Water breaks through along fractures is a major concern in tight sandstone reservoirs with a bottom aquifer. Analytical models fail to handle the three-dimensional two-phase flow problem for partially penetrating inclined fractures, so time-consuming numerical simulation are often used for this problem. This paper presents an efficient semianalytical model for this problem considering three-dimensional fractures and two-phase flow. In the model, the hydraulic fracture is handled discretely with a numerical discrete method. The three-dimensional volumetric source function in real space and superposition principle are employed to solve the model analytically for fluid flow in the reservoir. The transient flow equations for flow in three-dimensional inclined fractures are solved by the finite difference method numerically, in which two-phase flow and stress-dependent properties are considered. The eventual solution of the model and transient responses are obtained by coupling the model for flow in the reservoir and discrete fracture dynamically. The validation of the semianalytical model is demonstrated in comparison to the solution of the commercial reservoir simulator Eclipse. Based on the proposed model, the effects of some critical parameters on the characteristics of water and oil flow performances are analyzed. The results show that the fracture conductivity, fracture permeability modulus, inclination angle of fractures, aquifer size, perforation location, and wellbore pressure drop significantly affect production rate and water breakthrough time. Lower fracture conductivity and larger inclination angle can delay the water breakthrough time and enhance the production rate, but the increment tends to decline gradually. Furthermore, water breakthrough will occur earlier if the wellbore pressure drop and aquifer size are larger. Besides, the stress sensitivity and perforation location can delay the water breakthrough time.

In recent years, tight sandstone reservoirs have been discovered as unconventional oil and gas resources with great potential, and hydraulic fracturing technology has been very critical for improving the well productivity of this kind of reservoir [

Much research has been studied on the transient behavior of fractured wells in tight sandstone reservoirs. However, most of them are limited to the single-phase flow problem [

This paper proposed a semianalytical model to analyze production data and predict water breakthrough time in tight sandstone reservoirs with a bottom aquifer. The three-dimensional fracture characteristics and two-phase flow behavior and stress-dependent fracture permeability are all incorporated in the model. The fluid flow in the fracture system is numerically characterized by the finite difference method. In contrast, the fluid flow in the matrix system is analytically characterized based on the three-dimensional source function theory. The solution of the model is solved by coupling the transient flow in the fracture and the matrix. The proposed model is shown to efficiently simulate the behavior of tight sandstone reservoirs with a bottom aquifer. The accuracy of the proposed semianalytical model is validated by using numerical simulation, and the influences of several parameters on production rates and water breakthrough time are also analyzed. The novelty of the new model is in the ability to semianalytically obtain the performance of production wells in tight sandstone reservoirs with a bottom aquifer. Furthermore, it provides an efficient method for modeling bottom water coning, combining three-dimensional fracture characteristics and two-phase flow.

This paper mainly focuses on the oil and water production performance of a vertical well intersecting a rectangle-shaped partially penetrating fracture in tight sandstone reservoirs with a bottom aquifer. As shown in Figure

The reservoir is box-shaped, and the external boundaries of the top and side are assumed to be closed, and the reservoir is assumed to be homogeneous

The fracture is located at the center of the reservoir, and the inclination angle of the fracture is

This model considers the three-dimensional fracture characteristics, and the fluid flow in the fracture is considered a 2D flow

Darcy seepage is applicable in both the fracture and matrix systems, and stress-dependent fracture permeability is considered

The vertical well produces constant bottom hole pressure and fluid flow into the wellbore only through the fracture

Schematic diagram of the physical model used in this work.

As illustrated in the physical model, two distinct flow processes are governed by different mechanisms: partially penetrating inclined fracture flow and matrix flow. In this section, we first describe the theoretical model for fluid flow in the three-dimensional inclined fracture, and the finite difference method is used to obtain the solution of this system. Next, a three-dimensional volumetric source function in real space and superposition principle is applied to solve unsteady state flow in the matrix system. Then, the eventual solution of the model and transient responses are obtained by coupling the two systems with continuity conditions.

