Laminar and drag-reduced polymeric foam flo · In recent years the use of gas-energized aqueous...

Laminar and drag-reduced polymeric foam flow

W. Winkler, P. Valk6,a) and M. J. Economidesa)

Institute of Drilling and Production, Mining University Leoben, A-8700 Austria

(Received 2 June 1993; accepted 18 September 1993)

Synopsis

The volume-equalized power law model has heen introduced as a constitutive equation for polymer foams. Large-scale horizontal flowing experiments, performed in pipe sections with different diameters at varying temperatures, demonstrate the applicability of the model. The model parameters, estimated by a simple linear regression procedure, were also used for the description of flow regimes which obey the phenomenon of drag reduction. In contrast to already existing turbulent foam flow correlations the foam behavior is described by a traditional friction factor versus a specific definition of Reynolds number plot. The proposed model fits the laminar and drag-reduced flow regime data. The nonlaminar results are compared to an already available correlation.

I. INTRODUCTION

In recent years the use of gas-energized aqueous polymer [primarily hydroxypro- pylguar (HPG )] solutions for hydraulic fracturing treatments has increased steadily. The advantage of these nitrogen and/or carbon dioxideMPG foams, compared to incompressible liquid systems, is the capability to facilitate post-treatment cleanup. The gas expands aiding the fracturing fluid flowback through the proppant pack. Commonly, petroleum engineers call fluids mixed with small amounts of surfactants “foams,” if the gas content in the base fluid, characterized by the quality, T’ ( = the volume of gas divided by the total volume of the foam), is between 0.3 and 0.8. To perform such fracturing treatments successfully it is important to know the foam rheology and, consequently, to predict the pressure losses along vertical and/or horizontal pipes.

Substantial research in this area in the past has characterized foam behavior based on measurements with laboratory-scale viscosimeters. Correlations and models derived from small experimental setups describe the foam behavior for certain conditions, but they may be of questionable value for industrially interesting scales [Heller and Kuntamukkula ( 1987)]. Scale-up problems are not only true for foams, but also for other fluids. For instance, Lord ( 1987) compared three different types of scale-up correlations methods for one neat HPG solution data set: Bowen (Melton-Malone type), friction factor-Reynolds number and friction velocity procedures. The predic- tions of pressure losses differed significantly, using the correlations for pipe sizes considerably outside their development range.

%nrent address: Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843-3116.

@ 1994 by The Society of Rheology, Inc. J. Rheol. 38(l), January/February 1994 0148~6055/94/38( 1)/111/17/$4.00 111

112 WINKLER, VALKb, AND ECONOMIDES

Spiirker et ai. ( 1991) performed experiments in a vertical flow loop of industrial scale allowing considerable insights into the fundamental foam flow behavior. Subse- quently, Valki, and Economides ( 1992) attempted to obtain a single medium constitutive equation for foams. They introduced the principle of volume equalizing. For constant mass flow rates a volume-equalized constitutive equation provides constant Fanning friction factors in pipe flow, even in the case when the density changes because of the foam compressibility.

Assuming the laminar foam flow obeys a rheological curve on a plot of wall shear stress versus wall shear rate (at a given temperature), which is completely determined by one additional parameter, the volume equalized principle states that this parameter is the density. This is in contradiction to other approaches where the foam quality is the additional parameter. Quality-related correlations assume implicitly that the pressure and, hence, the density of the gas phase has no influence on the rheological behavior. Since the quality is pressure dependent these correlations are only valid for single pressure values. Valko and Economides ( 1992) showed that available correlations and volume-equalized models provide very similar results if the pressure conditions correspond to the pressures used for the development of the quality-related correlations. With the volume-equalized principle, quality effects can be accounted for. All findings concerning this new concept are based on the assumption that a foam is characterized by the ratio of liquid mass to gaseous mass, and this ratio is the same at every point of a tube or a tube system.

For the construction of a volume-equalized constitutive equation it is convenient to express all density-dependent parameters with respect to the liquid density. The specific volume expansion ratio l s was introduced

v Pl es=-=--,

VI P (1)

and, clearly, there is no one-to-one correspondence between the quality and the specific volume expansion ratio, even at a fixed temperature, if the pressure is not constant. With this variable the traditional power law model could be volume equalized (VEPL):

