research article a novel dynamic model for predicting...

Research ArticleA Novel Dynamic Model for Predicting Pressure Wave Velocityin Four-Phase Fluid Flowing along the Drilling Annulus

Xiangwei Kong1 Yuanhua Lin2 Yijie Qiu2 and Xing Qi2

1School of Chemistry and Chemical Engineering Daqing Normal University Daqing Heilongjiang 163712 China2State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University ChengduSichuan 610500 China

Correspondence should be addressed to Xiangwei Kong m13880214723163com and Yuanhua Lin yhlin28163com

Received 22 March 2014 Accepted 27 July 2014

Academic Editor Alexei Mailybaev

Copyright copy 2015 Xiangwei Kong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A dynamic pressure wave velocity model is presented based onmomentum equation mass-balance equation equation of state andsmall perturbation theory Simultaneously the driftmodel was used to analyze the flow characteristics of oil gas water and drillingfluid multiphase flow In addition the dynamic model considers the gas dissolution virtual mass force drag force and relativemotion of the interphase as well Finite difference and Newton-Raphson iterative are introduced to the numerical simulation ofthe dynamic model The calculation results indicate that the wave velocity is more sensitive to the increase of gas influx rate thanthe increase of oilwater influx rate Wave velocity decreases significantly with the increase of gas influx Influenced by the pressuredrop of four-phase fluid flowing along the annulus wave velocity tends to increase with respect to well depth contrary to thegradual reduction of gas void fraction at different depths with the increase of backpressure (BP) Analysis also found that thegrowth of angular frequency will lead to an increase of wave velocity at low range Comparison with the calculation results withoutconsidering virtual mass force demonstrates that the calculated wave velocity is relatively bigger by using the presented model

1 Introduction

In petroleum industry managed pressure drilling (MPD)is considered to be one of the most important techniqueswhich allows accurate control of bottom hole pressure (BHP)by controlling the flow rate drilling mud density and backpressure (BP) at the wellhead [1] As extensively used in thedrilling of huge risk uneconomical or even abnormal for-mation MPD has been of interest in the literature for at leastthe past decade especially the related issues about pressurecontrol inducing the dynamic well and kick detection [2]Due to those efforts research of the dynamic pressure wavevelocity is of great significance to the detection of gas influxand effective control of the pressure at the bottom of well[3] During the drilling operation of the so-called ldquomicrofluxcontrolrdquo an MPD technique developed by Santos et al [4]the return flow is monitored and adjusted to control fluid lossor gain In the light of control principal simulation studieswere performed to determine the most appropriate initialresponse to kicks arising due to MPD specific complications

caused by BHP fluctuations [5] During the MPD operationsall unsteady operating such as changing of pumping rateadjustment of choke and controls of BP at wellhead willgenerate a pressure wave and threaten the drilling equipment[6] For the same reason while tripping out of a drill string inthe wellbore bottom hole is submitted to a suddenly decreasein pressure leading to fluid expansion and movement outof the annulus The rapidly expanding fluids and dynamicpressure fluctuations can also lead to rock instability in areservoir [7] Particularly the effects are more important insystems inwhichmultiphase flows occur New kick-detectiontools are now available that are based on acoustic principleswhich are of great benefit to potentially earlier and moresensitive detection of a gas influx than pit-gain or paddleflow measurements [8] The study of propagation of pressurewave is also relevant to control of downhole tool such asintelligent well downhole control valves applied in differentfield formany purposesWith further development of oilfielddownhole tool technique for special casing wells is receivingmuch more attention [9] At the meantime an important

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 134102 17 pageshttpdxdoiorg1011552015134102

2 Mathematical Problems in Engineering

problem in deep drilling is the propagation of measurement-while-drilling (MWD) mud pulse transmitting real-timevarious data from sensors located down hole near the drillbit Hence the propagation behavior of pressure wave isconsidered to provide reference for the MPD operations

However we also noticed that the conventional theoriespresent are difficult to be employed in a systematic andaccurate prediction when influxes generates As the influxesfluid including gas oil and water will lead to variationsof physic characteristics parameter of fluid in the annulusthe distribution of pressure wave velocity in gas oil waterand drilling mud four-phase flow along the annulus will bedynamic changing with respect to time and well depthTheseeffects may influence the safe operation of devices

Pressure waves are disturbances that transmit energy andmomentum along the wellbore through drilling fluid withoutsignificant displacement of matter As migration of dispersedgas water and oil towards wellhead is quite complicated thefundamental characteristics of the four-phase flow are stillunknown and modeling results of pressure wave velocity arequestionable The proper treatment of propagation behaviorof pressure wave in the four-phase flow in the annulusrequires knowledge of description of influxes generationdevelopment and pressure wave propagationmodel in a two-phase mixture

In gasliquid two-phase flow two-phase medium inter-action greatly changes the structure characteristic of flow-ing fluid which results in greater compressibility of two-component flow than single-phase gas or liquid and fur-ther causes pressure waves propagating speed to be greatlyreduced This can be explained from the viewpoints ofmixture density and compressibility of two-phase fluid andthe pressure drop along the flow direction in the wellboreIn the low void fraction range the gas phase is dispersedin the liquid as bubble so the wave velocity is influencedgreatly by the added gas phase According to the EOS if gasinvades into the wellbore with a small amount in the bottomhole the density of the drilling mud has little variationwhile the compressibility increases obviously which makesthe wave velocity be decreased This phenomenon was firstproposed by Mallock [10] and attracts much attention for itsimportant role in the development of science and technologyapplications Extensive investigations involved in the issueof pressure wave propagation have been taken and somesignificant achievements associated with the theories havealso been made in the researches In early stage researchingWood [11] extended the researches of Mallock and presenteda succinct formula by assuming that the compressibility oftwo-component fluid is a function related to single-phasecompressibility and elastic modulus E Carstensen and Flody[12] measured the velocity of pressure wave under a lake Adispersion relation for pressure waves propagating through abubbly fluid was derived by using a linear scattering theorydeveloped by Foldy Under the hypothesis of homogeneousand adiabatic laminar flow Thuraisingham [13] took thetwo-phase media as a homogeneous fluid and derived thesolution model concerning the problem of wave velocitymodel in two-phase flow at low gas void fraction accordingto the analysis of state equation of mixture According to the

early stage investigations Hsieh and Plesset [14] and Murray[15] researched the influence of thermal conductivity andviscosity coefficient on pressure wave propagation Wallis[16] firstly studied the propagation mechanism of pressurewave and derived the propagation velocity in bubbly flow andseparated flow using the homogeneous model The proposedmodel is based on the respective compressibility of the vaporand the liquid Also the two-phase mixture is treated asa compressible fluid with suitably averaged properties Inthat expression mass and heat transfers are neglected soit can be applied to any gasliquid mixture in so far as nomajor effect due to vaporization or condensation must betaken into account It may not be valid for such complexmixtures that never reach equilibrium Assuming that noevaporation or condensation occurs when pressure wave istransporting Moody [17] developed a simple acoustic wavemodel for bubbly flow and annular flow and established arelationship between sonic velocity and two-phase criticalflow Similar model was developed by DrsquoArcy [18] In theresearches of Mcwilliam and Duggins [19] surface tensionand compressibility of liquid phase were also consideredHenry et al [20] calculated the velocity as a function ofvoid fraction using a correlation to account for the change inbubble shape with void fraction Martin and Padmanabhan[21] extended the simple model proposed by Henry byconsidering wave reflection and wave transmission at gas-liquid interfaces Researches ofMori et al [22 23] suggest thatimpact pipe elasticity on pressure wave propagation velocityis limited to the range of gas void of less than 1 At highgas void range the pressure wave velocity is in between thetwo velocities of single-phase Nguyen et al [24] proposedanother type of model for bubbly flow (diluted gas phase inthe liquid) The simple relations for prediction of the propa-gation of pressure disturbances in liquid-gaseous two-phasesystems are presented The model makes use of the well-known physical behavior that the wave velocity in a single-phase fluid is influenced by the elasticity of the confiningwalls The interface of the one phase is considered to act asthe elastic wall of the other phase and vice versa Mecredyand Hamilton [25] used the two-fluid model to predict thepressure wave propagation in vapor-liquid flow in detailHowever the analysis contained the important assumptionthat the evaporation or condensationwas governed by kinetictheory Michaelides and Zissis [26] developed a computa-tional method which yields the sound velocity in terms ofthe thermodynamic coordinates of the substance withoutthe use of diagrams Corresponding velocities of sound forthe four substances considered exhibit a certain similaritywhich is examined statistically The relationship between thesound velocity and the critical mass flux is also investigatedThuraisingham [13] studied the wave velocity in bubbly waterat megahertz frequencies (1sim10MHZ) Numerical analyticalresults indicate that volume concentrations and the radius ofthe bubble relative to the incident wavelength of sound arethe important parameters which determine the deviation ofsound speed form that of bubble-free water

Currently the two-fluid continuum model is the mostcommon and reliable method to describe the gasliquid two-phase flow phenomenon [27] In the model the governing

Mathematical Problems in Engineering 3

equation and phase interface relationship is established basedon the assumption that each phase satisfies the continuumconditions To obtain the practical flow equations reasonableassumptions and constitutive equations should be intro-duced In consequence the predicted wave velocities werefound to depend strongly on the introduced assumptions andequations In recent years the two-fluid model was appliedin determining the pressure wave propagation characteristics[28] Ruggles et al [29 30] firstly performed the experimentalinvestigation on the dispersion of pressure wave propagationin air-water bubbly flow and studied the propagation ofpressure disturbance based on the two-fluid model smallperturbation analysis methodThrough the comparison withexperimental data they found that the virtual mass forcecoefficient is a function of gas void Chung et al [31]calculated the sonic velocity versus angular frequency formthe concept of bubble compressibility in a two-componentbubbly flow regime He also extended such amodel to predictthe sonic velocity of a vapor-liquid system Lee et al [32] con-structed the two fluids model to determine the pressure wavepropagation speed for two-phase bubbly flow slug flow andstratified flow by using pressure disturbance instead of virtualmass and other phase interfacial terms The results fit wellwith the experimental in steam water and air water of Henryand theoretical analysis of Nguyen Zhao and Li [33] derivedthe general formula of sonic velocity in gas-liquid two-phaseflow linear analysis using the linear analysis of the closedfundamental equations of compressible gas-liquid two-phaseflow It is proposed that the appropriate formula for calculat-ing sonic velocity in gas-liquid two-phase flows under usualconditions may be Wood adiabatic sonic velocity formulaBy linearizing the conservation equations of two-fluidmodelLiu [34] derived a wave number equation of pressure wavefor adiabatic gas-liquid two-phase flow The effects of dragforce and virtual mass force on propagation and dispersionof pressure wave were investigated Xu and Chen [35] usedthe transient two-fluid model to develop a general relationfor acoustic waves with steam-water two-phase mixture inone-dimensional flowing system Both the mechanical andthermal nonequilibrium are considered Brennen [36] takenmass and heat exchanges into account and proposed morecomplete expressions of the speed of sound in two-phasemixture However calibration of themass and heat exchangesrequires some further experimental investigations Yeom andChang [37] numerically investigated the wave propagation inthe two-phase flows An assessment was made on the effectof interfacial friction terms Zhang et al [27] investigatedthe propagation of the pressure wave in the water-gas two-phase bubbly flow with a one-dimensional two-fluid modeland employing small perturbation analysis The governingequations are simplified and closured according to high-speed aerated flow characteristics in hydraulic engineeringThe effects of aerated concentration liquid pressure per-turbation frequency and interfacial forces on the acousticwave velocity and its attenuation in the aerated flow arealso explored With the application of thermal phase changemodel in computational fluid dynamics code CFX Li etal [38] proposed a pressure wave propagation model andinvestigated the pressure wave propagation characteristics

in two-phase fuel systems of liquid-propellant rocket Thepropagation of pressure wave during the condensation ofR404A and R134A refrigerants in pipe minichannels wasgiven by Kuczynski [39] Heat exchange between the phasesin the condensation process was calculated by using the one-dimensional form of Fourierrsquos equation

