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Prograding and Retrograding Hypo- and Hyper-Pycnal Deltaic Formations into Quiescent Ambients

  • Kolumban HutterEmail author
  • Yongqi Wang
  • Irina P. Chubarenko
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

Sediment transport from mountainous rivers into a quiescent ambient with simultaneous formation of deltas is reviewed. The focus is restricted to flow in vertical cross-sections with no changes perpendicular to the plane of the flow. The bed load transport in the river is derived for quasi-steady situations using sediment mass balance and the Mohr-Terzaghi shear stress-pressure relation with the angle of internal friction, \(\varphi \), as the essential frictional parameter. The emerging model is a diffusion equation for the upper surface of the moving sediment layer and corresponding boundary conditions. Its diffusivity is expressible in terms of the hydraulic discharge, the densities of the sediment and the turbid water, the angle of internal friction and a parameter characterizing the bed-parallel sediment velocity in terms of the average velocity in the turbid layer. When this river flow enters quiescent water, two different classes of deltas can be formed. When the entering water is either neutrally buoyant or lighter than the ambient water, the sudden reduction in tractive force along the bed generates a conspicuous avalanching flow to depth. This leads to steep-sloped foreset deposits with delta fronts inclined by the angle of internal friction. Such so-called Gilbert-type deltas are governed by a jump requirement of the sediment flux across the shore line and the geometry of the receiving basin. When the inflowing discharge is denser than the receiving ambient water, it will dive down as a turbulent under-current. The basal sediment transport in this subaqueous density current is analogous to the subaerial case and again described by a diffusion equation with similarly determined diffusivity. The combined dual subaerial-subaqueous sedimenting process is mathematically very similar to a (generalized) Stefan problem, e.g. the freezing of an ice cover on a lake. We present (mostly analytical) solutions for (1) bedrock-alluvial transitions, (2) overtopping failure of a dam, (3) topset-foreset diffusion processes for hypo- and hyper-pycnal deltas. Laboratory experiments demonstrate the adequacy of the models.

References

  1. 1.
    Abramowitz, M. and Stegun, I. A.: Handbook of Mathematical Functions, Dover Publ. Inc, New York, (1964)Google Scholar
  2. 2.
    Adams, E. W., Schlager, W. and Anselmetti, F. S.: Morphology and curvature of delta slopes in Swiss lakes: Lessons for the interpretation of clinoforms in seismic data. Sedientology, 48, 661–679 (2001)Google Scholar
  3. 3.
    Begin, Z. B., Meyer, D. F. and Schumm, S. A.: Development of longitudinal profiles of alluvial channels in reponse to base level lowering. Earth Surface Processes & Landforms, 6, 49–68 (1981)Google Scholar
  4. 4.
    Bell, H. S.: Density currents as agents for transporting sediments. J. Geology, 50, 512–547 (1942)Google Scholar
  5. 5.
    Bornhold, B. D. and Prior, D. B.: Morphology and sedimentary processes on the subaqeous Noeick River delta, British Columbia, Canada. In: Coarse-Grained Deltas, Spec. Publs. Int. Ass. Sediment., (Collela, A. and Prior, D. B., Eds.) 10, 169–181 (1960)Google Scholar
  6. 6.
    Capart, H., Bellal, M. and Young, D. L.: Self-similar evolution of semi-infinite alluvial channels with moving boundaries. J. Sedimentary Research, 77, 13–22, (2007)Google Scholar
  7. 7.
    Capart, H., Hsu, J. P. C., Lai, S. Y. J. and Hsieh, M.-L.: Formation and decay of a tributary-dammed lake, Laonong River, Taiwan, Water Resources Res., doi: 10.1029/2010WR009159 (2010)
  8. 8.
    Carslaw, H. S. and Jaeger, J. C.: Conduction of Heat in Solids. Oxford University Press, Oxford (1959)Google Scholar
  9. 9.
    Crank, J.: Free and Moving Boundary Problems, Cambridge University Press, Cambridge (1984)Google Scholar
  10. 10.
    Culling, W. E. H.: Analytical theory of erosion. J. Geology, 68, 336–344Google Scholar
  11. 11.
    Davis, W.M.: The lakes of California. Calif. J. Mines, 29, 175–236 (1933)Google Scholar
  12. 12.
    Ellison, T. H. and Turner, J. S.: Turbulent entrainment in stratified flows J. Fluid Mech. 6, 423–448 (1959)Google Scholar
  13. 13.
