Productivity for Horizontal Wells in Low-Permeability Reservoir with Oil/Water Two-Phase Flow

Abstract

This paper presents a semianalytical steady-state productivity of oil/water two-phase flow in low-permeability reservoirs with both top and bottom boundaries closed which takes the permeability stress-sensitive and threshold pressure gradient into account. Using the similar approach as Joshi’s (1988), the three-dimensional (3D) horizontal well problem is divided into two two-dimensional problems (2D), and then the corresponding nonlinear steady seepage mathematical models in vertical and horizontal planes are established. Through the separation of variables method and equivalent flow resistance principle, the productivity equation of horizontal well is obtained. The liquid and oil productivity with different influential factors are plotted, and the related effects are also analyzed. This paper expanded the conventional productivity equations of single phase into multiphase flow which have both theoretical and practical significance in predicting production behaviors in such reservoirs.

1. Introduction

With the development of drilling technology and the reduction of its cost, more and more horizontal wells have been used in low-permeability reservoirs, fractured reservoirs, multilayered reservoirs, and bottom water drive reservoirs. And the steady-state productivity of horizontal well is always a hot topic for the petroleum engineers. After several decades of development and research, many methods, including analytical method, conformal transformation method, potential superposition method, equivalent flow resistance method, and the point source function method, were proposed to calculate it [1–9].

Merkulov [1] and Borisov [2] derived analytical productivity equation of the horizontal well with single oil phase flow. Giger et al. [3, 4] and Karcher and Giger [5] developed a concept of replacement ratio, FR, which indicates the required number of vertical wells to produce at the same rate as that of a single phase from formation well to horizontal well. Reiss [6] proposed an equation to calculate the productivity index for horizontal well. Thereafter, through subdividing the 3D flow of horizontal well into two 2D problems (flow on horizontal plane and vertical plane, resp.), Joshi [7] derived an equation to calculate the productivity of steady-state horizontal well, which is the most popular method nowadays.

Babu and Odeh [8] proposed an equation to calculate the productivity of horizontal well under the assumption that the shape of the drainage volume is box and all the boundaries are closed. Renard and Dupuy [9] derived the flow efficiency equation for horizontal well in anisotropic reservoir with the consideration of skin factor. Then, Helmy and Wattenbarger [10] and Billiter et al. [11] also proposed their corresponding equations to calculate the productivity. Anklam and Wiggins [12] obtained the steady-state productivity with the consideration of the mechanical properties of fluid flow into the wellbore. Using the steady-state point source function theory, Lu [13] achieved the productivity equations for horizontal wells under different boundary conditions.

Most of the productivity equations of horizontal well mentioned above are mainly concentrated on single oil phase flow, but the ones related to multiphase are rare. In this paper, we employ the same method described by Joshi [7] and derived a steady-state productivity formula for a horizontal well with oil/water two-phase flow problem in low-permeability reservoir, which takes the permeability stress-sensitive and threshold pressure gradient into account.

2. Theoretical Analysis

2.1. Physical Model

Figure 1 is a schematic of a horizontal well drilled in low-permeability oil reservoirs. To make the problem more tractable, the following assumptions are made: (1) the reservoir is horizontal, homogeneous with uniform thickness of ℎ, and both the top and the bottom boundaries are closed; (2) the well length and radius are 𝐿 and 𝑟_𝑤, respectively, the well is located at the center of the formation, and the pressure at the drainage boundary is 𝑝_𝑖 with the radius of 𝑟_𝑒; (3) two-phase fluid flows from the reservoir to the well at a constant bottomhole pressure 𝑝_𝑤𝑓 and ignore the gravity and capillary pressure effects.

Figure 1 

Schematic of a horizontal well in a top and bottom boundaries closed reservoir.

