Estimation of Wellbore and Formation Temperatures_Mou Yang

8/12/2019 Estimation of Wellbore and Formation Temperatures_Mou Yang

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleEstimation of Wellbore and Formation Temperatures duringthe Drilling Process under Lost Circulation Conditions

Mou Yang,1Yingfeng Meng,1 Gao Li,1Yongjie Li,1Ying Chen,1

Xiangyang Zhao,2 and Hongtao Li1

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu , China Research Institute of Petroleum Engineering, SINOPEC, Beijing , China

Correspondence should be addressed to Yingeng Meng; [email protected] and Gao Li; @qq.com

Received February ; Revised June ; Accepted June

Academic Editor: Zhijun Zhang

Copyright Mou Yang et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Signicant change o wellbore and surrounding ormation temperatures during the whole drilling process or oil and gas resourcesofen leads by annulus uid uxes into ormation and may pose a threat to operational security o drilling and completionprocess. Based on energy exchange mechanisms o wellbore and ormation systems during circulation and shut-in stages underlost circulation conditions, a set o partial differential equations were developed to account or the transient heat exchange processbetween wellbore and ormation. A nite difference method was used to solve the transient heat transer models, which enables the

wellbore and ormation temperature proles to be accurately predicted. Moreover, heat exchange generated by heat convection dueto circulation losses to the rock surrounding a well was also considered in the mathematical model. Te results indicated that thelost circulation zone and the casing programme had signicant effects on the temperature distributions o wellbore and ormation.Te disturbance distance o ormation temperature was inuenced by circulation and shut-in stages. A comparative perectiontheoretical basis or temperature distribution o wellbore-ormation system in a deep well drilling was developed in presence o lostcirculation.

1. Introduction

Annulus uid uxed into ormation usually in presence olost circulation problem occurs in oil-gas and geothermalwells during the drilling stage with increasing well depth,

thus resulting in continuous variation o the temperatureo wellbore (inside drilling string uid, drilling pipe, andannulus) and surrounding ormation (casing, cement sheath,static drilling uid, and rock). Moreover, the determination otransient temperature distributions in and around oil-gas wellunder circulation and shut-in conditions is a complex taskbecause o the occurrenceo lost circulation due to the changeo drilling uid ow state and ormation property [].Tereore, it is important to obtain the accurate temperaturedistributions o wellbore and ormation underlost circulationconditions, which can develop the adequate drilling style anddesign the excellent property o drilling uids and cementslurries [,].

A reliable and accurate estimation o such temperaturedistribution requires a complete dynamic thermal studyrelated to the drilling uid ow in and around the wellbore,which includes a set o numerical models as well as boundaryand initial conditions. At present, the estimation temperaturemethod in and around oil-gas well is mainly classied intotwo classes. One deals with using classical analytical methodsbased on conductive heat ow in cylindrical coordinate [], exclusive o conductive-convective heat ow method []and the spherical and radial heat ow method []. Tesemodels have been considered as excellent methods in manyapplications due to their simplicity, whereas the ormationtemperature obtained by these methods is normally lowerthan the initial temperature []. Te other class attempts todescribe the transient heat transer processes usingnumericalmodels based on the energy balance principle in each regiono a well during circulation and shut-in stages [].

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Mathematical Problems in Engineering

Te previous estimation methods are mainly ocused onstudying the wellbore and ormation temperatures underno lost circulation condition. Tat is to say, these meth-ods cannot accurately estimate the wellbore and ormationtemperatures in presence o lost circulation. Recently, withregard to this, only a ew studies involved several aspects or

estimating the temperatures o uid and ormation when lostcirculation is being []. However, those studies have littleattention on studying the heat exchanged mechanism andlaw between wellbore and ormation under lost circulationconditions by the numerical model.

Tereore, in this work, the development o transientheat transer model or estimation o wellbore and ormationtemperatures in oil-gas wells during circulation and shut-in stages under lost circulation condition are presented.Here, the well-ormation interace is considered as a porousmedium through which uid lost by circulation []. More-over, to deeply analyze lost circulation process or radial heattranser equation, the model also takes the radial uid motionand the radial heat ow rom annulus to ormation intoconsideration. Tereby, under lost circulation, the compre-hensive model is applied to estimate each heat transer regionin a well according to the actual data o well drilling.

