Rock Mechanics and Rock Engineering

, Volume 52, Issue 10, pp 3675–3690

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

• Chao Jiang
• Gao-Feng Zhao
Original Paper

## 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.

## Keywords

Hydraulic fracturing Distinct lattice spring model Lattice Boltzmann method

## Roman alphabets

$$a$$

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

$$G$$

$$F$$

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

## Greek symbols

$$\Delta t$$

The time step

$$\Delta x$$

The lattice space width

## Notes

### Acknowledgements

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

## References

1. Adachi J, Siebrits E, Peirce A, Desroches J (2007) Computer simulation of hydraulic fractures. Int J Rock Mech Min Sci 44(5):739–757
2. Braza M, Chassaing P, Ha MH (1986) Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J Fluid Mech 165(165):79–130
3. Buxton GA, Verberg R, Jasnow D, Balazs AC (2005) Newtonian fluid meets an elastic solid: coupling lattice boltzmann and lattice-spring models. Physic Rev E. 71(5 Pt 2):056707
4. Chen S, Doolen GD (1998) Lattice boltzmann method for fluid flows. Annual Rev Fluid Mech 30(1):329–364
5. Faxén H (1946) Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes. In: Proceedings of the Royal Swedish Academy of Engineering Sciences, vol 187, p 1Google Scholar
6. Garcia M, Gutierrez J, Rueda N (2011) Fluid–structure coupling using lattice-Boltzmann and fixed-grid FEM. Finite Elem Analy Design 47(8):906–912
7. García-Salaberri PA, Gostick JT, Hwang G, Weber AZ, Vera M (2015) Effective diffusivity in partially-saturated carbon-fiber gas diffusion layers: effect of local saturation and application to macroscopic continuum models. J Power Sourc 296:440–453
8. Ghia U, Chia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys 48(3):387–411
9. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Month Not R Astron Soc 181(3):375–389
10. Gui Y, Zhao GF (2015) Modelling of laboratory soil desiccation cracking using DLSM with a two-phase bond model. Comput Geotech 69:578–587
11. Han Y, Cundall PA (2011) Resolution sensitivity of momentum exchange and immersed boundary methods for solid–fluid interaction in the lattice Boltzmann method. Int J Numer Meth Fluids 67(3):314–327
12. Han Y, Cundall PA (2013) LBM–DEM modeling of fluid–solid interaction in porous media. Int J Numer Analyt Meth Geomech 37(10):1391–1407
13. Holmes DW, Williams JR, Tilke P (2011) Smooth particle hydrodynamics simulations of low Reynolds number flows through porous media. Int J Numer Analyt Meth Geomech 35(4):419–437
14. Hoogerbrugge P, Koelman J (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. EPL 19(3):155
15. Hu HH, Joseph DD, Crochet MJ (1992) Direct simulation of fluid particle motions. Theor Comput Fluid Dyn 3(5):285–306
16. Inamuro T (2012) Lattice Boltzmann methods for moving boundary flows. Fluid Dyn Res 44(2):024001
17. Ji C, Munjiza A, Williams JJR (2012) A novel iterative direct-forcing immersed boundary method and its finite volume applications. J Comput Phys 231(4):1797–1821
18. Jiang C, Zhao G-F (2018) Implementation of a coupled plastic damage distinct lattice spring model for dynamic crack propagation in geomaterials. Int J Numer Analyt Meth Geomech 42(4):674–693
19. Jiang C, Zhao G-F, Zhu J, Zhao Y-X, Shen L (2016) Investigation of dynamic crack coalescence using a gypsum-like 3D printing material rock mech. Rock Eng 49(10):3983–3998
20. Jiang C, Zhao G-F, Khalili N (2017) On crack propagation in brittle material using the distinct lattice spring model. Int J Solid Struct 118–119:1339–1351Google Scholar
21. Kazerani T, Zhao G-F, Zhao J (2010) Dynamic fracturing simulation of brittle material using the distinct lattice spring method with a full rate-dependent cohesive law. Rock Mech Rock Eng 43(6):717–726
22. Kollmannsberger S, Geller S, Düster A, Tölke J, Sorger C, Krafczyk M, Rank E (2009) Fixed-grid fluid–structure interaction in two dimensions based on a partitioned lattice boltzmann and p-fem approach. Int J Numer Meth Eng 79(7):817–845
23. Krause MJ, Heuveline V (2013) Parallel fluid flow control and optimisation with lattice Boltzmann methods and automatic differentiation. Comput Fluids 80(1):28–36
24. Kwon YW (2008) Coupling of lattice Boltzmann and finite element methods for fluid-structure interaction application. J Press Vessel Tech 130:011302
