A multilayered sharp interface model of coupled ... - Wiley Online Library

WATER RESOURCES RESEARCH, VOL. 26, NO. 7, PAGES !431-1454, JUL,Y 1990

A Multilayered Sharp Interface Model of Coupled Freshwater and Saltwater Flow in Coastal Systems' Model Development and Application HEDEFF

I. ESSAID

U.S. GeologicalSurvey,Menlo Park, California

A quasithree-dimensional, finitedifference model,thatsimulates freshwater andsaltwaterflow separated by a sharpinterface, hasbeendeveloped to studylayeredcoastalaquifersystems. The modelallowsfor regionalsimulation of coastalgroundwater conditions, including the effectsof saltwaterdynamics on thefreshwater system. Verticallyintegrated freshwater andsaltwater flow equations incorporating theinterface boundary condition aresolved withineachaquifer.Leakage throughconfining layersis calculated by Darcy'slaw,accounting for densitydifferences across the

layer.The locations of the interface tip andtoe, withingridblocks,are trackedby linearly extrapolating theposition of theinterface. Themodel hasbeenverified usingavailable analytical solutions andexperimental results. Application ofthemodeltotheSoquel-Aptos basin,SantaCruz County, California, illustrates theuseofthequasi three-dimensional, sharp interface approach forthe examination of freshwater-saltwater dynamics in regionalsystems. Simulation suggests that the

interface, today, isstillresponding tolong-term Pleistocene sealevelfluctuations andhasnotachieved equilibriumwith presentday sealevelconditions.

flowing through theaquiferoutcrop and/orleakingthrough theconfining layers.Thenatureandtimeframeof aquifer Coastalaquifersare an importantwaterresourcein areas during thetransient period willdepend ontheflow bordering seas.In manycoastalsettings, aquifersystems are response conditions, boundaryconditions, and aquiferproperties. INTRODUCTION

characterized by sequences of layerswith varyinghydraulic The easewith whichsaltwatercan moveinto, or out of, an

properties. Well-known examples ofsuchsystems arefound system affects therateof interface movement in in the north Ariantic and Israeli coastal plains and the aquifer response to changes in freshwater discharge. To fully underLlobregat deltain Barcelona,Spain[Collinsand Gelhar, stand the behavior of coastal systems, it is necessaryto

1971;Schmorak,1967; Custodio,1981].A quasithree- examine thedynamics of boththefreshwater andsaltwater dimensional, numericalfinite differencemodelto simulate flow domains. freshwater andsaltwaterflowseparated by a sharpinterface

in layeredcoastalaquifersystems is presented here.The SALTWATERINTRUSIONMODELING APPROACHES modeling approach facilitates regional simulation of coastal groundwater conditions andincludes theeffects ofsaltwater Twogeneral approaches havebeenusedto analyze saltdynamics onthefreshwater system. Demonstration of the waterintrusion in coastalaquifers: thedisperse interfaceand utilityof thismodelis illustrated by an application to the thesharpinterface [ReillyandGoodman, 1985].Thedismultilayered aquifersystemin the Soquel-Aptos basin, perseinterface approach explicitly represents thepresence California.

of the transitionzone, wherethereis mixingof freshwater An idealized cross section through a layered coastal andsaltwaterdueto the effectsof hydrodynamic dispersion.

aquifersystemextending offshore to a submarine canyon The sharpinterface approach simplifies the analysis by

outcrop is shown in Figure1. Undernatural, undisturbed assuming thatthe transition zoneis thinrelativeto the conditions anequilibrium seaward hydraulic gradient exists dimensionsof the aquifer.

withineachaquifer,withexcess freshwaterdischarging to Simulationof the transitionzone separatingfresh water thesea.In theuppermost, unconfined aquifer thefreshwater andsaltwaterrequiressimultaneous solutionof the equaflows out to sea across the ocean floor. In the lower, tionsgoverning fluidflowandsolutetransport for a conserconfined aquifers thefreshwaterdischarges to theseaby vativechemical species [Bear,1979].In caseswherethe leaking upwardthroughthe overlying layersand/orby transition zoneis verydisperse andchlorideconcentration flowing outthecanyon outcrop. Withineachlayera wedge- gradients arelowintheareaofinterest theeffects ofvariable shaped bodyof denser seawaterwilldevelop beneath the density maybeneglected, allowing decoupling of theequalighter fresh water. tions[Reddell andSunada,1970;Bredehoeft andPinder, Anychange in the flowregimen withinthe freshwater1973; Andrews, 1981].In manycases, however, thevertical

region, caused bychanges in discharge orrecharge inland, density gradient is significant andmustbeaccounted for inducesmovementof the freshwater-saltwater interface. [Pinder andCooper, 1970; Volker andRushton, 1982; Segol Reduction in freshwater flowtowardtheseacauses intrusion et al., 1975;Lee andCheng,1974;Frind,1982a,b; Voss,

ofsaltwaterintotheaquifers astheinterface moves inland. 1984b]. Theincreased computational effortrequired to solve In a layered system, saltwatercanenteran aquifer by thedensity-dependent transport problem haslimitedmost to two-dimensional verticalcrosssections. ThreeThispaper isnotsubject toU.S.copyright. Published in1990 by solutions theAmericanGeophysicalUnion.

dimensional, density-dependent solutetransport codeshave

Papernumber90WR00200.

beendeveloped [INTERA,1979;Huyakornet al., 1987; 1431

1432

ESSAID.' MULTILAYER FRESH\VATER AND SALTWATER INTERFACE MODEL

tionof theflowequations overthevertical, canbeapplied areallyto largephysicalsystems.This approach doesnot give informationconcerningthe natureof the transition zone;however,it doesreproduce theregionalflowdynamics

of the systemand response of the interfaceto applied stresses. VolkerandRushton[1982],compared steadystate

solutions for boththe disperse and sharpinterface approachesand showedthat as the coefficientof dispersion

decreases the two solutions approach eachother.Sharp interface models which simulate flow in the /¾eshwater Fig. 1. Idealized cross sectionof a layered coastalaquifer system.

Kipp, 1987] but are limited in their application to regional coastal systemsby computationalconstraints.Andersen et al. [1986, 1988] have used this approachin the study of the Florida

coast.

The sharp interface approach facilitates regional scale studies of coastal areas. When the width

of the transition

zone is small relative to the thicknessof the aquifer, it can be assumed,for the purposeof analysis,that the salt water and fresh water are separated by a sharp interface. This approachreproducesthe generalposition, shape,and behavior of the interface. Sharp interface models couple the freshwater and saltwater flow domainsthrough the interfacial boundary condition of continuity of flux and pressure. In three dimensions this boundary condition is highly nonlinear [Bear, 1979], however, assuminghorizontal aquifer flow and integrating the flow equationsover the vertical simplifies the problem. Sharp interface models generally fall into two categories:those that model coupledfreshwater and sa!twater flow (two-fluid approach), and those that model freshwater flow only (one-dynamic-fluidapproach). In the two-fluid approach the coupled freshwater and saltwater flow equations are solved simultaneously[Shamir and Dagan, 1971; Bonnet and Sauty, 1975; Mercer et al., 1980a, b; Bear and Kapu!er, 1981; Polo and Ramis, 1983; Pinder and Page, 1977; Wilson and Sa da Costa, 1982; Contractor, 1983; Liu eta!., 1981]. The movement of the interface is dictated by the freshwater and saltwater flow dynamics. The sharp interface models which simulate flow in the freshwater region only [Fetter, 1972; Anderson, 1976; Ayers and Vacher, 1983; Voss, 1984a; Taigbenu et aI., 1984; Volker and Rushton, 1982] incorporate the GhybenHerzberg approximation [Badon-Ghyben, 1889; Herzberg, 1901] assumingthat at each time step salt water adjusts instantaneously to changes in the freshwater zone, and therefore an equilibrium interface position is achieved. Each of these approacheshas its advantagesand limitations and can be employed successful!yonly under the appropriate conditions. The disperseinterface is necessary in areas where

the transition

zone

is wide.

