Finite Difference Method of Modelling Groundwater Flow

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1 Jornal of Water Resorce and Protection, 20, 3, doi:0.4236/warp Pblished Online March 20 ( Finite Difference Method of Modelling Grondwater Flow Abstract Magns. U. Igboekwe*, N. J. Achi Department of Physics, Michael Okpara University of Agricltre, Umahia, Nigeria Received Janary, 20; revised Febrary 8, 20; accepted March 2, 20 In this stdy, finite difference method is sed to solve the eqations that govern grondwater flow to obtain flow rates, flow direction and hydralic heads throgh an aqifer. The aim therefore is to discss the principles of Finite Difference Method and its applications in grondwater modelling. To achieve this, a rectanglar grid is overlain an aqifer in order to obtain an exact soltion. Initial and bondary conditions are then determined. By discretizing the system into grids and cells that are small compared to the entire aqifer, exact soltions are obtained. A flow chart of the comptational algorithm for particle tracking is also developed. Reslts show that nder a steady-state flow with no recharge, pathlines coincide with streamlines. It is also fond that the accracy of the nmerical soltion by Finite Difference Method is largely dependent on initial particle distribtion and nmber of particles assigned to a cell. It is therefore conclded that Finite Difference Method can be sed to predict the ftre direction of flow and particle location within a simlation domain. eywords: Finite Difference Method, Grondwater Modelling, Particle Tracking, Algorithm, Discretization, Flow Rates, Hydralic Heads. Introdction Grondwater models are mathematical models derived from Darcy s law which is sed to calclate the rate and movement of grondwater throgh the aqifer and confining nits in the sb-srface []. Grond-water models which can also be sed to evalate impact assessment reqired for water in a reglated aqifer system has been of importance to agricltrists, environmentalists, hydrologists etc. It is necessary to stdy the grondwater resorce potentials of a site becase the simlation of grondwater flow reqires a thorogh knowledge and nderstanding of hydrogeologic characteristics of the site. Grondwater models are sed as tools for decision making in the management of a water resorce system. They may also be sed to predict some ftre grondwater flow. Some of the established soltion techniqes available for solving the governing eqations of the model are Finite Difference and Finite Element approximation or a combination of both provided that model parameters and initial and bondary conditions are properly specified. The nmerical soltion applied in this research work is the finite difference method. This is an old method made more sefl with the advent of high speed compters (digital compters). This method is an approach to comptational flid dynamics (CFD) and very effective in grondwater flow modelling. Grondwater is an important resorce in so many areas for its se as a sorce of drinking water and irrigation water. In many areas, grondwater is threatened by leaching of pesticides and other agricltral chemicals and the leaching of indstrial chemicals from hazardos-waste sites. Becase of the importance of this resorce, and becase the degradation of grondwater cannot be easily reversed, the assessment of threat to grondwater qality from hman activities is often reqired. So grondwater models are increasingly sed as part of this assessment. The maor aim of this research work is to discss the principles of Finite Difference Method and its application in grondwater modeling The word model is simply a representation of a real system or process. Bt model has variety of definitions that it is often difficlt to define [2]. A model is a hypothesis for how a system or process operates. These models in se today are deterministic mathematical models that are based on conservation of mass, momentm and energy. Modelling is the process by which a physical system is simplified to obtain a mathematically tractable sitation. The reslting simplified Copyright 20 SciRes.

