Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-20T15:26:01.310Z Has data issue: false hasContentIssue false
  • English
  • Français

CRITICAL SURFACE CONING DUE TO A LINE SINK IN A VERTICAL DRAIN CONTAINING A POROUS MEDIUM

Published online by Cambridge University Press:  19 July 2019

S. AL-ALI Open the ORCID record for S. AL-ALI [Opens in a new window] ,
G. C. HOCKING Open the ORCID record for G. C. HOCKING [Opens in a new window]  and
D. E. FARROW Open the ORCID record for D. E. FARROW [Opens in a new window]
S. AL-ALI
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, WA, Australia email [email protected], [email protected], [email protected] Mathematics Department, College of Sciences and Mathematics, Tikrit University, Saladin, Iraq
G. C. HOCKING*
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, WA, Australia email [email protected], [email protected], [email protected]
D. E. FARROW
Affiliation:
Mathematics and Statistics, Murdoch University, Perth, WA, Australia email [email protected], [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The withdrawal of water with a free surface through a line sink from a two-dimensional, vertical sand column is considered using the hodograph method and a novel spectral method. Hodograph solutions are presented for slow flow and for critical, limiting steady flows, and these are compared with spectral solutions to the steady problem. The spectral method is then extended to obtain unsteady solutions and hence the evolution of the phreatic surface to the steady solutions when they exist. It is found that for each height of the interface there is a unique critical coning value of flow rate, but also that the value obtained is dependent on the flow history.

Keywords

coningline sinkporous mediafree surface
Type
Research Article
Information
The ANZIAM Journal , Volume 61 , Issue 3 , July 2019 , pp. 249 - 269
Copyright
© 2019 Australian Mathematical Society 

References

Bear, J., Dynamics of fluids in porous media (McGraw-Hill, New York, 1972).Google Scholar
Bear, J. and Dagan, G., “Some exact solutions of interface problems by means of the hodograph method”, J. Geophys. Res. 69 (1964) 1563–1572; doi:10.1029/JZ069i008p01563.Google Scholar
Blake, J. R. and Kucera, A., “Coning in oil reservoirs”, Math. Sci. 13 (1988) 36–47; http://www.appliedprobability.org/data/files/TMS%20articles/13_1_6.pdf.Google Scholar
Childs, E. C., “A treatment of the capillary fringe in the theory of drainage”, J. Soil Sci. 10 (1959) 83–100; doi:10.1111/j.1365-2389.1959.tb00668.x.Google Scholar
Darcy, H., Fontaines publiques de la ville de Dijon (ed. Dalmont, V.), (Libraire des Corps imperiaux des ponts et chaussées et des mines, 1856).Google Scholar
Forbes, L. K. and McCue, S. W., “Optimal fluid injection-strategies for in situ mineral leaching in two-dimensions”, J. Eng. Math. 36 (1999) 185–206; doi:10.1023/A:1004406311441.Google Scholar
Giger, F. M., “Analytic 2-D models of water cresting before breakthrough for horizontal wells”, SPE Res. Eng. 4 (1989) 409–416; doi:10.2118/15378-PA.Google Scholar
Hinch, E. J., “The recovery of oil from underground reservoirs”, J. Physico-Chem. Hydrodynamics 6 (1985) 601–622; doi:10.1016/B978-0-444-87707-9.50017-8.Google Scholar
Hocking, G. C., “Supercritical withdrawal from a two-layer fluid through a line sink”, J. Fluid Mech. 297 (1995) 37–47; doi:10.1017/S0022112095002990.Google Scholar
Hocking, G. C. and Forbes, L. K., “Supercritical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth”, J. Fluid Mech. 428 (2001) 333–348; doi:10.1017/S0022112000002780.Google Scholar
Hocking, G. C. and Zhang, H., “A note on withdrawal from a two-layer fluid through a line sink in a porous medium”, ANZIAM J. 50 (2008) 101–110; doi:10.1017/S144618110800028X.Google Scholar
Hocking, G. C. and Zhang, H., “Coning during withdrawal from two fluids of different density in a porous medium”, J. Eng. Math. 65 (2009) 101–109; doi:10.1007/s10665-009-9267-1.Google Scholar
Hocking, G. C. and Zhang, H., “A note on axisymmetric supercritical coning in a porous medium”, ANZIAM J. 55 (2014) 327–335; doi:10.1017/S1446181114000170.Google Scholar
Letchford, N. A., Forbes, L. K. and Hocking, G. C., “Inviscid and viscous models of axisymmetric fluid jets or plumes”, ANZIAM J. 53 (2012) 228–250; doi:10.1017/S1446181112000156.Google Scholar
McCarthy, J. F., “Gas and water cresting towards horizontal wells”, ANZIAM J. 35 (1993) 174–197; doi:10.1017/S0334270000009115.Google Scholar
Muskat, M. and Wyckoff, R. D., “An approximate theory of water coning in oil production”, Amer. Inst. Mining Engineers, Petroleum Development Technol. 114 (1935) 144–163; doi:10.2118/935144-G.Google Scholar
Russell, P., Forbes, L. K. and Hocking, G. C., “The initiation of a planar fluid plume beneath a rigid lid”, J. Eng. Math. 106 (2017) 107–121; doi:10.1007/s10665-016-9895-1.Google Scholar
Van Deemter, J. J., “Bijdragen tot de kennis van enige natuurkundige grootheden van de grond” Versl. Landb. Onderz. 56 (1950) 1–67.Google Scholar
Youngs, E. G., “Effect of the capillary fringe on steady-state water tables in drained lands”, J. Irrigation Drainage Eng. 138 (2012) 803–814; doi:10.1061/(asce)ir.1943-4774.0000467.Google Scholar
Yih, C. S., Stratified Flows, 2nd edn (Academic Press, New York, 1980).Google Scholar
Yih, C. S., “On steady stratified flows in porous media”, Quart. J. Appl. Math. 40 (1982) 219–230; doi:10.1090/qam/666676.Google Scholar
Yu, D., Jackson, K. and Harmon, T. C., “Dispersion and diffusion in porous media under supercritical conditions”, Chem. Eng. Sci. 54 (1999) 357–367; doi:10.1016/S0009-2509(98)00271-1.Google Scholar
Zaradny, H. and Feddes, R. A., “Calculation of non-steady flow towards a drain in saturated-unsaturated soil by finite elements”, Agricultural Water Management 2 (1979) 37–53; doi:10.1016/0378-3774(79)90012-X.Google Scholar
Zhang, H. and Hocking, G. C., “Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface”, J. Eng. Math. 32 (1997) 365–376; doi:10.1023/A:1004227232732.Google Scholar
Zhang, H., Hocking, G. C. and Barry, D. A., “An analytical solution for critical withdrawal of layered fluid through a line sink in a porous medium”, ANZIAM J. 39 (1997) 271–279; doi:10.1017/S0334270000008845.Google Scholar
Zhang, H., Hocking, G. C. and Seymour, B., “Critical and supercritical withdrawal from a two-layer fluid through a line sink in a partially bounded aquifer”, Adv. Water Res. 32 (2009) 1703–1710; doi:10.1016/j.advwatres.2009.09.002.Google Scholar