NUMERICAL INVESTIGATION OF THE TURBULENT FLOW PARAMETERS DISTRIBUTION IN A PARTLY PERFORATED HORIZONTAL WELLBORE

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1 NUMERICAL INVESTIGATION OF THE TURBULENT FLOW PARAMETERS DISTRIBUTION IN A PARTLY PERFORATED HORIZONTAL WELLBORE Mohammed Abdulwahhab Abdulwahd Research Scholar, Department of Marne Engneerng, Andhra Unversty, AP, Inda Ass. Prof. Sadoun Fahad Dakhl Department of Fuel& Energy, Basrah Techncal College, Iraq Prof. Nranan Kumar Inet Department of Marn Engneerng, Andhra Unversty, AP, Inda Abstract The overall pressure drop n a horzontal wellbore used n the recovery of ol and gas ndustry was classfed nto four separate effects due to wall frcton, ncrease n momentum, perforaton roughness and type of flud mng. A perforated secton s followed by a plan secton for many horzontal wells. The addtonal pressure drop due to combned effect of perforaton roughness and the type of flud mng was analyzed through numercal CFD and the results were compared wth the epermental results of other researchers. The computatons were based on the fnte volume method wth the SIMPLE algorthm standard k ε model. The ppe was used geometrcally smlar to the real perforated wellbore wth 60 phasng, 6 SPF (shoot per foot) and the ptch of the perforatons 60 mm (the number of perforatons n ths paper are less than epermental ppe). The parameters that are beng nvestgated are pressure drops of the ppe and so far smulatons have been carred out for an nlet ppe Reynolds numbers rangng from 28,773 to 90,153 for the total flow rate rato rangng from 0% to 100%. Numercal smulatons were performed usng CFX of ANSYS FLUENT 13, where the governng equatons of mass and momentum were solved smultaneously, usng the two equatons of standard k-ε turbulence model. As the rate of flow through the perforatons ncreases.e. wth the ncrease n flow rate rato, the total pressure ncreases due to large acceleraton pressure drop for hgher flow rate through the perforatons. The ncreases n perforatons number ncrease the total pressure drop and vce versa. The numercal results agreed wth the epermental work. 372

2 Keywords: Pressure drop, perforaton, wellbore, Reynolds number, numercal, total flow rate rato 1: Introducton. Over the last decade horzontal wells have become a well-establshed technology for the recovery of ol and gas. Consderable amount of analytcal and epermental work has been publshed on varous aspects of horzontal well used n the producton whch ncludes transent flow models, productvty ndces, and crestng behavor. Although these methods provde nsght nto the behavor of horzontal wells, certan methods have consdered the pressure drop along the wellbore and essentally assumng nfnte conductvty. The pressure drop n the ppe can severely lmt the actual producton length of the ppe. It s clear from the above that frctonal effects wll lead to a sgnfcant drop n the pressure between the heel and the toe of the well. An accurate set of sngle-phase eperments n a perforated ppe wth radal nflow has been conducted by (Su and Gudmundsson, 1995) (SG). SG conducted the eperment to account for the total pressure drop n a perforated ppe s contrbuted by frctonal, acceleratonal pressure drops and pressure drop due to the effects of radal nflow and mng pressure drop. SG observed that the frctonal and acceleratonal pressure drops are sgnfcant parts of the total pressure drop. However, the mng pressure drop s sgnfcant and ts contrbuton to the pressure drop s often negatve. Addtonally, when the veloctes of the radal and aal are equal, the radal flow wll penetrate the aal ppe and blockage of the ppe. Ths wll lead to an ncrease of the pressure drop n the ppe. Ouyang et al., (1996) performed eperments to determne the frcton factors for ppe flow wth radal nflow n lamnar and turbulent flow. The frcton factor based on the Stanford horzontal wellbore data yelds a frcton factor n whch the correlaton due to radal nflow s dependent on the Reynolds number of the flow n the radal perforaton. Ouyang et al., (1996) also mentoned that for turbulent flow, nflow reduces the wall frcton. Ouyang et al., (1998) studed a sngle-phase wellbore flow model that ncorporated not only frctonal, acceleratonal, and gravtatonal pressure drop, but also the pressure drop caused by nflow. The new model was readly applcable to dfferent wellbore-perforaton patterns and well completons, and was easly ncorporated nto reservor smulators or analytcal, reservor-nflow models. It was found that the nfluence of ether nflow or outflow depended on the flow regme present n the wellbore. It was found that the acceleratonal pressure drop may or may not be mportant compared to the frctonal component, dependng on the specfc ppe 373