With consideration of the stress-sensitivity effect and oil-water two-phase flow behavior, the governing equation of a three-dimensional fracture system can be described as follows:

For the oil phase,

For the water phase,

The relationship between oil and water saturation is given by

The initial condition is

Under constant bottom-hole flowing pressure condition, a well production equation is given by [

We first divide the fracture into several fracture panels, as illustrated in Figure

Discretization of the three-dimensional fracture along both the horizontal axis and the inclined axis.

Geometric transmissibility (

The coefficients of

Bulk volume of the fracture panel (

For a slightly compressible fluid, the coefficients of

Applying the finite difference approximation to the production boundary conditions, Equations (

In this paper, the flowing material balance method is used to calculate the water influx rate (

Assuming that

Inserting Equation (

The water influx rate of each fracture panel connecting to the bottom aquifer is given by

Substituting the production condition Equation (

Taking Equation (

Equations (

There are

In our work, the three-dimensional macro fracture is discretized into

For an isotropic permeability reservoir, the matrix flow equation can be expressed as

Initial condition

The outer boundary is assumed to be closed and given by

The top boundary is assumed to be closed

And the bottom boundary is assumed to produce at constant pressure, which is given by

According to Jia et al.’s [

As we can see in Figure

Basic parameters of fracture panel (

Equation (

To simplify the form of the equations, let

Taking Equations (

Then, the pressure response of fracture segment g caused by all fracture panels at time level

We arrange Equation (

It should be noted that there are

The continuity conditions of pressure and flux along the fracture plane are given by

Based on the continuity condition, we combine the two-phase flow equations in the fracture and single-phase flow equations in the matrix to develop a system of equations to characterize the transient flow behavior of the three-dimensional inclined fracture.

Equation (

In this section, the proposed semianalytical method is benchmarked against the commercial numerical simulator Eclipse. As shown in Figure ^{-1}, the aquifer size is

Schematic diagram of fracture distribution and discretizing for numerical simulation.

Parameters used for model validation.

Parameters | Symbol | Unit | Value |
---|---|---|---|

Fracture permeability | mD | 500 | |

Fracture porosity | Dimensionless | 0.4 | |

Fracture half-length | m | 70 | |

Fracture half-height | m | 50 | |

Fracture width | m | 0.01 | |

Inclination angle of fracture | ° | 75 | |

Fracture compressibility | MPa^{-1} | ||

Permeability modules | MPa^{-1} | 0.05 | |

Matrix permeability | mD | 0.1 | |

Matrix porosity | Dimensionless | 0.1 | |

Formation thickness | m | 100 | |

Aquifer size | m^{3} | 1 × 10^{7} | |

Oil volume coefficient | m^{3}/m^{3} | 1.6 | |

Oil viscosity | mPa·s | 0.3 | |

Water volume coefficient | m^{3}/m^{3} | 1.0 | |

Water viscosity | mPa·s | 0.8 |

Oil-water relative permeability of the hydraulic fracture system.

Figure

Comparison of the results of the numerical simulator and the semianalytical method.

Oil production rate

Water production rate

Based on the proposed model, the influences of the following parameters on oil and water production performance and water breakthrough time are analyzed: fracture conductivity, fracture permeability modulus, inclination angle of fracture, aquifer size, perforation location, and wellbore pressure drop. Table

Parameters used for sensitivity analysis.

Parameters | Symbol | Unit | Value |
---|---|---|---|

Fracture conductivity | mD·m | 5, 10, 50 | |

Permeability modules | MPa^{-1} | 0.05, 0.1, 0.2 | |

Inclination angle of fracture | ° | 45, 60, 75 | |

Aquifer size | m^{3} | ||

Perforation location | m | 20, 60 | |

Wellbore pressure drop | MPa | 5, 10, 15 |

Figure

Effect of fracture conductivity on oil and water production rates.

Oil production rate

Water production rate

It can be seen from Figure

Effect of fracture permeability modulus on oil and water production rates.

Oil production rate

Water production rate

Figure

Effect of the inclination angle on oil and water production rates.

