SummaryThis paper presents an investigation of perforated horizontalwell performance in areally anisotropic reservoirs. Theoretical investigation is based on a 3D analytical IPR model. The analytical IPR model considers an arbitrary distribution of perforations along the completed segments. Changes in flow rate, pseudosteadystate productivity, and cumulative production can be computed using the solution. The analytical IPR model was compared with the models available in the literature and verified. It was then used to investigate the effects of well and reservoir parameters on the inflow performance of perforated horizontal wells.
Inflow Performance Relationship for Perforated Horizontal Wells Turhan Yildiz, SPE, Colorado School of Mines Summary This paper presents an investigation of perforated horizontal-well performance in areally anisotropic reservoirs Theoretical investigation is based on a 3D analytical IPR model The analytical IPR model considers an arbitrary distribution of perforations along the completed segments Changes in flow rate, pseudosteady-state productivity, and cumulative production can be computed using the solution The analytical IPR model was compared with the models available in the literature and verified It was then used to investigate the effects of well and reservoir parameters on the inflow performance of perforated horizontal wells Introduction Horizontal wells may be perforated in selected intervals for several reasons The most common reasons for selective completion are reducing the cost, delaying premature water/gas breakthrough, preventing wellbore collapse in unstable formations, and producing multiple zones with large productivity contrast effectively Open-completed horizontal wells with negligible wellbore pressure loss display a u-shaped influx profile; fluid velocities at the heel and toe end of the well are higher than those at the midsection of the well Several simulation studies have shown that water/gas prematurely breaks through at the heel end of the well and causes inefficient sweep Uniform influx along the horizontal wellbore is desirable to delay premature water/gas breakthrough and improve the sweep efficiency Water/gas breakthrough could be delayed by restricting the flow and communication between reservoir and wellbore at the intervals where local fluid velocities are higher Selective perforating with blank sections provides flexibility for future intervention and workover options and for shutting off the sections subject to excessive water/gas intrusion On the other hand, partial completion and enforcing uniform inflow along the wellbore by variable shot density reduce the well productivity Therefore, a complete engineering analysis is required to weigh the gains from improved sweep provided by uniform inflow against the loss in well productivity The orientation of perforations is also a concern in optimizing well productivity Perforations aligned with minimum stress direction produce more sand To reduce the risk of sand production, it may be better to orient the perforations vertically Additionally, subsurface rocks exhibit horizontal permeabilities that are higher than vertical permeabilities Therefore, perforation tunnels perpendicular to higher permeability would possess better flow efficiency On the other hand, debris resulting from perforation process has to be surged out of the tunnels to improve the productivity of the perforated completions It is more difficult to clean the perforations on the low side of the horizontal wells Solid debris in the low-side perforation tunnels may not be removed under the typical underbalance pressures applied Vertically oriented perforation tunnels at the top side of the horizontal wellbore are preferred for better perforation stability Copyright © 2004 Society of Petroleum Engineers This paper (SPE 88987) was revised for publication from paper SPE 67233, presented at the 2001 SPE Production Operations Symposium, Oklahoma City, Oklahoma, 24–27 March Original manuscript received for review 21 June 2002 Revised manuscript received 10 May 2004 Manuscript peer approved 17 May 2004 September 2004 SPE Journal and cleanup efficiency However, if the perforations are to be packed, it is difficult to transport the gravel into vertically oriented tunnels at the top side Background Perforating has been one of the most common completion methods for vertical wells requiring sand control and prevention of wellbore collapse and water/gas intrusion The performance of perforated vertical wells has been investigated extensively However, only a few modeling studies have dealt with the productivity of perforated horizontal wells Field applications of the perforated horizontal wells have preceded the modeling efforts Some of the field applications and research studies are summarized below to acknowledge the previous contribution Field Applications Horizontal wells are mostly completed openhole with slotted liners or prepacked screens However, in some fields, horizontal wells may need to be cased and perforated Selective perforating has been implemented in the horizontal wells drilled in many fields such as Bongkot,1 Joanne,2 Andrew,3 Oseberg,4,5 Wytch Farm,6 Statfjord,7 Elk Hills,8 Beryl,9 Dan,10 Alba,11 Yowlumne,12 and Prudhoe Bay.13 King14 pointed out the significance of gun clearance and positioning in perforating and gravel packing the horizontal wells Previous Modeling Studies The performance of perforated vertical wells has been the subject of many modeling studies.15–19 Perforation shot density and length and penetration ratio are found to be the dominant parameters controlling the flow efficiency of vertical wells Inflow performance models for horizontal open holes are relatively simple and well known.20–22 However, these models ignore the near-wellbore completion effects and consider that the total drilled horizontal length is open to flow In many cases, horizontal wells may be completed at selected intervals along the well axis The horizontal wells partially completed at multiple producing segments are referred as selectively completed wells Several modeling studies on the performance and pressure behavior of selectively-completed horizontal wells have appeared in the literature.23–26 Although these models account for selective completion effect, they all ignore the details of flow convergence because of the existence of perforations and slots in the near-wellbore region A few studies have addressed how the perforations and slots influence the flow into horizontal wells.27–34 Landman and Goldthorpe27 and Marett and Landman28 presented steady-state models for flow into perforations distributed around a horizontal wellbore They also proposed the use of variable perforation shot density to obtain a uniform influx along the wellbore Gonzalez-Guevara and Camacho-Velazques29 extended the model of Refs 27 and 28 to account for multiphase flow in the wellbore and reservoir and non-Darcy flow around the perforations Asheim and Oudeman30 presented a simple algorithm to predict the perforation shot density yielding uniform influx along a horizontal well They concluded that optimal perforation distribution is rate-dependent, and enforcing uniform flux along the wellbore reduces the well productivity/injectivity Thomas et al.31 incorporated the near-wellbore skin term and non-Darcy flow coefficient into their reservoir simulator and the existing semianalytical IPR models for horizontal wells They proposed the use of perforation pseudoskin nomograph developed for vertical wells by Locke.16 They defined a global mechanical skin factor (s), composed of perforation pseudoskin (sp), formation 265 damage skin (sd), and skin caused by rock crushing around the perforations (sdp) They treat the global mechanical skin factor as the sum of three individual skins listed However, it can be shown that three skin terms (sp, sd, and sd) are not additive Ozkan et al.32 presented a 3D transient-flow model for perforated horizontal and inclined wells and compared the pressure transient responses of openhole, partially open (selectively completed), and perforated horizontal wells The simulated results have shown that the transient component of convergent flow around perforations subsides very fast Beyond early-time flow regime, the convergent flow near perforations results in a stabilized additional pressure drop around the wellbore The model presented in Ref 32 may require extensive CPU time for long horizontal wells Recently, Goktas and Ertekin33 developed a numerical simulator for perforated horizontal wells Their model relies on flexible and locally refined grids to capture the details of the convergent flow around the perforation accurately They stated that their numerical model is limited to low shot densities, otherwise demanding extensive computation time Most recently, Tang et al.34 extended the perforation pseudoskin model proposed by Ozkan et al.32 and investigated the impact of perforation parameters on horizontal well performance They observed that the perforation densities higher than 0.5 spf yield a marginal increase in well productivity In a previous paper,35 it has been shown that, in fully perforated vertical wells, an ideal combination of perforation length, shot density, and phasing angle may result in a negative perforation pseudoskin Under steady-state flow conditions, this negative pseudoskin caused by perforating may yield