510 hydraulic engineering software - wit press · m. koch department of geohydraulics and...

Dupuit-Forchheimer formulation of a

transport finite element model: application to

remediation of a shallow, unconfmed aquifer

M. KochDepartment of Geohydraulics and Engineering Hydrology,

University ofKassel, Kurt-Wolters Strafie 3, 34109 Kassel, Germany

Email: [email protected]

Abstract

Groundwater contamination is frequently concentrated in the surflcial or water-tableaquifer. The first remediative step of aquifer clean-up consists of the extraction ofthe polluted water by pumping. Depending on the pumping rates, this may lead insituations where the surficial aquifer is shallow to significant drawdowns of thewater-table. Modeling the horizontal solute transport in such a pumped unconfinedaquifer by means of a classical 2D transport model that assumes a constant aquiferthickness, may lead to significant errors and, consequently, bias in the remediationstrategy. Thus there is a desire to have an improved 2D solute transport model forsimulating remediation in a phreatic shallow aquifer. In the present paper thewell-known 2D SUTRA model is modified to include a Dupuit-Forchheimerformulation for unconfined aquifers which takes into account the laterally varyingthickness of the water-bearing layer. The new model is validated by comparisonswith analytical solutions and through testing of asymptotic consistency with theclassical confined-aquifer solution. The model is used to simulate a possibleremediation action in a shallow contaminated phreatic aquifer in Central Florida andto compute effective resident and clean-up times. These are compared withpredictions obtained with the original SUTRA. The new Dupuit-SUTRA modelpredicts significantly shorter total clean-up times than the original model. The lattertherefore gives a more conservative estimate of the remediation time. Budgetcalculations for the two model-versions reveal large differences in the total amountof contaminants extracted from the aquifer during the pumping process.

Transactions on Ecology and the Environment vol 19, © 1998 WIT Press, www.witpress.com, ISSN 1743-3541

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1. Introduction

Groundwater pollution control has become a major issue of environmental and

public health agencies in recent years (e.g. Canter and Knox, 1985). Most of

the remediation action plans (RAP) proposed for the clean-up of a

contaminated aquifer include some sort of extraction of the polluted

groundwater by pumping, treatment and reinjection of the treated water into

the underground through installation of an infiltration gallery (Driscoll, 1986).

The design of optimal pumping strategies in an aquifer remediation isoften guided by cost considerations for the installation and the maintenance ofthe extraction and injection wells. This, and the desire to shorten clean-up

times, usually leads to the installation of only a few wells and applying

relatively high pumping rates to them. The movement of the contaminant plume

and the remediation times are then estimated to first approximation by

considering the volume-extent of the plume and the applied discharge rate. A

more precise estimate of the plume movement in response to pumping is

obtained through the use of a numerical solute transport model. Most of the

commercially available transport models are still two-dimensional (2D)

(Javandel etal, 1984; Maidment, 1993), though several 3D transport models,

have become available in the public domain in recent years. However, a fully3D model is often difficult to use, requires large computer resources andsufficient data for model validation (Koch, 1994). Therefore, a 3D transportmodel is often discarded by ordinary hydrologists for exploratory modelingstudies, and a classical 2D transport model is favored instead. In order to

capture the areal extent of the plume, such a 2D model is to be usedhorizontally, assuming a constant depth of the aquifer. Such a "confined

aquifer" approach will be inappropriate for a free-surface (phreatic) aquifer

- where most of the groundwater pollution occurs - particularly, whensignificant drawdowns are induced as a consequence of strong pumping. Thisis because flow in a free-surface aquifer is fully 3D and can only be treated as

approximately 2D under the so-called Dupuit assumptions (DA) (Bear, 1979).

While there exist several groundwater flow models (Javandel et al,1984) that incorporate the DA for modeling flow in phreatic aquifers, this isnot the case for 2D areal transport models. The present paper is to remedy thissituation by implementing a Dupuit-Forchheimer formulation into the well-

known finite element transport model SUTRA. After validation, the new model

will be employed to model solute transport in a shallow contaminated phreaticaquifer in Central Florida that is subject to remediative pumping. Results forthe effective resident and clean-up times predicted by the new Dupuit-SUTRAand the old model will be compared. This will answer the question whether"pump and treat" remediation for a phreatic aquifer can be modeled preciselyenough by means of a common "confined-aquifer" transport model.


