A New Skin-Factor Model for Perforated Horizontal Wells K. Furui,* D. Zhu,** and A.D. Hill,** University of Texas at Austin
Summary Using Usin g a com combina binatio tion n of ana analyti lytical cal cal calcula culation tionss and 3D fini finiteteelement eleme nt simula simulation, tion, we have developed developed a compr comprehen ehensive sive skinfactor fac tor mod model el for per perfor forate ated d hor horizo izonta ntall wel wells. ls. In thi thiss pap paper, er, we present prese nt the mathe mathematica maticall model development development and validation by comparison with finite-element simulation results. With the new perforation skin model, we then show how to optimize horizontal well perforating perforating to maxim maximize ize well productivity. productivity. A cased, perforated well may have lower productivity (as characterized by a positive skin factor) relative to the equivalent openhole completion because of two factors: the convergence of the flow flo w to the per perfor forati ations ons,, and the blo blocka ckage ge of the flow by the wellbo wel lbore re its itself. elf. Bec Becaus ausee of the ori orient entati ation on of a hor horizo izonta ntall wel welll relative to the anisotropic permeability field, perforation skin models for vertic vertical al wells that consider these effects, notably the Karakas and Tariq model (1991), are not directly applicable to perforated horizontal completions. Using appropriate variable transformations, we derived a skin-factor model for a horizontal perforated comple com pletio tion n tha thatt is ana analog logous ous to the Kar Karaka akass and Tar Tariq iq (19 (1991) 91) vertical-well model. The empirical parameters in the model were determined from an extensive 3D finite-element simulation study. The results of the new model show that the azimuth of a perforation (the angle between the perforation tunnel and the maximum permeability direction, usually thought to be in the horizontal direction) direc tion) affects the performance performance of perfo perforated rated completions completions in anisotropic reservoirs. When perforations are normal to the maximum-permeability direction, perforations will enhance horizontalwell flow compared with an openhole completion (a negative skin factor). But if perforations are in the same direction as the maximum permeability, significant positive skin will result. The new skin-factor model provides a clear guide to the shot density, perforation forati on orientation, orientation, and level of perforation perforation damag damagee that is toler toler-able to create high-productivity perforated completions in horizontal wells. Introduction A skin factor can be used to mathematically account for any deviations of the flow and pressure field in the near-well vicinity from perfectly perfectly radial flow to a wellbo wellbore re of radiu radiuss r w. A perforated completion obviously has a flow and pressure field near the perforations that is not perfectly radial. As shown by Karakas and Tariq (1991), the altered flow characteristics near perforations can be conveniently divided into three parts: the flow in a plane perpendicular to the wellbore, the blockage of flow to the perforations by the wellbore itself, and the fully 3D flow resulting from the asymmetric distribution of perforations along the wellbore. These effects effec ts on the near-well flow field and the corresponding corresponding perforation skin factor components are illustrated in Fig. 1. Perforation skin models for vertical wells (Karakas and Tariq 1991; Harris 1966; Locke 1981; Klotz et al. 1974) have already been presented in many papers. However, they are not directly applicable to a horizontal well because the reservoir anisotropy in a horizo horizontal ntal well creat creates es compl complex ex plane-flow geometry geometry norma normall to the well, which alters the flow efficiency of a perforated comple-
* Now with ConocoPhillips ** Now with Texas A&M University Copyright © 2008 Society of Petroleum Engineers Original SPE manuscript received for review 05 October 2004. Revised manuscript received for review 17 January 2008. Paper SPE 77363 peer approved 21 January 2008.
September 2008 SPE Drilling & Completion
tion. In this work, we present a new skin-factor model developed for a cased, perforated horizontal well. From our observations, the 2D plane flow skin, s 2 D , the wellbore blockage skin, s wb , and the 3D convergence skin, s 3 D , greatly depend on the magnitude of the permeability anisotropy and the perforation angle measured from a horizontal plane. Our model is based on the conventional perforation skin model for a vertical well presented by Karakas and Tariq (1991). Our perforation skin model is a semianalytical solution that is correlated with numerical simulation results. The reliability of any empirical correlation for perforation skin factors will depend on the acc accurac uracy y of the num numeric erical al simu simulati lations ons.. The fini finite-e te-eleme lement nt method (FEM), which is suitable for complex flow geometry problems, has been widely applied by many authors (Karakas and Tariq 1991; Klotz et al. 1974). In this study, we used the FEM to numerically model the performance of perforated horizontal wells. Our model uses an automatic and adaptive mesh generation program, GID (CIMNE 2006), to generate the finite-element grid. One of the great advantages of introducing a skin model for a perforated well is that it can be easily incorporated into any existing model of reservoir inflow performance or into a reservoir simulator. simula tor. The modif modified ied perfo perforation ration skin model developed developed here gives optimized perforation conditions and helps us to understand complex flow geometry in a horizontal perforated well. Using an accurate finite-element simulator, we also show a verification of the model model..
