Single-Phase Inflow Performance Relationship for Horizontal, Pinnate-Branch Horizontal, and Radial-Branch Wells

Transcription

1 Single-Phase Inflow Performance Relationship for Horizontal, Pinnate-Branch Horizontal, and Radial-Branch Wells Huiqing Liu and Jing Wang, Ministry of Education Key Laboratory of Petroleum Engineering, China University of Petroleum; Jian Zheng, Ministry of Education Key Laboratory of Petroleum Engineering, China University of Petroleum, and Institute of Oil Production Technology, Shengli Oil Field Company, SINOPEC; and Ying Zhang, Xinglongtai Oil Production Plant, Liaohe Oil Field Company, CNPC Summary Horizontal and multibranch wells are likely to become the major means of modern exploitation strategies; inflow performances for these wells are needed. Because this paper considers the finite conductivity of a horizontal well, it establishes the inflow performance relationships (IPRs) for different branch configurations of horizontal wells. We find that the IPR of a horizontal well presents nonlinear characteristics and is similar to Vogel s equation, which has been used extensively and successfully for analyzing the IPR of a vertical well in a solution-gas-drive reservoir. Instead of the effect of a two-phase (oil and gas) flow in a reservoir described by Vogel s equation, the nonlinear characteristics of horizontal wells are mainly the result of pressure drops caused by friction, acceleration, and gravity along the horizontal wellbore. The nonlinearity coefficient presents the pressure drop along the major branch, and it is a function of major-wellbore length, major-wellbore diameter, oil viscosity, and relative roughness. Then, the horizontal-well IPR is used to study the performance of the pinnate-branch horizontal well and the radial-branch (horizontal lateral) well. The branch number, branch length, major-wellbore length, major-wellbore diameter, oil viscosity, and relative roughness are combined into grouped parameters to present the effect on the deliverability incremental ratio J H and the nonlinearity coefficient ratio R v of the pinnate-branch horizontal well to the conventional horizontal well, which show regressioelationships with the grouped parameters for pinnate-branch horizontal wells. In addition, another binomial relationship between the deliverability incremental ratio J V and the grouped parameter combined by branch number, branch length, and equivalent oil drainage diameter is obtained for radial-branch (horizontal lateral) wells. The new IPR also covers conventional horizontal wells and vertical wells (with no branch) because the deliverability incremental ratios J H and J V in both cases are unity. The IPR is very valuable for calculating the productivity of horizontal wells, pinnatebranch horizontal wells, and radial-branch wells. Introduction The IPR is the basis of optimization design in production engineering because it can present a boundary condition for artificial lift designs by showing the relationship of productioates varying with bottomhole flowing pressure. The IPR of an oil well depends oeservoir rock-pore type (Brown 984), well configuration, and reservoir recovery means (Liu et al. 996; Liu and Zhang 999). A longer branch of the horizontal well will increase the contact area between wellbore and formation, which makes it easier for oil to enter the wellbore or for working fluids to be injected into the formation. However, when the horizontal well is in operation, the length of the horizontal segment plays an important role in the pressure drops of horizontal wells. In earlier studies, only the frictional pressure drop was taken into consideration Copyright VC 23 Society of Petroleum Engineers Original SPE manuscript received for review 26 November 2. Revised manuscript received for review 22 June 22. Paper (SPE 634) peer approved 6 July 22. (Dikken 99; Islam and Chakma 99; Ozkan et al. 992; Seines et al. 993), and the pressure drops of variable mass flow and fluid-mixing effect are neglected. Su and Gudmundsson (994) carried out pressure-drop experiments in perforated pipes to study the influences of friction, acceleration, gravity effects, and perforation. Ouyang et al. (998a) and Schulkes and Utvik (998) established the single-phase model for the horizontal well by considering friction, acceleration, and gravity effects. The productivity index will be affected by the pressure drop along the wellbore (Penmatcha et al. 997). If another strong drainage zone exists around the toe of the horizontal well, the productivity index will even decrease because of counterflow from heel to toe. With the development of drilling technique, wellbore trajectory will have more branches. Because of the complexities of well drilling and completion technology and the influence of branch configuration oeservoir fluid flow, the deliverability of multibranch horizontal wells will be greatly different from that of conventional horizontal wells, and reservoir fluid flow state, multibranch spacing, branch length, and branch angle will also affect well deliverability (Ozkan et al. 99; Zhao et al. 26). Therefore, systematic studies on the coupled reservoir flow and pipe flow are very important to evaluate the deliverability of multibranch horizontal wells appropriately and to achieve a highly effective reservoir production and management. IPR of Conventional Horizontal Wells Fluid flow in a wellbore as shown in Fig. is considered by assuming the single-phase flow of an incompressible Newtonian fluid under isothermal conditions with no heat transfer. The inflow direction angle from perforations to wellbore is equal to p/2. Then the Ouyang model will be as follows (Ouyang et al. 998a): Dp ¼ q B v2 s w SDx v2 2 qgsinhdx;... ðþ A where s w ¼ f qv2 2 ¼ f q 8 ðv þ v 2 Þ 2... ð2þ Kinney (968) numerically found that the ratio of the local friction factor f to the no-wall-flow friction factor f is dependent on the wall Reynolds number for laminar flow. Ouyang et al. (998a) proposed a correlation based on Kinney s data: f ¼ 6 ð þ :434NRe;w :642 Þ:... ð3þ N Re In addition, a new correlation for the local friction factor for turbulent flow was developed by Ouyang et al. (998a), f ¼ f ð :3N :3978 Re;w Þ:... ð4þ The f can be determined from the Colebrook-White equation or from one of its explicit approximations (Ouyang and Aziz 996; Colebrook and White 937), April 23 SPE Journal 29

2 h p wfi > p wfj > p wfn qwfi < q wfj < q wfn.8.6 = 33 m = 39 m = 4 m = m = 7 m (p wfn, q n ) (pwfj, q j ) (p wfi, q i ) pwf/pr.4 Fig. Sketch of pressure drop and local productioate as fluids flow along the horizontal wellbore. e p ffiffiffi ¼ 4:log f 3:7 þ :26 pffiffiffi :... ðþ N Re f For comparison, the reservoir parameters reported by Zhao et al. (26) are used for studying the deliverability in a conventional horizontal well. The formation depth is m, the pay zone is m, the well drainage area is m, the porosity is 2%, the permeability is 3 lm 2, the initial oil saturation is.7, the connate water saturation is.3, the oil viscosity is 2 mpa/s, and the initial pressure is MPa. Single-phase oil exists in the formation during production. The dissolved gas is neglected to investigate only the effect of pressure drop along the well on IPR. Constant pressures at inner and outer boundaries are used: pj r¼rwðat heel endþ ¼ p w pj r¼re ¼ p e :... ð6þ By changing the bottomhole flowing pressure, the deliverability at different completion lengths of the conventional horizontal well is calculated, and the simulatioesults are shown in Fig. 2. It is shown that, when the wellbore pressure drops are considered, the relationship between productioate and bottomhole flowing pressure of a horizontal well is nonlinear. In addition, the shapes of an IPR curve are similar to those of the solution-gas-drive reservoir. Therefore, Vogel s equation (Vogel 968) is referenced to obtain the dimensionless pressure and productioate. The dimensionless deliverability relationship is obtained by regression and is shown as follows (Bendakhlia and Aziz 989): q ¼ ð vþ p wf v p 2 wf :... ð7þ q CHmax p r p r It is very similar to Vogel s equation, but the mechanisms are different. The nonlinearity of Vogel s equation in a vertical well is caused by the effect of two-phase (oil and gas) flow (Vogel 968; Cheng 99; Retnanto and Economides 998), whereas the nonlinearity of Eq. 7 is caused not only by the pressure drop along the horizontal well that is a result of friction, acceleration, and gravity as the fluids flow from the toe to the heel but also by the lateral influx distributed along the horizontal well (Ouyang et al. 998b). The modeling results indicate that the length of the horizontal well, Reynolds number, and surface roughness have significant effects on the wellbore pressure drops and the nonlinearity of the IPR curve. These effects are simulated and shown in Fig. 3. It is shown that the longer the horizontal well, the larger the pressure differential between the toe and heel of the horizontal well, and the greater the nonlinearity degree of deliverability relationships. The horizontal well with a smaller wellbore diameter will generate a higher nonlinearity coefficient (v). In addition, a higher oil viscosity will also result in a bigger flowing resistance. However, when the bottomhole flowing pressure is constant, the production plays an important role in determining the nonlinearity coefficient (v). A lower production will generate a smaller v. A bigger relative roughness will result in a bigger friction drop; thus, the v is increased. To quantitatively describe the relationship between v and the parameters aforementioned, an orthogonal test plan [L 6 (4 )] is q/q CHmax Fig. 2 Bottomhole flowing pressures and productioates at different major wellbore lengths. established, as shown in Table. According to the simulation results under different conditions, a grouped parameter (N Hv ) could be obtained through the correlation between each parameter and v. Then, a nonlinear regression procedure is used to develop the relationship between v and N Hv ), and it can be characterized by the exponential variogram model. The fitting curves between experimental data and Eq. 8 are shown in Fig. 4. v ¼ :6 exp 3N Hv... ð8þ 7 N Hv ¼ L:8 m e:9 l o d 2 :... ð9þ From Eq. 8, with the increase of N Hv,vincreases gradually. When the value of N Hv is less than, v is less than., representing a lower nonlinearity of the IPR curve; when the value of N Hv is higher than, the increasing tendency of v becomes less pronounced, and maximum v equals.6. From Eq. 9, the sensitivities of v to the parameters can be arranged as d > > l o > e. Deliverability Characteristics of Pinnate-Branch Horizontal Wells Reservoir Pressure Distribution Along Pinnate-Branch Horizontal Well. The simulation area is a rectangle with aspect ratio 3:2; the length of the rectangle depends on horizontal well length. Reservoir properties are shown in Table 2. The configurations of pinnate-branch horizontal wells with different branch numbers are shown in Fig.. Corresponding to specific configurations the pressure distributions in the wellbore and the perforated grids of conventional horizontal wells and pinnate-branch horizontal wells with different branch numbers bottomhole flowing pressures are obtained (see Fig. 6). As shown, at a given bottomhole flowing pressure, the pressure in the major wellbore is reduced gradually from the toe to the heel. The flow rate near the heel is much higher, and the reducing rate of pressure is faster. With the increase of branch number, both the productioate and the flow rate increase. Thus, the pressure differences between the toe and the heel are much bigger. In perforated grids, because of the production of oil from branches, the formation pressure is reduced. Therefore, the pressure from the toe to the heel shows a tendency to decrease first and then increase. In addition, the greater the number of branches, the more distinct the tendency will be. At a given number of branches, when the bottomhole flowing pressure is much higher, the productioate is low, and the flow rate in wellbore is also low. Thus, the pressure drop is small, and the pressure drop from the toe to the heel is not 22 April 23 SPE Journal

