Application of Energy Concepts to Groundwater Flow: Time Step

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WATER RESOURCES RESEARCH, VOL. 27, NO. 12, PAGES 3225-3235,DECEMBER 1991

Applicationof EnergyConceptsto GroundwaterFlow' Time Step Controland IntegratedSensitivityAnalysis BRYAN W. KARNEY AND ASITHA SENEVIRATNE •

Departmentof CivilEngineering, University of Toronto,Toronto,Ontario,Canada

Whenfluidpasses intoor outof an aquifer,workis doneat theboundaries whichis usedpartlyto changetheinternalenergyof thesystemandpartlyto overcome resistance to flow.For a saturated medium,thechangein internalenergyis furtherpartitioned intotwo terms,the strainenergystored in the elastic soil matrix and the strain energy storedin the pore water due to compression.A

technique is developed in thispaperwhichinterprets thedynamicbehavior of an aquiferin termsof its energytransformations. The centralfeatureof thisapproach is the quantification of the physical processes intoindividual energyandworkparameters whichtogether characterize theresponse of an entireaquiferto a givensetof excitations. By thismeans,therateof change of strainenergyis shown to be a naturalindexof the unsteadiness of a system,an insightwhichleadsto an adaptivealgorithm

foradjusting thetimestepofa transient groundwater flowmodel.Further,theenergyapproach isused to assessdifferentialcompaction in a heterogeneous aquifer,therebyprovidinga basisfor efficient computationandfor rationalacquisitionof compressibility data.

THE ENERGY METHOD

INTRODUCTION

In the followingsections,a generalapproachis developed Groundwater-relatedproblemsoften requireconsiderable effort for both analysis and collection of data. Although from first principlesand then by mathematicalmanipulation governing equations. Toplace'these develprogresshas been made, there is need for further under- ofthetraditional opments in perspective, a brief review of energy principles in standingof how a porous medium respondsto various

applications is presented. development schemes as well as for computationally effi- gi'oundwater cient modelingtechniques.This paper continuesthe sea.r•ch

for better analysistechniquesby describingtransientflow Historical Background conditionsin a porousmediumin termsof compositeenergy

In his pioneeringarticle, "The Theory of Groundwater functions.The underlyingconceptis simple:Whenequilibrium flow Conditionsare disturbed,changesto a system's Motion," Hubbert [1940] applied energy principles to the energyare partitionedamongseveralcomponents, the rela- problemof describinggroundwatermotion. Hubbert introtive magnitudeof which indicatesthe dominanceof the ducedthe "fluid potential" as the work requiredto displace a unit mass of fluid from a reference state to a new state. The

differenteventsin a system.Unlike the traditional approach of focusingattentionon conditionsat a finite set of points, fluidpotentialß for a unit controJmassof the fluid was then the energy method provides an integrated view of the expressedas transient responseof the entire system.

One significant advantageof the energymethodis the ease

pdV+ pV+ -•fvVO v2

dO = 9'z- poVo +

(1)

with which the state of a system can be assessed.As subsequent developmentsshow,the rate of changeof strain energyis an excellentindex of the unsteadiness of a system. in which # is the accelerationdue to gravity, z is the The basicidea is that systemsundergoing rapid' changes elevationabovethe referencestate,p is the fluid pressure,v require smallertime incrementsthan systemsnear steady is the fluid velocity, V is the specificfluid volume, and the

state.Thenewapproach adjusts thetimestepbasedonthe subscript0 indicatesconditionsat the referencestate. Col9'zisthegravitational potential energy andv2/2is ratio of the rate of energy dissipationin a region to the rate lectively,

of changeof strain energy. This nondimensional procedure the kinetic energyper unit mass.Two distinctwork compoto theflow:thef[o pdV termis associated permitsdynamicadjustmentof the time stepin any domain nentscontribute with the compressibility of•the porous medium and the and can considerablyimprove execution times.

In thispreliminary paper,theenergyconcepts aredevel- so-called "flOw work .... is representedby p V - P0 V0.

opedfrom first principlesand appliedto a one-dimensional When integrated by parts, the definite integral in (1) becomes flow situationonly. However, direct extensionof the energy method to higher dimensionsis possibleand it is.in these more complexproblemsthat the energy approachhas the p dV=poVo-pV+ Vdp. greatestpotential as an interpretive tool. o

lNow at JacquesWhitfordEnvironment Limited,Fredricton, This relation transforms (1) into the total hydraulic energy potentialfor a barotropicfluid:

New Brunswick, Canada.

Copyright1991by the AmericanGeophysicalUnion.

dO= 9'z +

Paper number 91WR01909. 0043-1397/91/91WR-01909505.00

V dp + •.

2

0

3225

(2)

3226

KARNEYAND SENEVIRATNE' APPLICATION OF ENERGYCONCEPTS TO GROUNDWATER FLOW

Several references[e.g., Freeze and Cherry, 1979;Bear, Hence, rr' = hp9 + C• in which C• is a constant.By 1979]use (2) to motivateBernoulli'sequationand Darcy's selectinga datum appropriately,C• can be made zero. law. Yet, from an energypointof view, thissecondequation Substitutingfor • = hp9 in (5) gives

obscures theroleof compressibility. Partof thef•o V dp term is involvedwith flow processes that are presentwheneverflow occurs,evenif the flowis steady;the otherpartis associatedwith compressibility effectsthat are mostsignificant under transient conditions.

The physicalinsightsthat are possibleusingenergyconcepts in porous media flows have advanced little since

0Esoil -- -« o•VoO[(hp9)2].

