Abstract This paper presents an approximate model to forecast the productivity of selective perforated wells. The model includes algebraic equations and it could be easily computed using a programmable calculator or spreadsheet program. The model has been compared against the rigorous 3D semianalytical model and software existing in the literature. The approximate model compares well against the 3D model and the software. The model is useful for designing perforation parameters.
SPE 89414 A Simple Approximate Method to Predict Inflow Performance of Selectively Perforated Vertical Wells E Guerra, SPE, and T Yildiz, SPE, Colorado School of Mines Copyright 2004, Society of Petroleum Engineers Inc This paper was prepared for presentation at the 2004 SPE/DOE Fourteenth Symposium on Improved Oil Recovery held in Tulsa, Oklahoma, U.S.A., 17–21 April 2004 This paper was selected for presentation by an SPE Program Committee following review of information contained in a proposal 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 a proposal of not more than 300 words; illustrations may not be copied The proposal must contain conspicuous acknowledgment of where and by whom the paper was presented Write Librarian, SPE, P.O Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435 Abstract This paper presents an approximate model to forecast the productivity of selective perforated wells The model includes algebraic equations and it could be easily computed using a programmable calculator or spreadsheet program The model has been compared against the rigorous 3D semi-analytical model and software existing in the literature The approximate model compares well against the 3D model and the software The model is useful for designing perforation parameters Introduction Oil and gas wells are generally perforated at multiple intervals along the well trajectory The principal objective of perforating is to open flow channels across the casing for formation fluid entry The well completed with perforations at multiple segments along the wellbore are referred to as the selectively perforated well (SPW) The productivity of the wells is controlled by the well completion type and formation damage Formation damage is the result of the permeability impairment in the near wellbore region Formation damage decreases the well productivity The influence of formation damage is localized in the near wellbore region The formation damage effect is quantified in terms of the mechanical skin factor, s d In openhole completed vertical wells, the complete formation produces uniformly and the specific productivity index is constant along the wellbore On the other hand, the selective perforating results in multiple flow convergence regions along the wellbore This disturbance in the flow pattern makes flow modeling considerably more difficult The impact of selective perforating may be quantified in term of completion pseudoskin Similar to the formation damage effect, the influence of the well completion is intensified in the near wellbore region Therefore, the well flow models have to account for not only the individual impact of the formation damage and well completion but also the dynamic interaction between them in the near wellbore region The compounded effects of the well perforating and formation damage are expressed in terms of total skin factor, s t Given the total skin factor, the steady state pressure drop in the perforated wells could be expressed as p e − p wf = 141.2 q sct µ Bo [ ln ( re / rw ) + s t ] (1) kh If the well produces from a non-radial reservoir under pseudo steady state flow conditions then the pressure drop equation is given as 141.2 q sct µ Bo 2.2458 A ~ [ ln ( ) + s t ] (2) p − p wf = kh C A rw2 The flow equations expressed in Eqs and are simple and straightforward provided that the total skin factor is accurately related to the perforating parameters and the formation damage based skin factor Background In this section, we would like to review the most relevant studies in the literature and set the ground for the model development Partially Completed Wells In partially completed or penetrated wells (PCW), only a single segment along the wellbore is open to flow The completed segment is considered to be barefoot Partial penetration forms a twodimensional (2D) flow field in the formation around the wellbore The effect of partial penetration on well productivity has been investigated in details.1-8 Brons and Marting1 observed that the partial penetration generates a pseudo damage reducing the well productivity The others have confirmed that an additional pressure drop is created by the partial penetration effect.2-8 The additional pressure drop is measured in terms of partial penetration pseudoskin, s pp Graphical results1, analytical solutions2,3, numerical solutions4,5, and empirical equations6-8 have been proposed to compute the pseudoskin resulting from partial completion Among these different SPE 89414 methods, the empirical equations proposed by Odeh6, Papatzacos7, and Vrbik8 are the most popular due to their simplicity The analytical solutions are not usually preferred because of their infectivity with infinite series and special functions We have compared the empirical methods