2 M. U. IGBOEWE ET AL. 93 version of the real system is called model of the system or a mathematical model [3]. Nmerical modelling of grondwater is a relatively new field. It was not extensively prsed ntil the mid-960s when digital compters with adeqate capacity became generally available. Since then, significant progress has been made in the development and application of sch techniqes to grondwater flow. Nmerical models are sed in grondwater modelling becase it yields approximate soltions to the governing eqations throgh the discretization of space and time. It helps in assessing the impact of polltion on an aqifer. Grondwater models generally reqire the soltion of partial differential eqations. The eqations describing the grondwater flows are second order partial differential eqations which can be classified on the basis of their mathematical properties. There are basically three types of second order partial differential eqations: parabolic, hyperbolic and elliptic eqations [4,5]. The two main types of nmerical models that are accepted for solving the grondwater eqations are the Finite Difference Method and the Finite Element Method presented by [6,7]. Both of these nmerical approaches reqire that the aqifer be sb-divided into a grid and analyzing the flows associated within a single zone of the aqifer or nodal grid. The eqation describing the grondwater flow is a Partial Differential Eqation. It can be solved mathematically by analytic or nmerical soltions. Bt analytic soltions are very difficlt to apply becase it reqires that parameter and bondaries shold be highly idealized. The advantages of analytical soltion, if it is possible to apply, are that it yields an exact soltion to the eqation and it is simple and efficient to obtain. Many of them have been developed for the flow eqations, bt most of them are limited to well hydralics problems [8]. The continos variable is replaced by discrete variables that are defined at grid blocks. Also the continos differential eqations which define the hydralic head in the system, is replaced by a finite nmber of head at different grids [9]. A common method for soltion of this eqation in civil engineering and soil mechanics is to se the graphical techniqes of drawing flow nets, where contors of hydralic head and the stream fnction make a crvilinear grid, allowing complex geometries to be solved approximately. The grondwater flow eqation is the mathematical relationship which is sed to describe the flow of grondwater throgh an aqifer [6]. In the stdy of grondwater flow eqation, one may discss abot transient flows and steady state flows. The transient flow which is described by a form of diffsion eqation similar to that sed in heat transfer to describe heat condction is the change in flow condition from one steady-state to another. The steady-state flow is described by a form of Laplace eqation. It is a flow in which all conditions (velocity, pressre, etc) remain constant with respect to time. The grondwater flow eqation is often derived for a small representative elemental volme where the properties of the medim are assmed to be effectively constant. A mass balance is obtained on the water flowing in and ot of this small volme along with Darcy s law to arrive at the transient grondwater flow eqation. The flow eqation is based on the continity eqation [0]: Inflow Otflow = change of storage. () For a small portion of an aqifer it can be restated as: Sbsrface sm + net flow= change in storage (2) Combining Darcy s law with this continity eqation yields the general form of the eqation describing the transient flow h Ss q G (3) t where S s = specific storage q = flx h = Hydralic head t = time The statement indicates that the change in hydralic head with time eqals the negative divergence of the flx (q) and the sorce (G). Here the head and flx are nknown bt Darcy s law relates the flx to the head by sbstitting it in the flx, that is h Ss h G (4) t where G = external flow = hydralic condctivity Simplifying this we have h 2 Ss hg (5) t if is replaced by it s eqivalent T, Eqation (5) can be written as h h h Ss Txx Tyy t x x y y (6) h Tzz G z z where: T = transmissivity A steady-state may be reached if the aqifer has recharging bondary conditions (or it may be sed as an approximation in many cases). The eqation describing this flow is a form of the Laplacian eqation given as 2 h 0 (7) This eqation states that hydralic head is a harmonic fnction and has many analogs in other fields. The eqa- Copyright 20 SciRes.

3 94 M. U. IGBOEWE ET AL. tion above can be rewritten as h h h x y z The above grondwater flow eqations are valid for three dimensional flow. We also have two dimensional grondwater flow and the general governing eqation is given by h h h b x b y G Ss x x y y t where k = hydralic condctivity b = satrated thickness Bt, b = T, therefore: h h h Tx Ty G S s (0) x x y y t In finite difference form, eqn.0 can be expressed as: WiB J+I J+I J+I AS B B J+I J TiBhB hb ABGB hb h B L t () ib where W i and L i are bondary width and flow path length. A = Area of a single zone. Grondwater velocity is based on hydralic condctivity (k), as well as the hydralic gradient (I). Therefore the eqation determined by Darcy to describe the basic relationship between sb-srface material and the movement of water throgh them is Q = IA (2) where Q = volmetric flow rate (Discharge) = Hydralic condctivity A = Area that the grondwater is flowing throgh This relationship is known as Darcy s law []. Rearranging Eqation (2) we obtain the flx (V) which is known as the apparent velocity. That is: Q I V (3) A where V = Apparent velocity, m / sec The actal grondwater velocity which is called the Darcy s flx (V x ) is given by Q I Vx (4) A n where V x = Actal velocity m / sec n = Porosity This is the actal velocity of grondwater and does accont for tortosity of flow paths by inclding porosity in its calclation. The Finite Difference Method is a comptational procedre based on dividing an aqifer into a grid (see Figre ) and analysing the flows associated within a single (8) (9) zone of the aqifer [2]. It tilizes a time distance grid of nodes and a trncated Taylor series approach to determine the condition of flow at any particlar node. A brief coverage of the application of Taylor Series and nodal grid will illstrate several points fndamental to flow simlation. The flow at time t, the profile of variable y with x may be described by a trncated Taylor Series given as Y x y x y x 0 x 0 0 x 2 3 x x y x0 y x0 2! 3! (5) The vale at x of node i is x 0 and the + notation in eqation 5 allows it to be sed in forward or backward difference approach. The forward difference first order is Y x0 y x0 x yx0 (6) The backward difference first order is Y x0 y x0 x yx0 (7) The smmation of forward or backward difference also yields a central difference expression for the first order derivative of variable y at x 0. The central difference first order x Y x0 yx0i, yx0 x (8) 2x In compter notation, these eqations (6-8) can be re-written as i+ i xi h i i xi h i x i 2h Figre. Compter notation for finite difference grid. Copyright 20 SciRes.

