research article reservoir characterization during underbalanced...

Research ArticleReservoir Characterization during Underbalanced Drilling ofHorizontal Wells Based on Real-Time Data Monitoring

Gao Li1 Hongtao Li1 Yingfeng Meng1 Na Wei1 Chaoyang Xu1 Li Zhu1 and Haibo Tang2

1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploration School of Petroleum EngineeringSouthwest Petroleum University Chengdu Sichuan 610500 China

2 Engineering Supervision Center SINOPEC Southwest Oil amp Gas Company Deyang Sichuan 618000 China

Correspondence should be addressed to Hongtao Li lihongtao053yahoocomcn and Yingfeng Meng cwctmyfvipsinacom

Received 4 April 2014 Accepted 15 July 2014 Published 17 August 2014

Academic Editor Vijay Gupta

Copyright copy 2014 Gao Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In thiswork amethodology for characterizing reservoir pore pressure and permeability during underbalanced drilling of horizontalwells was presented The methodology utilizes a transient multiphase wellbore flow model that is extended with a transient wellinflux analytical model during underbalanced drilling of horizontal wells The effects of the density behavior of drilling fluid andwellbore heat transfer are considered in our wellbore flow model Based on Kneisslrsquos methodology an improved method with adifferent testing procedure was used to estimate the reservoir pore pressure by introducing fluctuations in the bottom hole pressureTo acquire timely basic data for reservoir characterization a dedicated fully automated control real-time data monitoring systemwas established The methodology is applied to a realistic case and the results indicate that the estimated reservoir pore pressureand permeability fit well to the truth values from well test after drilling The results also show that the real-time data monitoringsystem is operational and can provide accurate and complete data set in real time for reservoir characterization The methodologycan handle reservoir characterization during underbalanced drilling of horizontal wells

1 Introduction

Underbalanced drilling has been used with increasing fre-quency in horizontal wells as it has conspicuous technicalsuperiorities compared to overbalanced drilling in decreas-ing drilling costs preventing invasive formation damageachieving higher rates of penetration and reducing drillingproblems such as pressure differential pipe sticking and lostcirculation [1 2] During underbalanced drilling the bottomhole pressure is kept below the reservoir pore pressureAs a result the reservoir will transport fluids towards thewellbore generating a transient well influx during drillingThe influx behavior of reservoir fluids depends on thepressure difference between the reservoir pore pressure andthe bottom hole pressure and reservoir permeability andporosity in addition to other reservoir parameters such as theviscosity and compressibility of reservoir fluidsThe transientwell influx causes flow behavior variations in the wellboreOwing to the changes in well fluid composition and well

fluid flow rates the transient influx of reservoir fluids cau-ses variations in the flow in the wellbore Analogous to welltesting and transient reservoir analysis the reservoir char-acteristics close to the well causing the influx might beidentified on site by monitoring some of the fluid flow dataof the well such as pressure changes and rate changes [3 4]This is the principal idea that also is the basis for reservoircharacterization during underbalanced drilling Estimationof the near wellbore characteristics of the formation givesimportant information for planning well completion designwell pressure control and optimizing drilling parameters onsite such as the wellrsquos horizontal length gas injection volumeand orientation

Reservoir characterization during underbalanced drillingstill needs strong theories to be formulatedwith reliable smartexperiments The complexity of this subject arises from thefact that the boundary conditions of transient well inflowchange with time during the test and that the rates of trans-ient well influx are influenced by variation in underbalance

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 905079 19 pageshttpdxdoiorg1011552014905079

2 Journal of Applied Mathematics

Moreover the exhaustive and reliable monitoring data arenot easy to be acquired in real time To the best of theauthorsrsquo knowledge the idea that reservoir characterizationwhile underbalanced drilling was first presented by Kardolusand van Kruijsdijk [5] in 1997 In their paper a transientreservoir influx model of a vertical well under a constantunderbalance based on the boundary element method wasdeveloped They compared the model with a transient ana-lytical reservoir model and demonstrated that the transientanalytical reservoir model could be used for evaluatingcharacteristic parameters in the reservoir in particular thepermeability In a following study van Kruijsdijk and Cox [6]developed a method for identifying the permeability profilesalong the well trajectory of a horizontal well considering thevariable underbalance and themore complex outer boundaryconditions of a horizontal well

Reservoir characterization during underbalanced drillinghas been investigated by several other researchers in recentyears Larsen and Nilsen [7] proposed a simple quasi-stationary model for predicting the inflow at various levelsof complexity and accuracy for spreadsheet and calculatorapplications based on drilling speed and knowledge of theratio of vertical to horizontal permeability The influx pre-dictions calculated by their model and a numerical reservoirsimulator were compared giving a good agreement Azar-Nejad [8] presented a model for influx predictions using thediscrete flux element method In his study he mentionedfactors affecting influx rates and indicated that the solutionsof model are very sensitive to the reservoir thickness Huntand Rester [9 10] developed and solved new reservoirmodelsdescribing the underbalanced drilling problem and charac-terizing the dynamic changes in single layer andmultilayeredreservoirs They also presented a history matching methodfor predicting the reservoir permeability Biswas et al [11]proposed a genetic search algorithm for identifying thereservoir characterization accounting for storage effects inboth the wellbore and reservoir In a paper by Vefring et al[3] a transient reservoir model coupled to a transientwellbore flow model was used They estimated the reservoirproperties using novel Levenberg-Marquardt algorithm anonlinear least-squares optimization method In a follow-up research [4] the ensemble Kalman filter was also usedto predict reservoir pore pressure and reservoir permeabilitywhile applying active tests and comparedwith the Levenberg-Marquardt algorithm A paper by Kneissl [12] presented amethod of simultaneously deriving reservoir pore pressureand permeability profiles in real time during underbalanceddrilling by introducing fluctuations in the bottom holepressure during drilling However the variations of fluid flowbehavior in the wellbore might pose difficulties in estimatingthe influx rates which may lead to large uncertainties inthe estimate of reservoir pore pressure [13] Stewart et al[14] proposed an integrated transient model of the formationand the wellbore based on superposition The method ofmatching the cumulative production data for estimatingpermeability using iterative search was also developed intheir paper However an assumption that the pore pressureis known independently was made analogously like mostof above researchers The problem at hand is complex [11]

Although many models and methods for identifying reser-voir characterization while underbalanced drilling have beenproposed the interpretation of reservoir properties is difficultin real reservoirs particularly in real time except for thesimplest cases

This paper describes a new pursuit of the complexproblem of identifying the reservoir characterization duringunderbalanced drilling of horizontal wells A model oftransient well influx of horizontal wells is used together witha multiphase wellbore flow model The effects of the densitybehavior of drilling fluid and wellbore heat transfer duringunderbalanced drilling are considered in our wellbore flowmodel Based on the Kneisslrsquos methodology an improvedmethod of estimating reservoir pore pressure performedby introducing fluctuations in the bottom hole pressure bymanipulating the choke valve opening and changing fluidcomposition or pump rates is presented in this paper Aneffort is also made to monitor accurately and automaticallymeasurements required for reservoir characterization in realtime

The paper is organized by first presenting the real-timedata monitoring Then the transient wellbore flow modeland transient well influx model are presented respectivelyBased on the data monitoring and mathematical modelswe then propose the methodology for estimating reservoircharacterization In this paper the reservoir pore pressureand permeability profiles are the main reservoir propertiesneeded to be characterizedThemethodology is then appliedto a realistic case and results are discussed Thereafter aconclusion is made

2 Real-Time Data Monitoring

The accurate and automatic acquisition of basis data inreal time has been a challenge for reservoir characteriza-tion during drilling of horizontal wells which is crucialand is a prerequisite During a standard underbalanceddrilling operation lots of indispensable parameters can berecorded by the compound logging and other measuringequipment such as the density and rheology of the drillingfluid well depth penetration rate pump pressure outletrates trajectory of the well and wellbore diameter Howeverthese measurements are insufficient and not accurate enoughfor reservoir characterization Being more pivotal most ofthese measurements cannot be acquired in real time whichis indispensable to identifying reservoir characterizationduring drilling This paper has presented a dedicated fullyautomated control real-time data monitoring system basedon the integration and modification of existing devices orcomponents and the installation of some more indispensablesensitive and accurate measuring instruments In addition toinstalling somemeasuring equipment monitoring indispens-able parameters for reservoir characterization unrecordedgenerally in conventional drilling the data from com-pound logging and PMWDMWD are involved in our real-time monitoring system It is believed that the integrationis also beneficial for well control and drilling parameter

Journal of Applied Mathematics 3

optimization in underbalanced operations The measure-ments of interest here are as follows

(i) injection and outlet drilling fluid densities(ii) injection and outlet drilling fluid rates(iii) drilling fluid viscosity(iv) oil outlet rates and density(v) penetration rate in the formation(vi) injection and outlet gas rates pressure and tempera-

ture(vii) outlet gas component(viii) standpipe and surface casing pressure(ix) bottom hole pressure(x) measured depth and wellbore diameter(xi) BHA and hole trajectory(xii) formation porosity

Schematics of the location of the real-time data moni-toring system with respect to the well for aerated drillingand liquid-based underbalanced drilling are illustrated inFigures 1 and 2 respectively In order to explain the datamonitoring systems clearly it is necessary to first introducethe circulation process for underbalanced drilling Duringaerated drilling operations drilling fluid from mud pits andnitrogen from gas injection system consisting of compressornitrogen generator and booster are mixed and injected intothe drill strings under pressure via a fluid line as shown inFigure 1 The gasified drilling fluid is then circulated out withthe produced formation fluid through the annulus into theprimary flow line and enters the separator where gas and oilare separated The drilling fluid returns to the mud pits viaa return line after taking out the solids using shale shakersProduced oil is sent to an oil shimming tank Separated gasis sent to a combustion cell For liquid-based underbalanceddrilling the circulation process is somewhat different becauseonly drilling fluid is injected as shown in Figure 2 Thusthe measurements of the gas injection rates pressure andtemperature were not involved in the data monitoring systemfor liquid-based unbalanced drilling For the reason that theattenuation of drilling fluid pulse in multiphase flow restrictsthe use of down hole measuring devices with transmissionmedium of drilling fluid pulse in aerated drilling [16 17]PMWDMWDdatawere not included in the datamonitoringsystem for aerated drilling

