estimation of wellbore and formation temperatures during drillin process under lost circulation...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013, Article ID 579091, 11 pageshttp://dx.doi.org/10.1155/2013/579091

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

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

Xiangyang Zhao,2 and Hongtao Li1

1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China2 Research Institute of Petroleum Engineering, SINOPEC, Beijing 100101, China

Correspondence should be addressed to Yingfeng Meng; [email protected] and Gao Li; [email protected]

Received 4 February 2013; Revised 22 June 2013; Accepted 24 June 2013

Academic Editor: Zhijun Zhang

Copyright 2013 Mou Yang et al. This 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.

Significant change of wellbore and surrounding formation temperatures during the whole drilling process for oil and gas resourcesoften leads by annulus fluid fluxes into formation and may pose a threat to operational security of drilling and completionprocess. Based on energy exchange mechanisms of wellbore and formation systems during circulation and shut-in stages underlost circulation conditions, a set of partial differential equations were developed to account for the transient heat exchange processbetween wellbore and formation. A finite difference method was used to solve the transient heat transfer models, which enables thewellbore and formation temperature profiles 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. The results indicated that thelost circulation zone and the casing programme had significant effects on the temperature distributions of wellbore and formation.The disturbance distance of formation temperature was influenced by circulation and shut-in stages. A comparative perfectiontheoretical basis for temperature distribution of wellbore-formation system in a deep well drilling was developed in presence of lostcirculation.

1. Introduction

Annulus fluid fluxed into formation usually in presence oflost circulation problem occurs in oil-gas and geothermalwells during the drilling stage with increasing well depth,thus resulting in continuous variation of the temperatureof wellbore (inside drilling string fluid, drilling pipe, andannulus) and surrounding formation (casing, cement sheath,static drilling fluid, and rock).Moreover, the determination oftransient temperature distributions in and around oil-gaswellunder circulation and shut-in conditions is a complex taskbecause of the occurrence of lost circulation due to the changeof drilling fluid flow state and formation property [13].Therefore, it is important to obtain the accurate temperaturedistributions of wellbore and formation under lost circulationconditions, which can develop the adequate drilling style anddesign the excellent property of drilling fluids and cementslurries [4, 5].

A reliable and accurate estimation of such temperaturedistribution requires a complete dynamic thermal studyrelated to the drilling fluid flow in and around the wellbore,which includes a set of numerical models as well as boundaryand initial conditions. At present, the estimation temperaturemethod in and around oil-gas well is mainly classified intotwo classes. One deals with using classical analytical methodsbased on conductive heat flow in cylindrical coordinate [611], exclusive of conductive-convective heat flow method [12]and the spherical and radial heat flow method [13]. Thesemodels have been considered as excellent methods in manyapplications due to their simplicity, whereas the formationtemperature obtained by these methods is normally lowerthan the initial temperature [14]. The other class attempts todescribe the transient heat transfer processes using numericalmodels based on the energy balance principle in each regionof a well during circulation and shut-in stages [1519].

2 Mathematical Problems in Engineering

The previous estimation methods are mainly focused onstudying the wellbore and formation temperatures underno lost circulation condition. That is to say, these meth-ods cannot accurately estimate the wellbore and formationtemperatures in presence of lost circulation. Recently, withregard to this, only a few studies involved several aspects forestimating the temperatures of fluid and formation when lostcirculation is being [2022].However, those studies have littleattention on studying the heat exchanged mechanism andlaw between wellbore and formation under lost circulationconditions by the numerical model.

Therefore, in this work, the development of transientheat transfer model for estimation of wellbore and formationtemperatures in oil-gas wells during circulation and shut-in stages under lost circulation condition are presented.Here, the well-formation interface is considered as a porousmedium through which fluid lost by circulation [23]. More-over, to deeply analyze lost circulation process for radial heattransfer equation, themodel also takes the radial fluidmotionand the radial heat flow from annulus to formation intoconsideration. Thereby, under lost circulation, the compre-hensive model is applied to estimate each heat transfer regionin a well according to the actual data of well drilling.