    Fan, J. and Morris G.L.: Reservoir sedimentation VI: Delta and density current deposits. J. Hydraul. Eng bf 118(3), 354–369 (1992)Google Scholar
  14. 14.
    Fleming, P. B. and Jordan, T. E.: A synthetic stratigraphic model of foreland basin development. Geophys, Res., 94, 3851–3866 (1989)Google Scholar
  15. 15.
    Fracarollo, L. and Capart, H.: Riemann wave description of erosional dam-break flows. J. Fluid Mech., 461, 183–228 (2002)Google Scholar
  16. 16.
    Galay, V. J., Tutt, D. B. and Kellerhals, R.: The meandering distributary channels of the upper Columbia river. In: River Meandering, edited by Eliott, C. M., pp 113–125, Am. Soc. Civil. Engr., New York (1983)Google Scholar
  17. 17.
    Garcia, M. H.: Hydraulic jumps in sediment-driven bottom currents. J. Hydraul. Eng., 119, (10), 1094–1117 (1993)Google Scholar
  18. 18.
    Gilbert, G. K.: Geology of the Henri Mountains 170 pp. U.S. Government Print. Office, Washington, D. C. (1877)Google Scholar
  19. 19.
    Gilbert, G. K.: Lake Bonneville U.S. Geological Survey Monograph, no. 1, 438 p. (1890)Google Scholar
  20. 20.
    Gill, M. A.: Diffusion model for degrading channels J. Hydraul. Res., 21, 369–378 (1983)Google Scholar
  21. 21.
    Graf, W. H.: Hydraulics of Sediment transport, McGraw-GHill, New York (1971)Google Scholar
  22. 22.
    Grover, N. C. and Howard, C. S.: The passage of turbid water through Lake Mead. Transactions ASCE, Paper No. 1994, 720–732, (1937)Google Scholar
  23. 23.
    Hinderer, M.: Late quarternary denudation of the Alps, valley and lake fillings and modern river loads. Geodinamica Acta, 14, 231–263 (2001)Google Scholar
  24. 24.
    Hsu, J. P. C. and Capart, H.: Onset and growth of tributary dammed lakes. Water Resources Research, 44, W11201 (2008)Google Scholar
  25. 25.
    Hutter, K.: A tutorial on prograding and retrograding hypo- and hyper-pycnal deltaic formations into quiescent ambients. In: Contributions on Sediment Transport. Bericht des Lehrstuhls und der Versuchsanstalt für Wasserbau und Wasserwirtschaft der TU München Nr. 127, Herausgeber: Prof. P. Rutschmann (2013)Google Scholar
  26. 26.
    Hydon, P. E.: Symmetry methods for differential equations, Cambridge University press, CambridgeGoogle Scholar
  27. 27.
    Jordan, T. E. and Fleming, P. B.: From geodynamic models to basin fill, a stratigraphic perspective. In: Quantitative Dynamic Stratigraphy (Ed.: T.A. Cross), 149–163 (1990)Google Scholar
  28. 28.
    Kenyon, P. M. and Turcotte, D. L.: Morphology of a delta prograding by bulk sediment transport. Geol. Soc. Amer. Bull., 96, 1457–1465 (1985)Google Scholar
  29. 29.
    Kostic, S. Parker, G. and Marr, J. G.: Role of turbidity currents in setting the foreset slope of clinoforms prograding into standing fresh water. J. Sedimentary Res., 72 (3), 353–362 (2002)Google Scholar
  30. 30.
    Kostic, S. and Parker, G.: Progradational sand mud deltas in lakes and reservoirs. Part I. Theory and numerical modeling. J. Hydr. Res. 41(2), 127–140 (2003)Google Scholar
  31. 31.
    Kostic, S. and Parker, G.: Progradational sand-mud deltas in lakes and reservoirs. Part II. Experiment and numerical simulation, J. Hydr. Res. 41(2), 141–152 (2003)Google Scholar
  32. 32.
    Kreyszig, E.: Advanced Engineering Mathematics, Wiley (2006)Google Scholar
  33. 33.
    Lai, S. Y. J.: Self-similar delta formation by hyperpycnal flows: theory and experiments. M. S. Thesis, Graduate Institute of Civil Engineering, Natl. Taiwan University, Taiwan, June 2006Google Scholar
  34. 34.
    Lai, S. Y. J. and Capart, H.: Two-diffusion description of hypopycnal deltas. J. Geophys Res. Earth Surface, 112, art F05005, 1–20 (2007) doi:  10.1029/2006JF00617
  35. 35.
