groundwater pollution - symposium - pollution des eaux...

Groundwater Pollution - Symposium - Pollution des Eaux Souterraines (Proceedings of the Moscow Symposium, August 1971; Actes du Colloque de Moscou, Août 1971): IAHS-AISH Publ. No. 103, 1975.

Groundwater flow patterns in confined aquifers and pollution*

Don Kirkham and R. R. van der Ploeg

Abstract In studying groundwater pollution one needs to know the flow pattern of the groundwater which carries the pollutants. If one knows the flow pattern in a pumped aquifer one might be able to predict the effects of a polluting source somewhere present in the flow region on the quality of the pumped water. We have developed an analytical theory that enables one to prepare a flow net for an arbitrarily shaped confined aquifer from which water is drawn by means of a well. The theory requires a knowledge of the shape of the aquifer, and the hydraulic head on the boundaries of the aquifer that are not impervious. We have prepared flow nets for a number of elliptically shaped confined aquifers, by which naturally occurring aquifers may be approximated, and we show how flow nets can be prepared for irregularly shaped aquifers and give examples. The examples include aquifers having part of their boundaries impervious.

Résumé. L'étude de la pollution des eaux souterraines rend nécessaire la connaissance de la configuration de l'écoulement des eaux souterraines qui portent les matières pollutantes. Si l'on connaît la configuration de l'écoulement dans une couche aquifère pompée, on pourra peut-être prédire les effets d'une source de pollution située quelque part dans la région d'écoulement, sur la qualité des eaux pompées. Nous avons dévloppé une théorie analytique qui permet la préparation d'un réseau orthoganol des lignes courant et des courbes isopièzes pour une couche aquifère resserrée d'une forme arbitraire d'où l'eau est puisée au moyen d'un puits. La théorie rend nécessaire une connaissance de la forme de la couche aquifère, et de la hauteur hydraulique sur les limites de la couche aquifère qui ne sont pas imperméables. Nous avons préparé des réseaux orthogonaux de lignes de courant et de courbes isopièzes pour un certain nombre de couches aquifères resserrées de forme elliptique, au moyen desquels les couches aquifères naturelles peuvent être calculées approximativement, et nous montrons comment des réseaux orthogonaux de lignes de courant et de courbes isopièzes peuvent être préparés pour les couches aquifères de forme irrégulière et nous en donnons des exemples. Les exemples comprennent les couches aquifères dont une partie des bornes est imperméable.

INTRODUCTION

The following eight widely accepted categories of water pollutants have been listed by the United States Senate Select Committee on Water Resources (US Senate Select Committee, 1960): salts and minerals; sediment; plant nutrients; organic wastes; infectious agents; organic chemical exotics; radioactivity; heat.

These eight pollutants can in general be transported by groundwater (Goldberg, 1970; LeGrand, 1970), and it is therefore important to know the flow pattern in an aquifer, particularly if the aquifer is being pumped by a well. Sometimes wells are used to dispose of pollutants (Hoopes and Harleman, 1967a, b). Here again it is important to know the groundwater flow pattern. In the literature, confined flow to a well centred in a circular aquifer is often considered (Todd, 1967). An exception is a well at the centre of an elliptical aquifer for which Polubarinova-Kochina (1962) obtained the discharge formula.

In the following we shall obtain steady-state flow nets for an aquifer that may have almost any shape. Some of the aquifer boundaries may be impervious. The hydraulic head may vary on the aquifer boundaries. Some flow nets for a semi-confined circular pumped aquifer have been given by Khan and Kirkham (1971) and Khan et al. (1971).

* Journal Paper No. J -6957 of the Iowa Agricultural and Home Economics Experiment Station, Ames, Iowa, Projects 998 and 1805.

12 Don Kirkham and R. R. van der Ploeg

THEORY FOR ELLIPSE-SHAPED AQUIFERS FOR ARBITRARY POSITIONS OF A WELL

Figure 1(a) represents a vertical section of a horizontal confined aquifer of uniform thickness h being pumped steadily by a well. We let <f> represent the hydraulic head at any point in the aquifer and let A(f> represent the head difference across the aquifer. In

/ CONFINING LAYER -*- — -/-'•

T

PUMPING WELL

7;:±±:A

////A////////;/////////. IMPERMEABLE LAYER , ,

P t R . e i

FIGURE 1. Geometrical representation of the flow region: (a) cross-sectional view; (b) plan view.

