Beggs and Brill Method. The Beggs and Brill method was the first one to predict predict flow flow behavi behavior or at all inclin inclinati ation on angels angels,, inclu includin ding g direc directio tional nal wells. wells. Their Their test test facility was 1-and 1.5-in. section of acrylic pipe, 9 ft long. The pipe could be inclined at any angel from the hori!ontal. The fluids were air and water. "or each pipe si!e, li#uid and gas gas rates were varied so that, when the the pipe was hori!ontal, all flow patterns were observed. $fter a particular set of flow rates was established, the inclination of the pipe was varied through the range of angles so that the effect of angl anglee on hold holdup up and and pres pressu sure re grad gradie ient nt were were meas measur ured ed at angl angles es from from the the o
o
o
o
o
o
o
o
o
, ±5 , ±1 , ±15 , ±% , ±&5 , ±55 , ±'5 , and ± 9
hori!o hori!onta ntall of developed from 5() measured tests.
. The correlations were
Beggs and Brill proposed the following pressure-gradient-e#uation for inclined pipe. f ρ n vm % dp dL
%d
=
+ ρ s g sin θ
1 − E k
*here + is is given by +# ).5& and ρ s
=
ρ L -θ + ρ g /1 − H L -θ 0
"low-attern rediction. "ig. ).12 illustrates the hori!ontal-flow pattern considered by Beggs and Brill. 3n the basis of observed flow pattern for hori!ontal flow only, the prepared an empirical map to predict flow pattern. Their original flow-pattern map map has has been been modi modifi fied ed slig slight htly ly to incl includ udee a tran transi siti tion on !one !one betw betwee een n the the segregated- and intermittent-flow patterns. "ig ).1' shows both the original and the modified dashed lines flow-pattern maps. Beggs and Brill chose to correlate flow-pattern transition boundaries with no-slip li#uid holdup and mi4ture "roude number, given by Fr
=
vm % gd
The e#uations for the modified flow-pattern transition boundaries are
.&%
L1
= &12λ L
L%
=
.9%5λ L −%.&2(
L&
=
.1λ L −1.)5%
and L)
=
2.'&(
.5λ L −
The following ine#ualities are used to determine the flow pattern that would e4ist if the pipe were hori!ontal. This flow pattern is a correlating parameter and, unless the pipe is hori!ontal, gives no information about the actual flow pattern. 6egregated. λ L
<
.1 and N Fr < L1
≥
.1 and N Fr < L%
or
λ L
Transition.
λ L
≥
.1 and L% ≤ N Fr ≤ L&
Intermittent Intermittent .1 ≤ λ L < .) and L& < N Fr ≤ L1 or
λ L
≥ .) and
L& < N Fr ≤ L)
Distributed Distributed
λ L < .) and N Fr ≥ L1 or
λ L ≥ .) and N Fr > L)
The Beggs and Brill model has been identified to be applicable in this research as it e4hibits several characteristics that set it apart from the other multiphase flow models7 a) Slippage between phases is taken into aount 8ue to the two different densities and viscosities involved in the flow, the lighter phase tends to travel faster than the heavier one termed as slippage. This leads to larger li#uid hold-up in practice than would be predicted by treating the mi4ture as a homogeneous one. b) Flow pattern onsideration 8epending on the velocity and composition of the mi4ture, the flow behaviour changes considerably, so that different flow patterns emerge. These are categori!ed as follows7-segregated, intermittent and distributed. 8epending upon the flow pattern established, the hold-up and friction factor correlations are determined. ) Flow angle onsideration This model deals with flows at angles other than those in the vertical upwards direction. 6ome assumptions had been used in the development of this correlation7 1. The two species involved do not react with one another, thus the composition of the mi4ture remains constant. %. The gaseous phase does not dissolve into the li#uid one, and evaporation of the li#uid into gas does not occur. The Beggs and Brill model /%0 has the following pressure-gradient e#uation for an inclined pipe7
*here d:d; is the pressure gradient, f is the friction factor,
$ reliable estimation of the pressure drop in well tubing is essential for the solution of a number of important production engineering and reservoir analysis problems. Many empirical correlation and mechanistic models have been proposed to estimate the pressure drop in vertical wells that produce a mi4ture of oil, water and gas. $lthough many correlations and models are available to calculate the pressure loss, these models developed based on certain assumption and for particular range of data where it may not be applicable to be used in different sets of data
INTRODUCTION Multiphase flow in pipes is the process of simultaneous flow of two phases or more. =n oil or gas production wells the multiphase flow usually consist of oil, gas and water. The estimation of the pressure drop in vertical wells is #uite important for cost effective design of well completions, production optimi!ation and surface facilities. >owever, due to the comple4ity of multiphase flow several approaches have been used to understand and analysis the multiphase flow.
