.00

10.00

0.01 0.10 1.00 10.00

Fri

ctio

n F

act

or

f (N

um

eri

cal

So

luti

on

)


Fluid C1

Fluid C2

Fluid C3

Fluid C4

Fluid D1

Fluid D2

Fluid D3

Fluid D4

-10%

10%

36

Figure 3.8: Comparison between correlation and Newtonian model (Bourgoyne, 1986)

Predictions

For the case of power-law fluids, surge pressure predictions are compared

with the exact numerical solution presented by Schuh (1964). His model has been

validated by direct comparison with field measurements (Fontenot and Clark

1974). Predictions show excellent agreement with the numerical results. Most of

the predictions fall within ±10% error bars (Fig. 3.9). A detailed calculation

procedure of Schuh’s model is presented in Appendix A.

0.001

0.010

0.100

1.000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Su

rge

Pre

sure

Gra

die

nt

(psi

/ft

)

Tripping Speed(ft/sec)

μ=203.7cp (Regression Model)



μ=203.7cp (Bourgoyne, 1986)



37

Figure 3.9: Comparison of Schuh’s solution with regression model for different Power-Law

Fluids

To evaluate model performance with Bingham plastic fluid, model

predictions are compared (Fig. 3.10) with those obtained from Burkhardt’s model.

Predictions of the new model are predominantly between error bars,

demonstrating excellent agreement. As expected, surge and swab pressures

increase with tripping speeds.

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800

Su

rge

Pre

sure

Re

gre

ssio

n M

od

el

(psi

)

Surge Pressure Schuh's Solution (psi)

Fluid A1

Fluid A2

Fluid A3

Fluid A4

Fluid B1

Fluid B2

Fluid B3

Fluid B4

Fluid P1

Fluid P2

Fluid P3

Fluid P4

10%

-10%

38

Figure 3.10: Comparison of correlation predictions and solution by Burkhardt (1961) for

different Bingham-Plastic Fluids

Surge and swab measurements with YPL fluid are very scarce. For YPL

fluids, the new model predictions are compared (Fig. 3.11) with results of back

extrusion experiments (Osorio and Steffe, 1991) that were obtained using 2.0%

aqueous solution of Kelset (commercial sodium-calcium alginate). Despite very

low extrusion pipe speeds, good agreement is obtained between the model and

experimental observations.

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500 600 700 800 900 1000

Sw

ab

Pre

sure

Co

rre

lati

on

(p

si)

Swab Pressure Burkhardt Model (psi)

Fluid E1

Fluid E2

Fluid E3

Fluid E4

Fluid E5

Fluid E6

Fluid F1

Fluid F2

Fluid F3

10%

-10%

39

Figure 3.11: Comparison of the correlation predictions with the back extrusion technique

(Osorio and Steffe, 1991) for a specific Yield-Power-Law Fluid

Generally, it is considered that the narrow-slot modeling approach is

applicable when the annular diameter ratio is high (i.e. greater than 0.3). To test

this hypothesis, the new model predictions are compared with exact numerical

solutions (Schuh, 1964) as depicted in Fig. 3.12. Model predictions are consistent

with the numerical solutions at high diameter ratios. However, as the diameter

ratio approaches the value of 0.3, discrepancies become substantial. Models based

on the narrow slot approximation over predict the surge pressure. Similar results

have been obtained for different fluids. It is also shown that the predictions of the

regression model and Chukwu’s model are very close, as both rely on the narrow-

slot approximation. Model comparison for Bingham plastic fluids (Fig. 3.13)

0.5

1.0

1.5

2.0

2.5

0.000 0.005 0.010 0.015 0.020 0.025 0.030

Pre

ssu

re D

rop

(p

si)

Pipe Velocity (ft/sec)

Regression Model

Osorio and Steffe, 1991

40

shows good agreement with the existing model (Burkhardt, 1961). It is important

to note that the predictions of Burkhardt’s model were previously validated using

field measurements. A detailed calculation procedure of Burkhardt’s model is

presented in Appendix B.

Figure 3.12: Comparison of different models for Power-Law Fluid

10.0

100.0

1000.0

10000.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Su

rge

Pre

sure

(p

si)

Diameter Ratio (dp/dh)

Regression Model

Chukwu, 1989

Schuh, 1964

41

Figure 3.13: Comparison of different models for Bingham Plastic Fluid

3.4. PARAMETRIC STUDY

The relationship between pressure surges and pipe velocities depends on a

number of drilling parameters including fluid rheology and borehole geometry.

After validating the model, sensitive analysis was carried out to examine the

influence of fluid behavior index, yield stress and diameter ratio on these

30

300

3000

0.1 0.3 0.5 0.7 0.9

Su

rge

Pre

sure

(p

si)


Regression Model

Burkhardt, 1961

42

pressures under different conditions. Figure 3.15 is a plot of surge pressures

versus diameter ratio for different power-law fluids that have the same

consistency index. It is shown that surge pressure and its sensitivity to trip speed

decreases as the fluid becomes more shear thinning. Therefore, in addition to the

trip speed and fluid rheology, adjustment may be considered to mitigate excessive

downhole pressure surges. Results are also in agreement with the predictions of

Schuh’s model.

