Dispersion Transport Equation with Distance-Dependent Coefficients

Journal of Hydrology 390 (2010) 57–65

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Journal of Hydrology

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Analytical solution for one-dimensional advection–dispersion transport equation with distance-dependent coefﬁcients

J.S. Pérez Guerrero a, T.H. Skaggs b,* a Radioactive Waste Division, Brazilian Nuclear Energy Commission, DIREJ/DRS/CNEN, R. General Severiano 90, 22290-901 RJ-Rio de Janeiro, Brazil b US Salinity Laboratory, USDA-ARS, Riverside, CA, USA article info summary

Article history: Mathematical models describing contaminant transport in heterogeneous porous media are often formu- Received 15 April 2010 lated as an advection–dispersion transport equation with distance-dependent transport coefficients. In Received in revised form 16 June 2010 this work, a general analytical solution is presented for the linear, one-dimensional advection–dispersion Accepted 18 June 2010 equation with distance-dependent coefficients. An integrating factor is employed to obtain a transport equation that has a self-adjoint differential operator, and a solution is found using the generalized inte- This manuscript was handled by Geoff Syme, Editor-in-Chief gral transform technique (GITT). It is demonstrated that an analytical expression for the integrating factor exists for several transport equation formulations of practical importance in groundwater transport modeling. Unlike nearly all solutions available in the literature, the current solution is developed for a finite Keywords: Contaminant transport spatial domain. As an illustration, solutions for the particular case of a linearly increasing dispersivity are Heterogeneous porous media developed in detail and results are compared with solutions from the literature. Among other applica- Distance-dependent dispersion tions, the current analytical solution will be particularly useful for testing or benchmarking numerical Analytical solution transport codes because of the incorporation of a finite spatial domain. Integral transform Published by Elsevier B.V. Integrating factor

1. Introduction or distance-dependent coefficients (Yates, 1990; Chrysikopoulos et al., 1990; Yates, 1992; Huang et al., 1996; Logan, 1996; Zop- The literature contains many analytical solutions for solute pou and Knight, 1997; Hunt, 1998, 1999, 2002; Pang and Hunt, transport in homogeneous porous media. These solutions, which 2001; Al-Humoud and Chamkha, 2007; Liu and Si, 2008; Chen, have been collected in various compendiums (Codell et al., 1982; 2007; Chen et al., 2003, 2007, 2008a,b; Kumar et al., 2010). van Genuchten, 1982; Javandel et al., 1984; Wexler, 1992), were While most of these solutions are applicable to a range of prob- found by solving the linear advection–dispersion transport equa- lems involving (effectively) semi-infinite or infinite media, other tion with constant coefficients, subject to appropriate boundary applications require consideration of finite media, such as anal- and initial conditions. Solutions are available for one-, two-, and ysis of transport in lysimeters or columns, or benchmarking three-dimensional spatial domains, with the vast majority being numerical transport codes. In these types of problems, the effect applicable to semi-infinite or infinite media. However, the advec- of the exit boundary may not be negligible. tion–dispersion equation with constant coefficients may not be One reason for the lack of progress in developing solutions for appropriate for transport in heterogeneous media, where transport finite domains is that the solution procedures tend to be relatively coefficients can be variable in space and/or time. Relatively few complicated, requiring difficult or tedious mathematical deriva- analytical results are available for the case of non-constant coeffi- tions and manipulations. However, the advent of software such cients, especially for the case of finite media. as Mathematica (Wolfram Research, Inc., 2007) with capabilities Analytical solutions for heterogeneous porous media include for symbolic manipulation has made these solution procedures solutions obtained for transport equations with time-dependent more tractable. coefficients (Barry and Sposito, 1989; Basha and El-Habel, 1993; Also facilitating the development of new solutions are system- Aral and Liao, 1996; Marinoschi et al., 1999; Kumar et al., 2010) atized integral transform techniques (ITTs) (Mikhailov and Ozisik, 1984). As noted by Ozisik (1993), the classic ITT (CITT) provides a systematic, efficient, and straightforward approach for the analytical solution of both transient and steady problems, with both * Corresponding author. Address: US Salinity Laboratory, 450 W. Big Springs Rd., Riverside, CA 92507, USA. Tel.: +1 951 369 4853; fax: +1 951 342 4964. homogeneous and non-homogeneous boundary conditions. A large E-mail address: [email protected] (T.H. Skaggs). variety of heat and mass diffusion problems have been categorized

0022-1694/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jhydrol.2010.06.030 58 J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65