7 = (K&yj/ “-‘)f. (2) Similarly, it was possible to construct a volume-equalized Bingham plastic consti-

tutive equation. But, for this work the VEPL model was used. For large scale (industrial use) the advantages of the volume-equalized power law

model include a constant friction factor along the whole pipe at a given mass flow rate, and the possibility to define one set of parameters (K and n) for all foam qualities (at least for foams with qualities from 20% up to 80%). This means also, that the model fits all conditions along the pipe., although the foam quality at the inlet of the pipe may differ considerably from the quality at the pipe outlet. This is true for all pipe incli- nations from vertical to horizontal. However, there is a certain advantage of vertical flowing experiments, compared to horizontal, because of the higher sensitivity of foam flow to rheology. This increased sensitivity is caused by the fact that smaller pressures due to friction losses (p < ~1) yield increased foam qualities, which cause higher pressure drops. Moreover this steadily increasing quality is amplified by the nonlinear hydrostatic pressure gradient decrease ( Apf >fiycr). Additionally, vertical experiments provide the possibility to obtain flowing conditions where the inlet pressure is equal to the outlet pressure (Apf = hydr). This means that the quality is constant along the pipe. Therefore, exact pairs of friction factors with corresponding foam

FLOWS OF POLYMERIC FOAMS 113

quality are available, providing the possibility to check the postulated model. Unfor- tunately, in large-scale vertical experiments it is difficult to vary pipe diameter, because they are carried out in existing wells.

In this paper the volume-equalized power law model, adapted for linear parameter estimations, is applied to results from horizontal large-scale experiments to study the influence of geometry and temperature on the model parameters. In addition, results of experiments at superficial velocities, causing nonlarninar flow regimes, are inter- preted.

II. VOLUME-EQUALIZED FOAM FLOW DESCRIPTION

A. Linear estimation of the volume-equalized power law model parameters

Describing the rheological behavior of foams by the VEPL model requires the knowledge of the model parameters K and n. Valko and Economides ( 1992) described two possibilities to determine the model parameters, which require the use of nonlinear parameter estimation algorithm. Here a considerably simpler approach is suggested. For the “original” power law model, used widely for incompressible non-Newtonian fluids, it is easy to obtain the model parameters by a simple linear regression of the rheological curve on a logarithmic diagram, where the wall shear stress is plotted as a function of the wall shear rate. To use the same procedure for compressible flow it is necessary to get, for a given mass flow rate, a shear rate independent of pressure. From the volume-equalized power law model [Eq. (2 )] the following logarithmic form can be obtained

ln: = lnK+nlnl. ES Es

(3)

This means that shear rate changes resulting from pressure drop changes have a one-to-one relation to the shear stress when both are related to the specific volume expansion ratio. Using Eq. (3) it is easy to determine the VEPL model parameters, defining the properties of flowing compressible media for a certain basic fluid at one flowing temperature and for all foam qualities (at least between 20% and 80% ) . The logarithmic rheological curve plots the wall shear stress divided by the specific volume expansion ratio versus the wall shear rate divided by the specific volume expansion ratio and forms a straight line, providing the logarithm of the volume-equalized consistency index K at the intercept at p/es equal to one. The volume equalized flow behavior index, n, is the slope of the curve. This logarithmic presentation of rheological foam behavior can be used as a foam flow curve replacing the rheological flow curve for incompressible flow.

The wall shear rate (Pw) of any power law fluid can be expressed with the geometry-independent flow behavior index n as

VW= (da)

Equation (4a) leads to the well-known definition of the wall shear rate for New- tonian fluids, that is SC/D, if n is equal to unity. Therefore, if one uses the volume- equalized shear stress shear rate plot for the determination of the model parameters, it is convenient that, at first, the “Newtonian” wall shear rate divided by E is used as the abscissa, since n is not yet known. From a linear regression, the straight line described by the following equation is determined:

114 WINKLER, VALK6, AND ECONOMIDES

(4b)

From this linear regression the values of n = np* and J$ can be obtained. The latter is the volume-equalized geometry-dependent power law consistency index for pipe flow. Since the wall shear stress is given by

the volume-equalized power law parameter K can be obtained from g as follows:

(W

B. Description of horizontal foam flow behavior by the Fanning friction factor

Since a volume-equalized constitutive equation provides a constant friction factor along the whole pipe, if the cross section and the temperature are constant, the behavior description of flowing media by the Fanning friction factor is very useful for flow regime determinations and simple wall shear stress calculations.

