**AIR AND WATER REQUIREMENTS **

**FOR- FOAM DRILLING OPERATIONS**

**By**

ProQuest N um ber: 10781730

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## uest

P roQ uest 10781730

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**SUBMITTAL**

**A Thesis submitted to the Faculty-and the Beard of Trustees **
**of the Colorado School of Mines in partial fulfillment of the r e **
**quirements for the degree of Masters of Science, Petroleum **

**Engineering.**
**Signed:_______**
**4^'Student**
**G o lden , Colorado **
**Date ***:Q ( e 'i&tldJj/f*

**, 19**

***7/****« t Q n A D O S C H O O L O F m q u**

**-’O L D E N ,****C O L O R A D O**

**G o lde n, Colorado**

**J L :****19**

**, 7 1****Approved**

**.**

**Thesis/ Advisor**

*A*

**o**

**d**

**-**

**/****Head of Department**

**T-1383**

**ABSTRACT**

**This study investigates a mathematical-model and its solution for **
**the application of foam in a drilling system consisting cf a vertical- **
**concentric pipe-hole arrangement with flow down the pipe and up the **
**a nnulus.** **The mathematical model which was written for foam was the **
**Bingham plastic model (Mitchell, 1970b) applied to a compressible fluid, **
**Foam-consisting of air, a surface active agent, and water, with the **
**aqueous solution being the continuous phase and the air being discon**
**tinuous bubbles, shall be considered. ** **Two equations were written de**
**scribing the flow of foam: ** **foam flowing down the pipe, and foam with**
**rock chips flowing up the annulus. ^ I n order to maintain nearly constant **
**fluid properties, the two equations were written - explicitly in terms of **
**pressure and depth and solved by iterations with a constant pressure in**
**crement. ** **For each iteration, the Theological properties of foam were **
**evaluated at the average increment pressure. ** **Adjustments to account **
**for the rock chips volume and mass were made to the annular Bingham **
**plastic flow, equation,**

**Air volume rate, water volume rate, and surface annular pressure **
**are the variables in the drilling system. ** **Surface injection pressure, **
**drilling rate, hole size, and pipe size were kept constant for each **
**solution. ** **Boundary conditions of foam flow of 1.5 fps at the bottom **
**of the annulus and a quality (ratio of air volume to water plus air **
**volume) of 0.96 at the surface of the annulus were imposed upon the **
**solution of the mathematical model, ** **With these boundary conditions,**

**the air and water volume rates, and surface annular pressure were **
**calculated using a high-speed digital computer. ** **Solutions for various **
**hole and pipe combinations, drilling rates, and surface injection **
**pressures are listed in tabular form at the end of this paper.**

**T-1383**

**ACKNOWLEDGMENTS**

**The author wishes to express his gratitude to Dr. B. J. Mitchell **
**for suggesting this investigation, for giving valuable guidance and **
**helpful assistance, and for serving on this thesis committee.**

**Many thanks are due to Dr. R. R. Faddick and C. A. Kohlhaas for **
**serving on this thesis committee and their valuable help.**