(7)

The strainenergydensityof the soil matrix •soilmay be definedas E soil/Vb. With this substitution,the time rate of changeof the strainenergydensityof the soil becomes

soil 1

O[p2•72h 2]

Hubbert'swork.Thisis unfortunate sinceenergyarguments • =- . (8) ot 2 ot are not only usefulin derivingand motivatingrelationships, they are alsoan aid in interpretingmanykindsof phenomAs a practicalaside,it shouldbe notedthat the magnitude ena. In what follows,the physicalinsightsare developedin of Esoildependson the selectionof the piezometrichead parallel with what appearsto be a rather arbitrary set of datum. However, since only changesin strain energy are mathematicalmanipulations.This parallel derivationis essential.The physicalapproachdevelopsinsight,while the mathematical derivations show the connection between the

usuallyof interest,the piezometricdrawdown(h - h0) can replaceh in the computationof the variousenergyexpressions.As a result, Esoilis initially zero and any changeis

traditionalgoverningequationsand the integratedenergy about this initial condition. This substitution of drawdowns expressions.In this way, links are forged not onIy to for piezometricheads Simplifiesthe interpretationof the Hubbert's original work but also to the mathematical litera- energyequationin the sameway that energyexpressions in ture. In fact, the expressions that follow can be interpreted mechanicalsystemsare simplifiedby measuringdisplaceas a special case of Gurtin's [1964] energy functional. ments withrespect tøequilibrium. Gurtin's approachwas originallyderivedfor the heat conductionproblemand is difficultto apply to complexdomains.A more direct approachto the energyequationfor Strain Energy Density of Water flow in porousmediais presentedhere. If the saturatedsoil volume Vo has a porosity n, the volume of pore water is Vw = n Vo. The coefficientof

compressibility/3 of Watermaybedefined as,

Strain Energy Density of a Saturated Elastic Porous

Medium

When a fluid flows througha saturatedelasticsoil matrix, the net work done at the boundaryof a control volume is

1 OVw --. Vw Op

/3 =

(9)

partitionedamongthree events:the elasticcompression of Following a derivation parallel to that shown for the soil the soil matrix, the elasticcompressionof the fluid and the energyterm, the rate of changeof the strain energydensity energydissipationdue to friction. Expressionsfor the first in the water can be expressedas two of theseterms are derivedfrom first principlesin this section.Note, however, that other energyterms also must 0•wate r 1 O[p2•72h 2] = - n/3 . (10) be accounted for in morecomplexmedia.For example,in an Ot 2 Ot unsaturatedporous medium, energy terms must accountfor

surfacetensioneffectsand potentialenergydue to gravity. Considera saturatedsoil volumeV0, initiallyin equilib-

ENERGY RELATIONS: AN OVERVIEW

rium, compressed by an infinitesimal volumechangeOVo The laws of thermodynamicsstipulatethat the equilibrium underaneffective normalstressrr'(kN/m2).Thechange in state of a system is altered only if heat and/or work are strainenergyof the soilmatrix0Esoilis equalto the work of

compression•r'OVo;that is, 0Esoil= rr' 0 V O.

(3)

The constantcoefficientof compressibility(a) of the soil under undrainedconditionsmay be definedas 10Vo - =

V O Orr'

(4)

exchangedwith the system's environment. In addition, the laws imply that a different equilibrium state is associated with. each value of a system's energy; thus, if no heat or work are doneon the system,the equilibriumstatecannotbe changed.When written for mechanicalenergyin a saturated porousmedia, conservationof energy requiresthat the net

work doneon a systemdW be partitionedbetweenchanges in internalenergydE and frictionaldissipation(conversion to heat) in the system. Thus,

EliminatingOVo from (3) and (4) gives

OEsoil = -21C•VbO[(o")2].

dW= dE + dD

(11)

(5) in which dD is the mechanicalenergydissipatedto over-

comefriction. Rewriting(11) as a rate relationand recognizFor a constantoverburdenstress,the changein pore ing that water pressureOpis equal to the negativeof the changein dE srsoil •'water 1 O[p2g2h2] effectivestressOrr'.Sincep = (h - z) pt/, the changeof

d-•--dt •

effective stress can also be written as

a•r'= -a[(h + z)p9].

(6)

produces

dt

=-[a + n/3] 2

Ot

(12)

KARNEY AND $ENEVIRATNE: APPLICATION OF ENERGY CONCEPTSTO GROUNDWATER FLOW

V

S

3227

h(z,0)= 10m

I

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I

!

i

i

!

!

i

!

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!

I

!

i

SAND

K

= 0.864

a

= 1.728(10 -s) kPa-•

i

i

i

i

i

i

I

!

i

(m/day) i

i

i

!

!

i

L

i

=

!

300 m

Fig. 1. Aquifer bounded by two constant head reservoirs: flow case 1.

dW

•'soil •water dD

•-

dt

dt

+ •+

--.

dt

(13)

dt

Equation (13) is a genetic equation for the energy in a saturated soil volume. The next section provides an alternative interpretation of the energy equation which shows its

relation to the usual equationsgoverningflow in a porous medium. In particular, the energy equation is derived by

manipulating theusualcontinuity equation andDarcy'slaw in a one-dimensional

domain.

292 dx =LPgKx Vx 2dx+1fx: =L[a+nfi]p 0[h 2]

=0

2

=0

Ot =L

= -a[pvxh]x x o'

(19)

This equation summarizes the various energy transformations taking place in a saturated, one-dimensional, confined aquifer of constantthickness;in the SI system of units, each term in the equation has dimensions of joules per second or

watts.Thus,(19)corresponds to thegeneral energyequation (13). The first term in (19) represents the frictional dissipation. The second term represents the change in internal The energy equation is now developedfor one-dimension- energy due to compression of both the soil matrix and pore al flow in a saturated heterogeneous confined aquifer of water (compare with (8) and (10)). Finally, the right-hand constant thickness. The equation of continuity for a control side of (19) represents the net work at the domain boundvolume of length /Sx,unit width, and constantthickness/3 aries. Clearly, the sum of the energy transformations within can be written as [Bear, 1979], a control volume is equal to the net work at the boundaries. O(pvx) O(np) This observation is used subsequently to characterize flow •+•=0. (14) conditions in a porous medium. Ox Ot THE ONE-DIMENSIONAL ENERGY EQUATION

Note that (19) allows a natural classification of flow regimes' Oh 1. When the flow is steady, the rate of change of internal vx + Kx -- = 0. (15) energy of the system is zero; there is equilibrium between Ox the flow work and mechanical energy dissipation. where h is the piezometrichead, Vx the Darcy velocity, n is 2. If the rate of dissipation is dominant compared to the the averageporosity, Kx is the hydraulicconductivity,p is rate of change of internal energy, then the compressibility the fluid density and x and t are the spaceand time variables effectsof the systemare insignificant.The changesin such a respectivelY. system are "quasi-steady" and allow larger time steps to be Following a procedure similar to Karney [1990], (14) can used by the flow model, thereby improving the computabe multipliedby h and (15) by pv•/K• and the two equations tional efficiency. summed to produce 3. If the internal energy term is dominant compared to the

Further, Darcy's law may be expressedas,

O(pVx) h •+

Oh pv x

Ox

+

vx2

•xx P•x

O(rlp)

+ h •=

Ot

0.