As shown in Figures 1 and 2 the real-time data monitor-ing systems consist of various kinds measuring instrumentsan incoherent receptor a signal converser and a PC Toacquire various parameters in different types in real time themeasuring facilities used here include pressure transducerstemperature transducer gas and liquid flowmeters densitymeasuring unit and portable gas analyzer The strong tech-nology and sophisticated equipment were applied in dataacquisitions in this paperThe pressure transducers used herehave a frequency response of 01sim1000Hz and a maximumfull scale output error of 075 FSO over the 0 to 70∘Ccompensated rangeThe injection and outlet gas temperature

was measured by the temperature transducers with widetemperature range from minus55∘C to +150∘C and excellentlinearity of plusmn03∘C In order to assure measurement accuracydifferent types of portable type clip ultrasonic flowmeter wereclamped on to the existing pipeline to acquire respectivelythe drilling fluid oil and gas rates The density measuringunit proposed by Carlsen et al [18] consisting of threepressure transducers placed along the circulating path fromthe pump to the connection to the drill string was usedto identify automatically the injection drilling fluid densityAn advantage of this method is that it enables automaticupdates of the density Likewise the outlet drilling fluiddensity can readily be determined The density of outlet oilwas measured using a common method for the reason that itis relatively fixed for a particular reservoir The acquisition ofoutlet gas component can be completed using a portable gasanalyzer One of the technical superiorities of this system isthat a digital microwave transmission technique was used totransmit the data recorded by various measuring equipmentinstead of the use of the cable All the measuring instrumentsshown in Figures 1 and 2 were attached to a wireless radiatedelement respectively Measurement signals from pressuretransducers flowmeters and other instrumentswere receivedwirelessly by an incoherent receptor and then analyzed andrecorded using a signal analyzer after being converted intodigital signals It is noted that the incoherent receptiontechnique indicated in Spalding and Middleton [19] wasapplied here to tackle the reception of signals in the impulsiveinterference environment Data acquisition software calledLabVIEW was used to capture the measurement signalsThese measurements together with compound logging dataand PMWDMWD data were preprocessed to detect neces-sary information for reservoir characterization in standarddata formats usingMATLAB on a personal computer Owingto the use of wireless transmission technique the signalreceptor and data processing system no longer have to belocated near the well site which makes remote supervisoryand technical service possible The work above in this paperensures that the data are accurate real time and easilyavailable which is critical and considerable for reservoircharacterization during drilling

3 Transient Wellbore Flow Model

The wellbore transients have significant effect on actualwell inflow and therefore should be considered in reservoircharacterization The real-time data monitoring system canprovide bottom hole pressure in liquid-based underbalanceddrilling During aerated drilling the wellbore transientsmust be calculated by mathematical means owing to theserious attenuation of PMWD signals In this section a one-dimensional liquid-gas two-phase transient model governingwellbore flow with thermal effects in aerated drilling isdeveloped with variables resolved in the axial direction onlyBesides the effect of the cutting bed in horizontal sections isconsidered Owing to complexity of the liquid-gas two-phaseflow transient in wellbore the following assumptions madefor the model development are primarily given


Choke manifold Combustion cell

Three-wayvalve

ThrottleMud pumps

BlenderThrottle

Portable gasanalyzer

Liquidflowmeter

Liquid flowmeterShale

shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

Booster

PC Reservoir characterization Compound logging data

Oil skimming tank

11

Sampling

Signal converser Incoherent receptor

Density measuring unit

Temperature transducer

Pressure transducer

Signal analyzer

Outlet (oil)

Pressure transducer

Gas flowmeter

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Drillstrings

Annulus

Injection(gas)

Injection(mixture)

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)

Primary flow line

RBOP

Secondaryflow line

Nitrogengenerator

Compressor

Atmosphere

Solids disposal

Mud return line


deg100

50

minus0

minus10mV

Figure 1 A schematic of the placement of the real-time data monitoring system for aerated drilling

deg


Three-wayvalve

ThrottleMud pumps Portable gas

analyzer

Liquidflowmeter


shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

100

50


Oil skimming tank

11

Sampling



Pressure transducer

Signal analyzer

Outlet (oil)

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Injection (mud)

Mud return line Drillstrings

Annulus

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)

Temperature transducer Primary flow line

RBOP

Secondaryflow line

Solids disposal

minus0

minus10mV

Figure 2 A schematic of the placement of the real-time data monitoring system for liquid-based underbalanced drilling

(i) The liquid phase fluids are incompressible and New-tonian

(ii) Properties of fluids are uniform and constant

(iii) The bubbles are spherical in shape having diameter asthe characteristic

(iv) The transverse lift force is neglected

(v) The turbulent dispersion force of gas phase is neglec-ted

(vi) The effects of breakup and coalescence of droplets andbubbles are neglected

(vii) Mass transfer effects are neglected

31 Model Development In this paper a transient model ofliquid-gas two-phase flow in wellbore is proposed on thebasis of the three-dimensional model averaged in time andspace given by Lahey andDrew [20]The governing equationsconsist of continuity equations for each component and a


momentum conservation equation Such equations can bewritten as follows

gas continuity equation

120597 (120588

119892120572

119892)

120597119905

+

120597 (120588

119892120572

119892119906

119892)

120597119911

= 0

(1)

liquid continuity equation

120597 (120588

119897120572

119897)

120597119905

+

120597 (120588

119897120572

119897119906

119897)

120597119911

= 0(2)

gas momentum conservation equation

120597 (120588

119892120572

119892119906

119892)

120597119905

+

120597 (120588

119892120572

119892119906

2

119892

)

120597119911

+ 120572

119892

120597119901

120597119911

= minus120572

119892120588

119892119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119892

(3)

liquid momentum conservation equation

120597 (120588

119897120572

119897119906

119897)

120597119905

+

120597 (120588

119897120572

119897119906

2

119897

)

120597119911

+ 120572

119897

120597119901

120597119911

= minus120572

119897120588

119897119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119897

(4)

where 119901 is pressure 119892 is the acceleration due to gravity 120588is density 119906 is velocity 120572 is the volumetric fraction 120579 is thedeviation angle 119905 and 119911 are temporal and spatial coordi-nates 119865

119889is drag force 119865V119898 is virtual mass force 119865

119908is

wall shear force and the subscripts 119892 119897 represent gas andliquid respectively The indispensable closure relationshipsappearing in equations above are formulated and describedin Appendix A

Due to the mixture of gas-liquid two phases the dis-tribution relation of two-phase homogeneous fluid can benormalized as

120572

119892+ 120572

119897= 1 (5)

In this paper gas phase can be deemed as the ideal gas andcan be evaluated using the equation of state in the followingmanner

119901 = 119885120588

119892119877119879 (6)

where 119879 is temperature 119877 denotes the gas constant and 119885 isthe real gas deviation factor which is the function of pressuretemperature and the nature of the gas

In the transient liquid-gas two-phase flow model theheat transfer between wellbore and formation is consideredby introducing a wellbore heat transfer model during two-phase flow presented by Hasan and Kabir [21] based on arigorous energy balance equation for two-phase flow [22] Anexpression for fluid temperature in wellbore as a function ofwell depth and producing time is primarily given by

119879 = 119879

119890119894+ 120585 [1 minus 119890

119867120585

] (

119892

119892

119888

sin 120579

119869119862

119901

+ 120593 + 119892

119879sin 120579)

+ 119890

(119867minus119911)120585

(119879

119891119887ℎminus 119879

119890119887ℎ)

(7)

where 119879

119890119894is undisturbed formation temperature at given

depth 120585 is inverse relaxation distance 119867 is total welldepth from surface 119892

119888and 119869 represent conversion factors

119862

119901is the mean heat capacity of the multiphase fluid at

constant pressure 119892119879is geothermal temperature gradient

120593 is parameter combining the Joule-Thompson and kineticenergy effects 119879

119891119887ℎis fluid temperature at the bottom hole

and 119879

119890119887ℎis formation temperature at bottom hole Some

important parameters are discussed in detail in Appendix BIn the transient wellbore flow model we consider the

density behavior of drilling fluid under high pressure andhigh temperature conditions during underbalanced drillingoperation A more accurate analytical model for density-pressure-temperature dependence for drilling fluid in drillingproposed by Karstad and Aadnoy [23] was presented in thefollowing manner

120588

119897= 120588

119897119904119891119890

120574119901(119901minus119901119904119891)+120574119901119901(119901minus119901119904119891)2+120574119879(119879minus119879119904119891)+120574119879119879(119879minus119879119904119891)

2+120574119901119879(119901minus119901119904119891)(119879minus119879119904119891)

(8)

where 120588

119897119904119891is the drilling fluid density under standard con-

ditions 119879119904119891

is the standard temperature 120588119904119891

is the standardpressure and the values of 120574

119901 120574

119901119901 120574

119879 120574

119879119879 and 120574

119901119879are

essentially unknown and must be determined for differentmuds from density measurements at elevated pressures andtemperatures

Though we have made an assumption of gas liquid two-phase flow in wellbore the effect of cutting bed in horizontalwell on wellbore flow simulation has to be considered Thecutting bed existing in the horizontal annulus reduces theopen flow area of gas liquid two-phase fluid which is acritical factor in wellbore flow simulation In this paper weconsider the effect of cutting bed by evaluating the open flow

area of gas liquid two-phase fluid in horizontal annuluscausing the movement of cutting bed The cuttings trans-port in horizontal wellbore has been investigated by manyresearchers [24ndash30] In this paper we use a mechanics modelwith aerated drilling fluid proposed by Zhou [30] to predictthe movement of cutting bed and the cutting bed height

32 Numerical Solution In order to solve the model thefinite difference technique which is one of the most widelyused numerical methods is applied in this paper Thetechnique is implemented by replacing all derivatives bydifference quotients Owing to the introduction of analyticalexpression of temperature the model is easier to solve using


the technique Discretization of governing equations can beobtained applying a first order downstream implicit schemefor time derivations For the stability of numerical solutionsthe donor cell concept is used The fluid exit conditions areassumed to be the same as the fluid conditions in the nodeitself The discretized form of governing equations can bewritten in the following matrix form

D119895(k0119895

) k119905+Δ119905119895

= E119895(k0119895

k119905119895

k119905+Δ119905119895minus1

) (9)

whereD is the coefficient matrix E is the independent vector119895 is cell number the superscripts 119905 and 119905+Δ119905 are the dependentvariables calculated at the old and new times respectively thesuperscript 0 represents the dummy variables for the iterativemethod and v is a column vector of dependent variablesgiven as follows

v = (119901 120572

119892 119906

119892 119906

119897)

119879

(10)

where the superscript 119879 indicates the transposeMechanistic models for two-phase flow in drill string and

annuli [31ndash33] are used to provide the necessary informationfor numerical calculation about flow patterns pressure dropsphase volumetric fractions and phase velocities in each oldtime node The wellbore fluid temperature at each grid pointcan be estimated using (7) In this paper we use the algorithmbased on the factorization of a matrix using a version of theGaussian elimination with partial pivoting

It is important to note that the numerical solution istransient due to the fact that almost all the transient termsare considered in our model In addition the numericalsolution of two-phase wellbore flow under constantly chang-ing flowing environmental can be given easily owing to theestablishment of real-time data monitoring system

4 Transient Well Influx Model

In underbalanced drilling the transient well influx can bedeemed as a complex well test The complexity arises fromthe fact that the boundary conditions vary with times invalue and type during the test because the well is con-tinuously increasing in length In the drilled part of thereservoir we have a Dirichlet boundary condition Insteadthe Neumann boundary condition is dealt with along thefuture well trajectory in the reservoir Also keep in mindthat the transient influx rates are influenced by variation inunderbalance and penetration rateTherefore the traditionalwell test techniques are not fit for the purpose of modelingtransient well influx during underbalanced drilling Becauseof the complexity of the problems a realistic model is neededto predict the transient well inflow while drilling

41 Mathematical Model Themodel in this paper representsa simplification of the actual reservoir It is assumed thatthe reservoir is symmetric around the well and possessesconstant reservoir and parameters During the transientperiod we assume that there is no pressure response from theborder of the reservoir For transient flow of a single phase

the governing differential equation can be given in the formof diffusivity equation

nabla

2

119901 =

120601120583119888

119896

120597119901

120597119905

(11)

where 120601 is porosity 119888 is reservoir compressibility 120583 is theviscosity of reservoir fluid and 119896 is the reservoir permeability

