Critical Review of Drilling Foam Rheology

Transcription

1 ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 11, 2003 Critical Review o Drilling Foam Rheology Ahmed Ramadan 1, Ergun Kuru 2 and Arild Saasen 3 1 IMV Projects, Calgary, AB T2P 2V6, Canada 2 School o Mining and Petroleum Engineering, University o Alberta, AB T6G 2G7, Canada 3 Statoil, N-4035, Stavanger, Norway ABSTRACT In this paper, problems related to the rheology o drilling oams are addressed. Both empirical and analytical oammodeling approaches are considered. The eects o dierent luid parameters on the rheology o oams are thoroughly investigated. A rigorous model comparison is made to evaluate the predictions o the models. Finally, a semi-empirical modeling approach is recommended based on analysis o the analytical and empirical models. INTRODUCTION Foams are being used in a number o petroleum industry applications, which exploit their high viscosity and low density. Foam can be used as a circulation luid during drilling, well completion and production operations. Drilling oams have the greatest beneits during underbalanced drilling due to their ability to lit large quantities o produced liquids and drilled cuttings. Underbalanced drilling technique allows the drilling o producing zones with minimum ormation damage. The low annular velocity that is typical or oam drilling greatly reduces the possibility o borehole erosion. At the some time, a good knowledge o oam rheology and hydraulics is a must or sae and economical drilling. Drilling oams are complex mixtures o gas and liquid, whose rheological and hydraulic properties are largely inluenced by oam quality, liquid phase viscosity, temperature and pressure. The quality is the volume raction o gas within the oam. There have been numerous experimental studies 1,2,3,4,5 o drilling oam rheology, covering wide range o quality, liquid viscosity and low properties. Although there are dierences in the results o these studies, the ollowing general conclusions can be drawn: i) rheology o oams mainly depend on the quality and low rate; ii) suractant concentration has little eect on oam viscosity at concentration typical o drilling oams; and iii) the oam viscosity increases with increasing liquid phase viscosity. Beyer et al. 1 have ound that pressure and temperature inluence the oam rheology is mainly by regulating oam quality. Increasing the pressure signiicantly reduces the volume occupied by the gaseous phase, indirectly reducing the oam quality and viscosity. At constant pressure, increasing the temperature obviously decreases the viscosity o the liquid phase. Consequently, the oam viscosity increases as the temperature decreases. In addition to the rheology, oam quality has great inluence on the structure o oam. Figure 1 presents relative viscosity (oam viscosity/liquid phase viscosity) and structural variation as a unction o oam quality. Thus, the oam viscosity increases with increasing quality up to about 94%. Thereater, viscosity moderately decreases with increasing quality, which indicates the start o breaking

2 down o oam bubbles 6,7. As the oam quality approaches to 100%, the viscosity drops very rapidly. Relative Viscosity Bubbly Liquid Spherical Bubbles Foam Quality [%] Wet Foam Spherical & Polyhedral Bubbles Figure 1 Relative viscosity as a unction o oam quality or water based oam Foam quality is the undamental descriptive parameter or a macroscopic structure o oams. As we increase the quality o oam rom zero to unity, irst a bubbly liquid will be ormed at low oam quality. However, a critical transition occurs as oam quality increases to rigidity transition. As the term implies, the oam becomes rigid and ordered structurally, and will not low reely. The structure o oam bubbles appears spherical. This critical value diers slightly depending on the liquid phase compositions. For water-based oams the rigidity transition occurs at about 63% quality 8. Further increase in oam quality exhibits peculiar rheological property when