3il ? @as industry is needed to have a general method for forecasting and evaluating the multiphase flow in vertical pipes oettmann, ? Aarpenter, 195%. Multiphase flow correlations are used to determine the pressure drop in the pipes. $lthough, many correlation and models have been proposed to calculate pressure drop in vertical well, yet its still arguing about the effectiveness of these proposed models.
umerous correlations and e#uations have been proposed for multiphase flow in vertical, inclined and hori!ontal wells in the literature. +arly methods treated the multiphase flow problem as the flow of a homogeneous mi4ture of li#uid and gas. This approach completely disregarded the well-nown observation that the gas phase, due to its lower density, overtaes the li#uid phase resulting in CslippageD between the phases. 6lippage increases the flowing density of the mi4ture as compared to the homogeneous flow of the two phases at e#ual velocities. Because of the poor physical model adopted, calculation accuracy was low for those early correlations. $nother reason behind that is the comple4ity in multiphase flow in the vertical pipes. *here water and oil may have nearly e#ual velocity, gas have much greater one. $s a results, the difference in the velocity will definitely affect the pressure drop. Many methods have been proposed to estimate the pressure drop in vertical wells that produce a mi4ture of oil and gas. The study conducted by ucnell et al. 199& concludes that none of the traditional multiphase flow correlations wors well across the full range of conditions encountered in oil and gas fields. Besides, most of the vertical pressure drop calculation models were developed for average oilfield fluids and this is why special conditions such asE emulsions, non ewtonian flow behavior, e4cessive scale or wa4 deposition on the tubing wall, etc. can pose severe problems. $ccordingly, predictions in such cases could be doubtful. Taacs, %1 The early approaches used the empirical correlation methods such as >agedron ? Brown 1925 8uns ? Fos, 192&, and 3ris!ewsi 192'. Then the trend shift into mechanistic modelling methods such as $nsari 199) and $!i! et al 19'% and lately the researchers have introduced the use of artificial intelligence into the oil and gas industry by using artificial neural networs such as $youb %) and Mohammadpoor %1 and many others. The main purpose of this study is to evaluate and assess the current empirical correlations, mechanistic model and artificial neural networs for pressure drop estimation in multiphase flow in vertical wells by comparing the most common methods in this area. The parameters affecting the pressure drop are very important for the pressure calculation Therefore, it will also be taen into account in the evaluation.
EMPIRICAL CORRELATIONS
The empirical correlation was created by using mathematical e#uations based on e4perimental data. Most of the early pressure drop calculation was based on this correlations because of its direct applicability and fair accuracy to the data range used in the model generation. =n this study, the empirical correlations for pressure drop estimation in multiphase flow in vertical wells are reviewed and evaluated with consideration of its re#uired dimensions, performance, limitation and range of applicability. Beggs & Brill Correlation (197!" The Beggs and Brill method was developed to predict the pressure drop for hori!ontal, inclined and vertical flow. =t also taes into account the several flow regimes in the multiphase flow. Therefore, Beggs ? Bril 19'& correlation is the most widely used and reliable one by the industry. =n their e4periment, they used 9 ft. long acrylic pipes data. "luids used were air and water and 5() tests were conducted. @as rate, li#uid rate and average system pressure was varied. ipes of 1 and 1.5 inch diameter were used. The parameters used are gas flow rate, ;i#uid flow rate, pipe diameter, inclination angel, li#uid holdup, pressure gradient and hori!ontal flow regime. This correlation has been developed so it can be used to predict the li#uid holdup and pressure drop.