Problem Description Fig. 2 shows the key parameters of a perforated completion presumed in this study. The horizontal well is assumed to be on the x-axis x-a xis.. In for format mation ionss with no sig signif nifica icant nt for format mation ion dam damage age or perforation damage, the perforation skin will be a function of: • The number of perforations perforations per plane, plane, m [analogous to angular phasing, , in Karakas and Tariq’s (1991) model]. • Perfo Perforation ration length, length, l p. • Perfo Perforation ration radius, radius, r p. • Perfo Perforation ration shot shot density, n s (or spacing between perforations along a well, h). • Wellbo Wellbore re radius, radius, r w. • The principal principal permeabilities, permeabilities, k x ,k y ,k z. • Perfo Perforation ration orientation orientation,, , the angle between the perforation direction and the maximum permeability direction. The most important and significant difference between perforation skin models for vertical and horizontal horizontal wells is the effect of perforation perfora tion orienta orientation, tion, . For a hor horizon izontal tal wel welll com comple pleted ted in anisotropic reservoirs, the perforation skin factor is strongly influenced by . In this study, we considered four types of commonly used perforation phasing as shown in Fig. 3. The mathematical solution of the problem requires that a number of ass assump umptio tions ns be mad madee con concer cernin ning g the por porous ous media, the perforation perfo rations, s, and the fluid. • There is steady, viscous viscous flow of a single-phase, single-phase, incompressincompressible fluid. • The effect of gravity gravity is negligible. negligible. • The well is horizontal. horizontal. • Fluid is produced only through the casing perforation perforations. s. • The reservoir anisotropy anisotropy is uniform through through the entire reservoir. reservoir. • At some radius beyond beyond the wellbore, wellbore, the effect of the perforations is not felt. Beyond this radius, flow is radial (or elliptical in ani anisot sotrop ropic ic res reserv ervoir oirs) s) and can be des descri cribed bed by the nor normal mal logarithmic distribution. Inside this radius, streamlines are deviating from radial flow as the flow converges to individual perfora205
where J p is a productivity estimated by an FEM simulator under a particular perforation condition, and J o is the ideal openhole completion with the same fluid and formation properties as the simulation. Some radius where the effect of the perforation is not felt is r b. The simulations are executed for different perforation conditions and generate the perforation skin database to find appropriate correlations. A New Skin-Factor Model for Perforated Horizontal Wells
Following Karakas and Tariq ’s approach (1991), we divided the perforation skin into three components: the 2D plane flow skin, s2 D; the wellbore blockage skin, swb; and the 3D convergence skin factor, s3 D. The total perforation skin factor is then given by s p = s 2 D + s wb + s 3 D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2)
Fig. 1—Perforation geometry modeling by GID.
tions. This deviation is accounted for by the skin-factor model developed here. • The pressure drops inside the perforations are assumed to be negligible. The perforation skin factor, s p , indicates the relative efficiency of a perforated well, compared with an ideal openhole completion. FEM was used to solve the diffusivity equation in Cartesian coordinates for a specific set of boundary conditions. From the simulation results, we calculate the perforation skin factors by comparing the productivity determined by the simulation with the ideal openhole productivity, according to the following: s p =
J o J p
− 1 ln r b r w, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
2D Plane Flow Skin. A skin factor that accounts for flow in the y-z plane without the existence of the wellbore is s 2 D. The plane flow is considered as infinite perforation-shot densities, and under most practical perforation lengths and well radii ( l p>3 in. and r w<10 in.), any convergence effect in the direction along the wellbore (x-direction in this work) can be neglected. With this consideration, the 2D plane flow behavior into perforations is quite similar to that of an infinite-conductivity fractured well (Fig. 4). This skin factor can be negative or positive, depending on the perforation conditions and the reservoir anisotropy. The 2D plane flow skin equation is derived in Appendix A. For m 1 and 2,
s2 D = a m ln
+ ln
and for m
4
l pD
+ 1 − a m ln
1 1 + l pD
k y k z + 1 2cos2 + k y k z sin2 0.5
, . . . . . . . . . . . . . . . . . . ( 3)
3 and 4,
Fig. 2—Perforation geometry. 206
September 2008 SPE Drilling & Completion
Fig. 3—Perforation phasing.