a 10 to 15% productivity improvement over openhole completed vertical wells However, in most cases, productivity improvement diminishes with time if the reservoir is sealed at the outer lateral boundaries and the well produces at constant wellbore pressure The objectives of the present study are: (1) to build a model for selectively perforated horizontal wells in bounded reservoirs with no-flow external boundaries, (2) to examine the flow rate and cumulative production responses of perforated horizontal wells under constant wellbore pressure, and (3) to investigate the impact of different perforating schemes on the long-term performance of horizontal wells Flow Model for Perforated Horizontal Well In a previous study,32 it was shown that the transients caused by the convergence of streamlines around perforations dies out very fast Hence, the energy loss caused by convergent flow into perforation tunnels can be represented as an additional stabilized pressure drop beyond very early times To construct a simple and computationally fast model, the 3D flow into a perforated horizontal well is decomposed into two smaller subproblems: a transient 3D model for flow into selectively completed horizontal well, and a perforation totalpseudoskin model for accounting the flow convergence around the tunnels in near-wellbore region This model will be referred to as the “decoupled” model The modeling concept is illustrated in Fig The decoupled model concept has previously also been applied to selectively perforated vertical wells.19,35 Selectively Completed Horizontal Well (SCHW) Model A multisegment horizontal well in a rectangular parallelepiped reservoir with impermeable external boundaries is considered Multisegmentation allows us to account for the local changes around the wellbore The SCHW assumes that the completed intervals are fully open to flow all around the perimeter of the segment A variable local skin around each segment is also incorporated into the SCHW model The additional pressure change caused by perforations is superimposed on SCHW in terms of local skin A schematic of reservoir model for multisegment horizontal well is given in Fig The mathematical treatment of the SCHW model is presented in the Appendix 266 Model for Perforation Total Pseudoskin (PTP) Here, we make use of the PTP model developed for perforated vertical wells The justification for the use of vertical well perforation pseudoskin is that perforation length is very small compared to horizontal well length and the distance to the external reservoir boundaries Therefore, 3D convergent flow around the perforations is not influenced by far field-flow conditions However, PTP calculated for a vertical well has to be properly scaled for use in horizontal well models PTP model takes care of the flow into perforation tunnels, flow in the damaged zone around the wellbore, and flow across the crushed/compacted zone around the tunnels A schematic of the PTP model is displayed in Fig The mathematical model for PTP has been described in two previous publications.19,35 The reader is referred to Refs 19 and 35 for a complete description of the PTP model The perforation pseudoskin is calculated for only a unit formation thickness (for example, ft of thickness) Therefore, the PTP model is computationally very efficient Verification of the Decoupled Model First, we measured the SCHW model up to the single-segment models proposed by Babu and Odeh20 and Goode and Kuchuk21 under pseudosteady-state flow conditions and observed excellent agreement Then, the SCHW model was compared against the partially open horizontal well model presented by Goode and Wilkinson.23 The comparison is made in terms of productivity ratio for an 800-ft partially open horizontal well producing at constant flow rate under pseudosteady-state flow conditions Productivity ratio (PR) is defined as the ratio of the productivity index of a partially open well to the productivity index of a fully open horizontal well Three different completion scenarios are considered The comparison is displayed in Fig As can be seen, an excellent agreement between the models has been established Also, the proposed model was compared against the model presented in Ref 25 The current model is for SCHW in bounded reservoirs The well-testing model described in Ref 25 is for SCHW in infinite reservoirs The comparison, depicted in Fig 5, is for a 900-ft-long horizontal well with five 100-ft-long open segments producing at a constant flow rate The models agree very well until no-flow reservoir boundaries are felt at the wellbore The proposed model was also compared with the model of Retnanto et al.24 However, the results from our model and the model of Ref 