**TABLE OF CONTENTS**

**Page**

**Abstract ** **iii**

**Acknowledgements ** **v**

**List of Tables ** **vii**

**List of Figures ** **ix**

**I ** **Introduction ** **1**

**II ** **Review of Foam Rheology ** ***4**

**III ** **Method of Investigation ** **11**

**IV ** **Results ** **25**

**V ** **Discussion of Results ** **30**

**VI ** **Conclusions and Recommendations ** **32**

**Appendices**

**Appendix A Nomenclature ** **79**

**B Bingham Plastic Model ** **Flow Down**

**the Pipe ** **81**

**C Bingham Plastic Model ** **Flew Up**

**the Annulus ** **37**

**D Computer Program Listing ** **90**

**T-1383**

**LIST OF TABLES**

**Table ** **Page**

**1. ** **Shear Stress-Shear Rate Relationships For**

**Drilling Foam ** **5**

**Air and Liquid Requirements For Foam Drilling**
**Hole Size 6.75 in ** **Pipe ** **Size 2.87 in**

**2. ** **Injection Pressure 200 psi ** **34**

**3. ** **" ** **" ** **300 " ** **35**

**4. ** **" ** **" ** **400 " ** **36**

**Hole Size 6.75 in ** **Pipe ** **Size 3.50 in**

**5. ** **Injection Pressure 200 psi ** **37**

**6 .** **~ " ** **" ** **300 ** **" ** **38**

**7. ** **n ** **" ** **400 " ** **3Q**

**Hole Size 7.87 in ** **Pipe ** **Size 2.87 in**

**8 .** **Injection Pressure 200 psi' ** **40**

**9. ** **" ** **" ** **300 ‘ " ** **41**

**10. ** **" ** **” ** **400 " ** **42**

**Hole Size 7.87 in ** **Pipe ** **Size 3.50 in**

**11. ** **Injection Pressure 200 psi ** **43**

**12. ** **" ** **” ** **300 M ** **44**

**13. ** **" ** **" ** **400 " ** **45**

**Hole Size 7.87 in ** **Pipe ** **Size 4.50 in**

**14. ** **Injection Pressure 200 psi ** **46**

**15. ** **" ** **M ** **300 M ** **47**

**16. ** **" ** **" ** **400 " ** **48**

**Hole Size 9.00 in ** **Pipe ** **Size 3.50 in**

**17 ** **Injection Pressure 200 psi ** **49**

**18. ** **" ** **" ** **300 " ** **50**

**19. ** **" ** **" ** **400 " ** **51**

**Hole Size 9.00 in ** **Pipe,Size ** **4.50 in**

**20. ** **Injection Pressure 200 psi ** **52**

**21. ** **" ** **" ** **300 ” ** **53**

**22. ** **" ****■'****400 M ** **54**

**Hole Size 9.00 in ** **Pipe ** **Size 5,50 in**

**23. ** **Injection Pressure 200 psi ** **55**

**24. ** **" ** **" ** **300 " ** **56**

**25. ** **n ** **” ** **400 M ** **57**

**Table**
**26.**
**27.**
**28.**
**29.**
**30.**
**31.**
**32.**
**33.**
**35.**
**36. **
**37;**
**38.**
**39.**
**40.**
**41**
**42.**
**43.**
**44,**
**45,**
**46,**
**LIST OF TABLES**
**Page**
**Hole Size ** **9.87 in ** **Pipe Size 3.50 in**

**Injection Pressure ** **200 psi ** **58**

**” ** **" ** **300 " ** **59**

**" ** **» ** **400 ” ** **60**

**Hole Size ** **9.87 in ** **Pipe Size 4.50 in**

**Injection Pressure 200 psi ** **61**

**" ** **” ** **300 " ** **62**

**” ** **" ** **400 ?r ** **63**

**Hole Size ** **9.87 in ** **Pipe Size 5.50 in**

**Injection Pressure ** **200 psi ** **64**

**" ** **» ** **300 11 ** **65**

**" ** **" ** **400 " ** **66**

**Hole Size ** **12.50 in ** **Pipe Size 4.50 in**

**Injection Pressure ** **200 psi ** **67**

**" ** **” ** **300 " ** **58**

**" ** **" ** **400 " ** **69**

**Hole Size ** **12.50 in ** **ripe Size 5.50 in**

**Injection Pressure ** **200 psi ** **70**

**n ** **it ** **300 ** **71**

**ft ** **i-QO " ** **72**

**Hole Size ** **15.00 in ** **Pipe Size 4.50 in**

**Injection Pressure ** **200 psi ** **73**

**m ** **m ** **300 •' ** **74**

**” ** **" ** **400 M ** **75**

**Hole Size ** **15.00 in ** **Pipe Size 5.50 in**

**Injection Pressure 200 psi ** **76**

**" ** **» ** **300 " ** **77**

**T-1383**

**LIST OF FIGURES**

**Figure ** **Page**

**1. ** **Schematic Diagram of Idealized Foam System ** **3**

**2. ** **Yield Stress vs. ** **Foam Quality, ** **7**

**3. ** **Plastic Viscosity vs. ** **Foam Quality ** **8**

**4. ** **Foam System in Down Pipe ** **10**

**5. ** **Forces Acting on Downward Flow In a Pipe ** **12**
**6 . ** **Air and Liquid Requirements For Foam Drilling**

**Hole Size 9.00 in ** **Pipe Size ^.SO in**

**Injection Pressure 200 psi ** **25**

**7. ** **Air and Liquid Requirements For Foam Drilling**
**Hole Size 9.00 in ** **Pipe Size 5.50 in**

**Injection Pressure 200 psi ** **27**

**8 .** **Flow Of a Bingham Fluid In a Circular Tube ** **82**
**9. ** **Rectilinear Flow Between Fixed Parallel Plates ** **88**

**CHAPTER ONE **
**INTRODUCTION**

**The use of foam in drilling operations in the petroleum industry **
**dates back to the early trials of air drilling during the late lOM-O's. **

**(P. Moore, 1966, p. lM-A). ** **Foam was used in air drilling as a means of **
**stabilizing a sloughing formation and controlling natural water flows **
**in the w e l l b o r e during drilling operations. ** **If air.were used to drill **
**an unconsolidated formation, the hole would usually cave. ** **When a water **
**producing formation was drilled with air, the wetted cuttings often **
**stuck to the drillpipe and the wall of the hole until circulation was **
**lost and the pipe was packed in the hole. ** **Foam was used to minimise **
**these problems. ** **Foam has proven itself to have an excellent capacity **
**to stabilize a sloughing formation. ** **(Dresser Magcobar, 1970, p. i). **
**Foam, when used in the drilling of a water producing formation, has a **
**tendency to use the water which flows into the wellbore as part of the **
**water constituent of the foam. ** **The cuttings are not wetted by the **
**water and do not stick to the drillpipe or the wall of the hole. ** **At **
**the present time (1971), there is little published literature de**
**scribing the practical application for a foam drilling operation.**

**In this study, foam consisting of air, a surface active agent, **
**and wafer shall be considered. ** **The surface active agent and the **
**water are the continuous phase with the air appearing as discontin**
**uous bubbles, ** **Mist shall be defined as a fluid consisting of the **
**above components, except the air shall be the continuous phase and**

**T-1383**

**the aqueous solution shall appear as droplets.**

**In the practical application of foam, the following variables **
**must be controlled: ** **1) air volume, 2) aqueous volume, 3) injection**
**pressure, and 4) annulus choke-pressure,**

**It is the purpose of this study to (present a mathematical^ **

**model and its solution for the application of foam in a system having **
**the following components: ** **vertical hole, concentric annulus between**
**the hole and the pipe, and continuous circulation of foam down the **
**pipe and up the annulus carrying the wetted cuttings from the bottom **
**of the pipe. ** **(See Figure 1). ** **The solution of the mathematical model **
**was computed by selecting air and aqueous volumes for various pipe and **
**hole configurations. ** **Using an iteration process, the aqueous volume, **
**air volume, and surface annulus pressure have been calculated for a **
**depth range of 1,000 ft, to 4,000 ft., a surface injection pressure of **
**200 psi to 400 psi, and a drilling rate of 0 fps to 1.5 fps,**

**Fipe Flow of Foam**

**Annular Flow of Foam **
**and Cuttings**

**--D**

**FI CURS: ** **1 **

**SCHEMATIC DIAGRAM OF IDEALIZED **
**FOAM SYSTEM**

**T-1383**

**CHAPTER TWO **
**REVIEW OF FOAM RHEOLOGY **

**Mitchell (1970a, p. 117-123) has shown through laboratory **

**measurements that foam has a linear relationship of shear rate-shear **
**stress for all foam qualities and all shear rates above 20,000 sec ^ **
**For shear rates below 20,000 sec “ , 'the shear rate-shear stress rela**
**tionship is linearized by subtracting the proper constant from the **
**measured shear stress (x). ** **The power law model equation which de**
**scribes the rheology of foam is**

**(i ** **x ) = M ** **1**

**Y ** **P**

**where x = measured shear stress **
**x ** **= yield shear stress**

**y**

**n = power coefficient**
**ii.p_= .plastic viscosity **

**<p = shear rate **
**The log form of Equation (1) is**

**log (x -x ) = nlog ($) + log (pi ) ** **2**

**y ** **P**

**Power coefficients, plastic viscosities, and yield stresses are listed **
**in Table 1 for a foam quality range of 0 to 0.96.**

**Based upon the data shown in Table 1 , the power coefficient (n) **
**has been chosen to be unity for all foam qualities. ** **Therefore, **
**Equation (1) is equivalent to the shear rate-shear stress relation**
**ship for a Bingham plastic model ** **Figures 2 and 3 show th e: relation**
**ship between foam quality ( D , ratio of air volume to the air volume**

**TABLE 1: ** **SHEAR STRESS - SHEAR RATE RELATIONSHIPS'FOR DRILLING FOAM**
**(After Mitchell, 1970a)**

r

**p p**

T

y **n**

**Quality** **Viscosity** **Yield Stress** **Power-Coeff:**

**Range**
**%**
**cps**
**lb./Ft2 **
**r**
9 0- 3 6 1 4 . 3 8 2 . 5 1 . 0 2
8 6 - 9 0 9 . 5 8 1 . 0 .99
8 0 - 86 ' 7 . 2 1 ) .68 .97
7 5 - 8 0 5 . 7 6 . 48 1 . 0 2
7 0 - 7 5 5 . 0 + . 40 1 . 0 0
6 5 - 7 0 4 . 3 .23 1 . 0 0
6 0 - 6 5 3 . 7 .14 **1.00**
5 5 - 6 0 3 . 3 6 0 1 . 0 0
4 5 - 5 5 2 . 8 8 **0** 1 . 0 0
3 5 - 45 2 . 4 0 0 1 . 0 0
3 0 - 3 5 1 . 6 0 0 1 , 0 0
2 5 - 3 0 1 , 5 8 0 1 . 0 0
0 - 2 5 1 . 2 5 0 1 . 0 0
0 1 . 0 2

**The above shear rate ** **shear stress foam data were taken b}**
**Mitchell (1970a) and shown to fit the following equation: **

**(x - t ) = y <j>n**

**T-1383**

**plus water volume , yield stress ** **an<^ plastic viscosity **
**^4^)-(Mitchell, 1 9 7 0b).**

**Q u alit y.Regions**

**Figures 2 and 3 have been divided into three quality regions: **
**dispersed bubble, bubble interference, and bubble deformation. ** **The **
**dispersed bubble region exists for a quality range between 0 and**
**0.525. ** **Tha_vi.s.cjQjslpy of ^fqam in this region— is— no±-.a function of **
**shear rate. (Mitchell, 1970a, p. 57)**

**The bubble interference region exists for a quality range be**
**tween 0.525 and 0.74, ** **It is interesting to note that uniform spheres **
**packed loosely (cubically) produce a solids content of 0.52 by volume **
**and when they are tightly packed (rhcmbohedrally) without deformation **
**the concentration is 0.74, ** **The shear stre s s ,,„p 1 as1 1 c v 1 scos.i.ty-,— and **
**yield stress of the foam in thls...regiou..are a_f.unctlon of-shear_.r.ate **
**and foam ..quality-. ** **(Mitchell, 1970a, p. 57).**

**The bubble deformation region exists for a quality range b e **

**tween 0.74 and 0.95, ** **All the bubbles in this region are assumed to be **
**deformed. ** **As the quality increases above 0.74, the bubbles cannot **
**expand without deforming the surrounding bubbles. ** **Maximum resistance **
**to flow of foam exists in this region. ** **The shear stress, plastic v i s **
**cosity, and yield stress of the foam in this region are also functions**
**of shear rate and foam dua l i ty .... (.Mi£.cbe.lX.,....1970 a , p. 57), ** **For foam**
**qualities greater than 0.96, mist flow is assumed to exist.**

