Differential quadrature method for two-dimensional unsaturated water flow

Document Type : Article

Authors

1 Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran

2 Department of Engineering, Najaf Abad Branch, Islamic Azad University, Najaf Abad, Iran

Abstract

This paper presents numerical solution of Richards' equation for water flow through unsaturated porous media. Differential Quadrature Method (DQM) is employed for the first time to solve the governing equations in two space dimensions. The moisture content based of Richards' equation is considered. This equation is known as a highly nonlinear partial differential equation due to strong nonlinearity between hydraulic conductivity (and diffusivity) and moisture content. In order to investigate the robustness of DQM in dealing with such strong nonlinearities, two popular constitutive models i.e. White and Broadbrige (1988), and Van Genuchten (1980) models are investigated for the 2D case. Analytical solution based on Brooks and Coley model in a special 1D case is used to compare the results with those of DQM. For the 2D case, the study also demonstrates that DQM with considerably smaller number of grid points gives excellent results which are in close agreement with other numerical techniques such as multigrid approach reported in the literature.

Keywords

Main Subjects


References
 1.Sander,  G.C.,  Parlange,  J.Y.,  Kuhnel,  V.,  Hogarth,W.L.,   Lockington,   D.  and  O'Kane,   J.P.J.  Exactnonlinear  solution  for  constant   ux  in ltration",J.Hydrol.,83, pp. 341-346 (1988).2.Broadbridge, P. and White, I. Constant rate rain fallin ltration:  a versatile nonlinear model 1",Anal. Sol.Water. Res. Res.,24, pp. 145-154 (1988).3.Tracy, F.T. Three dimensional analytical solutions ofRichard's equation for a box shaped soil sample withpiecewise-constant  head  boundary  conditions  on  thetop",J. Hydrol.,336(3), pp. 391-400 (2007).4.Menziani, M., Puganghi, S. and Vincenzi, M. Analyt-ical solutions of the linearized Richard's equation fordiscrete arbitrary initial and boundary conditions",J.Hydrol.,332(1-2) pp. 214-225 (2007).5.Huang,   R.Q.   and   Wu,   L.Z.   Analytical   solutionsto  1-D  horizontal  and  vertical  water  in ltration  insaturated/unsaturated  soils  considering  time-varyingrainfall",J. Comp. Geotec.,39, pp. 66-72 (2012).6.Haverkamp,  R.,  Vauclin,  M.,  Touma,  J.,  Wiereng,P.J.  and  Vachaud,  G.  A  comparison  of  numericalsimulation models for one-dimensional in ltration",J.Soil Soc. Am.,41, pp. 285-293 (1977).7.G0ttardi,   G.  and  Venutelli,   M.  Two-dimensional nite-element  groundwater   ow  model  for  saturated-unsaturated  soils",J.  Comput.  Geosci.,27,  pp.  179-189 (2001).8.Narado, N., Braud, I., Ross, P.J. and Haverkamp, R.Assessment  of  an  ecient  numerical  solution  of  the1D Richard's equation on bare soil",J. Hydrol.,323(1-4), pp. 244-257 (2006).9.Taheri Shahraiyani, H. and Ataei Ashtiani, B. Com-parison  of   nite  di erence  schemes  for  water   ow  inunsaturated  soils",Int.  J.  Aerosp.  Mech.  Eng.,3(1),pp. 1-5 (2009).10.An,  H.,  Ichikawa,  Y.,  Tachikawa,  Y.  and  Shiiba,  M.A new Iterative Alternating Direction Implicit (IADI)algorithm for multi-dimensional saturated-unsaturated ow",J. Hydrol.,408, pp. 127-139 (2011).11.Bergamaschi, L. and Putti, M. Mixed  nite elementsand  Newton-type  linearizations  for  the  solution  of Richards",Int. J. Num. Methods. Eng.,45, pp. 1025-46 (1999).12.Gottardi,   G.  and  Venutelli,   M.  Two-dimensional nite-element  groundwater   ow  model  for  saturated-unsaturated  soils",J.  Comput.  Geosci.,27,  pp.  179-189 (2001).13.He,  X.  and  Ren,  L.  An  adaptive  multiscale   niteelement  method  for  unsaturated   ow  problems  inheterogeneous  porous  media",J.  Hydrol.,pp.  56-70(2009).14.Ginting, V. Time integration techniques for Richard'sequation",Procedia   Comput.   Sci.,9.  pp.  670-678(2012).15.Eymard, R., Gutnic, M. and Hilhorst, D. The  nitevolume  method  for  Richards'  equation",J.  Comput.Geosci.,12, pp. 259-294 (1999).16.Manzini, G. and Ferraris, S. Mass-conservative  nitevolume  methods  on  2-D  unstructured  grids  for  theRichards'  equation",J.  Adv.  Water.  Res.,27,  pp.1199-1215 (2004)17.Misiats,  O. and Lipnikov,  K. Second-order accuratemonotone   nite  volume  scheme  for  Richard's  equa-tion",J. Comput. Phys.,239, pp. 123-137 (2013).18.Zambra, C.E., Dumbser, M., Toro, E.F. and Moraga,N.O. A novel numerical method of high order accu-racy  for   ow  in  unsaturated  porous  media",Int.  J.Numer. Meth. Eng.,89, pp. 227-240 (2012).19.Caviedes   Voullieme,   D.,   Garcia   Navarro,   P.   andMurillo,  J.  Veri cation,  conservation,  stability  andeciency of a  nite volume method for the 1D Richardsequation",J. Hydrol.,480, pp. 69-84 (2013).20.Juncu, G., Nikola, A. and Popa, C. Nonlinear multi-grid methods for numerical solution of the unsaturated ow  equation  in  two  space  dimensions",J.  Transp.Porous. Med.,83, pp. 637-652 (2010).21.Carr, E.J., Moroney, T.J. and Turner, I.W. Ecientsimulation of unsaturated  ow using exponential timeintegration",J.  Appl.  Matt.  Comput.,217(14),  pp.6587-6596 (2011).22.Shu, C.Di erential Quadrature and Its Application inEngineering, Springer, Singapore (2000).23.Hashemi,  M.R.  and  Hatam,  F.  Unsteady  seepageanalysis using local radial basis function based di er-ential  quadrature  method",Appl.  Math.  Model.,35,pp. 4934-4950 (2011).
4.Hu,  Y.J.,  Zhu,  Y.Y.  and  Cheng,  C.J.  DQM  for  dy-namic response of  uid-saturated visco-elastic porousmedia",Int.   J.   Solids   Struct.,46,   pp.  1667-1675(2009).25.Chen, R.P., Zhou, W.H., Wang, H.Z. and Chen, Y.M.One-dimensional  nonlinear  consolidation  of  multi-layered  soil  by  di erential  quadrature  method",J.Comput. Geotec.,32, pp. 358-369 (2009).26.White, I. and Broadbridge, P. Constant rate rainfallin ltration:  a  versatile  nonlinear  model  2",J.  Appl.Sol. Water Res. Res., pp. 155-162 (1988).27.Van  Genuchten,  M.T.  A  closed-form  equation  forpredicting  the  hydraulic  conductivity  of  unsaturatedsoils",J. Soil Sci. Soc. Am.,44, pp. 892-898 (1980).28.Belman,  R.,  Kahef,  B.  and  Casti,  J.  Di erentialquadrature a technique for the rapid solution of nonlin-ear partial di erential equations",J.  Comput.  Phys.,10, pp. 40-52 (1972).29.Kuraz,  M.,  Mayer,  P.,  Leps,  M.  and  Trpkosoa,  D.An  adaptive  time  discretization  of  the  classical  andthe  dual  porosity  model  of  Richards'  equation",J.Comput. App. Math.,233, pp. 3167-3177 (2010).30.Asghari,  A.,  Bagheripour,  M.H.  and  Mollazadeh,  M.A  generalized  analytical  solution  for  a  nonlinearin ltration equation using the exp-function method",J. Scientia Iranica,18(1), pp. 28-35 (2011).31.Brooks, R.H. and Corey, A.T.Hydraulic Properties ofPorous Media, Colorado State University, Fort Collins,Hydrology Paper,3(1964).