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Numerical Calculation of Equivalent Permeability Tensor for Fractured Vuggy Porous Media Based on Homogenization Theory

Published online by Cambridge University Press:  20 August 2015

Zhaoqin Huang ,
Jun Yao ,
Yajun Li ,
Chenchen Wang  and
Xinrui Lv
Zhaoqin Huang*
Affiliation:
School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266555, China
Jun Yao*
Affiliation:
School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266555, China
Yajun Li*
Affiliation:
School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266555, China
Chenchen Wang*
Affiliation:
School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266555, China
Xinrui Lv*
Affiliation:
School of Petroleum Engineering, China University of Petroleum (Huadong), Qingdao 266555, China
*
Corresponding author.Email:huangzhqin@gmail.com
Email:yaojunhdpu@126.com
Email:yajunok@126.com
Email:wcc1220@163.com
Email:xinruiabc@gmail.com
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Abstract

A numerical procedure for the evaluation of equivalent permeability tensor for fractured vuggy porous media is presented. At first we proposed a new conceptual model, i.e., discrete fracture-vug network model, to model the realistic fluid flow in fractured vuggy porous medium on fine scale. This new model consists of three systems: rock matrix system, fractures system, and vugs system. The fractures and vugs are embedded in porous rock, and the isolated vugs could be connected via discrete fracture network. The flow in porous rock and fractures follows Darcy’s law, and the vugs system is free fluid region. Based on two-scale homogenization theory, we obtained an equivalent macroscopic Darcy’s law on coarse scale from fine-scale discrete fracture-vug network model. A finite element numerical formulation for homogenization equations is developed. The method is verified through application to a periodic model problem and then is applied to the calculation of equivalent permeability tensor of porous media with complex fracture-vug networks. The applicability and validity of the method for these more general fractured vuggy systems are assessed through a simple test of the coarse-scale model.

Keywords

35B2735Q3076S05Fractured vuggy porous mediaequivalent permeability tensorhomogenizationupscaling
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 1 , January 2011 , pp. 180 - 204
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Lucia, F. J., Carbonate Reservoir Characterization, Springer-Verlag Berlin Heidelbert, 2007.Google Scholar
[2]Eezeybek, S. and Akin, S., Pore netwpork modeling of multiphase flow in fissured and vuggy carbonates, SPE paper 113384, presented at the 2008 SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, USA, 19-23 April, 2008.Google Scholar
[3]Arbogast, T. and Lehr, L. H., Homogenization of a Darcy-Stokes system modeling vuggy porous media, Comput. Geosci., 10(3) (2006), 291–302.CrossRefGoogle Scholar
[4]Gulbransen, A. F., Hauge, V. L., and Lie, K. A., A multiscale mixed finite-element method for vuggy and naturally-fractured reservoirs, SPE paper 119104, presented at the 2009 SPE Reservoir Simulation Symposium, Woodlands, Texas, USA, 2-4 February, 2009.Google Scholar
[5]Arbogast, T., Brunson, D. S., Bryant, S. L.et al., A preliminary computational investigation of a macro-model for vuggy porous media, In Computational Methods in Water Resources XV, New York, Elsevier, 2004.Google Scholar
[6]Neale, G. H. and Nader, W. K., The permeability of a uniformly vuggy porous medium, SPE J., 13(2) (1973), 69–74.Google Scholar
[7]Arbogast, T. and Brunson, D. S., A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium, Comput. Geosci., 11 (2007), 207–218.CrossRefGoogle Scholar
[8]Arbogast, T. and Gomez, M. S. M., A discretization and multigrid solver for aDarcy-Stokes system of three dimensional vuggy porous media, Comput. Geosci., 13 (2009), 331–348.CrossRefGoogle Scholar
[9]Popov, P., Qin, G., Bi, L., et al., Multiscale methods for modeling fluid flow through naturally fractured carbonate karst reservoirs, SPE paper 110778, presented at the 2007 SPE Annual Technical Conference and Exhibition, Anaheim, California, USA, 11-14 November, 2007.Google Scholar
[10]Popov, P., Bi, L. F., Efendiev, Y., et al., Multiphysics and multiscale methods for modeling fluid flow through naturally fractured carbonate reservoirs,SPE paper 105378, presented at the 15th SPE Middle East Oil & Gas Show and Conference, Bahrain, 11-14 March, 2007.Google Scholar
[11]Popov, P., Efendiev, Y., and Qin, G., Multiscale Modeling and Simulations of Flows in Naturally Fractured Karst Reservoirs, Commun. Comput. Phys., 6(1) (2009), 162–184.Google Scholar
[12]Kang, Z. J., Wu, Y. S., , J.L, et al., Modeling multiphase flow in naturally fractured vuggy petroleum reservoirs, SPE paper 102356, presented at the 2006 SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 24-27 September, 2006.Google Scholar
[13]Wu, Y. S., Qin, G., Ewing, R. E., et al., A multiple-continuum approach for modeling multiphase flow in naturally fractured vuggy petroleum reservoirs, SPE paper 104173, presented at the 2006 SPE International oil & Gas Conference and Exhibition, Beijing, China, 5-7 December, 2006.CrossRefGoogle Scholar
[14]Wu, Y. S., Qin, G., Kang, Z. J., et al., A triple-continuum pressure-transient model for a naturally fractured vuggy reservoir. SPE paper 110044, presented at the 2007 SPE Annual Technical Conference and Exhibition, Anaheim, California, USA, 11-14 November, 2007.Google Scholar
[15]Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method Volume 3: Fluid Dynamics, New York, McGraw-Hill, 2000.Google Scholar
[16]Reddy, J. N. and Gartling, D. K., The Finite Element Method in Heat transfer and Fluid dynamics, New York, CRC, 2000.Google Scholar
[17]Donea, J. and Huerta, A., Finite Element Methods for Flow Problems, England, John Wiley & Sons Ltd, 2003.CrossRefGoogle Scholar
[18]Witherspoon, P. A., Wang, J. S. Y., Iwai, K., and Gale, J. E., Validity of Cubic Law for Fluid Flow in a Deformable Rock Fracture, Water Resour. Res., 16(6) (1980), 1016–1024.Google Scholar
[19]Kim, J. and Deo, M., Finite element discrete fracture model for multiphase flow in porous media, AIChE J., 46(6) (2000), 1120–1130.CrossRefGoogle Scholar
[20]Karimi-Fard, M. and Firoozabadi, A., Numerical simulation of water injection in 2D fractured media using discrete-fracture model, SPE J., 4 (2003), 117–126.Google Scholar
[21]Hoteita, H. and Firoozabadi, A., An efficient numerical model for incompressible two-phase flow in fractured media, Adv. in Water Resour., 31 (2008), 891–905.Google Scholar
[22]Beavers, G. S. and Joseph, D. D., Boundary condition at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197–207.CrossRefGoogle Scholar
[23]Beavers, G. S., Sparrow, E. M., and Magnuson, R. A., Experiments on coupled parallel flows in a channel and a bounding porous medium, J. Basic Eng., 92 D (1970), 843–848.Google Scholar
[24]Saffman, P. G., On the boundary condition at the surface of a porous medium, Stud. Appl. Math., 1 (1971), 93–101.CrossRefGoogle Scholar
[25]Jones, I. P., Low Reynolds number flow past a porous spherical shell, Cambridge Philosophical Society, Proceedings, 73 (1973), 231–238.Google Scholar
[26]Taylor, G. I., A model for the boundary conditions of a porous material, Part 1. J. Fluid Mech., 49 (1971), 319–326.Google Scholar
[27]Dagan, G., Generalization of Darcy law for nonuniform flows, Water Resour. Res., 15 (1979), 1–7.CrossRefGoogle Scholar
[28]Sahraoui, M. and Kaviany, M., Slip and no-slip velocity boundary conditions at interface of porous, plain media, Int. J. Heat and Mass Transfer, 35 (1992), 927–944.CrossRefGoogle Scholar
[29]Benssousan, A., Lions, J. L., and Papanicoulau, G., Asymptotic Analysis for Periodic Structures, Amesterdam, North Holland, 1978.Google Scholar
[30]Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics, Springer-Verlag Berlin, 1980.Google Scholar
[31]Oleinik, O. A., On homogenization problems, Trends and Applications of Pure Mathematics to Mechanics, Springer Berlin, 1984.Google Scholar
[32]Zhikov, V. V, Kozlov, S. M., and Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag Berlin, 1994.Google Scholar
[33]Hornung, U., Homogenization and Porous Media, Springer New York, 1997.Google Scholar
[34]Salinger, A., Aris, R. and Derby, J., Finite element formulations for large-scale, coupled flows in adjacent porous and open fluid domains, Int. J. Numer. Meth. Fluids, 18(12) (1994), 1185–1209.CrossRefGoogle Scholar
[35]Gartling, D. K., Hickox, C. E., and Givler, R. C., Simulation of coupled viscous and porous flow problems, Comput. Fluid Dyn., 7(1) (1996), 23–48.Google Scholar
[36]Layton, W. J., Schieweck, F., and Yotov, I., Coupling fluid flow with porous media flow, SIA M. J. Numer. Anal., 40 (2003), 2195–2218.Google Scholar
[37]Tan, H. and Pillai, K. M., Finite element implementation of stress-jump and stress-continuity conditions at porous-medium, clear-fluid interface, Computers and Fluids, 38 (2009), 1118–1131.Google Scholar