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Compared with conventional well, herringbone-like laterals wells can increase the area of oil release, and can reduce the number of wellhead slots of platforms, and also can greatly improve the development efficiency. Based on threshold pressure gradient in heavy oil reservoir, and the applied principle of mirror reflection and superposition, the pressure distribution equation of herringbone-like laterals wells is obtained in heavy oil reservoir. Productivity model of herringbone-like laterals wells is proposed by reservoir-wellbore steady seepage. The example shows that the productivity model is great accuracy to predict the productivity of herringbone-like laterals wells. The model is used to analyze the branching length, branching angle, branching symmetry, branching position and spacing and their effects on productivity of herringbone-like laterals wells. The principle of optimizing the well shape of herringbone-like laterals wells is proposed.

Compared with the onshore oilfields, the number of wellhead slots is limited in offshore oilfields. The herringbone-like laterals wells not only increase the drainage area and single well-controlled reserves, but also increase the production of oil wells; it has certain advantages in offshore oilfield development. Therefore, it is necessary to study the productivity prediction of herringbone-like laterals wells in reservoirs. In 1996, Salas [

In reference [

The law of non-Darcy seepage can be described by the following formula:

v = − K μ ( d p d r − λ ) , d p d r > λ (1)

where v is seepage velocity, m^{3}/s; K is reservoir permeability, mD; d p d r is pressure gradient, MPa/m; λ is threshold pressure gradient, MPa/m.

when Φ = K μ p , have:

v = − K μ d Φ d r + λ K μ (2)

Assuming that the formation is infinitely homogeneous and isotropic, there is a horizontal well in which the length of the horizontal well is L, the coordinates of the two ends of the well are (x_{1}, 0, z_{w}), (x_{1}, 0, z_{w}), setting the horizontal well is homogeneous line sink, the productivity of the well is Q, the productivity of per unit length L is q, selecting micro-element dx_{0} at x_{0} of horizontal well. It can be regarded as the seepage velocity of M(x_{0}, y_{0}, z_{0}) at any point:

v = q 4 π r 2 (3)

A new velocity potential function considering threshold pressure gradient at point M is obtained:

d Φ = − Q 4 π L d x 0 ( x 0 − x ) 2 + ( y 0 − y ) 2 + ( z 0 − z ) 2 + λ K μ d R D + C (4)

where R D is the shortest distance between the micro-element and the moving boundary.

According to the superposition principle of potential, the velocity potential caused by the whole horizontal well is as follows:

Φ = ∫ x 1 x 2 − Q 4 π L d x 0 ( x 0 − x ) 2 + ( y 0 − y ) 2 + ( z 0 − z ) 2 + λ K μ R D + c (5)

In this paper, a horizontal well is divided into N segments by mathematical discretization method. As the productivity of the well varies little in a small section, it can be assumed that the productivity of this section is a fixed value, and the production of each section is different:

Φ = ∑ i = 0 N − 1 ( − Q 4 π L i ∫ x i x i + 1 d x 0 ( x 0 − x ) 2 + ( y 0 − y ) 2 + ( z 0 − z ) 2 ) + λ K μ R D + c ′ (6)

As the horizontal well is parallel to the X-axis and on the XOY plane, so y 0 = y , z 0 = z w , the Formula (6) is changed to:

Φ ( M ) = ∑ i = 0 N − 1 − Q 4 π L i ln ( x i + 1 − x ) + r i + 1 ( x i − x ) + r i + λ K μ R D + c ′ (7)

where r i = ( x i − x ) 2 + y 2 + ( z w − z ) 2 .

The herringbone-like laterals wells have M branching (including main branching). The herringbone-like laterals wells are divided into several segments. And it is setted the flow rate of section s of the branching t. According to the superposition principle of mirror reflection and potential, potential produced by herringbone-like laterals wells system at any point M in reservoir:

Φ t s ( M ) = ∑ k = − ∞ + ∞ ∑ t = 0 M − 1 ∑ s = 1 N [ φ ( ( x [ t ] [ s ] , y [ t ] [ s ] , 4 k h + z w ) , ( x [ t ] [ s + 1 ] , y [ t ] [ s + 1 ] , 4 k h + z w ) , α t ) + φ ( ( x [ t ] [ s ] , y [ t ] [ s ] , 4 k h + 2 h − z w ) , ( x [ t ] [ s + 1 ] , y [ t ] [ s + 1 ] , 4 k h + 2 h − z w ) , α t ) − φ ( ( x [ t ] [ s ] , y [ t ] [ s ] , 4 k h − z w ) , ( x [ t ] [ s + 1 ] , y [ t ] [ s + 1 ] , 4 k h − z w ) , α t ) − φ ( ( x [ t ] [ s ] , y [ t ] [ s ] , 4 k h − 2 h + z w ) , ( x [ t ] [ s + 1 ] , y [ t ] [ s + 1 ] , 4 k h − 2 h + z w ) , α t ) ] + λ K μ ( R e − x ) 2 + y 2 + z 2 (8)

where:

