Rock Mechanics and Rock Engineering

, Volume 52, Issue 6, pp 1781–1801 | Cite as

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

  • Quansheng Liu
  • Lei Sun
  • Xuhai TangEmail author
  • Bo Guo
Original Paper


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.


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

List of Symbols

σx, σy and σz

In situ stress in x, y, and z direction


Boundary fluid pressure


Maximum principal stress at the fracture tip


Strength of the material


Stress tensor at the fracture tip


Stress tensor at the integration point i


Weight function associated with integration point i for the calculation of stress tensor at fracture tip


Maximum principal stress at the integration point i


Total number of integration points in a computational area


Distance from the integration point i to the fracture tip


A positive but small number

σ and ε

Cauchy stress tensor and strain tensor


Tetrahedral element with four vertex nodes

P1, P2, P3, P4

Vertex nodes of a tetrahedral element

ω1(x), ω2(x), ω3(x), ω4(x)

Weight functions at a computational point x of GFEM


Global approximation in domain Ω


Local approximation associated with node i

\(\chi _{\Omega }^{{{\text{vis}}}}\)

Visibility zone


Sub-weight functions


Shape functions of GFEM associated with node i

vol (P1, P2, P3, P4)

Volume of a tetrahedral element


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


Apertures of point x on fracture surface


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


Fracture element with nodes i, j and k

pi, pj and pk

Fluid pressures at nodes i, j and k


Centroid point of a fracture element


Fluid path between node i and node j


Fluid pressure gradient between node i and node j

zi and zj

Vertical coordinates of node i and node j


Distance between node i and node j


Fluid dynamic viscosity coefficient


Equivalent aperture of the flow path between node i and node j


Equivalent width of the fluid path between node i and node j


Flow rate from node i to node j


Node saturation

sn and sn−1

Node saturation at the current time step and previous time step


A function of saturation

pn and pn−1

Fluid pressure at the current time step and previous time step


Bulk modulus of fluid


Total flow rate


Time increment

Vn and Vn−1

Volumes of the node at the current time step and previous time step


Global stiffness matrix of GFEM


Nodal displacement vector of GFEM


Load vector of GFEM

Fsolid, Ffluid

Vectors of the forces for solid mechanics and forces for fluid mechanics


Shape functions for calculating fluid pressure


Young’s modulus of the rock mass


Poisson’s ratio of the rock mass


Radius of the fracture


Distance to the fracture central point


Ratio of the in situ stresses and hydraulic pressure

p1, p2, p3, p4

Monitoring points



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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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil EngineeringWuhan UniversityWuhanChina
  2. 2.State Key Laboratory of Water Resources and Hydropower Engineering ScienceWuhan UniversityWuhanChina
  3. 3.Department of Energy Resources EngineeringStanford UniversityStanfordUS

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