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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
Article
  • 91 Downloads

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

Ebth

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

kr

Phase relative permeability

K = [kij ]

Absolute permeability tensor

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

Inverse of permeability tensor

Keff

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

qo

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

Qinj

Total rate of injection

Sw

Saturation (volume fraction) of water phase

t

Time

tbh

Breakthrough time

tsf

Final simulation time, t sflΔt 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

xi

Cartesian coordinates (i = 1,2,3)

Greek symbols

α

Thresholding parameter

\(\varepsilon \)

Convergence error

δij

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

Δti

Sampling time step (here it is constant)

Δx1

Fine grid block size in the x 1 direction

Δx2

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

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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References

  1. Andrew Oghena, B.E.: Quantification of uncertainties associated with reservoir performance simulation, MS thesis, Department of Petroleum Engineering, Texas Tech University (2007)Google Scholar
  2. Aziz, K., Settari, A.: Petroleum reservoir simulation. Applied Science, London (1979)Google Scholar
  3. Begg S.H., Carter R.R. and Dranfield P. (1989). Assigning effective values to simulator gridblocks parameters for heterogeneous reservoirs. SPE Reserv. Eng. 4: 455–464 Google Scholar
  4. Castellini A. (2001). Flow base grids for reservoir simulation MS dissertation. The university of California, StanfordGoogle Scholar
  5. Chen Y., Durlofsky L.J., Gerritsen M. and Wen X.H. (2003). A coupled local—global upscaling approach for simulating flow in highly heterogeneous formations. Adv. Water. Res. 26: 1041–1060 CrossRefGoogle Scholar
  6. Chen Y. and Durlofsky L.J. (2006). Adaptive local-global upscaling for general flow scenarios in heterogeneous formations. Transport Porous Med. 62: 157–185 CrossRefGoogle Scholar
  7. Chen C.-Y. and Meiburg E. (1998). Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities. J. Fluid Mech. 371: 269–299 CrossRefGoogle Scholar
  8. Christie M.A. and Blunt M.J. (2001). Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Reserv. Eval. Eng. 4: 308–317 Google Scholar
  9. De Josselin de Jong G. (1960). Singularity distribution for the analysis of multiple-fluid flow through porous media. J. Geophys. Res. 65(11): 3739–3758 CrossRefGoogle Scholar
  10. Durlofsky L.J. (1998). Coarse scale models of two-phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques. Comput. Geosci. 2: 73–92 CrossRefGoogle Scholar
  11. Durlofsky, L.J.: Upscaling of geocellular models for reservoir flow simulation: a review of recent progress, 7th international forum on reservoir simulation Bühl/Baden-Baden, Germany, June 23–27 (2003)Google Scholar
  12. Durlofsky L.J., Behrens R.A., Jones R.C. and Bernath A. (1996). Scale up of heterogeneous three dimensional reservoir descriptions. SPE J. 1: 313–326 Google Scholar
  13. Durlofsky L.J., Jones R.C. and Milliken W.J. (1997). A nonuniform coarsening approach for the scale up of displacement processes in heterogeneous media. Adv. Water Res. 20: 335–347 CrossRefGoogle Scholar
  14. Edwards, M.G., Rogers, C.F.: A flux continuous scheme for the full tensor pressure equation. Proceeding fourth European conference on the mathematics of oil recovery, Roros, Norway, 7–10 June (1994)Google Scholar
  15. Farmer C.L. (2002). Upscaling: a review. Int. J. Numer. Methods Fluid 40: 63–78 CrossRefGoogle Scholar
  16. Farmer, C.L., Heath, D.E., Moody, R.O.: A global optimization approach to grid generation, 11th SPE Symposium on Reservoir Simulation, Anaheim, CA, February (1991)Google Scholar
  17. Garcia M.H., Journal A.G. and Aziz K. (1992). Automatic grid generation for modeling reservoir heterogeneities. SPE Reserv. Eng. 7: 278–284 Google Scholar
