SPE 106970 MS P

Copyright 2007, Society of Petroleum EngineersThis paper was prepared for presentation at the 2007 SPE Production and Operations Symposium held in Oklahoma City, Oklahoma, U.S.A., 31 March–3 April 2007.This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.AbstractThis paper presents a simple productivity equation for a horizontal well in pseudo-steady state in a closed anisotropic box-shaped reservoir, using a uniform line sink model. A new equation for calculating pseudo skin factor due to partial penetration is also proposed. Compared with the equations for horizontal wells in pseudo-steady state in the literature, the new equations are more practical and easy to use in the field practice. They are derived by solving analytically the involved three-dimensional partial differential equations and are extracted from a very complex general solution,The effects of the following parameters on productivity have been investigated: well location, pay zone thickness, well producing length, principal permeability, and dimensions of drainage volume. It is concluded that, though each of these parameters has an effect on productivity, the well length and the permeability perpendicular to the well in the horizontal plane have the strongest influence.IntroductionTo determine the economical feasibility of drilling a horizontal well, the engineers need reliable methods to estimate its expected productivity. There have been several attempts to describe and estimate horizontal well productivity indexes. Lu 1,2,3 presented steady state productivity equations for a horizontal well in an infinite extention, finite thickness reservoir and in a circular cylinder drainage volume reservoir.Babu and Odeh 4 introduced a complex equation to calculate productivity of horizontal wells which required that the drainage volume be approximately box-shaped, and all the boundaries of the drainage volume be sealed. They reduced their original infinite series solution intoequations for shape factor and partial penetration skin. Although their expression for the shape factor is quite simple, the expression for the partial penetration skin is very complicated. Thompson et al.5 proposed an algorithm to compute the horizontal well pressure response in bounded reservoirs. Their algorithm switches between two infinite series solutions (with different rate of convergence) to improve the overall convergence rate.Economides et al.6 proposed a pseudo-steady state computer model using the continuous point source solution. Their model also accounts for well orientation. Helmy and Wattenbarger 7 provided corrections to calculate the shape factor and partial penetration skin factor in pseudo-steady state. Their correlations were developed using nonlinear regression of more than 800 numerical simulation runs for different reservoir aspect ratios, well locations and well penetration ratios.All of the above pseudo-steady state equations are either complicated to use or require computer programming, or empirical relationships that lacks in theoretical bases. This paper provides analytical equations to calculate productivity and pseudo skin factor due to partial penetration of a horizontal well in pseudo-steady state in a closed anistropic box-shaped reservoir. Analytical solutions are derived by assuming uniform fluid withdrawal along the portion of the wellbore open to flow.Horizontal Well ModelFigure 1 is a schematic of a horizontal well. A horizontal well of length L drains a box-shaped reservoir with height H , length (x direction)a 2, and width (y direction) b 2. The well is parallel to the x direction with a length .2a L ≤The following assumptions are made: 1. The reservoir is a horizontal, homogeneous, anisotropic, and has constant z y x K K K ,,permeabilities, thickness H , porosity φ. All the boundaries of the box-shaped drainage volume are sealed.2. The reservoir pressure is initially constant. At time ,0=t pressure is uniformly distributed in the reservoir, equal to the initial pressure i P .3. The production occurs through a horizontal well ofSPE 106970A Simple Productivity Equation for a Horizontal Well in Pseudosteady State in a Closed Anisotropic Box-Shaped ReservoirJing Lu, and Djebbar Tiab, SPE, U. of Oklahoma2 SPE 106970radius w R , represented in the model by a uniform line sink located at a distance w z from the lower boundary, the length of the well is L .4. A single phase fluid, of small and constant compressibility f C , constant viscosity µ, and formation volume factor B , flows from the reservoir to the well at a constant rate w Q . Fluid properties are independent of pressure. Gravity forces are neglected.5. The magnitude of wellbore pressure drop is negligible.6. Any additional pressure drops caused by formation damage, stimulation or perforation are ignored. Only pseudo skin factor due to partial penetration is considered.Equations for Initial and Boundary ConditionsAs Figure 1 shows, the horizontal well is a uniform line sink in three dimensional space, the coordinates of the two ends are ),0,2/(w z L − and ),0,2/(w z L . Let),0(),(),(H b b a a ×−×−=Ω(1) Ω is the box-shaped drainage volume.All the boundaries of the box-shaped drianage volume are sealed; i.e the reservoir has the following outer boundary condition: 0|=Γ∂∂N P (2) where Γ∂∂|NP is the external normal derivative of pressureon the surface of box-shaped drainage volume Ω∂=Γ. The reservoir pressure is at initial condition,i t P P ==0|(3) The geometric average permeability K a is:3/1)(z y x a K K K K =(4) The vertical and horizontal permeabilities arerespectively: z v K K =2/1)(y x h K K K =(5) Let β be the permeability anisotropic factor:2/1)/(v h K K =β(6) And η be the p artially penetration factor:)2/(a L =η(7)Horizontal Well ProductivityA horizontal well in a sealed box-shaped reservoir in pseudo-steady state can be calculated from 8:Pw a z y SB P P L K K w Q +Λ−=)/()()(001127.02/1µ (8) where a P is average reservoir pressure throughout the box-shaped drainage volume, and)]}2/(sin[)/sin(4ln{()(216D wD D wD H b H R H z D Dπππ−=Λ(9) The pseudo skin factor due to partial penetration p S is:)}}coth(()/(cos){()}}/coth(){({)()1)(()]/1)(cosh[(01121)/sinh()]/1)(/cosh[(1412333222πµπππµµππππmn a L Mm d Mn D wD L H b a D D Mm b m a a L b m a m L H b a L a L Hb La p mn D D mn mnm D D D DD D D D D D D D DDD DD DDDD H z n b m a S −×+−×++−=−===−∑∑∑(10)sinh(.), cosh(.)and coth(.) are hyperbolic sine, hyperbolic cosine and hyperbolic cotangent, respectively.It has been shown that in Equation (10), the integer number 100=M is sufficiently big enough to reach the engineering accuracy for all practical purpose, and2/1,0;1,0=>==m m d m d m(11)2/122])/()/[(D D D D mn H na b ma +=µ(12)For a fully penetrating horizontal well, a L 2=,then0=p S(13) And Equation (8) reduces to Λ−=)/()()(001127.02/1B P P L K K w w a z y Q µ(14)Where Λ has the same meaning as in Equation (9).For a given well, the flow rate reaches maximum value when it is located in the middle height of pay zone 8.In order to calculate p S , we may separate Eq. 10 