the quality exceeds 0.8. The structure o oam bubbles begins to change rom spherical to polyhedral coniguration. Consequently, the bubbles deorm against their neighbors. But they remain separated by thin ilms o liquid phase that keeps the bubbles rom rupture. As the quality approaches 0.946, the bubbles acquire an increasingly polyhedral shape. As a result o this crowding, the oam system attains the highest viscosity close to (dry oam limit). Previous studies 9,10 on Dry Foam Mist oam rheology have ound that polyhedral bubbles predominately appear when the oam quality is between 88% and 95%. Additional increase in the quality o dry oam beyond the dry oam limit moderately decreases the viscosity until oam stability limit is reached. Further increase in quality beyond this limit convert the dry oam into mist, resulting a drastic viscosity reduction. The stability limit o water-based oam is about 97.5%. Addition o viscosiiers into the liquid phase increases the stability limit 6. Foam rheological studies typically adopt one o several distinct approaches. A purely empirical approach involves ormulating a constitutive law or a material on the basis o experimental data alone. Such laws can be very useul as they are based on experimental observation o real systems. Nevertheless, the orm o the equation generally has no physical basis. An alternative approach is to produce a mathematical model in which explicit account is taken or the oam structure and low properties o each phase to determine the low properties o oams. However, problems requently arise in such theories because o assumptions and idealizations that are necessary to simpliy the mathematics o the real problem. As a result, mathematical models oten bear little resemblance to the physical problem. Consequently, a semi-empirical approach, which has a physical basis but involves experimentally determined constants, is oten preerred. EMPIRICAL FOAM MODELS Underbalanced drilling can be perormed using dierent types o oams. Oten oam drilling are classiied as stable oam (water based oam) or sti oam (polymer oam) drilling. In stable oams the liquid phase can contain suractants, salts and corrosion inhibitors, none o which has a signiicant impact on the viscosity o the liquid phase. Sti oams contain viscosiiers in addition to these additives 6.

3 Stable Foams Ater measuring oam rheology using small diameter tubes, Mitchell 4 ound that the oam viscosity is related to its quality and the viscosity o the liquid phase by: ( Γ) = 6 (1) L where Γ and L denote the oam quality and liquid phase viscosity respectively. This relation is applicable when Γ For Γ 0.55, the study suggested another correlation, which is given by: = L (1 Γ ) (2) Sanghani and Ikoku 7 experimentally studied oam rheology with a concentric annular viscometer that closely simulated actual borehole conditions. They concluded that oam is a power-law luid with low behavior index n and low consistency K, which are both unctions o oam quality. Recently, Martins et al. 11 have determined the relationship between these parameters and the oam quality as: a2 1 Γ n = a1 (3) Γ b2 1 Γ K = b1 (4) Γ where a 1, a 2, b 1 and b 2 are empirical constants whose values are determined by oam bubble size and liquid phase properties. Sti Foams Sti oams usually made by adding a oaming agent to a airly thin, unweighted drilling mud. Essentially the same structures are seen in both sti and stable oams. However, sti oams can be created with a higher quality than the stable oams. The rheology o sti oam greatly depends on the liquid phase viscosity and the oam quality. Reidenbach et al. 2 investigated the low properties o oams at dierent qualities and viscosiying polymer concentrations. Results indicated that the rheological behavior o oam luid is primarily that o a yield power-law luid and can best be described by a Herschel-Bulkley model. Accordingly, the yield stress, τ y is given by: τ y = αγ, or Γ 0.6 (5) and τ = βe δγ y, or Γ 0.6 (6) where α, β and δ are empirical