Rates of Physical and Chemical Processes in a Carbonate Aquifer

3 Rates of Physical and Chemical Processes

Downloaded by UNIV OF MASSACHUSETTS AMHERST on October 7, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch003

in a Carbonate Aquifer WILLIAM BACK and BRUCE B. HANSHAW U. S. Geological Survey, Washington, D. C. 20242

For much

of the Tertiary

carbonate

ida, the velocity

of ground-water

meters per year.

Water

calcite

in about 4000 carbon-14

dolomite amount

it attains

produced

processes with carbon-14 the total entropy production The

equilibrium

values

from thermal

for the system as a function range from

about

Flor

undersatu-

As the

water

with respect

years and with years.

respect

Combining

chemical

ages provides

(excluding

area is

and dolomite.

in about 15,000 carbon-14 of entropy

system of

ranges from 2 to 8

in the recharge

rated with respect to both calcite moves downgradient,

aquifer

flow

and

to to the

physical

an approximation

of

energy from heat flow) of time and

—2 to 7

years for various flow paths of about 100

distance.

mcal/kg/°K/1000 km.

A g e n e r a l reference base for i r r e v e r s i b l e processes is p r o v i d e d b y e n t r o p y p r o d u c t i o n w h i c h serves as a u n i f y i n g c o n c e p t r e l a t i n g changes i n b o t h p h y s i c a l a n d c h e m i c a l energy. D i s t r i b u t i o n of e n t r o p y p r o d u c t i o n p r o v i d e s a n i n t e g r a t i n g v a r i a b l e for use i n e v a l u a t i n g the r e l a t i v e i m p o r tance of p h y s i c a l a n d c h e m i c a l processes at p o i n t s w i t h i n a system or b e t w e e n t w o h y d r o l o g i e systems. B e c a u s e the c o n c e p t of e n t r o p y is g e n e r a l l y not f a m i l i a r to h y d r o l o gists, a b r i e f i n t r o d u c t i o n is p r o b a b l y i n order. A t h o r o u g h a n d rigorous e x p l a n a t i o n c a n b e o b t a i n e d f r o m s t a n d a r d w o r k s s u c h as those b y F a s t ( I ) , Fitts (2), Katchalsky and Curran ( 3 ) , Klotz (4), L e w i s and R a n dall (5), and Prigogine (6).

A statement of the s e c o n d l a w of t h e r m o

d y n a m i c s is g e n e r a l l y u s e d as a d e f i n i t i o n of e n t r o p y of a system as f o l l o w s : dS ^ DQ/T,

w h e r e dS is a n i n f i n i t e s i m a l c h a n g e i n e n t r o p y for

a n i n f i n i t e s i m a l p a r t of a process c a r r i e d out r e v e r s i b l y , DQ

is the heat

a b s o r b e d , a n d Τ is the absolute t e m p e r a t u r e at w h i c h the heat is a b sorbed.

I n one sense, e n t r o p y is a m a t h e m a t i c a l f u n c t i o n for the t e r m 77 In Nonequilibrium Systems in Natural Water Chemistry; Hem, J.; Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

78

NONEQUILIBRIUM SYSTEMS IN N A T U R A L WATERS

DQ/T,

w h i c h is a n exact d i f f e r e n t i a l , whereas DQ

alone cannot b e i n t e -

g r a t e d w i t h o u t h a v i n g a p a t h specified (4, p. 1 0 1 ) . T h a t is, DQ/T

is b o t h

a n extensive v a r i a b l e a n d a t h e r m o d y n a m i c f u n c t i o n a n d merits a s y m b o l a n d name—i.e., e n t r o p y , w h i c h comes f r o m the G r e e k w o r d m e a n i n g "evolution." T h e s e c o n d l a w of t h e r m o d y n a m i c s is often stated to be the l a w of d i s s i p a t i o n or d e g r a d a t i o n of energy; h o w e v e r , this c a n l e a d to c o n f u s i o n Downloaded by UNIV OF MASSACHUSETTS AMHERST on October 7, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch003

because it seems to v i o l a t e the first l a w of t h e r m o d y n a m i c s , a statement of c o n s e r v a t i o n of energy.