    Lai, S. Y. J. and Capart, H.: Reservoir infill by hyperpycnal deltas over bedrock. Geophys. Res. Lett., 36 (2009) L08402, doi: 10.1029/2008GL037139(2009)
  36. 36.
    Lai, S. Y. J. and Capart, H.: Response of hyperpycnal deltas to steady rise in base level. 5\(^{th}\) IAHR Symposium on River, Coastal and Estuarine Morphodynamics in press (2011) Google Scholar
  37. 37.
    Lambert. A.: Turbidity currents from the Rhine River on the bottom of Lake Constance. Wasserwirtschaft, 72(4), 14 (in German), (1982)Google Scholar
  38. 38.
    Lane, E. W.: The importance of fluvial morphology in hydraulic engineering. Proceedings Am. Soc. Civ. Eng. 81, 745-1-745-17 (1955)Google Scholar
  39. 39.
    Lee, H.-Y., Lin, Y.-T., Chiu, Y. J. Quantitative estimation of reservoir sedimentation from three typhoon events. J. Hydraul. Eng. 11, 362–370 (2006)Google Scholar
  40. 40.
    Lorenzo-Trueba, J., Voller, V. R., Muto, T., Kim, W., Paola, C. and Swenson, J. B.: A similarity solution for a dual moving boundary problem associated with a coastal-plain depositional system. J. Fluid Mech., 628, 427–443 (2009)Google Scholar
  41. 41.
    Mackin, J. H.: Concept of the graded river. Bulletin Geol. Soc. America  59, 463–512 (1948)Google Scholar
  42. 42.
    Meyer-Peter, E. and Müller, R.: Formulas for bedload transport. 2nd Meeting, Int. Hydraul. Structures Research, Stockholm, 39–64 (1948)Google Scholar
  43. 43.
    Mitchell, N. C.: Morphologies of kinkpoints in submarine canyons. Geol. Soc. Amer. Bull., 118, 589–605 (2006)Google Scholar
  44. 44.
    Müller, G.: The new Rhine delta in Lake Constance. In: Deltas in their geologic framework, Shirley, M.L. and Ragsdale, J. E., eds. Houston Geological Society, 107–124 (1966)Google Scholar
  45. 45.
    Müller, G. and Förstner, U.: General relationship between suspended sediment concentration and water discharge in the Alpenrhein and some other rivers. Nature 217, 244–245 (1966)Google Scholar
  46. 46.
    Muto, T. Shoreline autoretreat substantiated in flume experiments. J. Sedimentary Res., 71(2), 246–254 (2001)Google Scholar
  47. 47.
    Muto, T., and Steel, R. J.: Retreat of the front in a prograding delta. Geology, 20, 967–970 (1992)Google Scholar
  48. 48.
    Muto, T. and Steel, R. J.: Autogenic attainment of large-scale alluvial grad with steady sea-level fall. Implications from flume-tank experiments. Geology, 32(5), 401–404; doi: 10.1130/G20269 (2004)Google Scholar
  49. 49.
    Muto, T. and Swenson, J. B.: Large-scale fluvial grade as a non-equilibrium state in linked depositional systems: Theory and experiment. J. Geophys. Res., 110(F3), art.no F03002 (2005)Google Scholar
  50. 50.
    Muto, T. and Swenson, J. B.: Autogenic attainment of large-scale alluvial grad with steady sea-level fall. An analog tank-flume experiment, Geology, 34(3), 161–164; doi: 10.1130/G21923.1 (2006)Google Scholar
  51. 51.
    Muto, T., Steel, R. J. and Swenson, J. B.: Autostratigraphy: A framework norm for genetic stratigraphy J. Sedimentary Res., 77, 2–12, doi: 10.2110/JSR2007.005 (2007)Google Scholar
  52. 52.
    Paola, C.: Subsidence and gravel transport in alluvial basins. In: New Perspectives in Basin Analysis (K. L. Kleinsphen and C. Paola, eds), 231–243 (1988)Google Scholar
  53. 53.
    Paola, G., Heller, P. L. and Angevine, C. L.: The large scale dynamics of grain-size variations in alluvial basins. 1: Theory. Basin Res. 4, 73–90 (1992)Google Scholar
  54. 54.
    Parker, G.: Basic principles of river hydraulics. Hydraul. Div. ASCE 103, 1077–1087 (1977)Google Scholar
  55. 55.
    Parker, G., Muto, T.: One-dimensional numerical model of delta response to rising sea-level. In: Proceedings third IAHR Symposium on River, Coastal and Estuarine Morphodynamics Editors: A Sánches-Arcilla and A. Bateman, pp. 558–570. Int Assoc. for Hydraulic Research, Madrid (2003)Google Scholar
  56. 56.