3 geometry, flud propertes and flow condtons. It was recommended that the new wellbore-flow model be ncluded n wellbore/reservor couplng models to acheve more accurate predctons of pressure drop and nflow dstrbuton along the wellbore, as well as the well producton or necton rates. Kloster, (1990) eperment studed flow resstance n a perforated ppe, both wth and wthout flow necton. The Reynolds numbers covered from 45,000 to 60,000. The frcton factor values were 25-70% hgher than those of regular commercal ppes. He also observed that small nectons through perforatons reduced the frcton factor. A new frcton factor correlaton for horzontal wellbore was proposed by (Ashem et al., 1992) whch ncluded acceleratonal pressure losses due to contnuous flud nflu along the wellbore. They assumed that the nected flud entered the man flow wth no momentum n the aal drecton. Ihara et al., (1994) performed eperments that studed the channel flow wth contnuous nflu nto the horzontal wellbore from the ol reservor. The pressure drop along the test channel ncreased due to the effect of nflu. The model of onedmenson momentum echange agreed wth the epermental data for relatvely large Reynolds numbers. The frcton factor of perforaton roughness was measured n ppes geometrcally on par wth casng used n horzontal wells. There was no flud flow through the perforatons n the work reported. The epermental data was analyzed usng the unversal velocty dstrbuton law and the concept of roughness functon. It was found that the roughness functon ncreased lnearly wth the perforaton/casng dameter rato. An emprcal relatonshp was obtaned to estmate the frcton factor n ppes wth perforaton roughness (Su and Gudmundsson, 1998). The am of ths work s to demonstrate the alternate use of CFD smulatons nstead of eperments to estmate the pressure drops, the pressure drop coeffcents at the horzontal wellbore and the effect of the perforaton densty for a range of locally changng flow ratos and Reynolds numbers. The total pressure drop ncorporates not only frctonal, acceleratonal pressure drops but also the pressure caused by nflow. The man dfference between the theoretcal study n ths paper wth the eperments carred out by SG are the dameter of perforatons and the perforaton densty of the ppe. SG has used a perforaton dameter of 3mm and 158 perforatons, where as n ths present study, a perforaton of 4mm dameter and 60 perforatons has been used. The paper s organzed as follows. The authors start wth some theoretcal background of the work. Ths s followed by detals of the geometry and numercal model and dscusson of the relevant parameters whch nfluence the pressure loss. Subsequently, numercal results are presented and a detaled data analyss s carred out. 374

4 2: Theoretcal Analyss Model Descrpton. The physcal model descrpton s that of a partly perforated ppe and a plan ppe wthout perforaton. The length of the ppe s equal to 1300 mm and 22 mm dameter as shown n fgure 1. The ppe s dvded nto two sectons wth equal length. The perforated secton s part of the ppe that has a 600 mm, perforaton phasng 60 and Smulaton Perforaton Densty 6 (SPF). The ptch of the perforatons s 60 mm wth a perforaton dameter of 4 mm. The other secton, 600 mm n length s dvded nto four equal sectons of 150 mm. Each one to nvestgate the pressure profle along the ppe when there s a perforated secton upstream. Ppe and perforaton geometry for epermental and theoretcal study s lsted n Table 2.1. (a) (b) Fgure (1) Confguraton of partly perforated test ppe (not to scale). The computatonal doman havng same dmensons s to be consdered as epermental rgs (Su and Gudmundsson, 1998). The geometry has been analysed usng 3 dmensonal Computatonal Flud Dynamcs (CFD). Fgure 2 s the unstructured computatonal grd, the mesh consst of nodes and elements wth fve boundary layers. The calculatons carred out were wth commercal fnte volume code ANSYS FLUENT 13 CFX5 usng a frst order scheme and a turbulent wth k epslon model. Fgure (2) The unstructured mesh for partly perforated ppe. 375