2. Physical Model

Te physical model o lost circulation during circulation stageis shown inFigure . Te process o circulation is consideredas three distinct phases: () uid enters the drill pipe with aspecied temperature (in) at the surace and passes downwith ow velocity V1 in the direction; () uid exits thedrill pipe through the bit and enters the annulus at thebottom; () uid passes up the annulus with ow velocityV3and exits the annulus with a specied temperature (out) atthe surace [, ]. I lost circulation is being in a certainormation, drilling uid would be own into surroundingormation so that it becomes hard to precisely dene thetemperature prole o a well.Tereore, to simulate thermalbehavioro uid during the circulation process, it is necessaryto develop a set o partial differential equations under theactual casing program and drill string assembly conditions,which is illustrated in Figure . Mass and energy conservationis considered as incompressibleow in the axial () and radial() directions. Meanwhile, the effect actors o boundaryconditions among each control unite are taken into account

in the solving model.During the circulation stage, the uid passes down the

pipe in the direction, and its temperature distribution isdetermined by the rate o heat convection down the drillingpipe and heat exchange with the metallic pipe wall. Atthe bottom, the uid temperature at the outlet o the drillpipe is the same as the uid temperature at the entranceo the annulus, and then the uid keeps on owing upin the annulus. Similarly, the annulus uid temperature isdetermined by the rate o heat convection up the annulus, therate o heat exchange between the annulus and drill pipe wall,and the rate o heat exchange between the wall o the welland annulus uid []. Meanwhile, well depth and ow rate

o lost circulation have great effects on annulus temperature.In addition, uid riction, rotational energy, and drill bitenergy also have signicant inuence on the overall energybalance o the wellbore []. During shut-in stage, abovethe lost depth, all wellbore uid will ow into ormation.Tereore, the temperatures in the wellbore which are decided

by uid ow state depend upon a number o different thermalprocesses involving conductive and convective mechanismsin different sections o well. When wellbore uid is in owingstate, the uid temperatures o inside drill string and annulusare strong dependent upon the rate o lost circulation in heatconvection way; i all o wellbore uid above the loss depthow into ormation, the heat exchange between wellbore andormation is only in a conduction way.

3. Mathematical Model

.. Assumption Conditions. Te mathematical model con-sisted o a set o partial differential equations used to

describe the temperatures o wellbore and ormation. Teundamental assumptions o numerical model include theollowing.

(i) Each control unit o wellbore and ormation system iscylindrical geometry.

(ii) Te physical properties o the ormation, cement, andmetal pipe are constant with the change o depths [].Te parameters include thermal conductivity, density,heat capacity, and viscosity.

(iii) Te radial temperature gradient within the uid maybe neglected.

(iv) Te heat conduction equation through surroundingwellbore is solved by using a two-dimensional tran-sient axial-symmetric temperature distribution.

(v) Viscous dissipation and thermal expansion effects areneglected.

Under these conditions, the governing equations andinitialand boundary conditions oreach region are as ollows.

.. Mathematical Formulation

... Transient Heat Transfer during Circulation Stage

() Inside the Drilling String.Te numerical model which cancalculate the temperature distribution o inside drilling stringis complemented by the ollowing three considerations: ()the inlet uid temperature is the boundary condition o themodel; () the ow velocity o uid in thedirection is alsodened by mass ow rate; and () heat generated by uidraction. Tereore, based on energy conservation principle,the model is expressed as ollows:

21 11121

211 21 = 111 . ()


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Surface

z

Casing

Cementing

Rockformation

Drill string

Annular

Fluid in annulus

Fluid inside

drill string

Bottom hole

r

Cementingsection

Lost circulation section

T1,j

T1,j

T1,j

Ti,j

Ti,j

T2,j

T3,j

T3,j

T3,j

T4,j

T5,j

r

r

z

z

Exit fluidTout

Entrance fluidTin

F : Physical model o drilling uid circulation under lost circulation condition.

Te boundary condition between uid and inner wall odrill string is written as ollows:

22=1

= 12 1 , ()

where1,2 are the temperatures o inside drilling stringuid and drilling string wall, respectively;

is the energy

source term o unit length inside the drilling string;1is thedensity o drilling uid; is the ow rate o inside drill string;1is the specic heat capacity o drilling uid;1is the radiuso inside drill string;2 is the thermal conductivity o drillstring; and1 is the convection coefficient o inside drillingstring wall.

