© 2004 Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 14th Europec Biennial Conference held in Madrid, Spain, 13-16 June 2005. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the SPE, their officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract This paper describes the use of vorticity to optimize the generation of coarse simulation grids from finely gridded geological models, thereby reducing (or removing) the need for two phase upscaling. Vorticity has been used in the simulation of flow in porous media since de Josselin de Jong1 first discussed it in 1960. However this is the first time, to our knowledge, that it has been used in the area of upscaling.

Vorticity maps are generated from simulations of single-phase flow in the finely gridded geological model of interest. A coarse simulation grid is then derived which is more refined in areas of high vorticity and coarser in areas of low vorticity. The method is demonstrated for simple, layered and lens heterogeneities as well as Model 1 from the SPE Comparative Study in Upscaling2. Good matches between fine and coarse grid simulations of waterflooding are obtained, compared with using a conventionally generated coarse grid model.

Introduction Reservoir simulation is widely applied in the oil industry for the management of reservoir performance and the prediction and evaluation of alternative development scenarios. A primary input to the reservoir simulator is the geological model of the reservoir, commonly built as one or more geostatistical realizations and containing petrophysical properties constrained to data of different types and scales. These geological models typically have a high resolution (up to 108 grid blocks) because it is well established that fine scale features can have a significant impact on reservoir flow performance3,4,5. In practice, however, these geological models usually cannot be directly input to flow simulation models due to memory and/or processing time constraints. Thus there is a requirement to coarsen or upscale the geological models whilst continuing to model the effects of important fine scale flow features.

The need to upscale reservoir flow has motivated the development of many different techniques, comprehensive

reviews of which can be found in Barker and Thibeau6 and Renard and de Marsily7. Although there is an extensive literature demonstrating that the coarse grid flow equations may not be the same as those on the fine scale (see for example Quintard and Whitaker8; Amaziane and Bourgeat9; Saez et al.10; Auriault11; Durlofsky12; Efendief et al.13), the most commonly applied methods assume the same flow equations apply on the coarse and fine scales and involve the use of an effective (equivalent) permeability and modified relative permeabilities (pseudoization techniques) often combined with specialized gridding techniques.

Indeed careful coarse grid generation can remove the need for upscaling multi-phase flow properties (for example Garcia et al.14; Durlofsky et al.15; Li et al.16; Durlofsky et al.17; Christie and Blunt2; Qi et al.18; Qi and Hesketh19). In essence all these methods attempt to minimize differences between fine and coarse grid simulations by optimizing the location of coarse grid block boundaries. Permeability based techniques such as those of (Garcia et al.14; Li et al.16; Soleng and Holden20; Qi et al.18; Qi and Hesketh19) preserve the variation of permeability within the coarse grid resulting in finer gridding around regions of extreme permeability. Flow-based techniques (such as those of Durlofsky et al.15; Durlofsky et al.17) refine the grid in areas of high flow-rate.

In this work a new grid coarsening technique is introduced that is based upon vorticity in the flows through heterogeneous porous media. We first discuss the causes and significance of vorticity in porous media and describe the governing equations. We then describe how we calculate permeability vorticity from fine grid, single-phase, flow simulations and use this to optimise the construction of the coarse grid for two phase flow simulations. Finally we demonstrate the successful application of the method to waterflooding simulations in a range of heterogeneous models.

Importance of vorticity in porous media Vorticity is a vector describing the rate and direction of rotation in fluid flow at any given location. Like the angular momentum of rotating solids it is a conserved quantity. It is calculated by taking the curl of the velocity field v:

( ), ,x y z

y yx xz zv vv vv v

y z z y y x

ω ω ω ω= = ∇×

∂ ∂ ∂ ∂ ∂ ∂= − + − + − ∂ ∂ ∂ ∂ ∂ ∂

v

i j k (1)

Although vorticity is not normally considered when modelling reservoir flows it was first used in 1960 by de Josselin de Jong1 to model multifluid displacements in porous