Neglecting radial distribution effects of velocity the differential mechanical energy balance for horizontal ( = no gravitational effects) isothermal circular pipe flow of compressible fluids can be written as

bp3 + ap* - abc*p - a*c*

2fb2c2p3+4fabc2+2fa2c2P dp= -;. (5)

The constants a, 6, and c, which depend on the gas mass and the gas behavior, are defined in the Appendix. For homogeneous foams the velocities of gas and liquid are assumed to be the same. If this mechanical energy balance for horizontal two-phase flow is integrated from the inlet point 1 to the outlet point 2, the constant Fanning friction factor can be calculated explicitly

h +a K;(pl-p2)-OSln~+K~ln- 1 bpz+a ’ (6)

1 K; =

z7

and

c*b*-a K;==-

III. EVALUATION OF EXPERIMENTAL RESULTS

A. Experiments and results

HPG/nitrogen foams were pumped through the flow loop, described in the Ap- pendix, with flow rates from 2.65 X 10 -3 m3/s up to 1.06~ lo-* m3/s at foam qual-


ities of 30%, 50%, and 70%, respectively. The volumetric flow rates and qualities were calculated for the conditions at the inlet of the first pipe section ( D = 0.0495 m): the inlet pressure was 2 MPa and the temperatures were 20 and 4O”C, respectively. Due to the constraint of the constant inlet pressure and the pressure drop in the smallest pipe section (D = 0.0213 m) experiments with foam flow rates higher than 9.275 x 10m3 m3/s could only be performed in the first two pipe sections (D = 0.0495 m and D = 0.0407 m) . In addition, all experiments were performed with two different concentrations of aqueous HPG solutions: 4800 and 7200 wppm of HPG in water. For these concentrations the Petroleum industry uses the designations HPG 40 and HPG 60, respectively (40 lb/1000 gal and 60 lb/loo0 gal).

The experimental series exhibited the expected pressure drop dependence on foam quality and velocity. The higher the quality and velocity the higher were the pressure losses. For the evaluation of the measurements it is important to stress, that the pipe inlet pressures, except for the first pipe section, depend on the pressure drop in the previous pipe, itself depending on the foam quality. In the case of quality-related models this would complicate the comparison of the results from these pipe flow experiments with different inlet pressures, but constant mass flow rates. The volume- equalized principle, hence, allows to use one plot for evaluations, although the measurements correspond to different qualities.

8. Determination and discussion of the VEPL parameters

In Fig. 1 the linear parameter estimation procedure is presented for the experimental series with HPG 40 foams in all four pipe sections at a flowing temperature of 20 “C. For this procedure only the experiments providing certainly laminar flow regimes are used. The average superficial velocity used to calculate the “Newtonian” shear rate is calculated for inlet conditions, whereby the volumetric behavior of the gas phase (nitrogen) is described by the virial equation of state truncated after the second term [Reid et al. ( 1987)]. For the determination of the specific volume expansion ratio this kind of gas phase description is also used, calculating the impact of the nitrogen on the overall foam density, whereby the dissolved quantity of nitrogen in the liquid phase is taken into consideration. The pipe wall shear stresses are calculated from the measured pressure differences (7~ = ApD/4L).

The straight line (Fig. 1) obtained from a linear regression procedure is

,

which [by virtue of Eq. (4d)] provides the VEPL consistency index K = 1.171 Pa s” and the flow behavior index n = 0.433. The volume-equalized principle allows for negligible differences in the estimated parameters when using outlet pressure or average pressure conditions for the determination of the shear rates.

The results of the parameter estimations for all experimental series, for HPG 40 and HPG 60 foams at 20 and 40 “C, respectively, are shown in Table I. Because of the higher viscosity of HPG 60 foams the consistency indices are higher than the ones for HPG 40 foams. The influence of the temperature on the flow behavior of foams will be discussed in Sec. III C 1.

A suitable definition of the Reynolds number corresponding to the VEPL model is [see Valkb et al. (1992)]:

116 WINKLER, VALK6. AND ECONOMIDES

100 71 c

c

c

iz L

e B P 10

L

1000

- - - - - - -

-

-

- - - - -

-

-

-

- - - -

-

-

- - - - -

-

-

-

-

- - - - -

-

-

- - - - -

-

-

-

10000

8u /DG [l/s1

FIG. 1. Volumecqualized plot of wall shear stress vs wall shear rate ( = flow curve for compressible HPG 4Ofoamsat2O”C).

RqEPL = D%2-npE~-1

K ’

and, consequently, the Friction factor can be computed from

(7)

(8)

Other possible volume-equalized Reynolds number definitions are given in the Appen- dix. The Fanning friction factors for the experimental series, calculated with Eq. (6), the theoretical friction factors, calculated with Rq. (8) and, the parameters determined from the flow curve of foams (Fig. 1) are plotted in Fig. 2 as a function of the volume-equalized Rower law Reynolds number. The points are calculated for inlet and

TABLE I. VEPL model parameters from linear regression procedures.