In drilling industry some scholars have been devoted tothis aspect In the late 1970s the former Soviet All-UnionDrilling Technology Research Institute [40] began to studycharacteristics of pressure wave velocity in gas-liquid two-phase flow to detect early gas influx and achieved someimportant results To study relationship between pressurewave velocity and gas void fraction Li et al [41] launcheda gas-drilling mud two-phase flow simulation experimentin vertical annulus It is proved that the two phase flow ofgas-liquid patterns and the velocity of gas migration canbe determined if the well depth mud properties and voidfraction in bottom are given The method which is fasterin detection time than the method of conventional kickdetectionwas proposed Startingwith the analysis of transientflow combining with the theory of transmission line Wang[42] obtained the calculating model of frequency domain forthe pulse velocity in drilling fluid built and the impendenceand transmission operator of drilling fluid Alternative initialresponses to kicks for various well scenarios during MPDoperations were also explored by Davoudi et al [43] Li etal [44] proposed a mathematical model for predicting theattenuation and propagation velocity of measurement whiledrilling (MWD) pressure pulses in aerated drilling using thetwo-phase flow model and considering the momentum andenergy exchange at the phase interface gravity of each phaseviscous pipe shear and other closing conditions Accordingto the theory of unsteady flow Xiushan [45] developedthe formulas of transmission velocity for mud pulse signalThe formulas which cover all kinds of boundary conditionsincluding thin wall pipes and thick ones and interactioninfluence of gas content and solid content on transmissionvelocity are suitable for positive and negative mud pulse andaccordwith drilling practice In a previouswork we proposeda united wave velocity model to predict the pressure wavevelocity in gas-drilling mud two-phase steady flowThe effectof well depth back pressure gas influx rate virtual massforce and angular frequency are all considered Howeverunder the effects of buoyancy and complicated turbulenceinteraction the existing theoretical solutions are not involvedin the dynamic model for predicting pressure wave velocityin four-phase fluid flowing along the drilling annulus wheninflux fluid migrate towards the top of wellbore

In this paper the drift model was used to analyze theflow characteristics of oil gas water and drilling fluidmultiphase flow As the important characteristics of influxdevelopment the relative motion of the interphase such asslippage of gas phase and oil phase is considered Moreoverto predict pressure wave velocity in gas-oil-water-drillingmud four-phase flow in the annulus duringMPD operationsa dynamic mathematical model is presented By computingthe influence factors of pressure wave velocity such as backpressure gas void fraction oil void fraction influx time


influx rate disturbance angular frequency and virtual massforce are analyzed

2 Mathematical Model

Before introducing the new dynamic model and to make thispoint clear this paper reviews the hydraulic system in MPDoperationsThe drilling system is a closed circulation with BPat wellhead The key equipments include the rotating controldevice dynamic well control system conventional pressurecontrol system industrial personal computer Coriolis meterchoke and pressure sensor First the drilling mud beginsto circulate from mud tank down the drill pipe and thedrill string and returns from the annulus travel back throughmud pit where drilling solids are taken away and then tosurface mud tank An important function of the drillingfluid is to provide pressure support to the wellbore wall Therock formation drilled through has some form of porosityfilled with formation fluids These fluids can be water or inthe case of a reservoir hydrocarbons The pressure in thesefluids is referred to as the pore pressure If the pore spacesare connected these formations will also have permeabilityFluids can flow through them in response to a pressuregradientThe pressure in the annulus is controlled by varyingBP to operate the fluid pressure in the wellbore The aim inMPD is thus to maintain the pressure in the annulus betweenthe two limits of pore pressure and fracture pressure [1]

21 The United Dynamic Model When the bottom holepressure is below the formation pressure formation fluid willinvade into the wellbore and the four-phase flow emerges inthe annulus constituted by the drill string and wellbore Asseen in Figure 1 take any cross section of the wellbore asan infinitesimal control volume In the infinitesimal controlvolume the four-phase drilling fluid is consisted by drillingmud (considered as a pseudohomogeneous liquid) influx oil(considered as oil phase) influx natural gas (considered asgas phase) and influx water (considered as water phase)Appropriate assumptions and governing equations are crit-ical to simulate realistic four-phase well-control operationsThe four-phase model was established based on the followingassumptions

(1) it is unsteady-state four-phase flow(2) the flow along the flow path is one-dimensional(3) the drilling mud is water-based(4) drilling mud is incompressibleIn the analysis of the multiphase flow characteristic

oil water and drilling mud which are all considered asliquid phase have great differences in physical and chemicalproperties As water-based medium water and drilling mudhave no substantive difference In the flow processes theyblend quickly and have no clear phase boundary Thus thewater-based fluid phase is discussed as water phase and theoil is considered as another phase

As the presence of oil and gas interphase mass transfer inthemultiphase flow systemwithin the wellbore the appropri-ate mass conservation equation can be listed according to oil

Liquid phase

Gas phase

OrOr

p

p + Δp

Δs

Slug flowBubble flow Annular flow

Figure 1 Infinitesimal control volume in effective wellbore

gas and water three components Given that 119909119894119896is the mass

fraction of 119896 components in 119894 phase the mass conservationequations for four-phase mixture are

119860

120597

120597119905

(120594

119900119896120588

119900120601

119900+ 120594

119892119896120588

119892120601

119892+ 120594

119908119894120588

119908120601

119908+ 120594

119898119894120588

119898120601

119898)

+

120597

120597119904

(120594

119900119896119860120588

119900120601

119900V119900+ 120594

119892119896119860120588

119892120601

119892V119892

+120594

119908119894119860120588

119908120601

119908V119908+ 120594

119898119894119860120588

119898120601

119898V119898) = 0

(1)

The momentum balance equation for the four-phasemixture is

120597

120597119905

(119860sum

119896

120588

119896120601

119896V119896) +

120597

120597119904

(119860sum

119896

120588

119896120601

119896V2119896)

+ 119860119892sum

119896

120588

119896120601

119896+

120597

120597119904

(119860119901) + 119860(

120597119901

120597119904

)

fr= 0

(2)

Equation (2) is a general momentum balance equationincluding hydrostatic pressure gradient frictional pressureloss gradient and acceleration loss gradient

Hence

sum

119894

120594

119900119894= 1 sum

119894

120594

119892119894= 1

sum

119894

120594

119908119894= 1 sum

119894

120594

119898119894= 1

(3)

where

120594

119908119900= 0 120594

119908119892= 0 120594

119908119908= 1 120594

119908119898= 0

120594

119898119900= 0 120594

119898119892= 0 120594mw = 0 120594

119898119898= 1

120594

119892119900= 0 120594

119892119892= 1 120594

119892119908= 0 120594

119892119898= 0

120594

119900119900=

119898

119900

119898

119900+ 119898

119892

120594

119900119892=

119898

119892

119898

119900+ 119898

119892

120594

119900119908= 0 120594

119900119898= 0

(4)


As propagation velocity is greatly affected by the gas voidfraction and angular frequency of the pressure disturbancethe superficial velocity of flowing medium has almost noeffect on the propagation velocity [46] oil water and drillingcan be considered as liquid phase for their similarity inmechanics According to the two-fluidmodel the flow can besupposed to be gas-liquid two-phase flow from amacroscopicview

To establish the wave velocity dispersion equation themass conservation equations for liquid and gas two phasescan be written individually as follows

120597

120597119905

(120601

119892120588

119892) +

120597

120597119904

(120601

119892120588

119892V119892) = 0

120597

120597119905

(120601

119871120588

119871) +

120597

120597119904

(120601

119871120588

119871V119871) = 0

(5)

Hence 120588119871= 120601

119900120588

119900+ 120601

119908120588

119908+ 120601

119898120588

119898

The gas momentum conservation equation is

120597

120597119904

(120601

119892120588

119892V119892) +

120597

120597119904

(120601

119892120588

119892V2119892)

= minus

120597

120597119904

(120601

119892120588

119892) +

120597

120597119904

[120601

119892(120591

fr119892+ 120591

Re119892)] +119872gi minus 4

120591

119892

119863

(6)

The liquid momentum conservation equation is

120597

120597119905

(120601

119871120588

119871V119871) +

120597

120597119904

(120601

119871120588

119871V2119871)

= minus

120597

120597119904

(120601

119871120588

119871) +

120597

120597119904

[120601

119871(120591

fr119871+ 120591

Re119871)] +119872Li minus 4

120591

119871

119863

(7)

The transfer of momentum 119872gi and 119872Li can be writtenby the following equations

119872gi = minus119872

ndLi minus119872

119889

Li + (120591frLi + 120591

ReLi )

120597120601

119871

120597119904

+

120597 (120601

119892120590

119904)

120597119904

+

120597 (120601

119892119901gi)

120597119904

minus 120601

119892

120597 (119901Li)

120597119904

119872Li = 119872

ndLi +119872

119889

Li + 119901Li120597 (120601

119871)

120597119904

minus (120591

frLi + 120591

ReLi )

120597120601

119871

120597119904

(8)

Virtual mass force is obtained by the equation in thefollowing form

119872

ndLi = 119888VM120601119892120588119871120572VM minus 01120601

119892120588

119871V119904

120597V119904

120597119904

minus 119888

1198981120588

119871V2119904

120597120601

119892

120597119904

(9)

where V119904= V119892minus V119871and 1198881198981

= 01The momentum transfer term is described as [47]

119872

119889

Li =3

8

119862

119863

119903

120588

119871119877

119902V2119904 (10)

The pressure difference between the liquid interface andliquid can be obtained by (11)

119901Li minus 119901119871 = minus119888

119901120588

119871V2119904 (11)

where 119888119901= 025

The gas interface pressure119901gi is defined as follows

119901gi minus 119901119892 asymp 0 (12)

The pressure of the liquid is

119901

119871= 119901 minus 025120588

119871120601

119892V2119904 (13)

The shear stress and the interphase shear stress can bedescribed as

120591

fr119892asymp 120591

frLi asymp 120591

fr119871asymp 120591

119892asymp 120591

Re119892

asymp 0 (14)

The Reynolds stress and interfacial average Reynoldsstress are

120591

Re119871

= minus119888

119903120588

119871V2119904

120601

119892

120601

119871

120591

ReLi = minus119888

119903120588

119871V2119904

(15)

Hence 119888119903= 02

The wall shear stress of liquid phase is expressed as [48]

120591

119871= 05119891

119871120588

119871V2119871 (16)

The pressure wave velocity of gas phase 119888119892and that of

liquid phase 119888119871can be expressed in the following form

119889119901

119871

119889120588

119871

= 119888

2

119871

119889119901

119892

119889120588

119892

= 119888

2

119892

(17)

Based on (17) the hydrodynamic equations of two-fluidmodel (5)ndash(7) can be written in the following matrix form

119860

120597120585

120597119905

+ 119861

120597120585

120597119904

= 119862120585(18)

Here A is the matrix of parameters considered in relationto time B is the matrix of parameters considered in relationto the spatial coordinate C is the vector of extractions Byintroducing the small disturbance theory the disturbance ofthe state variable 120585(120601

119892 119901 V119892 V119871)

119879 can be written as

120585 = 120585

0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (19)

where 119896 is the wave numberAccording to the solvable condition of the homogenous

linear equations that the determinant of the equations iszero the equation of pressure wave can be expressed in thefollowing form


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(120588

119892+ 119888

119901120601

119892120588

119871

V2119904

119888

2

119892

)119908

120601

119892

119888

2

119892

[1 minus 119888

119901120601

119871]

V2119904

119888

2

119871

119908 minus[120601

119892120588

119892119896 + 2119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908] 22119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908

minus120588

119871119908

1 minus 120601

119892

119888

2

119871

119908 0 minus119896 (1 minus 120601

119892) 120588

119871

120588

119871V2119903119896 (minus120601

119892119888

119901+ 119888

119903minus 119888

119894+ 119888

1198982) minus120601

119892119896 [1 minus 120601

119871

119888

119901V2119904

119888

2

119871

+ 119888

119894

V2119904

119888

2

119871

]

120601

119892(120588

119892+ 119888vm120588119871)119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119892119908120588

119892V119892)

minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871V2119904119896 (120601

119871119888

119901minus 2119888

119903minus 119888

1198982) minus119896(120601

119871+ 119888

119903120601

119892

V2119904

119888

2

119871

) minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871[120601

119871+ 120601

119892119888vm]119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119871120588

119871V119871)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= 0

(20)

where 119888119894= 03 119888

1198982= 01 119888

119903= 02 and 119888

119901= 025

The real value of wave number is determined as thepressure wave velocity c and pressure wave velocity in thefour-phase flow is

119888 =

1003816

1003816

1003816

1003816

119908119877

+(119896) minus 119908119877

minus(119896)

1003816

1003816

1003816

1003816

2

(21)

22 Physical Models To define the velocity of pressure wavepropagation in the four-phase flow the related physicalmodels are required such as equations of state temperaturedistribution model gas dissolution and oil phase volumefactor

221 Equations of State for Gas The equation of state (EOS)for gas can be expressed as

120588

119892=

119901

(119885

119892sdot 119877 sdot 119879)

(22)

For 119901 lt 35MPa the compression factor is obtained asfollows

119885

119892= 1 + (03051 minus

10467

119879

119903

minus

05783

119879

3

119903

)120588

119903

+ (05353 minus

026123

119879

119903

minus

06816

119879

3

119903

)120588

2

119903

(23)

where 119879119903= 119879119879

119888 119901119903= 119901119901

119888 120588119903= 027119901

119903119885

119866119879

119903

For 119901 ge 35MPa the compression factor under thecondition of high pressure is [49]

119885

119892=

006125119875

119903119879

minus1

119903exp (minus12 (1 minus 119879minus1

119903)

2

)

