# Horizontal Well Completion with Multiple Artificial Bottom Holes Improves Production Performance in Bottom Water Reservoir

Authors: Haidong Wang; Yikun Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The horizontal well completion with stinger is usually used to control the bottom water cone. Although the pressure profile and the inflow profile along the horizontal wellbore can be divided into two parts by the stinger, these profiles have not really flattened. In order to flatten the pressure distribution and inflow distribution further, it proposes a new technology. This new horizontal well has multiple artificial bottom holes (MABH) along the wellbore and it has application potential. In order to verify the effectiveness of MABH technology, a model of horizontal well completion with MABH was established, and the production performance of different water cone control technologies was analyzed: conventional horizontal well, stinger completion horizontal well, and MABH completion horizontal well. The results show that the MABH technology has more advantages than the stinger technology. The uniformity of pressure distribution of the 6-MABH horizontal well is 55% higher than that of the horizontal well with string technology, and the uniformity of inflow distribution is increased by 65.25%. At the same time, although the operation of MABH technology is very simple, it should follow a rule of MABH installation: the position of the first MABH should be set at 242.5 m from the heel hole of the horizontal wellbore, and the other interval is 92.4 m.


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## Body

## 1. Introduction

The water cone is formed due to the pressure loss from the toe to the heel of the horizontal well [1, 2]. It causes less inflow contribution from the upstream portion near the toe than from the downstream portion of the heel [3–5]. Once the bottom water breaks through into the wellbore, the water cut will rise rapidly, the production efficiency will drop significantly, and the life of the well will be greatly shortened. In order to solve this problem, some studies have been proposed as follows: the method of variable density perforation [2, 6] can delay the breakthrough time of water. Landman and Goldthorpe [4] established a predictive model that can calculate the inflow profile along the wellbore. In addition, the best variable density perforation technology [7] is also reported. Although this method has the effect of controlling the bottom water cone, it cannot solve the problem of pressure loss from the heel to the toe of the horizontal well.In the early 1990s, Norwegian Hydropower engineers first proposed the concept of an inflow control device (ICD) and developed a tortuous flow control device to balance the influx of various parts along the horizontal well. Subsequently, the oil service company developed pipeline channel ICD, orifice plate ICD, automatic mixing ICD, and nozzle ICD [8–12]. Now ICD technology has been well developed. However, ICD still has some problems, such as the problem of being eroded and blocked by particles in the fluid.On the other hand, technology of horizontal well completion with stinger was proposed [13]. It is pointed out that the stinger technology can delay the bottom water cone of long horizontal wells. The principle is to redistribute the pressure distribution of the hole by changing the actual position of the bottom hole. The optimal length of the stinger under various oil-water viscosity ratios and oil column thickness conditions were studied [5]. The stinger technology can make the wellbore pressure distribution flat, but the uniform effect is not ideal. Therefore, the unified problem of stinger technology should be studied in detail [14].In this paper, a new method is proposed to further flatten the pressure profile and inflow profile along the horizontal wellbore. This method is horizontal well completion with multiple artificial bottom holes (MABH). At the same time, the pressure profile and the inflow profile along the new horizontal wellbore have been theoretically studied.


## 2. Horizontal Well Completion with MABH

In this article, we set up some independent annulus tubes through MABH to connect the base pipe (see Figure1) and then use these MABH and annulus tubes to convert the unidirectional flow in the base pipe into multidirectional flow. Likewise, it is foreseeable that the pressure distribution along the base tube will exhibit periodic wavelets (red line) and that the red line will be flatter than the conventional pressure distribution (blue dotted line). Similarly, as can be seen from Figure 1, the pressure drop difference between the reservoir and the new wellbore (N1 ≈ N2 ≈ N3) is much better than the pressure drop difference between the reservoir and the conventional wellbore (C1 < C2 < C3). Then compare the new inflow profile (N11 ≈ N22 ≈ N33) with the traditional inflow profile (C11 > C22 > C33). Therefore, the heel-toe effect will be weakened.Figure 1
Technology of horizontal well completion with MABH.


