Abstract
The salt water intrusion phenomenon endangering the groundwater sources, particularly in coastal aquifers, is modeled by the coupled density-dependent flow and solute transport equations. This study focuses on the solution of these interdependent equations by means of the polynomial based Differential Quadrature Method (DQM). The Lagrange interpolation polynomials were selected as basis functions to obtain weighting coefficients for approximating the spatial derivatives and the 4-stage, 4th order Strong Stability Preserving linear Runge Kutta (SSP-RK) algorithm was employed as the time integrator. Two well-known benchmark cases; Henry and Elder problems were examined to test the accuracy and the reliability of the proposed numerical model. Apart from these theoretical benchmark problems, the numerical model was tested with real experimental data from a laboratoryscale study in the literature. The DQM model was observed to provide stable and highly precise results regarding the current semianalytical and numerical solution schemes.
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Atkinson, S.F., Miller, G.D., and Curry, D.S., 1986, Salt-water Intrusion, Status and Potential in the Contiguous United States. Lewis Publishers, Chelsea, 390 p.
Bardsley, K.J. and Sukop, M.C., 2008, Simulating density-dependent flows using the Lattice Boltzmann Method. Proceedings of the Salt Water Intrusion Meeting, Naples, July 23–27, p. 14–17.
Batayneh, A.T. and Al-Taani, A.A., 2016, Integrated resistivity and water chemistry for evaluation of groundwater quality of the Gulf of Aqaba coastal area in Saudi Arabia. Geosciences Journal, 20, 403–413.
Baxter, G.P. and Wallace, C.C., 1916, Changes in volume upon solution in water of halogen salts of alkali metals. IX American Chemical Society Journal, 38, 70–104.
Bear, J., 1972, Dynamics of Fluids in Porous Media. American Elsevier Publishing Company, New York, 764 p.
Bear, J. and Dagan, G., 1964, Some exact solutions of interface problems by means of the hodograph method. Journal of Geophysical Research, 69, 1563–1572.
Bellman, R., Bayesteh, K., Lee, E.S., and Vasudevan, R., 1975, Differential quadrature and splines. Computers & Mathematics with Applications, 1, 371–376.
Bellman, R., Kashef, B.G., and Casti, J., 1972, Differential quadrature; a technique for the rapid solution of nonlinear partial differential equation. Journal of Computational Physics, 10, 40–52.
Bert, C.W. and Malik, M., 1996, Differential quadrature method in computational mechanics: A review. Applied Mechanics Reviews, 49, 1–27.
Chapman, S.J., 2015, MATLAB Programming for Engineers. Cengage Learning, Boston, 682 p.
Chen, W. and Zhong, T., 1997, The study on the nonlinear computations of the DQ and DC methods. Numerical Methods for Partial Differential Equations, 13, 57–75.
Diersch, H.J.G., 2005, FEFLOW: Finite element subsurface flow and transport simulation system, Reference Manual. Institute for Water Resources Planning and System Research, Berlin, 292 p.
Elder, J., 1967, Transient convection in a porous medium. Journal of Fluid Mechanics, 27, 609–623.
Fein, E., 1998, D3F–A simulator for density driven flow modeling. User’s Manual, GRS, Braunschweig, Germany, 139 p.
Ghaheri, A. and Meraji, S., 2012, Numerical simulation of groundwater table falling in horizontal and sloping aquifers by differential quadrature method (DQM). Journal of Hydrologic Engineering, 17, 869–879.
Ghyben, W.B., 1888, Nota in verband met de voorgenomen putboring nabij Amsterdam. Tijdschrift van het Koninklijk Instituut van Ingenieurs, 9, 8–22.
Goswami, R.R. and Clement T.P., 2007, Laboratory-scale investigation of saltwater intrusion dynamics. Water Resources Research, 43, W04418.
Gottlieb, S., 2005, On high order strong stability preserving Runge-Kutta and multi step time discretizations. Journal of Scientific Computing, 25, 105–127.
Guo, W. and Langevin, C.D., 2002, User’s Guide to SEAWAT: A Computer Program for Simulation of Three-Dimensional Variable-Density Ground-Water Flow. Techniques of Water-Resources Investigations Book 6, Chapter A7, Tallahassee, Florida, 77 p.
Hashemi, M.R., Abedini, M.J., and Malekzadeh, P., 2007, A differential quadrature analysis of unsteady open channel flow. Applied Mathematical Modelling, 31, 1594–1608.
Henry, H.R., 1964, Effects of dispersion on salt encroachment in coastal aquifers. In: Cooper, H.H. (ed.), Sea water in coastal aquifers. US Geological Survey Water-Supply Paper, 1613-C, Washington D.C., p. 70–84.
Herzberg, A., 1901, Die Wasserversorgung einiger nordseebader. Journal Gasbeleucht und Wasserversorg, 44, 815–819.
Holzbecher, E., 1998, Modeling density-driven flow in porous media: principles, numerics, software. Springer-Verlag, Berlin, 286 p.
Huppert, H.E. and Woods, A.W., 1995, Gravity-driven flows in porous layers. Journal of Fluid Mechanics, 292, 55–59.
Huyakorn, P.S., Anderson, P.F., Mercer, J.W., and White, H.O., 1987, Saltwater intrusion in aquifer: Development and testing of a threedimensional finite-element model. Water Resources Research, 23, 293–312.
Jeen, S.W., Kim, J.M., Ko, K.S., Yum, B., and Chang, H.W., 2001, Hydrogeochemical characteristics of groundwater in a mid-western coastal aquifer system, Korea. Geosciences Journal, 5, 339–348.
