Modeling Foam Breakup in Batch Separators and

J194002 DOI: 10.2118/194002-PA Date: 3-October-18

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Modeling Foam Breakup in Batch Separators and Cylindrical Cyclones J. A. Moncayo, Bravo Natural Resources and Promonsa; R. Dabirian, R. Mohan, and O. Shoham, University of Tulsa; and G. Kouba, Chevon (ret.)

Summary Models are developed for foam breakup in a batch separator, as well as inlet-cyclone and gas/liquid-cylindrical-cyclone (GLCC) compact separators, by improving the Saint-Jalmes et al. (2000) “1-g” foam-batch-separator model. The modified batch-separator model shows a better performance with respect to experimental data than does the original model. An extension of the modified “1-g” foambatch-separator model to swirling foam flow in cyclones is developed, which accounts for the effect of higher g-forces in the cyclones. The extended model accurately predicts experimental foam-breakup data for both the inlet-cyclone and the GLCC compact separators. Introduction Foam is generated naturally in oil and gas production because of gas entrainment in the oil and brine phases (Karaaslan et al. 2018). For crude oil with a high gas/oil ratio (GOR), a reduction in pressure promotes gas liberation; when the free gas is mixed with available surfactants a foam layer is created on top of the flowing oil (Karaaslan 2009). Foam might also be created artificially to solve productionoperation problems, such as unloading gas wells, as well as elimination of severe slugging (Moncayo et al. 2018). Aqueous foams are composed of gas bubbles trapped in a liquid phase in the presence of a surfactant. Foams drain naturally under the effect of gravitational force by pulling the denser liquid phase downward, traveling throughout the total height of the foam sample and accumulating at the bottom. Gas bubbles are prone to travel upward, where they collide either with each other or with air in the atmosphere (Nitasari 2013; Nababan 2015). This causes a reduction of the foam volume, which is referred to as foam decay. The liquid drains in various ways, depending on the bubble-size distribution and bubble arrangement. Bubbles form a network of dimples (plateau borders and nodes), through which the liquid flows downward (Nitasari 2013). Fig. 1 shows the structure of the foam-bubble network. Previous models have been developed for foam-liquid drainage under gravity. Free-liquid drainage refers to foam samples that are left to drain naturally under “1-g” conditions. On the other hand, enhanced liquid drainage refers to models developed for foam drainage by liquid injection at the top of a dry-foam sample. Magrabi et al. (2001) developed a model assuming that all the liquid contained in the foam is trapped in the plateau borders (bubble lamellae), and that the liquid that is left behind in the lamellae is negligible. The liquid flow in the plateau borders is considered as a viscous dissipation with no flow through the nodes, whereby the average velocity (v) is calculated using Stokes law. A dimensionless bottom-liquid evolution (k b ) is derived from the conservation of mass over an infinitesimal element, " # cv VB e rC b LP np 0:5 0:5 1:5 @e qg ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð1Þ ð1 eÞ e v ¼ pffiffiffi @z 2 VB 20 3 l LP np 1 e kb ðsÞ ¼ kb ðs ¼ 0Þ

e2b ð1 eb Þ2

sþ

e0:5 b ð1 eb Þ2:5

ðs 0

@e ds: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð2Þ @k k¼kb

Plateau border ρG ρL ρG

ρG

Face or lamellae

Node or vertex

ρG Fig. 1—Structure of the foam-bubble network.

An enhanced-foam-drainage model was presented by Verbist et al. (1996), which predicts how the liquid travels and drains through a dry-foam column when artificially injecting liquid at the top of the foam sample. In the model, the liquid is considered to flow through the plateau borders, similar to flow in porous media, which is described physically by Darcy’s law. A continuity expression for the C 2018 Society of Petroleum Engineers Copyright V

Original SPE manuscript received for review 9 January 2018. Revised manuscript received for review 8 June 2018. Paper (SPE 194002) peer approved 14 June 2018.

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foam-liquid drainage through the cross-sectional area of the plateau borders, taken as idealized vertical channels and for 1D fluid flow in the downward direction, is proposed as pffiffi e @e @e @ þ e2 ¼ 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð3Þ 2 @k @s @z Eq. 3 allows for prediction of the liquid-height evolution as a function of time for enhanced foam-liquid drainage because it takes into account all the related physical phenomena of foam drainage. Saint-Jalmes et al. (2000) developed a model for foam-liquid drainage using a free-drainage approach. Applying a mass balance on the liquid phase yields @e @ ðuD eÞ þ ¼ 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð4Þ @t @z Eq. 4 is the traditional foam-drainage differential equation that is valid only for a constant cross-sectional area, A, along the z-direction. A generalized free-foam-liquid-drainage velocity below the drying front is proposed as rffiffiffiffiffi eC n @e m : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð5Þ u D ¼ vo e 1 e e @z Eq. 5 takes into account three drainage mechanisms: gravitational, capillary, and viscosity effects. The variables that the foam-liquiddrainage velocity depends on are the maximum characteristic flow speed, vo, which is the bubble-rise velocity in a liquid-phase medium; the capillary-rise-scale term, n, which takes into account capillarity effects; e, the liquid holdup in the foam; the m exponent, caused by the nature of the viscous dissipation through the foam; and eC, the critical liquid holdup, at which bubbles in the foam become randomly close-packed spheres. Saint-Jalmes et al. suggest this value to be eC ¼ 1–0.635. Finally, @e=@z is the change of the liquid holdup with the foam height in the positive downward direction. The liquid-drainage velocity uD , which is below the front, is a strong function of the corresponding liquid holdup, e (z,t). However, the capillary term in Eq. 5 was neglected by the authors, resulting in uD ¼ vo em : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð6Þ The bubble-rise velocity and the capillary-rise scale proposed by Saint-Jalmes et al. (2000) are given, respectively, by vo /