constants that vary with liquid and gas phase properties. The oam consistency coeicient and the low behavior index are represented by K = K L and n = αγ+βγ 2. Cawiezel and Niles 3 investigated the rheological properties o oams at downhole conditions. The result conirmed that the yield power-law model best describes the rheology o oams. However, a problem with this approach is that the empirical parameters need to be determined exactly or dierent oam systems encountered in a drilling operation. Kraynik 12 has suggested that empirical correlations based on oam-low data have limited predictive value when the connection between structure and rheology is lacking. Thereore, another method o oam rheology prediction is necessary. Valkó and Economides 13 have presented principle o volume equalization to describe the rheology o oams. The technique uses the speciic-volume expansion ratio, ε as the additional parameter representing the phase relation o the gas and liquid. This quantity is deined as the ratio o the liquid density to the oam density, which varies along the low path because o the change in the

4 pressure. Thus, the speciic-volume expansion ratio is given by: ρ L ε = (7) ρ The principle o volume equalization is derived rom an invariance requirement. It assumes that or a straight duct low o constant cross section, both compressible and incompressible lows posses the invariance property. This means that the loss o mechanical energy is proportional to the kinetic energy, in other words the Reynolds number is constant 14. I we demand the same invariance property or the low o a compressible non-newtonian luid, this restricts the orm o the constitutive rheological equation. Constitutive equations, providing the required invariance, are called volume equalized. The volume equalized power law equation is given by 14 : τ ε γ = K ε n (8) One o the advantages o this approach is that the volume equalized wall shear stress versus the volume equalized equivalent Newtonian shear rate plotted on a log-log scale results in a straight line or a given gas-liquid pair or a wide range o oam qualities and pressures. Recently, Ozbayoglu et al. 5 have perormed a comparative study to investigate the predictive ability o the available oam hydraulic models. They have conducted low experiments using stable oam at 70, 80 and 90 % oam qualities. The result illustrated that the power law model can better characterize the rheology o oams at 70 and 80 % qualities, whereas the rheology o oam at 90 % quality best its with the Bingham plastic model. Comparison o the model predictions with experimental pressure loss values indicated that model predictions o rictional pressure losses can be signiicantly dierent rom the actual values. In spite o this, Valkó and Economides 13 model gives relatively accurate pressures loss predictions. ANALYTICAL FOAM MODELS Mathematical modeling o oam rheology is a challenging task. Since microscopic structure and dynamics o oams are poorly understood and remain a subject o basic scientiic interest to physicists and engineers. Oten oam lows are characterized by bubble deormations and rupturing that have signiicant eect on the rheological properties o oams. In steady homogenous shear low, small change in bubble shape rom sphericity depends upon a capillary number Ca, which is given by: aγ Ca = (9) σ where a, σ and γ denote average bubble radius, interacial tension and shear rate respectively. The capillary number is a relative measure o viscous orces that tend to distort the bubble and interacial tension, which avors sphericity. Hatschek 15 attempted to relate concentrated emulsion viscosity to its structure by = L (1-Γ 0.33 ) -1. The ormula has been applied to high quality oams at low Ca. Sibree 16 experimentally measured viscosity o oam at high Ca and observed similar quantitative dependency upon quality. The liquid phase in his experiments was strongly non-newtonian and the oam quality was in the range o 0.52 to Recently, Llewellin et al. 17 have studied the rheology o bubbly liquids at low Ca and developed a semi-empirical model or Γ 0.5, which is based on theoretical analysis o Frankel and Acrivos 18. Accordingly, or steady low with small bubble deormations (i.e. small Ca) the oam viscosity is given by:

5 ( 1+ 9Γ) (10) = L The model predictions show good agreement with experimental observations. Based on a mathematical treatment o the behavior o liquid droplets with an elastic bounding membrane, suspended in a viscous liquid, Barthes-Biesel and Chhim 19 developed a constitutive equation or dilute suspensions that account or high values o Ca as: L 2 = 1+ (2.5 ψca ) Γ (11) where ψ ϕ + 60λ. The constant ϕ depends on non-linearity o the membrane material. Viscosity ratio λ is deined as the ratio o the viscosity o dispersed phase to that o continuous phase. Since the surace tension is constant, the constant ϕ is negligible or oam bubbles. The viscosity ratio or dry oam ranges rom 0.02 to Thereore, reasonable value o ψ is about 70. For Ca (2.5/ψ) 0.5 the viscosity o the oam will be less than the liquid phase viscosity. Moreover, or very small capillary number (Ca << 1), Eq. 11 becomes the Einstein s equation or dilute suspension o solid particles, where Γ is solid particle volume raction. Currently, widely cited oam modeling studies 20,21 are those o Khan and Armstrong. In these studies, a twodimensional oam model has been developed or dry oams (i.e. Γ 0.946) by assuming hexagonal and monodispersed oam cell structure. A general expression or the stress is obtained, which gives the relative viscosity in terms o Γ and Ca. Ater adoption and simpliication, the Khan and Armstrong 20 expression reads: L 0.31 = + 26 Γ 1 Ca ( Γ ) (12) Thus, or Newtonian base luid, the magnitude o the yield stress is directly proportional to the liquid surace tension and inversely proportional to the bubble size. Since Ca = aγ /σ. Harris 22 has measured oam bubble diameter at dierent pressure and ound that the size o the bubble diameters ranges rom 10 µm to 2.4 mm, while the mean diameter varies with pressure rom 800 µm at 6.5 MPa to 1.9 mm at 0.5 MPa. Plotting the result on pressure versus bubble volume graph indicates that the bubble size is correlated with the pressure according to the ideal gas equation. Thereore, the bubble size can be predicted or the actual borehole conditions i it is known at the surace. Bubble size at the surace mainly depends on oam production process and the properties o the liquid phase 23. According to equation 12, the interacial tension is another parameter that aects the yield stress o oams. Increasing temperature. Surace tension o ordinary water substance linearly decreases with temperature 23. In addition to the temperature, the concentration o suractant greatly aects the surace tension o water. This reduction in surace tension is normally seen below the critical micelles concentration (the concentration at which micelle ormation becomes signiicant). Above the critical micelles concentration, the inluence o concentration on surace tension becomes insigniicant 24. Princen and Kiss 25 presented a semiempirical oam model, which is applicable or high quality oams (i.e Γ 0.98) at low capillary number (10-4 Ca). The model is based on theoretical analysis o oam microstructure and experimental investigation o rheology o emulsions. Princen and Kiss 25 deined the yield stress as: 0.33 σγ τ y = F( Γ) (13) a

