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Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM

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Abstract

This paper presents a new numerical tool to model the two-dimensional contaminant transport through saturated porous media using a meshfree method, called radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. The advection–dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using 2D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Three numerical examples are presented and the results are compared with those obtained from the finite element method and analytical solutions. In order to test the practical applicability and performance of the RPIM, two case studies of contaminant transport through landfill liners are presented. A good agreement is obtained between the results of the RPIM and the field investigation data.

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Abbreviations

a i :

Coefficient for radial basis function

b j :

Coefficient for monomial basis function

C :

Contaminant concentration

C 0 :

Concentration of contaminant at the source

C i :

Initial concentration of contaminant

D :

Coefficient of hydrodynamic dispersion

g :

Concentration gradient at the exist boundary

K d :

Distribution coefficient

M :

Number of polynomial terms

N :

Number of nodes

θ :

Porosity of contaminant soil

n s :

Unit normal to domain

p j :

Basis function of monomial

P Q :

Moment matrix of monomial basis function

q :

Thin plate spline radial basis function shape parameter

Q :

Nodal mass flux

R :

Retardation coefficient

R i :

Thin plate spline radial basis function

r i :

Radial distance between x and x i

R Q :

Moment matrix of radial basis function

t :

Time

u :

Discharge velocity

v :

Seepage velocity

x :

Spatial coordinate

Ω:

Problem domain

\( \Upphi \) :

Meshfree shape function

ρ d :

Bulk density of contaminant soil

Γ :

Surface boundary of domain

ε:

Constant

References

  • Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127

    Article  Google Scholar 

  • Bear J (1979) Hydraulics of groundwater. McGraw-Hill, New York

    Google Scholar 

  • Beuth L, Benz T, Vermeer PA, Coetzee CJ, Bonnier P, Van Den Berg P (2007) Formulation and validation of a quasi-static material point method. In: Proceedings of the 10th international symposium on numerical methods in Geomechanics (NUMOG), Rhodes, Greece, pp 189–195

  • Boztosun I, Charafi A (2002) An analysis of the linear advection-diffusion equation using mesh-free and mesh-dependent methods. Eng Anal Bound Elem 26:889–895

    Article  Google Scholar 

  • Boztosun I, Charafi A, Zerroukat M, Djidjeli K (2002) Thin-plate spline radial basis function scheme for advection–diffusion problems. Electron J Bound Elem BETEQ 2001:267–282

    Google Scholar 

  • Boztosun I, Charafi A, Boztosun D (2003) On the numerical solution of linear advection-diffusion equation using compactly supported radial basis functions, vol 26. In: Griebel M, Alexander Schweitzer M (eds) Lecture notes in Comput Sci Eng Meshfree Methods for Partial Diff Eq. Springer, pp 63–73

  • Chinchapatnam PP, Djidjeli K, Nair PB (2006) Unsymmetric and symmetric meshless schemes for the unsteady convection-diffusion equation. Comp Meth Appl Mech Eng 195:2432–2453

    Article  Google Scholar 

  • Coetzee CJ (2004) The modelling of granular flow using the particle-in-cell method. PhD Thesis, University of Stellenbosch, South Africa

  • Coetzee CJ, Vermeer PA, Basson AH (2005) The modelling of anchors using the material point method. Int J Numer Anal Methods Geomech 29:879–895

    Article  Google Scholar 

  • Craig JR, Rabideau AJ (2006) Finite difference modelling of contaminant transport using analytic element flow solutions. Adv Water Resour 29:1075–1087

    Article  Google Scholar 

  • Crank J (1956) The mathematics of diffusion. Oxford Press, London

    Google Scholar 

  • Dolbow J, Belytschko T (1998) An introduction to programming the meshless element-free Galerkin method. Arch Comput Methods Eng 5:207–241

    Article  Google Scholar 

  • Domenico P (1987) An analytical model for multidimensional transport of a decaying contaminant species. J Hydrol 91:49–58

    Article  Google Scholar 

  • Durner W, Schultze B, Zurmühl T (1999) State-of-the-art in inverse modeling of inflow/outflow experiments. In: van Genuchten MTh, Leij FJ, Wu L (eds) Proceedings of international workshop on characterisation and measurement of the hydraulic properties of unsaturated porous media, University of California, Riverside, CA, pp 661–681

    Google Scholar 

  • Eldho TI, Rao BV (1997) Simulation of two-dimensional contaminant transport with dual reciprocity boundary elements. Eng Anal Bound Elem 20:213–228

    Article  Google Scholar 

  • Finsterle S (2004) Multiphase inverse modelling: review and iTOUGH2 applications. Vadose Zone J 3:747–762

    Article  Google Scholar 

  • Freiboth S, Class H, Helmig R (2007) Numerical simulation of multiphase multicomponent processes considering structural alterations of porous media—a phenomenological model. In: Schanz T (ed) Theoretical and numerical unsaturated soil mechanics. Springer, Berlin, pp 127–134

    Chapter  Google Scholar 

  • Freiboth S, Class H, Helmig R, Graf T, Ehlers W, Schwarz V, Vrettos C (2009) A model for multi-phase flow and transport in porous media including a phenomenological approach to account for deformation—a model concept and its validation within a code intercomparison study. Comput Geosci 13:281–300

