# Numerical Investigation for Three-Dimensional Multiscale Fracture Networks Based on a Coupled Hybrid Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{®}. Furthermore, an integrated workflow of simulation modeling for multiscale CFNs combined with a field example in Sichuan from China is used to analyzing the production information of fractured horizontal wells in shale gas reservoirs. Compared with the field production data from this typical well, it can be proved that the hybrid model has strong reliability and practicability.

## 1. Introduction

^{®}in Section 3. A field case of shale gas reservoirs from the Sichuan Basin of southwestern China is practiced by our model (in Section 4). Finally, some conclusions on the presented multi-scale fractures model are given.

## 2. Methods

#### 2.1. Classification for Multi-Scale Fractures

#### 2.2. Parameterization of 3D Ellipsoidal Macro-Fracture

_{0}at a certain point on the plane, that is, the reference vector

**α**

_{0}and two linearly independent (i.e., orthogonal) vectors

**α**

_{1}and

**α**

_{2}on the fracture plane. These two linearly independent vectors constitute two basis vectors of the local coordinate system with A

_{0}as the origin of the plane. The points on the fracture plane can be expressed as:

**r**represents the vector diameter of the point on the fracture plane;

**α**is the reference vector;

_{0}**α**and

_{1}**α**are two linearly independent vectors selected on the fracture plane; u and v are corresponding parameters, i.e., even pair (u,v) (i.e., coordinates in the local coordinate system {A

_{2}_{0};

**α**,

_{1}**α**}) and points on the fracture plane.

_{2}#### 2.3. Mathematical Models for Matrix

#### 2.3.1. Control Equations of Gas-Water Two-Phase Flow

- (1)
- Shale reservoir is homogeneous and equal thickness;
- (2)
- Compressible reservoir fluid is isothermal flow, and obeys Darcy’s law;
- (3)
- Mixed gas can be considered to be simplified as a single component CH
_{4}; - (4)
- Two-phase flow (gas & water) and the effect of dissolved gas in the water are considered;
- (5)
- The gravity term is considered on the fluids flow.

_{g}and B

_{w}represent gas and water volume factor respectively; μ represents corresponding fluid viscosity; p represents corresponding fluid pressure; s represents corresponding fluid saturation; ρ

_{gsi}and ρ

_{wsi}indicate gas and water density under standard ground conditions respectively; q

_{gsi}and q

_{wsi}represent gas and water production rate under standard ground conditions respectively. In addition, symbol g represents gravitational acceleration; D indicates reservoir depth; R

_{s}represents gas solubility; ϕ represents porous media porosity; δ is the Dirac function; q

_{diff}is the flux rate of gas diffusion into the matrix grid.

#### 2.3.2. Gas Desorption and Diffusion

_{L}indicates the Langmuir volume; P

_{L}represents the Langmuir pressure. The V

_{L}represents the maximum gas adsorption amount obtained by the experiment test, and P

_{L}is the pressure when the adsorption amount reaches half of the maximum adsorption amount. The Langmuir pressure in the Langmuir isothermal coefficient model can be obtained by fitting the experimental data.

_{m}is the matrix density; C

_{c}is the current adsorption concentration of matrix; ${C}_{c}^{\infty}$ is the current equilibrium adsorption concentration, and it can be obtained from the Langmuir isotherm model (Equation (6)). The diffusion velocity v is used to characterize the diffusion speed of desorption gas into macropores, which can be obtained by fitting the time curve of adsorption volume.

#### 2.4. Coupling Method for Multiscale Fractures

#### 2.4.1. Mass Transfer in Hydraulic Macro-Fractures

_{NNC}represents the effective permeability of the NNC; A

_{NNC}indicates the contact area of the NNC; ${\u2329d\u232a}_{\mathrm{NNC}}$ indicates the feature distances related to NNC.

_{m}represents the volume of a matrix cell.

- (1)
- Matrix–fracture mass transfer is the unsteady flow;
- (2)
- Matrix grid only flows to the fractures within the grid, and there is no fluid exchange with fractures in the adjacent grids;
- (3)
- The upstream weight is used to calculate the fluid mobility in the multiphase fluid exchange between matrix and fracture.