Kacimov, A.R. and Obnosov, Y.V., 2001, Analytical solution for a sharp interface problem in sea water intrusion into a coastal aquifer. Proceedings: Mathematical, Physical and Engineering Sciences, 457, 3023–3038.
Kaya, B., 2010, Solution of the advection-diffusion equation using the differential quadrature method. KSCE Journal of Civil Engineering, 14, 69–75.
Kaya, B. and Arisoy, Y., 2011, Differential quadrature solution for onedimensional aquifer flow. Mathematical and Computational Applications, 16, 524–534.
Kim, J.H., Yum, B.W., Kim, R.H., Koh, D.C., Cheong, T.J., Lee, J., Chang, H.W., 2003, Application of cluster analysis for the hydrogeochemical factors of saline groundwater in Kimje, Korea. Geosciences Journal, 7, 313–322.
Kipp, K.L., 1986, HST3D: A Computer Code for Simulation of Heat and Solute Transport in Three dimensional Groundwater Flow Systems. IGWMC, International Ground Water Modeling Center, USGS Water-Resources Investigations Report 86-4095, Reston, Virginia, 517 p.
Lee, J.Y. and Song, S.H., 2007, Groundwater chemistry and ionic ratios in a western coastal aquifer of Buan, Korea: implication for seawater intrusion. Geosciences Journal, 11, 259–270.
Lester, B., 1991, SWICHA, A Three-Dimensional Finite-Element Code for Analyzing Seawater Intrusion in Coastal Aquifers, Version 5.05. GeoTrans, Inc, Sterling, Virginia, 178 p.
Meral, G., 2013, Differential quadrature solution of heat- and mass-transfer equations. Applied Mathematical Modelling, 37, 4350–4359.
Naji, A., Cheng, A.H-D., and Ouazar, D., 1998, Analytical stochastic solutions of saltwater/ freshwater interface in coastal aquifers. Stochastic Hydrology andHydraulics, 12, 413–429.
Ngo, M.T., Lee, J.M., Lee, H.A., and Woo, N.C., 2015, The sustainability risk of Ho Chi Minh City, Vietnam, due to saltwater intrusion. Geosciences Journal, 19, 547–560.
Reeuwijk, M. van Mathias, S.A., Simmons, C.T., and Ward, J.D., 2009, Insights from a pseudospectral approach to the Elder problem. Water Resources Research, 45, W04416.
Robati, A. and Barani, G.A., 2009, Modeling of water surface profile in subterranean channel by differential quadrature method (DQM). Applied Mathematical Modelling, 33, 1295–1305.
Quan, J.R. and Chang, C.T., 1989, New insights in solving distributed system equations by the quadrature methods. Computers & Chemical Engineering, 13, 779–788.
Sauter, F.J., Leijnse, A., and Beusen, A.H.W., 1993, METROPOL’s User’s Guide. Report number 725205003, National Institute of Public Health and Environmental Protection, Bilthoven, the Netherlands, 375 p.
Segol, G., 1994, Classic groundwater simulations, Proving and improving numerical models. Prentice Hall, New York, 531 p.
Shamir, U. and Dagan, G., 1971, Motion of Seawater Interface in Coastal Aquifers–Numerical Solution. Water Resources Research, 7, 644–657.
Shu, C., 2000, Differential Quadrature and Its Application in Engineering. Springer-Verlag, Berlin, 340 p.
Shu, C. and Richards, B.E., 1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 15, 791–798.
Shu, C., Wang, L., and Chew, Y.T., 2003, Numerical computation of three-dimensional incompressible Navier-Stokes equations in primitive variable form by DQ method. International Journal for Numerical Methods in Fluids, 43, 345–368.
Shu, C. and Wee, K.H.A., 2001, Numerical simulation of natural convection in a square cavity by SIMPLE-GDQ method. Computers & Fluids, 31, 209–226.
Shu, C. and Wu, Y.L., 2007, Integrated radial basis functions-based differential quadrature method and its performance. International Journal for Numerical Methods in Fluids, 53, 969–984.
Simpson, M.J. and Clement, T.B., 2004, Improving the worthiness of the Henry problem as a benchmark for density-dependent groundwater flow models. Water Resources Research, 40, W01504.
Striz, A.G., Wang, X., and Bert, C.W., 1995, Harmonic differential method and applications to structural components. Acta Mechanica, 111, 85–94.
Voss, C.I., 1984, SUTRA–A finite element simulation for saturatedunsaturated, fluid-density dependent ground-water flow with energy transport or chemically reactive single-species solute transport. USGS Water-Resources Investigations Report 84-4369, Reston, Virginia, 429 p.
Voss., C.I. and Provost, A.M., 2010, SUTRA version 2.2-A Model for Saturated-Unsaturated, Variable-Density Ground-Water Flow with Solute or Energy Transport. Water-Resources Investigations Report 02-4231, Reston, Virginia, 291 p.
Voss, C.I. and Souza, W.R., 1987, Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater- saltwater transition zone. Water Resources Research, 23, 1851–1866.
Wilson, J.L. and Sa da Costa, A., 1982, Finite-element simulation of a saltwater fresh-water interface with indirect toe tracking. Water Resources Research, 18, 1069–1080.
Zhang, L., Guo, Y., and He, S., 2007, Numerical simulation of threedimensional incompressible fluid in a box flow passage considering fluid-structure interaction by differential quadrature method. Applied Mathematical Modelling, 31, 2034–2049.
Zhong, H., 2004, Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Applied Mathematical Modelling, 28, 353–366.
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Ciftci, E. Modelling coupled density-dependent flow and solute transport with the differential quadrature method. Geosci J 21, 807–817 (2017). https://doi.org/10.1007/s12303-017-0009-5
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DOI: https://doi.org/10.1007/s12303-017-0009-5