Abstract
Suspension flow through porous medium was studied using the Stokesian dynamics simulation method. Stokesian dynamics is an efficient tool to carry out numerical simulations for suspension of rigid particles interacting through hydrodynamic and nonhydrodynamic forces. After validating the simulation method for a single particle flowing through an array of fixed grain particles, we have analysed the suspension transport through porous medium. It was observed that the hydrodynamic interactions and the interparticle nonhydrodynamic forces between the moving and fixed grain particles have a strong influence on the particle trajectories. This was apparent from the change in particle flux with the fractional channel width in the presence of nonhydrodynamic forces. Hydrodynamic interaction between the suspension and grain particles was also studied for largescale porous system that was generated by a random arrangement of the particles in a periodic cell. It was found that the change in porosity leads to change in the average fluctuation velocity of the suspension. The fluctuation velocity was observed to vary linearly with the particle concentration and average suspension velocity. Finally, a comparative study was performed with suspension flow in a straight channel and it was observed that the shearinduced particle migration in porous medium is altered by the presence of grain particles.
This is a preview of subscription content, log in to check access.
Access options
Buy single article
Instant unlimited access to the full article PDF.
US$ 39.95
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
US$ 199
This is the net price. Taxes to be calculated in checkout.
Abbreviations
 \( a \) :

Radius of particle
 \( D \) :

Brownian diffusivity of an isolated particle
 \( E \) :

Strain tensors of the imposed flow
 \( \hat{e} \) :

Unit vector
 〈e〉:

Rate of strain tensor of disturbance flow
 \( F \) :

The matrix collectively contains hydrodynamic force and torque on the particles
 \( F_{0} \) :

Constant related to the magnitude of the force
 \( I \) :

Force density distributed over the particle surface
 \( l \) :

Lattice constant
 \( M \) :

Mobility tensor
 p :

Pressure
 \( {\text{Pe}} \) :

Peclet number based on particle size
 \( R \) :

Grand resistance tensor
 \( T \) :

Torque of the particles
 \( u \) :

Translational velocity of particles
 〈u〉:

The average velocity of the suspension
 \( U \) :

The matrix collectively contains the translational and rotational velocities of all particles
 \( \text{Re}_{\text{p}} \) :

Reynolds number based on particle size
 \( R_{\text{FU}}^{ij} \) :

The resistance matrix represents the interactions between \( i \) and \( j \) particles
 \( S \) :

The stresslet on the particles
 \( V^{'} \) :

Average rootmeansquare (RMS) velocity
 〈V〉:

Average velocity
 \( \overline{V} \) :

Timeaveraged RMS velocity
 \( W_{i} \) :

Normalized width of y interval of the channel \( i \)
 e:

External force
 FE:

Matrix correlates force and strain
 FU:

Matrix correlates force and velocity
 US:

Correlates velocity and stresslet
 ES:

Correlates strain and stresslet
 r:

Repulsive force
 \( R^{\infty } \) :

Far field
 s:

Suspended particle
 \( F_{\alpha \beta }^{\text{r}} \) :

Between sphere \( \alpha \) and sphere \( \beta \)
 w:

Grain particle
 \( \tau \) :

The parameter related to the range of force
 \( \phi \) :

Porosity the considered system
 \( \varepsilon \) :

The spacing between sphere \( \alpha \) and sphere \( \beta \)
 \( \varepsilon_{ijk} \) :

Levi–Civita symbol
 \( \varOmega \) :

Rotational velocity
 \( \dot{\gamma } \) :

Nominal shear rate
 \( \eta \) :

Viscosity of fluid
 \( \delta_{ij} \) :

Dirac delta function
 \( \rho \) :

Density of fluid
 DLVO:

Derjaguin–Landau–Verwey–Overbeek
 FCW:

Fractional channel width
 FPF:

Fractional particle flux
 SD:

Stokesian dynamics
References
Ahfir, N.D., Benamar, A., Alem, A., Wang, H.: Influence of internal structure and medium length on transport and deposition of suspended particles: a laboratory study. Transp. Porous Med. 76, 289–307 (2009)
Arshadi, M., Zolfaghari, A., Piri, M., AlMuntasheri, G.A., Sayed, M.: The effect of deformation on twophase flow through proppantpacked fractured shale samples: a microscale experimental investigation. Adv. Water Resour. 105, 108–131 (2017)
Batchelor, G.K., Green, J.T.: The hydrodynamic interaction of two small freelymoving spheres in a linear flow field. J. Fluid Mech. 56(2), 375–400 (1972). https://doi.org/10.1017/s0022112072002927
Bedrikovetsky, P.: Upscaling of stochastic micro model for suspension transport in porous media. Transp. Porous Med. 75, 35 (2008)
Benamar, A., Ahfir, N.D., Wang, H., Alem, A.: Particle transport in a saturated porous medium: pore structure effects. ScienceDirect 339(10), 674–681 (2007)
Boccardo, G., Marchisio, D.L., Sethi, R.: Microscale simulation of particle deposition in porous media. J. Colloid Interface Sci. 417, 227–237 (2014)
Bossis, G., Brady, J.F.: Dynamic simulation of sheared suspensions. I. General method. J. Chem. Phys. 80(10), 5141–5154 (1984)
Bradford, S.A., Torkzaban, S., Leij, F., Šimůnek, J., van Genuchten, M.T.: Modeling the coupled effects of pore space geometry and velocity on colloid transport and retention. Water Resour. Res. (2009). https://doi.org/10.1029/2008wr007096
Brady, J.F., Bossis, G.: Stokesian dynamics. J. Fluid Mech. 20, 111–157 (1988)
Bujurke, N.M., Madalli, V.S., Mulimani, B.G.: Laminar flow in a uniformly porous pipe. Indian J. Pure Appl. Math. 31(3), 341–352 (2000)
Chetti, A., Benamar, A., Hazzab, A.: Modeling of particle migration in porous media: application to soil suffusion. Transp. Porous Med. 113(3), 591–606 (2016)
Davis, R.H., Leighton, D.T.: Shearinduced transport of a particle layer along a porous wall. Chem. Eng. Sci. 42(2), 275–281 (1987). https://doi.org/10.1016/00092509(87)850571
De, N., Singh, A.: Numerical simulation of particle migration in suspension flow through heterogeneous porous media. Part. Sci. Technol. (2019). https://doi.org/10.1080/02726351.2019.1651806
Durlofsky, L., Brady, J.F., Bossis, G.: Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49 (1987)
Elimelech, M.: Predicting collision efficiencies of colloidal particles in porous media. Water Res. 26(1), 1–8 (1992). https://doi.org/10.1016/00431354(92)90104C
Elimelech, M., O’Melia, C.R.: Kinetics of deposition of colloidal particles in porous media. Environ. Sci. Technol. 24(10), 1528–1536 (1990). https://doi.org/10.1021/es00080a012
Fallah, H., Fathi, H.B., Mohammadi, H.: The mathematical model for particle suspension flow through porous medium. Sci. Res. 2, 57–62 (2012)
Galal, S.K., Elgibaly, A.A., Elsayed, S.K.: Formation damage due to fines migration and its remedial methods. Egypt. J. Pet. 25, 515–524 (2016)
Ghidaglia, C., Arcangelis, L.D., Hinch, J., Guazzelli, E.: Transition in particle capture in deep bed filtration. Phys. Rev. E 53, R3028–R3031 (1996)
Granger, J., Dodds, J., Leclerc, D., Midoux, N.: Flow and diffusion of particles in a channel with one porous wall: polarization chromatography. Chem. Eng. Sci. 41(12), 3119–3128 (1986). https://doi.org/10.1016/00092509(86)850497
Guazzelli, E., Morris, J.: A Physical Introduction to Suspension Dynamics. Cambridge University Press, Cambridge (2012)
Hassonjee, Q., Ganatoss, P., Pfeffers, R.: A stronginteraction theory for the motion of arbitrary threedimensional clusters of spherical particles at low Reynolds number. J. Fluid Mech. 197, 1–37 (1988)
Hou, B., Chen, M., Cheng, W., Diao, C.: Investigation of hydraulic fracture networks in shale gas reservoirs with random fractures. Arab. J. Sci. Eng. 41, 2681–2691 (2016)
Ikni, T., Benamar, A., Kadri, M., Ahfir, N.D., Wang, H.Q.: Particle transport within watersaturated porous media: effect of pore size on retention kinetics and size selection. C. R. Geosci. 345(9–10), 392–400 (2013)
