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The aim of this paper is to solve the problems that the existing method of critical production of gas cap reservoir is only suitable for single-phase flow, and the method of critical production of gas cap reservoir under water-flooding is still blank. In this paper, the relationships between dynamic and static equilibrium, plane radial flow theory, oil-water infiltration method and three-dimensional seepage field decomposition theory, were applied to study a calculation method for critical production of directional wells and horizontal wells. Furthermore, the effects of different factors on critical output were studied, such as horizontal permeability, ratio of horizontal permeability to vertical permeability, length of horizontal section, effective thickness, viscosity of crude oil and water content etc. Results show that the critical production increases with the increment of the horizontal permeability, the ratio of the vertical permeability to the horizontal permeability, the reservoir thickness and the horizontal well length; when the viscosity of crude oil is small, the critical production decreases first and then increases with the increase of water content; when the viscosity of crude oil is high, the critical production increases continuously with the increase of water content. This study could provide theoretical and technical guidance for changing of the working system of oil wells. It can avoid gas channeling and improve the development effect.

The reservoir with a gas cap is a special reservoir type. After the gas cap reservoir was developed, the original balance relationship between oil and gas was broken. Then the gas coning formed near the production well. And then the gas channeling occurred. Although the horizontal wells have an important advantage in delaying the gas cone penetration, a reasonable work system to production is still necessary. At present, scholars have carried out relevant research work. The total pressure drop in the equation of gravity balance was calculated by graphic method, and the critical production was calculated accordingly [

Critical production is the max production when the wells don’t have gas breakthrough. The critical condition means that the oil-gas interface stays on the top of well perforation interval steadily, as shown in

d p d z d z d A = − Δ ρ go g d z d A (1)

Then we obtain

d p = − Δ ρ go g d z (2)

where dp is the pressure difference, dz is the thickness of infinitesimal, dA is the area of infinitesimal, and Δ ρ g o is the difference of oil and gas density.

Before water breakthrough, only oil phase flow into the hole of production well. Based on the theory of radial flow, the migration velocity of fluid is defined by

v = k μ o d p d r (3)

where v is the migration velocity, k is the absolute permeability, and μ_{o} is the oil viscosity.

Production equation is expressed as

q = v A (4)

Cross section area is expressed as

A = 2 π r ( h − z ) (5)

Combining Equations (3)-(5), we obtain

q = k μ o d p d r 2 π r ( h − z ) (6)

From Equations (6) and (2), we can rewrite Equation (6) as

q = k μ o − Δ ρ go g d z d r 2 π r ( h − z ) (7)

Boundary conditions are considered.

{ r = r w z = h − b r = r e z = 0 (8)

By integrating Equation (7), we obtain surface critical production.

q = q c B o (9)

q c = π k Δ ρ go g ( h 2 − b 2 ) B o μ o ln r e r w (10)

where q_{c} is surface critical production, B_{o} is volume factor of oil, and b is the thickness of perforation.

Considering the well skin factor, finally we obtain the critical well production before the water breakthrough.

q c = π k Δ ρ go g ( h 2 − b 2 ) B o μ o ( ln r e r w + s ) (11)

where s is the skin factor.

After the breakthrough, both the oil phase and the water phase flow into the wellbore, Equation (11) is not applicable to calculate the production, but the phase production can be obtained with the same method.

For oil phase

q oc = π k k ro Δ ρ go g ( h 2 − b 2 ) B o μ o ( ln r e r w + s ) (12)

For water phase

q ow = π k k rw Δ ρ go g ( h 2 − b 2 ) B w μ w ( ln r e r w + s ) (13)

where q_{oc} is the critical production of oil phase, k_{ro} is the relative permeability of oil phase, q_{ow} is the critical production of water phase, k_{rw} is the relative permeability of water phase, B_{w} is volume factor of water, and μ_{w} is viscosity of water.