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2. Mathematical formulation

2,1 Governing equations for flow and transport

Flow and solute transport in both confined or unconfmed aquifers are governed

by the general groundwater flow equation

, g^ , (1)at

and the solute transport equation (assuming a conservative tracer)

- (D Vc) +g (2)at

(Bear, 1979). Notations in Eqs.l and 2 are: h = p/pg, + z, [L], piezometrichead; p [M/L/T ], pressure; g, [L/T ], gravity acceleration; z«,[L] elevationhead; p [M/L ], density; f [T], time; K [L/T], hydraulic conductivity tensor;

ty[T'], fluid source/sink term; SQ [L''], specific storativity (specific yield for

a phreatic aquifer); c\M JM ^ solute concentration; v [L/T], average pore

velocity (=v/«; v& Darcy velocity; n, porosity); q, [T*], solute-source/sink

term; D [LVT], hydrodynamic dispersion tensor defined as D = D* + a? \v\

8y + (UL - <*r) M V / 1^1 ' 4/ ' Kronecker symbol; D* [L /T], the coefficientof molecular diffusion; and (XL, 0% , the longitudinal and transverse

dispersivities, respectively.

2.2 Equations for horizontal flow in confined and unconfined aquifers

For a confined aquifer that is horizontally isotropic (K = K^ = Kyy)and has thickness B(x,y), vertical integration of Eq. (1) gives

( T ) + ( 7 ) + *

a, a* ay ay '

where T(xy)= KB [L /T], the transmissivity; qf = qfB [L/T], area-specific

flow sink/source term; S = SoB, storage coefficient of the aquifer.For an unconfined aquifer, using Dupuit's assumptions (DA) (small

inclination of the water-table of height h and horizontal flow, i.e. equipotentiallines A=const. are vertical) (Bear, 1979), Eq. (3) is replaced by the Dupuit-

Boussinesq (Forchheimer) equation



with qy* = qjh [L/T]. Because Eq. (4) is nonlinear, two major linearization

techniques are commonly being used (Bear, 1979):(1) Replacing the variable T=Kh, the effective local transmissivity of

the unconfmed layer by TQ = Kh^ where ho is the average water-table height,

results in an eq. similar to Eq. (3), with T=TQ=const.\ i.e. it amounts to

approximating the unconfined by an equivalent confined aquifer.(2) Rewriting the time-derivative term of Eq.(4) approximately as

d(tf/2)/dt, one obtains a linear equation in tf:

,5,

where ho is the average thickness of the water-bearing layer (Bear, 1972).

For steady state phreatic flow that will prevail for long-term

remediative pumping considered in the present paper, Eq. (5) reduces to

(6)ax ax dy dy

and this is the flow eq. to be solved by the new adaptation of the SUTRA.A numerical analysis of the two linearization approaches was carried

out by Zhang and Koch (1991) who found that the Dupuit Eq. (6) providesa smaller drawdown than the "confined-aquifer" approach based on Eq.(3).

3. Numerical implementation of Dupuit's theory

3.1 General description of the SUTRA model

The steady-state Dupuit-Forchheimer theory is implemented into the well-known SUTRA (Saturated-Unsaturated-TRAnsport) model (Voss, 1984).SUTRA is a 2D finite element model and can either be used in the horizontalx-y domain (employed here) or in a vertical x-z cross-section. The discretization

of the x-y domain is done by means of 4-node quadrilateral elements. A bilinearapproximation for h and c is used in each element and the classical Galerkinformulation of eqs. (1) and (2) is established. For each timestep the resultinglinear matrix systems for the nodal unknowns p or c are solved consecutivelyby an iterative conjugate gradient technique (Koch, 1992). The integration intime is performed using a fully implicit method which is unconditionally stable.

3,2 Adaptation of SUTRA to model transport in a phreatic aquifer

Both the flow and the transport part of SUTRA are to be adapted formodeling transport in an unconfined aquifer. As for the flow part the basic eq.



to be solved is the steady-state Dupuit Eq.(6). The latter is linear in tf and is,

after substitution by h' = tf and by #D/*=2gy formally equivalent to the

steady-state, "confined-aquifer" form of the flow Eq. (3), as solved in the

original SUTRA. While this modification in the flow part of SUTRA is

straightforward, there are several more complicated considerations to be taken

into account, as well for the flow part as for the transport part of the model.

One relates to the proper integration over the varying depth fi= h(x,y) of the

unconfined layer to account for the proper mass conservation for both fluid and

solute and another, very intricate one, to the correct specification of theDirichlet boundary conditions (be) h = h^ for flow and c = c^ for soluteconcentration under 'phreatic' conditions (see Koch, 1998, for details).

3.3 Restrictions and limitations of the Dupuit-SUTRA model

While providing a better approximation than the original "confined" SUTRA

model, the new Dupuit-SUTRA model is valid only within the DA. Therefore,

the same restrictions and limitations as for the DA apply, the most stringent of

which is that the water-table inclination, or the drawdown Ah relative to the

total thickness B should not exceed 15-20% (Bear, 1979). Less stringentlimitations apply for the validity of calculated discharge rates, however (seeZhang and Koch, 1991, for a review/ Another "non-Dupuit"-phenomenon isthe seepage face which occurs at an outflow surface of the model domain

(Fig. 1) and causes the true water table to lie above the 'Dupuit'-computed

surface. This has the effect that the true discharge will be up to 10 % largerthan the one computed by Dupuit's theory (Shamsai and Narasimhan 1991).

