Discussion of Papers/ Christopher Neuzil, Discussion Editor

‘‘On the Use and Error of Approximation in the Domenico(1987) Solution,’’ by Michael R. West, Bernard H. Kueper, andMichael J. Ungs, March–April 2007, v. 45, no. 2: 126–135.

Comment by Anand Prakash, URS Corporation, 604Raintree Rd., Buffalo Grove, IL 60089; (847) 634-3906;anandprakash@comcast.net

West et al. (2007) must be complimented for aninteresting and thought-provoking paper. In addition towell-known approximations, I also found mathemati-cal inconsistencies in the Domenico equations (Prakash2005). Nevertheless, the equations are relatively simpleto use and provide reasonable screening level results formost practical situations. My perusal of West et al. gen-erated the following comments:

1. Equation 3, as a solution of Equation 2 and anabbreviation of Equation 22 of the paper, shouldactually be written as:

C(x, y, z, t)/C0 = C1(x, t) × C2(y, t) × C3(z, t)

(26)

where C0 = source concentration; C1, C2, andC3 = solutions of Equation 2 in (x, t), (y, t), and(z, t) domains, respectively, with unit source con-centration and with the same initial and bound-ary conditions that are used to obtain Equation 22.Substitution would show that Equation 26 satisfiesEquation 2 or 10 and so conserves mass. Math-ematical induction would show that the productin Equation 26 will provide a valid solution ofa second-order partial differential equation (pde)(e.g., Equations 1, 2, or 10) only if there is nosecond-order differential term with respect to therepeating variable (e.g., t) and a consistent ini-tial condition and boundary conditions are used toobtain the individual solutions in different domains(Carslaw and Jaeger 1984).Note that the boundary conditions used to obtainthe three individual solutions in Equation 22 are

Copyright © 2009 The Author(s)Journal compilation ©2009NationalGroundWaterAssociation.doi: 10.1111/j.1745-6584.2009.00617.x

inconsistent. For instance, the boundary conditionfor the solution in the (x, t) domain is x = 0,C/C0 = 1 for all t > 0 (i.e., source concentra-tion is maintained at C0 for all times). On theother hand, the solutions in the (y, t) and (z, t)

domains are obtained for C/C0 = 1 at t = 0; thatis, C/C0 �= 1 if t �= 0 (i.e., source concentration isnot maintained at C0). Thus, the use of the prod-uct rule (i.e., Equation 26) in obtaining the solutionof Equation 22 is not valid. This is why the solu-tions in the (y, t) and (z, t) domains in Equation 24should be included within the integrand over time,as in Equation 23.

2. The transformation t = x/v, with truncation,results in Equation 15, which may be abbreviatedas:

C(x, y, z, t)/C0 = C1(x, t) × C2(x, y) × C3(x, z)

(27)

The repeating variable in Equation 27 or Equa-tion 15 is x. A second-order differential termwith respect to x occurs in the governing pde(Equation 10). Substitution of Equation 27 wouldshow that it does not satisfy Equation 10 anddoes not conserve mass. Thus, the productrule is not applicable and the resulting solution(i.e., Equation 15) is not valid. For some cases,the truncated solution of Equation 15 may bea reasonable numerical approximation. But, thistruncated solution does not satisfy the bound-ary condition of the problem at x = 0,because

C(x = 0, 0 < t < ∞)/C0 �= 1;−Y/2 < y < Y/2; −Z/2 < z < Z/2 (28)

As shown subsequently (see Equation 33), theresult of the transformation t = x/v (i.e., Equa-tion 15) may also be interpreted as combining anunsteady state solution in the (x, t) domain (e.g.,the truncated form of Equation 19) with steady-statesolutions in the (x, y) and (x, z) domains. This maybe why Equation 15 does not satisfy the governingpde (Equation 10).

3. West et al. (2007) did not fully explore the rea-sons for errors in the results of the following,

758 Vol. 47, No. 6–GROUND WATER–November-December 2009 NGWA.org

most commonly used, steady-state version of theDomenico solutions:

C(x, y, z)/C0

= (1/4) exp{(x/2αx)[1 − (1 + 4λαx/u)1/2]}× {erf[(y +Y/2)/(4αyx)1/2]

− erf[(y −Y/2)/(4αyx)1/2]}× {erf[(z + Z/2)/(4αzx)1/2]

− erf[(z −Z/2)/(4αzx)1/2]} (29)

Equation 29 can be abbreviated as:

C(x, y, z)/C0 = C1(x) × C2(x, y) × C3(x, z)

(30)

The governing pde is:

Dx∂2C/∂x2 + Dy∂

2C/∂y2

+ Dz∂2C/∂z2 –v∂C/∂x –λC = 0 (31)

The component C1(x) in Equation 29 is an exactsolution of the corresponding steady-state compo-nent in Equation 31; that is:

αx∂2C/∂x2 = ∂C/∂x + λC/v (32)

with C(x = 0)/C0 = 1 and C(x → ∞)/C0 = 0.Also, C2(x, y) is the solution of the followingsteady-state component of the governing equation(Domenico and Placiauskas 1982),

αy∂2C/∂y2 = ∂C/∂x (33)

with

C(x = 0, y)/C0 = 1; −Y/2 < y < Y/2= 0; y < −Y/2 or y > Y/2

C(y → ±∞) = 0∂C/∂y(y → ±∞) = 0.

Similarly, C3(x, z) is the solution of the correspond-ing steady-state component of Equation 31. Each ofthe three individual solutions in Equation 29 or 30is exact and the desired boundary conditions arealso satisfied. But, x is the repeating variable inEquation 30 and a term with second-order differ-ential with respect to x occurs in the governingpde (i.e., Equation 31). Substitution of Equation 30would show that it does not satisfy Equation 31 ormass balance. So, the application of the product ruleand the resulting solution given by Equation 29 or30 are not mathematically valid.

ReferencesCarslaw, H.S., and J.C. Jaeger. 1984. Conduction of Heat in

Solids. Oxford, UK: Clarendon.

Prakash, A. 2005. A practical analytical model for contaminanttransport in porous media. In Proceedings of the WatershedManagement Conference: Managing Watersheds for Humanand Natural Impacts. Reston, Virginia: ASCE.

West, M.R., B.H. Kueper, and M.J. Ungs. 2007. On the use anderror of approximation in the Domenico (1987) solution.Ground Water 45, no. 2: 126–135.

Note: Comment received May 7, 2007, accepted May16, 2007.

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