Numerical

instabilitiesand errors are encounteredwhen simulatingthe movement of a narrow concentrationfront, especially in zones where the transition zone approachesa sharp interface. Frind [1982a] has shown that when a velocitydependentdispersioncoefficientis used, this problem is encounteredin areasof stagnantsalt water. Vossand Souza [1986, 1987]indicatethat when flow is predominantlyhori-

regiononly,by incorporating theGhyben-Herzberg approximation,assume thatthesaltwaterdomainadjusts rapidly to appliedstresses.In manycases,to reproducethe short-term behavior of a coastal aquifer, it is necessaryto includethe influence of saltwater flow [Essaid, 1986]. MULTILAYER FRESHWATER-SALTWATERFLOW MODEL

Sharpinterfacesaltwaterintrusionmodelshavegenerally been limited to one-layer problems. Experimental models [Collins et al., 1972] and analytical models [Ruiner and

Shiau, 1968;Mualem, 1973;Collinsand Gelhat, 1971,1977; Mua!em and Bear, 1974]have beendevelopedfor a sharp interface in a layered or stratified aquifer. Several numerical modelsallow for an overlyingleaky aquitard[Mercer et aI., 1980a, b; Voss, 1984a]. Fetter [1972] and Anderson [1976] representedlayers with differenthydraulicconductivities by a single layer with an averaged conductivity value. Two numerical models have been presentedfor the multilayered sharp interface case. The model of Bear and Kapuler [1981] is restricted to the study of a vertical cross section through two aquifers separated by a thin impervious layer. Sapik [1988] has presented a model that simulates steady state conditions in a multiple-aquifer system having steadyfreshwater flow and static salt water (Ghyben-Herzberg conditions). Mercer et al. [1986] used the INTERA

[1979] codeto

build a quasi three-dimensional solute transport modelof Volusia County, Florida, by neglecting the density dependence of the problem. In order to develop a sharp interface model for freshwater and saltwaterflow in layered coastalaquifers,representation of the moving interface within a discretized systemis nec-

essary.The positionof the interfacetip (the intersection of the interface with the top of the aquifer) and the interfacetoe (the intersection of the interface with the bottom of the aquifer) will not always coincide with the block or element

boundaries.Shamir andDagan [1971]andBear and Kaplder [1981] overcame this for a vertical cross sectionby usinga

movingfinitedifferencegrid. WilsonandSa da Costa[1982] incorporated an indirecttoe trackingalgorithmintoa fixed grid finite elementmodel. Other sharpinterfacemodelshave

madeno attemptto track the interfacetip and toe positions. In this work a quasithree-dimensional, numericalfinite differencemodelto simulatecoupledfreshwaterandsaltwa-

ter flow separatedby a sharpinterfacein layeredcoastal aquifer systemsis presented. The interface tip and toe

positions aretrackedwithineachlayer.Detailsofthemodel development are givenby Essaid[ 1987].The modelaccommodates multiple aquifers with spatially variable porous media properties.The uppermostaquifer of the systemmay

zontal, the vertical discretization must be of the same order

be confined,semiconfined, or unconfined with areallydis-

of magnitudeas the transversedispersivity. The sharpinterfaceapproach,in conjunctionwith integra-

tributedrecharge.Temporalvariationsin recharge and pumping are accounted for by multiplepumping periods.

ESSA!D:MULTILAYER FRESHWATER ANDSALTWATER INTERFACE MODEL /??/?/////'?/////////'//?/////////////////?//'? I?////////?////?///ll

1433

Aquiferabove confininglayer

ql

Ca= Za+Pa/¾a K'/B'

Freshwater domain za

Aquiferbelowconfininglayer

•b= Zb+Pb/¾b Datum

'•_•_ _• Datum

Fig. 3. Leakage(ql) througha confininglayer(za, elevationof topof confining layer;Zb,elevationof bottomof confining layer;cI)a, Fig. 2. Freshwaterand saltwaterflow domainsin a singleaqui- headaboveconfininglayer; •,, headbelowconfininglayer). fer(•r0,elevation of bottomof aquifer;sr•,elevation of interface; st2, elevationof top of aquifer).

Ts are the freshwater and saltwater specific weights The boundary conditions, which may be simulated in the model, are no-flow boundaries, constant freshwater head and/or saltwater head boundaries, constant flux, and head-

(ML-1T-2);Sf andSsarethefreshwater andsaltwater specific storages (L-l); qf andqs are the freshwater and saltwater specific discharges (LT-1); st0,•rl,and•'2arethe elevationsof the base of the aquifer, the interface, and the

dependent flux boundariesin the uppermostaquifer.By use top of the aquifer,respectively(L); n is the effectiveporos-

of these boundary conditions it is possibleto accommodate ity;•f and•s arethevertically averaged freshwater and the variety of onshore and offshore settingsencounteredin saltwater heads(L); Bf = st2- srlis thethickness of the coastalsystems. Freshwater and saltwater massbalances freshwaterzone(L); Bs = •'l - st0is the thicknessof the are calculated for each aquifer. saltwater zone(L);qf:andqszarethevertical components of

freshwater and saltwater fluxes(L); K• and K• are the verticallyaveragedfreshwaterand saltwaterhydrauliccon-

VERTICAL INTEGRATION OF THE COUPLED

ductivitytensors (LT- •); and

FRESHWATER-SALTWATER FLOW EQUATIONS

Within each aquifer of a layered coastal system the freshwaterand saltwater domains are coupled by the com-

mon boundary at the interface (Figure 2). For each flow domainthe equationof continuitymay be integratedoverthe verticaldimension,reducingthe determinationof the position of the interfaceto a problemin two dimensions(x and

q' = qxlx + qyly o( ) •7'(. )=•lx+•ly Ox

a( ) Oy

y). The verticallyintegratedequationsof freshwaterand

of notationall furtherreferences to freshwater saltwater flow in each confined aquifer are, respectively For simplicity and saltwater heads ((I)œ, CI)s) imply the vertically averaged [Bear, 1979, pp. 389-390],

heads((•f, (bs). Invoking continuity of pressure at the interface,the interfaceelevationcan be calculatedfrom the

fr,

dz'-' -V' . (BfK•.V'•œ)

freshwater and saltwater heads:

(2)

+ (nt5+ SfBf)

In equations(!a) and (lb), respectively,the terms ot

- q[ ;2ßV'•2 + q•:l•:

+

-

v'g2 +qzzl;= =0

(la)

represent theboundary conditions at thetopandbottomof theaquifer,respectively. If theboundaries areimpermeable,

0 V'qs +Ss --•-jdz

thesetermswill equalzero. If, however,the boundariesare

leaky,thesetermswill be givenby the leakagethroughthe overlying andunderlying confining layers.Forthecaseof an unconfined aquifertheupperboundaryis a freesurface,and thedrainage fromthe watertableis givenby nOq)f/Ot.

- -V'-(BsK;' V'•s) + [SsBs + n(1+ $)]

-nS

Ot

ßv' + q$1c0

-

=0 qszlG

(lb)

where •œ= z + pf/•/fisthefreshwater head, assuming an

Leakage Terms

Theleakage througha semipervious confining layercanbe

by applyingDarcy'sLaw in onedimension (Figincompressible fluid(L);•s = z + Ps/Ts isthesaltwater head calculated

(L);z istheelevation (L);pœ andPsarethefreshwater and ure 3) if the effectsof storagewithinthe confininglayer are andflowthrough theconfining layeris essentially saltwater fluidpressures, respectively (ML-1T-2); •/œ and negligible

1434

ESSAID: MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL

vertical [Bredehoeft and Pinder, 1970]. When the waters on either side of the confining layer have the same density, Darcy's law can be formulated in terms of hydraulic head differences across the layer. However, if waters of different density occur on either side of the confining layer, vertical density gradientsbecome important and Darcy's Law must be formulated in terms of pressure:

q• = - -

+ og

(3)

Integrated Equations

Introducing theseboundary conditions andaccounting for

source/sink terms,the verticallyintegrated equations for freshwaterand saltwaterflow, respectively,become

(1)

(3)

0 BfKfx Orbfh +

whereq/is theverticalleakage (positive upward)(LT-•), k' is the verticalpermeability of the confining layer(L2), /.•is the dynamicviscosity(ML-• T-J), p is the fluidpressure (ML-• T-Z), p is the fluid density(ML-3), and g is the gravitational acceleration (LT-2). Evaluating the pressures above and below the confininglayer and making use of the definitionsfor hydraulic head, specificweight, and freshwater hydraulic conductivity of the confining layer (K' =

(2)

(4)

+Qf+Qtf (7a)

(4)

(5)

(6)

Orbs [ Orbs

SsBs • + n(l+8)-Ot- ns ,,, a;j (1)

(3)

k'•/f/lz, equation(3) canbe rewrittenas =-

!

[Ta(rba-- Za)-- Tb(rbb- Zb)q- •/B']

ql =

(4)

Ox

BsKs

+

BsKsy

x OxJ •yy

Oy /

+ Qs+ Qts

(7b)

(4) (4) (5) (6) where B' is the thicknessof the confininglayer (L), rbaand andKsxare the freshwaterandsaltwaterhydraurb•, are the hydraulic heads above and below the confining whereKœx inthex direction (LT-1),KfyandKsy are layer, respectively(L), and z, and zb are the elevationsof licconductivities the top and bottom of the confining layer (L). When fresh water occurs on one side of the aquitard and salt water occurs on the other, there is a transition from one type of water to the other through the confininglayer. The density above and below the aquitard is determinedby the type of water present; however, the density distribution within the aquitard depends on the direction of flow. This is unknown until the equationis solved. For simplicity,the gravity term

in (3) is approximated by pg = z/= ('Yaq- •/b)/2,where and (•/•,) are the specific weights above and below the confining layer. Rearranging,the general form of the leakage term be-

the freshwater and saltwater hydraulic conductivitiesin the

y direction (LT-•), Qf andQs arethe freshwater and saltwater source/sink terms (LT-•), QlfandQtsarethesums of freshwaterand saltwaterleakagetermsacrossthetopand bottom of the aquifer given by equation(5) (LT-•), the parameter a is equal to one for an unconfinedaquiferand0 for a confined aquifer, and all other variables are as defined earlier.

