Methods Note/

A Semi-Analytical Solution Based ona Numerical Solution of the Solute Transportof a Conservative and Nonreactive Tracerby Jim Zhang1, Jay Clare2, and Jia Guo3

AbstractIn the evaluation of potential risk from ingestion of groundwater near an impacted site, numerical simulation

of fate and transport processes of chemicals of concern is often required. If there is potential concern aboutmultiple chemicals, numerical simulation of each chemical separately is often needed. In this paper, a semi-analytical solution is presented based on a numerical solution of the transport of a conservative and nonreactivetracer. When multiple chemicals undergoing sorption and first-order degradation need to be modeled, we can avoidperforming individual numerical simulations for each chemical by applying the semi-analytical solution. Numericaltest runs were conducted to verify the semi-analytical solution; simulation results reveal that the concentrationsderived from the semi-analytical solution are identical to those derived from the individual numerical fate andtransport model simulations. The semi-analytical solution requires steady-state flow conditions, no continuingcontaminant source, and similar initial source concentration distributions.

IntroductionTo quantitatively evaluate the potential risk from

ingestion of groundwater near an impacted site, thespatial and temporal concentration distributions of con-taminants in groundwater underneath and downgradientof the impacted site are often of interest. Owing tothe heterogeneity of the subsurface soils, the surfacehydrogeologic features (irrigation, rivers and streams,drainage, evapotranspiration, etc.), groundwater with-drawals and recharge, etc., the groundwater flow condi-tions are often not uniform. Therefore, the well-knownanalytical solutions to the simpler forms of transport

1Corresponding author: URS Corporation, 1333 Broadway,Suite 800, Oakland, CA 94612; (510) 874-3154; fax: (510) 874-3268; jim_zhang@urscorp.com

2URS Corporation, 1333 Broadway, Suite 800, Oakland,CA 94612.

3URS Corporation, 100 South Wacker Drive, Suite 500,Chicago, IL 60606.

Received January 2011, accepted August 2011.© 2011, The Author(s)Ground Water © 2011, National Ground Water Association.doi: 10.1111/j.1745-6584.2011.00865.x

equation (Baetsle 1969; Ogata 1970; Domenico 1987) andother analytical solutions for heterogeneous porous media(Barry and Sposito 1989; Yates 1990, 1992; Aral and Liao1996; Huang et al. 1996; Marinoschi et al. 1999) are notapplicable for most sites. Consequently, a numerical fateand transport model is required for the evaluation of mostimpacted sites.

The contaminant transport processes in groundwatermay include advection, dispersion, sorption, and degra-dation. Advection describes the transport of a misciblechemical at the same velocity as the groundwater and isdependent on the groundwater conditions only. Dispersionis caused by mechanical dispersion (resulting from devi-ations of actual velocity from the average groundwatervelocity) and by molecular diffusion driven by concen-tration gradients. Molecular diffusion (chemical specific)is generally secondary and negligible and only becomesimportant when the groundwater velocity is very low(Anderson and Woessner 1992). Sorption refers to themass transfer process between the chemicals dissolvedin groundwater and the chemicals sorbed on the porousmedium and retards the chemical migration. Degradationrepresents the chemical’s mass loss over time, either due

NGWA.org GROUND WATER 1

to biochemical reaction, breakdown from parent prod-ucts to daughter products, or radioactive decay.

At some impacted sites such as refineries or under-ground storage tanks, a large number of potentialchemicals of concern (COCs) may exist. Due to theirvarious chemical properties (i.e., sorption and degrada-tion), separate transport simulations for each chemicalsare often required. If there are a large number of COCsat an impacted site, the evaluation of all chemicals can beextraordinarily time-consuming, especially if the fate andtransport model is relatively complex.

In this paper, we present a semi-analytical solutionbased on the numerical solution of the transport of a con-servative and nonreactive tracer (CNRT) in steady-stategroundwater flow. The solution can help us to improveour understanding of the concentration variation of chem-icals that undergo sorption and degradation. In the appli-cation to transport simulation, the numerical solution ofthe solute transport of a CNRT is obtained first, then thesolution of solute transport of chemicals with sorptionand degradation can be obtained analytically by a scal-ing method, thus avoiding the necessity of modeling eachchemical separately.

To validate the semi-analytical solution, numericaltest runs were conducted to simulate the transport pro-cesses of a CNRT and chemicals undergoing sorption anddegradation. The test runs show that the concentrationsderived from the semi-analytical solution are identical tothose derived from the transport model simulations.