3 .8.6 = 9 m = 2 m = 33 m = 4 m = 7 m = m = 27 m = 39 m = m.8.6 d =. m d =.2 m d =.4 m d =.6 m d =.8 m d =.2 m pwf/pr.4 pwf/pr q/q CHmax q/q CHmax (a) Horizontal Well Length (b) Inner Diameter pwf/pr = mpa s = 6 mpa s = mpa s = 6 mpa s = 2 mpa s = 26 mpa s pwf/pr e =. e =.2 e =.3 e =.4 e = q/q CHmax q/q CHmax.8 (c) Oil Viscosity (d) Relative Roughness Fig. 3 Effects of different parameters on IPR linearity. distinct. On the contrary, when the well s bottomhole flowing pressure is lower, the productioate is higher, and the flow rate of wellbore is also higher. Therefore, for the perforated grid, with decrease in bottomhole flowing pressure, the productioates of the major wellbore and branch wellbores increase, and the concave degree is also intensified. IPR of a Pinnate-Branch Horizontal Well. For a 33-m major wellbore with different branch lengths and branches with a.4- m diameter, reservoir numerical simulation was performed to calculate the productioates at different bottomhole flowing pressures. The oil viscosity was mpas, and the relative roughness was.3. Dimensionless pressure and productioate can be obtained in the form p wf /p r for pressure and q/q BHmax for production rate, as shown in Fig. 7a. In addition, if the dimensionless productioate is defined as a ratio of productioate to the maximum rate of conventional horizontal well, q/q CHmax, the IPR will be different, as shown in Fig. 7b). By observing Fig. 7, we find that because of the increasing of the branch length, the drainage area increases, and the deliverability and flow rate in the major wellbore will increase, which results in larger pressure loss and higher nonlinearity of the IPR curve. Compared with the conventional horizontal well at the same bottomhole flowing pressure, the productivity of a pinnate-branch horizontal well is higher. Similarly, the increase of production will also increase the wellbore pressure drop, resulting in an increase in nonlinearity. The IPR is similar to that of the conventional horizontal well, and Eq. 7 can be written as q ¼ ð v b Þ p wf p 2 wf v b... ðþ q BHmax p r p r Then, the deliverability incremental ratio J H is defined as the ratio of q BHmax to q CHmax, which represents the deliverability increase of the pinnate-branch horizontal well compared with that of the conventional horizontal well. In addition, the v ratio R v is defined as the ratio of v b to v, which represents the nonlinearity difference between the pinnate-branch horizontal-well IPR and the conventional-horizontal-well IPR. Then, Eq. can be written as " q ¼ J H ð R v vþ p wf R v v p # 2 wf :... ðþ q CHmax p r p r Besides the effects of horizontal-well length, Reynolds number, and surface roughness, J H and R v also depend on branch number and branch length. Table 3 lists five levels for each of the influencing parameters. The trends of J H and R v with the level changes of each influencing parameter are presented in Fig. 8. J H and R v are shown to increase with branch number, branch length, oil viscosity, and major-wellbore diameter but to decrease with the major wellbore length and relative roughness. This indicates that better improvement will be achieved by increasing branch April 23 SPE Journal 22