BE##S AND BRILL MET$OD %OR TO P$ASE %LO CALCULATIONS The Beggs and Brill correlation was developed from e4perimental data obtained in a small-scale test facility. The facility consisted of 1-inch and 1.5-inch sections of acrylic pipe 9 feet long. The pipe could be inclined at any angle. The parameters studied and their range of variation were7
1. @as flow rates of to & Mscf:8 %. ;i#uid flow rates of to & gpm &. $verage system pressure of &5 to 95 psia ). ipe diameter of 1 and 1.5 inches 5. ;i#uid hold-up of to .(' 2. ressure gradients of to .( psi:ft. '. =nclination angles of 9 degrees to G9 degrees "luids used were air and water. "or each pipe si!e, li#uid and gas rates were varied so that all flow patterns were observed. $fter a particular set of flow rates was set,
the angle of the pipe was varied through the range of angles so that the effect of angle on holdup and pressure gradient could be observed. ;i#uid holdup and pressure gradient were measured at angles from the hori!ontal at , plus and minus 5,1,15, %, &5, 55, '5, and 9 degrees. The correlations were developed from 5() measured tests $ORI'ONTAL %LO
8ifferent correlations for li#uid holdup are presented for each of the three hori!ontal flow regimes. The li#uid holdup, which would e4ist if the pipe were hori!ontal, is first calculated and then corrected for the actual pipe inclination. Three of the hori!ontal flow patterns are illustrated on "igure '-1. $ fourth, the transition region, was added by Beggs and Brill to produce the map shown on "igure '-%. The variation of li#uid holdup with pipe inclination is shown on "igure '-& for three of the tests. The holdup was found to have a ma4imum at appro4imately G5 degrees from the hori!ontal and a minimum at appro4imately 5 degrees. $ two-phase friction factor is calculated using e#uations, which are independent of flow regime, but dependent on holdup. $ graph of a normali!ed friction factor as a function of li#uid holdupand input li#uid content is given on "igure '-).
"igure '-17 >ori!ontal "low atterns
"igure '-%7 >ori!ontal "low attern Map
%igre 7)" Li*i+ $ol+, -s. Angle
"igure '-)7 Two-phase "riction "actor %LO RE#IME DETERMINATION
The following variables are used to determine which flow regime would e4ist if the pipe was in a hori!ontal position. This flow regime is a correlating parameter and gives no information about the actual flow regime unless the pipe is completely hori!ontal.
The hori!ontal flow regime limits are7
*hen the flow falls in the transition region, the li#uid holdup must be calculated using both the segregated and intermittent e#uations, and interpolated using the following weighting factors7
>;transition H $ 4 >;segregated G B 4 >;intermittent
/ERTICAL %LO
"or vertical flow, I H 1, and d; H dJ, soE
The pressure drop caused by elevation change depends on the density of the two phase mi4ture and is usually calculated using a li#uid holdup value. +4cept for high velocity situations, most of the pressure drop in vertical flow is caused by elevation change. The frictional pressure loss re#uires evaluation of the two-phase friction factor. The acceleration loss is usually ignored e4cept for high velocity cases. /ERTICAL %LO RE#IMES B00le lo2
=n bubble flow the pipe is almost completely filled with li#uid and the free gas phase is present in small bubbles. The bubbles move at different velocities and e4cept for their density, have little effect on the pressure gradient. The wall of the pipe is always contacted by the li#uid phase. Slg lo2
=n slug flow the gas phase is more pronounced. $lthough the li#uid phase is still continuous, the gas bubbles coalesce and form plugs or slugs, which almost fill the pipe cross-section. The gas bubble velocity is greater than that of the li#uid. The li#uid in the film around the bubble may move downward at low velocities. Both the gas and li#uid have significant effects on the pressure gradient.