the initial analysis we take A$ to be unity as measured between any point on the external boundary of the flow region and a point on the well boundary. Figure 1(b) represents the horizontal cross-section of the aquifer when it is of elliptical shape. The theory for a pumped well in an elliptical confined aquifer has been given by Van der Ploeg et al. (1971). This theory will be recalled here and developed further to include almost any shape of external boundary. For the figure, symbols are as follows:

$/A0 = dimensionless potential or head À0 = constant head difference between well and external boundary x,y = rectangular coordinates with origin 0 at centre of the ellipse r,8 = polar coordinates with origin at centre of the well rw = radius of well a,b = semimajor and semiminor axes of ellipse c,d = x and y coordinates of well centre K = hydraulic conductivity P(r,d) = a point in the flow medium P(R,d) = a point on boundary of the ellipse h = thickness of aquifer

Groundwater flow patterns 13

The dimensionless potential <£/A0 or <p is given by

ln(r/rw) " f ( r / f l r / 2 - [ £ / ( a r ) r / 2 \ m

+m 4...

ÂNm \—T^ïi^f^^-i sin ^ (1)

where the last value of the superscript TV on a summation sign may be either even or odd and indicates the order of approximation of the solution. For N -»• °°, the solution becomes exact. The constants ANmin (1) are obtained by a modified Gram-Schmidt method described by Van der Ploegetal. (1971).

The stream function \p corresponding to <p is

, JANoB " (rla)ml2 + [rll{ar)rl2

m

i> ~K\, , , . + L ANm sm^ô lln(a/rw) m =2,4, - , / 2 , , , » 2 l

^m 4 , . . . A»>» iZtfJïy^m cos — ej (2)

The stream function \p is such that if ^(rx, 61 ) and i//(r2, 62) are any two points in the flow medium, the amount of water per unit time that flows between the two streamlines which pass through these respective points is \p(r2 , 02) — 4>(f\, #i)-

When the well is on the major axis (similarly for the minor axis) the potential function, equation (1), reduces, because of symmetry, to the form

l n ( r / 0 £ f ( r / a r - [ ' i / ( « r ) ] m l <P = ANO .-77—. + L ANm \ — - , 2 . 2 — — \ cosmd (3)

ln(a/rw) m=i { [ 1 - (>i/a2 ) ] m J where m = 1,2,.... The stream function corresponding to 0 of equation (3) is given by

$=^J_^_ + x ANm< — 7 - j T v ^ n — r s i n m e r (4)

When the well is at the centre of the ellipse, there is additional symmetry, and equation (3) becomes (m is changed to 2m) the expression

Hr/rw) » fir/a)*" - Kl(ar)]*»

ln(a!rw)+m%ANm\ l^l/a2)

and similarly equation (4) becomes

- ANO r-rr-, + T ANm\ r - T ^ - r r ^ ; — f c o s 2md (5)

, J A™6 x Y „ J W 2 m + [4/(ar)]2™\ . 1

* = *•< — — + L 4v m 1 , 2 / 2 , 2 m f an 2m0f (6) Un(a/rw) m=i I l - ( r^ / a 2 ) 2 m J J

For all the flow configurations corresponding to equations (1)™(6) there is a single equation for the well flux. If we denote this flux by Q (and the thickness of the aquifer is h and head difference A0), then Q is given by

-2irKhAN0A(p

ln(a/rw)


where the minus indicates the flow is toward the well. In equation (7) we note that Q depends only on the zeroth coefficient ANO in the modified Gram-Schmidt method; coefficients ANI,AN2,~-ANN are n o t needed.

Before going over to the development of the theory for any shape of aquifer, we shall show a few flow nets calculated from equations ( l ) - (6) for some elliptical aquifers.

FLOW NETS FOR ELLIPSE-SHAPED AQUIFERS

Figure 2 shows the flow net for a well that is neither at the centre nor on an axis of the ellipse. Equal amounts of flow pass between adjacent streamlines. When the streamlines are far apart the velocity of the water is small. Thus pollutants near a point A will move much faster to the well than pollutants in the neighbourhood of point B.

FIGURE 2. Flow net for a confined elliptical aquifer, with the well not located in the centre of the ellipse nor on an axis, with a = 1, 6 = 0.5, c = d = 0.25 and rw = 0.0025.