## 3. Calculation Method

In order to explore the use of MABH technology to reduce the efficiency of the heel-toe effect, relevant theoretical calculations were carried out. The detailed calculation method is as follows.Wellbore flow model: divide the horizontal wellbore intoN segments, and remove one segment for force analysis (Figure 2).Figure 2
Flow analysis of the wellbore’s infinitesimal section.According to the law of conservation of mass, the equation can be expressed as follows:(1)ρ1A1v1−ρ2A2v2+πDΔxρvrγ=∂ρAΔx∂t,where ρ is the fluid density in the pipe, kg/m3; A is the cross-sectional area of the tube, m2; v is the flow velocity, m/s;γ is the opening coefficient of the base pipe; t is the flow time, s;Δx is the length of the infinitesimal segment, m; and D is the inner diameter of the pipe, m.WhenΔx becomes 0, according to the definition of derivative, equation (1) can be acquired as follows:(2)∂ρAv∂x+∂ρA∂t=πDρvrγ.Then, according to the principle of momentum, the equation can be expressed as follows:(3)P1A1−P2A2−ρgAΔxsinθ−τaπDΔxγ=ρ2A2v22−ρ1A1v12+ρrArvr2−∂vρAΔx∂t,where g is the gravitational acceleration, m/s2;θ is the angle between wellbore axis and horizontal direction, degrees; and τa is the average frictional stress, Pa.Assuming that the flow rate is stable, and when ∆x becomes 0, combining equations (2) and (3), the pressure drop ΔPi in the base pipe can be expressed as follows:(4)ΔPi=ρgsinθΔxi+ζ4τaγDΔxi+Ω64ρQiqriπ2D4γΔxi,where Qi is the flux of ith segment in the wellbore, m3/s; qri is the influx rate of the ith wellbore segment, m3/s; ζ is the correction coefficient caused by wear and corrosion in the wellbore, dimensionless; and Ω is the correction coefficient caused by variable mass pipe flow in the wellbore, dimensionless.Similarly, pressure dropΔPi in the annulus tube can be represented as follows:(5)ΔPi=ρgsinθΔxi+ζ4τaDΔxi.The average frictional stressτa can be represented as follows:(6)τa=fρv28.According to the research results of Ouyang [15], f is shown in Table 1.Table 1
Value off.
Flow stateVarious mass flowsConstant mass flowLaminarf=f01+0.043NRe,w0.6142f=16.0/NReTurbulentf=f01+0.0153NRe,w0.3978f−0.5=−4.0logε/3.7D+1.255/f−0.5NReThe following equation gives the friction factor:(7)1f0=−2logε3.7D−5.02NRelogε3.7D−5.02NRelogε3.7D+13NRe.Here,(8)NRe,w=4ρQiμπD,NRe=4ρqiμπD,where f0 is the correction coefficient caused by variable mass pipe flow in the wellbore, dimensionless; NRe is the Reynolds number, dimensionless; and ε is the friction factor, dimensionless.Reservoir inflow model: firstly, we make the following assumptions: ① The bottom water reservoir is homogeneous and infinite. ② There is a closed boundary at the reservoir top and a constant pressure boundary at the bottom. The distance between the upper and lower boundaries ish, and the distance between the horizontal wellbore and the lower boundary is zi. The reservoir temperature is constant. ③ The length of the horizontal wellbore is much larger than its diameter. ④ The model is assumed to be single-phase flow. If the pressure drops below the bubble point, the model is not available. ⑤ The flow in the reservoir follows Darcy’s law.The above assumptions will affect the calculation results of the total output. However, the focus of this article is to study the performance of the inflow profile and the pressure profile of horizontal well. Therefore, these assumptions do not affect the results of this