qL g R2B ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð7Þ lL

and n/

r : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð8Þ qL g RB

Guzmań (2005) proposed a different bubble-rise-velocity expression, vo ¼ K

qL g R2B ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð9Þ lL

where K represents a degree of freedom, which can be calculated using the collected experimental data, K¼

vSL vSG

1:318 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð10Þ

In this study, a modified and improved “1-g” model for foam breakup in a gravity batch separator, derived from Saint-Jalmes et al. (2000), is developed. This model is capable of predicting the liquid-drainage phenomena more accurately. In addition, an extension of the modified model to swirling-flow foam breakup in cyclones is performed, which accounts for the effect of higher g-forces in the cyclones. Modeling This section presents the modified model for foam breakup in a gravity batch separator, as well as the extension of the modified model to swirling-flow foam breakup in cyclones. Modified Foam-Batch-Separator Model. The process of free-liquid drainage vs. time is presented in Fig. 2. As shown, a foam sample is taken at t ¼ 0, under static conditions with initial foam and liquid heights of hFo and hLo, respectively, and initial average liquid holdup eo . The sample is assumed to be wet foam, and therefore as time passes the water drains to the bottom, leaving dry foam behind with liquid holdup less than eo . The boundary between the wet and dry foam is called “the drying front.” At time t, the increased liquid height is hL(t) and the total volume of collected liquid is VL(t), which is a function of the velocity uD below the drying front. Three assumptions are considered: no inertial forces, monodispersed bubble-size distribution, and no lateral drainage. The free-liquid-drainage equation for a foam sample is determined using a modified Eq. 5. The objective of this modification is to keep the original generalized Eq. 5 and include the gravitational, capillary, and viscosity mechanisms that were previously neglected by Saint-Jalmes et al. (2000), yet affect the drainage velocity: rffiffiffiffiffi eC n @e : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð11Þ u0D ¼ vo em 1 eo eo @z t¼0 2

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As can be seen, the liquid holdup e is replaced with the initial average liquid holdup eo. The term @e=@z is approximated as the change of the liquid holdup with the change in the foam height in the z-direction at t ¼ 0. This assumption is valid because it is observed experimentally that at t ¼ 0, there is a liquid-holdup profile in the foam column. The slope is approximated by a straight line depending on the boundary conditions e ðz ¼ 0; t ¼ 0Þ ¼ 0 e ðz ¼ hFo ; t ¼ 0Þ ¼ 1;

ð12Þ

yielding @e 1 ¼ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð13Þ @z t¼0 hFo Note that with Eqs. 12 and 13, the second term inside the parentheses in Eq. 11 becomes a constant, which considerably simplifies the solution. The exponent m, which takes into account the foam-viscosity-dissipation effect, is calculated using experimental data.

vDF

Dry foam ε < εo

Drying-front position

+Z vDF

hFo uD

uD hL(t)

Wet foam ε = εo

VL(t )

hLo t=0

t>0

Fig. 2—Schematic of free-liquid drainage in a batch separator.

We propose not to drop the capillary-rise-scale term, as performed originally by Saint-Jalmes et al. (2000), but to determine it using Eq. 8 with a proportionality constant J, which is determined from the experimental data, n¼J

r : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð14Þ qL g R B

The bubble-rise velocity, vo, can be determined using the kinematic theory of 1D sedimentation of ideal suspensions, which has been developed for suspensions with monosized perfect spheres under the effect of gravity. For this case, the bubble-rise velocity vo is a function of the local volumetric holdup and the rise velocity of a bubble in an infinite medium, vS. Two different equations from Einstein (1911) and Kumar and Hartland (1985) are used to calculate the bubble-rise velocity. These equations take into account the effect of density difference, qL–qG, and therefore can be extended to high-pressure systems, which Saint-Jalmes et al. (2000) and Guzmań (2005) did not consider. The bubble-rise velocities proposed by Einstein (1911) and Kumar and Hartland (1985) are given, respectively, by vo ¼ vS ð1 2:5 eÞ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð15Þ and vo ¼ v S

ð1 eÞ2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð16Þ ð1 þ 4:56 eÞ

The rise velocity of a bubble in an infinite medium, vS, is given by Stokes law, vS ¼

ðqL qG Þ g R2B : ................................................................... 18 lL

ð17Þ

In Eqs. 15 and 16, e is the liquid holdup, which is a function of time and space: e ¼ e (z,t). It is assumed that the liquid holdup e (z,t) is constant, which could be approximated by the initial average liquid holdup, eo. Eqs. 15 and 16 are used to evaluate the model performance. Substituting Eqs. 11 and 13 into Eq. 4 results in rffiffiffiffiffi eC n 1 @ vo emþ1 1 eo eo hFo @e þ ¼ 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð18Þ @t @z Taking the derivative of the second term in Eq. 18 results in rffiffiffiffiffi @e eC n 1 @e þ ðm þ 1Þ vo em 1 ¼ 0: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð19Þ @t eo eo hFo @z 2018 SPE Journal



