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

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.

\(q_{i}\) :

Heat flow rate in the i direction

\(k_{ij}\) :

Thermal conductivity tensor of rock matrix

\(T\) :

Temperature

\(M\) :

Mass

\(Q_{\text{net}}\) :

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

\(t\) :

Time

\(C_{\text{s}}\) :

Specific heat of rock matrix

\(A\) :

Area of a triangular element

\(n\) :

Outer normal vector

\(\bar{T}^{m}\) :

Average temperature of edge m

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

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

\(\in_{ij}\) :

Two-dimensional permutation tensor

\(q_{x}\) :

Heat flow rate along the x direction

\(q_{y}\) :

Heat flow rate along the y direction

\(n_{i}^{(n)}\) :

Outer normal unit vector of the edge opposite to node n

\(L^{(n)}\) :

Length of the edge opposite to node n

Q _Δ123 :

Heat flow flowing into node 1 from triangular element Δ123

\(Q_{\text{s}}\) :

Total heat flow into node 1 per unit time

\(T_{t}^{\text{s}}\) :

Nodal temperature at \(t\)

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

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

\(\rho_{\text{s}}\) :

Mass density of rock matrix

\(V_{\text{s}}\) :

Rock matrix volume of node 1

\(\Delta x\) :

Size of the smallest element

\(h\) :

Convective heat-transfer coefficient

\(\kappa\) :

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

\(T_{2}\) :

Temperature at node 2

\(k_{\text{f}}\) :

Thermal conductivity of fluid

\(Q_{{{\text{f}}1}}\) :

Total heat flow rate due to heat conduction

\(q_{\text{f}}\) :

Fluid flow rate between node 1 and node 2

\(p_{1}\) :

Pressure at node 1

\(p_{2}\) :

Pressure at node 2

\(\Delta p\) :

Pressure difference between node 1 and node 2

\(\mu\) :

Dynamic viscosity of fluid

\(a\) :

Aperture of fractures

\(Q_{\text{f2}}\) :

Total heat flow rate of node 1 due to heat convection

\(T_{t}^{\text{f}}\) :

Temperature of node 1 at \(t\)

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

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

\(C_{\text{f}}\) :

Specific heat of fluid

\(\rho_{\text{f}}\) :

Mass density of fluid

\(Q_{\text{f}}\) :

Total heat flow rate of node 1

Δt :

Time step

\(V\) :

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

\(T_{\text{f}}\) :

Temperature of fluid in a fracture

\(L\) :

Fracture length

\(Q_{\text{e}}\) :

Heat exchange between fluid and rock matrix per unit time

\(T_{\text{L}}\) :

Temperature of the left boundary

\(T_{\text{R}}\) :

Temperature of the right boundary

\(\hat{T}\) :

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

\(P_{\text{e}}\) :

Peclet number

\(v_{\text{f}}\) :

Flow velocity of fluid

\(k_{\text{s}}\) :

Thermal conductivity of rock matrix

\(T_{{{\text{s}}0}}\) :

Initial temperature of rock matrix

\(T_{{{\text{f}}0}}\) :

Fluid temperature at the left boundary

\(erfc\) :

Complementary error function

\(\mu\) :

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.

Authors and Affiliations

1. Faculty of Engineering, China University of Geosciences, Wuhan, 430074, China

   Chengzeng Yan & Yu-Yong Jiao

2. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 21116, China

   Shengqi Yang

Corresponding author

Correspondence to Yu-Yong Jiao.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Reprints and permissions