Numerical solution of one dimensional contaminant transport equation with variable coefficient (temporal) by using Haar wavelet

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1 Goba Journa of Pure and Appied Mathematics. ISSN Voume 1, Number (16), pp Research India Pubications Numerica soution of one dimensiona contaminant transport equation with variabe coefficient (tempora) by using Haar waveet A. C. Pate and V. H. Pradhan Department of Appied Mathematics and Humanities, S. V. N. I. T, Surat -3957, Gujarat, India. Abstract In the present paper Haar waveet method is impemented on advectiondispersion equation representing one dimensiona contaminant transport through a porous medium. Non uniform fow is considered by assuming veocity and dispersion varying with time as an exponentiay increasing function. Expressing the Haar waveets in advection-dispersion equation into Haar series provides the main advantage in the existing method where the simpicity of the Haar waveet is preserved. The obtained numerica resuts are compared with the exact soution of advection-dispersion equation with constants coefficients as there are very few anaytica soutions with the variabe coefficients. The computations are carried out with the aid of the MatLab program. It is concuded that Haar waveet method is easy, efficient and convenient. Keywords: Advection-dispersion equation, Haar waveet, temporay varying coefficients, Pecet number. Introduction During the past severa years the study was more emphasized towards water suppy probems and water suppy potentia of aquifers. However the studies are now emphasized more in water quaity probems in the past years. Consequenty the need arises to study the contaminant transport through the subsurface environment. In the earier study the deveoping methods where focused to anayze aquifers of high permeabiity. However the studies are now focused argey on the reactive and nonreactive soute transport in adsorbing and non-adsorbing porous media. An extensive iterature is avaiabe on the study of transport and dispersion processes with various

2 184 A. C. Pate and V. H. Pradhan kinds of contaminants. The deterministic mathematica modes used for most of the groundwater modes and its computer simuation has gained a technoogica growth in the soute transport in groundwater system. These types of modes are generay governed by the partia differentia equations (PDE) based on basic principes and aws. The contaminants being physicay, chemicay or bioogicay active the transport of contaminants becomes a subject of great importance in engineering and science. Usuay advection-dispersion equation (ADE) describes the contaminant transport through a medium and is a paraboic PDE. This type of equation is broady cassified as ADE with constant and variabe coefficients. The veocity and dispersivity are assumed to be constant in the ADE with constant coefficient, whereas veocity and dispersivity vary with either space or time in ADE with variabe coefficient. An extensive iterature is avaiabe with the exact soutions of ADEin one dimension with constant coefficients [6]. There are very few anaytica soutions with the variabe coefficients [1-] and consequenty the numerica soutions are compared with the exact soutions of ADE with constant coefficients. The anaytica soutions in genera are restricted with specific boundary conditions and may not have much practica importance or may be compex. As a resut the numerica soutions can be widey used to sove the ADE with non-uniform fow with respect todifferent types of initia and boundary conditions. In the present paper a numerica soution is obtained for the ADE with non-uniform veocity fieds and variabe dispersion coefficients using Haar waveet method. Dirichet boundary conditions at the infow and out fow ends of the fow system and initia condition in the Gaussian form is considered for various cases. Mode Formuation We consider the transport of a contaminant through a homogeneous finite aquifer of ength L under transient-state fow. Initiay it is assumed that the domain is not cean and may contain some contamination. The initia contamination may be assumed to be non-zero constant or in Gaussian form. The infow and out-fow conditions of the fow system are assumed to be time-dependent. Let c( x, t ) be the concentration of contaminants in the aquifer at position x and time t., D x, t are the veocity v x t and of the medium transporting the contaminants and the soute dispersion parameter D x, t is assumed to be constant if it is independent of position and time respectivey. and known as dispersion coefficient. Then the probem with first-order decay can be mathematicay formuated as foows: c c R Dx, t vx, tc R c, x L, t t (1) t x x D t D V t, v t v V t ()

