Changes in water table induced by transient boundary condition and ...

**Changes** **in** **water** **table** **in**duced **by** **transient** **boundary****condition** **and** space-dependent recharge- A Green element approachOO Onyejekwe*, AB Karama **and** DS TeshomeUniversity of Durban-Westville, Department of Civil Eng**in**eer**in**g, Durban 4000, South AfricaAbstractAn efficient **boundary** **in**tegral procedure has been developed to study ground**water** fluctuations for a well-aquifer systemcharacterised **by** a **transient** **boundary** **condition**. This technique, referred to as the Green element method (GEM) is a hybridprocedure which implements the Boundary element method (BEM) **in** such a way that doma**in** **in**tegration is carried out element**by** element. By adopt**in**g a procedure that is close to the F**in**ite element method (FEM), GEM is able to yield a solution that not onlyagrees with the physics of the problem **and** the accompany**in**g **boundary** **condition**s, but compares favourably with an availableclosed form solution.IntroductionThe problem of ground**water** movement through an underly**in**gporous medium is very important **in** several aspects of eng**in**eer**in**g.Ground**water** movement plays a role **in** agricultural, environmental**and** **in**dustrial processes. In all these cases, the eng**in**eeris primarily concerned with the problem of quantify**in**g or determ**in****in**gthe net movement of **water** due to losses **and** variousmeans of transport. Efficient management of aquifers requires anaccurate estimation of susta**in**able yield at which ground**water**can be exploited without stress**in**g the porous medium. Recently,this concern has acquired new dimensions due to ris**in**g concernsabout cont**in**ued **and** sometimes **in**tensive ground**water** exploitation**in** areas where surface **water** is becom**in**g **in**sufficient.The flow of **water** through porous media is a subject of great**in**terest **and** has attracted a number of researchers. Early **in**vestigationson ground**water** resources of the African Sahara wereperformed **by** Hammad (1969) on a series of wells arranged **in** l**in**ewith small spaces **in** between them. Some of his solutions couldnot be related to the **boundary** **and** **in**itial **condition**s proposed.Later, Gill (1981) improved on the solution of Hammad (1969) **by**provid**in**g a new set of **in**itial **and** **boundary** **condition**s **and**formulated a **transient** flow problem whose solution reduced to asteady state after a long period of time. Mustafa (1984), work**in**gon the same problem as Gill (1981) **in**troduced a leakage **and** arecharge source **and** applied Fourier series analysis to arrive attime-dependent solutions of his **transient** problem. Ram et al.(1994) assumed recharge as a l**in**e source **in** addition to a vertical**in**filtration. Their solutions agreed closely with those presented**by** Mustafa (1984). Onyejekwe (1994) arrived at an analyticsolution to the problem of an unsteady flow to an observation wellfrom a semi-conf**in**ed leaky aquifer **by** convert**in**g both the nonhomogeneousgovern**in**g equation **and** the **boundary** **condition**s toa set of Sturm-Liouville problems. Further work on a semiconf**in**edleaky aquifer subjected to a spatially vary**in**g recharge**and** **transient** **boundary** **condition** was presented **by** Onyejekwe(1998). Plausible agreements were achieved on comparison of his*To whom all correspondence should be addressed.