Transport in Porous Media

, Volume 75, Issue 2, pp 167–201 | Cite as

Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution

  • Mohammad Ali Ashjari
  • Bahar Firoozabadi
  • Hassan MahaniEmail author


An improved vorticity-based gridding technique is presented and applied to create optimal non-uniform Cartesian coarse grid for numerical simulation of two-phase flow. The optimal coarse grid distribution (OCGD) is obtained in a manner to capture variations in both permeability and fluid velocity of the fine grid using a single physical quantity called “vorticity”. Only single-phase flow simulation on the fine grid is required to extract the vorticity. Based on the fine-scale vorticity information, several coarse grid models are generated for a given fine grid model. Then the vorticity map preservation error is used to predict how well each coarse grid model reproduces the fine-scale simulation results. The coarse grid model which best preserves the fine-scale vorticity, i.e. has the minimum vorticity map preservation error is recognized as an OCGD. The performance of vorticity-based optimal coarse grid is evaluated for two highly heterogeneous 2D formations. It is also shown that two-phase flow parameters such as mobility ratio have only minor impact on the performance of the predicted OCGD.


Upscaling Vorticity Non-uniform grid Optimal coarse grid generation Porous media Two-phase flow 



Vorticity cut-off ratio


Breakthrough time error

\(E_{\rm{RMS}}^{opr} \)

Root mean squares of oil production rate error


Vorticity map preservation error


Fractional flow of water phase


Fraction of oil in the produced fluid


Discretization size

\(\vec{i}_i \)

Perpendicular unit vectors for i = 1,2,3


Information Loss Index


Phase relative permeability

K = [kij ]

Absolute permeability tensor

\(K^{-1}=[k_{ij}^{-1} ]\)

Inverse of permeability tensor


Effective permeability tensor


Constant absolute permeability value


Number of time steps


Euclidian norm


Number of fine cells or coarse grid blocks in the x 2 direction


Mobility ratio


Number of fine cells or coarse grid blocks in the x 1 direction


Number of fine cells with absolute normalized vorticity greater than α


Optimal coarse grid distribution




Pressure solver method to calculate K eff


Pore volume injected


Oil production rate at each time step non-dimensionalized by Q inj


Total rate of injection


Saturation (volume fraction) of water phase




Breakthrough time


Final simulation time, t sflΔt i


Upscaling ratio


Single-phase velocity vector \((\vec{V}=v_1 \vec{i}+v_2 \vec{j}+v_3 \vec{k})\)

\(\vec{V}_{\rm t} \)

Total velocity vector


Cartesian coordinates (i = 1,2,3)

Greek symbols


Thresholding parameter

\(\varepsilon \)

Convergence error


The Kronecker delta, equal to 1 for i = j and to 0 for i ≠ j

\(\varepsilon_{ijl} \)

The permutation symbol equal to \(\left\{\begin{array}{l}\\ {+1}\\ {-1}\\ 0\\ \end{array}\right\}\) According to whether \(i,j,l\left\{\begin{array}{l}\\ {\it form\ an\ even}\\ {\it form\ an\ odd}\\ {\it do\ not\ form\ a}\\ \end{array} \right\}\) Permutation of 1, 2, 3


Sampling time step (here it is constant)


Fine grid block size in the x 1 direction


Fine grid block size in the x 2 direction




Total mobility


Phase dynamic viscosity

\(\vec{\omega }\)

Single-phase vorticity vector \((\vec{\omega }=\omega_i \vec{i}_i )\)

Gradient operator


Divergent operator

∇ × 

Curl operator

Summation sign



Fine grid model


Refined-coarse grid model


Exact solution



Coarse grid model


Permeability of channel


Desired dimensions of coarse grid model


Oil phase


Water phase


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Mohammad Ali Ashjari
    • 1
  • Bahar Firoozabadi
    • 1
  • Hassan Mahani
    • 2
    Email author
  1. 1.Department of Mechanical EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran

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