W h e n the second l a w is stated i n the a b o v e

f o r m , i t is r e a l l y r e f e r r i n g to the d e g r a d a t i o n of the " u s e a b l e " energy of a system. E n t r o p y is therefore a n i n d i c a t i o n of the d e g r a d a t i o n of a system or a n i n d e x of the exhaustion of a system (4, p. 1 3 0 ) . It f o l l o w s l o g i c a l l y that the c o m b i n a t i o n of a l l spontaneous

reactions

w i t h i n a n a t u r a l system w i l l t e n d to increase the e n t r o p y of that system, a n d this is the basis for the statement that e n t r o p y of the u n i v e r s e is s t r i v i n g t o w a r d a m a x i m u m . A l t h o u g h energy a n d e n t r o p y are expressed i n s o m e w h a t s i m i l a r u n i t s , calories per m o l e for energy a n d calories p e r m o l e per degree for e n t r o p y , c o n f u s i o n arises i f t h e y are t h o u g h t of as h a v i n g s i m i l a r attributes or characteristics. A s K l o t z (4, p. 129)

points

out, one c a n t h i n k of energy as b e i n g a k i n d of m a t e r i a l f l u i d , a n d h e n c e it flows f r o m one area to another a n d is conserved.

E n t r o p y , o n t h e other

h a n d , m u s t b e v i e w e d as a n i n d e x of c o n d i t i o n o r character r a t h e r t h a n as the measure of content of some i m a g i n a r y f l u i d a n d is the i n d e x of c a p a c i t y for spontaneous change.

E n t r o p y s u m m a r i z e s i n a concise f o r m

the possible w a y s i n w h i c h the v a r i a b l e s of t e m p e r a t u r e , pressure, a n d c o m p o s i t i o n m a y c h a n g e i n n a t u r a l processes. O n e of the f u n d a m e n t a l tasks r e q u i r e d to a c h i e v e the u l t i m a t e g o a l of h y d r o g e o l o g y is to u n d e r s t a n d the controls o n energy d i s t r i b u t i o n a n d t r a n s f o r m a t i o n w i t h i n a n a q u i f e r system. If this is a c c e p t e d , it t h e n becomes the h y d r o l o g i s t s ' role to b r i n g together i n t o one c o n c e p t the fluxes a n d forces of the c h e m i c a l reactions, of the h y d r o d y n a m i c flow paths, a n d of heat.

T h i s idea was clearly articulated a n d developed b y G . B.

M a x e y (7, p. 1 4 5 ) , w h o stated i n p a r t : " A q u i f e r systems h a v e b e e n s t u d i e d b y three separate m e t h o d s of analysis: (1) h y d r o d y n a m i c , utilizing a distributed potential system; (2) h y d r o c h e m i c a l , u s i n g parameters of w a t e r q u a l i t y ; a n d ( 3 ) h y d r o t h e r m a l , u s i n g d i s t r i b u t i o n a n d gradients of t e m p e r a t u r e . T h e v a r i o u s approaches h a v e b e e n d i c t a t e d l a r g e l y b y the s p e c i a l i z e d t r a i n i n g a n d experience of the i n d i v i d u a l research w o r k e r . H o w e v e r , the c o m p l e x i t y of present h y d r o l o g i e p r o b l e m s n o w requires b r i n g i n g together the v a r i o u s aspects i n t o a single c o n c e p t of a f u n c t i o n i n g s y s t e m . " It f o l l o w s that one of the f u n d a m e n t a l objectives of

hydrogeochem-

istry is to evaluate the r e l a t i v e significance of v a r i o u s processes that c o n t r o l the t o t a l e n e r g y d i s t r i b u t i o n a n d energy d i s s i p a t i o n w i t h i n a h y -

In Nonequilibrium Systems in Natural Water Chemistry; Hem, J.; Advances in Chemistry; American Chemical Society: Washington, DC, 1971.

3.

BACK A N D HANSHAW

drologic

system.

Carbonate

79

Aquifer

Classical "thermostatics" c a n provide

description of the functioning of a hydrogeochemical

only a partial

system, a n d i t is

necessary to a p p l y t h e p r i n c i p l e s of i r r e v e r s i b l e or n o n e q u i l i b r i u m t h e r modynamics. I n t h e f u n c t i o n i n g of a carbonate a q u i f e r , r a i n f a l l infiltrates t h r o u g h the s o i l zone, becomes c h a r g e d w i t h c a r b o n d i o x i d e , moves to t h e w a t e r table, dissolves

soluble m i n e r a l s of t h e a q u i f e r , increases

i n chemical


c o n c e n t r a t i o n , a n d continues to m o v e to d e e p e r parts o f the a q u i f e r , e v e n t u a l l y to discharge to t h e ocean. A l l of these c h e m i c a l a n d p h y s i c a l p r o c esses a r e i r r e v e r s i b l e reactions a n d c a n b e t h o r o u g h l y u n d e r s t o o d

only

b y t h e a p p l i c a t i o n of p r i n c i p l e s of i r r e v e r s i b l e t h e r m o d y n a m i c s .