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We let the expression inside the brackets of Eq. 19 be rffiffiffiffiffi eC n 1 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð20Þ v0D ¼ ðm þ 1Þ vo em 1 eo eo hFo where v0D is the velocity below the drying front, which is a function of the liquid holdup e (z,t). Following the methodology of SaintJalmes et al. (2000), the drying front travels at a constant speed, and it is assumed that the corresponding liquid holdup e at the front can be approximated to the initial average liquid holdup eo . Thus, the velocity of the drying front, v0DF (constant), is given by rffiffiffiffiffi eC n 1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð21Þ v0DF ¼ ðm þ 1Þ vo em o 1 eo eo hFo The method of characteristics is used to solve the partial-differential equation, Eq. 19. The initial condition is eðz; 0Þ ¼ eo , where the liquid holdup e at t ¼ 0 is equal to the initial average liquid holdup eo . Solving the partial-differential equation for the liquid holdup yields a family of solutions, 8 eo > > v0DF t z > < eðz; tÞ ¼

> > > : eo

1=m

z v0DF

t

v0DF t > z: ð22Þ

The position of the drying front z can be obtained at any time t by z ¼ v0DF t: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð23Þ For times when v0DF t hFo, the front has not yet arrived at the bottom of the sample, whereas for times when v0DF t > hFo, the front reaches the bottom of the sample. As presented by Saint-Jalmes et al. (2000), the volume of drained liquid as a function of time can be determined by taking the integral of the liquid holdup as a function of the positive z-position, depending on the location of the drying front,

VL ðtÞ ¼

8 1=m ð v0 t ð hFo DF z > > > V e A dz eo dz; TOT o > > v0DF t < 0 vDF t

v0DF t hFo

> ð hFo > > > > : VTOT eo A

v0DF t > hFo ;

0

z v0DF

1=m t

dz;

ð24Þ

where VTOT ¼ eohFoA is the total liquid volume in the foam sample. Integrating and simplifying Eq. 24 yield 0 8 1 vDF t > > ; > v0DF t hFo ; ð25aÞ VL ðtÞ < m þ 1 hFo ¼ VTOT > m hFo 1=m > > :1 0 ; v0DF t > hFo ð25bÞ mþ1 v t DF

The evolution and growth of the liquid-accumulation height hL(t) in a batch separator can be easily determined by dividing Eq. 25 by the cross-sectional area A and substituting for ATOT, yielding 8 eo > v0DF t; > > < mþ1 " 1=m # hL ðtÞ ¼ > > h e 1 m hFo > ; : Fo o m þ 1 v0DF t

v0DF t hFo ;

ð26aÞ

v0DF t > hFo :

ð26bÞ

Extension To Swirling-Foam Flow. The modified “1-g” model presented previously is extended to swirling-foam flow in cyclones. This extension addresses the foam-flow behavior in a cyclone under higher g-forces. Foam Flow in a Cyclone. Fig. 3 shows a schematic of gas, foam, and liquid flow through an inlet-pipe section into a cyclone. Cross-sectional Area No. 1 shows the distribution of the phases in the pipe, and cross-sectional Area No. 2 represents the nozzle created by reducing the inlet area open to flow to 25%. It is assumed that the nozzle is completely filled by foam, with no water or gas. Thus, the initial foam thickness hFi at the nozzle exit (cyclone inlet) is approximated by hFi ¼ 0:25 dP ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð27Þ where dP is the diameter of the inlet-pipe section. It is also assumed that no expansion of foam flow occurs once it exits the nozzle into the cyclone, whereby the maximum foam-layer thickness in the cyclone is hFi. A schematic of the foam-breakup phenomena inside a cyclone is presented in Fig. 4. The model assumes a 1D problem, meaning that drainage occurs only in the radial direction and the effects of gravity pulling the swirl downward are neglected. As mentioned previously, the nozzle outlet is filled with foam, and thus the initial foam thickness in the cyclone is hFi (see Fig. 4). The foam flows around the cyclone wall at a tangential velocity vt, experiencing high g-forces that break the foam in the radial direction. The resulting radial 4



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force causes the liquid held in the plateau borders and nodes of the foam to flow in the radial direction toward the wall of the cyclone. The liquid-film thickness in the swirling flow, hL,SF(tR), is increased because of the foam breakup and the accumulation on the wall. The height is a function of tR, the foam-residence time in the cyclone, given by tR ¼

N p dC : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð28Þ vt

According to Karaaslan (2009), the tangential velocity vt depends on the cyclone-inlet-inclination angle and is given for the inlet cyclone and GLCC, respectively, by vt ¼ 4 vM ¼ 4 ðvSL þ vSG Þ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð29Þ and vt ¼ 4 vM cos ðhÞ ¼ 4 ðvSL þ vSG Þ cos ðhÞ: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð30Þ The number of turns the foam swirls around the cyclone, N, can be determined from an expression developed using the experimental data and as a function of the generated g-force, N ¼ 0:0142 G0:7042 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð31Þ The radial acceleration and the g-force created by the tangential velocity in the cyclone are given, respectively, by Ra ¼

ðvt Þ2 ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð32Þ RC

and G¼

Ra : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð33Þ g

1 2

ρG ρF ρL

Fig. 3—Schematic of foam flow in inlet pipe and cyclone.

+z direction

t

r

hL,SF (tR) dC

vt

hFi Fig. 4—Schematic of foam breakup in a cyclone.

