Abstract
Drilling mud filtration occurs during an overbalanced drilling activity and concurrently with mud loss through pore throats and fractures. Mud loss and filtration are increased when the wellbore fluid condition is in a dynamic mode (pipe rotation and/or fluid circulation), rather than static. Formation damage is a critical industry challenge that results from mud loss and filtration. There is a considerable amount of experimental studies with only a few modeling approaches for characterizing dynamic mud filtration. Most of these studies have not accounted for factors that can exacerbate mud filtration which includes but not limited to: temperature, pipe rotation, pip/wellbore geometry/eccentricity, and porous media complexity. In this study, two mathematical and computational modeling approaches that can be used to predict dynamic drilling mud filtration in a radial coordinate system are presented. In the first modeling approach, a mechanistic model that is based on a material balance of filter cake evolution is presented. Critical factors that impact dynamic–radial mud filtration (temperature, rotary speed, rock permeability, and rock porosity) and other factors (wellbore/reservoir dimensions, filter cake properties, and mud/filtrate rheological properties at reservoir temperature) were accounted for. The model was solved with a numerical approach and commercial software. In the second approach, a scanning electron microscopy image of selected dry core samples, combined with image processing, was used to estimate the pore size and porosity of the internal filter cake. The pore structure of the rock samples and filter cake was modeled using the bundle of curved tubes approach. The deposition probability of mud particles was considered through filtration theories. The modeling results were validated with dynamic–radial filtration experiments. The results from both models closely matched the experimental results. On average, the models revealed no more than 4% relative error in predicting dynamic mud filtration. The novelty in both approaches is the incorporation of critical parameters in the models over a wide range and their responses to cumulative filtrate invasion in different rock types.
Similar content being viewed by others
Abbreviations
 c _{p} :

Mass of particles per unit volume of carrier fluid in slurry (g/cm^{3}mud)
 h :

Height of formation/cake height (cm)
 K :

Permeability
 K _{f} :

Formation permeability (Darcy)
 K _{c} :

External mud cake permeability (Darcy)
 k :

Consistency index (dynes/cm^{2}/s^{n′})
 k _{d} :

Deposition coefficient (dimensionless)
 k _{e} :

Erosion coefficient (s/cm)
 L :

Length of the porous media
 n :

Flow behavior index (dimensionless)
 P :

Trapping probability of particles
 P _{c} :

Wellbore pressure at the cake surface (atm)
 P _{e} :

Formation pressure (atm)
 q :

Filtration rate (cm^{3}/s)
 q _{o} :

Initial filtration rate (cm^{3}/s)
 RS:

Rotary speed (RPM)
 R _{ps} :

Net mass rate of deposition of particles to form external filter cake (g/s/cm^{2})
 r _{r} :

Reduced pore radius
 r _{p} :

Pore radius
 r _{e} :

Reservoir radius (cm)
 r _{c} :

Filter cake radius (cm)
 r _{w} :

Wellbore radius (cm)
 ppg:

Pounds per gallon
 S :

Particles sphericity
 T _{f} :

Tortuosity of formation (dimensionless)
 T _{c} :

Tortuosity of filter cake (dimensionless)
 u _{c} :

Fluid flux at the cake surface (cm/s)
 V _{S} :

Volume of the deposited solid particles
 x :

Particle radius
 β _{f} :

Formation inertial flow coefficient (cm^{−1})
 µ :

Viscosity (cp)
 \(\delta\) :

Filter cake thickness (cm)
 ρ _{p} :

Particle mass density (g/cm^{3})
 Φ _{f} :

Cake porosity (volume fraction)
 φ :

Porosity
 θ :

Lumped parameter used in trapping probability calculation
 θ _{0} :

Lumped parameter of electrical charges
 υ :

Flow velocity in pores
 υ ^{*} :

Minimum fluidization velocity
 \(\tau_{\text{cr}}\) :

Critical shear stress (dynes/cm^{2})
 \(\tau_{\text{s}}\) :

Slurry shear stress at the cake surface (dynes/cm^{2})
 ε _{s} :