To simplify the mathematical solution, the 3D problem is subdivided into two 2D problems [7]. Figure 2 shows the following subdivision of the ellipsoidal drainage problem: (1) fluid flow into the horizontal well in the horizontal plane; (2) fluid flow into the horizontal well in the vertical plane.

Figure 2 

Schematic of the fluid flow to a horizontal well: (a) horizontal plane and (b) vertical plane.

2.2. Permeability Model in Low-Permeability Reservoir

In this paper, we use the exponential permeability model to describe the relationship between permeability and pore pressure [14–20]. This model is based on a permeability modulus, 𝛼_𝑘, which is defined as the following: 𝛼_𝑘=1𝑘𝜕𝑘.𝜕𝑝 (1)

Integrating of (1) with 𝑝 from 𝑝_𝑖 to 𝑝 yields 𝑘𝑘_𝑖=exp−𝛼_𝑘𝑝_𝑖−𝑝, (2) where 𝛼_𝑘 is the permeability modulus, MPa⁻¹; 𝑝 is the formation pressure, MPa; 𝑝_𝑖 is the initial pressure, MPa; 𝑘_𝑖 is the permeability at initial condition, D; 𝑘 is the permeability at current condition, D.

2.3. Threshold Pressure Gradient in Oil/Water Two-Phase Flow

According to the experimental investigations (the schematic of core sample experiments is shown in Figure 3), an appropriate form of Darcy’s law with a threshold gradient should be used as [21–28] −𝑘𝑣=𝜇𝜕𝑝,𝜕𝑟−𝐺𝜕𝑝𝜕𝑟>𝐺,0,𝜕𝑝𝜕𝑟≤𝐺, (3) where 𝑣 is velocity of the fluid flow, m/s; 𝜇 is the fluid viscosity, cp; 𝑟 is the radius, m; 𝐺 is the threshold pressure gradient, MPa/m.

Figure 3 

A schematic of core sample laboratory experiments in low-permeability reservoir.

2.4. Flow in a Horizontal Plane

Figure 2(a) shows a schematic of fluid flow to a horizontal well in a horizontal plane. The drainage area is an ellipse and introduces 𝑧=1𝐿/221𝜉+𝜉, (4)

by defining 𝑧=𝑥+𝑖⋅𝑦 and 𝜉=𝑢+𝑖⋅𝑣=𝑟_𝑒𝐻⋅𝑒^𝑖𝜃. Substituting them into (4) and then equating the real and imaginary parts yield 𝑥=1𝐿/22𝑟_𝑒𝐻+1𝑟_𝑒𝐻(𝜃),𝑦cos=1𝐿/22𝑟_𝑒𝐻−1𝑟_𝑒𝐻sin(𝜃). (5)

The real equation of the elliptical drainage area can be described as 𝑥²𝑎²+𝑦²𝑏²=1, (6) where 𝑎 and 𝑏 are the major and minor axes of the ellipse, m.

Combing (5) with (6), we have 𝑟_𝑒𝐻=𝑎+𝑏.0.5𝐿 (7)

Moreover, +𝐿/2 and −𝐿/2 represent foci of the ellipse, which have the following relationship with 𝑎 and 𝑏: 𝑎²−𝑏²=𝐿2². (8)

The drainage radius of horizontal well, 𝑟_𝑒, can be obtained by equaling the areas of a circle and ellipse, which is 𝑟_𝑒=√𝑎𝑏. (9)

Combing (9) with (8), we have 𝑎=0.5𝐿0.5+0.25+2𝑟_𝑒𝐿^40.5,𝑏=𝑎²−𝐿2². (10)