6 where the unction F can be approximated by F(Γ) = log(1-Γ). Thus, the relative viscosity is given as: L F( Γ) Γ = Ca Plug Flow 1/ 3 Γ / 2 Ca u s Slip Flow Shear Flow Plug Flow (14) SLIP AT THE WALL Slip at the wall is another interesting and important characteristic o oam lows. A convenient macro scale description o wall slip mechanism is based on the existence o a thin luid layer that does not itsel slip but wets the wall and lubricates the oam low. Experimental evidences or tube and rotational lows indicates that the wall slip occurs above inite values o wall shear stress called yield slip stress, τ sy. Because o this, various low behavior o the oams have been recorded, such as shear low, plug low and slip low as shown in Fig Shear Flow o oam with and without solid particles. But they did not observe slip in rough pipes, whereas it is commonly occurring in smooth ones. For low in smooth pipe, the slip velocity or dry oam is estimated by 12 : u Ca σ s S = (15) L where Ca s is modiied capillary number given by 27 : 1/ 3 6aτ 1 Γ Ca = w s (16) σ ( Γ) Like the ordinary capillary number, the modiied capillary number is a relative measure o viscous orces at the wall that tend to distort the oam bubble and interacial tension, which avors sphericity. The modiied capillary number becomes negative or oam quality less than Since Eq. 15 is applicable or dry oam when Ca s << 1. Slip velocities o water based dry oams with surace tension o N/m and bubble radius o 1 mm are estimated using the model and presented in Fig. 3. The igure shows that the slip τ sy <τ w <τ y (a) τ sy <τ y <τ w (b) τ y <τ w <τ sy (c) 100 Figure 2 Velocity Proiles in Foam Flows. In igure 2a we have a plug low with uniorm velocity proile. This type o low occurs when the wall shear stress, τ y is greater than the slip yield stress and less than the yield stress. Beyer et al. 1 reported the existence o plug lows during oam low experiment. However, when τ sy <τ y <τ w, we have slipping Bingham luid low with slipping velocity, u S as shown in Fig. 2b. Normal Bingham luid low with no-slip condition at the wall is shown in Fig. 2c, which occurs when τ y <τ w <τ sy. Thondavadi and Lemlich 26 investigated low properties Slip Velocity [m/s] 10 1 Γ Γ= Γ=0.97 Γ= Wall Shear Stress [Pa] Figure 3 Slip Velocities o Dry Foams as a Function o Wall Shear Stress. velocity decreases as the oam quality increases. This means that in dry oam Ca s

7 range, high quality oams require higher wall shear stress than low quality oams to have the same slip velocity. It is apparent rom the igure that the slip velocity signiicantly increases with increasing the wall shear stress. Figure 3 is in qualitative agreement with the observations o Beyer et al. 1, who also ound that slip velocity increases with liquid volume raction o oam. However, the model overestimates the slip velocity when Ca s is greater than 0.1. Practical importance o the slip velocity is to ind accurate relationship between the low rate and the pressure drop. As seen rom Fig. 2, the oam lows experience dierent types o velocity proiles. I we consider laminar low o power law luids and assume a dierential length o pipe throughout which the density may be considered constant, then mean low velocity U in a circular pipe can be expressed as 27 : 20%. It is apparent rom the igure that the two graphs cross at 60% oam quality, which is very close to the rigidity transition limit. Thus, the Hatschek 15 model, which is recommended or Γ 0.52 shows signiicant increase in the relative viscosity. This can be attributed to the structural change occurs at the rigidity limit. Thereore, the semiempirical model o Llewellin et al. 17 is more reliable than other models or bubbly liquid (Γ 0.6) at low Ca. The classical model o Hatschek 15 is still applicable or bubble liquids and oams (0.0 Γ 0.8) at low Ca. Relative Viscosity Hatschek 15 Llewellin et al. 17 U u dp D Re n = s + (17) dx 32ρ where dp/dx denotes pressure gradient and Re n is generalized Reynolds number given by: Re n = ρ ( U u ) s 2 n D K(3 + 1/ n) n n 2 3 n where D is diameter o the pipe. (18) MODEL COMPARISON So ar we have seen dierent oam models based on analytical and empirical observations, but it is very important to select the most appropriate oam models that can be used to predict low behaviors o drilling oams. Figure 4 presents relative viscosity proiles o oams based on the Hatschek 15 and Llewellin et al. 17 models. The two graphs ollow the same pattern or Γ 0.6 with maximum dierence o about Foam Bubbly Liquid Foam Quality [%] Figure 4 Relative Viscosities o Foams Using Hatschek 15 and Liewellin et al. 17 Models. In the upper part o a well, drilling oams can be in the dry oam limit, where the bubble shape is dominantly polygonal. Although several analytical dry oam models have been proposed, their usages are very limited. Since they are very complicated and inconvenient or practical use. Most widely accepted dry oam model is that o Khan and Armstrong 20. However, this model still suers several restrictions and idealizations. Thereore, it requires calibration to account or the limiting actors. For high quality oams Eq. 12 can be rewritten in a simple orm as:

8 Ratio o Yield Stresses L = Ca ( Γ) Γ (19) Thus, the dry oam plastic viscosity (13 L (1- Γ)) decreases with increasing the oam quality. This is in agreement with previous experimental observations, which is commonly seen when oam quality is over 94.6%. Previous studies 6,7 on dry oam rheology suggested that the viscosity moderately decreases with increasing quality up to the oam stability limit, which is 97.5% or water based dry oams. At low Ca (Ca < 0.001), the second term in the right hand side o Eq. 19 becomes negligible. This contradicts with the experimental observations 7 that ound moderate decrease in viscosity as the quality o dry oam increases. Figure 5 compares the yield stress predictions o Khan and Armstrong 20 and Princen and Kiss 25 model as a unction oam quality. The ratio o yield stresses is deined as the ratio o predictions o Khan and Armstrong model to the predictions o Princen and Kiss 25 model. As seen rom the igure, the predictions o Khan and Armstrong model are signiicantly higher than that o Princen and Kiss 25 model. However, very close to unity the predicted values are comparable, reairming again the restrictions o Khan and Armstrong model (i.e. applicable only or dry oams). Relative Viscosity 1e+5 1e+4 1e+3 1e+2 1e+1 Princen & Kiss 25 Ca=10-5 Princen & Kiss 25 Ca=10-4 Princen & Kiss 25 Ca=10-3 Princen & Kiss 25 Ca=10-2 Princen & Kiss 25 Ca=10-1 Barthes-Biesel & Chhim 19 Ca=0.02 Barthes-Biesel & Chhim 19 Ca=0.1 Hatschek 15 at low Ca 1e Quality Figure 5 Yield Stress Ratio Versus Foam Quality. Figure 6 presents relative viscosity predictions o oam models as a unction o quality and capillary number. The igure shows that the inluence o quality on relative viscosity is considerably high at high capillary number. It is also interesting to note that relative viscosity predictions o Figure 6 Relative Viscosity Predictions o Foam Models. Brathes-Biesel and Chhim 19, and Princen and Kiss 25 models are the same when oam quality is about 0.8. Moreover, the patterns o curves apparently indicate a sharp increase in relative viscosity, which is commonly seen during oam rheology measurement. Thereore, smooth relative viscosity predictions can be obtained i Brathes-Biesel and Chhim 19, and Princen and Kiss 25 models are employed or 0.0 Γ 0.80 and or 0.8 Γ 0.95 respectively. As the capillary number decreases, the relative viscosity curves o Brathes-Biesel and Chhim 19 move up, approaching the Hatschek 15 model curve, which is oten applied at low Ca. CONCLUSION The rheology o drilling oams can be estimated using dierent semi-empirical

9 models, which take account or both oam quality and capillary number. These two oam low parameters can successully describe the rheology o drilling oams. Purely empirical oam rheological corrections are unreliable beyond the original data upon, which they are based. However, simpliied analytical models can provide valuable physical insight to develop more reliable semi-empirical models. Flow and rheological properties o drilling oams greatly change as the oam quality increases. Thereore, obtaining a single oam rheological model can be practically diicult. Wall slip, and bubble intererence, deormation and rupturing are undamental characteristics o oam low, which aect both rheology and hydrodynamic. ACKNOWLEDGEMENTS The authors wish to acknowledge IMV Projects and Hardcastle Design LTD. o Canada or their assistance in the preparation o the manuscript. REFERENCES 1. Beyer, A.H., Millhone, R.S. and Foote, R.W. (1972) Flow Behavior o Foam as a Well Circulating Fluid, paper SPE 3986 presented at the 1972 SPE Annual Fall Meeting, San Antonio, Texas. 