# Numerical Simulation of Fracture Propagation during Refracturing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{h}+ σ

_{x}is higher than σ

_{H}+ σ

_{z}, where σ

_{H}and σ

_{h}denote the original maximum and minimum horizontal principal stress, respectively, and σ

_{x}and σ

_{z}denote the induced stress in the σ

_{h}and σ

_{H}directions, respectively. The hydro-mechanical coupling effect is the most important factor affecting stress reorientation before refracturing. The stress reorientation occurs when fluids flow out of the poroelastic media during production of wells, which causes depletion in pore pressure and rock deformation, finally creating the induced stress field. Palmer [1] suggested that both high pressure and confined fracture height are beneficial for stress reversal in the CBM wells. Elbel and Mack [19] proposed the theory of fracture reorientation in refracturing. Numerical simulation shows that there is an optimum time window of the refracturing operation, which depends on the reservoir properties and the production of wells. A long original fracture helps create a fracture with the maximum penetration into the un-stimulated region when fracture reorientation occurs. The fracture perpendicular to the original fracture is beneficial to following production of wells as the new fracture can propagate into the undrained area with the lower pressure depletion. Once the secondary fracture propagates beyond the isotropic point, the new fracture can be diverted to the original σ

_{H}direction, as illustrated in Figure 1 [16].

## 2. Method and Theory

#### 2.1. Enrichment Displacement Functions

_{I}(x) are the standard basis functions; u

_{I}is the standard displacement field; a

_{I}is the nodal enriched degree of the freedom (DoF) vector where the basis function support is intersected by the fracture interior; ${b}_{I}^{\alpha}$ is the enriched DoF vector where the basis function support is cut by the crack tip; H(x) is the associated discontinuous step function across the fracture surface and is expressed as

^{*}is the crack point closest to x, and n is the unit normal vector to point outward to the crack surface at x

^{*}.

_{α}(x) is expressed as

#### 2.2. Fluid Flow within HFs

_{t}is the tangential permeability along with HFs, and $\nabla p$ is the pressure gradient along with HFs.

_{L}is the leak-off rate; c

_{L}is the leak-off coefficient; p

_{i}and p

_{L}are the pressure within HFs and on the fracture surfaces, respectively.

_{ij}and ${\sigma}_{ij}^{0}$ are the Cauthy stress and original stress tensor, respectively; E is elastic modulus; p

_{w}and ${p}_{w}^{0}$ are the pressure in the rock matrix and the original formation pressure, respectively; ε

_{ij}and ε

_{kk}are the strain tensor and the volume strain, respectively; α is the Biot constant between 0–1; and δ

_{ij}is the Kronecker delta symbol.

#### 2.3. Traction–Separation Constitutive Behavior

_{n}and t

_{s}; δ is the separation vector with two components of δ

_{n}and δ

_{s}; K is the cohesive stiffness matrix with non-zero diagonal elements of K

_{nn}and K

_{ss}.

_{n}and t

_{s}, are, respectively, written as:

_{n}and T

_{s}are normal and shear stress components without damage, respectively. To describe the damage evolution across the fracture, an effective separation δ

_{m}is written as:

_{equiv}and G

_{equivC}are the equivalent and critical fracture energy release rate, respectively; G

_{I}and G

_{II}are the fracture energy release rate in the normal and shear directions, respectively; G

_{IC}and G

_{IIC}are the critical fracture energy release rate in the normal and shear directions, respectively; η is the power exponent constant. This model presents a power-law relationship that combines mixed-mode energy release rates with fracture criterion.

## 3. Numerical Simulation

_{HF}in Figure 5) propagates along the original fracture when the injected fluids flow into the wellbore in the first fracturing step. Then, the fractured wells are produced for some time, and the pore pressure depletion zone occurs due to the poroelastic effect. Meanwhile, the stress field is reorientated near the HF in the first fracturing. In the refracturing process, the HF interacts with a pre-existing fracture (PF) of different approaching angles (denoted as Γ

_{PF}) in the domain. The HF is re-initiated from the tips of pre-existing fractures (PFs) and propagates and diverts along a certain direction due to the production-induced stress field.

