Skip to main content
SpringerLink
Log in

Semi-analytical Approach to Modeling Forchheimer Flow in Porous Media at Meso- and Macroscales

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Darcy’s law (which states that a fluid flow rate is directly proportional to the pressure gradient) is shown to be accurate in a rather narrow range of flow velocities. Numerous studies show that at low pressure gradients gas slippage effect occurs, which gives overestimated flow rates compared to Darcy’s law. At higher flow rates, Darcy’s law is usually replaced by the Forchheimer equation which accounts for inertial forces including a quadratic term in the flow rate. Darcy’s and Forchheimer’s laws and the problem of detecting transitions between their ranges of applicability are discussed in this study. Analysis of experimental data shows that deviation from Darcy’s law is governed by the Forchheimer number, which is defined by the authors as a product of tortuosity and Reynolds number. The use of the Forchheimer number and semi-analytical approaches enables us to describe non-Darcy flow as a simple universal equation valid for any flow geometry. Comparison of the experimental data with predictions based on a semi-analytical model shows excellent agreement for a wide range of reservoir properties.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Code Availability

Code sharing is not applicable to this article as no software application or custom code was generated or analyzed during the current study.

Abbreviations

\({D}_{\mathrm{p}}\) :

Particles diameter, m

\({D}_{\mathrm{t}}\) :

Throat diameter, m

\(E\) :

Non-Darcy effect

\(F\) :

Formation resistivity factor

\(\mathrm{Fo}\) :

Forchheimer number

\({\mathrm{Fo}}_{\mathrm{D}}\) :

Forchheimer number related to Darcy flow conditions

\({\mathrm{Fo}}_{\mathrm{c}}\) :

Critical Forchheimer number

\({\mathrm{Fo}}_{\mathrm{c exp}}\) :

Experimentally measured critical Forchheimer number

\({\mathrm{Fo}}_{\mathrm{c sim}}\) :

Critical Forchheimer number obtained during the simulation

\(P\) :

Pressure, Pa

\({P}_{0}\) :

Standard pressure, Pa

\({P}_{1}\) :

Inlet pressure, Pa

\({P}_{2}\) :

Outlet pressure, Pa

\({P}_{\mathrm{D}}\) :

Pressure related to Darcy flow conditions, Pa

\({P}_{\mathrm{F}}\) :

Pressure related to Forchheimer flow conditions, Pa

\({P}_{\mathrm{e}}\) :

External boundary pressure, Pa

\({P}_{\mathrm{w}}\) :

Bottomhole pressure, Pa

\(\mathrm{Re}\) :

Reynolds number

\({\mathrm{Re}}_{\mathrm{D}}\) :

Reynolds number related to Darcy flow conditions

\({\mathrm{Re}}_{\mathrm{F}}\) :

Reynolds number related to Forchheimer flow conditions

\({\mathrm{Re}}_{\mathrm{c}}\) :

Critical Reynolds number

\(d\) :

Mean pore diameter, m

\({d}_{\mathrm{eqv}}\) :

Equivalent pore diameter, m

\(k\) :

Permeability, mD

\(q\) :

Gas flow rate, m3/d

\({q}_{\mathrm{D}}\) :

Volumetric flow rate related to Darcy flow conditions, m3/d

\({q}_{\mathrm{Dm}}\) :

Mass flow rate related to Darcy flow conditions, kg/d

\({q}_{\mathrm{F}}\) :

Volumetric flow rate related to Forchheimer flow conditions, m3/d

\({q}_{\mathrm{Fm}}\) :

Mass flow rate related to Forchheimer flow conditions, kg/d

\(r_{\text{e}}\) :

External drainage radius, m

\({r}_{\mathrm{w}}\) :

Wellbore radius, m

\(u\) :

Velocity, m/s

\({u}_{\mathrm{D}}\) :

Volumetric velocity related to Darcy flow conditions, m/s

\({u}_{\mathrm{Dm}}\) :

Mass velocity related to Darcy flow conditions, kg/(s m2)

\({u}_{\mathrm{F}}\) :

Volumetric velocity related to Forchheimer flow conditions, m/s

\({u}_{\mathrm{Fm}}\) :

Mass velocity related to Forchheimer flow conditions, kg/(s m2)

\(h\) :

Formation thickness, m

\(\mathrm{grad} \,P\) :

Pressure gradient, Pa/m

\(r\) :

Pore throat radius in Table 2, m

\(x\) :

Linear coordinate, m

\(R2\) :

Determination coefficient

\({B}_{1}\) :