( (031) 204-4749; fax (031) 204-4383; e-mail: okey@pixie.udw.ac.zaReceived 17 September 1997; accepted **in** revised form 25 June 1998results with those obta**in**ed **by** Gill (1981) **and** Mustafa (1984).Analytic solutions were also presented **by** Mustafa (1987), Mar**in**o(1974) **and** Lat**in**opoulos (1981) for various cases of ground**water**flow through porous media.In the work reported here**in**, we adopt the Green elementmethod (GEM) to determ**in**e changes **in** the ground **water** **table** **by**compar**in**g our numerical results with those obta**in**ed **by** Onyejekwe(1998).Problem formulationThe partial differential equation which describes the movementof **water** **in** a conf**in**ed aquifer coupled with recharge is given **by**(Mustafa, 1984):where:µ (the source term ) = x(1-x)u = u(x,t), is the height of **water** **table**v = T/ST = aquifer transmissivityS = the storage coefficientL = x 0- x Lis the length of the problem doma**in**.For Eq. (1) to be well posed, **in**formation is required at **in**itial time(t = 0) **and** at the boundaries.We propose a Green element numerical solution of Eq. (1).Most of the theoretical **and** computational aspects **in**volved **in** thedevelopment of GEM can be found **in** our previous papers(Onyejekwe, 1996; Taigbenu **and** Onyejekwe, 1997; Onyejekwe,1997), **and** as such no effort will be made to go through them **in**detail. As **in**dicated earlier, GEM is based essentially on the**boundary** **in**tegral theory, but its element-**by**-element implementationfollows that of the F**in**ite element method (FEM). Theresult**in**g hybrid procedure yields a method that is more adaptedto h**and**le those problems which give rise to considerable numericaldifficulty to Boundary element method (BEM) (Taigbenu **and**Onyejekwe, 1997; Onyejekwe, 1996)). We, however, hasten to(1)ISSN 0378-4738 = Water SA Vol. 24 No. 4 October 1998 309

comment that the whole idea of GEM is not to replace other“traditional” **and** more “entrenched” numerical techniques,but to enhance the numerical application of the **boundary** **in**tegraltheory to the solution of eng**in**eer**in**g problems. In this context, wetest the suitability of GEM on a ground**water** problem **in**volv**in**ga **transient** **boundary** specification, **and** a recharge function.Application of GEM to solve Eq. (1) essentially requires thefollow**in**g steps:• Integral replication of the govern**in**g partial differential equation.• Discretisation **and** representation of the result**in**g **in**tegralequation on a generic element of the problem doma**in**.• A f**in**ite element type solution to determ**in**e the field variables.We start **by** convert**in**g Eq. (1) **in**to its **in**tegral form. This isachieved more straightforwardly via the Green’s second identity.For two functions G **and** u which are twice differentiable, theGreen’s second identity is formally represented as:In order for us to be able to utilise, Eq. (2), we seek a complementarydifferential equation to Eq. (1). This is given **by**:(2)(3)differential equation which is solved for each element of theproblem doma**in** to yield both the primary variable **and** itsderivative. The hybridisation procedure resumes with the divisionof the problem doma**in** **in**to elements. If for example we havea total of M enumber of elements, Eq. (8) can be written for allthe elements as:where the length of a typical element is given **by** l (e) = x 2(e)- x 1(e),**and** x 2, x 1are the co-ord**in**ates of the end po**in**ts.Numerical procedureHav**in**g described the theoretical framework of GEM, the basicconcepts of its numerical implementation are now discussed.Unlike BEM, doma**in** discretisation is not considered a disadvantage;this aspect of FEM lends the GEM much of its robustness**and** versatility (Onyejekwe, 1996). In addition to divid**in**g theproblem doma**in** **in**to elements, the dependent variables arespecified **in** terms of the summation of the products of the**in**terpolation functions **and** the unknown nodal values of the fieldvariable.