# Bonding Strength Effects in Hydro-Mechanical Coupling Transport in Granular Porous Media by Pore-Scale Modeling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

#### 2.1. Lattice Boltzmann Method (LBM)

_{i}. A widespread LBM implementation in the literature is the lattice Bhatnagar–Gross–Krook (LBGK) model, where the collision operator is simplified as a linearized version by assuming that the collision operator relaxes the local particle distribution functions at a single rate [15]. The evolution equation of “LBGK” model can be written as [15]:

_{i}is the distribution function in the ith discrete velocity direction ${\mathit{e}}_{\mathit{i}}$, ${f}_{i}^{eq}$ is the corresponding equilibrium distribution function, ${\mathsf{\delta}}_{t}$ is the time step, and τ is the relaxation time related to the fluid kinematic viscosity:

#### 2.2. Discrete Element Method (DEM)

#### 2.3. Fluid and Solid Interaction

**R**is the mass center of the particle.

## 3. Benchmarks

#### 3.1. Single Particle Sedimentation

^{3}and 10

^{−6}m

^{2}/s, respectively. The particle with density of 1010 kg/m

^{3}is initially placed off the centerline of channel, and then released with a zero velocity. Because the particle is heavier than the fluid, it will settle down along the direction of gravity, and oscillate around the centerline owing to the influence of channel walls as in Figure 2b. Eventually a steady state can be achieved, where the particle reaches a terminal velocity with no lateral motion, and the terminal particle Reynolds number can be calculated:

#### 3.2. Two-Particle Sedimentation

^{−6}m

^{2}/s and density 1000 kg/m

^{3}. The diameter of particle is 0.2 cm, and particles density is 1010 kg/m

^{3}. The first particle is at 0.999 cm and 0.8 cm in Y and X direction, respectively. The second particle is at 1.0 cm and 1.2 cm in Y and X direction, respectively. For comparison, all these parameters are the same as those in [46,47]. Initially, the two particles are released with a zero velocity, and the upper one will settle in the wake of the other, which leads to the rearrangement mechanism called “drafting, kissing, and tumbling” or DKT motion [48]. Drafting means the particle in the wake will accelerate with an increasing acceleration owing to the low pressure in the wake. Then two particles contact with each other, which is called kissing. Due to the instability of contacting particles aligned in the direction parallel to the motion, they tend to “tumble” to another position. As a result, the center line of particles rotates an angle, and eventually the two particles depart from each other due to the unsymmetrical wake [48].

## 4. Numerical Results and Discussion

#### 4.1. Biaxial Compression Simulation

#### 4.2. Sand Production Simulation

#### 4.2.1. Physical Model

^{−5}m. From Equation (2) the time step can be calculated (dt = 1.6 × 10

^{−4}s) which is a safe value for a stable solution in DEM integration, because it is smaller than the DEM critical time step limit (9 × 10

^{−4}s) calculated by:

_{n}is the normal spring stiffness. Moreover, under this grid system the largest Mach number in sand production simulation is O(10

^{−3}) which is much smaller than 0.1, so the deleterious compressibility effect can be annihilated [15].

