IJRRAS 6 () February www.arpapress.com/volumes/vol6issue/ijrras_6 5.pdf TEMPORALLY EPENENT ISPERSION THROUGH SEMI-INFINITE HOMOGENEOUS POROUS MEIA: AN ANALYTICAL SOLUTION R.R.Yadav, ilip Kumar Jaiswal, Hareesh Kumar Yadav & Gulrana epartment of Mathematics and Astronomy Lucknow University, Lucknow-67, India Corresponding authors: yadav_rr@yahoo.co.in, dilip3jais@gmail.com ABSTRACT An analytical solution is obtained for two-dimensional dispersion through semi-infinite homogeneous porous medium. Point source concentration of pulse type is considered at the origin. Nature of pollutant is chemically nonreactive. Initially the domain is not solute free. The seepage velocities are considered exponentially decreasing function of time along both directions (longitudinal and lateral). ispersion coefficient is assumed proportional to seepage velocity. First order decay is also considered which is proportional to dispersion coefficient. Laplace technique is used to obtain the analytical solution. Key Words: Advection; ispersion; Groundwater; Point source; Seepage velocity.. INTROUCTION In last few years, modeling of solute transport in porous media remain a key issue in the area of soil physics, hydrogeology and environmental science, because anthropogenic chemicals frequently enter the soil, subsoil, aquifers and groundwater either by accident or by human activities, and resulting chemical residues pose hazards to the environment and groundwater. The advection-dispersion equation is the most common method of modeling solute transport in subsurface porous media. This equation includes terms that describe the physical process governing advective transport with flowing groundwater, molecular diffusion, hydrodynamic dispersion, retardation, first order decay, zero-order production and equilibrium or non-equilibrium exchange with the solid phase if reactive solutes are involved. The transport and mixing of contaminants in conduits is governed by advection, dispersion, and decay. Several models/solutions are available to trace the transport of such constituents and most assume that the principal mechanisms for transport are advection and reaction only. A number of analytical solutions have been developed to describe one-dimensional convective dispersive solute transport. Elder [8] by using Taylor's approach and assuming a logarithmic velocity distribution, derived an expression for the longitudinal dispersion coefficient for an infinitely wide open channel. Fischer [] derived another expression for longitudinal dispersion coefficient assuming that the velocity distribution in lateral direction was the primary mechanism responsible for longitudinal dispersion. Marino [6], van Genucheten [3], Banks and Jerasate [3], Rumer [8] and Yadav et al. [5] considered dispersion along unsteady flow. Al-Niami and Rushton [] considered uniform flow where as Kumar [4] took unsteady flow against the dispersion in finite porous media. Most of these works have included the attenuation effect due adsorption, first order radio-active decay and / or chemical reactions. Shen [] presented a generalized closed form solution for three-dimensional dispersion in saturated ambient porous media resulting from sources of finite extent with time-dependent input concentration. Winter et al. [4] defined one- and two-dimensional formation analytically, relate the dispersion parameter to the statistics of the hydraulic conductivity spatial distribution. Batu [5] discussed time-dependent linearized twodimensional infiltration and evaporation from non-uniform and non-periodic strip source. Latinopoulos et al. [5] studied the chemical transport in two-dimensional aquifer. Ellosworth and Butters [9] discussed three-dimensional solutions used for transport problems involving arbitrary Cartesian coordinate systems. Aral and Liao [] examined solutions to two-dimensional advection-dispersion equation with time-dependent dispersion coefficients. In particular, they developed instantaneous and continuous point source solutions for constant, linear, asymptotic, and exponentially varying dispersion coefficients. Stenbacka et al. [] employed a two-dimensional analytical model for estimating the first-order degradation rate constant of hydrophobic organic compounds (HOCs) in contaminated groundwater under steady-state conditions. Massabo et al. [7] gave some analytical solutions for a two-dimensional advection equation with anisotropic dispersion. Chemical decay or adsorption-like reaction inside the liquid phase is considered. Basha and Malaeb [4] presented a method for simulating the advection-dispersion-reaction process of constituent transport in water networks and using Eulerian-Lagrangian method where dispersion term in the governing equation is approximated using finite differences and the resulting first-order partial differential equation is then integrated using the method 58