24 significantly deviated The PTP model has already been previously compared against the experimental data and the Karakas and Tariq18 model, showing good agreement The details on the verification of PTP model can be obtained in Ref 19 Fig 1—Decoupled model for perforated horizontal wells September 2004 SPE Journal Fig 2—Multisegment selectively-completed horizontal well model After verifying the individual components of it, the decoupled model itself was tested against the results presented in Refs 32 and 34 Unlike the decoupled model, the pressure transient analysis model of Ozkan et al.32 considers all the individual perforations around the wellbore The comparison with the Ozkan et al.32 model is shown in Fig Except for the early-time flow period at which only the formation around individual perforations contributes to flow, very good agreement is observed between the models Finally, the proposed model was compared with the results given by Tang et al.34 We calculated the transient productivity index of the well described in the data set given in Table of Ref 34 The PI values predicted from the decoupled model are identical to the values given in Table 3A of Ref 34 The decoupled model is computationally efficient It takes less than a minute (typically 15 to 20 seconds) on a high-end PC to simulate the 10-year rate and cumulative production responses of the cases displayed in Figs through 20 Limitations of the Model There are several other factors that may influence the performance of a perforated well but are not considered in the model The pressure losses inside the perforations may be important in high-rate wells However, this factor has not yet been incorporated into the model In several numerical studies on perforated vertical wells, it has been shown that non-Darcy flow around the perforations could substantially reduce the flow efficiency The impact of non-Darcy flow is not considered in this study Near-wellbore heterogeneity and accumulation of fines around perforations may also contribute to perforation efficiency Heterogeneity and fines movement are not accounted for in the model Discussion In this section, using the decoupled model, we simulate different perforation schemes and discuss the impact of perforating param- Fig 3—Perforation total pseudoskin model.19 September 2004 SPE Journal 267 Fig 4—Comparison of SCHW model with Goode and Wilkinson model.23 Fig 5—Comparison of SCHW models in bounded and infinite reservoirs Fig 6—Comparison of the decoupled model to the “exact” perforated horizontal well model of Ozkan et al.32 Fig 7—Effect of shot density on flow-rate decline 268 September 2004 SPE Journal Fig 8—Effect of shot density on cumulative production eters on rate decline, cumulative production, and the productivity during boundary dominated flow period The data set given in Table is used in the simulations Effect of Shot Density To examine the impact of perforation shot density on well performance, we considered shot densities of 0.25, 0.5, 1, 2, 4, 8, 12, and 16 spf The other perforation parameters used in the simulations are given in Figs and First, we calculated perforation total skin for a fully perforated vertical well (sptV) Then sptV was scaled for horizontal well by multiplying it with h/Lh The scaled perforation total pseudoskin for horizontal wells will be referred to as sptH; sptV and sptH for each perforation scheme are listed in Table Shot densities higher than spf create a negative pseudoskin After predicting sptH, we introduced it into our SCHW model We simulated how the flow rate and cumulative production change as a function of time when the well produces at a constant wellbore pressure drop of 500 psi The results are shown in Figs and Initially, there is a considerable difference in flow rates However, initial high flow rates, because of negative perforation pseudoskin resulting from high shot density, diminish with time For each perforation scheme, the cumulative productions at the end of and years are given in Table The cumulative productions from a horizontal well perforated with spf are almost identical to those from openhole Higher shot densities improve the cumulative production slightly For each scenario, we also determined productivity ratio The productivity ratio is defined as the ratio of the productivity index of a perforated well to the productivity index of an open hole under pseudosteady-state flow conditions when the well produces at a constant flow rate The productivity ratios for different cases are listed in Table The productivity ratio for a perforated well with spf is 0.98 The results in Table and Figs and show that a perforated well with a shot density of spf performs as well as an open