(16)

This equation can be multiplied by g dx and integrated with respect to x over a length L. The result is 2

g[pvxhlj=• +flcLPg•xx •d xdx =0

+

:=LO(np) gh

=0

dx = 0.

(17)

ot

PARTITIONING OF ENERGY IN AQUIFERS

In this section, the energy approach is illustrated by considering the transient behavior of several aquifers subjected to time-dependentboundary conditions. Initially, the solution to two simple flow problems is discussedand the energy approachis contrastedto the more traditional piezotime-dependentboundary condition is considered.

Figure 1 shows a one-dimensionalsaturated confined

Oh

Ot - P2g[a +nfi]Ot Using this relation in (17) produces

significantand the system's response should be treated as fully transient.

metric head summaries.Following this•' a more-complex

In addition, Bear [1979] shows that

O(np)

dissipation, thenthecompressibility effects ofthesystem are

(18)

aquifer bounded by two constant head reservoirs. The equation which describesthe transient flow in this domain is usually written as

3228

KARNEY AND SENEVIRATNE'

APPLICATION

OF ENERGY CONCEPTS TO GROUNDWATER

h(z, 0) = 10rn

11 .o/•_ 10.O •__

4rn

STEADYSTATE B

s.o

SAND

= 0.864

a

= 1.728(10 -5) kPa-•

ooooo ANALYTICAl. SOLUTION ATDO M ANALYTICAl. SOLUTION AT 1DO M FINIT( (L(U(NT SOLUTION

••

V

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

K

-•



t

t

t

t

t

t

(m/day)

L =

[



FLOW •E

--

1

6.0 • .... FLOW •E - 2

300 rn

5.0• Fig. 2.

FLOW

I

I

1 O0

200

Aquifer bounded by constant head and no-flow bound-

3OO

TIME (DAYS)

aries: flow case 2.

Fig. 3.

Effect of different boundary conditions on aquifer response.

Oh



Ox

= s --

(2o)

ot

=

2••r=L =0

[h(x, O) - hs(x)] sin

an •

n7rx L dx.

(23)

in which T is the transmissivity and S is the storage coefficientof the aquifer. In this example, the aquifer material is assumed to be homogeneoussand with a hydraulic

The piezometric head distribution calculated from the finite element solution is compared with the sum of the first 150 terms of the Fourier series solution in Figure 3. Very good conductivity of 10-5 m/s(0.864m/day)anda compressibilityagreement is observed. of1.728 x 10-5kPa-• lithecompressibility ofwater is7.7x The generalizedFourier series solutionfor flow case 2 is

10-7kPa-•. Theresponse of thisbasicsystem isconsidered[Haberman,

for two different

excitations:

1. For flow case 1, the water level in both reservoirs and the piezometric head distribution in the aquifer are both initially set at 10 m. At the instant the simulation begins, the water level in reservoir B suddenly drops by 4 m. Thus, the initial and boundary conditions are respectively, h(0, t) = 10.0 m, h(L, t) = 6.0 m and h(x, 0) = 10.0 m. 2. For flow case 2, the same aquifer is bounded by only one constanthead boundary and a no-flow boundary (Figure 2). The initial and boundary conditions are respectively, Oh(O, t)/Ox = 0, h(L, t) = 6.0 m and h(x, 0) = 10.0 m. In both cases, the transient drawdown in the aquifer is monitored at distances 50 m and 150 m from the origin O. Once the flow equation is solved, the computedpiezometric head h and the Darcy velocity v can be substituted in the energy equation to obtain the magnitudes of the energy components.

1987],

h(x, t) = hs(x) +

E

n=1

a n exp -

(2n-1)rr.Kt 2L

(2n - 1)•rx ßsin

2L

.

(24)

In this case, hs(x) = h(L, t) while a n is given by

=

2fx• =0 =œ

[h(0, t)-

an •

hs(x)] sin

(2n -2L1)rrx dx.

(25)

The analytical solution generatedby summing 150 terms of the Fourier series is compared with the finite element solution in Figure 3. Again there is good agreementbetween the two solutions.

VERIFICATION

OF THE FLOW MODEL

Becauseof their simplicity, the head distribution obtained using a Galerkin's finite element program [e.g., Huyakorn and Pinder, 1983; Wang and Anderson, 1982; Pinder and Gray, 1977] can be compared with a closedform solution for the first two flow cases.

