Transport in Porous Media

, Volume 68, Issue 2, pp 153–173

# A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—part I. Theoretical development and asymptotic analyses

• Meng Lu
• Luke D. Connell
Article

## Abstract

In this paper a rigorous dual-porosity model is formulated, which accurately represents the coupling between large-scale fractures and the micropores within dual porosity media. The overall structure of the porous medium is conceptualized as being blocks of diffusion dominated micropores separated by natural fractures (e.g. cleats for coal) through which Darcy’s flow occurs. In the developed model, diffusion in the matrix blocks is fully coupled to the pressure distribution within the fracture system. Specific assumptions on the pressure behaviour at the matrix boundary, such as step-time function employed in some earlier studies, are not invoked. The model involves introducing an analytical solution for diffusion within a matrix block, and the resultant combined flow equation is a nonlinear integro-(partial) differential equation. Analyses to the equation in this text, in addition to the theoretical development of the proposed model, include: (1) discussion on the “fading memory” of the model; (2); one-dimensional perturbation solution subject to a specific condition; and (3) asymptotic analyses of the “long-time” and “short-time” responses of the flow. Two previous models, the Warren-Root and the modified Vermeulen models, are compared with the proposed model. The advantages of the new model are demonstrated, particularly for early time prediction where the approximations of these other models can lead to significant error.

## Keywords

Porous media Dual-porosity Adsorption Permeability Diffusion Asymptotic analysis Integro-differential equation Gas Reservoir Sequestration of CO2

## References

1. Aziz K. and Settari A. (2002). Petroleum Reservoir Simulation. Blitzprint Ltd, Canada, Calgrary, Alberta Google Scholar
2. Barenblatt G.I. and Zheltov Yu.P. (1960). Fundamental equations of filtration of homogeneous liduids in fissured rocks. J. Appl. Math. Engl. Transl. 24: 1286–1303 Google Scholar
3. Carslaw H.S. and Jaeger J.C. (1959). Conduction of Heat in Solids. Clarendon Press, Oxford Google Scholar
4. Coleman B.D. (1964). Thermodynamics of materials with memory. Arch. Ration. Mech. An. 17: 1–46 Google Scholar
5. De Swann A. (1977). Analytical solutions for determining naturally fractured reservoir properties by well testing. Soc. Petrol. Eng. J. 16: 117–122 Google Scholar
6. Dykhuizen R.C. (1990). A new coupling term for dual-porosity models. Water Resour. Res. 26: 351–356
7. Gasem, K.A.M., Pan, Z., Fitzgerald, J.E., Sudibandryio, M.: Adsorption modeling update. Present at the Second International Forum on Geological Sequestration of CO2 in deep unmineable coalseams (Coal-Seq II), Wahsington DC, 6–7 March (2003)Google Scholar
8. Gray, I.: Reservoir engineering in coal seams: part I—the physical process of gas storage and movement in coal seams. SPE Reservoir Engineering (February), pp. 28–34 (1987)Google Scholar
9. Kevorkian J. and Cole J.D. (1981). Perturbation Method in Applied Mathematics. Springe-Verlag, New York Google Scholar
10. King, G.R., Ertekin, T., Schwerer, F.C.: Numerical simulations of the transient behaviour of coal-seam degasification wells, SPE formulation Evaluation, pp. 165–183 (1986)Google Scholar
11. Kolesar, J.E., Ertekin, T., Obut, S.T.: The unsteady-state nature of sorption and diffusion phenomena in the micropore structure of coal: part 1—theory and mathematical formulations, SPE Formation Evaluation, pp. 81–88 (1990a)Google Scholar
12. Kolesar, J.E., Ertekin, T., Obut, S.T.: The unsteady-state nature of sorption and diffusion phenomena in the micropore structure of coal: part 2—solutions, SPE Formation Evaluation, pp. 89–97 (1990b)Google Scholar
13. Law. D.H.-S.: Numerical model comparison study for greenhouse gas sequestration in coalbes—an undate. Present at the 2nd International Forum on Geological Sequestration of CO2 in Deep Unmineable Coalseams (Coal-Seq II), Washington DC, 6–7 March (2002)Google Scholar
14. Law, D.H.-S., van der Meer, L.G.H.: Numerical simulator comparison study for enhanced coalbed methane recovery processes, part I: pure carbon dioxide injection, SPE Gas Technology Symposium. Calary, Alberta, Canada, 20 April–2 May (2002)Google Scholar
15. Li J.C. and Zhou X.C. (2002). Asymptotic Methods in Mathematical Physics. Science Press, Beijing Google Scholar
16. Lim K.T. and Aziz K. (1995). Matrix-fracture transfer shape factors for dual-porosity simulators. J. Petrol. Sci. Eng. 13: 169–178
17. Najurieta H.L. (1980). Theory for pressure transient analysis of naturally fractured reservoirs. J. Petrol. Technol. 32: 1241–1250 Google Scholar
18. Nayfey A.H. (1973). Perturbation Methods. John Wiley, New York Google Scholar
19. Özisik M.N. (1968). Boundary Value Problems of Heat Conduction. International Textbook Company, Scranton, Pennsylvania Google Scholar
20. Pruess, K., Narasimham, T.N.: A practical method for modelling fluid and heat flow in fractured porous media. SPE J. 14–26 (1985)Google Scholar
21. Pruess, K., Wu Y.-S.: A new semi-analytical method for numerical simulation of fluid and heat flow in fractured reservoirs. SPE-18426 Society of Petroleum Engineers, Dallas, USA (1989)Google Scholar
22. Saulsberry J.L., Schafer P.S. and Schraufnagel R.A. (1996). A Guide to Coalbed Methane Reservoir Engineering. Gas Research Institute, Chicago, USA Google Scholar
23. Shi J.Q. and Durucan S. (2003). A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel 82: 1219–1229
24. Vermeulen T. (1953). Theory of irresversible and constant-pattern solid diffusion. Ind. Eng. Chem. 45: 1664–1670
25. Warren J.E. and Root P.J. (1963). The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J. 3: 235–255 Google Scholar
26. Yang R.T. (1987). Gas Separation by Adsorption Processes. Imperial College Press, UK Google Scholar
27. Zimmerman R.W. and Bodvarsson G.S. (1989). Integral method solution for diffusion into a spherical block. J. Hydrology 111: 213–224
28. Zimmerman R.W., Chen G., Hadgu T. and Bodvarsson G.S. (1993). A numerical dual-porosity model with semianalytical treatment of fracture/matrix flow. Water Resour. Res. 29: 2127–2137