119884

(24)

where Y is given by the follow equations

minus 006125119875

119903119879

minus1

119903exp (minus12(1 minus 119879minus1

119903)

2

) +

119884 + 119884

2+ 119884

3+ 119884

4

(1 minus 119884)

3

= (1476119879

minus1

119903minus 976119879

minus2

119903+ 458119879

minus3

119903) 119884

2

minus (907119879

minus1

119903minus 2422119879

minus2

119903+ 424119879

minus3

119903) 119884

(218+282119879minus1

119903

)

(25)

222 Equations of State for Liquid With 119879 lt 130

∘C thedensity of drilling mud is expressed as follows

120588

119898= 120588

0(1 + 4 times 10

minus10119901

119871minus 4 times 10

minus5119879 minus 3 times 10

minus6119879

2) (26)

With 119879 ge 130

∘C the density of drilling mud is expressed asfollows

120588

119898= 120588

0(1 + 4 times 10

minus10119901


minus5119879

minus3 times 10

minus6119879

2+ 04(

119879 minus 130

119879

)

2

)

(27)

223 Temperature Distribution Model The temperature ofthe drilling mud at any depth of the wellbore is [50]

119879 = 119879

119890119894+ 119865 [1 minus 119890

(119911119887ℎ

minus119911)119860] (minus

119892 sin 120579119892

119888119869119888

119901119898

+ 119873 + 119892

119879sin 120579)

+ 119890

(119911119887ℎ

minus119911)119860(119879

119891119887ℎminus 119879

119890119887ℎ)

(28)

224 GasDissolution Assuming gas goes into and comes outof solution instantaneously gas solubility can be obtained by(29)

119877

119904= 0021120574gs[(119901 + 01757) 10

(17688120574osminus0001638119879)]

1205

(29)


225 Oil Phase Volume Factor The volume factor is calcu-lated as follows

119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)

23 Flow Pattern Prediction Models The flow regime theflow pattern and structure of the flow are some of theimportant parameters to describe two-phase gas-quid flowsidentify two phase gas-quid flow regimes and calculatethe dynamic pressure wave propagation velocity Thus thetransition among the three main flow regimes (bubbly slugand annular) is desirable to be known [51]

In the hydraulic calculation of annulus the effectivediameter of annulus [52] should be given as

119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)

The effective roughness of annulus can be calculated by

119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)

At low gas flow rate the liquid is continuous phase andthe gas bubbles are dispersed in the liquid phase Studies ofTaitel [53] give the minimum diameter necessary to formbubbly flow as

119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)

The critical condition for forming bubbly flow is

V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)

For slug flow the critical balance superficial flow rate ofgas carrying droplets needs to meet the condition [54] that

[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)

For annular flow the pattern transition criterions [55] is

Vsg gt [Vsg]cr (38)

231 Bubbly Flow Gas void fraction of four-phase flow isdescribed as

120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)

The value of the distribution factor 119878119892can be determined

by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)

Harmathy [56] established the calculation formula ofgas slip velocity in bubbly flow based on the study of themigration velocity of the bubble in a stationary liquid as

V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)

The average density of four-phase mixture flow is

120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)

The oil void fraction for four-phase flow is

120601

119900=

(1 minus 120601

119892) Vso

119878

119900(Vso + Vsw + Vsm) + (1 minus 120601119892) V119900119903

(43)

The value of the distribution factor is 119878

119900= 105 +

0371(119863

119894119863

119900)

On the basis of the total liquid fluid establish the oil phasevelocity relationship as

V119900= 119878

119900V119871+ V119900119903 (44)

According to cross-section flow rate phase distributionand slip mechanism of liquid phase we can the draw thefollowing relationship

V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)

Water void fraction is

120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)

Drilling mud void fraction is

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)

Due to the similar physical properties of water anddrilling fluids V

119908 V119898 and V

119908119887can be expressed by

V119908= V119898= V119908119887 (49)


The coefficient of virtual mass force 119862vm for bubbly flowcan be expressed as follows

119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)

The coefficient of resistance coefficientCD for bubbly flowcan be expressed by

119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)

The friction pressure gradient for bubbly flow can beobtained from the following equation

120591

119891= 119891

120588

119871V2119871

2119863

(52)

232 Slug Flow Thevoid fraction of the four phasesΦ119892Φ119900

Φ

119908 and Φ

119898 can be determined by (39) (43) (47) and (48)

the same as bubbly flowThe value of the distribution factor 119878

119892for slug flow can be

described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)

For slug flow the slip velocity can be calculated as followsFrom experimental studies Hasan and Kabir [57] estab-

lished the calculation formula of drift velocity for slug flowon the basis of research on Taylor bubble migration rule ofDavies and Taylor as

Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)

The coefficient of virtual mass force 119862vm for slug flow canbe expressed as follows

119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)

The coefficient of resistance coefficient 119862119863for slug flow

can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)

233 Annular Flow As for annular flow due to the miscibleflow state of gas at center the simplification can be 119881gr = 0

The void fraction of gas can be determined by

120601

119892= (1 + 119883

08)

minus0378

(57)

where119883 is defined as

119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3

Bottom hole WellheadΔsi

i minus 1 i + 1 N minus1N minus 2 Ni

Figure 2 Computational cells for semi-implicit difference solution

Oil void fraction is

120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)

The same as slug flow 119862vm and 119862119863can be determined by

(55) and (56)

3 Solution of the Dynamic Model

Now since obtaining the analytic solution of the aforemen-tioned theoretical model directly is impossible discretizationof the model to a numerical model is required [58] In thispaper the mathematical methods based on finite differencemethod provide a numerical solution approach for thedynamic model As for the solution of the pressure wavevelocity model spatial domain includes the entire wellboreand the formation node time domain is the time periodinflux fluid flowing from the bottom hole to the wellheadalong the wellbore Discretizing the domain of determinacythe entire spatial and time domain can be divided intodiscrete networked systems

According to finite difference scheme the four equations(5)ndash(7) are solved by using the finite difference methodwith computational cells shown in Figure 2 the differenceequation systems of which described the basic principles offour-phase fluid motion in wellbore is presented as follows

For the drilling mud phase

(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters

Meet demand Delete two roots

Assume influx time nmin nmax

Assume p of bottom

Assume nod i and i + 1

Calculate parameters of nod (i n + 1)

i + 1 lt nmax

Assume p(i + 1 n + 1)

Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No

|120601g(i + 1 n + 1) minus 120601c(i + 1 n + 1)| lt 10minus3

|p(i + 1 n + 1) minus pc(i + 1 n + 1)| lt 10minus3

Obtain c

Is wellheadi = i + 1

|pc(i + 1 n + 1) minus BP| lt 120576

Solve equation (2) for 120588k

Solve equation (2) for g(i + 1 n + 1)

Solve equation (2) for 120601c(i + 1 n + 1)

Solve equation (2) for pc(i + 1 n + 1)

Solve determinant equation (20) for four roots

Figure 3 Solution procedures for pressure wave velocity in MPD operations

= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)

For momentum balance equation

(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)

Besides finite-difference method the characterizedmethod and the Newton-Raphson iterative method areadopted to solve the united dynamic model in four phasesalong annulus On the basis of discretization the solution ofmodels was realized by applying personally complied codeon VB NET and the solution procedure of that are shown inFigure 3

The four-phase flow system is described completely byeight variables including pressure temperature gas voidfraction and liquid void fraction gas and liquid densitiesand gas and liquid velocities There are still five unknownssuch as gas and liquid velocities gas fraction pressure andgas density based on the above assumptions Therefore fiveequations are required to compute the unknown variableswith boundary conditions At different annulus depth wecan obtain pressure temperature gas velocity oil velocitywater velocity drilling mud velocity and gas void fraction oilvoid fraction andwater void fraction duringMPDoperationsby application of the finite-difference At initial time thewellhead BP wellbore structure well depths wellhead tem-perature gas-oil-water-drilling mud properties and so forthare known On node 119894 flow parameters such as pressuretemperature and void fraction of gas-oil-water-drilling mudmultiphase can be obtained by adopting finite differenceThen the determinant equation (20) is calculated based onthe calculated parameters Omitting the two unreasonableroots the pressure wave at different depth of annulus inMPD

operations can be solved by (21) Upon substitution of theactual magnitudes V turned out to be the velocity of lightThe process is repeated until the pressure wave velocity in thewhole annulus has been obtained

Thepublished experimental data presented by Li et al [41]is used to verify the new developed united dynamicmodel InFigure 4 the points represent Lirsquos experimental data and thelines represent the calculation results From the comparisonsit can be concluded that the developed united dynamicmodelagrees well with the experimental data The averaged errorbetween model and data in Figure 4 is 2853

4 Analysis and Discussion

One of the potentially most useful capabilities of transientwell control model is the ability to simulate various chokecontrol procedures Typically the control of choke to main-tain constant BHP depends on the experience and training ofthe hand on the chokeThe gas and liquid flow rate measuredby Coriolis meter and the BP measured by pressure senoris taken as the initial data for annulus pressure calculationThe well used for calculation is a gas well in Sichuan RegionChinaThewellbore structure of thiswell is shown in Figure 5and information about gas-oil-water-drillingmud propertieswell design parameters and operational conditions of calcu-lation well are listed in Table 1

In the following example the drilling mud mixed withgas oil and water is taken as a four-phase flowmediumwhenthe influx of gas oil and water occurs at the bottom of thewell In the calculation the BP is 02MPa and flow rate is119876

119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=

720m3h respectively The propagation velocity of pressurewave in the four-phase flow is calculated and discussed byusing the established model and well parameters

41 Effect of Well Depth on the Pressure Wave Velocity Withbottom hole pressure being constant the effect of well depthon pressure wave velocity is analyzed When gas-oil-waterinflux fluid invades into the wellbore at the bottom hole


Table 1 Parameters of the calculation well

Type Property Value

Mud Density (kgm3) 1460Viscosity (Pasdots) 20 times 10

minus3

Gas Relative density 065Viscosity (Pasdots) 114 times 10

minus5

Oil Density (kgm3) 800Viscosity (Pasdots) 40 times 10

minus3

Water Density (kgm3) 1000Viscosity (Pasdots) 10 times 10

minus3

StringElastic modulus of string (Pa) 207 times 10

11

Poisson ratio of string 03Roughness (m) 154 times 10

minus7

Surfacecondition

Surface temperature (K) 298Atmosphere pressure (MPa) 0101

0 5 10 15 200

500

1000

1500

2000

120601 = 001 this paper drilling mud120601 = 002 this paper drilling mud120601 = 004 this paper drilling mud120601 = 001 Li (1997)120601 = 002 Li (1997)120601 = 004 Li (1997)

c(m

s)

p (MPa)

w = 50Hz Cvm = Re

Figure 4 Comparison between calculation results of new dynamicmodel and the experimental data of Li

gas oil water and drilling mud four-phase flow appearsin the annulus and migrates from the bottom of the well(119867 = 4000m) to the wellhead along the flow direction Itis assumed that the initial length of the gas pillar is 500munder the bottom hole pressure and temperature conditionin the analysis of the migration of influx gas The migrationand change of gas column including the solubility and lengthchange lead to the variation of pressure wave velocity inthe annulus The formation and development of influx are adynamic process and the flow of fluid in the annulus is verycomplicated The flow condition and flow parameter at anytime are various

RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm

1206011778mm (12060178mm)Drill string

Figure 5 Structural parameters of the calculation well

Figure 6 illustrates the distribution of pressure wavevelocity at different well depth after 516 minutes and 1871minutesrsquo development of influx It is shown that the top ofaerated mud column arrives at the position of 119867 = 3000mand 119867 = 2000m respectively Due to the circulation ofdrilling mud and slip of gas phase the length of aeratedmud column that represents the rising height of gas willincrease with the gradual decrease of pressure along theannulus Besides in a segment of aerated mud columnthe gas void fraction of the top is higher than that of thebottom for the slop of gas phase So in the declining stagepressure wave velocity in aerated mud column presents anincreasing tendency along the flow direction As the gasmigrates upwards the casing pressure should be changeddynamically to keep the bottomhole pressure constant whichwill lead to an increase of annular pressure and back pressureAs the influx develops for 1871minutes the back pressurewillbe prompted to 239MPa increased by 029MPa comparedwith that of 516 minutes

42 Effect of BP on Pressure Wave Velocity In Figures 7 and8 the three-dimensional figures illustrate the distributionsof gas void fraction and variations of pressure wave velocityalong the flow direction in the annulus at different time whenBP is 01MPa 30MPa 60MPa and 90MPa respectivelyWhen influx occurs increase of applied BP is of the mostcommonly used approach for bottom hole pressure controlControl of back pressure has a great influence on the gasvoid fraction In the two graphs the two full lines give theshape of the void fraction and pressure wave velocity atdifferent back pressure In each graph the two clusters ofdotted lines at the two-dimensional plane are the projectionof the full lines which respectively represent the influenceof well depth and time on the distributions of gas void