^{−13}Pa/(m·s), and the first fracturing and refracturing steps last for 100 s and 86,400 s at the rate of 2 × 10

^{−3}m

^{2}/s and 2 × 10

^{−4}m

^{2}/s, respectively. A total of 1089 quadrilateral pore pressure elements (CPE4P) were generated in the computational domain. The keyword “propagation mode = merging” was added to the ABAQUS input file to simulate the fracture re-initiation and propagation. It is worth noting that the flow rate Q

_{inj}was positive in the production step and is negative in the first fracturing and refracturing steps. A constant pore pressure and the roller displacement boundary conditions were set on the outer boundary in Figure 5.

#### 3.1. Model Verification

_{m}, proposed by Bunger et al. [40], and expressed as:

_{m}> 4 and is viscosity-dominated when K

_{m}< 0.5.

^{2}/min. All the parameters of the XFEM model are given in Table 2. The dimensionless parameter K

_{m}in Equation (12) was 0.313, indicating HF propagation in the viscosity-dominated regime. The XFEM-based cohesive zone method (CZM) was applied in ABAQUS. The BK law is used to simulate the failure process in hydraulic fracturing [37]. The well-known Irwin formula was used to calculate the fracture energy according to the fracture toughness in Table 1 [41]. Figure 6 gives the numerical results of the fracture opening along with the HF and the injection pressure over the pumping time. The XFEM numerical results are consistent with the KGD analytical solution, indicating the reliability of the XFEM model.

#### 3.2. The Impact of Approaching Angle on Fracture Reorientation

_{H}direction at different approaching angles in Figure 8, stress reorientation occurred around HFs due to the poroelastic effect. With an approaching angle of 90°, the reversal stress zone appeared around the new fracture, and the local σ

_{H}direction was consistent with the original direction of the far-field σ

_{h}. When the fracture was diverted into the original far-field σ

_{H}direction, the stress reorientation zone was not observed. The numerical results are consistent with the previous analysis of refracturing [19,20,42].

#### 3.3. The Impact of Stress Difference on Fracture Reorientation

_{H}direction at various stress differences in Figure 11, stress reorientation occurred around HFs due to the poroelastic effect. With an approaching angle of 90 °, the reversal stress zone occurred around new fractures, and the local σ

_{H}direction was consistent with the original far-field σ

_{h}direction. The lower stress difference caused a relatively large stress reversal zone. The numerical results are consistent with the previous analysis of refracturing again [19,20,42].