Volume mass fraction of solid in the filter cake
References
Al Otaibi MB, NasrElDin HA, Hill AD (2008) Characteristics and removal of filter cake formed by formatebased drilling mud. In: SPE international symposium and exhibition on formation damage control, Lafayette, Louisiana, 13–15 February. SPE112427MS. https://doi.org/10.2118/112427MS
Allen D, Auzerais F, Dussan E, Goode P, Ramakrshnan TS, Schwartz L, Wilkinson D, Fordham E, Hammond P, Williams R (1991) Invasion revisited. Oilfield Rev
API 13 B1 (2003) Recommended practice for field testing water baseddrilling fluids, 3rd edn. API, Washington, DC
Basanta KS (1974) Determination of average grain sphericity in granular porous media. J Sedim Res 44(2):578–582. https://doi.org/10.1306/74d72a952b2111d78648000102c1865d
Chenevert ME, Dewan JT (2001) A model for filtration of waterbase mud during drilling: determination of mud cake parameters. Petrophysics 42(3):237–250 (SPWLA2001v42n3a4)
Churcher PL, French PR, Shaw JC et al (1991) Rock properties of Berea sandstone, baker dolomite, and Indiana limestone. In: SPE international symposium on oilfield chemistry, Anaheim, California, 20–22 February. SPE 21044. https://doi.org/10.2523/21044MS
Civan F (1994) A multiphase mud filtrate invasion and wellbore filter cake formation model. In: SPE international petroleum conference and exhibition of Mexico, Veracruz, Mexico, 10–13 October. SPE28709MS. https://doi.org/10.2118/28709MS
Civan F (1996) A multipurpose formation damage model. In: SPE international symposium and exhibition on formation damage control, Lafayette, Louisiana, 14–15 February. SPE31101MS. https://doi.org/10.2118/31101MS
Civan (2007) Reservoir formation damage, 2nd edn. Gulf Publishing Company, Waltham, pp 341–403–780–782
Ezeakacha CP (2018) Dynamic drilling fluid loss and filtration: impact of dynamic wellbore conditions and wellbore strengthening implications. PhD Dissertation, The University of Oklahoma, Norman, Oklahoma (December 2018). https://hdl.handle.net/11244/316304
Ezeakacha CP, Salehi S (2018) Experimental and statistical investigation of drilling fluids loss in porous mediaPart 1. J Nat Gas Sci Eng 51:104–115. https://doi.org/10.1016/j.jngse.2017.12.024
Ezeakacha CP, Salehi S, Hayatdavoudi A (2017) Experimental study of drilling fluid’s filtration and mud cake evolution in sandstone formations. ASME J Energy Resour Technol 139(2):022912–022918. https://doi.org/10.1115/1.4035425
Ezeakacha CP, Salehi S, Kiran K (2018a) Quantification of multiple factors and interaction effects on drilling fluid invasion. ASME 2018 37th international conference on ocean, offshore and arctic engineering, Madrid Spain, 17–22 June. OMAE201878328
Ezeakacha CP, Salehi S, Bi H (2018b) A new approach to characterize dynamic drilling fluids invasion profiles in application to nearwellbore strengthening effect. In: IADC/SPE drilling conference and exhibition forth worth, Texas. March 6–8, 2018. IADC/SPE189596MS. https://doi.org/10.2118/189596MS
Fakhreldin YE (2010) Novel fluid formulations to remove mud filtercake without affecting rock mineralogy. In: SPE production and operations conference and exhibition, Tunis, Tunisia, 8–10 June. SPE136093MS. https://doi.org/10.2118/136093MS
Forchheimer P (1901) Wasserbewegung durch Boden. Z Ver Deutsch Ing 45(1901):1782–1788
Gamwo IK, Kabir MA (2015) Impact of drilling fluid rheology and wellbore pressure on rock cuttings removal performance: numerical investigation. Asia Pac J Chem Eng 10(6):809–822. https://doi.org/10.1002/apj.1917
Hajra MG, Reddi LN, Glasgow LA et al (2002) Effects of ionic strength on fine particle clogging of soil filters. J Geotech Geoenviron Eng 128(8):631–639. https://doi.org/10.1061/(asce)10900241(2002)128:8(631)
Jiao D, Sharma MM (1994) Mechanism of cake buildup in crossflow filtration of colloidal suspensions. J Colloid Interface Sci 162(2):454–462. https://doi.org/10.1006/jcis.1994.1060
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 2(7):146–164. https://academicjournals.org/journal/JPGE/articlefulltextpdf/34FCA323950
Kiran R, Salehi S (2016) Thermoporoelastic modeling of timedependent wellbore strengthening and casing smear. ASME J Energy Resour Technol 139(2):022903–022907. https://doi.org/10.1115/1.4033591
Kozeny J (1927) Uber kapillare leitung der wasser in boden. R Acad Sci Vienna Proc Class I 136:271–306
Lanfrey PY, Kuzeljevic ZV, Dudukovic MP (2010) Tortuosity model for fixed beds randomly packed with identical particles. Chem Eng Sci 65(5):1891–1896. https://doi.org/10.1016/j.ces.2009.11.011
Lavrov A, Tronvoll J (2004) Modeling mud loss in fractured formations. In: 11th Abu Dhabi international petroleum exhibition and conference, Abu Dhabi, UAE, 10–13 October. SPE88700. http://dx.doi.org/10.2118/88700MS
Leontaritis K (1998) Asphaltene nearwellbore formation damage modeling. In: SPE international symposium and exhibition on formation damage control, Lafayette, Louisiana, 18–191 February. SPE39446MS. https://doi.org/10.2118/39446MS