Nomenclature

a slope of dispersivity–distance relationship t, T dimensionless and dimensional time A ðnÞ; A ðnÞ 00 01 coefficients in generic transport equation U mean transport velocity A10ðnÞ; A20ðnÞ x, X dimensionless and dimensional space coordinate B mathematical operator specifying boundary conditions k Yr Bessel function of the second kind and order r c, C dimensionless and dimensional solute concentration w(n) coefficient in self-adjoint equation cI(x), CI dimensionless and dimensional initial solute concentration Greek symbols C0 boundary (first-type) or influent (third-type) concentra- a; a dimensionless and dimensional dispersivity tion bi eigenvalue c steady state solute concentration (dimensionless) 1 dij Kronecker delta D; D dimensionless and dimensional dispersion coefficient gk coefficient in boundary condition dM,DM dimensionless and dimensional coefficient of molecular h unknown function with homogeneous boundary condi- diffusion tions f integral transformed initial condition i hi integral transform of the function h g(nk) source term in generic boundary condition k decay constant Hk coefficient in boundary condition l ratio of generic transport coefficients, A00(n)/A01(n) Ir modified Bessel function of first kind and order r n space variable in generic transport equation J Bessel function of first kind and order r r n0, n1 position of boundaries in generic transport equation Kr modified Bessel function of second kind and order r p 3.14... k index denoting position r equal to 1/a L advection–dispersion operator s time variable in generic transport equation L0 domain length u(x) auxiliary function Ni norm of ith eigenfunction wi eigenfunction p(n) integrating factor ~ wi normalized eigenfucntion q(n) coefficient in operator S x(n, s) dependent variable in generic transport equation Q(n) source-sink term in the generic transport equation x1(n) steady state (s ? 1)ofx(n, s) R retardation coefficient xI(n) initial condition of x(n, s) S self-adjoint operator

and treated systematically using this technique, creating a unified 2. General problem formulation approach for solving those problems (Mikhailov and Ozisik, 1984). Transport equations not immediately analytically solvable with the Consider a one-dimensional finite medium with space variable CITT can often be transformed into an amenable form using tech- n and time variable s. For some quantity x x(n, s), a generic niques such as algebraic substitution or integrating factor methods advection–dispersion transport equation with distance-dependent (e.g. Pérez Guerrero et al., 2009a,b, 2010). transport coefficients can be formulated as: Cotta (1993) extended the CITT to develop the generalized integral transform technique (GITT). The GITT permits the analytical or @x @ @x @x A01ðnÞ ¼ DðnÞ UðnÞ A00ðnÞx þ QðnÞ; semi-analytical solution of more general parabolic or elliptic prob- @s @n @n @n lems, both linear and nonlinear. In the GITT, the unknown function n0 < n < n1 ð1Þ is represented in terms of an eigenfunction series expansion. The solution is obtained in the following steps (Cotta, 1993): where n0 and n1 are the locations of the boundaries, A01(n), D(n), U(n), and A00(n) are distance-dependent coefficients, and Q(n)isa (a) Choose an appropriate auxiliary eigenvalue problem and source-sink term. On the right side of Eq. (1), the first term is the find the associated eigenvalues, eigenfunctions, norm, and dispersion term, the second is the advection term, and the third orthogonalization property. term represents processes such as the first-order decay or produc- (b) Develop the integral and inverse transforms. tion. Eq. (1) is a generic transport equation which may be either (c) Transform the partial differential equation into a system of dimensional or dimensionless. ordinary differential or algebraic equations. Expanding the dispersion term and grouping like terms gives, (d) Solve the ordinary differential or algebraic system. 2 (e) Use the inverse transform to obtain the unknown @x @ x @x A01ðnÞ ¼ A20ðnÞ þ A10ðnÞ A00ðnÞx þ QðnÞ; function. @s @n2 @n

n0 < n < n1 ð2aÞ The aim of the present study is to develop an analytical solution for solute transport in finite, heterogeneous porous media. For this where purpose, a general advection–dispersion transport equation with @DðnÞ distance-dependent coefficients is transformed through the use A ðnÞ¼DðnÞ; A ðnÞ¼ UðnÞð2b; cÞ 20 10 @n of an integrating factor and a solution is obtained with the GITT. We demonstrate that for a number of transport formulations of We define the advection–dispersion operator L with distance- practical importance, an analytical expression for the integrating dependent coefficients A00(n), A10(n), and A20(n)tobe factor exists. The particular case of a linearly increasing dispersiv- 2 ity coefficient is developed in detail and the solution is compared @ @ L A20ðnÞ 2 þ A10ðnÞ A00ðnÞð3Þ with results from the literature. @n @n J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65 59