The

processes a n d reactions c o u l d b e f o r m u l a t e d a n d expressed

i n energy

terms, b u t i t i n t u i t i v e l y a p p e a r e d m o r e s i m p l e to us to b r i n g

together

the p r o d u c t s o f these processes t h r o u g h t h e c o n c e p t o f e n t r o p y r a t h e r t h a n t h r o u g h a n energy f u n c t i o n .

Line of equal head above \ sea level, in meters Area of principal recharge

Figure

1.

Pnncipal artesian aquifer of central Florida, major recharge (after Ref. 9, plate 12)

showing


area of

80


Hydrogeology C e r t a i n p r i n c i p l e s of i r r e v e r s i b l e t h e i m o d y n a m i c s c a n be a p p l i e d b y c o n s i d e r i n g the interrelations b e t w e e n geology, g r o u n d - w a t e r

flow

pat-

tern, a n d c h e m i c a l character of w a t e r i n the F l o r i d i a n p e n i n s u l a .

The

p r i n c i p a l artesian a q u i f e r of F l o r i d a consists chiefly of T e r t i a r y limestone, w i t h m i n o r amounts of d o l o m i t e , a n d ranges i n age f r o m m i d d l e E o c e n e Downloaded by UNIV OF MASSACHUSETTS AMHERST on October 7, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch003

to m i d d l e M i o c e n e .

It is one of the most extensive limestone aquifers

i n the U n i t e d States. T h e T e r t i a r y limestones c r o p out i n n o r t h - c e n t r a l F l o r i d a and i n a broad belt extending from western F l o r i d a through southeastern A l a b a m a , G e o r g i a , a n d southeastern C a r o l i n a , a p p r o x i m a t e l y p a r a l l e l i n g the F a l l L i n e . T h e O c a l a L i m e s t o n e of late E o c e n e age is one of the most p r o d u c t i v e w a t e r - b e a r i n g formations of the p r i n c i p a l a q u i f e r (8, p. 3 1 ) . F i g u r e 1 shows the height of the energy surface i n meters a b o v e sea level.

T w o m o u n d s t e n d to d o m i n a t e the g r o u n d - w a t e r flow of c e n t r a l

F l o r i d a : one near the center of the m a p that is 40 meters a b o v e sea l e v e l a n d another smaller one to the west that is about 25 meters a b o v e sea level. T h e g e n e r a l p a t t e r n of flow is p r i m a r i l y d o w n the p o t e n t i o m e t r i c g r a d i e n t a n d p e r p e n d i c u l a r to the contours. A l s o s h o w n is the area of p r i n c i p a l recharge.

N o r t h of the

two

m o u n d s , the o v e r l y i n g sediments that f o r m the c o n f i n i n g b e d are t h i n to nonexistent, a n d because of exposed limestone i n this area, a large a m o u n t of recharge occurs; h o w e v e r , the p o t e n t i o m e t r i c surface is l o w o w i n g to r a p i d discharge of the w a t e r . T h i s is a r e g i o n i n w h i c h a great d e a l of w a t e r is d i s c h a r g e d t h r o u g h m a n y springs, s u c h as S i l v e r a n d R a i n b o w S p r i n g s , that exist i n the area of the g r o u n d - w a t e r saddle f o r m e d b y the c e n t r a l m o u n d a n d a p o t e n t i o m e t r i c h i g h n o r t h of the s t u d y area. A l t h o u g h the p o t e n t i o m e t r i c surface has essentially the same g r a d i e n t a n d shape n o r t h a n d south of the m o u n d s , less recharge occurs i n s o u t h ern parts of the elongated d o m e t h a n i n the n o r t h e r n part because of a t h i c k e r c o n f i n i n g b e d a n d l o w e r t r a n s m i s s i v i t y of the a q u i f e r to the south. W a t e r that flows s o u t h w a r d discharges u p w a r d t h r o u g h the c o n f i n i n g b e d a n d also to the ocean a n d gulf.

