Iraqi Journal of Chemical and Petroleum Engineering Vol.13 No.4 (December 2012) 35- 45 ISSN: 1997-4884 Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells RIYADH HAZIM FAWZI University of Baghdad, College of Engineering, Petroleum Engineering Department, Baghdad, Iraq Abstract Empirical equation has been presented to predict the optimum hydrodynamic pressure gradient with optimum mud flow rate (one equation) of five Iraqi oil wells to obtain the optimum carrying capacity of the drilling fluid ( optimum transport cuttings from the hole to the surface through the annulus). This equation is a function of mud flow rate, mud density and penetration rate without using any charts or graphs. The correlation coefficient accuracy is more than 0.9999. Keywords Correlation, drilling, pressure gradient, mud flow rate, penetration rate and mud density Introduction Saleh [1] has taken the actual data from Iraqi oil fields (five wells) in order to determine the optimum flow rate using graphical solution. Four of the wells are production wells (No.1, 2, 3, 4) and the fifth one is observation well (No.5). The hydrodynamic pressure gradient of the wells can be expressed by Saleh [4] Using the following equation: D Li L PT D Pbh i J i .)( 1     ...(1) And:    J i LiD 1 …(2) Saleh [1] draws five graphs of five wells by using the relation between the hydrodynamic pressure gradient and mud flow rate to choose the optimum mud flow rate. In the previous studies of Iraqi fields, the empirical equations were not used to find the hydrodynamic pressure gradient, but rather the derivation equations only. In this study, the empirical equation (one equation) has been performed to predict the hydrodynamic pressure gradient of five wells .Then the optimum mud flow rate can be calculated. Tables (1) and (2) show the range of data ( only important data were chosen ) and the special information of wells used for this study respectively also table (3) shows the data used for this study.All data are from Saleh[1] . Iraqi Journal of Chemical and Petroleum Engineering University of Baghdad College of Engineering Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells 36 IJCPE Vol.13 No.4 (December 2012) -Available online at: www.iasj.net Previous Investigation Chien [2] expressed the below equation of pressure gradient: )()/(1[ )Pr( 052.0052.0 2 VsVmDhDp c D PT       D P  …(3) Differentiate Equation (1) with respect to Vm, setting the differential from equal to zero, and then solving it for the optimum annular fluid velocity (Vm)opt.Chien[2] used four models such as Bingham, power-law, Casson and Robertson- Stiff to find the optimum fluid velocity. Most of researchers focused upon the relation of rheological models and optimum annular flow rate. H.N. Hall and Howard Thompson [3] show the drilled cuttings velocities computed through estimates of cuttings circulation time from bottom. The observed discontinuity in transport at the transition from laminar to turbulent flow was significant. Sifferman, T,R. el.[4].concluded that the cutting transport efficiency increases as fluid velocity increases. Also, they found that variables such as particle size, drill pipe rotation, drill pipe eccentricity and drilling rate, at most, only moderate effects on carrying capacity in their experiments. Thomas, R.P. el.