Skip to main content Accessibility help
×
Home
Hostname: page-component-558cb97cc8-mjrxc Total loading time: 1.432 Render date: 2022-10-06T11:18:23.417Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": true, "useSa": true } hasContentIssue true

Flow regimes for fluid injection into a confined porous medium

Published online by Cambridge University Press:  24 February 2015

Zhong Zheng
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Bo Guo
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Ivan C. Christov
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Michael A. Celia
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
Email address for correspondence: hastone@princeton.edu

Abstract

We report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behaviour is summarized in a diagram with five distinct dynamical regimes: a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

Type
Papers
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acton, J. M., Huppert, H. E. & Worster, M. G. 2001 Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440, 359380.CrossRefGoogle Scholar
Barenblatt, G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.CrossRefGoogle Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Boussinesq, J. V. 1904 Recherches theoretique sur l’ecoulement des nappes d’eau infiltrees dans le sol et sur le debit des sources. J. Math. Pures Appl. 10, 578.Google Scholar
Gasda, S. E., Bachu, S. & Celia, M. A. 2004 Spatial characterization of the location of potentially leaky wells penetrating a deep saline aquifer in a mature sedimentary basin. Environ. Geol. 46, 707720.CrossRefGoogle Scholar
Gunn, I. & Woods, A. W. 2011 On the flow of buoyant fluid injected into a confined, inclined aquifer. J. Fluid Mech. 672, 109129.CrossRefGoogle Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 7191.CrossRefGoogle Scholar
Hesse, M. A., Orr, F. M. Jr & Tchelepi, H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 3560.CrossRefGoogle Scholar
Hesse, M. A., Tchelepi, H. A., Cantwell, B. J. & Orr, F. M. Jr 2007 Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363383.CrossRefGoogle Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271311.CrossRefGoogle Scholar
Huppert, H. E. 1982a Flow and instability of a viscous current down a slope. Nature 300, 427429.CrossRefGoogle Scholar
Huppert, H. E. 1982b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.CrossRefGoogle Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 5569.CrossRefGoogle Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241282.CrossRefGoogle Scholar
Lake, L. W. 1989 Enhanced Oil Recovery. Prentice Hall.Google Scholar
Lenormand, R., Touboul, E. & Zarcone, C. 1988 Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165187.CrossRefGoogle Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631653.CrossRefGoogle Scholar
Lyle, S., Huppert, H. E., Hallworth, M., Bickle, M. & Chadwick, A. 2005 Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293302.CrossRefGoogle Scholar
MacMinn, C. W., Szulczewski, M. L. & Juanes, R. 2010 $\text{CO}_{2}$ migration in saline aquifers. Part 1. Capillary trapping under slope and groundwater flow. J. Fluid Mech. 662, 329351.CrossRefGoogle Scholar
MacMinn, C. W., Szulczewski, M. L. & Juanes, R. 2011 $\text{CO}_{2}$ migration in saline aquifers. Part 2. Combined capillary and solubility trapping. J. Fluid Mech. 688, 321351.CrossRefGoogle Scholar
Metz, B., Davidson, O., De Connick, H., Loos, M. & Meyer, L. 2005 IPCC Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press.Google Scholar
Neufeld, J. A., Vella, D., Huppert, H. E. & Lister, J. R. 2011 Leakage from gravity currents in a porous medium. Part 1. A localized sink. J. Fluid Mech. 666, 391413.CrossRefGoogle Scholar
Nordbotten, J. M. & Celia, M. A. 2006 Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307327.CrossRefGoogle Scholar
Pegler, S. S., Huppert, H. E. & Neufeld, J. A. 2014 Fluid injection into a confined porous layer. J. Fluid Mech. 745, 592620.CrossRefGoogle Scholar
Pritchard, D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415431.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95110.CrossRefGoogle Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312329.CrossRefGoogle Scholar
Saha, S., Salin, D. & Talon, L. 2013 Low Reynolds number suspension gravity currents. Eur. Phys. J. E 36, 10385.CrossRefGoogle ScholarPubMed
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 134.CrossRefGoogle Scholar
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012 Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 696, 175214.CrossRefGoogle Scholar
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 135.CrossRefGoogle Scholar
Vella, D. & Huppert, H. E. 2006 Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555, 353362.CrossRefGoogle Scholar
Verdon, J. & Woods, A. W. 2007 Gravity-driven reacting flows in a confined porous aquifer. J. Fluid Mech. 588, 2941.CrossRefGoogle Scholar
Yortsos, Y. C. & Salin, D. 2006 On the selection principle for viscous fingering in porous media. J. Fluid Mech. 557, 225236.CrossRefGoogle Scholar
Zheng, Z., Christov, I. C. & Stone, H. A. 2014 Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents. J. Fluid Mech. 747, 218246.CrossRefGoogle Scholar
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558568.CrossRefGoogle Scholar
47
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Flow regimes for fluid injection into a confined porous medium
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Flow regimes for fluid injection into a confined porous medium
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Flow regimes for fluid injection into a confined porous medium
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *