symmetry solutions for transient solute transport in unsaturated soils with realistic water profile

DOI 10.1007/s11242-004-6799-8Transport in Porous Media (2005) 61:109–125 © Springer 2005

Symmetry Solutions for Transient SoluteTransport in Unsaturated Soils with RealisticWater Profile

R. J. MOITSHEKI1, P. BROADBRIDGE2,∗ and M. P. EDWARDS1

1Institute for Mathematical Modelling and Computational Systems, University ofWollongong, Northfields Ave., Wollongong 2522, New South Wales, Australia2Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall,Newark DE 19716-2553, U.S.A.

(Received: 29 July 2003; accepted in final form: 11 November 2004)

Abstract. We examine solutions for solute transport using the convection-dispersion equa-tion (CDE) during steady evaporation from a water table. It is common, when solving theCDE, to first approximate the volumetric water content of the soil as a constant. Here,we assume a reasonable function for the water content profile and construct realistic non-linear hydraulic transport properties. Both classical and nonclassical symmetry techniquesare employed. Invariant solutions are obtained for the one dimensional CDE even witha nontrivial background profile for volumetric water content.

Key words: solute transport, unsaturated soil, analytical solutions, Lie point symmetries.

1. Introduction

Prediction of coupled water and solute movement through porous media isimportant to the theory of soil salinisation and movement of agriculturaland industrial contaminants.

A search for exact analytic solutions for water flow and solute transportequations continues to be of scientific interest. For example, exact analyticsolutions for solute transport in unsaturated soil have been considered byNachabe et al. (1994) in which the Broadbridge and White nonlinear modelwas used to solve the Richards’ equation for vertical water flow under a con-stant infiltration rate. The method of characteristics was then used to deter-mine the location of the solute front. Other notable contributors include vanGenunchten and Alves (1981) who constructed a compendium of solutionsfor the one dimensional constant-coefficient approximations to the convec-tive-dispersion equation. Perhaps the most notable challenge in the search for

∗Author for correspondence: Tel.: +1-302-831-2652; Fax: +1-302-831-4511; email:[email protected]

110 R. J. MOITSHEKI ET AL.

exact analytic solutions is that both water flow and solute transport in unsat-urated soils contribute to the transient phenomena (Nachabe et al., 1994).However if obtained, exact analytic solutions could be used as tools to verifythe accuracy of numerical schemes.

A steady-state flow problem of interest is the upward movement ofsaline groundwater from a water table, and subsequent evaporation at thesoil surface. Although the evaporation rate may be small, a significantquantity of salts may accumulate over a large time scale. Gardner (1958)derived soil hydraulic models which make it possible to construct exactsolutions of some steady-state unsaturated flow problems and approximatesolutions of some transient problems. Gardner (1958) gave solutions ofthe steady-state problem in one dimension for several different relationsbetween capillary conductivity and soil suction potential. However, thesesolutions for volumetric water content would then appear in the solutetransport equation as complicated nonconstant coefficients. It would thenbe unlikely to find exact solutions, in simple analytic form, for the cou-pled solute transport equation. Therefore, we adopt an alternative inverseapproach to this problem. We find suitable functions θ(z) that are physi-cally reasonable water content profiles and which lead to extra symmetry inthe coupled solute dispersion equation. We then construct the realistic non-linear soil hydraulic conductivity K(θ) and diffusivity D(θ) functions thatpermit the solution θ(z), and we are able to construct exact transient solu-tions to the convection–dispersion equation for solute transport. This haspreviously been achieved only for the very simplest of background waterflows.

In practical problems, it is normal to determine approximate analyti-cal solutions for relevant partial differential equations (e.g. Warrick et al.,1971; Smiles et al., 1978; De Smedt and Wierenga, 1978; Elrick et al.,1987). Elrick et al. (1987) claimed that approximate analytical solutionsbypass the more complicated exact analytic solutions and numerical tech-niques. De Smedt and Wierenga (1978) presented approximate analyti-cal solutions for solute flow during infiltration and redistribution. Theyassumed that the volumetric water content was dependent on time andindependent of the depth down to the transition zone of the solute penetra-tion depth. After this simplification, knowledge of the relationship betweenhydraulic conductivity and volumetric water content and between volumet-ric water content and soil water interaction potential was not required.However, in unsaturated water flow the essential properties are the hydrau-lic conductivity as a function of volumetric water content or soil waterinteraction potential, and the relationship between volumetric water con-tent and soil water interaction potential.