25. Kwon YW, Jo JC (2008) 3D modeling of fluid-structure interaction with external flow using coupled LBM and FEM. J Press Vessel Tech 130(2):021301
26. Leonardi A, Wittel FK, Mendoza M, Herrmann HJ (2014) Coupled DEM–LBM method for the free-surface simulation of heterogeneous suspensions. Comput Particle Mech 1(1):3–13
27. Li JC, Li HB, Zhao J (2015) An improved equivalent viscoelastic medium method for wave propagation across layered rock masses. Int J Rock Mech Min Sci 73(1):62–69
28. Li JC, Li NN, Li HB, Zhao J (2017) An SHPB test study on wave propagation across rock masses with different contact area ratios of joint. Int J Impact Eng 105:109–116
29. Lisjak A, Grasselli G, Vietor T (2014) Continuum-discontinuum analysis of failure mechanisms around unsupported circular excavations in anisotropic clay shales. Int J Rock Mech Min Sci 65:96–115
30. Liu M, Meakin P, Huang H (2007) Dissipative particle dynamics simulation of pore-scale multiphase fluid flow. Water Resour Res 43(4):244–247
31. Martel C, Iacono-marziano G (2015) Timescales of bubble coalescence, outgassing, and foam collapse in decompressed rhyolitic melts. Earth Planet Sci Lett 412:173–185
32. Men X, Tang CA, Wang S, Li Y, Yang T, Ma T (2013) Numerical simulation of hydraulic fracturing in heterogeneous rock: the effect of perforation angles and bedding plane on hydraulic fractures evolutions. In: Bunger AP, Mclennan J, Jeffrey R (eds) Effective and sustainable hydraulic fracturing. InTech, RijekaGoogle Scholar
33. Mohamad AA, Kuzmin A (2010) A critical evaluation of force term in lattice Boltzmann method, natural convection problem. Int J Heat Mass Trans 53(5–6):990–996
34. Mora P, Wang Y, Alonso-marroquin F (2015) Lattice solid/Boltzmann microscopic model to simulate solid/fluid systems—a tool to study creation of fluid flow networks for viable deep geothermal energy. J Earth Sci 26(1):11–19
35. Munjiza A, Owen DRJ, Bicanic N (1995) A combined finite-discrete element method in transient dynamics of fracturing solids. Eng Comput 12(2):145–174
36. Palabos 1.5R (2017) http://www.palabos.org/[Online]. Accessed April 25 2017
37. Richou AB, Ambari A, Naciri JK (2004) Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—part 1: poiseuille flow (numerical). Chem Eng Sci 59(15):3215–3222
38. Wang H (2015) Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone method. J Petrol Sci Eng 135:127–140
39. Wang M, Fen YT, Wang CY (2016) Coupled bonded particle and lattice Boltzmann method for modelling fluid–solid interaction. Int J Numer Analyt Meth Geomech 40(10):1383–1401
40. Xue S, Yuan L, Wang J, Wang Y, Xie J (2015) A coupled DEM and LBM model for simulation of outbursts of coal and gas. Int J Coal Sci Tech 2(1):22–29
41. Yin P, Zhao G-F (2015) Numerical study of two-phase fluid distributions in fractured porous media. Int J Numer Analyt Meth Geomech 39(11):1188–1211
42. Yu D, Mei R, Luo LS, Shyy W (2003) Viscous flow computations with the method of lattice Boltzmann equation. Prog Aerospace Sci 39(5):329–367
43. Zhang H, Tan Y, Shu S, Niu X, Trias FX, Yan GD, Li H, Sheng Y (2014) Numerical investigation on the role of discrete element method in combined LBM–IBM–DEM modeling. Comput Fluids 94(2):37–48
44. Zhao G-F (2015) Modelling 3d jointed rock masses using a lattice spring model. Int J Rock Mech Min Sci 78:79–90
45. Zhao G-F (2017) Developing a four-dimensional lattice spring model for mechanical responses of solids. Comput Meth Appl Mech Eng 315:881–895
46. Zhao G-F, Khalili N (2012) A lattice spring model for coupled fluid flow and deformation problems in geomechanics. Rock Mech Rock Eng 45(5):781–799Google Scholar
47. Zhao G-F, Fang J, Zhao J (2011) A 3D distinct lattice spring model for elasticity and dynamic failure. Int J Numer Analyt Meth Geomech 35:859–885
48. Zhao G-F, Russell A, Zhao X, Khalili N (2014) Strain rate dependency of uniaxial tensile strength in Gosford sandstone by the distinct lattice spring model with x-ray micro CT. Int J Solids Struct 51(7–8):1587–1600
49. Zhao G-F, Kazerani T, Man K, Gao M, Zhao J (2015) Numerical study of the semi-circular bend dynamic fracture toughness test using discrete element models. Sci China Tech Sci 58(9):1587–1595
50. Zhao G-F, Lian J, Russell A, Khalili N (2019) Implementation of a modified Drucker-Prager model in the lattice spring model for plasticity and fracture. Comput Geotech 107:97–109