against the analytical solutions It has been observed that the results from the Vrbik method and analytical solutions match very well.9 The Vrbik method was chosen to build the approximate model described in this paper The Vrbik model for partial penetration pseudoskin is recaped in Appendix A It should be noted that partially penetrating well models assume that the open interval allows fluid entry at every point on wellbore surface The partial penetration pseudoskin does not account for formation damage or additional flow convergence due to perforations and slots The simultaneous effects of formation damage and partial penetration have been modeled using analytical and numerical techniques.3-5 It has been observed that formation damage results in a greater productivity loss in partially penetrating wells The total skin factor combining the individual contribution of formation damage and partial penetration is given as below st = h s d + s pp (3) hp where sd is the formation damage skin factor formulated by Hawkins.10 s d = ( k / k d − 1) ln ( rd / rw ) (4) In the derivation of Eq 3, it is assumed that the flow convergence owing to partial penetration is completed outside the damaged zone If the flow convergence towards open segment happens partly outside and partly inside the damaged zone then the total skin factor is given as h st = s d + s pp (5) γ hp where γ is greater than Odeh3 and Jones and Watts4 proposed simple equations to compute the γ parameter Selectively Completed Wells In field applications, the wells are usually completed across several intervals along the well trajectory The well completed this way is referred as to the selectively completed well (SCW) Each completed segment on the SCW is barefoot Therefore, the flow pattern towards the SCW is 2D Brons and Marting1 indicated that if the well is completed symmetrically along the wellbore then the PCW models could be applied to each symmetrical unit and the productivity of the SCW could be estimated by multiplying the productivity of the symmetrical unit by the number of the symmetrical elements Since then, several 2D analytical models have been developed to simulate the fluid flow into the SCWs.11-13 These models are more complex than the analytical models for PCWs In addition to being contaminated with the special functions and infinite series, the analytical models for SCWs require matrix construction and inversion since the models compute not only the pressure drop but also the rate at each open segment In this study, we present an approximate model to replace the 2D analytical solutions and to avoid the matrix solution and complicated mathematical functions Fully Perforated Wells The well perforated completely all along the well trajectory is referred as to fully perforated well (FPW) The flow efficiency of the FPWs has been the subject of many investigations.13-25 The effect of ideal perforating has been quantified in terms of perforation pseudoskin, s p Numerical14-17, semi-analytical18-21, and analytical methods22 have been proposed to compute the perforation pseudoskin In the absence of formation damage and rock compaction around the perforation tunnels, the perforation pseudoskin is a function of perforation length, perforation radius, phasing angle, shot density, wellbore radius, and permeability anistropy s p = s p ( n spf , L p , rp , θ p , rw , k z / k r ) (6) It has been observed that the productivity of the FPW is not only controlled by the magnitude of perforation pseudoskin but also the skin factors due to formation damage and the rock compaction around the perforation tunnels The skin factor resulting from formation damage, sd , is characterized by the degree of permeability impairment ( k d / k ) and the extent of the damaged zone ( rd ) as shown in Eq The skin factor due to rock compaction around the perforation tunnels is expressed as s cz = ∆ zp Lp ( k k − ) ln( rcz / r p ) (7) k cz k d To predict the productivity of the FPW accurately, the interaction between the three skin terms ( s p , s d , and s cz ) has to be formulated properly The combined effect of perforation pseudoskin, formation damage, and rock compaction around perforations is referred as to perforation total skin, s pdc Among the many methods for estimating the perforation total skin, the methods proposed by McLeod18 and Karakas and Tariq19, 20 have been very popular due to their simplicity For the sake of completeness, the Karakas-Tariq method is summarized in Appendix B Additionally, Jones and Slusser21 proposed a simple but accurate method combining perforation pseudoskin and formation damage skin A three-dimensional (3-D) analytical model to determine the perforation total skin has been proposed in Ref 22 The main advantage of the 3D solution is that it could handle arbitrary perforation distribution and non-uniform perforation parameters SPE 89414 A software package called SPAN is also available for computing the productivity of the perforated wells.23 SPAN uses a modified version of the Karakas-Tariq algorithm described in Refs 19 and 20.24 Recently, the McLeod method, the Karakas-Tariq algorithm, Jones-Slusser method, SPAN software, and the 3D analytical solution were compared against the experimental