Extension of Foam-Drainage Equation. The free-liquid-drainage velocity for a “1-g” batch separator given by Eq. 11 is modified again, extending it to swirling flow under higher g-forces, rffiffiffiffiffi eC n eo : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð34Þ uD;SF ¼ vo em 1 eo eo 2018 SPE Journal



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As can be seen in Eq. 34, the change of the liquid holdup with the foam height in the z-direction at t ¼ 0, ð@e=@zÞt¼0 , is approximated by the initial average liquid holdup eo. This assumption is justified because in a cyclone the initial foam height hFo is relatively small. This is also true because the residence swirling time in the cyclone is considerably small in magnitude. Following the same procedure applied for the foam batch separator, the constant velocity of the drying front under swirling-flow conditions, vDF,SF, is given by rffiffiffiffiffi eC n vDF;SF ¼ ðm þ 1Þ vo em 1 e o : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð35Þ o eo eo We first attempted to use Eq. 35 with the solution for the liquid-height growth as a function of time, given by Eq. 26, to model the swirling-flow effect, replacing the acceleration of gravity “g” in these two expressions of vo and f with the higher cyclone g-force. However, it was determined that the calculated value for hL,SF (tR) is a weak function of the g-force. Thus, a new model is developed that incorporates the physical phenomena in the cyclone, in particular the swirling flow. Because the liquid-drainage equation for foam under “1-g” is a weak function of gravity, the solution of the forced-foam-liquid drainage in the cyclone, hL,SF, can be approximated by the product of two single-variable functions, hL;SF ðtR ; GÞ ¼ hL ðtR Þ XðGÞ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð36Þ where X is the swirl intensity in the cyclone, which is a unique function of the g-force. Note that hL(tR) is determined from Eq. 26 and vDF,SF is calculated using Eq. 35. The swirl intensity X is a dimensionless number defined as the ratio of the tangential momentum flux to the total momentum flux at any axial location, ðR 2 p q uz uh rdr X¼

0 2 p q R2C Uav

; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð37Þ

where Uav is the average axial velocity, RC is the radius of the cyclone, and q is the fluid density. Mantilla (1998) proposed the following correlation to calculate the swirl intensity for single-phase flow, " # Mt 2 0:93 Mt 4 0:35 0:16 y 0:7 I exp 0:5 I Re ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð38Þ X¼c dC MT MT where c is 1.48, Mt /MT is the ratio of the tangential to the total momentum flux at inlet conditions, I is an inlet-geometry factor, Re is the Reynolds number, y is the axial distance, and dC is the diameter of the cyclone. The collected experimental data show that Eq. 38 with c ¼ 119 predicts the swirl intensity with high accuracy for both cyclones. The momentum ratio at inlet conditions is given by Mt vt ¼ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð39Þ MT Uav The average axial mixture velocity is calculated by 2 dP : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð40Þ Uav ¼ ðvSL þ vSG Þ dC The Reynolds number is given by Re ¼

ðqL qG Þ Uav dC : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð41Þ lL

The inlet factor I is a function of the number of tangential inlets n, n I ¼ 1 exp : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð42Þ 2 The value of the axial distance y, the distance from the inlet nozzle to the liquid level in the cyclone, is held at approximately 6 to 7 in. for all experiments. By modifying Eq. 26 and replacing hFo with hFi, the incremental liquid level at a given swirling residence time tR under “1-g” conditions can be calculated as 8 eo > vDF;SF tR ; > > < mþ1 " 1=m # hL ðtR Þ ¼ > m hFi > > h e 1 ; : Fi o m þ 1 vDF;SF tR

vDF;SF tR hFi ;

ð43aÞ

vDF;SF tR > hFi :

ð43bÞ

Eq. 43 can then be combined with the foam-swirl-intensity correlation, Eq. 38, to yield the foam breakup under higher g-forces, hL;SF ðtR Þ ¼ hL ðtR Þ X: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð44Þ 6



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Foam-Breakup Efficiency. The efficiency for foam breakup in a batch separator is given by DhLo 100: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð45Þ gBatch Separator ð%Þ ¼ hFo;U An extension of this for the swirling flow in cyclones yields hL;SF 100: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð46Þ gCyclone ð%Þ ¼ hFi Substituting Eq. 44 into Eq. 46 yields the final equation for the foam-breakup efficiency in a cyclone, hL ðtR Þ X 100: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ð47Þ gCyclone ð%Þ ¼ hFi Experimental Program Fig. 5 shows a photograph of the foam-characterization rig (FCR). The flow loop is constructed from a transparent polyvinyl chloride pipe, which is operated up to 80 psig. As shown in Fig. 5, the mixture of the compressed air, tap water, and surfactant are introduced into the 1-in. flow loop, and the mixture flows upward through a vertical section (the foam-generation section). The section consists of the integration of 125-mm meshes at the top and bottom and a static mixer. A 60-hp compressor with a delivery-pressure range of 115 to 125 psig is used to introduce the air to the flow loop, and it operates with a capacity flow rate of 1,200 scf/min. The tap water is injected to the loop using a centrifugal pump with 1.5-hp motor rotating at 3,450 rev/min and delivering a pressure range of 0 to 100 psig. The flow rates of the air and water are measured by two rotameters. The fresh surfactant mixture is prepared in a 17-L cylinder and is pumped using a positive-displacement pump with an injection flow rate of 0.002 to 0.2 gal/hr and maximum delivery pressure of 50 psig.

6. Cyclone section

3. Foam-generation section 2. Metering section

7. Downstreamsampling section 4. Foam-flowdevelopment section

5. Upstreamsampling section 1. Storage section

Fig. 5—Foam-characterization rig (FCR) sections.