Liquid phase 4-I

HPG 40 20 0.437 1.171 HPG 60 20 0.429 1.324 HPG 40 40 0.439 0.901 HPG 60 40 0.421 1.259


measured data loutletl

100 1000

Re VEPL [-I

loo00

FIG. 2. Fanning friction factor as a function of the volume-equalized power law Reynolds’s number for HPG 40 foams at 20 ‘C.

outlet conditions of HPG 40 foams in all four pipe sections at a flowing temperature of 20 “C. The conformity of data points corresponding to different pressure conditions and, therefore, to different foam qualities demonstrates the applicability of the volume- equalized principle providing a constant friction factor. Experiments providing Rey- nolds numbers smaller than 1500 were assumed to be laminar since no significant increased deviation of the friction factors from the calculated values could be found.

In addition to the already presented results from linear parameter estimations it was also possible to determine the consistency index and flow behavior index for the VEPL model either with a nonlinear “two-step methodology” or with a more straightforward nonlinear “one-step procedure.” For the latter the outlet pressure was defined by the integrated mechanical energy balance, in which the constant model-dependent friction factor was determined, and, finally, the model parameters yielding the minimum sum of squares of the differences between computed and measured outlet pressures were estimated. In the two-step procedure the “experimental” friction factors were first derived [Eq. (6)] and then the least-squares principle was applied to the model- depending friction factor definition. Using the standard nonlinear least-squares parameter estimation algorithm of Marquardt the two model parameters (K,n) of HPG 40 foams where fund to be K = 1.168 and n = 0.433. Relative weighting was used to neutralize large variations in the measured values of the friction factors. The differences in the obtained model parameters for HGP 40 foams, compared to the values in

118 WINKLER, VALKd, AND ECONOMIDES

Table I, were very small. This shows that the application of the linear parameter estimation on the rheological curve for foams is suthcient.

It is of interest to compare the model parameters with the ones obtained from vertical large-scale HPG 4O/nitrogen foam experiments at a flowing temperature of 15 “C (K = 1.40 Pa s” and n = 0.47, see Valko and Economides, 1992). The flow behavior indices obtained from vertical and horizontal flow experiments are very near each other. Little differences are caused by the different parameter estimations, depending on the relative weighting. More significant variations are found in the consistency factors. But, these differences are not sufficiently conclusive to assume different rheological behavior of vertical and horizontal flowing foams. Different flowing temperatures and the variations in the HPG 40 concentrations might cause these differences, and, in addition, some phase segregation problems in the horizontal setup cannot be excluded completely.

C. Influence of foam temperature and pipe diameter on the VEPL model

1. Temperature influences on the foam flow behavior

Using foams with different base fluid gel concentrations (4800 and 7200 wppm) implies differences in the resulting viscosities. These differences are described by the consistency index, recognized in the variations of the K factors in Table I, whereby the impact on the flow behavior index is not significant. The change of the temperature has the same consequences on foam flow behavior as the varying of the gel concentration, because the increase of the foam temperature causes thinning effects on the HPG gel. Additionally, the temperature influences the gas behavior itself, resulting in quality changes. Hence, the viscosity changes due to different foam qualities are compensated by the volume-equalized principle. Therefore, higher foam temperatures provide lower K values. This can also be seen in Fig. 3, where the wall shear stress is plotted as a function of the “Newtonian” wall shear rate for HPG 40 and HPG 60 foams at temperatures of 20 and 40 “C ( P = 50%). The effective viscosity is defined as pefi = &L?/32L u and the pipe diameter and pipe length are constant. Therefore, at a certain shear rate ( = at one velocity) the pressure drops are lower at higher temperatures because viscosity decreases, resulting in smaller shear stresses. In terms of the volume-equalized power law model

The data for different temperatures and HPG concentrations presented in Fig. 3 suppport the observation of a negligible temperature influence on the flow behavior index of foams by the parallelism of their regression curves. The three slopes, corresponding to n, are nearly the same. Additionally, it was found that the temperature thinning effects are greater at low quality than at high quality foams. This means that the more liquid phase in the foam the more significant the influence of the temperature, which is in correspondence to the findings of Harris and Reidenbach ( 1987), who found that the viscosity loss of fracturing fluids caused by temperature increase is less with foams than with gelled fluids without gas content.

2. Geometry independence of the volume-equalized power law model

The purpose of a constitutive flow behavior equation is the possibility of application independent of geometry. This was also assumed for the volume-equalized power law model describing foam flow once the model parameters were estimated. The power law


10

HPG 40 foam, T=20°C . Hffi 40 foam, T=40°C

. HPG 60 foam, T=20°C

- HPG 40 (2OW rear.

- HPG 40 (4O’W regr.

