groundwater

A Unified Approach t o Regional Groundwater Management

Robert Willis Humboldt S t a t e University, Arcata, Cal ifornia 95521

Introduction

The management of groundwater resources and t h e evaluation of the

hydrologic and environmental impacts associated with groundwater

development is commonly approached using simulation o r optimization

models of the aquifer system. Simulation models a r e predict ive

models of t he hydraul ic response of the groundwater system. In

simulation modeling, a s e t of groundwater management pol ic ies is

analyzed t o determine a probable response of t h e aquifer system.

From t h i s information, a policy is then determined which best meets

t he objectives of t he management problem. However, i n simulation

t h e pol ic ies a r e inherent ly nonoptimal. They a r e nonoptimal i n an

operational sense i n t h a t only a l imited number of a l te rna t ives can

usually be analyzed. Furthermore, t he t r a d e o f f s associated with

t he system's economic or hydrologic object ives a r e d i f f i c u l t t o

determine. In cont ras t , however, optimization modeling represents

a unif ied approach t o groundwater management. Optimization modeling

iden t i f i e s t he optimal planning, design, and operat ional po l ic ies

and t h e t r a d e o f f s i n t h e system's objectives. Moreover, optimiza-

t i on modeling can a l s o generate t he s e t of noninferior solut ions

t o multiobjective groundwater planning problems.

The object ive of t h i s paper is t o present an optimization method-

ology f o r regional groundwater management. Spec i f ica l ly , it w i l l

be shown how the response equations fo r confined and unconfined

aquifer systems can be incorporated within t he framework of an

optimal planning model. A s a r e s u l t , t h e hydraulic response of t h e

Water Resources Monograph Groundwater Hydraulics Vol. 9

Copyright American Geophysical Union

Unified Approach t o RegionaZ Groundwater Management 393

aqui fer system is an in t eg ra l par t of t he optimization model. I n

t h e optimization methodology, t he groundwater planning problem i s

formulated a s a multiobjective optimization model. The methodology

is applied t o t he Yun Lin Basin, Taiwan, t o determine the optimal

groundwater extract ion pat tern.

Response Equations

The response o r t r ans fe r equations of t he groundwater system a r e

those equations r e l a t i ng the s t a t ed variables of t he aqui fer and the

proposed planning o r management pol ic ies . A s has been discussed

by Maddock [1972],Willis and Dracup [1973], and Aguado and Remson

[1974], t h e technique transforms t h e p a r t i a l d i f f e r e n t i a l equation

of t he groundwater system via Green's functions, f i n i t e d i f fe rence

o r f i n i t e element methods. These r e su l t i ng equations may be imbed-

ded within t h e constraint region of t h e planning o r design problem,

o r equivalently, t he problem can be formulated a s a problem i n

optimal control [Wil l is and Newman, 19771.

Confined o r Leaky Aquifer System

We assume t h a t t he surface-groundwater system may be represented

by the v e r t i c a l l y averaged continui ty equation f o r a leaky aqui fer

[Cooley, 19741 :

where T is t h e t ransmissivi ty tensor (L'/T), h is the hydraul ic

head (L), S is the s torage coef f ic ien t , and S* is a source o r s ink

term, e.g., leakage. 0 is an index s e t defining the locat ion of

a l l wells i n t he basin and 6( ) is the Dirac de l t a function.

The boundary conditions of t h e aqui fer system may be expressed

88



39 4 Groundwater HydrauZics

where u l and u2 def ine t h e boundary of t he basin, h* is t h e known

potent ia l , - n is t h e outward pointing uni t normal t o ~ 2 , and q* is

the given f lux. Generally, these equations a r e time-dependent

boundary conditions.

Equation (1) may be transformed i n t o a system of ordinary dif-

f e r e n t i a l equations with t h e Galerkin f i n i t e element method. The

transformed equations may be wr i t ten a s [Pinder and Frind, 19721

where h now represents t h e f i n i t e element approximation t o t h e

hydraulic head; 10 a r e t he i n i t i a l conditions f o r the problem.