With the above assumption that there is no pressureresponse from the border of the reservoir the outer boundaryis given in the following Neumann manner

120597119901

120597119899

= 0(12)

where 119899 is the normal of the boundary surfacesEssentially the main difference of transient influx during

drilling with convention well test operations lies in the innerboundary which changes with time For the part of thereservoir penetrated by a horizontal well we have a Dirichletboundary condition

119901 = 119901

119908ℎ (13)

where 119901119908ℎ

is the pressure at bottom holeFor the undisturbed part of the reservoir the influx rates

are zero in other words the inner boundary conditionslikewise can be captured by the above Neumannmanner It isimportant to note that as a result of the continuous increasein well length the inner boundary conditions do not onlychange in value but in types as well Every drill step the partof the reservoir that is dominated by the Neumann conditiondecreases while the zone governed by theDirichlet conditionincreases This means that we have to deal with a new set ofequations with different boundary conditions for every stepdrill stepThus we are in dire need of new realistic expressionto describe the transient well influx during underbalanceddrilling

42 Approximate Analytical Solution To allow analyticalsolutions of the transient influx problem some essentialhypotheses are made in this paper We assume that thereservoir consists of a number of independent elements orzones as shown in Figure 3 It is assumed that the fluidflows along planes perpendicular to the well During under-balanced drilling a new zone is added to the well everystep in time The size of the time step and the thickness ofdrilled zones must conform to the penetration rate which isconsidered to be constant here It is assumed that productionfrom new added zone starts at the moment when it is beingpenetrated by drilling The total production rate at a giventime can be obtained by adding the production response ofindividual zones that time Thus the total production rate asa function of time can be given by

119902

119905119900119905(119905) = int

119911(119905)

0

119902

119908(119911 119905) 119889119911

(14)


InjectionOutlet

Undisturbedreservoir

Opened reservoir elements

Permeability k1 k2 k3 k4

k1 k2 k3 k4

Figure 3 A schematic of a horizontal well drill into the reservoirconsists of a number of independent elements

where depth 119911 is a function of time 119905 and the penetrationrate 119906

119901 and 119902

119908(119909 119905) is the well influx from a unit thickness

Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)

Using Duhamelrsquos theorem van Kruijisdijk and Cox [6]proposed that the transient well influx for a given elementcan be rendered to a convolution of the underbalance andthe time derivative of the well influx for a constant andunit underbalance and rewrote the above equation in thefollowing manners

119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902

119908119906(119905) denotes the unit-underbalanced inflow of unit

reservoir segment at the given time Θ represents the setof reservoir parameters that define the flow geometry andproperties 119901

119903119890119904is the reservoir pore pressure and 119901

119908is a

function of time and is defined as

119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)

The function 119905open(119911) represents the time atwhich the drillbit penetrates point 119911 thereby opening that point along thewell trajectory up to flow

To obtain an analytical solution of transient well inflowat any given time we need a realistic expression of the unit-underbalance transient inflow of each unit thickness segment119902

119908119906(119905) Based on the appropriate source functions in Laplace

space van Kruijisdijik [34] proposed the following analyticalsolution

119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)

where ℎ is the reservoir thickness 119903119908is the radius of the

wellbore119861119900is the volume factor of the reservoir fluid 119878

119911is the

effective skin due to vertical flow 1198861and 119886

2are the coefficient

and have respectively the values of 0611 and 1083 in the earlytime of transient influx

For gas reservoir the expression of the transient inflowis a bit different from (16) the analytical solution for oilreservoir proposed by van Kruijisdijk and Cox [6] Thecompressibility of gas decides the influx difference betweenoil and gas reservoirWe define the pseudopressure as follows

= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901

0is the known pressure for reference 120583

119892is the gas

viscosity and 119885 is the compressibility factor of the gas Thenthe total influx rates of gas reservoir during underbalanceddrilling can be given

119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)

Analogously we rewrite the analytical expression of 119902119908119906

forgas reservoir in the following manner

119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)

where the subscript denotes the standard state

5 Methodology of EstimatingReservoir Characterization

After establishing datamonitoring system in real time a tran-sient liquid-gas two-phase flow model in wellbore and wellinflux model during drilling have been derived Hereuntowe have reliable data acquisition approach and theoreticalmodel for reservoir characterization In this section let usfocus on specific approach to estimate reservoir properties Atfirst we will propose a new methodology to acquire reservoirpore pressure and then discuss how to extract the reservoirpermeability profiles during drilling

51 Reservoir Pore Pressure Inspired by the interestingmethodology of identifying reservoir pore pressure proposedby Kneissl [12] a new improved method of testing reservoirpore pressure is developed using different testing procedureavoiding the uncertainties affected by variations of fluid flowbehavior in the annulus sections and unpredictable inflowof new added zones It is noted that the idea of identifyingreservoir pore pressure by actively fluctuating the bottomhole pressure is still the key kernel thought of our methodFocusing on the new methodology a principal assumptionhas to be made that the reservoir drilled consists of differentzones with constant formation pressure coefficient whichcorresponds with the most common circumstances Withthe continuous pressure system assumption it is easy to


obtain reservoir pore pressure of each zone by extracting thereservoir pressure coefficient when the reservoir is openedThis greatly simplifies the procedure of pore pressure testand avoids complex mathematical computation usually usedin other methods To eliminate the uncertainties of estimateresults introduced by continuous inflow of new zones it isneeded to stop drilling but keep circulation to begin thetesting operation when the first reservoir zone was openedwhich is the difference from Kneisslrsquos method in testingprocedure Taking into account the fact that only the first zonewas opened in the testing operation the transient inflow ofwell can be given by

119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)

It can be rewritten as

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)

At very short time the expression reduces to the followingmanner

lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)

which is entirely independent of the time and the downside of the equation can be considered constant for a givenreservoir It also can be interpreted that the transient inflowof well has a significant negative linear correlation with thepressure at bottom hole in the earliest hours of inflow Forthe inflow of well to become zero the reservoir pore pressureis approached by the pressure at bottom hole Thence wecan identify the reservoir pore pressure by searching forthe linear correlation between the inflow and pressure atbottom at the early stage of transient inflow This processis performed by introducing artificially fluctuations in thebottom hole pressure and by measuring the inflow rates andbottom hole pressure based on the real-time data monitoringsystem For aerated drilling the bottom hole pressure can beestimated using the transient wellbore flowmodel mentionedin this paper Figure 4 illustrates some details concerninghow to extract the reservoir pore pressure Here linear fittingof some measurement points is used to identify the linearcorrelation between thewell inflow and bottomhole pressureand the reservoir pore pressure can be read at the crossingof the best-fit regression line and the horizontal axis Forthe reliability of test results the number of testing pointsshould not be less than three Fluctuations of the bottom holepressure can be introduced by managing the choke valvechanging gas injection rates or pump rates To eliminatethe effects of variations of fluid flow in the wellbore onthe estimated results it is necessary to leave enough timeto capture relative steady inflow rates for each fluctuating

High

High

Low

0

qw

pw

Testing dataBest-fit regression line

Reservoir pressure

Figure 4 Reservoir pore pressure test by introducing fluctuation atthe bottom hole pressure

operation and the time needed lasts for a few minutes ormore than one hour which depends on well depth the typeof operations introducing the fluctuation in the bottom holepressure reservoir characteristics such as permeability andformation fluid properties and so on

Having acquired the reservoir pore pressure of the firstzone when the reservoir is opened the reservoir pressurecoefficient and pressure of subsequent zones can be obtainedduring drilling using a simple formula For the case wheremore than one reservoir with different pressure system wasdrilled new testing procedure should be repeated when newreservoir was opened The only difference is that the inflowmeasurements consist of the transient inflow response ofthe former reservoir and the transient well inflow of newreservoirrsquos first zone Here necessarymathematical treatmentof the inflow measurements is needed

52 Permeability Profiles To determine the reservoir perme-ability we make the simplifying assumption that the perme-ability is constant in specified zones of the well Transientinflux from each reservoir zone with different permeabilityat the time of interest constitutes the total production rates ofthewell which can be got from the real-time data acquisitionssystem The analytical expressions of total production ratesand unit-underbalance response of each zone which arerelated to the permeability have been established in the lastsection of this paper The reservoir pore pressure of eachreservoir zone can be calculated after testing the reservoirpressure coefficient just when the reservoir was opened Itis assumed that the porosity reservoir height and otherreservoir parameters apart from permeability are constantand known Theoretically the permeability of each reservoirzone can be easily determined using (18) and (21) Unfor-tunately the analytical expressions of the total productionrates are given in integral form which results in that theycannot readily be implemented in spreadsheet and calculator


applications To facilitate the calculation a full discretizationof the total production rates formula is given instead of theintegral form in the following manner

119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)

where Δ119905 represents the time needed to penetrate througha reservoir element which is constant with the assumptionof the fixed penetration rate 119902

119908119906119894(Θ

119894 119905 minus 119905

119894) denotes unit-

underbalanced inflow of unit reservoir segment from zone 119894and 119905

119894denotes the time the wellbore reaches zone 119894 The value

of 119902119908119906119894

(Θ

119894 119905 minus 119905

119894) for nonpositive arguments is set equal to 0

The above discretization equation was given on the basisof a reasonable assumption that we have a constant underbal-ance which is realistic and is of engineering significance forunderbalanced drilling of a horizontal well

First we discuss how to extract the permeability of the1st reservoir zone The total production rates from the real-time monitoring system at the first time step when reservoiris opened can be simply considered to be the influx from thefirst zone Then we have the following formula

119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)

Solving the equation above 1198961 the permeability of the 1st

reservoir zone can be given as follows

119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)

Analogously well influx from the reservoir element 119894 canbe given

119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)

where 119902tot(119905119894) represents the total production rates of the wellat time 119905

119894when the reservoir element is opened 119902

119908119906119899(119905

119894minus 119905

119899)

denotes the unit-underbalance transient influx response ofunit thickness reservoir from reservoir zone 119899 at time 119905

119894and

can be given using (21)Then we can extract the permeability of the reservoir

zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)

Using the methodology the permeability of a set ofreservoir zones can be successively extracted as the reservoiris being penetrated by a horizontal well To facilitate theprecise calculation and estimate permeability profiles in timea computer procedure was used in this paper The success ofreservoir permeability estimation profits from reliable real-time data monitoring system advanced wellbore flow andreservoir model and an innovative reservoir pore pressuretest approach

6 Results and Discussion

The methodology presented in this paper has been used toidentify the reservoir pore pressure and permeability profilesof approximately 10 wells during underbalanced drilling inChina In order to illustrate the above approach in detail anactual horizontal well and gas reservoir are considered in thissection It is appropriate to emphasize that the well picked ascase study is the first successful application of this method-ology More exhaustive data and offset information from thiswell facilitate the effectiveness evaluation of this methodThereservoir dealt with here is the Ordovician carbonate gasreservoir with natural fractures as the dominant gas storagespaces and flowing channels To avoid lost circulation andformation damage the well is drilled underbalanced usingthe aerated clay-free drilling fluid while this is also beneficialfor well control Here nitrogen is used as the injection gasThe hole geometry and drilling assembly of this well wereillustrated in Figure 5 Some basic parameters for reservoirand drilling fluid used in reservoir characterization wereshown in Table 1