    Article  Google Scholar 

  • Frind EO (1988) Solution of the advection–dispersion equation with free exit boundary. Numer Methods Partial Differ Equ 4:301–313

    Article  Google Scholar 

  • GeoSlope International Ltd (2007) Transport modelling with CTRAN/W 2007: an engineering methodology, student version 7.02, Alberta, Canada

  • Golberg MA, Chen CS, Bowman H (1999) Some recent results and proposals for the use of radial basis functions in the BEM. Eng Anal Bound Elem 23:285–296

    Article  Google Scholar 

  • King KS, Quigley RM, Fernandez F, Reades DW, Bacopoulos A (1993) Hydraulic conductivity and diffusion monitoring of the Keele Valley Landfill liner, Maple, Ontario. Can Geotech J 30:124–134

    Google Scholar 

  • Li J, Chen Y, Pepper D (2003) Radial basis function method for 1-d and 2-d groundwater contaminant transport modelling. Comput Mech 32:10–15

    Article  Google Scholar 

  • Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Eng 20:1081–1106

    Article  Google Scholar 

  • Liu GR, Zhang GY, Gu YT, Wang YY (2005) A meshfree radial point interpolation method (RPIM) for three-dimensional solids. Comput Mech 36:421–430

    Article  Google Scholar 

  • Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30:543–574

    Article  Google Scholar 

  • Ogata A, Banks RB (1961) A solution of the differential equation of longitudinal dispersion in porous media. USGS, Professional paper: 411-A, Reston, VA

  • Praveen Kumar R (2008) Modelling of 1d, 2d and 3d contaminant transport through saturated and unsaturated porous media using meshfree techniques. PhD Thesis, Indian Institute of Technology Madras, India

  • Praveen Kumar R, Dodagoudar GR (2008) Two-dimensional modelling of contaminant transport through saturated porous media using the radial point interpolation method (RPIM). Hydrogeol J 16:1497–1505

    Article  Google Scholar 

  • Praveen Kumar R, Dodagoudar GR, Rao BN (2007) Meshfree modelling of one-dimensional contaminant transport in unsaturated porous media. Geomech Geoeng An Int J 2:129–136

    Article  Google Scholar 

  • Priesack E, Durner W (2006) Closed-form expression for the multi-modal unsaturated conductivity function. Vadose Zone J 5:121–124

    Article  Google Scholar 

  • Quigley RM, Rowe RK (1986) Leachate migration through clay below a domestic waste landfill, Sarnia, Ontario, Canada: chemical interpretation and modelling philosophies. In: Lorenzen D, Conway RA, Jackson LP, Hamza A, Perket CL, Lacy WJ (eds) Hazardous and industrial solid waste testing and disposal, vol 6, ASTM STP 933. American Society for Testing and Materials, Philadelphia, pp 93–103

    Chapter  Google Scholar 

  • Quigley RM, Fernandez F, Yanful E, Helgason T, Margaritis A, Whitby JL (1987) Hydraulic conductivity of contaminated natural clay directly beneath a domestic landfill. Can Geotech J 24:377–383

    Article  Google Scholar 

  • Rowe RK, Booker JR (1985) 1-D pollutant migration in soils of finite depth. J Geotech Eng ASCE 111:479–499

    Article  Google Scholar 

  • Rowe RK, Quigley RM, Brachman RWI, Booker JR (2004) Barrier systems for waste disposal. E & FN Spon Press, London

    Google Scholar 

  • Sheu WHT, Chen YH (2002) Finite element analysis of contaminant transport in groundwater. Appl Math Comput 127:23–43

    Article  Google Scholar 

  • Sukumar N, Moran B, Semenov AY, Belikov VV (2001) Natural neighbour Galerkin methods. Int J Numer Methods Eng 50:1–27

    Article  Google Scholar 

  • Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196

    Article  Google Scholar 

  • van Genuchten MTh (1981) Analytical solutions for chemical transport with simultaneous adsorption, zero-order production and first-order decay. J Hydrol 49:213–233

    Article  Google Scholar 

  • Vermeer PA, Beuth L, Benz T (2008) A quasi-static method for large deformation problems in geomechanics. In: Proceedings of the 12th international conference of international association for computer methods and advances in geomechanics (IACMAG), Goa, India, pp 55–63

  • Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54:1623–1648

    Article  Google Scholar 

  • Zimmermann S, Koumoutsakos P, Kinzelbach W (2001) Simulation of pollutant transport using a particle method. J Comput Physics 173:322–347

    Article  Google Scholar 

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Authors and Affiliations

  1. Centre for Environmental Risk Assessment and Remediation (CERAR), University of South Australia, Mawson Lakes Campus, Mawson Lakes, SA, 5095, Australia

    R. Praveen Kumar

  2. Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, 600036, Tamil Nadu, India

    G. R. Dodagoudar

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  1. R. Praveen Kumar
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  2. G. R. Dodagoudar
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Correspondence to G. R. Dodagoudar.

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Praveen Kumar, R., Dodagoudar, G.R. Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM. Environ Earth Sci 61, 341–353 (2010). https://doi.org/10.1007/s12665-009-0346-4

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