SPE 94319

Improved Coarse Grid Generation Using Vorticity H. Mahani, SPE, Imperial College London, A.H. Muggeridge, SPE, Imperial College London

2 H. Mahani, A.H. Muggeridge SPE 94319

media. More recently Meiburg and co-workers have published a number of papers (Meiburg and Homsy21; Chen and Meiburg22) describing the use of vorticity-streamfunction methods to model miscible displacements. Sposito23 has shown that vorticity occurs in groundwater flows as a result of heterogeneity. Kapoor24 also found that dispersion and concentration fluctuations in random porous media can be quantified using the rotational characteristics of flow.

However the influence of vorticity on upscaling has not been investigated, to our knowledge, although White and Horne25 used vorticity to motivate the use of tensor effective permeabilities in upscaled simulation models. Ignoring gravity, for multi-phase flow through a heterogeneous porous medium, substituting Darcy’s Law into equation (1) gives:

ω = vT × ∇ ln k +d ln λT

d Sw

∇Sw

(2)

where k is the permeability, vT is the total velocity, λT is total mobility and Sw is the water saturation. It can be seen that vorticity in this case is a function of total velocity, the gradient of ln k, the rate of change of total mobility with saturation and the gradient in saturation. It will be largest in regions with large flow rates perpendicular to large gradients in permeability. In general, in heterogeneous rock the saturation

term, d ln λT

d Sw

∇Sw is negligible except where there are large

saturation gradients perpendicular to the principal flow direction.

As it is a conserved quantity, the vorticity distribution should be the same in fine and coarse grid simulation models if the coarse upscaled model is to correctly reproduce the average behaviour observed on the fine scale. This fact combined with equation (2) explains the success of the coarse grid generation methods based either on preserving permeability variance or geologic complexity (the second term on equation (2), see for example (Garcia et al.14; Li et al.16; Qi et al.18; Qi and Hesketh19) or areas of high flow rate and physics of flow in porous media (the first term in equation (2), see for example (Durlofsky et al.15,17). Equation (2) also explains why upscaling solutions are sensitive to flow direction – if the direction of flow changes then so does the vorticity.

We propose that the best coarse grids to represent the influence of small-scale heterogeneity on large-scale flows are those that preserve vorticity. New grid coarsening approach The technique entails 1) building a fine scale geological model of the reservoir, 2) performing a fine scale single-phase flow simulation

using the geological model, 3) generation of the vorticity map, 4) coarse grid construction based on preserving areas of high

vorticity, 5) upscaling permeability for the new coarse grid.

The fine grid, single-phase flow simulation provides the global velocity distribution in the system from which vorticity is calculated numerically and then mapped. We used a

commercial, finite difference, black oil simulator for this work (ECLIPSE™)26.

For 2D flow in the x-y plane the vorticity vector points in the z-direction. The x and y components of vorticity are zero, so:

y xz

v vx y

ω∂ ∂

= −∂ ∂

(3)

Thus the vorticity in grid block ( , )i j is given by:

1 1 1 1( , ) ( , ) ( , ) ( , )2 2 2 2

y y x x

z

v i j v i j v i j v i j

x yω

+ − − + − −= −

∆ ∆(4)

where, as velocity is a face-centred quantity, each velocity is the arithmetic average of the four neighboring velocities as depicted in figure 1.

1( , )2

1 1 1 1 1[ ( , 1) ( , 1) ( , ) ( , )]4 2 2 2 2

x

x x x x

v i j

v i j v i j v i j v i j

± =

− ± + + ± + − + +∑ (5)

1( , )2

1 1 1 1 1[ ( , ) ( 1, ) ( , ) ( 1, )]4 2 2 2 2

y

y y y y

v i j

v i j v i j v i j v i j

± =

− + ± − + + + ± +∑ (6)

Depending on the directional gradient of velocity field, vorticity can be negative as well as positive. As discussed in the previous section it will be largest in regions where there is a high permeability gradient perpendicular to the flow direction. The calculated vorticities are then normalized by dividing them by the maximum value found in the geological model being analysed to make them between 1,1− , so that the application of technique be more straightforward in practice.