So, as shown in Figure 4, the fluid flow in the 𝑢-𝑣 plane can be viewed as a unit radius vertical well produced at a circle drainage area with the radius of (𝑎+𝑏)/(0.5𝐿). Combining the modified Darcy flow equation with the steady-state governing equation, the fluid flow in the horizontal plane can be described as 1𝑟dd𝑟𝑟𝑒^{−𝛼𝑘(𝑝𝑖−𝑝)}𝜆_𝑡d𝑝d𝑟−𝐺^′=0,𝑝_𝑟=1=𝑝_𝑐,𝑝_{𝑟=𝑟𝑒𝐻}=𝑝_𝑖, (11)(12)(13) where 𝜆_𝑡 is the total mobility of the oil/water phase, 𝜆_𝑡=𝑘_𝑟𝑜/𝜇_𝑜+𝑘_𝑟𝑤/𝜇_𝑤, which is a function of the radius, and 𝐺^′ is the equivalent threshold pressure gradient in the horizontal plane, which is 𝐺^′=𝐺𝑟_𝑒−𝐿/2)).((𝑎+𝑏/0.5𝐿−1 (14)

Figure 4 

The schematic of fluid flow in horizontal plane after conformal transformation.

The solution of mathematical models of (11)–(13) is (the detailed derivations are showed in the Appendix) 𝑝(𝑟)=𝑝_𝑖+1𝛼_𝑘ln𝑒^{𝛼𝑘𝐺′𝑟}𝑒^{−𝛼𝑘(𝐺′+𝑝𝑖−𝑝𝑐)}+𝑒^{−𝛼𝑘𝐺′𝑟𝑒𝐻}−𝑒^{−𝛼𝑘(𝐺′+𝑝𝑖−𝑝𝑐)}∫^𝑟𝑒𝐻₁𝑒^{−𝛼𝑘𝐺′𝑟}⋅1/𝑟𝜆_𝑡(𝑟)⋅d𝑟_𝑟1𝑒^{−𝛼𝑘𝐺′𝑟}1𝑟𝜆_𝑡(𝑟)d𝑟. (15)

The liquid flow rate in horizontal plane can be calculated by 𝑞_𝐿𝐻=𝐶ℎ𝑘𝜆_𝑡(1)⋅𝑟d𝑝d𝑟_𝑟=1, (16) where 𝐶 is the unit conversion factor, 𝐶=542; 𝑞_𝐿𝐻 is the liquid flow rate at reservoir condition, m³/d.

Combining (15) with (16), the liquid flow rate in the horizontal plane is 𝑞_𝐿𝐻=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑖−𝑝𝑐)}ℎ⋅𝜆_𝑡(1)⋅𝐺^′+𝑒^{−𝛼𝑘[𝐺′𝑟𝑒𝐻−(𝑝𝑖−𝑝𝑐)]}−𝑒^{−𝛼𝑘𝐺′}𝛼_𝑘∫^𝑟𝑒𝐻₁𝑒^{−𝛼𝑘𝐺′𝑟}⋅1/𝑟𝜆_𝑡(𝑟).d𝑟 (17)

The corresponding oil production rate in the horizontal plane is 𝑞_𝑜𝐻=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑖−𝑝𝑐)}ℎ𝜆_𝑜(1)⋅𝐺^′+1𝜆_𝑡(1)⋅𝑒^{−𝛼𝑘[𝐺′𝑟𝑒𝐻−(𝑝𝑖−𝑝𝑐)]}−𝑒^{−𝛼𝑘𝐺′}𝛼_𝑘∫^𝑟𝑒𝐻₁𝑒^{−𝛼𝑘𝐺′𝑟}⋅1/𝑟𝜆_𝑡(𝑟),d𝑟 (18) where 𝑞_𝑜𝐻 is the oil flow rate at reservoir condition, m³/d; 𝜆_𝑡(𝑟_𝑤) is the total mobility of the oil/water phase in the well bottomhole, D/cp.

