Excel based finite difference modeling of ground water flow

Transcription

1 Journal of Himalaan Eart Sciences 39(006) Ecel based finite difference modeling of ground water flow M. Gulraiz Akter 1, Zulfiqar Amad 1 and Kalid Amin Kan 1 Department of Eart Sciences, Quaid-i-Azam Universit, Islamabad Petro Researc & Training Institute, Oil & Gas Development Compan Ltd., Islamabad Abstract Tis paper presents a simple ground water modeling spreadseet template developed in Ecel wit macros written in Visual Basic for Applications (VBA). It is based on Finite Difference metod and can be used to model an aquifer of an sape and size. Te given template can be epanded to build ground water flow models of virtuall unlimited number of nodes. Te final modeled eads, at eac node, can be output as XYZ file, wic can be girded b an contouring software to produce contour maps and 3D surfaces. 1. Introduction Several tpes of models ave been used to stud groundwater flow sstem. Tese can be divided into tree broad categories analog model, matematical models and analtical and numerical models (Prickett, 1975). Hdrogeological studies usuall involve matematical modeling of ground water flow. Suc models consist of a set of differential equations wic govern te flow of groundwater. Te ave been in use since te late 1800 s, but ave found widespread application wit te increase in available computing power. In computer programming tese models are implemented using different approaces among wic te finite difference and finite element metods are most common (Wang and Anderson, 198). In bot tese metods a sstem of nodal points is superimposed over te problem domain. Te difference between te two metods is in te distribution of nodes. Te finite difference nodes are in a regular grid order were nodes can be block-centered or mes-centered. Te finite element metods, on te oter and, can ave an irregular distribution of nodes wic are connected togeter to form triangular sub-areas called elements. Several commercial software applications for groundwater modeling eist (Anderson and Woessner, 199). To use tese applications properl, te user needs to be full trained. In tis paper we present an Ecel spreadseet template for modeling groundwater wic is ver eas to use and requires no training. Due to te grid nature of Ecel cells te finite difference metod is used were eac cell represents a grid node. Pre-programmed sample cells for interior nodes and noflow boundar nodes for different sides and corners are given in te template. Tese cells are copied to te design region according to te requirements of te aquifer to be modeled. Boundar values for constant ead nodes are defined and te iterative procedure is initiated to get te flow model. Tis spreadseet template can be used in teacing as well as real groundwater modeling proects.. Matematical background Te psics of groundwater flow in tree dimensions is defined b Darc s law as q K ; q K ; q z K z (1) were q, q, q z are te specific discarge in te,, z directions, K is te Hdraulic Conductivit, and is te ead wic is te function of all tree space coordinates and terefore represented b partial derivates. Te second important law is te continuit or conservation wic for stead state conditions states tat te amount of water flowing into a representative elemental volume must be equal to te amount flowing out. Matematicall it can be epressed b te continuit equation as q q qz 0 () z Now combining te above two equations and assuming K to be independent of,,z for a omogeneous and isotropic Te finite difference approimation to Laplace s equation (McDonald and Harbaug, 1998) for suc a grid is given b aquifer we get a single second-order partial differential equation z 0... (3) 49