2. Physical Model

Thephysicalmodel of lost circulation during circulation stageis shown in Figure 1. The process of circulation is consideredas three distinct phases: (1) fluid enters the drill pipe with aspecified temperature (in) at the surface and passes downwith flow velocity V

1in the direction; (2) fluid exits the

drill pipe through the bit and enters the annulus at thebottom; (3) fluid passes up the annulus with flow velocity V

3

and exits the annulus with a specified temperature (out) atthe surface [15, 18]. If lost circulation is being in a certainformation, drilling fluid would be flown into surroundingformation so that it becomes hard to precisely define thetemperature profile of a well. Therefore, to simulate thermalbehavior of fluid during the circulation process, it is necessaryto develop a set of partial differential equations under theactual casing program and drill string assembly conditions,which is illustrated in Figure 1.Mass and energy conservationis considered as incompressible flow in the axial () and radial() directions. Meanwhile, the effect factors of boundaryconditions among each control unite are taken into accountin the solving model.

During the circulation stage, the fluid passes down thepipe in the direction, and its temperature distribution isdetermined by the rate of heat convection down the drillingpipe and heat exchange with the metallic pipe wall. Atthe bottom, the fluid temperature at the outlet of the drillpipe is the same as the fluid temperature at the entranceof the annulus, and then the fluid keeps on flowing upin the annulus. Similarly, the annulus fluid temperature isdetermined by the rate of heat convection up the annulus, therate of heat exchange between the annulus and drill pipe wall,and the rate of heat exchange between the wall of the welland annulus fluid [23]. Meanwhile, well depth and flow rate

of lost circulation have great effects on annulus temperature.In addition, fluid friction, rotational energy, and drill bitenergy also have significant influence on the overall energybalance of the wellbore [24]. During shut-in stage, abovethe lost depth, all wellbore fluid will flow into formation.Therefore, the temperatures in thewellborewhich are decidedby fluid flow state depend upon a number of different thermalprocesses involving conductive and convective mechanismsin different sections of well. When wellbore fluid is in flowingstate, the fluid temperatures of inside drill string and annulusare strong dependent upon the rate of lost circulation in heatconvection way; if all of wellbore fluid above the loss depthflow into formation, the heat exchange between wellbore andformation is only in a conduction way.

3. Mathematical Model

3.1. Assumption Conditions. The mathematical model con-sisted of a set of partial differential equations used todescribe the temperatures of wellbore and formation. Thefundamental assumptions of numerical model include thefollowing.

(i) Each control unit of wellbore and formation system iscylindrical geometry.

(ii) The physical properties of the formation, cement, andmetal pipe are constant with the change of depths [5].The parameters include thermal conductivity, density,heat capacity, and viscosity.

(iii) The radial temperature gradient within the fluid maybe neglected.

(iv) The 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 andinitial and boundary conditions for each region are as follows.

3.2. Mathematical Formulation

3.2.1. Transient Heat Transfer during Circulation Stage

(1) Inside the Drilling String.The numerical model which cancalculate the temperature distribution of inside drilling stringis complemented by the following three considerations: (1)the inlet fluid temperature is the boundary condition of themodel; (2) the flow velocity of fluid in the direction is alsodefined by mass flow rate; and (3) heat generated by fluidfraction. Therefore, based on energy conservation principle,the model is expressed as follows:

2

1

(111)

2

121(1 2)

1

= (111)

. (1)

Mathematical Problems in Engineering 3

Surface

z

Casing

Cementing

Rockformation

Drill string

Annular

Fluid in annulusFluid insidedrill 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 uid Tout

Entrance uid Tin

Figure 1: Physical model of drilling fluid circulation under lost circulation condition.