into three parts:321Ψ+Ψ+Ψ=p S(15) Where:)1)((224121DD DDDDDD a L a L Hb La +−=Ψ(16))}}/coth(){(){(1)/sinh()]/1)(/cosh[(12332D D Mm b m a a L b m a mL H b b m a D D D D D D D D Dππππ−=Ψ∑=−(17)SPE 106970 3)}}coth(){()/(cos){()]/1)(cosh[(0112333πµππµµπmn a L Mm d Mn D wD L H b a mn D D mn mnm D D D DH z n −×=Ψ−==∑∑(18)Examples and Effects of Some Critical Parameters Examples are given below to compare the proposed equation with Babu and Helmy’s equations 4,7. These equations are also used to investigate the effects of some critical parameters on productivity index (PI).Example 1: Equation (8) is used to calculate productivity index of a horizontal well in pseudo-steady state in a sealed box-shaped drainage volume. Using reservoir and fluid properties data are in Table 1 we have :ft b ft a 2000,1000==)(99.125)50200200()(3/13/1mD K K K K z y x a =××==2)50/200()/(2/12/1===v h K K β5.02000/1000)2/(===a L ηUsing the definations of dimensionless variables in the appendix, we obtain:7937.0)200/99.125(2/1==D L7937.0)1000/1000()200/99.125(2/1=×=D a5874.1)1000/2000()200/99.125(2/1=×=D b07937.0)1000/50()50/99.125(2/1=×=wD z15874.0)1000/100()50/99.125(2/1=×=D H000265.0)10002/(25.0])200/50()50/200[(4/14/1=××+=wD R0/1,=−=D D D D a L L a5.0/,5.0/==D wD D D H z b a2/122)254/(n m mn +=µ392.2)]}15874.02/(000265.0sin[)2/sin(4ln{)()(2115874.065874.1=×××−=Λ×πππ00521.0)11)((17937.012=+−=Ψ54606.0))5.0coth((()5.0sinh()0cosh(100117937.015874.05874.12332−=−×=Ψ∑=××ππm m m m00495.0)]}5coth()[()]254/coth([][){(cos ][)0cosh(125122)254/sinh()0cosh(1001)254/(2100122)7937.0)(15874.0)(5874.1(7937.03222/32233−=−++−×=Ψ+=+=∑∑πππππn n m nn m m n m n nSo5458.0)00495.0()54606.0(00521.0−=−+−+=p SProductivity index of this well, i.e. the flow rate per unit pressure drop, is)//(54.59)/()11/()1000)(50200)(001127.0(psi D STB P Q PI w ==∆=××It is interesting to find that p S is negative in the abovecaculation. The denominator of Equation (8) for a partially penetrating well is smaller than the denominator for a fully penetrating well, (0=p S ). But the flow rate of the partially penetrating well still decreases due to smaller well length L in the numerator of Equation (8).Example 2: In this example, only fluid properties, reservoir width and length are constants, other reservoir parameters are variables. The effects of some critical4 SPE 106970parameters on productivity index investigate areinvestigated here.1. Effect of Pay Zone Thickness on PITable 3 and Figure 2 present the effect of pay zone thickness on PI calculated by the proposed new equation (Eq. 8) and Babu and Helmy's equations, with the following constant parameters:mDK K mD K K K ft L v z h y x 50,200,1000======and the horizontal well is at mid-height of the pay zone, .2/H z w =Table 3 and Figure 2 show that no significant differences among the PIs calculated by the threeequations. If the payzone thickness His not very big (ft H 50<),PI increases rapidly with the increasing H , but when ft H 50>, PI increases slowly with the increasing H . From ft H 10= to ft 50, PI increases about 4 times; from ft H 50= to ft 100, PI increases about 1.5 times. For this reason, horizontal wells are believed to perform better than their vertical counterparts in thin reservoirs.2. Effect of Well Location in Vertical Direction on PI Table 4 and Figure 3 present the effect of well location in vertical direction on PI , with the following constant parameters:ftH mD K K mD K K K ft L v z h y x 100,50,200,1000=======Table 4 and Figure 3 show that the location of horizontal well in vertical