Computational Geosciences

, Volume 22, Issue 5, pp 1231–1250 | Cite as

Analysis of subdiffusion in disordered and fractured media using a Grünwald-Letnikov fractional calculus model

  • Abiola D. Obembe
  • Sidqi A. Abu-Khamsin
  • M. Enamul Hossain
  • Kassem Mustapha
Original Paper


The increasing applications of fractional calculus in simulating the anomalous transport behavior in disordered and fractured heterogeneous porous media has grown rapidly over the past decade. In the present study, a temporal fractional flux relationship is employed as a constitutive equation to relate the volumetric flow rate to the gradient of the pore pressure. The novelty of this paper entails interpreting the time fractional derivative operator in the flux relationship by the Grünwald-Letnikov (G-L) definition as opposed to the Caputo interpretation which has been widely considered. Subsequently, a numerical scheme based on the block-centered finite-difference discretization is formulated to handle the resulting non-linear fractional diffusion model. In addition, a linear stability analysis is successfully performed to establish the stability criterion of the developed numerical scheme. An expression for the modified incremental material balance index was derived to assess the effectiveness of the numerical discretization process. Finally, numerical experiments were performed to provide qualitative insights into the nature of pressure evolution in a hydrocarbon reservoir under the influence subdiffusion. In summary, the results establish that subdiffusion regime results in the development of higher pressure drop in the reservoir. This paper will provide a strong foundation for researchers interested in investigating anomalous diffusion phenomena in porous media.


Fractional calculus Anomalous Grünwald-Letnikov Caputo interpretation Subdiffusion Finite difference 



Cross-sectional area of rock perpendicular to the flow of flowing fluid in x direction (ft2)


Cross-sectional area of rock perpendicular to the flow of flowing fluid in y direction (ft2)


Oil formation volume factor (bbl stb− 1)


Oil compressibility (psi− 1)


Total compressibility of the systems (psi− 1)


Formation rock compressibility of the systems (psi− 1)


Pseudo-compressibility, see Eq. 14


Reservoir height (ft)


Block centroid counter


Incremental material balance index


Absolute variable permeability (mD)


Pseudo-permeability (mD day1−)


Length of reservoir along x direction (ft)


Block counter for multi-dimensional flow


New time level

n − 1

Old time level


Pressure of the system (psia)


Initial pressure of the system (psia)


A reference pressure for the system (psia)


Flowing bottom hole pressure (psia)


Source term (stb day− 1)


Equivalent radius (ft)


Wellbore radius (ft)


Solution gas ratio (scf/stb)


Time (day)


Fluid transmissibility, see Eq. 14


Pseudo-transmissibility for wellbore model, see Eq. 28


Temperature (K)


Filtration velocity in x direction (ft/day)


Flow dimension at any point along the x-direction (ft)

Greek symbols


Volumetric conversion factor, 5.615


Conversion factor, 1.127× 10− 3


Fractional change in viscosity per unit change of pressure (psia− 1)


Finite difference kernel


fractional order of differentiation, dimensionless


Oil density (lb/ft3)


Reference density (lb/ft3)


Porosity, fraction


Initial porosity, fraction


Reference porosity, fraction


Oil specific gravity

\({\sigma _{r}^{2}}\)

Mean square displacement


fluid dynamic viscosity (cp)


Oil viscosity above bubble point pressure (cp)


Oil viscosity at bubble point pressure (cp)


Phenomenological coefficient; (mD day1− cp− 1)

Acronyms and field units


American Petroleum Institute


Reservoir barrel


Standard barrel


Standard cubic feet

Conversion factors

1 ft

0.3048 m

1 psia

6.894757 kPa

1 cp

0.001 Pa s− 1

1 bbl day− 1

0.1589873 std m3 day− 1


0.555556 K

1 lbm ft3 − 1

16.01846 kg/m3


0.9869233 × 10− 6 m2

1 bbl stb− 1

1 m3/std m3


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Funding Information

The authors would like to acknowledge the support provided via King Abdulaziz City for Science and Technology (KACST), through the Science & Technology Unit at King Fahd University of Petroleum & Minerals (KFUPM), for funding this work through project No. 11-OIL1661-04, as part of the National Science Technology and Innovation Plan (NSTIP).


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Petroleum Engineering, College of Petroleum Engineering and GeosciencesKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia
  2. 2.Department of Petroleum Engineering, School of Mining and GeosciencesNazarbayev UniversityAstanaRepublic of Kazakhstan
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia

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