CFD–DEM simulation of mud cake formation in heterogeneous porous medium for lost circulation control


The lost circulation is a problem commonly observed in well drilling operations under overbalanced pressure conditions. Among the geological formation characteristics that can intensify the loss of drilling fluid, the presence of highly permeable regions is one of the most critical. In this case, corrective measures are necessary, like the use of lost circulation materials (LCM) to build a mud cake in the porous substrate, sealing the pores, and controlling the lost circulation. This work presents a numerical study to understand the influence of fluid–particle interaction under dynamic filtration on the process of particle packing in porous media, resembling the use of LCMs to grow a mud cake and reduce the fluid invasion. The flow of an incompressible Newtonian fluid in a porous medium modeled in pore scale is considered, being the voids structured as an anisotropic array of staggered solid cylinders. The LCM is considered as solid and discrete spherical particles (dp= 0.75 mm) immersed in a water–glycerin fluid. A Euler–Lagrange approach to model the liquid–solid two-phase flow is employed, with the simulation being resorted via the dense discrete phase model coupled to the discrete element method (DEM). The DEM computes collision and friction forces present in the particle–particle and particle–wall interactions. Variation effects for different values of the Reynolds number in the vertical channel ReCH,i (125, 250, 500) and the lost circulation intensity measured by the initial fluid loss ratio Qloss (5, 10, 20%) are considered. By releasing solid particles, the fluid loss ratio, relative permeability, particle layer dimensions, and also the vertical channel pressure are altered. For an initial fluid loss ratio of 20%, the decay in Qloss after 60 s for ReCH,i values of 125, 250 and 500 is, respectively, 17%, 13% and 12%. The variation of the initial fluid loss ratio Qloss dramatically influences the number of particles that forms the bed. After 60 s, no particle has entered the porous matrix for Qloss = 5%. When Qloss = 10%, the particles build a bed, and for 20%, the plugging of the substrate occurs.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


A :

Surface area

C D :

Drag coefficient (–)

C lm :

Magnus lift coefficient (–)

C ls :

Saffman lift coefficient (–)

C vm :

Virtual mass coefficient (–)

C ω :

Rotational drag coefficient (–)

d :

Diameter (mm)

e PM :

Cylindrical array height (mm)

F d :

Drag force (N)


DEM-computed force (N)

F ls :

Saffman lift force (N)

F lm :

Magnus lift force (N)

F n :

Contact force in the normal direction (N)

F pg :

Pressure gradient force (N)

F t :

Contact force in the tangential direction (N)

F vm :

Virtual mass force (N)

f pβ :

Particle–fluid coupling source term (N/m3)

F t :

Contact force on the tangential direction (N)

g :

Gravity acceleration vector (m/s2)

h :

Length along x-direction (mm)

I :

Moment of inertia (kg m2)

k :

Permeability (m2)

k n :

Stiffness constant (N/m)

K :

Relative permeability (–)

l :

Length along y-direction (mm)

l z :

Geometric domain thickness (mm)

m :

Mass (kg)

\({\dot{m}}_{{\rm p,PM,i}}\) :

Particle mass flow rate through the control surface PM,i (kg/s)

n :

Unitary vector in the normal contact direction (–)

N :

Number of particles injected per second (Part./s)

Q :

Flow rate (m3/s)

Q loss :

Fluid loss ratio (–)

R :

Radius (mm)

p :

Pressure (Pa)

P m,CH,i :

Dimensionless pressure at the inlet surface control surface (–)


Pore throat dimension (mm)


Reynolds number (–)

t :

Time (s)

t :

Unitary vector in tangential contact direction (–)

T d :

Rotational drag torque (N m)


DEM-calculated torque (N m)

\({\bar{u}}\) :

Surface-averaged velocity (m/s)

u :

Velocity vector (m/s)

V :

Volume (m3)

x, y, z :

Spatial variables (mm)

δ n :

Particle overlap

Δ :


ε :

Phase volumetric fraction (–)

η n :

Damping coefficient (N s/m)

μ :

Friction coefficient (–)

μ g :

Sliding friction coefficient (–)