direction does not have significant effect on PI . For maximum PI , the well should be located at the center of the pay zone.3. Effect of Well Length on PITable 5 and Figure 4 present the effect of well length on PI , with the following constant parameters:ftz ft H mD K K mD K K K ft L w v z h y x 50,100,50,200,1000========Table 5 and Figure 4 show that no significant differences among the PI values calculated by the three equations. PI increases slowly with the increasing L . For every increase in L of ft 250, PI only increases about psi D STB //10. From ft 250 to ft 1000, PI increases 4 times. From ft 1000 to ft 2000, PI onlyincreases about 1.6 times. So PIis not sensitive to L and η.Table 6 presents the pseudo skin factor p S due to partial penetration caculated by the Equations (15), (16), (17) and (18). For a partially penetrating horizontal well, 1Ψis postive while 2Ψ and 3Ψ are negative.From Table 6 we find that when η increases, 1Ψ decreases, the absolute values of 2Ψ and 3Ψ also decreases. When 25.0≥η, the absolute value of 3Ψ is very small compared with 1Ψ and absolute value of 2Ψ. So, in many cases, 3Ψ is negligible when we calculate p S .4. Effect of Horizontal Permeability on PITable 7 and Figure 5 present the effect of horizontal permeability )(h y x K K K == on PI , with the following constant parameters:,25,,1000mD K K K K K ft L v z h y x =====.50,100ft z ft H w ==Table 7 and Figure 5 show that PI is a strong function of horizontal permeability (assuming )(h y x K K K ==. Horizontal permeability increases 8 times, while β only increases 2.8 times and PI calculated by the three equations increase about 5.5 times.5. Effect of Vertical Permeability on PITable 8 and Figure 6 present the effect of vertical permeability on PI , with the following constant parameters:v z h y x K K mD K K K ft L =====,100,1000.50,100ft z ft H w ==Table 8 and Figure 6 show that PI is a weak function of vertical permeability, vertical permeability increases 10 times, (β decreases about 3 times) PIs calculated by the three equations increase about 1.6 times.6. Effect of Permeability Perpendicular to Horizontal Well on PITable 9 and Figure 7 present the effect of permeability perpendicular to well in horizontal plane )(y K on PI , with the following constant parameters:v z x K K mD K ft L ===,50,1000.50,100ft z ft H w ==Table 9 and Figure 7 show that PI is a strong function of y K when mD K y 25≤. From 5 mD to 25 mD,y K increases 5 times, and PI calculated by the threeequations also increase about 5 times; but PI is a weak function of y K when mD K y 25>. From 25 mD to 250 mD, y K increases 10 times, but PI values calculated by the three equations only increase about 5 times.SPE 106970 57. Effect of Permeability Parallel to Horizontal Well on PITable 10 and Figure 8 present the effect of permeability parallel to horizontal well )(x K on PI , with the following constant parameters:mD K mD K ft L z y 25,50,1000===.50,100ft z ft H w ==Table 10 and Figure 8 show that PI is a weak function of x K , x K increases 50 times, (β increases about 3 times) PI values calculated by the three equations increase about 1.6 times.Example Three: In this example, we investigate the effects of reservoir size on productivity index (PI ).Fluid properties, other reservoir parameters are the same as in Table 1.8. Effect of Reservoir Width on PITable 11 and Figure 9 present the effect of reservoir width (RESW , the size of reservoir parallel to horizontal well) on PI , with the following constant parameters:ftz ft H mD K