#### 4.2.2. Damage Evolution

#### 4.2.3. Effect of Bonding Strength

#### 4.2.4. The Effect of Flow Rate

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Ranjith, P.G.; Perera, M.S.A.; Perera, W.K.G.; Wu, B.; Choi, K.S. Effective parameters for sand production in unconsolidated formations: An experimentl study. J. Petrol. Sci. Eng.
**2013**, 105, 34–42. [Google Scholar] [CrossRef] - Papamichos, E. Erosion and multiphase flow in porous media: Application to sand production. Eur. J. Environ. Civ. Eng.
**2010**, 14, 1129–1154. [Google Scholar] [CrossRef] - Ranjith, P.G.; Pereraa, M.S.A.; Pereraa, W.K.G.; Choib, S.K.; Yasarc, E. Sand production during the extrusion of hydrocarbons from geological formations: A review. J. Petrol. Sci. Eng.
**2014**, 124, 72–82. [Google Scholar] [CrossRef] - Van den Hoek, P.; Hertogh, G.M.M.; Kooijman, A.P.; de Bree, P.; Kenter, C.J.; Papamichos, E. A new concept of sand production prediction: Theory and laboratory experiments. SPE Drill. Complet.
**2000**, 15, 261–273. [Google Scholar] [CrossRef] - Tronvoll, J.; Skj, A.; Papamichos, E. Sand production: Mechanical failure or hydrodynamic erosion? Int. J. Rock Mech. Min. Sci.
**1997**, 34, 291. [Google Scholar] [CrossRef] - Vardoulakis, I.; Stavropoulou, M.; Papanastasiou, P. Hydro-mechanical aspects of the sand production problem. Transp. Porous Med.
**1996**, 22, 225–244. [Google Scholar] [CrossRef] - Boutt, D.F.; Cook, B.K.; Williams, J.R. A coupled fluid-solid model for problems in geomechanics: Application to sand production. Int. J. Numer. Anal. Methods Geomech.
**2011**, 35, 997–1018. [Google Scholar] [CrossRef] - Cundall, P.A.; Strack, O.D. A discrete numerical model for granular assemblies. Geotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - O’Connor, R.M.; Torczynski, J.R.; Preece, D.S.; Klosek, J.T.; Williams, J.R. Discrete element modeling of sand production. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1997**, 34, 231. [Google Scholar] [CrossRef] - Jensen, R.P.; Preece, D.S. Modeling Sand Production with Darcy-Flow Coupled with Discrete Elements; Sandia National Labs: Albuquerque, NM, USA; Livermore, CA, USA, 2000.
- Zhou, Z.Y.; Yu, A.B.; Choi, S.K. Numerical simulation of the liquid-induced erosion in a weakly bonded sand assembly. Powder Technol.
**2011**, 211, 237–249. [Google Scholar] [CrossRef] - Climent, N.; Arroyoa, M.; O’Sullivanb, C.; Gensa, A. Sand production simulation coupling DEM with CFD. Eur. J. Environ. Civ. Eng.
**2014**, 18, 983–1008. [Google Scholar] [CrossRef] - Li, L.; Papamichos, E.; Cerasi, P. Investigation of sand production mechanisms using DEM with fluid flow. In Proceedings of the International Symposium of the International Society for Rock Mechanics (Eurock’06), Liege, Belgium, 13–15 August 2014.
- Han, Y.; Cundall, P.A. LBM-DEM modeling of fluid-solid interaction in porous media. Int. J. Numer. Anal. Methods Geomech.
**2013**, 37, 1391–1407. [Google Scholar] [CrossRef] - Chen, S.; Doolen, G.D. Lattice boltzmann method for fluid flows. Annu. Rev. Fluid Mech.
**1998**, 30, 329–364. [Google Scholar] [CrossRef] - Wang, M.; Chen, S. Electroosmosis in homogeneously charged micro- and nanoscale random porous media. J. Coll. Interface Sci.
**2007**, 314, 264–273. [Google Scholar] [CrossRef] [PubMed] - Wang, M.; Pan, N. Predictions of effective physical properties of complex multiphase materials. Mater. Sci. Eng. R Rep.
**2008**, 63, 1–30. [Google Scholar] [CrossRef] - Aidun, C.K.; Clausen, J.R. Lattice-Boltzmann Method for Complex Flows. Annu. Rev. Fluid Mech.
**2010**, 42, 439–472. [Google Scholar] [CrossRef] - Ladd, A.J.C. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech.
**1994**, 271, 285–309. [Google Scholar] [CrossRef] - Aidun, C.K.; Lu, Y.; Ding, E.J. Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech.
**1998**, 373, 287–311. [Google Scholar] [CrossRef] - Chen, Y.; Kanga, Q.; Caia, Q.; Wanga, M.; Zhang, D. Lattice Boltzmann Simulation of Particle Motion in Binary Immiscible Fluids. Commun. Comput. Phys.
**2015**, 18, 757–786. [Google Scholar] [CrossRef] - Ghassemi, A.; Pak, A. Numerical simulation of sand production experiment using a coupled Lattice Boltzmann-Discrete Element Method. J. Petrol. Sci. Eng.
**2015**, 135, 218–231. [Google Scholar] [CrossRef] - Velloso, R.Q.; Vargas, E.A.; Goncalves, C.J.; Prestes, A. Analysis of sand production processes at the pore scale using the discrete element method and lattice Boltzman procedures. In Proceedings of the 44th US Rock Mechanics Symposium and 5th US-Canada Rock Mechanics Symposium, Salt Lake City, UT, USA, 27–30 June 2010.
- Tronvoll, J.; Papamichos, E.; Skjaerstein, A.; Sanfilippo, F. Sand production in ultra-weak sandstones: Is sand control absolutely necessary? In Proceedings of the Latin American and Caribbean Petroleum Engineering Conference, Rio de Janeiro, Brazil, 30 August–3 September 1997.
- Jiang, M.; Yu, H.S.; Leroueil, S. A simple and efficient approach to capturing bonding effect in naturally microstructured sands by discrete element method. Int. J. Numer. Methods Eng.