Figure 3.14: Effect of fluid behavior index on surge pressures for Power-Law Fluids at

different speeds

The sensitivity analysis for YPL fluids was performed considering a set of

field data (White and Zamora, 1997) as the base case input. Figure 3.16 presents

predictions of the new model showing the effect of yield stress on surge pressure.

10

100

1000

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Su

rge

Pre

sure

(p

si)


n=0.3 (Regression Model) n=0.4 (Regression Model)n=0.5 (Regression Model) n=0.6 (Regression Model)n=0.7 (Regression Model) n=0.8 (Regression Model)n=0.3 (Schuh, 1964) n=0.4 (Schuh, 1964)n=0.5 (Schuh, 1964) n=0.6 (Schuh, 1964)n=0.7 (Schuh, 1964) n=0.8 (Schuh, 1964)

43

As anticipated, at high yield stress values, the surge pressure increases and the

influence of pipe velocity diminishes as the fluid becomes more shear thinning.

This is consistent with the previous observation with power law fluids.

Figure 3.15: Surge pressures vs. tripping speed for different yield stresses

The diameter ratio is expected to have a stronger influence on pressure

surges. Model predictions shown in Fig. 3.17 indicate that surge and swab

pressures become higher when the annular clearance gets smaller. Moreover, at

high diameter ratios these pressures become more sensitive to the increase in trip

velocity indicating the severity of reciprocation of a fully closed drillstring in

wellbores with small annular clearance such as in the case of casing drilling.

10

100

1000

1 2 3 4 5

Sw

ab

Pre

ssu

re, P

si

Pipe Velocity, ft/sec

τ₀= 0 lbf/100 ft²

τ₀= 5 lbf/100 ft²

τ₀= 10 lbf/100 ft²

τ₀= 20 lbf/100 ft²

τ₀= 30 lbf/100 ft²

44

Figure 3.16: Surge pressure vs. diameter ratio for different tripping speeds

10

100

1000

10000

0.4 0.5 0.6 0.7 0.8

Su

rge

Pre

ssu

re (

psi

)


Vp = 1.0 ft/sec

Vp = 2.0 ft/sec

Vp = 3.0 ft/sec

Vp = 4.0 ft/sec

Vp = 5.0 ft/sec

45

4. EXPERIMENTAL STUDY

This investigation is aimed at studying both experimentally and

theoretically the effects of fluid properties and drilling parameters on surge and

swab pressures. To achieve the objectives of the investigation and validate the

predictions of the new model, experiments were carried out under fully controlled

laboratory conditions. All tests were conducted at ambient temperature and

pressure.

4.1. EXPERIMENTAL SETUP

The experimental study was conducted at the Well Construction

Technology Center (WCTC) of the University of Oklahoma. A test setup has been

developed (Fig. 4.1) to carry out the proposed experiments. The setup has the

capability to vary the tripping speed and accurately measure surge or swab

pressure. A schematic of the setup is shown in Fig. 4.2. It consists of: i) vertical

test section; ii) guiding rod; iii) variable speed motor; iv) pressure transducer; v)

data acquisition system; and vi) fluid mixing and collection tanks.

46

Figure 4.1: Schematic view of experimental Setup

Drillpipe

Transparent

Polycarbonate

Tube

Guiding Rod

Motor

Controller

Motor

Water

Mixing Tank

Collector Tank Computer

Pressure

Transducer

P

Polymer

Cable Guide

47

Figure 4.2: Actual view of experimental setup

4.1.1. Vertical Test Section

A 10-ft vertical test section is formed by a fully transparent polycarbonate

tubing (2 inches ID) acting as the casing or borehole and inner steel pipe (1.32

inches OD) acting as a drillstring (Fig. 4.3). The test section is clamped to a

supporting structure. It is vertically aligned to keep the inner pipe in concentric

48

configuration. Blind flange and drain valve are installed at the bottom of the test

section. The flange supports the guiding rod.

Figure 4.3: Vertical test section

4.1.2. Guiding Rod

In order to ensure concentric annular geometry as assumed by the

presented model, a thin guiding rod (0.25-in OD) is used (Fig. 4.3). The guiding

rod is bolted at the center of the blind flange. The bottom the pipe was plugged

Guiding Rod

Inner Pipe

Polycarbonate Tubing

Drainage

49

and 0.27-inch hole was made for the guide rod. The guide protects the lateral

movement of the pipe during the test.

4.1.3. Variable Speed Motor

A variable speed motor (Fig. 4.4) with a controller lift the inner pipe at the

desired speed (0 - 1.0 ft/s) with accuracy of 0.01 ft/sec. The motor has a pulley

with a thin hosting cable (1/16-in steel cable) to move the pipe upward or

downward by switching the direction of the motor rotation. The test setup allows

a maximum stroke of 4.0 ft.

