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

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Abstract

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.

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Abbreviations

CR:

Vorticity cut-off ratio

E bth :

Breakthrough time error

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

Root mean squares of oil production rate error

Evmp :

Vorticity map preservation error

f :

Fractional flow of water phase

Fo :

Fraction of oil in the produced fluid

h :

Discretization size

\(\vec{i}_i \) :

Perpendicular unit vectors for i = 1,2,3

ILI:

Information Loss Index

k r :

Phase relative permeability

K = [k ij ]:

Absolute permeability tensor

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

Inverse of permeability tensor

K eff :

Effective permeability tensor

K :

Constant absolute permeability value

l :

Number of time steps

L2:

Euclidian norm

m :

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

M :

Mobility ratio

n :

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

N :

Number of fine cells with absolute normalized vorticity greater than α

OCGD:

Optimal coarse grid distribution

P :

Pressure

PSM:

Pressure solver method to calculate K eff

PVI:

Pore volume injected

q o :

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

Q inj :

Total rate of injection

S w :

Saturation (volume fraction) of water phase

t :

Time

t bh :

Breakthrough time

t sf :

Final simulation time, t sf = lΔt i

UR:

Upscaling ratio

\(\vec{V}\) :

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

x i :

Cartesian coordinates (i = 1,2,3)

α :

Thresholding parameter

\(\varepsilon \) :

Convergence error

δ ij :

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

Δt i :

Sampling time step (here it is constant)

Δx 1 :

Fine grid block size in the x 1 direction

Δx 2 :

Fine grid block size in the x 2 direction

ϕ:

Porosity

λt :

Total mobility

μ:

Phase dynamic viscosity

\(\vec{\omega }\) :

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

∇:

Gradient operator

∇.:

Divergent operator

∇ × :

Curl operator

∑:

Summation sign

f :

Fine grid model

rc :

Refined-coarse grid model

*:

Exact solution

c :

Coarse grid model

ch :

Permeability of channel

d :

Desired dimensions of coarse grid model

o :

Oil phase

w :

Water phase

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Mohammad Ali Ashjari & Bahar Firoozabadi

  2. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran

    Hassan Mahani

Authors
  1. Mohammad Ali Ashjari
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  2. Bahar Firoozabadi
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  3. Hassan Mahani
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Correspondence to Hassan Mahani.

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Ashjari, M.A., Firoozabadi, B. & Mahani, H. Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution. Transp Porous Med 75, 167–201 (2008). https://doi.org/10.1007/s11242-008-9217-9

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