(9)where δ(x-x i) is the Dirac delta function. Let the solution ofEq. (3) be of the form:where k is an arbitrary constant, **and** G(x,x i) is known as thefundamental solution. It can be physically **in**terpreted as theresponse of a system governed **by** Eq. (3), when it is subjected toa unit **in**stantaneous **in**put. The derivative of Eq. (4) with respectto x is given **by**:(4)where:Ω 1(e)= 1- ξ, Ω 2(e)= ξ are **in**terpolat**in**g functionsξ = (x-x 1(e))/ l (e) is a local co-ord**in**ate.Substitut**in**g Eq. (10) **in**to Eq. (9) yields:(10)(5)where H(x,x i), the Heaviside function, is def**in**ed as:Introduc**in**g Eqs. (1) **and** (3) **in**to Eq. (2) yields:(6)(7)(11)Observe that we have substituted the longest spatial element size**in** the problem doma**in** for the arbitrary constant k; this guaranteesthat the coefficient matrix rema**in**s positive. We obta**in** twoequations from Eq. (11), **by** consider**in**g the positions of thesource node at x 1**and** x 2respectively. If the source node x iislocated at x 1we obta**in** a discretised equation given **by**:Eq. (7) can now be simplified to give:(8)(12)**in** which λ takes the value of unity if the source po**in**t x iis with**in**the computational doma**in** or half if it is located at the boundaries.Eq. (8) is the **in**tegral representation of the govern**in**g partialSimilarly when the source node is at x 2, the follow**in**g discretisedequation applies:310ISSN 0378-4738 = Water SA Vol. 24 No. 4 October 1998

(13)As a result of this procedure, we have discretised the govern**in**gpartial differential equation **and** replaced it with a set of quasiord**in**arydifferential equations at the nodes. These equations cannow be put **in**to a s**in**gle matrix equation given **by**:**and** α is a weight**in**g factor which varies from 0 to 1, **and** positionsthe time derivative at the time level which it will be evaluated.Conventional def**in**itions of these time levels **in** the f**in**ite difference**and** f**in**ite element methodologies describe α = 1.0 as thefully implicit scheme, α = 0 as the fully explicit scheme, α = 0.5is referred to as the Crank-Nicholson sheme, **and** α = 0.67 isgiven as the Galerk**in**’s scheme. In this work, The Galerk**in**’ssheme gives results that are closest to the closed form solutions.S**in**ce the time term has been evaluated at t m+ α∆t, we can now goahead **and** evaluate other terms of Eq. (14) at this time level. Wethen adopt a weighted average expression given **by**:**and** the elements of the coefficient matrix are given **by**:(14)(15)(16)(17)(19)The subscripts m **and** m+1 refer to the value of the coefficients attime t **and** t+ ∆t respectively. A global system of matrix equationrepresent**in**g the problem is given **by**:(20)where A is the coefficient matrix, **and** RHS is the right-h**and**vector. Eq. (20) is a l**in**ear algebraic equation **and** can be solved**by** any method based on Gaussian elim**in**ation.Approximation **in** timeVarious numerical techniques can be adopted for the time derivativeof Eq. (14). In the work reported here**in**, we shall adopt a2-level time discretisation scheme to approximate the temporalderivative. This procedure carries out the approximation of thetime derivative at t = t m+α= t m+ α∆t. This procedure is given **by**the follow**in**g f**in**ite difference expression:(18)where t mis the previous time level, ∆ t = t m+1- t mis the time step,**and** t m+1is the current time level where the solution is determ**in**ed,Example problem **and** discussionWith GEM formulation completed, we now compare our numericalsolution to the analytical solution obta**in**ed **by** Onyejekwe etal. (1997). The example problem is a leaky conf**in**ed aquifer oflength L rest**in**g on a horizontal **boundary** bounded **by** a well onone side **and** a river on the other as shown **in** Fig. 1. The aquiferconsidered is assumed to be homogeneous with respect to storage**and** isotropic with respect to transmissivity. The **water** level **in**the well is assumed to be U 0+ S o**in**itially **and** drops to U 0after along period of time (U 0= 1, S o= 0.5 **and** transmissivity, T =1,ν = 1 for this problem). The height of the **water** **in** the river isassumed to be constant **and** that at the well varies as a function oftime. In addition a parabolic recharge function R= x(1-x) isimposed on the system. The problem thus specified is desribed **by**the follow**in**g **in**itial **and** **boundary** **condition**s:Figure 1Schematicrepresentation of theproblemISSN 0378-4738 = Water SA Vol. 24 No. 4 October 1998 311