data.25 It has been shown that 1) the McLeod method underestimates the perforation total skin, 2) the Karakas-Tariq method for perforation pseudoskin works fine, however, the Karakas-Tariq algorithm overpredicts perforation total skin in the presence of formation damage and crushed zone, 3) the 3D analytical solution and SPAN software replicate the experimental data very well, and 4) the Jones-Slusser model, which does not consider the effect of compacted zone, agrees well with the 3D analytical solution and SPAN software when the crushed zone is also ignored in the 3D solution and SPAN In Ref 25, a modified version of the Jones-Slusser method, accounting for the skin factor due to rock compaction around the perforation tunnels, has been proposed The modified Jones-Slusser method could be analytically derived if it is assumed that the perforations are terminated inside the damaged zone and a radial flow geometry exists beyond the damaged zone The modified Jones-Slusser model compared well against the experimental results for the special case of short perforations terminated inside the damaged zone In the modified Jones-Slusser method, the perforation total skin is expressed as below s pdc = s d + k s p + s cz (8) kd Partially Perforated Wells If only a segment along a cased well is completed with perforations then this type of well is referred as to the partially perforated well (PPW) The flow convergence and the shape of the streamlines around the PPW are controlled by the combined effect of partial penetration and perforations If it is assumed that the flow convergence due to partial penetration is completed before the fluid feels the impact of the perforations and the damaged zone then an analytical expression could be derived to compute the total skin factor including the simultaneous effects of partial penetration, perforations, formation damage, and the crushed zone.13, 20-23, 25 The equation for the total skin factor is st = γ h s pdc + s pp (9) hp Eq has been verified against the 3D analytical solution.22 Selectively Perforated Wells To the authors’ knowledge, there exist two studies on the performance of the selectively perforated vertical wells (SPW) Ref 22 described a general 3D analytical solution considering arbitrary distribution of perforation and variable perforation properties The solution involves matrix construction, Bessel functions, numerical and analytical integration of Besses functions, and infinite series The size of the matrix is ( n p + 1) × ( n p + 1) where n p is the number of the perforations Therefore, if a large number of perforations are involved, the computation of the 3D analytical solution may demand long CPU time Ref 22 and 23 also presented a pseudo 3D model based on the SCW model of Ref 12 The pseudo 3D model is very efficient even for the wells with tens of thousands of perforations However, the pseudo 3D models of Refs 22 and 23 are still composed of matrix construction, Bessel functions, and infinite series The objective of the current paper is to develop a fast and easy-to-use approximate SPW model free of matrix setup, infinite series, and special functions Approximate Model The approximate model for selectively perforated wells is broken into two submodels; a perforation total skin model considering unit formation thickness of ft and a SCW with a variable total skin across the perforated intervals We will describe both models Perforation Total Skin Model A hybrid method is used to compute the total skin factor combining the individual contributions of perforation pseudoskin, formation damage, and compacted zone around the perforations First, we only determine the perforation pseudoskin, s p , by using the steps 1-4 of the Karakas-Tariq algorithm The Karakas-Tariq algorithm is appended At this stage, the effects of formation damage and rock compaction are not accounted for yet For the short perforations ending inside the damaged zone, we use the modified Jones-Slusser method to estimate the combined effects of perforation pseudoskin, formation damage, and rock compaction The modified Jones-Slusser methos is basically the expression in Eq For the long perforations reaching beyond the damaged zone, we use a modified version of the Karakas-Tariq method provided by Hegeman.24 The modified method is also used in SPAN software, version 6.0.23 There are basically two changes applied to the original Karakas-Tariq method The first modification is that true wellbore radius ( rw ) instead of the effective wellbore radius ( r′w ) is used in computing s H term The second modification is in the calculation of scz term True perforation length ( L p ) instead of the effective perforation length ( L′p ) is used in computing s cz term Approximate Model for Selectively Completed Wells For the simplicity, we will develop the approximate model considering steady state flow However, the methodology could be also applied to the well producing under pseudo steady state flow Consider a damaged vertical openhole producing under steady state flow conditions The specific productivity index, ~ J o , for such a well is obtained by rearranging Eq ~ Jo = q sct k = h ( p e − p wf ) 141.2 µ Bo (10) re ln + sd rw SPE 89414 In a vertical openhole, every