4 TABLE ORTHOGONAL ARRAY L 6 (4 ) Test No. Parameters (m) D (m) e l o (mpas) number and branch length for shorter major wellbore, smaller major-wellbore diameter, less relative roughness, and higher oil viscosity. To quantitatively describe the relationship of J H and R v with these parameters, based on the orthogonal test mentioned earlier (Table ), the values of J H and R v under different parameters (including the major wellbore length, branch number, and branch length) are calculated. According to the correlation of J H and R v with these parameters, a grouped parameter is obtained. Then, on the basis of regression method, the relationship between J H,R v, and their assembly parameter is obtained, and the fitting curves between experimental data and Eqs. 2 and 4 are shown in Figs. 9 and : J H ¼ þ 2:4N HJ 8:6NHJ 2 ; ðr2 ¼ :9476Þ... ð2þ N HJ ¼ n b L b lo :6 d :3 L :7 m... ð3þ e:2 Property TABLE 2 RESERVOIR PROPERTIES LIST Value Formation depth (m) Formation thickness (m) Initial pressure (MPa) Formation volume factor. Dead oil density (kg/m 3 ) 92 Initial oil saturation (%) 7 Initial water saturation (%) 3 Permeability ( 3 lm 2 ) 3 Porosity (%) 2 Total compressibility ( 4 /MPa) 3. v N Hv Fig. 4 Relationship between coefficient m and N Hv. R v ¼ expð:34n Hbv Þ; ðr 2 ¼ :933Þ... ð4þ N Hbv ¼ n:4 b L :2 b l :9 o d :3 L 2 m :... ðþ e:2 Later, by integrating Eqs. 8 and 4, one can obtain the value of v b as shown in Table 4. Thus, compared with that from the actual results that the calculation value plotted on the x axis and the actual value plotted on the y axis shown in Fig., the calculated data and the actual data are distributed in the surrounding region of this straight line, y=x,which demonstrates that the calculation values are close to the true values. Therefore, it is presented that this model is reliable. From Eqs. 2 through, for the parameter J H, representing the increase of horizontal-well productivity, the sensitivities of it to the parameters can be arranged as > d > L b ¼ n b > l o > e. For the other parameter, R v, representing the increase of wellbore pressure drop, the sensitivities of it to the parameters can be arranged as: > d > L b > l o > n b > e. It is clear that J H increases as the N HJ increases but slows gradually at higher N HJ.J H can be used to evaluate the validity of reservoir reconstruction projects or drilling branches for horizontal wells. The reasonable branch number and branch length can be obtained by evaluating J H and the corresponding cost. In addition, the grouped parameters will be zero for the conventional horizontal well (no branch), and both productivity incremental ratio J H and v ratio R v are unity in Eq. 2 and Eq. 4, and Eqs. 7 and are the same. The preceding studies are conducted under the same branch length. To study the influence of different branch lengths on the IPR curve, we simulate the IPR curves of these plans in Fig. 2 for different branch lengths and different distributions, as shown in Fig. 3. As determined from the simulatioesults, the differences in the IPR curves for the same average branch length but different branch-length combinations are very small. This is because the pressure drop is mostly from the major wellbore. The branches mainly provide drainage areas. N b and the average branch length L b can be used if the branches have different lengths. Deliverability of the Radial-Branch (Horizontal Lateral) Well IPR of Radial-Branch Well. The simulation area is a square (42 42 m), and formation thickness is m. Reservoir a b c d e Fig. The configurations of pinnate-branch horizontal wells. 222 April 23 SPE Journal

5 Pressure in the wellbore (MPa) n =, p wf = 8 MPa n =, p wf = 8 MPa n = 2, p wf = 8 MPa n = 3, p wf = 8 MPa n = 4, p wf = 8 MPa n =, p wf = 8 MPa Distance from heel (m) (a) Pressure distribution in the wellbore for different branch numbers (p wf = 8 MPa) Pressure in the wellbore (MPa) p wf = 4 MPa p wf = 3 MPa p wf = 2 MPa 3 p wf = MPa p wf = MPa p wf = 9 MPa p wf = 8 MPa p wf = 7 MPa p wf = 6 MPa p wf = MPa p wf = 4 MPa p wf = 3 MPa p wf = 2 MPa p wf = MPa p wf = MPa Distance from heel (m) (b) Pressure distribution in the wellbore for different bottomhole flowing pressures (n b = ) Pressure out the wellbore (MPa) n =, p wf = 8 MPa n =, p wf = 8 MPa n = 2, p wf = 8 MPa n = 3, p wf = 8 MPa n = 4, p wf = 8 MPa n =, p wf = 8 MPa Distance from heel (m) (c) Pressure distribution in the perforated grids for different branch numbers (p wf = 8 MPa) Pressure out the wellbore (MPa) wf = 4 MPa wf = 3 MPa wf = 2 MPa wf = MPa wf = MPa wf = 9 MPa wf = 8 MPa wf = 7 MPa wf = 6 MPa wf = MPa wf = 4 MPa p wf = 3 MPa p wf = 2 MPa p wf = MPa p wf = MPa Distance from heel (m) (d) Pressure distribution in the perforated grids for different bottomhole flowing pressures (n b = ) Fig. 6 Pressure distribution along pinnate-branch horizontal wells under different conditions..8.6 n =, L b = 27. m n = 2, L b = 27. m n = 27. m n = 4, L b = 27. m n =, L b = 27. m Convent. well.8.6 n = 42. m n = 8 m n = 27. m n = 7 m n = 22. m Convent.well pwf /pr.4 pwf/pr q/q CHmax q/q BHmax (a) p wf /p r vs. q/q BHmax (b) p wf /p r vs. q/q CH max Fig. 7 IPR curves of multibranch horizontal wells with different branch length and numbers. April 23 SPE Journal 223