4 M. U. IGBOEWE ET AL. 95 If there is recharge to the aqifer, a more sefl reslt is the smmation of the second order forward or backward difference forms of the trncated Taylor series to determine the second differential of variable y at x 0 i.e. 2 i+ i xi 2 - h i (9) A special notation is sed to describe the positions of the node in the finite difference grids. In order to solve the grondwater flow eqation we mst be able to specify the initial and bondary conditions. There are two basic types of bondary conditions. If the head is known at the bondary of the flow region, this is known as a Drichlet condition. Then if the flx across a bondary to the flow region is known, this is known as Nemann condition. For a steady-state problem, bondary conditions are reqired; whereas for a transient problem, bondary and initial conditions mst be specified. Bt in some cases the bondary conditions will be mixed with some portions having known head and some portions having known flx [3].Initial conditions and bondary conditions can be related to levels, pressre, and hydralic head on the one hand (head conditions), or to grondwater inflow and otflow on the other hand (flow conditions).as an example of bondary conditions, let s consider Figre 2 below. This is a sand and gravel aqifer overlying an impermeable basement rock. There are several flow regions present, each leading away from a grondwater divide towards a stream. The plane of W to X is a grondwater divide. There is no flow across the divide and the bondary condition along the divide is dh 0 dx (20) The plane of Z to Y is the centre of the stream and it is also a grondwater divide, across which no flow takes place. The bondary condition here is also: dh 0 dy (2) No flow occrs from the sand deposit into the impermeable bedrock, so along the X to Y plane there is no flow and the bondary condition is dh 0 dz (22) These three bondaries are Nemann bondaries where the flow conditions are specified. Figre 2. Bondary conditions for a cross section of a regional aqifer. grid. Several different hydralic heads and flx vales are specified along the bondaries. A river enters the aqifer at the northern bondary and exits at the soth. The prodction wells near the river withdraw water from the aqifer. Grondwater flow occrs mainly from the north to the soth of the domain. The aqifer is hetero- 2. Methods A hypothetical site sch as the one shown below (Figre 3) is chosen. The domain is discretized into an irreglar Figre 3. Discretized domain of a hypothetical site showing wells and river sorce. Copyright 20 SciRes.

5 96 M. U. IGBOEWE ET AL. geneos and anisotropic. A finite difference grid is laid over an aqifer. It is a block centred finite difference grid where the node points fall in the centre of the grid. The grid parameters are: Nmber of the grid colmns 43 Nmber of grid rows 65 Minimm i coordinate 0 Minimm coordinate 0 The basic grid is reglar, with the rows and colmns being normal to each other. The rows and colmns may be varied so that there are more node points in certain parts of the aqifer than others. Bt this is desirable in the area arond a well field. For nconfined aqifers the satrated thickness b i is a fnction of the hydralic heads. A priori both the hydralic heads and the satrated thickness are nknown. Mathematically, this leads to non-linear behavior. An initial gess for the satrated thickness is reqired in order to estimate the aqifer transmissivities T xxi, and T yyi,. An iterative scheme mst be sed to calclate the hydralic head distribtion, pdate the transmissivities and check if the discrepancy between the previosly estimated satrated thickness and the pdated one is greater than a specific tolerance i.e. New old hi, h, i max min h h > toleranc (23) The directional transmissivities between adacent nodes: T xxi + ½, T yyi, + ½ are calclated by mltiplying harmonic mean of hydralic condctivities and geometric mean of satrated thickness [4]. T h BT h BT xxi2 xxi2 i, i, i, i T h BT h BT yy2 yyi2 i, i, i, i (24) where BTi, = aqifer bottom elevation at node i, xxi + ½, yyi + ½ = Hydralic condctivities between blocks i, and i+,. i,, xxii, xxi, xxi 2 xxi, xi, xxi, xi, x xi y yi i,, yyi, yyi, yyi 2 yyi, yi, yyi, yi, (25) There are two different types of velocity vectors (a) Nodal velocities (b) Bondary velocities Bondary velocities are calclated directly from Darcy s law sing the hydralic head difference and material properties between two nodes, whereas the nodal velocities are interpolated vales at the grid nodes. The internodal components of the