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A Coupling Model of Distinct Lattice Spring Model and Lattice Boltzmann Method for Hydraulic Fracturing

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Abstract

In this work, the distinct lattice spring model (DLSM) and the lattice Boltzmann method (LBM) are coupled together to simulate hydraulic fracturing problems. As the DLSM and LBM are both lattice modelling methods, the lattice meshes in these two systems are simply overlapped, which results in the same resolution in both the DLSM and LBM. The momentum exchange bounce-back algorithm is used to evaluate the forces exerted on the solid particles. Moreover, the calculation step in the LBM and DLSM is synchronised for prompt updates of fluid–solid interactions. The coupled model is further validated through a series of benchmarks. Finally, the coupled model shows its ability to simulate hydraulic fracturing in formations with complex discrete fracture networks.

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Abbreviations

\(a\) :

One of the nine directions of the D2Q9 model

\(\tilde{a}\) :

The opposite direction of α

\(\alpha\) :

The velocity-coupling factor from the DLSM to LBM

\(\beta\) :

The force-coupling factor from the LBM to DLSM

\(b\) :

The half-channel width

\(c\) :

The basic speed on the lattice

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

The speed of sound in the lattice

\(D\) :

The spatial dimension of the analysis

\(G\) :

The pressure gradient

\(F\) :

The body force term

\(f\) :

The particle distribution function

\(f_{\text{a}}^{\text{eq}}\) :

The equilibrium distribution

\(\lambda\) :

The wall correction factor of the drag force

\(L_{\text{lid}}\) :

The length of the lid-driven cavity

\(L_{R}\) :

The length scale factor of the DLSM to a physical system

\(L_{\text{r}}\) :

The length scale factor of the LBM to a physical system

\(P\) :

The macroscopic pressure

\({\text{Re}}\) :

The Reynolds number

\(\rho\) :

The macroscopic density

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

The density scale factor of the DLSM to a physical system

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

The density scale factor of the LBM to a physical system

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

The initial density of the LBM

\(\sigma_{\text{NS}}\) :

The far-field stresses from north or south

\(\sigma_{\text{EW}}\) :

The far-field stresses from east or west

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

The time-scale factor of the DLSM to a physical system

\(t_{r}\) :

The time-scale factor of the LBM to a physical system

\(k\) :

The ratio of the diameter of the cylinder to the width of the channel

\(u_{\hbox{max} }\) :

The maximum velocity approaching the cylinder

\(\mu\) :

The dynamic viscosity

\(U_{\text{x}}\) :

The flow velocity in the X direction

\(\Delta t\) :

The time step

\(\Delta x\) :

The lattice space width

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Acknowledgements

This research is financially supported by the National Natural Science Foundation of China (Grant No. 1177020290).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin, China

    Chao Jiang & Gao-Feng Zhao

  2. School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia

    Chao Jiang

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  1. Chao Jiang
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  2. Gao-Feng Zhao
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Correspondence to Gao-Feng Zhao.

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Jiang, C., Zhao, GF. A Coupling Model of Distinct Lattice Spring Model and Lattice Boltzmann Method for Hydraulic Fracturing. Rock Mech Rock Eng 52, 3675–3690 (2019). https://doi.org/10.1007/s00603-019-01819-3

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