# 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.

## Keywords

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

- 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

- F
_{o} 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

*L*2Euclidian 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}=*l*Δ*t*_{ 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)

## *Greek symbols*

- 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

- Δ
*t*_{i} Sampling time step (here it is constant)

- Δ
*x*_{1} Fine grid block size in the

*x*_{1}direction- Δ
*x*_{2} Fine grid block size in the

*x*_{2}direction- ϕ
Porosity

- λ
_{t} 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

## *Superscripts*

*f*Fine grid model

*rc*Refined-coarse grid model

- *
Exact solution

## *Subscripts*

*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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