T h e m a x i m u m g r a d i e n t of the p o -

t e n t i o m e t r i c surface of c e n t r a l F l o r i d a is about 2.5 meters per k i l o m e t e r w i t h a n average g r a d i e n t of about 1 meter per k i l o m e t e r . T h e g r o u n d w a t e r of c e n t r a l F l o r i d a comprises one major h y d r o l o g i e system, a n d it has r e c e n t l y b e e n s h o w n that a g e o c h e m i c a l coexistent w i t h the h y d r o l o g i e system (10, 11).

system is

D e p t h s of w e l l s s a m p l e d

d u r i n g this s t u d y range f r o m about 100 to 500 meters.

A b o d y of salt

w a t e r that u n d e r l i e s the entire F l o r i d a p e n i n s u l a ranges i n d e p t h f r o m near sea l e v e l at parts of the shoreline to about 700 meters i n c e n t r a l F l o r i d a . T h e interface b e t w e e n the fresh w a t e r a n d salt w a t e r forms one


3.

BACK A N D HANSHAW

Carbonate

81

Aquifer

of the b o u n d a r i e s of the fresh-water system. G e o c h e m i c a l m a p p i n g , i n c l u d i n g d i s t r i b u t i o n of c h l o r i d e , sulfate, c a l c i u m , m a g n e s i u m , a n d c a r bon-14 concentrations, shows a systematic p a t t e r n of increase d o w n g r a dient. It w a s c o n c l u d e d that, a l t h o u g h the w e l l s h a v e a range of t o t a l depths a n d o p e n i n t e r v a l s , t h e y are s a m p l i n g parts of the same h y d r o -


l o g i c a l l y - c o n n e c t e d g e o c h e m i c a l system

Figure

2.

(11).

Principal artesian aquifer, showing areas of under saturation ground water with respect to calcite and dolomite

of

Chemical Reactions I n the carbonate a q u i f e r system of c e n t r a l F l o r i d a , t w o major controls o n the c h e m i c a l c h a r a c t e r of the w a t e r are s o l u t i o n of c a l c i t e a n d of dolomite.

O n e w a y to evaluate the significance of these reactions as

c h e m i c a l controls is to d e t e r m i n e the d e p a r t u r e f r o m e q u i l i b r i u m of the w a t e r w i t h respect to e a c h of the m i n e r a l s . T o c a l c u l a t e d e p a r t u r e f r o m e q u i l i b r i u m , s o l u b i l i t y p r o d u c t s of 10"

8 3 5

and 2 Χ

10"

17

w e r e u s e d for

calcite a n d d o l o m i t e , r e s p e c t i v e l y . T h e d e p a r t u r e f r o m e q u i l i b r i u m w i t h


82


respect to c a l c i t e is s h o w n i n F i g u r e 2. T h e area of u n d e r s a t u r a t i o n c o i n cides closely w i t h the area of m a j o r r e c h a r g e ( F i g u r e 1 ) . T h e e q u i l i b r i u m b o u n d a r y outlines the e l o n g a t e d d o m e o n the p o t e n t i o m e t r i c m a p , w h i c h i n d i c a t e s that some w a t e r is r e c h a r g e d i n t o the a q u i f e r a l o n g the t o p of the d o m e a n d is t h e r e b y l o w e r i n g the a m o u n t of s a t u r a t i o n i n this area. A p r e l i m i n a r y m a p of d e p a r t u r e f r o m e q u i l i b r i u m w i t h respect

to

d o l o m i t e is also s h o w n i n F i g u r e 2. T h e area of u n d e r s a t u r a t i o n is larger Downloaded by UNIV OF MASSACHUSETTS AMHERST on October 7, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch003

for d o l o m i t e t h a n for calcite. B e c a u s e the d o l o m i t e v a l u e is exactly one h a l f that of c a l c i t e , i t f o l l o w s that i n o r d e r for a w a t e r that is i n e q u i l i b r i u m w i t h c a l c i t e to b e c o m e saturated w i t h respect to d o l o m i t e , it is o n l y necessary f o r the m a g n e s i u m c o n c e n t r a t i o n to e q u a l the c a l c i u m c o n c e n t r a t i o n ( 1 2 , 13, 14).

T h e area of recharge a n d the area of highest

p o t e n t i o m e t r i c surface s h o w that the w a t e r is u n d e r s a t u r a t e d w i t h respect to

dolomite;

downgradient,

it p r o g r e s s i v e l y

attains e q u i l i b r i u m w i t h

d o l o m i t e a n d e v e n t u a l l y becomes supersaturated. I n m a k i n g t h e r m o d y n a m i c c a l c u l a t i o n s to d e t e r m i n e d e p a r t u r e f r o m equilibrium, thermodynamic dolomite were used.

d a t a for p u r e s t o i c h i o m e t r i c c a l c i t e

and

H o w e v e r , minéralogie a n d x-ray e x a m i n a t i o n of

a q u i f e r m a t e r i a l has s h o w n that the c a l c i t e m a y h a v e several m o l e percent m a g n e s i u m ; the d o l o m i t e t h a t occurs i n the system is g e n e r a l l y c a l c i u m r i c h (IS).