5 Table 2.1. Geometry of the test ppe. Item Epermental Theoretcal Outer Dameter 30 mm - Inner Dameter mm 22 mm Perfo. Dameter 3.0 mm 4.0 mm Total perfo. number Perfo. phasng Perfo. densty 12 SPF 6 SPF Smulaton Parameters. The flud consdered for the smulatons s water wth constant densty of kg/m 3 and dynamc vscosty of kg/m s. The flud s assumed as Incompressble flow. Three tests were carred out wth Reynolds number of the nlet flow rangng from 28,773 to 90,153. In each of the tests, the flow rate through the perforatons was ncreased from zero to mamum value. The roughness of the test ppe wall was 0.03 mm; the type of the test ppe was PVC. Test detals are summarzed n table 2.2. Table 2.2 Parameters of partly perforated ppe tests. Test Inlet Flow Rate (lter/hr) Perforaton nlet Flow Rate (lter/hr) Inlet Flow (Re) Test 1 5,157-5, ,756-90,153 Test 2 3,361-3, ,935-61,557 Test 3 1,793-2, ,773-37,198 Unform water mass flow s ntroduced at the nlet of a partally perforated ppe. Two boundary condtons are consdered. At the nlet, mass flow rate s taken nto consderaton both aally and radally where as at the et, outlet pressure s consdered as the boundary condton. It s assumed that no-slp boundary condtons occur along the walls. Water enters at a unform temperature (T) of 25 C. For the symmetry lnes both velocty and pressure s kept constant. 3. Theoretcal Smulatons. Over a long perod of tme the pressure drop n a fully developed turbulent ppe flow s beng studed by several researchers and nvestgators. The pressure drop n a straght ppe has been determned n numerous eperments. In ths study, the general model developed by (Su and Gudmundsson, 1998) was adopted to analyze the acqured data. They suggested pressure terms lke acceleraton, frcton, perforaton roughness and flud mng pressure drop components. The followng relatonshp gves the four pressure drop terms that make up the total pressure drop n a perforated horzontal well p = p p p p (1) acc. wall perfo. m. 376

6 The pressure drop caused by perforaton roughness and flud mng were lumped together and classfed as addtonal losses. Eq. 1 can then be rewrtten as p = pacc. pwall padd. (2) Applyng the conservaton of lnear momentum to the control volume n the aal drecton for each perforaton unt wthn equal length L, results n the sum of the forces actng on the control volume surfaces towards the downstream drecton of the ppe as.. F = moutuout mnun (3) where the mass flow rate s. m = ρau (4) When radal nflow occurs, the statc pressure n the ppe s not unform, and the velocty profle s not fully developed. In addton to the force contrbuted by the pressure dfference across the control volume and wall shear force, the sum of the forces actng on the control volume surface ncludes a force due to the combned effects of the rreversble process of flud mng and the presence of the perforaton hole, ncludng the effect of non-unformly dstrbuton of statc pressure and non-fully developed velocty profle, F = ( pn A pout A) τ w ( πd L) Fadd. (5) From the above equatons, ths can be rearranged to get the followng total pressure drop, 2 2 pn pout = ρ ( uout un ) pwall padd. (6) Eq. (6) ndcates that the total pressure drop conssts of three dfferent components: The pressure drop due to knematc energy change (acceleraton effects). Ths demonstrates the frst term on the rght sde of Eq. (6). The frctonal pressure drop due to wall frcton n a perforaton unt, p wall, the second term of Eq. (6), and can be calculated from the Darcy- Wesbach equaton (Whte, 1986 ), f L 2 p wall = ρv (7) 2 D When the relatve roughness of the ppe s known, an accurate and convenent relatonshp for the frcton factor n the turbulent ppe flow s the Haaland equaton (Haaland, 1983) ε f = 1.8log10 (8) Re 3.7D 377

