# A Hybrid Finite Volume and Extended Finite Element Method for Hydraulic Fracturing with Cohesive Crack Propagation in Quasi-Brittle Materials

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Conceptual Model

#### 2.2. Deformation Model

_{HF}is depicted in Figure 1. Considering that the deformation model satisfies the condition of equilibrium, the linear momentum balance for this problem can be expressed as:

**σ**is the stress tensor, ρ is the body density, and

**b**is the body force per unit volume. The boundary conditions of this balance equation can be expressed as:

**n**

_{Г}is the unit outward normal vector to the external boundary, ${\mathbf{n}}_{{\mathsf{\Gamma}}_{\mathcal{HF}}}^{-}$ and ${\mathbf{n}}_{{\mathsf{\Gamma}}_{\mathcal{HF}}}^{+}$ are the unit outward normal vectors on either side of the discontinuity (the + and − superscripts represent two sides of the discontinuity), p

^{+}and p

^{−}are the fluid pressures within the fracture surface on either side of the discontinuity, and

**t**

^{coh}is the cohesive traction acting at the fracture process zone. Considering that the difference between the corresponding values at the two fracture surfaces is small, it is, therefore, assumed that the fluid pressures and cohesive tractions are equivalent at both faces of the crack (i.e., p

^{+}= p

^{−}= p, n

^{+}= n

^{−}= n).

**D**is the fourth-order linear elastic stiffness tensor of the solid materials and

**ε**denotes the related strain tensor; the latter can be linked to displacement by:

**u**represents the displacement vector of the domain.

#### 2.3. Fracture Flow Model

#### 2.4. Fracture Propagation Model

_{ult}is the ultimate strength of the material and G

_{c}is the unit fracture energy.

#### 2.5. Discretization and Solution Algorithms

#### 2.5.1. Discretization of Equilibrium Equation

^{+}and Ω

^{−}. At any one time, the displacement field consists of two parts: a standard displacement and an additional displacement. It is assumed that the discontinuity must cross an element without accounting for the fracture tip singularity because the crack-tip singularity will disappear once the cohesive crack model is adopted anywhere in the mesh [50]. The XFEM-based displacement approximation can thus be written as:

_{I}is the regular shape function of node I corresponding to the regular dofs of the displacement field

**u**

_{I}, ${N}_{I}^{\mathcal{HF}}$ is the enriched shape function of node I associated with the enriched dofs of the displacement field ${\tilde{\mathbf{u}}}_{I}^{\mathcal{HF}}$, and ${H}_{{\mathsf{\Gamma}}_{\mathcal{HF}}}$ is the Heaviside step function typically used to characterize the strong discontinuities. Note that the Heaviside step function can be expressed as:

**N**(

**x**) is the matrix of the regular shape functions corresponding to the regular dofs of the displacement field

**U**and ${\tilde{\mathbf{N}}}^{\mathcal{HF}}(\mathbf{x})$ denotes the matrix of the enriched shape functions associated with the enriched dofs of the displacement field ${\tilde{\mathbf{U}}}^{\mathcal{HF}}$.

**K**and

**F**denote the global stiffness matrix and the global force vector, respectively, and

**u**represents the displacement vector.

**L**is the regular differential operator as defined in the classic FEM.

#### 2.5.2. Discretization of the Fracture Flow

_{i}and A

_{i}can be expressed as:

_{ic}is the distance between the control cell c and its neighbouring cell i, and l

_{ic}is the distance between the centroid of the interface and the centroid of the control cell c. It is worth noting that the face area A

_{i}is replaced by the fracture aperture w

_{i}for a 2D cell.

**M**is the stiffness matrix of the pressure field defined as follows:

_{1}and M

_{n}only consist of two terms as the boundary conditions at the fluid injection point and the crack tip are known. Hence, the integration of (34) over the first control volume cell can be written as:

_{0}is the injection pressure at the crack mouth and p

_{t}is the pressure of the crack tip, which for this case is zero, i.e., p

_{t}= 0.

#### 2.6. Fluid–Solid Coupling Procedure and the Picard Iteration Approach

## 3. Results and Discussion

#### 3.1. Khristianovic–Geertsma–de Klerk (KGD) Model

_{0}, and the fluid drove the fracture propagation. In consideration of the in situ stress state field, a confining far-field external force σ

_{0}= 3.7 MPa was vertically imposed at the top and bottom of the model, and the left side was constrained in the x-direction. As previously mentioned, the linear elastic constitutive law was considered, and the cohesive crack model was used in the fracture process. The material parameters used in this analysis are given in Table 1.

