Analytical solution of advection diffusion equation in heterogeneous infinite medium using Green s function method

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1 Analytical solution of advection diffusion equation in heterogeneous infinite medium using Green s function method Abhishek Sanskrityayn Naveen Kumar Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 22 5, India. Corresponding author. abhi.bhu28@gmail.com Some analytical solutions of one-dimensional advection diffusion equation ADE) with variable dispersion coefficient velocity are obtained using Green s function method GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially temporally dependent. The spatial dependence is considered to be linear temporal dependence is considered to be of linear, exponential asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant s mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.. Introduction The pollutants solute transport due to advection diffusion through a medium is described by a partial differential equation of parabolic type, derived on the principle of conservation of mass, known as advection diffusion equation ADE). In one dimension, it has two coefficients: diffusion coefficient velocity. ADE is efficiently used to describe solute transport through a medium away from its source over long distances in open media like rivers as well as in porous media like aquifers. In the latter case, the velocity satisfies the Darcy law. But solute transport through heterogeneous medium cannot be described properly by ADE with constant coefficients Simpson 978; Matheron de Marsily 98; Pickens Grisak 98a). Scale-dependent dispersion in heterogeneous hydrological systems was modelled by timedependent dispersion coefficient of linear, exponential, parabolic asymptotic forms Pickens Grisak 98b). The model was hled using finite element linear triangular) method. This theory has been strengthened by field experiments Anderson 979; Gelhar et al. 985) by theoretical developments Sposito et al. 986; Dagan 987). Theoretical deterministic analysis Güven et al. 984) has also established the temporal dependence of the dispersion coefficient in a stratified aquifer. A generalized analytical transient, one-, two-, /or three-dimensional AT 23D) computer Keywords. Advection; diffusion; heterogeneity; non-degenerate form; Green s function method. J. Earth Syst. Sci., DOI.7/s , 25, No. 8, December 26, pp c Indian Academy of Sciences 73

2 74 Abhishek Sanskrityayn Naveen Kumar code was developed for estimating the transport of wastes in the groundwater aquifer system Yeh 98) which contains 45 options of types of wastes, source configurations, source of release, boundary conditions, aquifer dimensions dimensions of the Cartesian frame of references. It is recognized as a pioneer work using a Green function method to develop solute transport model, though the coefficients of the ADE were assumed constant. The analytical solution of one-dimensional ADE with temporally dependent dispersion coefficient uniform velocity in an infinite heterogeneity medium was obtained by Basha El-Habel 993) using Green s function method for the instantaneous continuous point sources. Later this work was extended in two dimensions Aral Liao 996) the analytical solutions were obtained using the same method. In both the works, a transformation introducing new time-domain has been used to get the ADE with constant coefficients. The objective of the study Leij Van Genuchten 2) was to formulate Green s function for the equilibrium non-equilibrium ADE, to use Green s function method to derive analytical solution for solute movement. The work Park Zhan 2) provides analytical solution of contaminant transport from one-, two-, threedimensional finite sources in a finite-thickness aquifer, using Green s function. The analytical solution was obtained to predict the contaminant concentration along non-uniform groundwater flow in semi-infinite medium Singh et al. 28). The analytical solutions have been presented to onedimensional advection diffusion equation with spatially dependent dispersion along non-uniform flow through inhomogeneous medium in which solute dispersion is assumed proportional to the square of velocity Kumar et al. 29). A new analytical solution was presented for the cross-sectional integrated one-dimensional ADE with spatially dependent coefficients Zamani Bombardelli 22). A series of one- multidimensional solutions of ADE with or without accounting for zero-order production first order decay was presented Van Genuchten et al. 23), which has been useful for simplified analysis of contaminant transport in surface water, for mathematical verification of more comprehensive numerical transport models. To account the mixing caused by velocity fluctuations, dispersion coefficient was considered as a function of both space time variables in the porous media in a theoretical experimental work by Sternberg et al. 996). On the basis of this paper, the dispersion parameter has been expressed in both the independent variables but in degenerate form Su et al. 25; Yadav et al. 2). To accommodate the effects of winds upon solute transport in open tidal fjords channel, the velocity has been considered as function of depth variable sinusoidally varying time variable Wang et al. 977). In the present work, the heterogeneity of the medium is delineated in a more general way by considering velocity dependence on both the independent variables dispersion parameter dependence on the time variable. Analytical solutions of one-dimensional ADE with these two coefficients are obtained in an infinite domain subjected to the instantaneous continuous injected sources using Green s function method. To use this method, a moving coordinate transformation is developed. It helps to get the solution of the ADE in the general form from which solutions are obtained for different particular cases including those of earlier works. The next section comprises of the mathematical formulation of the problem its analytical solutions for instantaneous continuous sources, a subsection of particular cases. It is followed by a section on the discussion of the results based on their illustrations through figures. Lastly, concluding remarks a list of references are given. 2. Mathematical formulation analytical solution Let us consider linear ADE in one dimension in general form as: c t = D ψt) c x x u fx, t)c + qx, t), ) where c is the solute concentration at position x at time t, D u are uniform dispersion coefficient uniform velocity, respectively, in a homogeneous medium, qx, t) is the injection term for the solute mass in the infinite domain. As discussed in the previous section, the medium is considered heterogeneous, so the two coefficients of the ADE, the dispersion coefficient Dx, t)velocity ux, t) are considered temporally dependent spatio-temporally dependent, are written as D ψt) u fx, t), respectively, in equation ). This equation may be written in exped form as: c t = D ψt) 2 c x u fx, t) c 2 x u f c + qx, t). x 2) First objective is to reduce the above equation into a form, whose analytical solution is known in that process, the expression for fx, t) is obtained. Let equation 2) be transformed into an equation C t = D ψ t ) 2 C X u C 2 X αgt )C + q X, t ), 3)

