Modeling aspects of oil–water core–annular flows

Journal of Petroleum Science and Engineering 32 (2001) 127 – 143 www.elsevier.com/locate/jpetscieng

Modeling aspects of oil–water core–annular flows Antonio C. Bannwart Department of Petroleum Engineering, School of Mechanical Engineering, State University of Campinas-UNICAMP, Cidade Universitaria-B. Geraldo, Cx.P. 6122, 13083-970, Campinas, SP, Brazil

Abstract The annular flow pattern of two immiscible liquids having very different viscosities in a horizontal pipe—also known as ‘core – annular flow’—provides an attractive means for the pipeline transportation of heavy oils since the oil tends to occupy the center of the tube, surrounded by a thin annulus of a lubricant fluid (usually water). The correspondent pressure drop is comparable to the flow of water only in the same pipe at the total volumetric flow rate. Recently, successful experiments led by the author and his group indicated that the core flow technology might be even more attractive for heavy oil production in vertical wells, due to the symmetry of the flow and the favorable effect of buoyancy. In this paper, several aspects of core – annular flow modeling are analyzed and discussed in the light of experimental data. First, criteria for existence of stable core flow are proposed, which show the essential role played by interfacial tension. Phenomenological models, based on mass and momentum balances, are developed for volume fraction and pressure drop and compared with data for both horizontal and vertical oil water core flows. The very good agreement observed is encouraging. D 2001 Elsevier Science B.V. All rights reserved. Keywords: Oil – water flow; Core – annular flow; Heavy oil production; Modeling

1. Introduction The importance of heavy oil in the world oil market is rapidly increasing, in view of its significant reserves (estimated in three trillion barrels of oil in place) and the progressive exhaustion of light oil reserves in the next decades. This leads to a growing economic interest in the exploitation of larger heavy oil fields and the research of technologies capable of improving their recovery factor (Moritis, 1995). In the Brazilian scenario, the biggest oil reservoirs are offshore, especially under deep-water sheets (over 1500 m). Besides the mechanical problems related to

E-mail address: [email protected] (A.C. Bannwart).

the depth, the low temperature at the sea bottom may cause the oil viscosity to increase exponentially, thus making unfeasible its pumping up to surface facilities. In the development of any petroleum field, the main objective is to increase the economic productivity of the wells. In the heavy oil case, this objective is more difficult to attain and a better integration of the technological solutions in each stage of the development process is necessary (Fig. 1). This includes several actions, ranging from the improvement of the flow conditions inside the reservoir up to the analysis of the technical specifications required by the refinery. The usually proposed technologies include the addition of heat (thermal methods) and the injection of diluents or aqueous solutions of surfactants or dispersants (also known as ‘cold production’), all aiming at viscosity

0920-4105/01/$ - see front matter D 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 4 1 0 5 ( 0 1 ) 0 0 1 5 5 - 3

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Fig. 1. Integration of development stages in a heavy oil field (Vanegas Prada and Bannwart, 1999a).

reduction (Vanegas Prada and Bannwart, 1999a). The use of light diluents demands the existence of light oil in the same production area, while thermal methods such as steam injection seem unfeasible in deep water. A possible alternative for lifting heavy oils (viscosity about 102 mPa s and above) is based upon the great accumulated experience in the transport of highly viscous fluids by injection of small amounts of water, in such a way to create an appropriate lubrication of the oil and to establish an annular pattern of liquid – liquid flow called ‘core flow’ or ‘core –annular flow’. Since the series of studies carried out in Canada by Russel and Charles (1959), and notably Charles et al. (1961), the advantages of the core flow technology have been quite appreciated. In fact, this flow pattern is shown to require the smallest pumping power for unity oil mass since the highly viscous oil flows in the center and is surrounded by a water ring close to the tube wall. More recently, theoretical and experimental studies have been done for horizontal flow, directed to applications in heavy oil transportation (for example Oliemans et al., 1987; Arney et al., 1993; Ribeiro et al., 1994; Bannwart, 2000). Laboratory experiments (Vanegas Prada and Bannwart, 1999b) with a viscous fuel oil (17,600 mPa s) and water at room temperature confirmed that the pressure drop in vertical core flow is comparable with that expected for single phase water flow at mixture volumetric flow rate. Besides, the thin water annulus requires injection of small amounts of water, thus making the core flow pattern an attractive alternative for the lift and transportation of heavy crudes. However, the idea of applying the core flow technology in heavy and ultra-viscous oil production has not yet been tested.

This paper presents the author’s personal view on hydrodynamic aspects of core flow, including occurrence conditions, volume fraction and pressure drop. Horizontal and vertical flows are focused.

2. Existence of core– annular flow pattern The pipe flow of two immiscible liquids typically forms different spatial configurations or flow patterns of the two phases, according to the flow rate of each fluid. The flow patterns can be grouped into three categories: dispersed flow, separated flow and intermittent flow. In the case of oil – water systems, dispersed flow includes oil bubbles in water, water drops in oil, water-in-oil and oil-in-water emulsions. Separated flow flows comprise annular (oil in the core, water in the annulus), stratified – annular and stratified flow patterns. Intermittent flows include mainly large oil bubbles into water (Charles et al., 1961). Among these flow patterns, the annular configuration presents the greatest interest due to its effectiveness in the pipeline lift and transportation of viscous liquids. For instance, in the production and transportation of heavy oil, a thinner fluid (such as water) can be injected laterally to the pipe, through a special injection nozzle, so as to lubricate the oil flow, which flows by the center. Figs. 2 and 3 show typical oil –water core – annular flow patterns in vertical and horizontal pipes. The observed waves are the result of complex phenomena occurring at the interface separating the two fluids. This interface is an internal boundary whose position is unknown and should be determined as part of the solution of the hydrodynamic problem.

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respectively. Consider the stability of a viscous liquid flowing at the core, surrounded by a turbulent annulus flow. Assuming that the fluids are well separated (i.e. there is no fluid 2 in the core and no fluid 1 in the annulus), criterion (1) becomes q2 J2 D > 2000 ð2Þ l1 > l2 þ 0:0005q2 J2 D for l2 and e > 0:5

Fig. 2. Upward core – annular flow of a 500 mPa s, 0.925 g/cm3 oil and water (Bannwart et al., 2000).

