Classification of the Group Invariant Solutions for Contaminant Transport in Saturated Soils under Radial Uniform Water Flows

The transport of chemicals through soils to the groundwater or precipitation at the soils surfaces leads to degradation of these resources. Serious consequences may be suffered in the long run. In this paper, we consider macroscopic deterministic models describing contaminant transport in saturated soils under uniform radial water flow backgrounds. The arising convectiondispersion equation given in terms of the stream functions is analyzed using classical Lie point symmetries. A number of exotic Lie point symmetries are admitted. Group invariant solutions are classified according to the elements of the one-dimensional optimal systems. We analyzed the group invariant solutions which satisfy the physical boundary conditions.


Introduction
Prediction of water and contaminant movements in soils is important to the theory of soil salinity and underground water pollution.Among others, the sources of contamination are movement of agricultural and industrial contaminants.Some exact solutions are constructed for constraint assumptions; as such, investigation and study of these problems are extremely difficult and challenging [1] even when the problem is given in terms of linear partial differential equations [2].
Broadbridge et al. [2][3][4] applied the classical Lie point symmetries to analyze the convection-dispersion equations arising in solute transport theory.A compendium of exact (group invariant) solutions is given in [5].Other researchers used different techniques to obtain exact solutions (see, e.g., [6][7][8]).Chen et al. [9] constructed analytical solutions for two-dimensional advection-dispersion equation with transverse dispersivities depending linearly on the spatial variable.Yadav et al. [10] constructed analytical solutions for solute transport in semiinfinite porous domain.Symmetry reductions and group invariant solutions were constructed for models describing motion of a polytropic gas [11].
Exact analytical solutions for contaminant transport in porous media have been constructed, for example, in [12][13][14][15].Chrysikopoulos and Sim [12] developed the onedimensional stochastic model describing virus transport in homogeneous, saturated semifinite porous media.The ideas in [12] are extended to three-dimensional problems in [13,14].Experimental investigations of acoustically enhanced colloid transport in water-saturated packed columns are carried out in [15].In this paper we focus on theoretical macroscopic deterministic models given in terms of partial differential equations.
Numerical models are able to simulate complex reactive transport phenomena but can be time consuming to construct and subject to numerical discretization errors [7].Also, available packages have significant disagreement in their prediction of solute transport [16].Therefore, exact solutions are very important because they are needed both as validation tests for numerical schemes and also provide insight into the water and solute transport processes.
In this paper, we construct group invariant solutions and classify them according to the one-dimensional optimal systems.This is a significant advancement of the work by [4,6].

Advances in Mathematical Physics
We consider contaminant transport under radial uniform water flows.The resulting convection-dispersion equation given in terms of stream function is analyzed when dispersion coefficient is proportional to the water pore velocity.It turns out that the Lie point symmetries are admitted when the dispersion coefficient is a constant in one case and depending on water pore velocity in the other.In Section 2, we provide the mathematical models.We briefly discuss the symmetry methods in Section 3. The problem at hand is analyzed in Section 4 and the group invariant solutions are classified according to the one-dimensional optimal systems in Section 5 and some physically realistic solutions are discussed.Lastly, some concluding remarks are given in Section 6.

Mathematical Model
The theoretical background in this section is obtained from Hillel [17].The macroscopic deterministic model, which is based on local conservation laws, is given by where (V) denotes the coefficient of hydrodynamic dispersion and ∇ is given in Cartesian coordinates.Experimental and theoretical observations show that the dispersion coefficient can take the power law form (V) = V  with  being a proportionality constant and 1 ≤  ≤ 2 (see, e.g., [18]).
Here, V is the pore water velocity.Given uniform water flow in saturated soils, then the continuity equation is given by ∇ ⋅ V = 0. Uniform saturated water flow implies that the soil water content  =   , where   is the water content at saturation.By Darcy's law, which states that water flux is proportional to the gradient of the hydraulic pressure head, that is, V = −  ∇Φ, where Φ is the total hydraulic pressure head and   is the hydraulic conductivity at saturation (see, e.g., [17]), as such we obtain Laplace's equation ∇ 2 Φ = 0.Under these assumptions, (1) then becomes where  =   /  and V = |∇Φ|.Note that this problem becomes extremely difficult to solve exactly when V must be the modulus of the potential flow velocity field for an incompressible fluid (see, e.g., [2]).However, one may transform (1) from Cartesian coordinates (, ) to the streamline coordinates (, ) using Laplace preserving or conformal transformations [19]; see also [2].The resulting solute transport equation is given by where ∇ = (/, /).The velocity potential is  and  is a conjugate harmonic stream function such that   =   and   = −  .Also, the functions  and  satisfy the Laplace equation.In radial water flows, the velocity potential is given by  = −(/  ) log  and the stream function  = −(/  ) arctan(/), where  is the source strength (pumping rate).Introducing the normalized concentration, velocity potential, and time given by  = /  ,  = − log , and  = /  , respectively, with   ,   , and  being the concentration, time at soil saturation, and the distance or radius from the point source, we may write (3) as We refer to (4) as the governing equation.Equation ( 4) is susceptible to symmetry analysis (see, e.g., [3,4]).