Figure 4.4: Variable speed motor

50

4.1.4. Pressure Transducer

A digital Pressure transducer (Fig. 4.5) is connected to the test unit to

measure the pressure differential across the annular section. The maximum

differential pressure span for the transducers is 0 - 1.0 psi with accuracy of

0.005 psi.

Figure 4.5: Differential pressure transducers with pressure tapings

4.1.5. Data Acquisition System

A Data Acquisition System consists of a personal computer (Fig. 4.6) and

a data acquisition card (Omega DAQ 3000) was used to record test parameters

51

and control the pipe speed. Measurements are displayed and recorded as a

function of time using Visual Basic for Applications (VBA) program. The

tripping speed is set in the VBA program before the test. Then, the controller

switch is used to start the motor. As the motor turns, the pipe moves downward

while the pressure transducer readings (i.e. pressure drop across the annulus) are

being recorded by the VBA program at the rate of 5 samples/second.

Figure 4.6: Personal computer

4.1.6. Fluid Mixing and Collection Tanks

Newtonian and non-Newtonian test fluids were prepared and stored in a 2-

gallon mixing tank prior to transfer to the test section. After the experiments, the

52

test fluid is discharged from the test section through a drain valve (Fig. 4.3) for

appropriate disposal using a waste collector tank.

4.2. SYSTEM CALIBRATION

As previously mentioned, pipe speed is controlled using a motor speed

controller (Variable Frequency Drive) which receives analog signal from the

DAQ system. To carry the experiments at the desired pipe speed, the system was

first calibrated to develop a relationship between controller input voltage and

measured pipe speed. The calibration was conducted by varying the voltage and

measuring the travel time for full stroke (4 ft) using a digital chronometer (stop

watch) to determine the pipe velocity. The procedure was repeated three times per

voltage value and the respective average pipe speed was calculated. The result

shows that the speed linearly varying with voltage. Expressions for upward (Fig.

4.7) and downward (Fig. 4.8) pipe speed as a function of voltage were developed:

…….......…..…………………….. (4.1)

…….......……….……........….. (4.2)

Both expressions are implemented into the VBA program in the data acquisition

system to accurately control the pipe speed.

53

Figure 4.7: Upward pipe speed as a function of voltage

Figure 4.8: Downward pipe speed as a function of voltage

4.3. TEST PROCEDURE

Preliminary test were conducted to develop a test procedure to measure

surge pressures. After establishing reliable and accurate procedure, the main

Vp (ft/sec) = 0.2514*Volt(v) + 0.0279 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10

Sp

ee

d (

rpm

)

Volt (v)

Vp (ft/sec) = 0.2498*Volt(v) + 0.0302

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 2 4 6 8 10

Sp

ee

d (

rpm

)

Volt (v)

54

experiments were carried out. All tests were performed out using the same

procedure that consists of the following steps:

Fluid preparation: Each experiment begins by preparing the test fluid with

the desirable polymer concentrations. First, powder polymer and water were

mixed in a tank that has a variable speed agitator. Adequate time was

allowed for the mixture to fully hydrate. The fluid rheology was measured

using a Fann 35 rotational viscometer (Fig. 4.9).

Fluid Transfer: After the test fluid was prepared, it was transferred to the

test section. This process is done carefully so there is no formation of

bubbles along the annular space. The fluid was left for 15 minutes in the

cylinder to allow any air bubbles to escape. A fluid sample was collected

during and after the test to check for any possible change in rheology

under test conditions.

Surge Pressure Test: Surge test begins by moving the inner pipe downward

at the desired speed while measuring and recording the pressure loss.

55

Figure 4.9: Fann 35 rotational viscometer

4.5. TEST MATERIALS

Extensive experiments were performed using regular and light-weight

mineral oil (Newtonian fluids), and different concentrations of polyanionic

cellulose (PAC) and Xanthan Gum suspensions (XG). The rheological properties

of the fluids tested were measured using a standard rotational viscometer (Model

35) that has a diameter ratio of 0.936. The dial readings were converted to obtain

wall shear stress values, using the following equation:

..……….…………...……………...………….. (4.3)

56

where is the wall shear stress ( ), is the spring factor and is the dial

reading. Wall shear rates were calculated from the speeds of the sleeve using the

following equation:

…...…………...…………...………...……….. (4.4)

where, is the wall shear rate ( ). Logarithmic curve fitting of wall shear

stress versus wall shear rate were made to determine the rheological parameters of

the fluids. All rotational viscometer readings are presented in Appendix C.

As expected viscosities of regular and light weight mineral oils were

constant and 203.7 cP and 24.1 cP, respectively. PAC based fluids show

considerable shear thinning (Fig. 4.10). The flow behavior of the PAC based

fluids best fit the power-law rheology model. Behavior of Xanthan Gum

suspensions is best represented by the yield-power-law (Herschel-Bulkley) model

(Fig. 4.11). Three different concentrations of PAC (1.00%, 0.75% and 0.56% by

weight) and Xanthan Gum fluids (1.00%, 0.67% and 0.44% by weight) were

tested. Also, a polymer mix of Xanthan Gum and PAC (0.28% PAC and 0.22%

Xanthan Gum by weight) was considered in this study. Rheological properties of

the fluids are presented in Table 4.1.