For flow case 1, the generalized Fourier series solution is [Haberman, 1987]

The general energy equation (19) was also verified using the finite element code. This was done by comparingthe sum of the individual energy terms on the left-hand side of the energy equation with the computed work term on the right side at each time step. It was found that left and right sides of (19) agreedto 3 decimal placesfor both flow casestested. With these preliminaries completed, the finite element code can be used to illustrate the energy approach. ENERGY

h(x,t) = hs(x )+ E anexp -

Kt sin

n=l

L (21)

in which g = T/S, hs(x) is the steadystate solutionand a n are coefficients.Specifically,h s(x) is given by hs(x) = h(L, t) + while a n is obtainedfrom

[h(0, t)-

h(L, t)]x L

(22)

INTERPRETATION

The energy response of the aquifer in flow case 1 is depicted in Figures 4-6. Figure 4 depicts the piezometric head distribution at specifiedlocations while Figures 5 and 6 show the relative magnitudesof the different energy components. The rapid drawdown in head at one boundary results in large piezometric head gradients, thereby inducing high velocities and strain energies within the system. Figure 5 shows that the transient response of the system gradually diminishesand approachessteady state around 100 days. At this time, the rate of change of strain energy of the soil

KARNEYAND SENEVIRATNE:APPLICATIONOF ENERGYCONCEPTS TO GROUNDWATERFLOW

3229

0.5

RATE OF CHANGE OF STRAIN ENERGY (SOIL) RATE

10.0 X

60

X--

•., 8.0

X

I 25

6.0

--

I 50

M

150

"240

OF

DISSIPATION

F -- FIXED POINT A'I'rRACTOR



M

M

I 75

1 O0

0.0

0.0

TIMe (DAYS)

O. 1

0.2

0.5

0.4

RATE OF WORK DONE AT THe BOUNDARY

Fig. 6. Phaseplot of energypartitioningfor flow case 1.

Fig. 4. Piezometrichead distributionfor flow case 1.

matrix tendsto zero implyingthat compressibilityeffectsof the porousmatrix are not governingthe flow. Hence, all of

ena are integratedover any part of the domainin a way that enablesa comparisonof individualphysicaleffects.Another the work at the boundaries is used to overcome frictional applicationof this insightis when assessing the importance dissipation. of compressibility effectsin an elasticaquifer.If work at the Figure6 is a phaseplot which representsthe sameenergy control surface is used primarily to overcome frictional transformations for flow case 1. On the phase plot, the dissipation, the strainenergyeffectsof the elasticmatrixare responseof the whole systemat a particulartime is repre- insignificant. This impliesthat the compressibility effectsare sentedby a point. When the systemchangesthe pointtraces unimportantand that a more efficient"quasi-steadystate" a pathonthe phaseplot. Figure6 revealsthat a largefraction model can be used with little loss of accuracy. of the work at the boundaries is transformed into frictional

dissipationand showsthat, with time, the systemconverges TIME-DEPENDENT BOUNDARY CONDITIONS to the new steadystate(point F). At steadystate,the change In this section,energytransformationsin an aquiferdue to in strain energy is zero and the work done on the porous volume is entirely used to overcomefriction. Point F is a time-dependentboundarycondition are analyzed. The aquifer depicted in Figure 1 is excited by the variable called a fixed point attractor. Note that in Figure 3 the aquifer with the two constant boundaryconditionshownin the upperportionof Figure 7. head boundariesexhibits a faster responsethan the aquifer The resultingpiezometric head distributionsat specified with one no-flow boundary. In both cases,there is a rapid locationsare shown in the lower portion of Figure 7. The drawdown in head at one boundary and work interactions systemreachesequilibriumaround 100 days and remains graduallybringthe systemto a steadystate.However,in the steady until 200 days. On the day 200, a "flood wave" first case,both headboundariesbringthe systeminto steady increasesthe boundary head at reservoir B to 8 m and state,whilein the secondcasethe work doneat th.eno-flow remains constantfor 20 days. The suddenincreasein the boundaryis zero. Hence, only one boundaryinfluencesthe boundaryheadrechargesthe depleted(compressed)aquifer changein the systemand steadystatewill be reachedat a after which the head drops to 2 m again. slower rate than in the first case.

The energyterms spatiallyintegratethe responseof any specifiedsubdomain in an aquifer.Conventionalsummaries,

TIME (DAYS)

suchas thoseshownin Figure 4, depictthe piezometrichead variationat specificlocationsand do not reveal the response of the aquiferas a whole. Such singlepoint assessments can be misleadingbecausethe rate of changeof flow parameters varies from point to point. For example, in a highly compressiblemediumsuchas clay, steadystateconditionsmay

o

lO

prevailat somepointsin the domain,while othersmay still

....

I

be in a transientstate. In the energyapproach,flow phenom-

I

190

210

T•ME

0.2

RATE Of CHANGE Of STRAIN ENERGY (SOIL)

9.0

RATE OF' DISSIPATION

X" 60 M

RATE OF WORK DONE AT THE BOUNDARY 0.0

I

25

5.0

50

75

T•ME (DAYS)

Fig. 5. Energypartitioning for flowcase1.

1 oo

0

X

,-

150

M

X

,-

240

M

I 1 O0



I 200

.,'100

TIME (DA•)

Fig. 7. Headdistribution andthetime-dependent boundary.

3230

KARNEYAND SENEVIRATNE:APPLICATIONOF ENERGYCONCEPTS TO GROUNDWATERFLOW TIME (DAYS) 0

stepin a transientalgorithm.This procedureis describedin

10

20

30

40

i

I

I

I

this section.

Many different adaptive strategieshave been used in both groundwaterand petroleumengineeringapplications.Automatic time selectorproceduresfor oil reservoirmodeling were developed by Jensen [1980] and Mehra et al. [1982]. $ammon and Rubin [1983] developed an error-driven time step selectionschemebased on techniquesdevelopedby Lindberg[1977]. In this paper, the time stepis adjustedby assessing the dynamicenergyresponseof the system. When an aquifer is excited, either by changingboundary conditionsor by externalforcingfunctionssuchas groundwater pumping, the system undergoestransient conditions before reachinga new steady state. Such unsteadyconditionsare reflectedby the rate of changeof the strainenergy of a porous medium. As the systemapproachesthe new steadystate,changesin the strainenergygraduallydiminish. The changingstrain energy can be describedby the initial

2.0

0.0

1.2

value problem, O.8



dE



dt

0.4

= e(t)

(26)

I\ wheree(t) is the rate of changeof strainenergyat time t and E(t) is the strainenergystoredin the porousmatrix at t. The initialconditionE(t = O) = Eo is usuallyassumedto be zero sinceonly changesin the strainenergyare important. To developan adaptivemethodof choosingthe time step, let the minimumtime incrementusedduringperiodsof rapid

-0.8

-1.21

excitationin a time-stepping algorithmbe &base'This value is the time stepwhichis adoptedby defaultif adaptivetime

STRAIN ENERGY CHANGE DISSIPATION WORK AT THE BOUNDARY

I 22O

control is not used. However, under less severe transient 25O

TIME (DAYS)

Fig. 8. Energypartitioningfor the time-dependent boundary.

conditions, thetimestepcanbeincreased to•itadap t > •itbase. Using a first-orderapproximationof Taylor's seriesto express (26) in discrete form gives

E(t + /itbase) -- E(t) = e(t)litbase.