In this paper, we construct symmetry solutions for solute transport inunsaturated soils with a background water content that is appropriate for

SYMMETRY SOLUTIONS FOR TRANSIENT SOLUTE TRANSPORT 111

steady evaporation from a water table. In Section 3 we choose a reasonablefunction θ(z) to describe the volumetric water content profile and then con-struct the realistic diffusivity and hydraulic conductivity functions. Here, weneglect the hysteresis effects. Classical Lie symmetry analysis is employed inSection 4. In Section 5 we construct exact analytic solutions by nonclassi-cal symmetry techniques. However, when realistic boundary conditions areassumed, approximate solution methods must be used to approximate thetransient behaviour of the solute before it approaches the exact steady statesolution.

2. Governing Equations

In order to solve the equation governing solute transport in soils, oneneeds to first find the volumetric water content of the soil as a functionof space and time. This is achieved by solving the nonlinear Fokker-Planckdiffusion–convection equation describing the water flow in porous media.In one dimension the water flow equation, obtained by combining Darcy’slaw for unsaturated flow

q =−D(θ)∂θ

∂z+K(θ)

together with the equation of continuity for mass conservation

∂θ

∂t=−∂q

∂z,

is given by

∂θ

∂t= ∂

∂z

(D(θ)

∂θ

∂z

)− dK

dθ

∂θ

∂z, (1)

which is a version of the familiar Richards’ equation. Here, the nonlinearsoil water diffusivity is D(θ) = K(θ)(d�/dθ). The hydraulic conductivityK(θ) (Richards, 1931; Philip, 1969) as well as diffusivity D(θ) (Buckingham,1907) are typically strongly increasing and concave functions of volumetricwater content. Dependence of D and K on θ renders Equation (1) highlynonlinear. Here t is time, z is vertical depth measured positively downward,θ is the volumetric water content and � which in unsaturated soil is lessthan zero, is the soil water interaction potential. Analytical models havebeen developed to predict water movement in the unsaturated zone (e.g.Broadbridge and White, 1988; Sander et al., 1988; Philip, 1992). The pos-sibility of exact solutions of Equation (1) requires special choices for themodels K(θ) and D(θ). An extensive symmetry classification of these func-tions is given in Oron and Rosenau (1986), Sposito (1990), Edwards (1994)and Yung et al. (1994).


Movement of non-volatile, inert solutes in unsaturated soils is describedby the convection–dispersion equation

∂(θc)

∂t= ∂

∂z

(θDv(v)

∂c

∂z

)− ∂(qc)

∂z, (2)

where c is the concentration of the solute in solution, q is the Darcianwater flux, θ , t and z are as in Equation (1). Dv(v) is the dispersioncoefficient depending on pore water velocity v = |q|/θ and it includesboth the molecular diffusion and the mechanical dispersion (Bear, 1972).Dv(v) has frequently been observed to be linearly proportional to the porewater velocity (Biggar and Nielsen, 1967; Bear, 1972; Anderson, 1979). Weassume Dv(v)=δv, where δ is the dispersivity. Dispersivity has a significantimpact on the migration process and is a natural candidate for the spa-tial characteristic length. It ranges from about 0.5 cm for laboratory-scaledisplacement experiments involving disturbed soils to about 10 cm or morefor field scale experiments (Nielsen et al., 1986). For steady-state water flowand using equation of continuity, Equation (2) reduces to

θ(z)∂c

∂t= ∂

∂z

(δ|q|∂c

∂z

)−q

∂c

∂z. (3)

With application to evaporation from a water table at a constant Darcianwater flux q =−R, Equation (3) then becomes

θ(z)∂c

∂t= δR

∂2c

∂t2+R

∂c

∂z. (4)

3. Steady Water Flow Models and Hydraulic Properties

Before considering the soil hydraulic properties, we first choose an appro-priate function for the volumetric water content profile. For simplicity weconsider a steady one-dimensional profile that may represent steady evapo-ration from a water table. We choose

θ(z)= θs +β(1− e(d−z)/ l), (5)

with β and l constants, d being the depth to the water table and θs beingthe volumetric water content at saturation. In terms of scaled dimension-less variables,

�=1+B(1− e(D−Z)/L), (6)

where B =β/θs,L= l/λs,D =d/λs and Z = z/λs . Here λs is a macroscopicsorptive length scale, see e.g. White and Sully (1987). In Figure 1, we plotthe water content profile (6) for selected values of B,D and L.