57

Figure 4.10: Rheology of PAC based test fluids

Figure 4.11: Rheology of Xanthan Gum based test fluids

0

50

100

150

200

250

0 200 400 600 800 1000

Sh

ea

r S

tre

ss (

lbf/

10

0ft

2)

Shear Rate (1/s)

PAC 1.00%

PAC 0.75%

PAC 0.56%

Mix PAC 0.28% + Xantan Gum 0.44%

1

10

100

1000

3 30 300

Sh

ea

r S

tre

ss (

lbf/

10

0ft

2)

Shear Rate (1/s)

Xantan Gum 1.00%

Xantan Gum 0.67%

Xantan Gum 0.44%

58

Table 4.1: Rheology of test fluids

Test Fluids Fluid Type Temperature

(°F)

Rheological Parameters

(lbf/100ft2) K

(lbf.sn/100ft2) n

Mineral Oil Newtonian 74 0.0 0.43 1.00

Light Mineral Oil Newtonian 75 0.0 0.05 1.00

1.00% PAC Power-Law 75 0.0 4.72 0.57

0.75% PAC Power-Law 75 0.0 1.44 0.67

0.56% PAC Power-Law 75 0.0 0.36 0.80

0.28% PAC+0.22% XG Power-Law 75 0.0 1.62 0.50

1.0% Xanthan Gum Yield-Power-Law 75 38.9 1.61 0.52



4.4. RECORDED DATA PROCESSING

Surge pressure was measured at each tripping speed. To avoid the effects

of pipe acceleration, enough time was allowed to stabilized and reach steady-state

conditions. Fig. 4.12 shows a typical pressure loss measurement. As the pipe

begins to move, first the fluid particles accelerate and the pressure loss increase

with time for a short period. Then, the flow establishes steady state condition and

the pressure loss becomes constant. Average pressure reading under steady state

condition was determined for each tripping speed. Steady state flow conditions

were established during low tripping speed (0.1 ft/s to 0.5 ft/s) tests. As depicted

in Fig 4.13, at high tripping speeds, it was not possible to establish state flow

59

0.00

0.10

0.20

0.30

0 20 40 60 80 100

Su

rge

Pre

ssu

re (

psi

)

Time (s)

Readings

Average

conditions due stroke length limitation. As a result, experiments were limited to

the maximum trip speed of 0.7 ft/s.

Figure 4.12: Surge pressure measurement with established steady flow condition

(1% PAC; 0.2 ft/s)

Figure 4.13: Surge pressure measurement without established steady flow condition

(Mineral Oil; 0.7 ft/s)

0.00

0.10

0.20

0.30

10 20 30 40 50 60

Su

rge

Pre

ssu

re (

psi

)

Time (s)

Readings

Average

Steady-State

Condition

Unsteady-State Condition

60

5. RESULTS AND DISSCUSSION

The presence of the guiding rod slightly reduces the displaced fluid flow rate.

Therefore, the displaced fluid flow rate equation (Eqn. 3.21) needs to be modified

to account for the effect of the guiding rod as:

(

) (5.1)

Then, the dimensionless total fluid flow rate during the experiment is computed

as:

(5.2)

Experimental measurements obtained from the test fluids (Newtonian and

non-Newtonian fluids) have been analyzed and presented in Figure 5.1 as the

friction versus generalized Reynolds number (Re). Experimental results were

highly correlated with the regression model ( ). The strong correlation

between the regression model line and the experimental data points indicate that

61

laminar flow is observed during the measurements. It also shows that the accuracy

of the experimental measurements.

To perform a comparative study between test measurements and model

predictions, surge pressure predictions were made for all experimental data points.

Predictions from others studies have been included in the analysis. All recorded

surge measurements are presented in Appendix D.

Figure 5.1: Friction Factor vs. Generalized Reynolds Number for experimental data

5.1. NEWTONIAN FLUIDS

Newtonian test results have been compared with predictions obtained with the

theoretical and regression model predictions. Both sets of data show a satisfactory

0.01

0.10

1.00

10.00

100.00

0 1 10 100 1000

Fri

ctio

n F

act

or

f

Generalized Reynolds Number Re

Light Mineral Oil

Mineral Oil

1% PAC

0.75% PAC

0.56% PAC

Mix 0.22% PAC + 0.22 XG

1% Xantan Gum

0.64% Xantan Gum

0.22 Xantan Gum

f = 16 / Re*

62

agreement with measurements over wide range of tripping speeds. In order to

revalidate the results for Newtonian fluids, measurements are compared with

exact numerical solutions (Appendix E). Surge test results with regular mineral oil

and light mineral oil are depicted in Fig. 5.2 and Fig. 5.3, respectively.

Discrepancies between measurements and exact numerical solutions are very

small.

Figure 5.2: Surge pressure gradient vs. trip speed with regular mineral oil.