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**T-1383**

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**Velocity Profile**

**The velocity profile for flow of foam in a circular pipe is **
**based upon a Bingham model as shown'by-.Figure 4, ** **Three, flow regimes **
**are present: ** **Laminar, transition, and plug. ** **The pipe wall is assumed**
**to be coated with a static, thin layer of the aqueous solution. ** **The **
**shear rate and shear stress have maximum values at the wall of the pipe **
**and minimum values at the axis of the pipe. ** **The flow parallel to the **
**longitudinal axis of the pipe may be represented by concentric element**
**al thin cylinders., ** **Each cylinder has an individual shear rate and **
**shear stress. ** **Laminar flow will exist in those cylinders which have **
**shear rates greater than or equal to about 20,000 sec ^ ** **A transi**
**tional regime separates the laminar and plug flow regimes,** **The plug **
**regime will include all cylinders having a shear rate equal to zero.**

**T-1383**

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**FIGURE: 4**

**FOAM SYSTEM IK DOWN PIPE**

**CHAPTER THREE **
**METHOD. OF INVESTIGATION **
**Bingham Plastic Model ** **, „ ***

**The Bingham plastic model is described by the following **
**assumptions:**

**1. ** **Isothermal, steady-state flow through a vertical circular **
**pipe.**

**2. ** **Rate of shear is proportional to the excess of the shear **
**stress over a constant yield value, below which the material **
**behaves as a continuous unit.**

**3. ** **No slippage exists at the pipe wall,**

**Using these assumptions the Buckingham-Reiner equation may be developed. **
**(See Figure 5 and Appendix B):**

**0- V**
**1** **y + 1**
**- _{T ^}** n

**_y_****8y L**

**P**

**3t r**

**3**

_{-T}**r J _**

**where Q = Volumetric flow rate**

**L = Flow length**
**R = Pipe radius**

**t ** **= Shear stress at the pipe wall **
K

**P = P ** **PgL **
**p = Pressure**

**g = Gravitational constant **
**p = Fluid density**

## Tn

**P A.**

**n 1**

2 R

**i) Total Pipe Length**

**Elements ** **, L0 , •«.., L **

**1** **2** **n**

**are chosen s\ach that**
**Pn-P, = F -P. = ...P ,-P**

**0 1 ** **1 2 ** **n-1 n**

**b) Segment of Pipe Length**

**FIGURE * ** **5**

**FORGES ACTING ON DOWNWARD **
**FLOW IN A PIPE**

**model assumes that the fluid has a constant temperature. ** **However, in**
**a foam drilling application, the fluid temperature will increase with**
**depth. ** **As an approximation to isothermal flow, the pipe and annulus**
**are divided into incremental lengths and the temperature throughout**
**each increment is assumed to be constant and equal to the formation**
**temperature at any depth, ** **A surface temperature of 50°F. and a**

**o**

**geothermal gradient of 1 .S"F./100 f t , , as an accepted rule of thumb, **
**have been used to determine the formation temperature for each incre**
**mental depth. ** **No additional temperature correction has been made due **
**to the compression or expansion of the air, or the friction of flow **
**in the pipe or annulus,**

**A circular pipe and annulus are assumed. ** **No correction has been **
**attempted for an eccentric pipe-annulus arrangement. ** **In actual practice, **
**the drilled hole will not be truly circular and the pipe-annulus **

**arrangement will not be concentric. ** **Since the severity of eccentricity **
**is not known, the ideal case has been used for computational purposes.**

**Mitchell (1970a, p. 60) found through laboratory experimenta**
**tion that foam slippage at the pipe wall did not exist or was insig**
**nificant ,**

**The foam viscosities and yield stresses have only been measured **
**between 71°F* and 31°F ** **(Mitchell, 1970a, p. 37). ** **Temperature **

**varia-T-1383**

**tions and its effect on the foam viscosity and yield stress have not**
**been investigated. ** **No attempt has therefore been made to include**
**temperature-viscosity and temperature-yield stress relationships.**
**Numerical Integration Procedure**

**Flow Down the P i p e : ** **The pressure-depth relationship for the**
**flow o f foam down the pipe is calculated by Eq. (3). ** **The term **

q

**(Ty/T^) ** **^-s assumed to be relatively small, and is dropped from Eq.**
**(3), ** **Craft, et al, (1962, p. 38) state that the error is less than**
**2 per cent whenever tu exceeds 2.5 t .** **The more convenient anoroximate**

**R ** **y**

**expression describing the flow rate of a Bingham plastic fluid is then**

**Q =**

**TT R (P„-PT )**
**8y L **

**P** **LR**

**The volumetric flow rate (Q) is defined by**

*r\*

**Q = tt R^V **

**where V = Average fluid velocity.**

**Substituting Eq. (5) into Eq. (4) and solving for Tj, results ii**

**t R*-(P ~P_)- **
** V ** **0 L ___ _**
**R^(Pr-PT ) ** **8[i LV**

**0 ** **L-****P**

**The shear stress at the pipe wall (t ; is given by**

## <WR

**‘B ** **2L**

**Comparing Eq. (5) and Eq, (7) gives**

**r**

**(**

**p**

** -p ) **

**r r **

**r**

**_ 0 _ L i - il _X_ **

**" **

3 J
**21.**

**r i p ~p )**

### ^ '

### uo

### V

**Sii LV P**

**-J**

**14**

**and solving for the pipe length (L) results in**

**9**
**8i R t 24 Vy**

y **p**

**The statis pressure head is also expressed in terms of the pipe length**

**explicit equation describing the pressure-depth relationship of a**
**Bingham plastic fluid:**

**In order to apply Eq. (II) to foam, which is a compressible **
**fluid, the pressure dependent variables, -flowing density (p), yield**