Figure 1. Graphical representation of change in volumetric water content withdepth. Parameters used: D =2,B =0.55 and L=2.

We now use dimensionless variables; �∗ = �/λs and K∗ = K/Ks , Ks

being the hydraulic conductivity at saturation and D∗ =K∗(d�∗/d�). Thesteady-state version of Equation (1) may be written

0= ∂

∂Z

(K∗(�)

∂�∗∂Z

)− dK∗

d�

∂�

∂Z

or equivalently,

0= ∂

∂Z

(K∗(�∗)

∂�∗∂Z

)− dK∗

d�∗

∂�∗∂Z

,

which integrates to

K∗

(d�∗dZ

−1)

=R∗, (7)

where R∗ = R/Ks , R being the constant evaporation rate so that −R isthe uniform volumetric water flux. Gardner (1958) solved Equation (7)for various hydraulic models including the power law and the exponentialK =Kse

α� . Here, α is the sorptive number, regarded as the reciprocal ofthe sorptive length scale. In the current analysis, we aim to solve the cou-pled solute transport equation. Therefore, we devise an inverse method inwhich the function K∗(�) is deduced after specifying a relatively simplewater content profile. The specification of a simple water content profileallows us to construct some special solutions of the coupled convection–dispersion equation for solute transport.


Since K∗ =1 at �=1, It follows from (7) that

d�∗d�

∣∣∣∣�=1

= L(1+R∗)B

(8)

Assuming K ′∗(�)� 0 andK ′′

∗ (�) � 0, we have the necessary relationships

h(�)�g(�) (9)

and

h′ +h2 − g′ +gh

R∗�0, (10)

where

h(�)= � ′′∗ (�)

� ′∗(�)

and

g(�)= L

1+B −�.

We are now free to choose any function h(�). If h(�)=−2/� then

�∗(�)= L(1+R∗)B

(�−1

�

). (11)

Clearly water content is a single valued function of soil water interac-tion potential and here, we neglect the effects of hysteresis. Thus from (7)we obtain

K∗(�)= BR∗�2

(1+R∗)(1+B −�)−B�2(12)

and so

D∗(�)= LR∗(1+R∗)(1+R∗)(1+B −�)−B�2

. (13)

The graphs of K∗(�) and D∗(�) are depicted in Figures 2 and 3, respec-tively.

On the other hand, if we consider �∗ as the independent variable then

K∗(�∗)= R∗BL2(1+R∗)(1+B)[L(1+R∗)−B�∗]2 −F , (14)

where

F = (L(1+R∗))2 −BL(1+R∗)�∗ +BL2(1+R∗).

Graphical representation of K∗(�∗) is depicted in Figure 4.If the water profile is given as in Equation (6), then the soil must have

hydraulic properties given in (12)–(14).


Figure 2. Graphical representation of K∗(�) in (12). Parameters used: R∗ = 1 andB =0.55.

4. Classical Symmetry Analysis

A symmetry of a differential equation is a transformation mapping anarbitrary solution to another solution of the differential equation. Theclassical Lie groups of point invariance transformations depend on con-tinuous parameters and act on a system’s graph space whose coordinatesare the independent and dependent variables. If a partial differential equa-tion (PDE) is invariant under a point symmetry, one can often determinesimilarity or invariant solutions which are invariant under some subgroupof the full group admitted by the PDE. These solutions result from solvinga reduced equation in fewer variables.

We consider Equation (4) with θ given in (5). In terms of scaled dimen-sionless variables we have

[1+B

(1− e(D−Z)/L

)]∂C

∂T= ∂2C

∂Z2+ ∂C

∂Z, (15)

where C = c/cs and T = t/ts with cs being the suitable concentration andts = θsδ/R.