Ling K, Zhang H, Shen Z et al (2015) A new approach to estimate invasion radius of waterbaseddrillingfluid filtrate to evaluate formation damage caused by overbalanced drilling. SPE Drill Complet 30(01):27–37. https://doi.org/10.2118/168184PA
Liu X, Civan F, Evans RD (1995) Correlation of the nonDarcy flow coefficient. J Can Pet Technol 34:10. https://doi.org/10.2118/951005
McCabe WL (2005) Unit operations of chemical engineering, 7th edn. MC Graw Hill Chemical Engineering Series, New York
McDowell LM, Hunt JR, Sitar N (1986) Particle transport through porous media. Water Recourse Res 22(13):1901–1921. https://doi.org/10.1029/WR022i013p01901
Otsu N (1979) A threshold selection method from graylevel histograms. IEEE Trans Syst Man Cybern 9(1):62–69. https://doi.org/10.1109/tsmc.1979.4310076
Rabbani A, Salehi S (2017) Dynamic modeling of the formation damage and mud cake deposition using filtration theories coupled with SEM image processing. J Nat Gas Sci Eng 42:157–168. https://doi.org/10.1016/j.jngse.2017.02.047
Rabbani A, Ayatollahi S, Kharrat R et al (2016) Estimation of 3D pore network coordination number of rocks from watershed segmentation of a single 2D image. Adv Water Resour 94:264–277. https://doi.org/10.1016/j.advwatres.2016.05.020
Rege SD, Fogler HS (1988) A network model for deep bed filtration of solid particles and emulsion drops. AIChE J 34(11):1761–1772. https://doi.org/10.1002/aic.690341102
Salehi S, Kiran R (2016) Integrated experimental and analytical wellbore strengthening solutions by mud plastering effects. ASME J Energy Resour Technol 138(3):032904–032907. https://doi.org/10.1115/1.4032236
Salehi S, Hussmann S, Karimi M et al (2014) Profiling drilling fluid’s invasion using scanning electron microscopy: implications for bridging and wellbore strengthening effects. In: SPE deepwater drilling and completions conference, Galveston, Texas, 10–11 September. SPE170315MS. http://dx.doi.org/10.2118/170315MS
Salimi S, Andersen KI (2004) Enhancement well productivityinvestigating the feasibility of UBD for minimizing formation damage in naturally fractured carbonate reservoirs. In: SPE/IADC underbalanced technology conference and exhibition, Houston, Texas, 11–12 October. SPE/IADC91544MS. https://doi.org/10.2118/91544MS
Schembre JM, Kovscek AR (2005) Mechanism of formation damage at elevated temperature. ASME J Energy Resour Technol 127(3):171–180. https://doi.org/10.1115/1.1924398
Stein PC (1940) A study of the theory of rapid filtration of water through sand. Massachusetts Institute of Technology, Department of Civil and Sanitary Engineering
Tavanaei A, Salehi S (2015) Pore throat and grain detection for rock sem images using digital watershed image segmentation algorithm. J Porous Media 18(5):507–518. https://doi.org/10.1615/JPorMedia.v18.i5.40
Tien C, Bai R, Ramarao BV (1997) Analysis of cake growth in cake filtration: effect of fine particle retention. AIChE J 43(1):33–44. https://doi.org/10.1002/aic.690430106
Zitoun KB, Sastry SK, Guezennec Y (2001) Investigation of threedimensional interstitial velocity, solids motion and orientation in solidliquid flow using particle tracking velocimetry. Int J Multiph Flow 27(8):1397–1414. https://doi.org/10.1016/s03019322(01)000118
Acknowledgements
The authors would like to thank Raj Kiran, Dr. Mark Cutis, and Jeff McCaskill at the Mewbourne School of Petroleum and Geological Engineering, The University of Oklahoma for their support in this study.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Filter cake material balance is given by (Civan 1994; 1996) as:
The volume mass fraction of solids as a function of porosity is given as:
The net deposition rate of particles to form an external filter cake is given as:
where \(\tau_{s}\) is the mud (slurry) shear stress and it is given as:
In Eq. (18),
Substituting Eqs. (16) and (17) into (15) yields
In terms of the filtration rate, the radial volumetric flux at the external cake surface is:
Substituting Eq. (21) into (20) gives:
where
and
The Forchheimer’s (1901) equation for radial flow of the mud is given as:
Combining Eqs. (25) and (21) yields:
where \(\beta_{i}\) is the inertial flow coefficient for the rock or filter cake and per Liu et al. (1995), it is given as:
In Eq. (27), T_{f/c} means tortuosity of formation or filter cake. The same goes for the porosity and permeability symbols (please check the Greek symbols). Integrating Eq. (26) for conditions before and during evolution of external filter cake results in Eqs. (28) and (29), respectively:
Combining Eqs. (28) and (29) eliminates (P_{c}–P_{e}) and yields the quadratic Eq. (30) for nonDarcy mud filtrate flow rate:
whose solution is in Eq. (31) (Civan 2007)
where
The filter cake thickness (δ) is a function of wellbore radius and filter cake radius (δ = r_{w} − r_{c}).
Rights and permissions
About this article
Cite this article
Ezeakacha, C.P., Salehi, S. An Advanced Coupled Rock–Fluid Computational Model for Dynamic–Radial Mud Filtration with Experimental Validation. Rock Mech Rock Eng 52, 3757–3770 (2019). https://doi.org/10.1007/s0060301901804w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s0060301901804w