_{f}is the number of fracture grids contained in the matrix grid.

#### 2.4.2. Dual-Medium Equivalence of Induced Fractures

_{m}represents the permeability of matrix; P

_{m}and P

_{f}represent grid pressure of matrix and fracture respectively.

#### 2.4.3. Matrix Enhancement Equivalence of Natural Micro-Fractures

_{e}is the enhanced porosity of matrix grid; ϕ

_{m}and ϕ

_{f}represent porosity of matrix grid and fracture grid respectively; k

_{e}represents the enhanced permeability of matrix grid; k

_{f}denotes permeability of fracture grid.

#### 2.5. Workflow of Modeling for Multi-Scale CFNs

## 3. Model Validation and Analysis

#### 3.1. Simulation for Embedded Macro-Fractures

^{®}developed by a group named rock flow dynamics (RFD), which has become the industry standard. The induced fractures and natural fractures are removed in this comparison. The purpose of the first example is to verify whether the proposed method can generate arbitrary-placed fractures in different layers to achieve 3D reservoir simulation. The fractures are vertical in this case because it is relatively difficult for commercial software to characterize the fractures with arbitrary dip angles. A fractured inclined-well modeled by proposed method and the 3D reservoir model with fracture elements and the inclined wellbore is shown in Figure 6. The reservoir dimensions are 200 × 200 × 20 m and have two layers. The different grid divisions, such as 20 × 20 × 2, 50 × 50 × 2, and 100 × 100 × 2, are used to test the mesh sensitivity of proposed model. Modeling parameters of reservoir and fluid are shown in Table 1. There are four fractures, of which No. 1 and No. 3 are in the top layer, and No. 2 and No. 4 are in the bottom layer. The fracture parameters in example 1 are shown in Table 2. The technology of local grid refinement (LGR) is used to describe the fractures and the schematic of the grid system with LGR for the commercial simulator tNavigator

^{®}is illustrated as Figure 7. The fractured inclined-well is producing at a constant BHP of 10 MPa and the simulation time for production is 1000 days. The comparison of pressure profiles for different layers between tNavigator

^{®}and our model on the 1000th day is illustrated in Figure 8 and Figure 9, and the computational results of the two models are very similar. Figure 10 shows the difference of the gas production rate of three simulators (tNavigator

^{®}, original EDFM, and mimetic GEM-based HFM) simulating over 1000 days, in which the effect of mesh sensitivity is tested in this comparison between the original model and modified model. The comparison of grid number and average relative errors are shown in Table 3. The result of simulators under this benchmark proves that the modified model has higher precision and higher robustness than previous EDFMs that adopt a linear steady assumption. The HFM based on mimetic GEM achieves an accurate calculate of matrix–fracture mass transfer. Consequently, the mesh convergence and accuracy of proposed method in this paper are validated when we take the solution of tNavigator

^{®}as the exact solution.

^{®}. The second numerical example is supplemented to discuss the availability of the characterization model of inclined elliptic fractures. Figure 11 shows the 3D view and overhead view of the reservoir with a three-stage fractured horizontal well, in which it is obvious that some of the fractures are inclined and arbitrarily distributed. The reservoir size is 1000 × 500 × 40 m and has four layers. The modeling parameters of reservoir and fluid are the same as in Table 1. There are seven fractures with different azimuth angles, dip angles, lengths, heights, and the reference center point coordinates of fractures in this example are shown in Table 4. Fractures No. 1, No. 2, No. 3, No. 4, and No. 5 are completely penetrating the 2nd layer and 3rd layer, and partly penetrating the 1st layer and 4th layer in varying degrees. By contrast, the fracture No. 6 and No. 7 are not penetrating the top layer and the bottom layer because of the self-height and inclination degree. The constant BHP of 10 MPa. The production time is 1000 days, and the comparison of pressure profiles for different layers on the 1000th day is shown in Figure 12. It can be seen from this example that the proposed model can be adaptive to arbitrary inclined fracture with an elliptic shape. Compared with previous EDFMs, this is an innovative characterization method for ellipsoidal-shaped fractures in 3D-reservoir. Compared with the simulator tNavigator

^{®}, the presented approach can handle not only arbitrary-placed fractures but also characterize the fractures with arbitrary dip angles.