Stated in a different way, a steady state source of pollutant of a certain concentration and located along a unit length of aquifer boundary near point B would have a small effect on the water quality of the well as compared with the steady state source of a pollutant of the same concentration located along a unit length of aquifer boundary near point A. We can be more specific. Consider, in the figure the approximately equal distances CC' and DD' on the aquifer boundary. We keep in mind, Fig. 2 shows, that equal amounts of water per day reach the well between adjacent streamlines. If the water at CC' is polluted, then the streamline pattern shows that one-tenth of the water reaching the well per day will be polluted. (We are neglecting dispersion and diffusion phenomena.) If the water at DD' is polluted the streamline pattern shows that only one-fortieth of the water reaching the well per day will be polluted. Thus, the location of the source at DD' rather than at CC' decreases the pollution concentration in the well by a factor of four.

Figure 3 shows the flow net for a well on the major axis of the ellipse. The net shows that a polluting source near B where the streamlines are far apart would be much less dangerous than a similar steady state pollutant inside the aquifer near point A.

Figure 4 is a flow net for a well on the minor axis of the ellipse. Near point A the flow is fast; near point B the flow is slow. Pollutants near point B are less harmful than pollutants near point A.


FIGURE 3. Flow net for a confined elliptical aquifer with the well located on the major axis of the ellipse, where a = 1, b = 0.5, c = 0.5 and rw = 0.02.

/ \ a» I b-,5 rw-.0025

FIGURE 4. Flow net for a confined elliptical aquifer, with the well located on the minor axis of the ellipse, where a = 1, b = 0.5, d = 0.25 and rw = 0.0025.

Figure 5 is a flow net when the well is at the centre of the ellipse. The velocities are high near point A as compared with point B.

Figure 6 is a flow net for a well in a circular aquifer. A circle is a special case of an ellipse when the major and minor axis are equal. A pollutant near point A would be more dangerous than an equal pollutant near point B.

FLOW THEORY FOR AN IRREGULAR-SHAPED AQUIFER

To obtain the theory for an irregular-shaped aquifer with a constant head on the outer boundary, we may use the same form of potential function <t> as given by equation (1). One of the simpler forms, equations (3) and (5), cannot be used since there is no symmetry, as is evident from the example of Fig. 7.

In the modified Gram-Schmidt method we have been using (Kirkham and Powers, 1972) we need two quantities. The first quantity is the potential function <p(r, d) as it is found on the boundary. Because, on the boundary, the value of r, say R, for every


0.50

0.25

0.00

-0 .25

-

B

- I 0 0

+ *

-0.75

^~~~^C^ -el

^S<r 50 [ f

* ' f O

1 1 — ' i -0.50 - 0 . 2 5

y

Sr-

> *

A

/ 0 ~ ~~~7 ;<V /

>

/~~~ \ y *

L2?\ / /

^ \ /

—T °V w .8 \ o ^ A "

103 ^ \ j

=i \ * _ ^ _ — 4 -0.00 0.25

a = l b=.5 c = 0 d = 0

* =0

i 1 0.50 0 7 5

r w = 0 2

1 \ 1 \ 1

\l

f\ * / I

/ 1 / i

1

i

100

FIGURE 5. Flow net for a confined elliptical aquifer, with the well located at the centre of the ellipse, where a = 1, b = 0.5 and rw = 0.02.

0.50

0.00

-0.50

* * a = l b*l rw=,02 c =.5 d»0

-1.00 -0.50 0.00 0.50 1.00

FIGURE 6. Flow net for a confined circular aquifer, with the well located halfway between the centre of the circle and the outer boundary, with a = 1, rw = 0.02.

value of 6 must be assumed known, we may say that 0(r, 6) goes over to a function 4(B) which we denote by fid). OuxjXd) corresponds to the/(x) of Kirkham and Powers (1972). Because we have specified that the head on the outer boundary of the aquifer of Fig. 7 is a unit height greater than the head at the well (A0 = 1), we have the relation

/ ( 0 ) = 1 , O < 0 < 2 T T (8)

Equation (8) is the useful representation of the first quantity needed. The second quantity needed in the modified Gram-Schmidt method is a set of

coefficients umir, 6) of the A^m of equation (1). These coefficients must, however, be evaluated on the boundary of the flow medium where r becomes R (see Fig. 1(b)) so that the second quantity umir, 6) may be expressed as umid), m = 0,1,2, ...,N, 0< 6 <2n, corresponding to umix), a < x < |3, of Kirkham and Powers (1972).