0.0

0.1

0.2

0.3

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)

Pipe Velocity (ft/s)

Measurements

Regression Model

Theoretical Model

Newtonian Solution

63

Figure 5.3: Surge pressure gradient vs. trip speed with light mineral oil

5.2. POWER-LAW FLUIDS

For power law fluids (i.e. fluids with flow behavior that best fit the power law

rheology model), the performances of the regression and theoretical model are

evaluated (Figs. 5.4 to 5.6) using experimental results and exact numerical

solutions (Schuh, 1964). Model predictions show excellent agreement with

measurements and numerical results for thick test fluids (1.00% and 0.75% PAC

suspensions). However, for thin suspension (0.56% PAC), we observed

significant difference between predictions and test results. The flow behavior of

this fluid has been characterized by curve fitting the viscometeric measurements

to the power-law rheology model. However, it is observed that there is significant

deviation between the fitted curve and the actual data points at low shear rates.

0.00

0.01

0.02

0.03

0.04

0.05

0.3 0.4 0.5 0.6

Du

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

Newtonian Solution

64

These deviations could be the cause of discrepancies between surge pressure

measurements and predictions. For this case, it could be more appropriate to use

other constitutive equations such the Ellis model that best fits the rheology

measurements of polymeric fluids at low and medium shear rates (Matsuhisa and

Bird, 1965). For the polymer mix (0.28% PAC and 0.22% Xanthan Gum by

weight), results show good agreement between test results and predictions (Fig.

5.7).

Figure 5.4: Surge pressure gradient vs. trip speed with 1.0% PAC

0.0

0.1

0.2

0.3

0.4

0.5

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

Schuh, 1964

65



0.0

0.1

0.2

0.3

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

Schuh, 1964

0.00

0.05

0.10

0.15

0.20

0.2 0.3 0.4 0.5 0.6 0.7

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

Schuh, 1964

66

Figure 5.7: Surge pressure gradient vs. trip speed for polymer mix

5.3. YIELD-POWER-LAW FLUIDS

To further evaluate the performance the new model with YPL fluids,

experiemtal measurements obtained using Xanthan Gum suspensions are

compared (Figs. 5.8 to 5.10) with model predictions. For test fluid with the

lowest yield stress (0.44% Xanthan Gum suspension), a satisfactory agreement

between measurements and predictions has been observed. However, for fluids

with higher yield stresses (1.00% and 0.67% Xanthan Gum suspensions)

predictions are slightly higher (10% to 15%) than measurements. One possible

explanation for the discrepancies could be overestimation of the yield stress

resulting from the regression technique that uses very limited data points at very

0.00

0.05

0.10

0.15

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

Schuh, 1964

67

low shear rates. Accurate viscometric data is necessary for better validation of the

new model.

Figure 5.8: Surge pressure gradient vs. trip speed with 1.0% Xanthan Gum

0.0

0.1

0.2

0.3

0.4

0.5

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

68

Figure 5.9: Yield Surge pressure gradient vs. trip speed with 0.67% Xanthan Gum

Figure 5.10: Surge pressure gradient vs. trip speed with 0.44% Xanthan Gum.

0.00

0.05

0.10

0.15

0.20

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

0.00

0.05

0.10

0.15

0.1 0.2 0.3 0.4 0.5

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)


Measurements

Regression Model

Theoretical Model

69

It is important to note that experimental tests were not performed for

Bingham-Plastic fluids. However, results obtained for Xanthan Gum solutions

and correlation validation for this special case lead us to establish that the

correlation provides good results for this fluid rheology. Experimental

measurements are presented in tables in Appendix C for all cases.

5.4. DISCUSSION

New theoretical and regression models have been developed to predict

surge and swab pressures and optimum safe trip velocities for yield-power law

fluids. Thus, time and operational cost is reduced and the possibility of kick and

lost circulation in tripping process or running a casing can be prevented. These

simplified models can be applied to many common field conditions and only

require knowledge of moving pipe velocities, fluid rheological parameters, and

wellbore geometry.

Surge and swab pressures are strongly affected by the flow behavior of the

drilling fluid. The yield-power-law rheology model describes the flow behavior of

most of drilling fluids better than other commonly used constitutive equations

such as power-law and Bingham plastic models. Especially, at low shear rates (i.e.

low trip speeds), the discrepancies between measurements and predictions can be

substantial and the use of yield-power-law model results in relatively accurate

predictions. Furthermore, the new model is valid for special cases of YPL fluids

70

such as Newtonian, Bingham plastic and power-law fluids, which makes the

model more versatile and applicable.

In addition to the properties of fluid, bottomhole pressure variations

during tripping strongly depend on the borehole geometry. Particularly, the

diameter ratio or annular clearance has considerable effect on pressure surge. It

has been shown that high diameter ratios (i.e. low annular clearances) make the

pressure variations very sensitive to the change in tripping speed. This condition

can be commonly observed during slimhole and low-clearance casing operations.

For horizontal and inclined wells, eccentricity of the drillpipe and

thickness of the cuttings bed need to be considered in the analysis to optimize the

trip speed. Eccentricity has a significant effect on surge and swab pressures.