unit-thickness of the formation produces the same amount of the fluid Therefore, the specific productivity index is constant and uniform at the wellbore as well as inside the formation across its thickness It should be reminded that the specific productivity index is different from the flux at the wellbore and these two concepts should not be interchanged The flux at the wellbore is the rate per unit length along the wellbore Now consider a damaged partially penetrating well As shown on Fig 1, partial completion makes the streamlines converge around the open segment and creates a 2D flow field However, the effect of partial completion on the fluid streamline pattern is concentrated in the near wellbore region At the locations away from the wellbore and deep inside the formation, the fluid flow is 1D radial and the streamlines are parallel to each other and the upper and lower reservoir boundaries Let’s refer the distance at which the streamlines start to converge towards the open segment as the radius of flow convergence ( rc ) The flow towards a partially completed well beyond the radius of flow convergence is almost the same as that towards an openhole ~ The specific productivity index for a PCW, J pc , is ~ J pc = q sct k = h ( p e − p wf ) 141.2 µ Bo (11) re ln + st rw ~ J sc = q sct k = h ( p e − p wf ) 141.2 µ Bo (12) re ln + s tsc rw where stsc is the total skin factor representing the effects of selective completion and variable formation damage Analytical expressions to compute stsc could be found in Refs 12 and 22 In general, stsc = s tsc ( n s , hbi , h pi , h pt / h, rw , k z / k r , s di ) (13) Here, we would like to offer an alternative method to ~ predict J sc Consider a SCW with n s number of open intervals distributed symmetrically along the wellbore Also assume that the open intervals are subject to the same degree of formation damage In such a case, all the completed intervals produce at the same rate of qsci and flow induced no-flow boundaries parallel to the bedding plane are formed at the center of each uncompleted segment Due to flow and completion symmetry, we can decompose the SCW into ns number of fictitious partially penetrating wells producing from the same number of independent fictitious reservoirs/layers Let hi′ be the fictitious thickness of the ith fictitious reservoir Additionally, let h′bi , and h pi represent the rescaled location where st is the total skin factor as expressed in Eq or Notice that, even for a PCW, the specific productivity index is defined with respect to formation thickness not the length of the completed segment Typically, in a PCW, the flux along the wellbore is discontinous; it is zero at the uncompleted segments and it varies somewhat along the open segment On the other hand, if we examine the flux along the formation thickness at a location beyond the radius of flow convergence then it can be stated that the flux beyond rc is constant and uniform across the formation thickness Similarly, if we consider the specific productivity index as a measure of formation capacity and evaluate it at a location beyond rc not at the wellbore then the specific productivity index is expected be constant and uniform across the formation thickness as well At this stage, let’s examine the fluid flow into a selectively completed well Consider a SCW with n s number of open intervals and variable formation damage skin factor across each open segment as shown on Fig The selective completion yields multiple flow convergence regions in the near wellbore region However, the impact of the selective completion on the flow streamlines is localized Beyond the radius of flow convergence, the streamlines are parallel to each other and reservoir bedding plane In SCWs, although the flux distribution at the wellbore is discontinous and nonuniform, the flux and the specific productivity index evaluated at the locations beyond rc are constant and uniform across the formation thickness The specific productivity index for a ~ SCW, J sc , could be written as and the actual length of the ith partially completed well producing only from the ith fictitious reservoir, respectively The special productivity index for each fictitious PCW could be written as below ~ ′ = J pci q sci k = hi′ ( p e − p wf ) 141.2 µ Bo r ln e + s ti′ rw (14) where s′ti is the total skin factor, representing the influences of partial completion, perforations, formation damage, and rock compaction, for the ith fictitious PCW s′ti could be computed using Eqs 3, 5, or 9, depending on the completion design Additionally, due to geometry, h1′ + h2′ + h3′ + + hi′ + + hn′ s = ns ∑ hi′ = h (15) i =1 n s number of fictitous PCWs are part of the original whole SCW Now if we evaluate the specific productivity indecies for the SCW and the fictitous PCW beyond the radius of flow convergence then all the specific productivity indicies should be equal ~ ~ ~ ~ ~ ~ J ′pc1 = J ′pc = J ′pc = J ′pci = = J ′pcns = J sc (16) SPE 89414 A comparison of Eqs 12 through 16 reveals that the total skin factors for the SCW and the n s number of fictitous PCWs should be the same ′ s = s sc (17) st′1 = s t′2 = st′3 = sti′ = = s tn It is very likely that the initial fictitious thickness distribution based on