SummaryThis study presents analytical models to predict the productivity of selectively perforated vertical wells. The models consider arbitrary phasing angle, nonuniform perforation size and length, and formation damage around perforations. The accuracy of the models was verified against the results from the experimental studies, the semianalytical correlation, and the numerical models all available in the literature. Unique applications of the models are presented. A comprehensive sensitivity study showing the impact of different perforation schemes on well productivity is documented.
Productivity of Selectively Perforated Vertical Wells Turhan Yildiz,* SPE, U of Tulsa Summary This study presents analytical models to predict the productivity of selectively perforated vertical wells The models consider arbitrary phasing angle, nonuniform perforation size and length, and formation damage around perforations The accuracy of the models was verified against the results from the experimental studies, the semianalytical correlation, and the numerical models all available in the literature Unique applications of the models are presented A comprehensive sensitivity study showing the impact of different perforation schemes on well productivity is documented Introduction Since its invention in the 1930s, perforating has been one of the most commonly used well completion techniques in oil and gas wells Wells are often cased to prevent sand production and wellbore collapse and to delay gas/water coning The main objective in perforating is to create flow tunnels across the casing for formation fluid entry In many cases, oil and gas wells are selectively perforated at multiple intervals The fluid flow into perforated completions is three-dimensional (3D) and has a convergent flow geometry The nature of this complex flow pattern makes the flow modeling considerably more difficult compared to that for an openhole completion The objectives of this study are: • To develop general productivity models accounting for 3D nature of flow into perforations and nonuniform perforation properties • To verify the models • To investigate the impact of different perforation strategies on well productivity Background The influences of the reservoir, well, and perforation parameters on the productivity of perforated wells have been the subject of many investigations A great number of publications have concentrated on the flow efficiency of perforated vertical wells Some of these studies are summarized below to acknowledge their previous contribution For convenience, the previous models on well performance are divided into three categories: electrolytic models, semi-analytical and empirical models, and numerical models Electrolytic Models The earliest models of flow into perforations relied upon the experimental results from electrolytic analog apparatus.1–3 McDowell and Muskat1 measured the effect of perforation length and diameter and phasing angle on well productivity They concluded that if the perforations are long enough, the productivity of a perforated well might be even higher than that of an open hole In an independent study, Howard and Watson2,3 conducted similar experiments and reported similar results Most recently, Pan and Tang4 conducted a comprehensive set of experiments on a scaled electrolytic apparatus and developed empirical equations for perforation flow efficiency Numerical Models Mathematical models based on finitedifference and finite-element methods have been proposed in *Now with Colorado School of Mines Copyright © 2002 Society of Petroleum Engineers This paper (SPE 78665) was revised for publication from paper SPE 64763, first presented at the 2000 SPE International Oil and Gas Conference and Exhibition, Beijing, 7–10 November Original manuscript received for review 20 January 2001 Revised manuscript received 15 January 2002 Manuscript peer approved February 2002 158 many studies.5–13 The studies relying on the numerical solution techniques reported significantly different productivities An important drawback for the numerical solutions is the extensive computational time and effort Harris5 investigated the productivity of perforated completions considering a wedge-shaped perforation by a finite difference model Hong6 worked with a similar model and reported the impact of the formation damage and perforation pattern on well productivity It was recommended that the perforation density should be at least 12 shots per foot (spf) to establish reasonable flow efficiency in damaged formations It was later noted that the geometrical irregularities