Doymamış koşullarda toprak suyu hareketinin Modifiye Kübik B-Spline Diferansiyel Kuadratur yöntemi ile modellenmesi

Year 2018, Volume: 24 Issue: 6, 952 - 959, 18.12.2018

Emin Çiftçi

Abstract

Bu
çalışma, toprak suyunun doymamış bölgedeki hareketinin Modifiye Kübik B-Spline
Diferansiyel Kuadratur yöntemi ile modellenmesi üzerine kurulmuştur. Toprak
nemi hareketini tanımlamak için hidrojeoloji alanında sıklıkla kullanılan
Richards denkleminin sayısal çözümü irdelenmiştir. Bu denklemde yer alan
parametrelerin konum türevlerini hesaplayabilmek için diferansiyel kuadratur
yöntemi, problemi zamanda ilerletebilmek için 4. mertebeden Runge-Kutta yöntemi
kullanılmıştır. Geliştirilen modelin verdiği sonuçların doğruluğunun
ölçülebilmesi için farklı sınır koşullarına sahip üç örnek uygulama ele
alınmıştır. Elde edilen sonuçlar mevcut analitik ve diğer sayısal yöntemlerin
verdiği sonuçlarla büyük tutarlılık göstermektedir. Etkili, basit, verimli,
kolay uygulanabilir ve yüksek doğrulukta sonuç veriyor oluşu, ele alınan yöntemin
bu tip problemlerin sayısal çözümü için uygun bir tercih olabileceğine işaret
etmektedir.

Keywords

Richards denklemi, Diferansiyel kuadratur, Modifiye kübik b-spline

Modelling soil water movement under unsaturated conditions with the Modified Cubic B-Spline Differential Quadrature method

Abstract

This
study focuses on modelling moisture movement in the unsaturated soil zone with
the use of the Modified Cubic B-Spline Differential Quadrature method. The numerical
solution of the Richards Equation, which is commonly used in the field of
hydrogeology for describing soil moisture movement, was investigated. The
differential quadrature method was employed to evaluate the spatial derivatives
of the equation variables, and the 4^th order Runge-Kutta Method was
used to march the solution in time. To verify the accuracy of the developed
model, three test cases having different boundary conditions were taken into
consideration. The retrieved results are seen to be in good agreement with the
values provided by the analytical and other available numerical techniques. On
account of its simplicity, efficiency and high accuracy, this method can be an
appropriate option for the numerical solution of this type of problems. 

