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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

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

σx, σy and σz

In situ stress in x, y, and z direction

pw

Boundary fluid pressure

σtip,1

Maximum principal stress at the fracture tip

T0

Strength of the material

σtip

Stress tensor at the fracture tip

σi

Stress tensor at the integration point i

ηi

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

σi,1

Maximum principal stress at the integration point i

ng

Total number of integration points in a computational area

di

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

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

uh(x)

Global approximation in domain Ω

ui(x)

Local approximation associated with node i

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

Visibility zone

φi(x)

Sub-weight functions

φi(x)

Shape functions of GFEM associated with node i

vol (P1, P2, P3, P4)

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 x on fracture surface

δ0

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

Δijk

Fracture element with nodes i, j and k

pi, pj and pk

Fluid pressures at nodes i, j and k

O

Centroid point of a fracture element

Ωij

Fluid path between node i and node j

Jij

Fluid pressure gradient between node i and node j

zi and zj

Vertical coordinates of node i and node j

lij

Distance between node i and node j

µ

Fluid dynamic viscosity coefficient

δij

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

bij

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

qij

Flow rate from node i to node j

s

Node saturation

sn and sn−1

Node saturation at the current time step and previous time step

fs

A function of saturation

pn and pn−1

Fluid pressure at the current time step and previous time step

Kw

Bulk modulus of fluid

q

Total flow rate

t

Time increment

Vn and Vn−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

Fsolid, Ffluid

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

p1, p2, p3, p4

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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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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