involved in perforated completions couldn’t be easily built into the finite difference models Therefore, it became apparent that the finite difference models were not appropriate to simulate the flow into irregular perforations It was realized that the finite element methods would be more convenient to represent the irregular perforation geometry Klotz et al.7 used a 2D finite element model and investigated the impact of a crushed zone and formation damage around the perforations on the well productivity To come with more realistic perforation geometries and to account for spiral perforation distributions, Locke8 made use of a 3D finite element model He also presented a nomograph to predict the pseudoskin factor in perforated wells Tariq9 investigated the influence of nondarcy flow on the flow efficiency of perforated completions He remarked that Locke’s numerical model, and thus his nomograph, overestimated the perforated well productivity owing to insufficient grid size In a later study, Tariq et al.10 investigated flow into perforations under the influence of formation anisotropy, shale laminations, and natural fractures They concluded that perforated completion efficiency is strongly affected by near-wellbore heterogeneities, and high shot densities are required in anisotropic and laminated formations Recently, Tang et al.11 reported results from a similar finite element model Behie and Settari12 and Dogulu13 proposed the use of hybrid grids and local grid refinement to overcome the shortcomings of finite difference models Both studies reported that the results from their model not agree with those from Tariq’s work when nondarcy flow regime is considered Semi-Analytical and Empirical Models The semi-analytical models proposed by McLeod,14 Karakas and Tariq,15 and Bell et al.16 are extensively used because of the simplicity of the proposed equations However, the simple radial flow model proposed by McLeod14 only accounts for the flow across the crushed and damaged zones around perforation and assumes that all the perforations are equivalent By integrating the results of the finite element simulations with the analytical solutions for hydraulically fractured and horizontal wells, Karakas and Tariq15 constructed a semi-analytical model and presented empirical equations to compute the perforation pseudoskin To estimate the productivity index of a partially perforated well, Bell et al.16 combined a pseudoskin equation for a partially penetrating well with the semi-analytical equation of Karakas and Tariq.15 However, the proposed method is limited to only one perforated interval Ahmed et al.17 developed a 3D analytical solution and compared their model with electrolytic tank model of McDowell and Muskat.1 Their analytical model was in reasonable agreement with the electrolytic model However, the computation of the analytical model presented in Ref 17 is very cumbersome June 2002 SPE Journal 3D Semi-Analytical Model The fluid flow into perforations is 3D, and streamlines around perforations have a converging flow pattern in the near-wellbore region However, away from the wellbore, there exists a radial flow in the formation Because the perforations are far away from the external reservoir boundaries, it might be assumed that 3D flow convergence around perforations is not influenced by far field flow conditions at the reservoir boundaries Therefore, inflow performance of selectively perforated wells can be predicted using the productivity equation for vertical open holes, provided that the additional pressure loss/gain caused by perforation and mechanical skin because of formation damage are taken into account For steady-state flow conditions, the productivity of perforated wells can be written as Jp = 141.2Bo kh ( 1) ͓ln͑re ր rw͒ + st͔ For pseudosteady-state flow in noncylindrical reservoirs, the productivity index may be expressed as Jp = kh ( 2) 141.2Bo ͓ln͑2.2458A ր rw2 CA͒ ր + st͔ To quantify the impact of the perforating process, the productivity index of a perforated well is compared to that of an ideal undamaged openhole completion The productivity index of an ideal open hole can be determined using Eqs and with st=0 The comparison is usually made in terms of productivity ratio (PR) PR is defined as the productivity index of a perforated well (Jp) divided by that of an ideal open hole (Jv) PR = Jp ln͑re ր rw͒ ( 3) = Jv ln͑re ր rw͒ + st In Eqs through 3, stסthe perforation total skin composed of perforation pseudoskin and mechanical skin because of crushing around perforations and formation damage The challenge here is to predict