In (7a) and (7b) the type 1 terms represent the changein elastic storagewithin each domain. The type 2 term representsthe changein freshwater storagedue to drainageat the water table, and the type 3 terms represent the changein storage within each domain due to movement of the interface

(nO•/Ot). The divergence of the fluxes in the x and y directions is representedby the type 4 terms. Sourcesand sinks to the aquifer are given in the type 5 terms (recharge, pumpage) and type 6 terms (leakage). In general, the change in storagedue to fluid displacementat the water tableand interface (terms 2 and 3) is greater than that due to elastic

comes

rba--'Yb rb/> + ('Yb--Ta)(Zb+Za)] (5) K'[y•

qt= - •'

'yf

'¾f

2

andK'/B' is the leakanceof the confininglayer. The firsttwo termsin (5) representthe equivalentfreshwaterheadsabove storage (term 1). and below the confininglayer. The third term incorporates Equations(7a) and (7b) representtwo coupled,parabolic the effect of the vertical density gradient. For the case of partial differentialequationswhich must be solvedsimultawaters with equal density above and below the confining neously forthefreshwater head(rbf)andthesaltwater head layer, this equation reducesto (rbs)-Once these values are known the interfaceelevation q• = -

(rba- rbb)

(6)

It is assumedthat whenwaterof onetypeleaksinto a zone of water of anothertype, the amountof leakageis small relative to the water in place. Thus the water mixes instantaneously and is incorporatedinto the flow zone that it leaks

(sr•)can be obtainedfrom (2). In regionsaway from the interface,onlyonetype of fluidis presentin the aquifer,and the flow is described by a single equation without the interface (type 3) storage terms. FINITE

DIFFERENCE

FORM OF THE FRESHWATER

AND

SALTWATER FLOW EQUATIONS

into. This assumption is reasonable for aquifersystemswith predominantlyhorizontalflow components. In systemswith considerablevertical leakage relative to horizontal flow it

An implicitfinitedifference discretization scheme thatis centralin spaceandbackwardin timehasbeenadopted for thesolution of equations (7a)and(7b).Spatialdiscretization

will not yield good results.

is achievedusinga block-centered finite difference grid

ESSAiD:MULTILAYERFRESHVCATER ANDSALTWATER INTERFACE MODEL

'ax j-1/2

•

1435

j+l/2

I j-1 Freshwater

Saltwater

] -1

j

ax j- 1/2 Fig. 4.

3+1

•

j+l/2

Block-centered finite differencegrid.

Fig. 5. Interface tip projection (acI>j-/ax anda•t./ax,freshand

saltwater head derivatives at block boundaries, respectively' xl, interface tip projection distance).

which allows for variable grid spacing (Figure 4). The elevation of each layer (k) is taken as the midplane of the acI)•

aquifer.In the central differenceapproximations for the spacederivativesthe thicknesses at the gridblockbound-

aCI)s i-1/2] a%+(1- w)-•y i-1/2]

'-'(I+G)•o?

aries are linearly interpolated, and the conductivity terms are estimated using the harmonic mean of nodal values. At blockscontainingpumping wells the amount of fresh water and salt water extracted depends on the position of the interface relative to the elevation of the screened interval of the well. The rate of freshwater and/or saltwater extraction

i+

ay

1/2

a%

i+

(8b)

1/2

The weightingis necessaryto preventabruptchangesin

from a block, relative to the total fluid extraction rate, is slopeasthe interfacetip or toe crosses fromoneblockinto determinedlinearly on the basisof the proportionof screen another. To obtain smooth movement of the interface, the

the total open interval of the well.

weightingfactor(0 < to-< !) variesas the interfaceprojection distance(x/) varies (see Figure 5). The weight of a

Interface Tip and Toe Tracking

derivativeincreasesas the interfacetip or toe moves into the interval over which it is calculated, that is, when xl = O,

penetrating the freshwaterand saltwaterzonesrelativeto

is The positionof the interfacetip (the intersectionof the •o= 0 andwhenxl = Ax/+1, to = 1. Oncetheinterface located, the fraction of the top of a block in contact with interfacewith the top of the aquifer) and the interfacetoe fresh water and the fraction of the bottom of the block in (the intersectionof the interface with the bottom of the contact with salt water are determined. Using these fracaquifer)will not alwayscoincidewith the blockboundaries. To determine the net freshwater and saltwater leakage into a tions, the leakagesacrossthe overlyingand underlying layersare calculated.Finally, the net freshwater block,the extentof freshwaterandsaltwaterin contactwith confining andsaltwaterleakageinto a blockis givenby the sumof the thetop and bottomof the blockmustbe known.This is achieved by determining thepositions oftheinterfacetipand leakagesacrossthe top and bottom.

toe within the finite differencegrid for eachaquifer.The tip is locatedby linearly projectingthe interfaceuntil it inter- Discretized Flow Equations sectsthetopof the aquifer.Similarly,at thetoetheinterface Incorporating the discretizedapproximations into equa-

is projecteduntil it intersectsthe bottomof the aquifer.In tions(7a)and(7b),thefinitedifference equationat timelevel thevicinityof the tip andtoetheinterfaceslopein thex and n for each grid point becomes y directionscan be calculatedon the basisof weighted freshwaterand saltwater head derivatives, as follows (Figure Z•'• ._ 1+ B•'• _ I + D•'"j_ 1+ E•' + •'• j + t 5):

• + 1 = Q' + u.'7 +• + s* 'n ......... (1 + 8x

+

...... 8x

where Z, B, D, E, F, H, and S are 2 x 2 coefficient

Ox

submatrices,

+(1-to)OcI>s

j-1/2] I OdPs I oo,-j7

(9)

Ox

J + 1/2

are the unknown head values, and

(8a) j+ 1/2

oy

+

oy

Q2 are the known right-handside values.The expansionof

ocI> s

...........

j - 1/2

oy

thesematrix coefficientscan be found in the paperby Essaid

1436

ESSAID: MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL

[1987]. Equation (9) is nonlinear, as the values of the coefficients Z, B, D, E, F, H, and S are time-dependent, changingwith the position of the interface. The equations are linearized within the iterative solution technique by

TABLE 1. Parameters Usedin Simulations fora Rotating LinearInterface,an IntrudingandRetreating Interface, and an Interface in a Layered Aquifer Retreating

evaluating the coefficientsat the previous iteration level (picard iteration). The system of M coupled equationsrepresentingfreshwa-

Parameter

ter and saltwater flow at each node may be expressed in

D, m

matrix

pf,g/cm 3

1.0

Kf, m/s

notation

as

and

Rotating Interface 10.

(10)

Layered Aquifer

27.

0.15

1.025

1.0

1.030

1.0

4.52x 10-4

0.69

0.10

Ss,m-1

1.0x 10-4

1.025X 10-4

1.0x 10-4

n

0.3

1.0 0.1 5.0

Ps,g/cm3 AtI) n= Q

Intruding Interface

1.025

K', m/s

where A is an M by M septadiagonalblock matrix of the coefficient submatrices; • is a block column vector of the unknown freshwater and saltwater heads; and Q is a block column vector containing the known fight-hand side values. To reduce roundoff error and increase solution accuracy

8.0 x 10-4

Sf,m-1 Ax, m A t, s

1.03X 10-4

5.0 86400.0

0.1

0.05

during computations, these equations are solved in the residual

form:

a steady statevalueof3.9cm3/sto 18.8cm3/s(Figure 7).In

(11) the case of the intruding interface the initial steadystate freshwater fluxof 19.1cm3/swasstopped abruptly andthe where•:n= [ti)n_ (i)n-1],andRn-1 = Q - AtI)n-1. Equation transientinterface allowed to intrude (Figure 8). The numerA•" = g n- 1

(11) is solved using the strongly implicit procedure for three-dimensional, two-phase flow [Stone, 1968; Weinstein et al., 1969, 1970]. MODEL

VERIFICATION

To verify the numerical solution obtained from the finite difference model, numerical simulations have been compared to analytical solutions and experimental Hele-Shaw analogs. Model results for one-layer problems without leakage and interface tracking were verified for the motion of a

linear interface and a retreating and intruding interface. These results are summarized

below.

ical results of Shamir and Dagan [1971] and the present model both show interfaceswith a shapeless concavethan those observed experimentally and .some lag in interface translation.This can be attributedto the error introduced by the Dupuit assumptionof horizontal flow, sincethis approximation deterioratesas vertical flow becomesmore pronounced at the outflow face. In addition, there is also

difficulty in realistically representingthe seepageboundary condition at the outflow face. Shamir and Dagan [1971] introduced the actual experimental values for freshwater head and interface

Keulegan [1954] presented an analytical solution for the location of the toe of an initially vertical interface rotating toward an equilibrium position in a confined aquifer of uniform

-2

(tApKfDl 1/2 X, npf /

at the outflow

face into their

o

thickness:

L(t) =

elevation

solution to improve their results.

--

••,,•%

"•'Xx

ß=T= 12.28 days ß =T=22.28 days ays

(12)

-4

where L(t) is the distance to the interface toe from its initial

-6

position (L),Ap = Ps - Pf (ML-3),andD is theaquifer thickness(L). To facilitate comparisonof the model results to another numerical model, as well as the analytical solution, the parameters (Table 1) and initial conditions of Mercer et al. [1980a] were used. The initial position of the interfacewas set at L = 20 m, a positioncorrespondingto a time of 12.28daysof rotationfrom the vertical position.The results obtained for a simulation period of 20 days and the results of Mercer et al. are shown in Figure 6. Both numerical solutionsfit the analytical solution quite well, with the exceptionof someminor smearingat the interfacetip and

>

-10

o

0

uJ -2 Z

•

ßMidpoint weighting B-

•

-'•

oUp__stre•m weighting •

toe.