() Drill String Wall. Te ormulation calculates the tem-perature distribution o drill string wall, and the conditionshere are dened by the three sections: () the mass owrate o uid in the inside drill string and annulus; () the

vertical conduction o heat in the drill pipe; and () the radialexchange o heat between the drill pipe and the uid inside

andoutside the string. Te numericalmodel o the drill stringwall is given as ollows:

2 2 + 21122 211 2 22222 212 3 =

222 . ()

Te boundary condition inuenced the temperature dis-tribution o drillstring wall includes two parts: one is the heatexchange between uid o inside drill string and drill string,

which is expressed by (), and the other is heat exchangebetween annulus and the drill string which is written as

22=2

= 22 3 , ()where3 is the temperature o annulus uid;2 is thedensity o drill string wall;2 is the specic heat capacity odrill string;2 is the outer radius o drill string;2 is thethermal conductivity o drill string; and2 is the convectioncoefficient o outer drilling string wall.

() In the Annulus. Te actors that inuenced the annulustemperature are consisted o three sections: () the mass ow


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rate o uid; () the temperature distributions o drill stringand wellbore walls; and () heat generated by uid ractionand drilling string rotation. Te transient heat transer modelin the annulus is expressed as ollows:

333 23 22 +

2222 323 22

23e3 423 22+ 23 22=

333 .()

Te interace between annulus and wellbore wall isimportant since it mathematically couples the surroundingormation with the ow in the annulus. Tereore, to guar-antee continuity o heat ux during circulation and shut-incondition, the boundary conditions are

33=3

+ e4 3 = e4=3

, ()where4 is the interace temperature between annulus uidand borehole wall;is the energy source term o unit lengthinside annulus; is the ow rate o annulus;3is the radiuso borehole wall;3 is the thermal conductivity o annulusuid; effective thermal conductivityeconsiders the effecto porosity;3 is the convection coefficient o borehole wall;eis the effective heat transer coefficient which considerstheeffect porosity.

() Each Heat Transfer Region in Surrounding Wellbore. Teeffect o actor on heat exchange or the surrounding wellboreincludes our sections: () the vertical conduction o heat inthe medium; () the rate o heat exchange among volume

elements; and () the rate o heat exchange or ormation uidwhich can ow in the porous medium. Te energy balance ineach heat transer medium is

e +1e

= e+ V , ()

where

e= + 1 V= ,,, in, .

()

Te mathematical ormulation or the hydrodynamicmodel o rock ormation is based on one-dimensional

volume-averaging equations that govern the hydrodynamicphenomena o an incompressible uid across an isotropicporous medium [], which are represented as ollows:

V= ,

22+1 +in = 0,

()

where is different unit temperature o porous mediumin the ormation; is the radius o porous medium in theormation; the magnitude ois decided by casing program( 4); is an effective porosity o ormation; andrepresent rock and pore uid, respectively; Vis theradialow

velocity; uis ormation uid mass ow to annulus; is thedrilling uid o mass ow; is the lateral ow area;is thedynamic viscosity; is the intrinsic average pressure; is theabsolute permeability o the isotropic porous medium; isthe mass source term; andis the relative permeability.... Transient Heat Transfer during Shut-In Stage. Duringstop circulation stage, the heat exchange method can beclassied into two ways, which is dependent on the inter-ace between gas and liquid. Tereore, combined with thestudy o heat exchange mechanism between wellbore andormation during uid circulation stage and energy and massconservation principles, the description o heat exchangetypes during shut-in stage is presented by a set o partial

difference equations.

() In the Drill String

() Te transient heat transer model o above interacebetween gas and liquid and below lost ormation isexpressed as

2122 1221ln 1+ 2/21 + 121ln 22/ 1+ 2= 111

.

()

() Te transient heat transer model o below interacebetween gas and liquid and above lost ormation isdescribed as

21 11121

211 21 = 111 . ()

() Drill String Wall


2323 23ln 1+ 2/21 + 2ln 22/ 1+ 221 20+ 2

222 2122 12ln 0+ 1/20 + 1ln 21/ 0+ 121 20

= 222 .()


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() Te transient heat transer model o below interacebetween gas and liquid and above lost ormation isexpressed as

2

2

+ 211

22

21

1 2

22222 212 3 =

222 .

