Pre-Darcy Flow and Klinkenberg effect in Dense, Consolidated Carbonate Formations

  • Zohreh Farmani
  • Danial Farokhian
  • Amin Izadpanahi
  • Fatemeh seifi
  • Parviz Zahedizadeh
  • Zohreh Safari
  • Azita Ghaderi
  • Fatemeh Kazemi
  • Reza AzinEmail author
Original Paper


Experimental studies on pre-Darcy flow generally focus on unconsolidated porous media with high permeability. In this work, pre-Darcy flow is studied in consolidated carbonate rocks from different formations of Zagros basin. The core samples are calcite, dolomite and chalk-salt characterized by low-porosity (12.33–28.21%) and low permeability (0.011–18.53 md). Digital rock images were obtained using dental CT-scan with a resolution of 200 pixels per μm. A series of single-phase flooding experiments were conducted to evaluate the onset of pre-Darcy flow in different cores using N2 as gas phase. Superficial velocity versus pressure gradient was measured and different onset values were reported. Results indicate the presence of pre-Darcy flow in single phase gas experiments. The minimum observed Darcy velocity was 0.445 × 10−05 m/s, below which pre-Darcy flow prevails. In addition, different flow regimes such as pre-Darcy, Darcy and non-Darcy flow regimes were determined using reduced pressure drop analysis and Reynolds number analysis. The range of Reynolds number were reported for different flow regimes and different core permeabilities. The pre-Darcy regime was observed at velocities in the range of (0.445–5.88) × 10−05 m/s for all cores. For a carbonate core with k = 0.01 md, pre-Darcy flow was observed in the range of \(Re_{\sqrt k }\) < 2.1 × 10−07, Red < 0.168 and Rep < 0.025. It was also found that pre-Darcy flow arises due to Klinkenberg effect at low gas pressure in different core samples which show different behavior depending on their permeability.


Pre-Darcy flow Core flooding Consolidated tight core Superficial velocity Klinkenberg effect CT scan 

List of symbols


Absolute permeability (md)


Constant coefficients in equation (3)


Constant coefficients in equation (3)


Darcian velocity (m/s)




Empirical exponent in equation (4)


Forchheimer constant


Karmen–Cozeny friction factor


Particle diameter (mm)

\(f_{\sqrt k }\)

Permeability based friction factor

\(Re_{\sqrt k }\)

Permeability based Reynolds number


Pore Reynolds number


Porosity (%)


Reynolds number based on Darcy velocity


Superficial velocity (m/s)

\(\nabla P\)

Threshold pressure gradient


Viscosity (cp)



The authors acknowledge technical support from Sazand Kavan Forouzan (SKF) registered in Persian Gulf Science and Technology Park (PGSTP) for field trip and technical assistance in coring from rock samples. Also, the authors appreciate Dr. Narges Aria for supporting this research by preparing dental CT-scan images in her dental radiology center.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Zohreh Farmani
    • 1
  • Danial Farokhian
    • 1
  • Amin Izadpanahi
    • 1
  • Fatemeh seifi
    • 1
  • Parviz Zahedizadeh
    • 1
  • Zohreh Safari
    • 1
  • Azita Ghaderi
    • 1
  • Fatemeh Kazemi
    • 1
  • Reza Azin
    • 1
    • 2
    Email author
  1. 1.Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical EngineeringPersian Gulf UniversityBushehrIran
  2. 2.Gas Condensate and Carbon Management Research Core, Faculty of Petroleum, Gas and Petrochemical EngineeringPersian Gulf UniversityBushehrIran

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