The boundary condition between fluid and inner wall ofdrill string is written as follows:

2(2

)

=1

= 1(2 1) , (2)

where 1, 2are the temperatures of inside drilling string

fluid and drilling string wall, respectively; is the energy

source term of unit length inside the drilling string; 1is the

density of drilling fluid; is the flow rate of inside drill string;1is the specific heat capacity of drilling fluid;

1is the radius

of inside drill string; 2is the thermal conductivity of drill

string; and 1is the convection coefficient of inside drilling

string wall.

(2) Drill String Wall. The formulation calculates the tem-perature distribution of drill string wall, and the conditionshere are defined by the three sections: (1) the mass flowrate of fluid in the inside drill string and annulus; (2) thevertical conduction of heat in the drill pipe; and (3) the radialexchange of heat between the drill pipe and the fluid inside

and outside the string.The numericalmodel of the drill stringwall is given as follows:

(2

2

) +211

(2

2 2

1)(1 2)

222

(2

2 2

1)(2 3) = (222)

.

(3)

The boundary condition influenced the temperature dis-tribution of drill string wall includes two parts: one is the heatexchange between fluid of inside drill string and drill string,which is expressed by (2), and the other is heat exchangebetween annulus and the drill string which is written as

2(2

)

=2

= 2(2 3) , (4)

where 3is the temperature of annulus fluid;

2is the

density of drill string wall; 2is the specific heat capacity of

drill string; 2is the outer radius of drill string;

2is the

thermal conductivity of drill string; and 2is the convection

coefficient of outer drilling string wall.

(3) In the Annulus. The factors that influenced the annulustemperature are consisted of three sections: (1) the mass flow


rate of fluid; (2) the temperature distributions of drill stringand wellbore walls; and (3) heat generated by fluid fractionand drilling string rotation.The transient heat transfer modelin the annulus is expressed as follows:

(33

3)

(2

3 2

2) +222(2 3)

2

3 2

2

23ef (3 4)

2

3 2

2

+

(2

3 2

2)= (333)

.

(5)

The interface between annulus and wellbore wall isimportant since it mathematically couples the surroundingformation with the flow in the annulus. Therefore, to guar-antee continuity of heat flux during circulation and shut-incondition, the boundary conditions are

3(3

)

=3

+ ef (4 3) = ef(4

)

=3

, (6)

where 4is the interface temperature between annulus fluid

and borehole wall;is the energy source term of unit length

inside annulus; is the flow rate of annulus; 3is the radius

of borehole wall; 3is the thermal conductivity of annulus

fluid; effective thermal conductivity ef considers the effectof porosity;

3is the convection coefficient of borehole wall;

ef is the effective heat transfer coefficient which considers theeffect porosity.

(4) Each Heat Transfer Region in Surrounding Wellbore. Theeffect of factor on heat exchange for the surrounding wellboreincludes four sections: (1) the vertical conduction of heat inthe medium; (2) the rate of heat exchange among volumeelements; and (3) the rate of heat exchange for formation fluidwhich can flow in the porous medium.The energy balance ineach heat transfer medium is

(ef

) +1

(ef

)

= ()ef(

+ V

) ,

(7)

where

()ef= (

)

+ (1 ) ()

V= (,,

, in, ) .

(8)

The mathematical formulation for the hydrodynamicmodel of rock formation is based on one-dimensionalvolume-averaging equations that govern the hydrodynamicphenomena of an incompressible fluid across an isotropicporous medium [22], which are represented as follows:

V=

,

(2

2+1

) +

in= 0,

(9)

where is different unit temperature of porous medium

in the formation; is the radius of porous medium in the

formation; the magnitude of is decided by casing program( 4); is an effective porosity of formation; and represent rock andpore fluid, respectively; V

is the radial flow

velocity;fu is formation fluid mass flow to annulus; is thedrilling fluid of mass flow;

is the lateral flow area; is the

dynamic viscosity; is the intrinsic average pressure; is theabsolute permeability of the isotropic porous medium; isthe mass source term; and

is the relative permeability.