- HPG 60 (2OW

100 loo0 10000

y, Ills1

FIG. 3. Wall shear stress vs wall shear rate (calculated for pipe inlet conditions) of HPG foams (r = 50%) at dEerent temperatures.

model for a neat, time-independent, non-Newtonian, fluid was assumed to be geometry independent, since the rheological curve was independent of the flow channel geometry or pipe diameter.

To prove the geometry independence of the VEPL model for foams the experimental series were performed in pipes with different diameters. On a logarithmic plot of shear stress versus shear rate the data points for foams with a certain inlet quality pumped through four different diameters fitted to one single curve. Because of the small differences in the pipe sections, resulting in quality changes, the values scattered around this curve. The same tendency can be assumed for any gas content in the non-Newtonian base fluid if the shear stress divided by the specific volume expansion ratio versus the shear rate divided by the specitic volume expansion ratio is presented on a logarithmic plot (Fig. 4). The measured data points obtained from different geometries follow also one curve, although no matching parameters were used. Addi- tionally, the parameter estimations with data obtained form experiments in one pipe section were nearly identical to the model parameters determined from the data set of all experiments performed in all pipes. This shows that the VEPL model for foams with different qualities does not depend on the pipe geometry.

IV. FOAM FLOW BEHAVIOR IN DRAG-REDUCING FLOWING REGIMES

Plots of friction factors versus VE Reynolds number and values of standard deviations suggested that experiments resulting in VE Reynolds numbers higher than 1500


1 I /

n D=O.O495n

A D=0.0407 n

l D=0.0338 n

x D=0.0213 n

100 1000

&/&$ Ills1

FIG. 4. Volume-equalized wall shear stress YS volumc-cqualizcd wall shear rate of HFQ 40 foams of qualities from 30% up to 75% at pipe inlet conditions of different pipe sections.

cannot be described by the VE constitutive equation for laminar foam flow. The introduction of linear polymers in a flow system results above a certain velocity in a phenomenon which has been termed previously drag reduction. This phenomenon is defined as any mod&&on to a turbulent fluid flow system resulting in a decrease of the rate of energy loss. An extensive review of the subject was presented by Virk (1975). He derived a maximum drag reduction asymptote, insensitive to polymeric parameters. On a volume-equalized Moody diagram one recognizes that our foams at high Reynolds numbers exhibit also the properties of drag-reducing polymers. Cam- eron and Prud’homme (1989) presented correlations for turbulent foam flow of Blauer et al. ( 1974) and Reidenbach et al. ( 1986) concluding that Blauer’s attempt does not fit to drag-reducing properties of polymer foams. The correlation of Reiden- bath et al. ( 1986) is a modified Melton-Malone type:

rR = A’&‘(845 (10)

where A’ and S are constants depending on the HPG concentration, resulting in a shear stress description independent of the pipe diameter. Of course this correlation might be true for the environment where it was investigated, and using it for different scales may not be accurate. Additionally, there is no continuity to a constitutive equation for laminar flow. Therefore our intention was to find a description of drag- reducing foam flow with a traditional fluid mechanics description such as the relation between friction factor and a specified Reynolds number. On a graph of Fanning friction factor versus VEPL Reynolds number it could be seen that the data in the drag


reduction regime followed a tendency parallel to the maximum drag reduction asymp tote [MDRA, Eq. ( 1 1 )] deiined by Virk ( 1975).

1

T =

f 19.0 log Re \rf- 32.4. (11)

Additionally, we obtained an explicit approximation of the MDRA in the form of f = f (Re) providing sufficient accuracy but easy statistical error calculations. Roth attempts plotting the Farming friction factor either as a function of the VEPL Rey- nolds number or as a function of the volume-equalized Metzner-Reed generalized Reynolds number (R-R) exhibit the parallelism to the MDRA, but did not fit it. Similarly Keck ( 1991) obtained smaller Fanning friction factors for neat polymer solutions than predicted by the MDRA plotted versus the generalized R-R. Virk ( 1975) suggested that for shear thinning solutions, a Reynolds number Rew , formed with the apparent wall viscosity is preferable to the generalized power law Reynolds number. Since the volume-equalized apparent viscosity at the pipe wall can be defined as

TW P app,w = -

YW

the volume-equalized power law wall Reynolds number is

1

, (12)

w RevEw=-=

P ww

= --& Lwp( fy)Y (13)

This Reynolds number [Es. ( 13)] was used for drag-reducing flowing systems in the VE Moody diagram presented in Fig. 5. The VEPL Reynolds number of 1500 corresponds to a VEPL wall Reynolds number of approximately 5500. The Fanning friction factors used for the modeling of the laminar data of HPG 40 and HPG 60 foams in Fig. 5 is