Algebraic Techniques for Symmetry Reduction
In brief, a symmetry of a differential equation is an invertible transformation of the dependent and independent variables that does not change the original differential equation.Symmetries depend continuously on a parameter and form a group: the one-parameter group of transformations.This group can be determined algorithmically by hand or by computer software programs such as YaLie [20], Reduce [21], and Dimsym [22].The theory of application of Lie groups to differential equations may be found in texts such as those of [23][24][25].Given a second order partial differential equation such as (4) describing contaminant transport under radial water flows, we seek transformations of the form Note that the transformations in (5) are equivalent to the one-parameter Lie group of transformations that leaves the equation in question invariant.Since (4) is second order, then one may prolong the symmetry generator (6) accordingly.The invariance criterion is then given by  [2] (Equation ( 4))     Equation (4) = 0, where  [2] is the second prolongation given by Here   ,   , and   are the extended or prolonged infinitesimals (see, e.g., [24]).The invariance criterion results in the overdetermined system of linear homogeneous partial differential equations known as the determining equations, which may be solved even by interactive programs such as YaLie [20] and Reduce [21].

Lie Point Symmetry Analysis of (4)
We consider contaminant transport under steady radial water flow in saturated soils.In this case, the relevant normalized point source, the water velocity, and the dispersion coefficient are given by  = −log , V =   , and (V) =   , respectively (see, e.g.[3]).One may simply omit the dependence of contaminant concentration on the clockwise polar angle coordinate.In the initial symmetry analysis, (4) with arbitrary  and  admits the time translation and the scaling of , that is,  1 = / and  2 = (/), and the infinite symmetry generator where  is any solution of (4).Extra symmetries are admitted only when  = 0 and  = 2.

Constant Dispersion Coefficient.
Given  = 0, then the dispersion coefficient becomes a proportionality constant, .
Given dispersion coefficient as an arbitrary constant  ̸ = 0, (4) admits finite two extra symmetries Further, symmetries may be admitted when  = 1 and  = −1.Note that  = −1 implies negative dispersion coefficient which is not physically realistic, and as such we focus on the case  = 1.Given  = 0 and  = 1, then (4) admits finite four extra symmetries given by

Velocity-Dependent Dispersion Coefficient.
The case  = 2 is in agreement with solute transport theory.This implies that the dispersion coefficient is now given in terms of the water pore velocity.In this case, (4) with arbitrary proportionality constant  admits extra four finite Lie point symmetries given by Note that the admitted symmetry structure and number are not affected by the constant ; that is, we obtain the same symmetries up to the specified  value.

Classification of Group Invariant Solutions
In general it is possible to reduce the number of independent variables by one using any linear combination of the admitted base vectors such as those in (11) and (12).In other words, for each l-parameter subgroup (or equivalently l-dimensional subalgebra) of the full symmetry group (or symmetry algebra) there is a corresponding family of groupinvariant solutions, which may be infinite [23].Thus one needs a systematic means to classify these solutions such that none can be derived from the other.In order to classify these solutions one needs to construct a set of optimal systems.Definition 1 (see [23]).An optimal system of -parameter group-invariant solutions to a differential equation (or system of differential equations) is a collection of solutions with the following properties.
(i) Each solution in the list is invariant under some parameter symmetry group of the differential equation (or system of differential equations).
(ii) If there exists another solution which is invariant under -parameter symmetry group, then there is a further symmetry generator admitted by the equation (or system) which maps this old solution to the new one.
Clearly, an optimal system is a set of elements which lead to symmetry reductions that are not equivalent by any transformation.Since we aim to classify the group invariant solutions of (4), we construct the optimal system for the Lie point symmetries in (11) and (12).

Construction of the Optimal System of Subalgebras.
In this section we adopt the method in [23] to construct the onedimensional system of subalgebras of the algebra spanned by the base vectors in array of (11).To construct the optimal system we first need to determine the commutators of the admitted symmetries.
Definition 2 (see [26]).Given the generators admitted by a kth-order partial differential equation  () =  (x, ,  (1) , . . .,  (−1) ) , then the commutator of  1 and  2 is defined by One may list the commutators of symmetries in (11) together with the time translation and concentration scaling in Table 1.
Furthermore we construct a set of one-dimensional subalgebras which are equivalent to a unique element of the set under some element of the adjoint representation given by where the commutator of   and   is defined above.The adjoint representation of base vectors in ( 11) is given in Table 2. Let It remains to use the adjoint table to simplify as much as possible the constants in (17).The key here is the recognition of the function () =  2 4 − 4 1  6 , which is invariant under the full adjoint action [23].To begin the simplification we concentrate on the constants  1 ,  4 ,  6 .If  is given as in (17), then ã   = Ad (exp ( 1 )) ∘ Ad (exp ( 6 ))  (18) has the coefficients Three cases arise.