The C and H coef f ic ien t matrices contain t h e s torage coef f ic ien ts

and t ransmiss iv i t ies , respect ively. The f vector contains t h e

Dir ich le t and Newmann boundary conditions and importantly, t he

planning pol ic ies [Wil l is , 1976bl. Equation (2) can a l so be

e x p l i c i t l y wr i t ten a s a system of ordinary d i f f e r e n t i a l equations

i n time a s

h = A h + g - - - (3

where A = 4-1 H and g = -c-1 - f .

Unconfined Aquifer System

Assuming Dupuit assumptions a r e va l id f o r unconfined ground-

water, t h e ve r t i ca l l y averaged Boussinesqu equation can be expressed

a s [Cooley, 19741

where _k - t he hydraulic conductivity tensor [LIT], Sy is the spe-

c i f i c y ie ld , and R[L/T] i a t he recharge occurring i n t he aquifer.



Unified Approach t o Regional Groundwater Management 395

Equation (3) is, however, a nonlinear function of t h e hydraul ic

head. Boundary and i n i t i a l conditions f o r t he problem a r e again

sunnnarized i n (1). F i n i t e d i f fe rence o r f i n i t e element methods

may be used t o transform t h e p a r t i a l d i f f e r e n t i a l equation i n t o

a system of nonlinear ordinary d i f f e r e n t i a l equations. These

transformed equations may be expressed a s

where t h e coef f ic ien t matrices D and E contain t h e s p e c i f i c y i e ld

and conductivity. Planning or operat ional po l ic ies , t h e recharge,

and boundary conditions a r e contained i n t h e L vector. Again, 5 represents t h e vector of t h e hydraul ic head a t a l l nodal points

i n t h e system.

Simplifying (5) , we have

where now A= - D - ~ E and & = -D-lr. - A s w i l l be discussed, we choose

t o l i n e a r i z e these equations using quas i l inear iza t ion [Bellman and

Kalaba, 19651. Assuming a t r i a l so lu t ion t o ( 6 ) , hk, and expanding

about t he solut ion using a generalized Taylor s e r i e s , we have

where H~ is a diagonal matrix containing hk; t h a t is EIllk=hlk,

~ ~ ~ = h ~ ~ , etc . Simplifying, we have the l i n e a r system of ordinary

d i f f e r e n t i a l equations,

where ~k = and gk = - gk - ~ h ~ , ~ . -

Solution of t he Response Equations

The response equations of t he groundwater system a r e usual ly

solved using conventional f i n i t e d i f fe rence approximations. Here,



396 Groundwater Hydraulics

however, (3 ) o r (7) w i l l be solved ana ly t i ca l l y by using the matrix

calculus. The general so lu t ion of these equations is [Bellman,

Assuming tha t t he planning o r management pol ic ies and the system's

boundary conditions a r e constant over a period T,

The matrix exponential eAt can be evaluated by A=RQR-l. The matrix

R contains the eigenvectors of A, and Q is a diagonal matrix contain-

ing the eigenvalues of A. A s a r e su l t e ~ t = e ~ ~ R - l t is simply R~R-I ,

where 4 is again a diagonal matrix; however, t h e elements a r e now

eAit, where h i is the i t h eigenvalue of t he system. Simplifying,

we have

here,

Al(t) = RQR-l and A2(t) = A'-'(I-RGR-')c-'

For a s e r i e s of planning periods t l , t 2 , t m of equal length T, t h e

equations may be expressed a s

or , funct ional ly,




The Planning Model

We consider a groundwater system loca ted i n an a g r i c u l t u r a l r i v e r

basin . The planning problem is t o determine t h e optimal groundwater

pumping p a t t e r n t o s a t i s f y t h e a g r i c u l t u r a l water demands of t h e

basin . Assuming t h a t t h e planning hor izon c o n s i s t s of m opera t ing

per iods , t h e pol icy va r iab les of t h e model a r e t h e groundwater

e x t r a c t i o n r a t e s f o r each wel l s i t e i n t h e basin . Recognizing

t h a t t h e ob jec t ives of t h e system may r e f l e c t economic, hydrologic ,

and environmental cons ide ra t ions , t h e o b j e c t i v e func t ion of t h e mo-

d e l may be expressed a s

m man r= r G z hpfp ( z n , ~ n ) n n- 1 P

where f p i s t h e p th o b j e c t i v e and hp i s t h e weight o r p re fe rence

assoc ia ted with o b j e c t i v e p [Cohon and Marks, 19751. Qn is t h e

t o t a l groundwater discharge dur ing period n; a n is t h e discount

f a c t o r . The pol icy v a r i a b l e s - hn and Qn a r e constra ined t o s a t i s f y