Well critical production calculation after the water breakthrough is calculated by combining Equations (12) and (13).

q c = q oc + q wc = π k k ro Δ ρ go g ( h 2 − b 2 ) B o μ o ( ln r e r w + s ) + π k k rw Δ ρ go g ( h 2 − b 2 ) B w μ w ( ln r e r w + s ) (14)

Water cut is an important evaluation index after the water breakthrough. Equation (14) is used as a function of water cut for filed application. Relative permeability curve is the function of water saturation, and is inducted to calculate the relation between critical and water cut. The specific steps are as follows. First the relationship between water saturation and water cut was stabilized by calculating water cut with relative permeability carve. Then the inverse method was used to calculate the water saturation with practical well water cut and relevant oil/water relative permeability was obtained. Finally the critical rate was calculated by introducing relative permeability to Equation (14).

Calculation method of critical production for horizontal wells was studied by using 3D flow theory. A 3D flow field of horizontal wells in formation consists of two 2D flow regions (the inner region and outer region, as shown in _{e} is outer drainage radius, P_{e} is bounder pressure, R_{p} is equivalent well radius, and P_{wf} is bottom hole flowing pressure, as shown in _{w} is wellbore radius, and r_{e} is inner drainage radius, as shown in

For the outer flow region, critical production of horizontal well can be obtained from Equation (10).

q c1 = π k h Δ ρ go g ( h 2 − h p 2 ) B o μ o ( ln R e R p + s ) (15)

where R e = M π = a + a 2 − ( L 2 ) 2 2 , and R p = L / 4 . q_{c1} is the critical production

of outer flow region, M is drainage area, a is the long half axle of elliptical flow region, L is the length of horizontal well, k_{h} is the horizontal permeability, h_{p} is the distance between the wellbore and the reservoir bottom, R_{e} is the drainage radius, and R_{p} is the equivalent well radius of circular drainage area.

For inner flow region, considering the well length as thickness, the production of horizontal well in vertical cross-section can be expressed with horizontal radial fluid flow method.

q c2 = 2 π k v L Δ p B o μ o ( ln r e r w + s ) (16)

where r e = h p . q_{c2} is the critical production of inner flow region, r_{e} is the drainage radius of inner flow region, k_{v} is the vertical permeability, and Δ p is the pressure difference.

When the production of the outer region is the critical production, the pressure difference can be expressed as

Δ p = Δ ρ go g ( h 2 − h p 2 ) 2 h (17)

Combining Equation (16) and (17), we can obtain

q c2 = π k v L Δ ρ go g ( h 2 − h p 2 ) B o μ o h ( ln r e r w + s ) (18)

And the critical production of horizontal wells before water breakthrough can be expressed as

q c = q c1 + q c2 = π k h Δ ρ go g ( h 2 − h p 2 ) B o μ o ( ln R e R p + s ) + π k v L Δ ρ go g ( h 2 − h p 2 ) B o μ o h ( ln r e r w + s ) (19)

After water breakthrough, critical production of horizontal wells can be calculated using the same method as the directional wells.

Critical production of outer flow region is

q c1 = π k h Δ ρ go g ( h 2 − h p 2 ) ln R e R p + s ( k ro B o μ o + k rw B w μ w ) (20)

Critical production of inner flow region is

q c2 = π k v L Δ ρ go g ( h 2 − h p 2 ) h ( ln r e r w + s ) ( k ro B o μ o + k rw B w μ w ) (21)

Critical production is

q c = π k h Δ ρ go g ( h 2 − h p 2 ) [ k h ln R e R p + s + k v L h ( ln r e r w + s ) ] ( k ro B o μ o + k rw B w μ w ) (22)

The relationship between critical production and water cut can be established by the same method of directional wells.

Factor analysis was carried out using data from Bohai oilfield. The layer geological and fluid parameters are shown as

As shown in

As shown in

Parameters | Permeability/mD | Net thickness/ m | Oil viscosity/ (mPa・s) | Water viscosity/(mPa・s) | Oil density/(g・cm^{-3}) | Oil volume factor/(m・m^{-3}) | Water volume factor/(m・m^{-3}) | Horizontal well length/m |
---|---|---|---|---|---|---|---|---|

Values | 1000 | 10 | 3 | 0.5 | 850 | 1.14 | 1.01 | 400 |

the less likely it is for the gas channeling to occur. For horizontal wells, the longer the horizontal section is, the more stable the gas cone is, and the less likely it is for the gas channeling to occur. Directional wells also have similar laws in net thickness.