3 Analytical solution of ADE by GFM 75 in a new domain X, t ) through coordinate transformation equations X = Xx, t),t = t. 4) It may be noted that while doing so, the function of the temporally dependent coefficient of secondorder space derivative also changes. The analytical solution of a similar ADE is given by Basha El-Habel 993). In equation 3), the decay term is also assumed varying with time for generality, where α is a parameter, which has dimension of inverse of time variable. Equation 3) is dimensionally consistent as ψ t )gt ) are dimensionless expressions. It is evident in the later part of this paper. Using the transformation equation 4), equation 2) may be written as equation 5) now onwards in place of t we are using only t for the sake of convenience) C t = D ψt) + X x D ψt) 2 X x 2 ) 2 2 C X 2 u fx, t) X x X C t X fx, t) u C + q X, t). 5) x Equation 5) will become equation 3) if equations 6a, b c) are followed ) 2 X ψt) = ψ t), 6a) x D ψt) 2 X x 2 u fx, t) X x X t = u, 6b) fx, t) u = αgt). 6c) x Solving 6c), fx, t) may be obtained; hence, the expression for velocity may be written as: ux, t) =u fx, t) =αxgt)+u φt). 7) Integrating equation 6a) with respect to x, we have ψ t) X = ψt) dx + φ t). 8) Using equations 6a, 7 8) into equation 6b), we get: ψ t) u {αxgt)+u φt)} ψt) ) = ψ t) t ψt) dx + φ t). 9) Simplifying equation 9), we get: ) dφ dt u ψ t) + u φt) ψt) x ) + d ψ t) dt ψt) + αgt) ψ t) ψt) x =. ) Equating the coefficients of x x from both sides in equation ) we have: d ψ t) dt ψt) = αgt) ψ t) ψt), a) dφ dt = u u φt) From equation a) we have: where ψ t) ψt). b) ψ t) = ψt) β 2 t), 2) βt) =exp α gv)dv, 3) is a dimensionless expression. Further using ψ t) in equation b) we get: dφ dt = u u φt) βt). 4) Using equation 7) for fx, t), equation ) may be considered as: c t = D ψt) c x x {αxgt)+u φt)} c + qx, t); <x<, t>. 5) Equation 5) may be reduced into the form C t = D ψt) 2 C β 2 t) X u C 2 X αgt)c + q X, t), 6) by using the transformation X = x βt) + u φv) t u dv, 7) βv) where βt) isdefinedinequation3).thevariable X is also a space variable. To complete the formulation, an initial condition is assumed as: CX, ) = C i ωx); <X<. 8)

4 76 Abhishek Sanskrityayn Naveen Kumar The transformation equation η = X u t 9) is used to eliminate the convective term from equation 6). Another transformation equation Kη, t) =Cη, t)βt) 2) is used to eliminate the first-order decay term, lastly, the transformation equation Crank 975) T = v= ψv) dv 2) β 2 v) is used. As a result, equation 6) may be obtained in the form K T = D 2 K Qη, T) + β 3 t), 22) η2 ψt) where T is another time variable. The initial condition becomes Kη, T =)=C i ωη). 23) Analytical solution of the problem described by equations 22 23) may be obtained by Green s method Haberman 987) as: Kη, T)= T exp + C i β 3 ζ) Qχ, ζ) 4πD T ζ) ψζ) ) dχdζ η χ)2 4D T ζ) ωχ) 4πD T exp η χ) 4D T 2 ) dχ. 24) Using the transformations used earlier, the desired solution may be obtained as: Qχ, t ) βt cx, t)= ) 4πD T ζ) βt) exp 4D T ζ) x 2 t βt) u φv) βv) dv χ dχdt + C i βt) ωχ) 4πD T exp x 4D T βt) u 2 φv) βv) dv χ dχ, 25) where using equation 2), the new time variable, T may be expressed in terms of old time variable, t for different expressions of ψt). The above solution may be expressed in more concise form by considering ξ = χβt ), so dξ = βt )dχ may be used in the first integral. Finally, the analytical solution in equation 25) may be written as: cx, t)= βt) where x βt) u + C i βt) Q ξ,t ) 4πD T ζ) exp ωχ) 4πD T φv) βv) dv exp x 4D T βt) u 4D T ζ) 2 ξ dξdt βt ) 2 φv) βv) dv χ dχ; 26) ψv) t T = β 2 v) dv, ζ = ψv) β 2 v) dv, φv) Q ξ,t )=q ξ + u βt ) βv) dv, t. 27) The solutions describing the concentration distribution pattern from an instantaneous point injection a continuous source of solute mass are obtained below. 2. Instantaneous point injection Instantaneous non-dimensional injection of a solute mass is given by qx, t) =Mδx)δt), 28) where M is the injected mass, δx) is the Dirac delta function. Further, we assume that the infinite domain is initially pollutant free, i.e., cx, ) =. 29) Using the property of Dirac delta function fx) = fx )δx x)dx,

5 Analytical solution of ADE by GFM 77 ft) = ft )δt t)dt, solution in equation 26) will provide for the solute transport from the instantaneous source as: cx, t)= M βt) 4πD T exp x 2 t 4D T βt) u φv) βv) dv. 2.2 Continuous injection 3) Letitbedefinedas: qx, t) =C u f,t)δx), t >. 3) The solution for the solute transport from continuous source may be obtained from equation 26) as: cx, t)= C u βt) φt ) 4πD T ζ) exp x 4D T ζ) βt) u 2.3 Particular cases t 2 φv) βv) dv dt Spatially dependent velocity constant coefficient of decay term 32) Let gt) =φt) =. So, from equation 7), velocity has the expression ux, t) = u + αx, from equation 3), βt) =expαt). The respective solutions for instantaneous injection continuous injection may be obtained from equations 3 32) as: cx, t)=m exp αt) 4πD T exp + u α {exp αt) } ) 2 x exp αt) 4D T, 33) exp αt) cx, t)=c u 4πD T ζ) exp 4D T ζ) x exp αt)+ u α {exp αt) ) 2 exp αt )} dt. 34) Further, following four cases of temporally dependent dispersion coefficient are considered. In each case, using equation 2), respective expression for the new time variable T may be obtained in both the above solutions. i) Constant dispersion coefficient: Let ψt) =,sodt) =D. Using equation 2) we have: T = exp 2αt). 35) 2α As α, both dispersion coefficient velocity become constant. Then, equation 33) becomes the solution of Yeh 98), Haberman 987) Basha El-Habel 993) in case of instantaneous point source, equation 34) becomes that of Yeh 98), Beck et al. 992) Yeh Yeh 27) for continuous point source. ii) Linear dispersion coefficient: Let ψt) =t/k, sodt) =D t/k), T = { exp 2αt)} t 4kα2 2kα exp 2αt). 36) iii) Exponential dispersion coefficient: Let ψt) = exp t/k), so Dt) = D { exp t/k)} T = k { exp 2αt)} + 2α { ) 2αk + } 2αk +)t exp. 37) k iv) Asymptotic dispersion coefficient: Let ψt) = t t + k,sodt) =D t/t + k)) T = v v + k e 2αv dv. 38) As α, then solution given by equations 33 34) become the respective solutions for instantaneous source continuous source given by Basha El-Habel 993). The parameter k = in above form leads to constant diffusion coefficient discussed in case i). The integral 38) may not be evaluated directly, hence numerical integration is carried out using Simpson s rule Spatially temporally dependent velocity constant coefficient of decay term To discuss this case, dispersion coefficient is considered asymptotic that is Dt)=D t/t + k)), so the new time variable T has the expression given

6 78 Abhishek Sanskrityayn Naveen Kumar in equation 38). Also, let us consider gt) =,so from equation 7) spatially temporally dependent velocity has the expression ux, t) = αx + u φt). Three expressions of φt) are considered. For each one, the solutions in both the cases of injected mass are written. i) φt) = exp mt), where m is a temporaldependent parameter of dimension inverse of time variable. The two solutions 3) 32) become cx, t)=m exp αt) 4πD T exp x exp αt) 4D T + u α {exp αt) } + u m + α ) 2 { exp mt αt)}, 39) φt )exp αt) cx, t) =C u 4πD T ζ) exp x exp αt) 4D T ζ) + u α {exp αt) exp αt )} + u α + m {exp αt mt ) exp αt mt)}) 2 dt, 4) respectively. ii) φt) = mt. Using equations 3 32), we obtain the solutions for instantaneous continuous injection sources as: cx, t) =M exp αt) 4πD T exp 4D T x exp αt)+ mtu α exp αt)+ mu {exp αt) } α2 φt )exp αt) 4πD T ζ) cx, t) =C u exp 4D T ζ) x exp αt) ) 2, 4) mu α {t exp αt ) t exp αt)} + mu ) 2 α {exp αt) exp αt )} dt 2, respectively. 42) iii) φt) =expmt). The two solutions 3) 32) become cx, t)=m exp αt) 4πD T x exp αt)+ u exp 4D T ) 2 α m { exp αt + mt)}, 43) φt ) exp αt) cx, t)=c u 4πD T ζ) exp x exp αt) 4D T ζ) + u α m {exp αt + mt ) exp αt + mt)}) 2 dt, 44) respectively Spatially temporally dependent velocity temporally dependent decay term Let us consider φt) =, so from equation 7), ux, t) = u + αxgt), where a form, gt) = /+m t) is considered m is another temporal dependence parameter of the same dimension of m, but different values are considered for the both. From equations 3 2), we have βt) = + m t) α/m T = v + m v+k v) 2α/m dv, respectively. Using equation 3), we obtain the solution for instantaneous injection as: cx, t)=m + m t) α/m 4πD T exp 4D T x +m t) α/m u { + m t) α/m }) 2, 45) m α that for a continuous injection, may be obtained from solution 32) as: + m t) α/m cx, t)=c u 4πD T ζ) x exp 4D T ζ) + m t) α/m u { + m t) α/m m α + m t ) α/m } ) 2 dt. 46)

7 Analytical solution of ADE by GFM Discussion To discuss the different solutions obtained as in the previous section, they are illustrated in the domain xmeter) at different time. The input values are chosen same as considered in the paper Basha El-Habel 993) to compare the results figures of both the works. These are: injected mass, M =., dispersion coefficient velocity in homogeneous medium D m 2 /hr) =., u m/hr) =.25, respectively. Figures 2 illustrate the solution 33) for instantaneous point source which is discussed in subsection Figure is drawn at thr) = 5, 5, 2 for two.4.2 t = 5hr) Concentration..8.6 =. =.25 k =.4 t = 5hr).2 t = 2hr) 2 Distance meter) Figure. Concentration distribution from instantaneous injection source along homogeneous medium dotted curves) heterogeneous medium solid curves), represented by equation 33) for constant dispersion coefficient..7.6 k = 5.5 Concentration.4.3 k = 2 t = 5 hr) = k = 2 Distance meter) Figure 2. Comparison of concentration levels from instantaneous source due to constant dispersion asymptotic dispersion along heterogeneous medium represented by equation 33).

