Chapter 10 Aquifer Tests

291

CHAPTER 10 AQUIFER TESTS

101 INTRODUCTION The hydraulic properties of aquifers (T and S ) and aquitards (K' and S') must be determined for predicting groundwater movement so that an aquifer can be evaluated as a fully developed source of water.

Various laboratory

techniques are available for determining the hydraulic properties of aquifers and other materials below a water table (Todd, 1959; Bouwer, 19781, but the values thus obtained are less reliable than values obtained in field tests.

In

these field tests (called aquifer tests, or simply pumping tests), a well is

pumped at a usually constant rate and the drawdown of the piezometric surface or water table is measured at some distance from the pumped well. Ideally, the pumping well should be located at a considerable distance from any other pumped well.

It should be one that can be pumped at a

high rate and shut down at will.

The observation wells for measuring the

drawdown of water levels in response to pumping can be existing wells or piezometers installed for the purpose.

To obtain representative drawdown

data, several observation wells are desirable, located in all directions and at various distances from the pumped well.

Lohman (1972) suggests a pair of

observation wells at distances from the pumped well of one, two, and four times the thickness of the aquifer.

Each pair consists of a well penetrating

just the top of the aquifer and a second well penetrating the entire thickness of the aquifer. Economic considerations and the availability of existing wells, however, often dictate the location, depth of penetration, and number of observation wells. Water levels in observation wells are sometimes affected by factors other than the pumped well. One of these factors is a change in barometric pressure.

To correct for this, the response of water levels to barometric

292

pressure should be observed for some period prior t o pumping.

Barometric

readings a r e converted t o feet of water (feet of water = 1.13 x inches of mercury), inverted, and plotted on arithmetic paper. a r e also plotted on the same sheet of paper.

Observed water levels Significant barometric

fluctuations a r e used to compare changes in water level caused by changes in atmospheric pressure.

The rise in water level as a result of a decrease

in atmospheric pressure and t h e decline in water level as a result of an increase in atmospheric pressure are calculated.

The barometric efficiency

of the aquifer can then be computed with the equation (Jacob, 1950) BE = (Ah/pa)lOO

(10.1)

where BE is the barometric efficiency, in percent; Ah is the change in water

level resulting from a change in atmospheric pressure, in f t ; and pa is t h e change in atmospheric pressure, in f t of water.

Observed water levels are

corrected for atmospheric pressure changes during an aquifer test by obtaining a record of atmospheric pressure fluctuations and using the relation Ah = BE*Apa/lOO

(10.2)

Tides and other changes in surface-water levels can also affect water levels in observation wells.

Observed water levels can b e corrected for any

such changes that occur during an aquifer test by obtaining a record of tidal or surface-water fluctuations during t h e test and using t h e relations (Jacob, 1950) TE = (Ah/AH)100

(10.3)

Ah = TE-AWl00

(10.4)

where TE is the tidal efficiency, in percent; Ah is the change in water level resulting from a change in surface-water stage, in f t ; and A H is the change in surface-water stage, in ft. In addition t o changes in barometric pressure, tides, or other changes in surface-water levels, water levels in observation wells can be affected by

293

e a r t h tides, pumping from other wells in t h e aquifer, and recharge or depletion of

groundwater.

If

these factors are expected to a f f e c t drawdowns

significantly during aquifer tests, water levels should be measured for some time prior to pumping so t h a t trends can be extrapolated to t h e pumping period for correction of t h e measured drawdowns. 10.2

CONFINED AQUIFERS

Several procedures have been developed, steady-state as well as transient-state methods, to determine t h e T and S of confined aquifers from pumping-test data. 10.2.A

Thiem Semilogarithmic Method

The Thiem, or equilibrium, formula of Section 8.3.A T = Q I n ( r 2 / r 1 ) / 2 r ( s l - s2)

(10.5)

provides a means for determining T from t h e pumping rate Q and t h e equilibrium drawdowns s1 and s2 respectively measured in two observation wells at distances rl and r 2 from t h e pumped well.

Recall t h a t in eq. (10.5)

i t is assumed t h a t pumping has continued at a uniform rate for sufficient t i m e for t h e hydraulic system to reach equilibrium (i.e., no change in rate of drawdown as a function of time).

Although the water levels in t h e

observation wells will never reach equilibrium (see Section 8.3.A), they may approach t h e equilibrium position close enough to yield reasonably a c c u r a t e estimates of T. The procedure for applying eq. (10.5) is to select some pumping time, t, a f t e r reaching equilibrium, and on semilogarithmic paper plot for each

observation well t h e drawdowns (on t h e arithmetic scale) versus t h e distances (on t h e logarithmic scale). For eq. (10.5) to apply, t h e d a t a should lie on a straight line.

From this straight line a n arbitrary selection of s1 and s2 is

made and t h e corresponding values of r1 and r 2 recorded. Equation (10.5) can t h e n be solved f o r T. A f t e r T has been calculated, S c a n b e determined with t h e transientflow equations (9.25) or (9.30).