(27)

If the time is increased to •itadapt, an analogous relation

The energyrelationsfor this caseare shownin Figure8. applies' Figure8a showsthe energypartitioningfor the first50 days of simulation.The threepeaksdepictthe increasein the rate E(t + •itadapt) -- E(t) = e(t)•itadap t. (28) of strainenergythat occurswhenthe boundaryheaddrops (27)from(28)andsolving for •itadap t gives duringthe first periodof excitation(1-20 days).The system Subtracting reachessteadystatearound50 daysandremainssteadyuntil E(t + /itadapt) -- E(t + /itbase) 200 days.

The rechargewhich occursin the systembetween200 and 220 days "expands" the compressedaquifer, reversesthe sign of the rate of changeof strain energy, and causesa discontinuityin the energyplot shownin Figure 8b. The nonequilibriumconditions due to the rapid drop in the reservoir B create a hydraulic head gradienttoward reser-

/itadapt =

e(t)

+ /itbase.(29 )

If the rate of changeof strain energy e(t) is known, (29)

allows•itadap t to be chosenfor a specified discrepancy in internalenergyvaluebetweenthe two predictions.In effect, the rate of changeof internal energye(t) becomesan index of how quickly the systemis respondingwhile the deviation

voir B whichdischarges waterfromtheaquifer,compressing IE(t + •itadap t) -- E(t + •itbase ) indicates howmuchthe the system.The rate of changeof strainenergychangessign energy state of the system is permitted to change. The againas shownin the discontinuityin Figure8. Beyond220 relationis logical sincethe internal energyof a systeme(t) days, the reservoir head remains constant at 2 m and the only changesunder transientconditions;as steady state is systemgraduallyapproachessteadystate. ADAPTIVE TIME STEP CONTROL

approached,the permissibletime stepincreasesas the value of e(t) decreases.The magnitudeof the deviation determines how importanttransienteffectsare to a given systemand,

hence,how large a time stepcan be usedfor a givenrate of The examplesillustratehow the stateof an aquifercan be interpretedwith the energyapproach.Oneapplicationof the energymethod is immediatelysuggestedby this insight:to use the energymethodfor controllingor adjustingthe time

changee(t). Note that the selection of an appropriate value of the

deviationin (29) dependsnot only on the dynamicsof the porousmedium, but also on its size and dimensions.Thus, it

KARNEY AND SENEVIRATNE: APPLICATION OF ENERGY CONCEPTS TO GROUNDWATER FLOW

10.0

3231

120

TOLER

(DIM. LESS)

RUN

(SECS)

1.0 -

-

-

-



TIME

0.5 0.1 0.01

NON



56

90

ADAPTIVE

ADAPTIVE

RUN

RUN

TIME

== 40

$EC$

TIMES

45 71 102 z

30 0.0

I

I

0.2

0.4

6.0

I

I

0.6

0.8

1.0

TOLERANCE

Fig. 10. Run times for different degrees of tolerance. NO ADAPTIVE CONTROL

40

$ECS

depicts the piezometric head response at a hypothetical observation well 150 m from the origin, for different values TIME (DAYS) of TALER. It is observed that the accuracy of the adaptive solution improves when the tolerance is tightened. When Fig. 9. Accuracy of the solution for different levels of tolerance: TALER = 0.01, the resulting piezometric head response is flow case 1. very close to the solution without adaptive control. Figure 10 shows the execution times when the system is simulated using different values of TALER on a 386 PC appears to be difficult to select an optimal value for the (16-MHz clock and 30387 math coprocessor). The execution deviation without a prior knowledge of the dynamics of the time includes both the computation and the reading of the system. Fortunately, however, it is simple to scale the input data for a simulation time of 300 days. For the case deviation to produce a nondimensional tolerance TALER. with no adaptive control (base solution), the computational This is accomplishedby dividing the deviation by the time- time was 40 s. The "best fit" adaptive solutiontook longer to averaged dissipation duringtheprevious timestep&tt,.That run (102 s). The reason is obvious: The integration associis,theproduct&t, f (v2/K)dxreplaces thedeviation in(29) ated with the energy principle added a computationalburden to produce to the program that was not made up for by the longer time step. However, if energy summariesare computed for both v2 runs, or if longer periods of simulation are used, the run time for the adaptive procedure is often reduced compared to the nonadaptive solution. Flow case 1 is simulated for different time horizons using It is the physics of the flow problem that justifies this choice of scalingfactor. Under highly transient conditions, a both the base solutions and an adaptive solution with a of 10-2. Figure11depictstheruntimesin seconds large fraction of the work interactionsis used to changethe tolerance internal energy of the system. In this case, the dissipationis versusthe simulationperiod in years. When the time horizon relatively low, the change in internal energy high and the increases, the gain in computation efficiency also increases resulting time step small. However, as the system ap- when adaptive procedures are used, and it is in these proaches equilibrium the importance of the strain energy problems that the adaptive time-stepping procedures beterm diminishesand the work is used primarily to overcome come more relevant to improve the run times. frictional dissipation.The result is a high ratio of the rate of COMPRESSIBILITY EFFECTS IN HETEROGENEOUS dissipationto the changein internal energy and a large time MEDIA step. (In practice, it is required to limit the maximum time step to some multiple of the base step.) Note that the ratio of When a heterogeneousaquifer is excited by an external term in (30) is well-behaved in that the internal energy force such as pumping, different regions of the aquifer may cannot change unless there is flow and, thus, dissipation. exhibit different degrees of compression. The strain energy The only way both terms can be zero is if there is no flow of a porous soil is a function of both the pressure head and 4.0