7 For a hydraulcally smooth ppe, surface roughness ε should be set to zero. Ths equaton apples to both lamnar and turbulent flow. The addtonal losses pressure drop term was obtaned from the measured pressure drop after subtractng the pressure drop contrbuton due to wall frcton and flud acceleraton. The pressure drop due to mng effects arses from the nteracton between perforaton flow and wellbore flow whch s causng dsturbances n the boundary layer and hence affects the pressure drop. The flud enters radally to the wellbore through a perforaton. It mes wth the aal flow and ncreases mass nto the well, thereby ncreasng the flow velocty n the well. Hence, the acceleraton of the flow wll ncrease the velocty at the outer. Ths gves a pressure drop across the perforaton. Subtractng the acceleratonal and frctonal pressure drops from the total pressure drop that flows through the perforatons, the remanng pressure drop s a practcal representaton of the addtonal pressure drop Ρ add (perforaton roughness and mng effects). Fgure (3) Horzontal completon effect on reservor nflu and wellbore hydraulc (Yula, 2001). Fgure 3 llustrates the nterplay between the pressure and flu dstrbuton along the wellbore through the completon openngs. It shows the ncrease n flow rate from toe to heel but the decrease n pressure from toe to the heel. 4. Results and Dscussons Total Pressure Drop n Perforated secton. Theoretcally was carred out on the ppe that was smulated wth the epermental ppe. Three tests wth dfferent ppe flow rates for aal flow and radal flow were carred out and the results are shown n table 2.1. Fgure 4 represents the total pressure drop n the perforated secton wth total flow rate rato where q = total perforaton flow rate dvded by the total flow rate 378

8 at the ppe outlet for the three tests. The total pressure drop values are calculated usng equaton 1 and 2. It s observed that there s an ncrease n flow rate rato as the rate flow through perforaton ncreases. The total pressure drop ncreased due to the larger acceleraton pressure drop at hgher flow rate through the perforatons. The total pressure drop was found to be hgher for hgher Reynolds numbers. For hgher aal flow velocty there s larger frctonal pressure drop at the wall. The ncrease of radal flow through perforatons ncreases the total flow rate rato and ncreases the total pressure drop. The total pressure drop of 60 perforatons (for the present paper) s lesser than the total pressure drop of 158 perforatons (epermental ppe). Ths s because the perforaton densty of the epermental ppe s twce and a half larger than that of the ppe for the present paper. The total pressure drop accordng to Eq. 2 s equal to the acceleraton, frctonal pressure drops and the effect of flud mng and perforaton roughness. The total flow rate rato ncreased due to the ncrease n the flow through the perforatons that ncreased the total pressure drop. Ths s due to the larger acceleraton pressure drop for hgher flow rate through the perforatons. Fgure 5 depcts the total pressure drop n the entre ppe whch s smlar to the behavor as shown n fgure 4. The values of total pressure drop for the entre ppe s hgher than the total pressure drop n the perforated secton for all the tests. For test 1, the values of the total pressure drop n the whole ppe s larger than the values n the perforated secton n the range of 34.2% to 25.5% at 0% to 100% flow rate rato respectvely, for test 2, wthn the range of 33.9% to 22% and for test 3 wthn the range of 35.3% to 19.4%. The ncrease of the total pressure drop n the whole ppe s due to the wall frcton pressure drop n the plan secton. Fgure (4) Total pressure drop n perforated secton, wth dfferent tests condton. Fgure (5) Total Pressure drop n the whole ppe wth dfferent tests condton. 379

9 4.2. Pressure Drop n Perforated Secton. Three tests wth dfferent ppe flow rates were carred out on the perforated ppe. The pressure drop due to momentum change (acceleraton pressure drop) was calculated from Eq. 6 (the frst term on the rght sde). It s notced that the values of acceleraton pressure drop for partly perforated wellbore were hgh. The pressure drop due to momentum ncreased for each test. The nlet mean velocty at the nlet of the perforated secton and the outlet mean velocty at the outlet of the perforated secton, the veloctes and the pressure drop due to momentum were calculated. Fgure 6 represents the relatonshp between pressure drop due to momentum and total flow rate rato (q). It depcts that the ncrease n the pressure drop wth ncrease n the value of q for all tests s smlar to the behavor n fgure 4. Fgure 7 represents the behavor of addtonal pressure drop for the three tests. The addtonal pressure drop decreases as the flow rate rato ncreases. In the present numercal analyss all the values of the addtonal pressure drop are negatve. When the perforaton nflow ncreases the total pressure decreases and the frctonal pressure drop decreases too. The values of addtonal pressure drop are negatve. Fgure 7 llustrates the frctonal pressure drop that was reduced when the flow rate rato was ncreased. Fgure (6) Pressure drop wth effect of momentum forces n perforated secton. 380