Constant in Eq. 20

\(a\) :

Constants in Table 1

\(b\) :

Constants in Table 1

\(c\) :

Constant in Eq. 4

\({c}^{^{\prime}}\) :

Constant in Eq. 5

\(a,b,c\) :

Coefficients in Eq. 34

\({a}_{1},{b}_{1},{c}_{1}\) :

Coefficients in Eq. 57

\(\alpha \) :

Constant in Eq. 20

\(\beta \) :

Non-Darcy coefficient, m−1

\(\gamma \) :

Weak inertia factor

\(\lambda \) :

New turbulent flow correlation factor, ft

\(\mu \) :

Viscosity, Pa s

\(\rho \) :

Density, kg/m3

\({\rho }_{0}\) :

Density at standard conditions, kg/m3

\(\tau \) :

Tortuosity

\(\phi \) :

Porosity, fraction

\(\mathrm{D}\) :

Related to Darcy flow conditions

\(\mathrm{F}\) :

Related to Forchheimer flow conditions

\(\mathrm{c}\) :

Critical

\(\mathrm{e}\) :

External

\(\mathrm{eqv}\) :

Equivalent

\(\mathrm{exp}\) :

Experiment

\(\mathrm{m}\) :

Mass

\(\mathrm{p}\) :

Particles

\(\mathrm{sim}\) :

Simulation

\(\mathrm{t}\) :

Throat

\(\mathrm{w}\) :

Well

References

  • Ahlers, C.F., Finsterle, S., Wu, Y.S., Bodvarsson, G.S.: Determination of pneumatic permeability of a multi-layered system by inversion of pneumatic pressure data. In: Proceedings of the 1995 AGU Fall Meeting. San Francisco, California (1995)

  • Ansah, E.O., Vo Thanh, H., Sugai, Y., Nguele, R., Sasaki, K.: Microbe-induced fluid viscosity variation: field-scale simulation, sensitivity and geological uncertainty. J. Petrol. Explor. Prod. Technol. 10, 1983–2003 (2020). https://doi.org/10.1007/s13202-020-00852-1

    Article  Google Scholar 

  • Barak, A.Z.: Comments on ‘high velocity flow in porous media’ by Hassanizadeh and Gray. Transp. Porous Med. 2, 533–535 (1987). https://doi.org/10.1007/BF00192153

    Article  Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)

    Google Scholar 

  • Belhaj, H.A., Agha, K.R., Nouri, A.M., Butt, S.D., Vaziri, H.F. Islam M.R.: Numerical simulation of Non-Darcy flow utilizing the new Forchheimer’s diffusivity equation. SPE 81499. In: Middle East Oil Show. Bahrain (2003). https://doi.org/10.2118/81499-MS

  • Brown, K.E.: The Technology of Artificial Lift Methods, pp. 8–10. Petroleum Publishing Co., Tulsa (1980)

    Google Scholar 

  • Chilton, T.H., Colburn, A.P.: Pressure drop in packed tubes. Ind. Eng. Chem. 23(8), 913–919 (1931). https://doi.org/10.1021/ie50260a016

    Article  Google Scholar 

  • Choi, C.S., Song, J.J.: Estimation of the non-Darcy coefficient using supercritical CO2 and various sandstones. JCR Solid Earth. 124, 442–455 (2019). https://doi.org/10.1029/2018JB016292

    Article  Google Scholar 

  • Coles, M.E., Hartman, K.J.: Non-Darcy measurements in dry core and the effect of immobile liquid. SPE 39977. In: SPE Gas Technology Symposium. Calgary (1998). https://doi.org/10.2118/39977-MS

  • Cooper, J.W., Wang, X., Mohanty, K.K.: Non-Darcy flow studies in anisotropic porous media. Soc. Petrol. Eng. J. 4(4), 334–341 (1999). https://doi.org/10.2118/57755-PA

    Article  Google Scholar 

  • Cornell, D., Katz, D.L.: Flow of gases through consolidated porous media. Ind. Eng. Chem. 45(10), 2145–2152 (1953). https://doi.org/10.1021/ie50526a021

    Article  Google Scholar 

  • Darcy, H.P.G.: Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris (1856)

    Google Scholar 

  • Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48(2), 89–94 (1952)

    Google Scholar 

  • Fancher, G.H., Lewis, J.A.: Flow of simple fluids through porous materials. Ind. Eng. Chem. 25(10), 1139–1147 (1933). https://doi.org/10.1021/ie50286a020