61 Data Acquisition in Real Time The real-time data mon-itoring system for aerated drilling presented in this paperwas used to acquire basic data for reservoir characterizationin real time The application of this system guaranteesthat the indispensable parameters are accurate and timelywhichmakes it possible to identify reservoir characterizationDuring drilling this system succeed in the integration of thedata from compound logging and our sensitive measuringinstruments and the digital microwave transmission wasproved to be realizable and reliable to deliver wirelesslydata Some typical acquired measurements plotted againsttime are shown in Figures 6ndash10 In order to illustrate thetransient behavior for different parameters the measuringtime for different parameters in figures differs The standpipe


Drill bit

Cement

Casing

Drill strings

Wellhead

Bottom holeBarefoot interval

Annulus

Protective casing interval

Drill strings size

Diameter 01524m

Diameter 01524mDepth 4000ndash4960m

Inner diameter 022m

Outside diameter 01397mInner diameter 01214m



Depth 0ndash

Depth 0ndash1800m

Depth 1800ndash4930m


4000m

Figure 5 The hole geometry and drilling assembly of the well

Table 1 Basic parameters for reservoir and drilling fluid used inreservoir characterization

Depth evaluation of the reservoir m 4156Reservoir thickness m 15Porosity 12Reservoir fluid volume factor m3stdm3 0008Reservoir fluid viscosity mPasdots 0035Reservoir compressibility Paminus1 116 times 10minus9

Reservoir fluid compressibility Paminus1 1177 times 10minus9

Geothermal temperature Ksdotmminus1 003Drilling fluid density at standard conditions gsdotcmminus3 104Drilling fluid viscosity at standard conditions mPasdots 20Injection gas specific gravity 1 097Injection gas viscosity at standard conditions mPasdots 0019

pressure and casing pressure for different time are plottedrespectively in Figures 6 and 7 It can be seen that thetransient fluctuations of standpipe pressure are more acutethan that of casing pressure In general this difference canbe aroused by unsteady drilling fluid and gas injection checkcalve measuring apparatus installed along drill string anddrill string vibrationThe casing pressure changing with timein Figure 7 indicates the different outlet gas rates or variedchoke opening Figure 8 shows the drilling fluid density asa function of time for outlet and injection respectivelyContrary to the standard drilling operations more accurateand automatic measurements of drilling fluid density can beobtained in real time utilizing instrumented standpipe andseparator drilling fluid outlet pipe using different pressure

2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930

Figure 6 Data acquisitionmdashstandpipe pressure data versus time(2010-9-27)

1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000

Figure 7 Data acquisitionmdashcasing pressure data versus time (2010-9-27)

transducers not measured manually in intervals of 15 min-utes It is noted that the outlet drilling fluid density is slightlylower than that of injection fluid and fluctuates wildly Thistransient behavior of outlet drilling fluid density occurs as aresult of varied outlet gas rates and nonthorough degassing ofdrilling fluid In this study we monitor the total hydrocarboncontent of outlet gas and outlet gas rates to acquire thereservoir gas inflow rates in real timeThemeasured outlet gasrates and reservoir gas inflow rates are presented in Figure 9The real-time reservoir gas inflow rates absolutely criticalfor reservoir characterization are calculated using the outletgas rates and total hydrocarbon content in time of interestas shown in Figure 10 In sum the application of the newreal-time data monitoring system in this well helps drillingcontractor with good underbalanced drilling operations and


1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111

Outlet drilling fluid densitiesInjection drilling fluid densities

Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)

Figure 8 Data acquisitionmdashdrilling fluid density data versus time(2010-9-25)

0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time

Outlet gas ratesGas inflow rates

Gas

rate

s(m

3middoth

minus1)

Figure 9 Data acquisitionmdashoutlet gas rates and gas inflow ratesdata versus time (2010-9-25)

provides accurate and more complete data set in real time forreservoir characterization during drilling

62 Reservoir Characterization In this paper the reservoircharacterization we are interested in refers to the reservoirpore pressure and reservoir permeability It is assumed thatthe reservoir dealt with here consists of a number of differentzones each 1m in thickness With each zone the reservoirpore pressure and permeability are constant In this sectionwe discuss the details of identifying the reservoir porepressure and permeability and their comparison with theactual data

230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000

Figure 10 Data acquisitionmdashtotal hydrocarbon content data versustime (2010-9-25)

For this well the reservoir was opened at the well depthof 4880m when visible gas inflow was detected for the firsttime And then we stopped drilling but kept circulation tobegin the reservoir pore pressure test procedure Here thefluctuations of the bottom hole pressure were introduced bymanually accommodating the choke valve without drillingfluid and gas injection changes After trying for some timesthree testing points with relative steady inflow lasts for morethan one hour were observed and illustrated in Figure 11 Foraerated drilling used in this well the bottom hole pressurefor three testing points can be estimated using the transientwellbore flow model The important input parameters in thewellbore flow simulation such as surface casing pressureoutlet gas and drilling fluid rates and outlet drilling fluiddensity were provided by the data monitoring system shownin Figure 1 in real time For the three testing points thetime-averaged gas inflow rates and bottom hole pressure andtheir best-fit regression line are plotted in Figure 12 With thefitted regression equation in Figure 12 it is easy to extractthe reservoir pore pressure for the first zone to be 4191MPaHaving known the vertical depth of 41397m when the testwas conducted the reservoir pressure coefficient can beobtained to be 1032 whichmeets very closely the actual valueof 103 obtained from well-test data after drilling The mea-sured inclination at the well depth of 4870 before beginningunderbalance drilling operation is 885∘ For aerated drillingthe hole inclination cannot bemeasured in real-time with theuse ofMWD It is appropriate tomake the assumption that wehave the same hole inclination for the underbalance drillinginterval with length of 80m Considering the case here thatthe reservoir with consistent pressure system was drilled by ahorizontal well it is easy to obtain the reservoir pore pressureprofile of thewell as shown in Figure 13 It is noted thatwe usethe actual reservoir pressure coefficient from well-test dataafter drilling Thus the good match between the estimatedreservoir pressure and actual data indicates the feasibility


003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800

Steady inflow fornumber 3 point


Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)

Figure 11 Three segments of steady inflow obtained by introducingfluctuation at the bottom hole pressure

37 38 39 40 41 42 430

100

200

300

400

500

600

Pressure at bottom hole (MPa)Tested dataBest-fit regression line

Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)

Figure 12 Gas inflow rates versus pressure at bottom hole and thebest-fit regression line

and reliability of the reservoir pressure test methodologypresented in this paper

Having obtained the reservoir pore pressure profile wecan extract the permeability of every zone from real-timemonitoring data using the methodology presented in thispaper To be candid this is not a simple event For everyzone complex mathematical computations are required toidentify permeability Some necessary mathematical equa-tions needed for computation have been mentioned in theabove section in this paper Basic parameters for reservoirand drilling fluid and the hole geometry and drilling assemblyof the well required for reservoir characterization havebeen given in Table 1 and Figure 5 respectively The datamonitoring system provided the real-time data required such

4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)

Figure 13 Reservoir pore pressure profile estimated using theimproved methodology

4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)

Well depth (m)EstimateActual

1E minus 4

1E minus 3

Figure 14 A comparison between estimated and actual permeabil-ity profiles

as casing pressure injection and outlet gas rates injection andoutlet drilling fluid density and injection drilling fluid rateswhen performing reservoir characterization for every zoneThe estimated and actual permeability profiles are indicatedin Figure 14 and show a good agreement in magnitude onthe whole However we observe that there is a differencebetween the estimated and actual values of the permeabilityfor different zones In this example the actual permeabilitywas obtained using the multi-interval test after drillingand given in the form of average value for every intervalTo be different from the actual permeability the separatepermeability of every unit reservoir thickness was estimatedby dividing the reservoir into discrete zones with thickness of1m which is an outstanding originality of the methodology


4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5

UnderbalancePenetration rate

Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)

Figure 15 Underbalance and penetration rates profiles

presented in this paper This is the major element causingthe difference between the estimated and actual permeabilityprofiles in Figure 14 Beyond this it is also observed that thequality of the reservoir characterization varies in differentzones The unconstant underbalance and penetration rateshown in Figure 15 might result in the difference In thispaper the mathematical expressions estimating the reservoirpermeability were derived with the constant underbalanceand penetration rate assumption The variable penetrationrates and underbalance cause the results that the estimatedpermeability are slight less accurate than those obtained withconstant penetration rates and underbalance [6] Besides theestimated results of the near zones opened subsequently arealso affected However the quality of the reservoir character-ization is still acceptable Low underbalance and penetrationrates are a benefit when performing characterization whichare also important for thewell control In the drilling practicethe penetration rates are easy to control but it is hard tokeep constant underbalance and penetration rate through allthe process and the only thing we could do was to restrictthe variation of the underbalance In order to validate thereservoir characterizationmethod proposed in this study thegas simulated outlet rates at a time when a high permeablezone was penetrated at a depth of 4907m were comparedin Figure 16 with the monitoring data The estimated perme-ability for different reservoir zones at depths from 4880m to4907m shown in Figure 14 was used to obtain the bottomhole gas influx rates in the simulation The real-time surfacecasing pressure measured using the data monitoring systembefore and after the zone of interest being opened was usedas the outlet boundary condition Some other importantparameters in the wellbore flow simulation were given inTable 2 The comparison is good showing that the estimatedpermeability is accurate and the transient well influx modeland multiphase flow model can be used to simulate thetransient wellbore flow during underbalanced drilling It isnoted that often there is no need to predict the outlet gas rateswhen performing the reservoir characterizationThe purpose

0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200

Time (s)Simulated dataActual data

Gas

out

let r

ates

(m3middoth

minus1)

Figure 16 Simulated and actual gas outlet rates when the highpermeable zone at the depth of 4907m was penetrated

Table 2 Some required parameters used in the wellbore flowsimulation

Well depth of interest m 4907Injection drilling fluid density gsdotcmminus3 104Injection drilling fluid rates Lsdotsminus1 18Injection drilling fluid viscosity 20Gas injection rates m3

sdotminminus1 20Gas injection pressure MPa 154Gas injection temperature K 307Surface temperature K 300

of the outlet gas flow rates estimation in Figure 16 wasto reversely validate the reservoir characterization methodusing the estimated permeability profile In the standardprocedure of reservoir characterization the real-time outletgas rates and casing pressure measured using the datameasuring system are required in the transient wellbore flowsimulation as boundary outlet conditions to estimate thebottom hole pressure which is very important for extractingthe permeability of the reservoir zones

It is noted that a new multiphase flow model was appliedto simulate the wellbore flow behavior for aerated drillingwithout PWD data Effects of drilling fluid density behaviorand wellbore transfer were involved in the new model Thevalidity and practicability of new model are verified bygood agreement between simulated and monitoring resultsas shown in Figures 16 and 17 Here we discuss the effects ofdrilling fluid behavior and wellbore transfer on the wellboreflow pressure profile in the annulus During drilling opera-tions the annulus fluid being lower than the formation tem-perature receives heat from the formation In the meantimethe formation near the wellbore is also cooled by the flowingfluid in the annulusThe wellbore temperature in the annulushas a direct bearing on the drilling fluid density and the gas