article.First, the horizontal wellbore is divided intoN sections, and the penetration velocity of the ith section surface can be expressed as follows:(9)v=qiS=qi2πrΔxiγ,where qi is the production rate of the ith wellbore segment, m3/s; S is the perforated opening area on the pipe, m2; v is the seepage velocity, m/s; S is the seepage area, m2; γ is the opening coefficient of the base tube; Δxi is the length of ith segment, m; and r is the inner radius of the pipe, m.In addition, the percolation rate can be represented as follows [16, 17]:(10)v=−kdpμdr=dpdr,dpdr<dpdrc,−knondpdrm,dpdr>dpdrc,where k is the permeability in the reservoir, m2;μ is the viscosity, Pa⋅s; knon is nonlinear permeability coefficient near wellbore, m2; dp/dr is pressure gradient, MPa/m; m is the experimental constant, and it is between 1.2∼1.5; dp/drc represents the end point of linear pressure gradient, between 4∼10, MPa/m.Whendp/dr<dp/drc, the potential of the ith segment in the infinite space can be obtained as follows:(11)∅=qi2πΔxiγlnr+C.Through the mirror image method (see Figure3) and the principle of potential superposition, the potential at a random point β (x, y, z) can be obtained as follows:(12)∅β=∑∅+Q+∑∅−Q.Figure 3
Schematic of mirror image.Here,(13)∑n=−∞+∞∅+Q=qi4πΔxiγ∑n=−∞+∞lnx−xi2+y2+z−ω12x−xi2+y2+z−ω22+C,∑n=−∞+∞∅−Q=−qi4πΔxiγ∑n=−∞+∞lnx−xi2+y2+z−δ12x−xi2+y2+z−δ22+C.Furthermore, the potential ofβ (x, y, z) at a random position caused by entire horizontal wellbore can be obtained as follows:(14)φβ=∑n=1L∅β+C.Here,(15)ri,j,+Q=x−xi2+y2+z−ωj2,j=1,2,ri,j,−Q=x−xi2+y2+z−δj2,j=1,2,ω1=4nh+2h−zi,ω2=4nh+zi,δ1=4nh+2h+zi,δ2=4nh−zi,n=0,±1,±2,±3,…,±∞.Therefore, the reservoir flow inith segment can be described as follows:(16)Pe−Pwf,i=∑n=1L∅β,where Pe is the pressure at the constant pressure boundary, Pa; Pwf,i is the bottom hole pressure of the ith segment, Pa; ∅ is the potential, Pa; xi is the x-coordinate value of the ith segment, m; zi is the z-coordinate value of the ith segment, m.The vector representation is given by(17)α⟶=A⟶Q⟶.Here,(18)α⟶=Pe−Pwf,1Pe−Pwf,2·········Pe−Pwf,L−1Pe−Pwf,L,A⟶=a1,1⋯aL,1⋮⋱⋮a1,L⋯aL,L,Q⟶=q1q2·········qL−1qL,aM,i=∅β.Whendp/dr>dp/drc, combine equations (9) and (10), flowing near the wellbore of the ith segment can be described as follows:(19)Pe−Pwf,i=mm−1knonqi2πΔxiγrem−1/m−rwf,im−1/m.Therefore, the actual reservoir flow model is as follows:(20)Pe−Pwf,i=Δ∑n=1L∅β,dpdr<dpdrc,mm−1knonqi2πΔxiγrem−1/m−rwf,im−1/m,dpdr>dpdrc.In addition, there is oil-water flow interaction in production. Therefore, combining equations (4) and (20), the oil-water wellbore flow model and the reservoir flow model can be obtained as follows:(21)ΔPi=ρw+ρogsinθΔxi+ζ4τaγDΔxi+Ω64ρQw,i+Qo,iqw,ri+qo,riπ2D4γΔxi,Pe−Pwf,i=∑n=1L∅β,dpdr<dpdrc,mm−1kw,non+ko,nonqw,i+qo,i2πΔxiγrem−1/m−rwf,im−1/m,dpdr>dpdrc.where ρw is the water density in the pipe, kg/m3; ρo is the oil density in the pipe, kg/m3; Qw,i is the water flux in the ith section of the wellbore, m3/s; Qo,i is the water flux in the ith section of the wellbore, m3/s; qw,ri is the water inflow rate in the ith wellbore segment, m3/s; qo,ri is the ith water production rate of the wellbore segment, m3/s; kw,non is the nonlinear permeability coefficient of the water phase, m2; ko,non is the nonlinear permeability coefficient of the water phase, m2; qw,i is the ith water production rate of the wellbore segment, m3/s; qo,i is the oil production rate of the ith wellbore segment, m3/s.