**2007**, 69, 1158–1193. [Google Scholar] [CrossRef] - Servant, G.; Marchina, P.; Nauroy, J.F. Near Wellbore Modeling: Sand Production Issues. In Proceedings of the SPE Annual Technical Conference and Exhibition, Anaheim, CA, USA, 11–14 November 2007.
- Wang, M.; Kang, Q.J.; Pan, N. Thermal conductivity enhancement of carbon fiber composites. Appl. Ther. Eng.
**2009**, 29, 418–421. [Google Scholar] [CrossRef] - Wang, M.; Pan, N.; Wang, J.; Chen, S. Lattice Poisson-Boltzmann simulations of electroosmotic flows in charged anisotropic porous media. Commun. Comput. Phys.
**2007**, 2, 1055–1070. [Google Scholar] - Wang, M.; Kang, Q.; Viswanathan, H.; Robinson, B. Modeling of electro-osmosis of dilute electrolyte solutions in silica microporous media. J. Geophys. Res. Solid Earth
**2010**, 115. [Google Scholar] [CrossRef] - Wang, M.R.; Kang, Q.J. Electrokinetic Transport in Microchannels with Random Roughness. Anal. Chem.
**2009**, 81, 2953–2961. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Wang, M.R. Modeling of electrokinetic reactive transport in micropore using a coupled lattice Boltzmann method. J. Geophys. Res. Solid Earth
**2015**, 120, 2877–2890. [Google Scholar] [CrossRef] - Yang, X.; Mehmanib, Y.; Perkinsa, W.A.; Pasqualic, A.; Schönherrc, M.; Kimd, K.; Peregod, M.; Parksd, M.L.; Traske, N.; Balhoff, M.T. Intercomparison of 3D pore-scale flow and solute transport simulation methods. Adv. Water Resour.
**2015**. [Google Scholar] [CrossRef] - Kang, Q.J.; Zhang, D.X.; Chen, S.Y. Displacement of a two-dimensional immiscible droplet in a channel. Phys. Fluids
**2002**, 14, 3203–3214. [Google Scholar] [CrossRef] - Huang, H.B.; Lu, X.Y. Relative permeabilities and coupling effects in steady-state gas-liquid flow in porous media: A lattice Boltzmann study. Phys. Fluids
**2009**, 21, 092104. [Google Scholar] [CrossRef] - Huang, H.B.; Wang, L.; Lu, X.Y. Evaluation of three lattice Boltzmann models for multiphase flows in porous media. Comput. Math. Appl.
**2010**, 61, 3606–3617. [Google Scholar] [CrossRef] - Huang, H.B.; Huang, J.J.; Lu, X.Y. Study of immiscible displacements in porous media using a color-gradient-based multiphase lattice Boltzmann method. Comput. Fluids
**2014**, 93, 164–172. [Google Scholar] [CrossRef] - Chen, L.; Kang, Q.; Mu, Y.; He, Y.; Tao, W. A critical review of the pseudopotential multiphase lattice Boltzmann model: Methods and applications. Int. J. Heat Mass Transf.
**2014**, 76, 210–236. [Google Scholar] [CrossRef] - Chen, Y.; Cai, Q.; Xia, Z.; Wang, M.; Chen, S. Momentum-exchange method in lattice Boltzmann simulations of particle-fluid interactions. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2013**, 88, 013303. [Google Scholar] [CrossRef] [PubMed] - De Rosis, A. On the dynamics of a tandem of asynchronous flapping wings: Lattice Boltzmann-immersed boundary simulations. Phys. A Stat. Mech. Appl.
**2014**, 410, 276–286. [Google Scholar] [CrossRef] - Cao, C.; Chen, S.; Li, J.; Liu, Z.; Zha, L.; Bao, S.; Zheng, C. Simulating the interactions of two freely settling spherical particles in Newtonian fluid using lattice-Boltzmann method. Appl. Math. Comput.
**2015**, 250, 533–551. [Google Scholar] [CrossRef] - Galindo-Torres, S. A coupled Discrete Element Lattice Boltzmann Method for the simulation of fluid-solid interaction with particles of general shapes. Comput. Methods Appl. Mech. Eng.
**2013**, 265, 107–119. [Google Scholar] [CrossRef] - Verlet, L. Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev.
**1967**, 159, 98. [Google Scholar] [CrossRef] - Jiang, M.; Yan, H.B.; Zhu, H.H.; Utili, S. Modeling shear behavior and strain localization in cemented sands by two-dimensional distinct element method analyses. Comput. Geotech.
**2011**, 38, 14–29. [Google Scholar] [CrossRef] - Wen, B.; Li, H.; Zhang, C.; Fang, H. Lattice-type-dependent momentum-exchange method for moving boundaries. Phys. Rev. E
**2012**, 85, 016704. [Google Scholar] [CrossRef] [PubMed] - Feng, J.; Hu, H.H.; Joseph, D.D. Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid Part 1. Sedimentation. J. Fluid Mech.
**1994**, 261, 95–134. [Google Scholar] [CrossRef] - Feng, Z.G.; Michaelides, E.E. The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. J. Comput. Phys.
**2004**, 195, 602–628. [Google Scholar] [CrossRef] - Niu, X.D.; Shu, C.; Chew, Y.T.; Peng, Y. A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows. Phys. Lett. Sect. A Gen. Atom. Solid State Phys.
**2006**, 354, 173–182. [Google Scholar] [CrossRef] - Fortes, A.F.; Joseph, D.D.; Lundgren, T.S. Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech.
**1987**, 177, 467–483. [Google Scholar] [CrossRef] - Galindo-Torres, S.; Pedroso, D.M.; Williams, D.J.; Mühlhaus, H.B. Strength of non-spherical particles with anisotropic geometries under triaxial and shearing loading configurations. Granul. Matter
**2013**, 15, 531–542. [Google Scholar] [CrossRef] - Wang, Y.H.; Leung, S.C. A particulate-scale investigation of cemented sand behavior. Can. Geotech. J.
**2008**, 45, 29–44. [Google Scholar] [CrossRef] - Hazzard, J.F.; Young, R.P.; Maxwell, S. Micromechanical modeling of cracking and failure in brittle rocks. J. Geophys. Res. Solid Earth
**2000**, 105, 16683–16697. [Google Scholar] [CrossRef]