#### 3.2. Simulation for Multi-Scale CFNs

^{®}. The reservoir dimensions are 800 × 400 × 10 m and only have one layer. The modeling parameters of the reservoir and fluid are the same as in Table 1. To saving the computing cost, all of the fractures are completely penetrating the reservoir and without any arbitrary dip angles. The fracture parameters are shown in Table 5 and the noticeable differences between different scale fractures are length, aperture, and azimuth degree. The multi-scale CFNs are modeled by tNavigator

^{®}and illustrated in Figure 14. In tNavigator

^{®}, the LGR method and DPDK model are used for modeling the hydraulic fractures and induced fractures separately, and the matrix enhancement equivalent approach is adopted for micro-scale fractures. In this case, the two models keep the same size of grid-block, but the number of fracture elements in tNavigator

^{®}is more than that in our model and this is caused by LGR. The production time is 1000 days under a constant BHP of 10 MPa. The comparison of pressure profiles for multi-scale CFNs in different periods is shown in Figure 15, the comparison of production dynamics calculated by two simulators over 1000 days is Figure 16, and the comparison of grid number and average relative errors for two simulators is Table 6. The simulation results showed a good agreement for pressure profiles, daily production curve, and the average reservoir pressure curve of two simulators on the 1000 days. The results within a reasonable error range indicate the high accuracy of the proposed hybrid model.

## 4. Application Case

## 5. Conclusions

^{®}. The results prove that the proposed model has high precision and high robustness. The proposed hybrid model was applied for simulation of a typical well in Sichuan shale, indicating that the proposed method can provide theoretical guidance for productivity forecasting and optimization. The simulation results show that the multiscale CFNs is an essential consideration affecting the production of gas wells. Generally, the enlightenment for engineering technicians mainly includes two aspects. Firstly, it is very important to establish high-conductivity macro-fractures as preferential channeling, and secondly, it is necessary to increase the area and utilization rate of SRV.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic plot of ellipsoidal fracture model (modified from Rao et al. [36]).

**Figure 7.**(

**Left**) top view of the global grid configuration for macro-fracture. (

**Right**) a view of locally interconnecting fractures and surrounding matrix blocks with permeability distribution in Y-direction. The distribution of permeability is also presented in the schematic by using various colors and the permeability distribution in X-direction is the same as that in Y-direction.

**Figure 14.**(

**Left**) top view of the global grid configuration for multi-scale CFNs. (

**Right**) a view of locally interconnecting fractures and surrounding matrix grids with permeability distribution in Y-direction. The distribution of permeability is also presented in the schematic by various colors and the permeability distribution in X-direction is the same as that in Y-direction.

Properties | Value | Properties | Value |
---|---|---|---|

Initial reservoir pressure, MPa | 20 | Gas density, fraction | 0.72 |

Gas viscosity, mPa·s | 0.01 | Gas Z-factor, fraction | 0.8 |

Langmuir pressure, MPa | 4.8 | Langmuir volume, m^{3}/kg | 4 × 10^{−3} |

Matrix density, kg/m^{3} | 2400 | Matrix compressibility, MPa^{−1} | 1.07 × 10^{−4} |

Matrix porosity, fraction | 0.1 | Matrix permeability, mD | 1 × 10^{−4} |

Fracture compressibility, MPa^{−1} | 1 × 10^{−2} | Fracture permeability, mD | 500 |

Number | Reference Point Coordinates | Azimuth Angle, ° | Fracture Length, m | Fracture Height, m |
---|---|---|---|---|