3.2.2. Transient Heat Transfer during Shut-In Stage. Duringstop circulation stage, the heat exchange method can beclassified into two ways, which is dependent on the inter-face between gas and liquid. Therefore, combined with thestudy of heat exchange mechanism between wellbore andformation during fluid circulation stage and energy andmassconservation principles, the description of heat exchangetypes during shut-in stage is presented by a set of partialdifference equations.

(1) In the Drill String

(1) The transient heat transfer model of above interfacebetween gas and liquid and below lost formation isexpressed as

2

1

2(

2

1)

22

1ln [(

1+ 2) /21] +

12

1ln [2

2/ (1+ 2)]

=

(

1

1

1)

.

(10)

(2) The transient heat transfer model of below interfacebetween gas and liquid and above lost formation isdescribed as

2

1

(

1

1

1)

2

1

2

1(

1

2)

1

=

(

1

1

1)

. (11)

(2) Drill String Wall


2

3

2(

3

2)

(

3ln ((

1+ 2) /21) +

2ln (2

2/ (1+ 2))) (

2

1 2

0)

+

2

2

2

2

2

1

2(

2

1)

(

2ln ((

0+ 1) /20) +

1ln (2

1/ (0+ 1))) (

2

1 2

0)

=

(

2

2

2)

.

(12)


(2) The transient heat transfer model of below interfacebetween gas and liquid and above lost formation isexpressed as

(

2

2

) +21

1

(2

2 2

1)(

1

2)

22

2

(2

2 2

1)(

2

3) =

(

2

2

2)

.

(13)

(3) In the Annulus


2

3

ef (

4

3)

(

ef ln ((2 + 3) /22) +

3ln (2

3/ (2+ 3))) (

2

2 2

1)

2

3

2(

3

2)

(

3ln ((

1+ 2) /21) +

2ln (2

2/ (1+ 2))) (

2

2 2

1)

=

(

3

3

3)

.

(14)

(2) The transient heat transfer model of below interfacebetween gas and liquid and above lost formation isdescribed as

(

3

3

3)

(2

3 2

2) +

22

2(

2

3)

2

3 2

2

23

ef (

3

4)

2

3 2

2

+

(2

3 2

2)=

(

3

3

3)

,

(15)

where the meaning parameters of the shut-in stage definedfrom (10) to (15) are the same as that of circulation stage.

The form of transient numerical model for each heattransfer region surrounding wellbore during shut-in stage isalso the same as (7).

4. Numerical Solutions

To obtain the temperature distribution on the term of time,the solution of these equations is complicated. Developedmodels incorporate solution methods which are based onfinite difference techniques. The wellbore and the adjacentformation are represented by a two-dimensional, mesh gridincluding a number of radial elements due to casing programand a variable number of vertical elements depending on thewell depth. Each of radial elements corresponds to differentportion of the wellbore cross-section from inside drill stringinto the formation [21]. Therefore, a set of partial differentialequations can be presented as finite difference form usingfinite differences technique for each element of grid so as todescribe the transient heat exchange in each element for an

implicit form [25]. A set of nonlinear algebraic equations arethen solved using an iterative method until the error rangecan be accepted. The case of finite difference can be definedas follows.The spatial derivatives and the time derivatives arethe first-order as exhibited in (16):

+1

, +1

,1

. (16)

The second-order spatial derivatives are represented bythree-point central difference approximations [26, 27]:

2

21

(

+1

,+1 +1

,

+0.5

+1

, +1

,1

0.5

) . (17)

The time discretization at node is expressed in

+1

,

,

. (18)

Application of earlier definitions enables the equation foreach node to be written in single generalized vector form:

+1

1,+ +1

,+ +1

+1,+ +1

,1+ +1

,+1= .

(19)

Thematrix formof finite difference for each node is given:

+1

= . (20)

The finite difference equations are solved by fast succes-sive overrelaxation (SOR) iteration method; the followinggeneral form for each node is expressed as follows:

V+1+1

,=

,

[,

,+ ,

V+1+1

1,

+,

V+1

+1,+ ,

V+1+1

,1+ ,

V+1

,+1]

+ (1 )

V+1

,.