6 TABLE 3 FIVE LEVELS FOR EACH OF THE INFLUENCING PARAMETERS OF J H AND R V Level No. L b (m) n b (m) l o (mpas) e d(m) properties used in this case are the same as in Table 2. The oil viscosity is mpas, the relative roughness is.3, and the wellbore diameter is.4 m. Both a constant pressure outer boundary and a bottomhole flowing pressure at the center outflow point are used. Dimensionless pressure and productioate can be obtained in the form p wf /p r for pressure and q/q BVmax for productioate, as shown in Fig. 4a. In addition, if the dimensionless production rate is defined as a ratio of productioate to the maximum rate of 3 J H R v L b n b e d Level No L b n b e d Level No. Fig. 8 The trends of J H and R v with the level changes of each influencing parameter. a vertical well, q/q CVmax, the IPR will be different, as shown in Fig. 4b. As seen in Fig. 4, because the individual branch length in a radial-branch well is generally smaller, according to the effect of pressure drop on well deliverability shown in Fig., the deliverability drop will not be significant; thus, the IPR of a radialbranch well is basically linear. However, the dimensionless productioate is much larger. The deliverability has a linear relationship for a vertical well with single-phase flow (Brown 984). The IPR of a radial-branch well can also be described by the following model: J H 2 R v N HJ Fig. 9 Relationship between J H and N HJ N Hbv Fig. Relationship between R v and N Hbv. 224 April 23 SPE Journal

7 TABLE 4 TESTING PARAMETERS FOR THE INFLUENCING FACTORS OF J V No. n b L b d e l o Actual v b Calculated v b April 23 SPE Journal 22

8 TABLE 4 (continued) TESTING PARAMETERS FOR THE INFLUENCING FACTORS OF J V No. n b L b d e l o Actual v b Calculated v b April 23 SPE Journal

9 TABLE 4 (continued) TESTING PARAMETERS FOR THE INFLUENCING FACTORS OF J V No. n b L b d e l o Actual v b Calculated v b Actual v b y = x R 2 = Calculational v b a c Fig. 2 The configurations of pinnate-branch horizontal wells with different branch lengths and distributions. b d Fig. Comparison of actual v b and calculated v b. q ¼ p wf :... ð6þ q BVmax p r Then, the deliverability incremental ratio J V is defined as the ratio of q BVmax to q CVmax, which represents the deliverability increase of the radial-branch well compared with that of the conventional vertical well, and Eq. 6 will be: q q CVmax ¼ J v p wf p r :... ð7þ J V depends on branch number, branch length, oil viscosity, wellbore diameter, and surface roughness. Table lists five levels for each of the influencing parameters. The trends of J V with the level changes of each influencing parameter are presented in Fig.. J V is demonstrated to increase with branch number and branch length but changes little with oil viscosity, wellbore diameter, and relative roughness. This indicates that better improvement will be achieved by increasing branch number and branch length. To quantitatively describe the relationship between J V and the parameters mentioned, the orthogonal test plan [L 6 (4 )] is used again, as shown in Table 6. According to the simulatioesults under different conditions, a grouped parameter (N Hv ) could be obtained through the correlation between each parameter and J V. Because the oil viscosity, wellbore diameter, and relative roughness have almost no effect on J V, the N Hv is only combined by branch number and branch length. Then, the following correlation can be obtained by regression analysis as shown in Fig. 6: J V ¼ þ 8:48N VJ :8NVJ 2 ðr2 ¼ :939Þ... ð8þ N VJ ¼ n:6 r L b ;... ð9þ R d where, p ffiffiffiffiffiffiffiffi R d ¼ 2L p =p :... ð2þ 2 a b c d.8 a b c d p wf (MPa) 9 6 p wf /p r q (t/d) q/q BHmax (a) p wf vs. q (b) p wf /p r vs. q/q BHmax Fig. 3 IPR curves of pinnate-branch horizontal wells with equivalent or different branch lengths. April 23 SPE Journal 227