Eq. (2) can then be written more compactly as: 3.2. Obtaining an equation with a self-adjoint operator @x A ðnÞ ¼ Lx þ QðnÞ; n < n < n ð4Þ Depending on the functional form of the coefficients A (n), 01 @s 0 1 00 A10(n), and A20(n), the operator L may or may not be self-adjoint. The transport problem is completed by specifying initial and A second-order differential operator, S, is self-adjoint if and only boundary conditions, respectively, as: if it has the form (Sagan, 1961) xðn; 0Þ¼xIðnÞ; n0 < n < n1 ð5Þ @ @ S pðnÞ þ qðnÞð14Þ @n @n Bkxðnk; sÞ¼gðnkÞ; k ¼ 0; 1 ð6Þ where p(n) is differentiable. The operator B can represent a first, second, or third-type k The form of S in Eq. (14) indicates that a self-adjoint second-or- boundary condition depending on the specification of the coeffi- der transport equation is equivalent to a purely diffusive problem cients g and H : k k such as commonly encountered in heat conduction. Such diffusion @ equations have been categorized by Mikhailov and Ozisik (1984) B g þ H ð7Þ k k @x k into seven classes, and the GITT can be used to obtain formal ana- lytic solutions for each of them (Mikhailov and Ozisik, 1984). When the operator L is non-self-adjoint, it may be possible to 3. General analytical solution transform the equation to obtain an equivalent problem with a self-adjoint operator. For this purpose, we use an integrating factor An analytical solution of Eqs. (4)–(6) is obtained in several to transform Eq. (11). Eq. (11) can be written as: steps. First the boundary conditions are homogenized using a ‘‘filter function” which is found by solving the transport problem in @h @2h @h the steady state regime. Next, the general non-self-adjoint trans- A01ðnÞ ¼ Lh ¼ A20ðnÞ 2 þ A10ðnÞ A00ðnÞh ð15Þ @s @n @n port equation is transformed into an equivalent self-adjoint equa- We multiply each term by pðnÞ , where p(n) is the unknown inte- tion through the use of an appropriate integrating factor. Finally, A20ðnÞ the resulting self-adjoint equation is solved using the GITT. grating factor: pðnÞ @h @2h pðnÞ @h pðnÞ 3.1. Steady state problem and homogenization of the boundary A01ðnÞ ¼ pðnÞ 2 þ A10ðnÞ A00ðnÞh ð16Þ conditions A20ðnÞ @s @n A20ðnÞ @n A20ðnÞ The idea of the integrating factor is that Eq. (16) should reduce For the asymptotic condition s ? 1, the quantity x in Eqs. (4)– to: (6) will tend to the steady state regime, x(n, s) ? x1(n). The asymptotic system is given by: pðnÞ @h @ @h pðnÞ A01ðnÞ ¼ pðnÞ A00ðnÞh ð17Þ A20ðnÞ @s @n @n A20ðnÞ Lx1 þ QðnÞ¼0; n0 < n < n1 ð8Þ This requires that: B x ðn Þ¼gðn Þ; k ¼ 0; 1 ð9Þ k 1 k k @pðnÞ pðnÞ ¼ A10ðnÞð18Þ Solving Eqs. (8) and (9) gives x1(n), which may be used as a fil- @n A20ðnÞ ter function to obtain a problem with homogeneous boundary con- Thus the integrating factor is given by: ditions. To this end, we express the unknown x(n, s) as the sum of Z the filter x (n) and an unknown function h(n, s): A ðnÞ 1 pðnÞ¼exp 10 dn ð19Þ A20ðnÞ xðn; sÞ¼x1ðnÞþhðn; sÞð10Þ Note that the integrating factor depends only of the dispersion The unknown quantity (n, ) is thus found by determining x s and advection coefficients. h(n, ). A system of equations for h(x, t) is obtained by substituting s With the integrating factor defined, Eq. (11) can be written in Eq. (10) into Eqs. (4)–(6). The resulting equations for h(n, ) are: s terms of the self-adjoint operator S Eq. (14): @h @h @ @h A01ðnÞ ¼ Lh; n0 < n < n1 ð11Þ wðnÞ ¼ Sh ¼ pðnÞ þ qðnÞh ð20Þ @s @s @n @n hðn; 0Þ¼xIðnÞx1ðxÞ; n0 < n < n1 ð12Þ where the coefficients are given by:

B hðn ; tÞ¼0; k ¼ 0; 1 ð13Þ pðnÞ pðnÞ k k wðnÞ¼ A01ðnÞ; qðnÞ¼ A00ðnÞð21a; bÞ A20ðnÞ A20ðnÞ where the boundary conditions Eq. (13) are homogeneous. For situations where A00ðnÞ ¼ l is a constant, a simpliﬁcation is It is important to note that the use of the integrating factor does A01ðnÞ possible. In this case, the last term in the operator L (as deﬁned not alter the boundary and initial conditions, Eqs. (12) and (13). in Eq. (3)) can be eliminated if the following expression is used in- Therefore, the unknown h(n, s) is found by solving Eq. (20) subject stead of Eq. (10): to Eqs. (12) and (13). For the conditions leading to Eqs. (100) and (110), the last term in Eq. (20) vanishes (q(n) = 0). 0 xðn; sÞ¼x1ðnÞþexpðlsÞhðn; sÞð10 Þ 3.3. Application of the GITT Substituting Eq. (100) into Eq. (4) yields the following equation instead of Eq. (11): Having obtained a transport equation (Eq. (20)) with a self-adjoint differential operator and homogeneous boundary conditions @h 0 Eq. (13), the unknown h(n, s) is found by applying the systematized A01ðnÞ ¼ Lh with A00ðnÞ¼0; n0 < n < n1 ð11 Þ @s GITT procedure. 60 J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65 ÀÁ 2 3.3.1. The auxiliary eigenvalue problem hiðsÞ¼f i exp bi s ð31Þ Eigenvalue problems with self-adjoint operators have several interesting properties, including (Zwillinger, 1992): the eigen- Finally, the original unknown x(n, s) can be calculated analytically values are real; the eigenvalues are enumerable (with no cluster from the inverse transform rule and Eq. (10): point); the eigenfunctions corresponding to distinct eigenvalues X1 ÀÁ ~ 2 are orthogonal; and the set of eigenfunctions are complete. xðn; sÞ¼x1ðnÞþ wiðnÞf i exp bi s ð32Þ Many eigenvalue problems can be associated with Eq. (20) (sub- i¼1 ject to Eqs. (12) and (13)). We choose the following one which per- For situations in which Eqs. (100) and (110) are applicable, the mits an exact integral transform: solution is given by: d dw X1 ÀÁ pðnÞ þ qðnÞw þ wðnÞb2w ¼ 0 ð22aÞ xðn; sÞ¼x ðnÞþexpðlsÞ w~ ðnÞf exp b2s ð33Þ dn dn 1 i i i i¼1