The grid optimization is accomplished by first selecting a maximum aggregation ratio (coarsening degree) for the coarse grid model, just to control and minimize numerical dispersion introduced in the coarse grid results. Next, regions of high and low vorticity are identified within each coarse grid block by employing a search window algorithm, akin to the one that Li et al..16 used but with different principals. In this approach the two search windows in 2-D moves independently in x and y direction (for 3-D in three directions; x, y and z) which their sizes can be as large as one fine grid cell, although it can take larger sizes. This window starts searching from the first cell in each direction and sweeps along and across the fine grid model and detects the fine grids whose absolute vorticity is above or below the specified vorticity cut-off between zero and one. If absolute vorticity of a grid cell is below the cut-off, boundaries of that grid are marked as vertical or horizontal boundaries of potential coarse grids in x and y direction respectively. Typically the larger is the cut-off, the coarser the grids are. Upscaled grids will tend to be coarser in areas of low vorticity and more refined in the area of high vorticity. Finally the effective permeability for each coarse grid is calculated using the pressure solver technique27.

SPE 94319 Improved Coarse Grid Generation Using Vorticity 3

Figure 1: The calculation of vorticity in each grid blocks requires the velocity along a grid block boundary to be calculated from the average of four face-centred velocities. A finite difference is then used to calculate the gradient of x-direction velocity with respect to y and the gradient of the y-direction velocity with respect to x.

Numerical Results and Discussion The first system investigated was a simple 2-D, 2-layered model with permeability ratio of 1 2: 1: 0.1k k = (horizontal) and zero vertical permeability. It is well known that the best way to upscale layered systems for two phase flow is to preserve the layers and minimize the number of grid blocks within each layer. Thus this system provides a sinple test of validity of our approach.

We generated the vorticity map shown in figure 2 from a fine grid single-phase flow simulation. The simulation used a 50 ×40 grid with a constant rate, injection well on the left and a production well on the right, both completed over the entire thickness of model. Table 1 gives the fluid viscosities and relative permeabilities.

As expected, vorticity exists only at the boundary of layers and is zero elsewhere. Applying the coarsening algorithm suggests the coarse grid boundary should coincide with the boundary between the two layers i.e. the best grid to use is one with 2 layers, one for each permeability layer. A coarse grid waterflood simulation was then performed on the fine grid model, a 20 2× grid with vorticity preserved and a 20 1× grid in which the layers were homogenized and vorticity was not preserved. The number of grid blocks in the x-direction was preserved in order to minimize the effects of numerical dispersion on the results. Figure 3 compares the water fractional flow for the fine grid model and the two coarse grid models. As expected, the 20 2× grid matches the fine grid results and the 20 1× grid does not.

A vorticity map was also calculated for flow perpendicular to the layers (i.e. flow in the y-direction). As expected, vorticity was zero everywhere suggesting that for this case the only consideration in generating a coarse grid is the control of numerical dispersion. Thus a1× 40 grid will perfectly match the fine grid results. The grid can only be refined further in the direction of flow if pseudo relative permeablities are used to compensate for numerical dispersion.

Table 1: Fine Grid Rock and Fluid Properties –

two layered model Porosity, φ 0.3 Grid block size, m 1.0

0.1xy

∆ =∆ =

Viscosity, cp 2.01.0

o

w

µµ

==

Density, 3/kg m 1.0o wρ ρ= = Relative permeability 2

2

(1 )o

w

r w

r w

k s

k s

= −

=

Figure 2: Vorticity map for the two layer model. There is highest vorticity at the boundary between the two permeability layers. Vorticity is zero elsewhere.

Figure 3: Comparision of fine and coarse grid water cut for the 2- layered model (mobility ratio of 2). The best match is obtained by preserving the layering seen in the fine grid model as suggested by inspecting the vorticity map.