During the experimental investigation, it was observed that when the inner pipe

was eccentric, surge pressures measurements were reduced as much as 42%

percent compared with a fully concentric test (Fig. 5.11). Adequate modeling of

eccentricity effects on surge and swab pressures can reduce significantly

unnecessarily low tripping speeds, reducing non-productive time and drilling cost

considerably.

71

Figure 5.11: Surge Pressures at different annular eccentricities (1.0% PAC; 0.2 ft/s)

The analysis the present investigation is based on steady state flow

assumption; hence, the surge and swab pressure predictions are only valid when

the tripping speed remains constant. In real drilling operations, pressure spikes

resulting from drillstring acceleration during the starting and ending periods of the

trip are observed. Therefore, transient flow (unsteady) models should be used in

order to estimate these pressure spikes. Also, in order to minimize pressure surge,

changes in trip speed should be gradual.

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 20 40 60 80 100

Pre

ssu

re (

psi

)

Time (s)

Concentric Annulus

Eccentric Annulus

72

The effects of gel strength (static time) on Xanthan Gum suspensions were

also studied. Surge pressure tests were run at 0.05 ft/s after shearing the fluid at

high speeds by reciprocation of the pipe and allowing it to rest for short (15

seconds) and long (6 minutes) periods. The same surge pressure values were

measured in both cases for different Xantham Gum based fluids (Figure 5.12).

Results show minimal gelling effect on surge pressure with Xantham Gum fluids.

Figure 5.12: Effect of static time on surge pressure measurements (1.0% Xanthan Gum; 0.05

ft/sec)

It was also observed that when the pipe was brought to rest the pressure

transducers did not record zero pressure drop across the test section as other test

fluids. This is due to the effect of the yield stress of the test fluid that generates

static pressure difference between pressure tapings.

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0 2 4 6 8 10 12

Su

rge

Pre

ssu

re G

rad

ien

t (p

si/

ft)

Time (min)

Test After 6 min

Test After 15 s

Yield Stress Effect

73

A complete dynamic modeling should include drillstring elongation

caused by axial loading, drilling fluid and formation compressibilities, and other

mechanisms such as wellbore ballooning that may have some influence on the

bottomhole pressure response of the wellbore during tripping operations. Open-

end pipe geometry also needs to be included in surge and swab pressure analysis.

5.5. PRACTICAL IMPLICATIONS

Surge and swab pressures are very critical in designing slim holes, low

clearance casing operations and deepwater applications. In these cases, specifying

and maintaining a safe maximum tripping speed is an important part of the

drilling program. When running casing in these scenarios, excessive surge

pressure in the wellbore can occur. Thus, the bottomhole pressure can be

increased sufficiently to exceed the formation fracture gradient and often results

in fluid losses and well control issues. The use of a diverter valve above the liner

has been suggested as a possible solution to this problem. This surge reduction

tool mitigates this problem by diverting the fluid through the ports into the

annulus, allowing the casing or liner to be run much faster without the risk of

surging the formation excessively.

Ignoring the effect of eccentricity on surge and swab pressures may lead to

overestimation of tripping speeds. Inefficient tripping speeds increase non-

74

productive time and operation costs. Therefore, eccentricity effects must be taken

into account to minimize non-productive time and drilling cost.

Rheological properties of the fluid must be monitored during drilling to

avoid excessive surge and swab pressures. A highly gelled drilling fluid can

create significant swab and surge pressures even if pipe movement is minimal

(Ward and Beique, 2000). Continuous control of drilling fluid rheology

considering physical characteristics of bottomhole assembly (BHA) is the main

point for correct estimation of optimal pipe running speed during tripping

operations.

75

6. CONCLUSIONS AND RECOMMENDATIONS

6.1. CONCLUSIONS

New theoretical and regression models have been developed to predict

surge and swab pressure for yield-power law fluids. The theoretical model is

based on the narrow slot flow approximation. The regression model has been

developed from numerical solution of the theoretical model. To validate the

models and better understand surge and swab phenomena, experimental

investigation was carried out using different fluids. Based on the investigation, the

following conclusions can be made:

The present model accurately predicts surge and swab pressures in

comparison with other existing models that are only valid for Newtonian,

power-law and Bingham plastic fluids;

The model provides reasonable predictions when the diameter ratio is greater

than 0.4 due to the use of narrow slot approximation;

Rheology parameters such as yield stress, fluid behavior index and

consistency index have substantial effects on surge and swab pressures;

Tripping speeds and diameter ratio have substantial effects on surge & swab

pressures;

76

In horizontal and inclined wells, pipe eccentricity can reduce significantly the

value of surge and swab pressure.

For fluids with high yield stress, the influence of trip speed on surge and swab

pressures diminishes considerably.

6.2. RECOMMENDATIONS

This analysis is based on steady state flow assumption. It also uses narrow

slot approximation to represent concentric annular geometry. In order to improve

the accuracy of surge pressure predictions, the following effects must be

incorporated in the development of surge and swab pressure models:

Open-end pipe geometry;

Transient (unsteady-state) flow; and

Eccentricity effects.