**small constant value of 5 psi and the incremental pipe length will be **
**calculated, ** **Solution of Eq. (11) with constant pressure differentials **
**of 2.5, 5.0, 7 .5, and 10 psi showed that the pressure dependent **

**variables did not vary significantly for this range. ** **Five psi was **
**therefore used for the calculations, ** **Writing Eq. (11) in an iteration **
**form and summing from the surface to the bottom of the pipe, the total **
**pressure-depth relationship for foam flow down a vertical pipe is **
**determined:**

**(L) as**

**P 0 ** **PL ** **p 0 ~ P L + PgL** 10

**Substituting Eq, (10) into Eq. (9) and solving for L results in an**

**11**

**stress (t** **), and plastic viscosity (u ** **), must be maintained at nearly **

y p

**constant values. ** **The pressure change i p -p ) will be maintained at a**

**T-1383**
12
**C T . ** **O V . J J .**
**^ p>g _ __y±.**
**3R ** **p 2**
**i=l**

**The following discussion explains the evaluation of the second**
**ary variables for each iteration of Eq. (12).**

**The volume flow rate for water at a depth z, Qw z > i-3 equal to the **
**volume flow rate for water at standard conditions s , Q**

**ws**

**Q ** **= Q. ** **13**

**wz ** **ws**

**The volume flow rate for air at a depth z , Q ** **, is determined**
**az**

**with the ideal gas law. The ideal gas law mathematically expresses**
**F Vol**

**that the pressure-volume-temperature ratio at state 1 . (—** **—** **)., , is**

**equal to the pressure-volume-temperature ratio at state 2 , (** **^ :**

**P V o l ** **p Vol**

**(£_12£) = (s_lE±.) **

**in.**

**' ** **rp ** **T ** **2**

**where Vol = Volume of gas **
**T = Temperature**

**If the expansion or compression of a gas from state 1 to state 2 **
**does not fulfill the relationship of Eq. ( I T ) , ** **the gas is considered **
**to be non-ideal, ** **Joule and Thompson have measured the non-ideality **
**of gases (Barrows, 1966, p. 138-140) and expressed it by**

**/ 3T x ** 1**c**

**yJT “ ** **3P H**

**where y** **= Joule-Thompson coefficient**

**U 1**

3T

**(w7r)u = Change of temperature with respect to a change of**

**or n**

**pressure at constant enthalpy **

**H = Enthalpy**

**For the expected pressures and temperatures of a drilling operation, **
**the Joule-Thompson, coefficient for air is close to zero. ** **Air will **
**therefore he assumed to behave as an ideal gas and follow the mathe**
**matical relationship of Eq. (14), ** **Hence**

**Q ** **= Q (P/T) (T/P) ** **16**

**az ** **as ** **s ** **z**

**The total volume flow rate at a depth z, Q ** **, is**
**tz**

**= Q ** **+ Q ** **17**

**tz ** **wz ** **az**

**At a depth z, the mass flow rate of air (M ** **), water (M ** **), and **

**^ ** **az ** **wz**

**the total (II ) are **
**tz**
**M ** **= Q ** **p \ ** **IS**
**az ** **az^az)**
**M ** **= Q p ** **,** **19**
**W Z ** **ItiZ w z****M ** **= M + M ** **20**
**tz ** **as ** **wz**

**The density (p) of flowing foam at a depth z is**

pz = % / ( Q t z ) 21

**In order to determine the pressure cha.nge attributable to friction, **
**the plastic viscosity, yield str e s s , and velocity must be. kbAHEU. The **
**foam quality at z must first be determined in order to evaluate the **
**yield sjtress,.. and .plastic ,viscps.ify.;„**

**Foam quality (T) is defined as**

**r _______ air volume_________ ** **22**

**T-1383**

**The foam quality at z may be found by the following calculation: **

**r **

## ^

**rz a 0 - ** **23**

**'tZ**

**Once the quality is known, the yield stress and plastic -viscosity **
**may be determined from Figures 2 and 3.**

**The velocity of foam at z is**

**Qtz **
**z ** **A.**

**l**

**where A. = Pine flow area.**
**l**

**In order to solve Eq. (12) this procedure must be followed:**
**I) the volume flow rate at z for water (Q ** **), air (Q ** **), and total**

**W Z ** **O.Z**

**( Q ^ ) are calculated with Eas. '(13), (15), and (17): 2) the mass**
**flow rate at z for air (M ** **), water (M ** **), and total (M^ ) are **

**calcu-az 5 ** **wz r ** **tz**

**latea with E q s . (13), (19), and (20): ** **3) the foam flowing density**
**(p^)' is calculated with Eq. (21): ** **4) the quality at z, F , is **
**cal-culatea with Eq. (23): ** **5) the yield stress at z, Ty Z s and plastic**
**viscosity at z, ** **are evaluated from Figures 2 and 3 -with the use**

**p z ' ** **-.. - — .** **•**

**of the quality at z : ** **5) the velocity at z, V , is calculated with**
**Eq. (24), ** **All the variables in Eq. (12) are now known and the incre**
**mental length for foam flow down the pipe may be determined for one **

**iteration.**

**Flow Up the Annulus: ** **Melrose, et al ** **(1958, p. 315-324) have**
**shown that the flow equation for the isothermal, laminar flow of a **
**Bingham plastic fluid in a concentric pipe-annulus arrangement could**