10 p wf /p r Convent. well = 3 m = 7 m = m = 4 m = 7 m p wf /p r Convent. well = 3 m = 7 m = m = 4 m = 7 m q/q BVmax q/q CVmax Fig. 4 IPR curves of radial-branch wells with different branch lengths. From Eq. 8, we can see that the productivity incremental ratio J V increases with branch number and branch length and that the J V of radial-branch well is much larger than that of a pinnate-branch horizontal well. In other words, the deliverability of a radial branches of a vertical well improves much more than that of pinnate branches of a horizontal well. Because the disturbance among branches will also increase gradually as branch number increases, the binomial term in Eq. 8 mainly reflects the disturbance among branches. In addition, J V can be used to evaluate the validity of reservoir reconstruction projects or drilling radial branches for vertical wells. The grouped parameters will be zero for the conventional vertical well (no branch), and productivity incremental ratio J V is unity in Eq. 8, and Eqs. 6 and 7 are the same. The studies preceding are conducted under a same-branch length. To study the TABLE FIVE LEVELS FOR EACH OF THE INFLUENCING PARAMETERS OF J V Level No. L b (m) n b l o (mpas) e d(m) influence of different branch lengths on IPR curves, we simulate the IPR curves of these plans in Fig. 7 for different branch lengths and different distributions, as shown in Fig. 8. According to the simulatioesults, the differences in the IPR curves for the same average branch length but different branchlength combinations are very small. N r and the average branch length L b can be used if the branches have different lengths. Validation of the IPR for Multibranch Wells Pinnate-Branch Horizontal Wells. Wells X-H3 and X-H32 are located in Xinggu reservoir block in Liaohe oil field. The buried depth of the formation is 4 m, and the pay zone is 2.7 m. Oil viscosity is approximately 3. mpas. Initial oil saturation is.67, and connate water saturation is.33. Reservoir pressure is 4 MPa. The rock porosity is 7.%, and permeability is approximately 3 lm 2. Wells X-H3 and X-H32 are pinnatebranch horizontal wells with three and four branches, respectively. The parameters of the two wells are shown in Table 7. Several test points exist for each well. First, we obtain the values of v, J H, and R v with the relevant parameters (v 3 ¼.9, J 3 ¼.3, R v3 ¼.383; v 4 ¼.4, J 4 ¼.424, R v3 ¼.398), and use one of the test points (q t3 ¼ 76.4 t/d, p wft3 ¼ 2. MPa; q t4 ¼ 89.6 t/d, p wft4 ¼ 29. MPa) to calculate the q CHmax by Eq. 2. Thus, the IPR can be obtained using Eq.. Then, all the test points are compared with the IPR, as shown in Fig. 9. As can be seen, the agreement is good. Therefore, the IPR can be used to predict the performance in the early development stage. J V L b d e Level No. Fig. The trends of J V with the level changes of each influencing parameter. 228 April 23 SPE Journal

11 2 TABLE 6 ORTHOGONAL ARRAY L 6 (4 ) 2 Test No. L b Parameters (m) D (m) e l o (mpas) J V N VJ Fig. 6 Relationship between J V and N VJ. a b c Fig. 7 IPR curves of radial-branch wells with equivalent or different branch lengths q CHmax ¼ ( " J H ð R v vþ p wft R v v p # 2 ): wft p r p r q t ð2þ 2 a b c Radial-Branch Wells. Wells Y-C8 and Y-C2 are located in the Y reservoir block in Shengli oil field. The buried depth of the formation is 348 m, and the pay zone is 3.9 m. Oil viscosity is approximately.2 mpas. Initial oil saturation is.7, and connate water saturation is.29. Reservoir pressure is 3. MPa. The rock porosity is 2.2%, and permeability is approximately 68 3 lm 2. Wells Y-C8 and Y-C2 are radial-branch wells with three and four branches, respectively. The well spacing of the radial-branch well group is defined as the distance between two trunk wells, and the well space of the well group where the two wells are located is 483 and 2 m, respectively. The parameters of the two wells are shown in Table 8. In addition, several test points exist for each well. The J V values of Y-C8 and Y-C2 are and 4.83, respectively. The test points (q t3 ¼ 44.2 t/d, p wft3 ¼ 26. MPa; q t4 ¼ 7.4 t/d, p wft4 ¼ 28.6 MPa) are used to calculate the q CVmax by Eq. 22. Thus, the IPR can be obtained with Eq. 7. Then, all the test points are compared with the IPR, as shown in Fig. 2. As can be seen, the agreement is also good. Therefore, the IPR can be used to predict the performance in the early development stage. q CVmax ¼ q t J V p wft :... ð22þ p r p wf (MPa) q (t/d) Fig. 8 The configurations of radial-branch wells with different branch lengths and distributions. Conclusions The single-phase IPR of a horizontal well is similar to Vogel s equation. However, the mechanism of this nonlinearity is the pressure drop as fluids flow along the horizontal well, not the effect of a two-phase (oil and gas) flow. The nonlinearity of IPR is related to the length of horizontal wellbore, wellbore diameter, oil TABLE 7 PARAMETERS OF PINNATE-BRANCH HORIZONTAL WELLS X-H3 AND X-H32 Well (m) L b (m) L b2 (m) L b3 (m) L b4 (m) Inner Diameter (m) Relative Roughness X-H3 (Three Branches) X-H32 (Four Branches) April 23 SPE Journal 229