In the initial symmetry analysis of Equation (4), in which θ(z) is an arbi-trary function, the symmetry finding package DIMSYM (Sherring, 1993)under the algebraic manipulation software REDUCE (Hearn, 1991) finds


Figure 3. Graphical representation of D∗(�) in (13). Paremters used: B = 0.55 andR∗ =1.

only trivial symmetries admitted by (4), namely, translations in t , scaling ofc and the infinite dimensional symmetry for linear superposition. These typesof simple symmetries that are shared by all forms of the governing equation,are often referred to as the principal Lie algebra for the equation in ques-tion (Ibragimov, 1994). Furthermore, DIMSYM (Sherring, 1993) reports thatextra finite symmetries may be obtained if seventy seven coefficient functionsand derivatives (not listed here) are linearly dependent. Full classification ofthese cases would be a major task and we have not completed it. We note thatextra symmetries could be obtained for the cases θ =θsez−d, θ = constant andθ =θs((z+γ )/(d +γ ))−1. No extra symmetries were obtained in the classicalsymmetry analysis of Equation (15). However a reduction of (15) is possi-ble using a linear combination of the translation in T and the scaling inC. Recent accounts on the theory and applications of continuous symmetrygroups may be found in many excellent texts such as those of Olver (1986),Bluman and Kumei (1989) and Hill (1992). Previous applications to hydrol-ogy may be found in Oron and Rosenau (1986), Sposito (1990), Edwards


Figure 4. Graphical representation of K∗(�∗). Parameters used: B =0.55,R∗ =1 andL=2.

(1994), Yung et al. (1994), Edwards and Broadbridge (1995), Baikov et al.(1997), Broadbridge et al. (2000) and Broadbridge et al. (2002).

4.1. classical symmetry reductions

Some time dependent solutions for Equation (15) could be constructed byassuming

Cs −C = e−λT F (Z), (16)

where Cs is the steady state concentration. In fact (16) is the functionalform obtained from invariance under linear combination of translation inT and scaling in C admitted by Equations (15) and (16) could be obtainedby separation of variables. In a steady state, the solute transport Equation(15) reduces to an ODE

C ′′s (Z)+C ′

s(Z)=0. (17)

The steady solution satisfying the boundary conditions

C =1, Z =D; (18)

C + ∂C

∂Z=0, Z =0. (19)

is given by


Cs = eD−Z. (20)

Therefore if solute concentration approaches steady state during a pro-longed period of evaporation, the long-term concentration at the surfacewill be C0 =eD. For example, for a sandy loam sorptive length 0.15 m withthe water table at 1.5 m depth, the salt concentration will be amplified bya factor e10, around 22,000. In practice, in arid regions with an artesianbasin, the concentration of salt will often reach saturation level well belowthe soil surface and salt will be precipitated. However, we will see later thata typical time scale for the approach to steady state will be tens of years,as observed in arid zones that are artificially irrigated.

We now solve for the transient C1 =Cs −C with λ>0. We observe thatF satisfies,

F ′′(Z)+F ′(Z)+(λ(1+B)−λ Be(D−Z)/L

)F(Z)=0. (21)

Under the transformation ζ =2L√

λ BeD/L e−Z/2L Equation (21) becomes

ζ 2 d2F

dζ 2+ (1−2L)ζ

dF

dζ+(

4λL2(1+B)− ζ 2)F =0. (22)

Hence,

C1 =Cs −C = e−λT ζL{a1Iv(ζ )+a2Kv(ζ )}, (23)

where Iv and Kv are modified Bessel functions of the first and the thirdkind (Abramowitz and Stegun, 1972; Watson, 1958) of the order

ν =2L√

1−4λ (1+B).

and a1 and a2 are constants. This exact solution, however, does not sat-isfy the familiar meaningful boundary conditions such as those discussedin Section 5.1. In the next section, we will find if Equation (4) has non-classical symmetry reductions that yield other solutions.