**Figure 1.**The computation of lattice Boltzmann method (LBM) and discrete element method (DEM) coupling.

**Figure 2.**The benchmark case of single particle sedimentation. (

**a**) The geometry of the simulation; (

**b**) the comparison of particle trajectories.

**Figure 3.**The benchmark case of two particles sedimentation. (

**a**) The geometry of the simulation; (

**b**) the settling trajectory of two-particle sedimentation.

**Figure 5.**The DEM sample for biaxial compression simulation. (

**a**) The hexagonal packing of the granular media; (

**b**) the boundary condition for biaxial compression simulation.

**Figure 6.**The deviatoric stress–strain responses of cemented granular media in biaxial compression simulation.

**Figure 7.**Shear band after macroscopic failure of cemented granular media in the biaxial compression simulation.

**Figure 8.**The deviatoric stress–strain response and number of accumulative bond breakage during biaxial compression: (

**a**) bonding strength is 0.0078 N; (

**b**) bonding strength is 0.039 N; and (

**c**) bonding strength is 0.078 N.

**Figure 9.**The cemented granular media used in sand production simulation. (

**a**) The cross section of a single perforation in cemented reservoir; (

**b**) the geometry of sample used in sand production simulation.

**Figure 10.**The process of sand production in cemented granular media. (

**a**) Initially, the sample is intact with no bonding breakage; (

**b**) the bond near the cavity begins to be broken; (

**c**) small cracks propagate into the sample leading to the localized failure near the cavity; (

**d**) the flow erodes the localized failure zone.

**Figure 11.**The number of accumulative bond breakage for three cemented samples under the pressure gradient of 2.86 Pa/m.

**Figure 12.**The state of cemented granular media with different bond strengths under the same effective stress and flow erosion: (

**a**) continuous sand production for low bonding strength 0.0078 N; and (

**b**) the stable cavity for high bonding strength 0.078 N.

**Figure 13.**The number of accumulative bond breakage for synthetic cemented materials: (

**a**) with high bonding strength 0.078 N; and (

**b**) with low bonding strength 0.0078 N under different flow rate.

Parameter | Value |
---|---|

Number of particle | 600 |

Diameter of particle | 2 mm |

${k}_{n}$ | 2.5 × 10^{4} N/m |

${k}_{s}$ | 1.0 × 10^{4} N/m |

μ | 0.2 |

Low bonding strength | 0.0078 N |

Middle bonding strength | 0.039 N |

High bonding strength | 0.078 N |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Xie, C.; Chen, Y.; Wang, M.
Bonding Strength Effects in Hydro-Mechanical Coupling Transport in Granular Porous Media by Pore-Scale Modeling. *Computation* **2016**, *4*, 15.
https://doi.org/10.3390/computation4010015

**AMA Style**

Chen Z, Xie C, Chen Y, Wang M.
Bonding Strength Effects in Hydro-Mechanical Coupling Transport in Granular Porous Media by Pore-Scale Modeling. *Computation*. 2016; 4(1):15.
https://doi.org/10.3390/computation4010015

**Chicago/Turabian Style**

Chen, Zhiqiang, Chiyu Xie, Yu Chen, and Moran Wang.
2016. "Bonding Strength Effects in Hydro-Mechanical Coupling Transport in Granular Porous Media by Pore-Scale Modeling" *Computation* 4, no. 1: 15.
https://doi.org/10.3390/computation4010015