( 1, )i j− ( 1, )i j+

( , 1)i j +

( , 1)i j −

xv

yv( , )i j+

x∆

y∆x

y

1( , )2yv i j−

1( , )2yv i j+

1( , )2xv i j−

1( , )2xv i j +

A x y∆ =∆ ∆

(b) (a)

4 H. Mahani, A.H. Muggeridge SPE 94319

The second synthetic example is a five-layered model, with permeability contrast ranging from 0.001 to 10 (see table 2). Figure 4 represents a typical vorticity map for one arrangement of layers. Figure 5 demonstrates the improvement in the performance of upscaled model when constructed using vorticity map and confining coordinates of coarse grids between layers boundaries, such that vorticity is preserved in the coarse grid.

Table 2: Fine Grid Rock and Fluid Properties – five-layered model

Porosity, φ 0.3 Grid block size, m 1.0

0.05xy

∆ =∆ =

Viscosity, cp 2.01.0

o

w

µµ

==

Density, 3/kg m 1.0o wρ ρ= = Relative permeability 2

2

(1 )o

w

r w

r w

k s

k s

= −

=

Permeability, mD Layer 1 = 0.1 Layer 2 = 0.01 Layer 3 = 1.0 Layer 4 = 0.001 Layer 5 = 10

Figure 4: Vorticity map for the five layer model. There is highest vorticity at the boundary between the each two rmeability layers. Vorticity is zero elsewhere.

The next example considered was a lens heterogeneity in which a very low permeability, rectangular region was embedded in a high permeability background. The fluids in this model tend to move through the high permeability sand and around lens, leaving unswept oil within the lens and downstream of it. The location and geometry of this lens has a significant impact on flow and makes it difficult to upscale.

Figure 5: Comparision of fine and coarse grid water cut for the 5- layered model (mobility ratio of 2). The best match is obtained by preserving the layering seen in the fine grid model as suggested by inspecting the vorticity map.

Figure 6 shows the velocity and vorticity maps for a fine grid 50 100× model. The background has a permeability of 1D and the lens has a permeability of 10mD. Vorticity here is concentrated around the top and base of lens, having equal magnitude but different signs (direction of rotation). As expected from equation (2) and the results for flow perpendicular to layer, vorticity is almost zero at the front and rear of the lens because the flow direction is parallel to the permeability gradient.

Figure 7 compares the water cut obtained from the fine grid model, the uniformly coarsened model and the coarse grid generated from the vorticity map. It can be seen that the non-uniform coarse grid model gives a better prediction of fine grid fractional flow compared with uniformly coarsened 5 10× model. In this example the aggregation ratio is almost 102 and vorticity cut-off is 0.75. Here grid blocks are larger in the high and low permeability areas, but finer around the boundaries of lens.

Further application – SPE 10th Comparative Solution Project (Punq Model 1)

The final example we consider is Model 1 from the SPE 10th Comparative Solution Project2. This is a 2D, uniformly gridded, 100 1 20× × vertical cross-section with no faults and a correlated permeability distribution (figure 8). The model is initially fully saturated with oil (there is no connate water) which is displaced by a low viscosity, incompressible, immiscible fluid. Gravity and capillary effects are ignored. A summary of properties for this model is given in Table 3. The fine grid relative permeabilities are shown in figure 9.

The vorticity map generated for this model is shown in figure 10. Figure 11 compares the 7 14× coarse grid generated from the vorticity map with the 20 5× uniform coarse grid used for comparison. The final aggreagation ratio for both models is about 20. It can be seen that the coarse grid generated from the vorticity map is more refined vertically than horizontally, capturing boundaries of high vorticities.

SPE 94319 Improved Coarse Grid Generation Using Vorticity 5

(a)

(b)

Figure 6: 3-D Vorticity (a) and 2-D velocity (b) map of lens heterogeneity obtained from single phase flow simulation

Figure 7: Comparision of fine and coarse grid water fractional flow for lens heterogeneity. The coarse grid generated from the vorticity map gives a better match to the fine grid than a uniform coarse grid.