460 690 920 1150 13804000

3000

2000

1000

0

L = 500m

H = 2000m T = 1871min BP = 239MPaH = 3500m T = 516min BP = 218MPa

H(m

)

c (ms)

Figure 6 Wave velocity variations at different depth

fraction or variations of pressure wave velocity at differentback pressure Increasing the casing pressure will make thecompressibility decrease obviously and energy dissipationwith the propagation of pressure wave decrease as well Thusit can be seen that with the reduction of back pressure voidfraction shows an increasing trend and that the pressure wavevelocity has a decrease at different depth of well Meanwhileat the high temperature and high pressure bottom holepressure wave velocity change in a comparatively smoothlevel in a consequence of compressibility of the four-phasefluid changing only in a small degree As a result as aerateddrillingmudmigrating from the bottom hole to the wellheadgas void fraction significantly increases first slightly and thensharply along the flow direction a drop of annular pressureConversely the pressure wave velocity shows a gradual fistand then remarkable decreasing tendency after the influx gasreaches the position close to the wellhead Also with thedevelopment of influx the influence of BP on the gas voidfraction and pressure wave velocity is larger over time

43 Effect of Gas Influx Rate on Pressure Wave VelocityFigures 9 and 10 graphically interpret the effect of gas influxrate on the distribution of gas void fraction and variations ofpressurewave velocity along the flowdirection in the annulusThe same as Figures 7 and 8 the two clusters of dottedlines at the two-dimensional plane are the projection of thefull lines The pressure wave velocity at different well depthis the transient value with respect to time It is noted thatthe changes of pressure wave velocity are obvious with themigration of gas which is extremely significant at wellheadeven at low gas influx rate With the decreasing of pressure atthe position near the wellhead rapid expansion of gas volumeappears and as a result the gas void fraction increases sharplyand the pressure wave velocity decrease obviously at the

60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(

)Figure 7 Effect of BP on the gas void fraction

BP = 01MPaBP = 30MPa


840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)

Figure 8 Effect of BP on the pressure wave velocity

same time However the gas void fraction and pressure wavevelocity have no obvious change at the bottom of well dueto the high pressure at the bottom of well and insignificantcompressibility of gas When the mainly reflect fundamentalfactors initial gas influx rate is increased pressure wavevelocity shows an obvious descending trend with the increaseof initial gas influx rate the value of pressure wave velocity atthe position close to the wellhead is much greater than that atbottom hole At the gas void fraction range of about 5sim80as gasmigrating along the annulus the pressure wave velocity


43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)

Figure 9 Gas void fraction distribution at different gas influx rate

0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)

Figure 10 Wave velocity variations at different gas influx rate

is decreased to a minimum at the wellhead with the gas voidfraction increasing to a maximum

44 Effect of Oil Influx Rate on Pressure Wave VelocityFigure 11 shows the effect of oil influx on the gas void fractionin gas-oil-water-drilling mud four-phase flow The increaseof oil influx rate has little influence on the gas void fractiondistribution which is because gas void fraction is mainlyimpacted by pressure temperature and gas influx rateWhenthe oil influx rate increased from 119876

119892= 036m3h to 119876

119892=

360m3h the change of mixture density is inconspicuous

60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(

)Figure 11 Gas void fraction distribution at different oil influx rates

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)

Figure 12 Wave velocity variations at different oil influx rates

compared with drilling mudTherefore the gas void fractionkeeps almost unchanged Similarly as shown in Figure 12with the oil influx rate increase the change of pressure wavevelocity can be neglected due to the little influence of oil influxrate on the gas void fraction Also when the water influxinvades the wellbore the influx water will dissolve in drillingmud and reduce the density of drillingmud as thewater phaseand water base drilling mud have similar physical propertiesBut the influence of changing water influx rate on pressurewave velocity is too little to be neglected as the oil influx ratedoes


H (m) T(min)

c(m

s)

w = 50Hz Cvm = Rew = 50Hz Cvm = 0

w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

Figure 13 Pressure wave velocity variations at different angularfrequency

45 Effect of Disturbance Angular Frequency on PressureWaveVelocity Figure 13 provides an illustration of the influence ofdisturbance angular frequency on the pressure wave velocitywhen gas oil and water influx generates It is indicated thatan increase in angular frequency results in a rise of pressurewave velocity In addition the influence on dynamic pressurewave velocity is enhanced with the continuation of timeand the decrease of well depth as the gas migrates from thebottom hole to the wellhead Accordingly the disturbanceangular frequency has a significantly distinct influence onthe pressure wave velocity at the position near the wellheaddue to the phenomenon that sharply increases of gas voidfraction For an increase in angular frequency from 1500Hzto 10000Hz the influence on wave velocity is not as great asthe increase form 50Hz to 1500HzThis is consistent with thefact that the effect of angular frequency on the pressure waveappears at a low angular frequency

5 Conclusions

Solution of dynamic pressure wave velocity in gas-oil-water-drilling mud four-phase flow is a lot more complicated thanthat in two-phase flow The dynamic model which takesconsideration of gas solubility oil compression coefficientsperturbation frequency virtual mass force and drag forceis established Owing to the solution of the migration ofdrilling mud which contains influx fluid (gas water and oil)the dynamic pressure wave varied with time and well depthis solved and the main conclusions can be summarized asfollows

(1)Agreeingwell with the previous experimental data thedeveloped untied dynamicmodel can be used to calculate thewave velocity in the annulus The model can accurately givethe dynamic at different depth and time during themigration

of influx which will be beneficial to provide a reference forwell control and MWD

(2) In the dynamic migration process the pressure wavevelocities at different well depth are changing constantly dueto the circulation of drilling mud and slip of gas phase Thelength of aerated mud column which represents the risingheight of gas increases with the gradually decreasing pressurealong the annulus In the segment of aerated mud columnpressure wave velocity presents an increasing tendency alongthe flow direction

(3) The flow condition and flow parameter at any timeare various Both the static pressure and void fraction distri-bution along the annulus changed with the position in theannulus and time and have direct impact on the pressurewave velocityThe pressure wave velocity happens to decreaseafter an elapsed period of time in which influx fluid migratesalong the annulus The decreasing tendency will last untilthe aerated mud column pass through that point of annuluscompletely Also the time required is gradually shortenedduring process of influx migration and growth

(4) At the bottom hole the void of gas water and oilis very low relative to drilling mud With the migrationof influx fluid the variation of gas void fraction is muchgreater than other phases in particular a sharp increase ofgas void fraction presents at the position near the wellheadfor the rapid expansion of gas volume while the voidfractions of water oil and drilling mud are decreased for thenearly negligible variations of physical properties and drasticexpansion of gas at the position near the wellhead

(5) When the BP at the wellhead increases gas voidfraction at any position of annulus decreases instead thepressure wave velocity increases Moreover the influenceof BP on the gas void fraction is larger at position closeto the wellhead than that at the bottom of well With thedevelopment of influx the influence of BP on the gas voidfraction and pressure wave velocity is larger over time

(6)Thepressure wave velocity is sensitive to the gas influxrate in particular at the position close to the wellhead Evenat low range the rise of gas influx rate also will result in asignificantly downward trend of pressure wave velocity Onthe contrary the influence of changing water influx rate andoil influx rate on pressure wave velocity is too little to neglect

(7) Ignoring virtual mass force the pressure wave velocityhas an obvious dispersion characteristic at different positionof well The effect of angular frequency on the pressure waveappears at a low angular frequency and the pressure wavevelocity increases together with the growth of the angularfrequency

Nomenclature

119860 Annulus effective cross area (m2)119861

119900 Oil phase volume factor

119888

119892 Pressure wave velocity in gas (ms)

119888

119871 Pressure wave velocity in liquid phase

(ms)119862

119863 The coefficient of drag force

119863 Annulus effective diameter (m)


119863

119894 Outer diameter of the inner pipe (m)

119863

119900 Diameter of the wellbore (m)

119863min The minimum diameter to bubbly flow(m)

119891

119866 Shear stresses coefficient of gas interface

119891

119871 Shear stresses coefficient of liquid phase

interfacefr Coefficient of friction resistance119865

119911 Pressure applied on the two-phase flow

(N)119892 Acceleration due to gravity (m2s)119892

119888 Conversion factor

119892

119879 Geothermal temperature gradient (Km)

ℎ One discrete length of annulus (m)119896

119890 Annulus effective roughness (m)

119896

0 Roughness of outer pipe (m)

119896

119894 Roughness of inner pipe (m)

119867 The depth of annulus (m)119871

119902 Air bubbles length of slug (m)

119872gi Momentum transfer in gas interface(Nm3)

119872Li Momentum transfer in liquid interface(Nm3)

119872

ndLi The nondrag force (virtual mass force)

(Nm3)119872

119889

Li The momentum transfer term (Nm3)119873 Parameter combiningThompson energy

effects119901gi Gas interface pressure (MPa)119901Li Liquid interface pressure (MPa)119901 Pressure (MPa)119901

0 Initial pressure of annulus (MPa)

119901

119892 Gas pressure (MPa)

119901

119888 Critical pressure (kPa)

119901

119871 Pressure of liquid phase (MPa)

119901

119903 Reduced pressure

119902

119898 Velocity of drilling mud (m3s)

119902

119892 Velocity of gas phase (m3s)

119902

119871 Velocity of liquid phase (m3s)

119876

119871 Velocity of liquid phase within the

annulus (m3h)119903 Average diameter of the bubble (m)119903os Standard oil density (kgm3)119903gs Standard gas density (kgm3)119877 Constant of EOS (JKgK)119877

119902 Air bubbles radius of slug (m)

119877

119887 The bubble diameter (m)

119878

119892 The distribution coefficient of gas velocity

119879 Temperature (K)119905 Time(s)119879

119888 Critical temperature (K)

119879

119903 Reduced temperature

119879

119890119894 Undisturbed formation temperature at an

depth (K)119879

119890119887ℎ Undisturbed formation temperature of

wellhead (K)119879

119891119887ℎ Undisturbed temperature at the bottomhole (K)

V119898 Gas and drilling mud flow velocity (ms)

V119908 Water mud flow velocity (ms)

V119900 Oil flow velocity (ms)

V119892 Gas flow velocity (ms)

V119892119903 The gas phase drift velocity

V119904 Slip velocity (ms)

119881

119872cr Critical flow rate of miscible phase (ms)Vsg Superficial gas velocity (ms)V119871 Liquid phase flow velocity (ms)

Vso Superficial oil velocity (ms)Vsm Superficial drilling mud velocity (ms)[Vsg]cr Critical balance superficial flow rate of gas

(ms)119882

119898 Mass flow velocity (kgm3)

119911

0 Initial calculation height of annulus (m)

119911

119887ℎ Total well depth from surface (m)

Greek Letters120588

119892 Gas density (kgm3)

120588

119903 Reduced density

120588

0 Density under atmospheric pressure (kgm3)

120588

119900 Oil density (kgm3)

120588

119898 Drilling mud density (kgm3)

120588

119908 Water density (kgm3)

120588

119871 Liquid density (kgm3)

120588

119872 Mixture density (kgm3)

120591

frLi Shear stresses of noninterfacial mud (Nm2)

120591

fr119871 Shear stresses of liquid phase interface (Nm2)

120591

Re119871 Reynolds stress of liquid phase (Nm2)

120591

ReLi Reynolds stress of liquid phase interface (Nm2)120591

Re119892 Reynolds stress of gas interface (Nm2)

120591

fr119892 Shear stresses of gas interface (Nm2)

120572VM Acceleration virtual force coefficient (ms2)120591

119871 Shear stresses of liquid phase along wall (Nm2)

120591

119892 Shear stresses of gas along wall (Nm2)

120591

Re119892 Reynolds stress of gas (Nm2)

120591

119891 Frictional pressure gradient (Pam)

120591

119908 Frictional pressure coefficient

120588

119871 Liquid phase density (kgm3)

120590

119904 Surface tension (Nm2)

120590wo Surface tension of water phase and oil phase (Nm2)Φ

119892 Gas void fraction

Φ

119898 Drilling mud void fraction

Φ

119896 119870 phase void fraction

Φ

119900 Oil void fraction

Φ

119908 Water void fraction

Φ

119871 Liquid void fraction

120583

119871 Viscosity of liquid phase (Pasdots)

120594

119900119896 Oil mass fraction

120594

119892119896 Gas mass fraction

120594

119908119896 Water mass fraction

120594

119898119896 Drilling mass fraction


120574os The relative density of oil phase120574gs The relative density of gas phase

SubscriptsEOS Equations of stateMPD Managed pressure drillingRCD The rotating control deviceDWC Dynamic well control systemCPC Conventional pressure control systemIPC Industrial Personal Computer

Subscripts of GraphBP Back pressure (MPa)119888 Pressure wave velocity (ms)119862vm The coefficient of virtual mass force119867 Depth of annulus (m)119876 Gas influx rate at the bottom hole (m3h)Re The value of virtual mass force coefficient119908 Angle frequency (Hz)Φ