8 72 Abhishek Sanskrityayn Naveen Kumar values of spatial dependence parameter α =.25 αhr) =.. The dispersion coefficient is considered constant first in the four forms discussed in this subsection). Solid curves dotted curves are drawn for the higher lower values of α, respectively. As the former value of α is much higher as compared to its latter value so it represents the velocity varying linearly with position the latter value represents the uniform velocity, i.e., u = u see section 2.3.). The dotted curves are exactly the same curves at k =of figure 2 of the paper Basha El-Habel 993). It may be noted that the solution in equation 23) of Basha El-Habel 993) for which they have drawn figure 2 may be obtained from our solution given by equation 33) under the limit α. The resemblance of solid curves with the dotted curves validates the correctness of the solution in equation 33) of the present work. The figure reveals that increasing value of velocity causes faster solute transport characterized by lower concentration peak as compared to the homogeneous medium represented by uniform velocity dispersion). Figure reveals the concentration distribution pattern originating from an instantaneous source. The concentration peak is the highest proximal to the injected location at t = 5hr). It attenuates drifts forward with time position. In rest of the figures, αhr) =.25 is taken, which implies heterogeneity of the medium. Figure 2 exhibits the same solution in equation 33) of section 2.3., but for the fourth form of the dispersion coefficient which is asymptotic. This figure compares the concentration pattern represented by the solution in equation 33) for constant dispersion coefficient k = ), with those for asymptotic dispersion coefficient for khr) = 2 5 at thr) = 5. This figure reveals the expected pattern of the peak concentration increasing significantly approaching towards the source location, with k. Figure 3 illustrates the solutions given by equations 39), 4) 43) discussed in section for three expressions of φt), for asymptotic dispersion coefficient using k =2atthr) = 5. The pattern of concentration dispersion in each case is very much congruous with the expressions of φt). The value of temporal dependence parameter mhr) is taken as.. The curve for φt) = in figure 3 may also be visualized in figure 2, the case for which the solution is given in equation 33). Figure 4 depicts the concentration values evaluated from the solution in equation 45) discussed in section at thr) = 5, 5, 2. The dispersion t)= t) = expmt) t) = -exp-mt) t) = exp-mt) Concentration t = 5 hr) k = 2..5 Distance meter) Figure 3. Comparison of concentration levels from instantaneous source due to asymptotic dispersion coefficient represented by equations 33, 39, 4 43).

9 Analytical solution of ADE by GFM t = 5hr).3.25 Concentration.2.5 gt) = gt) = /+m t) t) =. k = 2.5 t = 5hr) t = 2hr) 2 Distance meter) Figure 4. Comparison of concentration levels from instantaneous source due to asymptotic dispersion coefficient represented by equations 33 45)..9 t = 2hr).8.7 Concentration =. =.25 k = t = 5hr) t = 5hr).2. 2 Distance meter) Figure 5. Concentration distribution from continuous source along homogeneous medium dotted curves) heterogeneous medium solid curves) represented by equation 34). coefficient is again considered asymptotic using k = 2. The temporal dependent decay term occurring in the expression of the velocity is assumed as gt) =/+m t), i.e, velocity decreases with time hence, same is the pattern of the concentration level. Another temporal dependence parameter is