Figure 12 compares the recoveries and oil rates obtained from the coarse grid simulations with the reference fine grid model. Disappointingly neither the coarse grid generated from vorticity mapping nor the uniformly coarsened model match the fine grid simulation. Furthermore the vorticity based

coarsening does not perform any better than the uniformly coarsened grid. We attribute this poor performance to the high levels of numerical dispersion in the coarse grid models (Christie and Blunt2 note that the best results obtained by other workers used pseudoisation of the relative permeabilities). Only the original rock relative permeabilities were used in all studies in this paper.

(a) Fine Grid

(b) Non-unioform coarse grid

Figure 8: Permeability distribution of model 1 from SPE 10th Comparative Solution Project, compared with the permeability distribution of the coarse grid model determined from the vorticity map.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8Sg

Kr

Krg

Kro

Figure 9: Relative permeabilities for SPE 10th Comparative Solution Project.

6 H. Mahani, A.H. Muggeridge SPE 94319

Table 3: Fine Grid Properties – SPE 10th Comparative Solution Project – Model 1

Porosity, φ 0.2 Grid block size, m 7.620

7.6200.762

xyz

∆ =∆ =∆ =

Viscosity, cp 1.00.01

o

g

µµ

=

=

Density, 3 3/ [ / ]kg m lb ft 1.0[43.68]0.7[0.0624]

o

g

ρρ

=

=

Injection rate = Production rate, 3 /m day

6.97

Immiscible, incompressible, negligible capillarity and gravity

Figure 10: Vorticity map for SPE 10th Comparative Solution Project, Punq Model 1.

Figure 11: Generated coarse grids: (a) applying coarsening algorithm, 7x14 grid (b) uniform coarsening, 20x5 grid.

We therefore decided to remove the effects of numerical dispersion from our simulations by retaining the fine grid but homogenizing the permeabilities using the grids shown in figure 11. As observed by Muggeridge28 there are two factors which influence the accuracy of coarse grid simulation models when compared with fine grid simulation models:

1. homogenization of the heterogeneity 2. numerical dispersion resulting from the use of finite

difference techniques on coarse grids. The purpose of using vorticity maps is to optimize homogenization. The issue of minimizing numerical dispersion can be addressed either by higher order numerical schemes or pseudo relative permeabilities.

The results are shown in figure 13. It can be seen that the

non-uniform grid obtained from vorticity mapping gives a significantly improved match to the fine grid model. The uniform model does not match the fine grid model. In this case at least, vorticity mapping results in improved homogenization of a heterogeneous model.

Figure 12: SPE 10th Comparison Study, Model 1. Comparison of the results obtained using a 7 x 14 non-uniform grid generated from vorticity mapping with those obtained from a 20x5 uniform grid. Neither grid results in a good match to the 100x20 fine grid simulations. This is attributed to numerical dispersion in both the coarse grid models.

Figure 13: SPE 10th Comparison Study, Model 1. Comparison of the results obtained using a non-uniformly homogenised grid generated from vorticity mapping with those obtained from a uniformly homogenised grid. The homogenization derived from vorticity mapping gives a very good match to the fine grid heterogeneous simulation, whilst the homogenization to a 20x5 grid does not. Conclusions We have presented a novel grid homogenisation/coarsening technique based upon the preservation of zones of high vorticity. This is the first time that the conservation of vorticity has been applied to the upscaling problem, to our knowledge. The need to conserve vorticity explains many

(b) (a)

SPE 94319 Improved Coarse Grid Generation Using Vorticity 7

issues that have been discussed in the upscaling literature. For example examination of equation (2) explains • the success of coarse grid generation algorithms based on

total velocity and preserving permeability variation • the importance of large variations in permeability ( ∇ ln k

term in vorticity) • the fact that it is important to preserve permeability

gradients perpendicular to flow • the sensitivity of upscaling to flow direction (vorticity

depends upon velocity and permeability gradient) • the fact that two phase flow upscaling is secondary to

single phase flow upscaling except when it is necessary to compensate for numerical dispersion