Eqs 20 and 21 will not satisy the conditions expressed in Eqs 16 – 18 In such a case, in the following iteration, we reallocated the fictitious thickness based on the ratio of specific productivity indicies for the individual PCW and SCW Additionally, ns ~ ~ ′ hi′ (18) J sc = ∑ J pci h i =1 If the open intervals are symmetrically distributed then all the fictitious layers has the same thickness h1′ = h2′ = h3′ = = hi′ = = hn′ s = h / n s (19) Now let’s go back and re-consider a SCW with arbitrary distribution of open segments and different degree of formation damage across them as displayed in Fig In such a case, the actual reservoir and SCW cannot be divided into n s number of equivalent layers and equivalent PCWs, respectively However, even in the case of asymmetric segment and contrasting formation damage distributions, it is expected that each completed segment will establish its own drainage volume; therefore, flow induced no-flow boundaries will emerge somewhere along the uncompleted segments between the completed ones not at the center of uncompleted segments as in case of symmetric completion In the asymmetric completions, the flow induced no-flow boundaries may not be completely parallel to reservoir bedding and as well defined as those in the symmetric completions Regardless, in case of asymmetric segment and unequal damage distribution, we could still decompose the actual SCW in the real reservoir into n s number of fictitious PCWs in n s layers However, each fictitous layer will have a different fictitious thickhness of h′i assigned to it In asymmetric completions, the fictitious PPWs are still part of the real SCW; therefore, when we evalute the specific productivity indecies beyond the radius of flow convergence, the actual SCW and the fictitious PPWs all should possess the same specific productivity index value In other words, Eqs 12 through 18 are also valid for the asymmetric completions Now, the remaining unresolved issue is how to assign the fictitious thickness to each fictitious layer Assignment of individual layer thickness requires an iterative procedure We suggest allocating the fictitious thickness based on the ratio of the segment height to the total penetration ratio initially hi′ = h pi / h pt (20) ns h pt = ∑ h pi (21) i =1 hi′ ( k +1) ~k J ′pci hi′ k (22) = ~ k J sc ~k where J sc is estimated from Eq 18 We presented the iterative approximate model (Eqs 12-18 and 20-22) for a selectively completed well and steady state flow conditions However, the same alghorithm also applies to selectively perforated wells and pseudo steady state flow For selectively perforated wells, we use Eq to estimate the total skin factor instead of Eq or which is for selectively completed wells To invoke the pseudo steady state flow condition, we just need to use Eq in the specific productivity index computation In Appendix C, a stepwise procedure is given for the iterative solution of the approximate model Verification of the Approximate Model As mentioned previously in the text, in the literature, there are 2D and 3D analytical solutions for selectively completed/perforated wells Also, the software SPAN could be used to predict the productivity of the partially perforated wells To verify the approximate model proposed in the current study, we compared it against the analytical solutions of Refs 12, 13, and 22 and the software SPAN Table shows the comparison of the 2D analytical and approximate models for SCWs only Three completed intervals and two different cases of formation damage were considered in the comparison As can be seen on the table, the results from the simple approximate model compare very well against those from the 2D analytical solution Besides the results shown in Table 1, we also conducted additional extensive comparison of the models For the majority of the cases, the approximate model replicated the results from the 2D analytical model However, for some negative skin values less than -2.3, the approximate model did not work well when the interval with negative skin was very short The approximate model was also tested extensively against SPAN software by considering PPWs with different completion/perforation schemes An example comparison is shown in Fig Table presents the data used in the comparison depicted in Fig As can be seen on the figure, the results from the approximate model and the software agree very well In some other comparisons, we observed small deviations between the compared models The average difference between results from the approximate model and SPAN was 6% Although the results are not shown, we also compared the approximate solution against the 3D solution presented in Ref 22 for selectively perforated wells and observed good agreement 6 Discussion In this section, we present the application of the approximate model to selectively perforated wells and investigate the effects of different perforation designs on the well performance Table lists the completion and perforation data considerd for the SPW The rest of the basic data set is the same as that tabulated in Table The well is perforated across three intervals The length and location of each interval are printed in Table First, we kept perforation length constant and