Experiments generally indicate that the following conditions should be satisfied in order to the core – annular flow pattern occur in a pipe: (a) the core phase must be much thicker than the annulus; (b) the input fraction of the thinner fluid must be such as to maintain a continuous core phase of the thicker fluid and at the same time to keep it from touching the pipe wall. Theoretical studies have been done to establish physical criteria for existence of core flow. Joseph et al. (1984) used the hydrodynamic stability theory and concluded that a fully developed laminar – laminar core flow of two liquids having the same density is stable when the more viscous fluid is placed at the core and occupies most of the cross-section. This criterion can be extended to turbulent flow in either phase by using the effective viscosity, as proposed by Joseph et al. (1996). Accordingly, this criterion can be expressed as l1;eff > l2;eff

and

e > 0:5

ð3Þ

where q2 is the density of the annulus fluid, J2 its superficial velocity (i.e. flow rate divided by pipe cross-sectional area) and D is the pipe diameter. Condition (2) expresses a necessary condition that the annulus turbulence is not capable of breaking up the core, which remains continuous. Condition (3) can be put in a more convenient form in terms of the input ratio J2/J1: J1 > sJ2

ð4Þ

where s is the slip ratio (i.e. the ratio of the velocity of fluid 1 to fluid 2). Consider for example the data from the equal density experiments of Charles et al. (1961) in a 2.54-cm i.d. pipe using oils of viscosity l1 = 6.29, 16.8, 65 mPa s and water (l2 = 0.894 mPa s, q2 = q1 = 998 kg/m3). Application of condition (2) gives J2 < 0.42, 1.2 and 5 m/s, respectively, and from condition (4), one obtains J1 > 0.42, 1.2 and 5 m/s, respectively (assuming no slip). These compares reasonably with the results summarized in Fig. 6 (l1 = 6.29 and 16.8 mPa s) and Fig. 7 (l1 = 65 mPa s) of their paper, where the transition to oil slugs corresponding to a midrange value J2 = 0.42 m/s occurs for J1 ffi 0.4 m/s. Criterion (2) predicts that in a 5-cm i.d. pipe and J2 = 0.4 m/s, oil –water core flow will be stable for oils of viscosity l1>10 cP. However, conditions (2) and (3) are not sufficient to ensure existence of core –annular flow, since so far, the effect of the density difference on the flow pattern has not been taken into account. Brauner and Moalem Maron (1998) proposed a flow

ð1Þ

where leff is the effective viscosity (i.e. the sum of the absolute molecular viscosity with the turbulent viscosity), e is the volume fraction of the core and subscripts ‘‘1’’ and ‘‘2’’ stand for the core and annulus properties,

Fig. 3. Horizontal core – annular flow of a 17,600 mPa s, 0.96 g/cm3 oil and water (from Vanegas Prada, 1999).

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concept, interfacial tension determines the average stable wavelength of the waves (Rodriguez and Bannwart, 2000). In fact, while the buoyant force stretches the core against shear, interfacial tension acts oppositely to stabilize the core as a continuous phase. Another reason to apply criterion (5) in vertical flow arises from the Kelvin Helmholtz stability criterion (see for example, Ishii, 1982): the difference in velocity of the two-phases is destabilizing, whereas surface tension always tends to stabilize a separated flow. Since in vertical core flow the difference in velocities increases with the density difference, a Eo¨tvos number-based stability criterion makes sense. Table 2 shows a comparison of criteria (2) and (5) with the observations of different authors for vertical upward flow. Note that the oil employed by Flores et al. (1997) seems too light, violating both the proposed criteria. Last, but not least, the core flow pattern is lost if the input fraction of the thicker fluid is too high: a continuous annulus of the thin fluid cannot be maintained and is dispersed as drops in the thicker phase, which sticks on the wall, causing an abrupt change in pressure drop. For oil –water systems, Charles et al. (1961) report the transition from core flow to a dispersion of water drops in oil at high oil flow rates, for all water flow rates tested. The exact point where this transition starts is unknown and depends on the relative wettability of the wall by the oil. For example, Oliemans (1986) reports that core flow was observed with only 2% water in the test section, but he used an additive with water, in order to make the wall oleophobic. Fortunately, the transition to a dispersion of

classification for horizontal flow based on an Eo¨tvos number defined as DqgD2/(8r), where r is the liquid –liquid interfacial tension: for Eo¨tvos number, much greater than unity, the flow will tend to stratify, whereas low Eo¨tvos numbers favor the annular flow pattern. The present author derived the theoretical interface shape in fully developed core flow and showed that capillary effects may be of the same order as the net buoyancy force on the core (Bannwart, 1999a). In a later work (Bannwart, 2001), the effect of peripheral flow in the annulus was included and stabilization of core flow in a pipe was shown to be possible when pDqgD2 e <8 4r

ð5Þ

Criterion (5) was originally derived for horizontal flow and represents the limit condition that the force associated with interfacial curvature gradients balances the buoyant force on the core, as in sessile bubbles (Clift et al., 1978). If this condition is not obeyed, then stratification is most likely to occur instead of core flow. A comparison of the results of different researchers with criteria (2) and (5) is presented in Table 1. Note that while criterion (2) is typically satisfied in practice, criterion (5) is not. It is possible that a criterion similar to criterion (5) be also valid for vertical flow. Like in ascending bubbles, interfacial tension keeps the core from breaking up into slugs, preserving it as a continuous phase. Besides, by analogy with the critical bubble diameter

Table 1 Predictions and observations of core flow in horizontal pipes System

Pipe size and properties

RHS of condition (2) for J2 = 1 m/s (mPa s)

LHS of condition (5) at e = 0.5

Core flow observed

Mineral oil – water (Trallero et al., 1997) Fuel oil – water with additive (Oliemans, 1986) Fuel oil – water (Bannwart, 1998) Fuel oil – water (Vanegas Prada, 1999) Crude oil – water (Bannwart et al., 2000)

D = 5 cm; l1/l2 = 30; q1/q2 = 0.85; r = 36 dyn/cm D = 5 cm; l1/l2 = 3000; q1/q2 = 0.975 D = 2.25 cm; l1/l2 = 2700; q1/q2 = 0.989; r = 30 dyn/cm D = 2.5 cm; l1/l2 = 18,000; q1/q2 = 0.96; r = 30 dynes/cm D = 2.84 cm; l1/l2 = 500; q1/q2 = 0.925; r = 29 dyn/cm

26

40

No

26

8 (assuming r ffi 30 dyn/cm)

Yes

11

0.7

Yes

14

3

Yes

14

7.7

Yes

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131

Table 2 Predictions and observations of core flow in vertical pipes System

Pipe size and properties

RHS of condition (2) for J2 = 1 m/s (mPa s)