the total skin accurately, given the perforation parameters and the extent and degree of rock crushing and formation damage The crux of the current study is the construction of a 3D semi-analytical model to calculate the total skin for wells perforated in multiple intervals Model for Perforation Total Skin Recently, a 3D analytical model to simulate the transient flow into a perforated vertical well was presented.22 A brief description of the model is given below All the details of the transient flow model can be found in Ref 22 Ref 22 shows that the transients caused by the convergent flow into perforations stabilize very quickly During the transient flow into a perforated well, once the radius of drainage expands beyond the near wellbore region where flow streamlines congregate into perforations, the perforations result in an additional stabilized pressure loss/gain around the wellbore While the transients caused by 3D convergent flow in the near wellbore are dampened, the tranJune 2002 SPE Journal sients away from the near wellbore region continue to spread out further into the reservoir Compared to an open hole, the stabilized additional pressure loss/gain caused by the perforations may be considered as a pseudoskin To confirm the stabilization of transients because of the perforations in a short time, we simulated the unsteady-state flow into a perforated well and compared it with that into an open hole with and without pseudoskin The results are displayed in Fig The relevant data for the perforated well were Lpס1 ft, spfס8, pס90o, rpס0.1 in., and rwס0.25 ft, which yield a perforation pseudoskin of−0.8626 The pressure response of the openhole well was simulated using the cylindrical source solution presented by van Everdingen and Hurst.23 Fig shows the pressure and derivative behavior of three cases considered The pressure-derivative response of the perforated well and the open hole becomes identical for tD>100, indicating the stabilization of transients because of the perforations A comparison of pressure behavior of the perforated well and the open hole with pseudoskin cases demonstrates that, beyond tD>100, the pressure drop in the perforated well is equal to that in the open hole with a pseudoskin value of−0.8626 Perforation pseudoskin model is based on the large-time approximation to the 3D transient flow model of Ref 22 The model assumes that the 3D diffusivity equation governs the convergent flow into a perforated vertical well The perforated well model is sketched in Figs and The well with np perforations produces at a constant flow rate The formation is sealed at the top and bottom The well communicates with the reservoir only through the perforations The perforations are represented as line sources The model does not consider the vertical section of the well The casing landed before perforating creates a cylindrical no-flow surface between wellbore and formation The true boundary condition at the wellbore should be defined with a hollow cylinder with an impermeable surface However, in the model, the space occupied by the hollow cylinder (wellbore) is treated as if it is a part of porous media In other words, in the model, the hollow space occupied by the wellbore is replaced by the porous material This assumption would yield optimistic productivity index values for the wells with zero-phased perforations The reservoir is homogeneous but anisotropic kx, ky, and kz are the principal permeabilities that are assumed to coincide with the directions of the Cartesian coordinate system A single-phase fluid of constant viscosity and compressibility is assumed The perforations deviate (in x-y plane) from the direction of the principal permeability kx (the x-direction) by an angle of pi Each perforation may have a different length, diameter, and phasing angle A skin factor around each perforation owing to formation damage and/or crushed zone is also considered The model is capable of considering a variable skin factor along each perforation It should be noted that, if needed, each perforation could be divided into multiple smaller segments in the model Such a need may arise, when perforation diameter and degree of rock crushing/formation damage around a given perforation vary along the perforation spf 1E+0 pD and dpD / d( ln tD ) Carnegie18 reported software that could be used to estimate productivity of perforated wells To calculate the perforation total skin, he implemented the semi-analytical model presented in Ref 15 Most recently, Brooks19 proposed simple empirical equations based on nondimensional analysis One of the limitations