assigned different values of the shot density, L p = 12 " and n spf = 4, 8, 12 However, in all three cases, all the perforated intervals had the same shot density The results, in terms of productivity index, total skin factor, and fractional segment rates, are summarized in Table For comparison purposes, the results for SCW and damaged and undamaged openhole completions are also listed in Table The results show that as the perforation length increases, well productivity is improved The well with n spf = 12 has 1.8 times higher productivity than that with n spf = It should be also noticed that the SPW with n spf = 12 performs slightly better than SCW We also investigated the impact of perforation length on the well performance The results for this investigation are reported in Table For L p = " , the total skin factors are substantially higher The skin factors for L p = " are about four times larger than those for L p = 12 " As a result of high total skin factors due to short perforations, the productivity index values for L p = " are approximately three times lower than those for L p = 12 " It should be noticed that, since the total skin factors are high, the changes in the skin factors not affect the fractional rate distribution The results in Table verifies that the deep penetrating perforations extending beyond the formation damage zone may improve the well productivity significantly Summary and Conclusions A simple approximate model to predict the inflow performance of selectively completed and selectively perforated wells has been developed The model is based on an iterative procedure and uses simple algebraic equations The model has been compared against the 2D analytical solution for selectively completed wells, SPAN software for partially perforated wells, and 3D analytical solution for selectively perforated wells In general, the approximate model agrees very well with the more complicated solutions and the software The accuracy of the approximate model has been verified by conducting an extensive comparison study Several novel applications of the approximate model have been presented The brief sensitivity study presented verifies that well productivity may be markedly improved if the perforations pierce through SPE 89414 the formation damage zone and communicate with the undamaged formation beyond the damaged zone Nomenclature A = drainage area, ft2 Bo = formation volume factor, dimensionless, rbbl/stb CA = reservoir shape factor h = formation thickness, L, ft hb= the distance between the bottom of the completed interval and reservoir, L, ft hp= the length of the completed interval, L, ft hpt= the length of the total completed interval, L, ft h ′ = formation thickness of the fictitious layer, L, ft Jc= productivity of completed well, stb/day/psi Jo= productivity index of open hole, stb/day/psi ~ J o = specific productivity index of open hole, stb/day/psi/ft ~ J pc = specific productivity index of partially completed wells, stb/day/psi/ft ~ J ′pc = specific productivity index of the fictitious partially completed wells, stb/day/psi/ft ~ J sc = specific productivity index of selectivley completed wells, stb/day/psi/ft k = permeability, L2, md kcz= permeability of crushed zone, L2, md kr= permeability in radial direction, L2, md, kx = ky = kz = Lp= ns= nspf= p= PR = qsct = qsci = rcz = re = rp = rw = scz = s′cz = sd = sp = spc = spd = spdc = spp = st = kxk y permeability in x-direction, L2, md permeability in y-direction, L2, md permeability in z-direction, L2, md perforation length, L, ft number of completed segments number of shots per foot pressure, m/Lt2, psi productivity ratio, dimensionless, fraction total well flow rate at surface, L3/t, stb/day flow rate across the ith segment, L3/t, stb/day radius of crushed zone around perforation, L, ft reservoir radius, L, ft perforation radius, L, ft wellbore radius, L, ft skin due to rock compaction around perforations in the presence of formation damage skin due to rock compaction around perforations in the absence of formation damage skin due to formation damage/stimulation pseudoskin due to perforating total skin combining flow convergence towards perforations and crushed zone skin total skin combining flow convergence towards perforations and formation damage total skin including perforation pseudoskin, formation damage, and rock compaction around perforation tunnels pseudoskin due to partial penetration total skin factor SPE 89414 stsc = total skin factor for selectively completed/perforated well µ = viscosity, m/Lt, cp pe = reservoir boundary pressure, m/Lt2, psi pwf = flowing wellbore pressure, m/Lt2, psi ~ p = average reservoir pressure, m/Lt2, psi ∆rcz= thickness of the crushed zone, L, ft ∆rd= damaged zone thickness around wellbore, L, ft ∆zp= the vertical distance between perforations, L, ft θp= perforation phasing angle Subscripts cz = crushed zone d = wellbore damage p = perforation t = total w = wellbore Acknowledgment The authors would like to thank Pete Hegeman for providing the information about the modified Karakas and Tariq method and a copy of SPAN software References Brons, F and Marting, V.E.: “The Effect of Restricted Fluid Entry on Well Productivity”, JPT (February 1961) 172 