(21)

Using implicit form of finite difference method, (1) and(2) are, respectively, shown as follows:

11

2

1

+1

1,1+ (11

+11

2

1

+21

1

) +1

1,

21

1

+1

2,=1

2

1

+11

1,

(22)

211

(2

2 2

1) +1

1,+2

0.5

+1

2,1

+2

+0.5

+1

2,+1+222

(2

2 2

1) +1

3,

(2

0.5

+2

+0.5

+211

(2

2 2

1)

+222

(2

2 2

1)+22

) +1

2,= 22

2,,

(23)


5000

4000

3000

2000

1000

0

Well

dep

th (m

)

Circulation 5 hrCirculation 10 hrInitial formation temperature

0 25 50 75 100 125Temperature profile (C)

Circulation 1 hr

Figure 2: Annulus temperature profiles at different circulation timeunder no lost circulation conditions.

where is 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 of

coefficients; SOR is the Gauss-Seidel iterative method if isequal to 1 in (21); the SOR is overrelaxation method if ismore than 1; SOR is under relaxation method if is less than1.

The calculation accuracy depends on the meshing ele-ments and the size of the interval values. In general, it isobserved that the vertical element size is less than 3% of thetotal well depth to ensure that the annulus temperature profileremains independent of the vertical element size [6].

5. Model Solution Procedure

5.1. Basic Data. The basic data of calculation in this studyare the referenced literatures reports [6], which are shown inTables 1, 2, and 3. The flow rate and depth of lost circulationare assumed as 4.0 l/s and 3500m, respectively.

5.2. Numerical Model Application

5.2.1. Example Analysis in Circulation Operation Condition.As shown in Figure 2, the annulus temperature distributionsas a function of depth at different circulation time underno lost circulation conditions are presented. As intermediatecasing depth is 3000m (Table 1), the annulus temperaturesof cementing section vary under different circulation time,whereas the annulus temperature of open-hole section grad-ually decreases with the increase of the circulation time.Thatis because casing thermal conductivity coefficient is 19.4 timesthan that of formation, resulting in more amount of heatexchange between annulus energy of cementing section andformation compared with that of open hole. Meanwhile, the

0 2 4 6 8 1020

22

24

26

28

30

32

34

Circulation time (hr)

Exit

tem

pera

ture

(C)

Figure 3: Outlet temperature distribution as a function of circula-tion time under lost circulation condition.

5000

4000

3000

2000

1000

0

Well

dep

th (m

)0 25 50 75 100 125

Temperature profile (C)

Circulation 5 hCirculation 10 hInitial formation temperature

Circulation 1 h

Figure 4: Annulus temperature profiles as a function of circulationtime under lost circulation conditions.

annulus heat quantity is gradually carried to surface withthe increase of the circulation time and thus results in thedecrease of annulus temperature of open-hole section [18].

The relationship between outlet temperature and circu-lation time under lost circulation condition is also investi-gated. As shown in Figure 3, the outlet temperature rapidlydecreases within the initial circulation (0.5 h) and then grad-ually increases during the latter circulation. One plausibleexplanation is thatmore heat quantity is carried towellmouthat initial circulation compared to that at latter circulation andthus leads to the wellbore wall of well mouth heat.

Under lost circulation conditions, the effect of alterationsin circulation time on the annulus temperature distributionis shown in Figure 4. It is found that the annulus temperatureof open-hole section continuously decreases as the increase of


Table 1: Basic data of drill string assembly and casing program.

Drill pipe Drill collar First casing Second casing Third casingInner diameter (mm) 151 70 486 318 224Outer diameter (mm) 168 171 508 340 244Depth (m) 4000 600 1500 3000Depth to cement (m) 0 300 1400

Table 2: Basic data of thermal physical parameters.