[5]. There Experimental results shows that although increasing rotary speed generally improves particle transport, it more pronounced at lower annular fluid velocities and appears to be negligible at high velocities. Newitt, D.M. Richarrdson,J.F. and Gleddon, B.J. [6] and Toda, M. et al [7].concluded that pressure losses due to solids –wall friction, pressure losses due to solids – solids collision and pressure losses due to solids – fluid friction can be neglected compared to the total pressure drop. Lummus J. L. [8]. resulted in a better understanding of the effect of drilling variables and their interactions. He defined the optimized drilling as the "mathematical treatment of the most important controllable drilling variables to develop a comprehensive minimum - cost drilling program". The variables involved in rotary drilling are classified as alterable or unalterable and the variables selected for mathematical optimization are herein described. The alterable variables: 1. Fluid properties such as: fluid density's solid content, viscosity, fluid loss and fluid type. 2. Hydraulic fluid system such as: pump pressure, jet velocity, circulating rate and annular velocity 3. Weight on bit. 4. Rotary speed 5. Bit type and bit size. The unalterable variables (uncontrolled) such as: weather, location, rig conditions, corrosive gas, bottom hole temperature, depth, round trip time formation characteristics, hole problems, water availability and crew efficiency. Correlation (This Study) The following general relation of hydrod-dynamic pressure gradient is assumed: Hydrodynamic Pressure Gradient = f( ρ , Pr , Qm ) …(4) The chaos of these important parameters is due to their direct effect on hydrodynamic pressure gradient, and these parameters are designed at the beginning of drilling. Hydrodynamic pressure gradient  (ρ, Pr, 1/Qm ) …(5) RIYADH HAZIM FAWZI -Available online at: www.iasj.net IJCPE Vol.13 No.4 (December 2012) 37 Table (3) shows data of sixty nine points used to write the following correlation equation in this study by using statistical program. Non-linear model will be chosen to develop the following equation: D Pbh = a1 + a2*ρ + a3*Pr + a4/Qm + a5*ρ 2 + a6*Pr 2 + a7/Qm 2 … (6) Where: a1= 1.833331 a2= - 0.299532 a3= - 0.063143 a4= - 0.825984 a5= 0.020372 a6= 0.002062 a7 =187.782992 Statistical Analysis Naji Tawfik. and Rashid Al – Salihi, [9] and W.H. Aleen, [10]. The relative deviation error in percent from the data:    N i Ei N APRE 1 1 …(7) Where: Ei is the relative deviation in percent of an estimated value from the compute value: ,,...2,1100 Ni Xcal XcalXest PRE         …(8) The average absolute percent relative error is given as:    N i PRE N AAPRE 1 1 …(9) The minimum and maximum values to know the range of error for this correlation are given as: PREE N i min 1 min   …(10) PREE N i max 1 max   …(11) Standard deviation (Sx) is the measure of the data dispersion around zero deviation as follows: 2 11 1    N i PRE N Sx …(12) Correlation coefficient (r) is the degree of success in reducing the standard deviation as follows:                      N i i N i i XcalX calXestX r 1 2 1 2 2 )( )( 1 …(13) Where    N i i calX N X 1 )( 1 …(14) Results and Discussions Fig(1) presents the relation between the calculated hydrodynamic pressure gradient Saleh [1] and the estimated hydrodynamic pressure gradient of this study by using Eq(6). Most of