Foam-liquid-drainage experimental data are acquired in a batch separator, as well as foam-breakup-efficiency data in a 2-in. inlet cyclone and GLCC. The SI-403 surfactant is used to create stable foam, where the critical micelle concentration (CMC) of the surfactant has a value of 0.5 vol%. In this paper, the data collected for the surfactant concentration of 0.01 vol% are provided. Comparison Study In this section, the developed models for the batch separators and the cyclones are tested against the acquired experimental data. Modified Batch-Separator Model. This section presents a comparison between model predictions for foam-liquid drainage and the acquired data from the batch separators. Foam samples were withdrawn from ports located upstream and downstream of a 2-in. inlet cyclone, and experimental data and the model are presented. Model Input Data. The required input data for the Saint-Jalmes et al. (2000) foam-drainage model are given in Table 1. Because the capillary-rise term and the effect of the gas density are not taken into account, the coefficient J and the gas density qG are not needed. The bubble-rise-velocity coefficient K and the viscous-dissipation index m must be calibrated depending on the type of fluids 2018 SPE Journal



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used to generate the foam (liquid, gas, and surfactant properties), the surfactant concentration CS, the average value of the bubble radius RB, and the type of foam-generation device. For this model, it was found that the values for K and m are 150 and 2.60, respectively. Table 2 summarizes the input data required for the modified batch-separator model. Similarly, the viscous-dissipation index m and the capillary-rise coefficient J must be calibrated accordingly. For this study, the determined values of m and J are 1.30 and 0.15, respectively. Variable

Symbol

Units

Values

Initial foam height

hFo

cm

–

Initial average liquid holdup

εo

–

–

Viscous-dissipation index

m

–

2.60

Surfactant-solution surface tension

σ

dynes/cm

Liquid density

ρL

kg/m

Liquid viscosity

μL

kg/m·s

0.001

Average bubble radius

RB

m

0.00032

Bubble-rise-velocity coefficient

K

–

150

Symbol

Units

Values

hFo

cm

–


εo

–

–


m

–

1.30


σ

dynes/cm

3

42 1000

Table 1—Saint-Jalmes et al. (2000) model input data. Variable Initial foam height

42

Liquid density

ρL

kg/m

3

Gas density

ρG

kg/m

3

Liquid viscosity

μL

kg/m·s

0.001


RB

m

0.00032

J

–

0.15

Capillary-rise-scale coefficient

1000 1.15

Table 2—Modified-model input data.

Model Comparison With Experimental Data. Fig. 6 shows a comparison of the foam-liquid drainage between the modified “1-g” model predictions and experimental data, in the form of liquid height vs. time, for both upstream and downstream samples. Also shown is the effect of the two bubble-rise velocity vo models considered: Einstein (1911) and Kumar and Hartland (1985). As can be seen, the predictions from both methods are in good agreement with the experimental data; because the Einstein (1911) method exhibits an overall slightly better performance, it is therefore used for all model predictions. 3.0 Batch separator

Liquid Height, hL (cm)

2.5

2.0 Experimental data upstream Experimental data downstream

1.5

This study and Einstein (1911), vo This study and Kumar and Hartland (1985), vo

1.0

0.5

0.0 0

5

10

15

20

25

Time, t (minutes) Fig. 6—Performance comparison of different bubble-rise-velocity equations (vSL 5 0.05 m/s, vSG 5 1.5 m/s, eo,U 5 0.16, hFo,U 5 10.80 cm, eo,D 5 0.11, hFo,D 5 4.70 cm).

Figs. 7 through 12 show comparisons between the experimental data, the Saint-Jalmes et al. (2000) original model, and the modified model using the Einstein (1911) vo expression. The comparisons demonstrate a better performance of the modified foam-batchseparator model for all flow conditions. As can be seen, at early times (t < 5 minutes), the Saint-Jalmes et al. (2000) method overpredicts the liquid-height growth as a function of time for both upstream and downstream conditions, whereas at later times, the model approaches closer to the experimental data. On the other hand, the proposed modified model shows better agreement for early times, 8



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when most of the liquid drainage takes place, and also at late times. Thus, it predicts the experimental data more accurately because of the proposed foam-drainage-velocity equation (Eq. 11), which takes into account the effect of the capillary rise and the liquid-holdup profile. In addition, the Saint-Jalmes et al. (2000) model neglects the capillary-rise scale, and thus the drying-front-drainage-velocity equation, vDF, is simplified, becoming only a function of the initial average liquid holdup eo and the bubble-rise velocity vo. Moreover, the equation for vo used by Saint-Jalmes et al. (2000) depends only on the liquid properties and does not include the effects of gas density and the liquid-holdup correction, which are taken into account by the modified model proposed in this study. 3.0 Batch separator


2.5


1.5

Saint-Jalmes et al. (2000) This study

1.0

0.5

0.0 0

5

10

15

20

25

Time, t (minutes) Fig. 7—Comparison between model predictions and experimental data (vSL 5 0.05 m/s, vSG 5 1.5 m/s, eo,U 5 0.160, hFo,U 5 10.80 cm, eo,D 5 0.11, hFo,D 5 4.70 cm). 3.0 Batch separator


2.5


1.5


1.0

0.5

0.0 0

5

10

15

20

25

Time, t (minutes) Fig. 8—Comparison between model predictions and experimental data (vSL 5 0.05 m/s, vSG 5 2.0 m/s, eo,U 5 0.163, hFo,U 5 10.20 cm, eo,D 5 0.16, hFo,D 5 4.13 cm).