BkwðnkÞ¼0; k ¼ 0; 1 ð22bÞ 4. Application of the general analytical solution to Eq. (22) is the Sturm–Liouville eigenvalue problem, where w w(n) heterogeneous groundwater systems is the eigenfunction and b is the eigenvalue. The orthogonality property for the set of linearly independent eigenfunctions, w (n), i The general analytical solution developed above for the advec- associated with Eq. (22) is given by: Z tion–dispersion equation with distance-dependent coefficients de- n1 pends on the existence of an analytical expression for the wðnÞwiðnÞwjðnÞdn ¼ di;jNi ð23Þ integrating factor defined in Eq. (19). Such an expression cannot n0 be guaranteed for arbitrary functional forms of the dispersion where Ni is the norm and di,j is the Kronecker delta. The normalized and advection coefficients. However, in this section we show that eigenfunction is defined: for many cases of practical interest in groundwater hydrology, the integrating factor exists. ~ wiðffiffiffiffiffinÞ wiðnÞ¼ p ; i ¼ 1; 2; 3; ... ð24Þ Contaminant transport in heterogeneous aquifer systems is of- N i ten modeled in terms of a constant average transport velocity, linear equilibrium sorption, and first-order decay. For a finite medium

3.3.2. Development of the integral transform pair of length L0, this problem can be formulated as: The unknown function h(n, s) is represented as a series expan- @C @ @C @C sion in terms of the eigenfunctions wi(x), R ¼ DðXÞ V kRC; 0 < X < L ð34Þ @T @X @X @X 0 X1 hðn; sÞ¼ w~ ðnÞh ðsÞðInverseÞð25Þ i i @Cð0; TÞ i¼1 Cð0; TÞ¼C or Dð0Þ þ VCð0; TÞ¼VC ð35; 36Þ 0 @T 0 where hiðsÞ is the transformed ‘‘potential”. Eq. (25) is the inverse transform rule. The corresponding transform rule is obtained by fol- @CðL ; TÞ 0 ¼ 0 ð37Þ lowing the procedureR of Ozisik(1993) and Cotta(1993), i.e. applying @X n1 ~ the operator wðnÞwjðnÞðÞdn to both sides of Eq. (20) and using Eq. n0 (23) (the orthogonality property) and (25) to obtain: CðX; 0Þ¼0 ð38Þ Z n1 3 ~ where C(X, T) is the dimensional concentration (M L ), C0 is the hiðsÞ¼ wðnÞwiðnÞhðn; sÞdn ðTransformÞð26Þ boundary Eq. (35) or influent Eq. (36) concentration (M L3), R is n0 the constant retardation coefficient (–), V is the average constant flow velocity (L T1), k is the first-order decay constant (T1), and 3.3.3. Integral transform of the differential equation DðXÞ is the longitudinal hydrodynamic dispersion coefficient Substituting the inverse formula into Eq. (20) results in: (L2 T1). The hydrodynamic dispersion coefficient is defined as the @ X1 X1 sum of the mechanical dispersion and the molecular diffusion, DM ~ ~ 2 1 wðnÞ wiðnÞhiðsÞ¼S wiðnÞhiðsÞð27Þ (L T ): @s i¼1 i¼1 R DðXÞ¼VaðXÞþDM ð39Þ n1 ~ Applying the operator wjðnÞðÞdn to both sides of Eq. (27) and n0 regrouping terms gives: where aðXÞ is the dispersivity (L). Z Z X1 n X1 n We define the following dimensionless variables: @h ðsÞ 1 1 i wðnÞw~ ðnÞw~ ðnÞdn ¼ h ðsÞ w~ ðnÞSw~ ðnÞdn ð28Þ @s i j i j i X T C kRL0 i¼1 n0 i¼1 n0 x ¼ ; t ¼ ÀÁ; c ¼ ; c ¼ ð40a—dÞ L RL0 C V 0 V 0 This equation can be simplified using Eqs. (22a), (23), and (26): Eqs. (34)–(38) can then be written in dimensionless form: dhi 2 ¼b hi ð29Þ @c @ @c @c ds i ¼ DðxÞ cc; 0 < x < 1 ð41Þ @t @x @x @x Analogously the initial condition is also transformed: Z n1 @cð0; tÞ ~ cð0; tÞ¼1or Dð0Þ þ cð0; tÞ¼1 ð42; 43Þ hið0Þ¼f i ¼ wðnÞwiðnÞðxIðnÞx1ðnÞÞdn ð30Þ @x n0 @cð1; tÞ ¼ 0 ð44Þ 3.3.4. Analytical solution for the transformed and original problems @x Eqs. (29) and (30) are a set of decoupled ordinary differential equations, whose analytical solution is: cðx; 0Þ¼0 ð45Þ J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65 61 where the dimensionless distance-dependent dispersion parameter case, with entrance boundary given by a third-type condition Eq. D(x) is given by: (43) and exit boundary by a second-type condition Eq. (44),is called ‘‘Case 3–2”. DðxL Þ aðxL Þ D DðxÞ¼ 0 ¼ 0 þ M ¼ aðxÞþd ð46aÞ VL L L V M 0 0 0 5.1. Steady transport aðxL Þ D aðxÞ 0 ; d M ð46b; cÞ Applying the general discussion of Section 3.1 to the current L M L V 0 0 problem, the dimensionless steady state concentration is found where a(x) is the dimensionless dispersivity and dM the dimension- by solving: less diffusion coefficient. d dc dc For heterogeneous aquifers, Pickens and Grisak (1981) proposed DðxÞ 1 1 cc ¼ 0 ð47Þ dx