77

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85

NOMENCLATURE

= Constant

= Geometric parameter

= Geometric parameter

= Bingham number

= Hole/Casing diameter

= Pipe diameter

= Guiding rod diameter

= friction factor

= Slot Thickness

= Consistency Index

= Diameter ratio ( ⁄ )

L = length of the wellbore

n = Fluid behavior index

N = Spring factor

86

= Flow rate

= Modified total flow rate

= Total flow rate

= Radius

= Dimensionless pressure gradient

= Surge/Swab pressure

Re = Modified Reynolds number

= Generalized Reynolds number

= Hole radius

= Pipe radius

= Fluid velocity in Region I

= Fluid velocity in Region

= Fluid velocity in Region III

= Velocity component due to displacement

= Downward pipe velocity

87

= Surface effective pipe velocity

Volt = Voltage

= Pipe velocity

= Total fluid velocity

= Upward pipe velocity

= Velocity component due to viscous drag

= Dimensionless velocity of Region I

= Dimensionless velocity of Region II

= Dimensionless velocity of Region III

= Slot width

= x-coordinate

= Dimensionless x-coordinate

= Lower limit of Region II

= Upper limit of Region II

= Dimensionless lower boundary limit of Region II

88

= Dimensionless upper boundary limit of Region II

Greek Letters

= Dimensional parameter

= Dimensional parameter

= Yield stress factor

= Conductance number

= Wall shear rate

= Fluid density

= Pi

= Dimensionless pressure

= Dimensionless plug thickness

= Dial reading

= Shear stress

= Shear stress at the wall

= Yield stress

89

= Shear rate

= Slot length/Wellbore depth

= Pressure drop

Subscripts

h = Hole

= Pipe

= Total

r = Radius

90

APPENDIX A

SCHUH’S MODEL TO PREDICT SURGE AND SWAB PRESSURES FOR

POWER-LAW FLUIDS

In the annular space between the drillstring and the borehole, the fluid velocity is

given by:

(B-1)

where is the velocity component due to fluid displacement given by:

(

) (B-2)

and is the velocity component due to viscous drag. This velocity depends on

whether the velocity in the annular section results in laminar flow or turbulent

flow condition. If the flow is laminar, then the velocity due to viscous drag is

given by:

(

( ) ) (B-3)

For turbulent flow, the velocity due to viscous drag is calculated as:

(B-4)

91

Applying the Dogde and Metzner (1959) method, a modified Reynolds number

for the annular flow can be calculated with the following expression:

( )

*

+

(B-5)

The friction factors are calculated using the expressions:

For – (laminar flow):

(B-6)

For – (fully developed turbulent flow):

*

(

)

+ (B-7)

For ( – ) ( – ) (transitional flow):

*

( )

+ ( ) (B-8)

where is the friction factor for Reynolds Number at the end of the laminar flow

regime ( – ) and is the friction factor for Reynolds Number

at the beginning of the turbulent flow regime ( – ). The

friction factor must be calculated by trial and error using Eqn. (B-7). Then, the

surge/swab pressure is:

92

( ) (B-9)

93

APPENDIX B

BURKHARDT’S MODEL TO PREDICT SURGE AND SWAB

PRESSURESFOR BINGHAM PLASTIC FLUIDS

According to Melrose et al (1958) the Bingham number and the

conductance number can be obtained from the expressions:

( )

(C-5)

*

+

(C-6)

The friction factor is calculated as follows:

For (laminar flow):

(C-7)

where the Reynolds number is given by:

( )

(C-9)

Where is calculated from Eqn. (B-1)

For (turbulent flow):

[ ( √ ) ]

(C-8)

94

Eqns. (C-6) and (C-8) must be calculated by trial and error. Finally, the friction

factor is used to obtain the surge/swab pressure as given by:

( ) (C-10)

95

APPENDIX C

ROTATIONAL VISCOMETER MEASUREMENTS OF TEST FLUIDS

Table D.1: Fann model 35 (#1/5 spring) measurements for light mineral oil

N Θ τ

(rpm) (reading) (1/sec) (lbf/100 ft^2)

3 0.3 5.1 0.32

6 1.1 10.2 1.17

30 2.6 51.1 2.77

60 5.1 102.2 5.44

90 7.6 153.3 8.10

100 8.3 170.3 8.85

180 15.0 306.5 15.99

200 16.4 340.6 17.48

300 24.4 510.9 26.01

600 47.8 1021.8 50.95

Table D.2: Fann model 35 (#1 spring) measurements for mineral oil

N θ τ


3 1.8 5.1 1.87

6 3.8 10.2 4.00

100 62.5 170.3 66.63

200 135.5 340.6 144.44

300 206.0 510.9 219.60

96

Table D.3: Fann model 35 (#1 spring) measurements for 1.0% PAC

N Θ τ


3 10.0 5.1 10.66

6 17.5 10.2 18.66

100 95.0 170.3 101.27

200 129.0 340.6 137.51

300 152.0 510.9 162.03

600 194.0 1021.8 206.80


N Θ τ


3 3.5 5.1 3.73

6 7.0 10.2 7.46

100 51.0 170.3 54.37

200 73.5 340.6 78.35

300 89.0 510.9 94.87

600 119.0 1021.8 126.85


N θ τ


3 1.0 5.1 1.07

6 2.5 10.2 2.67

100 25.5 170.3 27.18

200 40.0 340.6 42.64

300 50.5 510.9 53.83

600 72.0 1021.8 76.75

97

Table D.6: Fann model 35 (#1 spring) measurements for mix 0.28% + 0.22% Xanthan Gum