s2 D = am ln
4
l pD
+ 1 − am ln
1 1 + l pD
for m
1,
, . . . . . . . . . . . . . . . . (4)
l pD,eff = l pD
where l pD = l p r w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)
The numerical values of am given in Table 1 were generated by FEM simulation results. In our simulations, the wellbore was replaced by a permeable formation, the permeability of which is the same as the reservoir, to account for the true locations of the perforations in the 2D plane. On the other hand, in Karakas and Tariq s work (1991), they set a very small wellbore radius to neglect the effect of wellbore. Therefore, the skin values estimated by the two models are somewhat different. ’
Wellbore Blockage Skin. Also estimated for the 2D plane flow geometry is swb. Because of the distortion of the flow into the perforation (Fig. 5) by the presence of the wellbore, the perforation skin simulated with the wellbore included, s FEM, will be comparatively greater than that given by Eq. 3 for m 1 and 2 or Eq. 4 for m 3 and 4. The difference of skin factor between the two is s wb; that is,
and for m
k y k z sin
2
k y k z cos
2
+ cos
2
+ sin
2
0.675
, . . . . . . . . . . . . . . . . . ( 8)
2,
l pD,eff = l pD
And for m
k y k z cos
2
0.625
1
+ sin
2
, . . . . . . . . . . . . . . . . . ( 9)
3 and 4,
l pD,eff = l pD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)
The numerical values of bm and cm given by Table 2 are generated by FEM simulations for 0.1< l pD. Because the definition of s2 D assumed by the Karakas and Tariq model (1991) and our model are different, the wellbore blockage skin factors given by both models are also different. However, the sum of the 2D plane flow skin and the wellbore blockage skin represents the same flow geometry. As a result, both models give almost the same numerical values.
The wellbore blockage skin will be positive for any perforation condition. The sum of s2 D and s wb is interpreted as a limit of the perforation skin factor for infinite perforation shot density. The wellbore-blockage skin correlation equation is derived on the basis of FEM simulation results (Appendix B).
3D Convergence Skin. For low perforation-shot densities, the flow geometry around a perforation becomes extremely complicated. Therefore, a 3D-FEM analysis is required. According to Karakas and Tariq s work (1991), the 3D wellbore blockage effect may be approximated by s wb , defined by the 2D analysis for practical perforation conditions. A skin factor estimated by the 3DFEM simulation, s FEM, is used to estimate an additional flow convergence effect into the perforations in the x-direction, denoted by s3 D , and s 3 D can be estimated by
swb = b m ln cm l pD,eff + exp − cm l pD,eff , . . . . . . . . . . . . . . . . . ( 7)
s3 D = s FEM − s 2 D − swb, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)
swb = s FEM − s 2 D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)
’
Fig. 4—2D plane flow approximation. September 2008 SPE Drilling & Completion
207
Examples Using the new skin correlation equations, Fig. 6 was generated for a particular perforation condition: l p1.0 ft, r w0.328 ft, r p0.0208 ft, k x k y2 measured depth (md), k z0.5 md, ns4 shots per foot (spf). The permeability anisotropy creates and amplifies the effect of perforation orientation, . Higher I ani makes perforation skin lower at 90 and higher at 0 . Perforation skin factors for 360 - and 180 -perforation phasing are greatly improved at 90 . On the other hand, skin factors for 120 - and 90 -perforation phasing do not change with respect to the perforation orientation. This is because of the particular perforation geometry of multidirectional perforations. The reservoir anisotropy makes some of the perforations lengthen and the others shorten in an equivalent isotropic system. Therefore, the overall effects cancel out. Fig. 7 was created with the same perforation conditions except ns0.5. The difference of the perforation skin factor between m2 and m 4 becomes 1.459 at 90 (0.750 in Fig. 8). Low perforation shot density will make the difference significant. Even with unidirectional perforation ( m1), the productivity will be better than multidirectional perforations ( m>3). As long as it is perforating from 75 to 90 , the perforation skin factors do not change significantly. Hence, failure of perforating in the vertical direction (the direction of minimum permeability) may not be a serious problem within the range of ±15 . One of the interesting topics of perforated horizontal wells is the selection of perforation phasing, m, to obtain the highest productivity. As we discussed, perforating in the vertical direction (at 90 ) provides the minimum perforation skin factors in anisotropic reservoirs. Fig. 8 shows the relationship between the reservoir anisotropy and the perforation skin factor perforating at 90 . As shown in Fig. 8, the best perforation phasing will strongly depend on the reservoir anisotropy. For slightly anisotropic reservoirs ( I ani≈1), multidirectional perforations ( m3 and 4) will provide higher perforation productivity than the other two. For anisotropic reservoirs ( I ani>1), 180 perforation phasing ( m2) will be the best phasing method as long as perforating in the vertical direction (the direction of minimum permeability). For highly anisotropic reservoirs, 360 perforation phasing ( m1) will also be a good perforating technique compared with multidirectional perforations (m>3). °