dx dx 1 that the spatial dependence of the dispersivity, aðXÞ, may be linear, parabolic, exponential, or asymptotic. Table 1 presents these model dc1ð0Þ dispersivity functions along with the corresponding dimensionless c1ð0Þ¼1or Dð0Þ þ c1ð0Þ¼1 ð48; 49Þ representations, a(x). Note that in each case aðXÞ and a(x) have the dx same functional form. dc ð1Þ To see if the integrating factor exists for the four model func- 1 ¼ 0 ð50Þ tions, Eq. (19) was evaluated analytically for each of the dimen- dx sionless models (with the evident variable equivalencies being where D(x)=ax + dM. A solution to this problem was obtained using n = x, A20(n)=D(x), and A10(n) = 1). Table 2 shows that indeed the the ‘‘DSolve” function of the Mathematica software package, version integrating factor exists for all four cases, with analytical results 6.0 (Wolfram Research, Inc., 2007). The closed-form analytical solu- being given for both non-negligible (dM – 0) and negligible tions for Cases 1–2 and 3–2 are, respectively: (d = 0) molecular diffusion (the latter situation yielding less com- M r=2 plex mathematical expressions). Hence for each of the forms pro- DðxÞ Ir½uðxÞKr1½uð1Þ þ Ir1½uð1ÞKr½uðxÞ c1ðxÞ¼ ð51Þ posed by Pickens and Grisak (1981) for practical groundwater Dð0Þ Ir½uð0ÞKr1½uð1Þ þ Ir1½uð1ÞKr½uð0Þ transport problems, the integrating factor exists and the general analytical solution may be applied. In the remainder of this paper, 1 1=2 DðxÞ r=2 c ðxÞ¼ we apply the general analytical solution to the particular case of 1 cDð0Þ Dð0Þ groundwater contaminant transport with a linearly increasing Ir½uðxÞKr1½uð1Þ þ Ir1½uð1ÞKr½uðxÞ dispersivity. ð52Þ Ir1½uð1ÞKrþ1½uð0Þ Irþ1½uð0ÞKr1½uð1Þ In the event that an analytical expression for the integrating factor is not available, the solution presented in Section 2 may be where regarded as a formal solution. In this case, it is possible to evaluate pffiffiffiffiffiffiffiffiffiffiffiffi 1 the integrating factor numerically. This formal analytical approach uðxÞ¼2r cDðxÞ; r ¼ ð53a; bÞ is the subject of ongoing research and will be documented in a fu- a ture publication. and where Ir and Kr are modified Bessel functions of the first and second kinds, respectively. 5. Example: linearly increasing dispersivity 5.2. Transient transport Using the Laplace transform technique, Yates (1990) obtained an analytical solution for advective–dispersive transport in heter- To apply the general solution developed in Section 2, we make ogenous, semi-infinite media in which dispersion increases line- the following variable equivalencies (with n ? x and s ? t): arly with transport distance. The equation solved by Yates (1990) A01ðxÞ¼1; A20ðxÞ¼DðxÞ; is equivalent to our Eq. (41) with D(x)=ax + dM. As a demonstration of the current methodology, we apply our general analytical solu- dDðxÞ A10ðxÞ¼ 1; A00ðxÞ¼c ð54a—dÞ tion to the specific problem solved by Yates (1990), except that the dx solution is obtained in the present work for a finite medium of Because the ratio A00ðxÞ ¼ c is a constant, c(x, t) will be represented length L . A01ðxÞ 0 according to Eq. (100). For Cases 1–2 and 3–2, the resulting system Two different problems can be formulated depending on the of equations for the unknown h is: boundary conditions imposed. The first case has the entrance boundary specified by a first-type condition Eq. (42) and the exit @h @2h @h ¼ DðxÞ A ðxÞ ð55Þ boundary by a second-type condition Eq. (44). Signifying the @t @x2 10 @x boundary types, we call this formulation ‘‘Case 1–2”. The second @hð0; tÞ hð0; tÞ¼0or Dð0Þ þ hð0; tÞ¼0 ð56; 57Þ @x Table 1 Functional forms for the dimensional (a) and dimensionless (a) dispersivities.a @hð1; tÞ ¼ 0 ð58Þ Type a(X) a(x) Coefficients @x Linear aX ax n Parabolic bX n bx n1 b ¼ bL0 hðx; 0Þ¼c1ðxÞð59Þ Exponential A 1 B A 1 B A ¼ A ; B ¼ B XþB xþB L0 L0 The integrating factor p(x) and the coefficient w(x) are (Table 2): Asymptotic E½1 expðFxÞ E E½1 expðFXÞ E ¼ ; F ¼ FL0 L0 11=a 1r r pðxÞ¼ðax þ dMÞ ¼ DðxÞ ; wðxÞ¼DðxÞ ð60; 61Þ a a, b, A, B, E, and F are dimensionless constants, whereas the same symbols with overbars are dimensional constants; n is a dimensionless constant; L0 is the domain The Sturm–Liouville problem for Cases 1–2 and 3–2 takes the length. form: 62 J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65

