Upscaling Algorithm for Heterogeneous Porous Media by Using
the Finite Analytic Method

doi:10.11885/j.issn.1674-5086.2015.03.04.06

Abstract

Abstract:

An algorithm for solving upscaled tensor permeability by using the finite analytic method is proposed，which is
suitable for the flow field described by the linear Darcy′s law. Adoptingthe double periodic boundary condition proposed by
Durlofsky，the proposed algorithm can provide the upscaled full tensor permeability with high efficiency. Only with 2×2 or
3×3 subdivisions，the proposed algorithm can provide rather accurate solutions，where the convergent speed is independent of
the permeability heterogeneity. In contrast，when using the traditional numerical schemes to simulate flow through a strong
heterogeneous porous medium，the refinement ratio for the grid cell needs to increase dramatically to get an accurate result.

Key words: seepage, reservoir simulation, heterogeneous porous meadia, upscaling for permeability, grid mesh

Liu Zhifan*, Liu Zhifeng, Wang Xiaohong. Upscaling Algorithm for Heterogeneous Porous Media by Using
the Finite Analytic Method[J]. 西南石油大学学报(自然科学版), DOI: 10.11885/j.issn.1674-5086.2015.03.04.06.

0
/ / Recommend

Add to citation manager EndNote|Ris|BibTeX

URL: http://journal15.magtechjournal.com/Jwk_xnzk/EN/10.11885/j.issn.1674-5086.2015.03.04.06

http://journal15.magtechjournal.com/Jwk_xnzk/EN/Y2015/V37/I3/109

References

[1] Cardwell W T，Parsons R L. Average permeabilities of

heterogeneous oil sands[J]. AIME，Transactions of the

AIME，1945，160（1）：34 42.

[2] Shah N，Ottino J M. Effective transport properties of

disordered，multi-phase composites：Application of realspace

renormalization group theory[J]. Chemical Engineering

Science，1986，41（2）：283 296.

[3] King P R. The use of renormalization for calculating

effective permeability[J]. Transport in Porous Media，

1989，4（1）：37 58.

[4] Gautier Y，Noetinger B. Preferential flow-paths detection

for heterogeneous reservoirs using a new renormalization

technique[J]. Transport in Porous Media，1997，26（1）：

1 23.

[5] Sanchez-Palencia E. Non homogeneous media and vibration

theory[M]. Lecture Notes in Physics 127，Springer，

1980.

[6] Bourgeat A. Homogenized behavior of two-phase flows in

naturally fractured reservoirs with uniform fractures distribution[

J]. Computer Methods in Applied Mechanics and

Engineering，1984，47（1）：205 216.

[7] Whitaker S. Flow in porous media I：A theoretical derivation

of Darcy’s law[J]. Transport in Porous Media，1986，

1（1）：3 25.

[8] Quintard M，Whitaker S. Ecoulement monophasique en

milieu poreux：Effet des hétérogénéités locales[J]. Journal

de mécanique théorique et appliquée，1987，6（5）：

691 726.

[9] Dagan G. Flow and transport in porous formations[M].

Springer，1989.

[10] Wen Xianhua，Gómez-Hernández J J. Upscaling hydraulic

conductivities in heterogeneous media：An overview[J].

Journal of Hydrology，1996，183（1 2）：ix xxxii.

[11] Renard P，De Marsily G. Calculating equivalent permeability：

A review[J]. Advances in Water Resources，1997，

20（5）：253 278.

[12] Begg S H，Carter R R. Assigning elective values to simulator

grid-block parameters for heterogeneous reservoirs[C].

SPE 16754，1987.

[13] Durlofsky L J. Numerical calculation of equivalent grid

block permeability tensors for heterogeneous porous

media[J]. Water Resources Research，1991，27（5）：

699 708.

[14] Aziz K，Setttari A. Petroleum reservoir simulation[M].

Applied Science，1979.

[15] Yeo I W，Zimmerman R W. Accuracy of the renormalization

method for computing effective conductivities of heterogeneous

media[J]. Transport in Porous Media，2001，

45（1）：129 138.

[16] Romeu R K，Noetinger B. Calculation of intermodal transmissibilities

in finite difference models of flow in heterogeneous

porous media[J]. Water Resour. Res.，1995，31：

943 959.

[17] Liu Z F，Wang X H. Finite analytic numerical method

for two-dimensional fluid flow in heterogeneous porous

media[J]. Journal of Computational Physics，2013，235：

286 301.