Odeh, A.S.:“Steady-State Flow Capacity of Wells with Limited Entry to Flow,” SPEJ (March 1968) 43; Trans., AIME, 243 Odeh, A.S.:”Pseudosteady-state Flow Capacity of Oil Wells with Limited Entry and an Altered Zone around the Wellbore,” SPEJ (August 1977) 271 Jones, L.G and Watts, J.W.:”Estimating Skin Effect in a Partially Completed Damaged Well,” JPT (February 1971) 249 Saidowski, R.M.:“Numerical Simulations of the Combined Effect of Wellbore Damage and Partial Penetration,” paper SPE 8204 presented at the 1979 SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, September 23-26 Odeh, A.S.:“An Equation for Calculating Skin Factor Due to Restricted Entry,” JPT (June 1980) 964 Papatzacos, P.:”Approximate Partial-Penetration Pseudoskin for Infinite-Conductivity Wells,” SPERE (May 1987) 227 Vrbik, J.:“A Simple Approximation to the Pseudoskin Factor Resulting from Restricted-Entry,” SPEFE (December 1991) 444 Guerra, E., Inflow Performance of Selectively Perforated Vertical Wells, MS Thesis, Colorado School of Mines, Golden, Colorado (May 2004) 10 Hawkins, M.F.:”A Note on the Skin Effect,” Trans AIME, (1956) 207 11 Larsen, L.:“The Pressure-Transient Behavior of Vertical Wells with Multiple Flow Entries,” paper SPE 26480 presented at the 1993 SPE Annual Technical Conference and Exhibition in Houston, October 3-6 12 Yildiz, T and Cinar, Y.:”Inflow Performance and Transient Pressure Behavior of Selectively Completed Vertical Wells,” SPE Reservoir Eng (October 1998) 467 13 Yildiz, T.:”Impact of Perforating on Well Performance and Cumulative Production” Journal of Energy Resources Technology, (September, 2002) 163 14 Hong, K.C.:”Productivity of Perforated Completions in Formations With or Without Damage,” JPT (August 1975) 1027, Trans AIME, 259 15 Locke, S.: “An Advanced Method for predicting the Productivity Ratio of a Perforated Well,” JPT (December 1981) 2481 16 Tariq, S.M.:”Evaluation of Flow Characteristics of Perforations Including Nonlinear Effects With the Finite Element Method,” SPEPE (May 1987) 104 17 Dogulu, Y.S.”Modeling of Well productivity in perforated Completions,” paper SPE 51048 presented at the 1998 SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, November 9-11 18 McLeod, H.:”The Effect of Perforating Conditions on Well Performance,” JPT (January 1983) 31 19 Karakas, M and Tariq, S.M.:”Semianalytical Productivity Models for Perforated Completions,” SPEPE (February 1991) 73 20 Bell, W.T., Sukup, R.A., and Tariq, S.M., Perforating, SPE Monograph Volume 16, Richardson, TX, 1995 21 Jones, L.G and Slusser, M.L.:”The Estimation of Productivity Loss Caused by Perforations – Including Partial Completion and Formation Damage,” paper SPE 4798 presented at the 1974 SPE Second Midwest Oil and Gas Symposium, Indianapolis, Indiana, March 28-29 22 Yildiz, T.:“Productivity of Selectively Perforated Vertical Wells,” SPEJ (June 2002) 158 23 SPAN user guide, Version 6.0, Schlumberger Perforating and Testing, 1999 24 Hegeman, P., Personal Communication, Schlumberger Product Center, Sugarland, Texas 25 Yildiz, T.: “Assessment of Total Skin Factor in Perforated Wells,” paper SPE 82249 presented at the 2003 SPE European Formation Damage Conference, The Hague, The Netherlands, May 13-14 Appendix A – Vrbik Model for Partial Penetration Pseudoskin The approximate model for the selectively perforated well is partially based on the partial penetration pseudoskin model proposed by Vrbik.? The Vrbik model is summarized below The details on the Vrbik model can be found in the original publication by Vrbik.? s pp = (1 / h pD − 1) (1.2704 − ln rwD ) − F / h 2pD (A-1) F = f (0) − f ( h pD ) + f ( z1 ) − 0.5[ f ( z ) + f ( z )] (A-2) f ( y ) = y ln y + (2 − y ) ln( − y ) + g ( y ) (A-3) g ( y ) = rwD ln [ sin (π y / 2) + 0.1053 rwD ] / π (A-4) z1 = − D (A-5) z = − D + h pD (A-6) z = − D − h pD (A-7) h pD = h p / h (A-8) hbD = hb / h (A-9) D = (1 − h pD ) / − hbD (A-10) rwD = rw k z / k r / h (A-11) SPE 89414 Appendix B – Karakas-Tariq Model for Perforation Pseudoskin Karakas and Tariq23 proposed the stepwise procedure below to estimate the pseudoskin due to perforating ' s cz = ∆ zp Lp ( k − 1) ln( rcz / r p ) (B-12) k cz Compute the pseudoskin due to flow convergence in the horizontal plane The simultaneous effects of flow convergence toward perforations and the permeability impairment around the perforations are formulated as below s H = ln( rw / rwe ) (B-1) ′ (B-13) s pc = s p + s cz rwe (θ p ) = α (θ p ) ( rw + L p ) (B-2) Add formation damage effect Eq B-2 is valid for all the phasing angles except zero α ( θ p ) is tabulated as a function of the phasing angle For α ( θ p ) = , rwe ( θ p = ) = L p / If the perforations are short and terminated inside the damaged zone then the total skin factor is given as below s pdc = s d + k ′ ) (B-14) ( s p + s x + s cz kd Estimate the pseudoskin due to cylindrical wellbore rwD = rw /( rw + L p ) (B-3) s wb (θ p ) = c1 (θ p ) exp [c (θ p ) rwD ] (B-4) c1 and c are tabulated as functions of the phasing angle where s x is negligible for most cases If the perforations are long and extend beyond the damaged zone, the perforation length and wellbore radius in steps through are replaced with the effective perforation length and effective wellbore radius