the data points are close to the perfect line 45 o (0.79 rad.). Table (4) shows the comparison of average percent relative error between the Saleh [1] calculations and this study correlation for the hydrodynamic pressure gradient with very small different between them for all data. The correlation coefficient for this study of Eq (6) presents high value of 0.9999 with the lowest value of Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells 38 IJCPE Vol.13 No.4 (December 2012) -Available online at: www.iasj.net standard deviation of 0.9380 as shown in table (5). Fig (2) shows the histogram distribution of calculated (Saleh [1]) and the correlation value of hydrodynamic pressure gradient. It also shows the behavior similarity of this study and Saleh [1]. Fig(3)presents the average percent relative error ( APRE ) for this study from  3%. It is Possible to use a simple computer program to extract the optimum mud flow rate ( optimum Qm) . The program can calculate the hydrodynamic pressure gradient (Eq(6)) by increasing the mud flow rate (Qm) as the input data (suggest data such as 100,150,200,...etc) in the computer program. It is known if mud flow rate is increased, the hydrodynamic pressure gradient decreases. When the hydrodynamic pressure gradient reaches a point which starts to increase or the previous value equal or increase with the present value ( with accuracy such as 1E-4 ), the point that precedes the previous value represents the optimum mud flow rate. Conclusions 1. Hydrodynamic pressure gradient of five Iraqi drilling wells have been estimated from equation (6). this equation has high correlation coefficient of 0.9999. 2. Equation (6) can be used for all mud flow rate, mud density and penetration rate instead of using many charts. 3. The importance of Equation (6) is to estimate the hydrodynamic pressure gradient by the directly using of the practical field factors. Nomenclatures AAPRE = average absolute percent relative error, (%), Eq.(9) APRE = average percent relative error, (%), Eq. (7) D = total depth of the well, (ft) max E = maximum absolute percent relative error, (%), Eq.(11) min E = minimum absolute percent relative F = function J = number of sections along the annulus. L = length, (ft) Li = length of the i th section of annulus, (ft) N = number of variables Pbh/D = hydrodynamic pressure gradient, (psi/ft) Pr = penetration rate, (ft/hr) PRE = percent relative error, (%) Eq.(8) Qm = mud flow rate, gal/min. r = correlation coefficient, Eq.(13) SX = Standard deviation, Eq. (12) Vm = average fluid velocity,ft/min. (Vm)opt = optimum fluid velocity, ft/min. Vs = particle slip velocity, ft/min. X = average value of x exp, Eq.(14) X cal = calculated value of X (Pbh/D) X est = estimated value of X (Pbh/D) ρ = mud density, (ib/gal). ρc = drilled cutting density, (ib/gal).  Pt = total pressure drop, (psi). Subscripts cal calculated from Saleh [1] est estimated from correlation (this study) max maximum min minimum SI Metric Conversion Factors atm  1.013 250* E + 05 = Pa Psi 6.894 757 E + 00 = kP References 1. Mohammad Midhat Saleh (1989). Carrying capacity design for RIYADH HAZIM FAWZI -Available online at: www.iasj.net IJCPE Vol.13 No.4 (December 2012) 39 rotary drilling operations.M.sc Thesis Baghdad University, College of Engineering, Department of Petroleum Engineering, Iraq. 2. Chien, S. (1972). Annular velocity for rotary drilling operations.Intl. J. Rock Mech. Min. Sci. 9, 403-16. 