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Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations

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Abstract

We present a novel unstructured coarse grid generation technique based on vorticity for upscaling two-phase flow in permeable media. In the technique, the fineness of the gridblocks throughout the domain is determined by vorticity distribution such that where the larger is the vorticity at a region, the finer are the gridblocks at that region. Vorticity is obtained from single-phase flow on original fine grid, and is utilized to generate a background grid which stores spacing parameter, and is used to steer generation of triangular and finally Voronoi grids. This technique is applied to two channelized and heterogeneous models and two-phase flow simulations are performed on the generated coarse grids and, the results are compared with the ones of fine scale grid and uniformly gridded coarse models. The results show a close match of unstructured coarse grid flow results with those of fine grid, and substantial accuracy compared to uniformly gridded coarse grid model.

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Abbreviations

A :

External area of control volume

A i :

Area of control volume i

C :

Compressibility

d max :

Maximum spacing parameter

d min :

Minimum spacing parameter

f :

Fluid cut

E:

Error

G:

Source term

\({\vec{g}}\) :

Gravity acceleration

h :

Gridblock length

\({\vec{i}, \vec{j}, \vec{k}}\) :

Unit vectors

K :

Permeability tensor

k :

Permeability

k r :

Phase relative permeability

\({\bar{{k}}}\) :

Upscaled permeability

L :

Spacing parameter

M :

Number of block edges

N :

Number of source points

\({\vec{n}}\) :

Outward normal unit vector

P :

Fluid pressure

q p :

Well flow of phase p

S:

Fluid saturation

t :

Time

\({\vec{u}}\) :

Velocity vector (\({\vec{u}=u_x \vec{i}+u_y \vec{j}+u_z \vec{k}}\))

UL:

Upscaling level

V :

Control volume

β :

Relaxation factor

Φ:

Phase potential

\({\varphi}\) :

Porosity

λ:

Mobility

μ :

Fluid viscosity

ρ :

Fluid density

\({\vec{\omega}}\) :

Vorticity vector

ψ :

Intensity of source element

is:

Internal source element

r :

Iteration number

t :

Time level

c:

Coarse

f:

Fine

n :

nth Source

o:

Oil

p:

Phase

T:

Total

w:

Water

wf:

Well flowing in P wf

x :

x direction

y :

y direction

z :

z direction

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Authors and Affiliations

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

    M. Evazi

  2. Shell International Exploration & Production B.V., Kessler Park 1, Rijswijk, 2288 GS, The Netherlands

    H. Mahani

Authors
  1. M. Evazi
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  2. H. Mahani
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Correspondence to H. Mahani.

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Evazi, M., Mahani, H. Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations. Transp Porous Med 83, 541–572 (2010). https://doi.org/10.1007/s11242-009-9458-2

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