2 Tis is Laplace s equation wic governs te flow of groundwater troug an isotropic, omogeneous aquifer under stead state conditions. Tis equation simpl states tat te sum of partial derivates of ead () wit respect to,, and z is zero. Te solution of Laplace s equation requires specification of boundar conditions suc as Diriclet conditions and Neumann conditions (Neuman, 1973). As currentl we are dealing wit ground water flow in two dimensions terefore te Laplace s equation reduces to 0...(4) Fig. 1. Finite difference grid of nodes. Now consider a regularl spaced grid of nodes represented b i columns and rows and orizontall and verticall spaced b distances and respectivel as sown in figure 1. Te ead at node i, is,. i 1, 1 ( ) ( ) i 1, 1 i Adding tese two terms according to Laplace s equation and considering for a square grid we ave 0 (5) i 1, i 1, Te above equation is te most widel used equation in finite difference solutions of stead-state flow problems and forms te eart of tis program. It is iterativel used in te form of a kernel or Laplacian operator for computation of eads at eac node in te grid. Suc operators are also used in satellite image processing for edge detection and image smooting (El- Seim et al., 005). 3. Working procedure Te complete working procedure of te presented modeling template is given in Figure. Full details for using tis template are given as a elp workseet in te Ecel Modeling file. To design a new model te user simpl needs to cop and paste te required sample cells into te design region of te template (Fig. 3) according to te sape, size and tpe of boundaries tat define te aquifer. Te values for constant ead nodes at te boundaries are also defined. If required te iteration parameters; Maimum iterations (default 100) and Maimum cange (default.001) are set using te Ecel menu; Tools > Options > Calculation Tab. During eac iteration te value of eads at all nodes are calculated. Te program keeps on iterating unless convergence occurs i.e. te absolute difference in eads at eac node from previous and current iteration is less tan te Maimum cange value. Tis parameter is often critical to te modeling process. Too large a value will degrade te final solution, because convergence will not reall ave occurred. Too small a value will lead to ver long eecution times and often result in no convergence because computational errors ma cause te difference in eads to remain larger tan te Maimum cange parameter. For tis purpose te Maimum iteration value is also given so tat te program does not enter into an infinite iterative loop. Tus iterations stop wen conditions for eiter of tese two parameters are satisfied. Now to start te iterative process te user simpl needs to press F9 ke. Te values in eac cell representing a grid node will keep on canging unless convergence occurs and a stead state flow is attained. Te modeled ead values for te grid nodes can be saved as a XYZ file (X, Y are columns, rows and Z is ead) using te Visual Basic for Applications macro. It is activated b pressing Ctrl-S kes for entering te output filename. Te output file can be used b Surfer or an oter contouring software for girding and subsequent contouring or surface generation. Te final contour map represents te variation of ead in te modeled region, were flow lines are considered to be moving perpendicular to te contour lines from iger to lower potential. Simulation results of tis spreadseet can be utilized in different commercial software s like PROCESSING MODFLOW (Ciang and Kinzelbac, 1998) etc. to develop a complete numerical ground water model. 50

3 Fig.. Processing flow cart of te modeling template along wit Surfer. Fig. 3. Ecel spreadseet template for modeling groundwater. Cells (gra to dark gra) at te top are preprogrammed for interior nodes and various tpes of no-flow boundar nodes. Te empt cells below form te design region were te model is created. 51

4 4. Practical eamples To demonstrate te design, working and output of te modeling spreadseet template two eamples are sown in Figures 4 and 5. Te also sow te final grapical outputs generated b contouring software. Tese eamples are also included in te Ecel Modeling file as eample workseets. Te elp te user in designing teir own groundwater models in te empt spreadseet template. 5. Conclusions It is concluded tat Ecel based spreadseet modeling can be effectivel utilized to develop te equipotential surface features of te groundwater flow regimes. Directions of groundwater flows are monitored b assigning te vector lines at rigt angle to tese equipotential surfaces. Groundwater volume in active storage can be ascertained from te complete flow-net based on Ecel based spreadseet modeling. In addition it can also be used, along wit contouring software, to generate small to large groundwater flow models. Fig. 4. Eample of a small groundwater flow model wit contour map and 3D surface from Surfer. 5

5 Fig. 5. Eample of a larger groundwater flow model wit contour map from Surfer. References Anderson, M.P., Woessner, W.W., 199. Applied Groundwater Modeling, Academic Press, USA, 0-1. Ciang, W.H., Kinzelbac.W., (PMPATH 98). An advective transport model for Processing Modflow and Modflow. El-Seim, N., Valeo, C. Habib, A., 005. Digital Terrain Modeling, Acquisition, Manipulation and Applications, Artec House Inc., -4. McDonald, M. G., Harbaug, A, W., A modular tree-dimensional finite difference ground water flow model. Modeling Tecniques, Book 6, USGS open-file report, Neuman, S. P, Calibration of distributed parameters groundwater flow models viewed as a multipleobective decision process under uncertaint. Water Resources Researc 9(4), Prickett, T.A., Modeling Tecniques for groundwater evaluation. In advance in Hdroscience. New York, Academic Press, Wang, H.F., Anderson, M.P., 198, Introduction to groundwater modeling finite difference and finite element metods. W.H. Freeman and Compan,