LHS of condition (5) at e = 0.5

Core flow observed

Mineral oil – water (Flores et al., 1997) Motor oil – water with additive (Bai, 1995) Fuel oil – water (Vanegas Prada, 1999) Crude oil – water (Bannwart et al., 2000)

D = 5 cm; l1/l2 = 20; q1/q2 = 0.85; r = 33.5 dyn/cm D = 0.95 cm; l1/l2 = 600; q1/q2 = 0.91; r = 22.5 dyn/cm D = 2.5 cm; l1/l2 = 18,000; q1/q2 = 0.96; r = 30 dyn/cm D = 2.84 cm; l1/l2 = 500; q1/q2 = 0.925; r = 29 dyn/cm

26

44

No

the thinner phase is not very important to design the pipeline since the correspondent input fraction of the lubricant is much lower than the observed for maximum reduction in pressure drop, which is around 8% in heavy oil – water flow in horizontal pipes (see Section 3.2). To summarize, core – annular flow will tend to occur in a pipe when the two fluids have very dissimilar viscosities but relatively close densities. This situation is often satisfied by heavy oils, crude or refined, whose viscosity is greater than 102 mPa s and density is close to water. Criteria represented by conditions (2), (3) and (5) provide necessary requirements for existence of core – annular flow. Lubricant input fractions lower than a few percent may cause transition to a dispersion of the lubricant in the viscous phase, which will wet the pipe wall.

3. Volume fraction Since core – annular flow usually involves a very viscous fluid, the measurement of its volume fraction is much more difficult than in gas – liquid flow. For example, intrusive methods such as probes are clearly unfeasible in core flow. Also, the quick-closing valves method presents some problems related to high viscosity (often thousands of centipoise), adhesion to the tube wall (oil) and the low density difference of the fluids, all that making it difficult for the fluids to stratify. The successful development of non-intrusive methods based on electric or acoustic impedance (ultra-sound) has not yet been found in the technical literature. The method based on photographs of the flow (Oliemans, 1986) seems appropriate; however, it

6

1.4

Yes

14

3

Yes

14

7.7

Yes

requires the assumption of axisymmetry of the core. In a situation such as the one shown in Fig. 3, this method can give somewhat inaccurate results. In view of the above considerations, the present author proposed a holdup determination method based on the measurement of the interfacial wave speed (Bannwart, 1998). In fact, in stable core –annular flow, the interfacial waves can be described through the mass balances alone and are called kinematic waves (see for example Whitham, 1974 or Wallis, 1969, who prefers the term ‘‘continuity waves’’). The speed of a kinematic wave is defined by
 qJ1 a¼ ð6Þ qe J where J is the mixture superficial velocity. Eq. (6) suggests working with a triangular relationship of the form f (e, J1, J ) = 0. Such relationship can be derived by eliminating the pressure gradient from the onedimensional momentum equations of each phase. The result is 

Si s i Sw s w v e þ eð1  eÞðq1  q2 Þgz ¼ 0 þ A A

ð7Þ

where si and sw are the interfacial and wall shear p stresses, Si ( = pD e) and Sw ( = pD) are the interface and wall perimeters, A ( = pD2/4) is the cross-sectional area of the pipe and gz is the component of gravity acceleration in the axial direction (negative for upflow, positive for down-flow). Eq. (7) is similar to that proposed by Brauner (1991). However, the parameter v is introduced in this equation to allow for negligible wall effects on interfacial phenomena: v is set to zero when wall shear can be neglected (Wallis, 1969, p. 90)

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and v = 1 otherwise. Setting v = 0 seems justified when the core is nearly concentric and the annulus flow is turbulent: wall effects become minor and a direct balance between drag and buoyancy can be applied. In order to transform Eq. (7) into the form f (e, J1, J ) = 0, interaction laws are needed to express si and sw. For the usual case of Newtonian fluids, these laws can be expressed in the following form fi;1 q AV1  Vi AðV1  Vi Þ si ¼ si ðV1 ; V2 ; eÞ ¼ 2 1

ð8Þ

or, si ¼

fi;2 q AVi  V2 AðVi  V2 Þ 2 2 pffiffi 1 þ ðq1  q2 Þgz D ekðeÞ 4

ð9Þ

fw q V2 AV2 A 2 2

1  ðq1  q2 Þgz DekðeÞ 4

8 < laminar flowðRek < 2000Þ : :

ð10Þ

where Vi is the velocity at the interface, fi,1, fi,2, fw are friction factors that depend on the flow regime in each phase and can be expressed in the following way

ak ¼ 16

nk ¼ 1

ak ¼ 0:079

nk ¼ 0:25

k ¼ 1; i; 2

turbulent flowðRek  2000Þ :

ð17Þ The function k(e) appearing in Eqs. (9) and (10) represent the effect of buoyancy on shear and applies to inclined or vertical flow only. This function can be derived for laminar annulus flow and smooth interface as 

and sw ¼ sw ðV2 ; eÞ ¼

and the parameters a1, ai, a2, n1, ni, n2 are given by

1 þ e þ 2elne 1e kðeÞ ¼ 1e

 ð18Þ

For turbulent annulus flow and wavy interfaces, k(e) f (1  e)m where the exponent m can be experimentally determined, as will be shown later. The velocity Vi at the interface appearing in Eqs. (8) and (9) can generally be obtained by elimination between these two equations, but the result has little practical use. With Eqs. (8) – (18), Eq. (7) allows determination of the volume fraction e in terms of the fluid properties and the superficial velocities. Some special cases are discussed.

fi;1 ¼ a1 ðRe1 Þn1

ð11Þ

3.1. Equal density or horizontal core flow

fi;2 ¼ ai ðRei Þni

ð12Þ

fw ¼ a2 ðRe2 Þn2

ð13Þ

When the fluids are in laminar flow and have the same density, replacing Eqs. (8) and (10) into Eq. (7) gives (taking v = 1)

where the Reynolds numbers Re1, Rei and Re2 are based on the hydraulic diameters i.e. pffiffi q AV1  Vi AD e ð14Þ Re1 ¼ 1 l1

Rei ¼

Re2 ¼

q2 AVi  V2 ADð1  eÞ pffiffi l2 e

q2 V2 Dð1  eÞ q2 J2 D ¼ l2 l2

ð15Þ

e¼

1 h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 þ JJ21 1 þ 1 þ ll2 JJ12 1