Simulationsfor a retreating and intruding interface were compared to observed interface behavior in a Hele-Shaw

experimentcarded out by Bear and Dagan [1964]. In both cases the same parameters were used (Table 1) and the outflow,or seepageface, wasapproximatedby assigningthe

-8

-10 0

20

40

60

80

100

DISTANCE (m)

boundarynodea highleakancevalue(3.3 s-1) and a zero Fig. 6. Simulation of a rotatinglinearinterface(pointsrepresent headin the overlyingaquifer.For the retreatinginterfacethe numerical results,curvesrepresent analyticalsolutions); (a) results seaward freshwater dischargewas suddenlyincreasedfrom

of presentmodel, (b) resultsof Mercer et al. [1980a].

ESSAID: MULTILAYER FRESH%'ATER ANDSALTWATER INTERFACE MODEL

1437

A -6

-12

_

Observed (Hele-Shaw Analog)

_---Simulated T=0•

T=20•

_

T=20

E

•

-18

z

o

Impervious

> -24 T=240

,

,

Fig. 9. The interface in a coastalaquifer with a thin horizontal semiperviouslayer.

Lu 0

B

analytical solutionfor the steady state shape of an interface in a coastal aquifer when a thin horizontal semipervious layer is present (Figure 9). Their solution is based on the -12 Dupuit assumptionand a linearization of the flow equations. In addition, Mualem and Bear [1974] made two simplifying -18 assumptionsregarding the leakage conditions in the region where fresh water in the lower aquifer is overlain by salt water in the upper aquifer (Figure 9): (1) (I),, = D/& = -24 -' '" T=13." constantalong the semiperviouslayer in this region, and (2) the fresh water leaking throughthe semiperviouslayer from below was incorporatedinto the freshwater flow zone above. DISTANCE (cm) The geometry of the test problem is shown in Figure 10, Fig. 7. Simulation of a retreating interface: (a) resultsof present and the parametersused in the simulation are given in Table model and (b) results of Shamir and Dagan [1971]. 1. Initially, to facilitate comparisonof the numerical solution with the analytical solution, Maulem and Bear's simplifying Incorporation of interface tip and toe tracking and accu- assumptionswere incorporated into the model. Figure 1la rate leakage calculations makes comparisonof the numerical showsgood agreement between the two solutions. The same modelto an analytical solution for a layered problem possi- problemwas then simulatedusingthe modelleakagecondible. Mualem and Bear [1974] presented an approximate tions, and the results are shown in Figure 1lb. In this case, when fresh water leaks through the semiperviouslayer into overlying salt water it mixes with the salt water and flows to the boundary. Under these conditionsthe interface below the semiperviouslayer is slightly deeper while above the Observed (Hele-Shaw Analog) A layer the interface extendsfurther inland. This result is due Simulated T=$2S -6 _ --to (1) the leakageof fresh water into the overlying salt water, and (2) flow in the saltwaterzone, which is actually a mixing u.I

-6

T=20•

.

zone.

-12

-

E

I_.."-

•//.•11

This exampleillustratestwo interpretationsof the position of the sharpinterfaceand its relation to the mixing of fresh water and salt water due to leakage. In the first solution, freshwaterleakageis always incorporatedinto the overlying

/

•o -18 z

o

_--' >

uJ

-..--...--...--, I

-24

/ , .-•

.-/.-'/X

/1

freshwater zone, and loss of fresh water due to mixing with salt water is not accounted for. In the second solution, fresh water that undergoes any mixing is excluded from the freshwaterdomain and incorporatedinto the saltwater zone.

/-'./,',

0

o

-

B

Thus in the first case the mixed water is included in the u.! z

-6

-

T=525

_

T=225•

•.

'..,•z•,'///

-

.-.

•

. 0

.:,/..X 40

80

. 120

160

200

DISTANCE (cm)

•i•. 8. Simu]•tJoa of •n j•t•din• j•te•ace: (•) resultsof present model•d (b) •esuRsof Sh•mJra•d Da•an [•½?•].

4

15 cm

,•,

4o,&xxx//Xx///,•,• 2 0

Semipervi0us layer ,,.

,J

'

)

50 DISTANCE (m)

.,,I

1 cm3/s

" 100 cm

Fig. !0. The geometryfor the caseof a layeredaquifer.

1438

ESSAID: MULTILAYER FRESHWATERAND SALTWATER INTERFACE MODEL

•%Incorporating assumptions -"e.

Semipervious layer

37ø

07' l'•'"//"///•/-///N':J'•)'///:Jc/'p'/////•/• 30" !

•-

-E

•

• •0-

,

" •

I

L •l

-

I •%

•

I

t

• Map•

•

"%

Area% California-.

?% o•

• •gf

IISco•s Valley .•

/

•San FranCisco

"

[

•

I•

Analytical solution

•

•'

•

•,

.•'

' '....

-

•[ STUDY AREA• ' '

'

_

•0 -

{

0

]..... -

122 ø00'

121 o45"

100

DISTANCE (cm) Fig. 11. Comparisonof numericaland analytical solutions(a) incorporatingassumptionsof analyticalsolutionand (b) with model leakageconditions.The dashedcurve representsthe positionof the interface in the absence of a semiperviouslayer.

freshwater domain, whereas in the second case it is excluded from the freshwater domain. The latter approachresultsin a conservative estimate of the position of the interface separating fresh water and salt water [Hill, 1988]. The assumptions of this approachare appropriatefor systemsdominated by horizontal flow componentsbut deteriorate when vertical flow approachesthe magnitudeof horizontal flow. In these casesleakage must be handled in a manner that is intermediate to the above two cases. Freshwater leakage into overlying salt water must be apportionedto both the freshwater and saltwater zones, approximating the effects of mixing.

Fig.12.Location oftheSoquel-Aptos along the coast of Monterey Bay. Today the Soquel-Aptos region is mainly an urban area with very little active agriculture or industry. As a result of the projected growth and increase in water demand, the Soquel Creek Water District (Figure 13) has been involved in evaluating and assessingthe groundwater resourcesin the basin as well as potentialwater quality problems. Several studies of the Soquel-Aptos area have been undertaken to determine the location, extent, and nature of the aquifers; the quantity of water flowing through the aquifers; and the potential for saltwater intrusion. Saltwater has not yet intruded onshore in the Soquel-Aptos area, and the

position of the interface offshore is not known. Previous

1220

370

55'

/ /

APPLICATION TO THE SOQUEL--APTOSBASIN,

southwest of theZayantefault,whichcoversabout150km2. Physiographically,the area rangesfrom very steep valley slopesand angularlandformsin the Santa Cruz Mountains to nearly flat marine terraces, sea cliffs, and narrow beaches

'• • \

/- Purisima production well ]

'• ! 4,Aromas production well I

•duYn•;•//// • • •:ulti-level mønitørin well 1 / '

CALIFORNIA

Applicationof the modelto the Soquel-Aptosbasin, Santa Cruz County, California, illustrates the use of the quasi three-dimensional,sharp interface approachfor the examination of freshwater-saltwaterdynamicsin coastal systems. This application demonstratesthe utility of the modeling approach for the study of a multilayered aquifer system on a regional scale. The model is used to examine the freshwatersaltwater dynamicsin the basin and the effect of groundwater development on the potential for saltwater intrusion. The Soquel-Aptos area lies betweenlatitudes36ø55'Nand 37ø10'N, and longitudes121ø45'Wand 122ø05'W(Figure 12), approximately 100 km south of San Francisco. The present study concentrateson the part of the Soquel-Aptos basin

1210 50'

05'

/ 370

I

--

/ / ..... C:&ap•ethorpe•/•Made•ine ß Tanne•

ß

HOSe•ale ßsc%/•c,ff H,,,cest ••-• '-:• /

Monterey .j

••i

District Boundary •O•te•

[

Aptos Cr.ß

/

Seas•pe

[

LaSelva .I

Fig. 13. TheSoquel CreekWaterDistrict areaandlocation of productionand monitoringwells.

ESSAID: MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL 52'30"

121ø 45'

I

I

1439

layerfreshwater-saltwater flow modelof the Soquel-Aptos basinis developedto examinetheseissues.

37ø

W. BranchSoquel

07' 30"

HYDROGEOLOGIC SETTING

The Soquel-Aptosarea has a mild Mediterraneanclimate

with relatively cool, dry summersand winters that are characterized by precipitationand mild temperatures.Precipitationis mainly rainfall, and approximately90% of it occursfrom November through April. Rainfall increases rapidly with increasingelevationfrom approximately500

Branciforte

Soquel

Aptos

mm at the coast, to approximately 1300 mm at the crest of

the SantaCruz mountains.Precipitationis the predominant Valencia

forte Creek;SoquelCreek with its tributary,West Branch SoquelCreek; and Aptos Creek (Figure 14). Table 2 summarizesthe locations,drainageareas,andperiodsof record for the gagingstationsin the area. Branciforte,Soquel, and AptosCreeksare perennialstreamshavinga componentof

37ø

Porter

Arana-Rodeo

o

MontereyBay

I

sourceof groundwaterrechargein the basin. Three majorcreeksdrain the Soquel-Aptosarea: Branci-

2mi

base flow discharge throughout the year. Groundwatersurfacewater interactionis importantin the system.