hole Shot densities larger than spf marginally improve the long-term well performance Fig 9—The influence of perforation length on transient-rate response September 2004 SPE Journal 269 Fig 10—Effect of perforation length on cumulative production Influence of Perforation Length To investigate the impact of perforation length on well performance, we considered perforation lengths of 3, 6, 12, 18, and 24 in and shot density of spf sptV and sptH for each perforation scheme are listed in Table 3- and 6-in perforations cause positive perforation pseudoskin of 0.2114 and 0.0677, respectively Longer perforations yield small negative pseudoskin values Flow rate and cumulative production responses are displayed in Figs and 10 Longer perforations initially produce at somewhat higher flow rates However, except for the completion with 3-in perforations, all the completion schemes result in approximately the same production rate after one month The cumulative production at the end of and years and productivity ratios are tabulated in Table The completion with 24-in perforations recovers 10% more fluid than that with 3-in perforations at the end of years The cumulative production and productivity for the horizontal well shot with 6-in perforations are slightly lower than those for openhole Marginally higher recov- eries and productivity ratios are obtained if the well is shot with longer perforations Impact of Phasing Angle To examine the effect of phasing angle, we considered 0°, 180°, and 90° phasing The other perforation parameters were fixed at spfס4, Lpס12 in., and dpס0.2 in Figs 11 and 12 display the insensitivity of transient flow rate and cumulative production to phasing angle The cumulative productions at the end of and years and productivity ratio under pseudosteady-state flow conditions are listed in Table All the results indicate that phasing angle has negligible impact on the performance of perforated horizontal wells in isotropic formations Although not shown here, we also investigated the effect of perforation diameter and observed that the productivity of perforated horizontal wells is insensitive to perforation tunnel diameter It should be stated that this conclusion applies to the case for which the pressure losses inside the perforation are negligible Fig 11—Impact of phasing angle on transient rate response 270 September 2004 SPE Journal Fig 12—Effect of phasing angle on cumulative production Combinations of Perforation Parameters In the discussion above, we only scrutinized the effect of one parameter by keeping the rest of the perforation parameters constant Here, we change all the perforation parameters and consider pessimistic, reasonable, and optimistic combinations of them The perforation parameters of each case are given in Table The simulated rates and cumulative productions for each case are displayed in Figs 13 and 14, respectively Table lists perforation pseudoskin, productivity ratio, and cumulative productions for each case The optimistic case yields a small negative pseudoskin, while the pessimistic case generates a positive pseudoskin of 1.37 There is a significant difference in rate and cumulative production responses of pessimistic and optimistic combinations of perforation parameter The optimistic perforation scheme recovers 44% more oil than the pessimistic case at the end of the 5-year period The rate, cumulative production, and productivity ratio responses to reasonable combination of perforation parameters are almost identical to those for openhole This exercise shows that although well productivity may be weakly sensitive or insensitive to a given perforation parameter, if all the perforation parameters act in the same direction, then well performance and cumulative production may be strongly influenced by perforation design The Impact of Formation Damage Many studies have concentrated on how formation damage and rock compaction impact the flow efficiency in perforated vertical wells Here, we investigate the influence of permeability reduction caused by formation damage around the horizontal wellbore We considered that the permeability in the damaged zone was 25% of the original Additionally, spfס2, Lpס12 in., dpס0.2 in., and pס180° The radius of damaged zone around the wellbore was varied from to 15 in The results are shown in Figs 15 and 16 and Table It is observed that as long as the perforation penetrates beyond damaged zone (rwdס6, 9, and 12 in.), the loss in productivity is below 10% Otherwise, if the perforation tunnel does not extend beyond the damaged zone, the productivity impairment is more severe When Fig 13—Transient-rate response to pessimistic, reasonable, and optimistic combinations of perforation parameters