Drill pipe/casing Drill string Drill fluid Cement Formation Formation fluidDensity (kg/cm3) 8000 8900 1200 2140 2640 1050Heat capacity (J/kgC) 400 400 1600 2000 800 4200Thermal conductivity (w/mC) 43.75 43.75 1.75 0.70 2.25 0.50

0 1 2 3 4 577

84

91

98

105

112

119

126

Formation radial distance (m)

Circulation 1 hCirculation 5 hCirculation 10 hCirculation 1 h

Circulation 5 hCirculation 10 hInitial formation temperature

Radi

al te

mpe

ratu

re (

C)

Figure 5: Formation radial temperature distributions of 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 of Figure 2. Meanwhile, the annulustemperature profiles of circulation 5 h and 10 h are both lowerthan the formation temperature below 1500m. Compared toFigure 2, Figure 4 indicates that the annulus temperature ofcement section decreases under lost circulation conditions.The reason is that the annulus fluid temperature at 3500mis higher than that of annulus fluid at any depth of cementsection.Herein, heat quantity of annulus fluid at 3500mflowsinto the formation increased, which can result in the decreaseof the annulus temperature.

Similarly, Figure 5 also reflects the formation radial tem-perature distributions of lost depth and bottom hole underdifferent circulation time. Noticeably, the formation radialtemperature decreases with the increase of the circulationtime, whereas the decrease of the surrounding formationtemperature at lost depth is less than that of at bottom hole

5000

4000

3000

2000

1000

0

Well

dep

th (m

)

4 2 0 2 4 6 8

Circulation 5 hrCirculation 10 hr

Circulation 1 hr

Temperature difference profile (C)

Figure 6: Temperature difference profiles between annulus andinside pipe fluid during different circulation time.

during the latter circulation. The surrounding formation iscontinuously heated by annulus fluid at lost depth duringinitial circulation stage and then leads to its temperature risebeyond its initial formation temperature as shown in Figure 5(circulation 1 h). Meanwhile, the annulus temperature islower than formation temperature after long circulation timeand thus leads to formation temperature decrease. However,the formation is continuously cooled by circulation fluidat bottom hole and then leads to the temperature of thesurrounding formation decrease below the initial formationtemperature.

To get a deep insight of the heat transfer mechanismfor wellbore during the circulation stage, the temperaturedifference distribution between annulus and inside pipe fluidunder different circulation time is studied. As shown inFigure 6, the temperature difference remarkably decreasesfrom bottom-hole to casing shoe with increasing the circu-lation time. Meanwhile, the temperature difference changes


Table 3: Other basic data of bottom hole.

Depth (m) Total well diameter(mm)Open hole diameter

(mm) Flow rate (l/s)Surface temperature

(C)Geothermal gradient

(C/100m)4600 660 213 13.2 16 2.23Inlet temperature(C)

Outlet temperature(C)

Plastic viscosity(mPas)

Yield value(mPa)

Consistencycoefficient (mPas) Fluidity point

24 32 34 10 0.34 0.65

0 2 4 6 8 1019

20

21

22

23

24

Shut-in time (hr)

Tem

pera

ture

of w

ell m

outh

(C)

Figure 7: Temperature of 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 fluid temperature inthe wellbore except for well mouth section during the wholecirculation stage.

5.2.2. Example Analysis in Shut Condition. As shown inFigure 7, the temperature of well mouth continuouslydecreases during the whole shut-in stage. The result ofFigure 3 shows that the outlet temperature increases duringthe latter circulation, resulting in surrounding formationcontinuously heated. However, during shut-in stage, as gasis instead of fluid at well mouth, heat exchange betweenwellbore and formation is less due to heat conductivity of gas.Therefore, the temperature of well mouth gradually decreaseswith shut-in time increased.