3. H.N. Hall, Howard Thompson, and Frank Nuss (February 1950). Journal of Petroleum Technology.Spe.Vol.2.N.2. pages 35-46. 4. Sifferman, Thomas R., Myers, George M., Haden, Elard L., Wahl, Harry A.(November 1974). Drill Cutting Transport in Full Scale Vertical Annuli. Spe- Journal of Petroleum Technology. Vol.26. N.11. Pages1295-1302. 5. Thomas, R.P., Azar, J.J. and Becker, T.E.(September 1982). Drillpipe Eccentricity Effect on Drilled Cuttings Behavior in Vertical Wellbores .Spe- Journal of Petroleum Technology. Vol.34.N.9. Pages 1929-1937. 6. Newitt, D.M. Richarrdson,J.F. and Gleddon, B.J. (1961). Hydraulic conveying of solids in vertical pipes.Trans. ICHE.Vol.39.pages 93- 100. 7. Toda, M. et al. (July 1969). Hydraulic conveying of solids through horizontal and vertical pipes. Int1. Chem. Eng. Vol.9.Pages 553-560. 8. Lummus J. L.(Oct-Dec 1969). Factors to be Considered in Drilling Optimization. Spe. Journal of Canadian Petroleum Technology. Vol.8. N.4. Pages 138-146. 9. Naji Tawfik. and Rashid Al – Salihi, (1989). Engineering Statistics. Baghdad University, College of Engineering. Iraq: Higher Education Publication. 10. W.H. Aleen and Co., Ltd. (1973). Hayslett, H.T., Statistics Made Simple. London. Table 1, Range of data Depth, ft 1325 to 3027 Mud density , ib/gal 9.66 to 10.41 Penetration rate, ft/hr 6.56 to 24.6 Pbh/D , psi/ft 0.45986 to 0.577299 Qm (suggestion) , gal/min 100 to 2000 Table 2, Selected information of wells. Saleh[1] Well NO. 1 2 3 4 5 Formation type Limestone Anhydrite Marle &Siltstone Anhydrite Limestone Depth, ft 1896 1407 3027 1906 1325 Mud density , ib/gal 10.41 9.58 10.41 8.75 9.66 Penetration rate, ft/hr 6.56 7.87 24.60 6.56 9.84 Applied mud flow rate, gal/min 385 400 570 460 265 Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells 40 IJCPE Vol.13 No.4 (December 2012) -Available online at: www.iasj.net Table 3, Data used for this study (five drilling wells). Saleh[1] Case No. Qm, gal/min Pbh/D, psi/ft ρ, ib/gal Pr, ft/hr 1 111.111 0.5553330 11.110 ..5.1 2 111.111 0.5248800 0.5.1 ....1 3 111.111 0.5735020 11.111 21..1 1 111.111 0.4853680 ...51 ..5.1 5 111.111 0.5276230 3...1 0..11 . 151.111 0.5526360 11.110 ..5.1 . 151.111 0.5138320 0.5.1 ....1 . 151.111 0.5690950 11.111 21..1 0 151.111 0.4665940 ...51 ..5.1 11 211.111 0.5513890 11.110 ..5.1 11 211.111 0.5098780 0.5.1 ....1 12 211.111 0.5676160 11.111 21..1 13 211.111 0.4633030 ...51 ..5.1 11 211.111 0.5330560 3...1 0..11 15 251.111 0.5505350 11.111 ..5.1 1. 251.111 0.5078590 0.5.1 ....1 1. 251.111 0.5665440 11.111 21..1 1. 251.111 0.4619740 ...51 ..5.1 10 311.111 0.5500250 11.110 ..5.1 21 311.111 0.5066380 0.5.1 ....1 21 311.111 0.5664550 11.111 21..1 22 311.111 0.4612680 ...51 ..5.1 23 311.111 0.5375840 3...1 0..11 21 111.111 0.5496300 11.110 ..5.1 25 111.111 0.5052400 0.5.1 ....1 2. 111.111 0.5658740 11.111 21..1 2. 111.111 0.4605550 ...51 ..5.1 2. 