φ ( ( x [ t ] [ s ] , y [ t ] [ s ] , 4 k h + 2 h − z w ) , ( x [ t ] [ s + 1 ] , y [ t ] [ s + 1 ] , 4 k h + 2 h − z w ) , α t ) = − q t s 4 π L t s ln r t s + r t s + 1 + L t s r t s + r t s + 1 − L t s + c t s

r t s = ( x t s − x ) 2 + ( y t s − y ) 2 + ( 4 k h + 2 h − z w − z ) 2 , m; L t s is the length of the s segment of the branching t, m; k is infinite well row reflected by mirror image of branching in Z direction.

The wellbore pressure distribution equation of herringbone-like laterals wells wells under bottom water reservoir is obtained according to Equation (8):

p w t s = p e + μ K ( Φ e t s − Φ t s ) (9)

For bottom water reservoirs, there is:

Φ e t s = 0 (10)

The FORMULA (9) is changed to:

p w t s = p e − μ k Φ t s (11)

where μ is oil viscosity, mPa・s; K is reservoir permeability, mD; p w t s is the well bottom hole flow pressure of the s segment of the branching t, MPa; p e is initial reservoir pressure, MPa.

The main wellbore and branching wellbore are divided into many units. Considering that the length of each unit is enough small, the pressure drop of the second section of the branching t of herringbone-like laterals wells is calculated in reference [

Δ p w t s = 2 f h w ρ π 2 D 5 ( 2 Q t s + q t s ) 2 Δ x + 16 ρ q t s π 2 D 4 ( 2 Q t s + q t s ) (12)

where q t s is radial inflow of micro-element section in the s segment of the branching t, m^{3}/s; Q t s is the upstream flow of the mainstream for the s segment of branching t, m^{3}/s; f h w is friction resistance coefficient of the well tube wall with radial inflow, f; ρ is fluid density, kg/m^{3}; D is the wellbore diameter, m.

While the fluid of the main wellbore and branching wellbore is confluencing, a mixed loss will occur, and generated local pressure drop. Based on the principle of fluid mechanics, a calculation model of local pressure drop at the confluence point of main wellbore and branching wellbore is established.

Assuming that the fluid in the wellbore flows steadily, adiabatically and isothermally, without considering the friction between the fluid and the pipe wall, and ignoring the influence of gravity, the confluence flow diagram of the branching wellbore is shown in

The momentum equation of the main wellbore direction of the fluid at the confluence point is as follows:

p 1 π 4 D 2 − p 2 π 4 D 2 + F x = ρ Q 2 V 2 − ρ Q 1 V 1 (13)

Continuity equation:

V 1 π 4 D 2 + q t = V 2 π 4 D 2 (14)

Energy equation:

p 1 ρ g + V 1 2 2 g = p 2 ρ g + V 2 2 2 g + h 12 (15)

where p 1 , p 2 are pressure at the inflow and outflow ends along the direction of the main wellbore at the confluence point, MPa; D is main wellbore and branching wellbore diameter, m; q t is from the branching wellbore t to main wellbore, m^{3}/s.

The force F x of the wall acting on the gas at the junction point can be derived from the momentum equation:

F x = ρ q t V 3 cos φ (16)

Combination the Formula (13) and (16):

p 1 − p 2 = 4 ρ π D 2 ( Q 2 V 2 − Q 1 V 1 − q t V 3 cos φ ) (17)

Combination the Formula (14), (15) and (17), pressure drop equation at the confluence point of main and branching wellbore is:

Δ p w t = p 1 − p 2 = 16 ρ q t 2 π 2 D 4 + 8 V 1 ρ q t π D 2 − 4 V 3 ρ q t π D 2 cos φ (18)

Substitute V 1 = 4 Q t π D 2 , V 3 = 4 q t π D 2 into the Formula (18):

Δ p w t = p 1 − p 2 = 16 ρ q t 2 π 2 D 4 ( 1 − cos φ ) + 32 ρ Q t q t π 2 D 4 (19)

where Q t is flow from upstream end of the point of main wellbore t, m^{3}/s; q t is flow from branching wellbore t to the main wellbore, m^{3}/s; φ is the angle of branching, degrees; D is diameter of the main wellbore and branching wellbore, m.