10 Fgure (7) Addtonal pressure drop for dfferent tests. The pressure drop ncreases wth the ncrease of Reynolds numbers whch contrbutes to the larger wall frcton pressure drop wth hgher flow veloctes. The ordnary wall frcton pressure drop of a perforated secton was calculated usng the Darcy-Wesbach equaton. For the turbulent flow, the aal velocty gradent near the ppe wall decreases and hence the wall frcton shear stress also decreases accordngly. On the contrary, outflow lowers and reduces the boundary layer and thus decreases the average velocty outsde the layer but ncreases the velocty nsde the layer, whch results n an ncrease of the aal velocty gradent near the ppe wall and hence the wall frcton shear stress (Ouyang et al., 1997) Pressure Drop Coeffcent. Pressure drop n a perforated secton s the functon of the flow rate n the ppe. Therefore, the numercal results from dfferent tests are nterested n the pressure drop coeffcent parameter. A pressure drop coeffcent represents an mportant parameter whch s defned as the pressure drop across the perforated secton dvded by the knetc energy at the outlet of the ppe. Ρ k = (9) 2 0.5* ρ * U where U s the average flow velocty at the outlet of the test ppe. The pressure drop coeffcents were calculated for total and addtonal pressure drops for perforated secton and for the whole ppe for all the three tests. The data ponts for each test follow a straght lne as shown n fgure 8. Data ponts of the three tests for total pressure drop followed parallely and closely for some ponts of those tests conducted wth Reynolds number range from 82,756 to 90,153 and from 53,935 to 61,557. The pressure drop coeffcents ncrease lnearly wth ncreasng total flow rate rato. 381

11 Fgure 9 represents the pressure drop coeffcent n the whole ppe.e. perforaton secton as well as plan secton. In fgure 11 there s a drastc change n the values between the three tests. The values of pressure drop coeffcent for test 3 (aal 1793 lt/hr and radal from 0 to 899 lt/hr) are much hgher compared to test 1 and 2. Ths s because the value of aal velocty s less than the other tests. Fgure (8) Pressure drop coeffcent n perforated secton, wth dfferent tests condton. Fgure (9) Pressure drop coeffcent n the whole ppe, wth dfferent tests condton. Fgure (10) Pressure drop coeffcent n plan secton, wth dfferent tests condton. Fgure 10 represents the results of pressure drop coeffcent n the plan secton for the three tests. The pressure drop coeffcent decreases wth the ncrease n total flow rate rato. The pressure drop coeffcent for test 3 s hgher than the values of the other tests. Ths s due to lower values of aal velocty at the entry of the plan secton of the ppe; the pressure drop coeffcent s hgh. On the contrary, wth the hgher values of the aal velocty, there s a drop n the values of the pressure drop coeffcent. 382

12 Fgure 11 represents the velocty dstrbuton contour for aal flow rate of 5618 lt/hr and radal flow of 841 lt/hr n the perforated secton. The flow velocty ncreases due to the mng of the aal flow wth radal flow at the uncton. It s notced that there s a drop n pressure and rse n flow rate at the perforated secton. Fgure (11) Velocty dstrbuton contour for perforated secton aal 5618 lt/hr & radal 841 lt/hr. (a) At frst perforatons. (b) At last perforatons. Fgure (12) Velocty streamlnes for perforated secton, aal 5618 lt/hr & radal 841 lt/hr. Fgure 12 llustrates velocty streamlnes for perforated secton when the aal flow s 5618 lt/hr and the radal flow s 841 lt/hr. It s observed that there s a rse n aal velocty at the end of the perforated secton due to flow through perforatons Turbulence Intensty. The turbulent ntensty T s lnked to the knetc energy and a reference mean flow velocty as follows: k 3 T = (10) U The equatons to be solved for ncompressble flow are the conservaton of mass Eq. (11) and momentum Eq. (12) n Cartesan coordnate. U = 0 (11) 383