3 Numerica soution of one dimensiona contaminant transport equation 185 dkd In equation (1) R 1 is the retardation factor. kd is distribution coefficient. n n is the porosity, d is the density and is the decay constant. In equation () D is the initia dispersion coefficient and v is the initia veocity. Here, we made foowing assumptions: a. Fuid is of constant density and viscosity. b. Soute is subject to first-order nonreactive transformation. c. No adsorption, kd. Based on the above assumption, equation (1) reduces to c c Dx, t v x, tc, x L, t t t x x mt Here we considered V t e where, m is the fow resistance coefficient. (3) Initia and Boundary Conditions We assume that there is some contamination in the fow system initiay. Thus in the initia condition the initia concentration of the contaminant can be assumed with nonzero constant vaue or function of space variabe. In the present paper we assume that the initia contaminant profie took on a Gaussian shape. We assume c x, f x, x L (4) To sove equation (3)competey different boundary conditions are associated with it. In genera Dirichet, Neumann and Cauchy boundary conditions can be appied at the infow and out fow ends of the fow system. The above three conditions are aso known as the boundary condition respectivey. The vaue of the concentration is represented by first-type, whereas the gradient and fux are represented by second and third type boundary condition respectivey. In the existing paper the first-type boundary condition is impemented at the infow and out fow of the fow system as the time dependent functions of sufficient smoothness as foows c,t g t t t (5), 1 c L t g t (6) where g 1 and g are the known sufficient smoothness. Haar Waveet A system of square wave is known as Haar waveet [3, 4, 5] hi orthogona and orthonorma. In this system h is a scaing function, h1 known as mother waveet and a other waveet transform are given by which is is

4 186 A. C. Pate and V. H. Pradhan h 1 j n h k, n j k, j, k j The Haar waveet famiy for,1 is defined as foows: k k.5 1,,, m m k.5 k1 hi 1,,, m m, esewhere. j Here, m, j,1,..., J(Maximumeveof resoution), k,1,..., m 1 are known as eve of waveets and transation parameter respectivey. In equation(7), the index i is given by i m k 1where the minima vaue is i = and the maximum vaues 1 i J. Let, P h d Q P d i i i i (8) We consider the coocation points given by (7).5 where 1,,..., M, through which we get the coefficient matrices H, P, Q with matrices. Equation (9) represents the matrix equation for cacuating the matrix P of order m [13] 1 P H m m P m 1 (9) Hm O m m in whicho is a nu matrix of order, Hmm h m x, hm x1,..., hm xm1 (1) i T and x i and Hmm Hmmdiag(r) m m m Once P m and Hm are cacuated the same can be used for soving various differentia equations. Method of Soution D mt Case I: We consider De (varying temporay), assuming v constant then equation (3) reduces to C mt C C De v (11) t x x x vt Using the dimensioness variabe and T equation(11) becomes L L

5 Numerica soution of one dimensiona contaminant transport equation 187 mt C e C C, 1, T T (1) T Pe vl ml where Pe is the Pecet number and m is dimensioness constant. D v with initia and boundary condition C, 8 e (13) C, T e T C 1, T e T 5T 5 T 54T 1 T Dividing the time interva into N equa parts, C,T expressed as, i1 i i (14) (15) in terms of Haar waveets is C T a h (16) where( ) and ( ' ) represents differentiation with respect to time and space variabe T respectivey, am is the row vector in the subintervat Ts, Ts 1 is constant. In equation(16), T and are integrated from T s to T and to respectivey, the variabe C,T, C,T C,T and s i i s i1 C,T can be successivey obtained. C, T T T a h C, T (17) C, T T T a P C, T C, T C, T (18) s i i s s i1,,,, C T T T a Q C T C T C T s i i s s i1,, C T C T,,, s (19) i i () i1 C T a Q C T C T Setting 1 in equation (19)and(), we have C, T C, T T T a Q 1 u 1, T C 1, T C, T C, T (1) s s i i s s i1 i i () i1, 1, 1, C T a Q C T C T Substituting equation (1)and ()into equation (17)-()and discretizing to and T tot s 1we get

6 188 A. C. Pate and V. H. Pradhan a Q C T a Q C T C T mt ' s 1, 1 1,, i i s1 i i s1 s1 i1 i1 e C, Ts 1 C, Ts 1 Pe T Using equation(3)thehaar coefficients a can be successivey obtained. (3) mt Case II: We consider v ve (varying temporay), assuming D constant in equation (3)we get C C mt C D v e (4) t x x x vt Using the dimensioness variabe and T equation(4) reduces to L L C 1 C mt ' C e 1, T T (5) T Pe vl ml where Pe is the Pecet number and m is dimensioness constant. D v Using initia condition(13)and boundary conditions (14)-(15)and appying Haar waveet method as discussed in case I to equation(5) we get aiqi C, Ts 1 aiqi 1 C 1, Ts 1 C, Ts 1 i1 i1 (6) 1 mt ' s 1 C, Ts 1 e C, Ts 1 Pe T Using equation(6)the Haar coefficients a can be successivey obatined. mt mt Case III: We consider, D De and v ve (both varying temporay) in equation(3) we get C mt C mt C De v e (7) t x x x vt Using the dimensioness variabe and T equation(7) reduces to L L mt ' C e C mt ' C e 1, T T (8) T Pe vl ml where Pe is the Pecet number and m is dimensioness constant. D v Using initia condition (13) and boundary conditions (14)-(15) and simiary appying Haar waveet method to equation(8) we get