°
°
°
°
°
°
This is also positive for any perforation condition. The 3D convergence skin correlation equation taking into account the reservoir anisotropy and the perforation orientation is derived in Appendix C. s3 D = 10 1h D2−1r pD2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 12)
with 1 = d m logr pD + e m, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 13) 2 = f mr pD + gm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 14)
For m1 and 2, h D =
r pD =
2 2 l pk x k z sin + k x k y cos
2h
cos −
k x k y
sin2 +
, . . . . . . . . . . . . . . . ( 15)
k x k z
cos2 + 1 ,
. . . . . . . . . . . . . . . . . . . . . . . . . . ( 16) where = arctank y k z tan , . . . . . . . . . . . . . . . . . . . . . . . . . . ( 17) = arctank z k y tan . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 18)
r pD =
h l p r p
2h
°
°
°
°
°
For m3 and 4, h D =
°
°
h
r p
°
k yk z
0.5
k x
k x
k y k z
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 19) 0.5
+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . ( 20)
The numerical values of d m , em , f m , and g m given by Table 3 were presented in Karakas and Tariq s paper (1991). These equations can be used to estimate the perforation skin factor for most practical ranges of system parameters ( h D10 and r pD0.01). ’
Relationship to Overall Skin Factor of a Horizontal Well The perforation skin factor presented here accounts for the effects of localized flow convergence on well performance. The manner in which this skin factor is incorporated into an overall skin factor for a horizontal well depends on the way in which the overall well performance is modeled. For partially completed horizontal wells (i.e., selectively perforated wells that include blank pipe intervals),
Fig. 5—Concept of wellbore blockage effect. 208
September 2008 SPE Drilling & Completion
4. The 180 perforation phasing (m 2) will be the best completion technique for horizontal perforated wells because all the perforations can be oriented in the direction of minimum permeability. As a result, multidirectional perforation techniques (m>3) applied for anisotropic reservoirs may not be efficient, unlike for isotropic reservoirs. °
the definition of skin factor will be different, depending on the inflow model used in the calculation. For an analytical inflow model or a nonsegmented wellbore model (with the use of a reservoir simulation program), these models assume uniform inflow along the entire lateral, the skin assigned to the inflow equation must include partial completion effects. For this particular case, the perforation skin factor should be multiplied by the ratio of reservoir length in the well direction ( L) to completed length, Lw , and added to the partial completion skin factor, s c (Pucknell and Clifford 1991): s =
L
Lw
s p + s c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 21)
When a multisegmented inflow modeling approach is used, because the horizontal wellbore is segmented so one can assign flow and no-flow intervals separately, the reservoir flow convergence effects to the open intervals will be taken into account by the reservoir grid system (or method of superposition). Thus the original s p (without the multiplier) should be used.
Nomenclature
am aw bm bw cm d m em f m gm h h D I ani
Conclusions
1. Our perforation skin equations for, s2 D , swb , and s3 D provide useful insight into the role played by the number of perforations per plane, m, the perforation length, l p , perforation radius, r p , perforation shot density, ns, the wellbore radius, r w , and perforation orientation, , on the productivity of perforated horizontal completions. 2. The major difference between perforation performances in a horizontal well compared with a vertical well is the influence of permeability anisotropy in the horizontal-well case. 3. The horizontal-well perforation skin model developed here shows that perforations should be oriented parallel to the direction of minimum permeability to give the minimum perforation skin factor (the maximum perforation productivity). For most horizontal wells, this means that perforations should be vertical, extending from the upper or lower sides of the wellbore. With this advantage, the reservoir anisotropy will make the perforation skin factor decrease and result in favorable production.
