Zhijun Zhou,Chunxue Cui,Da Zheng,Kai Li,Yinglong Xuan,Chunyao Wang
Transient Productivity Analysis and Completion Parameter Optimization of Perforating Herringbone Wells
Herringbone wells are of great significance in tapping the potential of remaining reservoirs, improving recovery efficiency, and reducing development costs and have been widely used in oilfields, especially offshore oilfields. Due to the complex structure of herringbone wells, there is mutual interference between wellbores during seepage, resulting in complex seepage problems, and it is difficult to analyze the productivity and evaluate the perforating effect. In this paper, considering the mutual interference between branches and perforations, a prediction model of transient productivity of perforated herringbone wells was derived based on the theory of transient seepage, and the complex structure with any number of branches, arbitrary configurations, and orientations in three-dimensional space could be considered in this model. The formation pressure in reservoir points at different production times, IPR curves, and the radial inflow of herringbone well were studied by the line-source superposition method, which directly reflected the process of productivity and pressure change, avoiding the one-sidedness caused by using a point source to replace the line source in the stability analysis. Then, through the productivity calculation of different perforation schemes, the influence curves of perforation density, perforation length, perforation phase angle, and perforation radius on unstable productivity were obtained. Orthogonal tests were performed to obtain the degree of impact of each parameter on productivity. Lastly, the selective completion perforation technology was adopted. When the shot density at the end of the wellbore was increased, the productivity of herringbone wells could be improved economically and effectively. According to the above study, a scientific and reasonable construction scheme is recommended for oil well completion construction, which provides a theoretical basis for the improvement and development of perforation completion technology.
Transient Productivity Analysis and Completion Parameter Optimization of Perforating Herringbone Wells

Herringbone wells are of great significance in tapping the potential of remaining reservoirs, improving recovery efficiency, and reducing development costs and have been widely used in oilfields, especially offshore oilfields. Due to the complex structure of herringbone wells, there is mutual interference between wellbores during seepage, resulting in complex seepage problems, and it is difficult to analyze the productivity and evaluate the perforating effect. In this paper, considering the mutual interference between branches and perforations, a prediction model of transient productivity of perforated herringbone wells was derived based on the theory of transient seepage, and the complex structure with any number of branches, arbitrary configurations, and orientations in three-dimensional space could be considered in this model. The formation pressure in reservoir points at different production times, IPR curves, and the radial inflow of herringbone well were studied by the line-source superposition method, which directly reflected the process of productivity and pressure change, avoiding the one-sidedness caused by using a point source to replace the line source in the stability analysis. Then, through the productivity calculation of different perforation schemes, the influence curves of perforation density, perforation length, perforation phase angle, and perforation radius on unstable productivity were obtained. Orthogonal tests were performed to obtain the degree of impact of each parameter on productivity. Lastly, the selective completion perforation technology was adopted. When the shot density at the end of the wellbore was increased, the productivity of herringbone wells could be improved economically and effectively. According to the above study, a scientific and reasonable construction scheme is recommended for oil well completion construction, which provides a theoretical basis for the improvement and development of perforation completion technology.

The herringbone lateral well is an effective production and recovery method, and it is the main development trend of oil and gas reservoir development. It has been successfully applied in oilfields due to its large oil drainage area, reduction of development costs, and economical and effective development of reservoirs. Due to the complex structure of herringbone wells, there is mutual interference between wellbores during seepage, resulting in complex seepage problems, and it is difficult to analyze the productivity and evaluate the perforating effect. Due to the complex structure of herringbone wells and the influence of perforations on the inflow, it is difficult to predict and evaluate the transient productivity, resulting in a certain gap between theoretical yield and actual yield. Since Ozkan introduced the theory of the point-source function method, extensive experimental, numerical simulation, and theoretical studies have been conducted on the reservoir pressure in single horizontal wellbores. Wang presented an unsteady seepage model for a horizontal well in a homogeneous anisotropic reservoir. The pressure distribution of the horizontal well on the plane is derived by applying the integration method. In this model, the method is only applicable to establish the well in two-dimensional planes. Moreover, for the pressure in a multibranch horizontal wellbore, both the coupling effect of the wellbore flow and the inflow from the reservoir formation and the mutual interference of the branch and the main wellbore should be considered. Therefore, the model of the multibranch wellbore is more complex than that of a single horizontal well. Yang derived a productivity model of herringbone gas wells and studied the changes in production in different periods. Fan and Han established a semianalytical production model of the herringbone well and analyzed the flow characteristics and the influence of branch parameters on productivity. Gladkikh and Wu used the finite element method to establish a herringbone well productivity model with perforating completion and studied the effect of perforating parameters on productivity. Yue focused on the pressure drop in the wellbore, established the herringbone well productivity model, and analyzed the sensitive factors of productivity, but for the evaluation of production based on the steady flow, the production time was not considered. At present, the study on transient productivity and its influence on perforated herringbone wells is less. Therefore, in this paper, according to the principles of pressure superposition and mirror reflection and the concept of infinitesimal line congruence, the transient productivity model of perforated herringbone well was derived, where interference between the branch wellbore and the main wellbore and the confluence from the branch to main wellbore are considered. By solving the model, we obtained the distribution of formation pressure at different times. Lastly, we used numerical simulation to cross-validate the results. The results indicate that the accuracy of the calculation results is reasonable and reliable. It can serve as a preliminary guide to optimize the design of herringbone well structures and tubing parameters in actual applications and provides a theoretical basis for dynamic perforation optimization of herringbone lateral wells.

