Rock Mechanics and Rock Engineering

, Volume 52, Issue 2, pp 611–627 | Cite as

Modeling Three-Dimensional Fluid-Driven Propagation of Multiple Fractures using TOUGH-FEMM

  • Xuhai TangEmail author
  • Jonny Rutqvist
  • Mengsu Hu
  • Nithin Manohar Rayudu
Original Paper


In this paper, a new numerical simulation tool named TOUGH-FEMM is presented and applied to model three-dimensional (3D) hydraulic fracturing in porous rock. The fluid flow in both fractures and porous rock is modeled using TOUGH2, which is a well-established code for analysis of multiphase and multi-component fluid flow. Rock deformations associated with fracture propagation are modeled using finite element-meshfree method (FEMM). FEMM is an approach to simulate fracture propagation without remeshing, in which the fracture path does not need to be predetermined. Fracture mechanics with mixed-mode stress intensity factors are employed to detect fracture instability and determine the direction of fracture propagation. TOUGH-FEMM is verified for modeling fluid-driven fracture propagation in 3D through a number of simulation examples, including modeling of hydraulic fracturing laboratory experiments and by comparison to independent numerical simulation results for multiple interacting hydraulic fractures at ten to hundred meter scale.


Hydraulic fracturing TOUGH-FEMM TOUGH2 Partition of unity method FE-Meshfree method 

List of Symbols


a tetrahedral element

{P1, P2, P3, P4}

The vertices of a tetrahedral element


Global approximation in FEMM


Local approximation associated to node i in FEMM


Weight function associated to node i in FEMM

\(\varphi _{i}^{\prime }\)

Sub-weight-function associated to node i in FEMM

\({\psi _\Omega }\)

a set of nodes related to an element domain in FEMM

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

Visibility zone related to an element domain in FEMM

\({M^\kappa }\)

Mass per unit volume of fluid phase \(\kappa\)


Porosity of fluid phase


Saturation of water


Density of water

\(X_{\psi }^{\kappa }\)

Mass fraction of component \(\kappa\) within fluid phase\(\psi\)

\(q_{\psi }^{\kappa }\)

Mass flux of component \(\kappa\) within fluid phase\(\psi\)

\({\mathbf{i}}_{\psi }^{\kappa }\)

Diffusive flux of component \(\kappa\) within fluid phase\(\psi\)


An effective molecular-diffusion coefficient of fluid phase

\({\theta _0}\)

Fracture propagation angle in a local coordinate system


Stress intensity factor

\({\bar {K}_I}\)

Equivalent stress intensity factor


Critical stress intensity factor

r, θ

Cylindrical co-ordinates at the fracture tip

\(\Delta {\vec {v}_n}\)

Propagation vectors



This work was supported by the National Natural Science Foundation of China (Grant No. 41602296) and by the US Department of Energy under contract No. DE-AC02-05CH11231 to the Lawrence Berkeley National Laboratory.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringWuhan UniversityWuhanChina
  2. 2.Lawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.Department of Mechanical EngineeringBirla Institute of Technology and ScienceGoaIndia

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