ðequal density laminar  laminar flowÞ

ð19Þ

whereas using Eqs. (9) and (10), Eq. (7) gives Vi ¼ si;o ¼ 2 V2

ð16Þ

ðequal density laminar annulus flowÞ

ð20Þ

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133

If wall shear effects are negligible, Eqs. (7) – (9) with v = 0 give V1 ¼ Vi ¼ V2 Z si;o ¼ 1 ðequal density with no wall shear effectsÞ

ð21Þ

The above no-slip result was experimentally found by the author in horizontal core flow (Bannwart, 1998), when the buoyancy term in Eq. (7) also disappears. A somewhat more general equation for determination of e in equal density flow for any flow regime in each phase is eðn1 1Þ=2




1e e

2 1  si;o

e J2 ð1  eÞJ1

2n1

ðequal density; any flow regimeÞ

¼ vX2

ð22Þ

where X 2 is the traditional Lockhart – Martinelli parameter defined as
a2 X2 ¼


a1

q 2 J2 D l2 q 1 J1 D l1

n2 n1

1 1 þ si;o JJ21

zontal core flow from wave speed measurements. Replacing J2 by J  J1 in the above equation and performing the partial derivative as in Eq. (6), the speed of kinematic waves is

a¼

J22

ðJ1 þ si;o J2 Þ2 si;o J

ð25Þ

ð23Þ J12

It can be shown that Eq. (22) reproduces Eq. (19) when n1 = n2 = 1, a1 = a2 = 16 and si,o = 2. Fig. 4 shows the relationship e( J2/J1, X 2, si,o, n1) for laminar flow in the core (i.e. n1=) and two si,o values, namely si,o = 1 and si,o = 2. As can be seen in the figure, the curves for si,o = 1 (continuous lines) differs substantially from si,o = 2 (dotted lines). The case of an infinitely viscous core and any annulus flow regime can be easily obtained from Eq. (22) by making X 2 = 0. The result is simply e¼

Fig. 4. Volumetric fraction of the core, e, as a function of the input ratio J2/J1 and X 2, assuming laminar flow in the core. The continuous lines stand for si,o = 1 and the dotted lines si,o = 2.

ðinfinitely viscous core;

any annulus flow regimeÞ

ð24Þ

Eq. (24) can be used in connection with Eq. (6) to determine the volume fraction and slip ratio in hori-

Two sets of experiments were performed, where the interfacial wave speed was measured using a highspeed camera at several J1 (oil) and J2 (water) values. In the first set, the oil was a no. 6 fuel (viscosity = 2700 mPa s, density = 0.989 g/cm3), flowing in core flow mode with water in a 2.25-cm i.d. pipe. Calculation of the Reynolds number Re2 as defined in Eq. (16) revealed that the annulus flow was turbulent and the interface was wavy. The best si,o value to fit the data was (Bannwart, 1998) si;o ¼ 1

ðheavy oil  water core flow in

2:25 cm i:d: horizontal pipeÞ

ð26Þ

This result is consistent with the assumption of negligible wall shear effects (Eq. (21)) and means that waves are ‘‘frozen’’ on the core (since it implies a = J = V1 = V2). The second data set was obtained very recently, using a heavy crude oil (viscosity = 500 mPa s, density = 0.925 g/cm3) and water inside a 2.84-cm

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where Vref ¼

gz ðq1  q2 Þ ni12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a Agz AD Agz ðq1  q2 ÞA i


ADqA q2

ð27Þ

a¼

Fig. 5 shows a comparison between measured and correlated wave speed. It can be concluded in this case that wall shear effects are not negligible in Eq. (7). This is probably due to the high value of the Eo¨tvos number introduced in Eq. (5), which is about 10 times higher than the first set of experiments, as can be seen in Table 1, making the core much more off-centered and wall effects significant.

J1 þ si;o J2 þ cVref eð1  eÞn1 ½2ð1  eÞ  ne 1 þ ðsi;o  1Þe

si;o ¼ 1:5 c ¼ 0:02

n¼2

ð33Þ

ð28Þ with the velocity Vref calculated using ai = 16 and ni = 1 in Eq. (30). A comparison between the volumetric fraction obtained from Eq. (31) with Bai’s data is

The use of Eqs. (9) and (10) into Eq. (7) gives the general result

turbulent annulus flow

ð31Þ

The above relation can be reduced to Eq. (25) when the densities are the same or the flow is horizontal. Bai (1995) reports wave speed and holdup measurements for vertical flow of Motor Oil (density = 905 kg/m3; viscosity = 600 mPa s) and water at room temperatures inside a 0.9525 cm i.d. tube. The input ratio was in the range 0.6 < J1/J2 < 8. He observed axisymmetric bamboo waves for up-flow and corkscrew waves for downflow. From his wave speed data, the best values of parameters si,o, k and n are

For infinitely viscous core, it can be concluded from replacement of Eqs. (8) and (10) into Eq. (7) that

:

ð30Þ

ð32Þ

3.2. Vertical flow with infinitely viscous core

8 < laminar annulus flow

ni pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2n i Agz ADD l2

The constants c and m can be adjusted to fit experimental wave speed data. The correspondent speed of the kinematic wave defined by Eq. (6) is given by

ðheavy oil  water core flow in

Vi ¼ V1

q2

J1 ð1  eÞ  si;o J2 e  cVref e2 ð1  eÞm ¼ 0

i.d. pipe (Bannwart and Vara, submitted for publication). Again, the annulus flow was mostly turbulent and the best si,o value was (Bannwart and Vara, submitted for publication)

2:84 cm i:d: horizontal pipeÞ

i

and ai and ni are given by Eq. (17). Eq. (29) represents the drift between the core and annulus due to buoyancy and annulus velocity profile effects. Since even in laminar annulus flow the interface is observed to be wavy, Eq. (29) can be written in a single approximate and convenient form as

Fig. 5. Plot of a/J2 versus J1/J2 for heavy oil – water core annular flow inside a 2.84-cm i.d. pipe. The continuous line corresponds to Eq. (25) with si,o = 1.23.

si;o ¼ 1:23

 2n1

  J1 ð1  eÞ  2J2 e þ Vref e2 ð1  eÞ 2 þ 1þe 1e lne ¼ 0 J1 ð1  eÞ  si;o J2 e þ Vref e

