Seepage Analysis through Earth Dam Based on Finite Difference Method

Transcription

1 J. Basic. Appl. Sci. Res., (11)111-1, 1 1, TetRoad Publication ISSN -44 Journal of Basic and Applied Scientific Researc Seepage Analysis troug Eart Dam Based on Finite Difference Metod E. Fadaei ermani 1,, G. A. Barani 1 1 Department of Civil Engineering, Said Baonar University, erman, Iran Young Researcer Society, Said Baonar University, erman, Iran ABSTRACT In te present paper, te finite metod (FDM), te five-point approimation tecnique, as been presented to deal wit seepage problem in eart dams. Te grid system, wit computational boundary being coincident wit te pysical boundary, was numerically obtained by solving Laplace equation. Te metod was applied to analye te steady seepage in an eart dam. In tis study, tree different grid types were considered, and te results were compared wit ones obtained by analying wit 7 software. It sowed tat by coosing small enoug increments, te results are satisfactory. EYWORDS: Seepage, Eart dam, Finite metod,. 1. INTRODUCTION Problems concerning movement of water troug soil as porous medium are very broad. In most proects of Civil engineering especially in water and soil trends, irrigation and Oil engineering, te fluid flow is discussed in a porous medium and tis topic is one of te proposed problems in all ydraulic structures and especially in eart dams and embankments. Water moving in a permeable soil media produces a force on te volume of soil wic is proportional to ydraulic gradient in desired direction and is called te seepage force. Determination of te seepage force is required to calculate stability of dams and oter ydraulic structures [1]. Anoter penomena tat is raised about dams, is piping. Tis penomenon is an erosion process tat may occur in te body and also under foundation of te dam. It occurs in places tat tere is concentrated seepage and output ydraulic gradient eceeds te critical ydraulic gradient. In oter word, resistance force of soil against erosion is less tan destructive force of seepage at tat point. Tis may cause to sudden destruction of te dam []. Analying te water flow troug a permeable media was first begun in 18 by introducing Darcy s law. Ten it was sown tat water flow in isotropic permeable media can be discussed by Darcy s law and tis law forms foundations of studies of water seepage in permeable soil media. Te Laplace equation wic governs water seepage cannot be solved analytically, ecept for cases wit very simple and special boundary conditions. Terefore, researcers ave invoked to empirical, grapical and recently numerical metods []. Recent developments in computer science ave advanced te use of numerical tecniques in seepage problems, and new works sow te capability of tese tecniques [4-7]. Te finite approimation metod is a convenient metod used to solve te Laplace equation wic governs water seepage troug soil media. In tis paper, te five-point approimation metod is applied to deal wit steady seepage analysis in omogeneous isotropic medium. Governing Equations Te -D governing equation describing te water flow troug a porous medium in steady state and obeying te Darcy s law cab be written as: ( ) ( ) (1) were and are ydraulic conductivities in te oriontal and vertical directions and is te water ead. If te soil is omogeneous wit respect to te ydraulic conductivity, Equation 1 simplifies to () * Corresponding Autor: Esan Fadaei erman Said Baonar University of erman, Department of Civil Engineering, erman, Iran. esanard@gmail.com 111

2 ermani and Baran 1 Te Finite Difference Metod Te finite metod is a numerical procedure used to solve a partial differential equation by discretiing te continuous pysical domain into a discrete finite grid, approimating te individual eact partial derivatives te partial differential equation by algebraic finite approimations, substituting tese approimations into te partial differential equation to obtain an algebraic finite equation, and solving te resulting algebraic equations for te dependent variable [8]. Te finite approimations developed by writing Taylor series for te depended variable at several neigboring grid points using grid point () as te base point, and combining tese Taylor series to solve for te desired partial derivatives. Replacing te derivatives in te -D seepage equation, Equation, by te second-order centered approimations at grid point (), yields i 1, i 1, i, 1 i, i, 1 ( ) ( ) i, i, () Equation is a second-order centered- approimation of Equation. Equation can written as were 1 1 (1 ) (4). Equation 4 is called Te approimation of te Laplace equation. Solving Equation 4 for, yields i 1 (1 ) 1 () Te finite stencil for te five-pint metod is sown in Figure 1. Figure 1. Finite stencil for te five-pint metod. 4 Boundary Conditions Boundary conditions are required at te boundaries of solution domain. Te boundary conditions governing te solution of te Laplace equation in seepage analysis troug eart dam could be divided into te following types. Diriclet boundary condition: In tis case te function values are specified on boundaries. In seepage analysis troug te eart dam, te Diriclet boundary condition is dominant wen te water ead is specified on boundaries, for eample upstream and downstream of te dam or on Preatic surface (saturated line). Neumann boundary condition: In tis case te function derivative values are specified on boundaries. For eample in seepage analysis, at te last flow line (impervious layer), ydraulic gradient is equal to ero in vertical direction. 11

