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
  • 9 Downloads

Abstract

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.

Keywords

Fractional calculus Anomalous Grünwald-Letnikov Caputo interpretation Subdiffusion Finite difference 

Nomenclature

Ax

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

Ay

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

Bo

Oil formation volume factor (bbl stb− 1)

co

Oil compressibility (psi− 1)

ct

Total compressibility of the systems (psi− 1)

cs

Formation rock compressibility of the systems (psi− 1)

C

Pseudo-compressibility, see Eq. 14

h

Reservoir height (ft)

i

Block centroid counter

IMB

Incremental material balance index

K

Absolute variable permeability (mD)

Kγ

Pseudo-permeability (mD day1−)

L

Length of reservoir along x direction (ft)

m

Block counter for multi-dimensional flow

n

New time level

n − 1

Old time level

p

Pressure of the system (psia)

pi

Initial pressure of the system (psia)

p0

A reference pressure for the system (psia)

pwf

Flowing bottom hole pressure (psia)

qsc

Source term (stb day− 1)

reqv

Equivalent radius (ft)

rw

Wellbore radius (ft)

Rs

Solution gas ratio (scf/stb)

t

Time (day)

T

Fluid transmissibility, see Eq. 14

Tgw

Pseudo-transmissibility for wellbore model, see Eq. 28

Temp

Temperature (K)

u

Filtration velocity in x direction (ft/day)

x

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

Greek symbols

αc

Volumetric conversion factor, 5.615

βc

Conversion factor, 1.127× 10− 3

cμ

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

δ

Finite difference kernel

γ

fractional order of differentiation, dimensionless

ρo

Oil density (lb/ft3)

ρ0

Reference density (lb/ft3)

ϕ

Porosity, fraction

ϕi

Initial porosity, fraction

ϕ0

Reference porosity, fraction

γo

Oil specific gravity

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

Mean square displacement

μ

fluid dynamic viscosity (cp)

μab

Oil viscosity above bubble point pressure (cp)

μob

Oil viscosity at bubble point pressure (cp)

η

Phenomenological coefficient; (mD day1− cp− 1)

Acronyms and field units

API

American Petroleum Institute

bbl

Reservoir barrel

stb

Standard barrel

scf

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

1R

0.555556 K

1 lbm ft3 − 1

16.01846 kg/m3

1mD

0.9869233 × 10− 6 m2

1 bbl stb− 1

1 m3/std m3

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Notes

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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Copyright information

© 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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