If we do not take into account the effects of permeability stress-sensitive and threshold pressure gradient on the productivity equation in horizontal plane (𝛼_𝑘=𝐺=0) and assume only oil flow in the formation (𝜆_𝑡(𝑟)=1/𝜇_𝑜), then (17) becomes 𝑞_SOH=0.542×10⁻³𝑘_𝑖ℎ𝑝_𝑖−𝑝_𝑐𝐵_𝑜𝜇_𝑜(2()),ln𝑎+𝑏/𝐿 (19) where 𝑞_SOH is the oil flow rate at surface condition, m³/d; 𝐵_𝑜 is the oil volume factor, sm³/m³.

Equation (19) is the same as the productivity equation in horizontal plane of (1) derived by Joshi [7].

2.5. Calculation of Flow in a Vertical Plane

The fluid flow in the vertical plane with top and bottom boundaries closed reservoir can be viewed as a vertical well with the radius of 𝜉_𝑤=2𝜋𝑟_𝑤/ℎ produced in a unit circle area after the conformal transformation (as shown in Figure 5).

Figure 5 

The schematic of fluid flow in vertical plane after conformal transformation.

The mathematical models to describe the steady-state fluid flow of oil/water flow in the vertical plane are 1𝑟dd𝑟𝑟𝑒^{−𝛼𝑘(𝑝𝑖−𝑝)}𝜆_𝑡d𝑝d𝑟−𝐺^′′𝑝=0,_{𝑟=𝜉𝑤}=𝑝_𝑤𝑓,𝑝_𝑟=1=𝑝_𝑐, (20) where 𝐺^′′ is the equivalent threshold pressure gradient in the vertical plane, which is 𝐺^′′=(ℎ/2)−𝑟_𝑤1−2𝜋𝑟_𝑤/ℎ𝐺. (21)

The solution of the pressure distribution along the radius can be solved with the same method showed in the Appendix. Then, the productivity equation in the vertical plane can be calculated by the following equation: 𝑞_𝐿𝑉=𝐶𝐿𝑘𝜆_𝑡𝜉_𝑤⋅𝑟d𝑝d𝑟_{𝑟=𝜉𝑤}. (22)

The liquid and oil productivity in the vertical plane are 𝑞_𝐿𝑉=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑐−𝑝𝑤𝑓)}𝐿⋅𝜆_𝑡𝜉_𝑤⋅𝐺^′′⋅𝜉_𝑤+𝑒^{−𝛼𝑘[𝐺′′−(𝑝𝑐−𝑝𝑤𝑓)]}−𝑒^{−𝛼𝑘𝐺′′𝜉𝑤}𝛼_𝑘∫¹_𝜉𝑤𝑒^{−𝛼𝑘𝐺′′𝑟}⋅1/𝑟𝜆_𝑡(𝑟),d𝑟𝑞_𝑜𝑉=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑐−𝑝𝑤𝑓)}𝐿𝜆_𝑜𝜉_𝑤×𝐺^′′⋅𝜉_𝑤+𝑒^{−𝛼𝑘[𝐺′′−(𝑝𝑐−𝑝𝑤𝑓)]}−𝑒^{−𝛼𝑘𝐺′′𝜉𝑤}𝜆_𝑡𝜉_𝑤𝛼_𝑘∫¹_𝜉𝑤𝑒^{−𝛼𝑘𝐺′′𝑟}⋅1/𝑟𝜆_𝑡(𝑟.)d𝑟 (∗)(∗∗)

Assuming the reservoir with single phase flow and neglecting the permeability stress-sensitive and threshold pressure gradient (𝛼_𝑘=𝐺=0, 𝜆_𝑡(𝑟)=1/𝜇_𝑜) yield 𝑞_𝑜𝑉=𝐶𝑘_𝑖𝐿𝑝_𝑐−𝑝_𝑤𝑓𝐵_𝑜𝜇_𝑜lnℎ/2𝑟_𝑤. (23)

Equation (23) is similar to the productivity equation in vertical plane of (D-3) derived by Joshi [7], which proves the correctness of the productivity equation in this paper.

2.6. Horizontal Well Eccentricity

Figure 5 and (∗)-(∗∗) are obtained under the assumption that the horizontal well is located at the center of the reservoir in the vertical plane. According to Muskat’s [29] formulation for off-centered wells, the liquid production rate of a well placed at a distance 𝛿 from the mid-height of the reservoir in a vertical plane is 𝑞_𝐿𝑉=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑐−𝑝𝑤𝑓)}ℎ⋅𝜆_𝑡𝜉_𝑤⋅𝐺^′⋅𝜉_𝑤+𝑒^{−𝛼𝑘[𝐺′𝛽−(𝑝𝑐−𝑝𝑤𝑓)]}−𝑒^{−𝛼𝑘𝐺′𝜉𝑤}𝛼_𝑘∫^𝛽_𝜉𝑤𝑒^{−𝛼𝑘𝐺′𝑟}⋅1/𝑟𝜆_𝑡(𝑟).d𝑟 ∗^′

The oil production rate is 𝑞_𝑜𝑉=𝐶𝑘_𝑖𝑒^{−𝛼𝑘(𝑝𝑐−𝑝𝑤𝑓)}ℎ𝜆_𝑜𝜉_𝑤×𝐺^′′⋅𝜉_𝑤+𝑒^{−𝛼𝑘[𝐺′′𝛽−(𝑝𝑐−𝑝𝑤𝑓)]}−𝑒^{−𝛼𝑘𝐺′′𝜉𝑤}𝜆_𝑡𝜉_𝑤𝛼_𝑘∫^𝛽_𝜉𝑤𝑒^{−𝛼𝑘𝐺′′𝑟}⋅1/𝑟𝜆_𝑡(𝑟),d𝑟 ∗∗^′ where 𝛿 is the vertical distance between the reservoir center and horizontal well location, m; 𝛽=1−(2𝛿/ℎ)².

3. The Solving Method of 𝜆_𝑡(𝑟)

In order to correctly calculate the productivity of horizontal well with oil/water phase flow, we must determinate the expression of total mobility, 𝜆_𝑡(𝑟), along the radius 𝑟. The following steps are the procedure to calculate it.

Step 1. According to the relative permeability curves and the viscosity of oil and water, the relationship between 𝜆_𝑡(𝑟) and 𝑆_𝑤 can be obtained.

Step 2. When the water displacement front breaks through the oil well, the distribution of water saturation along the well radius satisfies the following Buckley-Leverett equation: 𝜋ℎ𝜙𝑟²_𝑒−𝑟²=d𝑓_𝑤d𝑠_𝑤⋅𝑄_𝑙, (24) where ℎ is the formation thickness, m; 𝜙 is the porosity, fraction; ∑𝑄_𝑙 is the cumulative fluid production, m³/d; 𝑓_𝑤 is the water ratio, fraction, 𝑓_𝑤=(𝑘_𝑟𝑤/𝜇_𝑤)/(𝑘_𝑟𝑜/𝜇_𝑜)+(𝑘_𝑟𝑤/𝜇_𝑤).
In (24), the expressions of the d𝑓_𝑤/d𝑆_𝑤 versus 𝑠_𝑤 can be calculated by the relative permeability curves.

Step 3. Combining the relationship of 𝜆_𝑡(𝑟) versus 𝑆_𝑤 and 𝑆_𝑤 versus 𝑟 derived in Steps 1 and 2, the relationship of 𝜆_𝑡(𝑟) with 𝑟 can be obtained.

Statistical results show that the relationships of 𝑆_𝑤 versus 𝑟, 𝜆_𝑡(𝑟) versus 𝑆_𝑤, and 𝜆_𝑡(𝑟) versus 𝑟 can be approximated by the following analytical functions: 𝑆_𝑤(𝑟)=𝑎₁+𝑏₁𝑟²,𝜆_𝑡𝑆_𝑤=𝑎₂+𝑏₂𝑆_𝑤+𝑐₂𝑆²_𝑤,𝜆_𝑡(𝑟)=𝑎₃+𝑏₃𝑟²+𝑐₃𝑟⁴. (25)

4. The Productivity Calculation Process

Using the electrical analog concept, the well production rate in the horizontal plane must equal the production rate in the vertical plane. Because of the nonlinearity of (17)-(18) and (∗)-(∗∗), we cannot obtain productivity expressions similar to Joshi [7]. With the aid of computer programs, the results can be obtained and the calculation diagram is showed in Figure 6.