The method has been applied successfully to a layered system, a lens heterogeneity as well as the more realistic Model 1 taken from the 10th SPE Comparative Study on Upscaling. However it is clear that although the method is successful for homogenisation, further techniques are required to compensate for numerical dispersion. In addition further testing is required on more complex 2D and 3D systems (e.g. Model 2 from the SPE Comparative Study. Acknowledgments Hassan Mahani gratefully acknowledges PhD scholarship from British Petroleum plc (BP) and financial support from British Council. Prof E. Meiburg from University of California and Prof Lou Durlofsky from Stanford are also appreciated for their constructive comments. The authors would also like to thank GeoQuest for providing ECLIPSE for use in this project. Nomenclature

2

-2

-1

, = grid block indices = permeability, L

= relative permeability of phase

= fluid pressure, FL = water saturation

= velocity, LT = horizontal coordinate, L

pr

w

i jkk p

pS

vxy

2

= vertical coordinate, L = coordinate perpendicluar to surfuce, L = area, L

zA

Greek symbols

T-2

-1

= porosity total mobility

= dynamic viscosity of fluid, FTL = vorticity, T

φλ

µ

ω

=

= difference operator = gradient operator

= curl

∆∇∇×

Subscripts = gas phase = oil phase

T = total

go

= water phase = mean value

w

References 1. de Josselin de Jong, G.: “Singularity Distribution for the Analysis

of Multiple-Fluid Flow through Porous Media,” Journal of Geophysical Research (1960) 65: 3739-3758.

2. Christie, M.A. and Blunt, M.J.: “Tenth SPE comparative solution project: A comparison of upscaling techniques,” SPE Reservoir Evaluation and Engineering (2001) 4: 308-317.

3. Jones, A.D.W., Verly, G.W. and Williams, J.K.: “What Reservoir Characterization is Required for Predicting Waterflood Performance in a High Net-to-Gross Fluvial Environment?,” North Sea Oil and Gas Reservoirs-III (1994) 223-232.

4. Jones, A.D.W. et al., D.: “Which Subsurface Heterogeneities Influence Waterflood Performance? A Case Study of a Low Net to Gross Fluvial Reservoir,” New Developments in Improved Oil Recovery (edited by De Haan H.J.), Geological Society Special Publication (1995) 84: 5-18.

5. Durlofsky, L.J. et al.: “Scale-up of Heterogeneous Three Dimensional Reservoir Descriptions,” paper SPE 30709 presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, October 22-25.

6. Barker, J.W. and Thibeau, S.: “A Critical Review of the Use of Pseudo Relative Permeabilities for Upscaling.” SPE Reservoir Engineering (1997) 12: 138-143.

7. Renard, P. and de Marsily, G.: “Calculating Equivalent Permeability: A Review,” Advances in Water Resources (1997) 20: 253-278.

8. Quintard M. and Whitaker, S.: “Two-Phase Flow in Heterogeneous Porous Media: The Method of Large-Scale Averaging,” Transport inn Porous Media (1988) 3: 357-413.

9. Amaziane B. and Bourgeat, A.: “Effective Behavior of Two-Phase Flow in Heterogeneous Reservoir,” In: M.F. Wheeler (Editor), Numerical Simulation in Oil Recovery, Springer-Verlag (1989).

10. Saez A.E., Otero, C.J. and Rusinek, I.: “The Effective Homogeneous Behavior of Heterogeneous Porous Media,” Transport inn Porous Media (1989) 4: 213-238.

11. Auriault J.L.: “Heterogeneous Medium. Is an Equivalent Macroscopic Description Possible?,” Int. J. Engng. Sci. (1991) 29: 785-795.

12. Durlofsky, L.J.: “Use of Higher Moments for the Description of Upscaled, Process Independent Relative Permeabilities,” SPE Journal (1997) 2: 474-484.