In addition to flowing along the length of horizontal wellbore, reservoir fluid also flows into wellbore along the horizontal wellbore direction. There is a coupling relationship between seepage flow in reservoir and in the wellbore.

The pressure distribution in wellbore can be calculated by the pressure drop calculation model as:

p w t s = p w t ( s − 1 ) + 0.5 ( Δ p w t ( s − 1 ) + Δ p w t s ) ( 0 ≤ t ≤ M − 1 , 1 ≤ s ≤ N ) (20)

Δ p w t 0 = 0 , p w t 0 = p w f t

where p w f t is the flow pressure at the heel of the branching wellbore t, MPa; p w t s is the flow pressure of s segment of the branching t, MPa; p w t ( s − 1 ) is the flow pressure of s^{−1} segment of the branching t, MPa; Δ p w t ( s − 1 ) is the pressure drop of s^{−1} segment of the t branch, MPa; Δ p w t s is the pressure drop of s segment of the branching t, MPa.

According to the principle of material balance, the inflow of each branching wellbore equals the sum of the inflow of each small section at the upstream:

Q t s = ∑ t = 0 M − 1 ∑ s = 1 N q t s (21)

As can be seen from the above, the reservoir seepage model has m × n equation, the wellbore pressure drop model has m × n equation, there are 2 m × n equations. The variables to be solved are q t s and p w t s ( 0 ≤ t ≤ M − 1 , 1 ≤ s ≤ N ) , which are also 2 m × n , so the equations are closed.

The coupling model is solved by iteration method. The specific steps are as follows: 1) Assuming that the initial value of p w t s is p w t s 0 , in actual calculation it can be assumed p w t s 0 = p w f t ; 2) Substitute p w t s into the Formula (11), used gauss elimination method to find q t s ; 3) Substitute q t s into the Formula (21), find Q t s ; 4) Substitute q t s and Q t s into the Formula (12) and (19), find Δ p w t s ; 5) Substitute Δ p w t s into the Formula (20), to update p w t s . This value is taken as the initial value of the next iteration; 6) repeat (2) - (5), comparing p w t s n + 1 , q t s n + 1 after n iterations with p w t s n and q t s n after n iterations, when both of them satisfied certain accuracy, the iteration stops, otherwise repeat (2) - (5) steps until the accuracy is satisfied; 7) Last, the Formula (21) can be used to calculate the total production of herringbone-like laterals wells.

The herringbone-like laterals wells in a heavy oil reservoir in Bohai Oilfield as an example, the well pattern is shown in ^{−1}/m.

The productivity coupling model deduced by the author is used to predict the productivity of herringbone-like laterals wells. Compared with the actual production data, as shown in

Well | A50H | |
---|---|---|

The method | Productivity (m^{3}/d) | Relative error |

Formula of Liu Xiangping | 267 | 7.2 |

Formula in this paper | 254 | 2.4 |

Actual data | 250 | - |

The reservoir-wellbore steady seepage coupling model is used to optimize the shape of herringbone-like laterals wells, for give full play to the advantages of herringbone-like laterals wells.

1) Branching length optimization

The optimization of branching length mainly studies during the total wellbore length is equal, the productivity difference between equal and unequal branching length. In the study, the structure of two branching shown in

2) Branching angle optimization

Using the three branching structures as an example, the effect of branching angle on productivity is studied. As can be seen from

From

3) Branching symmetry optimization

In order to study the effect of branching symmetry on productivity of the well, four branching structures are designed as shown in

As can be seen from

4) Branching location and spacing optimization

In order to study the influence of branching location and spacing on the productivity of the well, four branching structures are designed as shown in

As can be seen from

Based on the threshold pressure gradient, the productivity coupling model of herringbone-like laterals wells is established in heavy oil reservoir-wellbore steady seepage. The productivity coupling model is suitable for predicting the productivity of herringbone-like laterals wells in heavy oil reservoir. Using the productivity coupling model in this paper, the well shape parameters of the well are optimized, and the principle of optimizing the well shape of herringbone-like laterals wells is proposed.

The authors declare no conflicts of interest regarding the publication of this paper.

Sun, E.H., Tan, J., Zhang, D., Wang, W. and Mu, S.R. (2019) Study on Productivity Model of Herringbone-Like Laterals Wells and Optimization of Morphological Parameters Considering Threshold Pressure Gradient in Heavy Oil Reservoirs. World Journal of Engineering and Technology, 7, 302-313. https://doi.org/10.4236/wjet.2019.72021