September 2004 SPE Journal 271 Fig 14—Cumulative productions for pessimistic, reasonable, and optimistic combinations of perforation parameters the whole tunnel is inside the damaged zone (rwdס15 in.), then the loss in productivity is 20% Therefore, the perforations should be designed to penetrate beyond the damage zone around the wellbore The Impact of Crushing Around Perforation Tunnels Many experimental studies on the API RP 43 setup have concluded that there exists a crushed zone around the perforation tunnel.36 The permeability of the crushed zone is about 10% of the original formation permeability Although the extent of the crushed zone varies along the perforation tunnel, an average and constant crushed-zone thickness of 0.5 in is accepted The results showing the combined impact of rock crushing around the tunnels and formation damage around the wellbore are summarized in Figs 17 and 18 and Table We considered kcp/kס0.1 and kd/kס0.25 in the simulations It can be observed that the rock crushing reduces the well productivity much more severely When the perforation has a crushed zone and does not extend beyond damaged zone around the wellbore, the total loss in well productivity is 32% The rock crushing alone results in a 25% productivity impairment Thus, it can be stated that as long as there exists a crushed zone around the perforation tunnel with a permeability lower than damaged-zone permeability, then the productivity impairment caused by the crushed zone is more severe and formation damage has a lesser influence The productivity loss caused by formation damage is marginal, especially if the perforation tunnel extends beyond the damaged zone Effect of Phasing in Anisotropic Formations We investigated the impact of perforation phasing angle in an anisotropic formation with kz/kxס0.1 and kyסkx Additionally, we considered spfס2, Lpס1 ft, and dpס0.2 in The results are given in Figs 19 and 20 Fig 15—Effect of formation damage on transient-rate response of perforated horizontal wells 272 September 2004 SPE Journal Fig 16—Impact of formation damage on long-term cumulative productions from perforated wells and also in Table 10 First, formation anisotropy significantly impacts the performance of open completed horizontal well While the horizontal open hole in isotropic formation produces 310,000 STB oil in 12 months, the same well in anisotropic formation recovers 226,200 STB oil in 13 months In general, when perforations are oriented vertically, well performance is enhanced because the perforations are normal- to high-permeability directions When all the perforations are vertical and phasing angle is 180°, the well performs the best In this case, the productivity of the perforated well is approximately 15% better than that of the openhole 90° phasing, which orients half of the perforations vertically and half of them horizontally, yields the second-best performance Vertical perforations with 0o phasing result in the third-best recovery However, these results are only valid for spfס2 We also simulated the effect of phasing in anisotropic formations with less dense perforations For spfס0.125, vertical perforations with 0° phasing produce more fluid than 90° phasing At low shot densities, perforations not interfere with each other In such a case, negative impact of 0° phasing diminishes The well shows the worst performance when all the perforations are horizontally oriented with a phasing angle of 0° Conclusions Based on the discussion presented above, the following conclusions are reached It should be emphasized that the conclusions listed pertain to the range of the data used in the simulations An analytical IPR model for perforated horizontal wells has been developed The model is compared against the several models in the literature and verified The perforation pseudoskin model for vertical wells can be used to predict the performance of perforated horizontal wells How- Fig 17—Influence of crushed zone and formation damage on transient-rate decline in perforated wells September 2004 SPE Journal 273 Fig 18—Cumulative production affected by crushed zone and formation damage ever, the perforation pseudoskin for vertical wells has to be properly rescaled for use in horizontal-well performance calculations In undamaged formations, a horizontal well perforated with a shot density of spf and 1-ft-long tunnels performs as effectively as the well-completed open hole Denser perforations yield slightly better cumulative recovery and productivity As long as it is not too short (longer than in.), perforation length has marginal impact on the long-term cumulative recovery and well productivity in undamaged formations In isotropic formations, perforation phase