13. Efendief, Y.R. and Durlofsky, L.J.: “Accurate Subgrid Models for Two-Phase Flow in Heterogeneous Reservoirs,” SPE Journal (2004) 219-226.

14. Garcia, M.H., Journel, A.G. and Aziz, K.: “Automatic Grid Generation for Modeling Reservoir Heterogeneities,” SPERE (1992) 7: 278-284.

15. Durlofsky, L.J., Jones, R.C. and Milliken, W.J.: “A New Method for the Scale up of Displacement Processes in Heterogeneous Reservoir,” Proceeding of the 4th European Conference on the Mathematics of Oil Recovery, Roros, Norway (1994).

16. Li, D., Cullick, A.S. and Lake, L.W.: “Global scale-up of reservoir model permeability with local grid refinement,” Journal of Petroleum Science and Engineering (1995) 14: 1-13.

8 H. Mahani, A.H. Muggeridge SPE 94319

17. Durlofsky, L.J., Behren, R.A. and Jones, R.C.: “Scale Up of Heterogeneous Three Dimensional Reservoir Description,” SPE Journal (1996) 1: 313-326.

18. Qi, D.S., Wong, P.M. and Liu, K. Y.: “An improved global upscaling approach for reservoir simulation,” Petrol. Sci. Technol. (2001) 19: 779-795.

19. Qi, D.S., and Hesketh, T.: “REV grid technique for reservoir upscaling,” Petroleum Science and Technology (2004) 22: 1595-1624.

20. Soleng, H.H. and Holden, L.: “Gridding for Petroleum Reservoir Simulation,” In: Numerical Grid Generation in Computational Field Simulations, edited by Cross, M., Soni, B.K., Thompson, J.F., Hauser, J. and Eiseman, P.R., Missisipi State University (1998).

21. Meiburg, E. and Homsy, G.M.: “Vortex Methods for Porous Media Flows,” In: M.F. Wheeler (Editor), Numerical Simulation in Oil Recovery, the IMA Volumes in Mathematics and Its Applications. Springer-Verlag (1986) 199-225.

22. Chen, C.Y. and Meiburg, E.: “Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities,” Journal of Fluid Mechanics (1998) 371: 269-299.

23. Sposito, G.: “Steady Groundwater Flow as a Dynamical System,” Water Resources Research (1994) 30: 2395-2401.

24. Kapoor, V.: “Vorticity in Three-Dimensionality Random Porous Media,” Transport in Porous Media (1997) 26: 109-119.

25. White, C.D. and Horne, R.N.: “Computing Absolute Transmissibility in the Presence of Fine-Scale Heterogeneity,” paper SPE 16011 presented at the SPE Symposium on Reservoir Simulation, San Antonio, TX, February 1 – 4, 1987.

26. ECLIPSE 100 Reference Manual, Version 2003a, GeoQuest Reservoir Technologies (2003).

27. Begg, S. H., Carter, R.R. and Dranfield, P.: “Assigning Effective Values to Simulator Grid block Parameters for Heterogeneous Reservoirs,” SPE Reservoir Engineering (1989) 4: 455-463.

28. Muggeridge, A.H.: “Generation of Effective Relative Permeabilities from Detailed Simulation of Flow in Heterogeneous Porous Media,” In: H.B.C. Larry W. Lake, Thomas C. Wesson (Editor), Reservoir Characterization II, Academic Press Inc. (1991).

Appendix I - Flowchart of New Grid Coarsening Technique

Start

Permeability Model construction

Running single phase flow simulation

Vorticity map generation

Coarse grid generation Input: normalized vorticity map, cut-off value for vorticity, maximum size of coarse grids Output: Coarse grid model

Upscaling Permeability for new coarse grid using

pressure solver method

Multi-phase flow simulation: Fine & Coarse

Comparing fine and coarse grid results and quality of match. acceptable?

Change vorticity cut-off

Stop

Yes

No