5. Nonclassical Symmetry Analysis

We now investigate the possibility of nonclassical symmetries admitted byEquation (4) with arbitrary θ(z). By the nonclassical method, it is possibleto find further types of explicit solutions by the same reduction techniquethat is commonly used in the classical method. Moreover, there exist PDEswhich possess symmetry reductions not obtainable via classical Lie groupmethods (Clarkson and Kruskal, 1989; Arrigo et al., 1994; Clarkson andMansfield, 1994; Goard and Broadbridge, 1996). We consider the generalequation given in scaled dimensionless variables, namely;


�(Z)∂C

∂T= ∂2C

∂Z2+ ∂C

∂Z. (24)

The nonclassical method, introduced by Bluman and Cole (1969), seeks theinvariance of the system of PDEs composed of the given Equation (24)with its invariant surface condition (ISC)

τ(T ,Z,C)∂C

∂T+ ξ(T ,Z,C)

∂C

∂Z=η(T ,Z,C), (25)

where the coefficients τ , ξ and η are the infinitesimals of the transforma-tions

T∗ =T + ετ(T ,Z,C)+O(ε2), (26)

Z∗ =Z = εξ(T ,Z,C)+O(ε2), (27)

C∗ =C + εη(T ,Z,C)+O(ε2). (28)

ε is a real parameter. If we demand the invariance of Equation (24) subjectto the constraints of the ISC with τ =1, we obtain

ξ = ξ(T ,Z), (29)

η=A(T ,Z) C +B(T ,Z) (30)

where

2AZ +�(Z)ξT − ξZZ +2�(Z)ξZξ + ξZ +�′(Z)ξ 2 =0, (31)

�(Z)AT −AZZ −AZ +2�(Z)ξZA+�′(Z)ξA =0, (32)

�(Z)BT −BZZ −BZ +2�(Z)ξZB +�′(Z)ξB =0. (33)

The prime denotes differentiation with respect to the indicated argument.Assuming ξ(T ,Z) to be a nonzero constant we thus obtain nonclassicalsymmetries admitted by Equation (24) for special cases of �(Z) and theseare summarized in Table I below. ξ(T ,Z) = 0 leads to the recovery ofthe classical symmetries or the principal Lie algebra admitted by Equation(24). A complete solution to this system is yet to be achieved.

5.1. nonclassical symmetry reductions

A symmetry reduction enables a simplification in the governing equation,with the aim of finding an exact analytic solution. In the case of theboundary or initial value problem, the admitted infinitesimal generatormust also leave all the conditions invariant. We consider as an exampleequation,(

( 12 +A1)+ (A1 − 1

2)e2Al(Z+c1)

1− e2A1(Z+c1)

)∂C

∂T= ∂2C

∂Z2+ ∂C

∂Z. (34)


Table I. Some nonclassical symmetries for Equation (24)

θ(z) τ ξ η

arbitrary 1 ξ(T ,Z) A(T,Z)C+B(T,Z)(A1+( 1

2 )+(A1− 12 )e2Al(Z+c1)

1−e2Al(Z+c1)

)1 2 −2

((A1+ 1

2 )+(A1− 12 )e2Al(Z+c1)

1−e2Al(Z+c1)

)C

A1 tan(A1(Z + c2))+ 12 1 2 −2

(A1 tan(A1(Z + c2))+ 1

2

)C

12 − 1

Z+c31 2 −2( 1

2 − 1Z+c3

)C

Equation (34) admits a nonclassical symmetry generator

� = ∂

∂T+2

∂

∂Z−2

(( 1

2 +A1)+ (A1 − 12)e2A1(Z+c1)

1− e2A1(Z+c1)

)C

∂

∂C, (35)

which leads to a functional form

C =[

exp((

A1 − 12

)(Z+c1)

)−exp

(−(

12

+A1

)(Z+C1)

)]×F(Z−2T ),

(36)

where F satisfies the ODE

F ′′ +F ′ +(

A12 − 1

4

)F =0. (37)

Thus we obtain solutions

case (a) A12 = 1

2

C =[

exp((

A1 − 12

)(Z + c1)

)− exp

(−(

12

+A1

)(Z +C1)

)]×

× exp(T − z

2

)× (k1(Z −2T )+k2) (38)

case (b) A12 < 1

2

C =[

exp((

A1 − 12

)(Z + c1)

)− exp

(−(

12

+A1

)(Z +C1)

)]×

×k1 exp

((√2−4A1

2 −12

)(Z −2T )

)+

+k2 exp

((−√

2−4A12 −1

2

)(Z −2T )

)(39)


case (c) A12 < 1

2

C =[

exp((

A1 − 12

)(Z +C1)

)− exp

(−(

12

+A1

)(Z +C1)

)]×

× exp(

T − Z

2

)×k1 sin

(√4A1

2 −22

(Z −2T )

)+

+k2 cos

(√4A1

2 −22

(Z −2T )

)(40)

We have provided nonclassical symmetry reductions and hence exact ana-lytic solutions which are not obtainable via the classical symmetry analysis.However, the obtained solutions do not satisfy the familiar boundary con-ditions with D = 2. In Figure 5, we compare approximate method of linessolution (Melgaard and Sincovec, 1981) and exact symmetry solution (39)subject to the conditions

C =−0.0042k1e(0.414−0.414T ) −0.0042k2e(−2.414+2.414T ), Z =2

and

C + ∂C

∂Z=−0.0075k1e−0.414T +0.0042k2e2.414T , Z =O.