Darcy s flx q x and q y are obtained by differentiation of the calclated hydralic heads. hi, h i qxi 2, xxi2, xi, 2 (26) hi, hi qyi 2, yyi 2, yi, 2 where, x i+ ½,, y i, + ½ = distance between nodes i, and i +, and nodes i, and i, + respectively xxi + ½,, yyi + ½, = Hydralic condctivities between i + i, at x direction and nodes i, and i, + at y direction respectively These vales are taken as the weighted harmonic mean between the hydralic condctivities. xxi,+ + 2,, = yyi + 2 Δ x + Δx = Δ x + Δ x i, i+, xxi, xxi+, xxi, i, xxi+ I, i+, Δ y + Δy Δ y + Δy i, i+, yyi, yyi, + yyi, i, yyi, + i, + (27) From the Darcy s flxes the pore velocities are calclated as: V V xxi2, yyi, 2 qx n qy n i2, i2, i2, i, 2 (28) where n i + ½,, n i, + ½ = effective porosities between nodes i, and i+, ; and nodes i, and i, + respectively. These vales are taken as the weighted arithmetic mean between the porosities of adacent blocks. nixi, ni, xi+, ni+ 2, xi, xi+, (29) niyi, ni+, yi+, ni, + 2 yi, y i+, Particle tracking provides a clearer description of grondwater flow within an aqifer. In steady-state flow field with no recharge, pathline (particle trace) coincides with streamlines. The two dimensional eqation of pathlines is given by: P x, y P x, y v d t (30) 0 0 where P = Vector containing the x, y coordinates of the pathline. P(x o,y o ) = The starting point of the pathline (initial condition). V = Average linear velocity t = Time Copyright 20 SciRes.

6 M. U. IGBOEWE ET AL. 97 Eqation 30 is written in discretized form as an explicit time stepping scheme X + Δ = X V Δt t t t xt Y+ t Δt =Yt VytΔt where Δ t = time increment, V xt, V yt, are the pstream components of the average linear velocity of the crrent particle locations at time t. In this scheme, a set of niformly distribted particles is assigned to each calclation cell. With each time step, every particle is moved to a new location based on velocity of the particle in the x and y directions. The velocity component at particle location is to be obtained from iterative scheme. In this scheme the particle representative area is assigned to each particle by dividing the cell into a set of eqal nmber of sqared sb cells with each particle at its centre. The temporal weighted velocity is to be obtained by forth order classical Rnge tta method [5]. A weighted velocity is based on its vales evalated at for points in time, and then it is sed to move the particle to a new direction. Figre 4 shows the particle tracking algorithm. The aqifer is discretized sing eqally sized calclation cells. In the particle tracking scheme, one particle per cell is sed. velocity component at particle location is obtained from iterative schemes. It is also fond that the accracy of the nmerical soltion by Finite Difference Method is largely dependent on initial particle distribtion and nmber of particles assigned to a cell. This scheme provides improved accracy over interpolation scheme especially for block heterogeneos aqifer bt with increased comptational efforts. 4. Conclsions Nmerical modelling has fond interesting application in grondwater flow and transport since the mid-960 s, when digital compters with adeqate capacity became generally available. The need for a nmerical model cannot be over emphasized since analytical models assme a homogenos aqifer. 3. Reslts and Discssion The hydrodynamics of grondwater flow in a hypothetical aqifer chosen for this stdy has been fond to depend mainly on the srface water flow within the site. The srface water flow is from north to soth. So also is the grondwater. The aqifer is heterogeneos and anisotropic. In the limiting case of a model calclation of an infinitely small grid space, the soltion approaches the exact soltion. Nodal and bondary velocities are calclated as the weighted harmonic mean between hydralic condctivities and the arithmetic mean of the porosities of adacent blocks. Initial and bondary conditions are sed to specify the flow condition. A nmerical problem had been taken which allows the testing of nmerical modelling techniqe called Finite Difference Method for the simlation of grondwater flow. The region is bonded by two no flow bondaries as show in Figre 2. The aqifer is discretized sing eqally sized 2 x 2 calclation cells. In the scheme, one particle per cell is sed. Reslts from the flow chart of a particle tracking algorithm show that in a steady-state field with no recharge, pathlines coincide with streamlines. This means that the Figre 4. Flowchart of the comptational algorithm for particle tracking. Copyright 20 SciRes.