T h e r e f o r e , b o t h of these m i n e r a l s i n the n a t u r a l state have a

h i g h e r free e n e r g y a n d h e n c e a s o m e w h a t h i g h e r s o l u b i l i t y t h a n the p u r e minerals.

T h u s , p a r t of the s u p e r s a t u r a t i o n that w e

have

calculated

m a y be more apparent than real. Rates of Flow and Chemical Reactions F o r the past several years, w e h a v e b e e n w o r k i n g to evaluate the r a d i o c a r b o n t e c h n i q u e for d a t i n g g r o u n d w a t e r ; that is, to d e t e r m i n e the a m o u n t of t i m e the w a t e r has b e e n out of contact w i t h the ( 1 5 , 1 6 , 17).

atmosphere

T h i s is d o n e b y means of the carbon-14 a c t i v i t y of the d i s -

s o l v e d c a r b o n a t e species.

Results of p a r t of this w o r k give t h e age

w a t e r as a f u n c t i o n of p o s i t i o n i n the a q u i f e r system.

of

I n the recharge

area, there are waters of m i x e d o r i g i n , a n d the age varies a c c o r d i n g to the a m o u n t of m i x i n g of e x c e e d i n g l y y o u n g w a t e r w i t h s o m e w h a t

older

w a t e r . H o w e v e r , d o w n g r a d i e n t f r o m the area of p r i n c i p a l r e c h a r g e , the age of the w a t e r increases i n a systematic m a n n e r . T h u s , b y c o m b i n i n g results f r o m r a d i o c a r b o n c o n c e n t r a t i o n w i t h changes i n the c h e m i c a l a n d p h y s i c a l parameters of the system, rates of c h e m i c a l a n d p h y s i c a l p r o c esses w h i c h o c c u r w i t h i n a system m a y be d e r i v e d . W i t h i n the recharge area, the m a x i m u m a p p a r e n t age of the m i x e d w a t e r is a p p r o x i m a t e l y 5000 years.

D o w n g r a d i e n t f r o m the recharge a r e a , the w a t e r increases

to a p p r o x i m a t e l y 30,000 years before present, w h i c h is the oldest f o u n d i n that p a r t of the a q u i f e r system.


age

3.

BACK A N D HANSHAW


83°

83°

Carbonate 82°

83

Aquifer 81°

82°

801

81°

80°

Figure 3. Residence time of water within the aquifer and velocity of ground-water flow; values at the arrow tips are averages for entire flow paths

T h e values o n the flow lines i n F i g u r e 3 w e r e c a l c u l a t e d f r o m r a d i o c a r b o n dates a n d i n d i c a t e velocities i n meters per year for v a r i o u s segments a l o n g p a r t i c u l a r flow p a t h s , w h i c h are s h o w n b y the h e a v y lines w i t h a r r o w s . T h e average values for the entire p a t h r a n g e f r o m

about

2.5 meters per year to 6.5 meters per year. A l o n g short reaches, the range of velocities is a b o u t 1.5 to 8.5 meters per year. I n a d d i t i o n to e s t i m a t i n g v e l o c i t y of g r o u n d - w a t e r

flow,

carbon-14

concentrations p e r m i t e s t i m a t i o n of the rate of s o l u t i o n a n d p r e c i p i t a t i o n of c a r b o n a t e m i n e r a l s . T h e a q u i f e r is c o m p o s e d of a p p r o x i m a t e l y

2/3

calcite a n d 1 / 3 d o l o m i t e d i s t r i b u t e d t h r o u g h o u t the section. S a t u r a t i o n w i t h respect to c a l c i t e occurs rather r a p i d l y , a n d it is o n l y i n areas of p r i n c i p a l recharge that u n d e r s a t u r a t e d waters are g e n e r a l l y f o u n d ( F i g ures 1 a n d 2 ) .