4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)

Well depth (m)Surface casing pressureEstimated standpipe pressure Measured standpipe pressure

Figure 17 Time-averaged measured and estimated standpipe pres-sure and surface casing pressure when every zone were beingpenetrated

compressibility distributions along the annulus which arevery important for thewellbore flow simulation In this paperthe effects of the drilling fluid density were considered byinvoking the temperature and drilling fluid density using theanalytical models in (7) and (8) respectively The estimatedwellbore flow pressure profiles in the annulus for differentdrilling fluid circulation time were plotted in Figure 18 Itis noted that the drilling fluid circulation time here wasonly used for estimating the wellbore temperature profilesin the annulus In the simulation we fixed the well depth at1960m outlet drilling fluid rates at 16 Lsdotsminus1 and the casingpressure at 18MPa respectively The outlet gas rates wereassumed at a very low level with 001m3sdotsminus1 to eliminatethe effect of gas compressibility induced by the differenttemperatures Some important parameters for drilling fluidcan be read in Table 1 In contrast with constant drilling fluiddensity in traditional model the simulated results of wellborepressure using new model are observed being slightly lessin the lower part of the wellbore indicated in Figure 18The results also show that the wellbore pressure profiles aredifferent for different circulation time especially in bottomhole We can explain this by inspecting the temperatureprofiles change in Figure 19 Table 3 reports the values of thethermophysical properties used in the wellbore temperaturecalculation Caused by the cooling effect the rate of wellboreheat transfer diminishes with increasing circulation timeConsequently the temperature in wellbore declines withcirculating and engenders eventually the changes of densityprofile in Figure 20 which determines the wellbore pressuredirectly Data of a water-based drilling fluid from Isambourget al [15] reproduced in Table 4 are used here to calculate thedrilling fluid density

The results of the example show that the methodologyproposed here can estimate the reservoir pore pressure and

0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)

Well depth (m)Constant density

4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h

Figure 18 Wellbore pressure profiles simulated using traditionalmodel and new model for different circulation time

0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)

Well depth (m)24h240h

1200 h

Figure 19 Temperature profiles simulated using new wellbore flowmodel for different circulation time

the permeability in real time and the estimated resultsare able to match the actual data to an acceptable levelHowever owing to the complexity of the subject someessential facets are necessary to provide to improve the qualityof the reservoir analysis when performing estimation ofpermeability and reservoir pore pressure during underbal-anced drilling First and most important the reservoir porepressure testing procedure should be initiated immediatelyonce visible gas flow from the reservoir is detected by thereal-time data monitoring system which indicates that gasreservoir has been opened For the reason that the linearcorrelation between the inflow and pressure at bottom exists


Table 3 The values of the thermophysical properties for the calculation of temperature profile

119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888

kgsdotmsdotNminus1sdotsecminus2119869

msdotNsdotJminus1119908

kgsdotsminus1119896

119890

Wsdotmminus1sdotKminus1119862

119901

Jsdotkgminus1sdotKminus1ℎ

119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)

Figure 20 Drilling fluid density profile simulated using newwellbore flow model for different circulation time

Table 4 Empirical constants used in the drilling fluid densitycalculation (from Isambourg et al [15])

120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1

2977 minus2293 minus1957 minus16838 0686

only at the earliest hours of transient inflow it is hard toacquire accurate estimation of the pore pressure once the bestopportunity is missed Moreover it is important to maintainthe wellbore in underbalance conditions while the reservoiris being penetrated and the underbalance should be smalland relatively stable The fluids can flow into the wellborefrom reservoir and be circulated out from the bottom holeonly when the well is drilled underbalance This is thefundamental premise of the reservoir characterization Thevariable underbalance introduces a further complication inthe reservoir analysis and produces less accurate estimatedresults of the permeability Thus the underbalance should bekept relatively stable by changing gas drilling fluid injectionrates or regulating the choke valve For better estimation ofthe permeability and well control low underbalance is alsoneeded In addition keeping constant and low penetrationrates is a benefit when performing reservoir characterizationduring underbalanced drilling For drilling operation thepenetration rates are relatively easy to control by regulatingartificially bit-weight in real time

7 Conclusion

In this study a methodology for estimating reservoir charac-terization during underbalanced drilling of horizontal wellshas been presented The main reservoir characterizationconsidered here is the reservoir pore pressure and per-meability profiles The methodology is established using amultiphase wellbore flowmodel merged with a transient wellinflux model of a horizontal well on the basis of the real-time data monitoring system for aerated drilling and liquid-based underbalanced drilling The methodology is appliedto estimate reservoir pore pressure and permeability Animportant aspect of themethodology presented is that we usean improved method to identify the reservoir pore pressurebased on the Kneisslrsquos methodology using a different testingprocedure

This methodology has been applied in approximately10 wells in China The estimated reservoir pore pressureand permeability are compared with the actual values fromwell test data and show a good agreement Owing to theintroduction of the models simulating the drilling fluiddensity behavior and wellbore heat transfer the multiphasewellbore flow model can be used to estimate the wellborebehavior more accurately which is very critical for reservoircharacterization during aerated underbalanced drilling

Appendices

A Closure Relationships of the TransientWellbore Flow Model

A1 Drag Forces In this paper the drag force acting on agas particle under steady state conditions can be expressedas follows

119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862

119889119892is the drag coefficient given by Ishii and Mishima

[35] as the following expressionsFor the bubbly flow

119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877

119901119892is the gas particle ratio and 120590

119904is the gas-liquid

interfacial tension given by Park et al [31]

120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)


According to the work of Lahey [36] the bubble radius119877

119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)

where 119881 is the total volume of every numerical nodeFor the slug flow

119862

119889119892= 98(1 minus 120572

119892)

3

(A5)

In this paper the flow patterns are confirmed using theflow pattern transition criteria from the work of Barnea [37]

A2 Virtual Mass Force The virtual mass force expresses themomentum exchange caused by relatively acceleratedmotion According to Lahey [36] the most common virtualmass force terms for gas phase can be expressed as

119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)

where 119862V119898 denotes the virtual mass coefficient of gas phasegiven by Ishii and Mishima [35]

for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871

119905119887are respectively the long bubble diameter

and length of the bubble Taylor shown in Figure 21 Twoparameters can be calculated according thework ofKaya [38]

A3 Wall Shear Force In the present paper the wall shearforces in both bubbly flow and slug flow are discussed Forthe bubbly flow thewall shear force for gas phase is neglectedand the liquid wall shear stress is presented byWongwised etal [39] as follows

Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)

Then the liquid wall shear force can be given by

119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)

where119860 is the cross sectional area of the pipe 119878119897is the liquid-

wall wetted perimeter and 119889 is the diameter of the pipeFor the slug flow the effect of gas wall shear is no longer

neglected According to the work of Cazarez-Candia et al

u tbuuuuuuuuuuuuuuuu tttttttttttttttbbbbbbbbbbbbbbbbbb

Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw

Figure 21 A schematic of key geometrical parameters in horizontalslug flow

[40] the wall shear forces for gas and liquid phase are cal-culated respectively as

119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)

where 119860 is the area 119878 is the perimeter and Γ is the shearstress Parameters 119871

119897119904 119871119904119906

are slug and slug unit lengthsrespectively The parameter 120595

1is used to modify the model

according to the deviation angle The subscripts 119892 119897 119894 and119908 denote gas liquid interface and wall respectively In thesame way the above geometrical parameters according to theregion in Figure 21 can be calculated using the means in theKayarsquos paper [38]

Shear stresses for angles from 0

∘ to 45∘ are given by Issaand Kempf [41]

Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)

where 119891 is friction factor can be given by

119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892

minus02 for Re119892gt 2300

119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)


Gas liquid and interfacial Reynolds numbers are givenrespectively by

Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889

119892are the liquid and gas hydraulic diameter

respectively given as follows

119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)

For deviation angle over 45∘ the liquid wall shear stress isgiven by

Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)

where the friction factor is given by the followingfor laminar flow

119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)

B Parameters of (7)The inverse relaxation distance 120585 is defined by

120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)

where 119908 is total mass flow rate 119903

119905119900is outside radius of

drill string 119896119890is formation conductivity 119880

119905119900is overall heat

transfer coefficient and119879

119863is dimensionless temperatureThe

expression for 119880119905119900can be obtained from

119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ

119888is convective heat transfer coefficient for annulus

fluid 119903119908119887

is outside radius of wellbore 119903119888119900is outside radius

of wellbore and 119896

119888119890119898is cement conductivity

The value of dimensionless temperature can be obtainedusing the following algebraic approximation presented byHasan and Kabir [42]

119879

119863=

(11281radic119905

119863) times (1 minus 03radic119905

119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)

where 119905119863is dimensionless time

The undisturbed formation temperature 119879119890119894 is generally

assumed to vary linearly with depth given as follows

119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879

119890119894119887ℎis undisturbed formation temperature at the

bottom hole which can be easily obtained by invoking theundisturbed formation temperature at the wellhead 119879

119890119894119908ℎ

In this work we use the empirical expression developedby Sagar et al [43] to estimate the value of 120593 as follows

120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865

119892119897is the ratio of gas and liquid and 120574

119892is gas specific

gravity

Conflict of Interests

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

Acknowledgments

The authors are grateful for the support of the NationalNatural Science Foundation of China (no 51104124 no51204140 and Grant no 51134004) Major state science andtechnology special project of China (no 2011ZX05021-003)and PhD Programs Foundation of Ministry of Education ofChina (Grant no 20125121110001)

References

[1] S P Smith G A Gregory N Munro and M Muqeem ldquoAppli-cation ofmultiphase flowmethods to horizontal underbalanceddrillingrdquo Journal of Canadian Petroleum Technology vol 39 no10 pp 52ndash60 2000

[2] K A Fattah S M El-Katatney and A A Dahab ldquoPotentialimplementation of underbalanced drilling technique in Egyp-tian oil fieldsrdquo Journal of King Saud UniversitymdashEngineeringSciences vol 23 pp 49ndash66 2011

[3] E H Vefring G Nygaard K K Fjelde R J LorentzenG Naeligvdal and A Merlo ldquoReservoir characterization duringunderbalanced drilling methodology accuracy and necessarydatardquo in Proceedings of the SPEAnnual Technical Conference andExhibition San Antonio Tex USA September-October 2002Paper SPE 77530

[4] E H Vefring G Nygaard R J Lorentzen G Nœvdal andK KFjelde ldquoReservoir characterization during underbalanced dril-ling (UBD) methodology and active testsrdquo SPE Journal vol 11no 2 pp 181ndash192 2006

[5] C B Kardolus and C P J W van Kruijsdijk ldquoFormation testingwhile underbalanced drillingrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition pp 521ndash528 San AntonioTex USA October 1997

[6] C P J W van Kruijsdijk and R J W Cox ldquoTesting whileunderbalanced drilling horizontal well permeability profilesrdquo


in Proceedings of the SPE European Formation Damage Confer-ence Paper SPE 54717 The Hague The Netherlands May-June1999

[7] L Larsen and F Nilsen ldquoInflow predictions and testing whileunderbalanced drillingrdquo in Proceedings of the SPE Annual Tech-nical Conference and Exhibition Houston Tex USA October1999 Paper SPE56684