4. Model verification

The new Z)w/?«/Y-SUTRA model is verified for various aquifer set-ups

including regional flow and local pumping. The most stringent test of themodel has to be (1) the comparison with possible analytical solutions and (2)asymptotic consistency, whereby the solutions of the new and the old SUTRAmodels should approach each other for large thicknesses B of the phreaticaquifer when horizontal variations Ah of the free surface become negligible.

4.1 Radial flow towards a well

For axisymmetric phreatic flow towards a well (Fig. 1) the analytical solutionfor the head h(r) is given by the Dupuit-Thiem formula (Bear, 1979). Fig. 2shows the Thiem solution together with the radial drawdown extracted from

the numerical solution of the DwpwY-SUTRA model on a 30 by 30 element

mesh which is strongly refined towards the well. Except in the very vicinity

of the well where Thiem's formula is singular a nearly perfect match of thenumerical with the analytical solution is obtained.



4.2 Comparative simulation of solute breakthrough-curves (ETC)

Fig. 3 shows two BTC's obtained by employing either the original (confined)

or the new Dupuit-SUTRA model. A regional head gradient of Ah = 2 m over

a horizontal distance of 240m is imposed through Dirichlet bc's at x=0 and

x=240m. One observes from Fig.Sa that for an aquifer thickness 5= 10m the

fronts of the Dupuit's BTC's are consistently trailing behind the onescomputed with the "confined" model (see Koch, 1998, for an explanation). As

is to be expected from the DA - and is also demonstrating the consistency ofthe Dupuit-SUTRA model with the original SUTRA for small JA/B-ratios,

when the "confmed"-aquifer approach is more valid) - this lag tends towardszero for larger B. For 5=5Om (Fig. 3b) the BTC's are practically identical.

5. Remediation of a shallow phreatic aquifer

5.1 Motivation

Groundwater pollution in Florida tends to concentrate in the surficial aquifer,which in many locations has only a thickness of the order of 10m. Simulation

of a remediation strategy through pumping should take into account thewater-table conditions of such a shallow aquifer. The advantage of using the

new Dw/w/7-SUTRA over the original "confined aquifer" version of SUTRAwill be illustrated in this section for a contaminated site in Central Florida.

5.2 Site history and contamination assessment

A contamination assessment report (CAR) of the site (referred henceforth as

the "DRT" site (DeMouy, 1990)) has been prepared by Woodward & Clyde

Consultants (WCC) (1990) for the Florida Department of EnvironmentalRegulation (FDER). The site history of the report characterizes the DRT siteas an unpermitted hazardous waste storage facility operating from the late

1970's until its closing in 1982. A systematic assessment of the pollution was

done in 1986 and 1989. Volatile organic compounds (VOC's), consisting

mainly of halogenated aliphatic hydrocarbons, acetone and small amounts ofBTX were detected The total VOC concentration values detected for the 1986and 1989 sampling events are shown in (Fig.4a) and (Fig.4b), respectively.

5.3 Modeling plume migration

The 'confined aquifer' SUTRA model-version of Koch and Zhang (1990) isused to simulate conservative (non-reactive) transport of the 1986 plume (Fig.4a) over the three-year sampling period. The flow part of the model has been

calibrated assuming B in the range of 8-12 m, as is supported from severalborehole drillings, and a pump-test inferred transmissivity T of 10"* nf/s.

For the transport part of the problem, the interpolated concentration



isolines for the total VOC of Fig. 4a for the 1986 sampling event were used as

initial conditions. They are shown in Fig,5b. c , a? are chosen as 10 and 1m,

respectively, which are representative for a regional aquifer (Neutnan, 1990).

Fig. 5c illustrates the modeled migrated plume after three years of simulationtime and should be compared with the observed plume ofFig.4b..

5.4 Modeling of resident or clean-up times under pumping conditions

The DRT site is presently subject of a remediation action plan (RAP) to becleaned up down to a level of total VOC which in Florida by law is between

50 and 200 ppb. Regardless of the secondary remedial steps to be taken,

extraction of the polluted groundwater by pumping through one or severalwells has to be the primary task.