## 4. Coupling Model Solution

The focus of this article is to study the performance of the inflow profile and pressure profile along the horizontal well. Therefore, single-phase flow or two-phase flow does not affect the results of this article and then the coupling solution takes single-phase flow as an example.Step 1: if a set of bottom hole pressures along eachith section of the horizontal wellbore are given Pwf,iPwf,1,Pwf,2,…,Pwf,L, the corresponding radial inflow qii=1,2,…,L can be calculated by equation (20) through the Gaussian elimination method.Step 2: divide the horizontal wellbore intoMn segments (Figure 4). The Mn part of the wellbore flow will be controlled by the heel hole. Assuming that the radial inflow along the horizontal wellbore is qjj=1,2,…,M1, the corresponding pressure difference of all pipe sections in the base pipe can be calculated by formula (4), ΔPwf,jj=1,2,…,M1.QM1 is known, and the value of QM1 can be represented as follows:(22)QM1=∑j=1N1qj,The centre pressure of theith segment of M1 wellbore is represented as follows:(23)Pwf,j′=Pwf,j−0.5+0.5ΔPwf,j.The end pressure of theith segment of M1 wellbore is represented as(24)Pwf,j+0.5=Pwf,j−0.5+ΔPwf,j,wherePwf,1′=Pwf,heel. A new group of bottom hole pressures can be obtained in section M1, Pwf,j′j=1,2,…,M1.Step 3: theM2 section is controlled by the first artificial bottom hole. In section M2, it is divided into two parts: M21 is the left part, and M22 is the right part.(1)
For segmentM21, firstly assume a group of radial influx, qjj=M1+M21−1,M1+M21−2,…,M1+M21−M21. Then the corresponding differential pressure ΔPwf,jj=M1+M21−1,M1+M21−2,…,M1+M21−M21 of random segments in the wellbore M21 can be calculated by equation (4).From segment (M1 + M21) to segment (M1), the centre pressure of the ith segment is represented as follows:(25)Pwf,j′=Pwf,j−0.5+0.5ΔPwf,j.From segment (M1 + M21) to segment M1, the end pressure of the ith segment is represented as follows:(26)Pwf,j+0.5=Pwf,j−0.5+ΔPwf,j,where(27)Pwf,M1+M21−1′=Parwf,M1+M21.The pressure ofPwf,M1+M21′ can be calculated as follows:(28)Pwf,M1+M21′=Pwf,heel+ρgsinθM1+M21+4τDM1+M21.A new set of bottom hole pressures can be obtained in sectionM21, Pwf,j′j=M1+M21−1,M1+M21−2,…,M1+M21−M21.Note. If the value of Pwf,M1+M21−M21′ calculated in Step 3 is different from that of Pwf,M1′ calculated in Step 2, there will be two kinds of discussions:①
Pwf,M1+M21−M21′≤Pwf,M1′. Part of Pwf,j′j=1,2,…M1 in Step 2 is not accurate and it should be recalculated. Calculation should be started from segment (M1 + M21) to (M1+M21−M21−i) until Pwf,M1+M21−M21−i′=Pwf,N1−i′. Then, an accurate set of bottom hole pressures can be obtained in section M1+M21, Pwf,j′j=1,2,3,…,N1−i−1,M1+M21−M21−i,M1+M21−M21−i+1,…,M1+M21.②
Pwf,M1+M21−M21′>Pwf,M1′. Part of Pwf,j′j=M1+M21−1,M1+M21−2,…,M1+M21−M21 in Step 3 is not accurate and it should be recalculated. Calculation should be started from segment M1 to segment (M1 + i) until Pwf,M1+i′=Pwf,M1+M21−N21−i′. Then, an accurate set of bottom hole pressures can be obtained in section M1+M21, Pwf,j′j=1,2,3,…,M1,M1+1,…,M1+i,M1+M21−N21−i+1,…,M1+M21−1,M1+M21.(2)
For segmentM22, first assume a group of radial influx, qjj=M1+M21+1,M1+M21+2,…,M1+M21+M22. The corresponding differential pressure ΔPwf,jj=M1+M21+1,M1+M21+2,…,M1+M21+M22 of random segments in the wellbore M22 can be calculated by equation (4).The centre pressure of theith segment of M22 wellbore is represented as follows:(29)Pwf,j′=Pwf,j−0.5+0.5ΔPwf,j.The end pressure of theith segment of M22 wellbore is represented as follows:(30)Pwf,j+0.5=Pwf,j−0.5+ΔPwf,j,where(31)Pwf,M1+M21′=Parwf,M1+M21+1.The pressure ofPwf,M1+M21′ can be calculated as(32)Pwf,M1+M21′=Pwf,heel+ρgsinθM1+M21+4τDM1+M21.A new set of bottom hole pressures can be obtained in sectionM22, Pwf,j′j=M1+M21+1,M1+M21+2,…,M1+M21+M22.The main flowQM2 of M22 segment is known and can be expressed as follows:(33)QM2=∑j=M1+1M1+M21+M22qjj=M1+1,M1+2,…,M1+M21+M22.Step 4: repeat Step 2 and Step 3 to calculate corresponding pressure distribution controlled by the next annulus bottom hole.Step 5: compare the pressurePwf,j′ with Pwf,i, if the inaccuracy meets the engineering inaccuracy requirement, and stop calculating or replacing Step 1 until the desired value is reached. The pressure profile and the inflow profile will be obtained.Figure 4
Schematic of horizontal wellbore flow with MABH.