111.111 0.5469220 3...1 0..11 20 511.111 0.5491470 11.111 ..5.1 31 511.111 0.5044690 0.5.1 ....1 31 511.111 0.5660530 11.111 21..1 32 511.111 0.4602170 ...51 ..5.1 33 511.111 0.5609960 3...1 0..11 31 .11.111 0.5491380 11.110 ..5.1 35 .11.111 0.5039850 0.5.1 ....1 3. .11.111 0.5672710 11.111 21..1 3. .11.111 0.4600370 ...51 ..5.1 3. .11.111 0.5491150 11.111 ..5.1 30 .11.111 0.5036570 0.5.1 ....1 11 .11.111 0.5687570 11.111 21..1 11 .11.111 0.4599380 ...51 ..5.1 12 .11.111 0.5489590 11.111 ..5.1 13 .11.111 0.5034220 0.5.1 ....1 11 .11.111 0.5686420 11.111 21..1 15 .11.111 0.4598860 ...51 ..5.1 RIYADH HAZIM FAWZI -Available online at: www.iasj.net IJCPE Vol.13 No.4 (December 2012) 41 Table 3, Data use for this study (five drilling wells) (continue). Saleh[1] Case No. Qm, gal/min Pbh/D, psi/ft ρ, ib/gal Pr, ft/hr 64 099.999 0.5489660 09.609 4.549 64 099.999 0.5032480 0.599 4.949 69 099.999 0.5710700 09.609 06.49 60 099.999 0.4598630 9.459 4.549 59 0999.999 0.5489250 09.609 4.549 50 0999.999 0.5031160 0.599 4.949 50 0999.999 0.5729180 09.609 06.49 53 0999.999 0.4598600 9.459 4.549 56 0059.999 0.5487720 09.609 4.549 55 0059.999 0.5028960 0.599 4.949 54 0059.999 0.5753300 09.609 06.49 54 0059.999 0.4599700 9.459 4.549 59 0599.999 0.5494640 09.600 4.549 50 0599.999 0.5027710 0.599 4.949 49 0599.999 0.5737500 09.609 06.49 40 0599.999 0.4600830 9.459 4.549 40 0459.999 0.5492910 09.609 4.549 43 0459.999 0.5026980 0.599 4.949 46 0459.999 0.5743210 09.609 06.49 45 0459.999 0.4603990 9.459 4.549 44 0999.999 0.5496250 09.609 4.549 44 0999.999 0.5026550 0.599 4.949 68 0999.999 0.5772990 10.410 24.60 69 0999.999 0.4611720 8.750 6.560 Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells 42 IJCPE Vol.13 No.4 (December 2012) -Available online at: www.iasj.net Table 4, Hydrodynamic pressure gradient estimated correlation for this study Case No. Calculated hydrodynamic pressure gradient(psi/ft), (Saleh,[1]) Estimated hydrodynamic pressure gradient(psi/ft), (this study) Deviation % Estimated hydrodynamic pressure gradient (APRE) Pbh/D(estimated) - Pbh/D(calculated) (Psi/ft) 0 0.555333 0.557735 0.432453 0.002402 0 0.524880 0.514002 - 2.072468 - 0.010878 3 0.573502 0.578067 0.796000 0.004565 6 0.485368 0.470196 - 3.125784 - 0.015172 5 0.527623 0.547815 3.826934 0.020192 4 0.552636 0.552088 - 0.099111 - 0.000548 4 0.513832 0.508356 - 1.065765 - 0.005476 9 0.569095 0.572421 0.584402 0.003326 0 0.466594 0.464550 - 0.438031 - 0.002044 09 0.551389 0.550379 - 0.183233 - 0.001010 00 0.509878 0.506646 - 0.633847 - 0.003232 00 0.567616 0.570711 0.545298 0.003095 03 0.463303 0.462841 - 0.099812 - 0.000462 06 0.533056 0.540459 1.388767 0.007403 05 0.550535 0.549709 - 0.149992 - 0.000826 04 0.507859 0.505977 - 0.370630 - 0.001882 04 0.566544 0.570042 0.617386 0.003498 09 0.461974 0.462171 0.042672 0.000197 00 0.550025 0.549411 - 0.111554 - 0.000614 09 0.506638 0.505679 - 0.189306 - 0.000959 00 0.566455 0.569744 0.580620 0.003289 00 0.461268 0.461873 0.131229 0.000605 03 0.537584 0.539492 0.354857 0.001908 06 0.549630 0.549206 - 0.077109 - 0.000424 05 0.505240 0.505474 0.046247 0.000234 04 0.565874 0.569539 0.647619 0.003665 04 0.460555 0.461668 0.241681 0.001113 09 0.546922 0.539286 - 1.396101 - 0.007636 00 0.549147 0.549172 0.004575 0.000025 39 0.504469 0.505440 0.192400 0.000971 30 0.566053 0.569505 0.609774 0.003452 30 0.460217 0.461634 0.307901 0.001417 33 0.560996 0.539252 - 3.875901 - 0.021744 36 0.549138 0.549187 0.008838 0.000049 35 0.503985 0.505454 0.291478 0.001469 34 0.567271 0.569519 0.396293 0.002248 34 0.460037 0.461648 0.350281 0.001611 39 0.549115 0.549215 0.018212 0.000100 30 0.503657 0.505482 0.362445 0.001825 69 0.568757 0.569548 0.138992 0.000791 60 0.459938 0.461677 0.378072 0.001739 RIYADH