Oil production rate

Water production rate

Figure ^{5} m^{3}, ^{7} m^{3}, and ^{9} m^{3} for the case study in this section. As illustrated in Figure

Effect of the aquifer size on oil and water production rates.

Oil production rate

Water production rate

The effect of the perforation location on production rate and water breakthrough time is analyzed in this part. We denote by

Effect of the perforation position on oil and water production rates.

Oil production rate

Water production rate

Figure

Effect of the wellbore pressure drop on oil and water production rates.

Oil production rate

Water production rate

This paper’s main contribution is the provision of a semianalytical method to efficiently simulate two-phase flow behavior in three-dimensional large-scale fractures and accurately predict the production profiles and water breakthrough time of fractured wells in tight sandstone reservoirs with a bottom aquifer. The main conclusions of this work are as follows:

The proposed semianalytical model can be used to evaluate the two-phase flow performance of a three-dimensional incline fracture with satisfying precision, and much computational time can be saved compared with fully numerical simulation

The important properties of the fracture system, including fracture conductivity, inclination angle, and stress-dependent permeability, have great effects on the production rates and water breakthrough time. A larger fracture permeability modulus and inclination angle of fracture can delay the water breakthrough time and enhance the oil production rate. A more significant fracture conductivity will lead to an earlier water breakthrough time

Aquifer size has a significant impact on the production rate, and a smaller aquifer size can efficiently delay the water breakthrough time

The perforation location and wellbore pressure drop play an important role in production performance. Reasonable perforation position and producing pressure difference are excellent for delaying the water breakthrough time and improving the production efficiency during the development of tight sandstone reservoirs with a bottom aquifer

Oil formation volume factor (Sm^{3}/m^{3})

Oil flow rate entering the fracture from the matrix per unit volume at standard conditions (1/d)

Water formation volume factor (Sm^{3}/m^{3})

Oil production rate per unit volume at standard conditions (1/d)

Fracture compressibility (MPa^{-1})

Water production rate per unit volume at standard conditions (1/d)

Oil compressibility (MPa^{-1})

Bottom water influx during its coning through the fracture (1/d)

Total compressibility of matrix system (MPa^{-1})

Radial distance of side external boundary (m)

Water compressibility (MPa^{-1})

Wellbore radial (m)

Geometric transmissibility of fracture panel

Oil saturation (dimensionless)

Formation thickness (m)

Water saturation (dimensionless)

Water invasion index

Time (day)

Permeability of fracture (mD)

Transmissibility of oil phase between fracture panel

Initial permeability of fracture (mD)

Transmissibility of water phase between fracture panel

Permeability of matrix (mD)

Transmissibility of oil phase between fracture panel

Relative permeability to oil (dimensionless)

Transmissibility of water phase between fracture panel

Relative permeability to water (dimensionless)

Bulk volume of fracture panel (^{3})

Number of panels of the discretized fracture

Aquifer size (m^{3})

Number of panels of the discretized fracture directly connected with the bottom aquifer

Cumulative water influx rate (m^{3})

Pressure of the bottom aquifer (MPa)

Fracture pressure (MPa)

Center position of the plane source

Initial formation pressure (MPa)

Reservoir dimension (m)

Matrix pressure (MPa)

Height of fracture panel (

Wellbore pressure (MPa)

Length of fracture panel (

Withdrawal rate at standard condition (m^{3}/d)

Time step size (d).

Porosity (dimensionless)

Fluid viscosity (mPa·s)

Inclination angle of fracture (°)

Differential operator

Conversion factor

Difference calculation

Conversion factor

Permeability modulus (MPa^{-1})

Direction of the fracture

Direction of the fracture.

Time level

Average parameter.

External

Oil

Fracture

Relative

Initial

Standard condition

Matrix

Wellbore.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors acknowledge that this study was partially funded by the CNOOC Research Institute Co., Ltd., Science and Technology Projects (No. CNOOC-YXKJ-QZJC-TJ-2020-01).