#### 3.4. The Impact of Production Time on Fracture Reorientation

_{H}direction at different production time is illustrated in Figure 14. We observed that stress reorientation occurred around HFs due to the poroelastic effect, similar to the results illustrated above. This stress reversal zone was beneficial for the penetration of the fracture into unstimulated reservoirs.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Palmer, I.D. Induced Stresses Due to Propped Hydraulic Fracture in Coalbed Methane Wells. In Proceedings of the Society of Petroleum Engineers, Oklahoma City, OK, USA, 1 January 1993. [Google Scholar]
- Too, J.L.; Cheng, A.; Linga, P. Fracturing Methane Hydrate in Sand: A Review of the Current Status. In Proceedings of the Offshore Technology Conference, Houston, TX, USA, 20 March 2018. [Google Scholar]
- Gale, J.F.W.; Laubach, S.E.; Olson, J.E.; Eichhubl, P.; Fall, A. Natural Fractures in Shale: A Review and New Observations. AAPG Bull.
**2014**, 98, 2165–2216. [Google Scholar] [CrossRef] - Economides, M.J.; Nolte, K.G. (Eds.) Reservoir Stimulation, 3rd ed.; Wiley: Chichester, UK; New York, NY, USA, 2000; ISBN 978-0-471-49192-7. [Google Scholar]
- Zhang, Q.; Hou, B.; Lin, B.; Liu, X.; Gao, Y. Integration of Discrete Fracture Reconstruction and Dual Porosity/Dual Permeability Models for Gas Production Analysis in a Deformable Fractured Shale Reservoir. J. Nat. Gas Sci. Eng.
**2021**, 93, 104028. [Google Scholar] [CrossRef] - Hou, B.; Chang, Z.; Fu, W.; Muhadasi, Y.; Chen, M. Fracture Initiation and Propagation in a Deep Shale Gas Reservoir Subject to an Alternating-Fluid-Injection Hydraulic-Fracturing Treatment. SPE J.
**2019**, 24, 1839–1855. [Google Scholar] [CrossRef] - Dahi-Taleghani, A. Fracture Re-Initiation As a Possible Branching Mechanism during Hydraulic Fracturing. In Proceedings of the American Rock Mechanics Association, Salt Lake City, UT, USA, 1 January 2010. [Google Scholar]
- Dahi-Taleghani, A.; Olson, J.E. How Natural Fractures Could Affect Hydraulic-Fracture Geometry. SPE J.
**2014**, 19, 161–171. [Google Scholar] [CrossRef] - Dahi-Taleghani, A.; Olson, J.E. Numerical Modeling of Multistranded-Hydraulic-Fracture Propagation: Accounting for the Interaction between Induced and Natural Fractures. SPE J.
**2011**, 16, 575–581. [Google Scholar] [CrossRef] - Maxwell, S.C.; Urbancic, T.I.; Steinsberger, N.; Zinno, R. Microseismic Imaging of Hydraulic Fracture Complexity in the Barnett Shale. In Proceedings of the Society of Petroleum Engineers, San Antonio, TX, USA, 1 January 2002. [Google Scholar]
- Yu, H.; Taleghani, A.D.; Lian, Z. On How Pumping Hesitations May Improve Complexity of Hydraulic Fractures, a Simulation Study. Fuel
**2019**, 249, 294–308. [Google Scholar] [CrossRef] - Yu, H.; Dahi Taleghani, A.; Lian, Z.; Lin, T. On How Asymmetric Stimulated Rock Volume in Shales May Impact Casing Integrity. Energy Sci. Eng.
**2020**, 8, 1524–1540. [Google Scholar] [CrossRef] - Zhou, F.; Liu, Y.; Yang, X.; Zhang, F.; Xiong, C.; Liu, X. Case Study: YM204 Obtained High Petroleum Production by Acid Fracture Treatment Combining Fluid Diversion and Fracture Reorientation. In Proceedings of the Society of Petroleum Engineers, Scheveningen, The Netherlands, 1 January 2009. [Google Scholar]
- Palisch, T.T.; Vincent, M.C.; Handren, P.J. Slickwater Fracturing: Food for Thought. In Proceedings of the Society of Petroleum Engineers, Alexandria, VA, USA, 1 January 2008. [Google Scholar]
- Warpinski, N.R.; Branagan, P.T. Altered-Stress Fracturing. J. Pet. Technol.
**1989**, 41, 990–997. [Google Scholar] [CrossRef] - Siebrits, E.; Elbel, J.L.; Detournay, E.; Detournay-Piette, C.; Christianson, M.; Robinson, B.M.; Diyashev, I.R. Parameters Affecting Azimuth and Length of a Secondary Fracture During a Refracture Treatment. In Proceedings of the Society of Petroleum Engineers, New Orleans, LA, USA, 1 January 1998. [Google Scholar]
- Roussel, N.P.; Sharma, M.M. Quantifying Transient Effects in Altered-Stress Refracturing of Vertical Wells. SPE J.
**2010**, 15, 770–782. [Google Scholar] [CrossRef] - Sneddon, I.N.; Elliot, H.A. The Opening of a Griffith Crack under Internal Pressure. Q. Appl. Math.
**1946**, 4, 262–267. [Google Scholar] [CrossRef] [Green Version] - Elbel, J.L.; Mack, M.G. Refracturing: Observations and Theories. In Proceedings of the Society of Petroleum Engineers, Oklahoma City, OK, USA, 1 January 1993. [Google Scholar]
- Li, X.; Wang, J.; Elsworth, D. Stress Redistribution and Fracture Propagation during Restimulation of Gas Shale Reservoirs. J. Pet. Sci. Eng.
