Transport in Porous Media

, Volume 129, Issue 3, pp 633–652 | Cite as

A Mathematical Model for Determining Oil Migration Characteristics in Low-Permeability Porous Media Based on Fractal Theory

  • Yongchao Zhang
  • Jianhui ZengEmail author
  • Jianchao Cai
  • Sen Feng
  • Xiao Feng
  • Juncheng Qiao


Oil migration processes are crucially important for determining oil reserve levels and distributions in low-permeability reservoirs. Previous researches have proved that the macroscopic parameters of oil migration processes are controlled by microscopic pore properties in low-permeability media. To analyze the relationship between pore-scale parameters and the oil migration process in low-permeability formations, a mathematical model combining the capillary bundle model and fractal theory is developed in this work. The accuracy of the proposed model is validated via comparisons with three well-designed oil charging experiments using natural core samples. Based on the validated model, the influence of four factors (pore fractal dimension, tortuosity fractal dimension, the wettability of the rock surface, and the formation water viscosity) on oil migration processes in low-permeability media is analyzed. Oil saturation and effective permeability are used as the output parameters for reflecting changes in the oil migration process. The calculation results indicate that oil saturation decreases as pore fractal dimension, tortuosity fractal dimension, the hydrophilicity of the rock surface, and brine viscosity increase. On the other hand, effective permeability decreases as tortuosity fractal dimension and formation water viscosity increase, but increases with pore fractal dimension. Moreover, the wettability dependence of effective permeability is relatively weak.


Fractal Oil migration Low-permeability formation Oil saturation Effective permeability 

List of Symbols


Sectional area of the rock


Pore fractal dimension

\(D_{\tau }\)

Tortuosity fractal dimension

h1, h2, h3

Coefficients in Eq. (1)


Effective permeability of the rock


Length of the rock sample


Length of a single capillary with radius r


Number of pores


Capillary pressure

\(\nabla p\)

Pressure gradient

\(\Delta p\)

Pressure difference


Flow rate of a single capillary


Flow rate in the rock sample


Pore radius


Critical capillary radius


Maximum pore radius


Minimum pore radius


Water saturation


Oil saturation


Mercury saturation in the rock sample


Total pore volume of the rock sample


Volume of the invaded oil


Thickness of the boundary layer


Viscosity of the brine in pores in Eq. (1)


Viscosity of oil


Viscosity of water

\(N_{( > r)}\)

Number of pores with radii larger than r


Total number of pores


Tortuosity of the capillary

\(\tau^{{\prime }}\)

Average tortuosity of capillaries


Average radius of capillaries


Porosity of the rock sample


Interfacial tension between oil and water



This work was financially supported by the National Natural Science Foundation of China (No. 41330319), and the Foundation of State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing (No. PRP/open-1601). Y. Zhang would show his sincere thanks to the Heriot-Watt University for the help in their preparation of the manuscript.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of PetroleumBeijingPR China
  2. 2.College of GeosciencesChina University of PetroleumBeijingPR China
  3. 3.Hubei Subsurface Multi-scale Imaging Key Laboratory, Institute of Geophysics and GeomaticsChina University of GeoscienceWuhanPR China
  4. 4.Institute of Petroleum EngineeringHeriot-watt UniversityEdinburghUK

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