2. Reidenbach, V.G., Harris, P.C., Lee, Y.N. and Lord, D.L. (1986) Rheology study o oam racturing luids using nitrogen and carbon dioxide, SPE Production Engineering, January Cawiezel, K. E. and Niles, T. D. (1987) Rheological Properties o Foam Fracturing Fluids Under Downhole Conditions paper presented at SPE Hydrocarbon and Economics and Evaluation Symposium held in Dallas, March Mitchell, B.J. (1971) Test Data Fill Theory Gap on Using Foam as a Drilling Fluid, Oil and Gas J., September 1971, pp Ozbayoglu, M.E., Kuru, E., Miska, S. and Takach, N. (2002) A comparative Study o Hydraulic Models or Foam Drilling J. Canadian Petroleum Technology, Vol. 41, pp Gas Research Institute (1997) Underbalanced Drilling Manual Chicago, pp Sanghani, V. and Ikoku, C. U., (1983) Rheology o Foam and Its Implications in Drilling and Cleanout Operations, ASME AO-203, presented at the 1983 Energy- Sources Technology Conerence and Exhibition held in Houston, Texas, January 30-February Holt, R.G., McDaniel, J.G. (2000) Rheology o Foam Near the Order-Disorder Phase Transition Proc. o the 5 th Microgravity Fluid Physics and Transport Phenomena Con., NASA Glenn Research Center, Cleveland, OH, CP , pp , August Debrégeas, G., Tabuteau, H. and di Meglio, J.-M. (2001) Deormation and Flow o a Two-Dimensional Foam under Continuous Shear Phys. Rev. Lett. 87, pp Gopal, A. D. and Durian, D. J. (1998) Shear-induced Melting o an Aqueous Foam March Meeting o The American Physical Society, Los Angeles, CA, March Martins, A.L., Lourenço, A.M.F., Sá, C.H.M. and Silva Jr., V. (2001) Foam Rheology Characterization as a Tool or Predicting Pressures While Drilling

10 Oshore Wells in UBD Conditions, paper SPE presented at the 2001 SPE/IADC Drilling Conerence, Amsterdam, 27 February-1 March. 12. Kraynik A.M. (1988) Foam Flows Ann. Rev. Fluid Mech., 20, pp Valkó, P. and Economides, M. J. (1992) Volume equalized constitutive equations or oamed polymer solutions Journal o Rheology, vol. 36, No. 6, pp Valkó, P. and Economides, M. J. (1994) Foam Proppant Transport paper SPE presented at the Western Regional Meeting, held in Long Beach, Caliornia, March Hatschek, E. (1911) Die viskosität der dispersoide Kolloid-Z, pp Sibree, J. O. (1934) The viscosity o roth Trans. Faraday Soc. 28, pp Llewellin, E.W., Mader, H.M. and Wilson, S.D.R. (2002) The rheology o a bubbly liquid Proceedings o the Royal Society, A 458, pp Frankel, N. A. and Acrivos, A. (1970) The constitutive equation or a dilute emulsion J. Fluid Mech. 44, Barthes-Biesel, D. and Chhim, V. (1981) The constitutive equation o a dilute suspension o spherical microcapsules Int. J. Multiphase Flow 7, pp having gas raction approaching unity J. Non-Newtonian Fluid Mech. 25, pp Harris, P.C. (1985) Eects o Texture on Rheology o Foam Fracturing Fluids paper SPE presented at the 60 th SPE Annual Technical Conerence and Exhibition held in Las Vegas. 23. Perry, R.H., Green, D.W. and Maloney, J.O. (1984) "Chemical Engineer's Hand Book," sixth edition, McGraw-Hill, Inc. Tokyo, pp and Schramm, L. L. (1994) Foams: Fundamentals and Applications in the Petroleum Industry American Chemical Society, Washington, pp Princen, H.M. and Kiss, A.D. (1989) Rheology o Foams and Highly Concentrated Emulsions J. o Colloid and Interace Science, Vol. 128, No. 1, pp Thondavadi, N. N. and Lemlich, R. (1985) Flow Properties o Foam with and without Solid Particles, Ind. Eng. Chem. Process Des. and Dev., 24, Kozicki, W., Chou, C. H. and Tiu, C. (1966) Non-Newtonian low in ducts o arbitrary cross-sectional shape Chemical Engineering Science, Vol. 21, pp Khan, S.A. and Armstrong, R.C (1986) Rheology o Foams: I Theory o dry oams J. Non-Newtonian Fluid Mech. 22, pp Khan, S.A. and Armstrong, R.C (1987) Rheology o Foams: II Eect o polydispersity and liquid viscosity or oams