Discrepancy Analysis. The model accuracy is quantified by conducting statistical analysis with the R2 value, which accounts for the error between the model predictions and experimental data. This method provides the coefficient of determination, which is a measure of how well the data are predicted by the model. This dimensionless number ranges between zero and unity, whereby the model is more reliable and shows less error when the R2 value approaches unity. Tables 3 and 4 show the results for the coefficient-of-determination values for both developed models at both sampling ports. Each R2 value corresponds to the model predictions shown in Figs. 7 through 12. As can be seen in Tables 3 and 4, the error between the “1-g” modified model and the experimental data is less than the error between the original Saint-Jalmes et al. (2000) model and the experimental data, exhibiting R2 values closer to unity for every condition. This demonstrates the better performance of the proposed model. Swirling-Flow Model. This section presents a comparison between the developed swirling-flow-foam-breakup efficiency model and the acquired experimental data for the 2-in. inlet cyclone and GLCC. Extension Input Data. The developed swirling-flow model is derived from the extension of the modified batch-separator model. Thus, most of the input data of both models are similar. Table 5 shows the required input data for the swirling-flow model. Breakup-Efficiency-Data Comparison. Comparisons between the swirling-flow-model predictions and the experimental data for the inlet cyclone and the GLCC are presented in Figs. 13 and 14, respectively. In general, good agreement is observed, whereby the model follows the experimental data closely. As can be seen, the proposed swirling-foam-flow model is not sensitive to the type of cyclone, predicting the breakup efficiency closely for both cyclone configurations used. The coefficients of determination (R2 values) for the inlet cyclone and the GLCC are 0.75 and 0.77, respectively. 2018 SPE Journal



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3.0 Batch separator


2.5


1.5


1.0

0.5

0.0 0

5

10

15

20

25


3.0 Batch separator


2.5


1.5


1.0

0.5

0.0 0

5

10

15

20

25


3.0 Batch separator


2.5

2.0

Experimental data upstream Experimental data downstream

1.5


1.0

0.5

0.0 0

5

10

15

20

25

Time, t (minutes) Fig. 11—Comparison between model predictions and experimental data (vSL 5 0.05 m/s, vSG 5 4.0 m/s, eo,U 5 0.169, hFo,U 5 8.10 cm, eo,D 5 0.22, hFo,D 5 1.43 cm). 10



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3.0 Batch separator


2.5 Experimental data upstream

2.0

Experimental data downstream Saint-Jalmes et al. (2000)

1.5

This study 1.0

0.5

0.0 0

5

10

15

20

25


J = 0.15; m = 1.30

Upstream

This study and Einstein (1911) vo

Sampling Port

Parameter vSL (m/s)

Downstream

0.05

0.05

2

vSG (m/s)

R

1.5

0.962

2.0

0.977

2.5

0.987

3.0

0.987

4.0

0.932

5.0

0.815

1.5

0.956

2.0

0.817

2.5

0.791

3.0

0.585

4.0

0.699

5.0

0.822

2

Table 3—R values for modified model.

K = 150; m = 2.60

Downstream

Saint-Jalmes et al. (2000)

Upstream

Sampling Port

Parameter vSL (m/s)

0.05

0.05

2

vSG (m/s)

R

1.5

0.979

2.0

0.953

2.5

0.851

3.0

0.869

4.0

0.730

5.0

0.543

1.5

0.912

2.0

0.580

2.5

0.098

3.0

0.381

4.0

0.459

5.0

0.150

2

Table 4—R values for the Saint-Jalmes et al. (2000) model. 2018 SPE Journal



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Variable

Symbol

Units

Values

Diameter of the inlet pipe

dP

in.

1


εo

–

–


σ

dynes/cm

42

Liquid density

ρL

kg/m

3 3

Previous “1-g” foam-batch-separatormodel variables

New swirling-flow variables

Stage:

1000

Gas density

ρG

kg/m

Liquid viscosity

μL

kg/m·s

0.001


RB

m

0.00032

1.15


m

–

1.30

Capillary-rise-scale coefficient

J

–

0.15

Superficial liquid velocity

vSL

m/s

–

Superficial gas velocity

vSG

m/s

–

Diameter of the cyclone

dC

in.

2

Inlet-section-inclination angle

θ

–

0 or 27

Foam-swirl-intensity constant

γ

–

116

Distance from nozzle to liquid leg

y

in.

6 to 7

Number of cyclone inlets

n

–

1

Table 5—Swirling-flow-model input data. 100 Inlet cyclone

Breakup Efficiency, η (%)

90 80 70 60

2-in. inlet cyclone

50

Swirling-flow extension 40 30 20 10 0 0

500

1,000

1,500

2,000

Cyclone g-Force, G Fig. 13—Swirling-flow-model predictions for the 2-in. inlet cyclone.

100 GLCC

Breakup Efficiency, η (%)

90 80 70 60 2-in. GLCC

50

Swirling-flow extension

40 30 20 10 0 0

200

400

600

800

1,000

1,200

1,400

Cyclone g-Force, G Fig. 14—Swirling-flow-model predictions for the 2-in. GLCC.

Conclusions A modified Saint-Jalmes et al. (2000) model is developed for a foam batch separator. The model takes into account the parameters neglected by the original study. J, the capillary-rise-scale coefficient, and m, the viscous dissipation through the foam index, are determined using the acquired experimental data. The modified foam-batch-separator model predicts the experimental data with a muchlower average discrepancy of 14%, compared with 38% for the original model. 12