H o w e v e r , the k i n e t i c s of d o l o m i t e f o r m a t i o n a n d d i s s o l u -

t i o n are q u i t e slow, a n d the area of u n d e r s a t u r a t i o n extends

farther

d o w n g r a d i e n t t h a n does the area of calcite u n d e r s a t u r a t i o n . B y

com-

b i n i n g age of w a t e r f r o m F i g u r e 3 w i t h s a t u r a t i o n b o u n d a r i e s of c a l c i t e a n d d o l o m i t e f r o m F i g u r e 2, a n a p p r o x i m a t i o n c a n be o b t a i n e d for the


84


t i m e r e q u i r e d for w a t e r to b e c o m e saturated w i t h these m i n e r a l s .

The

results s h o w that w a t e r attains e q u i l i b r i u m w i t h respect to calcite i n a b o u t 4000 carbon-14 years a n d w i t h respect to d o l o m i t e i n about

15,000

carbon-14 years. Rate of Entropy

Production


T h u s f a r i n this d i s c u s s i o n , these

fluid-filled

formations of

Florida

h a v e b e e n c o n s i d e r e d as p a r t of a geologic system, a h y d r o l o g i e system, a n d as a c o e x i s t i n g g e o c h e m i c a l system. It w o u l d seem d e s i r a b l e to c o m b i n e the results of the v a r i o u s n a t u r a l processes i n t o one u n i f y i n g concept. T h e i n p u t to the c o m b i n e d system occurs o n the p o t e n t i o m e t r i c h i g h s i n the f o r m of r a i n f a l l c o n t a i n i n g m i n o r amounts of t o t a l d i s s o l v e d solids. I n i t i a l changes i n w a t e r c h e m i s t r y o c c u r w i t h i n the s o i l zone w h e r e the w a t e r is c h a r g e d w i t h large amounts of C 0

2

gas.

This C 0 - r i c h water

percolates into the g r o u n d - w a t e r system w h e r e the C 0

2

2

attacks the car

bonate m i n e r a l s . T h i s is a n i r r e v e r s i b l e c h e m i c a l process w h e r e b y C0

2

the

i n the w a t e r reacts w i t h the m i n e r a l s a n d b r i n g s t h e m i n t o solution. Likewise, simple gravitational movement

of w a t e r f r o m

potentio

m e t r i c highs to oceanic base l e v e l is a n i r r e v e r s i b l e p h y s i c a l process w h i c h p r o d u c e s a loss of p o t e n t i a l energy.

T h e basis for e v a l u a t i n g energy d i s

t r i b u t i o n of a g r o u n d - w a t e r system is the p o t e n t i a l theory best e x p l a i n e d in a classical paper b y H u b b e r t ( 8 ) .

P o t e n t i a l is c o m p o s e d of the s u m

of t w o terms, a g r a v i t a t i o n a l p o t e n t i a l energy a n d a pressure energy.

Po

t e n t i a l is e q u a l to the w o r k r e q u i r e d to t r a n s f o r m a u n i t of mass of

fluid

f r o m a n a r b i t r a r i l y chosen s t a n d a r d state to the state at the p o i n t u n d e r c o n s i d e r a t i o n (18,

p. 7 9 7 - 8 ) .

F o r the s t a n d a r d state, it is c o n v e n i e n t to

use a n e l e v a t i o n of zero, a pressure of 1 a t m , a n d a v e l o c i t y of zero.

Po

t e n t i a l , φ, for g r o u n d w a t e r c a n be expressed as f o l l o w s (19, p. 1959) Φ =

gz +

7

(1)

w h e r e g is a c c e l e r a t i o n o w i n g to g r a v i t y , ζ is e l e v a t i o n , Ρ is gage pressure, a n d ρ is density.

I n almost a l l instances, the k i n e t i c energy of

g r o u n d w a t e r is n e g l i g i b l e because of the l o w velocities of

flow.

flowing "Total

h e a d " as u s e d b y h y d r o l o g i s t s is r e l a t e d to " p o t e n t i a l " b y the expression φ =

gh, w h e r e h is h e a d . A l t h o u g h it m a y i n t u i t i v e l y seem that the t o t a l

p o t e n t i a l c o u l d i n c l u d e terms other t h a n g r a v i t y a n d pressure to reflect c h e m i c a l a n d t h e r m a l energy changes,

H u b b e r t s p o t e n t i a l c o n c e p t is

r e s t r i c t e d to m e c h a n i c a l energy o n l y a n d is so u s e d i n this p a p e r . H e a d is a n intensive state v a r i a b l e a n d is i n d e p e n d e n t of the process that p r o d u c e s a c h a n g e i n h e a d . T h u s , to c a l c u l a t e energy loss f r o m

flow,

a k n o w n r e v e r s i b l e process c a n replace the u n k n o w n i r r e v e r s i b l e process.