    Petter, A. L. and Muto, T.: Sustained alluvial aggradation and autogenic detachment of the alluvial river from the shoreline in response to steady fall of relative sea level. J. Sedimentary Res., 78, 98–111; doi:  10.2110/JRS.2008.012 (2008)Google Scholar
  57. 57.
    Pitman, W. C.: Relationship between eustacy and stratigraphic sequences on passive margins. Geol. Soc. Am. Bull. 89, 1389–1403 (1978)Google Scholar
  58. 58.
    Roethlisberger, H.: Eislawinen und Ausbrüche von Gletscherseen. Jahrbuch der Schwei- zerischen Naturforschenden Gesellschaft, wissenschaftlicher Teil, 170–211 (1978)Google Scholar
  59. 59.
    Roth, M. M., Weber, M. and Bezzola, G. R.: Physical modeling of sediment deposits in a river delta: The Alpenrhein Delta in Lake Constance. Proc. XXIX IAHR Congress, Beijing, China, Theme E., 187–194 (2001)Google Scholar
  60. 60.
    Schumm, S. A.: The Fluvial System, 338 pp. John Wiley, Hoboken, N. J. (1977)Google Scholar
  61. 61.
    Smith, W. O., Vettere, C. P. and Cummings, G. B.: Comprehensive survey of sedimentation in Lake Mead, 1948–1949. U.S. Geol. Survey Prof. Papier 295, 253 p. (1960)Google Scholar
  62. 62.
    Sokolnikoff, I. S. and Redheffer, R. M.: Mathematics of Physics and Modern Engineering, McGraw Hill Book Company, New York, etc. (1966)Google Scholar
  63. 63.
    Soni, J. P.: Unsteady sediment teransport law and prediction of aggradation parameters Water Resour. Res., 17, 33–40 (1981)Google Scholar
  64. 64.
    Swenson, J. B., Voller, V. R., Paola, C., Parker, G. and Marr, J. G.: Fluvio-deltaic sedimentation: A generalized Stefan problem. Euro J. of Applied Mathematics, 11, 433–452 (2000)Google Scholar
  65. 65.
    Tonioli, H. and Schultz, J.: Experiments on sediment trap efficiency in reservoirs. Lakes and Reservoirs  10, 13–24 (2005)Google Scholar
  66. 66.
    Turner, T. H.: Buoyancy effects in fluids. Cambridge University Press, Cambridge UK (1973)Google Scholar
  67. 67.
    Voller, V. R., Swenson, J. B., Kim, W. and Paola, C.: An enthalpy method for moving boundary problems on the Earth’s surface. Int. J. Numerical Methods for Heat & Fluid Flow, 16(5), 641–654, (2006)Google Scholar
  68. 68.
    Wright, H. E., Jr., Lease, K. and Johnson, S. Glacier River Warren, Lake Pepin, and the environmental history of southeastern Minnesota. In: Contributions to Quarternary Studies in Minnesota, Minn. Geol. Surv. Rep. Invest. Vol 49, (C.J. Patterson and H.E. Wright, Jr., eds., pp. 131–140, Univ. of Minn. Press, Minneapolis, Minn (1998)Google Scholar
  69. 69.
    Write, L. D., Wiseman, W. J., Bornhold, B. D., Prior, D. B., Suhayda, J. N., Keller, G. H. Yang, Z.-S. and Fan, J. B.: Marine dispersal and deposition of Yellow River silts by gravity driven underflows. Nature, 332, 629–632 (1988)Google Scholar
  70. 70.
    Yu, W.-S., Lee, H.-Y. and Hsu, S. M.: Experimental study on delta formation in a reservoir. J. Chin. Inst. Civ. Hydraul. Eng., 12(1), 171–177 (2000) [in Chinese]Google Scholar
  71. 71.
    Zumberge, J. H.: The lakes of Minnesota, their Origin and Classification Univ. of Minn. Press, Minneapolis, Minn. (1952)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kolumban Hutter
    • 1
    Email author
  • Yongqi Wang
    • 2
  • Irina P. Chubarenko
    • 3
  1. 1.c/o Laboratory of Hydraulics, Hydrology and Glaciology at ETHZürichSwitzerland
  2. 2.Chair of Fluid Dynamics, Department of Mechanical EngineeringTU DarmstadtDarmstadtGermany
  3. 3.P.P. Shirshov Institute of OceanologyRussian Academy of SciencesKaliningradRussia

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