According to the principles of pressure superposition and mirror reflection and the concept of infinitesimal line congruence, the transient productivity model of perforated herringbone well was derived, where interference between the branch wellbore and the main wellbore and the confluence from the branch to main wellbore are considered. This model could realize pressure analysis or productivity evaluation of the herringbone well with perforated completion in a bottom water reservoir. The present work is organized as follows: (1) description of the herringbone well seepage model, including the physical model, assumptions, definitions, mathematical model, and solutions; (2) verification of the proposed model by comparing it with field application; and (3) the study of the impact of the perforation parameters related to productivity on the transient productivity of the model.

(a) The herringbone laterals are located in a homogeneous, anisotropic reservoir, and the anisotropic permeability is given by the following formula: Kx = Ky = Kh, Kz = Kv. The top and bottom boundaries of the pay zone are parallel. The top boundary is a closed boundary and the bottom boundary is a constant-pressure boundary. (b) The complex structure with any number of branches, arbitrary configurations, and orientations in three-dimensional space can be considered in this model. The main wellbore of the model is perforated and the branch wells have an open hole. (c) The fluid in the reservoir is the weakly compressible single-phase fluid, conforming to the Darcy law. (d) The pseudopressure is in equilibrium and equal to the initial formation pressure pini before production, and the pseudopressure drop Δp of the formation after production changes with time and space. All perforations have infinite conductivity. Figure 1 shows the schematic diagram of a perforated herringbone well. The flow pattern of fluids in an infinite reservoir flowing to any perforation in the main wellbore conforms to the three-dimensional partial differential equation of seepage flow, as shown by eq 1The initial and boundary conditions are given as eq 2where p(x, y, z) is the pressure at the point of (x, y, z); pini is the initial formation pressure, MPa; xw, yw, zw are the coordinates of any point in the wellbore; rw is the radius of the wellbore; and pwf(i) is the pressure of the bottom hole. For the convenience of derivation and calculation, we selected the half length of the main wellbore as the characteristic length, obtaining the dimensionless parameter as eq 3where Kh and Kv are horizontal and vertical permeabilities, respectively, mD; Ct is the total compression coefficient, MPa–1; t is time, days; h is the reservoir thickness, m; μ is the viscosity of fluids, mPa·s; φ is porosity, %; lp is the perforation length, mm; and dp is the perforation diameter, mm. Considering the trajectory of the herringbone well, the 3D spatial distribution characteristics of the herringbone well production section are shown in Figure 2 and the trajectory described is shown in eqs 4a and 4b. The herringbone well has N branches, the length of the main wellbore is L and divided into M microsegments, and the jth branch length is Lbj and divided into T microsegments. Assuming that the original coordinate of the herringbone well production section is M0(x0, y0, z0), the spatial description of any perforation v of the ith microsegment of the main wellbore is shown as eq 4aThe kth infinitesimal segment of the jth branch wellbore with any point Rjk(xjk, yjk, zjk) satisfies the following mathematical expressions, as shown in eq 4bwhere np is the perforation density, holes/m; ΔLm and ΔLbj are the infinitesimal segment lengths of the main wellbore and the jth branch, respectively; (xmi,v, ymi,v, zmi,v) is any point of on vth perforation of the ith microsegment of the main wellbore; θmk and αmk are the deviation angle and the altitudinal angle of the ith infinitesimal segment of the main wellbore, deg; θjk and αjk are the deviation angle and the altitudinal angle of the kth infinitesimal segment of the jth branch, deg; γi,v is the angle between the z axis and the vth perforation of the ith infinitesimal segment of the main wellbore, γi,v = γi,v(θi, αi, ω, v); χi,v is the angle between the x axis and the projection line of the vth perforation of the ith infinitesimal segment in the xoy plane, χi,v = χi,v (θi, αi, ω, v); ω is the perforation phase angle, deg; s is the integer part of ; and δxj, δyj, and δzj are the projections onto the x-axis, y-axis, and z-axis of (Lmfj – t·ΔLm).