Fig. 6—Effect of perforation orientation on perforation skin factor (n =4). s
September 2008 SPE Drilling & Completion
J o J p k l p l pD L Lw m ns q r b r p r pD r w sc sFEM s p
correlation constant vertical semiaxis of an ellipse, ft [m] correlation constant horizontal semiaxis of an ellipse, ft [m] correlation constant correlation constant correlation constant correlation constant correlation constant perforation spacing, ft [m] dimensionless perforation spacing, dimensionless k y index of anisotropy, k z ideal openhole completion productivity permeability, md [m2] perforation length, ft [m] dimensionless perforation length, dimensionless well direction completed length the number of perforations per plane perforation shot density, spf [shots/m] flow rate, STB/D, [m3 /s] radius where the effect of perforation is not felt, ft [m] perforation radius, ft [m] dimensionless perforation radius, dimensionless wellbore radius, ft [m] partial completion skin factor skin estimated by an FEM simulator, dimensionless perforation skin, dimensionless
Fig. 7—Effect of perforation orientation on perforation skin factor (n =0.5). s
209
Fig. 8—Effect of reservoir anisotropy at
=90°.
wellbore blockage skin, dimensionless 2D plane flow skin, dimensionless 3D convergence skin, dimensionless perforation orientation, degree azimuth angle measured from the y ’ axis in the equivalent isotropic system, degree azimuth measured from the z ’ axis to the vertical plane containing the transform of the original circular perforation cross section in the equivalent isotropic system, degree 1 correlation constant 2 correlation constant perforation phasing angle, degree
swb s2 D s3 D
Subscripts eff effective eq equivalent o openhole p perforated x x-coordinate y y-coordinate z z- coordinate
Harris, M.H. 1966. The Effect of Perforating on Well Productivity. JPT 18 (4) 518–528; Trans., AIME, 237. SPE-1236-PA. DOI: 10.2118/1236PA. Karakas, M. and Tariq, S.M. 1991. Semianalytical Productivity Models for Perforated Completions. SPEPE 6 (1): 73 –82. SPE-18247-PA. DOI: 10.2118/18247-PA. Klotz, J.A., Krueger, R.F., and Pye, D.S. 1974. Effect of Perforation Damage on Well Productivity. JPT 26 (11): 1303–1314; Trans., AIME, 257. SPE-4654-PA. DOI: 10.2118/4654-PA. Kucuk, F. and Brigham, W.E. 1979. Transient Flow in Elliptical Systems. SPEJ 19 (6): 401 –410; Trans., AIME, 267. SPE-7488-PA. DOI: 10.2118/7488-PA. Locke, S. 1981. An Advanced Method for Predicting the Productivity Ratio of a Perforated Well. JPT 33 (12): 2481–2488. SPE-8804-PA. DOI: 10.2118/8804-PA. Prats, M. 1961. Effect of Vertical Fractures on Reservoir Behavior— Incompressible Fluid Case. SPEJ 1 (2): 105-17; Trans., AIME, 222. SPE-1575-G. DOI: 10.2118/1575-G. Pucknell, J.K. and Clifford, P.J. 1991. Calculation of Total Skin Factors. Paper SPE 23100 presented at Offshore Europe, Aberdeen, 3–6 September. DOI: 10.2118/23100-MS. Spivey, J.P. and Lee, W.J. 1999. Estimating the Pressure-Transient Response for a Horizontal or a Hydraulically Fractured Well at an Arbitrary Orientation in an Anisotropic Reservoir. SPEREE 2 (5): 462 –469. SPE-58119-PA. DOI: 10.2118/58119-PA.
Appendix A—Development of 2D Plane Flow Skin Equation in an Anisotropic Medium The 2D plane flow behavior into perforations is quite similar to that of an infinite-conductivity fractured well. The equipressure lines are close to confocal ellipses, with some distortion from the presence of the wellbore. The flux distribution at the perforation, as for a fracture, is significantly higher at the perforation tip and root. Under plane flow conditions, and assuming negligible wellbore effects, we can introduce the “effective well radius ” concept developed by Prats (1961) for vertically fractured wells. According to his work, the effective well radius is given by l p /4 for a single fracture without a pressure drop inside the fracture (Fig. A-1). Because of the geometric similarity between a unidirectional perforation (m1) and a fracture, the perforation skin factor equation can be analytically derived by s2 D = ln
Acknowledgments The authors thank the sponsors of the Improved Well Performance Research Program of the Center for Petroleum & Geosystems Engineering at the University of Texas at Austin for providing the financial support for this study. References CIMNE. 2006. GiD 8—Reference Manual. Barcelona, Spain: International Center for Numerical Methods in Engineering. http://www.gidhome. com/support_team/su04.html.
4r w l p
= ln
4
l pD
,
for m = 1, . . . . . . . . . . . . . . . . ( A-1)
where l pD = l p r w. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( A-2)
The dimensionless perforation length is l pD. As m approaches infinity as shown in Fig. A-1, the effective well radius approaches to r w+l p. The skin factor is also analytically given by s2 D = ln
r w r w + l p
= ln
1 1 + l pD
,
for m =
. . . . . . . . . . . ( A-3)
Fig. A-1—Effective well radius. 210
September 2008 SPE Drilling & Completion
Fig. A-2—2D plane flow skin factor in an isotropic medium.