3.

BACK A N D HANSHAW

Carbonate

Aquifer

85

H e a d loss thus represents useable energy lost f r o m the system a n d m a y be t h o u g h t of as a measure of c h a n g e i n e n t r o p y . L i k e w i s e , the i r r e v e r s i b l e process of d i s s o l v i n g m i n e r a l s i n the a q u i f e r system has a n e n t r o p y c h a n g e associated w i t h i t . O n e w a y i n w h i c h t h e p h y s i c a l a n d c h e m i c a l processes w i t h i n s u c h a system c a n be c o m p a r e d is t h r o u g h use of e n t r o p y concepts. B e c a u s e none of the p o t e n t i a l energy of a g r o u n d - w a t e r system is Downloaded by UNIV OF MASSACHUSETTS AMHERST on October 7, 2015 | http://pubs.acs.org Publication Date: June 1, 1971 | doi: 10.1021/ba-1971-0106.ch003

c o n v e r t e d to k i n e t i c energy, a l l the e n e r g y is t r a n s f o r m e d to heat w h i c h is a b s o r b e d b y the system. T h e r e f o r e , changes i n e n t r o p y o w i n g to h e a d loss, w h i c h c a n b e t r e a t e d as a r e v e r s i b l e process, are o b t a i n e d b y c a l c u l a t i n g the changes i n p o t e n t i a l energy associated w i t h flow t h r o u g h the system. T h i s p r o v i d e s a d e t e r m i n a t i o n of m i n i m u m e n t r o p y p r o d u c t i o n c a u s e d b y c h a n g e i n a l t i t u d e . W h e n i t becomes possible to separate a l l sources of heat to the system ( e a r t h heat flow, solar r a d i a t i o n , heats of s o l u t i o n a n d p r e c i p i t a t i o n , a n d f r i c t i o n a l heat p r o d u c t i o n ) , the a d d i t i o n a l

Figure

4.

Distribution

of entropy change resulting from head loss within the aquifer


86


e n t r o p y p r o d u c t i o n c a n b e c o m b i n e d w i t h the m i n i m u m to o b t a i n t o t a l e n t r o p y p r o d u c e d f r o m p h y s i c a l processes. K i l o g r a m - m e t e r s c a n b e c o n v e r t e d r e a d i l y to m i l l i c a l o r i e s ( m e a l ) p e r k i l o g r a m as f o l l o w s 1 k g - m e t e r = 2.34 Χ 1 0 m e a l

(2)

3


T o convert to e n t r o p y b e t w e e n a n y t w o p o i n t s Ammeters] X 2.34 Χ 1 0 m e a l 3

Entropy [mcal/kg/°K] =

O

R

k

g

"

m

e

t

e

r

(3)

S

w h e r e Ah is loss i n h e a d b e t w e e n t w o p o i n t s . T h e t e m p e r a t u r e of g r o u n d w a t e r i n this system ranges f r o m a b o u t 23 ° C i n the r e c h a r g e area to a m a x i m u m of a b o u t 28 ° C i n the deepest p a r t of the system. A s i m p l e sensitivity test c a n b e m a d e as f o l l o w s : If Τ = =

30 m , t h e n f o r Τ =

m e a l ; for Τ = a n d for Τ =

298.16° db 2 ° K a n d h e a d

296.16°K ( 2 3 ° C ) , the c a l c u l a t e d e n t r o p y is 237.0

298.16°K ( 2 5 ° C ) , the c a l c u l a t e d e n t r o p y is 235.5 m e a l ; 300.16°K ( 2 7 ° C ) , the c a l c u l a t e d e n t r o p y is 234.0 m e a l .

T h i s suggests t h a t o v e r the n a r r o w r a n g e of o b s e r v e d t e m p e r a t u r e s , the e n t i r e system m a y b e a p p r o x i m a t e d b y a s s u m i n g a n i s o t h e r m a l system at 25 ° C . F o r this p r e l i m i n a r y s t u d y , the a s s u m p t i o n of a n i s o t h e r m a l system p e r m i t s n e g l e c t i n g t h e r m a l energy transfer f r o m sources m e n t i o n e d above. T h i s topic w i l l be

rigorously

e v a l u a t e d i n a subsequent s t u d y .