## 5. Results and Discussion

This article focuses on a new technology to improve the impact of horizontal wellbore pressure profile and production profile, so a single-phase flow model is used. Here, we have designed three schemes (Table2) to predict production performance. The basic data of well XU-2 is given in Table 3. The calculation method of scheme 2 is based on Permadi and Wibowo [5].Table 2
Schemes of different water cone controlling method.
SchemeMethod1234567Table 3
Parameters related to XU-2 horizontal well.
ParametersValuePe (MPa)12k (10−3 μm2)500Lh (m)800Dbase (mm)100Pwf,heel (MPa)11ρ0 (g/cm3)0.95μ0 (MPa·s)70h (m)30γ0.3kro (dimensionless)0.82Dstinger/Da (mm)40/20krw (dimensionless)0.26


## 6. Optimal Water Control Method

Figure5 shows three pressure distributions along the horizontal well. Here, (4), (5), and (6) represent the difference between the maximum pressure and the minimum pressure along the horizontal well (blue line). As shown in Scheme 1, the pressure distribution along the horizontal well increases monotonously, and the value of (4) is 0.0964 MPa. In Scheme 2, it can be seen that the pressure distribution along the horizontal well has been divided into two parts by the stinger and is funnel-shaped. The value of (5) is 0.0889 MPa. In Scheme 3, the pressure distribution of MABH horizontal wells shows wavelets, indicating that the pressure distribution of horizontal wells is divided into several wavelets using MAHB. Here, the value of (6) is only 0.04 MPa. In summary, the pressure curve of Scheme 2 is better than the pressure curve of Scheme 1, and the uniformity of the former is 7.8% higher than that of the latter. The pressure curve in Scheme 3 is better than the pressure curve in Scheme 2, and the uniformity of the former is improved by 55% compared with the latter. Obviously, the new pressure profile of Scheme 3 is production desired.Figure 5
Pressure profile and inflow profile with different completions.Figure5 also shows the distribution of inflow profiles along horizontal wells. Here, (1), (2), and (3) represent the difference between the maximum inflow and the minimum inflow along the horizontal wellbore, respectively. As can be seen from Scheme 1, the value of (1) is 0.0561 (m3/d)/m. It can be seen from Scheme 2 that the difference can be reduced by using a stinger, but the inflow fluctuation range between the funnel point and the end of the wellbore is still very large, and the value of (2) is 0.0468 (m3/d)/meter. It is worth noting that in Scheme 3, by using MABH, the value of (3) is only 0.01626 (m3/d)/m. Compared with (3) and (2), the uniformity of the former is improved by 65.25% than the latter. Therefore, it can be concluded that MABH can further flatten the inflow profile.


## 7. Optimal Installation Interval of MABH

Figure6 shows the pressure profile and inflow profile. The inflow profile gradually flattens and periodically flattens as the number of MABH increases. In addition, when the same amount of MABH is installed at the toes, the uniformity of the pressure profile and the inflow profile will be more uniform than the installation at the heel. When the length of the base pipe controlled by the wellbore hole is less than 242.5 m and the length of the MABH controlled base pipe is less than 92.4 m, the difference between the maximum inflow and the minimum inflow along the horizontal wellbore does not exceed 3% of the average value. Therefore, we set the first MABH to 242.5 m along the horizontal well, and other installation intervals to 92.4 m. But this rule only applies to wellbore with the above parameters. For other wellbores, we can recalculate the optimal length of the base pipe controlled by MABH according to the above method.Figure 6
Pressure profile and inflow profile with different installation conditions.