For other values of m such as m 2, 3, and 4, the skin factors should range between those given by Eqs. A-1 and A-3. Fig. A-2 shows FEM simulation results for different perforation phasing (m 1, 2, 3, and 4). The following interpolation can approximately give s2 D;
s2 D = am ln
+ 4
l pD
1 − am ln
+ 1
1 l pD
, . . . . . . . . . . . . . . (A-4)
where a m is a constant given in Table 1 for different values of m. In this study, the numerical values for am were obtained through FEM simulations for all the phasing except for m 1 and . As shown in Fig. A-2, Eq. A-4 matches the FEM results almost exactly. The performance of perforated completions in anisotropic reservoirs is greatly controlled by the azimuth of a perforation (the angle between the perforation tunnel and the maximum permeability direction, usually thought to be the horizontal direction). Fig. A-3 shows the impact of perforation orientation, , on s2 D. For a particular condition ( l pD 1.0 and √k y / k z 5), the perforation skin factor was reduced from 2.48 to 0.88 for m 1 and from 1.17
to –0.56 for m 2 by changing the orientation from horizontal ( 0°) to vertical ( 90°). The contrast of skin factor will increase for a higher anisotropic ratio. For m 1, we can analytically calculate skin factor by using a coordinate transformation into the equivalent isotropic space. The effective perforation length and the equivalent wellbore radius are given by
k z k y cos2
, . . . . . . . . . (A-5)
r w,eq = r wk y k z + k z k y 2. . . . . . . . . . . . . . . . . . . . (A-6) 4
4
Substituting the previous equations into Eq. A-1 gives s2 D = ln
4
l pD
+ ln
k y k z + 1 2
2cos
+ k y k z sin2 0.5
. . . . . . (A-7)
The second term represents the effect of the reservoir anisotropy and perforation orientation. For a fixed perforation orientation (constant ), the reservoir anisotropy makes the skin equation simply move up or down depending on the orientation. As
Fig. A-3—Effect of perforation orientation, September 2008 SPE Drilling & Completion
0.5
l p,eff = l pk y k z sin 2 +
,
on s 2D . 211
Fig. A-5 —Comparison with FEM simulation database for Fig. A-4—Comparison with FEM simulation database for
=2.
m
approaches 0 (the direction of the maximum permeability), the perforation skin increases. On the other hand, as approaches 90 (the direction of the minimum permeability), the perforation skin decreases. As with to m 1, the 2D plane flow skin equation for m 2 in an anisotropic medium is approximately given by °
°
s2 D = a2 ln
+ ln
+ 4
l pD
1 − a2 ln
2cos
1
1 l pD
+ k y k z sin2 0.5
Appendix B Development of Wellbore Blockage Skin Equation in an Anisotropic Medium —
The wellbore is a complete barrier to the flow into perforations. This wellbore blockage effect can be quite significant, especially in the case of m 1. As a result of the additional pressure drop, well productivity would be less than that estimated only with s 2 D. Thus, this effect can be quantified in terms of a wellbore pseudoskin, swb , which is always positive:
s p = s 2 D + s wb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)
+
k y k z + 1 2
=3.
m
. . . . . . . . . . . . . . . . (A-8)
As shown in Fig. A-4, Eq. A-8 shows good agreements with the FEM simulation results. For multidirectional perforation cases, especially m>3, the estimation of the effective well radius is not simple because the coordinate transformation into an equivalent isotropic system gives different effective perforation lengths for each direction. According to the FEM simulation results (Fig. A-3), the influence of the orientation for m 3 and 4 is not as significant as those for m 1 and 2. Therefore, we can conclude that the effect of the reservoir anisotropy and perforation orientation for m>3 are negligible, and Eq. A-4 is directly applied even for an anisotropic medium. The comparisons of Eq. A-4 with FEM simulation results are shown in Figs. A-5 and A-6.