95ni 42ni

ð1  eÞ

ni 2ni

 3e 2

þ

lne 1e

ni  2n

ð29Þ i

¼0

A.C. Bannwart / Journal of Petroleum Science and Engineering 32 (2001) 127–143

Fig. 6. Plot of e versus J1/J2 for core annular up flow of oil – water inside a 0.9525-cm i.d. vertical tube. The points stand for Bai’s (1995) experimental data; the dash-dot line corresponds to Bai’s average slip correlation (Eq. (34)) and the shaded area corresponds to Eq. (31).

shown in Fig. 6. He expressed the results for up flow by an average slip ratio of 1.39. This corresponds to eexp ¼

1 1 þ 1:39 JJ21

ð34Þ

which is also shown in Fig. 6. The shaded region represents Eq. (31). The agreement between experimental and calculated holdup values can be considered very good. The difference between the holdup calculated from Eqs. (31) and (34) is less than 10%.

upon the lubrication theory, which takes into account the waviness of the interface but assumes that the wave parameters are known a priori. Oliemans et al. (1987) further included the turbulence in the annulus flow. Brauner (1991) proposed a two-fluid one-dimensional model, which can be used for any flow regime in either phase. As usual in this type of model, the pressure drops of each phase flowing alone in the pipe are required to form the traditional Lockhart – Martinelli parameters. Arney et al. (1993) suggested that all friction factor data could be correlated with a two-phase Reynolds number defined according to the PCAF theory. The oil holdup was determined from an empirical correlation. These features and the effects of fouling (mentioned by the authors) and possibly the roughness of the pipe walls might explain the relatively large spread of part of the data used in their Fig. 12. The wave speed and holdup experiments described in the previous section for viscous oil and water showed that both phases move at close or even equal velocities, despite their strongly different viscosities. This suggests that core flow can be appropriately represented by a simple drift flux model with mixture properties. Considering first the case of a fully developed laminar –laminar concentric core –annular flow with smooth interface, the pressure gradient C can be expressed as C¼

4. Pressure drop 4.1. Horizontal flow The most attractive feature of the core – annular flow pattern is that the correspondent pressure drop is comparable to the flow of the thinner fluid alone in the same pipe since it keeps in contact with the pipe wall. This can be advantageously used in pipeline transportation of very viscous fluids. Russel and Charles (1959) proposed a simple analysis of pressure drop assuming fully developed laminar axisymmetric flow with a smooth circular interface. This approach was recently also employed by Bobok et al. (1996). It is sometimes called the ‘‘perfect core – annular flow’’ approach (hereafter PCAF). Ooms et al. (1984) proposed a more sophisticated model, based

135

128l2 Q 128l2 Q h   iffi l pD4 ð1  e2 Þ pD4 1  1  l2 e2

ð35Þ

1

where Q is the mixture flow rate ( Q = Q1 + Q2) and the volumetric fraction e is given by Eq. (19). The approximation in Eq. (35) is valid for l2/l1KQ2/Q1; this condition is often true in applications. It is important to note that Eq. (35) can be interpreted as the laminar pressure drop of a pseudo-fluid at flow rate Q and mixture viscosity lm defined by 1 e2 1  e2 1  e2 ¼ þ ffi lm l1 l2 l2

ð36Þ

According to author’s measurements (Bannwart, 1999b), pressure drop data are poorly represented by the above model. This is due to three phenomena not considered in the PCAF theory: turbulence in annulus

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flow, interface waviness and wall properties. The latter are related to roughness and fouling behavior of the wall (oil adhesion). Here, it is proposed the incorporation of these effects into the traditional expression of the pressure gradient:
C¼b

qm JD lm

n

qm J2 2D

ð37Þ

where J is the mixture superficial velocity ( = Q/A) and qm is the mixture density. qm ¼ eq1 þ ð1  eÞq2

ð38Þ

If b = 64, n = 1 and Eq. (36) is used to express lm, then Eq. (35) is found back. However, the annulus flow is most often turbulent (Oliemans, 1986; Bannwart 1999b) as the annulus Reynolds number Re 2 = (q2J2D)/(l2) > 2000. In this regime, a modified expression of the mixture viscosity lm is recommended 1 e 1e 1e ¼ þ ffi lm l1 l2 l2

ð39Þ

where again, the approximation holds for small annulus-to-core viscosity ratio. The main reason for adopting the above expression instead of Eq. (36) is the behavior of shear in turbulent flow. In fact, using a parallel layer model, it can be shown that the usual consideration that the shear stress is constant close to the wall leads to Eq. (39) [see, for example, Wallis, 1969, Chap. 3], whereas a linearly varying stress leads to Eq. (36). Since the annulus flow is usually turbulent, the assumption of constant shear is justified. These two equations give significantly different results for e 1.

Accordingly, Eq. (37) can be expressed in a more general form as C ¼ C2 ðQÞ/2o

ð40Þ

where C2( Q) is the pressure gradient for the annulus fluid flowing alone at total flow rate Q in the same pipe (at current pipe wall conditions) i.e.
 q2 JD n q2 J2 ð41Þ C2 ðQÞ ¼ b l2 2D and /2o is a two-phase multiplier given by
1n
n qm l2 /2o ¼ q2 lm

ð42Þ

Furthermore, according to the results of the former section, the volume fraction e is determined from Eq. (24) along with si,o = constant. In terms of the input fraction C2 = Q2/Q of the lubricant, this gives e¼

1  C2 1 þ ðsi;o  1ÞC2

ð43Þ

The proper choice of parameters b and n in Eqs. (37), (41) and (42) is now considered. The Blasius set b = 0.316, n = 0.25 can be recommended only if the pipe wall is known to be smooth and clean. For rough and clean pipes, a power law approximation of Colebrook’s friction law is still valid for limited Re2 range, using a different set of constants than Blasius’, as shown in Table 3. In the case of oil –water flows, the possible fouling of the wall by oil adhesion should be considered. In fact, oleophobic – hydrophilic walls such as provided

Table 3 Power law representation of Colebrook’s friction law Relative roughness (e/D)

Re2 range

Power law approximation to Colebrook’s law

Maximum percentage error

0

2 103 – 104 2 103 – 105 2 103 – 105 2 103 – 104 2 103 – 105 2 103 – 104 2 103 – 105

b = 0.443; n = 0.29 b = 0.308; n = 0.248 b = 0.316; n = 0.25 (Blasius) b = 0.411; n = 0.279 b = 0.170; n = 0.186 b = 0.383; n = 0.269 b = 0.19; n = 0.19