2+6n f=2 - WEW ( 1 n ’

(14)

The filled data points obtained from friction factor calculations with Fq. (6) and the Reynolds number defined in Eq. ( 13)) which are obviously under the influence of drag reduction, follow very closely the MDRA defined by Eq. ( 11). For the determination of the VEPL wall Reynolds number of parameters (n,K) computed from the volume-equalized rheological curve were used. There are small deviations from the MDRA at higher Reynolds numbers. In principle one could introduce a new Reynolds number using Vi&s ideas concerning the velocity distribution in drag reduction cases. Our work, hence, intend not to increase the variety of available Reynolds numbers. Therefore, we used the suggested wall Reynolds number following strictly the approach of Virk in handling the drag-reduced data of non-Newtonian fluids. As shown in Eqs. ( 12) and ( 13 ), respectively, the Rew is defined by the wall shear rate, which is determined at a given mean velocity once the constitutive equation is known. This wall shear rate is the actual shear rate at the wall in laminar flow. Virk uses the same definition of wall shear rate in turbulent flow. For such a flow regime, however, this wall shear rate is just a fictious parameter. Therefore, it is clear that the more non-


100 1000 10000 looooo

ReVEW f-1

FIG. 5. F&g friction factors of HPG foams with different qualities as a function of volume-equalized wall Reynolds number for laminar and drag-reduced flow experiments.

Newtonian a fluid is the more significant the deviation of the fictious and the actual shear rate. This might cause a problem when handling non-Newtonian drag-reduced data and might also explain some minor discrepancies in our results.

In addition the dotted curve in Fig. 5 represents the von Mn equation for Newtonian flow in smooth pipes with the constants determined form Nikuradse. This curve does not fit the measured values. Again, the significant influence of drag reduction can be seen. Chakrabarti ef al. ( 1991) found that neat 100 wppm HPG solutions follow over a wide range the von Kiumh curve until drag reduction causes marked deviation. Hence, they also found that HPG solutions, with concentrations in the range of the concentrations used for the experiments in this paper, show a more continuous transition from laminar towards Virk’s asymptote. From the results of our foam experiments (Fig. 5) no significant transition zone between K&-m&s and Virk’s curve could be seen. Therefore, different onsets of the drag reduction for HPG 40 and HPG 60 foams could also not be established.

In Fig. 6 the measured wall shear stresses of HPG 40 foams with different gas contents obtained from experiments with drag-reducing flow regimes are compared to the wall shear stresses calculated from different correlations. Texture differences caused by various surfactant compositions and foam-creation procedures seem to be unimportant in our pipe scales. Reidenbach et al. seem to support this statement. The VE-Virk points are calculated with the help of Eq. ( 11) when the Reynolds number is defined by Eq. ( 13 ). The Reidenbach ef al. values correspond to Eq. ( lo), where the constants are given in the Appendix. As seen from the figure, the VE-Virk approach is satisfactory for the whole range of wall stresses, the Reidenbach et al. correlation


t.! 60

20

0

. . T 20°C

. VE-Virk

* Reidenbach

0 50 100

T, (measured) [Pal 150 200

FIG. 6. Wall shear stresses estimated by the VEMDRA approach and by the Reideobach correlation as a function of measured wall shear stresses of HPG 40 foams at 20 “C.

gives higher errors at higher wall stresses. This suggests that Reidenbach’s correlation is of questionable value for environments other than the one from where it was derived.

From the limited number of “turbulent” experiments in the three larger pipe sections no obvious dependence of the results on pipe diameter was recognizable. This proves again the applicability of Virk’s maximum drag reduction asymptote for nonlaminar flowing foams, if the quality effects are compensated by the volume-equalized principle.

V. CONCLUSION

The recently developed constitutive equation, namely, the volume-equalized power law model, was found to be capable to describe isothermal horizontal compressible non-Newtonian flowing systems, independent of the pipe geometry. The presentation of the experimental data on a volume-equalized logarithmic plot of the wall shear stress versus the wall shear rate provided rheological flow curves yielding the two model parameters. The increase of the foam temperature had no significant influence on the foam flow behavior, but on the viscosity and, hence, on the volume-equalized power law consistency index. Experiments at high superficial velocities obeyed the phenomenon of drag reduction. Describing these data with the well-known maximum drag reduction asymptote in connection with the volume-equalized power law wall Reynolds number led to good agreements with the measured values. Further inten- tions using the volume-equalized approach are investigations checking the possibility to simplify matters of two-phase flow description in general.