1 | (65, 100, 15) | 90 | 100 | 10 |

2 | (135, 100, 5) | 90 | 100 | 10 |

3 | (100, 65, 15) | 180 | 100 | 10 |

4 | (100, 135, 5) | 180 | 100 | 10 |

Modeling Simulator | Number of Grids | Average Relative Errors, % |
---|---|---|

tNavigator^{®} (220 × 220 × 2) | 96,800 | - |

Original EDFM (20 × 20 × 2) | 800 | 48.11 |

Original EDFM (50 × 50 × 2) | 5000 | 21.81 |

Original EDFM (100 × 100 × 2) | 20,000 | 4.16 |

Proposed model (20 × 20 × 2) | 800 | 28.24 |

Proposed model (50 × 50 × 2) | 5000 | 9.59 |

Proposed model (100 × 100 × 2) | 20,000 | 1.08 |

Number | Center Point Coordinates | Azimuth Angle, ° | Dip Angle, ° | Fracture Length, m | Fracture Height, m |
---|---|---|---|---|---|

1 | (305, 255, 20) | 70 | 95 | 300 | 30 |

2 | (505, 255, 20) | 93 | 70 | 280 | 30 |

3 | (705, 255, 20) | 87 | 75 | 295 | 30 |

4 | (305, 195, 20) | 15 | 60 | 200 | 25 |

5 | (335, 345, 20) | 30 | 90 | 180 | 25 |

6 | (515, 315, 20) | 20 | 88 | 190 | 20 |

7 | (695, 155, 20) | 20 | 90 | 210 | 15 |

Fracture Type | Azimuth Angle, ° | Fracture Length, m | Fracture Aperture, mm | Permeability, mD |
---|---|---|---|---|

Hydraulic fracture | 90 | 200 | 1 | 500 |

Induced fracture | 180 | 100 | 0.1 | 50 |

Natural fracture | 30 | 50 | 0.01 | 1 |

Modeling Simulator | Number of Grids | Average Relative Errors, % |
---|---|---|

tNavigator^{®} (830 × 440 × 1) | 365,200 | - |

Proposed hybrid model (800 × 400 × 1) | 320,000 | 4.09 |

Properties | Value | Properties | Value |
---|---|---|---|

Reservoir volume, m | 1000 × 500 × 30 | Grid number | 100 × 50 × 3 |

Depth, m | 2800 | Initial reservoir pressure, MPa | 69 |

Horizontal wellbore length, m | 805 | Well radius, m | 0.18 |

Skin factor | 0.1 | Gas density, fraction | 0.72 |

Gas viscosity, mPa·s | 0.01 | Gas Z-factor, fraction | 0.8 |

Langmuir pressure, MPa | 4.8 | Langmuir volume, m^{3}/kg | 4 × 10^{−3} |

Matrix density, kg/m^{3} | 2400 | Matrix compressibility, MPa^{−1} | 1.07 × 10^{−4} |

Matrix porosity, fraction | 0.1 | Matrix permeability, mD | 6 × 10^{−4} |

Fracture number | 15 | Fracture spacing, m | 50 |

Fracture half-length, m | 125 | Fracture compressibility, MPa^{−1} | 1 × 10^{−2} |

Macro-fracture aperture, mm | 1 | Macro-fracture permeability, mD | 500 |

Induced-fracture aperture, mm | 0.1 | Induced-fracture permeability, mD | 50 |

Micro-fracture aperture, mm | 0.01 | Micro-fracture permeability, mD | 1 |

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**MDPI and ACS Style**

Du, X.; Cheng, L.; Chen, J.; Cai, J.; Niu, L.; Cao, R.
Numerical Investigation for Three-Dimensional Multiscale Fracture Networks Based on a Coupled Hybrid Model. *Energies* **2021**, *14*, 6354.
https://doi.org/10.3390/en14196354

**AMA Style**

Du X, Cheng L, Chen J, Cai J, Niu L, Cao R.
Numerical Investigation for Three-Dimensional Multiscale Fracture Networks Based on a Coupled Hybrid Model. *Energies*. 2021; 14(19):6354.
https://doi.org/10.3390/en14196354

**Chicago/Turabian Style**

Du, Xulin, Linsong Cheng, Jun Chen, Jianchao Cai, Langyu Niu, and Renyi Cao.
2021. "Numerical Investigation for Three-Dimensional Multiscale Fracture Networks Based on a Coupled Hybrid Model" *Energies* 14, no. 19: 6354.
https://doi.org/10.3390/en14196354