()

() In the Annulus


23e4 3e

ln 2+ 3/22 + 3ln 23/ 2+ 322 21 2323 23ln 1+ 2/21 + 2ln 22/ 1+ 222 21

= 333 .()

() Te transient heat transer model o below interacebetween gas and liquid and above lost ormation isdescribed as

333

2

3 2

2 +2222 3

2

3 2

2 23e3 4

2

3 2

2

+ 23 22= 333 ,

()

where the meaning parameters o the shut-in stage denedrom () to () are the same as that o circulation stage.

Te orm o transient numerical model or each heattranser region surrounding wellbore during shut-in stage isalso the same as ().

4. Numerical Solutions

o obtain the temperature distribution on the term o time,the solution o these equations is complicated. Developedmodels incorporate solution methods which are based onnite difference techniques. Te wellbore and the adjacentormation are represented by a two-dimensional, mesh gridincluding a number o radial elements due to casing programand a variable number o vertical elements depending on thewell depth. Each o radial elements corresponds to differentportion o the wellbore cross-section rom inside drill stringinto the ormation []. Tereore, a set o partial differentialequations can be presented as nite difference orm usingnite differences technique or each element o grid so as todescribe the transient heat exchange in each element or an

implicit orm []. A set o nonlinear algebraic equations arethen solved using an iterative method until the error rangecan be accepted. Te case o nite difference can be denedas ollows. Te spatial derivatives and the time derivatives arethe rst-order as exhibited in ():

+1,

+1,1

. ()Te second-order spatial derivatives are represented by

three-point central difference approximations [,]:

22 1 (+1,+1 +1,+0.5

+1, +1,10.5 ) . ()Te time discretization at node is expressed in

+1, , . ()

Application o earlier denitions enables the equation or

each node to be written in single generalized vector orm:

+11,+ +1,+ +1+1,+ +1,1+ +1,+1= .()

Te matrix orm o nite difference oreach node is given:

+1 = . ()Te nite difference equations are solved by ast succes-

sive overrelaxation (SOR) iteration method; the ollowinggeneral orm or each node is expressed as ollows:

V+1+1, =,

,,+ , V+1+11,+, V+1+1,+ , V+1+1,1+ , V+1,+1

+1 V+1, .()

Using implicit orm o nite difference method, () and() are, respectively, shown as ollows:

1121 +11,1+ 11+1121 +

211 +11,

211

+12,

=1

21 +

11

1,

()

21122 21 +11,+ 20.5

+12,1

+ 2+0.5 +12,+1+ 22222 21

+13,

20.5 + 2+0.5 +

21122 21+22222 21+

22 +12,= 22 2,,

()


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5000

4000

3000

2000

1000

0

Welldepth(m)

Circulation 5 hr

Circulation 10 hr

Initial formation temperature

0 25 50 75 100 125

Temperature prole (C)

Circulation 1 hr

F : Annulus temperature proles at different circulation timeunder no lost circulation conditions.

whereis the variable temperature;is the step incrementin the space coordinate; is the time node; indicates thenode number in the direction; is the node number inthe direction;,,,, and are the matrices ocoefficients; SOR is the Gauss-Seidel iterative method iisequal to in (); the SOR is overrelaxation method i ismore than ; SOR is under relaxation method iis less than.

Te calculation accuracy depends on the meshing ele-

ments and the size o the interval values. In general, it isobserved that the vertical element size is less than % o thetotal well depth to ensure that the annulus temperature proleremains independent o the vertical element size [].

5. Model Solution Procedure

.. Basic Data. Te basic data o calculation in this studyare the reerenced literatures reports [], which are shown inables,, and. Te ow rate and depth o lost circulationare assumed as . l/s and m, respectively.

.. Numerical Model Application

... Example Analysis in Circulation Operation Condition.As shown inFigure , the annulus temperature distributionsas a unction o depth at different circulation time underno lost circulation conditions are presented. As intermediatecasing depth is m (able ), the annulus temperatureso cementing section vary under different circulation time,whereas the annulus temperature o open-hole section grad-ually decreases with the increase o the circulation time. Tatis because casing thermalconductivity coefficient is . timesthan that o ormation, resulting in more amount o heatexchange between annulus energy o cementing section andormation compared with that o open hole. Meanwhile, the

0 2 4 6 8 10

20

22

24

26

28

30

32

34

Circulation time (hr)

Exittemperature(C)

F : Outlet temperature distribution as a unction o circula-tion time under lost circulation condition.