[8] F Azar-Nejad ldquoFormation evaluation while underbalanced dri-lling of horizontal wells with coiled tubing technologyrdquo inProceedings of the CSPG and Petroleum Society Joint ConventionDigging Deeper Finding a Better Bottom Line Paper SPE99-53Alberta Canada June 1999

[9] J L Hunt and S Rester ldquoReservoir characterization duringunderbalanced drilling a new modelrdquo in Proceedings of theSPECERI Gas Technology Symposium Paper SPE 59743 Cal-gary Canada April 2000

[10] J L Hunt and S Rester ldquoMultilayer reservoir model enablesmore complete reservoir characterization during underbal-anced drillingrdquo in Proceedings of the SPESPE UnderbalancedTechnical Conference and Exhibition pp 25ndash26 Houston TexUSA 2003 Paper SPE 81638

[11] D Biswas P V Suryanarayana P J Frink and S RahmanldquoAn improved model to predict reservoir characteristics duringunderbalanced drillingrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition pp 1111ndash1121 Denver ColoUSA October 2003 Paper SPE 84176

[12] W Kneissl ldquoReservoir characterization whilst underbalanceddrillingrdquo in Proceedings of the SPEIADC Drilling ConferenceAmsterdamTheNetherlands February 2001 Paper SPEIADC67690

[13] W Kneissl and Y Kuhn de Chizelle ldquoA method of determiningproperties relating to an underbalanced wellrdquo Patent Applica-tion 00243311

[14] G Stewart S Lakshminarayanan J Villatoro and S BoutalbildquoVariable rate UBD production data facilitates reservoir char-acterizationrdquo in Proceedings of the SPEIADCManaged PressureDrilling and Underbalanced Operations Conference and Exhibi-tion pp 32ndash43 Kuala Lumpur Malaysia February 2010

[15] P Isambourg B T Anfinsen and C Marken ldquoVolumetricbehavior of drilling muds at high pressure and high tempera-turerdquo in Proceedings of the European Petroleum Conference pp157ndash165 Milan Italy October 1996

[16] G Aithoff A A Arian A B Kavaipatti G L Varsamis andL T Wisniewski ldquoMWD ultrasonic caliper advanced detectiontechniquesrdquo in Proceedings of the the SPWLA 39th AnnualLogging Symposium May 1998

[17] 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

[18] L A Carlsen GNygaard andR Time ldquoUtilizing instrumentedstand pipe formonitoring drilling fluid dynamics for improvingautomated drilling operationsrdquo in Proceedings of the 1st IFACWorkshop on Automatic Control of Offshore Oil and Gas Pro-duction (ACOOG rsquo12) pp 217ndash222 Trondheim Norway June2012

[19] A D Spalding and D Middleton ldquoOptimum reception in animpulsive interference environmentmdashpart II incoherent recep-tionrdquo IEEE Transactions on Communications vol 25 no 9 pp924ndash934 1977

[20] R T Lahey Jr andDADrew ldquoThe three-dimensional time andvolume averaged conservation equations of two-phase flowrdquoAdvances in Nuclear Science and Technology vol 20 no 1 pp1ndash69 1989

[21] A R Hasan and C S Kabir ldquoAspects of wellbore heat transferduring two-phase flowrdquo SPE Production amp Facilities vol 9 no3 pp 211ndash216 1994

[22] I N Alves F J S Alhanati andO Shoham ldquoAunifiedmodel forpredicting flowing temperature distribution in wellbores andpipelinesrdquo SPE Production Engineering vol 7 no 4 pp 363ndash3671992

[23] E Karstad and B S Aadnoy ldquoDensity behavior of drilling fluidsduring high pressure high temperature drilling operationsrdquo inProceedings of the IADCSPE Asia Pacific Drilling TechnologyPaper SPE 47806 Jakarta Indonesia September 1998

[24] H Cho S N Shah and S O Osisanya ldquoA three-segmenthydraulic model for cuttings transport in coiled tubing hori-zontal and deviated drillingrdquo Journal of Canadian PetroleumTechnology vol 41 no 6 pp 32ndash39 2002

[25] J Li and S Walker ldquoSensitivity analysis of hole cleaning para-meters in directional wellsrdquo SPE Journal vol 6 no 4 pp 356ndash363 2001

[26] G Espinosa-Paredes and O Cazarez-Candia ldquoTwo-regionaverage model for cuttings transport in horizontal wellbores Itransport equationsrdquo Petroleum Science and Technology vol 29no 13 pp 1366ndash1376 2011

[27] G Espinosa-Paredes and O Cazarez-Candia ldquoTwo-regionaverage model for cuttings transport in horizontal wellbores IIinterregion conditionsrdquo Petroleum Science and Technology vol29 no 13 pp 1377ndash1386 2011

[28] L Zhou R M Ahmed S Z Miska N E Takach M Yu andM B Pickell ldquoExperimental study and modeling of cuttingstransport with aerated mud in horizontal wellbore at simulateddownhole conditionsrdquo in Proceedings of the SPE Annual Tech-nical Conference and Exhibition pp 1051ndash1062 Houston TexUSA September 2004 Paper SPE 90038

[29] R J Avila E J Pereira S Z Miska and N E Takach ldquoCor-relations and analysis of cuttings transport with aerated fluidsin deviated wellsrdquo SPE Drilling and Completion vol 23 no 2pp 132ndash141 2008

[30] L Zhou ldquoHole cleaning during underbalanced drilling inhorizontal and inclined wellborerdquo SPE Drilling and Completionvol 23 no 3 pp 267ndash273 2008

[31] 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

[32] N Petalas and K Aziz ldquoA mechanistic model for multiphaseflow in pipesrdquo Journal of Canadian Petroleum Technology vol39 no 6 13 pages 2000

[33] A C V M Lage Two-phase flow models and experiments forlow-head and underbalanced drilling [PhD thesis] StavangerUniversity College Stavanger Norway 2000

[34] C P J W van Kruijsdijk ldquoSemianalytical modeling of pressuretransient in fractured reservoirsrdquo in Proceedings of the SPEAnnual Technical Conference and Exhibition Houston TexUSA October 1988 Paper SPE 18169


[35] M Ishii and K Mishima ldquoTwo-fluid model and hydrodynamicconstitutive relationsrdquo Nuclear Engineering and Design vol 82no 2-3 pp 107ndash126 1984

[36] R T Lahey Boiling Heat Transfer Modern Developments andAdvances Elsevier Amsterdam The Netherlands 1992

[37] D Barnea ldquoA unified model for predicting flow-pattern tran-sitions for the whole range of pipe inclinationsrdquo InternationalJournal of Multiphase Flow vol 13 no 1 pp 1ndash12 1987

[38] A S Kaya Comprehensive mechanistic modeling of two-phaseflow in deviated wells [MS thesis] The University of TulsaTulsa Okla USA 1998

[39] S Wongwises A Pornsee and E Siroratsakul ldquoGas-wall shearstress distribution in horizontal stratified two-phase flowrdquoInternational Communications in Heat and Mass Transfer vol26 no 6 pp 849ndash860 1999

[40] O Cazarez-Candia O C Benıtez-Centeno and G Espinosa-Paredes ldquoTwo-fluid model for transient analysis of slug flow inoil wellsrdquo International Journal of Heat and Fluid Flow vol 32no 3 pp 762ndash770 2011

[41] R I Issa and M H W Kempf ldquoSimulation of slug flowin horizontal and nearly horizontal pipes with the two-fluidmodelrdquo International Journal of Multiphase Flow vol 29 no 1pp 69ndash95 2003

[42] A R Hasan and C S Kabir ldquoHeat transfer during two-phase flow in wellboresmdashpart I Formation temperaturerdquo inProceedings of the the SPE Annual Technical Conference andExhibition vol 22866 Dallas Tex USA 1991

[43] R Sagar D R Doty and Z Schmidt ldquoPredicting temperatureprofiles in a flowing wellrdquo SPE Production Engineering vol 6no 4 pp 441ndash448 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Moreover the exhaustive and reliable monitoring data arenot easy to be acquired in real time To the best of theauthorsrsquo knowledge the idea that reservoir characterizationwhile underbalanced drilling was first presented by Kardolusand van Kruijsdijk [5] in 1997 In their paper a transientreservoir influx model of a vertical well under a constantunderbalance based on the boundary element method wasdeveloped They compared the model with a transient ana-lytical reservoir model and demonstrated that the transientanalytical reservoir model could be used for evaluatingcharacteristic parameters in the reservoir in particular thepermeability In a following study van Kruijsdijk and Cox [6]developed a method for identifying the permeability profilesalong the well trajectory of a horizontal well considering thevariable underbalance and themore complex outer boundaryconditions of a horizontal well

Reservoir characterization during underbalanced drillinghas been investigated by several other researchers in recentyears Larsen and Nilsen [7] proposed a simple quasi-stationary model for predicting the inflow at various levelsof complexity and accuracy for spreadsheet and calculatorapplications based on drilling speed and knowledge of theratio of vertical to horizontal permeability The influx pre-dictions calculated by their model and a numerical reservoirsimulator were compared giving a good agreement Azar-Nejad [8] presented a model for influx predictions using thediscrete flux element method In his study he mentionedfactors affecting influx rates and indicated that the solutionsof model are very sensitive to the reservoir thickness Huntand Rester [9 10] developed and solved new reservoirmodelsdescribing the underbalanced drilling problem and charac-terizing the dynamic changes in single layer andmultilayeredreservoirs They also presented a history matching methodfor predicting the reservoir permeability Biswas et al [11]proposed a genetic search algorithm for identifying thereservoir characterization accounting for storage effects inboth the wellbore and reservoir In a paper by Vefring et al[3] a transient reservoir model coupled to a transientwellbore flow model was used They estimated the reservoirproperties using novel Levenberg-Marquardt algorithm anonlinear least-squares optimization method In a follow-up research [4] the ensemble Kalman filter was also usedto predict reservoir pore pressure and reservoir permeabilitywhile applying active tests and comparedwith the Levenberg-Marquardt algorithm A paper by Kneissl [12] presented amethod of simultaneously deriving reservoir pore pressureand permeability profiles in real time during underbalanceddrilling by introducing fluctuations in the bottom holepressure during drilling However the variations of fluid flowbehavior in the wellbore might pose difficulties in estimatingthe influx rates which may lead to large uncertainties inthe estimate of reservoir pore pressure [13] Stewart et al[14] proposed an integrated transient model of the formationand the wellbore based on superposition The method ofmatching the cumulative production data for estimatingpermeability using iterative search was also developed intheir paper However an assumption that the pore pressureis known independently was made analogously like mostof above researchers The problem at hand is complex [11]

Although many models and methods for identifying reser-voir characterization while underbalanced drilling have beenproposed the interpretation of reservoir properties is difficultin real reservoirs particularly in real time except for thesimplest cases

This paper describes a new pursuit of the complexproblem of identifying the reservoir characterization duringunderbalanced drilling of horizontal wells A model oftransient well influx of horizontal wells is used together witha multiphase wellbore flow model The effects of the densitybehavior of drilling fluid and wellbore heat transfer duringunderbalanced drilling are considered in our wellbore flowmodel Based on the Kneisslrsquos methodology an improvedmethod of estimating reservoir pore pressure performedby introducing fluctuations in the bottom hole pressure bymanipulating the choke valve opening and changing fluidcomposition or pump rates is presented in this paper Aneffort is also made to monitor accurately and automaticallymeasurements required for reservoir characterization in realtime

