Correction to â•œAn efficient finite element ... - Wiley Online Library

WATER RESOURCESRESEARCH,VOL. 30, NO. 8, PAGES2485-2486, AUGUST 1994

Correction to "Anefficient finiteelement method formødeling multiphase flow" bYJ. J. Kaluarachchi andJ. C. Parker In the paper" An efficient finite element method for modeling multiphaseflow" by J. J. Kaluarachchiand J. C. Parker(Water ResourcesResearch, 25(1), 43-54, 1989), a finite element model for water and oil flow with gas at constantpressurewas presented. The governingequations wereexpressedin termsof water andoil pressureswith mass storage termsexpandedastheproductsof pressure-time and saturation-pressure derivatives. The split time derivative formulationused by Kaluarachchi and Parker [1989] has the form

Ot=•xi + ) OhwOho O[ [Ohw

(1)

Cww Ot, + +C

=•

Ko

+

O$oo

[/ oho+ )]

ckat=•xiKoo• ox i Prollj in which the saturation time derivatives

(5)

are treated as

implicit functionsof pressures.Details of the finite element formulation and the treatment of nonlinearitiesusing the

Newton-Raphson methodhave been reportedby Katyal et al. [1991], Katyal and Parker [1992], and Environmental Systemsand Technologies,Inc. [1992]asimplementedin the computer models MOFAT 2.0 and MOTRANS. Upon revisiting the results of Kaluarachchi and Parker [1989] with the mixed formulation, severe shortcomingsof the split time derivative formulation implemented in MOFAT

1.0 were noted which will be discussed below. We

will demonstratethe effectivenessand accuracy of the mixed formulation by simulatingexamples 1, 2, and 3 of Kaluarachchi and Parker [1989]. Readers should refer to Kaluarachchi and Parker [1989] for details on input parameters,

(2)

Cow Ot ooOt Oxi o'[ Ox i PrøUj

where Cpqisthepq phase fluidcapacity forp = o (oil)orw physicaldomain, and initial and boundaryconditions. (water); t istime;xi is the/-direction coordinate; Kpijisthe Kaluarachchi and Parker [1989] investigated various

p phase conductivity tensor(i, j = 1, 2) for phasep; h•, =

methodsof computingfluid capacitiesand obtainedthe best

Pp/Pw# is the waterheightequivalent headof p phase, massbalanceresultsusing a modified chord slope scheme. where P•,isthep phase pressure, Pwisthedensity ofwater, The results presentedhere for the split time derivative

andg is gravitationalacceleration;Prois the specificgravity formulationcorrespondto the modifiedchord slopescheme

ofoil; uj is the unit gravitational vectorwherez is the elevation;and

of KaIuarachchi and Parker [1989].

Example1. This probleminvolvesinfiltrationand redis-

C•q= •

OSe

........ Ohq

tribution of oil into a 100-cm vertical homogeneousand

(3) isotropicsoilcolumn.The watertableis 75 cm fromthetop surface. Oil was allowed into the column under a water

equivalent oilheadof 3.0cmuntil5 cm3 cm-2 ofoilhad

where qbis porosity, S•,is thep phasefluidsaturation, and infiltrated.Subsequently,oil was permittedto redistribute p,q=o,w.

Recentwork by Katyal et al. [1992]andKatyal andParker [1992]hasdemonstratedthat a mixedform of the multiphase flowequations,which avoidsexpandingthe time derivative term,givesresultsthat are sfiperiorto thosefrom the split timederivative form of the governingequations.Similar conclusions have beenreportedby Celia et al. [1990]and CeliaandBinning[1992]for unsaturated flow. Faustet al. [1989], Forsyth[1989],andSleepand Sykes[1993a,b] used acombination of fluidpressures andsaturations asindependentvariables and obtained good mass balance results. L. M. Abriola and K. Rathfelder (Mass balanceerrors in

for 100hourswith a zero-fluxconditionfor oil imposedon all boundaries.Lenhard and Parker [ 1987]obtainedthree phase

saturation-pressure constitutiverelationsby introducing fluiddependent scaling factors,•ao and/3ow, betweenair-oil andoil-waterphases,respectively. As reportedby Lenhard andParker[1987],ajumpconditionoccursduringtransition froma two-to a three-phase systemwhen1/13ao + 1/!3ow < 1. Thisexample involves testingthemodelfor case1Awith a no-jumpcondition (1/lgao+ 1//3o• = 1) andcaselB in whicha jumpcondition accompanies phasetransition. The resultspresented in Table1 demonstrate the superiority of the mixedform of theformulation,whichgivesmassbalance

modeling two-phase immiscible flows:Causes andremedies, errors of 0.10% and 0.25% for case 1A and case lB, submittedto Advances in Water Resources, 1993) demon-

respectively, compared to errorsof 3.4%and6.4%forthe

stratedthat mass conservationwith split time derivative splittimederivativeformulation.

formulations depends criticallyonthemethodof evaluating fluidcapacityterms.In the mixedformulation of Katyalet al. [1992]andKatyal andParker[1992]the flowequations Table 1. Mass BalanceError (Percent) arewritten for the off-water flow problem as

SplitTimeDerivative

c•Ot=OX---•. wiJ[•xi +UJ

(4)

Example 1A lB 2 3B

Copyright 1994by theAmerican Geophysical Union. Paper number94WR01000. 0043-1397/94/94 WR-01000502.00 2485