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The modified foam-batch-separator model is a weak function of the acceleration; therefore, its extension to swirling foam flow by simply changing the g value does not properly predict the foam breakup occurring in a cyclone with higher g-forces. Therefore, the extension is performed using X, the swirl intensity in the cyclone. The extended model accurately predicts the experimental foambreakup-efficiency data in both the inlet cyclone and the GLCC with respective average discrepancies of 25 and 23%. Nomenclature ap ¼ plateau-border cross-sectional area, m2 b ¼ averaging constant for pentagonal dodecahedra bubble structure, 3/15 bi ¼ individual systematic uncertainty bR ¼ systematic error source cv ¼ velocity coefficient C ¼ supplementary constant, {[30.5–(p/2)]0.5} CS ¼ surfactant SI-403 concentration in FCR flow loop, vol% dB ¼ bubble diameter, m dC ¼ cyclone diameter, m, in. dGLCC ¼ GLCC diameter, in. dP ¼ diameter of the inlet-pipe section, in. f ¼ supplementary constant g ¼ acceleration caused by gravity, 9.81 m/s2 G ¼ cyclone g-force hD ¼ dimensionless height hF ¼ foam-height evolution with time, cm hFi ¼ initial foam height at the cyclone’s nozzle, cm hF,j ¼ foam height at t ¼ j, cm hFo ¼ initial foam height at t ¼ 0, cm hFo,D ¼ initial foam height at t ¼ 0 at downstream conditions, cm hFo,U ¼ initial foam height at t ¼ 0 at upstream conditions, cm hL ¼ liquid-height evolution with time, cm hL,j ¼ liquid height at t ¼ j, cm hLo ¼ initial liquid height at t ¼ 0, cm hLo,D ¼ initial liquid height at t ¼ 0 at downstream conditions, cm hL,SF ¼ liquid-height evolution with time under swirling flow, cm hLo,U ¼ initial liquid height at t ¼ 0 at upstream conditions, cm I ¼ swirl-intensity-inlet geometric factor j ¼ time integer J ¼ capillary-rise-scale-term coefficient K ¼ Saint-Jalmes et al. (2000) bubble-rise-velocity coefficient LP ¼ plateau-border length, m m ¼ exponent of nature of the viscous dissipation through the foam Mt ¼ tangential momentum flux at inlet conditions, kgm/s MT ¼ total momentum flux at inlet conditions, kgm/s n ¼ number of tangential inlets in a cyclone np ¼ number of plateau borders per bubble N ¼ number of foam revolutions in a cyclone NB ¼ number of bubbles per unit volume in foam pb ¼ bubble pressure, Pa rc ¼ plateau-border radius of curvature, m R2 ¼ coefficient of determination Ra ¼ radial acceleration, m/s2 RB ¼ bubble radius, m RC ¼ cyclone radius, m, in. Re ¼ Reynolds number SSE ¼ sum of squares caused by error SST ¼ total sum of squares SX,i ¼ standard deviation SX;R ¼ random-error source t ¼ testing time, minutes tFILLING ¼ testing-cylinder filling time, minutes to ¼ time-parameter constant, seconds tR ¼ foam-residence time in the cyclone, seconds u ¼ fluid velocity, m/s uD ¼ simplified liquid-drain velocity below the drying front, m/s u0D ¼ modified liquid-drain velocity below the drying front, m/s uD,SF ¼ modified liquid-drain velocity below the drying front under swirling flow, m/s uz ¼ fluid velocity in z-direction, m/s uh ¼ fluid velocity in h-direction, m/s Uav ¼ bulk average axial mixture velocity in a cyclone, m/s v ¼ plateau-border average velocity, m/s vD ¼ new form of liquid-drain velocity below the drying front, m/s v0D ¼ modified new form of liquid-drain velocity below the drying front, m/s vDF ¼ velocity of the drying front, m/s 2018 SPE Journal



v0DF ¼ vDF,SF ¼ vM ¼ vNOZZLE ¼ vo ¼ vS ¼ vSG ¼ vSL ¼ vt ¼ VB ¼ VG ¼ VL ¼ VTOT ¼ w¼ y¼ z¼ zo ¼ a¼ b¼ Dr ¼ c¼ e¼ eb ¼ eC ¼ eo ¼ eo,D ¼ eo,U ¼ gBatch Separator ¼ gCyclone ¼ h¼ k¼ kb ¼ l¼ leff ¼ lL ¼ q¼ qG ¼ qL ¼ r¼ rC ¼ s¼ Ui ¼ X¼ n¼

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modified velocity of the drying front, m/s velocity of the drying front under swirling flow, m/s mixture velocity, m/s tangential velocity at the cyclone-inlet nozzle, m/s bubble-rise velocity in liquid-phase medium, m/s bubble-rise velocity in an infinite medium, m/s superficial gas velocity, m/s superficial liquid velocity, m/s tangential velocity, vNOZZLE, m/s bubble volume, m3 gas volume in foam, m3 liquid volume in foam, m3 foam total volume, m3 minimum velocity required for primary foam breakup, m/s swirl-intensity axial distance, m vertical-downward direction in foam samples, cm vertical-direction-parameter constant, m characteristic length of acceleration, m supplementary constant, 30.5–(p/2) surface-tension difference, N/m foam-swirl-intensity constant foam-liquid holdup liquid holdup at the bottom of foam column, 0.26 critical liquid holdup, 1–0.635 initial average foam-liquid holdup initial average foam-liquid holdup at downstream conditions initial average foam-liquid holdup at upstream conditions foam-breakup efficiency of batch separator, % foam-breakup efficiency of cyclone, % GLCC inlet-section inclination angle from the horizontal, degrees nondimensional space coordinate location of the foam/surfactant-solution interface fluid viscosity, kg/ms effective viscosity, kg/ms liquid viscosity, kg/ms fluid density, kg/m3 gas density, kg/m3 liquid density, kg/m3 surfactant/water-mixture surface tension, dynes/cm surface tension, N/m nondimensional time coordinate ith-measure parameter cyclone-swirl intensity capillary-rise-scale term, m