I

0

100

I

200

t•tadap t--t•tbase q-t•tp TALER' e(t) '

300

(30)

(trivial solution). In summary, the ratio dissipation/rate of strain energy change is a natural index of the "state of the system." The

adaptive timestept•tadap t is evaluated by thesystemat each time level based on (30). For a specified tolerance, the adaptive time step is inversely related to the rate of change of the strain energy. When the system approaches steady state, the adaptive algorithm computes larger time steps becauseof the diminishingmagnitudeof the rate of changeof strain energy. The piezometric head at any point is not to be used as a controlling parameter, since it represents only a specificpoint and does not represent the entire system. To illustrate the use of adaptive time control, consider again the flow situation described in Figure 7. Figure 9

360

--

__BASE RUN TIME

2ao 200

120

4O

'" •,•• '" 2

I

4

I

6

TIME (YEARS)

Fig. 11. Computational times with and without adaptive control.

3232

KARNEY

AND SENEVIRATNE:

V

APPLICATION

OF ENERGY CONCEPTS TO GROUNDWATER

FLOW

h(a:,0) = 10 m

s

S•-•TE•DY••[-[4m V illiilllilllllllllllllllllllllll

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

L =

i

i

i

i

i

I

i

i

i

i

i

i

i

i

i

i

300 m

I - SILT

2-

SAND

0.864 (m/day)

1.728 (m/day)

1.000(10 -4) kPa-1

1.728(10 -5) kPa-1

Fig. 12. Aquifer with heterogeneousmaterial: flow case 1.

the compressibility of that medium, and can be used to assessthe degree of compaction of the different materials in the aquifer. The application of the strain energy concept is illustrated through the following examples. Flow case 1 is analyzed under two different soil conditions. In the first instance, the aquifer material is homogeneous sand while in the second the aquifer material consists of sand and silt as shown in Figure 12. The first 150-m region contains silt and the second region contains sand. Silt is assumed to have a hydraulic

Flow case 2 is also analyzed under identical soil conditions as in Figure 12 with the exception that the constant head reservoir A is replaced with an impermeable boundary. This means that the final steady state in flow case 2 is a 4-m head drop everywhere. The piezometric head distribution at a well 180 m from the origin is plotted in Figure 15. Figure 16 showsthat the final strain energy in both soil layers becomes equal, indicating no long-term differential compaction between the regions 1 and 2 in the homogeneous case. For the heterogeneous case, the sand showed a higher conductivity of 0.864m/dayanda compressibility of 10-4 degree of compaction during the initial periods while silt kPa-• andtheproperties of sandarethesameaspreviously compacted more after 200 days. The observed strain enerreported. Figure 13 depicts the piezometric head variation at gies are higher than in flow case 1 due to its larger drawthe observation well located at 180 m from the origin O. down. Further, silt shows a higher degree of compressionat The strain energies of the different soil layers are plotted steady state. When Figures 14 and 16 are compared, it is against time in Figure 14. For the homogeneous case, seen that under homogeneoussoil conditions at steady state, portion 2 exhibits a much higher strain energy storage (=5 flow case 2 results in no differential compaction between kJ) compared to portion 1 (• 1 kJ). This is expected since zones 1 and 2 while flow case 1 shows a substantial differportion 2 is close to the falling head boundary. In the ential compaction. However, under heterogeneous condiheterogeneouscase, sand shows a higher energy up to about tions the reverse occurs and flow case 1 has a smaller 260 days. Hence, the sand layer compactsmore than the silt compaction at steady state. layer during this period even though the silt has a higher Such an analysis helps to determine the areas which may compressibility. However, beyond 300 days, the silt is be severely affected by the falling water level of a river or compressed more than the sand. The homogeneous case due to pumping of groundwater. Further, it indicates the reaches steady state in close to 100 days while flow condi- importance of compressibility effects in a flow domain and tions are not steady in the heterogeneous case even at 500 helps to determine whether the system requires steady state days. Beyond 100 days, compressibility effects in the homo- or transient simulation. This feature may be particularly geneous aquifer are insignificant, and the system may be important in the transient analysis of heterogeneousaquifers modeled by using a steady state model. However, for the since such systemsrequire large quantities of both hydraulic period up to 100 days the system needs full transient conductivity and compressibility data. The present trend in modeling. groundwater modeling is to extrapolate localized data ob-

•ø1\

.OMOGEaEOUS SOIL

•.I\\• /

8

SII.•

HETEROGENEOUS SOIl-

HOMOGENEOUS SOIL (2)

HOMOGENEOUS SOIL (1)-- .......

^T - ,,,oI

I

200

•500

4OO

500

TIME (DAYS)

Fig. 13. Comparisonof drawdown for the different soil conditions: flow case 1.

O0

I 200

I 300

400

500

TIME (DAYS)

Fig. 14. Comparison of strain energy storage in different soil layers: flow case 1.

KARNEY AND SENEVIRATNE: APPLICATION OF ENERGY CONCEPTS TO GROUNDWATER FLOW

3233

i=M HOMOGENEOUS HETEROGENEOUS

SOIL SOIL

E=Z (otiTB) •••ic=Li =0 Oh Ot h --

dx.