  18. Gerritsen, M.G., Lambers, J.V., Mallison, B.T.: A variable and compact MPFA for transmissibility upscaling with guaranteed monotonicity, IAPCO, ECMOR X., 10th European Conference on the Mathematics of Oil Recovery, Amsterdam, Netherlands, September 4 – 7 (2006)Google Scholar
  19. Guzman, R.E., Giodano, D., Fayers, J., Godi, A., Aziz, K.: The use of dynamic pseudo functions in reservoir simulation, Proceeding fifth international forum on reservoir simulation, Muscat, Oman, December 10–14 (1994)Google Scholar
  20. Kapoor V. (1997). Vorticity in three-dimensionality random porous media. Transport Porous Med. 26: 109–119 CrossRefGoogle Scholar
  21. Lambers, J., Gerritsen, M.: An integration of multi-level local-global upscaling with grid adaptivity, paper SPE 97250, presented at SPE annual technical conference and exhibition, Dallas, October 2005Google Scholar
  22. Lambers, J.V., Gerritsen, M.G., Mallison, B.T.: Accurate local upscaling with variable compact multi-point transmissibility calculations. Comput. Geosci. doi: 10.1007/s10596-007-9068-4 (2008)
  23. Lee, S.H., Durlofsky, L.J., Lough, M.F., Chen, W.H.: Finite difference simulation of geologically complex reservoirs with tensor permeabilities. SPE Reserv. Eval. Eng. 567–574 (1998)Google Scholar
  24. Li D., Cullick A.S. and Lake L.W. (1995). Global scale-up of reservoir model permeability with local grid refinement. J. Petroleum Sci. Eng. 14: 1–13 CrossRefGoogle Scholar
  25. Lodder, R., Hieftje, G.: Quantile analysis: a method for characterizing data distributions. Appl. Spectrosc. 42(8), 1512–1520 (1988)CrossRefGoogle Scholar
  26. Mahani, H.: Upscaling and optimal coarse grid generation for the numerical simulation of two-phase flow in porous media, PhD thesis. Department of Earth Science and Engineering, Imperial College London, London (2005)Google Scholar
  27. Mahani, H., Muggeridge, A.H.: Improved coarse grid generation using vorticity. SPE paper 94319, presented at the 14th annual SPE/EAGE conference. Madrid, Spain, June 13–16 (2005)Google Scholar
  28. Meiburg E., Homsy G.M.: Vortex methods for porous media flows. Numerical simulation in oil recovery. In: Wheeler, M.F. (ed.), pp. 199–225. Springer-Verlag (1986)Google Scholar
  29. Qi D. and Hesketh T. (2005). An improved p-normal transformation approach in reservoir upscaling. Petroleum Sci. Technol. 23: 1291–1302 CrossRefGoogle Scholar
  30. Riaz A. and Meiburg E. (2002). Three-dimensional vorticity dynamics of miscible porous media flows. J. Turbulence 3: 061 CrossRefGoogle Scholar
  31. Renard Ph. and de Marsily G. (1997). Calculating effective permeability: a review. Adv. Water Res. 20: 253–278 CrossRefGoogle Scholar
  32. Robey T.H. (1995). An adaptive grid technique for minimizing heterogeneity of cells or elements. Math. Geol. 27(6): 706–729 CrossRefGoogle Scholar
  33. Sposito G. (1994). Steady groundwater flow as a dynamical system. Water Resour. Res. 30(8): 2395–2401 CrossRefGoogle Scholar
  34. Sposito G. (2001). Topological groundwater hydrodynamics. Adv. Water Resour. 24: 793–801 CrossRefGoogle Scholar
  35. Tran, T.T.: Stochastic simulation of permeability field and their scale-up for flow modeling. PhD thesis, Stanford University, CA (1995)Google Scholar
  36. Wallstrom T.C., Hou S., Christie M.A., Durlofsky L.J. and Sharp D.H. (1999). Accurate scale up of two-phase flow using renormalization and non-uniform coarsening. Comput. Geosci. 3: 69–87 CrossRefGoogle Scholar
  37. White, C.D., Horne, R.N.: Computing absolute transmissibility in the presence of fine-scale heterogeneity, SPE Paper 16011 presented at ninth symposium on reservoir simulation, pp. 209–220. San Antonio, Texas (1987)Google Scholar
  38. Wen X.H. and Gomez-Hernandez J.J. (1996). Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. 183: ix–xxxii CrossRefGoogle Scholar
  39. Wen X.H., Durlofsky L.J. and Edwards M.G. (2003a). Upscaling of channel systems in two dimensions using flow-based grids. Transport Porous Med. 51: 343–366 CrossRefGoogle Scholar
  40. Wen X.H., Durlofsky L.J. and Edwards M.G. (2003). Use of border regions for improved permeability upscaling. Math. Geol. 35(5): 521–547 CrossRefGoogle Scholar

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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