5000

4000

3000

2000

1000

0

Welldepth(m)

0 25 50 75 100 125


Circulation 5 hCirculation 10 h


Circulation 1 h

F : Annulus temperature proles as a unction o circulationtime under lost circulation conditions.

annulus heat quantity is gradually carried to surace withthe increase o the circulation time and thus results in the

decrease o annulus temperature o open-hole section [].Te relationship between outlet temperature and circu-

lation time under lost circulation condition is also investi-gated. As shown inFigure , the outlet temperature rapidlydecreases within the initial circulation (. h) and then grad-ually increases during the latter circulation. One plausibleexplanation is that more heat quantity is carriedto well mouthat initial circulation compared to that at latter circulation andthus leads to the wellbore wall o well mouth heat.

Under lost circulation conditions, the effect o alterationsin circulation time on the annulus temperature distributionis shown in Figure . It is ound that the annulus temperatureo open-hole section continuously decreasesas the increase o


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: Basic data o drill string assembly and casing program.

Drill pipe Drill collar First casing Second casing Tird casing

Inner diameter (mm)

Outer diameter (mm)

Depth (m)

Depth to cement (m)

: Basic data o thermal physical parameters.

Drill pipe/casing Drill string Drill uid Cement Formation Formation uid

Density (kg/cm)

Heat capacity (J/kgC) Termal conductivity (w/mC) . . . . . .

0 1 2 3 4 577

84

91

98

105

112

119

126

Formation radial distance (m)

Circulation 1 h

Circulation 5hCirculation 10 h

Circulation 1 h

Circulation 5 hCirculation 10 hInitial formation temperature

Radialtemperature(C)

F : Formation radial temperature distributions o lost depthand bottom hole.

the circulation time. Additionally, the closer the bottom holeis, the less decrease the temperature will be, which is in accor-dance with the result oFigure . Meanwhile, the annulustemperature proles o circulation h and h are both lowerthan the ormation temperature below m. Compared to

Figure ,Figure indicates that the annulus temperature ocement section decreases under lost circulation conditions.Te reason is that the annulus uid temperature at mis higher than that o annulus uid at any depth o cementsection. Herein, heat quantity o annulus uid at m owsinto the ormation increased, which can result in the decreaseo the annulus temperature.

Similarly,Figure also reects the ormation radial tem-perature distributions o lost depth and bottom hole underdifferent circulation time. Noticeably, the ormation radialtemperature decreases with the increase o the circulationtime, whereas the decrease o the surrounding ormationtemperature at lost depth is less than that o at bottom hole

5000

4000

3000

2000

1000

0

Welldepth(m)

4 2 0 2 4 6 8

Circulation 5 hr

Circulation 10 hr

Circulation 1 hr

Temperature difference prole (C)

F : emperature difference proles between annulus andinside pipe uid during different circulation time.

during the latter circulation. Te surrounding ormation iscontinuously heated by annulus uid at lost depth duringinitial circulation stage and then leads to its temperature risebeyond its initial ormation temperature as shown in Figure (circulation h). Meanwhile, the annulus temperature is

lower than ormation temperature afer long circulation timeand thus leads to ormation temperature decrease. However,the ormation is continuously cooled by circulation uidat bottom hole and then leads to the temperature o thesurrounding ormation decrease below the initial ormationtemperature.

o get a deep insight o the heat transer mechanismor wellbore during the circulation stage, the temperaturedifference distribution between annulus and inside pipe uidunder different circulation time is studied. As shown inFigure , the temperature difference remarkably decreasesrom bottom-hole to casing shoe with increasing the circu-lation time. Meanwhile, the temperature difference changes


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: Other basic data o bottom hole.

Depth (m) otal well diameter

(mm)Open hole diameter

(mm) Flow rate (l/s)

Surace temperature(C)

Geothermal gradient(C/ m)

. .

Inlet temperature(C)

Outlet temperature(C)

Plastic viscosity(mPa

s)

Yield value(mPa)

Consistencycoefficient (mPa

s)

Fluidity point

. .

0 2 4 6 8 1019

20

21

22

23

24

Shut-in time (hr)

Temperatureofwellmouth(C)

F : emperature o well mouth during shut-in stage underlost circulation condition.

at the lost circulation point. Additionally, the annulus tem-perature is higher than the inside pipe uid temperature in

the wellbore except or well mouth section during the wholecirculation stage.