(33)

i=1

Differentiating(32) with respect to the dimensionlessquan-

tity aj 7B gives, WELL

AT

x



180

,o'o

0O(ajTB) Oh ]

M

"

200

,&

300

d(ajTB) - (øtit dEi 2)f•r =0 =Lih

,o

TIME (DAYS)

Fig. 15. Comparisonof drawdownfor the differentsoil conditions'

ot

flow case 2.

oh

oh

+•

.

dx

Ot O(aj'yB) tainedthroughpumpingtestsusingstandardtechniquessuch as kriging. However, if the aquifer region of interest is insensitive to the detailed heterogeneities of the flow domain, more cost-effectiveacquisitionof data may be possible. This conceptis explored more completelyin the following section. INTEGRATED

SENSITIVITY

+-[•j•/B] fX:--L, B T0O[aiTB] = o h--dx Oh Ot where

O[otiYB] •= 15ij= I

ANALYSIS

OF POROUS MEDIA

Heterogeneitiesof the subsurfacehave always been difficult to model. However, when simulating such aquifers, the heterogeneitieshave been treated deterministically or stochastically depending on the extent of knowledge of the

subsurfaceproperties.In the absenceof detaileddata, it is better to understandthe sensitivity of the given systemto the unknown parameters. The traditional methods such as the influence coefficient method, the variational method and the method of sensitivityequations[Yeh, 1987]all computethe sensitivity of the aquifer at specifiedpoints on the flow domain. Such analyses are an inevitable starting point; hOWever,the interpretationof theseresultsfor efficientdata acquisitionis both tedious and indirect. The approach developed in this paper indicatesthe sensitivityof an entire aquifer regionor a subregionto a given parameterin contrast to the pointwise analysisof the traditional approaches.The techniqueis developednext from the basic expressionfor the rate of changeof strain energyof a soil matrix with the

i=j (34)

O[otiYB]

O[aj yB] 15 iJ 0 i•:j in which15ijis the Kronecker's deltafunction.The index i (i = 1, ßßß , M) in (34) representsany genericregionwithin the domain out of M such different regions. Index j (j = 1, ... , M) representsthe region with respect to which the

sensitivityanalysisis carded out.

To computedEi/d(aj•/B) in (34) it is alsonecessary to knowOh/O(aj •/B).Following a procedure recommended by Yeh [1987], the original flow problem is first differentiated

withrespect to aj7B;thenthetransformed boundary value problemin whichthe dependent variableis Oh/O(ajTB)is solved.

The flow equation (20) together with the specifiedhead boundary and initial conditionscan be written as

assumption thatthespatialvariation in thefluiddensityp is

OT

Oh

negligible.Hence, p is considereda constantin the derivations to follow and is combined with g to give % the unit weight of the fluid.

Ox

- Bpg[ai + n13] •

Ot

(35)

The rate of strainenergychangeof an aquiferreachof lengthLi is 24

1 2 =Li Oh 2dx.

Ei= • tz'y

=o

ot

(31)

2O

SILT SAND

This equation can be rewritten as

Ei (a•/B) ••••r=Li =o Oh Ot =

h•dx

(32)

in which B is the saturated thickness. For a heterogeneous

aquifer, the rates of strain energy changein the individual

O0

2

,0

TIME (DAYS)

subdomains may be superimposed to givethe total strain Fig. 16. Comparisonof strain energy storage in different soil layers: flow case 2. energychangein the aquifertOproduce

3234

KARNEY

AND SENEVIRATNE:

,

APPLICATION

OF ENERGY CONCEPTS TO GROUNDWATER

0.4

FLOW

2.9 24

0ø,3

0.2

•1

0.1



z

J 1Io

o.o

:

I SANDI I 30 40

20

:

53

SILT

11o

TIME (DAYS)

2O

TIME (DAYS)

Fig. 17. Sensitivity of strain energies to bulk compressibilityof

Fig. 18. Sensitivity of strain energies to bulk compressibilityof

silt.

sand.

while that of sand is of the order 10-5 m2/kN. The compressibility of silt is first increasedby an order of magnitude

and

h(O, t) = h(O)

to 10-3 m2/kN;theresulting percentage errorof thepiezo-

h(L, t) = h(L)

m

h(x, O) = h(x)

m

(36)

metric head at the end of 50 days is approximately 3%. However, when the compressibility of sand is increased by

anorderof magnitude to 10-4m2/kN,theresulting percentage changeincreased to 22%.

The exampleshowsthat if drawdowns in the s,andregion

respectively. Differentiating (35)with respectto aj •/B, 0 T•

Oh

O(aj'yB) Ox

= Bp•7[oti +

are of importance, then compressibility data of the sand layer shouldbe acquired with precision. Hence, the inte-

Oh

OO(ajyB)

gratedsensitivity approach provides a basisto identifythose regionswhich contribute mostly to the dynamic effects of a given study area of the aquifer.

Ot

O(oti'YB)

+ •/5

0(ye)

Oh

(37)

The transformed boundary and initial conditions are respectively,

Oh(O, t) •=0 O(ayB)

(38)

CONCLUSIONS

The concept of energy transformation presented in this paper provides a comprehensiveway of analyzing dynamic aquifers. Central to the energy approach is its ability to evaluate the different physical phenomena occurring within an aquifer. For example, When water is pumped out of a

confinedaquifer,the porousmatrixundergoes compresssion

which is measuredby the strain energy stored in the aquifer Oh(L, t) skeleton. Further, the rate of change of strain energy quan•=0 (39) tifies the transient nature of the aquifer. Integration of the O(ayB) individual processeswithin an aquifer into a representative Oh(x, O) single parameter makes it possible to compare the domi= o (40) nanceof the flow phenomenain the aquifer. In summary, the energy method is an effective way of collapsingthe dynamic Equation(34)represents the marginalchangein E i with behavior of aquifers into individual parameters describing respect to •3,B. Therelative magnitudes of dEi/d(aj•/B) the state of the system. indicate the influence of the bulk compressibilityon different The contribution of this approach is its ability to assess regions of the aquifer. and interpret flow conditions in a porous medium. For This sensitivity technique is illustrated for the system example, a model can select a variable time step based on

shownin Figure 12. The objec. tive is to evaluatethe sensi-

tivity of the responseof the sand layer to errors in the bulk compressibility data for either zone. Figure 17 depicts the sensitivity of the rate of strain energy changeof the silt and sand zones to variations in the bulk compressibility of silt. The sensitivity of the rate of change of strain energy of the sand zone is much lower compared to the silt zone, for

10.0 9.5

•'

9.o

• 8.5

variationsin silt compressibility.However, Figure .18shows that sand is highly sensitive to variations in its own bulk compressibility. The indications of Figures 17 and 18 are

verifiedin Figure19whichdisplays thepiezometric headat a well 240 m from the origin. The head distribution for the first 50 days is evaluated for different orders of magnitudeof the bulk compressibility of both sand and silt. In the base

case,silt hasa compressib!lity of the order10?-4m2/kN

ß• • •-

7.0 6.5

•[:

SILT(Ir--o-•)

s•cx([-•).