119892 Void fraction ()

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research work was cofinanced by the National Nat-ural Science Foundation of China (no 51274170) andImportant National Science amp Technology Specific Projects(2011ZX05022-005-005HZ)Without their support this workwould not have been possible

References

[1] W A Bacon Consideration of compressibility effects for applied-backpressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Tex USA 2011

[2] S Saeed and R Lovorn ldquoAutomated drilling systems for MPD-the realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012

[3] A Malekpour and B W Karney ldquoRapid filling analysis ofpipelines with undulating profiles by the method of character-isticsrdquo ISRN Applied Mathematics vol 2011 Article ID 93046016 pages 2011

[4] H Santos C Leuchtenberg and S Shayegi ldquoMicro-flux controlthe next generation in drilling processrdquo inProceedings of the SPELatin and Caribbean Petroleum Engineering Conference PaperSPE 81183 Port-of-Spain Trinidad West Indies 2003

[5] H Guner Simulation study of emerging well control methodsfor influxes caused by bottomhole pressure fluctuations duringmanaged pressure drilling Louisiana State University BatonRouge La USA 2009

[6] G M D Oliveira A T Franco C O R Negrao A LMartins and R A Silva ldquoModeling and validation of pressurepropagation in drilling fluids pumped into a closed wellrdquo

Journal of Petroleum Science and Engineering vol 103 pp 61ndash71 2013

[7] I Zubizarreta ldquoPore pressure evolution core damage and trip-ping out schedulesf a computational fluid dynamics approachrdquoin Proceedings of the SPEIADC Drilling Conference and Exhibi-tion Amsterdam The Netherlands 2013

[8] J I Hage and D ter Avest ldquoBorehole acoustics applied to kickdetectionrdquo Journal of Petroleum Science and Engineering vol 12no 2 pp 157ndash166 1994

[9] P Thodi M McQueen M Paulin G Lanan and INTECSEAldquoTheory and Application of Probabilistic Method of DesigningCustomized Interval Control Valves Choke Trim for MultizoneIntelligent Well Systemsrdquo 2007

[10] A Mallock ldquoThe damping of sound by frothy liquidsrdquo Pro-ceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 84 no 572 pp 391ndash395 1910

[11] A BWoodATextbook of Sound G Bill and Sons London UK1941

[12] E L Carstensen and L L Foldy ldquoPropagation of sound througha liquid containing bubblesrdquoTheJournal of theAcoustical Societyof America vol 19 no 3 pp 481ndash501 1941

[13] R A Thuraisingham ldquoSound speed in bubbly water at mega-hertz frequenciesrdquo Ultrasonics vol 36 no 6 pp 767ndash773 1998

[14] D Hsieh and M S Plesset ldquoOn the propagation of sound in aliquid containing gas bubblesrdquo Physics of Fluids vol 4 no 8 pp970ndash975 1961

[15] J D Murray ldquoNote on the propagation of disturbances in aliquid containing gas bubblesrdquo Applied Scientific Research vol13 no 1 pp 281ndash290 1964

[16] G B WallisOne Dimensional Two Phase Flow vol 7 McGraw-Hill New York NY USA 1969

[17] F J Moody ldquoA pressure pulse model for two-phase critical flowand sonic velocityrdquo Journal of Heat Transfer vol 91 no 3 pp371ndash384 1969

[18] D F DrsquoArcy ldquoOn acoustic propagation and critical mass flux intwo-phase flowrdquo Journal of Heat Transfer vol 93 no 4 pp 413ndash421 1971

[19] D McWilliam and R K Duggins ldquoSpeed of sound in a bubblyliquidsrdquo Proceedings of the Institution ofMechanical Engineers Cvol 184 no 3 pp 102ndash107 1969

[20] R E Henry M A Grolmes and H K Fauske ldquoPressure-pulsepropagation in two-phase one- and two-component mixturesrdquoReactor Analysis and Safety Division Argonne National Labo-ratory 1971

[21] C SMartin andM Padmanabhan ldquoPressure pulse propagationin two-component slug flowrdquo Journal of Fluids Engineering vol101 no 1 pp 44ndash52 1979

[22] Y Mori K Hijikata and A Komine ldquoPropagation of pressurewaves in two-phase flowrdquo International Journal of MultiphaseFlow vol 2 no 2 pp 139ndash152 1975

[23] Y Mori K Huikata and T Ohmori ldquoPropagation of a pressurewave in two-phase flow with very high void fractionrdquo Interna-tional Journal of Multiphase Flow vol 2 no 4 pp 453ndash4641976

[24] D L Nguyen E R F Winter and M Greiner ldquoSonic velocityin two-phase systemsrdquo International Journal ofMultiphase Flowvol 7 no 3 pp 311ndash320 1981

[25] R C Mecredy and L J Hamilton ldquoThe effects of nonequilib-rium heat mass and momentum transfer on two-phase soundspeedrdquo International Journal of Heat and Mass Transfer vol 15no 1 pp 61ndash72 1972


[26] E E Michaelides and K L Zissis ldquoVelocity of sound in two-phase mixturesrdquo International Journal of Heat and Fluid Flowvol 4 no 2 pp 79ndash84 1983

[27] H W Zhang Z P Liu D Zhang and Y H Wu ldquoStudy onthe sound velocity in an aerated flowrdquo Journal of HydraulicEngineering vol 44 no 9 pp 1015ndash1022 2013

[28] L Y Cheng D A Drew and R T Lahey Jr ldquoAn analysis ofwave propagation in bubbly two-component two-phase flowrdquoJournal of Heat Transfer vol 107 no 2 pp 402ndash408 1985

[29] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoAn investigation of the propagation of pressure perturbationsin bubbly airwater flowsrdquo Journal of Heat Transfer vol 110 no2 pp 494ndash499 1988

[30] A E Ruggles R T Lahey Jr D A Drew and H A ScartonldquoThe Relationship between standing waves pressure pulsepropagation and critical flow rate in two-phase mixturesrdquoJournal of Heat Transfer vol 111 no 2 pp 467ndash473 1989

[31] N M Chung W K Lin B S Pei and Y Y Hsu ldquoA model forsound velocity in a two-phase air-water bubbly flowrdquo NuclearTechnology vol 99 no 1 pp 80ndash89 1992

[32] S Lee K Chang and K Kim ldquoPressure wave speeds fromthe characteristics of two fluids two-phase hyperbolic equationsystemrdquo International Journal of Multiphase Flow vol 24 no 5pp 855ndash866 1998

[33] J F Zhao and W Li ldquoSonic velocity in gas-liquid two-phaseflowsrdquo Journal of Basic Science and Engineering vol 7 no 3 pp321ndash325 1999

[34] L Liu Y Wang and F Zhou ldquoPropagation speed of pressurewave in gas-liquid two-phase flowrdquo Chinese Journal of AppliedMechanics vol 16 no 3 pp 22ndash27 1999

[35] J L Xu and T K Chen ldquoAcoustic wave prediction in flowingsteam-water two-phase mixturerdquo International Journal of Heatand Mass Transfer vol 43 no 7 pp 1079ndash1088 2000

[36] C E Brennen Fundamentals of Multiphase Flows CambridgeUniversity Press 2005

[37] G Yeom and K Chang ldquoThe wave characteristics of two-phaseflows predicted byHLL scheme using interfacial friction termsrdquoNumerical Heat Transfer A Applications vol 58 no 5 pp 356ndash384 2010

[38] Y Li C Li E Chen and Y Ying ldquoPressure wave propaga-tion characteristics in a two-phase flow pipeline for liquid-propellant rocketrdquoAerospace Science and Technology vol 15 no6 pp 453ndash464 2011

[39] W Kuczynski ldquoCharacterization of pressure-wave propagationduring the condensation of R404A and R134a refrigerantsin pipe mini-channels that undergo periodic hydrodynamicdisturbancesrdquo International Journal of Heat and Fluid Flow vol40 pp 135ndash150 2013

[40] Z S Fu ldquoThe gas cut detect technology during drilling in SovietUnionrdquo Petroleum Drilling Techniques vol 18 no 1 pp 19ndash211990

[41] X F Li C X Guan and X X Sui ldquoThe theory of gas influxdetection of pressure wave and its applicationrdquo Acta PetroleiSinica vol 18 no 3 pp 128ndash133 1997

[42] X Wang ldquoTransmission velocity model in frequency domainof drilling-fluid consecutive pulse signalrdquo Journal of SouthwestPetroleum University vol 31 no 3 pp 138ndash141 2009

[43] M Davoudi J R Smith B Patel and J E Chirinos ldquoEvaluationof alternative initial responses to kicks taken during managedpressure drillingrdquo in Proceedings of the IADCSPE DrillingConference and Exhibition SPE-128424-MS New Orleans LaUSA February 2010

[44] H Li G Li Y Meng G Shu K Zhu and X Xu ldquoAttenuationlaw of MWD pulses in aerated drillingrdquo Petroleum Explorationand Development vol 39 no 2 pp 250ndash255 2012

[45] L Xiushan ldquoMultiphase simulation technique of drilling fluidpulse transmission along well borerdquo Acta Petrolei Sinica vol 27no 4 pp 115ndash118 2006

[46] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005

[47] J-W Park D A Drew and R T Lahey Jr ldquoThe analysisof void wave propagation in adiabatic monodispersed bubblytwo-phase flows using an ensemble-averaged two-fluid modelrdquoInternational Journal ofMultiphase Flow vol 24 no 7 pp 1205ndash1244 1998

[48] G B Wallis One Dimensional Two Phase Flow vol 7 McGrawHill New York NY USA 1969

[49] L Yarborough and K R Hall ldquoHow to solve equation of statefor z-factorsrdquoOil and Gas Journal vol 72 no 7 pp 86ndash88 1974

[50] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86ndash87 pp 127ndash136 2012

[51] M Al-Oufi Fahd An investigation of gas void fraction andtransition conditions for two-phase flow in an annular gap bubblecolumn [PhD thesis] Loughborough University 2011

[52] M J Sanchez Comparison of correlations for predicting pressurelosses in vertical multiphase annular flow [MS thesis] TheUniversity of Tulsa 1972

[53] Y Taitel D Bornea and A E Dukler ldquoModelling flow patterntransitions for steady upward gas-liquid flow in vertical tubesrdquoAIChE Journal vol 26 no 3 pp 345ndash354 1980

[54] G Costigan and P B Whalley ldquoSlug flow regime identificationfrom dynamic void fraction measurements in vertical air-waterflowsrdquo International Journal of Multiphase Flow vol 23 no 2pp 263ndash282 1997

[55] V Casariego and A T Bourgoyne ldquoGeneration migration andtransportation of gas-contaminated regions of drilling fluidrdquoin Proceedings of the SPE Annual Technical Conference andExhibition Houston Tex USA October 1988

[56] T Z Harmathy ldquoVelocity of large drops and bubbles in mediaof re-stricted or infinite extentrdquo AIChE Journal vol 6 no 2 pp281ndash288 1960

[57] A R Hasan and C S Kabir ldquoPredicting multiphase flowbehavior in a deviated wellrdquo SPE Production Engineering vol3 no 4 pp 474ndash482 1988

[58] X W Kong Y H Lin and Y J Qiu ldquoA new method forpredicting the position of gas influx based on PRP in drillingoperationsrdquo Journal of Applied Mathematics vol 2014 ArticleID 969465 12 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


problem in deep drilling is the propagation of measurement-while-drilling (MWD) mud pulse transmitting real-timevarious data from sensors located down hole near the drillbit Hence the propagation behavior of pressure wave isconsidered to provide reference for the MPD operations

However we also noticed that the conventional theoriespresent are difficult to be employed in a systematic andaccurate prediction when influxes generates As the influxesfluid including gas oil and water will lead to variationsof physic characteristics parameter of fluid in the annulusthe distribution of pressure wave velocity in gas oil waterand drilling mud four-phase flow along the annulus will bedynamic changing with respect to time and well depthTheseeffects may influence the safe operation of devices

Pressure waves are disturbances that transmit energy andmomentum along the wellbore through drilling fluid withoutsignificant displacement of matter As migration of dispersedgas water and oil towards wellhead is quite complicated thefundamental characteristics of the four-phase flow are stillunknown and modeling results of pressure wave velocity arequestionable The proper treatment of propagation behaviorof pressure wave in the four-phase flow in the annulusrequires knowledge of description of influxes generationdevelopment and pressure wave propagationmodel in a two-phase mixture