13 384 = u u U U p U U t U ρ µ ρ ρ (12) where: u s man aal velocty and U s bulk velocty. The Transport equatons for ε k model are for k, ( ) ( ) k k b k k t S Y p p k ku k t = ρε σ µ µ ρ ρ (13) And For ε, ( ) ( ) ( ) ε ε ε ε ε ε ρ ε ε σ µ µ ρε ρε S k C P C P k C u t b k t = (14) Fgure (13) Turbulent Intensty for Fgure (14) Turbulent Intensty for 5618 lt/hr wth dfferent radal values. dfferent Aal flows wth radal 899 lt/hr. Fgure 13 represents turbulent ntensty for a fed aal flow of 5618 lt/hr wth dfferent radal flow values of 93.44, 467 and 841 lt/hr. As a consequence, usng an optmzed power curve ft s the best ft for every ndvdual case over the new upper and lower lmts. It has a hgher value of turbulent ntensty at the nlet of the wellbore for all the three tests, and lower value of turbulent ntensty n the flow along the wellbore. Fgure 14 represents turbulent ntensty for a fed radal flow of 899 lt/hr wth dfferent aal flow values of 2318, 3836, and 5618 lt/hr. Power curve ft was appled, and resultant curves were shown and compared wth the curves from fgure 15. The curves for varyng aal flows wth fed radal flow show that there s a sharp drop n turbulent ntensty. Fgure 15 shows the comparson between the numercal smulatons (present work) wth epermental work (Su and Gudmundsson, 1998). It s observed that the graphs drawn for the epermental work and the numercal smulatons are appearng to have smlar behavor but the values are

14 dfferent between the three tests. The percentage error for test 2 wth rangng from 42.44% at zero of the total flow rate rato and decreasng to 1.4% at for total flow rate rato and then the values of total pressure drop of the present work ncreasng so that the percentage error ncreases from 4.4% to 7.4%. The percentage error for test 2 wth rangng from 42.39% at zero of the total flow rate rato and decreasng to 3.1% at for total flow rate rato and then, the values of total pressure drop of present work ncreasng so that the percentage error ncreases from 5.9% to 16.5%. Fnally, for test 3 rangng from 42.24% at zero of the total flow rate rato and decreasng to 14.4% at for total flow rate rato and then the values of total pressure drop of the present work ncreasng so that the percentage error ncreases from 2.8% to 33.8%. The total pressure drop values of the epermental work were obvously larger than those of present work. After a certan value of the total flow rate rato, the values of the total pressure drop ncreased. Ths was because the perforaton densty of epermental work [9] was twce and a half larger than of the present work. Fgure (15) Comparson of numercal smulaton and epermental data (Su and Gudmundsson, 1998). 5. Concluson Numercal smulatons have been carred out on the flow n a partly perforated ppe wth nflow through the perforatons. The geometry of the ppe that was used appromately smlar to the ppe used n the eperment (Su and Gudmundsson, 1998), ecept the ptch of the perforatons was 30 mm wth a perforaton dameter of 4 mm and perforaton densty of 6 (SPF) nstead of a perforaton dameter of 3 mm and 12 (SPE) as adopted n the epermental test rg. 1- The total pressure drop n the perforated secton of the ppe ncreased wth ncrease n the flow rate rato, but the value of the total pressure drop n the whole ppe was greater than the value n the perforated secton. 385