0.2 2.1 2.7 0.6 0.9 0.6 3.5

0.0005 0.001

A.C. Bannwart / Journal of Petroleum Science and Engineering 32 (2001) 127–143

by cemented pipes (Ribeiro, 1994; Arney et al., 1996) remain clean, yet they are rough. On the other hand, oleophilic walls such as steel pipes’ are comparatively smooth, yet they foul with oil. It is quite uncertain that oil fouling may be adequately represented by Colebrook’s law, which was validated with experiments in sand-roughened pipes. While incipient oil fouling is unstable and unevenly distributed on the wall, severe fouling usually creates a more or less uniform oil coat that may reduce significantly the useful cross-sectional area of the pipe and the amount of oil in the core, increasing drastically the pressure drop. For these reasons, in order to account for wall conditions (fouling and roughness) in the actual twophase flow, use of the simple power-law representation of friction factor is proposed, with coefficients b and n determined so as to fit experimental pressure drop data. Accordingly, the two-phase multiplier in Eq. (42) becomes C /2o ¼ C2 ðQÞ
 1n
 n q l ¼ 1 1 1 e 1 1 2 e q2 l1 ð44Þ and the pressure gradient can be expressed as
 1n
 n q l C ¼ kQ2n 1  1  1 e 1 1 2 e q2 l1 ð45Þ

137

989 kg/m3. The original tube sizes were 26.7 mm for the carbon-steel tube and 23.9 mm for the cementlined tube. As mentioned in the previous section, wave speed measurements gave si,o = 1 for this system. Although the runs were short-term (15 –20 min) and both tubes were cleaned with water before each run, the steel tube was observed to be affected by oil fouling, whereas the cemented tube remained clean. The best data fit was obtained for b ¼ 0:305 n ¼ 0:159 cemented pipe ðroughÞ b ¼ 0:066 n ¼ 0:047

steel pipe ðfouledÞ ð47Þ

with Re2 in the range 103 < Re2 < 104. Note that since l2Kl1, q2ffiq1 and using Eq. (39), the annulus and mixture Reynolds numbers become in this case equivalent, i.e. Rem ¼

qm JD q2 J2 D ffi ¼ Re2 lm l2

ð48Þ

The goodness of fit can be observed in Figs. 7 and 8, which show plots of the pressure gradient as a function of a modified flow rate Q* defined by n=ð2nÞ

Q ¼ QC2

ðq1 ffi q2 ; l2 Kl1 Þ

ð49Þ

as suggested by Eq. (45). The standard deviation of the data to the calculated values was F 20% for both pipes.

with the holdup determined from Eq. (43). Adjusting k and n for best fit ensures that the current pipe wall conditions present in the actual flow are taken into account in the friction law. The dimensionless constant b is determined from k by the relation b ¼ 2k

 p 2n 4

n 5n qn1 2 l2 D

ð46Þ

The experimental pressure drop data obtained by the author for heavy oil– water core –annular flow in steel and cemented pipes (Bannwart, 1999b) can be reinterpreted using the method just described. The oil tested was a fuel oil of viscosity 2700 cP and density

Fig. 7. Pressure gradient as a function of the flow rate Q * — cemented tube. The solid line represents Eq. (45) with b = 0.305, n = 0.159.

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Fig. 8. Pressure gradient as a function of the flow rate Q * —steel tube. The solid line represents Eq. (45) with b = 0.066, n = 0.047.

The present method can also be used as a predictive model when the wall conditions and flow regime are previously known. As an illustrative example, consider the core – annular flow of oil and water inside a 5 cm horizontal smooth tube investigated by Oliemans (1986). The oil properties were l1 = 3 Pa s, q1 = 975 kg/m3. Oliemans employed a transparent pipe and water with additive to get an oleophobic pipe wall behavior. Accordingly, the pressure drop can be predicted by (assuming si,o = 1 for simplicity)
C¼b

q2 JD l2

n



1n
 q2 J 2 q 1  1  1 ð1  C2 Þ 2D q2



n
 l 1  1  2 ð1  C2 Þ l1

ð50Þ

using the Blasius set b = 0.316, n = 0.25. The measured pressure drop is given on p. 64 of his thesis. Fig. 9 shows the comparison between calculated and experimental values. The standard deviation between calculated and measured values is F 14%. This can be considered a very encouraging result, even assuming a probably inaccurate si,o value. Note that, for the range 0.05 < C2 < 0.2 of Oliemans’ experiments, the effect of si,o on holdup in Eq. (43) is minor. Fig. 10 shows a plot of the two-phase multiplier defined by Eq. (44) as a function of the input fraction C2. Different pipe wall conditions and flow regimes can be represented in a single diagram. The result using the PCAF approach (i.e. n = 1 and Eq. (36) replaced in Eq. (42)) is also shown for comparison. Both cemented and steel tubes have lower multipliers

Fig. 9. Comparison between experimental pressure drop (Oliemans, 1986) and calculated using Eq. (50). Ccalc = Cexp on the straight line.

in comparison with smooth and clean tubes, although their pressure drops are higher due to roughness and fouling. This is because rougher tubes have higher degrees of turbulence in comparison with smooth tubes, and this fact can be associated with lower n values. The same applies to fouled tubes. Fig. 10 also shows that pressure drop in a real core – annular flow (turbulent regime) is comparable to the flow of the lubricant alone in the pipe at total flow rate. A word must be said about the existence of an optimum input ratio Q2/Q1, as reported by many authors, e.g. Russel and Charles (1959) and more recently by Arney et al. (1993). This happens because the addition of water helps the oil flow but at the same time increases the total flow rate. This result can also be reproduced with Eq. (50) and shown in Fig. 11,

Fig. 10. Two-phase multiplier as a function of the input fraction for different wall conditions and flow regimes.