(1 ) t h e water demand i n each i r r i g a t e d a r e a R., o r

(where Dt represen t s t h e demand i n i r r i g a t i o n system g i n per iod

n demand l e s s e f f e c t i v e p r e c i p i t a t i o n and s u r f a c e water a v a i l a b i l -

i t y ) , ( 2 ) t h e balance c o n s t r a i n t s ,

(3) t h e response equations (equat ions (10d)) and, poss ib ly , lower

bounds o r head g rad ien t c o n s t r a i n t s t o minimize subsidence o r sea-

water in t rus ion . These c o n s t r a i n t s may be w r i t t e n a s compactly a s

where X is an index s e t de f in ing t h e l o c a t i o n of t h e c o n t r o l p o i n t s



398 Groundwater HydrauZics

i n t h e bas in and h* a r e t h e des i red bounds on t h e head. j

We a l s o have t h e w e l l capac i ty r e s t r i c t i o n ,

where Qi,,,, is t h e maximum pumping r a t e a t w e l l s i t e 1. F i n a l l y ,

t h e nonnegat ivi ty r e s t r i c t i o n s of t h e dec i s ion v a r i a b l e s ,

The planning-optimization model has s e v e r a l important a t t r i b u t e s .

F i r s t , t h e c o n s t r a i n t s e t is a convex s e t . This was e s s e n t i a l l y

t h e r a t i o n a l f o r l i n e a r i z i n g t h e unconfined flow equat ions . Second,

i f t h e ob jec t ives a r e separab le concave ( o r convex i f minimizing)

func t ions of t h e dec i s ion v a r i a b l e s , then g l o b a l l y optimal s o l u t i o n s

w i l l be obtained t o t h e planning problem. Third, f o r t h e l i n e a r i z e d

unconfined flow problem, a s e r i e s of opt imizat ion problems w i l l be

solved. The head d i s t r i b u t i o n from one s o l u t i o n i s then t h e b a s i s

f o r updating t h e response equat ions i n t h e next s o l u t i o n of t h e

planning model. This convergence and t h e o r e t i c a l p r o p e r t i e s of t h e

a lgor i thm a r e presented by Rosen [I9661 and Meyer [1970]. An appl i -

c a t i o n of t h e procedure t o parameter es t imat ion problems is d i s -

cussed by Willis [1976a].

Model Appl icat ion

Over t h e pas t 2 yea rs , a s p a r t of an i n t e r n a t i o n a l cooperat ive

research program, t h e mult l o b j e c t i v e planning model has been appl ied

t o t h e water resources problems of t h e Yun Lin Basin, Taiwan. The

over r id ing ob jec t ives of t h e resea rch program a r e t o develop (1)

planning and opera t iona l p o l i c i e s a l l o c a t i n g s u r f a c e and groundwater

resources t o a g r i c u l t u r a l water demands wi th in t h e basin , (2) t o

determine t h e t rade-offs a s s o c i a t e d with a d d i t i o n a l groundwater

development and a g r i c u l t u r a l water demands, and (3 ) t o minimize

t h e p o t e n t i a l impacts of s a l t w a t e r i n t r u s i o n . We consider he re ,



Unified Approach t o RegionaZ Groundwater Management 399

--- IRRIGATION SYSTEM

Fig. 1. The Yun Lin groundwater basin.

however, one pa r t i cu l a r applicat ion of t h e planning model involving

t h e determination of t h e optimal pumping pa t te rn f o r two d i f f e r en t

scenarios regarding groundwater development. In t h e f i r s t , ground-

water extract ions a r e determined assuming a well capacity r e s t r i c -

t i o n of 15,000 m3/d ( the current maximum). In t h e second case,

t h i s bound is increased t o 50,000 m3/d t o r e f l e c t t he poten t ia l

f o r addit ional groundwater development. Other uses of t he model

a r e presented by Willis [I9811 and Willis and Liu [1981].