Figure 6 

The productivity calculation diagram.

5. Results and Their Sensitive Analysis

In this section, the liquid and oil production rate are calculated, and the essential parameters of well, reservoir, and fluid properties are listed in Table 1, and the relative permeability curves are showed in Figure 7.

Table 1 

The parameters of reservoir and fluid properties.

Figure 7 

The relative permeability curves.

According to the relative permeability curves and the parameters in Table 1, the relationship of 𝜆_𝑡(𝑆_𝑤) versus 𝑆_𝑤 and d𝑓_𝑤/d𝑆_𝑤 versus 𝑆_𝑤 can be plotted when water cut ratio reaches 0.6 (as shown in Figures 8 and 9). The corresponding regression curve equations can be obtained as follows: 𝜆_𝑡=0.4917−2.3096𝑆_𝑤+3.2547𝑆²_𝑤,d𝑓_𝑤d𝑆_𝑤=−6.0226𝑆_𝑤+4.3193,𝑆_𝑤>0.6. (26)

Figure 8 

The relationship between 𝜆𝑡(𝑆𝑤) and 𝑆𝑤.

Figure 9 

The relationship between d𝑓𝑤/d𝑆𝑤 and 𝑆𝑤.

Taking (26) as well as the cumulative fluid production into (24), the expressions between the 𝜆_𝑡(𝑟) and 𝑟 can be obtained, which is 𝜆_𝑡=0.2+1.53×10⁻⁶𝑟²+6.8×10⁻¹³𝑟⁴. (27)

Combining (27) and other parameters in Table 1 with (17)-(18) and (∗)-(∗∗), the steady-state fluid productivity can be calculated.

Figure 10 shows the effect of permeability modulus 𝛼_𝑘 on liquid and oil productivity of horizontal well in low permeability reservoir. It can be seen from the figure that the permeability stress-sensitive has a significant effect on the productivity; the bigger the 𝛼_𝑘 is, the smaller the liquid and oil productivity are, which is mainly because, with the same pressure drop of the reservoir, big 𝛼_𝑘 will lead to a serious permeability decreasing. When we do not take into account the permeability stress sensitive (𝛼_𝑘=0), the liquid and oil productivity can be calculated with the limit of 𝛼_𝑘 tending to zero for (17)-(18) and (∗)-(∗∗).

Figure 10 

The effect of permeability modulus (𝛼𝑘) on liquid production rate.

Figures 11 and 12 show the effect of threshold pressure gradient (𝐺) and well length (𝐿) on liquid and oil productivity. It can be seen from the chart that the threshold pressure gradient has small effect on the productivity of horizontal well for big drainage volume. In general, the bigger the 𝐺 is, the smaller the liquid and oil productivity are. When the reservoir has both threshold pressure gradient and permeability stress sensitive, the longer the well length is, the bigger the productivity is.

Figure 11 

The effect of threshold pressure gradient (𝐺) on liquid production rate.

Figure 12 

The effect of well length (𝐿) on liquid production rate.

Figure 13 shows the liquid productivity with different bottomhole pressure when 𝛼_𝑘=0.1, 𝐺=0.001 and 𝛼_𝑘=0, 𝐺=0, the corresponding values are listed in Table 2. It can be clearly seen that the permeability stress-sensitive and threshold pressure gradient have significant effects on the well productivity, and the bigger 𝛼_𝑘 and 𝐺 are, the more obvious the effect is. And when the pressure drop is small, the fluid cannot flow for the existing of threshold pressure gradient, which is mainly because only the fluid can flow when the pressure drop overcomes the threshold pressure for multiphase flow.

