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Finite-Difference Approximation for Fluid-Flow Simulation and Calculation of Permeability in Porous Media

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Abstract

We introduce a finite-difference method to simulate pore scale steady-state creeping fluid flow in porous media. First, a geometrical approximation is invoked to describe the interstitial space of grid-based images of porous media. Subsequently, a generalized Laplace equation is derived and solved to calculate fluid pressure and velocity distributions in the interstitial space domain. We use a previously validated lattice-Boltzmann method (LBM) as ground truth for modeling comparison purposes. Our method requires on average 17 % of the CPU time used by LBM to calculate permeability in the same pore-scale distributions. After grid refinement, calculations of permeability performed from velocity distributions converge with both methods, and our modeling results differ within 6 % from those yielded by LBM. However, without grid refinement, permeability calculations differ within 20 % from those yielded by LBM for the case of high-porosity rocks and by as much as 100 % in low-porosity and highly tortuous porous media. We confirm that grid refinement is essential to secure reliable results when modeling fluid flow in porous media. Without grid refinement, permeability results obtained with our modeling method are closer to converged results than those yielded by LBM in low-porosity and highly tortuous media. However, the accuracy of the presented model decreases in pores with elongated cross sections.

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Abbreviations

3D:

Three-dimensional

FDGPA:

Finite-difference geometrical pore approximation

LBM:

Lattice-Boltzmann method

NS:

Navier–Stokes

\({\vec{\vec{A}}}\) :

A septa-diagonal matrix, representing the relevant \({\vec{w}}\) for all grids, s

\({\vec{B}}\) :

Boundary condition representing the inlet and outlet pressures, Pa s

d :

Digital equivalent of r, dimensionless

d max :

Digital equivalent of r max, dimensionless

f c(d max):

Calibration function, dimensionless

J, J z :

Mass flux, kg m−2 s−1

J tot :

Total mass flux, kg m−2 s−1

K :

Permeability, m2[1 D = 1 Darcy = 9.869 × 10−13 m2]

L :

Porous media (tube) length, m

\({P, P_{1}, P_{2}, \vec{P}}\) :

Pressure, Pa

P avg :

Average pressure, Pa

R :

Tube radius, m

r :

Radial distance from the inner wall, m

r max :

The largest inscribed radius, m

S(d max):

Area of the smallest possible cross-section for a given d max, dimensionless

V :

Volume flux, m s−1

\({\vec{w}}\) :

Weighting factor in the generalized Laplace equation, s

μ :

Viscosity, Pa s

ν :

Local fluid velocity, m s−1

ρ :

Fluid density, kg m−3

ρ avg :

Average fluid density, kg m−3

τ :

Relaxation parameter, dimensionless

τ rz :

Z-momentum across a surface perpendicular to the radial direction, kg m s−1

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Authors and Affiliations

  1. Department of Petroleum and Geosystems Engineering, The University of Texas at Austin, 1 University Station C0300, Austin, TX, 78712-0228, USA

    Vahid Shabro, Carlos Torres-Verdín & Kamy Sepehrnoori

  2. Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin, University Station, Box X, Austin, TX, 78713-8924, USA

    Farzam Javadpour

Authors
  1. Vahid Shabro
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  2. Carlos Torres-Verdín
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  3. Farzam Javadpour
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  4. Kamy Sepehrnoori
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Correspondence to Vahid Shabro.

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Shabro, V., Torres-Verdín, C., Javadpour, F. et al. Finite-Difference Approximation for Fluid-Flow Simulation and Calculation of Permeability in Porous Media. Transp Porous Med 94, 775–793 (2012). https://doi.org/10.1007/s11242-012-0024-y

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