References Einstein, A. 1911. Bemerkung zu dem Gesetz von Eo¨tvo¨s. Annalen der Physik 339 (1): 165–169. https://doi.org/10.1002/andp.19113390109. Guzmań, N. 2005. Foam Flow in Gas-Liquid Cylindrical Cyclone (GLCCV) Compact Separator. PhD dissertation, University of Tulsa, Oklahoma. Karaaslan, M. 2009. A Study of Foam Break-Up in Inlet Cyclones. Master’s thesis, University of Tulsa, Oklahoma. Karaaslan, M., Dabirian, R., Mohan, R. et al. 2018. Foam Breakup in Inlet Cyclones. J. Nat. Gas Sci. Eng. 55 (July): 89–105. https://doi.org/10.1016/ j.jngse.2018.04.024. Kumar, A. and Hartland, S. 1985. Gravity Settling in Liquid/Liquid Dispersions. Can. J. Chem. Eng. 63 (3): 368–376. https://doi.org/10.1002/ cjce.5450630303. Magrabi, S. A., Dlugogorski, B. Z., and Jameson, G. J. 2001. Free Drainage in Aqueous Foams: Model and Experimental Study. AIChE J. 47 (2): 314–327. https://doi.org/10.1002/aic.690470210. Mantilla, I. 1998. Bubble Trajectory Analysis in GLCC Separators. Master’s thesis, University of Tulsa, Oklahoma. Moncayo, J. A., Dabirian, R., Mohan, R. S. et al. 2018. Foam Break-Up Under Swirling Flow in Inlet Cyclone and GLCC. J. Pet. Sci. Eng. 165 (June): 234–242. https://doi.org/10.1016/j.petrol.2018.02.027. Nababan, A. 2015. Foam Breakup in CFC/GLCC System. Master’s thesis, University of Tulsa, Oklahoma. Nitasari, P. D. 2013. Utilization of Churn Flow Coalescer (CFC) for Improving Foam Breakup in GLCC. Master’s thesis, University of Tulsa, Oklahoma. Saint-Jalmes, A., Vera, M. U., and Durian, D. J. 2000. Free Drainage of Aqueous Foams: Container Shape Effects on Capillarity and Vertical Gradients. Europhys. Lett. 50 (5): 695–701. https://doi.org/10.1209/epl/i2000-00326-y. Verbist, G., Weaire, D., and Kraynik, A. M. 1996. The Foam Drainage Equation. J. Phys. Condens. Mat. 8 (21): 3715–3731. https://doi.org/10.1088/ 0953-8984/8/21/002. C

Jose ´ Antonio Moncayo is a reservoir-engineer consultant at Bravo Natural Resources and a project manager at Promonsa. Previously, he worked with Bravo and its affiliates for more than 7 years. Moncayo’s current interests include reservoir management, petroleum geology, strength of materials, and petrophysical/geomechanical modeling of oil and gas shales. He holds bachelor’s and master’s degrees in petroleum engineering, both from the University of Tulsa. Ramin Dabirian is a research associate at the University of Tulsa in the field of petroleum engineering, with a primary focus in production engineering. He has conducted several projects in the field of petroleum engineering, including multiphase compact 14



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separators, flow assurance, multiphase emulsion characterization, and foam characterization and breakup. Dabirian has authored or coauthored more than 30 refereed-journal and conference papers, and holds two pending patents. He holds a PhD degree in mechanical engineering and a master’s degree in petroleum engineering, both from the University of Tulsa. Dabirian is a member of SPE and the American Society of Mechanical Engineers. Ram Mohan is a professor of mechanical engineering at the University of Tulsa, and has more than 27 years of experience, of which 21 years are in academia and 17 years in industry (partly concurrently). He teaches and conducts research in the areas of control systems, multiphase-flow separation, oil/water-dispersion characterization, instrumentation and signal processing, manufacturing processes, and high-pressure-fluid applications. Mohan has served as the codirector of the Tulsa University Separation Technology Projects (TUSTP) since 1996, and has experience in successfully directing projects funded by the US Department of Energy, the National Science Foundation, the Oklahoma Center for the Advancement of Science and Technology, and several oil and gas companies. He has authored or coauthored more than 75 refereed publications in the areas of his research, holds six patents/invention disclosures, and has received several best-paper awards. Mohan holds a bachelor’s degree in mechanical engineering from the University of Kerala, India, and master’s and PhD degrees in mechanical engineering from the University of Kentucky. He is a member of SPE and a fellow of the American Society of Mechanical Engineers. Ovadia Shoham is the F. M. Stevenson Distinguished Presidential Chair Professor of Petroleum Engineering at the University of Tulsa. Since 1994, he has directed or codirected TUSTP and has conducted research on compact separators and in several projects supported by industry. Shoham has authored or coauthored more than 90 refereed publications in the areas of multiphase flow, multiphase separation, and production operations, and he holds six patents/disclosures. He holds bachelor’s and master’s degrees in chemical engineering from the Technion in Israel and the University of Houston, respectively, and a PhD degree in mechanical engineering from Tel Aviv University. Shoham is a member of SPE and has authored a widely referenced SPE book, Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. He is the recipient of the 2003 SPE International Production and Operations Award, and has received several best-paper awards. Gene Kouba is retired from Chevron, where he was a research consultant for the Chevron Energy Technology Company. His research interests primarily include issues in multiphase-flow transport and separation. Kouba has authored or coauthored more than 100 technical papers and holds nine US patents. He holds a PhD degree in petroleum engineering from the University of Tulsa. Kouba was the recipient of the 2014 SPE Projects, Facilities, and Construction Award.

2018 SPE Journal


Modeling Foam Breakup in Batch Separators and

Modeling Foam Breakup in Batch Separators and

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