## 8. Operation and Challenges

The annulus tube can be oval and can be extruded through a seamless steel tube. It is locked between the base pipe and the screen pipe by the support ring and crossover. Here, you can set a MABH on the cross ring and connect the MABH to the annulus and the base pipe (see Figures1 and 7(e)), respectively. Then, Figures 7(a)–7(e) show the ground installation process of the production unit with MABH. Install the screen spacers, wire wrap screen, crossover with MABH and support rings, annulus tubes, and screen shroud at the corresponding positions of the base pipe until the production unit shown in Step 6 is formed. Then, it is possible to form horizontal wells with MABH by connecting each production unit in accordance with the steps given in Figures 7(g), 7(h), and 7(i). In particular, after tightening the connection of the base pipe, annulus tubes are automatically further inserted upstream of annulus tubes on both sides, which has a good sealing and self-locking function. The completion structure of MABH horizontal well is complicated, but the operation is simple and convenient. Similarly, when the length of the annulus tube is designed to be 10 m and the cross section is designed to be 314 mm2, the cost of an annulus tube is about $ 610. This is the challenge of developing low-yield fields. But it is suitable for high-yield bottom water oil fields or gas fields.Figure 7
Operation process of horizontal well completion with MABH.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 9. Further Discussion

The reason for the multidirectional flow in the base pipe can be obtained from the following equation:(34)ΔP=ΔPg+ΔPs+ΔPacc,various mass flowsbase pipe flow,ΔPg+ΔPs,constant mass flowannulus tube flow,where ΔP is the pressure drop along the wellbore, MPa; ΔPg is the gravity pressure drop, MPa; ΔPs is the friction pressure drop, MPa; and ΔPacc is the acceleration pressure drop, MPa.The results show that, compared with the constant mass flow in the annulus tube, the various mass flows in the base pipe overcome the greater flow resistance. The reason is because there are perforated openings in the base pipe, but not in the annulus tube, so the additionalΔPacc in the base pipe will be formed by radial inflow acceleration. Therefore, the lower pressure in the base pipe at the toe can be controlled by using MABH, and then the fluid at the toe can easily bypass the heel (as shown in Figures 1 and 8). The influx of toes in the wellbore has increased, which is beneficial for reducing the heel-toe effect of horizontal wells. In short, MABH technology should be considered when controlling the water cone problem of horizontal wells for better reservoir management.Figure 8
MABH technical mechanism.
(a)(b)(c)


## 10. Conclusions

Here are some suggestions and conclusions:(1)
A new MABH completion method is established to solve the water coning problem. The main mechanism of MABH technology is to reduce the original pressure in the conventional horizontal wellbore by using MABH. The original streamline from the toe to the heel in a conventional horizontal wellbore is changed by multidirectional flow. Then, the pressure distribution along the base pipe can be flattened, and the inflow profile along the horizontal wellbore can be made uniform.(2)
A coupling model of horizontal well with MABH and reservoir is proposed, which can be used to calculate the pressure loss and inflow difference along the horizontal well. Then, the dynamics of water control technologies are compared based on the output. The influence of the type of wellbore on the development effect is also discussed. Both methods can effectively control the pressure profile and the inflow profile, but the system with 6-MABH has significantly improved the pressure profile and the inflow profile than the system with stinger. The uniformity of the pressure distribution of the horizontal well using 6-MABH is 55% higher than that of the horizontal well using the stinger, and the uniformity of the inflow profile is increased by 65.25%.(3)
Although the MABH structure is complex, the operation is simple and convenient. The cost of a MABH and an annulus tube with a length of 10 m and a cross section of 314 mm2 is only $ 610, which is a challenge for the development of low-yield oil fields. At the same time, the first MABH should be set at 242.5 m away from the heel hole of the horizontal wellbore, and the other interval is 92.4 m.



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*Source: 7247480-2020-07-24.xml*