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Arabian Journal for Science and Engineering

, Volume 43, Issue 11, pp 6333–6353 | Cite as

Numerical Simulation of Viscous Fingering and Flow Channeling Phenomena in Perturbed Stochastic Fields: Finite Volume Approach with Tracer Injection Tests

  • Eric Thompson Brantson
  • Binshan Ju
  • Dan Wu
Research Article - Petroleum Engineering

Abstract

Modeling viscous fingering perturbations in stochastic fields have always been a formidable task for geoscientists. Although fingering and channeling are often used synonymously to describe uneven front displacements, in fact, both phenomena represent different bypassing effects on hydrocarbon ultimate recovery. However, fingering perturbations in stochastic fields to invoke them during simulation often lead to severe channeling. Hence, the focus of this paper is to develop a finite volume numerical simulator with a level of certainty where fingering perturbations in stochastic fields do not result into severe channeling. The simulator development involves discretization of convection–diffusion and pressure equations with total variation diminishing and central differencing schemes, respectively. Correlated stochastic fields for simulation runs were created using convolution with a Gaussian filter achieved through 2D fast and inverse Fourier transform, Dykstra–Parsons coefficient, random porosity generator, and autocorrelation lengths. The obtained pressure and tracer concentration profiles were solved by iterative implicit pressure and explicit concentration approach, respectively. Numerical results of concentration profiles indicated that, unlike flow channeling, viscous fingering occurred in both homogeneous and heterogeneous fields. The obtained flow pattern map captures these flow regimes, namely fluid mobility fingering, channeling, and heterogeneous fingering. Sensitivity analyses of diffusion coefficient contributions to channeling and fingering phenomena are of extreme importance. The simulator shows a good validation with analytical solution while insensitive to grid refinements and spurious oscillations. The simulator is an improvement on first-order scheme simulators with the capability of computationally tracking fingering perturbations in stochastic fields which often result into channeling anomalies.

Keywords

Gaussian filter Total variation diminishing scheme Central differencing scheme Iterative implicit pressure explicit concentration Tracer concentration profiles 

List of symbols

u

Darcy velocity (\(\hbox {m/s}\))

k

Permeability (\(\hbox {m}^{2}\))

\(\phi \)

Average porosity (–)

\(\mu \)

Viscosity of fluid (Pa\(\cdot \)s)

p

Reservoir pressure (Pa)

c

Concentration of tracer (\(\hbox {mol/m}^{3}\))

\(c_o \)

Initial concentration of fluid (\(\hbox {mol/m}^{3}\))

\(c_{\mathrm{analytical}} \)

Analytical concentration of tracer (\(\hbox {mol/m}^{3}\))

\(c_L \)

Concentration of tracer at the end of the medium (\(\hbox {mol/m}^{3}\))

D

Diffusion coefficient (\(\hbox {m}^{2}/\hbox {s}\))

f(x)

Function of \(\hbox {x}\)

\(f(x_i)\)

Function of known derivative

\(\frac{\mathrm{d}^{n}f}{\mathrm{d}x^{n}}\)

Higher derivative of a function

\(\phi (r)\)

Flux limiter function

\(\eta \)

Mesh ratio

f

Flux (\(\hbox {m}^{3}/\hbox {s}\))

A

Anti-diffusive flux (\(\hbox {m}^{3}/\hbox {s}\))

r

Ratio of second-order terms

t

Final timestep (s)

\(\Delta t\)

Timestep change (s)

\(t_o \)

Initial timestep (s)

\(\sigma \)

Sample standard deviation

\({I}_{\mathrm{H}}\)

Heterogeneity index (\({I}_{\mathrm{H}}\))

\(\sigma _{\ln (k)}^2 \)

Variance of log permeability fields,

\(\lambda _R \)

Dimensionless correlation length (–)

\(\lambda \)

Correlation length along flow direction (\(\hbox {m}\))

L

System length of the porous medium (\(\hbox {m}\))

\(C_\mathrm{V} \)

Coefficient of permeability variation

\(\hbox {SD}\)

Standard deviation

E(X)

Population mean

\({H}_{\mathrm{K}}\)

Koval heterogeneity factor

Subscripts and Superscripts

x

Space (m)

t

Time (s)

i

Initial value

n

Derivative value

n!

Factorial of a value

n

Old timestep

\(n+1\)

Current timestep

\(i\pm \frac{1}{2}\)

Numerical flux interface

Abbreviations

\(\hbox {TVD}\)

Total variation diminishing

\(\hbox {TV}\)

Total variation

\({V}_{\mathrm{DP}}\)

Dykstra–Parsons coefficient

\(\hbox {FFT}\)

Fast Fourier transform

\(\hbox {IFFT}\)

Inverse Fourier fast transform

\(\hbox {IH}\)

Heterogeneity index

\(\hbox {IMPEC}\)

Implicit pressure explicit concentration

\(\hbox {ETAC}\)

Ethyl acetate

\(\hbox {ETOH}\)

Ethanol

\(\hbox {DLA}\)

Diffusion-limited aggregation

\(\hbox {PSD}\)

Pore size distributions

\(\hbox {CD}\)

Convection–diffusion

Unit conversion

1 Darcy

\(= 9.86923 \times 10^{-13}\,\hbox {m}^{2}\)

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Notes

Acknowledgements

The work was supported by the Fundamental Research Funds for National Science and Technology Major Projects (2016ZX05011-002) and the Central Universities (2652015142). The tireless assistance from Dr. Ali Akbar Eftekhari, Dr. Yao Yevenyo Ziggah, Addo Bright Junior, and Akwensi Perpetual Hope are duly appreciated for their effort in carrying out this novel research work. We would also like to thank the anonymous reviewers for their comments and suggestions that were helpful in improving the manuscript.

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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.School of Energy Resources, Key Laboratory of Marine Reservoir Evolution and Hydrocarbon Accumulation Mechanism, Ministry of EducationChina University of GeosciencesBeijingChina
  2. 2.Patent Examination Cooperation CentreSIPOBeijingChina

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