Juanes, R., Sharma, M.M.: Modeling fracpacks and fracture propagation in poorly consolidated sands. Paper presented at the SPE annual technical conference, Texas (2005)
Kampel, G., Goldsztein, G.H., Santamarina, J.C.: Particle transport in porous media: the role of inertial effects and path tortuosity in the velocity of the particles. Appl. Phys. Lett. 95(19), 194103 (2009)
Kim, S., Karrila, S.J.: Microhydrodynamics: Principles and selected Application. Butterworth  Heinemann Series in Chemical Engineering. ButterworthHeinmann, Stoneham (1991)
Lee, J., Koplik, J.: Microscopic motion of particles flowing through a porous medium. Phys. Fluid. 11, 76–87 (1999)
Leighton, D., Acrivos, A.: The shearinduced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415–439 (1987)
Liu, Q., Prosperetti, A.: Pressuredriven flow in a channel with porous walls. J. Fluid Mech. 679, 77–100 (2011). https://doi.org/10.1017/jfm.2011.124
Nabovati, A., Sousa, A.C.M.: Fluid flow simulation I in random porous media at pore level using the Lattice Boltzmann method. J. Eng. Sci. Technol. 2(3), 226–237 (2007)
Nott, P.R., Brady, J.F.: Pressuredriven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157–199 (1994)
Otomo, R., Harada, S.: End effect on permeability of loose particulate bed having different internal structures. Part. Sci. Technol. 29(1), 2–13 (2011). https://doi.org/10.1080/02726351.2010.488715
Otomo, R., Harada, S.: Fluid permeability in stratified unconsolidated particulate bed. Transp. Porous Media 96(3), 439–456 (2013). https://doi.org/10.1007/s1124201200986
Ranjith, P.G., Perera, M.S.A., Perera, W.K.G., Wub, B., Choi, S.K.: Effective parameters for sand production in unconsolidated formations: an experimental study. J. Pet. Sci. Eng. 105, 34–42 (2013)
Sahimi, M., Imdakm, A.O.: Hydrodynamics of particle motion in porous media. Phys. Rev. Lett. 66(9), 1169–1172 (1991)
Santos, A., Barros, P.H.L.: Multiple particle retention mechanisms during filtration in porous media. Environ. Sci. Technol. 44(7), 2515–2521 (2010). https://doi.org/10.1021/es9034792
Santos, A., Bedrikovetsky, P.: A stochastic model for particulate suspension flow in porous media. Transp. Porous Media 62(1), 23–53 (2006). https://doi.org/10.1007/s1124200551757
Santos, A., Bedrikovetsky, P., Fontoura, S.: Analytical micro model for size exclusion: pore blocking and permeability reduction. J. Membr. Sci. 308(1), 115–127 (2008)
Schwarzer, S.: Sedimentation and flow through porous media: simulating dynamically coupled discrete and continuum phases. Phys. Rev. E 52(6), 6461–6475 (1995)
Sen, T.K., Mahajan, S.P., Khilar, K.C.: Colloidassociated contaminant transport in porous media: 1. Experimental studies. AIChE J. 48(10), 2366–2374 (2002)
Shapiro, A.A., Bedrikovetsky, P.G., Santos, A., Medvedev, O.O.: A stochastic model for filtration of particulate suspensions with incomplete pore plugging. Transp. Porous Media 67(1), 135–164 (2007). https://doi.org/10.1007/s1124200600295
Sharma, M.M., Yortos, Y.C.: Fine migration through porous media. AIChE J. 33(10), 1654–1662 (1987)
Sharma, M.M., Yortsos, Y.C.: Transport of particulate suspensions in porous media. AIChE J. 33(10), 1636–1643 (1987)
Shen, W., Xu, Y., Li, X., Huang, W., Gu, J.: Numerical simulation of gas and water flow mechanism in hydraulically fractured shale gas reservoirs. J. Nat. Gas Sci. Eng. 35, 726–735 (2016)
Singh, A.: Rheology of Noncolloidal Suspension. Phd. Thesis, Indian Institute of Science (2000)
Singh, A., Nott, P.R.: Normal stresses and microstructure in bounded sheared suspensions via Stokesian dynamics simulations. J. Fluid Mech. 412, 279–301 (2000)
Siyyam, H., Hamdan, M.H.: Analysis of particulate behaviour in porous media. Paper presented at the international conference on FLUID MECHANICS, Mexico, January 25–27 (2008)
Song, W., Jinzhou, Z., Yongming, L.: Hydraulic fracturing simulation of complex fractures growth in naturally fractured shale gas reservoir. Arab. J. Sci. Eng. 39, 7411–7419 (2014)