Green’s function and Newman’s product principle are used to solve the differential equation; the obtained infinite spatial point-source solution in the Laplace domain is shown as eq 5where Δp(xD, yD, zD) is the dimensionless pressure drop at the point of (x, y, z); tD is the dimensionless time; τD is dimensionless at the τ time; φ is the point-source strength; and RD is the dimensionless distance between (xD, yD, zD) and (xwD, ywD, zwD).For the pressure distribution of herringbone wells in an infinite formation, the method of the mirror image and the theory of potential superimposition can be utilized to superimpose the pressure generated by each microsegment of the wellbore and to obtain the total pressure in the Laplace domain generated by the entire herringbone well, as shown in eq 6Using the Poisson summation formula, we attained the point-source solution, as shown in eq 7where s is the variable corresponding to t in Laplace’s domain, q̅(s) is the productivity in Laplace’s domain, and K0 is the Bessel function. Model I is the perforated reservoir flow model of the main wellbore, the production section shows all perforations, and the pressure drop generated by the main wellbore at any point in the reservoir is the result of all perforations. The main wellbore production interval is divided into N infinitesimal intervals by the idea of infinitesimals. Regarding the perforation as a line source, by integrating the point-source solution along the perforation depth, we obtained the pressure drop generated by a single perforation at the point M (xw, yw, zw), as shown in eq 8Taking account of the interference between perforations, the pressure distribution of the main wellbore of the herringbone well at the point M (xw, yw, zw) in the reservoir is obtained as eq 9, according to the pressure drop superposition principlewhere qmr(i) is the inflow into the wellbore from all perforations in the ith infinitesimal segment of the main wellbore and v is any perforation.Model II is the branch wellbore reservoir flow model with an open hole; for the branch wellbore reservoir seepage model of the herringbone well, the production section is in direct contact with the reservoir. Therefore, the branch wellbore can be regarded as a line source, which is composed of many persistent point sources, and the fluid flow direction of the reservoir to the wellbore conforms to the three-dimensional seepage partial differential equation. According to the study of the point-source solution, the pressure drop generated by the kth element of the jth branch at the point M (xw, yw, zw) in the reservoir is shown by eq 10When all of the branch wellbores are produced at the same time, the pressure drop generated at any point M (xw, yw, zw) in the reservoir can be expressed by eq 11where qbr(j, k) is the radial inflow of the kth infinitesimal segment of the jth branch. Considering the interference between wellbores and perforation, according to the principle of pressure drop superposition, the pressure drop generated at the point M (xw, yw, zw) in the reservoir during the simultaneous production of the main wellbore and the branch wellbore of the herringbone well is shown in eq 12Therefore, as long as the herringbone production section track can be described by eqs 4a and 4b, the relationship between pressure, productivity, and time can be solved by eq 12 under the condition of infinite diversion.

It was assumed that the flow in the wellbore is infinite and the bottom hole pressure of each well interval is equal. The middle point of each infinitesimal segment is taken as the reference point; based on the transient pressure drop model of the herringbone well, the transient productivity equations can be obtained, as shown in eq 13where pm(M,v)m(M,v) is the interference between the vth perforation of the Mth microsegment in the main wellbore; pm(M,v)b(N,TN) is the interference between the vth perforation of the Mth microsegment in the main wellbore and the N branch Tth microsegment; pb(N,TN)(N,TN) is the interference between the N branch Tth microsegment; and the meanings of other parameters are the same as before. When the total pseudopressure drop of the wellbore wall is known, the productivity of the herringbone well and the inflow distribution along each well interval can be obtained by solving the above equation.