10 722 Abhishek Sanskrityayn Naveen Kumar chosen as m hr) =. to draw this figure. Figure 5 is drawn for solution 34) obtained for continuous source discussed in section 2.3. for the constant dispersion coefficient, at thr) = 5, 5 2 for the same two values of α. The solid curves are drawn for α =.25 showing the effect of increasing velocity with position on the concentration pattern. The dotted curves α =.) match with the curves of figure 3 for k =Basha El-Habel 993). 4. Conclusions In a previous paper Basha El-Habel 993), the heterogeneity of the medium has been described by time-dependent dispersion coefficient of the ADE velocity has been considered uniform. The present work extends this work by considering velocity as spatially temporally dependent dispersion coefficient as temporally dependent. This assumption is based on the conclusions of the paper Sternberg et al. 996). It describes the heterogeneity of the medium in more general way than in previous works. A moving coordinate transformation is introduced to reduce the ADE of the present formulation into the one considered in the paper Basha El-Habel 993). The results of both the works are matched through particular solutions figures for instantaneous continuous sources. The resemblance between the results figures of both the works validates our formulation the solutions obtained through the transformation derived. Analytical solutions of ADE with variety of expressions in terms of position time variables for its both the coefficients with the help of appropriately derived transformation equations will be the main task of the author s future works. Acknowledgements The first author expresses his gratitude to University Grants Commission, Government of India, for financial academic assistance in the form of Senior Research Fellowship. The authors express their gratitude to the reviewers for their valuable comments suggestions which improved the present work to a great extent. List of symbols c: Solute concentration in domain x C: Solute concentration in new domain X C i : Initial concentration C : Reference concentration D : Dispersion coefficient in homogeneous medium u : Velocity in homogeneous medium K: Solute concentration in the new domain η M: Injected pollutant mass k: Asymptotically varying parameter ν, t, χ, ς, ξ: Dummy variables m: Temporal dependence parameter m : Another temporal dependence parameter x: Position variable T : Another new time variable t: Time variable t : Time in new domain X: Position in new domain η: A new space variable α: Coefficient in the decay term δ ): Dirac delta function References Anderson M 979 Modeling of groundwater flow systems as they relate to the movement of contaminants; CRC Crit. Rev. Environ. Control Aral M M Liao B 996 Analytical solutions for twodimensional transport equations with time-dependent dispersion coefficients; J. Hydrol. Eng Basha H A El-Habel F S 993 Analytical solution of the one-dimensional time dependent transport equation; Water Resour. Res. 299) Beck J V, Cole K D Litkouhi B 992 Heat conduction using Green s function; Hemisphere Publishing Co., Washington, D.C. Crank J 975 Mathematics of Diffusion; Oxford University Press, New York. Dagan G 987 Theory of solute transport by groundwater; Ann. Rev. Fluid Mech Gelhar L W, Mantoglou A, Welty C Rehfeldt K R 985 A review of field-scale physical transport processes in saturated unsaturated porous media; EPRI Rep. EA-49, Elec. Power Res. Inst., Palo Alto, Calif. Güven O, Molz F J Melville J G 984 An analysis of dispersion in a stratified aquifer; Water Resour. Res. 2) Haberman R 987 Elementary applied partial differential Equations; Prentice-Hall, Englewood Cliffs, New Jersey. Kumar A, Jaiswal D K Kumar N 29 Analytical solutions of one-dimensional advection diffusion equation with variable coefficients in a finite domain; J. Earth Syst. Sci. 85) Leij F J Van Genuchten M 2 Analytical modeling of nonaqueous phase liquid dissolution with Green s functions; Transp. Porous Media Matheron G de Marsily G 98 Is transport in porous media always diffusive? A counter example; Water Resour. Res Park E Zhan H 2 Analytical solutions of contaminant transport from finite one-, two-, three-dimensional sources in a finite-thickness aquifer; J. Contam. Hydrol Pickens J F Grisak G E 98a Scale-dependent dispersion in stratified granular aquifer; Water Resour. Res. 74) 9 2.

11 Analytical solution of ADE by GFM 723 Pickens J F Grisak G E 98b Modeling of scaledependent dispersion in hydrogeologic systems; Water Resour. Res. 76) 7 7. Simpson E S 978 A note on the structure of the dispersion coefficient; Geol. Soc. Am. Abstr. Programs, 393p. Singh M K, Mahato N K Singh P 28 Longitudinal dispersion with time-dependent source concentration in semi-infinite aquifer; J. Earth Syst. Sci. 76) Sposito G W, Jury W A Gupta V K 986 Fundamental problems in the stochastic convection dispersion model of solute transport in aquifers field soils; Water Resour. Res Sternberg S P K, Cushman J H Greenkorn R A 996 Laboratory observation of nonlocal dispersion; Trans. Porous Media Su N, Ser G C, Liu F, Anh V Barry D A 25 Similarity solutions for solute transport in fractal porous media using a time- scale-dependent dispersivity; Appl. Math. Model. 299) Van Genuchten M Th, Leij F J, Skaggs T H, Toride N, Bradford S A Pontedeiro E M 23 Exact analytical solutions for contaminant transport in rivers.. The equilibrium advection dispersion equation; J. Hydrol. Hydromech. 62) Wang S T, McMillan A F Chen B H 977 Analytical model of dispersion in tidal fjords; J. Hydraul. Div. ASCE 3HY7) Yadav S, Kumar A, Jaiswal D K Kumar N 2 Onedimensional unsteady solute transport along unsteady flow through inhomogeneous medium; J. Earth Syst. Sci. 22) Yeh G T 98 AT23D: Analytical transient one-, two-, three-dimensional simulation of waste transport in the aquifer system. Envir. Sci. Div. 439, ReportORNL- 562, Oak Ridge, Tennessee, USAp. Yeh G T Yeh H D 27 Analysis of point-source boundary-source solutions of one-dimensional groundwater transport equation; J. Environ. Eng. 33) Zamani K Bombardelli F A 22 One-dimensional, mass conservative, spatially-dependent transport equation: New analytical solution; 2th Pan-American Congress of Applied Mechanics, 2 6 January, 22, Port of Spain, Trinidad. MS received 4 December 25; revised 27 August 26; accepted 2 September 26 Corresponding editor: Subimal Ghosh