The Soquel-Aptos area is within the California Coast Ranges Province, which emcompassesthe Santa Cruz Fig. 14. Drainagebasinsof the Soquel-Aptosarea. mountains,and has undergonea complex Tertiary tectonic history. The geology of the Santa Cruz Mountains in the vicinity of the Soquel-Aptos area has been studied by Allen studies havegivenvaluesfor through-flow or potentialyield [1946], Cummingset al. [1962], Clark [1966], and Clark' and for the PurisimaFormation aquifersrangingfrom 5.4 to 16. Rietman [1973]. More detailed studiesof the geology of the million m3/yr[AkersandHickey,1967;Hickey,1968;Muir, Soquel-Aptosarea have been carded out by Hickey [1968], !980; Thorup, 1981; Luhdorffand Scalmanini, 1981, 1984, Luhdorff and Scalmanini [1984], and Johnson [1980]. The 1985].Disparityin the resultsof previouswork indicatesthat offshoregeologyof the Monterey Bay has been investigated

the methodsof analysisappliedto the Soquel-Aptosbasin by Martin [1964] and Greene [1970, 1977]. have not been sophisticatedenough to adequatelycharacPre-Tertiary crystalline basementrocks are overlain by a terize the groundwater system and its interaction with offshoreconditions.To determine the quantity of water that can be developed without inducing groundwater quality degradationdue to seawater intrusion, severalissuesmust be addressed:the amount of freshwater flow through the system,the quantity of natural freshwater outflow to the sea, the undisturbedposition of the interfaceoffshore,the quan-

tity of dischargethat mustbe maintainedin orderto keepthe interfaceat or near the shore, and the rate at which the interfacewill move due to onshoredevelopment.A multi-

TABLE 2.

Station

successionof regressiveand transgressivesedimentaryunits which have been offset by contemporaneous strike-slip movement along the faults of the San Andreas fault system. Onshore,the major faults which have displacedthe strata in the Soquel-Aptos region are the San Andreas fault to the northeastand the Zayante fault to the southwest(Figure 15). Offshore, two major fault zones are present: the Monterey Bay fault zone and the Palo Colorado-San Gregorio fault zone. Southwest of the Zayante fault, Tertiary and Quaternary sedimentsare tilted and dip gently toward the southeast

GagingStationsin the Soquel-AptosArea and AverageDischargesfor Period of Record

Location

Drainage Area,

Period of Record

Average Discharge,

km2

(WaterYear)

m3/s

104.

1952-present

1.3

82.9

1969-1970 and 1972

'"

SoquelCreek at Soquel SoquelCreek near Soquel

36ø59'29"N 121ø57'17"W 37ø02'02" 121ø56'35"W

WestBranchSoquel CreeknearSoquel

37ø03'OY'N 121o56 ' 17"W

31.6

1959-1972

0.35

Aptos Creek at Aptos Aptos Creek near Aptos BranciforteCreek at Santa Cruz

36ø58'33"N 121ø54'05"W 37ø00'06" 121ø54'18"W 36ø59'10" 122000'48"W

31.9

!959-1972

0.22

26.4

1972-present

0.31

44.8

1940-1943

0.59

DatafromU.S. Geological SurveyandCalifornia Department of WaterResources records.

1440

ESSAID.' MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL

% %

Cruz

Elkhorn 'h

43

%--qooo m N

o

•

0

5 km

,

• mi

36

Fig. 15. General structural features and model grid area.

(3 to 5 degrees) except in the regions which have been disruptedby faulting. The offshore Monterey Canyon is one of the largest submarinecanyoncomplexesin the easternPacificOcean. It is similarin dimensionsto the Grand Canyonof the Colorado River, having a maximum relief of 1800 m and a maximum width of about 20 km [Martin, 1964]. Formation of the modem Monterey canyon may have begunduring the late

the Monterey Formation appears to have been removedby erosion.

In the Soquel-Aptos area the upper Miocene to Pliocene sequence is represented by the Purisima Formation. The Purisima Formation is a shallow to deep water marineunit consisting of shale, fine-grained sandstone, and minor conglomerate. It has been described in detail in the northern Santa Cruz Mountains by Cummings et al. [1962]. The Pliocene,with continueddevelopmentof the canyonduring Purisima Formation is exposed at the surface over mostof the Pleistocene [Green, 1977]. the Soquel-Aptos area, and offshore it crops out on the Cenozoicsedimentaryrocksunconformablyoverlie crys- ocean floor and along the walls of the Monterey Canyon tallinebasementrocksin the Soquel-Aptosareaandbeneath (Figure 17). It pinches out at the northwestern border of the

the Monterey Bay. Figure 16 is a compositestratigraphic sectionof the northernpart of MontereyBay eastof the Palo Colorado-SanGregariofault [Greene, 1977].The basement complex of the Soquel-Aptos area is composedof Cretaceousgraniticrocksand associatedmetasedimentary rocks. In the northwesternand westernpart of the area the basement complexis closeto the surfacebut increasesin depth to the southeast,reachinga depthof more than 600 m below sealevel at the easternboundaryof the area. Onshorewell

area but increases in thickness toward the southeast, where

it reaches a thicknessof greater than 600 m. Onshoreit lies directly on the granitic basement, whereas offshore it overlies the Monterey Formation.

The quaternarysequencein the area is representedby the AromasSand,terracedeposits,and alluvium[Dupre,1975]. The Aromas, which is made up of interbeddedfluvial, marine, and eoliandepositsof Pleistoceneage, overliesthe Purisimain the easternpart of the area, both onshoreand

information[Hickey, 1968]and offshoreseismicprofiles offshore, and increases in thickness toward the southeast [Greene, 1977] indicate that the basement surface is an undulating erosional surface. The sequenceof lower Eocene to lower Miocene sedimen-

(Figure 17). The terrace depositsof late Pleistoceneagerest unconformably on the Purisima Formation along the coast.

Unconsolidated silt, clay, sand,andgravelalluvialdeposits tary rocksis absentin the Soquel-Aptosarea southwestof of Holoceneage occurin the valleysof the majorcreeks the Zayante fault. The middle Miocene sedimentaryse- onshore and cover the ocean floor offshore. quenceis representedby the thick bedded,siliceousorganic CONCEPTUAL MODEL shaleand siltstoneof the MontereyFormation.In the study areait is presentoffshorewhereit overliesgraniticbasement For simulation purposes the hydrogeologic conditions of rocks [Greene,1977].Onshore,in the Soquel-Aptosarea, the Soquel-Aptosbasinhavebeentranslatedinto theframe-

ESSAID: MULTILAYER FRESHWATER ANDSALTWATER INTERFACE MODEL

1441

Thick-

Age

Sequence Formation Lithology ness ........

•e

ßc o • õ •. • 0 '• •• •' • = c

Surficial

'-= o

./ Aromas Sand Deltaic material

e

•

•.

Purisima

ß o •

,..

;,_o..;.

•_•;

40

670

"'•'0"• 370 •.,,

e •e

Sandandmud;madne, deltaic

Greenish-gray, semi-consolidated to

consolidated sandstone, siltstone, ....

200

,_

Eli 1•' / It/ I,

•

'"Well sorted, cross-bedded, qu•.rtz•'se'

andshale; marine, generally fossiliferous

Sandsto•••...... • 1•27..•.-: '•._.

o

'

some broken consolidated material

Santa Cruz Mudstone

Santa Margarita •

Gravel, sand, and mud.

............

•

o.

t: e

Recently deposited sand and mud ' ' Submarine landslide and slump matedal

,',,,•300 sand; nonmarine, eolian

Formation"•-

O

• _9 • •-

40 .•.• 240 •:--... .-:'., •;;,.' 50 i::•,

deposits

Description

(meters)

,

Monterey Formation

'"'_•--_'",d:.• 550 _:.T•Z• •.

:•x

Siliceous, organic mudstone; madne

Bedded arkosic sandstone (?) Light olive-gray, rhythmically bedded, organic mudstone, diatomaceous and silliceousshale and siltstone;madne

x

•lXx ,• Xx

Granitic rocks txxxXx x Mesozoic orolder(crystalline basement) /x xxX xx •X xXX IX X,

Probably predominantly grandiorite

Xx

Fig. 16. Compositestratigraphic sectionof the northernpartof MontereyBayeastof the PaloColorado-SanGregario fault zone (modifiedfrom Greene [1977]).

work of a conceptual model representing the onshore and System Geometry offshoregeometry, boundary conditions, and physical paThe principalhydrogeologicunit of interest in the Soquelrameters. Figure 15 shows the extent of the modeled area Aptos basin is the Purisima Formation. Figure 17 is a and finite difference grid (43 rows by 36 columns). In the generalized geologic map of the area showing the areal onshorearea the grid blocks have dimensionsof 610 by 610 rn, with increasing spacingtoward the southeastand south- extent of the Purisima Formation and the boundaries of the modeled area. From exploratory wells, geophysical logs, westfacilitating incorporation of the offshoreboundaries.