General Transport EquationThe general partial differential equation describing

the fate and transport of a chemical in a three-dimensional(3D) transient groundwater flow systems can be writtenas (Zheng and Wang 1999):

∂

∂xi

(θDij

∂C

∂xj

)− ∂

∂xi

(θviC)

+ qsCs +∑

Rn = ∂(θC)

∂t(1)

where xi is distance along the respective Cartesiancoordinate axis [L], θ is the porosity of the subsurfacemedium [dimensionless], Dij is hydrodynamic dispersioncoefficient tensor [L2/T], C is the dissolved concentrationof the chemical [M/L3], vi is the seepage or linear porewater velocity [L/T], qs is volumetric flow rate per unitvolume of aquifer representing fluid sources (positive)and sinks (negative) [1/T], Cs is the concentration of thesource or sink flux of the chemical [M/L3], �Rn is thechemical reaction term [M/L3T], and t is time [T].

Under a steady-state flow condition, if there is nosource or sink, and the porosity (θ ) does not change overspace, Equation 1 becomes:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) +∑

Rn

θ= ∂C

∂t(2)

The hydrodynamic dispersion coefficient occurs as aconsequence of two different processes: mechanical

dispersion and diffusion. These two contributions tohydrodynamic dispersion are represented mathematicallyas (Domenico and Schwartz 1998):

D = D′ + D∗d (3)

where D′ is the coefficient of mechanical dispersion[L2/T] and D∗

d is the bulk diffusion coefficient [L2/T]. D′is proportional to the groundwater velocity and indepen-dent of the chemical, whereas D∗

d is chemical specific. Forgeneral groundwater conditions, mechanical dispersionoften dominates the dispersion process and the diffu-sion process is negligible (Anderson and Woessner 1992).Consequently, the hydrodynamic dispersion coefficient ispractically independent of chemical properties. This is trueexcept for cases in which groundwater moves very slowlythrough a very low permeable medium (such as clay).

The chemical reaction term �Rn in Equations 1and 2 can be used to include the effects of generalbiochemical and geochemical reactions or radioactivity.Considering only two basic types of chemical reactions,aqueous-solid surface reaction (sorption) and first-orderrate degradation, the chemical reaction term can beexpressed as (Zheng and Wang 1999):∑

Rn = −ρb∂S

∂t− λ1θC − λ2ρbS (4)

where ρb is the bulk density of the subsurface medium[M/L], S is the concentration sorbed on the subsurfacesolids [M/M], and λ1 and λ2 are the first-order reactionrates for the dissolved phase and the sorbed (solid) phase,respectively [1/T]. For radioactive decay, the reactiongenerally occurs at the same rate in both phases (λ1 = λ2).For biodegradation, however, it has been observed thatcertain reactions occur only in the dissolved phase (i.e.,λ2 = 0).

Taking Equation 4 into Equation 2, Equation 2 canbe rearranged and written as:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) − λ1θC + λ2ρbS

θ

= ∂C

∂t+ ρb

θ

∂S

∂t(5)

Equation 5 is essentially a mass balance statement,that is, the change in the mass storage (both dissolved andsorbed phases) at any given time is equal to the differencein the mass inflow and outflow due to dispersion,advection, sink/source, and chemical reactions.

Sorption refers to the mass transfer process betweenthe dissolved contaminants and the sorbed contaminants.Local equilibrium is generally assumed for the sorptionprocesses. In general, although there are three typesof equilibrium sorption isotherms (linear, Freundlich,and Langmuir) used in transport modeling (Zheng andWang 1999), the linear sorption isotherm is the mostcommonly used. The linear sorption isotherm assumesthat the concentration sorbed on the solids (S) is directlyproportional to the dissolved concentration (C):

S = KdC (6)

2 J. Zhang et al. GROUND WATER NGWA.org

where Kd is the distribution coefficient [L3/M]. The sorp-tion process causes the advance rate of the contaminantplume to be retarded. The retardation factor R is thusgiven as:

R = 1 + ρb

θ

∂S

∂C= 1 + ρbKd

θ(7)

When a linear adsorption isotherm is invoked,Equation 5 can be expressed in the following form:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC)

−(

λ1 + λ2ρbKd

θ

)C = R

∂C

∂t(8)

Equation 8 is the general transport equation for a chemicalundergoing sorption (a linear adsorption isotherm) andfirst-order degradation under steady-state groundwaterflow without concentration sources or sinks.

Semi-Analytical SolutionThe purpose of the study is to derive a semi-analytical

solution that can be used to derive the modeling results ofa CNRT to the modeling results of chemicals undergoingsorption and degradation.