In gasliquid two-phase flow two-phase medium inter-action greatly changes the structure characteristic of flow-ing fluid which results in greater compressibility of two-component flow than single-phase gas or liquid and fur-ther causes pressure waves propagating speed to be greatlyreduced This can be explained from the viewpoints ofmixture density and compressibility of two-phase fluid andthe pressure drop along the flow direction in the wellboreIn the low void fraction range the gas phase is dispersedin the liquid as bubble so the wave velocity is influencedgreatly by the added gas phase According to the EOS if gasinvades into the wellbore with a small amount in the bottomhole the density of the drilling mud has little variationwhile the compressibility increases obviously which makesthe wave velocity be decreased This phenomenon was firstproposed by Mallock [10] and attracts much attention for itsimportant role in the development of science and technologyapplications Extensive investigations involved in the issueof pressure wave propagation have been taken and somesignificant achievements associated with the theories havealso been made in the researches In early stage researchingWood [11] extended the researches of Mallock and presenteda succinct formula by assuming that the compressibility oftwo-component fluid is a function related to single-phasecompressibility and elastic modulus E Carstensen and Flody[12] measured the velocity of pressure wave under a lake Adispersion relation for pressure waves propagating through abubbly fluid was derived by using a linear scattering theorydeveloped by Foldy Under the hypothesis of homogeneousand adiabatic laminar flow Thuraisingham [13] took thetwo-phase media as a homogeneous fluid and derived thesolution model concerning the problem of wave velocitymodel in two-phase flow at low gas void fraction accordingto the analysis of state equation of mixture According to the

early stage investigations Hsieh and Plesset [14] and Murray[15] researched the influence of thermal conductivity andviscosity coefficient on pressure wave propagation Wallis[16] firstly studied the propagation mechanism of pressurewave and derived the propagation velocity in bubbly flow andseparated flow using the homogeneous model The proposedmodel is based on the respective compressibility of the vaporand the liquid Also the two-phase mixture is treated asa compressible fluid with suitably averaged properties Inthat expression mass and heat transfers are neglected soit can be applied to any gasliquid mixture in so far as nomajor effect due to vaporization or condensation must betaken into account It may not be valid for such complexmixtures that never reach equilibrium Assuming that noevaporation or condensation occurs when pressure wave istransporting Moody [17] developed a simple acoustic wavemodel for bubbly flow and annular flow and established arelationship between sonic velocity and two-phase criticalflow Similar model was developed by DrsquoArcy [18] In theresearches of Mcwilliam and Duggins [19] surface tensionand compressibility of liquid phase were also consideredHenry et al [20] calculated the velocity as a function ofvoid fraction using a correlation to account for the change inbubble shape with void fraction Martin and Padmanabhan[21] extended the simple model proposed by Henry byconsidering wave reflection and wave transmission at gas-liquid interfaces Researches ofMori et al [22 23] suggest thatimpact pipe elasticity on pressure wave propagation velocityis limited to the range of gas void of less than 1 At highgas void range the pressure wave velocity is in between thetwo velocities of single-phase Nguyen et al [24] proposedanother type of model for bubbly flow (diluted gas phase inthe liquid) The simple relations for prediction of the propa-gation of pressure disturbances in liquid-gaseous two-phasesystems are presented The model makes use of the well-known physical behavior that the wave velocity in a single-phase fluid is influenced by the elasticity of the confiningwalls The interface of the one phase is considered to act asthe elastic wall of the other phase and vice versa Mecredyand Hamilton [25] used the two-fluid model to predict thepressure wave propagation in vapor-liquid flow in detailHowever the analysis contained the important assumptionthat the evaporation or condensationwas governed by kinetictheory Michaelides and Zissis [26] developed a computa-tional method which yields the sound velocity in terms ofthe thermodynamic coordinates of the substance withoutthe use of diagrams Corresponding velocities of sound forthe four substances considered exhibit a certain similaritywhich is examined statistically The relationship between thesound velocity and the critical mass flux is also investigatedThuraisingham [13] studied the wave velocity in bubbly waterat megahertz frequencies (1sim10MHZ) Numerical analyticalresults indicate that volume concentrations and the radius ofthe bubble relative to the incident wavelength of sound arethe important parameters which determine the deviation ofsound speed form that of bubble-free water

Currently the two-fluid continuum model is the mostcommon and reliable method to describe the gasliquid two-phase flow phenomenon [27] In the model the governing














Liquid phase

Gas phase

OrOr

p

p + Δp

Δs





119860

120597

120597119905

(120594

119900119896120588

119900120601

119900+ 120594

119892119896120588

119892120601

119892+ 120594

119908119894120588

119908120601

119908+ 120594

119898119894120588

119898120601

119898)

+

120597

120597119904

(120594

119900119896119860120588

119900120601

119900V119900+ 120594

119892119896119860120588

119892120601

119892V119892

+120594

119908119894119860120588

119908120601

119908V119908+ 120594

119898119894119860120588

119898120601

119898V119898) = 0

(1)


120597

120597119905

(119860sum

119896

120588

119896120601

119896V119896) +

120597

120597119904

(119860sum

119896

120588

119896120601

119896V2119896)

+ 119860119892sum

119896

120588

119896120601

119896+

120597

120597119904

(119860119901) + 119860(

120597119901

120597119904

)

fr= 0

(2)


Hence

sum

119894

120594

119900119894= 1 sum

119894

120594

119892119894= 1

sum

119894

120594

119908119894= 1 sum

119894

120594

119898119894= 1

(3)

where

120594

119908119900= 0 120594

119908119892= 0 120594

119908119908= 1 120594

119908119898= 0

120594

119898119900= 0 120594

119898119892= 0 120594mw = 0 120594

119898119898= 1

120594

119892119900= 0 120594

119892119892= 1 120594

119892119908= 0 120594

119892119898= 0

120594

119900119900=

119898

119900

119898

119900+ 119898

119892

120594

119900119892=

119898

119892

119898

119900+ 119898

119892

120594

119900119908= 0 120594

119900119898= 0

(4)




120597

120597119905

(120601

119892120588

119892) +

120597

120597119904

(120601

119892120588

119892V119892) = 0

120597

120597119905

(120601

119871120588

119871) +

120597

120597119904

(120601

119871120588

119871V119871) = 0

(5)

Hence 120588119871= 120601

119900120588

119900+ 120601

119908120588

119908+ 120601

119898120588

119898


120597

120597119904

(120601

119892120588

119892V119892) +

120597

120597119904

(120601

119892120588

119892V2119892)

= minus

120597

120597119904

(120601

119892120588

119892) +

120597

120597119904

[120601

119892(120591

fr119892+ 120591

Re119892)] +119872gi minus 4

120591

119892

119863

(6)


120597

120597119905

(120601

119871120588

119871V119871) +

120597

120597119904

(120601

119871120588

119871V2119871)

= minus

120597

120597119904

(120601

119871120588

119871) +

120597

120597119904

[120601

119871(120591

fr119871+ 120591

Re119871)] +119872Li minus 4

120591

119871

119863

(7)


119872gi = minus119872

ndLi minus119872

119889

Li + (120591frLi + 120591

ReLi )

120597120601

119871

120597119904

+

120597 (120601

119892120590

119904)

120597119904

+

120597 (120601

119892119901gi)

120597119904

minus 120601

119892

120597 (119901Li)

120597119904

119872Li = 119872

ndLi +119872

119889

Li + 119901Li120597 (120601

119871)

120597119904

minus (120591

frLi + 120591

ReLi )

120597120601

119871

120597119904

(8)


119872

ndLi = 119888VM120601119892120588119871120572VM minus 01120601

119892120588

119871V119904

120597V119904

120597119904

minus 119888

1198981120588

119871V2119904

120597120601

119892

120597119904

(9)



119872

119889

Li =3

8

119862

119863

119903

120588

119871119877

119902V2119904 (10)


119901Li minus 119901119871 = minus119888

119901120588

119871V2119904 (11)

where 119888119901= 025


119901gi minus 119901119892 asymp 0 (12)


119901

119871= 119901 minus 025120588

119871120601

119892V2119904 (13)


120591

fr119892asymp 120591

frLi asymp 120591

fr119871asymp 120591

119892asymp 120591

Re119892

asymp 0 (14)


120591

Re119871

= minus119888

119903120588

119871V2119904

120601

119892

120601

119871

120591

ReLi = minus119888

119903120588

119871V2119904

(15)

Hence 119888119903= 02


120591

119871= 05119891

119871120588

119871V2119871 (16)



119889119901

119871

119889120588

119871

= 119888

2

119871

119889119901

119892

119889120588

119892

= 119888

2

119892

(17)


119860

120597120585

120597119905

+ 119861

120597120585

120597119904

= 119862120585(18)


119892 119901 V119892 V119871)


120585 = 120585

0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (19)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(120588

119892+ 119888

119901120601

119892120588

119871

V2119904

119888

2

119892

)119908

120601

119892

119888

2

119892

[1 minus 119888

119901120601

119871]

V2119904

119888

2

119871

119908 minus[120601

119892120588

119892119896 + 2119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908] 22119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908

minus120588

119871119908

1 minus 120601

119892

119888

2

119871

119908 0 minus119896 (1 minus 120601

119892) 120588

119871

120588

119871V2119903119896 (minus120601

119892119888

119901+ 119888

119903minus 119888

119894+ 119888

1198982) minus120601

119892119896 [1 minus 120601

119871

119888

119901V2119904

119888

2

119871

+ 119888

119894

V2119904

119888

2

119871

]

120601

119892(120588

119892+ 119888vm120588119871)119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119892119908120588

119892V119892)

minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871V2119904119896 (120601

119871119888

119901minus 2119888

119903minus 119888

1198982) minus119896(120601

119871+ 119888

119903120601

119892

V2119904

119888

2

119871

) minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871[120601

119871+ 120601

119892119888vm]119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119871120588

119871V119871)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= 0

(20)

where 119888119894= 03 119888

1198982= 01 119888

119903= 02 and 119888

119901= 025


119888 =

1003816

1003816

1003816

1003816

119908119877

+(119896) minus 119908119877

minus(119896)

1003816

1003816

1003816

1003816

2

(21)



120588

119892=

119901

(119885

119892sdot 119877 sdot 119879)

(22)


119885

119892= 1 + (03051 minus

10467

119879

119903

minus

05783

119879

3

119903

)120588

119903

+ (05353 minus

026123

119879

119903

minus

06816

119879

3

119903

)120588

2

119903

(23)

where 119879119903= 119879119879

119888 119901119903= 119901119901

119888 120588119903= 027119901

119903119885

119866119879

119903


119885

119892=

006125119875

119903119879

minus1


119903)

2

)

119884

(24)


minus 006125119875

119903119879

minus1


119903)

2

) +

119884 + 119884

2+ 119884

3+ 119884

4

(1 minus 119884)

3

= (1476119879

minus1

119903minus 976119879

minus2

119903+ 458119879

minus3

119903) 119884

2

minus (907119879

minus1

119903minus 2422119879

minus2

119903+ 424119879

minus3

119903) 119884

(218+282119879minus1

119903

)

(25)



120588

119898= 120588

0(1 + 4 times 10

minus10119901



minus6119879

2) (26)

With 119879 ge 130


120588

119898= 120588

0(1 + 4 times 10

minus10119901


minus5119879

minus3 times 10

minus6119879

2+ 04(

119879 minus 130

119879

)

2

)

(27)


119879 = 119879

119890119894+ 119865 [1 minus 119890

(119911119887ℎ

minus119911)119860] (minus

119892 sin 120579119892

119888119869119888

119901119898

+ 119873 + 119892

119879sin 120579)

+ 119890

(119911119887ℎ

minus119911)119860(119879

119891119887ℎminus 119879

119890119887ℎ)

(28)


119877

119904= 0021120574gs[(119901 + 01757) 10

(17688120574osminus0001638119879)]

1205

(29)



119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)



119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)


119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)


119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)


V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)


[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)


Vsg gt [Vsg]cr (38)


120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)


by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)


V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)


120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)


120601

119900=

(1 minus 120601

119892) Vso

119878


(43)


119900= 105 +

0371(119863

119894119863

119900)


V119900= 119878

119900V119871+ V119900119903 (44)


V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)


120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)


120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)


119908 V119898 and V


V119908= V119898= V119908119887 (49)



119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Liquid phase

Gas phase

OrOr

p

p + Δp

Δs





119860

120597

120597119905

(120594

119900119896120588

119900120601

119900+ 120594

119892119896120588

119892120601

119892+ 120594

119908119894120588

119908120601

119908+ 120594

119898119894120588

119898120601

119898)

+

120597

120597119904

(120594

119900119896119860120588

119900120601

119900V119900+ 120594

119892119896119860120588

119892120601

119892V119892

+120594

119908119894119860120588

119908120601

119908V119908+ 120594

119898119894119860120588

119898120601

119898V119898) = 0

(1)


120597

120597119905

(119860sum

119896

120588

119896120601

119896V119896) +

120597

120597119904

(119860sum

119896

120588

119896120601

119896V2119896)

+ 119860119892sum

119896

120588

119896120601

119896+

120597

120597119904

(119860119901) + 119860(

120597119901

120597119904

)

fr= 0

(2)