POLLUTANTS ABOUT POINT A ARE LESS DANGEROUS THAN ABOUT POINT

WELL RADIUS = 0 .0025

DISTANCE FROM WELL CENTRE TO POINT P

= I.O

FIGURE 7. Flow net for an irregular shaped confined aquifer. The distance of the well centre to the point P is taken as 1, the well radius rw = 0.0025 times this unit distance.

From our definition of um(6) and equation (1) we can write for the irregular shaped aquifer problem the expression

"o(0) = In(*//W)

ln(a/rw) ' (m = 0)

for m = 0; and

J(RlaT!2~{rll(aR)r,2}cos™Ô

for m = 2, 4, ...; and

,(e): [r2wl(aR)](m+1)'2 m + l

— sin

(9)

(10)

(11) l - (4 /a 2 )< m + 1 >/ 2 2

for m = 1,3, .... In equations (9), (10) and (11) as applied to Fig. 7, Ris the distance from the well

centre to a point on the boundary of angular coordinate value 6. The angle 8 is measured counterclockwise from a line connecting the well centre to the point P shown.

With the two functions fid) and um(6) we can now evaluate the A^m of equation (1) and these ANm can then be related to the um (0) (see Kirkham and Powers (1972) appendix 2, equation (i)) by

(12) fN(d)=AN0u0(8) + AN1 Ui(6) + ... +A]wvUpf(d)

where/JV(0) is the Mh approximation to fid) of equation (8). For A^^-°°the right side of equation (12) becomes equal to fid) of equation (1) for every value oft?.

The way to find the ANm of equations (12) and (1) is shown in Kirkham and Powers (1972) and need not be discussed here except to say that two sets of constants are first to be obtained fiomfiB) and um(9). The first set of constants is denoted by wm defined by


?" wm= j f(d)um(e)de (13)

0

and the second set of constants is denoted by umn defined by

umn = ) um(e)_un(d) de (14) 0

n = 0,1,2, ...,N, n<m (15)

In equation (13)/(0) is replaced, in accordance with equation (8), by one and in equations (13) and (14) the functions um(d) and un{0) are replaced by equations (9), (10) and (11) as they are applicable. When the aquifer is of irregular shape the integrations needed to get the wm and «m„are carried out by digital computation numerically. For the flow net of Fig. 7, the interval 0 < 0 < 2ir was divided into increments of 1 °.

With the wm and umn determined (the approximation N = 15 was found to be good enough for Fig. 7), the A^m were determined, also by the digital computer, from the formulas given at the end of appendix 2 of Kirkham and Powers (1972). With the ANm known, the potential function (j> and stream function i of equations (1) and (2) were, in turn, computed by the digital computer for a sufficient number of values to plot the equipotentials shown in Fig. 7. The cost of computing the wm, the umn, ANm

sufficient values of 4>(r,d) and 4/(r,9) for the flow net, and for the plotting of the flow net itself by the automatic plotter was, for Fig. 7, 16.32 US dollars.

The method we have described for getting the flow net of Fig. 7 should work when the outer boundary of the flow region has an arbitrary potential value <t>(6) =f(d), other than the value f(d) = 1 given by equation (8). For arbitrary /(#), one would place each known head value f{6) in equation (8), instead of the value 1, for the numerical integration, and let the digital computer go to work.

FLOW THEORY FOR AN AQUIFER WHEN SOME OF THE BOUNDARIES ARE IMPERVIOUS

When parts of the aquifer boundary (in the vertical direction) are impervious, the boundary conditions become more complicated than in the problems we have considered so far. When a part of the boundary is impervious we have the zero flow regulation —K 90/9n = 0 or

d<j>/dn = 0 (16)

where $ is the hydraulic head as given by equation (1) and n is a distance measured in the normal direction to the boundary at the boundary point in question.

We may also write equation (16) in vector dot product notation, as V<t>-n = Q (17)

where V<t> (the gradient of <t>) is to be evaluated in the polar coordinate form from equation (1) and the unit normal must also be expressed in the polar coordinate form. In polar coordinates we have gradient of <f> given by

**•%** 7$>> ( 1 8 )

where r0 is a unit vector in the outward r direction and 80 is a unit vector (a distance not an angle) in the 6 (counterclockwise) direction. Let a be the angle between, the normal to a boundary point in question and the outward r direction.