The paper is organized by first presenting the real-timedata monitoring Then the transient wellbore flow modeland transient well influx model are presented respectivelyBased on the data monitoring and mathematical modelswe then propose the methodology for estimating reservoircharacterization In this paper the reservoir pore pressureand permeability profiles are the main reservoir propertiesneeded to be characterizedThemethodology is then appliedto a realistic case and results are discussed Thereafter aconclusion is made

2 Real-Time Data Monitoring

The accurate and automatic acquisition of basis data inreal time has been a challenge for reservoir characteriza-tion during drilling of horizontal wells which is crucialand is a prerequisite During a standard underbalanceddrilling operation lots of indispensable parameters can berecorded by the compound logging and other measuringequipment such as the density and rheology of the drillingfluid well depth penetration rate pump pressure outletrates trajectory of the well and wellbore diameter Howeverthese measurements are insufficient and not accurate enoughfor reservoir characterization Being more pivotal most ofthese measurements cannot be acquired in real time whichis indispensable to identifying reservoir characterizationduring drilling This paper has presented a dedicated fullyautomated control real-time data monitoring system basedon the integration and modification of existing devices orcomponents and the installation of some more indispensablesensitive and accurate measuring instruments In addition toinstalling somemeasuring equipment monitoring indispens-able parameters for reservoir characterization unrecordedgenerally in conventional drilling the data from com-pound logging and PMWDMWD are involved in our real-time monitoring system It is believed that the integrationis also beneficial for well control and drilling parameter












Three-wayvalve

ThrottleMud pumps

BlenderThrottle


Liquidflowmeter


shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

Booster


Oil skimming tank

11

Sampling




Pressure transducer

Signal analyzer

Outlet (oil)

Pressure transducer

Gas flowmeter

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Drillstrings

Annulus

Injection(gas)

Injection(mixture)

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)

Primary flow line

RBOP

Secondaryflow line

Nitrogengenerator

Compressor

Atmosphere

Solids disposal

Mud return line


deg100

50

minus0

minus10mV


deg


Three-wayvalve


analyzer

Liquidflowmeter


shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

100

50


Oil skimming tank

11

Sampling



Pressure transducer

Signal analyzer

Outlet (oil)

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Injection (mud)


Annulus

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)


RBOP

Secondaryflow line

Solids disposal

minus0

minus10mV













120597 (120588

119892120572

119892)

120597119905

+

120597 (120588

119892120572

119892119906

119892)

120597119911

= 0

(1)


120597 (120588

119897120572

119897)

120597119905

+

120597 (120588

119897120572

119897119906

119897)

120597119911

= 0(2)


120597 (120588

119892120572

119892119906

119892)

120597119905

+

120597 (120588

119892120572

119892119906

2

119892

)

120597119911

+ 120572

119892

120597119901

120597119911

= minus120572

119892120588

119892119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119892

(3)


120597 (120588

119897120572

119897119906

119897)

120597119905

+

120597 (120588

119897120572

119897119906

2

119897

)

120597119911

+ 120572

119897

120597119901

120597119911

= minus120572

119897120588

119897119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119897

(4)



119908is



120572

119892+ 120572

119897= 1 (5)


119901 = 119885120588

119892119877119879 (6)



119879 = 119879

119890119894+ 120585 [1 minus 119890

119867120585

] (

119892

119892

119888

sin 120579

119869119862

119901

+ 120593 + 119892

119879sin 120579)

+ 119890

(119867minus119911)120585

(119879

119891119887ℎminus 119879

119890119887ℎ)

(7)

where 119879




119862





and 119879




120588

119897= 120588

119897119904119891119890


2+120574119901119879(119901minus119901119904119891)(119879minus119879119904119891)

(8)

where 120588


ditions 119879119904119891



119901 120574

119901119901 120574

119879 120574

119879119879 and 120574

119901119879are







D119895(k0119895

) k119905+Δ119905119895

= E119895(k0119895

k119905119895

k119905+Δ119905119895minus1

) (9)


v = (119901 120572

119892 119906

119892 119906

119897)

119879

(10)








nabla

2

119901 =

120601120583119888

119896

120597119901

120597119905

(11)



120597119901

120597119899

= 0(12)



119901 = 119901

119908ℎ (13)

where 119901119908ℎ




119902

119905119900119905(119905) = int

119911(119905)

0

119902

119908(119911 119905) 119889119911

(14)


InjectionOutlet




k1 k2 k3 k4



119901 and 119902


Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902




119908is a


119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)





119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)



119911is the





= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901


119892is the gas


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)



119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)







119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Three-wayvalve

ThrottleMud pumps

BlenderThrottle


Liquidflowmeter


shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

Booster


Oil skimming tank

11

Sampling




Pressure transducer

Signal analyzer

Outlet (oil)

Pressure transducer

Gas flowmeter

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Drillstrings

Annulus

Injection(gas)

Injection(mixture)

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)

Primary flow line

RBOP

Secondaryflow line

Nitrogengenerator

Compressor

Atmosphere

Solids disposal

Mud return line


deg100

50

minus0

minus10mV


deg


Three-wayvalve


analyzer

Liquidflowmeter


shakers

Pressure transducer

Reserve pit

Mud pits

Pressuretransducer

Sampling

100

50


Oil skimming tank

11

Sampling



Pressure transducer

Signal analyzer

Outlet (oil)

Density measuring

unit

Throttle

Separator

Liquid flowmeter

Gas flowmeter

Injection (mud)

Injection (mud)


Annulus

Outlet

Outlet (mud)

Outlet (gas)

Outlet (mud)


RBOP

Secondaryflow line

Solids disposal

minus0

minus10mV













120597 (120588

119892120572

119892)

120597119905

+

120597 (120588

119892120572

119892119906

119892)

120597119911

= 0

(1)


120597 (120588

119897120572

119897)

120597119905

+

120597 (120588

119897120572

119897119906

119897)

120597119911

= 0(2)


120597 (120588

119892120572

119892119906

119892)

120597119905

+

120597 (120588

119892120572

119892119906

2

119892

)

120597119911

+ 120572

119892

120597119901

120597119911

= minus120572

119892120588

119892119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119892

(3)


120597 (120588

119897120572

119897119906

119897)

120597119905

+

120597 (120588

119897120572

119897119906

2

119897

)

120597119911

+ 120572

119897

120597119901

120597119911

= minus120572

119897120588

119897119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119897

(4)



119908is



120572

119892+ 120572

119897= 1 (5)


119901 = 119885120588

119892119877119879 (6)



119879 = 119879

119890119894+ 120585 [1 minus 119890

119867120585

] (

119892

119892

119888

sin 120579

119869119862

119901

+ 120593 + 119892

119879sin 120579)

+ 119890

(119867minus119911)120585

(119879

119891119887ℎminus 119879

119890119887ℎ)

(7)

where 119879




119862





and 119879




120588

119897= 120588

119897119904119891119890


2+120574119901119879(119901minus119901119904119891)(119879minus119879119904119891)

(8)

where 120588


ditions 119879119904119891



119901 120574

119901119901 120574

119879 120574

119879119879 and 120574

119901119879are







D119895(k0119895

) k119905+Δ119905119895

= E119895(k0119895

k119905119895

k119905+Δ119905119895minus1

) (9)


v = (119901 120572

119892 119906

119892 119906

119897)

119879

(10)








nabla

2

119901 =

120601120583119888

119896

120597119901

120597119905

(11)



120597119901

120597119899

= 0(12)



119901 = 119901

119908ℎ (13)

where 119901119908ℎ




119902

119905119900119905(119905) = int

119911(119905)

0

119902

119908(119911 119905) 119889119911

(14)


InjectionOutlet




k1 k2 k3 k4



119901 and 119902


Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902




119908is a


119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)





119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)



119911is the





= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901


119892is the gas


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)



119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)







119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597 (120588

119892120572

119892)

120597119905

+

120597 (120588

119892120572

119892119906

119892)

120597119911

= 0

(1)


120597 (120588

119897120572

119897)

120597119905

+

120597 (120588

119897120572

119897119906

119897)

120597119911

= 0(2)


120597 (120588

119892120572

119892119906

119892)

120597119905

+

120597 (120588

119892120572

119892119906

2

119892

)

120597119911

+ 120572

119892

120597119901

120597119911

= minus120572

119892120588

119892119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119892

(3)


120597 (120588

119897120572

119897119906

119897)

120597119905

+

120597 (120588

119897120572

119897119906

2

119897

)

120597119911

+ 120572

119897

120597119901

120597119911

= minus120572

119897120588

119897119892 sin 120579 minus 119865

119889119892minus 119865V119898 minus 119865

119908119897

(4)



119908is



120572

119892+ 120572

119897= 1 (5)


119901 = 119885120588

119892119877119879 (6)



119879 = 119879

119890119894+ 120585 [1 minus 119890

119867120585

] (

119892

119892

119888

sin 120579

119869119862

119901

+ 120593 + 119892

119879sin 120579)

+ 119890

(119867minus119911)120585

(119879

119891119887ℎminus 119879

119890119887ℎ)

(7)

where 119879




119862





and 119879




120588

119897= 120588

119897119904119891119890


2+120574119901119879(119901minus119901119904119891)(119879minus119879119904119891)

(8)

where 120588


ditions 119879119904119891



119901 120574

119901119901 120574

119879 120574

119879119879 and 120574

119901119879are







D119895(k0119895

) k119905+Δ119905119895

= E119895(k0119895

k119905119895

k119905+Δ119905119895minus1

) (9)


v = (119901 120572

119892 119906

119892 119906

119897)

119879

(10)








nabla

2

119901 =

120601120583119888

119896

120597119901

120597119905

(11)



120597119901

120597119899

= 0(12)



119901 = 119901

119908ℎ (13)

where 119901119908ℎ




119902

119905119900119905(119905) = int

119911(119905)

0

119902

119908(119911 119905) 119889119911

(14)


InjectionOutlet




k1 k2 k3 k4



119901 and 119902


Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902




119908is a


119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)





119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)



119911is the





= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901


119892is the gas


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)



119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)







119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











D119895(k0119895

) k119905+Δ119905119895

= E119895(k0119895

k119905119895

k119905+Δ119905119895minus1

) (9)


v = (119901 120572

119892 119906

119892 119906

119897)

119879

(10)








nabla

2

119901 =

120601120583119888

119896

120597119901

120597119905

(11)



120597119901

120597119899

= 0(12)



119901 = 119901

119908ℎ (13)

where 119901119908ℎ




119902

119905119900119905(119905) = int

119911(119905)

0

119902

119908(119911 119905) 119889119911

(14)


InjectionOutlet




k1 k2 k3 k4



119901 and 119902


Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902




119908is a


119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)





119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)



119911is the





= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901


119892is the gas


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)



119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)







119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










InjectionOutlet




k1 k2 k3 k4



119901 and 119902


Substitution of 119889119911 = 119906

119901119889119905 yields

119902

119905119900119905(119905) = int

119905

0

119902

119908(119911 120591) 119906

119901119889120591 (15)