    Article  Google Scholar 

  • Firoozabadi, A., Katz, D.L.: An analysis of high-velocity gas flow through porous media. J. Petrol. Technol. (1979). https://doi.org/10.2118/6827-PA

    Article  Google Scholar 

  • Firouzi, M., Alnoaimi, K., Kovscek, A., Wilcox, J.: Klinkenberg effect on predicting and measuring helium permeability in gas shales. Int. J. Coal Geol. 123, 62–68 (2014). https://doi.org/10.1016/j.coal.2013.09.006

    Article  Google Scholar 

  • Forchheimer, P.: Wasserberwegung durch Boden. Z Vereines deutscher Ing. 45(50), 1782–1788 (1901)

    Google Scholar 

  • Frederick, D.C., Grave, R.M: New correlations to predict non‐Darcy flow coefficients at immobile and mobile water saturation. SPE 28451. In: SPE Annual Technical Conference and Exhibition. New Orleans (1994). https://doi.org/10.2118/28451-MS

  • Friedel, T., Voigt, H.D.: Investigation of non-Darcy flow in tight-gas reservoirs with fractured wells. J. Pet. Sci. Eng. 54, 112–128 (2006). https://doi.org/10.1016/j.petrol.2006.07.002

    Article  Google Scholar 

  • Fuente, M., Munoz, E., Sicilia, I., Goggins, J., Hung, L.C., Frutos, B., Foley, M.: Investigation of gas flow through soils and granular fill materials for the optimisation of radon soil depressurisation system. J. Environ. Radioact. 198, 200–209 (2019). https://doi.org/10.1016/j.jenvrad.2018.12.024

    Article  Google Scholar 

  • Geertsma, J.: Estimating the coefficient of inertial resistance in fluid flow through porous media. Soc. Petrol. Eng. J. (1974). https://doi.org/10.2118/4706-PA

    Article  Google Scholar 

  • Ghane, E., Fausey, N.R., Brown, L.C.: Non-Darcy flow of water through woodchip media. J. Hydrol. 519, 3400–3409 (2014). https://doi.org/10.1016/j.jhydrol.2014.09.065

    Article  Google Scholar 

  • Gidley, J.L.: A method for correcting dimensionless fracture conductivity for non-Darcy flow effect. SPE 20710. SPE Prod. Eng. J. (1991). https://doi.org/10.2118/20710-PA

    Article  Google Scholar 

  • Gjengedal, S., Brøtan, V., Buset, O.T., Larsen, E., Berg, O.A., Torsæter, O., Ramstad, R.K., Hilmo, B.O., Frengstad, B.S.: Fluid flow through 3D-printed particle beds: a new technique for understanding, validating, and improving predictability of permeability from empirical equations. Transp. Porous Med. 134, 1–40 (2020). https://doi.org/10.1007/s11242-020-01432-x

    Article  Google Scholar 

  • Green, L.J., Duwez, P.: Fluid flow through porous metals. J. Appl. Mech. 18, 39–45 (1951)

    Google Scholar 

  • Holditch, S.A., Morse, R.A.: The effects of non-Darcy flow on the behavior of hydraulically fractured gas wells. J. Pet. Technol. 28(10), 1179–1196 (1976). https://doi.org/10.2118/5586-PA

    Article  Google Scholar 

  • Homuth, S., Götz, A.E., Sass, I.: Physical Properties of the geothermal carbonate reservoirs of the Molasse Basin, Germany-Outcrop Analogue vs. Reservoir Data. World Geothermal Congress, Melbourne (2015)

    Google Scholar 

  • Huang, H., Ayoub, J.A.: Applicability of the Forchheimer equation for non‐Darcy flow in porous media. SPE 102715. In: SPE Annual Technical Conference and Exhibition. Texas (2008). https://doi.org/10.2118/102715-MS

  • Hubbert, M.K.: Darcy law and the field equations of the flow of underground fluids. Trans. Am. Inst. Min. Mandal. Eng. 207, 222–239 (1956). https://doi.org/10.2118/749-G

    Article  Google Scholar 

  • Irmay, S.: On the theoretical derivation of Darcy and Forchheimer formulas. J. Geophys. Res. 39, 702–707 (1958). https://doi.org/10.1029/TR039i004p00702

    Article  Google Scholar 

  • Janicek, J.D., Katz, D.L.: Applications of unsteady state gas flow calculations. In: Proceedings of University of Michigan Research Conference. June 20 (1955)