Table 2 Integrating factor p(x) for various model dispersivity functions a(x).a

a(x) p(x) (non-negligible diffusion, dM – 0) p(x) (negligible diffusion, dM =0)

11/a 11/a ax (ax + dM) x hi n bx n n n x1n ðbx þdM Þ x n 1 1 bx x exp exp 2 ½bx F1 1; 1 þ ; 2 þ ; dM ðn þ 1Þ bðn1Þ dM n 1 d 2 n n dM ð þ Þ M hi 1 AB B 2 xþðBAÞ lnðxÞþA lnðBþxÞ A 1 ðAþdM Þ exp xþB ½AxþðBþxÞdM exp x A Bþx AþdM 1 1 1 1 E½1 expðFxÞ FðEþd Þ EF expðFxÞ½ðE þ dMÞ expðFxÞE M ½expðFxÞ1 expðFxÞ

a dM is the coefﬁcient of molecular diffusion; a, b, A, B, E, and F are numerical constants; 2F1() is the hypergeometric function.

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi d dw 2 1r r 2 ½1 þ rDð0Þ Yr½2rbi Dð1ÞYr1½2rbi Dð1ÞYrþ1½2rbi Dð1Þ DðxÞ þ DðxÞ b w ¼ 0 ð62Þ N 4 pffiffiffiffiffiffiffiffiffiffi dx dx 12i ð Þ¼ 2 Yr½2rbi Dð0Þ 70e @wð0Þ ð Þ wð0Þ¼0or Dð0Þ þ wð0Þ¼0 ð63; 64Þ @x X4 N ¼ N ðmÞð71aÞ @wð1Þ i 32i ¼ 0 ð65Þ m¼1 @x pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 rDð0Þ Jr1½2rbi Dð0ÞJrþ1½2rbi Dð0ÞJr½2rbi Dð0Þ This eigenvalue problem was solved in Mathematica using the N 1 pffiffiffiffiffiffiffiffiffiffi 32i ð Þ¼ 2 ‘‘DSolve” function in combination with other commands for alge- Jrþ1½2rbi Dð0Þ braic simplifications. The set of eigenfunctions are given in terms ð71bÞ of Bessel functions of the first (Jr) and second (Yr) kind. The result pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi for Case 1–2 is: 2 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi !½1 þ rDð0Þ Jr½2rbi Dð1ÞJr1½2rbi Dð1ÞJrþ1½2rbi Dð1Þ N 2 pffiffiffiffiffiffiffiffiffiffi 32i ð Þ¼ 2 r=2 Jr½2rbi DðxÞ Yr½2rbi DðxÞ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Jrþ1½2rbi Dð0Þ wiðxÞ¼DðxÞ ; Jr½2rbi Dð0Þ Yr½2rbi Dð0Þ ð71cÞ i ¼ 1; 2; 3; ... ð66Þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 rDð0Þ Yr1½2rbi Dð0ÞYrþ1½2rbi Dð0ÞYr½2rbi Dð0Þ whereas for Case 3–2 it is: N32 ð3Þ¼ pffiffiffiffiffiffiffiffiffiffi i Y2 ½2rb Dð0Þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! rþ1 i J ½2rb DðxÞ Yr½2rb DðxÞ ð71dÞ w ðxÞ¼DðxÞr=2 r i pffiffiffiffiffiffiffiffiffiffi þ i pffiffiffiffiffiffiffiffiffiffi i J ½2rb Dð0Þ Yrþ1½2rb Dð0Þ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi rþ1 i i 2 ½1 þ rDð0Þ Yr½2rbi Dð1ÞYr1½2rbi Dð1ÞYrþ1½2rbi Dð1Þ i ¼ 1; 2; 3; ... ð67Þ N 4 pffiffiffiffiffiffiffiffiffiffi 32i ð Þ¼ 2 Yrþ1½2rbi Dð0Þ