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (119901

119903119890119904minus 119901

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(16)

where 119902




119908is a


119901

119908(119905) =

119901

119903119890119904for 119905 le 119905open (119911)

119901

119908ℎ(119905) for 119905 gt 119905open (119911)

(17)





119902

119908119906(119905) =

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(18)



119911is the





= 2int

119901

1199010

119901

120583

119892(119901)119885 (119901)

119889119901 (19)

where 119901


119892is the gas


119902

119905119900119905(119905) = int

119905

0

119906

119901int

119905

120591

120597 (

119903119890119904minus

119908(120577))

120597120577

119902

119908119906(Θ 119905 minus 120577) 119889120577 119889120591

(20)



119902

119908119906(119905) =

120587119896ℎ

119861

119892

119885

119904119888119879

119904119888120588

119892119904119888

119901

119904119888119879

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(21)







119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902

119908(119905) =

2120587119896ℎ (119901

119903119890119904minus 119901

119908(119905))

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(22)


119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

(23)


lim119905rarr0

119902

119908(119905)

119901

119903119890119904minus 119901

119908(119905)

= lim119905rarr0

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896119905120601120583119888119903

2

119908

=

2120587119896ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911

(24)


High

High

Low

0

qw

pw


Reservoir pressure







119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902

119905119900119905(119905) = 119906

119901(119901

119903119890119904minus 119901

119908)

119899=119905Δ119905

sum

119894=1

119902

119908119906119894(Θ

119894 119905 minus 119905

119894) Δ119905 (25)


119908119906119894(Θ

119894 119905 minus 119905




of 119902119908119906119894

(Θ

119894 119905 minus 119905




119902

119905119900119905(119905

1) = 119902

1199081(119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) 119902

1199081199061(Θ

1 119905

1)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

1ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896Δ119905120601120583119888119903

2

119908

(26)



119896

1= (119898

2119902

1199081(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

1199081

(119905

1) Δ119905 +

119902

1199081(119905

1) 119878

119911

119898

1

)

2

(27)

where

119898

1=

2119906

119901Δ119905 (119901

119903119890119904minus 119901

119908) ℎ119886

1

120583119861

119900119903

119908

119898

2=

119886

2

2119898

1119903

119908

radic120601120583119888

(28)


119902

119908119894(119905

1)

= 119902

119905119900119905(119905

119894) minus

119894minus1

sum

119899=1

119902

119908119899(119905

119894minus119899)

= 119902

119905119900119905(119905

119894) minus 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

119894minus1

sum

119899=1

119902

119908119906119899(119905

119894minus 119905

119899)

= 119906

119901Δ119905 (119901

119903119890119904minus 119901

119908)

2120587119896

119894ℎ

120583119861

119900

119886

1 (120587119903

119908)

119878

119911+ 119886

2

radic119896

119894Δ119905120601120583119888119903

2

119908

(29)



119908119906119899(119905

119894minus 119905

119899)


119894and


zone 119894 by

119896

119894= (119898

2119902

119908119894(119905

1)radicΔ119905 +

radic119898

2

2

119902

2

119908119894

(119905

1) Δ119905 +

119902

119908119894(119905

1) 119878

119911

119898

1

)

2

(30)






Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Drill bit

Cement

Casing

Drill strings

Wellhead


Annulus


Drill strings size

Diameter 01524m


Inner diameter 022m




Depth 0ndash

Depth 0ndash1800m



4000m







2030 2118 2207 2254 234110

11

12

13

14

15

16

17

18

Stan

dpip

e pre

ssur

e (M

Pa)

Time1930


1151 1342 1527 1715 1901 2052 2230250

275

300

325

350

375

400Ca

sing

pres

sure

(MPa

)

Time1000




1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1320 1340 1400 1420 1440 1500093

096

099

102

105

108

111


Time1300

Dril

ling

fluid

den

sitie

s(gmiddot

cmminus3)


0230 0454 0716 0933 11530

500

1000

1500

2000

2500

000Time


Gas

rate

s(m

3middoth

minus1)




230 454 716 933 115320

25

30

35

40

45

Tota

l hyd

roca

rbon

cont

ent (

)

Time000




003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










003 102 157 251 346 440 533 6270

100

200

300

400

500

600

700

800



Time2302


Gas

inflo

w ra

tes(

m3middoth

minus1)


37 38 39 40 41 42 430

100

200

300

400

500

600


Reservoir pressure

Gas

inflo

w ra

tes

y = minus12883x + 539944

R2 = 097611

(m3middotminminus1)




4880 4890 4900 4910 4920 4930 4940 4950 49604190

4191

4192

4193

4194

Rese

rvoi

r pre

ssur

e (M

Pa)

Well depth (m)


4880 4900 4920 4940 4960

001

01

1

10

100Pe

rmea

bilit

y (m

D)


1E minus 4

1E minus 3




4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4880 4890 4900 4910 4920 4930 4940 4950 49600

1

2

3

4

5


Well depth (m)

Und

erba

lanc

e (M

Pa)

0

2

4

6

8

10

Pene

trat

ion

rate

(mmiddoth

minus1)



0 1000 2000 3000 4000 5000 6000 70000

640

1280

1920

2560

3200


Gas

out

let r

ates

(m3middoth

minus1)








4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4880 4890 4900 4910 4920 4930 4940 4950 49600

5

10

15

20

25

Pres

sure

(MPa

)





0 1000 2000 3000 4000 50000

6

12

18

24

30

36

42

Pres

sure

(MPa

)


4800 4840 4880 4920 4960 5000400

402

404

406

408

410

Pres

sure

(MPa

)

Well depth (m)

24h240h1200 h


0 1000 2000 3000 4000 5000300

320

340

360

380

400

420

440

Tem

pera

ture

(K)


1200 h





119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879

119890119894119887ℎ

K119879

119890119894119908ℎ

K119892

119879

Ksdotmminus1119892

119888



kgsdotsminus1119896

119890


119901


119888


119888119890119898


427 303 003 100 100 3982 144 167472 909 402

0 1000 2000 3000 4000 5000100

101

102

103

104

105


1200 h

Dril

ling

fluid

den

sity

(gmiddotcm

minus3)



120574

119901

sdot1010

Paminus1120574

119901119901

sdot1019

Paminus2120574

119879

sdot104Kminus1

120574

119879119879

sdot107Kminus2

120574

119901119879

sdot1013

Kminus1Paminus1



7 Conclusion



Appendices



119865

119889119892=

3

8

120572

119892120588

119897119862

119889119892

(119906

119892minus 119906

119897)

1003816

1003816

1003816

1003816

1003816

119906

119892minus 119906

119897

1003816

1003816

1003816

1003816

1003816

119877

119901119892

(A1)

where 119862



119862

119889119892=

4119877

119901119892

3

radic

119892 (120588

119892minus 120588

119897)

120590

119904

[

[

1 + 1767[119891 (120572

119892)]

67

1867119891(120572

119892)

]

]

2

119891 (120572

119892) = (1 minus 120572

119892)

15

(A2)

where 119877




120590

119904= minus03120588

119897(119906

119892minus 119906

119897)

2

(A3)



119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901119892 is given by

119877

119901119892= [(

2

32

)(

120572

119892119881

120587

)]

13

(A4)


119862

119889119892= 98(1 minus 120572

119892)

3

(A5)



119865V119898 = 120572

119892120588

119897119862V119898 (

120597119906

119892

120597119905

minus

120597119906

119908

120597119905

+ 119906

119892

120597119906

119892

120597119911

minus 119906

119897

120597119906

119897

120597119911

) (A6)


for the bubbly flow

119862V119898 =

1

2

120572

119892

1 + 2120572

119892

1 minus 120572

119892

(A7)

for the slug flow

119862V119898 = 5120572

119892[066 + 034 (

1 minus 119889

119905119887119871

119905119887

1 minus 119889

1199051198873119871

119905119887

)] (A8)

where 119889

119905119887and 119871




Γ

119897119908=

0046

2

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A9)


119865

119908119897=

Γ

119897119908119878

119897

119860

=

0046119878

119897

2119860

120588

119897119906

119897

1003816

1003816

1003816

1003816

119906

119897

1003816

1003816

1003816

1003816

(

119906

119897119889

120592

119897

)

minus02

(A10)





Lsu

Ltb Lls

120591iultb ulls

Sl

Sl

Sg

Si

Sltb

Al

Al

Ag

120591gw



119865

119908119892=

Γ

119892119908119878

119892120595

119897

4119860

119892

+

Γ

119892119894119878

119894

4119860

119892

119871

119905119887

119871

119904119906

119865

119908119897=

Γ

119897119908

4119871

119904119906

(

119878

119897119871

119897119904

119860

119897

+

119878

119897119905119887119871

119905119887

119860

119897119905119887

) +

Γ

119897119894119878

119894

4119860

119897

119871

119905119887

119871

119904119906

(A11)


119897119904 119871119904119906






Γ

119892119908=

119891

119892120588

119892119906

2

119892

2

Γ

119897119908=

119891

119897120588

119897119906

2

1

2

Γ

119897119894=

119891

119894120588

119892(119906

119892minus 119906

119897)

2

2

(A12)


119891

119892=

16

Re119892

for Re119892le 2300

0046Re119892


119891

119897=

24

Re119897

for Re119897le 2300

00262(120572

119897Re119897)


119891

119894=

16

Re119892

for Re119894le 2300

0046Re119894


(A13)



Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Re119897=

119889

119897119906

119897120588

119897

120583

119897

Re119892=

119889

119892119906

119892120588

119892

120583

119892

Re119894=

119889

119892(119906

119892minus 119906

119897) 120588

119892

120583

119892

(A14)

where 119889

119897 119889



119889

119897=

4119860

119897

119878

119897

119889

119892=

4119860

119892

119878

119892+ 119878

119894

(A15)


Γ

119897119908=

119891

119897120588

119897119906

1

2

2119889

119897

120587119889 (A16)


119891

119897=

16

Re119897

(A17)

for turbulent flow

119891

119897= 0079Re

119897

minus025

(A18)

where

Re119897=

119889119906

119897120588

119897

120583

119897

(A19)


120585 =

119862

119901119908

2120587

(

119896

119890+ 119903

119905119900119880

119905119900119879

119863

119903

119905119900119880

119905119900119896

119890

) (B1)








119880

119905119900= [

1

ℎ

119888

+

119903

119905119900ln (119903

119908119887119903

119888119900)

119896

119888119890119898

] (B2)

where ℎ


fluid 119903119908119887





119879

119863=

(11281radic119905


119863)

for 10

minus10

le 119905

119863le 15

(04063 + 05 ln 119905

119863) times (1 +

06

119905

119863

)

for 119905

119863gt 15

(B3)




119879

119890119894= 119879

119890119894119887ℎminus 119892

119879119911 (B4)

where 119879



119890119894119908ℎ


120593 =minus 0002978 + 1006 times 10

minus6

119901

119908ℎ

+ 1906 times 10

minus4

119908 minus 1047 times 10

minus6

119865

119892119897

+ 3229 times 10

minus5

119860119875119868 + 0004009120574

119892minus 03551119892

119879

(B5)

where 119865



gravity



Acknowledgments


References






















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article reservoir characterization during underbalanced...

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