Acta Geotechnica

, Volume 14, Issue 2, pp 403–416 | Cite as

A 2D coupled hydro-thermal model for the combined finite-discrete element method

  • Chengzeng Yan
  • Yu-Yong JiaoEmail author
  • Shengqi Yang
Research Paper


Based on the combined finite-discrete element method (FDEM), a two-dimensional coupled hydro-thermal model is proposed. This model can simulate fluid flow and heat transfer in rock masses with arbitrary complex fracture networks. The model consists of three parts: a heat conduction model of the rock matrix, a heat-transfer model of the fluid in the fracture (including the heat conduction and convection of fluid), and a heat exchange model between the fluid and rock at the fracture surface. Three examples with analytical solutions are given to verify the correctness of the coupled model. Finally, the coupled model is applied to hydro-thermal coupling simulations of a rock mass with a fracture network. The temperature field evolution, the effect of thermal conductivity of the rock matrix thermal conductivity and the fracture aperture on the outlet temperature are studied. The coupled model presented in this paper will enable the application of FDEM to study rock rupture driven by the effect of hydro-thermo-mechanical coupling in geomaterials such as in geothermal systems, petroleum engineering, environmental engineering and nuclear waste geological storage.


Finite-discrete element method (FDEM) Fracture flow Heat conduction and convection Hydro-thermal coupling Numerical simulation 

List of symbols


Heat flow rate in the i direction


Thermal conductivity tensor of rock matrix






Net heat flowing into mass \(M\) per unit time




Specific heat of rock matrix


Area of a triangular element


Outer normal vector


Average temperature of edge m

\(\Delta x_{j}^{m}\)

Difference between the coordinate components of the two vertices at edge m


Two-dimensional permutation tensor


Heat flow rate along the x direction


Heat flow rate along the y direction


Outer normal unit vector of the edge opposite to node n


Length of the edge opposite to node n


Heat flow flowing into node 1 from triangular element Δ123


Total heat flow into node 1 per unit time


Nodal temperature at \(t\)

\(T_{t + \Delta t}^{\text{s}}\)

Nodal temperature at \(t + \Delta t\) and \(t\)


Mass density of rock matrix


Rock matrix volume of node 1

\(\Delta x\)

Size of the smallest element


Convective heat-transfer coefficient


Thermal diffusion coefficient (\(k/\rho C_{\text{p}}\) when \(k_{x} = k_{y}\))

\(\Delta T\)

Temperature difference between node 1 and node 2

\(T_{ 1}\)

Temperature at node 1


Temperature at node 2


Thermal conductivity of fluid


Total heat flow rate due to heat conduction


Fluid flow rate between node 1 and node 2


Pressure at node 1


Pressure at node 2

\(\Delta p\)

Pressure difference between node 1 and node 2


Dynamic viscosity of fluid


Aperture of fractures


Total heat flow rate of node 1 due to heat convection


Temperature of node 1 at \(t\)

\(T_{t + \Delta t}^{\text{f}}\)

Temperature of node 1 at \(t + \Delta t\)


Specific heat of fluid


Mass density of fluid


Total heat flow rate of node 1


Time step


Half of the volume of all the broken joint elements that connect to node 1

\(T_{\text{s}}^{ + }\), \(T_{\text{s}}^{ - }\)

Temperature of rock matrix at both sides of a fracture


Temperature of fluid in a fracture


Fracture length


Heat exchange between fluid and rock matrix per unit time


Temperature of the left boundary


Temperature of the right boundary


\((T - T_{\text{L}} )/T_{\text{L}}\)


Peclet number


Flow velocity of fluid


Thermal conductivity of rock matrix


Initial temperature of rock matrix


Fluid temperature at the left boundary


Complementary error function


Dynamic viscosity of fluid



This work was supported by the National Natural Science Foundation of China under the Grant Number 11602006; the Beijing Natural Science Foundation under the Grant Number 1174012; the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan); the Chaoyang District Postdoctoral Science Foundation funded project under the Grant Number 2016ZZ-01-08; and the National Natural Science Foundation of China under Grant Number 41731284, 11672360 and 51479191.


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Copyright information

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

Authors and Affiliations

  1. 1.Faculty of EngineeringChina University of GeosciencesWuhanChina
  2. 2.State Key Laboratory for Geomechanics and Deep Underground EngineeringChina University of Mining and TechnologyXuzhouChina

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