Groundwater flow patterns

Then the unit vector n of equation (17) may be expressed as

n = l(cos a)r0 + l(sin a)60

Using (18) and (19) to obtain the vector dot product, we find the result

39 1 3 V0 • n = — c o s a + ~ r r sin a

or r do

Equations (16), (17) and (20) give for our boundary condition on the impervious barrier

1 dé — = —- cos a + — —- sin a :

dr dr r do 0

19

(19)

(20)

(21)

The way to introduce the boundary condition, equation (21), into the modified Gram-Schmidt method of Kirkham and Powers (1972) may be illustrated by considering the circular aquifer of Fig. 8 where one-quarter of the

POLLUTANTS MOVE FASTER NEAR POINTS A THAN NEAR POINT B

WELL RADIUS = 0 . 0 0 2 5

EXTERNAL RADIUS -1.0

BARRIER

FIGURE 8. Flow net for a confined circular aquifer, of which one quadrant of the outer boundary is impervious, and in which a = 1, rw = 0.0025.

circular boundary is impervious. Here it is convenient to take half of the impervious arc in the fourth quadrant and half in the first. Because of symmetry we need consider only the flow in the upper half of the circle. Taking the circle to be of radius a the boundary conditions are seen to be

(22)

(23)

(24)

(25)

(26)

The appropriate potential is given by equation (3) where we see that boundary condition equations (24), (25) and (26) are satisfied for any values of ANO

a nd

dèjdn = 0,

0 = 1 ,

dèjde = o,

30/30 = 0,

0 = 0 ,

r = a,

r = a,

0=0,

d=n,

r = rw,

o<e<Tr/4

TT/4<d<n

rw <r<a

rw <r <a

0<6 <n


the setANm. We shall now evaluate t h e ^ o and the/ljv>„ using the modified Gram-Schmidt method.

Equations (21) and (22) now give, because a is 0, the result

be ; r = 0 , r = a, 0<d<ir/4 (27a) dr

and we rewrite the boundary condition of equation (23) as 0 = 1 , r = a, nl4<e<ir (27b)

In potential flow theory, equations (27a) and (27b) represent a mixed boundary condition.

Differentiating equation (3) with respect to r gives

9é Mr " m{\lr)[{rlar+{rllarr\ V=ANO7TT\ Z* ÂNm ^ T T T T T T ^ cosmd (28) dr ln(fl/rw) m=i [ l - ( 4 / « ) ]

Using boundary condition equation (27a) in equation (28) (put r = a and put the left side 0 in equation (28)) yields, after multiplying both sides of the result by a the expression

1 N m\\ +(r2 la2Y"\ 0 = 4 o r 7 r - / I ANm (/L2L- cosmfl, 0 « ? < W / 4 (29)

m(alrw) m=i i.—{rw/a ) Using boundary condition equation (27) in equation (3) yields

N

\=AN0+ X ANm cosm9, n/4<9 <n, (30) m=l

To use the modified Gram-Schmidt method we look at the left sides of equations (29) and (30) and specify the boundary condition function f{9) as

0<d<n/4 (31)

ff/4<0 <7T

And we look at the right sides of equations (29) and (30) and find the um{6) function (the coefficient of Apfm) as

fl/ln(fl/rw), 0<8<nl4 Uo(e)=< (32)

[l, Tr/4<dir

for m - 0; and for m = 1,2, ... (with D as defined below)

f{m[l+(rl/a2)m]ID] cosmd, 0<d<n/4 um(d)=< (33)

[cosmd, Tt/4<6<n

D=l-fâla2y" (33a) With equations (31), (32) and (33) providing/(0) and um(6), we can now find by

equations (13) and (14a) the wm and umn and hence the A^m as before. With the A^m

found (with the help of the digital computer), the digital computer then evaluates the 6 function of equation (3) and the \p function of equation (4) at enough points so that the plotter of the digital computer can draw the flow net that has been traced and presented already as Fig. 8. For the flow net of Fig. 8, N in equations (3) and (4), etc., was taken to be 100. The high value of N was found necessary because of the


discontinuities in the right sides of equations (32) and (33) at 6 = ir/4 and r = a. The cost (at N= 100) for all the computations and automatic plotting of Fig. 8 was 23.98 US dollars. This rather high expense gives d and \p to good accuracy at points where the impermeable boundary meets the non-impermeable boundary. If one is not interested in details at the boundary discontinuity one may use N = 15 with a cost of about 2.00 US dollars for all the computer work.

Figure 9 gives a flow net like that of Fig. 8 except that half the circle rather than one quarter is impervious.

FIGURE 9. Flow net for a confined circular aquifer, of which half of the outer boundary is impervious, and in which a = 1, rw = 0.0025.