#### 3.2. Hydraulic Fracturing Test

#### 3.3. Hydraulic Fracturing in a Concrete Gravity Dam

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The definition and boundary conditions of a hydraulic fracturing body within a geomechanical discontinuity.

**Figure 7.**A hydraulic-driven fracture propagation in an imperious domain. A schematic representation of the Khristianovic–Geertsma–de Klerk (KGD) problem, the geometry, the boundary conditions, and the finite element mesh.

**Figure 8.**Comparison of the numerical and analytical solutions regarding: (

**left**) the fracture width at the crack mouth and (

**right**) the fracture width profile.

**Figure 9.**Comparison of the numerical and analytical solutions regarding: (

**left**) the fracture half-length and (

**right**) the fracture mouth pressure.

**Figure 10.**Relative error for the fracture width, the fracture half-length, and the fracture mouth pressure using: (

**left**) the FVM-XFEM method and (

**right**) the Carrier et al. [22] model.

**Figure 11.**A schematic illustration of the hydraulic fracturing (HF) test: (

**left**) Problem definition and (

**right**) the XFEM meshes.

**Figure 12.**The test process (see

**a**,

**b**) and chart of the fracture propagation path containing both the test results (see

**c**–

**h**) and the numerical results (

**i**).

**Figure 14.**Comparison of the numerical and test results regarding: (

**left**) the fracture width and (

**right**) the fluid pressure distribution.

**Figure 15.**A concrete gravity dam under hydrostatic pressure: the geometry, boundary conditions (

**left**), and XFEM meshes (

**right**).

**Figure 16.**The crack mouth opening displacement (

**left**) and the water pressure distribution along the crack (

**right**).

**Figure 17.**The contour of maximum principal stress using the coupling FVM-XFEM method (

**left**) and the patterns of crack growth in the concrete dam using the FVM-XFEM method and the “constant pressure algorithm” (

**right**).

**Table 1.**Parameters for the bulk material and the cohesive zone (from [22]).

Young’s modulus | E = 17.0 GPa |

Poisson’s ratio | ν = 0.2 |

Tensile strength | f_{t} = 1.25 MPa |

Cohesive fracture energy | G_{c} = 120 N/m |

Water density | ρ_{w} = 1000 kg/m^{3} |

Water dynamic viscosity | μ = 1.00 × 10^{−4} Pa s |

Fluid injection rate | Q_{0} = 5.00 × 10^{−4} m^{2}/s |

Young’s modulus | E = 27.25 GPa |

Poisson’s ratio | ν = 0.173 |

Tensile strength | f_{t} = 2.01 MPa |

Cohesive fracture energy | G_{c} = 140 N/m |

Water density | ρ_{w} = 1000 kg/m^{3} |

Water dynamic viscosity | μ = 1.00 × 10^{−6} kPa s |

Young’s modulus | E = 25.0 GPa |

Poisson’s ratio | ν = 0.167 |

Tensile strength | f_{t} = 2.05 MPa |

Cohesive fracture energy | G_{c} = 150 N/m |

Water density | ρ_{w} = 1000 kg/m^{3} |

Water dynamic viscosity | μ = 1.00 × 10^{−6} kPa∙s |

Solid density | ρ_{w} = 2400 kg/m^{3} |

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**MDPI and ACS Style**

Liu, C.; Shen, Z.; Gan, L.; Jin, T.; Zhang, H.; Liu, D.
A Hybrid Finite Volume and Extended Finite Element Method for Hydraulic Fracturing with Cohesive Crack Propagation in Quasi-Brittle Materials. *Materials* **2018**, *11*, 1921.
https://doi.org/10.3390/ma11101921

**AMA Style**

Liu C, Shen Z, Gan L, Jin T, Zhang H, Liu D.
A Hybrid Finite Volume and Extended Finite Element Method for Hydraulic Fracturing with Cohesive Crack Propagation in Quasi-Brittle Materials. *Materials*. 2018; 11(10):1921.
https://doi.org/10.3390/ma11101921

**Chicago/Turabian Style**

Liu, Chong, Zhenzhong Shen, Lei Gan, Tian Jin, Hongwei Zhang, and Detan Liu.
2018. "A Hybrid Finite Volume and Extended Finite Element Method for Hydraulic Fracturing with Cohesive Crack Propagation in Quasi-Brittle Materials" *Materials* 11, no. 10: 1921.
https://doi.org/10.3390/ma11101921