Hence

sum

119894

120594

119900119894= 1 sum

119894

120594

119892119894= 1

sum

119894

120594

119908119894= 1 sum

119894

120594

119898119894= 1

(3)

where

120594

119908119900= 0 120594

119908119892= 0 120594

119908119908= 1 120594

119908119898= 0

120594

119898119900= 0 120594

119898119892= 0 120594mw = 0 120594

119898119898= 1

120594

119892119900= 0 120594

119892119892= 1 120594

119892119908= 0 120594

119892119898= 0

120594

119900119900=

119898

119900

119898

119900+ 119898

119892

120594

119900119892=

119898

119892

119898

119900+ 119898

119892

120594

119900119908= 0 120594

119900119898= 0

(4)




120597

120597119905

(120601

119892120588

119892) +

120597

120597119904

(120601

119892120588

119892V119892) = 0

120597

120597119905

(120601

119871120588

119871) +

120597

120597119904

(120601

119871120588

119871V119871) = 0

(5)

Hence 120588119871= 120601

119900120588

119900+ 120601

119908120588

119908+ 120601

119898120588

119898


120597

120597119904

(120601

119892120588

119892V119892) +

120597

120597119904

(120601

119892120588

119892V2119892)

= minus

120597

120597119904

(120601

119892120588

119892) +

120597

120597119904

[120601

119892(120591

fr119892+ 120591

Re119892)] +119872gi minus 4

120591

119892

119863

(6)


120597

120597119905

(120601

119871120588

119871V119871) +

120597

120597119904

(120601

119871120588

119871V2119871)

= minus

120597

120597119904

(120601

119871120588

119871) +

120597

120597119904

[120601

119871(120591

fr119871+ 120591

Re119871)] +119872Li minus 4

120591

119871

119863

(7)


119872gi = minus119872

ndLi minus119872

119889

Li + (120591frLi + 120591

ReLi )

120597120601

119871

120597119904

+

120597 (120601

119892120590

119904)

120597119904

+

120597 (120601

119892119901gi)

120597119904

minus 120601

119892

120597 (119901Li)

120597119904

119872Li = 119872

ndLi +119872

119889

Li + 119901Li120597 (120601

119871)

120597119904

minus (120591

frLi + 120591

ReLi )

120597120601

119871

120597119904

(8)


119872

ndLi = 119888VM120601119892120588119871120572VM minus 01120601

119892120588

119871V119904

120597V119904

120597119904

minus 119888

1198981120588

119871V2119904

120597120601

119892

120597119904

(9)



119872

119889

Li =3

8

119862

119863

119903

120588

119871119877

119902V2119904 (10)


119901Li minus 119901119871 = minus119888

119901120588

119871V2119904 (11)

where 119888119901= 025


119901gi minus 119901119892 asymp 0 (12)


119901

119871= 119901 minus 025120588

119871120601

119892V2119904 (13)


120591

fr119892asymp 120591

frLi asymp 120591

fr119871asymp 120591

119892asymp 120591

Re119892

asymp 0 (14)


120591

Re119871

= minus119888

119903120588

119871V2119904

120601

119892

120601

119871

120591

ReLi = minus119888

119903120588

119871V2119904

(15)

Hence 119888119903= 02


120591

119871= 05119891

119871120588

119871V2119871 (16)



119889119901

119871

119889120588

119871

= 119888

2

119871

119889119901

119892

119889120588

119892

= 119888

2

119892

(17)


119860

120597120585

120597119905

+ 119861

120597120585

120597119904

= 119862120585(18)


119892 119901 V119892 V119871)


120585 = 120585

0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (19)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(120588

119892+ 119888

119901120601

119892120588

119871

V2119904

119888

2

119892

)119908

120601

119892

119888

2

119892

[1 minus 119888

119901120601

119871]

V2119904

119888

2

119871

119908 minus[120601

119892120588

119892119896 + 2119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908] 22119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908

minus120588

119871119908

1 minus 120601

119892

119888

2

119871

119908 0 minus119896 (1 minus 120601

119892) 120588

119871

120588

119871V2119903119896 (minus120601

119892119888

119901+ 119888

119903minus 119888

119894+ 119888

1198982) minus120601

119892119896 [1 minus 120601

119871

119888

119901V2119904

119888

2

119871

+ 119888

119894

V2119904

119888

2

119871

]

120601

119892(120588

119892+ 119888vm120588119871)119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119892119908120588

119892V119892)

minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871V2119904119896 (120601

119871119888

119901minus 2119888

119903minus 119888

1198982) minus119896(120601

119871+ 119888

119903120601

119892

V2119904

119888

2

119871

) minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871[120601

119871+ 120601

119892119888vm]119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119871120588

119871V119871)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= 0

(20)

where 119888119894= 03 119888

1198982= 01 119888

119903= 02 and 119888

119901= 025


119888 =

1003816

1003816

1003816

1003816

119908119877

+(119896) minus 119908119877

minus(119896)

1003816

1003816

1003816

1003816

2

(21)



120588

119892=

119901

(119885

119892sdot 119877 sdot 119879)

(22)


119885

119892= 1 + (03051 minus

10467

119879

119903

minus

05783

119879

3

119903

)120588

119903

+ (05353 minus

026123

119879

119903

minus

06816

119879

3

119903

)120588

2

119903

(23)

where 119879119903= 119879119879

119888 119901119903= 119901119901

119888 120588119903= 027119901

119903119885

119866119879

119903


119885

119892=

006125119875

119903119879

minus1


119903)

2

)

119884

(24)


minus 006125119875

119903119879

minus1


119903)

2

) +

119884 + 119884

2+ 119884

3+ 119884

4

(1 minus 119884)

3

= (1476119879

minus1

119903minus 976119879

minus2

119903+ 458119879

minus3

119903) 119884

2

minus (907119879

minus1

119903minus 2422119879

minus2

119903+ 424119879

minus3

119903) 119884

(218+282119879minus1

119903

)

(25)



120588

119898= 120588

0(1 + 4 times 10

minus10119901



minus6119879

2) (26)

With 119879 ge 130


120588

119898= 120588

0(1 + 4 times 10

minus10119901


minus5119879

minus3 times 10

minus6119879

2+ 04(

119879 minus 130

119879

)

2

)

(27)


119879 = 119879

119890119894+ 119865 [1 minus 119890

(119911119887ℎ

minus119911)119860] (minus

119892 sin 120579119892

119888119869119888

119901119898

+ 119873 + 119892

119879sin 120579)

+ 119890

(119911119887ℎ

minus119911)119860(119879

119891119887ℎminus 119879

119890119887ℎ)

(28)


119877

119904= 0021120574gs[(119901 + 01757) 10

(17688120574osminus0001638119879)]

1205

(29)



119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)



119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)


119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)


119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)


V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)


[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)


Vsg gt [Vsg]cr (38)


120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)


by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)


V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)


120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)


120601

119900=

(1 minus 120601

119892) Vso

119878


(43)


119900= 105 +

0371(119863

119894119863

119900)


V119900= 119878

119900V119871+ V119900119903 (44)


V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)


120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)


120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)


119908 V119898 and V


V119908= V119898= V119908119887 (49)



119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597

120597119905

(120601

119892120588

119892) +

120597

120597119904

(120601

119892120588

119892V119892) = 0

120597

120597119905

(120601

119871120588

119871) +

120597

120597119904

(120601

119871120588

119871V119871) = 0

(5)

Hence 120588119871= 120601

119900120588

119900+ 120601

119908120588

119908+ 120601

119898120588

119898


120597

120597119904

(120601

119892120588

119892V119892) +

120597

120597119904

(120601

119892120588

119892V2119892)

= minus

120597

120597119904

(120601

119892120588

119892) +

120597

120597119904

[120601

119892(120591

fr119892+ 120591

Re119892)] +119872gi minus 4

120591

119892

119863

(6)


120597

120597119905

(120601

119871120588

119871V119871) +

120597

120597119904

(120601

119871120588

119871V2119871)

= minus

120597

120597119904

(120601

119871120588

119871) +

120597

120597119904

[120601

119871(120591

fr119871+ 120591

Re119871)] +119872Li minus 4

120591

119871

119863

(7)


119872gi = minus119872

ndLi minus119872

119889

Li + (120591frLi + 120591

ReLi )

120597120601

119871

120597119904

+

120597 (120601

119892120590

119904)

120597119904

+

120597 (120601

119892119901gi)

120597119904

minus 120601

119892

120597 (119901Li)

120597119904

119872Li = 119872

ndLi +119872

119889

Li + 119901Li120597 (120601

119871)

120597119904

minus (120591

frLi + 120591

ReLi )

120597120601

119871

120597119904

(8)


119872

ndLi = 119888VM120601119892120588119871120572VM minus 01120601

119892120588

119871V119904

120597V119904

120597119904

minus 119888

1198981120588

119871V2119904

120597120601

119892

120597119904

(9)



119872

119889

Li =3

8

119862

119863

119903

120588

119871119877

119902V2119904 (10)


119901Li minus 119901119871 = minus119888

119901120588

119871V2119904 (11)

where 119888119901= 025


119901gi minus 119901119892 asymp 0 (12)


119901

119871= 119901 minus 025120588

119871120601

119892V2119904 (13)


120591

fr119892asymp 120591

frLi asymp 120591

fr119871asymp 120591

119892asymp 120591

Re119892

asymp 0 (14)


120591

Re119871

= minus119888

119903120588

119871V2119904

120601

119892

120601

119871

120591

ReLi = minus119888

119903120588

119871V2119904

(15)

Hence 119888119903= 02


120591

119871= 05119891

119871120588

119871V2119871 (16)



119889119901

119871

119889120588

119871

= 119888

2

119871

119889119901

119892

119889120588

119892

= 119888

2

119892

(17)


119860

120597120585

120597119905

+ 119861

120597120585

120597119904

= 119862120585(18)


119892 119901 V119892 V119871)


120585 = 120585

0+ 120575120585 exp [119894 (119908119905 minus 119896119905)] (19)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(120588

119892+ 119888

119901120601

119892120588

119871

V2119904

119888

2

119892

)119908

120601

119892

119888

2

119892

[1 minus 119888

119901120601

119871]

V2119904

119888

2

119871

119908 minus[120601

119892120588

119892119896 + 2119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908] 22119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908

minus120588

119871119908

1 minus 120601

119892

119888

2

119871

119908 0 minus119896 (1 minus 120601

119892) 120588

119871

120588

119871V2119903119896 (minus120601

119892119888

119901+ 119888

119903minus 119888

119894+ 119888

1198982) minus120601

119892119896 [1 minus 120601

119871

119888

119901V2119904

119888

2

119871

+ 119888

119894

V2119904

119888

2

119871

]

120601

119892(120588

119892+ 119888vm120588119871)119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119892119908120588

119892V119892)

minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871V2119904119896 (120601

119871119888

119901minus 2119888

119903minus 119888

1198982) minus119896(120601

119871+ 119888

119903120601

119892

V2119904

119888

2

119871

) minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871[120601

119871+ 120601

119892119888vm]119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119871120588

119871V119871)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= 0

(20)

where 119888119894= 03 119888

1198982= 01 119888

119903= 02 and 119888

119901= 025


119888 =

1003816

1003816

1003816

1003816

119908119877

+(119896) minus 119908119877

minus(119896)

1003816

1003816

1003816

1003816

2

(21)



120588

119892=

119901

(119885

119892sdot 119877 sdot 119879)

(22)


119885

119892= 1 + (03051 minus

10467

119879

119903

minus

05783

119879

3

119903

)120588

119903

+ (05353 minus

026123

119879

119903

minus

06816

119879

3

119903

)120588

2

119903

(23)

where 119879119903= 119879119879

119888 119901119903= 119901119901

119888 120588119903= 027119901

119903119885

119866119879

119903


119885

119892=

006125119875

119903119879

minus1


119903)

2

)

119884

(24)


minus 006125119875

119903119879

minus1


119903)

2

) +

119884 + 119884

2+ 119884

3+ 119884

4

(1 minus 119884)

3

= (1476119879

minus1

119903minus 976119879

minus2

119903+ 458119879

minus3

119903) 119884

2

minus (907119879

minus1

119903minus 2422119879

minus2

119903+ 424119879

minus3

119903) 119884

(218+282119879minus1

119903

)

(25)



120588

119898= 120588

0(1 + 4 times 10

minus10119901



minus6119879

2) (26)

With 119879 ge 130


120588

119898= 120588

0(1 + 4 times 10

minus10119901


minus5119879

minus3 times 10

minus6119879

2+ 04(

119879 minus 130

119879

)

2

)

(27)


119879 = 119879

119890119894+ 119865 [1 minus 119890

(119911119887ℎ

minus119911)119860] (minus

119892 sin 120579119892

119888119869119888

119901119898

+ 119873 + 119892

119879sin 120579)

+ 119890

(119911119887ℎ

minus119911)119860(119879

119891119887ℎminus 119879

119890119887ℎ)

(28)


119877

119904= 0021120574gs[(119901 + 01757) 10

(17688120574osminus0001638119879)]