124 WINKLER, VALKO, AND ECONOMIDES

pi, p4, p5, p8 . . . . . Inlet Pressures p2, p3, p6, p7 . . . . . Outlet Pressures

FIG. 7. Horizontal flow loop sections.

ACKNOWLEDGMENTS

The authors wish to thank Western Petroleum Services International, and 6MV for partial support of this research. Also, thanks are due to J. Handler, J. Brandegger, H. F. Spijrker, and G. Prieler for help with the experiments.

APPENDIX

1. Horizontal flow loop design

To perform the horizontal foam llow tests a loop consisting of five main compo- nents was constructed These were the fluid storage tanks, the pumping and measuring equipment, the gas storage, injection and metering equipment, which were described by Spiirker et al. ( 199 1 ), the heating system and the horizontal loop section. With the heater, the fluid temperature could be increased from 15 to 40 “C via a water-filled primary circuit from the heater to a heat exchanger and a secondary circuit circulating the tank fluid through the heat exchanger. The flow loop consisted of four tubing sizes, as shown in Fig. 7. The pipe inner diameters were: 0.0495, 0.0407,0.0338, and 0.0213 m. It was possible to circulate either through all four pipes or through the biggest pipe section and one of the other three sections as a return line.

The inlet and outlet pressure transducers were spaced exactly 30.5 m ( 100 ft) apart and placed roughly 2.5 m from the actual inlet and outlet point. This distance was chosen because of the findings mentioned by Govier and Aziz ( 1972). In their work they suggested estimations with correlations for non-Newtonian fluids since there are no confirmed methods for multiphase mixtures to predict reliably the length of pipe in which pressure gradients change because of entrance and end effects. The suggested graphical solution ( WdDReG, vs n) resulted in entrance lengths up to 1.5 m for HPG solutions. Because for turbulent flow regimes entrance lengths of non-Newtonian fluids are lower than for water and the differences of the pipe diameters are small the entrance length of 2.5 m was assumed to be sufficient to avoid the influence of any entrance effects. The foam temperature was measured at the inlet of the first pipe and at the outlet of the last pipe for recalculation purposes of the actual foam properties. The pipe inner surfaces were smooth. Neat water experiments showed that the relative roughness of the pipes was 0.0005, resulting in a limiting friction factor f m = 0.0043.

Gaseous nitrogen was injected in a continuous manner into the flowing aqueous HPG solutions mixed with 2000 wppm (weight part per million) surfactant to reduce phase separation. The homogeneous foam was formed in a 0.114-m o.d., 1.2-m-long


mechanical foam generator filled with curled steel chips. For the exact calibration of the gas flow meter, necessary for adjusting the desired foam quality, a certain gas temperature was assumed. This temperature was obtained and kept constant during all tests with the help of a heated water bath before the flow meter.

For each test series, after a steady-state operation condition was reached, the following variables were measured with a computer-controlled data acquisition system with a sampling rate of l/s for 2 min: four inlet pressures, four outlet pressures, the inlet and outlet foam temperature, the liquid flow rate, and the gas rate and temperature. The average values of the 120 measured values have been used for the data evaluation.

2. Mechanical energy balance for horizontal foam flow

For solving the differential mechanical energy balance the following definitions derived by Valk6 et al. ( 1992) were applied. They used the Virial equation of state for volumetric gas behavior description.

Constants:

RT

a=wgMRp

wgRTB’ b=-

ME? +(1-w,+,

pg

The superficial velocity is defined by

(Ala)

(Alb)

(A2a)

and, hence

du = -;dp. P

(A2b)

3. Relation of the different volume-equalized power law Reynolds numbers

The VEPL Reynolds number R~~EPL and the VEPL wall Reynolds number RevEw are defined by Eqs. (7) and ( 13)) respectively. The volume-equalized generalized power law Reynolds number is defined by Eq. (A3) :

WEMR= DV-“p

Peff = --&$%-~p(~)-: (A31

where peff is given by Eq. (9).

126 WINKLER, VALKd, AND ECONOMIDES

Reynolds number relations:

(A4b)

4. Values used for the correlation [Eq. (lo)] of Reidenbach et al. (1996)

For HPG 40 foams: A’ = 10e3 ( lbf/ft2) (ft3/lbm)s-1 (s/ft)” and s = 1.23, whereby the density is inserted in [lbm/ft3] and the velocity in [ft/s] resulting in a wall shear stress in [lbf/ft2]. To obtain [Pa] multiply the stress by 47.88.