Under the same flow conditions, chemicals migratedifferently because of their different chemical proper-ties (sorption and degradation). However, the transportprocesses of advection and dispersion depend only onflow (i.e., independent of chemicals, again assuming themechanical dispersion dominates the dispersion process),making a semi-analytical solution possible.

Equation 8 can be further simplified as:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) − λC = R∂C

∂t(9)

Where λ = λ1 + λ2ρbKd

θis the comprehensive first-order

decay rate accounting degradations in both dissolved andsorbed phases.

In the case that a chemical behaves as a CNRT(i.e., R = 1.0 and λ1 = λ2 = 0), Equation 9 becomes theadvection-dispersion equation:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) = ∂C

∂t(10)

To derive the solution C(xi, t) to the general transportequation (Equation 9) from the solution to the precedingadvection-dispersion equation, we first solve the equationfor chemicals undergoing adsorption process only. WhenR > 1.0 and λ1 = λ2 = 0, Equation 9 becomes theadvection-dispersion-sorption equation:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) = R∂C

∂t(11)

Equation 11 can be re-written as:

∂

∂xi

(Dij

∂C

∂xj

)− ∂

∂xi

(viC) = ∂C

∂τ(11a)

where τ = t/R [T] is a scaled time. It is noted that withthe scaled time τ , Equation 11a becomes identical to theadvection-dispersion equation (Equation 10).

Assume that with a specific initial concentrationcondition C0(xi), the advection-dispersion Equation 10has a unique numerical solution C1(xi , t) under specificgroundwater conditions vi(xi). Then for a chemical havingthe same initial concentrations C0(xi) but undergoing asorption process, the solution C1(xi , τ ) will satisfy theadvection-dispersion-sorption Equation 11a. That is:

∂

∂xi

(Dij

∂C1(xi,

tR

)∂xj

)− ∂

∂xi

(viC1

(xi,

t

R

))

= ∂C1(xi,

tR

)∂

(tR

) (12)

Conceptually, the degradation process is not actuallya “transport” process because the concentration decreasesassociated with degradation are not related to the chem-ical’s migrations. Consequently, for a chemical that hasthe same initial concentrations C0(xi) and undergoes sorp-tion and/or degradation, we may assume that the solutionC(xi , t) to the general transport equation (Equation 9) is acombination of the solution C1(xi , t/R) and a degradationprocess C2(t) as the following relationship (Domenico andSchwartz 1998):

C(xi, t) = C1

(xi,

t

R

)× C2(t) (13)

Note that C2 is a function of time only and C2(0) =1.0. Taking Equation 13 into Equation 9 and re-arrangingthe equation, we have

C2(t)∂

∂xi

(Dij

∂C1(xi,

tR

)∂xj

)− C2(t)

∂

∂xi

(viC1

(xi,

t

R

))

− λC1

(xi,

t

R

)C2(t) = RC1

(xi,

t

R

)dC2(t)

dt

+ RC2(t)∂C1

(xi,

tR

)∂t

(14)

Dividing both sides of Equation 14 by C2(t) results in

∂

∂xi

(Dij

∂C1(xi,

tR

)∂xj

)− ∂

∂xi

(viC1

(xi,

t

R

))

− λC1

(xi,

t

R

)= R

C1(xi,

tR

)C2(t)

dC2(t)

dt

+ R∂C1

(xi,

tR

)∂t

(15)

NGWA.org J. Zhang et al. GROUND WATER 3

Subtracting Equation 12 from Equation 15, we have

−λC1

(xi,

t

R

)= R

C1(xi,

tR

)C2(t)

dC2(t)

dt(16)

Dividing both sides of Equation 16 by C1(xi , t/R)

and re-arranging Equation 16, we have

dC2(t)

dt= −λC2(t)

R(17)

The general solution to Equation 17 is:

C2(t) = Ae− λR

t (18)

where A is an integration constant. Since C2(t = 0) =1.0, we have A = 1.0. Consequently, the solution to thegeneral transport equation (Equation 9) becomes:

C(xi, t) = C1

(xi,

t

R

)e− λ

Rt (19)

Since λ = λ1 + λ2ρbKd

θ, the solution of Equation 19

can be written as,

C(xi, t) = C1

(xi,

t

R

)exp

(−λ1θ + λ2ρbKd

θ + ρbKdt

)(20)

Equation 20 indicates that under the conditions ofsteady-state flow and no contaminant sources and/or sinks,for any chemical undergoing sorption and degradation, thechemical’s concentration at any location and at any timecan be derived from that of a CNRT having the sameinitial concentrations. Consequently, if multiple chemi-cals have the same initial concentration distribution at animpacted site, only one model simulation of a CNRT isneeded. The spatial and temporal concentration distribu-tions of the multiple chemicals can be derived by scalingthe modeled results of the CNRT using Equation 20.