K mD K K K ft L w v z h y x 50,100,50,200,1000========And reservoir length (RESL ) is a constant, 4000ft.Table 11 and Figure 9 show that PI is a weak function of reservoir width (RESW ), from RESW=1000ft to 2200ft, (the partially penetrating factor η drops from 1.00 to 0.455,)PIs calculated by the three equations increase slowly;from RESW =2200ft to 4000ft, ( thepartially penetrating factor ηdrops from 0.455 to 0.250,) PIs calculated by the three equations decrease slowly.Table 12 presents the pseudo skin factor p S due to partial penetration due to different reservoir width. We obtain the same conclusion as that in Table 6 in Example 2. For a partially penetrating horizontal well, 1Ψ is postive while 2Ψ and 3Ψ are negative. When 25.0≥η, the absolute value of 3Ψ is very small compared with 1Ψ and absolute value of 2Ψ.9. Effect of Reservoir Length on PITable 13 and Figure 10 present the effect of reservoir length (RESL , the size of reservoir perpendicular to horizontal well) on PI , with the following constant parameters:ftz ft H mD K K mD K K K ft L w v z h y x 50,100,50,200,1000========And reservoir width (RESW ) = 2000 ft.Table 13 and Figure 10 show that PI is a weak function of reservoir length (RESL ).Conclusions1. The proposed new productivity equation for ahorizontal well in a sealed box-shaped drainage system in pseudo-steady state is based on solving three dimensional partial differential equations, and has a solid mathematical base in the theory of flow in porous media; it is reliable and easy to use in oil field.2. For maximum productivity, the horizontal wellshould be located at the center of the pay zone. 3. The horizontal well productivity is a weakfunction of well length, well location, vertical permeability, the permeability parallel to the well in the horizontal plane, and reservoir size;4. The horizontal well productivity is a strongfunction of pay zone thickness, the permeability perpendicular to the well in the horizontal plane.NomenclatureB = formation volume factor, STB RB / fC = fluid compressibility, psi /1cosh(.)= hyperbolic cosine function coth(.) = hyperbolic cotangent function H = payzone thickness, ft K = permeability, mD L = well length, ft P = pressure, psiPI = productivity index, psi D STB // w Q = well flow rate, D STB / w R = wellbore radius, ft RESL = reservoir length, ftRESW = reservoir width, ftp S = pseudo skin factor due to partial penetration,fractionsinh(.) = hyperbolic sine functionGreek Symbolsβ= permeability anisotropic factor, fraction η= partially penetrating factor, fraction µ= fluid viscosity, cpΛ= a function defined by Equation (8). φ= porosity, fraction1Ψ= a function defined by Equation (16). 2Ψ= a function defined by Equation (17). 3Ψ= a function defined by Equation (18).Subscriptsa = average D = dimensionless e = external h = horizontal i = initial6 SPE 106970v = verticalw = wellz y x ,,= coordinate indicators References[1] Lu, J.: “New Productivity Formulae of Horizontal Wells.” Journal of Canadian Petroleum Technology, 40(10):55-67, Oct., 2001.[2] Lu, J.: “Productivity Formulae for Horizontal Wells in a Circular Cylinder Drainage Volume.” Journal of Canadian Petroleum Technology, 42(10):28-31, Oct., 2003.[3] Lu, J., Lin. T, Rogers. B, Lu. T .: “The Supplements and Discussions of Productivity Formulae of Horizontal Wells.” Journal of Canadian Petroleum Technology, 42(10):23-27, Oct., 2003.[4] Babu, D.K, Odeh, A.S. “Productivity of a Horizontal Well .” SPE Reservoir Engineering, 4(4), 417- 421, Nov. 1989.