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In order to improve the validity of the previous models on calculating flow pressure for oil well with partially perforating fracture, a new physical model that obeys the actual heterogeneous reservoir characteristics was built. Different conditions, including reservoir with impermeable top and bottom borders or the reservoir top which has constant pressure, were considered. Through dimensionless transformation, Laplace transformation, Fourier cosine transformation, separation of variables, and other mathematical methods, the analytical solution of Laplace domain was obtained. By using Stephenson numerical methods, the numerical solution pressure in a real domain was obtained. The results of this method agree with the numerical simulations, suggesting that this new method is reliable. The following sensitivity analysis showed that the pressure dynamic linear flow curve can be divided into four flow streams of early linear flow, midradial flow, advanced spherical flow, and border controlling flow. Fracture length controls the early linear flow. Permeability anisotropy significantly affects the midradial flow. The degree of penetration and fracture orientation dominantly affect the late spherical flow. The boundary conditions and reservoir boundary width mainly affect the border controlling flow. The method can be used to determine the optimal degree of opening shot, vertical permeability, and other useful parameters, providing theoretical guidance for reservoir engineering analysis.

Low permeability carbonate reservoirs generally have the characteristics of large thickness and natural fractures. It is well known that the perforation completion is used in well completion, and the hydraulic fracturing is used to increase the well production. However, the degree of perforation is relatively small (a few meters), which can affect the wellbore’s pressure transmission and production wells’ productivity [

However, hydraulic fractures are assumed to be fully penetrating the formation in the previous studies. Limited efforts have been made to investigate the effects of partially penetrating fracture height on the performance of wells. In practice, fully penetrating fractures may lead to an early or immediate water or gas breakthrough in a reservoir with bottom water or gas cap in contact, whereas partially penetrating fractures provide a better way to prevent the early breakthrough. What is more, most of the previous methods are based on Gringarten and Ramey’s point source solution and Green function, whereas the original physical model established by Gringarten and Ramey only considered the upper and lower bounds, limiting the scope of the application. Therefore, it is necessary to study flow pressure for low permeability oil well with partially penetrating fracture.

This paper first presents the physical model and the mathematical model of the unstable seepage flow in the three-dimensional anisotropic rectangular reservoir under certain conditions, and then the solution of the mathematical model is obtained, which provides a new way to calculate flow pressure for low permeability oil well with partially penetrating fracture.

The reservoir formation is fractured by hydraulic fracturing to form a plurality of fractures as shown in Figure

The oil production is a constant, and the formation of the reservoir is bounded and nonhomogeneous, with equal thickness.

Before producing, the reservoir pressure is equal to the original formation pressure.

The reservoir fluid is microcompressible, occurring single-phase and unstable seepage.

The intercrack interference is ignored, and the fluid flow in the fracture obeys the Darcy law.

The gravity of the fluid and capillary force is ignored, and the porosity and fluid viscosity is constant.

The fracture is partially penetrating the formation, and the reservoir fluid flows to the wellbore in a limited range.

Crossflow between the fracture and the matrix is ignored, and the fracture has infinite diversion capability.

The schematic of multistage fracturing vertical wells.

Single fracture schematic of section penetration.

The center of one single fracture is located at (

The initial conditions are written as follows:

According to the literature survey, there are mainly three methods to solve equation (

The dimensionless transformation is a method of converting the seepage equation into a conventional mathematical equation. By dimensionless transformation, the number of comparisons can be greatly reduced, which makes the mathematical physics equation simple, neat, and easy to analyze and solve and help to check and verify the seepage equation [

So (

Laplace transform can eliminate the partial derivative of time from the unstable seepage equation and has been widely used to solve the problem of unstable seepage [

The mathematical model established in this paper can solve the pressure of multifracture system. For the sake of simplification, it is assumed that the number of fracture is 10. The numerical simulation is used to calculate the seepage field, and then the pressure value (simulated solution) at different time and different positions is output and compared with the numerical solution of the seepage pressure calculated by Stehfest numerical inversion (this work).