The eigenvalues, bi, for Cases 1–2 and 3–2 must be calculated, ð71eÞ respectively, from the following transcendental equations: The normalized eigenfunction is computed according to pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Eq. (24). J 2 b D 1 Y 2 b D 1 The transformed initial condition Eq. (30) was determined ana- r1½ r pi ffiffiffiffiffiffiffiffiffiffið Þ r1½ r pi ffiffiffiffiffiffiffiffiffiffið Þ ¼ 0 ð68Þ lytically using the ‘‘Integrate” function in Mathematica. Eqs. (72) Jr½2rbi Dð0Þ Yr½2rbi Dð0Þ and (73) give expressions of the transformed initial conditions pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi for Cases 1–2 and 3–2, respectively: J ½2rb Dð1Þ Yr1½2rb Dð1Þ r1 ipffiffiffiffiffiffiffiffiffiffi ipffiffiffiffiffiffiffiffiffiffi ¼ 0 ð69Þ pffiffiffiffiffiffiffiffiffiffi auð1ÞT þ Dð1ÞT b Jrþ1½2rbi Dð0Þ Yrþ1½2rbi Dð0Þ hihi1 p 2 i f i ¼ ÀÁffiffiffiffiffi ð72aÞ uð0Þ uð0Þ r=2 2 p pffiffi pffiffi pðT1 þ T2ÞJr c bi Yr c bi uð1ÞDð0Þ bi þ c Ni The analytical expression for the norm (see Eq. (23)) is given by Eq. (70) for Case 1–2 and by Eq. (71) for Case 3–2: T1 ¼ Ir½uð0ÞKr1½uð1Þ þ Ir1½uð1ÞKr½uð0Þ ð72bÞ X4 uð1Þ uð0Þ uð0Þ uð0Þ N ¼ N ðmÞð70aÞ T2 ¼ J pffiffiffi b Yr pffiffiffi b J pffiffiffi b Yr1 pffiffiffi b ð72cÞ i 12i r1 c i c i r c i c i m¼1 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! pffiffiffi hihi2 c Jr1½2rbi Dð0ÞJrþ1½2rbi Dð0Þ f i ¼ ffiffiffiffiffi ð73Þ N 1 D 0 pffiffiffiffiffiffiffiffiffiffi 1 70b uð0Þ uð0Þ r=2 2 p 12i ð Þ¼r ð Þ ð Þ pffiffi pffiffi pJrþ1 bi Yrþ1 bi uð0ÞDð0Þ biðb þ cÞ Ni Jr½2rbi Dð0Þ c c i pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 uðxÞ¼2 cDðxÞ: ð74Þ ½1 þ rDð0Þ J ½2rbi Dð1ÞJ ½2rbi Dð1ÞJrþ1½2rbi Dð1Þ r r rp1 ffiffiffiffiffiffiffiffiffiffi N12i ð2Þ¼ 2 Jr½2rbi Dð0Þ Finally, the solute concentration is found according to Eq. (33). ð70cÞ 6. Results for test-case pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! Yr1½2rbi Dð0ÞYprffiffiffiffiffiffiffiffiffiffiþ1½2rbi Dð0Þ N12 ð3Þ¼rDð0Þ 1 ð70dÞ The GITT analytical solution developed in the previous section i Y ½2rb Dð0Þ r i was evaluated for a hypothetical test-case whose parameter values J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65 63

Table 3 1 Data values for test-case. GITT, finite domain 0.1 Yates (1990), semi-infinite domain Parameter Value Domain length, L 100 m 0 0.01 Retardation factor, R 1 4 2 1 Molecular diffusion, 10 m d a = 1 dM 0.001 Average velocity, u 0.25 m d1 1 Decay coefﬁcient, k 0.05 d 0.0001 a = 0.5

a = 0.1 1E-005 Relative Concentration Relative are given in Table 3. The solution was implemented in Mathemat- ica. The eigenvalues were calculated from the transcendental Eqs. 1E-006 (65) and (66) using the Mathematica function ‘‘FindRoot”. Solu- 020406080100 tions were computed using default settings for Mathematica sys- Distance (m) tem parameters. The GITT solution was programmed such that dimensionless concentration values less than 1010 were taken to Fig. 1. Steady state concentration profiles for finite (GITT) and semi-infinite (Yates, be zero. 1990) domains and for various values of dispersion spatial dependence parameter a. Inlet boundary is a first-type condition. For comparison purposes, the formal solution of Yates (1990) for a semi-infinite domain was also implemented in the Mathem- atica platform. The integrals in the Yates (1990) solution were Table 5 evaluated numerically using the Mathematica function Concentration profiles for t = 0.1 and a=0.1.

‘‘NIntegrate”. X (m) Dimensionless concentration Table 4 presents the dimensionless steady state concentration GITT Yates (1990) GITT Yates (1990) c with the linear dependence of the dispersion coefficient speci- 1 Case 1–2 Case 1–2 Case 3–2 Case 3–2 fied by a= 0.1. For both Cases 1–2 and 3–2, the results for the finite 0 1. 1. 9.99911E 11 9.99911E 11 domain (the current GITT solution) and the semi-infinite domain 10 9.41422E2 9.41422E2 9.41263E2 9.41263E2 (the Yates (1990) formal analytical solution) are identical at the en- 20 7.8846E4 7.8846E4 7.8804E4 7.8804E2 trance boundary and for over more than half of the spatial domain. 30 1.05935E6 1.05936E6 1.05834E6 1.05835E6 As expected, however, the solutions diverge further along the do- 40 5.71943E10 5.71614E10 5.71138E10 5.70811E10 main where the effect of the finite boundary condition becomes 50 0 3.28618E13 0 3.23334E13 60 0 2.14595E13 0 1.32517E13 apparent. Fig. 1 shows steady state results for different values of 70 0 3.86535E13 0 3.6215E13 a. Field observations of contaminant dispersion, as well as theoret- 80 0 6.37506E14 0 7.92379E14 ical considerations, indicate that the value of a should be in the 90 0 3.91891E13 0 7.30922E13 range 0 6 a 6 1(Huang et al., 1996). Fig. 1 demonstrates that the 100 0 9.99429E14 0 1.95371E15 effect of the finite domain, and hence the disagreement with the semi-infinite solution, increases when the value of a increases. Table 6 Tables 5–7 show results for the transient concentration at the Concentration profiles for t = 0.1 and a=0.5. dimensionless time t = 0.1 and various values of the parameter a. The parameter values in Table 4 were again used. Results for finite X (m) Dimensionless concentration and semi-infinite domains are presented. The numbers of terms, N, GITT Yates (1990) GITT Yates (1990) required for convergence of the GITT solution depended on the va- Case 1–2 Case 1–2 Case 3–2 Case 3–2 lue of a: N = 50 for a= 0.1 (Table 5); N = 30 for a= 0.5 (Table 6); 0 1. 1. 9.9984E1 9.9984E1 and N = 10 for a= 1(Table 7). Increasing a has the effect of increas- 10 1.15427E1 1.15427E1 1.15402E1 1.15402E1 ing dispersion relative to advection, and consequently the number 20 1.93139E2 1.93139E2 1.93076E2 1.93076E2 30 3.21145E3 3.21145E3 3.20998E3 3.20998E3 of terms required for convergence is lower. As was the case with 40 5.19991E4 5.19991E4 5.19677E4 5.19677E4 the steady state results, the concentration profiles in Tables 5–7 50 8.22096E5 8.22096E5 8.21477E5 8.21477E4 for the finite and semi-infinite domains differ only in the vicinity 60 1.2749E5 1.2749E5 1.27375E5 1.27375E5 of the exit boundary. 70 1.94688E6 1.94685E6 1.9448E6 1.9448E6 80 2.93698E7 2.93614E7 2.9332E7 2.93258E7 90 4.45395E8 4.38308E8 4.44581E8 4.3771E8 100 1.25334E8 6.48763E9 1.24949E8 6.47779E9 Table 4 Steady state concentration profiles for a = 0.1.