Similarly to s2 D , the wellbore blockage effect will depend on the dimensionless perforation length, l pD ; the perforation orientation, ; and the anisotropy ratio, √k y / k z . For isotropic reservoirs, swb is assumed to depend only on the dimensionless perforation length. Fig. B-1 illustrates this dependency for the case of m 1, 2, 3, and 4. As shown in Fig. B-1, the wellbore skin for a given dimensionless perforation length is significantly larger for m 1 than for other cases ( m 2, 3, and 4). Compared with the FEM simulation results, the wellbore blockage skin, swb , can be closely approximated by
swb = bm lncm l pD + exp −cm l pD, . . . . . . . . . . . . . . . . . . . (B-2)
where b m and cm are constants given in Table 2. For anisotropic reservoirs (Fig. B-2), swb depends also on the perforation orientation and on the anisotropy ratio. To derive a wellbore blockage skin equation, taking into account the effect of perforation orientation, a coordinate transformation into the equivalent isotropic space is applied. The effective perforation length is now given by Eq. A-5. As shown in Fig. 5, the wellbore itself is characterized as a complete barrier against the flow into the perforation. We concluded that the effective wellbore radius (the barrier height) would be a key parameter, which can be transformed as 0.5
r w,eff = r wk y k z sin 2 + 2 + k z k y cos 2 + 2 0.5
= r wk z k y sin 2 + k y k z cos2 . . . . . . . . . . (B-3) From Eqs. A-5 and B-3, the ratio of the effective perforation length to the effective well radius can be calculated by l p,eff r w,eff
= l pD , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-4)
where
Fig. A-6—Comparison with FEM simulation database for 212
=4.
m
=
+ cos2 . . . . . . . . . . . . . . . . . . . . . . . (B-5) 2 2 k y k z cos + sin k y k z sin
2
September 2008 SPE Drilling & Completion
Fig. B-1—Wellbore blockage effect for an anisotropic reservoir.
The symbol represents the effect of the reservoir anisotropy and perforation orientation. Compared with the simulation results, a good agreement (Fig. B-3) was obtained by setting the effective dimensionless perforation length as 1.35
l pD,eff = l pD
for m = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . (B-6)
The wellbore blockage skin is calculated by inserting Eq. B-6 into Eq. B-2 swb = b m ln cm l pD,eff + exp −cm l pD,eff . . . . . . . . . . . . . . . . (B-7)
Fig. B-4 shows FEM simulation results for m is empirically set as
2. The symbol, ,
=
0.5
k y k z cos2 + k z k y sin2
and the effective dimensionless perforation length is empirically estimated by 1.25 l pD,eff = l pD
for m = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . ( B-9)
As shown in Fig. B-2, the effect of perforation orientation for m 3 and m 4 are comparatively small; that is, the effect of anisotropy is not significant. Therefore, we can conclude that the wellbore skin equation for m>3 is independent of the anisotropy ratio and (i.e., Eq. B-2 can be directly used).
Fig. B-2—Effect of perforation orientation on September 2008 SPE Drilling & Completion
, . . . . . . . . . . . . . . ( B-8)
s 2D .
213
Fig. B-3—Wellbore blockage effect in an anisotropic reservoir (m =1).
Appendix C Development of 3D Convergence Flow Skin Equation in an Anisotropic Medium According to Karakas and Tariq s (1991) correlation equation, an additional 3D convergence flow skin factor is mainly estimated by dimensionless perforation spacing and dimensionless perforation radius. —
’
h D = r pD =
heff l p,eff r p,eff heff
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-2)
Appropriate coordinate transformations give the effective perforation spacing, perforation length, and perforation radius.
k
heff = h
k x
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-3)
k k y cos2
l p,eff = l p
+ k k z sin
2
, . . . . . . . . . . . . . . . ( C-4)
where k =
3 k x k y k z.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-5)
An additional consideration is needed to estimate the effective perforation radius (Spivey and Lee 1999). Fig. C-1 shows a perforation in an anisotropic reservoir. The circular perforation is assumed to lie at an arbitrary azimuth in the maximum and minimum principle axes of permeability, respectively. The axis of the perforation lies at an angle, , to the y-axis. In order to specify the problem completely in the equivalent isotropic system, we must also transform the perforation cylinder. The perforation in the anisotropic system is considered as a right circular cylinder. In the equivalent isotropic system, the cylinder becomes an elliptical cylinder, with the base no longer perpendicular to the axis. We approximate this ellipse by a right ellipse that has the same perpendicular cross section, but the length of which is given by Eq. C-4. The vertical semiaxis of the elliptical cross section, a , is given by w
Fig. B-4—Wellbore blockage effect in an anisotropic reservoir (m =2). 214
September 2008 SPE Drilling & Completion
Fig. C-1—Details of the perforation in the equivalent isotropic reservoir.
aw = r pk k x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-6)
If we transform the original circular cross section, the horizontal semiaxis becomes
k k y sin2 + k k z cos2 .