[5] Thompson, L.G, Manrique, J.L, Jelmert, T., A : “Efficient Algorithms for Computing the Bounded Reservoir Horizontal Well Pressure Response .” Paper SPE 21827 presented at the 1991 SPE Rock Mountain Regional Meeting and Low- Permeability Reservoir Symposium, Denver, Colorado, October 1991.[6] Economides, M.J, Brand, C.W, Frick, T.P. “Well Configuration in Anisotropic Reservoirs.” Paper SPE 27980 presented at the 1994 University of Tulsa Centennial Petroleum Engineering Symposium, Tulsa, Oklahoma, August 1994.[7] Helmy, M. W., Wattenbarger, R. A.: “Simplified Productivity Equations for HorizontalWells Producing at Constant Rate and Constant Pressure.” Paper SPE 49090 presented at the 1998 Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 1998.Appendix: Definition of Dimensionless Variables2/12/12/1))(())(())((za ya xa K z D K KL y D K K L x D z y x ===(A-1) 2/12/12/1)(())(())((za w ya xa K KL z wD KKL b D K K L a D z b a ===(A-2)2/12/1))(()(za xa KH D K KD H L ==(A-3)2L C tK D t a t φµ=(A-4)qBP P L K D i a P µ)(−=(A-5))2/(])/()/[(4/14/1L R K K K K R w x z z x wD +=(A-6)Table 1: Reservoir and Fluid Propertiesfor Example 1Table 2: Reservoir and Fluid Propertiesfor Example 2Table 3: Effect of Payzone Thickness on PIPI (STB/D/psi)Babu Lu and Tiab Helmy H=10(ft)8.4259.3259.609H=20(ft)16.16117.79917.358H=30(ft)23.15625.37623.886H=40(ft)29.44832.11529.527H=50(ft)35.09638.10134.490H=60(ft)40.16543.42138.917H=70(ft)44.71748.15642.911H=80(ft)48.81152.37946.546H=90(ft)52.49956.15549.880H=100(ft)55.82859.53852.9581000ft 100ft 4000ft 2000ft 50ft 0.25ft 200mD 200mD 50mD 1.0cp 1.0RB/STBOil Viscosity, µFormation Volume Factor, BReservoir Length, RESL Reservoir Width, RESW Permeability in X direction, Kx Permeability in Y direction, Ky Permeability in Z direction, KzWell Length, L Pay Zone Thickness, H Well Vertical Location, Zw Wellbore Radius, R w4000ft 2000ft 0.25ft 1.0cp 1.0RB/STBFormation Volume Factor, BReservoir Length, RESL Reservoir Width, RESW Wellbore Radius, R wOil Viscosity, µSPE 106970 7Table 4: Effect of Well Location in Vertical Directionon PIPI (STB/D/psi)Babu Lu and Tiab Helmy Zw=10(ft)51.09754.45350.807Zw=20(ft)53.58357.15651.962Zw=30(ft)54.91058.57352.556Zw=40(ft)55.60859.30852.862Zw=50(ft)55.82859.53852.958Zw=60(ft)55.60859.30852.862Zw=70(ft)54.91058.57352.556Zw=80(ft)53.58357.15651.962Zw=90(ft)51.09754.45350.807Table 5: Effect of Well Length on PIBabu Lu and Tiab Helmy 21.22222.34217.99434.30636.21430.47245.44448.27741.80055.82859.53852.95865.87070.18364.18975.71079.99475.27885.33788.42885.60894.63794.33594.259PI (STB/D/psi)L=250(ft), η=0.125L=500(ft), η=0.250L=750(ft), η=0.375L=1000(ft), η=0.500L=1250(ft), η=0.625L=1500(ft), η=0.750L=1750(ft), η=0.875L=2000(ft), η=1.000Table 6: Pseudo Skin Factor CalculationsTable 7: Effect of Horizontal Permeability on PIBabu Lu and Tiab Helmy 8.6269.3798.56215.74416.95215.19322.09423.65321.07127.91429.76926.47933.33335.45131.55038.43140.79036.36143.26445.84840.96347.87050.66945.390PI (STB/D/psi)K h =25(mD), β=1.000K h =50(mD), β=1.414K h =75(mD), β=1.732K h =100(mD), β=2.000K h =125(mD), β=2.236K h =150(mD), β=2.449K h =175(mD), β=2.646K h =200(mD), β=2.828Table 8: Effect of Vertical Permeability on PIBabu Lu and Tiab Helmy 22.60923.89021.52826.66628.36125.24028.90230.89727.50130.39432.62229.12431.48933.90530.38532.34034.91331.41233.02835.73332.27633.59936.42033.02034.08537.00733.67134.50437.51634.248PI (STB/D/psi)K v =10(mD), β=3.162K v =20(mD), β=2.236K v =30(mD), β=1.826K v =40(mD), β=1.581K v =50(mD), β=1.414K v =60(mD), β=1.291K v =70(mD), β=1.195K v =80(mD), β=1.118K v =90(mD), β=3.162K v =100(mD), β=1.000Table 9: Effect of Permeability Perpendicular toHorizontal Well on PIBabu Lu and Tiab Helmy 2.527 2.609 2.4256.316 6.626 6.0359.4209.9799.02115.74416.95015.19320.93822.78720.33925.45927.94024.86829.51832.61428.96633.23336.92632.73936.67840.95036.25539.90544.73639.56042.94948.32242.68745.83851.73445.662PI (STB/D/psi)K y =5(mD), β=0.795K y =15(mD), β=1.047K y =25(mD), β=1.189K y =50(mD), β=1.414K y =75(mD), β=1.565K y =100(mD), β=1.682K y =125(mD), β=1.778K y =150(mD), β=1.861K y =175(mD), β=1.934K y =200(mD), β=2.000K y =225(mD), β=2.060K y =250(mD), β=2.115ψ1ψ2ψ3Sp0.03988-1.15038-0.02031-1.130800.05859-0.88431-0.01015-0.835870.06104-0.69534-0.00677-0.641070.05208-0.54607-0.00500-0.498980.03662-0.41720-0.00406-0.384640.01953-0.29477-0.00338-0.278620.00570-0.16434-0.00291-0.161550.000000.000000.000000.00000L=250(ft), η=0.125L=500(ft), η=0.250L=750(ft), η=0.375L=2000(ft), η=1.000L=1000(ft), η=0.500L=1250(ft), η=0.625L=1500(ft), η=0.750L=1750(ft), η=0.8758 SPE 106970Table 10: Effect of Permeability Parallel toHorizontal Well on PIBabu Lu and Tiab Helmy 9.60611.77310.00912.93914.66013.05514.20015.73614.12215.74416.95015.19316.54217.53115.64117.05617.88915.89417.42618.13616.05717.71018.32016.17117.93718.46316.25618.12418.57916.32218.28218.67416.37418.41818.75516.417PI (STB/D/psi)K x =5(mD), β=0.795K x =15(mD), β=1.047K x =25(mD), β=1.189K x =50(mD), β=1.414K x =75(mD), β=1.565K x =100(mD), β=1.682K x =125(mD), β=1.778K x =150(mD), β=1.861K x =175(mD), β=1.934K x =200(mD), β=2.000K x =225(mD), β=2.060K x =250(mD), β=2.115Table 11: Effect of Reservoir Width on PIBabu Lu and Tiab Helmy 47.31847.16847.12952.20554.07350.79254.64657.57652.39755.68259.24652.92655.91359.85752.87655.68259.85352.50455.19059.49051.95354.55358.92151.30153.84258.23450.59653.09957.48649.86552.34956.70749.125PI (STB/D/psi)RESW=1000(ft), η=1.000RESW=1300(ft), η=0.769RESW=1600(ft), η=0.625RESW=1900(ft), η=0.526RESW=2200(ft), η=0.455RESW=2500(ft), η=0.400RESW=2800(ft), η=0.357RESW=3100(ft), η=0.323RESW=3400(ft), η=0.294RESW=3700(ft), η=0.270RESW=4000(ft), η=0.250Table 12: Pseudo Skin Factor Calculationsψ1ψ2ψ3Sp 0.000000.000000.000000.000000.00721-0.30983-0.00508-0.307690.02344-0.45286-0.00508-0.434500.04441-0.52898-0.00508-0.489660.06818-0.57218-0.00508-0.509080.09375-0.59761-0.00508-0.508940.12054-0.61292-0.00508-0.497460.14819-0.62226-0.00508-0.479160.17647-0.62801-0.00498-0.456530.20524-0.63157-0.00498-0.431320.23438-0.63378-0.00498-0.40439RESE=3100(ft),η=0.323RESE=3400(ft), η=0.294RESE=3700(ft), η=0.270RESE=4000(ft), η=0.250RESE=1900(ft), η=0.526RESE=2200(ft), η=0.455RESE=2500(ft), η=0.400RESE=2800(ft), η=0.357RESE=1000(ft), η=1.000RESE=1300(ft), η=0.769RESE=1600(ft), η=0.625Table 13: Effect of Reservoir Length on PIPI (STB/D/psi)Babu Lu and Tiab Helmy RESL=1000(ft)78.01186.88581.143RESL=1500(ft)75.22381.77774.390RESL=2000(ft)70.33676.26968.654RESL=2500(ft)66.04571.28863.810RESL=3000(ft)62.24866.88959.671RESL=3500(ft)58.86362.99456.091RESL=4000(ft)55.82859.52852.958RESL=4500(ft)53.09056.42350.188RESL=5000(ft)50.60853.62547.720Figure 1: Horizontal Well Model.Figure 2: Effect of Payzone Thickness on PISPE 1069709Figure 3: Effect of Well Location inVertical Direction on PIFigure 4: Effect of Well Length on PIFigure 5: Effect of Horizontal Permeability on PI)(h y x K K K ==Figure 6: Effect of Vertical Permeability on PIFigure 7: Effect of y K on PIFigure 8: Effect of x K on PI10 SPE106970Figure 9: Effect of Reservoir Width on PIFigure 10: Effect of Reservoir Length on PI。