Mo•el bou dary

•

n

Ki'•L•f,• •/'•0oo a meters ••Q• , . ii

%

Sea

•

level

•

900

0 5km

1 •i8. l?. Gcncr•]izcd •colo8•c m•pshowin• onshore •ndo•shor•sur•cia] •½olo•y •ndcross sections (•, •uatcrn•ry deposits' Tp, •urJsima •ormadon; Tin,Monterey Formation; Ei, •r•niticb•s•mcnt).

1442

ESSAID: MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL

Fig. 18. Geologic cross sectionsshowing Purisima subunits (modified from Luhdorff and Scalmanini [1984]).

of 183 m.

By intersecting the structural contours for the Purisima subunits with the land surface topography, the extent of the subunit outcrops has been delineated (Figure 21). The older

and other geologic.evidence, Luhd.or.ffand Sc.almanini [1984] units a. reby exposed inthe we. stern area and are progressively overlmn theyounger .umts toward the southeast. Onthe wereableto delineate seven distinct Punsims .subunits basis oftheabove geologic conceptual model, thethickness, framework was

throughout the basin (Figure 18). This

adopte. d forthenum.encal model, however, theAAandA subun. ltSwerecombined asweretheE andF subunits, resulting in five model layers. Distinct, finite thickness

confining layers could not be identified between the sub-

units. Thus the system has been modeled with five layers of

differinghydraulicpropertiesand with the verticalleakance

•4J• -•7 50

50

representing the hydraulic connection between these layers. Basin-wide structural contour and thickness maps were N , -- __•__ constructed by integrating onshore and offshore geologic information. The system's geometry was delineated by in-

.--

corporating outcrop data [Hickey, l.968], w.ell logs [Luhdorff •x•;--[! -370• and Scalmanini, 1984], and seismic stratigraphic sections [Greene, 1977]. The.base oftheflowsystem wastaken tobe

\\

l

the CretaceousgranltlCbasementonshoreand the relatively impermeable Monterey Formation offshore. Basement to-

mapforthe top ofPurls•ma subumt Athe issho.wn inF!gu. re20. pography is shown in.Figure 19, .and stmct.ural contour Structural contour maps of the other subumtsare slmfiar to

subunit .A. . . . . Subumt A hasvarla. blethickness (maximum 250m),since thebasement was.originally anune.ven surf. a.ce oferos!on that was later buried by the overlying Punslma deposits.

Overlying subunits (B,C, andD) haverelatively uniform

thicknessesthroughoutthe basin, except in the outcrop areas where the original thicknesses have been reduced due to erosion. Subunit B has a thickness of 76 m, subunit C of 37 m, and subunit D of 30 m. Subunit E increases in

--

- coastline 0 2kin !

I

2mi

Fig. 20.

Structuralcontourmap for top of subunitA, contoursare elevation (in meters) relative to sea level.

ESSAID: MULTILAYER FRESHWATER ANDSALTVCATER INTERFACE MODEL

1443

elevation,and areal extent of each layer were discretized overthe 43 by 36 grid.

Boundary Conditions.

ß

Thelate.ral boundaries. ofthesystem have been delineated

....

on the bas•.s .of the. physical features affecting the hydrogeo-

logiccondotrans m the basra(Figure 22). The Purisima

Formation decreases inthickness westward and pinches out alongthe northwesternborder of the area. This was pre-

,•

N

' [---7

'J-?

--

scribedas a no-flow boundaryfor all layers. The Zayante

Production

faultwas set as a no-flow boundary, as was the offshorePalo

Colorado-San Gregoriofault zone.At the southeastern edge of the Soquel-Aptosarea there was no physicalboundary, therefore the model boundary was extended 8 km toward the

Boundary Conditions Constant

Head Nodes

No-flow Boundary

interest. The canyon outcrop was represented by constant

0 2kin

heads were fixed at zero at this

boundary. Freshwater heads were fixed at the equivalent freshwaterhead representing the column of salt water above the outcrop of each subunit. The base of the system is relatively impermeable, however, water enters or leaves the system across the top. Recharge enters each layer through its onshore outcrop areas, where groundwater also dischargesto streams as base

Well

Monitoring Well

southeast,and an estimatedflow line was representedby a no-flowboundary. This boundary was placed far enough away so as not to influence the solutionin the area of head nodes. Saltwater

'.....

Stream

2mi

Fig. 22. Boundary conditions and major features of the model area.

fixed at the freshwater head equivalent to the overlying

flow. Offshore, freshwater discharges totheseathrough column ofsaltwater. Byspecifying the. upper boundary condition in this manner, water could leak •nto or out of each

ocean floor outc.rops. Itis.ver. yd.ifficult accurately allowing use. ofthe model toestimate the quantity of mine the quantity and d•stnbutlon of to these fluxes.deterToth layer, [1963] showed thattopography iscommonly thedriving recharge to,and d•scharge from, thesystem.

force for groundwater .flow and that the water table is Inthe southeastern part ofThis the model area, subunit Eis generally asubdued rephca oftheland surface. Inthemodeloverlain bytheAromas Sand. unit has undergone more the. onshore outcrops ofea. chsubunit were .simulate. dbytha. n60m.of drawdown insome areas due top. umpage inthe fix•n.g the water table elevation above the aq.uifer and •nt.roPajaro basin [Bond and Bredehoeft, 1987]. Atime-dependen water table elevation was used to represent this condition, duc•ng aleakance factor represent vertical ithalinear decrease inheads from predevelopment condibetween the aquifer andto water tablethe (Figure 23).connection Onshore, w. theelevation ofthewater table wastaken astheaverage nons topresent daywater levels. topography.Offshore, the head in the overlyinglayer was

I

MODEL

CALIBRATION

The model values of aquifer parameters (conductivity, storativity, leakance, and effective porosity) were initially

I I

I

chosen on the basis of available data and then adjusted to obtain simulated water levels which satisfactorily matched observed water levels. The parameter values obtained by this trial and error process of model calibration are not

!

uniquebut can be constrainedon the basis of additional hydrologicinformation.The calibrationprocessalsoleadsto

E

an understandingof the factors determining the system's N

responseandbehavior,that is, the parametersand boundary conditionsto which the system is most sensitive. In April

Fixed Water Table

K7B' o 2kin

2mi

Fig.21. Discretized outcrop areas ofmodel layers.

Recharge

Discharge

Active Aquifer

Fig.23. Fixedwater tableboundary condition.

1444

ESSAID:MULTILAYER FRESHWATERAND SALTWATERINTERFACEMODEL

1220

370

55'

121050'

05'

I /, •'x' '' •

Study area // /

xXX]}•

Historical Pumpage

Table3 liststhe SoquelCreekWaterDistrictproduction ßWell location wellstappingthe PurisimaFormation,theirperforated inter-

-',-Flow direction

vals, subunitstapped,andyear of installation.From 1967to %Water leVrel contours present, actual pumpagevalues were availablefrom the

boundary!/' "F• "•\•/J• .ix• me• •

waterdistrict,butfor theperiodpriorto 1967,pumpage had to be estimated. The California State Water Resources Board [1953]estimatedthe draft from the Soquel-Aptos

basinin 1949to be about0.024m3/s.An exponential

functionwas fit to the known valuesto obtain estimates of

37 0

ß

thetotalannualpumpage for the intermediate years(Figure 25). The presentday distributionof pumpageamongthe wellsin thebasinwasusedto apportion historical pumpage

.

to individual wells as they went into production. Wells

penetratingseverallayerswere simulatedby increasing the leakancebetweenthe layers in those blocks, allowingthe model solution to determine the flow contribution of each

A•lO•terey Z?ay

layer. In the Aptos Creek well it was necessaryto apportion 20% of the total pumpage to subunit A in order to match heads.

Fig. 24. Approximate groundwaterlevel contours (in meters), April 1981,for the Soquel-Aptosarea (modifiedfrom Bloyd [1981]).

Recharge and Base Flow Estimates

The quantity of recharge to the Soquel-Aptos basinwas estimated from a water balance. Unfortunately, the water

balance approach could only give an order of magnitude estimate of groundwater recharge because of uncertaintyin flux values. Estimates of average annual precipitation and runoff for Branciforte, West Branch Soquel, Soquel, and from the shallow Purisima subunits, except near the coast Aptos creeks were taken from Rantz [1974]. There is conwhere productionwells tap the deeperunits.Bloyd observed siderable uncertainty associatedwith the estimate of evapothat the general direction of groundwatermovementin the transpiration from the basin. This is a critical parameterfor area was from the higher elevationsin the northern part of evaluating the amount of water rechargingthe groundwater the area toward the stream valleys and the coast. Bloyd's basin. Evapotranspirationwas initially calculatedusingthe map representsthe most comprehensivemeasurementof estimates of Blaney and Ewing [1953] for normal consumpwater levels in the basin and has been used for model tive use of natural vegetation on the Pajaro Valley floor, calibration. In addition, estimates of recharge to the system which is adjacent to the Soquel-Aptos area. Table 4 sum-

1981,Bloyd [1981]compiledwater level measurements for about 150 wells in the Soquel-Aptosarea and constructeda groundwaterlevel map representingthe near-surfaceflow system (Figure 24). These wells representedwater levels

and base flow

to the

streams were

made and used to

marizes

the estimated

values

for these calculations and

thatthe totalrecharge to thebasinis 0.35m3/s. constrainthe calibrationprocess.Conditionswere simulated indicates for the period from 1930 to April, 1981, using annual time Johnson [1980] estimatedthat the evapotranspirationin the steps through 1980 and monthly time stepsthrough April, northern part of the Soque!-Aptos basin (north of the 1981, for comparison with to Bloyd's map of observed Zayantefault) was from 52 to 59% of meanannualprecipitation. Usingan intermediatevalue of 55% of meanrainfall

heads.