A.C. Bannwart / Journal of Petroleum Science and Engineering 32 (2001) 127–143

139

rates, the system was run with pure water until the pressure drop in the test section became low enough so as it could be assumed to be clean from any fouling action by the oil. The frictional pressure gradient in core flow (Cf) can be defined as the total pressure gradient minus the gravity term of the mixture (Arney et al., 1993) and was determined from the measurements of pressure difference in the vertical test section, in the following way: Cf ¼ Fig. 11. Pressure drop in horizontal core – annular flow as a function of the input ratio Q2/Q1, for a fixed Q1.

again assuming l2Kl1 and q2ffiq1. For a fixed flow rate of fluid 1, the optimum input ratio corresponds to n/(2  n) and n/(3  n) for minimum pressure drop and minimum pumping power, respectively. These values are much lower than those predicted by PCAF model (0.5 and 0.3, respectively). However, such low ratios must be seen with care if fouling is a possibility. 4.2. Vertical upward flow In contrast with the horizontal case, where the net buoyancy force (Archimedes force) causes the oil core to be eccentric, in vertical flow this force favors the flow of the (lighter) oil by the center and thus the stabilization of the flow itself. An experimental investigation of vertical core flow performed at UNICAMP (Vanegas Prada and Bannwart, in press) allowed the development of a pressure drop correlation, which takes into account the effect of buoyancy on the frictional pressure gradient. The results were compared with measurements done by Bai (1995) in a 0.9525-cm i.d. glass tube. In the experiments at UNICAMP, the oil was a 17,600 mPa s, 0.963 g/cm3 fuel oil at room temperature. From the water-injection point, the oil –water mixture flowed into a 2.76-cm i.d. test section pipe made on galvanized steel, through vertical and horizontal segments, returning to the separator tank. Pressure drop in an 84-cm segment of the vertical upward test section was measured by means of a Validyne differential pressure transducer (accuracy 3% of full scale). Before setting each pair of flow

DPfriction DPdpt ¼  ðq1  q2 Þge H H

ð51Þ

where DPdpt is the pressure difference read at the differential pressure transducer, H is the length between pressure taps and g is the gravity acceleration. The oil fraction e was determined from Eq. (31) using the set of constants given in Eq. (33). Pressure drop was measured for nine oil flow rates in the range 0.297 –1.045 l/s, with different water flow rates ranging from 0.063 to 0.315 l/s. The total number of runs was 65. The measured values of the frictional pressure gradient are plotted in Fig. 12 as a function of the water – oil ratio ( Jw/Jo), for each fixed oil superficial velocity ( Jo). The existence of a minimum pressure gradient for a certain input ratio, at a given oil flow rate, can be

Fig. 12. Frictional pressure gradient in upward oil – water core flow as a function of input ratio, at different oil superficial velocities.

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A.C. Bannwart / Journal of Petroleum Science and Engineering 32 (2001) 127–143

where m = l2/l1 and the approximation holds for small m. The volume fraction in this model is determined from the laminar formula in Eq. (29). Fig. 13 shows a plot of the sum of measured frictional pressure gradient Cf with the buoyancy term in the RHS of Eq. (52) (i.e. the irreversible part of Cf), as a function of the flow rate Qpcaf = Q/(1  e2). A linear relationship, suggested by Eq. (52), is also indicated in the same figure. Clearly, the PCAF model is not effective to describe the data. As for horizontal flow, this fact can be attributed to the presence of waves on the interface and turbulent water flow, since the annulus Reynolds was in the range 2000 < Re2 < 16,000. In order to embody the wavy character and annulus turbulence effects together with the buoyancy effect, the following frictional pressure drop model is proposed: Fig. 13. Irreversible part of the pressure gradient as a function of the equivalent flow rate Qpcaf. The straight line stands for Eq. (52).

clearly observed. This result has been reported for horizontal flow and is also confirmed for upward flow (Bai, 1995). The optimum input ratio, however, depends on the superficial velocity of the oil, and is observed to be in the range 0.07– 0.5. When the superficial oil velocity increases, the minimum pressure gradient point moves toward lower values of input ratio. In other words, that the largest oil flow rates need, proportionally, lower amounts of water to reach the minimum frictional pressure gradient. This is indeed a very attractive feature of this flow pattern. In the perfect core – annular flow (PCAF) model the two Newtonian immiscible fluids flow inside a vertical pipe in a concentric configuration with a smooth circular interface. The frictional pressure gradient can be expressed as

Cf ¼

q JD Cf ¼ b m lm

n

qm J 2  Cðq2  q1 Þgeð1  eÞ 2D ð53Þ

where the irreversible part is analogous to Eq. (37) and the mixture density qm and viscosity lm are given by Eqs. (38) and (39). The volume fraction e was

128l2 Q  e2 ð1  mÞ

pD4 ½1 

ffi




ðq2  q1 Þgeð1  eÞ½1  eð1  mÞ ½1  e2 ð1  mÞ

128l2 Q ðq  q1 Þgeð1  eÞ  2 4 2 pD ð1  e Þ ð1 þ eÞ

ð52Þ

Fig. 14. Calculated and experimental irreversible pressure gradient versus modified flow rate Q*.

A.C. Bannwart / Journal of Petroleum Science and Engineering 32 (2001) 127–143

141

determined from Eq. (31) using the set of constants given in Eq. (33). The constants b, n and C are parameters to be adjusted from experiments. The parameter n was set to 0.25 (turbulent flow in smooth walled pipe); thus, only b and C were adjusted. The following values were found b ¼ 0:257 ðn ¼ 0:25Þ

ð54Þ

C ¼ 0:159 Fig. 14 shows a comparison of the experimental and theoretical values of the irreversible pressure gradient (first term on the RHS of Eq. (53)) as a function of the modified flow rate Q* defined by Q ¼

Q n

ð1  eÞ 2n

ð55Þ

This plot is, in fact, similar to Fig. 13 and shows the great improvement obtained through the use the turbulent annulus flow picture over PCAF model. A comparison of the calculated and measured friction pressure gradients Cf is shown in Fig. 15. The agreement between both is within F 25%. The present model was also compared with frictional pressure gradient data by Bai (1995), who studied the vertical core – annular flow inside a

Fig. 16. Friction pressure gradient calculated by Eq. (53) versus the experimental data by Bai (1995).

0.9525-cm i.d. glass tube using an oil – water system with a much higher density difference than the present study (q1 = 0.905 g/cm3, l1 = 600 mPa s at 22 C). This comparison, as seen in Fig. 16, shows an excellent agreement between calculated and measured frictional pressure gradients. In fact, this agreement is even better than our pressure drop data, because the correlation used to determine e, i.e. Eq. (31) was previously validated with Bai’s wave speed data in the same system. Eq. (53) can be expressed in a more general form as
1n
n q l2 Cf ¼ Cf ;2 ðQÞ m q2 lm Cðq2  q1 Þgeð1  eÞ

ð56Þ

where Cf,2( Q) is the frictional pressure gradient for single phase flow of fluid 2 at mixture flow rate. This generalization is similar to Eq. (40) for horizontal flow, but buoyancy effects are taken into account.

5. Conclusions

Fig. 15. Frictional pressure gradient calculated by Eq. (53) versus experimental values (our data).