μ r :

Rolling friction coefficient (–)

μ s :

Stick friction coefficient (–)

μ β :

Fluid dynamic viscosity (Pa s)

φ :

Porosity (–)

ω :

Angular velocity (1/s)


Vertical channel


Vertical channel inlet control surface


Vertical channel outlet control surface


Vertical channel downstream of the porous region


Outlet channel of the porous region

i :

SEPC index


Isotropic region


Particle injection

j :

Particle index


Particle, solid phase


Porous region inlet control surface


Porous region outlet control surface


Porous region anisotropic region


Porous region isotropic region


Square periodic elementary cell dimension


Vertical channel bounded by the porous region


Vertical channel upstream of the porous region

β :

Fluid phase

βp :

Fluid–particle relative quantity

ω :

Rotational related


  1. 1.

    Skalle P (2011) Pressure control during oil well drilling. P. Skalle & Ventus Publishing, London

    Google Scholar 

  2. 2.

    Lavrov A (2016) Lost circulation: mechanisms and solutions. Gulf Professional Publishing, Cambridge

    Google Scholar 

  3. 3.

    Civan F (2007) Reservoir formation damage: fundamentals, modeling, assessment, and mitigation, 2nd edn. Gulf Professional Publishing, Cambridge

    Google Scholar 

  4. 4.

    Aadnoy BS, Looyeh R (2011) Petroleum rock mechanics: drilling operations and well design. Gulf Professional Publishing, Oxford

    Google Scholar 

  5. 5.

    Whitfill DL, Hemphill T (2004) Pre-treating fluids with lost circulation materials. Drilling Contract May/June, pp 54–57

  6. 6.

    Gray GR, Darley HCH (1980) Composition and properties of oil well drilling fluids, 4th edn. Gulf Publishing Company, Houston

    Google Scholar 

  7. 7.

    Ferguson CK, Klotz JA (1954) Filtration from mud during drilling. J Pet Technol 6:30–43.

    Article  Google Scholar 

  8. 8.

    Simpson JP (1974) Drilling fluid filtration under simulated downhole conditions. In: AIME symposium on formation damage control. Society of Petroleum Engineers, New Orleans, LA

  9. 9.

    Fisher KA, Wakeman RJ, Chiu TW, Meuric OFJ (2000) Numerical modelling of cake formation and fluid loss from non-Newtonian muds during drilling using eccentric/concentric drill strings with/without rotation. Chem Eng Res Des 78:707–714.

    Article  Google Scholar 

  10. 10.

    Civan F (1996) Interactions of the horizontal wellbore hydraulics and formation damage. In: 1996 Permian Basin oil & gas recovery conference. Midland, TX

  11. 11.

    Liu X, Civan F (1996) Formation damage and filter cake buildup in laboratory core tests: modeling and model-assisted analysis. Soc Pet Eng.

    Article  Google Scholar 

  12. 12.

    Horner V, White M, Cochran C, Deily F (1957) Microbit dynamic filtration studies. Pet Trans AIME 210:183–189

    Article  Google Scholar 

  13. 13.

    Krueger RF (1963) Evaluation of drilling-fluid filter-loss additives under dynamic conditions. J Pet Technol 15:90–98.

    Article  Google Scholar 

  14. 14.

    Vaussard A, Martin M, Konirsch O, Patroni JM (1986) An experimental study of drilling fluids dynamic filtration. In: 61st annual technical conference and exhibition of the society of petroleum engineers. New Orleans, LA

  15. 15.

    Jiao D, Sharma MM (1994) Mechanism of cake buildup in crossflow filtration of colloidal suspensions. J Colloid Interface Sci 162:454–462.

    Article  Google Scholar 

  16. 16.

    Dick MA, Heinz TJ, Svoboda CF, Aston M (2000) Optimizing the selection of bridging particles for reservoir drilling fluids. In: 2000 SPE international symposium on formation damage, Lafayette, LA

  17. 17.

    Zinati FF, Farajzadeh R, Currie PK, Zitha PLJ (2009) Modeling of external filter cake build-up in radial geometry. Pet Sci Technol 27:746–763.