Keywords

Richards equation, Differential quadrature, Modified cubic b-spline

References

• Buckingham E. “Studies on the Movement of Soil Moisture”. USDA Bureau of Soils, Bulletin 38, Washington, DC, 1907.
• Richards LA. “Capillary conduction of liquids through porous mediums”. Physics, 1(5), 318-333, 1931.
• Philip JR. “General method of exact solution of the concentraton-dependent diffusion equation”. Australian Journal of Physics, 13(1), 1-12, 1960.
• Sander GC, Parlange JY, Kuhnel V, Hogarth WL, Lockington D, O’Kane JPJ. “Exact nonlinear solution for constant flux infiltration”. Journal of Hydrology, 97(3-4), 341-346, 1988.
• Warrick AW, Lomen DO, Islas A. “An analytical solution to Richards' equation for a draining soil profile”. Water Resources Research, 26(2), 253-258, 1990.
• Parlange JY, Hogarth WL, Barry DA, Parlange MB, Haverkamp R, Ross PJ, Steenhuis TS, DiCarlo DA, Katul G. “Analytical approximation to the solutions of Richards' equation with applications to infiltration, ponding, and time compression approximation”. Advances in Water Resources, 23(2), 189-194, 1999.
• Chen, JM, Tan, YC, Chen, CH. “Analytical solutions of one-dimensional infiltration before and after ponding. Hydrological Processes”. 17(4), 815-822, 2003.
• Celia MA, Bouloutas ET, Zarba RL. “A General mass-conservative numerical solution for the unsaturated flow equation”. Water Resources Research, 26(7), 1483-1496, 1990.
• Clement TP, William RW, Molz FJ. “A physically based two-dimensional, finite-difference algorithm for modeling variably saturated flow”. Journal of Hydrology, 161(1-4), 71-90, 1994.
• Miller, CT, Williams, GA, Kelley, CT, Tocci, MD. “Robust solution of Richards' equation for non uniform porous media”. Water Resources Research, 34(10), 2599-2610, 1998.
• Kumar CP. “A numerical simulation model for one-dimensional ınfiltration”. Journal of Hydraulic Engineering, 4(1), 5-15, 1998.
• Shahraiyni HT, Ashtiani BA, “Comparison of finite difference schemes for water flow in unsaturated soils”. International Scholarly and Scientific Research & Innovation. 2(4), 226-230, 2008.
• Forsyth PA, Wu YS, Pruess K. “Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media”. Advances in Water Resources, 18(1), 25-38, 1995.
• Bergamaschi L, Putti M. “Mixed finite element and Newton-type linearizations for the solution of Richards’ equation”. International Journal for Numerical Methods in Engineering, 45(8), 1025-1046, 1999.
• He X, Ren L. “An adaptive multiscale finite element method for unsaturated flow problems in heterogeneous porous Media”. Journal of Hydrology, 374(1-2), 56-70, 2009.
• Eymard R, Gutnic M, Hilhorst D. “The finite volume method for Richards equation”. Computational Geosciences, 3(3-4), 259-294, 1999.
• McBride D, Cross M, Croft N, Bennett C, Gebhardt J. “Computational modeling of variably saturated flow in porous media with complex three-dimensional geometries”. International Journal for Numerical Methods in Fluids, 50(9), 1085-1117, 2006.
• Wencong L, Ogden FL. “A mass-conservative finite volume predictor-corrector solution of the 1D Richards’ equation”. Journal of Hydrology, 523, 119-127, 2015.
• Bellman R, Kashef BG, Casti J. “Differential quadrature: a technique for the rapid solution of nonlinear differential equations”. Journal of Computational Physics, 10(1), 40-52, 1972.
• Quan JR, Chang CT. “New insights in solving distributed system equations by the quadrature method-I. Analysis”. Computers & Chemical Engineering, 13(7), 779-788, 1989.
• Shu C, Richards RE. “Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations”. International Journal for Numerical Methods in Fluids, 15(7), 791-798, 1992.
• Striz AG, Wang X, Bert CW. “Harmonic differential quadrature method and applications to analysis of structural components”. Acta Mechanica, 111(1-2), 85-94. 1995.
• Zhong H. “Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates”. Applied Mathematical Modelling, 28(4), 353-366, 2004.
• Shu C, Wu YL. “Integrated radial basis functions-based differential quadrature method and its performance”. International Journal for Numerical Methods in Fluids, 53(6), 969-984, 2007.
• Korkmaz A, Dağ İ. “Cubic B-spline differential quadrature methods for the advection-diffusion equation”. International Journal of Numerical Methods for Heat & Fluid Flow, 22(8), 1021-1036, 2012.
• Arora G, Singh BK. “Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method”. Applied Mathematics Computation, 224, 166-177, 2013.
• Mittal RC, Dahiya S. “Numerical simulation on hyperbolic diffusion equations using modified cubic B-spline differential quadrature methods”. Computers & Mathematics with Applications, 70(5), 737-749, 2015.
• Mittal RC, Rohila R. “Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method”. Chaos, Solitons & Fractals, 92, 9-19, 2016.
• Shukla HS, Tamsir M. “Numerical solution of nonlinear sine-Gordon equation by using the modified cubic B-spline differential quadrature method”. Beni-Suef University Journal of Basic and Applied Sciences, (in press) doi:10.1016/j.bjbas.2016.12.001.
• Brooks RJ, and Corey AT. “Hydraulic properties of porous media”. Hydrol. Pap. 3, Colo. State Univ., Fort Collins, USA, 1964.
• Haverkamp R, Vaculin M, Touma J, Wierenga PJ, Vachaud G. “A comparison of numerical simulation models for one-dimensional infiltration”. Soil Science Society of America Journal, 41(2), 285-294, 1977.
• van Genuchten MT. “A closed form equation for predicting the hydraulic conductivity of unsaturated soils”. Soil Science Society of America Journal, 44(5), 892-898, 1980.
• Shu C. Differential Quadrature and Its Application in Engineering. Springer Science & Business Media, 2000.
• Jain MK. Numerical Solution of Differential Equations, John Wiley & Sons (Asia) Pte. Ltd. 1984.
• Zhu J, Mohanty BP. “Analytical solutions for steady state vertical infiltration”. Water Water Resources Research, 38(8), 20-1-20-5, 2002.
• Simunek J, Sejna M, Saito H, Sakai M van Genuchten, MT. The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. Department of Environmental Sciences, University of California Riverside, 2009.