This paper intends to provide petroleum engineers with tools for the design and operation of heavy oil pipelines operating in core – annular flow mode, for

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application in production and transportation of heavy oil. It presents several modeling topics on core – annular flows based on investigations performed personally by the author and by his group. Besides occurrence conditions, correlations for volume fraction and pressure drop for this flow pattern are proposed. The main conclusions can be summarized as follows: (a) the core – annular flow pattern requires two immiscible liquids of very different viscosities and relatively small density difference, a situation often satisfied by heavy oils; the interfacial tension plays a prominent role in favoring the annular configuration, however its importance decreases at large pipe sizes; (b) modeled volume fraction is based on a drift equation where buoyancy is accounted for; this equation was adjusted to fit wave speed data and satisfies the no slip condition observed in horizontal flow; (c) pressure drop in horizontal core –annular flow should take into account the turbulence in the annulus and waviness of the interface; these aspects lead to a correlation very different from the PCAF model (laminar – laminar flow with smooth interface); (d) pressure drop in vertical flow should further include buoyancy effects that favor the flow of a lighter core. Comparisons of the proposed correlations show very good agreement with different sources of data. The conclusion that pressure drop in oil –water core flow is comparable to water alone in the pipe at total volumetric flow rate is confirmed. References Arney, M.S., Bai, R., Guevara, E., Joseph, D.D., Liu, K., 1993. Friction factor and holdup studies for lubricated pipelining. I: Experiments and correlations. International Journal of Multiphase Flow 19, 1061 – 1076. Arney, M.S., Ribeiro, G.S., Guevara, E., Bai, R., Joseph, D.D., 1996. Cement-lined pipes for water lubricated transport of heavy oil. International Journal of Multiphase Flow 22, 207 – 221. Bai, R., 1995. Traveling waves in a high viscosity ratio and axisymmetric core annular flow. PhD thesis, University of Minnesota.

Bannwart, A.C., 1998. Wavespeed and volumetric fraction in core annular flow. International Journal of Multiphase Flow 24, 961 – 974. Bannwart, A.C., 1999a. The role of surface tension in core – annular flow. 2nd International Symposium on Two-Phase Flow Modeling and Experimentation—ISTP’99, vol. 2. Pisa, Italy, pp. 1297 – 1302. Bannwart, A.C., 1999b. A simple model for pressure drop in horizontal core annular flow. Journal of the Brazilian Society of Mechanical Sciences 21 (2), 233 – 244. Bannwart, A.C., 2001. Bubble analogy and stabilization of core – annular flow. Journal of Energy Resources Technology ASME 123, 127 – 132. Bannwart, A.C., Vara, R.M.O., 2001. One dimensional waves and stability criteria for annular liquid – liquid flow in horizontal pipes. XVI Brazilian Congress of Mechanical Engineering— COBEM’2001, Uberlandia, MG, Paper PROD-17060, Houston, Texas, 8 p. Bannwart, A.C., Rodriguez, O.M.H., Carvalho, C.H.M., Wang, I.S., Vara, R.M.O., 2000. Flow patterns in heavy crude oil – water core annular flow. Proceedings of the Engineering Technology Conference of Energy—ETCE 2001, CR-ROM, Feb. 5 – 7, Houston, Texas. Bobok, E., Magyari, D., Udvardi, G., 1996. Heavy oil transport through lubricated pipeline, SPE European Petroleum Conference, Milan, Italy, vol. 1, paper SPE 36841, pp. 239 – 244. Brauner, N., 1991. Two-phase liquid – liquid annular flow. International Journal of Multiphase Flow 17 (1), 59 – 76. Brauner, N., Moalem Maron, D., 1998. Classification of liquid – liquid two-phase flow systems and the prediction of flow pattern maps. 2nd International Symposium on Two-Phase Flow Modeling and Experimentation—ISTP’99, vol. 2. Pisa, Italy, pp. 747 – 754. Charles, M.E., Govier, G.W., Hodgson, G.W., 1961. The horizontal pipeline flow of equal density oil – water mixtures. Canadian Journal of Chemical Engineering 39 (1), 27 – 36. Clift, R., Grace, J.R., Weber, M.E., 1978. Bubbles, Drops and Particles. Academic Press, New York. Flores, J.G., Chen, X.T., Sarica, C., Brill, J.P., 1997. Characterization of oil – water flow patterns in vertical and deviated wells, paper SPE 38810, SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5 – 8 October. Ishii, M., 1982. Wave phenomena and two-phase flow instabilities Chap. 2.4. In: Hetsroni, G. (Ed.), Handbook of Multiphase Systems. Hemisphere, Washington, pp. 2-95 – 2-98. Joseph, D.D., Renardy, Y., Renardy, M., 1984. Instability of the flow of immiscible liquids with different viscosities in a pipe. Journal of Fluid Mechanics 141, 309 – 317. Joseph, D.D., Bannwart, A.C., Liu, Y.J., 1996. Stability of annular flow and slugging. International Journal of Multiphase Flow 22, 1247 – 1254. Moritis, G., 1995. Heavy oil expansions gather momentum worldwide. Oil & Gas Journal, August 14. Oliemans, R.V.A., 1986. The lubricating-film model for core – annular flow, PhD Thesis, Delft University, The Netherlands. Oliemans, R.V.A., Ooms, G., Wu, H.L., Duijvestijn, A., 1987. Core – annular oil/water flow: the turbulent-lubricating-film

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Vanegas Prada, J.W., 1999. Estudo experimental do escoamento anular o´leo-a´gua (‘‘core-flow’’) na elevacßa˜o de o´leos ultraviscosos, (in portuguese) MSc Thesis, School of Mechanical Engineering, State University of Campinas, Campinas, Brazil. Vanegas Prada, J.W., Bannwart, A.C., 1999a. Core flow lift: a new alternative for heavy oil production. XV Brazilian Congress of ´ guas de Lindo´ia, SP, Mechanical Engineering, COBEM’99. A Brazil, 10 pp., CD-ROM. Vanegas Prada, J.W., Bannwart, A.C., 1999b. Pressure drop in vertical core – annular flow. XV Brazilian Congress of Mechanical ´ guas de Lindo´ia, SP, Brazil, 10 pp., Engineering, COBEM’99. A CD-ROM. Vanegas Prada, J.W., Bannwart, A.C., 2001. Modeling of vertical core annular flows and application to heavy oil production. Journal of Energy Resources Technology, ASME (in press). Wallis, G.B., 1969. One-Dimensional Two-phase Flow. McGrawHill, New York. Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York.