3 J. Basic. Appl. Sci. Res., (11)111-1, 1 Using a second-order centered- approimation for te derivative boundary condition yields i. 1 1 ( ) () Substituting Equation into Equation 4 yields i, 1 i 1, (1 ) (7) Equation 7 applies at derivative boundary conditions to determine water ead values. Eample of Practice Figure sows a definition sketc for an isotropic and omogeneous eart dam wit a toe drain at downstream built up on an impervious oriontal base. Te ydraulic conductivity of materials from wic te dam is constructed is equal to.*1 m/ s. Solution and Results Figure. Geotecnical section of te dam. Te first flow line (Preatic surface) is calculated according to Casagrande s metod, and te closed solution domain is created in space. Te finite grids are created for different spacings Δ and Δ. Te water ead values are calculated at grid points by solving governing equations (Equations and 7) simultaneously according to boundary conditions. Te preceding eart dam was modeled and analyed wit 7 software. Figure sows te model analyed wit software. 11

4 ermani and Baran 1 Figure. Presentation of te model analyed wit software. Te water ead values obtained by te five-point approimation metod for different grid sies, Δ and Δ, are summaried in Tables 1 and. Tese values were compared wit te results obtained by software. Distance Table 1. Water ead values for Δ= m and Δ=m. 1 Metod Distance Table. Water ead values for Δ=.m and Δ=m. 1 Metod

5 J. Basic. Appl. Sci. Res., (11)111-1, 1 Distance Table. Water ead values for Δ= 1m and Δ=1m. Metod Conclusions Seepage analysis is necessary for eart dam design. Irregular seepage troug te eart dam may be tread to te integrity and stability of te structure and could lead to te failure of te dam. In tis paper, te five-point metod based on te finite approimations is presented to deal wit te steady seepage troug an eart dam. An isotropic and omogeneous eart dam wit a toe drain at downstream built up on an impervious oriontal base is considered in tis paper. Te solution closed domain is created, and te grid system is numerically obtained wit computed boundary coinciding wit te pysical boundary. Te solution of seepage problem by te five-point metod is compared wit te solution obtained by software. It sows tat wit small enoug Δ and Δ, te results are satisfactory. It is to be mentioned tat some oter numerical metods, including finite volume metod, finite element metod and FEM, ave been utilied before to deal wit seepage problems. Tese metods are more complicated compared wit te finite metod. Te evaluation of capability of te five-point metod for seepage analysis troug an isotropic and omogeneous eart dam proved to be successful. However, it is suggested tat tis metod applied wen te medium is inomogeneous. REFERENCES 1. Das, B.M, 18. Advanced Soil Mecanics. McGraw-Hill. New York.. Cedergren, H.R, 1. Seepage, Drainage and Flow Nets. Wiley. New York.. Desa C.S. and Cristain, J.T, 177. Numerical Metods in Geotecnical Engineering. McGraw-Hill. New York. 4. Toufig, M.M,. Relationsip seepage wit nonlinear permeability by least square EEM. International Journal of Engineering. (), pp Yuin, J., Guanou, J., Zeyu, M. and Guangin, L, 4. A seepage analysis based on boundary-fitted coordinate transformation metod. Journal of Computers and Geotecnics. 1 (4), pp Benmebarek, N., Benmebarek, S., and astner, R,. Numerical studies of seepage failure of sand witin a cofferdam. Journal of Computers and Geotecnics. (4), pp Junfeng, F.U. and Seng, J,. A study on unsteady seepage flow troug eart dam. Journal of Hydrodynamics., 1 (4), pp Hoffman, J.D, 1. Numerical Metods for engineers and Scientists. McGraw-Hill. New York. 1