of the semi-analytical and empirical equations is that these models assume that all the perforation parameters, such as diameter, shot density, length, and degree of crushing/damage around perforations, are distributed uniformly However, the experiments conducted by Mason et al.20 on sandstone blocks have indicated that even the perforation parameters of two neighboring perforations could differ significantly Hofsaess and Kleinitz21 pointed out the importance of nonuniform perforation properties on injectivity decline The second major limitation of the semi-analytical equations is that these models cannot account for the effect of multiple intervals, which are selectively perforated Lp = ft rp = 0.1 in rw = 0.25 ft sp = Ð0.8626 1E-1 Openhole Openhole with sp = Ð0.8626 Perforated well 1E-2 1E-2 1E-1 1E+0 tD 1E+1 1E+2 1E+3 Fig 1—Comparison of transient flow into perforated and open completed wells 159 O sdcpj Lpj r cpj θpj N rw Fig 2—The perforated well model considered in the study.22 Ref 22 shows that, at late times, the transient pressure response at the jth perforation segment can be expressed as ppjD = kD ͑ln tvD + 0.80907͒ + spj ( 4) np ͚q spj = D + piD Aji ( 5) i= D = kD ln͑kD͒ ր ( 6) Aji = ͐ kD LpiD LpjD LojD+ LpjD LojD ͐ LoiD+ LpiD LoiD ⌿ji du drD ( 7) ϱ ͚K ͑ R ⍀ji͑z* jD ͒ ( 8) ⍀ji͑z*jD͒ = cos͑n zpiD͒cos͑n z* jD ͒ ( 9) ⌿ji = − ln͑RjiD͒ + o n jiD ͒ n= RjiD = ͌r D + u2 − 2rD u cos͑˜ pj − ˜ pi͒ ( 10) n = n ր hD ( 11) The dimensionless variables are defined as follows: ppjD = kh ͑p − ppj͒ ( 12) 141.2qtBo o − tvD = 2.63679 × 10 kt ր ct L ( 13) ͌k2 ր kxky ( 15) qpiD qpi ր qt ( 16) ͱ ͱ LojD = rw L LpjD = Lpj L k k cos2pj + sin2pj ( 17) kx ky k k cos2pj + sin2pj ( 18) kx ky ˜ pj = arctan ͑͌kx ր ky tanpj͒ ( 19) kpj = ky cos2pj + kx sin2pj ( 20) z*jD = zpjD + rpejD ( 21) zpjD = zpj ͌k ր kz ր L ( 22) rpejD = 160 rpj ͓͑k ր k ͒0.25 + ͑kpj ր kz͒0.25͔ ( 23) 2L z pj 141.2 qpj Bo ⌬pspj = sdcpj = Lpj͌kzkpj ͩ͌ ͪ sdcpj ( 24) kzkpj rdpj − ln + kdpj rcpj ͩ͌ ͪ kzkpj rcpj − ln , ( 25) kcpj rpj where sdcpjסthe skin factor, including the effect of permeability reduction in damaged and crushed zones around the jth perforation segment kcpj and rcpjסthe permeability and radius of crushed zone and kdpj and rdpjסthe permeability and radius of damaged zone, respectively As stated previously, rdpj may be approximated as the half-distance between the neighboring perforations If the additional pressure drop caused by skin is expressed in terms of dimensionless pressure, ⌬pspjD = qpjD h qpjD s = s , ( 26) LpjD L dcpj LpjD dcpjD where sdcpjDסthe scaled skin factor hD = h͌k ր kz ր L ( 14) kD = In the preceding definitions, Lסa reference length in the system and kסthe geometric mean permeability, k͌סkxkykz In this study, we choose L as the vertical well radius The pseudoskin expression given in Eq does not include the effect of formation damage or crushed zone around the perforation segments To incorporate these effects into the perforated-well solution, the flow model depicted in Fig is envisioned Fig displays the details of a perforated well with damage around the well and the perforations To precisely predict the additional pressure drop caused by damage, a composite model composed of two reservoir regions around the wellbore is needed However, in this study, the geometry of the damaged zone is simplified/ distorted to add the influence of damage and crushing around the jth perforation on the pressure drop in the jth perforation The circular damaged zone characterized by rdw around the wellbore is exchanged with a circular damaged zone defined with rdpj around the jth perforation rdpj can be approximated as the half-distance or the full distance between the neighboring perforations Additionally, it is momentarily assumed that the thickness of the damaged zone around the wellbore is equal to the length of the jth perforation This assumption will be removed soon Also, it is hypothesized that the flow in the damaged/crushed zone around the perforation segment is normal to the perforation axis (i.e., the flow around perforation segments is radial with respect to the perforation axis) It is expected that the crushed zone does not extend beyond damaged zone around the perforations Under the assumptions stated, the additional pressure drop resulting from formation damage/crushing around the jth perforation segment is written as sdcpjD = h s ( 27) L dcpj If the jth perforation penetrates beyond the damaged zone, then the jth perforation is divided into two smaller segments The first segment covers the region from the wellbore face to the radius of the damaged zone, rw