**be approximated with good precision by the equation which describes **
**laminar flow through a narrow slot. ** **The flow equation can be written **
**as**
**9**
**EW x**
**Q =** **w**
**6y**
**( ****\**** 31**
**3x** *T * *\*
**i - o _{2t }7 + i_{2}**

**' J L \**

_{x 1}**w**

**W /**

**V**

**/**

**-Where E = Lateral extent of the slot**

**= Mean annular circumference, tt**

**D + D.**
**o ** **i**
**25**
**W = Width of slot**
**= Annular width, ~(D -D.)**
**2 o ** **l**
**D = Annulus outside diameter**

**0**

**D. = Annulus inside diameter**
**1**

**t ** **= Shear stress at the slot wall **
**w**

**For the derivation of Eq. (25), refer to Appendix C.**

**The pressure-depth relationship for the flow of a Bingham **

**plastic fluid in a pipe-annulus arrangement is calculated with the use**
**3**

**of Eq. (25), ** **The last term in Eq. (25), (t /x ) ** **, is assumed to be **

**^ ** **^ ** **y w ** **5**

**small and is dropped. ** **Craft, et al. (1962, p. 43) state that the**

**error is less than 7 per cent whenever x exceeds 2.5 x .** **The **

**approxi-w ** **y**

**mate equation describing the annular flow rate, Q, is**

**Q =**
2
**EW x**
**w**
**6u**
**3X**
**1** **V** **26**

**The volumetric flow rate, 0, is **

**T-1.383**
**Thus**
**V =**
**Wt**
**Sy** **1 **
**-3t**
**2T** **28**
**w**

**Solving Eq. (28) for the shear stress at the slot wall (t )**
**w**
**12Vy**

**2t** **+' 3 t**

**w ** **W ** **y**

**The shear stress at the slot wall is**
**W(p0-pL )**

**29**

**2L** **30**

**Comparing Eq. (29) and Eq. (30) give;**
**W(P -P_) ** **12Vy **

**0 L ** **_ ** **p**

**+ 3t**

**L ** **W ** **y**

**and solving for the pipe length (L)**

**31**
**L = (Pn-P. )**
**0 ** **jj**
**12Vy** **OT**
**P** _{y}**w** **w**
**32**

**The static pressure head, however, is also expressed in terms of the **
**pipe length (L)**

**L = P 0 ** **pL ** **PgL**

**The slot width (W) is analogous to the annular width and is **
**given by**

**W = ****- A V**** -D.)**

**2 o ** **i** **34**

**Substituting Eq. (33) and Eq. (34) into Eq. (32) and solving for L **
**results in an explicit equation describing the pressure-depth rela**
**tionship for the flow of a Bingham plastic fluid up a concentric **
**annulus:**

**■P-D -■P-D**
**3t**
y___
**|<D -D.)**
**I****O ** **.L**
**+ Pg**

**Eq. (35) is applied to foam, a compressible fluid, by maintain**
**ing the pressure-dependent variables, flowing density (p), yield **
**stress ( ) , and plastic viscosity (y ) ** **nearly constant values.**
**The total pressure difference (p -p ) is maintained again at 5 psi and**

**U ■“ Li**

**the incremental annulus length is calculated. ** **Writing Eq. (35) in an **
**iteration form and summing from the bottom of the pipe to the annulus **
**surface, the total pressure-depth relationship for the flow of foam **
**up a concentric annulus is calculated:**

**n**
V f Lo
/ t
**2**. - .**(p.., _{‘ 1+1 }**

**P-)**

_{c}

_{i}### /

**i=l**

**-**

**j'****'18V.P**

**.**

*■J*. 1 D1

**p g +**

**+**

**r ^**

**5t**

**V I**

**( D - D . d**

**< V V |**

**36**

**o i**

**Because the particles are being lifted by the foam in the annulus, **
**the flowing densit^r, quality, and velocity calculations must be ad **
**justed for the mass and volume of the sand cuttings. ** **A constant **
**drilling or cleaning rate will be assumed.**

**The density of foam flowing in an annulus at a depth z will **
**become**

**tz** **n**

**(Q**

**tz,** **n r**

**37**

**where M = Mass rate of rock **
**r**

**0 = Volume flow rate of rock**
**r**

T -1333

**The velocity of the foam and rock chips in the annulus is **

**Qtz + ****%**** '**

**where A = Annular flow area.**

**Since the calculation.of foam quality requires the knowledge of **
**the air and water volumes, a correction must be made for the rock **
**chip volume in the annulus. ** **Writing a volume balance on a section **
**of the annulus:**

**Vol ** **+ Vol ** **= Vol ** **Vol ** **39**

**az ** **wz ** **ann ** **rock**

**where Vol ** **= Volume of air at z **
**az**

**Vol ** **= Volume of water at z **
**wz**

**Vol ** **= Volume of annulus **
**ann**

**Vol ** **, = Volume of rock**
**rock**

**The annular volume for a unit length is**

**7****7**

**tt ( D -D . )1**

**Vol ** **= --- 2 * —** **40**

**ann ** **4**

**Assuming that rock particle slippage does not occur, then the rock **
**volume flowing in a unit length of annular volume is **

**(Vol ** **)Q**

**Vol ** **, = ** **41**

**rocK ** **CL + Q **
**'tz ** **r**

**Substitution of Eq. (40) and Eq. (41) into Eq. (39( gives the air and **
**water volumes in a unit length of the annulus:**