defined below L′p = L p − (1 − kd / k ) ∆ rd (B-15) Compute the pseudoskin due to flow convergence in the vertical plane rw′ = rw + (1 − k d / k )∆ rd (B-16) ∆ z p = / n spf (B-5) Appendix C – Alghorithm for the Approximate Model Assume that the reservoir and fluid properties, pressure drop, the number of open segments, the location and length of the completed segments, and the degree of formation damage and perforation variables for all the completed segments are available Given ∆ p , µ , Bo , k r , k z , re , rw , h , n s , hbi , h pi , (k d / k ) i , ∆ rdi , n spfi , L pi , rpi , θ pi , (k cz / k ) i , and ∆ z pD = ∆ z p k r / k z / L p (B-6) rpD = ( rp / ∆ z p ) (1 + k z / k r ) (B-7) a = a1 (θ p ) log(rpD ) + a (θ p ) (B-8) ∆ rczi b = b1 (θ p ) rpD + b2 (θ p ) (B-9) a1 , a , b1 , and b2 are all tabulated as functions of the phasing angle Divide the SCW or SPW into n s number of PCWs or PPWs For the first iteration, estimate the initial values of h′i for ≤ i ≤ n s using hi′ = h pi / h pt (C-1) −1 b s v = 10 a ∆ z bpD rpD (B-10) ns h pt = ∑ h pi (C-2) i =1 Determine the perforation pseudoskin s p = s H + s v + s wb (B-11) Recalculate the location of the open segments in the fictitious layers Add crushed zone effect First, estimate the skin factor due to crushed zone around the perforation tunnels ′ = hbi − hbi i −1 ∑ hi′ , j =1 hb′ = hb1 (C-3) SPE 89414 ′ + h pi ) > hi′ then If (hbi ′ = hi′ − h pi (C-4) hbi ~ ~ J ′pci = J ′pci hi′ = J sc hi′ (C-12) ′ = then If hbi ~ J sc = J sc h .(C-13) hi′ = h pi (C-5) and ∆ rdi , compute s ′ppi and s di for ≤ i ≤ n s using the Vrbik method and Hawkins equation, respectively Then, estimate the total skin factor, s′ti , for all the fictitious PCWs q sct = J sc ( p e − p wf ) (C-15) γ h s pdci + s ′ppi (C-7) hp Compute the specific productivity indecies for all the fictitious PCWs/PPWs and the SCW ~ ′ = J pci k 141.2 µ Bo ln re + s ti′ rw (C-8) ns ~ ~ ′ hi′ (C-9) J sc = ∑ J pci h i =1 Check if ~ ~ ~ ~ ~ ′ J ′pc1 ≈ J ′pc ≈ J ′pci ≈ ≈ J pcn ≈ J sc (C-10) s ′ (C-11) st′1 ≈ s t′2 ≈ s ti′ ≈ ≈ stn s If the convergence criteria are not satisfied then recalculate the thickness values for each fictitious layers for the next iteration ~k J ′pci hi′ k (C-16) hi′ ( k +1) = ~ k J sc If the intervals are perforated then estimate the perforation total skin, s pdci , for ≤ i ≤ n s by s ti′ = and hi′ s di + s ′ppi (C-6) γ h pi using the hybrid method described in the body of the text Combine the influences of partial penetration and perforation total skin as displayed below q sci = J ′pci ( p e − p wf ) (C-14) If the well is selectively completed then, for the given sets of k z / k r , rw , h′i , h′bi , h pi , (k d / k ) i , s ti′ = If the convergence criteria are satisfied then compute the productivity indicies or the rates for individual segments and the SCW Go to step Iterate on the steps 2-8 until the convergence criteria given in step are satisfied SI Metric Conversion Factors bbl cp ft ft3 in lbf lbm mD × × × × × × × × 1.589 873 1.0* 3.048* 2.831 685 2.54* 4.448 222 4.535 924 9.869 233 E-01 E-03 E-01 E-02 E+00 E+00 E-01 E-04 * Conversion factor is exact = = = = = = = = m3 Pa ì s m m3 cm N kg àm2 10 SPE 89414 TABLE – COMPARISON OF APPROXIMATE AND 2D ANALYTICAL MODELS FOR SCWs k r = k z = 100 md , Bo = 1.3 rbbl/stb , µ = 0.8 cp , TABLE – DATA USED FOR THE SPW EXAMPLE WITH THREE PERFORATED SEGMENTS ns re = 1,000' , rw = 0.25' , h = 100' , h p1 = hb1 = 10' , h p = 30' , hb2 = 50' , h p3 = 5' , hb3 = 95' , n s = h, ft s di = s d 1,2,3 = 6, 4, This study 100 h p1 , ft hb1 , ft h p , ft 2D This study 2D J sc 5.608 5.628 3.45 3.43 hb , ft 50 q sc1 / q sct 0.259 0.249 0.175 0.164 h p3 , ft 12 q sc / q sct 0.619 0.626 0.590 0.584 hb3 , ft 80 q sc3 / q sct 0.122 0.126 0.235 0.252 h pt / h st 3.849 11.443 s′t1 3.794 11.582 s′t 3.811 11.568 s′t 3.802 11.527 0.2 n spf 4, 8, 12 L p , inches 3, 12 TABLE – THE IMPACT OF SHOT DENSITY ON SPW PERFORMANCE, L p = 12 " Fractional segment rate TABLE – DATA USED TO COMPARE THE APPROXIMATE MODEL AND SPAN FOR PPWs 1.3 Bo , rbbl/stb µ , cp 0.8 h, ft 120 h p , ft 10 hb , ft 90 re , ft 745 rw , ft 0.25 k r , md 20 kz / kr ∆ rd , ft 0.5 k d , md n spf L p , inches 0.2-12 rp , inches 0.1 θ p , degrees 90 ∆ rcz , inches 0.5 k cz , md n spf J sc 0.386 27.2 0.580 12 st 2nd 3rd 0.108 0.295 0.597 15.5 0.114 0.292 0.594 0.697 11.5 0.117 0.290 0.592 SCW 0.688 11.7 0.130 0.295 0.575 OH* 1.100 4.4 OH** 1.703 * Formation damage, 1st ** No formation damage TABLE – THE IMPACT OF PERFORATION LENGTH ON SPW PERFORMANCE Fractional segment rate J sc st Lp n spf 1st 2nd 3rd 12 0.386 27.2 0.108 0.295 0.597 12 0.580 15.5 0.114 0.292 0.594 0.110 115 0.103 0.300 0.597 0.180 67.5 0.106 0.299 0.595 SPE 89414 11 0.25 This study, spf=8 This study, spf=12 This study, spf=16 SPAN, spf=8 SPAN, spf=12 SPAN, spf=16 Productivity Ratio 0.20 Damaged + crushed zones 0.15 0.10 0.05 Crushed zone only 0.00 0.0 2.0 4.0 6.0 8.0 10.0 12.0 Lp, in Fig – Flow convergence towards a partially penetrating well Fig – Flow convergence towards a selectively completed well Fig – The approximate model - Selectively completed well replaced by n s fictitious partially completed wells Fig – Comparison of the approximate model and SPAN software for a partially perforated well