**2017**, 154, 150–160. [Google Scholar] [CrossRef] - Bruno, M.S.; Nakagawa, F.M. Pore Pressure Influence on Tensile Fracture Propagation in Sedimentary Rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1991**, 28, 261–273. [Google Scholar] [CrossRef] - Berchenko, I.; Detournay, E. Deviation of Hydraulic Fractures through Poroelastic Stress Changes Induced by Fluid Injection and Pumping. Int. J. Rock Mech. Min. Sci.
**1997**, 34, 1009–1019. [Google Scholar] [CrossRef] - Boone, T.J.; Wawrzynek, P.A. Exploiting Poroelastic Effects and Permeability Contrasts to Control Fracture Orientation. In Proceedings of the Society of Petroleum Engineers, Delft, The Netherlands, 1 January 1994. [Google Scholar]
- Dalkhaa, C.; Azzolina, N.A.; Chakhmakhchev, A.; Kurz, B.A.; Sorensen, J.A.; Gorecki, C.D.; Harju, J.A. Refracturing in the Bakken—An Analysis of Data from Across North Dakota. In Proceedings of the OnePetro, Houston, TX, USA, 20 June 2022. [Google Scholar]
- Wright, C.A.; Conant, R.A. Hydraulic Fracture Reorientation in Primary and Secondary Recovery from Low-Permeability Reservoirs. In Proceedings of the Society of Petroleum Engineers, Dallas, TX, USA, 1 January 1995. [Google Scholar]
- Wang, Y.; Yao, Y.; Wang, L.; Hu, Y.; Wu, H.; Wang, H. Case Study: Analysis of Refracturing Crack Orientation-Angle and Extension-Length in Tight Gas Reservoir, Sulige Gasfield of China. In Proceedings of the OnePetro, Madrid, Spain, 6 June 2022. [Google Scholar]
- Wang, D.-B.; Zhou, F.-J.; Li, Y.-P.; Yu, B.; Martyushev, D.; Liu, X.-F.; Wang, M.; He, C.-M.; Han, D.-X.; Sun, D.-L. Numerical Simulation of Fracture Propagation in Russia Carbonate Reservoirs during Refracturing. Pet. Sci.
**2022**, in press. [Google Scholar] [CrossRef] - Chen, J.; Zhang, Q.; Zhang, J. Numerical Simulations of Temporary Plugging-Refracturing Processes in a Conglomerate Reservoir Under Various In-Situ Stress Difference Conditions. Front. Phys.
**2022**, 9, 825. [Google Scholar] [CrossRef] - Melenk, J.M.; Babuška, I. The Partition of Unity Finite Element Method: Basic Theory and Applications. Comput. Methods Appl. Mech. Eng.
**1996**, 139, 289–314. [Google Scholar] [CrossRef] [Green Version] - Belytschko, T.; Black, T. Elastic Crack Growth in Finite Elements with Minimal Remeshing. Int. J. Numer. Methods Eng.
**1999**, 45, 601–620. [Google Scholar] [CrossRef] - Abaqus 6.14-1 Documentation; Dassault Systemes Simulia Corporation: Providence, RI, USA, 2014.
- Song, J.; Areias, P.M.A.; Belytschko, T. A Method for Dynamic Crack and Shear Band Propagation with Phantom Nodes. Int. J. Numer. Methods Eng.
**2006**, 67, 868–893. [Google Scholar] [CrossRef] - Ma, T.; Zhang, K.; Shen, W.; Guo, C.; Xu, H. Discontinuous and Continuous Galerkin Methods for Compressible Single-Phase and Two-Phase Flow in Fractured Porous Media. Adv. Water Resour.
**2021**, 156, 104039. [Google Scholar] [CrossRef] - Xu, Y.; Sheng, G.; Zhao, H.; Hui, Y.; Zhou, Y.; Ma, J.; Rao, X.; Zhong, X.; Gong, J. A New Approach for Gas-Water Flow Simulation in Multi-Fractured Horizontal Wells of Shale Gas Reservoirs. J. Pet. Sci. Eng.
**2021**, 199, 108292. [Google Scholar] [CrossRef] - Rice, J.R.; Cleary, M.P. Some Basic Stress Diffusion Solutions for Fluid—Saturated Elastic Porous Media with Compressible Constituents. Rev. Geophys.
**1976**, 14, 227–241. [Google Scholar] [CrossRef] - Barenblatt, G.I. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. In Advances in Applied Mechanics; Dryden, H.L., von Kármán, T., Kuerti, G., van den Dungen, F.H., Howarth, L., Eds.; Elsevier: Amsterdam, The Netherlands, 1962; Volume 7, pp. 55–129. [Google Scholar]
- Benzeggagh, M.L.; Kenane, M. Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus. Compos. Sci. Technol.
**1996**, 56, 439–449. [Google Scholar] [CrossRef] - Geertsma, J.; De Klerk, F. A Rapid Method of Predicting Width and Extent of Hydraulically Induced Fractures. J. Pet. Technol.
**1969**, 21, 1571–1581. [Google Scholar] [CrossRef] - Detournay, E. Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks. Int. J. Geomech.
**2004**, 4, 35–45. [Google Scholar] [CrossRef] - Bunger, A.P.; Detournay, E.; Garagash, D.I. Toughness-Dominated Hydraulic Fracture with Leak-Off. Int. J. Fract.
**2005**, 134, 175–190. [Google Scholar] [CrossRef] - Irwin, G. Fracture Strength Relative to Onset and Arrest of Crack Propagation. In Proceedings of the Proc ASTM, Philadelphia, PA, USA, 1 January 1958; Volume 58, pp. 640–657. [Google Scholar]
- Roussel, N.P.; Sharma, M.M. Role of Stress Reorientation in the Success of Refracture Treatments in Tight Gas Sands. SPE Prod. Oper.
**2012**, 27, 346–355. [Google Scholar] [CrossRef]