Taking horizontal wells for example, critical productions of different oil viscosity on different water cut stages were plotted as shown

BZ oilfield is a complex fault-block oil field with gas cap on Bohai bay yellow river estuary sag. The reliability of the method was illustrated by an example of a well. The critical production of a production well was calculated by using Equations (19) and (22). The critical production before water breakthrough was 139 m^{3}/d. But when the water cut is 20%, the critical production is 92 m^{3}/d. Then the working system of the well was adjusted according to the calculated results. The daily fluid production was limited from 110 m^{3}/d to 80 m^{3}/d. Production proration of the other wells was optimized using the study results. The steam oil ratio of oilfield goes down and oil production goes steadily after optimization as shown in

This method is also applied in other wells. For the wells that are not degassed and whose production is lower than the critical production, raising the pump frequency can be used to increase oil production. For the wells that have been degassed and whose production exceeds the critical production rate, reducing the pump frequency or reducing the choke size can be used to decrease the liquid rate and control the gas channeling. Accordingly, the working system of 18 wells near gas cap in BZ oilfield was optimized, and the measurement was established with “single well customization”. Five wells were treated with increasing the pump frequency, and six wells were treated with decreasing the chock size or pump frequency. The daily oil of 11 wells was increased by 94 m^{3}/d by working system optimization (

Well No. | Oil Viscosity /mPa・s | Before optimization | Measure | After optimization | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Liquid /(m^{3}・d^{−1}) | Oil /(m^{3}・d^{−1}) | Water cut/% | GOR /(m^{3}・m^{−3}) | Calculated critical production /(m^{3}・d^{−1}) | Liquid /(m^{3}・d^{−1}) | Oil /(m^{3}・d^{−1}) | Water cut /% | GOR /(m^{3}・m^{−3}) | Oil increment /(m^{3}・m^{−3}) | |||

A01h | 10 | 80 | 23 | 71.3 | 38 | 116 | Pump frequency increasing | 115 | 33 | 71.4 | 42 | 10 |

A10h | 3 | 100 | 98 | 2 | 61 | 160 | 157 | 155 | 1.3 | 63 | 57 | |

A13h | 5 | 61 | 29 | 52.5 | 50 | 93 | 90 | 41 | 54.4 | 53 | 12 | |

A37h | 10 | 71 | 16 | 77.5 | 46 | 120 | 120 | 25 | 79.2 | 47 | 9 | |

F22h | 3 | 73 | 45 | 38.4 | 65 | 105 | 105 | 64 | 39 | 75 | 19 | |

A02h | 5 | 110 | 108 | 1.8 | 167 | 100 | Pump frequency decreasing | 100 | 99 | 1 | 122 | −9 |

D02h | 10 | 58 | 30 | 48.3 | 90 | 45 | 41 | 25 | 39 | 61 | −5 | |

A19h | 1.5 | 166 | 98 | 41 | 166 | 145 | 140 | 103 | 26.4 | 108 | 5 | |

A29h | 3 | 122 | 20 | 83.7 | 187 | 90 | 90 | 23 | 74.4 | 120 | 3 | |

A33h | 3 | 100 | 71 | 29 | 221 | 80 | Shrink the nozzle | 80 | 66 | 17.5 | 155 | −5 |

F32h | 1.5 | 167 | 101 | 39.5 | 149 | 140 | 140 | 99 | 29.3 | 117 | −2 | |

Summation | 94 |

1) New critical production calculation method of directional wells in a water flooding reservoir with gas cap was established according to fluid hydraulics equilibrium, radial fluid flow theory and oil/water relative permeability relationship). The critical production after water breakthrough can be obtained by the method.

2) New critical production calculation method of horizontal wells was set up using 3D flow field decomposition theory of horizontal wells.

3) Oil viscosity affects the relationship between critical production and water cut. When the oil viscosity is low, the critical production decreases firstly and then increases with the increment of the water cut. Well working system should be adjusted to avoid gas channeling. And when the oil viscosity is high, the critical production increases with the increment of the water cut. Liquid production of wells can be raised properly.

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

Chen, C.L., Yang, M., Liu, X., Shi, F. and Liu, M.J. (2019) Paper Title. Open Journal of Yangtze Gas and Oil, 4, 31-42. https://doi.org/10.4236/ojogas.2019.41003