N θ τ


3 3.5 5.1 3.73

6 4.8 10.2 5.06

100 19.5 170.3 20.79

200 28.5 340.6 30.38

300 35.0 510.9 37.31

600 48.0 1021.8 51.17

Table D.7: Fann model 35 (#1 spring) measurements for 1.0% Xanthan Gum

N θ τ


3 39.0 5.1 41.57

6 42.5 10.2 45.31

100 59.0 170.3 62.89

200 67.5 340.6 71.96

300 75.0 510.9 79.95

600 92.0 1021.8 98.07


N θ τ


3 19.0 5.1 20.25

6 21.0 10.2 22.39

100 31.0 170.3 33.05

200 38.0 340.6 40.51

300 43.0 510.9 45.84

600 53.0 1021.8 56.50

98


N θ g τ


3 8.0 5.1 8.53

6 9.5 10.2 10.13

100 17.0 170.3 18.12

200 21.0 340.6 22.39

300 25.0 510.9 26.65

600 32.5 1021.8 34.65

99

APPENDIX D

SURGE PRESSURE MEASUREMENTS

Table E.1: Surge pressure gradient readings for mineral oil

Vp ΔP/ΔL ΔP/ΔL

ft/s inH₂O/ft psi/ft

0.1 1.70 0.0613

0.2 2.95 0.1064

0.3 4.40 0.1587

0.4 5.80 0.2093

0.5 7.00 0.2526

Table E.2: Surge pressure gradient readings for light mineral oil

Vp ΔP/ΔL ΔP/ΔL


0.3 0.29 0.0207

0.4 0.38 0.0271

0.5 0.46 0.0334

0.6 0.56 0.0406

Table E.3: Surge pressure gradient readings for 1.0% PAC

Vp ΔP/ΔL ΔP/ΔL


0.1 4.80 0.1732

0.2 7.20 0.2598

0.3 9.00 0.3247

0.4 10.60 0.3824

0.5 11.20 0.4041

0.6 12.20 0.4402

100


Vp ΔP/ΔL ΔP/ΔL


0.1 2.30 0.0830

0.2 3.40 0.1227

0.3 4.40 0.1587

0.4 5.30 0.1912

0.5 5.80 0.2093

0.6 6.50 0.2345


Vp ΔP/ΔL ΔP/ΔL


0.1 1.30 0.0469

0.2 1.80 0.0649

0.3 2.40 0.0866

0.4 2.80 0.1010

0.5 3.20 0.1155

0.6 3.60 0.1299

0.7 3.80 0.1371

Table E.6: Surge pressure gradient readings for mix 0.28% + 0.22% Xanthan Gum

Vp ΔP/ΔL ΔP/ΔL


0.1 1.30 0.0469

0.2 1.80 0.0649

0.3 2.20 0.0794

0.4 2.50 0.0902

0.5 2.70 0.0974

0.6 2.90 0.1046

101

Table E.7: Surge pressure gradient readings for 1.0% Xanthan Gum

Vp ΔP/ΔL ΔP/ΔL


0.1 6.15 0.2219

0.2 6.90 0.2489

0.3 7.10 0.2562

0.4 7.60 0.2742

0.5 7.80 0.2814


Vp ΔP/ΔL ΔP/ΔL


0.1 3.40 0.1227

0.2 3.60 0.1299

0.3 4.00 0.1443

0.4 4.20 0.1515

0.5 4.25 0.1533


Vp ΔP/ΔL ΔP/ΔL


0.1 1.90 0.0685

0.2 2.20 0.0794

0.3 2.35 0.0848

0.4 2.50 0.0902

0.5 2.60 0.0938

102

APPENDIX E

SURGE AND SWAB PRESSURES IN CONCENTRIC ANNULAR

GEOMETRY WITH NEWTONIAN FLUIDS

For a concentric annular, the velocity profile is expressed as (Bourgoyne,

1986):

(A-1)

Applying the boundary conditions ( ( ) and ( ) ), the following

equations are obtained to determine the values of and and

(

)

( )

( ) (A-2)

*

(

)

( )

( ) + (A-3)

Then, the flow rate is obtained upon integration of the of the velocity profile:

{

(

)

*

+

(

)

} (A-4)

For closed pipe geometry, the flow rate is equal to the rate at which the

fluid is being displaced by the inner pipe. Hence:

103

(A-5)

The surge and swab pressure is obtained by solving simultaneously Eqns.

(A-2), (A-3), (A-4) and (A-5) numerically for a given combination of annular

geometry and fluid rheology.

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