TABLE 3.

Production Wells in the Soquel-Aptos Basin Estimated

Hydraulic Well

Perforated Interval, m

Subunits

Opal # 1 Opal #4

to - 39 -39 to -64

A A

Hillcrest

- 17 to - 56

Aptos

3 to -76

E E E A

Seacliff

Monterey

to - 88

to - 118

Mar Vista Cliff

-40

Maplethorpe Aptos Creek

-71 to - 150 -66 to -209

Tannery Madeline Rosedale

'" to - 101

-66 to - 145 -82 to - 152, - 191 to -246 -24

to - 130

Data from Luhdorff and Scalmanini [1984].

ABCD

E A BCDE A ABCD ABC

On-line

to present to present

1923 to 1930 to 1935 to 1950 to

present present present present

1950 to 1968

1961 to 1965 to 1965 to 1971 to 1973 to 1984 to

present present present present present present

Grid Location

18,17,1 18,17,1 14,21,5 11,24,5 13,22,5 15,17,1 13,20,1 14,26,5 14,17,1 13,22,5 14,17,1 13,20,1 15,16,1

Conductivity, m/s

1. x 1. x 2. x 3. x 2. x

10-4 10-4 10-5 10-5 10-5

8. x 10-5 1. x 8. x 8. x 8. x 8. x 1. x

10-5 10-5 10-6 10-'5 10-6 10-4

ESSAID: MULTILAYER FRESHWATER ANDSALFWATER INTERFACE •{ODEL TABLE 5.

1445

Estimated Mean Annual Base Flow for the Main

DrainageBasinsof the SoqueI-AptosArea Total

Area

Mean

Drainage Within g

Model,

km 2

km2

m3/s

m3•'s

Soquel West BranchSoquel

104. 31.6

35.2 11.7

0.37 0.09

0.12 0.03

Aptos Branciforte Total

31.6 44.8 212.

16.8 26.7 90.4

0.10 0.07 0.63

0.05 0.04 0.24

Basin

E

i

1920

i

i "'

1940

i

i

Base Flow,

Base Flow,

The followingregressionequationswere used to estimate mean annualbaseflow for yearswithoutrecord (baseflow in cubic meters per second,precipitationin millimeters)'West BranchSoquelbase

i

!960

Proportioned

Area,

•- o.l_

0.0

Annual

1980

Fig. 25. Annual pumpagefrom the PurisimaFormation:crosses flow = exp (-2.007 + 1.17 log (Soquelbaseflow) -0.406 (Santa representmeasuredvalues and the dashed curve is the fitted Cruzprecipitation)); R2 = 90.5%.Aptosbaseflow= exp(- 1.764+

function (pumpage = exp(0.0503(year)-98.1), R2 = 0.93).

0.881log(Soque! baseflow)-0.0132(SantaCruzprecipitation)): R2 = 92.2%. Brancifortebaseflow = exp (-0.629 + 0.391 log (Soquel

baseflow)-0.0099(SantaCruzprecipitation))' R2 = 64.0%. to calculateevapotranspiration,the total rechargeto the

basin becomes 0.66m3/s(Table 4),nearly twicetheprevious

Early simulationsshowed that hydraulic conductivity and

estimate.

In orderto obtainadditionalinformationabouttheground- leakancewerethe criticalpropertiescontrollingthe behavior

water flow through the system, an analysisof the stream of the system. Changes in conductivity affected the head hydrographswas carried out to estimate base flow. For the gradients in the basin and the drawdowns at the wells. streamrecordsavailable(Branciforte,WestBranchSoquel, Leakance values controlled the amount of water moving Soquel, and Aptos creeks), estimates of annual base flow through the system (rechargeand discharge)and the areal wereobtainedby a simplestraightline hydrographsepara- extent of the conesof depression(amount of vertical leakage tion. To calculate mean base flow, annual base flow esti- to the wells). From these simulationsit also became apparmateswere made for the creeks having incompleterecords. ent that the freshwater flow system achieved a nearly steady state freshwater head distribution within the annual time Recordsof equal length (1953-1984) were generatedby step, and therefore storativity was not a critical parameter. correlatingthe logarithm of the base flow at each creek to the In addition, simulations were insensitive to changes in logarithmof the base flow at Soquel Creek and the annual precipitationat Santa Cruz. The regressionequationsare effectiveporosity which was assignedas a constant value of given in Table 5. From these estimates of annual base flow for the period from 1953 to 1984 the mean annualbase flow was calculated for each drainage basin and also for the proportion of the drainage basin within the model area (Table 5).

0.1 in all units.

Specific capacity measurements (discharge/drawdown) are made on a regular basis at the production wells [Luhdorff and Scalmanini, 1985]. Table 3 lists the mean conductivities at each well estimatedfrom specificcapacity measurements [Theis et al., 1963]. A trend of decreasing conductivities Table 6 summarizes the estimates of fluxes that have been made for the Soquel-Aptos basin and the method used. toward the east and in the younger subunitsis noticeable in There is a wide range in the values; however, the estimates these data. The conductivities of all layers were initially set

tendto bewithin0.2 to 0.6 m3/s.Thesevalues,in additionto

at 3.0 x 10-5 m/s and the leakanceat 10-•ø s-2. Simulation

the water levels, were used to constrainthe parameter showed, however, that the values had to be reduced by

approximately an order of magnitude to obtain a head distribution and flux through the system that resembled observed conditions. This discrepancy between measured

calibration.

Parameter Calibration

values and calibrated

values can be attributed

to the fact that

The aquifer parameters for which the model was cali- the specificcapacity tests measure the hydraulic conductivbratedare hydraulicconductivity,leakance,andstorativity. ity of the more permeable horizons and also include the TABLE 4. Mean

Water Balance Estimates for the Soquel-Aptos Basin

Mean

ppt,* mm

Runoff',* mm

Soquel

1020

340

620

0.13

559

WestBranchSoquel

1120

394

650

0.074

615

914 991

240 371

589 612

0.086 0.0068

503 546

Basin

Aptos Branciforte Total

Evapotranspiration,? mm

Recharge,? m3/s

Evapotranspiration,$ mm

0.35

*Rantz[1974]. ?Amount basedon normalconsumptive useof naturalvegetation on Pajarovalleyfloor,BlaneyandEwing[1953].

:•Fifty-five percentof meanprecipitation.

Recharge,$ m3/s 0.27 0.11 0.17 0.11 0.66

1446

ESSAID:MULTILAYERFRESHWATER AND SALTWATER INTERFACE MODEL RESULTS OF 1930-1981

TABLE 6. Estimatesof Flux Throughthe Soquel-AptosBasin

SIMULATIONS

Flux,

Study

m3/s

Methodof Estimation

Hickey[1968]

0.40

application of Darcy'slaw to the primary water bearingunits

Muir [1980] Thorup[1981]

0.18 0.50

methodof Todd[1964] basedona waterbalance

Luhdorffand

0.49

application of Darcy'slaw

0.36-0.67*

basedon a waterbalance

0.24-0.63?

for the system based on estimates of

Scalmanini [1984]

PresentStudy

for the basin

at the coast

mean annual base flow

*Lower valuecalculatedusingevapotranspiration of Blaneyand Ewing [1953],upper value calculatedwith evapotranspiration as 55% of mean precipitation.

?Lower value is baseflow proportionedto the area within the model, uppervalue is total baseflow.

The initial conditionsfor the 1930-1981 period were obtained by simulating predevelopment conditions and allowing the system to achieve steady state both onshoreand offshore. The simulated onshore predevelopment-waterlevels are shown in Figure 30a. Figure 30a is a compositewater level map constructed by taking the water levels from the unit outcroppingin each area (e.g., unit A in the west, unitE in the east). As can be observed from this map, in the near-surfacegroundwater system, water flows mainly to the

streamsandtowardthe coastunderthe influenceof topography. Early reports of conditions in the area indicate that wells in the Capitola area were naturally flowing [California State Water Resources Board, 1953] when first drilled. The

simulated water levels of approximately 30 m at the coastin the Capitola area would reflect naturally flowing well conditions.

The simulated composite water levels for April 1981and the observed water levels of Bloyd [1981] are shown in

effectsof vertical leakage.Both of thesewould lead to an overestimation of the bulk (averaged) effective hydraulic conductivityof the units. Table 7 summarizesthe aquifer parametersusedin the model.Maps of subunitconductivities and leakance values are given in Figures 26 and 27, respectively. To check and fine tune the calibrated model, calculated

monthly water levels were comparedto thosemeasuredat the coastal multilevel monitoring wells from 1983 to 1985. Simulationswere begunin 1982in an attemptto includethe transienteffectsdue to conditionsprior to 1983.The plots of observed and simulated freshwater heads at two representa-

Figure 30b. In the northern part of the area, water levels have remained relatively the same from predevelopment conditions to 1981. At the coast, however, pumpage has

modified the predevelopment flow field. The cones of depression,which are below sea level, reflect the headsin the lower subunits tapped by the deep production wells. The cones of depressionnow capture some of the water that previouslyflowedto the streamsand offshore.Table 8 shows the predevelopmentand 1981baseflowsto the creeks,which

compare favorably to the estimatesobtained above from hydrograph data.