The Yun Lin groundwater system is e s sen t i a l l y a semiconfined

aquifer . The aquifer , which is located i n t h e Cho Shui a l l u v i a l fan,

i s composed primarily of unconsolidated sand and gravel materials .

The aquifer depth ranges from 40 m i n t h e eastern portion of t h e

basin t o more than 1000 m i n t he Peikang area. Approximately 76% of

t h e t o t a l groundwater recharge occurs v i a i n f i l t r a t i o n of precipi-

t a t i o n and seepage from the numerous streams i n t h e basin [Water

Resources Planning Commission (WRPC), 19761. The Cho Shui River,

which forms t h e northern boundary of t h e study area , is t h e princi-

pa l recharge boundary of t he system. The Peikang River i n t h e



Groundwater Hydraulics

Fig. 2. F i n i t e element g r i d : Yun Lin groundwater bas i n .

south, does not however i n t e r a c t wi th t h e Yun Lin a q u i f e r system

(Figure 1).

Water resources i n t h e bas in a r e d i s t r i b u t e d v i a four i r r i g a t i o n

systems: t h e Cho Shui, Fu Wei, S i Lo, and Tou Liu systems. Each

i r r i g a t i o n d i s t r i c t is administered by t h e Yun Lin I r r i g a t i o n

Association. The a s s o c i a t i o n c o n t r o l s t h e a l l o c a t i o n of s u r f a c e

water , o r i g i n a t i n g from t h e Cho Shui River, and groundwater from

t h e 500 assoc ia t ion we l l s i n t h e basin. Current ly , t h e t o t a l

i r r i g a t e d a r e a i n t h e basin is approximately 43,260 ha.

The hydrology of t h e bas in i s charac te r i zed by d i s t i n c t r a iny and

d ry seasons. The ra iny per iod, which extends from May through

October, is dominated by typhoon-producing thunderstorms. Seventy

s i x percent of t h e t o t a l r a i n f a l l occurs during t h i s period [Water

Resources Planning Commission, 19801. During t h e dry season, north-

e a s t monsoons produce t h e major i ty of t h e p r e c i p i t a t i o n . However,

streamflow i n t h e dry season is i n s u f f i c i e n t t o supply t h e a g r i -



Unified Approach to Regional Groundwater Management 401

TABLE 1. Recharge Parameters f o r t h e Yun Lin Basin

Recharge Rates . Recharge x 1?i4 m/d ~

Zone November December January February March Apr i l

c u l t u r a l water demands of t h e basin. Typical ly , groundwater extrac-

t i o n s account f o r more than 90% of t h e t o t a l water usage during

t h e dry season. A s a r e s u l t , it is during t h e d ry season t h a t t h e

groundwater system i s most highly s t r e s s e d . For t h i s reason, t h e

groundwater pumping p a t t e r n is determined during t h i s period.

A Galerkin f i n i t e element s imulat ion model was developed t o

p red ic t t h e hydrau l ic head d i s t r i b u t i o n i n t h e Yun Lin system

and t o generate t h e response equations f o r t h e opt imizat ion a n a l y s i s

[Tsao e t a l . , 19801. The system was d i s c r e t i z e d i n t o 78 (4 by 4 km)

l i n e a r q u a d r i l a t e r a l elements; t h e system has 101 nodal points. The

f i n i t e element g r i d f o r t h e basin is d e t a i l e d i n Figure 2.

The va l ida t ion and c a l i b r a t i o n of t h e model is discussed by

W i l l i s [1981], and Tsao et a l . [1980]. The model's groundwater and

hydrologic parameters a r e summarized i n Tables 1, 2 , and 3. The

TABLE 2. Mean Dry Season Hydrology

Mean Mean I r r i g a t i o n P r e c i p i t a t i o n , Surface Water,* Water Target,*

System mm m3/dry season m3 /dry season

Cho Shui 194. S i Lo 232. Fu Wei 212. Tou Liu 355.



Groundwater HydrauZics

TABLE 3. Hydraulic Parameters of t h e Yun Lin Basin

Transmiss ivi ty , Storage Mate r ia l Zone m2/d Coef f i c ien t

demand d a t a , which represen t a v a r i e t y of cropping p a t t e r n s i n t h e

Yun Lin Basin, was obtained from t h e Yun Lin I r r i g a t i o n Associa t ion

[KO, personal communication, 19811.

