Excel Based Finite Difference Modeling of Ground Water Flow

This is Laplace’s equation which governs the flowof groundwater through an isotropic, homogeneousaquifer under steady state conditions. This equationsimply states that the sum of partial derivates of head(h) with respect to x, y, and z is zero. The solution ofLaplace’s equation requires specification of boundaryconditions such as Dirichlet conditions and Neumannconditions (Neuman, 1973). As currently we aredealing with ground water flow in two dimensionstherefore the Laplace’s equation reduces to2 2h h0 …………...(4)2 2x yFig. 1.Finite difference grid of nodes.Now consider a regularly spaced grid of nodesrepresented by i columns and j rows and horizontally andvertically spaced by distances x and y respectivelyas shown in figure 1. The head at node i ,j is h ,.2h2x2h2yhhi 1, ji,j 12hi,j( x)2h(2i,jy)2hhi 1, ji,j 1i jAdding these two terms according to Laplace’sequation and considering x y for a square gridwe haveh h h h h 0 …(5)i 1,j i 1, j i,j 1 i,j 14 i,jThe above equation is the most widely usedequation in finite difference solutions of steady-stateflow problems and forms the heart of this program. It isiteratively used in the form of a kernel or Laplacianoperator for computation of heads at each node in thegrid. Such operators are also used in satellite imageprocessing for edge detection and image smoothing (El-Sheimy et al., 2005).3. Working procedureThe complete working procedure of the presentedmodeling template is given in Figure 2. Full details forusing this template are given as a help worksheet in theExcel Modeling file. To design a new model the usersimply needs to copy and paste the required samplecells into the design region of the template (Fig. 3)according to the shape, size and type of boundaries thatdefine the aquifer. The values for constant head nodesat the boundaries are also defined.If required the iteration parameters; Maximumiterations (default 100) and Maximum change (default.001) are set using the Excel menu; Tools > Options >Calculation Tab. During each iteration the value ofheads at all nodes are calculated. The program keeps oniterating unless convergence occurs i.e. the absolutedifference in heads at each node from previous andcurrent iteration is less than the Maximum changevalue.This parameter is often critical to the modelingprocess. Too large a value will degrade the finalsolution, because convergence will not really haveoccurred. Too small a value will lead to very longexecution times and often result in no convergencebecause computational errors may cause thedifference in heads to remain larger than theMaximum change parameter. For this purpose theMaximum iteration value is also given so that theprogram does not enter into an infinite iterative loop.Thus iterations stop when conditions for either ofthese two parameters are satisfied. Now to start theiterative process the user simply needs to press F9key. The values in each cell representing a grid nodewill keep on changing unless convergence occurs anda steady state flow is attained.The modeled head values for the grid nodes can besaved as a XYZ file (X, Y are columns, rows and Z ishead) using the Visual Basic for Applications macro. Itis activated by pressing Ctrl-S keys for entering theoutput filename. The output file can be used by Surferor any other contouring software for girding andsubsequent contouring or surface generation. The finalcontour map represents the variation of head in themodeled region, where flow lines are considered to bemoving perpendicular to the contour lines from higherto lower potential.Simulation results of this spreadsheet can beutilized in different commercial software’s likePROCESSING MODFLOW (Chiang and Kinzelbach,1998) etc. to develop a complete numerical groundwater model.50