As it is seen from Figure 8, the annulus temperaturecontinuously increases with the increase of 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 8.5 h on allfluids of inside pipe and annulus above lost depth flows intoformation.Therefore, the heat exchange between thewellboreand formation by heat conduction is more than that of heatconvection during 8.5 h of shut-in. After that, the formationenergy fluxes into annulus in the heat conduction way aswellbore fluid is in the static state beyond 8.5 h, thus resultingin the improvement of temperature. Furthermore, the tem-perature eventually increases to be equal to the formation

5000

4000

3000

2000

1000

0

Wel

l dep

th (m

)

0 25 50 75 100 125Annulus temperature profile (C)

Shut-in 1 hrShut-in 5 hr

Shut-in 10 hrInitial formation temperature

Figure 8: Annulus fluid temperature profiles during different shut-in time.

temperature. However, when the temperature is above thelost depth, annulus energy which arose from formation isnearly equal to that of the annulus gas transferring to thesurrounding formation and the inside pipe drilling fluid bythe heat convective way, which leads to the temperature hardchange.

Figure 9 indicates that the radial formation temperaturedecreases with increasing the shut-in time at lost circulationdepth and bottom hole, and both of them are lower thanthat of initial temperature. However, the radial formationtemperature at lost depth slowly decreases with the increaseof the shut-in time, and temperature change at the bottomhole is larger than that of at lost depth. The reasonableexplanation is that the temperature difference between annu-lus and formation at bottom hole is larger than that of atlost circulation point during circulation stage. Comparedto Figure 5, it is surprising that Figure 9 implies that theformation temperature disturbance distance in shut-in stageis larger than that of circulation stage. It is derived from thatthe starting point of disturbance distance for radial formationtemperature is atwell wall during the circulation stage, but thestarting point of disturbance distance is at inside formationduring the shut-in stage which is the destination point ofdisturbance distance for circulation stage.

As shown in Figure 10, the annulus temperature fromtop hole (500m) to bottom hole is higher than the inside


0 1 2 3 4 580

88

96

104

112

120

Formation radial distance (m)

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

Shut-in 5 hrShut-in 10 hrInitial formation temperature

Radi

al te

mpe

ratu

re (

C)

Figure 9: Formation radial temperature distributions of lost circu-lation point and bottom hole during shut-in stage.

5000

4000

3000

2000

1000

0

Well

dep

th (m

)

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

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

Figure 10: Temperature difference profiles between annulus andinside pipe during shut-in stage.

pipe temperature during the whole shut-in stage, and onlythe temperature difference between annulus and inside pipefrom well mouth to the depth of 500m 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 10, it is observed that the temperature differencebetween annulus and inside pipe is greatly influenced by thelost depth, make up of string, and casing program.

As shown in Figures 11 and 12, annulus temperaturechanges of 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

Ann

ulus

tem

pera

ture

of l

ost p

oint

(C)

Figure 11: Annulus temperature distributions of lost circulationdepth during circulation and shut-in stages.

0 4 8 12 16 2090

96

102

108

114

120

126

132

Circulation and shut-in time (hr)

Circulation temperatureShut-in temperatureFormation temperature

Ann

ulus

tem

pera

ture

of b

otto

m h

ole (

C)

Figure 12:Annulus temperature distributions of bottomhole duringcirculation and shut-in stages.

circulation and the shut-in stages. It can be seen that theannulus temperature rapidly decreases during the initialcirculation stage and slowly varies in the latter circulation andshut-in stage, followed by the change of annulus temperatureshowing the same tendency under the two conditions earlier.Meanwhile, the annulus temperatures at initial circulationstage are both higher than that of the initial formationtemperature at lost depth and bottom hole, and then bothof them are lower than initial formation temperature duringlatter circulation and shut-in stages.However, beforewellborefluid above the lost depth flows into formation (8.5 h), it isinterestingly noted that the annulus temperature gradually


increases with the increase of shut-in time at lost depth byheat convection, followed by quickly decreasing, and thenslowly increases at bottom hole by heat conduction. Figures11 and 12 also show that if the annulus temperature aftercirculation recovers to the initial formation temperature,shut-in time can be longer than that of circulation time [28].The phenomenon accounts for the reason why long time forshut-in is needed to obtain initial formation temperature.

