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Determination of optimum relaxation coefficient using finite difference method for groundwater flow

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Abstract

Solution of Laplace’s equation can be done by iteration methods likes Jacobi, Gauss–Seidel, and successive over-relaxation (SOR). There is no new knowledge about the relaxation coefficient (ω) in SOR method. In this paper, we used SOR for solving Laplace’s differential equation with emphasis to obtaining the optimum (minimum) number of iterations with variations of the relaxation coefficient (ω). For this purpose, a code in FORTRAN language has been written to show the solution of a set of equations and its number of iterations. The results demonstrate that the optimum value of ω with minimum iterations is achieved between 1.7 and 1.9. Also, with increasing β = ∆x/∆y from 0.25 to 10, the number of iterations reduced and the optimum value is obtained for β = 2.

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Abbreviations

ω :

Relaxation coefficient

Δx :

Computational grid interval in the x direction

Δy :

Computational grid interval in the y direction

β :

\( \Delta x/\Delta y\,{\text ratio} \,of\,\Delta x\,{\text{per}}\,\Delta y \)

K :

Hydraulic conductivity

c :

Slope of the water table

h :

Groundwater head

h i,j m :

Head in i,j node and time interval m

y 0 :

Assumed initial head at x = 0

s :

Maximum length in the x direction

q :

Unit width discharge

q x , q y :

Discharges per unit width in the x and y directions, respectively

Δt :

Time increment

g :

Gravitational acceleration

j :

jth grid point value

i :

ith grid point value

References

  • Bennett GD (1976) Introduction to ground water hydraulics: a programmed text for self-instruction. Techniques of water resources investigations, book 3, chapter B2. U.S. Geological Survey, 172 pp

  • Freeze RA, Witherspoon PA (1966) Theoretical analysis of regional groundwater flow: 1. Analytical and numerical solutions to the mathematical model. Water Resour Res 2(4):641–656

    Article  Google Scholar 

  • Freeze RA, Witherspoon PA (1967) Theoretical analysis of regional groundwater flow: 2. Effect of water-table configuration and subsurface permeability variation. Water Resour Res 3(2):623–634

    Article  Google Scholar 

  • Freeze RA, Witherspoon PA (1968) Theoretical analysis of regional groundwater flow: 3. Quantitative interpretations. Water Resour Res 4(3):581–590

    Article  Google Scholar 

  • McBride MS, Pfannkuch HO (1975) The distribution of seepage within lakebeds. US Geol Surv J Res 3(5):505–512

    Google Scholar 

  • Remson I, Hornberger GM, Molz FJ (1971) Numerical methods in subsurface hydrology. Wiley Interscience, New York, 389 pp

    Google Scholar 

  • Singh TN, Pradhan SP, Vishal V (2012) Stability of slopes in a fire-prone mine in Jharia Coalfield, India. Arab J Geosci. doi:10.1007/s12517-011-0341-4

  • Surinaidu L, Gurunadha Rao VVS, Ramesh G (2012) Assessment of groundwater inflows into Kuteshwar Limestone Mines through flow modeling study, Madhya Pradesh, India. Arab J Geosci. doi:10.1007/s12517-011-0421-5

  • Tamma Rao G, Gurunadha Rao VVS, Surinaidu L, Mahesh J, Padalu G (2012) Application of numerical modeling for groundwater flow and contaminant transport analysis in the basaltic terrain, Bagalkot, India. Arab J Geosci. doi:10.1007/s12517-011-0461-x

  • Toth J (1962) A theory of groundwater motion in small drainage basins in Central Alberta, Canada. J Geophys Res 67(11):4375–4387

    Article  Google Scholar 

  • Wang HF, Anderson MP (1982) Introduction to groundwater modeling: finite difference and finite element methods. Academic, New York, 237 pp. ISBN-13: 978–0127345857

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Authors and Affiliations

  1. Department of Water Engineering, Faculty of Agriculture, University of Tabriz, Tabriz, Iran

    Farzin Salmasi

  2. River Engineering and Urban Drainage (REDAC), Universiti Sains Malaysia, Seri Ampangan, Malaysia

    Hazi Mohammad Azamathulla

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  1. Farzin Salmasi
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Correspondence to Farzin Salmasi.

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Salmasi, F., Azamathulla, H.M. Determination of optimum relaxation coefficient using finite difference method for groundwater flow. Arab J Geosci 6, 3409–3415 (2013). https://doi.org/10.1007/s12517-012-0591-9

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  • DOI: https://doi.org/10.1007/s12517-012-0591-9

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