Framework for Coupling Flow and Deformation of a Porous Solid
Publication: International Journal of Geomechanics
Volume 15, Issue 5
Abstract
In this paper, the flow of an incompressible fluid in a deformable porous solid is considered. A mathematical model using the framework offered by the theory of interacting continua is presented. In its most general form, this framework provides a mechanism for capturing multiphase flow, deformation, chemical reactions, and thermal processes, as well as interactions between the various physics, in a conveniently implemented fashion. To simplify the presentation of the framework, the results are presented for a particular model, which can be seen as an extension of Darcy’s equation (which assumes that the porous solid is rigid) and that takes into account the elastic deformation of the porous solid. The model also considers the effect of deformation on porosity. It is shown that by using this model identical results can be recovered as in the framework proposed in the literature. Some salient features of the framework are as follows: (1) it is a consistent mixture theory model, and adheres to the laws and principles of continuum thermodynamics; (2) the model is capable of simulating various important phenomena, such as consolidation and surface subsidence; and (3) the model is amenable to several extensions. Numerical coupling algorithms used to obtain a coupled flow-deformation response are also presented. Several representative numerical examples are presented to illustrate the capability of the mathematical model and the performance of the computational framework.
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Acknowledgments
This research was supported in part by Sandia National Laboratories through the Laboratory Directed Research and Development program (Contract No. C11-00239). Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. K. B. N. also acknowledges the support of the National Science Foundation under Grant No. CMMI 1068181. M. J. M was supported in part by the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DESC0001114. The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the sponsors.
References
Atkin, R. J., and Craine, R. E. (1976). “Continuum theories of mixtures: Basic theory and historical development.” Q. J. Mech. Appl. Math., 29(2), 209–244.
Bachu, S. (2003). “Screening and ranking of sedimentary basins for sequestration of in geological media in response to climate change.” Environ. Geol., 44(3), 277–289.
Barus, C. (1893). “Isotherms, isopiestics and isometrics relative to viscosity.” Am. J. Sci., 45(266), 87–96.
Bedford, A., and Drumheller, D. S. (1983). “Theories of immiscible and structured mixtures.” Int. J. Eng. Sci., 21(8), 863–960.
Benson, S. M., and Cole, D. R. (2008). “ sequestration in deep sedimentary formations.” Elements, 4(5), 325–331.
Berli, M., and Or, D. (2006). “Deformation of pores in viscoplastic soil material.” Int. J. Geomech., 108–118.
Bobeck, P. (2004). The public fountains of the city of Dijon, Kendall Hunt, Dubuque, IA.
Bowen, R. M. (1976). “Theory of mixtures.” Continuum physics, A. C. Eringen, ed., Vol. III, Academic, New York.
Bowen, R. M. (2010). “Porous elasticity: Lectures on the elasticity of porous materials as an application of the theory of mixtures.” Texas A&M Univ., College Station, TX, 〈http://repository.tamu.edu/handle/1969.1/2500〉 (Nov. 14, 2011).
Bridgman, P. W. (1931). The physics of high pressure, MacMillan, New York.
Brinkman, H. C. (1947). “A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles.” Appl. Sci. Res., 1(1), 27–34.
Coussy, O. (2004). Poromechanics, Wiley, New York.
Coussy, O. (2010). Mechanics and physics of porous solids, Wiley, New York.
Darcy, H. (1856). Les fontaines publiques de la ville de Dijon, Victor Dalmont, Paris.
Dean, R. H., Gai, X., Stone, C. M., and Minkoff, S. E. (2003). “A comparison of techniques for coupling porous flow and geomechanics.” Proc., SPE Reservoir Simulation Symp., Vol. 11, Society of Petroleum Engineers, Richardson, TX, 132–140.
de Boer, R. (2000). Theory of porous media: Highlights in the historical development and current state, Springer, New York.
de Boer, R., Ehlers, W., and Liu, Z. (1993). “One-dimensional transient wave propagation in fluid-saturated incompressible porous media.” Arch. Appl. Mech., 63(1), 59–72.
de Lyra Nogueira, C., de Azevedo, R. F., and Zornberg, J. G. (2009). “Coupled analyses of excavations in saturated soil.” Int. J. Geomech., 73–81.
Diebels, S., and Ehlers, W. (1996). “Dynamic analysis of a fully saturated porous medium accounting for geometrical and material non-linearities.” Int. J. Numer. Methods Eng., 39(1), 81–97.
Fick, A. (1855). “Ueber diffusion.” Ann. Phys., 170(1), 59–86.
Forsyth, P. A. (1990). “A control-volume, finite-element method for local mesh refinement in thermal reservoir simulation.” SPE Reservoir Eng., 5(04), 561–566.
Forsyth, P. A. (1991). “A control volume finite element approach to NAPL groundwater contamination.” SIAM J. Sci. Stat. Comput., 12(5), 1029–1057.
Fuego. (2008). “SIERRA/Fuego 2.7 user’s manual.” Rep. SAND 2006-6084P, Sandia National Laboratories, Albuquerque, NM.
Helmig, R. (1997). Multiphase flow and transport processes in the subsurface: A contribution to the modeling of hydrosystems, 2nd Ed., Springer, Berlin.
Lax, P. D. (2002). Functional analysis, Wiley, New York.
Leach, A., Mason, C. F., and van’t Veld, K. (2009). “Co-optimization of enhanced oil recovery and carbon sequestration.” NBER Working Paper No. 15035, National Bureau of Economic Research, Cambridge, MA, 〈http://www.nber.org/papers/w15035〉 (Jan. 1, 2010).
Macfarlane, A. M. (2007). “Energy: The issue of the 21st century.” Elements, 3(3), 165–170.
Nair, R., Abousleiman, Y., and Zaman, M. (2005). “Modeling fully coupled oil–gas flow in a dual-porosity medium.” Int. J. Geomech., 326–338.
Nakshatrala, K. B., and Rajagopal, K. R. (2011). “A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium.” Int. J. Numer. Methods Fluids, 67(3), 342–368.
Notz, P., Subia, S. R., Hopkins, M., Moffat, H. K., and Noble, D. R. (2007). “Aria 1.5 user’s manual.” Technical Rep. SAND2007-2734, Sandia National Laboratories, Albuquerque, NM.
Rajagopal, K. R., and Tao, L. (1995). Mechanics of mixtures, World Scientific, River Edge, NJ.
Schrag, D. P. (2007). “Confronting the climate-energy challenge.” Elements, 3(3), 171–178.
Truesdell, C. (1957a). “Sulla basi della thermomechanica.” Rendiconti Lincei, 22(8), 33–38.
Truesdell, C. (1957b). “Sulla basi della thermomechanica.” Rendiconti Lincei, 22(8), 158–166.
Truesdell, C. (1984). Rational thermodynamics, Springer, New York.
Truesdell, C., and Toupin, R. A. (1960). “The classical field theories.” Handbuch der physik, Vol. III/1, Springer, New York, 226–793.
Voyiadjis, G. Z., and Song, C. R. (2006). The coupled theory of mixtures in geomechanics with applications, Springer, New York.
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Published In
International Journal of Geomechanics
Volume 15 • Issue 5 • October 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Nov 14, 2013
Accepted: Apr 22, 2014
Published online: Jun 5, 2014
Published in print: Oct 1, 2015
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