Combined Finite-Discrete Element Method for Simulation of Hydraulic Fracturing

Hydraulic fracturing is widely used in the exploitation of unconventional gas (such as shale gas).Thus, the study of hydraulic fracturing is of particular importance for petroleum industry. The combined finite-discrete element method (FDEM) proposed by Munjiza is an innovative numerical technique to capture progressive damage and failure processes in rock. However, it cannot model the fracturing process of rock driven by hydraulic pressure. In this study, we present a coupled hydro-mechanical model based on FDEM for the simulation of hydraulic fracturing in complex fracture geometries, where an algorithm for updating hydraulic fracture network is proposed. The algorithm can carry out connectivity searches for arbitrarily complex fracture networks. Then, we develop a new combined finite-discrete element method numerical code (Y-flow) for the simulation of hydraulic fracturing. Finally, several verification examples are given, and the simulation results agree well with the analytical or experimental results, indicating that the newly developed numerical code can capture hydraulic fracturing process correctly and effectively.

M :

Lumped mass diagonal matrices

C :

Damping diagonal matrices

F :

Nodal force vector

X :

Vector of nodal coordinates

Δp :

Pressure differential between two adjacent nodes

p ₁ :

Pressure of Node 1

p ₂ :

Pressure of Node 2

ρ _w :

Mass density of fluid

g :

Gravitational acceleration

y ₁ :

Vertical coordinate of Node 1

y ₂ :

Vertical coordinate of Node 2

q :

Flow rate

μ :

Dynamic coefficient of viscosity

a :

Average aperture of crack element

a ₀ :

Initial value of aperture

a _res :

Minimum value of aperture

a _max :

Maximum value of aperture

u _n :

Normal displacement of crack element

L :

Length of crack element

f _s :

Function of saturation

f _n :

Total fluid pressure on edge of triangular element

s :

Saturation of current time step

s ₀ :

Saturation of previous time step

Q :

Total flow rate

q ₂₁ :

Flow rate from Node 1 to 2

q ₂₃ :

Flow rate from Node 3 to 2

q ₂₄ :

Flow rate from Node 4 to 2

p :

Pressure of the node at the current time step

p ₀ :

Pressure of the node at the previous time step

K _w :

Bulk modulus of the fluid

k _i :

Permeability factor of the ith crack element

Δt :

Time step

Δt _f :

Critical time step

V :

Volume of the node at the current time step

V ₀ :

Volume of the node at the previous time step

ΔV :

Volume change

V _m :

Average volume

V ₂ :

Volume of Node 2

V _j21 :

Volume of crack element j21

V _j23 :

Volume of crack element j23

V _j24 :

Volume of crack element j24

h :

Height of free surface

h ₁ :

Water level on the left

h ₂ :

Water level on the right

x :

X-axis coordinate component

B :

Width of the dam

P :

Hydraulic pressure at the location with a distance to the impermeable boundary of (l − x)

P ₀ :

Hydraulic at the left end

T :

Non-dimensional time

t :

Time

l :

Length of single crack

ζ :

\(\frac{l - x}{l}\)

This work has been supported by the National Natural Science Foundation of China under the Grant number 11202223; and the Natural Basic Research Program of China under the Grant numbers: 2011CB013505 and 2014CB047100. Special thanks to the anonymous reviewer and editor-in-chief who have given their time and expertise to improve this article.

Authors and Affiliations

1. Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing, 100124, China

   Chengzeng Yan

2. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, China

   Hong Zheng, Guanhua Sun & Xiurun Ge

Corresponding author

Correspondence to Hong Zheng.

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