To understandhow developmenthas modified the fluxes through the system, the annual recharge, discharge,and relatively good. Sensitivityof the model to aquifer parametervalues was pumpagerelationsfor the period from 1930 to 1985,as examinedby systematicallyvaryingthe parametersin each calculatedby the model, are shown in Figure 31. Priorto therecharge to thesystem was0.50m3/s, of individuallayer and comparingthe computedvaluesof head development,

tive monitoringwells (Figure 28) show that the match is

to the observed values (273 measurements).The standard error of estimate was calculated for each case, and Figure 29 shows the result of these runs. The system is most sensitive to changesin subunitsA and E, because these are the thickest units from which most of the groundwater with-

which 0.47 m3/s dischargedonshoreto creeksandto the

overlying Aromas Sand.Only0.03m3/sof thewaterflowed offshoreto the sea, yet this was sufficientto maintainthe freshwater-saltwater interfacepositionoffshore.With devel-

opmentandan increasein pumpagein the basinthe 1981

drawal occurs. The behavior of the other subunits is deter-

onshore andoffshore discharges haddecreased to0.43m3/s

minedby'the conditionsin A and E. The stress(pumpage)on the systemis the dominantfactor controllinghead distributions. Changesin conductivityaffectthe magnitudeof drawdown due to pumpageat the productionwells, significantly influencingthe heads at the coast. Variations in leakance do not have as large an impact on the head distributionbut do significantlyalter the amount of flux through the system.

and 0.01 m3/s,respectively.The decreasein groundwater dischargeto the streamsis shownin Figure 31 by the decreasein base flow to Soquel Creek with increased

TABLE 7.

Summary of Aquifer ParametersUsed in the Model

Hydraulic Conductivity,

Maximum Thickness,

Storativity,

Leakance,

Layer

m/s

m

m- 1

s- 1

A B C D E

10-6 to 10-5 10-6 10-6 10-6 10-6

250 76 37 30 183

10-6 to 10-7 10-7 to 10-6 10-6 10-7 to 10-6 10-5

10-9 10-9 10-9 10-9 10-9

to 10-13 to 10-11 to 10-10 to 10-10 to 10-11

groundwater pumpage.

These resultsdemonstratethat most of the water entering

the Soquel-Aptosbasinis beingdischarged to the streams

onshore,and only a smallcomponent of the recharge is flowingoffshore. Thefreshwater flowsystem ismostactive onshorewherethereis strongtopographic reliefdrivingflow toward the streams.Water that is not dischargedto the

streamsflowsoffshore.Owingto the gentleslopeof the continentalshelf, low conductivity,and increasein the

overlying equivalent freshwater head,thisflowis small but hasbeensufficient to prevent saltwater intrusion. Mostof the water that has been developedfrom the basincomes

from capturedbaseflow and additionalinducedrecharge.

Thisphenomenon hasimplications forfuturedevelopment and managementof the basin'sgroundwaterresources.

ESSAID: MULTILAYER FRESHWATER ANDSALTX•, •,TERIIX'IERF-kCE •1ODEL

1447

I'

A

C

"

'-

I

-[

D N

[-'-q 1.x 10'6

I---] 5. x10 -6

o 2km

J

Fig. 26.

Maps showingthe areal extent and distributionof hydraulicconductivitytin metersper second)in each model layer.

FUTUREGROUNDWATER DEVELOPMENT AND THE

solutetransportmodel. Their analysisshowedthat although

POTENTIAL FOR SALTWATER INTRUSION

lateral flow of salt water through the offshore outcrop

Thesimulation of conditions from1930 to 1985showedcontaminates theaquifer atahigher rate,thiswater hadnot

almost no moveme. ntoftheinterface offshore, although yet moved onshore. The main pathway forsaltwater intrusignificant changes mtheflowsystem occurred. Thesimu-sion intotheonshore portion oftheaquifer wasvertical lated 1985 freshwater heads, saltwater heads, andinterface leakage; thisleakage hasoccurred primarily beneath the tipandtoe positionsfor the five Purisimasubunitsare shown estuaries and sloughs as well as through the near-shore ocean floor. As indicated from the 1930-1985 simulation of in Figure 32. Simulation results suggestthat interface response is quite slow and takesplace over longtime frames. the Soquel-Aptos basin, interface movement is very slow andtakesplaceover very longtime frames.Vertical leakage

As Bredehoeft. et•h/•t [1982] state, th.e magnitude ofon pumpage from a basin canbe sustained depends the of saltwaterintotheshallow aquifers, however, maybe amount ofnatural discharge thatcanbecaptured and,toa possible overtheshortterm.For1985conditions, fresh lesser degree, theamount ofadditional recharge which can wateris flowing offshore fromalllayers, andthereis no beinduced.The acceptablemagnitudeof developmentde- leakageof salt water into the aquifer(see Figure 32). In order to examine the consequencesof increased pumppends on the hydrologicconsequences thatcanbetolerated, ing in the basin,a simulationwas carriedout with pumpage andin manycasesit takeslongperiodsof timebeforea new equilibriumis achieved.The amountof developmentthat the

of twice the 1985 rates at all wells. The 1985 heads were used

Soquel-Aptos basin cansustain isdependent ontheamountasinitialconditions, anda period of 10years wassimulated 0fbase flowthatcanbecaptured byproper location ofwells, (inannual timesteps). Theresults areshown in Figure33. theresulting decrease in surface waterflowthatis accept- Increased pumpage haslittleeffect ontherateof movement

able, andthepotential forsaltwater intrusion asa result of oftheinterface. It does, however, cause thedevelopment of groundwater pumpage. Saltwatermayentertheaquifer by larger cones ofdepression which, intheupper units,extend intrusionof thefreshwater-saltwater interface, orbyvertical offshore.As a result,verticalleakageof seawaterinto

leakage ofseawater intothefreshwater zonethrough ocean subunit E is induced. Thusthemostimmediate potential orestuary floors. BondandBredehoeft [1987] investigated pathway forsaltwater intrusion would appear tobevertical the pathways ofsaltwater intrusion intotheprimary confinedleakage induced byoverpumping in theshallow subunits aquifer inthePajaro Valleyusing anareal two-dimensional nearthecoast. Thesystem • especially susceptible during

1448

ESSAID.' MULTILAYER FRESHWATER AND SALTWATER INTERFACE MODEL

I

A

I

I.

I I

I .... I I I

N

[---] < lo-ll j-"-] _> 10-11 o 2km

J--'-J>_. 10-10 '.:':• >_. 10-9

r

2mi

Fig. 27.

Maps showing distribution of leakance (per second) for the model layers.

dry seasons when water levels are low. Therefore it would be beneficial to pump from the lower aquifers near the coast and/or to place shallow wells inland.

MONITORING

WELL

SC-1

Subunit 2O

INTERFACE MOVEMENT

ß

A

o

B

AND SEA I.EVEL CHANGE

The previous simulations have shown that the interface responseis quite slow and takes place over longtime frames. This slow responsesuggestthat the offshore interface position in the Soquel-Aptos basin today could still be influenced by the Pleistocene sea level changes. This hypothesis has been tested using a long-term simulation incorporating sea level change. The Monterey Bay region has undergone a complex Quaternary history of sea level changesaccompaniedby tectonic downbowing in the lowland areas of the Pajaro Valley and uplift of Pleistocene terraces in the northern area (greatestin the Aptos-Capitola area) [Dupre, 1975;Bradley

and Griggs, 1976; Greene and Clark, 1979]. Dupre et al. [1980] examinedthe vertical sequenceof sedimentarystruc-

10

E

I

LU

--

I

MONITORING WELL SC-3

Subunit O A /•

:x; xx

...

B

xc

'X'XN.•.. ,-,XX • ,,• .X,?"'-X-

lO

tures preserved inmarine terrace deposits inthe northern•o Monterey Bay region. Their analysisshowed that the Quaternary deposits in the Watsonville region reflect at least 11 cyclesof glacioeustaticsealevel changesincethe emergence of the Purisima. The last major low stand of sea level occurred about 18,000 years before present during the Wis-

1983

1984

1985

Fig. 28. Simulated andmeasured monthly waterlevels attwoof

consinglacial maximum,and sealevel has beenrisingsince the coastalmonitoringwells. Pointsare observedlevelsandcurves then [Schwartz et al., 1986]. Fossil molluscs ha,'e been

are simulated

levels.

ESSAID: MULTILAYER FRESHWATER ANDSALTWATER INTERFACE MODEL

1449

CONDUCTIVITY

k.

,.-- .,?

A[' ..; | t...

Subunit

l/

ß

----0 C

/.,

t

x

\

ht_h -'

/,

• I

i

I 1 I I Ill

I

•