... Example Analysis in Shut Condition. As shown inFigure , the temperature o well mouth continuouslydecreases during the whole shut-in stage. Te result oFigure shows that the outlet temperature increases duringthe latter circulation, resulting in surrounding ormationcontinuously heated. However, during shut-in stage, as gasis instead o uid at well mouth, heat exchange betweenwellbore and ormation is less due to heat conductivity o gas.Tereore, the temperature o well mouth gradually decreaseswith shut-in time increased.

As it is seen rom Figure , the annulus temperaturecontinuously increases with the increase o shut-in timewhen the well depth is beyond casing shoe, whereas theannulus temperature hardly varies when the well depth isabove the casing shoe point. It spends about . h on alluids o inside pipe and annulus above lost depth ows intoormation.Tereore, the heatexchange between the wellboreand ormation by heat conduction is more than that o heatconvection during . h o shut-in. Afer that, the ormationenergy uxes into annulus in the heat conduction way aswellbore uid is in the static state beyond . h, thus resultingin the improvement o temperature. Furthermore, the tem-perature eventually increases to be equal to the ormation

5000

4000

3000

2000

1000

0

Welldepth(m)

0 25 50 75 100 125

Annulus temperature prole (C)

Shut-in 1 hr

Shut-in 5 hr

Shut-in 10 hr


F : Annulus uid temperature proles during different shut-in time.

temperature. However, when the temperature is above thelost depth, annulus energy which arose rom ormation isnearly equal to that o the annulus gas transerring to thesurrounding ormation and the inside pipe drilling uid bythe heat convective way, which leads to the temperature hardchange.

Figure indicates that the radial ormation temperaturedecreases with increasing the shut-in time at lost circulationdepth and bottom hole, and both o them are lower thanthat o initial temperature. However, the radial ormationtemperature at lost depth slowly decreases with the increaseo the shut-in time, and temperature change at the bottomhole is larger than that o at lost depth. Te reasonable

explanation is that the temperature difference between annu-lus and ormation at bottom hole is larger than that o atlost circulation point during circulation stage. Comparedto Figure , it is surprising that Figure implies that theormation temperature disturbance distance in shut-in stageis larger than that o circulation stage. It is derived rom thatthe starting point o disturbance distance or radial ormationtemperature is at well wallduring thecirculationstage, but thestarting point o disturbance distance is at inside ormationduring the shut-in stage which is the destination point odisturbance distance or circulation stage.

As shown in Figure , the annulus temperature romtop hole ( m) to bottom hole is higher than the inside


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0 1 2 3 4 580

88

96

104

112

120

Formation radial distance (m)

Shut-in 1 hr

Shut-in 5 hr

Shut-in 10 hr

Shut-in 1 hr

Shut-in 5 hrShut-in 10 hr


Radialtemperature(C)

F : Formation radial temperature distributions o lost circu-lation point and bottom hole during shut-in stage.

5000

4000

3000

2000

1000

0

Welldepth(m)

3.0 1.5 0.0 1.5 3.0 4.5 6.0Temperature difference prole (C)

Shut-in 1 hrShut-in 5 hrShut-in 10 hr

F : emperature difference proles between annulus andinside pipe during shut-in stage.

pipe temperature during the whole shut-in stage, and onlythe temperature difference between annulus and inside piperom well mouth to the depth o m is negative value.Furthermore, the temperature difference between annulusand inside pipe gradually decreases with increasing shut-intime and then trends to thermodynamic equilibrium state.From Figure , it is observed that the temperature differencebetween annulus and inside pipe is greatly inuenced by thelost depth, make up o string, and casing program.

As shown in Figures and , annulus temperaturechanges o lost depth and bottom hole are related to the

0 4 8 12 16 2080

85

90

95

100

105

110

Circulation and shut-in time (hr)

Circulation temperature

Shut-in temperature

Formation temperature

Annulustemperatureoflostpoint(C)

F : Annulus temperature distributions o lost circulationdepth during circulation and shut-in stages.