  • Jones, S.C.: Using the inertial coefficient, β, to characterize heterogeneity in reservoir rock. SPE 16949. In: SPE Annual Technical Conference and Exhibition. New Orleans (1987). https://doi.org/10.2118/16949-MS

  • Kadi, K.S.: Non-Darcy flow in dissolved gas-drive reservoirs. SPE 9301. In: SPE Annual Fall Technical Conference. Dallas (1980). https://doi.org/10.2118/9301-MS

  • Klinkenberg, L.J.: The permeability of porous media to liquids and gases, drilling and production practice. Am. Petrol. Inst. 200–213 (1941)

  • Kollbotn, L., Bratteli, F.: An alternative approach to the determination of the internal flow coefficient. In: International Symposium of the Society of Core Analysts. Toronto (2005)

  • Kutasov, I.M.: Equation predicts non-Darcy flow coefficient. Oil Gas J. 91(11), 66–67 (1993)

    Google Scholar 

  • Li, D., Engler, T.W.: Literature review on correlations of the non-Darcy coefficient. SPE 70015. In: SPE Permian Basin Oil and Gas Recovery Conference. Midland (2001). https://doi.org/10.2118/70015-MS

  • Liu, X., Civan, F., Evans, R.D.: Correlation of the non-Darcy flow coefficient. J. Can. Pet. Tech. 34(10), 50–54 (1995). https://doi.org/10.2118/95-10-05

    Article  Google Scholar 

  • Ma, H., Ruth, D.W.: The microscopic analysis of high Forchheimer number flow in porous media. Transp. Porous Med. 13, 139–160 (1993). https://doi.org/10.1007/BF00654407

    Article  Google Scholar 

  • Macdonald, I.F., El-Sayed, M.S., Mow, K., Dullien, F.A.L.: Flow through porous media: the Ergun Equation revisited. Ind. Eng. Chem. Fundam. 18, 189–208 (1979). https://doi.org/10.1021/i160071a001

    Article  Google Scholar 

  • Macini, P., Mesini, E., Viola, R.: Laboratory measurements of non-Darcy flow coefficients in natural and artificial unconsolidated porous media. J. Pet. Sci. Eng. 77(3–4), 365–374 (2011). https://doi.org/10.1016/j.petrol.2011.04.016

    Article  Google Scholar 

  • Martins, J.P., Milton-Taylor, D., Leung, H.K.: The effects of non-Darcy flow in propped hydraulic fractures. SPE 20790. In: Proceedings of the SPE Annual Technical Conference. New Orleans (1990). https://doi.org/10.2118/20709-MS

  • Mei, C.C., Auriault, J.L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991). https://doi.org/10.1017/S0022112091001258

    Article  Google Scholar 

  • Millionshikov, M.D.: Degeneration of homogeneous isotropic turbulence in a viscous incompressible fluid. Rep. Acad. Sci. USSR. 22(5), 236–240 (1935) (in Russian)

    Google Scholar 

  • Morad, S., Al-Ramadan, K., Ketzer, J.M., De Ros, L.F.: The impact of diagenesis on the heterogeneity of sandstone reservoirs: a review of the role of depositional facies and sequence stratigraphy. AAPG Bull. 94(8), 1267–1309 (2010). https://doi.org/10.1306/04211009178

    Article  Google Scholar 

  • Muljadi, B.P., Blunt, M.J., Raeini, A.Q., Bijeljic, B.: The impact of porous media heterogeneity on non-Darcy flow behaviour from pore-scale simulation. Adv. Water. Resour. 95, 329–340 (2016)

    Article  Google Scholar 

  • Noman, R., Shrimanker, N., Archer, J.S.: Estimation of the coefficient of inertial resistance in high-rate gas wells. SPE 14207. In: SPE Annual Technical Conference. Las Vegas (1985). https://doi.org/10.2118/14207-MS

  • Pavlovsky, N.: Theory of Movement of Groundwater Under Hydraulic Engineering Constructions and Its Main Applications. Publishing House of Science and Melioration Institute, Petrograd (1922) (in Russian)

    Google Scholar 

  • Rasoloarijaona, M., Auriault, J.L.: Nonlinear seepage flow through a rigid porous medium. Eur. J. Mech. B/Fluids 13, 177–195 (1994)

    Google Scholar 

  • Ruth, D., Ma, H.: On the derivation of the Forchheimer equation by means of the average theorem. Transp. Porous Med. 7(3), 255–264 (1992). https://doi.org/10.1007/BF01063962

    Article  Google Scholar 

  • Scheidegger, A.E.: The Physics of Flow Through Porous Media. University of Toronto Press, Toronto (1974)