The reservoir E300 module in Eclipse 2011 is developed for fractured heterogeneous reservoirs. E300 is used to simulate the pressure variation around a fractured vertical well in a rectangular heterogeneous reservoir. In order to meet the assumptions of (

The width and the length of the rectangular heterogeneous reservoir are 1 km, and there is an oil production well in the center of the reservoir, which is showed as in Figure

The geometry information representation of the reservoir.

In order to describe the formation fluid heterogeneity, the triangular network of grid system is used to ensure that each crack at least has 3 grids, which is showed in Figure

Basic data of the system.

parameters | value |
---|---|

saturation pressure | 25 MPa |

oil viscosity | 1.21 mPa·s |

oil density | 0.79 g/cm^{3} |

water compressibility | 4.9 × 10^{-4 }MPa^{−1} |

oil volume coefficient | 1.21 m^{3}/m^{3} |

porosity | 0.12 |

injection pressure | 46 MPa |

effective thickness | 5 m |

formation temperature | 158°F |

water viscosity | 1.6 mPa·s |

dissolved gas and oil ratio | 22.31 m^{3}/m^{3} |

oil compressibility | 8.1 × 10^{-4 }MPa^{−1} |

rock Compressibility | 4.5 × 10^{-4 }MPa^{−1} |

permeability | 1.2 mD |

original formation pressure | 27 MPa |

injecting water intensity | 0.044 m^{3}/(d·MPa·m) |

The results of comparative table.

production time | Oil production^{3}/d) | Pressure of this work | Pressure of E300 | relative error |
---|---|---|---|---|

30 | 17.492 | 22.984 | 22.249 | 3.201 |

60 | 17.202 | 22.603 | 21.870 | 3.244 |

90 | 16.951 | 22.274 | 21.539 | 3.298 |

120 | 16.729 | 21.982 | 21.251 | 3.323 |

150 | 16.532 | 21.723 | 20.993 | 3.363 |

180 | 16.355 | 21.490 | 20.767 | 3.369 |

210 | 16.194 | 21.279 | 20.558 | 3.390 |

240 | 16.048 | 21.087 | 20.366 | 3.421 |

270 | 15.913 | 20.910 | 20.196 | 3.412 |

300 | 15.770 | 20.723 | 19.912 | 3.913 |

330 | 15.627 | 20.538 | 19.731 | 3.929 |

360 | 15.486 | 20.353 | 19.450 | 4.436 |

The fracture center is located on the centerline of the wellbore axis, and the partially penetrating degree is 50%. Moreover,

The flow division schematic of partial penetration fractured vertical wells.

Based on the control variable method, the parameters affecting pressure and pressure derivative (template curve) such as fracture orientation, fracture scale, the degree of penetration in the reservoir, permeability anisotropy, reservoir boundary condition, and reservoir scale were analyzed by using the parameters of Table

In this paper, the orientation of fractures is divided into two aspects, namely, in the reservoir center (as shown in Figure

Schematic diagram of fracture orientation.

The effect of fracture orientation on template curves.

Figure

With different fracturing scale, the length of the fractures is not the same. With different fracture length, the seepage area exposed to the reservoir is not the same, so the pressure transmission trend is different, as shown in Figure

The effect of fracture length on template curves.

Reasonable degree of penetration in the reservoir not only can save the cost of perforation, but also can get the maximum yield. The effect of the degree of penetration in the reservoir on fluid pressure is obvious, as shown in Figure

The effect of opening shot degree on template curves.

Since the permeability varies little in the horizontal direction, the permeability in

The effect of reservoir anisotropy on template curves.

The combination of different reservoir boundary conditions has different effect on the template curve, as shown in Figure

The effect of reservoir boundary conditions on template curves.

By setting different reservoir widths, the influence of reservoir width on the template curve is studied, as shown in Figure

The effect of reservoir width on template curves.

The authors declare that they have no conflicts of interest.