1205

(29)



119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)



119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)


119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)


119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)


V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)


[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)


Vsg gt [Vsg]cr (38)


120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)


by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)


V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)


120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)


120601

119900=

(1 minus 120601

119892) Vso

119878


(43)


119900= 105 +

0371(119863

119894119863

119900)


V119900= 119878

119900V119871+ V119900119903 (44)


V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)


120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)


120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)


119908 V119898 and V


V119908= V119898= V119908119887 (49)



119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(120588

119892+ 119888

119901120601

119892120588

119871

V2119904

119888

2

119892

)119908

120601

119892

119888

2

119892

[1 minus 119888

119901120601

119871]

V2119904

119888

2

119871

119908 minus[120601

119892120588

119892119896 + 2119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908] 22119888

119901120601

119892120601

119871120588

119871

V119904

119888

2

119871

119908

minus120588

119871119908

1 minus 120601

119892

119888

2

119871

119908 0 minus119896 (1 minus 120601

119892) 120588

119871

120588

119871V2119903119896 (minus120601

119892119888

119901+ 119888

119903minus 119888

119894+ 119888

1198982) minus120601

119892119896 [1 minus 120601

119871

119888

119901V2119904

119888

2

119871

+ 119888

119894

V2119904

119888

2

119871

]

120601

119892(120588

119892+ 119888vm120588119871)119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119892119908120588

119892V119892)

minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871V2119904119896 (120601

119871119888

119901minus 2119888

119903minus 119888

1198982) minus119896(120601

119871+ 119888

119903120601

119892

V2119904

119888

2

119871

) minus119888vm120601119892120588119871119908 + 119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904)

120588

119871[120601

119871+ 120601

119892119888vm]119908

minus119894 (

3

4

119888

119863

119903

120588

119871120601

119892V119904+

4

119863

119891

119871120588

119871V119871)

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

= 0

(20)

where 119888119894= 03 119888

1198982= 01 119888

119903= 02 and 119888

119901= 025


119888 =

1003816

1003816

1003816

1003816

119908119877

+(119896) minus 119908119877

minus(119896)

1003816

1003816

1003816

1003816

2

(21)



120588

119892=

119901

(119885

119892sdot 119877 sdot 119879)

(22)


119885

119892= 1 + (03051 minus

10467

119879

119903

minus

05783

119879

3

119903

)120588

119903

+ (05353 minus

026123

119879

119903

minus

06816

119879

3

119903

)120588

2

119903

(23)

where 119879119903= 119879119879

119888 119901119903= 119901119901

119888 120588119903= 027119901

119903119885

119866119879

119903


119885

119892=

006125119875

119903119879

minus1


119903)

2

)

119884

(24)


minus 006125119875

119903119879

minus1


119903)

2

) +

119884 + 119884

2+ 119884

3+ 119884

4

(1 minus 119884)

3

= (1476119879

minus1

119903minus 976119879

minus2

119903+ 458119879

minus3

119903) 119884

2

minus (907119879

minus1

119903minus 2422119879

minus2

119903+ 424119879

minus3

119903) 119884

(218+282119879minus1

119903

)

(25)



120588

119898= 120588

0(1 + 4 times 10

minus10119901



minus6119879

2) (26)

With 119879 ge 130


120588

119898= 120588

0(1 + 4 times 10

minus10119901


minus5119879

minus3 times 10

minus6119879

2+ 04(

119879 minus 130

119879

)

2

)

(27)


119879 = 119879

119890119894+ 119865 [1 minus 119890

(119911119887ℎ

minus119911)119860] (minus

119892 sin 120579119892

119888119869119888

119901119898

+ 119873 + 119892

119879sin 120579)

+ 119890

(119911119887ℎ

minus119911)119860(119879

119891119887ℎminus 119879

119890119887ℎ)

(28)


119877

119904= 0021120574gs[(119901 + 01757) 10

(17688120574osminus0001638119879)]

1205

(29)



119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)



119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)


119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)


119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)


V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)


[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)


Vsg gt [Vsg]cr (38)


120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)


by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)


V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)


120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)


120601

119900=

(1 minus 120601

119892) Vso

119878


(43)


119900= 105 +

0371(119863

119894119863

119900)


V119900= 119878

119900V119871+ V119900119903 (44)


V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)


120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)


120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)


119908 V119898 and V


V119908= V119898= V119908119887 (49)



119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119861

119900= 0976 + 000012[5612(

120574gs

120574os)

05

119877

119904+ 225119879 + 40]

12

(30)



119863 =

120587 (119863

2

119900minus 119863

2

119894) 4

120587 (119863

119900+ 119863

119894) 4

= 119863

119900minus 119863

119894

(31)


119896

119890= 119896

0

119863

119900

119863

119900+ 119863

119894

+ 119896

119894

119863

119894

119863

119900+ 119863

119894

(32)


119863min = 19[

(120588

119871minus 120588

119892) 120590

119904

119892120588

2

119871

]

05

(33)


V112119872cr = 588119863

048[

119892 (120588

119871minus 120588

119892)

120590

119904

]

05

(

120590

119904

120588

119871

)(

120588

119872

120583

119871

)

008

(34)

119863 gt 119863min

120601

119892le 025 V

119872le V119872cr

120601

119892le 052 V

119872gt V119872cr

(35)


[Vsg]cr = 31[

119892120590 (120588

119871minus 120588

119892)

120588

2

119892

]

025

(36)

120601

119892gt 025 V

119872le V119872cr

120601

119892gt 052 V

119872gt V119872cr

Vsg le [Vsg]cr

(37)


Vsg gt [Vsg]cr (38)


120601

119892=

Vsg119878

119892(Vso + Vsg + Vsw + Vsm) + V

119892119903

(39)


by

119878

119892= 120 + 0371 (

119863

119894

119863

119900

) (40)


V119892119903= 153[

119892120590

119904(120588

119871minus 120588

119892)

120588

2

119871

]

025

(41)


120588

119872= 120601

119871120588

119871+ 120601

119892120588

119892 (42)


120601

119900=

(1 minus 120601

119892) Vso

119878


(43)


119900= 105 +

0371(119863

119894119863

119900)


V119900= 119878

119900V119871+ V119900119903 (44)


V119900119903= 153[

119892120590

119908119900minus 120588

119900

120588

2

119908119887

]

2

(45)

where

120588

119908119887= 120601

119908120588

119908+ 120601

119898120588

119898 (46)


120601

119908=

(1 minus 120601

119892minus 120601

119900) Vsw

Vsw + Vsm

(47)


120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908 (48)


119908 V119898 and V


V119908= V119898= V119908119887 (49)



119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862vm = 05

1 + 2120601

119892

1 minus 120601

119892

(50)


119862

119863=

4119877

119887

3

radic

119892 (120588

119871minus 120588

119892)

120590

119904

[

1 + 1767120601

97

119871

1867120601

15

119871

]

2

(51)


120591

119891= 119891

120588

119871V2119871

2119863

(52)


Φ

119908 and Φ




described as

119878

119892= 1182 + 09 (

119863

119894

119863

119900

) (53)



Vgr = (035 + 01

119863

119894

119863

119900

)[

119892119863

119900(120588

119871minus 120588

119892)

120588

119871

]

05

(54)


119862vm = 33 + 17

3119871

119902minus 3119877

119902

3119871

119902minus 119877

119902

(55)


can be expressed as

119862

119863= 110120601

3

119871119877

119887 (56)



120601

119892= (1 + 119883

08)

minus0378

(57)


119883 = radic

(119889119901119889119904

119871)fr

(119889119901119889119904

119892)

fr

(58)

1 2 3





120601

119900=

(1 minus 120601

119892) Vso

Vso + Vsw + Vsm

(59)


120601

119908=

(1 minus 120601

119892) Vsw

Vso + Vsw + Vsm

120601

119898= 1 minus 120601

119892minus 120601

119900minus 120601

119908

(60)


(55) and (56)





(119860Vsm)119899+1

119894+1minus (119860Vsm)

119899+1

119894

Δ119904

=

(119860120601

119898)

119899

119894+ (119860120601

119898)

119899

119894+1minus (119860120601

119898)

119899+1

119894minus (119860120601

119898)

119899+1

119894+1

2Δ119905

(61)

For the water phase

(119860Vsw)119899+1

119894+1minus (119860Vsw)

119899+1

119894

Δ119904

=

(119860120601

119908)

119899

119894+ (119860120601

119908)

119899

119894+1minus (119860120601

119908)

119899+1

119894minus (119860120601

119908)

119899+1

119894+1

2Δ119905

(62)

For the oil phase

(119860 (Vso119861119900))119899+1

119894+1minus (119860 (Vso119861119900))

119899+1

119894

Δ119904


Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

End

Initial parameters



Assume p of bottom



i + 1 lt nmax


Assume 120601c(i + 1 n + 1)

No

No

No

No

No

No



Obtain c


|pc(i + 1 n + 1) minus BP| lt 120576







= ((119860

120601so119861

119900

)

119899

119894

+ (119860

120601so119861

119900

)

119899

119894+1

minus (119860

120601so119861

119900

)

119899+1

119894

minus(119860

120601so119861

119900

)

119899+1

119894+1

) times (2Δ119905)

minus1

(63)

For the gas phase

[119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894+1minus [119860 (120588

119892Vsg + 120588gs119877119904Vso119861119900)]

119899+1

119894

Δ119904


=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894

2Δ119905

+

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899

119894+1

2Δ119905

minus

[119860 (120588

119892120601

119892+ 120588gs119877119904120601119900119861119900)]

119899+1

119894

2Δ119905

minus

[119860 (120588

119892120601

119892+ (120588gs119877119904120601119900119861119900))]

119899+1

119894+1

2Δ119905

(64)


(119860119901)

119899+1

119894+1minus (119860119901)

119899+1

119894= 120585

1+ 120585

2+ 120585

3+ 120585

4 (65)

where

120585

1=

Δ119904

2Δ119905

[

[

[

(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+ (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899

119894+1

minus(119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894minus (119860 (120588

119898Vsm + 120588

119908Vsw + 120588119900Vso + 120588119892Vsg))

119899+1

119894+1

]

]

]

(66)

and

120585

2= [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894

minus [119860(

120588

119898V2sm120601

119898

+

120588

119908V2sw120601

119908

+

120588

119892V2sg120601

119892

+

120588

119900V2so120601

119900

)]

119899+1

119894+1

120585

3= minus

119892Δ119904

2

[(119860120588

119872)

119899+1

119894+ (119860120588

119872)

119899+1

119894+1]

120585

4= minus

Δ119904

2

[(119860(

120597119901

120597119904

))

119899+1

fr119894+ (119860(

120597119901

120597119904

))

119899+1

fr119894+1]

(67)








119892= 36m3h 119876

119908= 36m3h 119876

119900= 36m3h and 119876

119898=





Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Type Property Value


minus3


minus5


minus3


minus3


11


minus7

Surfacecondition


0 5 10 15 200

500

1000

1500

2000


c(m

s)

p (MPa)

w = 50Hz Cvm = Re



RCD

OilGasWater

Meter

Formation

ChokeSensors

annulus

0m

3600m

4000m

Casing

1206011270mm (1206011086mm)Drill pipe

1206012445mm






460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










460 690 920 1150 13804000

3000

2000

1000

0

L = 500m


H(m

)

c (ms)




60

45

30

15

0

0 40

30

20

100

BP =

BP = 30

BP = 60

BP = 90

H (m)T

(min)

4000

30002000

1000

120601(




840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










43

0 40

30

20

100

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

52

39

26

13

0

H (m) T(min)

120601(

)


0

40003000

2000

10000 0

10

20

3040

300

600

900

1200

Qg = 036m3hQg = 108m3h

Qg = 180m3hQg = 360m3h

H (m) T(min)

c(m

s)




119892= 036m3h to 119876

119892=


60

45

30

15

0

0 40

30

20

100

40003000

2000

1000

Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

H (m) T(min)

120601(


Qo = 036m3hQo = 108m3h

Qo = 180m3hQo = 360m3h

840

630

420

210

0

40003000

2000

10000 0

10

20

3040

H (m) T(min)

c(m

s)




H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H (m) T(min)

c(m

s)


w = 1500Hz Cvm = 0

w = 10000Hz Cvm = 0

840

630

420

210

0

40003000

2000

10000 0

10

20

3040



5 Conclusions










Nomenclature



119888


119888


(ms)119862




119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119863


119863



119891


119891






119892




119896


119896






119872


(Nm3)119872

119889




119901


119901


119901


119901


119902


119902


119902


119876




119877


119878




119879


119879


depth (K)119879


wellhead (K)119879








119881



(ms)119882


119911


119911


Greek Letters120588


120588


120588


120588


120588


120588


120588


120588


120591


120591


120591


120591



120591




120591


120591


120591


120591


120588


120590




Φ


Φ


Φ


Φ


Φ


120583


120594


120594


120594


120594









Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Acknowledgments


References




































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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