In Figs. 8 and 9 one sees that a source of pollutant at a point A would be more dangerous than at point B.

A comment may be made for the well discharge for Figs. 8 and 9 and for a circular aquifer with no portion of the boundary impervious. We let Q(0), Q(irl4) and Q(ir/2) be the flux when rw = 0.0025, and a = 1 and for an impervious boundary of 0, cnr/4 and air/2 length. We find

0(0) : (Kit 14) : ÔO/2) as 1.049:1.020:0.938

so the comment is: the well discharge is not as much affected by the impermeable boundaries as is the flow net.

The aquifers that we have dealt with in Figs. 2-9 are, except for Fig. 7, more or less idealized, to illustrate the procedure. We are using our procedures further to analyse the Ames aquifer, located at Ames, Iowa, where steady state conditions are supplied by two rivers (Ver Steeg, 1968; Akhavi, 1970). A source of pollution in this confined aquifer is being studied.

We mentioned at the outset that flow patterns in aquifers being used for waste


disposal through a well would be of interest. In Fig. 9 if the well were being used to dispose of radioactive wastes, radioactivity at points B might be small as compared with points A because of the longer time of travel and hence longer decay times for the activity reaching points B.

SUMMARY AND CONCLUSIONS

Analytical theory has been worked out for obtaining flow patterns under steady state conditions in horizontal confined aquifers of any shape that are pumped by a well. Part of the aquifer boundary may be impervious and the part of the boundary that is not impervious may have a variable head distribution. By use of the theory a number of flow nets have been presented and interpreted in regard to pollution hazards of the pumped well water. At points in the aquifer where the velocities are low (streamlines far apart) a source of pollution will not be as dangerous as an equal source placed at a location in the pumped aquifer where the velocities are high (streamlines are close together). The theory is valid for any shape of aquifer and any diameter of well and for any location of the well in the aquifer. The steady state results indicate approximately what happens under non-steady state conditions.

Acknowledgements. Work supported in part by Project B-019-Ia of the US Department of the Interior, Office of Water Resources Research.

REFERENCES

Akhavi, M. S. (1970) Occurrence, movement and evaluation of shallow groundwater in the Ames, Iowa area. Unpublished Ph.D. thesis on file at Iowa State University Library, Ames, Iowa.

Goldberg, M. C. (1970) Sources of nitrogen in water supplies. In Agricultural Practices and Water Quality, 94-124. (Editors T. L. Willrich and G. E. Smith). The Iowa State University Press, Ames, Iowa.

Hoopes, J. A. and Harleman, D. R. F. (1967a) Dispersion in radial flow from a recharge well. /. geophys. Res. 72, 3595-3607.

Hoopes, J. A. and Harleman, D. R. F. (1967b) Wastewater recharge and dispersion in porous media. Proc. Amer. Soc. Civ. Eng. 93, No. HY5, 51-71.

Khan, M. Y. and Kirkham, Don (1971) Spacing of drainage wells in a layered aquifer. Water Resour. Res. 7, 166-183.

Khan, M. Y., Kirkham, Don and Toksôz, Sadik (1971) Steady state flow around a well in a two-layered aquifer. Water Resour. Res. 7, 155-165.

Kirkham, Don and Powers, W. L. (1972) Advanced Soil Physics, chap. 4 and appendix 2, John Wiley and Sons, Inc., New York.

LeGrand, H. E. (1970) Movement of agricultural pollutants with groundwater. In Agricultural Practices and Water Quality, 303-313. (Editors T. L. Willrich and G. E. Smith). The Iowa State University Press, Ames, Iowa.

Polubarinova-Kochina, P. Ya. (1962) Theory of Ground Water Movement, 366-368. Translated from the Russian by J. M. Roger De Wiest. Princeton University Press, Princeton, New Jersey.

Todd, D. K. (1967) Groundwater Hydrology, 82, 6th ed. John Wiley and Sons, Inc., New York. US Senate Select Committee on National Water Resources (1960) Water Resources Activities in

the United States: Pollution Abatement. Comm. Print, no. 9, 86th Congress, 2nd Sess., 38 p.

Van der Ploeg, R. R., Kirkham, Don and Boast, C. W. (1971) Steady-state well flow theory for a confined elliptical aquifer. Water Resour. Res.

Ver Steeg, D. J. (1968) Electric analog model of the regolith aquifer supplying Ames, Iowa. Unpublished M.S. thesis on file at Iowa State University Library, Ames, Iowa.

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