Transition lo2
The change from a continuous li#uid phase to a continuous gas phase is called transition flow. The gas bubbles may Koin and li#uid may be entrained in the bubbles. $lthough the li#uid effects are significant, the gas phase effects are predominant. Mist lo2 =n mist flow, the gas phase is continuous and the bul of the li#uid is entrained as droplets in the gas phase. The pipe wall is coated with a li#uid film, but the gas phase predominantly controls the pressure gradient. =llustrations of bubble, slug, transition, and mist flow are shown below.
"igure '-57 Lertical "low atterns $ typical two-phase flow regime map is shown on "igure '-2.
"igure '-27 "low Fegime Map *here the following are the procedure e#uations for calculating vertical flow with Beggs and Bill method
SET UP CALCULATIONS
1 a. b. c. d. %
@iven =nlet mv, m;,
) Aalculate Lolumetric "low rates 5
2 Aalculate 6uperficial Lelocities Lsg, Lsl,Lm
8+T+FM=+ ";3* F+@=M+ 5. Aalculate "roude o. '
2. Aalculate o-slip li#uid holdup
'. Aalculate dimension less C;D parameters
(. $pply flow regime rules a. =s N; 〈 .1 and "r 〈 ;1 O P o Qes R 6egregated flow b. =s N; S .1 and "r 〈 ; % O P R o Qes 6egregated flow c. =s N; S .1 and ; % "r ;& O
P o Qes R Transition flow d. =s . 1 N; 〈 . ) and ; & 〈 " r ; 1 O P o Qes R =ntermittent flow e. =s N; S .) and ;& 〈 "r ; ) O P o Qes R =ntermittent flow f. =s N; 〈 .) and "r S ;1 O P o Qes R 8istributed flow g. =s N; S .) and "r 〉 ; ) O 3utside the range of the Beggs - Brill method DETERMINE LI3UID $OLDUP & TO)P$ASE DENSIT4 9.Aalculate > ; o the holdup which would e4ist at the same conditions in a hori!ontal pipe.
"or transition flow calculate > ; segregated and >; =ntermittent. 1. Aalculate ;L the li#uid velocity number
11. Aalculate c
"or transition uphill flow, calculate c segregated and c intermittent. A must be S 1%. Aalculate U
where V is the actual angle of the pipe from hori!ontal. "or vertical flow W H 9. 1&. Aalculate the li#uid holdup, >; W "or segregated, intermittent, and distributed flow7
"or transition flow7 >; transition H $ X > ; segregated G B X >; intermittent
1). Aalculate >g H 1 > ; W 15. Aalculate two phase density ; W G g 8+T+FM=+ +;+L$T=3 T+FM
8+T+FM=+ "F=AT=3 T+FM
1'. Aalculate no-slip two-phase density
%. Aalculate no-slip friction factor
%1. Aalculate
%%. Aalculate
%&. Aalculate the two phase friction factor
%). Aalculate the friction loss term,
A$;AZ;$T+ T>+ $AA+;+F$T=3 T+FM %5.
A$;AZ;$T+ T>+ T3T$; F+66ZF+ @F$8=+T
B+@@6 $8 BF=;; M+T>38 3M+A;$TZF+ $ "low area 8 8iameter + Beggs and Brill acceleration term "r "roude number f "riction factor g $cceleration due to gravity gc @ravitational constant > >oldup h enthalpy ; ;ength M Molecular weight pressure ;* ;osses due to irreversible processes # >eat loss input [ Lolumetric flow rate Fe Feynolds number 6 +ntropy 6@ 6pecific gravity 6T 6urface tension Z =nternal energy L Lolume v Lelocity * Mass flow rate \ 8imensionless flow pattern region coefficient Q "low pattern region coefficient J +levation N 6lip Liscosity ] ;ochartMartinelli correlating parameter < 8ensity ^ 6hear force %] Two phase