    Google Scholar 

  • Shelkachev, V.N., Lapuk, B.B.: Groundwater Hydraulics. p. 736. ISBN: 5-93972-081-1, Moscow—Igevsk (2001) (in Russian)

  • Skjetne, E.: High-velocity flow in porous media; analytical, numerical and experimental studies. Doctoral Thesis at Department of Petroleum Engineering and Applied Geophysics, Faculty of Applied Earth Sciences and Metallurgy, Norwegian University of Science and Technology (1995)

  • Skjetne, E., Auriault, J.L.: New insights on steady, non-linear flow in porous media. Eur. J. Mech. B/Fluids 18(1), 131–145 (1999). https://doi.org/10.1016/S0997-7546(99)80010-7

    Article  Google Scholar 

  • Tek, M.R., Coats, K.H., Katz, D.L.: The effect of turbulence on flow of natural gas through porous reservoirs. J. Petrol. Technol. AIME Trans. 222, 799–806 (1962). https://doi.org/10.2118/147-PA

    Article  Google Scholar 

  • Tessem, R.: High Velocity Coefficient’s Dependence on Rock Properties: A Laboratory Study. Thesis Pet. Inst., NTH. Trondheim (1980)

  • Thauvin, F., Mohanty, K.K.: Network modeling of non-Darcy flow through porous media. Transp. Porous Med. 31, 19–37 (1998). https://doi.org/10.1023/A:1006558926606

    Article  Google Scholar 

  • Torsæter, O., Tessem, R., Berge, B.: High Velocity Coefficient’s Dependence on Rock Properties. SINTEF report, NTH. Trondheim (1981)

  • Whitaker, S.: The Forchheimer equation: a theoretical development. Transp. Porous Med. 25, 27–61 (1996). https://doi.org/10.1007/BF00141261

    Article  Google Scholar 

  • Wodie, J.C., Levy, T.: Correction non lineaire de la loi de Darcy. C. R. Acad. Sci. 2(312), 157–161 (1991)

    Google Scholar 

  • Wong, S.W.: Effect of liquid saturation on turbulence factors for gas-liquid systems. J. Can. Petrol. Tech. 9(4), 274–278 (1970). https://doi.org/10.2118/70-04-08

    Article  Google Scholar 

  • Wu, Y.S., Pruess, K., Persoff, P.: Gas flow in porous media with Klinkenberg effects. Transp. Porous Media 32(1), 117–137 (1998). https://doi.org/10.1023/A:1006535211684

    Article  Google Scholar 

  • Zeng, Z., Grigg, R.: A criterion for non-Darcy flow in porous media. Transp. Porous Med. 63, 57–69 (2006). https://doi.org/10.1007/s11242-005-2720-3

    Article  Google Scholar 

  • Zolotukhin, A.B., Gayubov, A.T.: Machine learning in reservoir permeability prediction and modeling of fluid flow in porous media. IOP Conf. Ser. Mater. Sci. Eng. 700, 012023 (2019). https://doi.org/10.1088/1757-899X/700/1/012023

    Article  Google Scholar 

  • Zolotukhin, A.B., Ursin, J.R.: Fundamentals of Petroleum Reservoir Engineering. Høyskoleforlaget AS – Norwegian Academic Press, Kristiansand (2000b)

    Google Scholar 

  • Zolotukhin, A.B., Ursin, J.R.: Introduction to Petroleum Reservoir Engineering. Norwegian Academic Press, Høyskoleforlaget (2000a)

    Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Oil and Gas Field Development, Gubkin Russian State University of Oil and Gas (National Research University), Moscow, Russia

    A. B. Zolotukhin & A. T. Gayubov

Authors
  1. A. B. Zolotukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. T. Gayubov
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by ABZ and ATG The first draft of the manuscript was written by ABZ and ATG, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. ABZ conceived the study and contributed to project administration, supervision, and validation. ABZ and ATG curated the data, performed the formal analysis, and contributed to investigation, methodology, resources, and writing—review and editing. Funding acquisition is not applicable. ATG contributed to software, visualization, and writing—original draft.

Corresponding author

Correspondence to A. B. Zolotukhin.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zolotukhin, A.B., Gayubov, A.T. Semi-analytical Approach to Modeling Forchheimer Flow in Porous Media at Meso- and Macroscales. Transp Porous Med 136, 715–741 (2021). https://doi.org/10.1007/s11242-020-01528-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-020-01528-4

Keywords

Search

Navigation