# Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Stress Criterion

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## Abstract

Accurate simulation of the propagation of hydraulic fractures under in situ stress conditions in three dimensions (3D) is critical for the enhanced design and optimization of hydraulic fracturing in various engineering applications, such as shale gas/oil production and geothermal utilization. To model fracture propagation for geotechnical applications numerically, a maximum principal stress criterion (MPS-criterion) with a weighted average approximation is conventionally applied. However, it is found that the weighted average approximation is inappropriate for hydraulic fracturing under in situ stress conditions, where the presence of both hydraulic pressure and in situ stress can lead to sharp changes of the stress field in the vicinity of the fracture tips. When both hydraulic pressure and in situ stress are considered, the simulated results with the weighted average approximation are inaccurate and are sensitive to the radius of the computational area. In this paper, we present numerical tests to identify this limitation of the weighted average approximation and propose a novel point-based approximation for the MPS-criterion. The performance of the MPS-criterion with the point-based approximation for hydraulic fracturing under in situ stress conditions is confirmed by a numerical test. It can be seen that, compared to the traditional weighted average approximation, the MPS-criterion with the point-based approximation is more stable and accurate for modelling hydraulic fracturing under in situ stress conditions.

## Keywords

Hydraulic fracturing Point-based approximation Maximum principal stress criterion (MPS-criterion) Generalized finite element method (GFEM) Hydro-mechanical coupling## List of Symbols

*σ*_{x},*σ*_{y}and*σ*_{z}In situ stress in

*x, y*, and*z*direction*p*_{w}Boundary fluid pressure

*σ*^{tip,1}Maximum principal stress at the fracture tip

*T*_{0}Strength of the material

**σ**^{tip}Stress tensor at the fracture tip

**σ**_{i}Stress tensor at the integration point

*i**η*_{i}Weight function associated with integration point

*i*for the calculation of stress tensor at fracture tip*σ*^{i,1}Maximum principal stress at the integration point

*i*- ng
Total number of integration points in a computational area

*d*_{i}Distance from the integration point

*i*to the fracture tip*ε*A positive but small number

**σ**and**ε**Cauchy stress tensor and strain tensor

**Ω**Tetrahedral element with four vertex nodes

*P*_{1},*P*_{2},*P*_{3},*P*_{4}Vertex nodes of a tetrahedral element

*ω*_{1}(**x**),*ω*_{2}(**x**),*ω*_{3}(**x**),*ω*_{4}(**x**)Weight functions at a computational point

of GFEM**x***u*^{h}(**x**)Global approximation in domain

**Ω***u*_{i}(**x**)Local approximation associated with node

*i*- \(\chi _{\Omega }^{{{\text{vis}}}}\)
Visibility zone

*φ*_{i}^{’}(**x**)Sub-weight functions

*φ*_{i}(**x**)Shape functions of GFEM associated with node

*i*- vol (P
_{1},*P*_{2},*P*_{3},*P*_{4}) Volume of a tetrahedral element

*S*Fracture surface

*S*^{+}and S^{−}The actual upper side and lower side of fracture surface

*u*^{+}(**x**) and*u*^{−}(**x**)Point displacements on the upper and lower sides of the fracture surface

*δ*(*x*)Apertures of point

on fracture surface**x***δ*_{0}Initial aperture of fracture

- \(\bar {p}_{n}^{{i+1}}\) and \(\bar {p}_{n}^{i}\)
Nodes of fracture tips at steps

*i*+ 1 and*i*- \(\overrightarrow {{\Delta _n}}\)
Fracture propagation vector

- Δ
*ijk* Fracture element with nodes

*i, j*and*k**p*_{i},*p*_{j}and*p*_{k}Fluid pressures at nodes

*i, j*and*k**O*Centroid point of a fracture element

*Ω*_{i− j}Fluid path between node

*i*and node*j**J*_{i− j}Fluid pressure gradient between node

*i*and node*j**z*_{i}and*z*_{j}Vertical coordinates of node

*i*and node*j**l*_{i− j}Distance between node

*i*and node*j**µ*Fluid dynamic viscosity coefficient

*δ*_{i− j}Equivalent aperture of the flow path between node

*i*and node*j**b*_{i− j}Equivalent width of the fluid path between node

*i*and node*j**q*_{i− j}Flow rate from node

*i*to node*j**s*Node saturation

*s*^{n}and*s*^{n−1}Node saturation at the current time step and previous time step

*f*_{s}A function of saturation

*p*^{n}and*p*^{n−1}Fluid pressure at the current time step and previous time step

*K*_{w}Bulk modulus of fluid

*q*Total flow rate

- ∆
*t* Time increment

*V*^{n}and*V*^{n−1}Volumes of the node at the current time step and previous time step

**K**Global stiffness matrix of GFEM

**a**Nodal displacement vector of GFEM

**F**Load vector of GFEM

**F**^{solid},**F**^{fluid}Vectors of the forces for solid mechanics and forces for fluid mechanics

**N**Shape functions for calculating fluid pressure

*E*Young’s modulus of the rock mass

*v*Poisson’s ratio of the rock mass

*a*Radius of the fracture

*r*Distance to the fracture central point

- λ
Ratio of the in situ stresses and hydraulic pressure

*p*_{1},*p*_{2},*p*_{3},*p*_{4}Monitoring points

## Notes

### Acknowledgements

This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC1501300) and the National Natural Science Foundation of China (Grant No. 41602296).

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