T h e results of c a l c u l a t i n g e n t r o p y p r o d u c t i o n f r o m h e a d values r a n g i n g b e t w e e n elevations of 40 meters to a b o u t sea l e v e l are s h o w n o n F i g u r e 4. N o t e that the h i g h p o i n t o n the p o t e n t i o m e t r i c surface is d e s i g n a t e d as h a v i n g a zero e n t r o p y l e v e l . T h i s is the i n p u t b o u n d a r y of the system, a n d b y o u r d e f i n i t i o n , the e n t r o p y of the w a t e r a t t r i b u t e d to p o s i t i o n is zero at this p o i n t . T h e r e f o r e i n o r d e r to d e p i c t t h e e n t r o p y increase a t t r i b u t e d to d o w n g r a d i e n t flow, the e q u a t i o n w a s m o d i f i e d to , . , Change in entropy = p

,, (ft x ma

, ν 2.34 Χ 1 0 m e a l K) 0^

, . (4)

3

A s the w a t e r flows d o w n the p o t e n t i o m e t r i c surface, e n t r o p y is p r o g r e s s i v e l y p r o d u c e d b y this p h y s i c a l process to a b o u t 300 m c a l / k g / ° K . T h e m i n e r a l o g y of the F l o r i d i a n a q u i f e r consists of a p p r o x i m a t e l y 6 5 % c a l c i t e a n d 3 4 % d o l o m i t e , w i t h m i n o r amounts of g y p s u m scattered t h r o u g h the f o r m a t i o n . G y p s u m m a y b e l o c a l l y a b u n d a n t i n some parts o f the a q u i f e r system. T h e r e f o r e , o n l y three c h e m i c a l reactions n e e d b e c o n s i d e r e d to d e s c r i b e the m a j o r c h e m i c a l changes i n this system. is the s o l u t i o n of c a l c i t e b y means of w a t e r a n d s o i l C 0

2

First

gas; s e c o n d is

the s o l u t i o n of d o l o m i t e , also b y means of w a t e r a n d s o i l C 0

2


gas; a n d

3.

Carbonate

BACK A N D HANSHAW

Table I.

87

Aquifer

Standard Entropy Values" s°

Cal/°K/Mole -13.2 -28.2 22.7 29.0 16.716 4.1 22.2 37.09 46.36

Ca + aq. Mg + aq. 2

2

HCO3-

C 0 aq. H 0 S0 C a C 0 [calcite] C a M g ( C 0 ) [dolomite] C a S 0 - 2 H 0 [gypsum] 2


2

4

2

3

3

4

6

2

2

Values from Rossini et al. (20) except where noted. Value for dolomite from Stout et al. {21).

0

6

t h i r d is s o l u t i o n of g y p s u m ( C a S 0

· 2H 0)

4

t o f o r m c a l c i u m ions, s u l

2

fate ions, a n d w a t e r . A l t h o u g h sulfate r e d u c t i o n occurs w i t h i n t h e system, the s i m p l i f y i n g a s s u m p t i o n has b e e n m a d e t h a t t h e decrease i n sulfate c o n c e n t r a t i o n is n o t significant f o r these c a l c u l a t i o n s . I n o r d e r to deter m i n e t h e c h e m i c a l e n t r o p y p r o d u c t i o n of t h e system, t h e e n t r o p y of e a c h of these three reactions w a s c a l c u l a t e d u s i n g values i n T a b l e I as f o l l o w s : Calcite CaC0

+ H 0 + C0

3

2

=

Abaction

-

= Ca + + 2 H C 0 2

2 a q

(5)

3

35.7 c a l / ° K / m o l e

Dolomite CaMg(C0 ) 3

2

+ 2H 0 + 2C0 2

=

^reaction

-

= Ca + + Mg + + 4 H C 0 2

2 a q

2

3

(6)

79.9 c a l / ° K / m o l e

Gypsum CaS0

+ 2 H 0 = Ca + + S 0

4

2

2

AS eaction = r

-

4

2

" +

2H 0 2

(7)

22.1 c a l / ° K / m o l e

T h e c h a n g e i n e n t r o p y at a n y p o i n t , i , i n t h e system o w i n g to the a b o v e three equations is g i v e n b y t h e f o l l o w i n g r e l a t i o n s h i p s ASi.calcite =

AStf.calcite [m

^'St'.dolom ite ASt.gypeum ^ Ο chem

=

=

AS, alcite

Ca

—

(^Mg +

omite X AStf,gypsum

^S0 )] 4

Wjiir

X m

Rates of Physical and Chemical Processes in a Carbonate Aquifer

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