Taking the perforating completed herringbone well in block ChengBei 31–37, which is an offshore oilfield in Shengli oilfield, the well was put into production on June 15, 2019, and the production layer was located in zone III. On June 16, 2003, the daily production was 120.2 m3/d. Figure 3 shows the layout of the herringbone well. We used the ECLIPSE 100 to simulate the cases of microsegments 5, 50, and 100. Taking the 75 microsegment numerical model as an example, the grid system is 35 × 15 × 33. The block size DX = 10 m and DY = 10 m and the DZ in the oil zone (z < 33) is 1 m and in the water zone (z = 33) is 10 m. Therefore, the reservoir model size is similar to the data given in Table 1. We also attached a big enough Fetkovich Aquifer (volume is 1012 m3) under the water cell to simulate the constant pressure of bottom water. The four-branch horizontal well in the model is shown in Figure 4. The actual production data of the herringbone well in block ChengBei 31–37 is shown in Figures 4 and 5. We can see that the productivity has an average of 118.17 m3/d. According to the derived productivity formula, we calculated the productivity of the ChengBei 31–37 block to be 110.75 m3/d. The relative error is small. When the given production time is 100 days, the simulation results and the theoretical results are shown in Figure 6. The numerical simulation of ECLIPSE 100 considers the frictional pressure loss and uses the Fanning friction factor to calculate the frictional pressure. Based on the Reynolds number Re, in the “uncertain region” (2000 < Re < 4000), E100 uses a linear interpolation between the values at Re = 2000 and 4000. Our model was derived under the condition of an infinite reservoir and does not consider the frictional pressure loss and other kinds of pressure losses in the wellbore. Thus, our results have smaller pressure losses than the simulation results and the productivity is bigger than that of the simulation results. Compared with the reservoir numerical simulation, the productivity model proposed in the present work can calculate the pressure and productivity of the herringbone well (Figure 6) and also shows that the productivity and the pressure drop affected by the microsegment number can be ignored when the segment number is above 70 for the case presented in this paper. The case study shows that the accuracy of the calculation results is reasonable and reliable. The theoretical method can conveniently and rapidly evaluate the parameters that influence the herringbone wells (main and branch wellbore structure, tubing size, and perforation parameters), as well as the influence of the reservoir parameters on productivity. It can serve as a preliminary guide to optimize the design of herringbone well structures and tubing parameters in actual applications.

When the production differential pressure was 1 MPa, we selected the reservoir points (0, 0, 10) and (50, 0, 5) and the wellbore wall points (50, 0, 10) and (100, 0, 10) in the reservoir. The change of transient pressure in the reservoir over time was studied, and the points were selected to represent the changes of pressure at different positions in the reservoir and wellbore; the results are shown in Figure 7. Figure 7a shows the herringbone well model in the grid system. Figure 7b shows the changes in transient pressure over time at the reservoir point. With the increase in production time, the pressure at the same position of the reservoir gradually increases and eventually tends to stabilize, which also indicates that when the pressure becomes stable, there are differences in the pressure at different positions; the pressure at the point of the wellbore (50, 0, 5) is greater than the point of (0, 0, 10). Figure 7c shows the changes in transient pressure over time at the wellbore wall point. When tD ≤ 1, the pressure increases with the changes of time; when tD > 1, the pressure tends to stabilize, and the pressure at the middle point of the wellbore (100, 0, 10) is greater than that at the end of the wellbore (50, 0, 10); then, the transient seepage reaches to a stable flow. Figure 8 and Table 2 show the transient productivity of the herringbone well at different times. With the increase in production time, the output of the herringbone well decreased continuously, then tended to a certain value, and the flow reached a pseudosteady state. This is because early seepage was limited to the vicinity of herringbone wells, the flow state was unstable, and the production decreased rapidly; with the increase in time, the pressure wave propagated to the area far away from the reservoir, and the flow tended to be stable. The model constructed in this paper is of great significance in analyzing the changes in the wellbore inflow and productivity over time. Figure 9 shows the wellbore inflow distribution of herringbone wells under a steady flow. The radial inflow of the main wellbore was high at both ends and low in the middle, and the radial flow suddenly decreased at the intersection due to the interference between the main wellbore and the branch wellbore. The degree of interference varied with the position of the branch wellbore. The radial flow near the heel ends of branches 1 and 2 was greater than that of branches 3 and 4, and the toe ends were opposite.