IJRRAS 6 () February of characteristics. Essa et al. [] investigated the dispersion of pollutants from a point source, analytically taking into consideration the vertical variation of both wind speed and eddy diffusivity. Shapiro and Bedrikovetsky [9] proposed a new approach to transport of the suspensions and tracers in porous media. The approach is based on a modified version of the continuous time random walk (CTRW) theory. In the framework of this theory they derived an elliptic transport equation. The new equation contains the time and the mixed dispersion terms expressing the dispersion of the particle time steps. Jaiswal et al. [] and Kumar et al. [3] obtained analytical solutions for temporally and spatially dependent solute dispersion in one-dimensional semi-infinite media. The objective of the present paper is to obtain an analytical solution for prediction of concentration distribution in shallow aquifer with time-dependent dispersion coefficient. Porous domain is considered homogeneous, isotropic, semi-infinite and non-reactive. Both components (longitudinal and lateral) of dispersion coefficient and flow velocity are considered. Seepage velocities are function of time. Point source and pulse type input concentration is considered at origin. Initially the domain is not solute free. There is no solute flux at end of both boundaries. ispersion is proportional to seepage velocity which is taken as time-dependent. First order decay term which is proportional to dispersion coefficient and retardation factor are also considered. Analytical solutions are obtained with the help of Laplace technique. Two-dimensional problem is used to demonstrate the different water and solute profile, for surface and surface line / non-point irrigation. Realistic approach of the obtain results are illustrated by different graphs.. MATHEMATICAL FORMULATION OF THE PROBLEM The two-dimensional parabolic partial differential equation describing hydrodynamic dispersion in adsorbing homogenous, isotropic porous medium can be written as (Schiedegger, []), S + c = t t x + y u t v t γ(t)c () where S is the adsorbed concentration, c is the dissolved concentration and γ is first order decay term. From relation S = K d c, where K d is distribution coefficient which is defined as ratio of the adsorbed contaminant concentration to the dissolved contaminants. In case no adsorption K d =. So that, S = K t d Eq. () becomes, c t ( + K d ) c t = x R c t = x + y + y u t u t v t γ(t)c (3) v t γ(t)c (4) where + K d = R, which is the retardation coefficient accounting for equilibrium linear sorption processes. Let u = u exp mt, v = v exp ( mt) (5) where u and v are initial velocity components along x and y axis respectively and m is flow resistance coefficient. Its dimension is inverse of time. Ebach and White [7], have established that the dispersion coefficient vary approximately directly to flow velocity, for different types of porous medium. So let us considered x = au and y = av (6) where a is a constant depends upon pore geometry of the medium. ispersion and decay term can be written by using expression from (5) and (6), as x = x exp mt, y = y exp mt and γ = γ exp ( mt) (7) where x = au and y = av are initial dispersion coefficient components along two respective directions and γ is the first order decay constant. Initial and boundary conditions for the present paper are, c = C i, t =, x, y (8) where C i is resident concentration in domain. c = C, < t t, x =, y = (9), t > t c c and =, = t, x, y () A pulse-type input condition represented by (9) in which t is the time span of the contaminant release (assuming release starts at time zero) and C is the constant concentration of the pulse at the inlet boundary. When considering the contamination release and discharge problem in industrial sites, this situation is realistic in the sense that generally industries or waste sites release pollution in a finite time period, either because industrial firms have a finite life or the pollution problem is controlled after a certain time with the awareness of the contamination or government regulation. Boundary condition (), indicate that there are no solute flux at end of the both boundaries. Using Eqs. (5) and (7) the differential equation (4) can be written as, () 59
IJRRAS 6 () February V t = t x + y u v γ c () exp mt where V t =. R Introducing the new time variable T by following transformation (Crank, [6]) t T = V t dt t or T = exp mt /R dt = exp ( mt) () mr For an expression exp mt which is taken such that exp mt = for m = or t =, the new time variable obtained from Eq. () satisfies the conditions T = for t = and T = for m =. The first condition ensures that the nature of the initial condition does not change in the new time variable domain. Thus Eq. () can be written as, = T x + y u v γ c (3) Let us introduced new variable as, X = x + y y (4) Eq. (3) reduces into c = c c T X U γ X c (5) where = x + y ; U = u + v y The initial and boundary conditions (8)-() in terms of new variables become, c = C i, T =, X (6) c = C, < T T, T > T, X = (7) c =, T, X (8) X Using the following transformation c X, T = K X, T exp U U X + γ 4 T (9) and applying Laplace technique, we get following boundary value problem, pk C i exp U X = d K () dx K X, p = C exp p p α α T ; X =, () and dk dx + U K =, X () where p is Laplace parameter, α = U + γ 4 and β = U. 4 The solution of boundary value problem ()-() in Laplacian domain may be written as, K X, p = C p α α T exp X p β (3) Applying inverse Laplace transformation on (3) and using Eq. (9), the solution in terms of c X, T written as, c X, T = C i F X, T + C F X, T, < T T (4a) c X, T = C i F X, T + C F X, T C F X, T T, T > T (4b) where F X, T = exp γ T X UT erfc T X F X, T = exp β β +γ exp UX C i p β X+UT erfc T erfc X U +4γ / T T, p + C U iexp X + exp β + β +γ / X β = U, = 4 x + y, U = u + v y erfc X+ U +4γ / T T, T = Rm exp ( mt) and X = x + y y,. 6