HAZIM FAWZI -Available online at: www.iasj.net IJCPE Vol.13 No.4 (December 2012) 43 Table 5, Statistical accuracy of the hydrodynamic pressure gradient of this study Table (5) - Statistical accuracy of the hydrodynamic pressure gradient of this study. Average relative error , % 0.0210 Average absolute relative error , % 0.5579 Minimum absolute relative error , % 0.0046 Maximum absolute relative error, % 3.8759 Standard deviation, % 0.9380 Correlation coefficient (r) 0.9999 Table 4, Hydrodynamic pressure gradient estimated correlation for this study (continue) Case No. Calculated hydrodynamic pressure gradient(psi/ft) from Saleh, [1] Estimated hydrodynamic pressure gradient(psi/ft) (this study Deviation % Estimated hydrodynamic pressure gradient (APRE) Pbh/D(estimated) - Pbh/D(calculated) Psi/ft 12 0.548959 0.549246 0.052335 0.000287 13 0.503422 0.505514 0.415512 0.002092 11 0.568642 0.569579 0.164747 0.000937 15 0.459886 0.461708 0.396227 0.001822 1. 0.548966 0.549277 0.056567 0.000311 1. 0.503248 0.505544 0.456238 0.002296 1. 0.571070 0.569609 - 0.255826 - 0.001461 10 0.459863 0.461738 0.407823 0.001875 51 0.548925 0.549304 0.069123 0.000379 51 0.503116 0.505572 0.488140 0.002456 52 0.572918 0.569637 - 0.572690 - 0.003281 53 0.459860 0.461766 0.414544 0.001906 51 0.548772 0.549363 0.107688 0.000591 55 0.502896 0.505630 0.543739 0.002734 5. 0.575330 0.569695 - 0.979353 - 0.005635 5. 0.459970 0.461825 0.403256 0.001855 5. 0.549464 0.549408 - 0.010206 - 0.000056 50 0.502771 0.505675 0.577679 0.002904 .1 0.573750 0.569740 - 0.698833 - 0.004010 .1 0.460083 0.461870 0.388368 0.001787 .2 0.549291 0.549443 0.027662 0.000152 .3 0.502698 0.505710 0.599251 0.003012 .1 0.574321 0.569775 - 0.791462 - 0.004546 .5 0.460399 0.461905 0.327073 0.001506 .. 0.549625 0.549471 - 0.028055 - 0.000154 .. 0.502655 0.505738 0.613399 0.003083 68 0.577299 0.569803 -1.298404 - 0.007496 69 0.461172 0.461933 0.164949 0.000761 Hydrodynamic Pressure Gradient Correlation of Some Iraqi Oil Wells 44 IJCPE Vol.13 No.4 (December 2012) -Available online at: www.iasj.net Fig. 1, Cross plot of hydrodynamic pressure gradient, (psi/ft) Saleh[1] SALEH[1] RIYADH 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 Hydrodynami c pre ssure gradi e nt (psi /ft) Fig(2):Histogram distribution of calculated and correlation values of hydrodynamic pressure gradient. 0 2 4 6 8 10 12 14 16 N o o f o b s Hi stogram .x 6v*69c)Fig(2): Histogram distribution of calculated and correlation values of hydrodynamic pressure gradient(محمد مدحت النهائي تعديل) Fig(1):Cross plot of hydrodynamic pressure gradient, (psi/ft) 0.45 0.48 0.51 0.54 0.57 0.60 0.45 0.48 0.51 0.54 0.57 0.6 Calculate d hydrodynamic pre s s ure gradie nt, (ps i/ft) Sale h[1] E st im at ed h yd ro dy na m ic pr es su re g ra di en t, (p si /ft ) (t hi s st ud y) Fig. 2, Histogram distribution of calculated and correlation values of hydrodynamic pressure gradient RIYADH HAZIM FAWZI -Available online at: www.iasj.net IJCPE Vol.13 No.4 (December 2012) 45 Average relative error % Fi g(3):Error di s tri bu ti on of h ydrodyn am i c pre s s u re gradi e n t corre l ati on (th i s s tu dy) F r e q u e n c y <= -4 (-4,-3] (-3,-2] (-2,-1] (-1,0] (0,1] (1,2] (2,3] (3,4] > 4 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% Fig. 3, Error distribution of hydrodynamic pressure gradient correlation (this study)