Note that in the derivation of Equation 20, for sim-plification, it is assumed that the chemical undergoingsorption and degradation has the same initial concentra-tions C0(xi) as the CNRT. However, only the same initialconcentration distribution is required, the magnitude of theinitial concentrations can be different. That is, the chem-ical can have an initial concentrations a × C0(xi) wherea is a constant. For example, if a uniform concentrationin a specific area (such as a source area) is consideredappropriate to represent the initial concentrations for mul-tiple chemicals, a unit concentration can be specified forthat area for the transport simulation of a CNRT. The con-centration of any specific chemical is the product of thescaled concentration from Equation 20 and the chemical’sinitial concentration.

Numerical ValidationTo numerically validate the semi-analytical solution,

a 3D groundwater flow and transport model was devel-oped for a hypothetical heterogeneous aquifer havingnonuniform steady-state flow conditions. The flow wasmodeled using MODFLOW (McDonald and Harbaugh1988; Harbaugh et. al. 2000; Hill et. al. 2000), and the

Figure 1. Model domain, boundary conditions, pumpingwell location, and simulated steady-state heads.

chemical transport was modeled using MT3DMS (Zhengand Wang 1999).

Flow ModelThe hypothetical aquifer is assumed to be 2000 feet

(610 m) long (parallel to groundwater flow direction) and1000 feet (305 m) wide, as shown in Figure 1. The modeldomain was uniformly discretized into 100 rows and 200columns with a uniform grid of 10 × 10 feet (3.05 ×3.05 m) square. Vertically, the aquifer was assumed tohave a uniform thickness of 30 feet (9.14 m), and wasdivided into two model layers, each having a uniformthickness of 15 feet (4.57 m).

Heterogeneous hydraulic conductivity ranging from10 ft/day (3.05 m/day) to 250 ft/day (76.2 m/day) wasspecified over the model domain for both model layers,and the bottom layer (layer 2) was assumed to be morepermeable. To simplify the problem, a constant verticalanisotropic ratio of 3.0 is assumed over the entire modeldomain, and the aquifer is assumed to be isotropic in thehorizontal direction.

A hypothetical pumping well is placed downgradientof the current contaminant plume. The pumping well wasassumed to be screened over the entire thickness of theaquifer and has a constant pumping rate of 2400 ft3/d(68 m3/d). To simplify the flow model setup, no-fluxboundary conditions were specified for both the upper andbottom boundaries, whereas constant-head boundary con-ditions were specified for the left and right boundaries,with hydraulic head of 105 and 90 ft (32.0 and 27.4 m),respectively, as shown in Figure 1. In addition, it wasassumed that no other hydrologic features such as sur-face water recharge and evapotranspiration exist insidethe model domain.

The steady-state groundwater flow condition asso-ciated with the specified boundary conditions andgroundwater pumping was modeled, and the modeledsteady-state hydraulic head distributions are also shownin Figure 1.

Transport ModelThe flow model was augmented with a transport

model, and the transport model has the same modeldomain and spatial finite-difference grid (model cells)defined earlier. The transport was modeled under the pre-viously simulated steady-state, nonuniform flow condition.

4 J. Zhang et al. GROUND WATER NGWA.org

Figure 2. Initial concentration distributions (layer 1) andobservation locations in transport model.

A hypothetical chemical plume was assumed to bepresent in the upper portion of the aquifer (model layer 1),whereas the bottom portion of the aquifer (model layer2) was assumed to be initially clean. The hypotheticalconcentrations ranging from 1.0 to 250 μg/L (Figure 2)were used as the initial concentration conditions for thetransport simulation. It was also assumed that the hypo-thetical plume was the only contaminant source (depletingsource).

The following transport parameters were specified inthe transport model: longitudinal, transverse, and verticaldispersivity of 25, 0.25, and 0.05 feet, respectively, wereused for all scenarios. When sorption was modeled, aneffective porosity of 0.25, a soil bulk density of 1.67 kg/L,and a distribution coefficient 0.6 L/kg were used. Theseparameters result in a retardation factor of 5.0. When first-order degradation was modeled, a half-life of 500 d wasused for degradation in the dissolved phase and/or solidphase.

Four observation points were set along the pathway ofchemical migration to record the simulated concentrationvariation. The locations of the four observation points areshown in Figure 2. Points A and B were set in modellayer 1, and points C and D were set in model layer 2.