Model P r e l i m i n a r i e s

I n i t i a l l y , t h e dynamic response equat ions of t h e a q u i f e r system

a r e generated using a s e r i e s of Matr ix Eigensystem Routines [1976].

The response equat ions analyzed t h e hydrau l i c response of t h e aqui-

f e r system during t h e November through Apr i l dry season. The res-

ponse equat ions incorporated t h e time-dependent boundary cond i t ions ;

t h e s e condi t ions were expressed a s piecewise l i n e a r func t ions of

t ime over t h e 180-day planning period.

Two ob jec t ives were considered i n t h e ana lys i s : (1) maximize

t h e sum of t h e hydrau l i c heads over a l l t h e planning per iod and

(2) minimize t h e t o t a l water d e f i c i t f o r a l l i r r i g a t i o n systems.

The f i r s t o b j e c t i v e is a l i n e a r s u r r o g a t e f o r minimizing t h e ground-



Unified Approach to Regional Groundwater Management 403

TABLE 4. Pumping Rates

Well S i t e (Node Number)

22 5 0 58 6 6 8 4 9 2 9 5

Jan .-Feb.

15000 15000 2821. 6837. 2804. 14739 869.

March-April

15000 15000 7630. 15000 14842. 15000 3333.

Constant Pumping

15000 12270. 1834. 5594. 3383. 15000. 716.

water ex t rac t ion c o s t s ; t h e l a t t e r o b j e c t i v e r e f l e c t s t h e l o s s e s

from decreased a g r i c u l t u r a l production. I n t h i s prel iminary analy-

sis t h e o b j e c t i v e weights were both s e t t o one, i n d i c a t i n g equal

preference f o r t h e ob jec t ives . The hydrau l i c head was a l s o bounded

a t -20 m t o r e f l e c t cu r ren t groundwater condi t ions .

A g r i c u l t u r a l Production

The r e s u l t i n g l i n e a r opt imizat ion model has 225 c o n s t r a i n t s and

438 dec i s ion v a r i a b l e s (no t including upper and lower bounds on t h e

head values and pumping r a t e s ) . The APEX-111 la rge-sca le optimiza-

t i o n package was used t o s o l v e t h e model [Control Data Corporation,

19801. Typical s o l u t i o n t imes averaged 800 CPU seconds; c e n t r a l

memory requirements a r e approximately 200K ( o c t a l ) .

TABLE 5. Pumping Rates

Well S i t e (Node Number) Nov.-Dec.

22 50000. 50 2982 58 0. 6 6 9612. 8 4 5244. 9 2 16327. 9 5 0.

Mar ch-April

50000. 1830. 2555. 7663. 11716. 18688. 3415.

Constant Pumping

50000. 2986. 2124. 6334. 3716. 15683. 765.



404 Groundwater HydrauZics

TABLE 6 . I r r i g a t i o n D e f i c i t s

Constant Pumping Nov.-Dec. Jan.-Feb. March-April

Cho Shui 3761255. 3761255. 3723352. 3648026. S i Lo 2943389. 2943389. 2943389. 2943389. Fu Wei 739575. 789575. 789575. 789575. Tou Liu 5414156. 5414156. 5411427. 5411427.

Tot a 1 12908375. 12908375. 12872743. 12792417.

Model Resu l t s

The r e s u l t s of t h e opt imizat ion analyses f o r t h e two d i f f e r e n t

combinat ions of pumping upper bounds a r e summarized, f o r s e l e c t e d

w e l l s and s i t e s i n Tables 4 and 5 . Several th ings a r e apparent from

t h e t a b l e s . F i r s t , given t h e opportuni ty t o pump more, t h e model

increased pumpage i n those regions which a r e more h igh ly permeable.

A s a r e s u l t , ex t rac t ions a r e increased i n c e r t a i n a reas , whi le they

a r e reduced i n t h e l e s s permeable regions of t h e aqu i fe r . For ex-

ample, consider node 92. The pumping r a t e has been increased i n

t h e f i r s t and t h i r d per iods wi th a minimal change i n t h e pumping

occurr ing during t h e second planning period. This is with t h e

i d e n t i c a l lower bound r e s t r i c t i o n on t h e head values.