NOMENCLATURE

: B’ C

f” K

Kl K2 L

Mg mg.ml n

nP P Pl P2 kf Phydr R R~VEPL

RQEW

RWEMR

Rew T U

[m2/s2] p$l

a [kg/(s m2)l

pj [Pa sn]

[Pa sn]

[Pa s”] [Pa s”] [ml

[kg/m4 [Wsl 1-l L-1 Pal Pal Pal [PaI Pal [J/(mol K)] 1-l

1-l

[-I

1-l WI Wsl

constant coefficient constant coefficient modified second virial coefficient constant coefficient pipe inner diameter Fanning friction factor volume-equalized power law consistency index volume-equalized geometry-dependent power law consistency index for pipe coefficient coefficient pipe length = distance between two pressure transducers molar mass of the gas phase mass rates of gas and liquid, respectively power law flow behavior index pipe flow power law flow behavior index pressure pressure at the pipe inlet pressure at the pipe outlet pressure loss due to friction hydrostaic pressure universal gas constant volume-equalized power law Reynolds number volume-equalized power law wall Reynolds number volume-equalized generalized power law Rey- nolds number wall Reynolds number foam temperature superficial velocity


*g Ll Es r i/ Yw V

VI

pg

P 7

TW

mass fraction of gas in foam length coordinate travel distance for the development of the ul- timate velocity profile specific volume expasion ratio foam quality (void fraction of gas) shear rate wall shear rate specific volume of foam specific volume of liquid phase gas density foam density shear stress wall shear stress

References

Blauer, R. E., B. J. Mitchell, and C. A. Kohlhaas, “Determination of Laminar, Turbulent, and Transitional Foam Flow Losses in Pipes,” paper SPE 4885 presented at the SPE California Regional Meeting, San Francisco, April 4-5 ( 1974).

Cameron, J. R. and R. K. Prud’homme, “Fracturing Fluid Behavior, *’ in Recent Advances in Hydraulic Fracturing, SPE Monogmph Series, edited by J. L. Gidley, S. A. Holditch, D. E. Nierode, and R. W. Veatch. Jr. (SPE, Richardson,TX, 1989), pp. 177-209.

Chakrabarti, S., B. Seidl, J. Vonverk, and P. 0. Brunn, “The rheology of Hydroxy-propyl-guar (HPG) solutions and its influence on the Bow through a porous medium and turbulent tube flow, respectively (Part I),” Rheol. Acta 30, 114-123 (1991).

Govier, G. W. and K. Aziz, The Flow of Complex Mixtures in Pipes (R. E. Krieger, Huntington, NY, 1977).

Harris, P. C. and V. G. Reidenbach, “High-Temperature Rheological Study of Foam Fracturing Fluids,” I. Pet. Technol. (May) 613619 (1987).

Heller, J. P. and M. S. Kuntamukkula, “Critical Review of the Foam Rheology Literature,” lnd. Eng. Chem. Res. 26, 318-325 (1987).

Kcck, R. G., “The Effects of Viscoelasticity on Friction Pressure of Fracturing Fluids,” paper SPE 21860 presented at the SPE RMRM/LPR Symposium, Denver, CO, Apr. 15-17 (1991).

Lord, D. L., “Turbulent Flow of Stimulation Fluids: An Evaluation of Friction Loss Scale-Up Methods,” paper SPE 16889, presented the SPE Annual Technical Conference and Exhibition, Dallas, TX, Sept. 27-30 (1987).

Reid, R. C., J. M. Prausnitz, and B. P. Poling, The Pmperties of Gases and Liquids, 4th cd. (McGraw-Hill, New York, 1987).

Reidenbach, V. G., P. C. Harris, Y. N. Lee, and D. L. Lord, “Rheological Study of Foam Fracturing Fluids Using Nitrogen and Carbon Dioxide,” SPE Prod. Eng. (Jan) 3141 (1986).

Spijrker, H. F., F. P. Trepess, P. Valk6, and M. J. Economides, “System Design for the Measurement of Downhole Dynamic Rheology for Foam Fracturing Fluids,” paper SPE 22840 presented at the 66th Ann. Techn. Conf. and Exh. of SPE, Dallas, TX. Oct. 6-9 (1991).

Valk6, P. and M. J. Economides, “Volume equalized constitutive equations for foamed polymer solutions,” J. Rheol. 36, 1033-1055 (1992).

Valk6, P., M. J. Economides, S. A. Baumgartner, and P. M. McElfresh, “The Rheological Properties of Carbon Dioxide and Nitrogen Foams,” paper SPE 23778 presented at the SPE Int. Symposium on Formation Damage Control, Lafayette, Louisiana, Feb. 26-27 (1992).

Virk, P. S., “Drag Reduction Fundamentals,” AIChE J. 21, 625456 (1975).

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