X (m) Dimensionless concentration Fig. 2 shows the solute concentration proﬁles for Case 1–2, cal- Eq. (51) Yates (1990) Eq. (52) Yates (1990) culated with different values of the parameter a and at various Case 1–2 Case 1–2 Case 3–2 Case 3–2 dimensionless times t. Results for Case 3–2 are also included in

0 1. 1. 9.99911E1 9.99911E1 Fig. 2, but at the scale of the figure, they superimpose on the results 10 1.35955E1 1.35955E1 1.35943E1 1.35943E1 for the first case and cannot be distinguished. For realistic levels of 20 2.46503E2 2.46503E2 2.46481E2 2.46481E2 dispersion (0 6 a 6 1), the different inlet conditions led to only 30 5.32019E3 5.32019E3 5.31971E3 5.31971E3 minor differences in the computed solution. Also shown in Fig. 2 40 1.29868E3 1.29868E3 1.29856E3 1.29856E3 are results computed for constant dispersion, D = a + d . In com- 50 3.48248E4 3.48248E4 3.48217E4 3.48217E4 M 60 1.00682E4 1.00682E4 1.00673E4 1.00673E4 paring the plots of Fig. 2, it can be seen that increasing the value 70 3.09787E5 3.09779E5 3.09759E5 3.09751E5 of a increases dispersion and creates a more diffuse concentration 80 1.00565E5 1.00476E5 1.00556E5 1.00467E6 front that penetrates more quickly into the finite domain. Also, the 90 3.49118E6 3.4108E6 3.49086E6 3.4105E6 divergence from the constant dispersivity case is greater for larger 100 1.82775E6 1.20497E6 1.82759E6 1.20487E6 values of a. 64 J.S. Pérez Guerrero, T.H. Skaggs / Journal of Hydrology 390 (2010) 57–65

Table 7 7. Summary and conclusions Concentration proﬁles for t = 0.1 and a=1.

X (m) Dimensionless concentration A general analytical solution for solute transport in ﬁnite, het- GITT Yates (1990) GITT Yates (1990) erogeneous porous media was developed. The linear advection– Case1–2 Case 1–2 Case 3–2 Caes 3–2 dispersion equation with distance-dependent coefﬁcients was

0 1. 1. 9.99336E1 9.99336E1 solved using an integrating factor in combination with the gen- 10 1.22758E1 1.22757E1 1.2266E1 1.2266E1 eralized integral transform technique (GITT). For a number of 20 3.41618E2 3.41612E2 3.41274E2 3.41274E2 parameterizations of practical importance to groundwater con- 30 1.08658E2 1.08656E2 1.08525E2 1.08525E2 taminant transport, we demonstrated that an analytical expres- 40 3.6527E3 3.65272E3 3.64748E3 3.64748E3 sion for the integrating factor exists. For the particular case of 50 1.26362E3 1.26374E3 1.26164E3 1.26164E3 60 4.4454E4 4.44671E4 4.43865E4 4.4383E4 a linearly increasing dispersivity, solutions were developed in 70 1.58159E4 1.58164E4 1.57997E4 1.57829E4 detail and compared with solutions from the literature obtained 80 5.71682E5 5.66704E5 5.7204E5 5.65375E5 for semi-inﬁnite media. As expected, the results differed only in 90 2.25485E5 2.04106E5 2.26559E5 2.03582E5 the vicinity of the exit boundary. Among other applications, the 100 1.42596E5 7.3792E6 1.43845E5 7.35863E6 new solution will be particularly useful analyzing problems were the exit boundary is signiﬁcant, such as transport in soil columns or lysimeters, or testing and validating numerical transport codes.

1 Acknowledgments

t = 1 0.8 The ﬁrst author acknowledges support by the Brazilian National t = 0.05 Research Council, CNPq; the Department of Environmental Sci- t = 0.025 ences at the University of California, Riverside; and the United 0.6 States Salinity Laboratory, USDA-ARS.

0.4 References

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