12 4 4 3 Computed IPR Test data Computed IPR Test data 3 3 p wf (MPa) 2 2 p wf (MPa) q (t/d) q (t/d) (a) Pinnate-branch horizontal well X-H3 with three branches Fig. 9 Validation of the pinnate-branch horizontal-well IPR. (b) Pinnate-branch horizontal well X-H32 with four branches TABLE 8 PARAMETERS OF RADIAL-BRANCH WELLS Y-C8 AND Y-C2 Well L p (m) R d (m) L b (m) L b2 (m) L b3 (m) L b4 (m) Y-C8 (Three Branches) Y-C2 (Four Branches) viscosity, and relative roughness. The longer the horizontal well, the larger the relative roughness, and when the viscosity of crude oil is lower and the wellbore diameter is smaller, v is larger. When N Hv is less than, the nonlinear characteristic can be neglected. However, when N Hv is greater than, v is almost stable, and the maximum value equals.6. The single-phase IPR curves of pinnate-branch wells are obtained by modifying the horizontal-well IPR. A deliverability incremental ratio J H exists when the dimensionless pressure is defined as q/q CHmax. In addition, a v ratio R v exists when the dimensionless pressure is defined as q/q BHmax. Both the deliverability incremental ratio J H and v ratio R v are influenced by branch number, branch length, major-wellbore length, wellbore diameter, oil viscosity, and relative roughness, and binomial relationships with the grouped parameters N HJ and N Hbv are proposed for the pinnate-branch horizontal well. J H can be used to evaluate the economic validity of reservoir reconstruction projects or drilling branches for horizontal wells. The single-phase IPR curves of radial-branch wells are also obtained. A deliverability incremental ratio J V also exists when the dimensionless pressure is defined as q/q CVmax. The deliverability incremental ratio J V shows a binomial relationship with the grouped parameter N VJ for the radial-branch wells. It also can be used to evaluate the validity of reservoir-reconstruction projects or drilling radial branches for vertical wells. By contrast, the deliverability incremental ratio J V of radial-branch wells is much larger than that of pinnate-branch horizontal wells. Further work on the multibranch-well IPR will focus on twophase IPR, which will consider both the two-phase flow and the pressure drop in the wellbore. Nomenclature A ¼ pipe cross-sectional area, m 2 B ¼ momentum correction factor, dimensionless d ¼ pipe diameter, m 3 3 Computed IPR Test data 3 3 Computed IPR Test data 2 2 p wf (MPa) 2 p wf (MPa) q (t/d) q (t/d) (a) Radial-branch well Y-C8 with three branches (b) Radial-branch well Y-C2 with four branches Fig. 2 Validation of the radial-branch well IPR. 23 April 23 SPE Journal