To model the effects of remediative pumping numerous computationswith mainly varying mass pumping rates Q=Qf (= pqj of the well anddifferent aquifer thicknesses B are performed, employing both the original

"confined" SUTRA and the new Dupuit-SUTRA model. Fig. 6 shows the

results of the two models for B= 8m and Q = 2.5 kg/s. The two panels a) and

b) reveal the strong pumping drawdown. One observes that near the well thelatter is larger for the "phreatic" Dupuit-SUTRA model than for the

"confined" SUTRA model. This results in larger flow velocities towards the

well for the Dupuit-model. The effects of this and of the dipping water-table

on the movement of the contaminant plume after four different time-elapses ofpumping are depicted in the panels (c) to (e). After the same time the plume ofthe Dupuit-SUTRA model is significantly smaller in size than that predicted bythe original "confined" SUTRA model. Eventually, this will lead to shorter

clean-up times for the Dupuit- SUTRA model (4 versus 5 months for theoriginal SUTRA to reach a clean-up level < 100 ppb).

References

Bear, J., Hydraulics of groundwater, Me Graw-Hill Inc., New York, 1979.Canter, L.W., and R.C. Knox, Groundwater Pollution Control, Lewis Publisher,

Inc., Chelsea, MI, 526pp., 1985.DeMouy, I, Application of a finite-element groundwater model: A case study, M.S.

thesis, Florida State University, Tallahassee, Fl, 1990.Driscoll, F.G., Groundwater and Wells, Johnson Division, St. Paul, MN., 1986.Javandel, I., C. Doughty, and C.F. Tsang, Groundwater Transport: Handbook of

Mathematical Models, Water Resour. Monogr. Series, vol. 10, AGUWashington, D.C., 1984.

Koch, M., Preconditioned conjugate gradient solvers for the SUTRA solutetransport model, Technical report, SCRI, FSU, Tallahassee, FL., 1992.

Koch, M., Application of the MT3D solute transport to the MADE-2 site, TechnicalReport, Armstrong Laboratory, Tyndall AFB, Panama City, FL, 1994.

Koch, M., Modelling of clean-up times of a shallow unconfined aquifer using aDupuit-Forchheimer approach, submitted to Ground Water, 1998.



Koch, M. and G. Zhang, Numerical simulations of solute transport by means of theSUTRA-model, Technical report, SCRI, FSU, Tallahassee, FL., 1990.

Maidment D.R., Handbook of Hydrology, McGraw Hill, New York, NY, 1993.Neuman, P.S., Universal scaling of hydraulic conductivities and dispersivities in

geological media, Water Resour. Res., 26, 1749-1758, 1990.Shamsai, A., and T.N. Narasimhan, A numerical investigation of free surface-

seepage face relationsliip under steady state flow conditions, Water Resour.Res., 27, 409-422, 1991.

Voss, C.I., SUTRA, A finite element simulation model for saturated-unsaturatedfluid-density-dependent ground-water flow with energy transport orchemically reactive single species solute transport, U.S. Geological SurveyWater Resources Investigations Report 84—4369, 409pp., 1984.

Woodward & Clyde Consultants, Contamination assessment report, DRT site,Florida, Technical report, FDER, Tallahassee, FL, 1990.

Zhang, G. and M. Koch, The Dupuit assumption and its reliability, Technicalreport, SCRI, FSU, Tallahassee, FL., 29 pp., 1991.

OuDull cone

Fig.l: Scheme of drawdown in a phreatic aquifer due to pumping. Notations: HQ,initial water-table; h, h^ Dupuit- elevations at distance r and at wellradius r ; h# elevation of seepage face; Qj> pumping rate.

Sulrn (Dup )

K-10 '* ni/s

20 40 GO 00 100Radius (in)

Fig. 2: Numerical (SUTRA (Dup)) versus analytical solution for the hydraulic headfor axisymmetric phreatic flow towards a well.



X (m)

FJg.3: Breakthrough curves after 5 and 15 months using the new Dupuit-SUTRAmodel and the original confined' SUTRA; (a) B=10 m; (b) B=50 m.

Fig.4: VOC concentrations at the DRT site for the 1986 (a) and the 1990 samplingevent (b). The rectangular box depicts the borders of the numerical modelused in the simulations (adapted from WCC, 1990).



CCMCENIfUllCN T ... 0 t

l Observed (dash lines) and modeled (solid lines) heads (in meters) at theDRT site; (b) initial concentration isolines (in ppb); (c) finalconcentration isolines after three years of simulation time.

(I)

Fig. 6: Remediation models for the DRT site, (a) heads (m) of the original 'confinedaquifer' SUTRA; (b) heads of the Dupuit-SUTRA model; (c) initialconcentration (ppb) prior to pumping; (d); (e) and (J), concentrations (ppb)after 0.5; 3 and 4 months, respectively. Solid lines: "confined aquifer'SUTRA model. Hatched lines: new Dupuit-SUTRA model. Contourintervals for head and concentration are 0.2 m and 500 ppb, respectively..


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