Simulation ScenariosTo validate the semi-analytical solution, four transport

simulation scenarios were conducted. The simulatedconcentration variations over time (breakthrough curve)were recorded at the four selected observation points, andthe simulated breakthrough curves were compared withthose derived from the semi-analytical solution. The fourscenarios are:

Scenario 1: advection and dispersion only (no chem-ical reactions).

Scenario 2: Scenario 1+ sorption (retardation factor=5.0).

Scenario 3: Scenario 2 + degradation in aqueousphase (half-life = 500 d), and

Scenario 4: Scenario 3 + degradation in solid phase(half-life = 500 d).

Scenario 1 represents the transport of a CNRT,and it serves as a baseline simulation in deriving theconcentrations of Scenarios 2–4 using the semi-analyticalsolution. Specifically, Scenario 1 is modeled first, andthe modeled concentrations are scaled to derive the con-centrations of Scenarios 2–4 using Equation 20. Then

Scenarios 2–4 are also modeled and the modeled con-centrations are compared with those derived from thescaling method. For Scenario 1, the transport simulationwas conducted for 1000 d. For Scenarios 2–4, due to theretardation factor of 5.0, the transport simulation timeswere extended to 5000 d.

Simulation Results and ScalingModeled chemical concentrations vs. time (i.e.,

breakthrough curve) for Scenarios 1–4 were recorded atthe four selected observation points as shown in Figure 2.The recorded breakthrough curves of Scenario 1 werescaled into the breakthrough curves of Scenarios 2–4using Equation 20. For Scenarios 2–4, the breakthroughcurves derived from the scaling method are compared withthose recorded from model simulations as mentioned inthe following paragraph.

Figure 3a–3c shows the modeled breakthrough curves(solid lines) and those derived from scaling method (cir-cular hollow dots) for Scenarios 2 through 4, respectively.The comparison indicates that the modeled concentrationsand the concentrations derived from the scaling methodare identical for all three scenarios. Note that due to thedegradation in both dissolved and sorbed phases of Sce-nario 4, the concentrations quickly drop to very low (closeto zero); thus, a log-scale was used for the concentrationin Figure 3c (Scenario 4).

Overall, the preceding numerical test runs validateEquation 20. Specifically, the numerical tests prove thatonce the numerical solution of the transport of a CNRT isobtained, the concentrations of multiple chemicals under-going sorption and/or degradation can be analyticallyderived using Equation 20, based on the retardation fac-tors and half-lives of those chemicals.

Discussion and ConclusionFor the cases where there are a large number of COCs

at an impacted site, a fate and transport model is oftendeveloped and transport simulation for each individualchemical is often required. Conducting transport simula-tions for a large number of chemicals often requires greateffort and is sometimes not practical. In addition, the eval-uation of current spatial concentration distributions (usedas initial concentrations in transport simulation) for eachchemical is often not practical due to limited data availableand/or the large number of required man-hours to com-plete the evaluation. To simplify the model simulations, auniform concentration distribution (or any specific spatialdistribution) is often assumed over a specific area (sourcearea) for all chemicals. However, with the same sourceconcentration distribution, chemicals often have differentconcentration values, depending on the chemicals’ prop-erties and the site conditions.

In many cases, although groundwater conditions fluc-tuate seasonally, steady-state flow conditions (uniform ornonuniform) representing the average groundwater flowconditions are often used for transport simulation. Underthis condition, if there is no continuing contaminantsource, a scaling method based on Equation 20 seems

NGWA.org J. Zhang et al. GROUND WATER 5

Figure 3. Comparison of simulated and scaled breakthroughcurves for: (a) Scenario 2, (b) Scenario 3, and (c) Scenario 4.

ideal for the evaluation of transport of a large numberof chemicals.

In addition to the steady-state flow conditions,having no continuing contaminant source, and havingthe same initial source concentration distribution forall chemicals, the application of Equation 20 has thefollowing requirements which are satisfied in many cases:

1. The aquifer has uniform effective porosity and bulkdensity distribution (i.e., they do not change overspace) such that the retardation factor does not changeover space.

2. The groundwater velocity is not very low and themechanical dispersion dominates the dispersion pro-cess (i.e., molecular diffusion is negligible) so that thehydrodynamic dispersion coefficient is independent ofthe chemicals.

3. The sorption is a linear sorption isotherm such that theretardation factor is independent of the concentrationsand does not change over space, and

4. The degradation follows a first-order decay in whichthe decay rate is independent of the concentration.

The scaling method based on Equation 20 has beenproven to be effective and accurate in deriving spatial and

temporal concentration distributions of multiple chemi-cals undergoing sorption and/or degradation. When thepreceding constraints are satisfied, the scaling method isvery useful when a large number of chemicals need to bemodeled.

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