**Figure 6.**Numerical results of the analytical solution and the XFEM model: (

**a**) injection pressure at the fracture mouth; (

**b**) fracture opening.

**Figure 7.**The impact of approaching angle on fracture reorientation: (

**a**) 15°; (

**b**) 30°; (

**c**) 60°; and (

**d**) 90°. Deformation factor = 20.

**Figure 8.**Direction of σH at different approaching angles: (

**a**) 15°; (

**b**) 30°; (

**c**) 60°; and (

**d**) 90°. Deformation factor = 20.

**Figure 9.**Injection pressure at different approaching angles: (

**a**) 15°; (

**b**) 30°; (

**c**) 60°; and (

**d**) 90°.

**Figure 10.**The impact of stress difference on fracture reorientation: (

**a**) 0 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa; (

**d**) 9 MPa. Deformation factor = 20.

**Figure 11.**Direction of σ

_{H}at different stress differences: (

**a**) 0 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa; (

**d**) 9 MPa. Deformation factor = 20.

**Figure 12.**Injection pressure over the injection time at different stress differences: (

**a**) 0 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa; (

**d**) 9 MPa.

**Figure 13.**The impact of production time on fracture reorientation: (

**a**) 21,600 s; (

**b**) 43,200 s; (

**c**) 86,400 s; (

**d**) 129,600 s. Deformation factor = 20.

**Figure 14.**Direction of σ

_{H}at different production time: (

**a**) 21,600 s; (

**b**) 43,200 s; (

**c**) 86,400 s; (

**d**) 129,600 s. Deformation factor = 20.

**Figure 15.**Injection pressure over injection time at different production time: (

**a**) 21,600 s; (

**b**) 43,200 s; (

**c**) 86,400 s; (

**d**) 129,600 s.

Parameters | Values |
---|---|

Elastic modulus, E | 15,000 MPa |

Poisson’s ratio, ν | 0.25 |

Critical fracture energy, G_{C} | 250 Pa·m |

Injection rate, Q_{inj} | 0.001 m^{2}/s |

Fluid viscosity, μ | 1 mPa·s |

Tensile strength, T_{max} | 3 MPa |

Filtration coefficient, c_{L} | 5.879 × 10^{−13} Pa/(m·s) |

Rock porosity, $\varphi $ | 0.1 |

Rock permeability, k | 0.01 mD |

Original pore pressure, p_{p} | 30 MPa |

Far-field stress, σ_{H}/σ_{h}/σ_{v} | 15/12/18 MPa |

Injection rate, Q_{inj} | 2 × 10^{−3} m^{2}/s |

Injection time, t_{inj} | 100 s |

Production rate, Q_{prod} | 2 × 10^{−4} m^{2}/s |

Production time, t_{prod} | 86,400 s |

Parameters | Values |
---|---|

Elastic modulus, E | 20,000 MPa |

Poisson’s ratio, ν | 0.22 |

Fracture toughness, K_{IC} | 100 kPa·m^{1/2} |

Flow rate, Q_{inj} | 0.06 m^{2}/min |

Fluid viscosity, μ | 100 mPa·s |

Dimensionless parameter, K_{m} | 0.313 |

Time, t | 0.5 min |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, D.; Dahi Taleghani, A.; Yu, B.; Wang, M.; He, C.
Numerical Simulation of Fracture Propagation during Refracturing. *Sustainability* **2022**, *14*, 9422.
https://doi.org/10.3390/su14159422

**AMA Style**

Wang D, Dahi Taleghani A, Yu B, Wang M, He C.
Numerical Simulation of Fracture Propagation during Refracturing. *Sustainability*. 2022; 14(15):9422.
https://doi.org/10.3390/su14159422

**Chicago/Turabian Style**

Wang, Daobing, Arash Dahi Taleghani, Bo Yu, Meng Wang, and Chunming He.
2022. "Numerical Simulation of Fracture Propagation during Refracturing" *Sustainability* 14, no. 15: 9422.
https://doi.org/10.3390/su14159422