Formulation 3.40 6.40 5.00 6.00

Mixed

Formulation 0.10 0.25 0.00 0.25

2486

KALUARACHCHI AND PARKER: CORRECTION

withthemixedformulation is 0.25%compared to 6%with the splittime derivativeformulation. 50 days

The resultsindicatethat massbalanceerrorwiththe mixedformulation is consistently muchlowerthanwiththe splittime derivativeformulation.Especiallyundercondi.

tionsofhighNAPLmobility, mass balance errors inthesplit time derivativeformulationmay lead to highlyerroneous

fluidsaturation distributions. Themixedformulation isnot subjectto these difficulties. References

Celia, M. A., and P. Binning,A massconservative numerical solution for two-phase flowin porousmediawith application to unsaturated flow,WaterResour.Res.,28(10),2819--2828, 1992. Celia,M. A., E. T. Bouloutas,and R. L. Zabra,A general mass Figure 1.

Oil saturati01tcontoursat 50 and 100days during

redistribution

conservative numerical solution for the unsaturated flowequa-

tion, Water Resour. Res., 26(7), 1483-1496, 1990.

for 10 x 34.5 m domain.

Environmental Systemsand Technologies,Inc., MOTRANS:^ finite elementmodel for multiphaseorganic chemicalflowand

multispecies transport, technical anduser'sguide,Blacksburg, Va., 1992.

Example 2. This example was designedto test the model performance under conditions of severe nonlinearity and heterogeneityin material properties.The geometryand the

initialmadboundary conditions ofthisproblem arethesame as those of example 1, but the soil profile is assumedto be

Faust, C. R., J. H. Guswa, and J. W. Mercer, Simulationof three-dimensional flow of immisciblefluids within and belowthe unsaturated zone, WaterResour.Res., 25(12), 2449-2464,1989. Forsyth, P. A., Adaptive-implicitcriteria for two phaseflowwith

gravityandcapillarypressure,SIAM J. Sci. Stat. Cornput., 10; 227-252, 1989.

Composedof three layers. The mass balance error for both Kaluarachchi, J. J., and J. C. Parker, An efficient finite element methodfor modelingmultiphase flow, WaterResour.Res.,25(1), formulations is presented in Table 1. There was no mass 43-54, 1989.

balanceerror with the mixed formulationcomparedto 5% Katyal, A. K., and J. C. Parker, An adaptive solution domain error with the split time derivative formulation. algorithmfor solvingmultiphaseflow equations,Cornput.Geosci., I8(1), 1-9, 1992. Example3. The objective of this examplewas to evaluate the modelfor a field scaleproblemand analyzethe effect Katyal, A. K., J. J. Kaluarachchi, and J. C. Parker, Evaluationof methods for improving the efficiency and robustnessof mulof fluid densityand viscosityon nonaqueous phaseliquid tiphaseflow equations,in SubsurfaceContaminationby Immis(NAPL) flow. The domain is 23 x 10 m with a water table

cibleFluids, editedby K. U. Weyer, A. A. Balkema, Rotterdam,

Netherlands, 1992. gradientof 0.87 rrdm.CaseB (highd•nsityandlow viscosity) exhibitsthe mostseverenonlinearitydue to highoil mobil- Katyal, A. K., J. J. Kaluarachchi,and J. C. Parker, MOFAT: A two dimensional finiteelement program formultiphase flowandmulity. In the interest of brevity, only this case has been

selectedfor comparison. The splittimederivativeformulation Kaluarachchiand Parker [1989]resultedin physically incorrect results for case B. Their results show that the

densenonaqueousphase liquid (DNAPL) saturationeven

ticomponenttransport, program documentation,version2.0,Final Rep. Proj. CR-814320, R. S. Kerr Environ. Res. Lab., U.S. Environ. Prot. Agency, Ada, Okla., 1991.

Lenhard,R. J., and J. C. Parker, Measurementand prediction of saturation-pressure relationshipsin three-phaseporousmedia systems,J. Contarn. Hydrol., 1,407-424, 1987.

after200daysofredistribution was5%attheaquiferbottom Sleep,B. E., andJ. F. Sykes,Compositional simulation of groundand 20% 2 m abovethe lower boundary.Usingthe mixed water contaminationby organic compounds, 1, Model develop-

formulation, we found that the DNAPL

saturation at the

ment and verification, Water Resour. Res., 29(6), 1697-1708,

1993a. aquifer bottom after 50 days of redistributionwas more than simulation of ground25%. Figure 1 shows the simulated DNAPL distribution Sleep,B. E., andJ. F. Sykes,Compositional

usingthe mixedformulationat 50 and 100daysof redistribution.DNAPL reachedthe rightboundaryafter about150

water contaminationby organiccompounds,2, Model application, Water Resour.Res., 29(6), 1709-1718, 1993b.

daysof theredistribution. Therefore we increased thephys-

J. J. Kaluarachchi,Utah Water ResearchLaboratory, UtahState University, Logan, UT 84322-8200. Inc., 34.5 m. All other modelinputswere the sameas described J. C. Parker, EnvironmentalSystemsand Technologies, 2701RambleRoad, Suite2, Blacksburg,VA 24062-0457.

ical dimension of the domain in the x direction from 23 to

by Kaluarachchiand Parker [1989].Table 1 showsthe mass balance error for both formulations. The massbalance error

(ReceivedFebruary22, 1994.)