In this paper, we studied the perforating parameters including perforation density, perforation diameter, perforation depth, and perforation phase angle. The study of the effect of perforating parameters on transient productivity provides a theoretical basis and reasonable construction scheme for herringbone well completion construction. The relationship between perforation parameters and productivity ratio is shown in Figure 10. The productivity ratio is defined as the ratio of the theoretical value of herringbone well productivity of perforated completion to that of open-hole completion. Figure 10a shows the change curves of the productivity ratio with the perforation depth at different production times; with the increase in the perforation depth, the productivity ratio increases, and the gap between the different perforation depths becomes smaller as production time increases. The curve shows a convergence phenomenon. Figure 10b shows the change curves of the productivity ratio with the perforation diameter at different production times; with the increase in the perforation diameter, the productivity ratio increases, and when the diameter is larger than 15 mm, the increase range is less. Figure 10c shows the curves of the productivity ratio with perforation density at different production times; it can be seen from the figure that the perforation density has a great influence on productivity. With the increase in perforation density, the contribution of perforation density to productivity increases, and when the perforation density is more than 18 holes/m, the increase range is little; this is because the interference between perforations increases. Figure 10d shows the curves of the productivity ratio with the perforation phase angle at different production times; the calculation results show that the productivity ratio at the 60 and 90° phase is equivalent; 45, 120, and 180° phases is similar, the best is 90°, and the worst is 360°. Tables 3a and 3b show the orthogonal test data and results, and the effect of completion parameters on herringbone well productivity was analyzed by designing orthogonal tests. Based on the analysis of the test results and the extreme difference, the effect of perforating parameters on productivity is as follows: perforating density > perforating diameter > perforating depth > perforating phase angle. This provides guidance for optimizing perforating parameters.

The technology of selective perforation is economical and efficacious to improve herringbone well production. Through selective perforation design, the cost can be saved and the lateral wells can have higher productivity. The main wellbore of the herringbone well was divided into five sections for selective perforation to study the changes in perforation inflow distribution and productivity along each section. The design of the selective perforation scheme is shown in Table 4. The radial inflow rate of each microsegment of the wellbore can be obtained via the proposed coupling model, as shown in Figure 11. The productivity of the selective perforation scheme is shown in Figure 12 and Table 5. The total productivity can be obtained by adding up the radial flow of each microsegment using eq 14where Qsc is the total productivity, m3/d; nΔt is the t time; and qi(nΔt) is the productivity of the ith microsegment in t time. Figure 11 shows the radial inflow profile of the wellbore during selective perforation of the main wellbore. It can be seen that the perforation inflow distribution along each horizontal section of the main wellbore was in a “U” shape. The productivity of the selective perforation scheme is shown in Figure 12 and Table 5. We can see that increasing the density of a perforated interval can improve the productivity of herringbone wells; when the perforation density at the heel end of the branch wellbore increases, the lateral well productivity would significantly increase, and the productivity of perforation along the whole interval can be basically achieved.

According to the principles of pressure superposition and mirror reflection and the concept of infinitesimal line congruence, the transient productivity model of perforated herringbone well was derived, where interference between the branch wellbore and the main wellbore and the confluence from the branch to main wellbore are considered; the theoretical method can conveniently and rapidly evaluate the parameters that influence the herringbone wells. It can serve as a preliminary guide to optimize the design of herringbone well structures and tubing parameters in actual applications. With the increase in production time, the pressure at the same position of the reservoir gradually increases and eventually tends to stabilize; the pressure at the middle point of the wellbore is greater than that at the end of the wellbore. With the increase in production time, the productivity of the herringbone well decreased continuously, then tended to a certain value, and the radial inflow of the main wellbore was high at both ends and low in the middle. The radial flow suddenly decreased at the intersection, the radial flow near the heel ends of branches 1 and 2 was greater than that of branches 3 and 4, and the toe ends were opposite. With the increases in perforation density, diameter, and depth, the productivity ratio also increases. When the diameter is larger than 15 mm, the extent of increase is small; when the perforation density is more than 18 shot/m, the increase range is less. The productivity ratio at the 60 and 90° phase is equivalent; 45, 120, and 180° phases is similar, the best is 90°, and the worst is 360°. The effect of perforating parameters on productivity are as follows: perforating density > perforating diameter > perforating depth > perforating phase angle. When the selective completion perforation technology is adopted, the shot density at the end of the wellbore is increased; the productivity of herringbone wells can be improved economically and effectively. The study provides a theoretical basis for the improvement and development of perforation completion technology.