WELL AT X m 240

•([-o•) M

TIME (•YS)

Fi•. ]9.

Drawdown sensitivity to diCerent comprcssibiliticsof sand and silt.

KARNEY AND $ENEVIRATNE:

APPLICATION

OF ENERGY CONCEPTS TO GROUNDWATER

The•nergymethodalsoevaluates the sensitivity of the

3235

control.

the rate of change of strain energy of the system. When the systemundergoesrapid changes,the changein strain energy is also rapid and the time step small; as the system approachessteady state, the changesin strainenergygradually

diminish and the size of the time step is increased. The adaptive control of the time step can improve the computational efficiency of the algorithm with a controllable loss of accuracyof the final solution. The computationalefficiency of the adaptive algorithm increaseswith longer simulations. The adaptive control proceduresare particularly suitableat a screening level when large numbers of simulations are required to narrow down the range of alternatives.

FLOW

•ij Kroneckerdeltafunction. ¾ unit weight of water. K

T/S.

ß fluid potential. p density of water. •r'

effective

stress in the soil.

•soil strain energy storedin a unit volume of th• soil. •water strainenergystoredin the pore watei' per unit bulk volume

of soil.

Acknowledgments.The authorswish to thank the anofiymous

reviewers for their helpful comments. The computingsupport proflow model to the relative compressibilitiesof heterogeneous vided by the Department of Civil Engineering of the University of mediaundera givensei of hydraulicconditions.This feature Toronto is acknowledged.

may be particularly important in identifyingwhich areasmay be affected due to aquifer developmentor due to a severe drawdown.

REFERENCES

Bear,J., Hydraulics of Gro•tndwater, Ser.in WaterRbsour. and Environ.Eng. "•569pp., McGraw-Hill, New York, 1979. NOTATION

B

dD

constant

saturated

thickness.

mechanicalenergy dissipatedto overcome friction.

dE changein internalenergy. dW

Esoil Ewate r E e(t)

785-944, 1940. Huyakorn, P.S.,

heterogeneoushydraulicconductivity. length of the one-dimensionalaquifer. number of distinct aquifer regions. porosity. fluid pressure.

T heterogeneou• transmissivity. user-defined dimensionless tolerance. time.

V volume of fluid in a genericsoil sample.

V0 volume of fluidat thereference state. Vb bulkvolume ofthesoilsample. Vw volume of waterwithinthesoilsample. z• average fluid velocity within the soil pores. z•x flow velocity in the x direction. x space dimension. z

WithFoUrier.SeriesandBoundaryValueProblems,2nded., 533 pp., Prentice-Hall, EnglewoodCliffs, N.J., 1987. •

Hubbert,K. M., Theoryof groundwater motion,J. Gebl., 48,

S storagecoe•cient.

t

Haberman, R..,. Elementary Applied PartialDifferential •Equations

strain energy storedin the soil matrix. strain energy stored in the pore water. strainenergyof soil and water combined. rate of strain energy changein the soil matrix.

P0 fluidpressure at thereference state. TOLER

Prentice-

Hall, Englewood Cliffs, N.J., 1979. Gurtin, M. E., Variational principlesfor linear initial value problems, Q. Appl. Math., 22,252-256, 1964. .....

net work interactions at the control surface.

•/ gravitational acceleration. h piezometric head. i a generic region with distinct aquifer properties. j region with respect to which the sensitivity analysis is done.

Kx L M n p

Freeze,R. A., and A. J. Cherry,Groundwater,6(}4•.,

datum head.

and G. F. Pinder, COmputational Methods in

Subsurface Flow,473pp.,Academic, SanDiego,, Calif.,1983. Jensen, O. K:-,An automatic timestepselection scheme forreservoir simulation,paperpres_ented at the 55th annt;alfall meetingof

SPE-AIME, Soc.of Pet. Eng., Dallas,Tex., Sept.21-24, 1980. Karney,B. W., Energymethods in transientclosedconduitflow,J.

Hydraul. Eng.Div.Am.Soc.Civ.Eng.,116(i0),i 180-1196, 1990. Lindberg:B., Cliaracterization of optimalstep-size sequences for methodsfor stiff differential equations, SIAM J. Nume•. Anal., 14, 859-887, 1977.

Mehra, R. K., M. Hadjitofi,findJ. K. Donnelly,An:automatictime step selector for reservoir models,•paper presented.at the 6th Symposiumon Reservoir Simulation, Soc. of Pet. Eng., New Orleans, La., Feb. 1-3, 1982.

Pinder,G. F.-,andW. G. Gray,FiniteElement Simulation in Surface a"•d Subsurface Hydrology, 295 pp., Academic, San Diego, Calif., 1977.

Sammon, P. H., andB. Rubin, .Practical contr61 of timestep selectionin thermal simulation,paper presentedat the 7th Sym-

posiumon ReservoirSimulation,Soc: of Pet. Eng., San Francisco, Calif., Nov. 15-18, 1983.

Wang, .H.F., and M.P. AnderSon,Ititroductionto Groundwater Modelling:Finite Differenceand Finite ElementMethods,237 pp., W. H. Freeman, New York, 1982. Yeh, W. W. G., Groundwater SystemsPlanning and Management, 416 pp., Prentice-Hall, Englewood Cliffs, N.J., 1987.

B. Karney, Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4.

A. Seneviratne, JacquesWhitford Environment Limited, P.O. Box 1116, 711 Woodstock Road, Fredricton, New Brunswick, Canada

E3B 5C2.

a coefficientof bulk compressibilityof the soil. /3 coefficientof bulk compressibilityof the pore water.

•tadap t adaptivetimestep. •tbase base time step used when there is no time

(Received July 9, 1990; revised July 8, 1991' accepted July 19, 1991.)