**^ Vol ** **+ Vol ** **(D2-D2 ) ** **1 4 - / --- ~— r— \ ** **42**

**—** **-az ** **wz) V ** **o i ** **Qtz +**

**The volume of air at a depth z in the unit length of the annulus is **
**Q Vol**
**Vol ** **= -^2--- 4.3**
**aZ ** **Qtz + r**
**The quality at z is**
**Vol v/.**
**= --- -2iE**— —
**z) ** **Vol ** **+ Vol**
**az ** **wzy**

**In order to solve Eq, (36) this procedure must be followed:**
**1) the density, p^, of flowing foam at z is calculated with Eq, (37):**
**2) the velocity in the annulus, V' , is calculated with Eq. (38): 3)**

**z**

**in order to evaluate the quality, T , at z, E a s . (42) and (43) must be **
**solved and substituted into Eq. (44): ** **4) the yield stress at z,**

**an<3 plastic viscosity at z, y** **, are evaluated from Figures 2 and **
**3 with the use of the quality at z. ** **All the variables in Eq. (36) are **
**now known and the incremental length for fcarn flow up the annulus may **
**be determined for one iteration.**

**Two differential equations result from the mathematical model **
**describing the flow of foam in a drilling operation: ** **one for the pipe**
**flow and one for the annular flow. ** **The differential equations are **
**explicitly expressed in terms of pressure and depth and are numerically **
**integrated over the full length of the pipe with a constant pressure **
**increment. ** **An iteration process develops because at the bottom of the **
**pipe the pressure within the pipe must equal the pressure within the **
**annulus. ** **Further boundary conditions are: ** **1) a minimum annular**
**velocity at the bottom of the pipe: ** **2) a maximum foam quality at the**

**T-1383**

**surface in the annulus. ** **At the bottom of the pipe, foam velocity will **
**be the controlling quantity for removal of cuttings. ** **The foam velocity **
**must be kept at a minimal value in order to reduce the enlargement of **
**the hole. ** **Foam quality should be a maximum at the annulus surface.**

**The viscosity- of foam does reach a maximum value at the beginning of the **
**mist region. ** **(See Figure 3). ** **If the quality is allowed to enter the **
**mist region, the viscosity will approach that of air, ** **This annular **
**boundary condition at the surface will therefore control the maximum **
**quality which is desirable for the most efficient cutting removal.**

**In order to maintain good hole properties, a bottom-hole annular **
**velocity of 1.5 fps (Wolke, 1970), and a foam quality of 0.96 in the **
**annulus at the surface will be imposed upon the system as boundary **
**conditions. ** **Field experience indicates that a bottom-hole velocity **
**of 1.5 fps will clean the hole of all solid particles while using foam. **
**If the quality becomes greater than 0.96, the foam viscosity will **

**decrease to that of mist and the solid particles will perhaps not be **
**carried from the annulus, ** **With the stipulated boundary conditions, **
**the required air volume, aqueous volume, and the annulus choke- **
**pressure are determined.**

**CHAPTER FOUR **
**RESULTS**

**With the use of the Colorado School of Mines PDP-10 computer, **
**the Bingham plastic model was applied to a foam drilling system. **
**Various pipe and hole sizes were used in the drilling simulation. **
**Surface injection pressures of 200, 300, and 4-00 psi, depths of 1,000, **
**2,000, 3,000, and M-,000 ft., and drilling rates of 0.0. 0.5, 1.0, and **
**1.5 fps were run on each pipe and hole combination. ** **The tabulated r e **
**sults are listed in the Tables section of this paper. ** **The surface **
**injection pressure, hole size, and tubing s i z e .determine which of the **
**tables should be used for foam drilling. ** **After the proper table is **
**selected, the required air volume rate, water volume rate, and surface **
**annulus pressure for various depths and drilling rates are read for **
**a stable foam drilling operation. ** **The engineer may match the output **
**of the on-site equipment with the tabulated results of foam theory **
**and thereby ensure a stable foam drilling operation.**

**Figures 6 and 7 represent a typical application of the tabulated **
**results. ** **Prior to drilling a hole with foam, the hole and drillpipe **
**size have been selected, and the surface injection pressure is usually **
**dictated by the source of the gas: ** **i.e., field gas or compressor.**
**With the use of the proper tables, the gas and liquid volume rates and **
**surface-annuius pressures are plotted as shown by the data in Figure 6. **
**Table 20 shows that as the depth increases, the air and liquid volume **
**rates must be increased while the surface-annuius pressure must be**

**.SUBFACE ANNULUS ’PRESSURE ■ ( psi )**
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**T-1383**

**decreased. ** **As the drilling rate increases, the air and liquid volume **
**rates must be increased, and the surface-annuius pressure must be **
**decreased, ** **By displaying the volume and pressure requirements in **
**graphical form, the adjustments in the field equipment may be deter**
**mined easily and quickly for the changing depth and drilling rate.**
**The air volume rate in Figures 6 and 7 represent a confidence interval **
**for the drilling rates, ** **The air volume rate is changing with depth **
**and drilling rate, but the amount of change is too small to be seen **
**on the figures.**

**Calculations show that the minimum hydraulic horsepower for **
**foam drilling will be attained when the lowest available surface in**
**jection pressure and the largest drillpipe applicable to a hole is **
**used. ** **At higher surface injection pressures, more air is required; **
**therefore the hydraulic horsepower is increased. ** **The use of the 1 cLP gest **
**drillpipe will minimize the hydraulic horsepower since the large drill**
**pipe has less frictional losses- than small pipe and will therefore **
**require less horsepower, ** **In order to minimize hydraulic horsepower **
**in a foam drilling operation, the minimum surface injection pressure **
**and the largest drillpipe should be used.**