angle has an insignificant effect on transient rate, cumulative production, and productivity of perforated horizontal wells Even when well performance may be weakly sensitive to a given single perforation parameter, if all the perforation param- eters act in the same direction, then well performance and cumulative production may be strongly influenced by perforation design If the perforation tunnel terminates inside the damaged zone around the wellbore, up to 20% loss in the productivity may be encountered If the tunnel created by perforating penetrates beyond the damaged zone, then the detrimental effect of formation damage is considerably less The productivity of perforated horizontal well is significantly reduced because of permeability impairment in the crushed zone around the perforation tunnels As long as there exists a crushed zone around the perforation tunnel with a permeability lower than damaged-zone permeability, then formation damage has a lesser influence Formation anisotropy has a significant impact on well produc- Fig 19—Effect of permeability anisotropy on transient-rate decline in perforated wells 274 September 2004 SPE Journal Fig 20—Effect of permeability anisotropy on cumulative production tivity The well has the best performance when all the perforations are vertically oriented with a phasing angle of 180° The well productivity is the lowest when all the perforations are horizontally oriented, with a phasing angle of 0° September 2004 SPE Journal 275 Nomenclature Bo סformation volume factor, RB/STB ct סtotal compressibility, psi−1 h סheight, ft k סpermeability, md kcp סpermeability of crushed zone around perforation tunnel, md kd סpermeability of damaged zone, md K0 סzero order modified Bessel function of the first kind L סreference length Lh סhorizontal well length Ls סsegment length nseg סnumber of selectively completed segments p סpressure, psi pi סinitial reservoir pressure, psi PR סproductivity ratio q סflow rate, STB/D qj סflow rate at the jth segment, STB/D qw סtotal well flow rate, STB/D r סradius, ft rw סwellbore radius, ft rwe סequivalent wellbore radius, ft s סLaplace space variable sptH סperforation total pseudoskin rescaled for horizontal wells sptV סperforation total pseudoskin for vertical wells t סtime, hours xe סlength of the reservoir in x-direction xs סlocation of the segment center in x-direction ye סwidth of the reservoir in y-direction yw סlocation of the well in y-direction z סvertical direction, ft zw סlocation of the well in vertical plane, ft סviscosity, cp סporosity Subscripts d סwellbore damage D סdimensionless variable i סinitial 276 r s w x y z ס ס ס ס ס ס radial direction segment wellbore x direction y direction vertical direction References Horn, M.J., Plathey, D.P., and Ibrahim, O.: “Multilateral Horizontal Well Increases Liquids Recovery in the Gulf of Thailand,” SPEDC (June 1998) 78 Thomson, D.W and Nazroo, M.F.: “Design and Installation of a CostEffective Completion System for Horizontal Chalk Wells Where Multiple Zones Require Acid Stimulation,” SPEDC (September 1998) 151 Kusaka, K et al.: “Underbalance Perforation in Long Horizontal 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al.: “Performance of Horizontal Wells Completed With Slotted Liners and Perforations,” paper SPE/CIM 65516 presented at the 2001 SPE/CIM International Conference on Horizontal Well Technology, Calgary, Alberta, 6–8 November 35 Yildiz, T.: “Impact of Perforating on Well Performance and Cumulative Production,” Journal of Energy Resources Technology (September 2002) 163 36 Karacan, C.O and Halleck, P.M.: “Mapping of Permeability Damage Around Perforation Tunnels,” paper ETCE 2000-10036 presented at the 2000 Engineering Technology Conference on Energy, New Orleans, 14–17 February Appendix—Transient Model for Multisegment Horizontal Well The 3D diffusivity equation governs the flow into a multisegment horizontal well The governing differential equation is as follows: Ѩ2pD ѨxD2 + Ѩ2pD ѨyD2 SD = GD + Ѩ2pD Ѩz2D nseg qiD i=1 siD ͚L + SD = ѨpD (A-1) ѨtD ͓H͑xD − i−͒ − H͑xD − i+͔͒ (A-2) GD = 2hD kD ␦͑zD − zwD͒␦͑yD − ywD͒ (A-3) One initial and six boundary conditions are needed to solve the differential equation given above The initial and boundary conditions are given in the following: pD ͑xD ,yD ,zD,0͒ = (A-4) ѨpD ͑0,yD ,zD ,tD͒ = (A-5) ѨxD ѨpD ͑x ,y ,z ,t ͒ = (A-6) ѨxD eD D D D ѨpD ͑x ,0,z ,t ͒ = (A-7) ѨyD D D D ѨpD ͑x ,y ,z ,t ͒ = (A-8) ѨyD D eD D D ѨpD ͑x ,y ,0,t ͒ = (A-9) ѨzD D D D ѨpD ͑x ,y ,h ,t ͒ = (A-10) ѨzD D D D D 277 Additionally, for constant flow rate at the wellbore, nseq ͚q = 1, (A-11) iD͑tD ͒ i=1 and for the production under constant wellbore pressure condition, nseg ͚q qwD ͑tD͒ = iD ͑tD ͒ (A-12) i=1 pwD ͑tD͒ = (A-13) The dimensionless variables are defined in the following: For constant flow rate, pD = kh ͓ p − p͑x,y,z,t͔͒ (A-14) 141.2qBo i qD = qs͑x,y,z,t͒Lh ր qw (A-15) For constant wellbore pressure case, pD = pi − p͑x,y,z,t͒ (A-16) pi − pw qwD͑tD͒ = 141.2qw͑t͒Bo ր kh͑ pi − pw͒ (A-17) NpD͑tD͒ = Np Bo ր 1.119 ct hL2͑ pi − pw͒ (A-18) tD = 2.63679 × 10−4kt ր ct L2 (A-19) thD = 2.63679 × 10−4kt ր ct ͑Lh ր 2͒2 (A-20) ͑͌ ⍀mnkl = Ko yD = y͌k ր ky ր L (A-23) zD = z͌k ր kz ր L (A-24) hD = h͌k ր kz ր L (A-25) kD = ͌k2 ր kx ky (A-26) LhD = Lh͌k ր kx ր L (A-27) i− = xsiD − LsiD ր (A-28) i+ = xsiD + LsiD ր (A-29) nseg LtD = ͚L iD (A-30) rw rweD = ͓͑kz ր ky͒0.25 + ͑ky ր kz͒0.25͔ (A-31) 2L In the definitions listed above, k and Lסthe characteristic permeability and characteristic length, respectively In this study, we choose k(סkxkykz)1/3 and LסLh/2 We have solved Eqs A-1 through A-13 using the Laplace and Finite Fourier Cosine transformations First, we assume uniform but unknown and different flux along each completed segment; then, we use the pressure-averaging technique The final Laplace space solution for segment j is as follows: nseg ͚A ͑s͒ q˜ ji (A-32) iD͑s ͒ i=1 Aji͑s͒ = F1ji͑s͒ + F2ji͑s͒ (A-33) ϱ ϱ ͚͚͚⌿ kD hD F1ji͑s͒ = xeD n=1 F2ji͑s͒ = 2kD hD xeD 0nkl (A-34) ϱ ϱ k=0 l=−ϱ ϱ ͚ ͚ ͚ ͚⍀ LhiD LhjD n=1 278 m 2 ͒ (A-37) 22 = ͑2ywD − 2l yeD͒2 + zkn (A-39) zk1 = 2hD k + rweD (A-40) zk2 = 2hD ͑k + 1͒ − rweD (A-41) zk3 = 2hD k + ͑2zwD + rweD͒ (A-42) zk4 = 2hD ͑k + 1͒ − ͑2zwD + rweD͒ (A-43) m = ͌s + ͑m ր xeD͒2 (A-44) Rxmi = sin ͑mi+ ր xeD͒ − sin ͑mi− ր xeD͒ (A-45) The additional pressure change caused by perforating, formation damage, and rock crushing around each segment is incorporated into the solution by assuming that the flow around each segment is normal to segment axis This yields qjBo ⌬ psj = 141.2 ͌kz ky Lsj sptVj, (A-46) where sptVj=the perforation total pseudoskin across the jth segment The dimensionless pressure change caused by local skin effect is k h sptVj = qjD sptHj (A-47) L ͌kz ky sj ⌬ psjD = qjD sptHj = k h sptVj (A-48) L ͌kz ky sj If the permeability anisotropy is accounted for in the calculation of perforation pseudoskin, then, in Eq A-48, k/(kzky)1/2ס1 should be set Adding the additional pressure-change expression into the solution in Eq A-32, we obtain nseg p˜jD͑s͒ = ͚A ͑s͒ q˜ ji iD͑s ͒ + q˜jD͑s͒sptHj (A-49) i=1 The solution up to this point is general and does not include any assumption related to inner-wellbore flow condition For infiniteconductivity and constant flow rate at the wellbore, pjD͑tD͒ = pwD͑tD͒ (A-50) k=0 l=−ϱ m=1 RxmiRxmj (A-35) m2 o ͚q iD͑tD ͒ = (A-51) i=1 Rewriting Eq A-49 for each completed segment, we obtain aji = −Aji (A-52) ajj = −Ajj − sptHj (A-53) ΄ 1 и 1 a11 a12 и a1ns a21 a22 и a2ns и и и и и ans1 ans2 и ansns ΅΄ ΅ ΄ ΅ p˜wD 1րs q˜1D q˜2D = и q˜nsD (A-54) The solution of the matrix in Eq A-54 gives the wellbore pressure and fractional-flow rates across each segment For constant pressure in an infinite-conductivity wellbore, the solution becomes pwD͑tD͒ = (A-55) mnkl ͑͌s  ͒ + K ͑͌s  ͒ (A-36) ⌿0nkl = Ko ͑͌ nseg i=1 p˜jD͑s͒ = ͒ 1 + K o 21 = ͑2l yeD͒2 + zkn (A-38) txeD = 2.63679 × 10−4kt ր ct xe2 (A-21) xD = x͌k ր kx ր L (A-22) m nseg qwD͑tD͒ = ͚q iD͑tD ͒ (A-56) i=1 September 2004 SPE Journal Combining Eqs A-49, A-55, and A-56, the solution for constantwellbore pressure drop is found: ˜ ͑s͒ = q˜ ͑s͒ ր s (A-63) N ptD wD aji = Aji (A-57) ajj = Ajj + sptHj (A-58) ΄ מ1 מ1 и מ1 a11 a12 и a1ns a21 a22 и a2ns и и и и и ans1 ans2 и ansns ΅΄ ΅ ΄ ΅ q˜ wD q˜ 1D 1րs q˜ 2D и q˜ nsD = 1րs (A-59) 1րs 1րs The solution of the matrix described in Eq A-59 results in wellbore flow rate and rate distribution along the wellbore Once the wellbore and segment flow rates are known, the cumulative production from the wellbore and each segment could be predicted as follows: NpiD͑tD͒ = ͐ tD qiD͑͒d (A-60) N˜ piD͑s͒ = q˜ iD͑s͒ ր s (A-61) NptD͑tD͒ = ͐ tD qwD͑͒d (A-62) September 2004 SPE Journal SI Metric Conversion Factors bbl × 1.589 873 E–01 סm3 cp × 1.0* E–03 סPaиs ft × 3.048* E–01 סm E–02 סm3 ft3 × 2.831 685 in × 2.54* E+00 סcm lbf × 4.448 222 E+00 סN lbm × 4.535 924 E–01 סkg md × 9.869 233 E–04 סmm2 *Conversion factor is exact Turhan Yildiz is an associate professor in the Petroleum Engineering Dept at the Colorado School of Mines e-mail: tyildiz@mines.edu Previously, he worked for the U of Tulsa, Simulation Sciences, Istanbul Technical U., and Louisiana State U Currently he is involved in modeling of complex reservoir flow problems and in intelligent/multilateral well design Yildiz holds a BS degree from Istanbul Technical U and MS and PhD degrees from Louisiana State U., all in petroleum engineering He serves on the SPE Editorial Committee 279