Second, t h e a b i l i t y t o shu t o f f t h e pumps t o a l low recovery of

t h e head l e v e l s , a l s o is e f f e c t i v e i n inc reas ing t h e y i e l d of t h e

aqu i fe r . This, i n conjunction with increased pumping from t h e

more permeable regions of t h e a q u i f e r , has t h e e f f e c t of inc reas ing

t h e groundwater y i e l d without v i o l a t i n g t h e minimum head r e s t r i c -

t i o n s i n t h e basin.

Third , i n comparison with a constant dry season pumping p a t t e r n ,

t h e groundwater y i e l d can be s i g n i f i c a n t l y increased. For example,

Tables 4 and 5 show t h e optimal constant pumping schedule [Wi l l i e

and Liu, 19811. The corresponding water d e f i c i t s , f o r a l l p o s s i b l e

cropping p a t t e r n s , a r e represented i n Tables 6 and 7 . I n comparison



Unified Approach t o Regional Grounduater Management 405

TABLE 7. I r r i ga t ion Def ic i t s

Constant Pumping Nov.-Dec. Jan.-Feb. March-April

Cho Shui 3691368. 3699342. 3621014. 3495427. S i LO 2663389. 2663389. 2663389. 2663389. Fu Wei 613735. 607227. 582389. 579575. Tou Liu 4994501. 4989748. 4994958. 4999813.

Tot a1 11962993. 11959756 11861750. 11738204.

with t he constant pumping pat tern, t he t rans ien t schedule reduces

t h e overal l d e f i c i t i n t he second and th i rd operat ional periods by

36,000 and 116,000 d i d . The s i t ua t ion is more dramatic when t h e

pumping upper bound is increased t o 50,000 m3/d. The water d e f i c i t

i s reduced i n each operational period. In t he f i r s t period, t h e

d e f i c i t decreases by 3200 m3/d (Tou Liu and Fe Wei regions).

This is balanced by an increase i n t h e d e f i c i t i n t he Cho Shui

area. The poten t ia l d e f i c i t i n t he second period, however, is

reduced by over 100,000 d / d . Pumping has increased i n t he Cho

Shui and Fe Wei i r r i g a t i o n d i s t r i c t s f i n a l l y during the t h i r d

period. The d e f i c i t has been decreased by 224,000 m3/d, again

primarily from increased extract ions i n Cho Shui and Fe Wei. The

s igni f icant r e su l t is t h e increased y ie ld does not degrade t h e

aqui fer below the current groundwater conditions, even with t h e

increased well capacity of t h e system.

Conclusions

This paper has presented a unif ied approach t o groundwater

management using an optimization methodology. The optimal planning

models a r e predicated on the response equations of t h e aqui fer

system. These same equations, which normally would be used i n a

simulation approach, can be incorporated d i r e c t l y within t h e frame-

work of optimization modeling. In contrast t o simulation modeling,



t h e optimization approach i d e n t i f i e s t h e optimal planning o r opera-

t i o n a l pol icies . In conjunction with mult iobject ive programming

techniques, t h e system trade-offs and t h e s e t of noninf e r i o r solu-

t i ons can a l s o be iden t i f ied . The methodology has been appl ied t o

Yun Lin Basin, Taiwan. Groundwater ex t rac t ion r a t e s were determined

f o r two scenarios , r e f l ec t i ng a l t e r n a t i v e groundwater development

scenarios. The r e s u l t s demonstrate t he u t i l i t y of optimization

modeling i n ident i fying t h e po t en t i a l s a f e y ie ld of regional

groundwater systems.

References

Aguado, E., and I. Remson, Ground-water hydraulics i n aqu i f e r management, J. Hydraul. Div. Am. Soc. Civ. Eng., S ( H Y l ) , 103- 118, 1974.

Bellman, R., Introduct ion t o Matrix Analysis, McGraw-Rill, New York, 1960.

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Willis, R., and P. Liu, Optimization model f o r groundwater planning, J. Water Resour. Plann. Manage. Div. Am. Soc. Civ. Eng., i n press, 1981.

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