0 4 8 12 16 20

90

96

102

108

114

120

126

132

Circulation and shut-in time (hr)

Circulation temperature

Shut-in temperature

Formation temperature

Annulus

temperatureofbottomhole(C)

F : Annulus temperaturedistributionso bottom holeduringcirculation and shut-in stages.

circulation and the shut-in stages. It can be seen that theannulus temperature rapidly decreases during the initialcirculation stage andslowly varies in the lattercirculation andshut-in stage, ollowed by the change o annulus temperatureshowing the same tendency under the two conditions earlier.Meanwhile, the annulus temperatures at initial circulationstage are both higher than that o the initial ormationtemperature at lost depth and bottom hole, and then botho them are lower than initial ormation temperature duringlatter circulation and shut-in stages. However, beore wellboreuid above the lost depth ows into ormation (. h), it isinterestingly noted that the annulus temperature gradually


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increases with the increase o shut-in time at lost depth byheat convection, ollowed by quickly decreasing, and thenslowly increases at bottom hole by heat conduction. Figures and also show that i the annulus temperature afercirculation recovers to the initial ormation temperature,shut-in time can be longer than that o circulation time [].

Te phenomenon accounts or the reason why long time orshut-in is needed to obtain initial ormation temperature.

6. Discussion

o demonstrate the applicability o the methodology devel-oped in this work, the OM-C geothermal well was consid-ered. Tis well is in Kenya, which was drilled with boreholediameters o , ., ., and . in. Te casing has ,., ., and in diameters at , , , and mdepths, respectively. emperatures in and around the OM-C geothermal well during circulation and shut-in stageswere estimated by the transient heat transer models. TeHorner (Dowdle andCobb,)and Hasan andKabir()methods were used to compare our numerical results [,].

Te input data to simulate the geothermal well are drilling

uid ow rate o . m3/h, surace temperature o . C,and drilling uid properties which include the thermal

conductivity o . W/mC, the density o kg/m3, theviscosity o . Pa.s, and specic heat o J/kg.C.Circulation time was h.

A compilation o main results obtained in these thermalstudies was presented in Figure . We reckoned that thelogged temperature o shut-in days was approximatelyequal to the static ormation temperature (SF) due tothermal recovery conditions during the long time shut-in.As shown inFigure , it can be observed that the measuredtemperature was satisactorily matched with the simulatedtemperature (continuous line). Figure also showed the SFcalculated by means o Horner (Dowdle and Cobb, ) andHasan and Kabir () methods [, ]. Obviously, as showninFigure , the Hasan-Kabir method is closer to the SFcompared to the Horner method. Differences between thecomputed data (or simulated days) and measured valueswere estimated andexpressed as a percentage deviation basedon the result oFigure , and the percentage deviationbetween the simulated SF and analytical methods as alsocomputed romFigure . It can be observed the maximumdeviation between measured and simulated data is .% and

minimum deviation is .%, which corresponded to thecontrol error in engineering. Te best approximation tothe simulated SF values corresponded to the Hasan-Kabir,which presented minimum differences o .%, .%, and.%. Tereore, the simulated SF method in this work isbetter than that o the analytical methods (Horner and Hasanand Kabir).

7. Conclusions

In this study, a set o numerical models have been developedto study the transient heat transer processes which occurs

3000

2500

2000

1500

1000

500

0

Depth(m)

Shut-in 9 hrShut-in 27days

Simulated SFT

Dowdle-Cobb (1975)

Hassan and Kabir (1994)

0 50 100 150 200 250 300 350


F : Simulated and logged temperature proles in OM-C geothermal well during shut-in stages. Also shown the SFsestimated with the Horner (Dowdle and Cobb, ) and Hasan andKabir () methods and with this work [,].

in oil-gas or geothermal well during circulation and shut-in stages under lost circulation conditions. Te equationsproperly account or the energy conservation in each regiono a well, and mass balances are perormed at any numericalnode where annulus uid uxes into ormation. Heat transercoefficients and thermophysical properties (gas instead ouid) in the annulus and the surrounding ormation changedue to lost circulation. Simulation results show that the

temperature distributing characters o the wellbore andsurrounding ormation are remarkably inuenced by thelost depth and casing program during the whole circulationand shut-in stage. Additionally, the disturbance distance oormation temperature at shut-in stage is larger than that o atcirculation stage at the same time. Moreover, it is necessary toprolong shut-in time than circulation time in order to obtainaccurate initial ormation temperature. Te present work canprovide a new way to improve the present methodology.

Acknowledgments

Te authors appreciate the nancial support by China

National Natural Science Foundation (no. ;; ; ), Major State Science andechnology Special Project o China (no. ZX-),Ph.D. Programs Foundation o Ministry o Educationo China (no. ), and Southwest PetroleumUniversity o Young Scientic Research Innovation eamFoundation (no. XJZ). Te authors would also liketo appreciate their laboratory members or the generoushelp.

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