7 Numerica soution of one dimensiona contaminant transport equation 189 a Q C T a Q C T C T mt ' s 1, 1 1,, i i s1 i i s1 s1 i1 i1 e mt ' s 1 C, Ts 1 e C, Ts 1 Pe T Using equation (9) the Haar coefficients a can be successivey obtained. (9) Resuts and Discussion The numerica resuts of equation (3)with respect to initia and boundary conditions (13)-(15) are obtained using equation (3) for case (I), equation (6) for case (II) and equation (9) for case (III) using vaues D.1, v.8 and m.1. The corresponding graphica representations are shown in Figure (1), () and(3). Figures reveas that the contaminant concentration decreases with the time and increases with the space variabe. This shows that the contamination concentration which is distributed initiay in the Gaussian form wi vanish after sufficient time with respect to considered time varying boundary conditions at the infow and out-fow of the fow system. The tabe (1) depicts a comparative study of ADE with constant coefficient and variabe coefficients. Three cases are considered for non-uniform fow. In case (I) dispersivity is varying with time keeping veocity constant, in case (II) veocity is varying with time keeping dispersivity constant and in case (III) the veocity and dispersivity are varying with time. It is found that the contaminant concentration is ess case (II) and (III) in comparison to advection dispersion with constant coefficient. However the contaminant concentration is found more in case (I) for exponentiay increasing dispersion in uniform fow. Tabe 1 Comparison of one dimensiona contaminant transport equation with constant coefficients and variabe coefficients (Tempora) at T =. 5(P =8). Constant Tempora Tempora D constant Tempora D and coefficient [6] mt D De mt and v ve v both variabe and v constant = = = = = = = = =

8 19 A. C. Pate and V. H. Pradhan C (,T ) T = T =. T =.4 T =.6 T =.8 T = Figure 1: Dimensioness contaminant concentration profiesfor various vaues of T with dispersivity varying with time and veocity as constant T= T =. T =.4 T =.6 T =.8 T = 1.7 C (,T) Figure : Dimensioness contaminant concentration profies for various vaues of T with veocity varying with time and dispersivity as constant T = T =. T =.4 T =.6 T =.8 T = 1.7 C(, T) Figure 3: Dimensioness contaminant concentration profie for various vaues of T with dispersivity and veocity both varying with time

9 Numerica soution of one dimensiona contaminant transport equation 191 List of symbos R x t retardation factor position time D initia dispersion coefficient v initia veocity k distribution coefficient d n porosity decay constant P Pecet number e m fow resistance coefficient m eve of waveets k J L t T transation parameter maximum eve of resoution ength time duration dimensioness space variabe dimensioness time variabe T dimensioness time duration m constant a T M Haar coefficients Concusion In the present paper the ADE is numericay discussed using Haar waveet method for non-uniform fow. Three cases are considered to observe the effect of dispersion (D) and veocity(v) varying with time in comparison to ADE with constant coefficients. Exponentiay increasing function is considered for D and v varying with time. From the obtained resuts it is found that when D is varying with time keeping v constant, the contaminant concentration wi be more in comparison to contaminant concentration for the ADE with constant coefficients. Thus the method is convenienty appied in a the three cases to observe the corresponding effects. It is concuded that the method is easy, efficient and accurate. Since the Haar waveets are orthonorma the L convergence in (16)is unconditiona and the method is aways stabe [7], the method can be equay appied to other partia differntia eqautions with various types of initia and boundary conditions.

10 19 A. C. Pate and V. H. Pradhan References [1] Zoppou C. and Knight J. H. (1999), Anaytica soution of a spatiay variabe coefficient advection-diffusion equation in up to three dimensions, Appied Mathematicamodeing, 3, [] Jaiswa D. K., Kumar A., Yadav R. R. (11), Anaytica soution to the onedimensiona advection-diffusion equation with temporay dependent coefficients, Journa of water resources and protection, 3, [3] Chen C. F. and Hsiao C. H. (1997), Haar waveet method for soving umped and distributed parameter systems, IEEE Proceeding: Part D, 144, [4] HariharanG. et. a. (1), Haar waveet soution for a few reaction diffusion probems, Ph. D thesis, Sastra university, Thanjavur, Tami Nadu. [5] LepikU., Hein H. (14), Haar waveets with appications, Springer, Switzerand. [6] Shi Z., Deng Li-Y., Chen Q.-J. (7), Numerica soution of differentia equations by using Haar waveets, Proc. Int. Conf. Waveet Ana. An Pattern Recog., [7] um. (6), Function approximation methods for optima contro probems, Ph. D. thesis, Saint Louis, Missouri.