Table 2 

The productivity of liquid and oil in different bottomhole pressure.

Figure 13 

The productivity of horizontal well in different bottomhole pressure.

6. Conclusions

In this paper, a semianalytical productivity equation of horizontal well in low-permeability oil reservoir with oil/water two-phase flow is established with the consideration of permeability stress-sensitive and threshold pressure gradient. Based on the above study, the following conclusions can be summarized.(1)The steady-state percolation mathematical models of horizontal well with oil/water two-phase flow are established and the corresponding solutions are solved by the method of separation of variables.(2)For low-permeability reservoir, there always exists the phenomenon of permeability stress-sensitive (𝛼_𝑘), which has a significant influence on the well productivity; the bigger the 𝛼_𝑘 is, the smaller the productivity is.(3)Due to the existence of capillary pressure of two-phase flow, there always is threshold pressure gradient (𝐺) in the fluid seepage process. Although the 𝐺 has a smaller effect on the productivity of the horizontal well for a big drainage volume, we cannot neglect its effect on the productivity.

Appendix

From the expressions of (11), we have 𝑟𝜆_𝑡𝑒^{−𝛼𝑘(𝑝𝑖−𝑝)}d𝑝d𝑟−𝐺^′=𝑐, (A.1) where 𝑐 is a constant.

We define the following expression: 𝑦=𝑒^{−𝛼𝑘(𝑝𝑖−𝑝)}. (A.2)

Substituting (A.2) into (A.1) and (12)-(13) yields d𝑦d𝑟=𝛼_𝑘𝐺^′𝑦+𝑐𝛼_𝑘𝑟𝜆_𝑡,𝑦_𝑟=1=𝑒^{−𝛼𝑘(𝑝𝑖−𝑝𝑐)},𝑦_{𝑟=𝑟𝑒𝐻}=1. (A.3)(A.4)(A.5)

According to the general solution of the Bernoulli differential equation [30], the solution of (A.3) can be obtained as 𝑦(𝑟)=𝑒^{𝛼𝑘𝐺′𝑟}⋅𝑒^{−𝛼𝑘𝐺′}𝑒^{−𝛼𝑘(𝑝𝑖−𝑝𝑐)}+^𝑟_𝑟𝑤𝑒^{−𝛼𝑘𝐺′𝑟}𝑐𝛼_𝑘𝑟𝜆_𝑡(𝑟)d𝑟. (A.6)

Substituting (A.6) into (A.5), the value of constant 𝑐 can be solved, and then combining it with (A.6), we have 𝑝(𝑟)=𝑝_𝑖+1𝛼_𝑘ln𝑒^{𝛼𝑘𝐺′𝑟}⋅𝑒^{−𝛼𝑘(𝐺′+𝑝𝑖−𝑝𝑐)}+𝑒^{−𝛼𝑘𝐺′𝑟𝑒𝐻}−𝑒^{−𝛼𝑘(𝐺′+𝑝𝑖−𝑝𝑐)}∫^𝑟𝑒𝐻_𝑟𝑤𝑒^{−𝛼𝑘𝐺′𝑟}⋅1/𝑟𝜆_𝑡(𝑟)⋅d𝑟_𝑟𝑟𝑤𝑒^{−𝛼𝑘𝐺′𝑟}1𝑟𝜆_𝑡(𝑟)d𝑟. (A.7)

Equation (A.7) is the pressure distribution relation along the radius 𝑟 with the oil/water two-phase flows in the horizontal plane of the horizontal well.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant no. 51374181) and the project of National Science Fund for Distinguished Young Scholars of China (Grant no. 51125019). The authors would also like to thank the reviewers and editors for their patience to read this paper and valuable comments.