Construction of Group-Invariant
Solutions.The groupinvariant solutions are constructed in this section.We further classify the group-invariant solutions according to the elements of the one-dimensional optimal systems.The reductions and group invariant solutions are listed in Tables 3 and 4.
Definition 3 (see [24]). = (x) is a group-invariant solution of a th-order PDE corresponding to the appropriate admitted symmetry generator if and only if  = (x) satisfies That is, The solution of ( 23) is obtained from the characteristics equation Classifications of the group-invariant solutions by the elements of the optimal system in ( 20) and ( 21) are listed in Tables 3 and 4, respectively.Wherever they appear,  1 and  2 are arbitrary constants, 1  1 (, ; ) and (, ; ) are the confluent hypergeometric functions, Ai() and Bi() are the Airy functions, while   () and Γ() represent the Bessel function of the first kind and Euler gamma function, respectively, and erf() represents an error function.A well-documented review of such functions is presented by Abramowitz and Stegun [27].

Given Constant Dispersion Coefficient
Example a. Suppose a concentration  0 of a solute is supplied to a single point in an instant of time.We require to determine the subsequent concentration of the pollution at various distances from where it was released (see, e.g., [28]).We would expect concentration to vanish at large distance; that is,  The generator  4 in (11) (also, an element of the onedimensional optimal systems given  = 0) leads to an invariant solution in functional form given by where  = √  and  satisfies the equation and hence Here erf() = (2/√) ∫  0  −] 2 ] is the error function [27].In terms of the original variable and subject to the boundary conditions, we obtain Here, the erfc(⋅) is the complement error function defined by (1 − erf(⋅)).Solution (29) is depicted in Figure 1. and  satisfies .
where  =  and . ) ; ; where  = 4 2 and  satisfies where  =  and  satisfies Total flux across  =   is given by The contaminant flux (30) is depicted in Figure 2. Total flux across   = 4 increases and flattens at large time.
Example b.The  6 -invariant solution is given in functional form as where  =   ,  satisfies the ODE   +   = 0.
We impose the boundary conditions Infinite concentration at the origin implies that there is a high supply of contaminants at this point.Furthermore, contaminant concentration vanishes when time evolves.In terms of the original variables we obtain the exact (groupinvariant) solution given by Solution (34) is depicted in Figures 3 and 4. In Figure 3, a sharp peak of concentration is observed shortly after  = 0 and decreases at later stage.This may be interpreted as an injection of contaminants at a single point; that is, the concentration at a single point increases but due to diffusion at larger time it smoothes out.Note that here we have restricted our analysis using symmetry generator  6 .This symmetry generator leads to simpler and realistic exact solution.In Figure 4, we observe that concentration at the origin decreases with time.Furthermore, this concentration vanishes at large distances.

Given Velocity-Dependent Dispersion Coefficient
Example a (steady-state solution).The time translation leads to the analysis of the steady-state contaminant transport.Steady-state solutions may be constructed and subjected to the following imposed boundary conditions: The boundary condition (35) implies that pollutants are supplied at the origin, and boundary condition (36) correspond to the assumption that pollutants are not carried though at some distance   , rather it accumulates here.We obtain the exact solution where Δ is given by The solution (37) is depicted in Figures 5, 6, and 7. We observe in Figure 5 that concentration starts decreasing and converges to some value at large distance for  = 2 than for lower values of , whereas  has an opposite effect as shown in Figure 6.

Example b (transient-state solution)
. It is quite difficult to construct exact solutions for transient contaminant transport subject to these boundary conditions (35) and (36).However, if one assumes that at an initial time, say  = 1, the concentration at the point source is given by a constant and that this concentration vanishes at large distances and prolonged periods, then using the symmetry combination  1 +  2 from Table 4, the group invariant solution is given by )  2 } , ∀ < 0,  > 0.
In his work, Philip [6] considered the instantaneous point source for contaminant dispersion during radial water flow in porous media.Exact solutions were constructed for the two- and three-dimensional models with dispersion coefficient depending on Péclet number.Here, we consider models in stream functions coordinates.Exact close-form (similarity) solutions are constructed using the elements of the onedimensional optimal systems.These new solutions may be viewed as representing the continual supply of contaminant at a point (source) which are dispersed radially.

Some Concluding Remarks
In this paper, we have focused only on the two-dimensional solute concentration field within water from a single injection well.The considered problem is a significant improvement in the study of solute transport under radial water background since we analyze the convection-dispersion equation in stream functions.We have observed that extra Lie point

Table 1 :
Commutators of the admitted symmetries.

Table 2 :
Adjoint representation for the base vectors.