IJRRAS 6 () February.. Particular cases (i) For steady flow The solution for steady in horizontal plane, can be obtained by putting m = in Eq. (4) c X, t = C i F X, t + C F X, t, < t t c X, t = C i F X, t + C F X, t C F X, t t, t > t (5a) (5b) where F X, t = exp γ t/r RX Ut erfc Rt UX exp RX+Ut erfc Rt, F X, t = exp β β +γ X erfc RX U +4γ / t Rt + exp β + β +γ / X erfc RX+ U +4γ / t Rt. (ii) One-dimensional solution along unsteady flow The solution for one-dimensional dispersion along unsteady flow through semi-infinite porous medium can be obtained by putting y = and v = in (4), c x, T = C i F x, T + C F x, T, < T T (6a) c x, T = C i F x, T + C F x, T C F x, T T, T > T (6b) where F x, T = exp γ T F x, T = exp β β +γ erfc x u T T exp u x x erfc x+u T T, erfc x u +4γ / T T where β = u 4 T will remain same as in (4). + exp β + β +γ / x erfc x+ U +4γ / T T,, u and x are initial velocity and dispersion coefficient along longitudinal direction (x-axis) and 3. RESULT AN ISCUSSION To illustrate the concentration distribution of the obtained analytical solution (4) in two-dimensional homogeneous porous medium in semi-infinite domain, an example has been chosen in which the different variables are assigned numerical values, where u =.5 (m/day), v =.5 (m/day), x =.5 (m /day) and y =.5 (m /day) respectively. The lateral velocity dispersion component are considered one-tenth of the longitudinal component. Lateral dispersion is also an essential factor to predict the temporal behavior of the contaminant. The ratio of longitudinal to lateral dispersivity in an aquifer is an important factor to determine the behavior of the contaminant. The flow resistance coefficient m =. (day - ), retardation coefficient R =.4 and first order decay constant γ =.4 have been chosen. Figure () and () are drawn for solution (4a) and (4b) with different time t =.3,.6 and.9 (days) in time domain t t and at t = 3., 3.4 and 3.7 (days) in time domain t > t, respectively. 6
IJRRAS 6 () February t =.3 t =.6 t =.9 c/c R =.4.5..4.6.8.6.4. x y.8 Figure. istribution of solute concentrations of f mt = exp ( mt) at different time for solution (4a)..8 t = 3. R =.4 t = 3.4 t = 3.7 c/c.4..4 x.6.8..4 y.6.8 Figure. istribution of solute concentrations of f mt = exp ( mt) at different time for solution (4b). In domain t t, the concentration behavior of solute at particular position increases with increasing time while in time domain t > t, solutes concentration decreases with increasing time. It means as the time increases dispersion process goes on dominating over the transport due to convection. Concentration distribution behavior are shown for different R =.4 and.8 at time t =.6 (days) in domain t t and t = 3.6 in domain t > t by Figs. (3) and (4) respectively. 6
IJRRAS 6 () February t =.6 R =.8 R =.4 c/c.5.4.6 x.8..4 y.6.8 Figure 3. Comparison of solute concentration for different retardation coefficient at time t =.6 (days) for solution (4a)..9.6 t = 3.6 R =.8 c/c R =.4.3..4.4.6. x.8 y.6.8 Figure 4. Comparison of solute concentration for different retardation coefficient at time t = 3.6 (days) for solution (4b). We observe that the contaminant concentration decreases with increasing retardation coefficient. The time-dependent behavior of solutes in subsurface is of interest for many practical problems where the concentration is observed or needs to be predicted at fixed positions. These solutions have practical application for many field problems. For example, many pipe or conduits carrying contaminated fluids exits in the country. Leakage from those pipe/conduits can contaminate the aquifer/soil below them. These contaminations problems where pipe/conduits are simulated as line/point source. Problems of solute transport in two-dimension involving sequential first order decay reactions and retardation coefficient frequently occurs in soil and groundwater systems, for example the migration of simultaneous movement of interacting nitrogen species, organic phosphate transport and the transport of pesticides and their metabolites. The accuracy of the numerical method is validated by direct comparisons with the analytical results. For a complex source area or non-steady-state plume, a superposition of analytical models that incorporate longitudinal and transverse dispersion and time may be used at sites where the centerline method would not be applicable. 4. CONCLUSION In this work, an analytical solution is obtained for two-dimensional dispersion through semi-infinite homogeneous porous medium. Point source concentration of pulse type is considered at the origin. Initially the domain is not solute free. The seepage velocities are considered exponentially decreasing function of time along both directions (longitudinal and 63
IJRRAS 6 () February lateral). ispersion coefficient is assumed proportional to seepage velocity and first order decay is also considered which is proportional to dispersion coefficient. The analytical solutions of two-dimensional dispersion problem may help to determine the position and time to reach the minimum / maximum or harmless concentration and it is very useful for design and interpretation of experiments in laboratory and possibly some homogeneous aquifers and for the verification of complicated numerical models. Obtained result may also be useful as a predictive tool in environmental / groundwater management and applies to real-world applications in which retardation rates are much faster than the rates of advection and dispersion. Multi-dimensional solutions are more widely applicable to transport problems encountered in the field because of the fact that a solute will disperse in any direction that a concentration gradient is present. 5. ACKNOWLEGEMENT Second author is gratefully acknowledged for giving financial assistance as UGC-r. S. Kothari Post octoral Fellowship. 6. REFERENCES []. Al-Niami, A.N.S. and K.R. Rushtom (977). Analysis of flow against dispersion in porous media. J. of Hydrology, 33, 87-97. []. M. M. Aral and B. Liao. Analytical solutions for two dimensional transport equation with time dependent dispersion coefficients. J. of Hydrologic Eng.,-,-3, (996). [3]. R.B. Banks and S. Jerasate. ispersion in unsteady porous media flow. J. of Hydroul.iv.,9,3-3, (96) [4]. H. A., Basha and L. N. Malae. Eulerian Lagrangian Method for Constituent Transport in Water istribution Networks. Journal of Hydraulic Engineering, 33(), 55 66, (7). [5]. V. Batu. Time-dependent. linearized two-dimensional infiltration and evaporation from non-uniform and non-periodic strip sources. Water Resources Research, 8, 75-733, (98). [6]. J. Crank. The Mathematics of iffusion. Oxford Univ. Press, London, nd edition, (975). [7]. E.H. Ebach and R.White. Mixing of fluids flowing through beds of packed soils. J.ofAmer. Inst.Chem. Eng., 4, 6-64, (958). [8]. J. W. Elder. The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544-56, (959). [9]. T.R. Ellsworth and G.L. Butters. Three-dimensional analytical solution to the advection- dispersion equation in arbitrary cartesian coordinate. Water Resour. Res., 337-354, (993). []. K. S. M. Essa, S. M. Etman and M. Embaby. New analytical solution of the dispersion equation Atmospheric Research, 84, 337 344, (7). []. H. B. Fischer. The mechanics of dispersion in natural streams. J. Hydraul. iv. ASCE, 93 (HY6), 87-6, (967). []..K. Jaiswal, A. Kumar, N. Kumar and R.R. Yadava. Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one dimensional semi-infinite media. Journal of Hydro-environment Research,, 54 63, (9). [3]. A. Kumar,.K. Jaiswal and N. Kumar. Analytical solutions to one-dimensional advection-diffusion with variable coefficients in seme-infinite media. Journal of Hydrology, 38(3-4), 33-337, (). [4]. N. Kumar. Unsteady flow against dispersion in finite porous media with unsteady velocity distribution. Nordic Hydrol., 4,67-78, (983). [5]. P.. Latinopoulos, Tolikas and Y. Mylopoulos. Analytical solution for two dimensional chemical transport in aquifer. J. of Hrdrol., 98, -9, (988). [6]. M.A. Marino. isrtibution of contaminants in porous media flow. Water Resour. Res., (5), 3-8, (974). [7]. M. Massabo, R. Cianci and O. Paladino. Some analytical solutions for two- dimensional Convection dispersion equation in cylindrical geometry. Environmental Modelling and Software,, 68-688, (6). [8]. R. Rumer. Longitudinal dispersion in steady flow. J. of Hydrul. iv., 88(4), 47-7, (96). [9]. A. A. Shapiro and P. G. Bedrikovetsky. Elliptic random-walk equation for suspension and tracer transport in porous media. Physica A, 387, 5963 5978, (8). []. A. F. Scheidegger. General theory of dispersion in porous media. J. of General Theory Res., 66,, (96). []. H. T. Shen. Transient dispersion in uniform porous media flow. J. of Hydraul. iv., (6), 77-75, (976). []. G. A. Stenbacka, S. K. Onga, S. W. Rogersa, and B. H. Kjartanson. Impact of transverse and longitudinal dispersion on first order degradation rate constant estimation. Journal of Contaminant Hydrology, 73, 3 4, (4). [3]. van Genuchten, M. Th. Analytical Solutions for chemical transport with Simultaneous adsorption, zero-order production and first order decay. J. of Hydrology, 49, 3-33, (98). [4]. C.L. Winter, C.M. Newman, and S.P. Newman. A perturbation expansion for diffusion in randam velocity field. SIAM J. Appl. Math., 44(), 4-44, (984). [5]. R. R. Yadava, R. R. Vinda, and N. Kumar. One dimensional dispersion in unsteady flow in an adsorbing porous medium: An analytical solution. Hydrological Processes, 4, 89 96, (99). 64