In this comparison, we assume parameter values k1 =−153.46, k2 =−156.012,A1 =−0.01, c1 =5. Although here we have used a small value for time, similarresults still hold at large T .

Now we consider more meaningful boundary conditions (18) and (19),and initial condition

C =0, T =0, 0≤Z ≤D. (41)

Condition (18) corresponds to the store of solute at fixed concentration atthe underlying ground water reservoir. Condition (19) corresponds to theassumption that the solute is not carried through the soil surface, with theevaporating water, rather it accumulates here.

Figure 6 depicts the numerical solution for Equation (15) subject to theinitial and boundary conditions (41), (18) and (19), approaching the exactsteady state solution.

6. Results and Discussion

The Fokker-Planck diffusion–convection equation describing a steady waterflow and the convection–dispersion equation (CDE) are simultaneouslysolved. In solving the diffusion–convection equation, we first assumed a


Figure 5. Numerical and exact anlaytic solution at T =0.01.

Figure 6. Numerical solution for equation (15) subject to conditions (41) to (19).B = 0.55, L = 2 and 1) = 2. C is the steady state solution given in (20) and thevolumetric water content in (6).


realistic function of z to depict the volumetric water content and con-structed hydraulic properties which resemble those of real soils. We notethat �∗(�) in (11) does not show the inflection point. However we are ableto obtain exact analytic solutions for the CDE using the smooth function�(Z) in (6). The problem has proved to be more difficult than antici-pated with Laplace transform resulting in an intractable analytic inversionand the method of self adjoint linear operators being inapplicable. Theonly two classical symmetries admitted by CDE with a nontrivial volumet-ric water content background led to an invariant solution which could beobtained by method of separation of variables. It is notable that Equation(4) may be transformed into a diffusion equation. However, this does notsimplify the boundary value problem.

We have let the infinitesimal ξ to be a nonzero constant in order tosolve the system of nonlinear determining equations arising from nonclas-sical symmetry analysis of the CDE (24) with arbitrary volumetric watercontent. Some nonclassical symmetries are obtained for special cases ofvolumetric water content. A full nonclassical symmetry classification forthis equation would be a major task if not impossible. Choosing the real-istic functions from the obtained special cases we could again constructthe realistic hydraulic properties. We obtained some nonclassical reductionswhich could not be obtained via the classical techniques. Figure 6 showssolute concentration build-up at the surface of the soil as time evolves.This concentration approaches the steady state concentration which couldbe obtained exactly in (20). Also from Figure 6, it is seen that the timetaken to approach steady state may be of the order of 10θsδ/R. For exam-ple, for a mean evaporation rate of 0.3 cm per day, over a water tableof depth 4 m covered by soil with porosity 0.4, this time is around 15years.

7. Conclusion

Solutions for the coupled solute transport equations are presented. Weemployed the inverse method to solve the nonlinear Richards equation.These solutions then appeared in the solute dispersion equations as nonconstant coefficients. Symmetry analysis of solute dispersion equation ledto invariant solutions which are useful to test the accuracy of the numeri-cal software when meaningful initial and boundary condition are imposed.We can bridge the exact steady state solution with validated approximatetransient solutions.

From our analytic and numerical solutions, under the prolonged peri-ods of evaporation in arid regions over an artesian basin, the salt contentis likely to reach saturation level, with precipitation occurring beneath thesurface. This could occur even if the underlying ground water had a low


salt concentration. The precipitation front would be an interesting subjectfor future mathematical modelling.

Acknowledgements

Raseelo J. Moitsheki is grateful to the University of Wollongong, Australia,to the University of Delaware, USA and to the National Research Foun-dation of South Africa for the generous financial support.

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