13 e ¼ relative roughness, dimensionless f ¼ wall-flow friction factor, dimensionless f ¼ no-wall-flow friction factor, dimensionless J H ¼ deliverability incremental ratio in horizontal wells, dimensionless J V ¼ deliverability incremental ratio in vertical wells, dimensionless L b ¼ branch length, m ¼ major-wellbore length, m L p ¼ well spacing, m n b ¼ branch number of pinnate well ¼ branch number of radial well N HJ ¼ grouped parameter N HV ¼ grouped parameter N Hbv ¼ grouped parameter N VJ ¼ grouped parameter N Re ¼ Reynolds number, dimensionless N Rew ¼ wall Reynolds number, based on pipe diameter and equivalent inflow/outflow velocity, dimensionless p wf ¼ bottomhole flowing pressure, MPa p wft ¼ tested bottomhole flowing pressure, MPa p r ¼ reservoir average pressure, MPa q ¼ oil-productioate, t/d q BHmax ¼ maximum oil-productioate at p wf ¼ for pinnatebranch horizontal well, t/d q BVmax ¼ maximum oil-productioate at p wf ¼ for radial branch well, t/d q CHmax ¼ maximum oil-productioate at p wf ¼ for conventional horizontal well, t/d q CVmax ¼ maximum oil-productioate at p wf ¼ for conventional vertical well, t/d q max ¼ maximum oil-productioate at p wf ¼, t/d q t ¼ tested oil-productioate, t/d R d ¼ equivalent oil-drainage diameter, m R v ¼ nonlinearity coefficient ratio, dimensionless S ¼ wellbore perimeter, m v ¼ average velocity, m/s h ¼ wellbore-inclination angle (from horizontal) l o ¼ oil viscosity, mpas q ¼ density, g/cm 3 s w ¼ wall-friction shear, N/m 2 v ¼ nonlinearity coefficient, dimensionless v b ¼ nonlinearity coefficient for pinnate-branch horizontal well, dimensionless Acknowledgments The authors would like to express appreciation to the management of the Fourth Oil Production Factory of Zhongyuan Oil Field Company, SINOPEC, who provided well data and financial support for this research, for their permission to publish this paper. The authors also thank Yuping Fan, a doctoral student, for his suggestions and insightful remarks on this research. References Bendakhlia, H. and Aziz, K Inflow Performance Relationships for Solution-Gas Drive Horizontal Wells. Paper SPE9823 presented at the 64th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, San Antonio, Texas, 8 October. dx.doi.org/.28/9823-ms. Brown, K.E The Technology of Artificial Lift Methods. Tulsa, Oklahoma: Penn Well Publishing Company. Cheng, A.M. 99. Inflow Performance Relationships for Solution-Gas- Drive Slanted/Horizontal Wells. Paper SPE272 presented at the 6th Annual Technical Conference and Exhibition, New Orleans, Louisiana, September. Colebrook, C.F. and White, C.M Experiments with Fluid Friction Roughened Pipes. Proc. R. Soc. Lond. A 6: dx.doi.org/.98/rspa Dikken, B.J. 99. Pressure Drop in Horizontal Wells and Its Effect on Production Performance. J. Pet Tech 42 (): dx.doi.org/.28/9824-pa. Islam, M.R. and Chakma, A. 99. Comprehensive Physical and Numerical Modeling of a Horizontal Well. Paper SPE2627 presented at the 6th SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September. Kinney, R Fully Developed Frictional and Heat-Transfer Characteristics of Laminar Flow in Porous Tubes. Int. J. Heat Mass Transfer (9): Liu, H.Q., Wu, X.D., and Zhang, Q Methodology of IPR Study for Steam Stimulation in Horizontal Well. Paper SPE3746 presented at the SPE International Conference on Horizontal Well Technology, Calgary, Canada, 8 2 November. Liu, H.Q. and Zhang, Q Inflow Performance Relationship for Steam Stimulation Well. Journal of University of Petroleum, China (in Chinese, abstract in English) 23 (3): Ouyang, L. and Aziz, K Steady-State Gas Flow in Pipes. J. Petroleum Science and Engineering 4 (3 4): Ouyang, L.B., Arbabi, S., and Aziz, K. 998a. General Wellbore Flow Model for Horizontal, Vertical, and Slanted Well Completions. SPE J. 3 (2): Ouyang, L.B., Petalas, N., Arbabi, S. et al. 998b. An Experimental Study of Single-Phase and Two-Phase Fluid Flow in Horizontal Wells. Paper SPE4622 presented at the SPE Western Regional Meeting, Bakersfield, California, 3 May. Ozkan, E., Sarica, C., Haciislamoglu, M. et al Effect of Conductivity on Horizontal Well Pressure Behavior. Paper SPE presented at the 67th SPE Annual Technical Conference and Exhibition, Washington, DC, 4 7 October. Ozkan, E., Sarica, C., Haciislamoglu, M. et al. 99. Effect of Conductivity on Horizontal Well Pressure Behavior. SPE Advanced Technology Series 3 (): Penmatcha, V.R., Arbabi, S., and Aziz, K Effects of Pressure Drop in Horizontal Wells and Optimum Well Length. Paper SPE37494 presented at the SPE Production Operations Symposium, Oklahoma City, Oklahoma, 9 March. Retnanto, A. and Economides, M.J Inflow Performance Relationships of Horizontal and Multibranched Wells in a Solution-Gas-Drive Reservoir. Paper SPE69 presented at the European Petroleum Conference, The Hague, The Netherlands, 2 22 October. dx.doi.org/.28/69-ms. Schulkes, R.M.S.M. and Utvik, O.H Pressure Drop in a Perforated Pipe With Radial Inflow: Single-Phase Flow. SPE J. 3 (): Seines, K., Aavatsmark, I., Lien, S.C. et al Considering Wellbore Friction Effects in Planning Horizontal Wells. J. Pet Tech 4 (): Su, Z. and Gudmundsson, J.S Pressure Drop in Perforated Pipes: Experiments and Analysis. Paper SPE288 presented at the SPE Asia Pacific Oil and Gas Conference, Melbourne, Australia, 7 November. Vogel, J.V Inflow Performance Relationships for Solution-Gas Drive Wells. J. Pet Tech 2 (): PA. Zhao, L.X., Jiang, M.H., and Zhao, X.F. 26. Research on Deliverability Relationship of Complicated Horizontal Well. Journal of China University of Petroleum (in Chinese with abstract in English) 3 (3): Huiqing Liu is a professor of petroleum engineering at the School of Petroleum Engineering, China University of Petroleum (Beijing). Liu previously studied at University of Wyoming as a visiting scholar. Now, he is the Assistant President of the School of Petroleum Engineering, China University of Petroleum (Beijing). Liu s research focuses on numerical simulation, thermal recovery, and enhanced oil recovery (EOR). He holds BS, MS, and PhD degrees from China University of Petroleum. Jing Wang is a PhD candidate in oil- and gasfield development at the School of Petroleum Engineering, China University of Petroleum (Beijing). Now, Wang is studying oil- and gas-field development, combining MS and PhD degrees. His research interests are in numerical simulation and EOR. He holds a BS degree from China University of Petroleum (East China). April 23 SPE Journal 23

14 Jian Zheng is a senior reservoir engineer at Shengli Oil field of Sinopec. His research interests are in numerical simulation, profile control, and EOR. He holds a BS degree from Northwest University in China. Zheng also holds an MS degree from China University of Petroleum (East China). Ying Zhang is a professor and senior reservoir engineer at Liaohe Oil Field of CNPC. Now, Zhang is the manager of Xinglongtai Block at Liaohe Oil Field. His research interests are in thermal recovery and oil-production engineering. He holds BS and MS degrees from China University of Petroleum. 232 April 23 SPE Journal