Taheri, S., Ghomeshi, S., Kantzas, A.: Permeability calculations in unconsolidated homogeneous sands. Powder Technol. 321, 380–389 (2017)
Tufenkji, N., Elimelech, M.: Deviation from the classical colloid filtration theory in the presence of repulsive DLVO interactions. Langmuir 20(25), 10818–10828 (2004). https://doi.org/10.1021/la0486638
Vaz, A., Maffra, D., Carageorgos, T., Bedrikovetsky, P.: Characterisation of formation damage during reactive flows in porous media. J. Nat. Gas Sci. Eng. 34, 1422–1433 (2016)
Vilarrasa, V., Koyama, T., Neretnieks, I., Jing, L.: ShearInduced flow channels in a single rock fracture and their effect on solute transport. Transp. Porous Med. 87, 503–523 (2010)
Vittal, S., Sharma, M.M.: A Stokesian dynamics model for particle deposition and bridging in granular media. J. Colloid Interface Sci. 153, 313–336 (1992)
Wang, F., Pan, Z., Zhang, Y., Zhang, S.: Simulation of coupled hydromechanicalchemical phenomena in hydraulically fractured gas shale during fracturingfluid flowback. J. Pet. Sci. Eng. 163, 16–26 (2018)
Wang, M., Brady, J.F.: Shorttime transport properties of bidisperse suspensions and porous media: a Stokesian dynamics study. J. Chem. Phys. 142(9), 094901 (2015). https://doi.org/10.1063/1.4913518
Wua, K., Chen, Z., Li, X., Xu, J., Li, J., Wang, K., Wang, H., Wang, S., Dong, X.: Flow behavior of gas confined in nanoporous shale at high pressure: real gas effect. Fuel 205, 173–183 (2017)
Yang, X.M., Sharma, M.M.: Formation damage caused by cement filtrate in sandstone cores. SPE Prod. Eng. 23, 399–406 (1991)
Yousif, O.S.Q., Karakouzian, M., Rahim, N.O.A., Rashed, K.A.: Physical clogging of uniformly graded porous media under constant flow rates. Transp. Porous Med. 643(3), 643–659 (2017)
Zhai, Z., Sharma, M.M.: A new approach to modelling hydraulic fractures in unconsolidated sands. Paper presented at the SPE annual technical conference, University of Texas at Austin (2005)
Zhang, Q., Prosperetti, A.: Pressuredriven flow in a twodimensional channel with porous walls. J. Fluid Mech. 631, 1–21 (2009). https://doi.org/10.1017/s0022112009005837
ZhaoQin, H., Jun, Y., YueYing, W., Ke, T.: Numerical study on twophase flow through fractured porous media. Sci. China Technol. Sci. 54(9), 2412–2420 (2011)
Acknowledgements
We would like to acknowledge the anonymous reviewers for providing valuable suggestions. The authors also acknowledge the Indian Institute of Technology Guwahati for providing the high performance computing facilities to carry out the simulations.
Author information
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Formulation of Grand Mobility Matrix
Appendix: Formulation of Grand Mobility Matrix
The velocity field in the fluid around a particle moving at low Reynolds number is given by the Stokes equation:
where \( p \) is the pressure, \( \eta \) is the viscosity of the fluid, and \( u \) is particle velocity. The momentum equation for a singlepoint force of strength F located at x is:
where \( \delta \) is Dirac delta function. The disturbance velocity field at any point in the fluid due to the presence of point force is obtained by the solution of Eqs. (14) and (15):
where \( \zeta \left( x \right) \) is the freespace Green function or propagator for Stokes flow, known alternatively as the Stokeslet or Oseen tensor, and is given by:
Here \( r = \left {x_{i}  x_{j} } \right \) is the resultant distance between the source and field point. The velocity field due to a finitesized particle can be considered as equivalent to the Stokeslets distributed over the surface of that particle. Now, the disturbance velocity field at any point in the fluid due to N such particles is simply:
where \( u_{i}^{\infty } (x) \) is the velocity field at a point \( x \) in the absence of particles, \( S_{\alpha } \) denotes the surface of the particle \( \alpha \), and \( y \) is the position of the point force on the particle surface. \( \sigma_{jk} (y) \) is the fluid stress on the surface, and \( n_{k} (y) \) is the surface normal vector pointing into the fluid. When the particles are in close arrangement, there will be large numbers of surfaces and calculation of velocity will be computationally difficult. To avoid this difficulty, Stokesian dynamics does not evaluate the integral in Eq. (18); instead, a multipole expansion of the integral about the centre of each particle is carried out. This technique starts with the Taylor series expansion of Eq. (18) about the centre of each particle:
where \( I_{j} \) is the force density, and \( \eta \) is fluid viscosity. The force density at any point y can be expressed in terms of fluid stress tensor:
The zeroth moment of the multipole expansion is the total force (\( F^{\alpha } \)) exerted by a particle \( \alpha \) on the fluid:
The first moment has both symmetric and antisymmetric parts. The symmetric part is called as stresslet \( \left( {S_{jk} } \right) \), and the antisymmetric component is the total torque \( (T_{i}^{\alpha } ) \):
where \( \varepsilon_{ijk} \) is Levi–Civita symbol. In the case of dilute suspension, the hydrodynamic interactions are well represented by the force–torque–stresslet approximation. Nonetheless, the series in Eq. (19) can be easily extended to include the higher order multipole moments for the case of a dense suspension. The nthorder multipole moment for particle α is given as:
Thus, the above formulations can be used to determine the disturbance velocity in the fluid due to the presence of large number of particles. Now, the interest is to find out the velocity field of the particles in the fluid due to the presence of other particles. The motion of a spherical particle \( \alpha \) immersed in the suspension is given by Faxén’s formulae (Batchelor and Green 1972):
where \( u_{i}^{'} (x^{\alpha } ) \) is the velocity disturbance at the centre of particle \( \alpha \) caused by the other particles, i.e., other than the particle \( \alpha \) itself, and relative to the imposed flow \( u_{i}^{\infty } \). Also, \( e_{ij}^{'} \) is the rate of strain of the disturbance flow, \( E_{ij}^{\infty } \) is the rate of strain of the imposed flow, and \( \varOmega^{\infty } \) is the angular velocity of the imposed flow. \( u^{\alpha } \) and \( \varOmega^{\alpha } \) are the translational and angular velocities of particle \( \alpha \), respectively.
Thereafter, applying Eqs. (25), (26), and (27) on each particle in the suspension, a grand mobility matrix, \( M \), can be obtained which relate the translational velocity, angular velocity, and rate of strain tensor of each particle to the force, torque, and stresslet due to N number of particles:
where \( U  U^{\infty } \) is a 6N vector containing the translational and angular velocities of all particles relative to the disturbed flow, and \(  E^{\infty } \) is a 9N vector that represents the rate of strain for the flow field at the centre of the particles. Besides, \( F \) is a 6N vector containing the force and torque exerted by the particles on the fluid and \( S \) is the 9N stresslet vector. The grand mobility matrix is a positive definite and symmetric matrix, and it can be decomposed into the following submatrix:
where the correlation between various components is indicated by the subscripts. \( M_{\text{UF}} \) relates particle velocity to the forces, \( M_{\text{US}} \) relates velocity to stresslet, \( M_{\text{EF}} \) relates the rate of strain tensor to forces, and \( M_{\text{ES}} \) relates the rate of strain tensor to stresslet.
Rights and permissions
About this article
Cite this article
De, N., Singh, A. Stokesian Dynamics Simulation of Suspension Flow in Porous Media. Transp Porous Med 131, 473–502 (2020). https://doi.org/10.1007/s11242019013543
Received:
Accepted:
Published:
Issue Date:
Keywords
 Stokesian dynamics
 Lubrication forces
 Porous media
 Particle trapping
 Suspension flow