w = r p
= arctank y k z tan . . . . . . . . . . . . . . . . . . . . . . . . . . (C-8)
The transformation of the original circular cross section lies at an angle to the z-axis, where is given by = arctank z k y tan . . . . . . . . . . . . . . . . . . . . . . . . . . (C-9)
The desired cross section of the transformed perforation lies at an angle to the z -axis. Thus, we must project the distance given in Eq. C-7 through an angle -. Finally, we obtain the horizontal semiaxis of the elliptical cross section of the transformed perforation from
k k y sin2 + k k z cos2 cos − .
bw = r p
. . . . . . . . . . . . . . . . . . . . . . . ( C-10) Kucuk and Brigham (1979) showed that the equivalent wellbore radius for an elliptical wellbore having semiaxis a and b is given by the arithmetic mean. Similarly, the perforation radius is given by w
r p
2
k k y
sin2 +
k k z
cos2 cos − +
w
k
k x
.
. . . . . . . . . . . . . . . . . . . . . . . ( C-11) Then, Eq. C-1 and C-2 for anisotropic media are given by h D =
r pD =
h
k x k z sin2 + k x k y cos2
, . . . . . . . . . . . . ( C-12)
l p
r p
2h
k x k y
sin2 +
k x k z
cos2 cos − + 1 . . . . . . . . . . . . . . . . . . . . . . . . ( C-13)
From Karakas and Tariq ’s (1991) correlation equation, the vertical pseudoskin is given by
2−1
s3 D = 10 1h D
r pD 2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-14)
with 1 = d m log r pD + em, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-15) 2 = f mr pD + gm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-16) September 2008 SPE Drilling & Completion
m
. . . . . . . . . . . . . . . . (C-7)
However, this is not the horizontal semiaxis of the transformed elliptical cylinder because, after the transformation, the original circular cross section is no longer perpendicular to the axis of the cylinder. Thus, we must find the projection of the distance given in Eq. C-7 along a direction perpendicular to the axis of the transformed perforation. The axis of the transformed perforation lying at an angle to the y -axis is given by
r p,eff =
The numerical values of d , e , f , and g are obtained from Table 3 developed by Karakas and Tariq (1991). As we discussed in 2D perforation skin analysis, the effect of perforation orientation and the reservoir anisotropy for m>3 is not significant and can be neglected. Therefore, Eqs. C-12 and C-13 can be simplified to h D = r pD =
h l p r p
m
m
m
2h
k y k z
0.5
k x
k x
k y k z
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( C-17) 0.5
+ 1 . . . . . . . . . . . . . . . . . . . . . . . ( C-18)
SI Metric Conversion Factors
in. ft
× ×
2.54* 3.048*
E+00 E−01
cm m
*Conversion factor is exact
Kenji Furui is a senior completions engineer at ConocoPhillips’ c o m p le t i o ns t e c h no l o g y g r o u p i n H o u st o n . e m a i l:
[email protected]. He is an expert in the areas of rock mechanics (sand production), well performance analysis, and intelligent well completions. Furui holds a BS degree in mineral resources and environmental engineering from Waseda University in Japan and MS and PhD degrees in petroleum engineering from the University of Texas at Austin. Ding Zhu is assistant professor of petroleum engineering and holds the W.D. Von Gonten Faculty Fellowship in Petroleum Engineering at Texas A&M University. She worked at the University of Texas as a research scientist for 11 years before joining Texas A&M University. She is an expert in the areas of production engineering, well stimulation (acidizing and fracturing), and complex well production (horizontal, multilateral, and intelligent wells) and has authored more than 70 technical papers. Zhu holds a BS degree in mechanical engineering from Beijing University of Science and Technology and MS and PhD degrees in petroleum engineering from the University of Texas at Austin. She served as SPE Austin Section Program Chair, Chairperson, and Scholarship Chair in 2002–2004, and has been a member and chairperson of numerous SPE committees. A.D. Hill is associate department head of petroleum engineering at Texas A&M University and holds the Robert L. Whiting Endowed Chair. Previously, he taught for 22 years at the University of Texas at Austin. He is an expert in the areas of production engineering, well completions, well stimulation, production logging, and complex well performance (horizontal and multilateral wells), and has presented lectures and courses and consulted on these topics throughout the world. He is the author of the SPE monograph Production Logging: Theoretical and Interpretive Elements, coauthor of Petroleum Production Systems, author of more than 110 technical papers, and he holds five patents. Hill also holds a BS degree from Texas A&M University and MS and PhD degrees from the University of Texas at Austin, all in chemical engineering. He currently serves on the SPE Editorial Review Board, has been an SPE Distinguished Lecturer, has served on and chaired numerous SPE committees, and was founding chairperson of the SPE Austin Section. He was named a Distinguished Member of SPE in 1999. 215