**The range of the maximum Reynolds number in the pipe is 609 to **
**104,000 for the various pipe and hole combinatipns.** **The range of the **
**maximum Reynolds number in the annulus is 1 to 22,000. ** **The maximum **
**Reynolds number for the pipe and annulus is at the bottom of the pipe.**
**At this point, the effective foam viscosity is at a minimum, the density**

**is at a maximum, and the velocity is at a minimum. ** **The effective vi s **
**cosity is extremely low and is the controlling number in the Reynolds **
**calculation. ** **The Reynolds number at this point is therefore a maxi**
**mum. ** **It should be noted that the majority of the Reynolds numbers **
**are less than 2,000 and the flow.is assumed to be in the laminar **
**flow regime. ** **The equation which was used to calculate the Reynolds **
**number in the pipe is **

**D.'Yp**

**Re = — —** **45**

**T ** **D.**
**where y ** **= y +**

**e ** **p ** **5V **

**and in the annulus is**

**Re - a** **46**

**v e**

**T W**
**where y ** **= y ** **+ Trrr- **

**e ** **p ** **4 V**

**It should be noted that a Reynolds number criteria of 2,000 for de**
**fining turbulent flow of foam has not been verified by experimental **
**work.**

**T-1383**

**CHAPTER FIVE **
**DISCUSSION OF RESULTS **

**The data show that for an increase in depth, the water and **
**air volumes must be increased while the surface annulus pressure **
**must be decreased if the boundary conditions of 0.96 surface-annular **

**quality and 1.5 fps bottom-hole velocity are to be maintained for **
**the various depths. ** **As the depth increases, the bottom-hole pressure **
**will also increase while the gas volume will decrease. ** **The velocity **
**will therefore decrease. ** **In order to maintain a velocity of .1.5 **
**fps at the bottom of the pipe, the quantity of air must be increased. **
**With the increase of air volume, there must be an associated increase **
**of 'water volume if the surface quality Is to be 0.96, ** **The annulus **
**surface pressure also affects the foam velocity and quality. ** **If the **
**annular pressure is not decreased with an increase in depth, larger **
**water and air volumes will be required if the imposed boundary **
**conditions are to be satisfied, ** **The reported surface-annular pres**
**sures are the minimum pressures required if minimal air and water **
**volumes are to be used in a foam drilling system.**

**For a specific depth, as the drilling rate is increased, the **
**air and water volume rates must be increased and the annulus surface **
**pressure must be decreased if the boundary conditions are to be met. **
**As the rock chips are picked up and carried out of the annulus, **
**they will add to the f oa m ’s density and increase the bottom-hole **
**pressure. ** **For higher drilling rates, the bottom-hole pressure will**

**be larger; larger volumes of air and water will be required. ** **As **
**was stated in the last paragraph, the required air and water volumes **
**are minimal volumes, based upon the minimum annulus surface pressure **
**for each group of data.**

**T-1383**

**CHAPTER SIX **

**CONCLUSIONS AND RECOMMENDATIONS**

**The purpose of this study was to present a mathematical model **
**and its solution for the application of foam in a drilling system.**
**The mathematical model of foam is the Bingham plastic fluid model **
**applied to a compressible fluid. ** **The Bingham model has been written **
**in an iterative form and solved with a constant pressure iteration **
**for a concentric vertical hcle-annulus arrangement, ** **Two equations **
**have been written: ** **one describing flow down the pipe and one d e **

**scribing flow up the annulus. ** **The following conclusions may be stated **
**about the mathematical model for foam:**

**1. The foam plastic viscosity is defined for a range of 1 to **
**14.7 cp for all qualities from 0 to 0.96.**

**2. The foam yield stress is defined for a range of 0 to 359 **
**cp for all qualities from 0.584 to 0.96.**

**3. Foam plastic viscosity^ and yield stress_ar.e„undef ined in **
**the mist region.**

**4. Maximum annular foam quality should not exceed 0.96.**

**5-. All bottom-hole annular velocities may be used in the model,**
**but computer time increases with higher velocities.**

**6. As the velocity increases, the foam properties will change **
**such that smaller pipe lengths are required for the same **
**change in pressure.**

**7. The optimum bottom-hole velocity has not been determined.**

**8. ** **Temperature effects on foam stability have not been **
**investigated,**

**9. ** **Rock chip volume and mcjss have been accounted for in the **
**quality calculation.**

**10. ** **Solutions for pipe sizes of 2.87 to 4-.50 in O.D. in hole **
**sizes of 5.75 to 15.0 inches have been calculated.**

**Since the results of this study are based upon ideal drilling **
**and hole conditions, the best test of the results would be an actual **
**drilling application. ** **It is recommended that the following items be **
**investigated:**

**1 ** **The bottom-hole velocity of 1.5 fps should be checked to **
**determine if the velocity is sufficient to clean the rock **
**chips from the bottom of the h o l e .**

**2, ** **The actual pressure drop in the system should be checked **
**against the theoretical calculated pressure change. ** **In this **
**manner, the Reynolds number for foam may be calculated and- **

**its effect upon the plastic viscosity and yield strength**
**be determined,**

**3. ** **Since the slip velocity in foam was assumed not to exist, **
**a laboratory investigation should be made to verify this **
**assumption,**

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