    Article  Google Scholar 

  18. 18.

    Kabir MA, Gamwo IK (2011) Filter cake formation on the vertical well at high temperature and high pressure: computational fluid dynamics modeling and simulations. J Pet Gas Eng 27:146–164.

    Article  Google Scholar 

  19. 19.

    Zhu HP, Zhou ZY, Yang RY, Yu AB (2007) Discrete particle simulation of particulate systems: theoretical developments. Chem Eng Sci 62:3378–3396.

    Article  Google Scholar 

  20. 20.

    De Lai FC, Franco AT, Junqueira SLM (2014) Numerical simulation of liquid–solid flow in a channel with transversal fracture. In: 15th Brazilian congress of thermal sciences and engineering—ENCIT 2014, Belém, PA

  21. 21.

    Barbosa MV, De Lai FC, Franco AT, Junqueira SLM (2015) Parametric analysis of particulate flow for fluid loss control to sealing fractured channels. In: 23rd ABCM international congress of mechanical engineering—COBEM 2015, Rio de Janeiro, RJ

  22. 22.

    Barbosa MV, De Lai FC, Junqueira SLM (2016) Numerical analysis of particulate flow applied to fluid loss control in fractured channels. In: ASME 2016 heat transfer summer conference, HT 2016, collocated with the ASME 2016 fluids engineering division summer meeting and the ASME 2016 14th international conference on nanochannels, microchannels, and minichannels. American Society of Mechanical Engineers, Washington, DC

  23. 23.

    Barbosa MV, De Lai FC, Junqueira SLM (2019) Numerical evaluation of CFD–DEM coupling applied to lost circulation control: effects of particle and flow inertia. Math Probl Eng.

    Article  Google Scholar 

  24. 24.

    Lima GH, De Lai FC, Junqueira SLM (2017) Numerical simulation of particulate flow for static mud cake process over heterogeneous porous media. In: 24th ABCM international congress of mechanical engineering—COBEM 2017. ABCM, Curitiba, PR

  25. 25.

    Bourgoyne AT Jr, Millheim KK, Chenevert ME, Young FS Jr (1991) Applied drilling engineering, 2nd edn. SPE, Richardson

    Google Scholar 

  26. 26.

    Civan F (2007) Formation damage mechanisms and their phenomenological modeling: an overview. In: European formation damage conference. Society of Petroleum Engineers, Scheveningen, The Netherlands

  27. 27.

    Popoff B, Braun M (2007) A Lagrangian approach to dense particulate flow. In: 6th international conference on multiphase flow, ICMF 2007. Leipzig, pp 510–521

  28. 28.

    Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Géotechnique 29:47–65.

    Article  Google Scholar 

  29. 29.

    Jaffal HA, El Mohtar CE, Gray KE (2017) Modeling of filtration and mudcake buildup: an experimental investigation. J Nat Gas Sci Eng 38:1–11.

    Article  Google Scholar 

  30. 30.

    Welahettige P, Lundberg J, Bjerketvedt D, Lie B, Vaagsaether K (2020) One-dimensional model of turbulent flow of non-Newtonian drilling mud in non-prismatic channels. J Pet Explor Prod Technol 10:847–857.

    Article  Google Scholar 

  31. 31.

    Crowe C, Sommerfeld M, Tsuji Y (1998) Multiphase flow with droplets and particles. CRC Press, Boca Raton

    Google Scholar 

  32. 32.

    De Lai FC (2013) Numerical simulation of particulate flow for filling of fractured channel (in Portuguese). Dissertation, Federal University of Technology, Parana

  33. 33.

    Morsi SA, Alexander AJ (1972) An investigation of particles trajectories in two-phase flow systems. J Fluid Mech 55:193–208.

    Article  MATH  Google Scholar 

  34. 34.

    Happel J, Brenner H (1983) Low Reynolds number hydrodynamics with special applications to particulate media. Martinus Nijhoff Publishers, The Hague

    Google Scholar 

  35. 35.

    Dennis SCR, Singh SN, Ingham DB (1980) The steady flow due to a rotating sphere at low and moderate Reynolds numbers. J Fluid Mech 101:257–279.