6. Discussion

To demonstrate the applicability of the methodology devel-oped in this work, the OM-16C geothermal well was consid-ered. This well is in Kenya, which was drilled with boreholediameters of 26, 17.5, 12.25, and 8.5 in. The casing has 20,13.375, 9.625, and 7 in diameters at 60, 300, 750, and 2680mdepths, respectively. Temperatures in and around the OM-16C geothermal well during circulation and shut-in stageswere estimated by the transient heat transfer models. TheHorner (Dowdle andCobb, 1975) andHasan andKabir (1994)methods were used to compare our numerical results [6, 12].

The input data to simulate the geothermal well are drillingfluid flow rate of 125.3m3/h, surface temperature of 21.6 C,and drilling fluid properties which include the thermalconductivity of 0.85W/mC, the density of 1100 kg/m3, theviscosity of 0.052 Pa.s, and specific heat of 1200 J/kg.C.Circulation time was 16 h.

A compilation of main results obtained in these thermalstudies was presented in Figure 13. We reckoned that thelogged temperature of shut-in 27 days was approximatelyequal to the static formation temperature (SFT) due tothermal recovery conditions during the long time shut-in.As shown in Figure 13, it can be observed that the measuredtemperature was satisfactorily matched with the simulatedtemperature (continuous line). Figure 13 also showed the SFTcalculated by means of Horner (Dowdle and Cobb, 1975) andHasan and Kabir (1994) methods [6, 12]. Obviously, as shownin Figure 13, the Hasan-Kabir method is closer to the SFTcompared to the Horner method. Differences between thecomputed data (or simulated 27 days) and measured valueswere estimated and expressed as a percentage deviation basedon the result of Figure 13, and the percentage deviationbetween the simulated SFT and analytical methods as alsocomputed from Figure 13. It can be observed the maximumdeviation between measured and simulated data is 3.1% andminimum deviation is 2.24%, which corresponded to thecontrol error in engineering. The best approximation tothe simulated SFT values corresponded to the Hasan-Kabir,which presented minimum differences of 3.50%, 4.12%, and4.08%. Therefore, the simulated SFT method in this work isbetter than that of the analytical methods (Horner andHasanand Kabir).

7. Conclusions

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

3000

2500

2000

1500

1000

500

0

Dep

th (m

)

Shut-in 9 hrShut-in 27 daysSimulated SFT

Dowdle-Cobb (1975)Hassan and Kabir (1994)

0 50 100 150 200 250 300 350Temperature profile (C)

Figure 13: Simulated and logged temperature profiles in OM-16C geothermal well during shut-in stages. Also shown the SFTsestimated with the Horner (Dowdle and Cobb, 1975) and Hasan andKabir (1994) methods and with this work [6, 12].

in oil-gas or geothermal well during circulation and shut-in stages under lost circulation conditions. The equationsproperly account for the energy conservation in each regionof a well, and mass balances are performed at any numericalnode where annulus fluid fluxes into formation. Heat transfercoefficients and thermophysical properties (gas instead offluid) in the annulus and the surrounding formation changedue to lost circulation. Simulation results show that thetemperature distributing characters of the wellbore andsurrounding formation are remarkably influenced by thelost depth and casing program during the whole circulationand shut-in stage. Additionally, the disturbance distance offormation temperature at shut-in stage is larger than that of atcirculation stage at the same time.Moreover, it is necessary toprolong shut-in time than circulation time in order to obtainaccurate initial formation temperature.The present work canprovide a new way to improve the present methodology.

Acknowledgments

The authors appreciate the financial support by ChinaNational Natural Science Foundation (no. 51134004;51104124; 51204142; 51204140), Major State Science andTechnology Special Project of China (no. 2011ZX05021-003),Ph.D. Programs Foundation of Ministry of Educationof China (no. 20125121110001), and Southwest PetroleumUniversity of Young Scientific Research Innovation TeamFoundation (no. 2012XJZT003). The authors would also liketo appreciate their laboratory members for the generoushelp.

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