Figure 2Variation of head with distanceFigure 3Variation of discharge with time312ISSN 0378-4738 = Water SA Vol. 24 No. 4 October 1998

Initial **condition**: u(x,0) = U 0+S 0Boundary **condition**s: u(0,t) = U 0+S 0e -Pt t>0u(L,t) = U 0+S 0t>0 (21)where for this problem, P the time coefficient is unity , the timestep ∆ t =.01 **and** the total number of elements M e= 10. The closedform solution to this problem as presented **by** Onyejekwe (1998)is:where A n(t) is given **by**:(22)Results **and** conclusionsResults obta**in**ed **by** GEM **and** those presented **by** Onyejekwe(1998) are shown **in** Figs. 2 **and** 3. It can be noted that the two plotsare **in** agreement with the physics of the problem **and** the specified**in**itial **and** **boundary** **condition**. In Fig. 2 when x=0, i.e. at the well,the height of **water** was found to decrease with **in**crease **in** time.After a long time **in**terval, this value was found to be constant atU 0. At x=L, the height of **water** **in** the river rema**in**s fixed atU 0+ S o. Similarly for Fig. 3, there is a steep rise **in** flux for valuesof t less than 3, but for longer periods, the flow rate stabilises as**in**dicated **by** Eq. (25).A **boundary** **in**tegral procedure has been developed for determ**in****in**gthe changes **in** **water** **table** exposed to a **transient** **boundary****condition** **and** space-dependent recharge. This technique wascompared with the closed form solution obta**in**ed **by** Onyejekwe(1998) **and** excellent results were obta**in**ed. Extension of GEM to2-D application is straightforward (Taigbenu **and** Onyejekwe,1995).References(23)where:λ n= nπ/L are eigenvalues correspond**in**g to the Fourierdecomposition of the solution.Us**in**g Darcy’s equation, the net flow rate **in** the ditch can bedeterm**in**ed from:(24)Differentiat**in**g Eq. (24) with respect to x **and** sett**in**g x to be zero,the rate of flow is:(25)GILL MA (1981) Unsteady free flow to a ditch from an artesian aquifer.J. of Hydrol. 52 39-45.HAMMAD YH (1969) Future of ground **water** **in** African Sahara. J. Irrig.Dra**in** Div. Proc. Am. Soc. Civ. Eng. 95(IR4) 563-580.LATINOPOULOS P (1981) The response of ground **water** to artificialrecharge schemes. Water Resour. Res. 17 1712-1714.MARINO MA (1974) Rise **and** decl**in**e of the **water** **table** **in**duced **by**vertical recharge. J. Hydrol. 23 289-298.MUSTAFA S (1987) Water flow **in** a semi-conf**in**ed aquifer for spatiallyvary**in**g recharge functions. Nigerian J. of Eng. 4 28-34.MUSTAFA S (1984) Unsteady flow to a ditch from a semi-conf**in**ed **and**leaky aquifer. Adv. Water Res. 7 81-84.ONYEJEKWE OO (1994) Unsteady free flow to an observation wellfrom a semi-conf**in**ed leaky aquifer. Adv. **in** Eng. Software 19 173-175.ONYEJEKWE OO (1996) Green element description of mass transfer **in**react**in**g systems. Num. Heat Transfer Part B 30 483-498.ONYEJEKWE OO (1997) Green element computation of the Sturm-Liouville equations. Adv. Numerical Software 28 615-620.ONYEJEKWE OO (1998) Personal communication. (**Changes** **in** **water****table** **in**duced **by** **transient** **boundary** **condition** **and** space dependentrecharge).RAM S, JAISWAL CS **and** CHAUHAN HS (1994) Transient **water** **table**rise with canal seepage **and** recharge. J. Hydrol. 163 197-202.TAIGBENU AE **and** ONYEJEKWE OO (1995) Green element simulationsof the **transient** nonl**in**ear unsaturated flow equation. Appl.Math. Modell**in**g 19 675-684.TAIGBENU AE **and** ONYEJEKWE OO (1997) Transient ID transportequation simulated **by** a mixed Green element formulation. Int. J.Num. Methods Fluids 25 437-454.ISSN 0378-4738 = Water SA Vol. 24 No. 4 October 1998 313

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