    Article  MATH  Google Scholar 

  36. 36.

    Odar F, Hamilton S (1964) Forces on a sphere accelerating in a viscous fluid. J Fluid Mech 18:302–314.

    Article  MATH  Google Scholar 

  37. 37.

    Anderson TB, Jackson R (1967) A fluid mechanical description of fluidized beds: equations of motion. Ind Chem Eng Fundam 6:527–539.

    Article  Google Scholar 

  38. 38.

    Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Fluid Mech 22:385–400.

    Article  MATH  Google Scholar 

  39. 39.

    Oesterlé B, Dinh TB (1998) Experiments on the lift of a spinning sphere in a range of intermediate Reynolds numbers. Exp Fluids 25:16–22.

    Article  Google Scholar 

  40. 40.

    Tsuji Y (2006) Particle–particle collision. In: Crowe CT (ed) Multiphase flow handbook. CRC Taylor & Francis, Boca Raton

    Google Scholar 

  41. 41.

    Kruggel-Emden H, Simsek E, Rickelt S, Wirtz S, Scherer V (2007) Review and extension of normal force models for the discrete element method. Powder Technol 171:157–173.

    Article  Google Scholar 

  42. 42.

    Kruggel-Emden H, Wirtz S, Scherer V (2008) A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behavior. Chem Eng Sci 63:1523–1541.

    Article  Google Scholar 

  43. 43.

    Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22:745.

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

    Patankar SV (1980) Numerical heat transfer and fluid flow. McGraw-Hill, New York

    Google Scholar 

  45. 45.

    Vasquez SA, Ianov VA (2000) A phase coupled method for solving multiphase problem on unstructured meshes. In: 2000 ASME fluids engineering division summer meeting, Boston, MA

  46. 46.

    FLUENT (2016) ANSYS FLUENT v17 - Theory Guide, ANSYS Inc.

  47. 47.

    Almagro SPB, Frates C, Garand J, Meyer A (2014) Sealing fractures: advances in lost circulation control treatments. Oilf Rev 26:4–13

    Google Scholar 

  48. 48.

    Schneider CL, De Lai FC, Junqueira SLM (2017) Experimental set-up for lost circulation control in fractured channel by particulate flow. In: 24th ABCM international congress of mechanical engineering. ABCM, Curitiba, PR

  49. 49.

    Poletto VG (2017) Modelling and numerical simulation of particle deposition in porous medium: a study of the mud cake formation in oil well drilling (in Portuguese). Dissertation, Federal University of Technology, Parana

  50. 50.

    Mordant N, Pinton J-F (2000) Velocity measurement of a settling sphere. Eur Phys J B 18:343–352.

    Article  Google Scholar 

  51. 51.

    Gondret P, Lance M, Petit L (2002) Bouncing motion of spherical particles in fluids. Phys Fluids 14:643–652.

    Article  MATH  Google Scholar 

  52. 52.

    Lima GH, De Lai FC, Junqueira SLM (2017) Numerical simulation of particulate flow over heterogeneous porous media. In: IV journeys in multiphase flows: JEM 2017, São Paulo, SP

  53. 53.

    Barbosa MV, De Lai FC, Franco AT, Junqueira SLM (2015) Eulerian–Lagrangian approach applied to particulate flow using dense discrete phase model. In: IV journeys in multiphase flows: JEM 2015, Campinas, SP

Download references


The authors would like to express their gratitude to the National Council for Scientific and Technological Development (CNPq) and Petrobras (which also provided technical assistance) for the financial support.

Author information



Corresponding author

Correspondence to Silvio L. M. Junqueira.

Additional information

Publisher's Note

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

Technical Editor: Celso Kazuyuki Morooka.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Poletto, V.G., De Lai, F.C. & Junqueira, S.L.M. CFD–DEM simulation of mud cake formation in heterogeneous porous medium for lost circulation control. J Braz. Soc. Mech. Sci. Eng. 42, 376 (2020).

Download citation


  • Dynamic filtration
  • Lost circulation
  • LCM
  • Mud cake