7 98 M. U. IGBOEWE ET AL. The nmerical method applied in this stdy is the Finite Difference Method, which in its simplicity provides necessary aids in finding soltion to grondwater problems. The finite difference grid overlain over the aqifer improves the accracy in the calclation of flow rate and direction. It is also an advantage in particle tracking within the aqifer domain. The modelling stdies have been carried ot in a hypothetical site with the aim of determining rate and direction of flow of grondwater throgh an aqifer domain. The domain is discretized with bondary conditions considered for easy calclations. In the flow model it was observed that water flows from region of high hydralic head to region of low hydralic head and that an exact soltion cold be obtained if the grid spacing is small enogh say 5 cm. Reslts show that in a steady-state flow field with no recharge, pathlines coincide with streamlines. It is therefore conclded that Finite Difference Method can be sed to predict the ftre direction of flow and particle location within a simlation domain. 5. References [] R. A. Freeze, Three Dimensional Transient Satrated Unsatrated Flow in a Grondwater Basin, Water resorces Research, 979, pp [2] L. F. onikow and J. D. Bredehoeft, Compter Model of Two Dimensional Solte Transport and Dispersion in Grondwater, U.S. Geological Srvey, Techniqes of Water Resorces Investigation Book 7, Chapter C2, 992, p. 90. [3] V. V.Hng and Ramin S. Estandiari, Dynamic Systems: Modelling and Analysis, McGraw-Hill, New York, 997 p. 62. [4] J. M. McDonald and A. W. Harbagh, A Modlar Three-Dimensional Finite-Difference Grondwater Flow M- ode, Techniqes of Water Resorces Investigations of the U. S. Geological Srvey Book.6, 988, p [5] T. Narasimha Reddy and V. V. S. Grnadha Rao, Water Balance Model and Grondwater Flow Model of Dlapally Basin, Granitic Terrain, A.P., Research Series No. 9, 99. [6] M. P. Anderson and W. W. Woessner, Applied Grondwater Modeling, Simlation of Flow and Advective Transport, Academic Press, San Diego, C. A, 992. [7] M. U. Igboekwe, V. V. S. Grnadha Rao and E. E. Okweze, Grondwater Flow Modelling of wa Ibo River Watershed Sotheastern Nigeria, Hydrological Processes, Vol. 22, No. 0, 2008, pp doi:0.002/hyp.6530 [8] M.. Hbbert, Darcy s Law and Field Eqations of Flow of Undergrond Flids Transaction, American Institte of Mining and Metallrgical engineers, 956, pp [9] T. A. Prickette and C. G. Longist, Selected Digital Compter Techniqes for Grondwater Resorce Evalation, Bll.55, Illinois State Water Srvey, Urbana, 98. p. 62. [0] W. J. Gray and J. L. Hoffman, A Nmerical Model Stdy of Grond-Water Contamination from Price's Landfill, New Jersey I. Data Base and Flow Simlation, Grondwater, Vol. 2, No., 983, pp doi:0./ tb00699.x [] D.. Todd, Grondwater Hydrology, 2 nd Edition, John Wiley and Sons, New York, 200. [2] Peter. C. Trescott and S. P. Larson, Finite Difference Model for Aqifer Simlation in Two-Dimensions with Reslts of Nmerical Experiments, U.S. Geological Srvey Techniqes at Water-Resorces Investigations, Book 7, Chapter C, 976, p. 6. [3] C. W. Fetter, Applied Hydrogeology: Finite Difference Model, CBS Pblishers and Distribtors, New Delhi, India, 2000, p [4] M. U. Igboekwe, Geoelectrical Exploration for Grond- Water Potentials in Abia State, Nigeria, An Unpblished PhD dissertation. Michael Okpara University of Agricltre, Umdike, 2005, p. 3. [5] R. T. Peter, Nmerical Analysis: Rnger tta Method, ISBN , 994, pp Copyright 20 SciRes.