15 2- A large amount of numercal data was acqured. The Reynolds numbers were n the range appromately from 28,000 to 90, The addtonal pressure drop decreases as the flow rate rato ncreased. The addtonal pressure drop was of a negatve value after 0.04 of total flow rate rato (q) whch resulted from a total pressure drop, but the addtonal pressure drop resultng from the pressure drop was stll of a postve value wth ncrease n the total flow rate rato. 4- The aal velocty ncreased at the end of the perforated secton due to flow through perforatons. 5- The values of the total pressure drop n the whole ppe were larger than the values n the perforated secton. 6- The total pressure drop values of the epermental work were obvously larger than those of the present work due to the dfference n perforaton densty. 7- Numercal results have demonstrated that the number of perforatons have relatonshp wth ncrease or decrease n the total pressure drop. The ncreases n perforatons number are ncreased n the total pressure drop and vce versa. 8- The numercal results agreed wth the epermental work (Su and Gudmundsson, 1998). Acknowledgements To all the people who helped me, thank you. Thank you for my specal apprecaton to Prof. T.V.K. Bhanuprakash for provdng numerous answers to my countless questons. Nomenclature C 1ε, Standard k-epslon model constant u velocty (fluct. th comp.) [m/s] [-] C 2ε Standard k-epslon model constant u average aal velocty at nlet 1 [-] [m/s] C k Standard k-epslon model constant u average aal velocty at outlet 2 [-] [m/s] D ppe nner dameter [m] U bulk velocty [m/s] length between perforatons [m] f frcton factor [-] L k turbulent knetc energy [m 2 /s 2 ] Ρ total pressure drop [Pa] P b effect of buoyancy [-] P k producton of k [-] S t modulus of the mean rate of stran tensor [-] tme [s] Ρ pressure drop due to momentum acc. [Pa] Ρ addtonal pressure drop [Pa] add Ρ Press. drop due to perforaton perfo roughness Ρ pressure drop due to flud mng mng [Pa] 386

16 u man aal velocty [m/s] Ρ pressure drop due to wall frcton wall [Pa] Greek conventons ε turbulent dsspaton rate [m 2 /s 3 ] μ dynamc vscosty [kg/ms] σ k turbulent Prandtl number for k [-] µ t turbulent vscosty [kg/ms] σ ε turbulent Prandtl number for ε [-] ρ Densty [kg/m 3 ] References: Su, Z. and Gudmundsson, J.S. Pressure Drop n Perforated Ppes, PROFIT Proected Summary Reports, Norwegan Petroleum Drectorate, Stavanger (1995). Su, Z. and Gudmundsson, J.S. Pressure Drop n Perforated Ppes, report, Department of Petroleum Engneerng and Appled Geophyscs, U. Trondhem, Norway (1995). L-B Ouyang, S. Arbab, and K. Azz. General Wellbore Flow Model for Horzontal, Vertcal and Slanted Well Comletons. In The 71 st Annual SPE Techncal Conference Ehbton, Denver, Colorado, Paper SPE L-B Ouyang, S. Arbab, and K. Azz. A Sngle Phase Wellbore Flow Model for Horzontal, Vertcal, and Slanted Wells. SPE Journal, pp , Kloster, J. Epermental Research on Flow Resstance n Perforated Ppe, Master thess, Norwegan Int. of Technology, Trondhem, Norway, Ashem, H. et al. A Flow Resstance Correlaton for Completed Wellbore, J. Petrol. Sc. Eng., 1992, 8 (2), pp Ihara, M. et al. Flow n Horzontal Wellbores wth Influ through Porous Walls, paper SPE presented at the 1994 SPE Annual Techncal Conference and Ehbton, New Orleans, September. Su, Z. and Gudmundsson, J.S. Frcton Factor of Perforaton Roughness n Ppes, SPE Presented at the SPE 68 th Annual Techncal Conference and Ehbton, Houston, TX, USA, 3-6 October, Su, Z. and Gudmundsson, J.S. Perforaton Inflow Reduces Frctonal Pressure Loss n Horzontal Wellbore, Journal of Petroleum Scence and Engneerng, 19, pp , Whte, F. M. Flud Mechancs. McGraw-Hll, Inc., Haaland, S.E., Smple and Eplct Formulas for the Frcton Factor n Turbulent Ppe Flow. Journal of Fluds Engneerng, Vol. 105, March Yula Tang. Optmzaton of Horzontal Well Completon. Ph.D. Dssertaton, The Unversty of Tulsa, Lang-Bao Ouyang, Sepehr Arbab, and Khald Azz. Sngle Phase Flud Flow n a Wellbore. Annual Techncal Report, Stanford Unversty,