When using eq. (9.30), for example, t h e

coordinates of any point on t h e s-vs.-log

r plot previously described can be

294

used to solve eq. (9.30) for S. Other procedures for calculating S a r e given by Lohman (1972). 10.2.B

Theis Type-Curve Method

Recall from Section 9.3.A that the Theis equation is given by (10.6) where 2 u = r S/4Tt

(10.7)

If parameters T, S, and Q a r e constants, then s is proportional to W(u) and 2 u is proportional t o r /t. Thus it follows from eq. (10.6) and (10.7) that log s - log W(u) = log ( Q / 4 f l ) = c o n s t a n t

(10.8)

and log(t/r

2

- log ( l / u ) = log (S/4T) = c o n s t a n t

(10.9)

It is evident from eqs. (10.8) and (10.9) that a logarithmic plot of W(u) vs. l/u (called a type curve) is similar to a logarithmic plot, on t h e same scale, 2 of s vs. t / r (called a data curve). That is, t h e superposition of t h e d a t a curve on t h e segment of the type curve corresponding t o t h e data curve , displaces the s and W(u) scales by t h e constant amount log Q / ~ I T Tand displaces the t / r 2 and l/u scales by the constant amount log S/4T.

This

makes possible the explicit solution of eq. (10.6) for the unknown values of parameters T and S using pumping-test data collected from one or more observation wells. The procedure of the Theis method is as follows: 1)

Plot a type curve of W(u) vs. l/u on logarithmic paper, as in Fig.

2)

Plot a data curve consisting of observed values of s vs. t/r

9.6.

2

(or

vs. t for the case of one observation well) on logarithm paper of the same

scale as that of t h e type curve.

295

3) Superimpose t h e d a t a curve on t h e type curve (or vice versa) and move horizontally and/or vertically, keeping t h e coordinate axes of t h e two curves parallel, to a position t h a t gives t h e best f i t of t h e observed d a t a to t h e type curve. 4) Choose a n arbitrary point (called t h e matching point) anywhere on 2 t h e overlapping s h e e t s and record t h e values of W(u), l/u, s, and t/r from t h e coordinates of t h e matching point.

For computational convenience, t h e

matching point is chosen such t h a t its type-curve coordinates or data-curve coordinates are both 1 or 0.1. 5) Substitute t h e coordinates of t h e matching point into eqs. (10.6) and (10.7) and solve f o r T and S. Example.

Consider a well fully penetrating a confined aquifer and

discharging uniformly at a rate of 300 gal/min (0.668 cfs).

Drawdown

variations with t i m e are recorded in an observation well 310 f t from t h e pumped well (Table 10.1). paper (data curve).

These values of s and t are plotted on logarithmic

Values of W(u) and l/u from Table 9.2 are plotted on

another s h e e t of logarithmic paper, and a curve is drawn through t h e points Table 10.1

Measured drawdown at observation well 310 f t from pumped

well. Time a f t e r pumping started ( m i d

Drawdown in observation well (ft)

15 20 30 40 50 60 70 80 90 100 130 160 200 260

1.5 1.7 2.8 3.4 4.2 4.5 4.7 5.3 5.8 6.2 6.7 7.2 8.3 8.5

Time a f t e r pumping s t a r t e d (min) 320 38 0 500 620 740 860 1200 1500 1800 210 0 2400 3000 4000 5000

Drawdown in observation well (ft) 9.8 10 .o 10.5 ll.8 12.4 12.5 13.9 14.0 14.3 15.1 15.9 16.7 17.3 17.7

296

0.1

10

I

100

1000

VALUE OF l / u

Figure 10.1

Example of Theis type-curve method.

(type curve).

The two sheets are superposed and shifted with coordinate

axes parallel until t h e best fit of t h e data curve and t h e type curve is obtained, shown in Fig. 10.1.

An arbitrarily selected matching point gives

W(u) = 1, l/u = 10, s = 3.1 f t , and t = 90 min.

Thus from eq. (10.61,

2 (o*668 c f s ) ( l ) T = -Q w ( u ) 0.017 f t /see = 10,980 g p d / f t 4lTs - (4)(3.14)(3.1 f t ) -

and from eq. (10.7)

s = -4Ttu r

10.2 .C

2

-

2 ( 4 ) ( 0 . 0 1 7 f t / s e c ) ( 5 4 0 0 s e c ) ( O . l ) - o.ooo3

96,100 f t 2 Cooper- Jacob Semilogar i t hm ic Method

Recall t h a t for relatively large values of t and/or relatively small values of 2 r (i.e., t/r > 5S/T or u < 0.05), eq. (10.6) can be expressed as

s = (2.3Q/4rT) log ( 2 . 2 5 T t / r 2 S)

(10.10)

It is evident t h a t a plot of s vs. log r 2 /t (or vs. log t for t h e ease of one well) gives a straight line with a slope As/Alog (r 2 /t), with an absolute

297

value m = 2.3Q/41rT

(10.11)

2 and an (r /t)-intercept 2 ( r / t ) o = 2.25T/S

(10.12)

For computational convenience, t h e slope m can be selected as As/one Therefore, if a straight line is passed through t h e d a t a points t h a t

cycle.

appear to define a straight-line variation on t h e semilogarithmic plot, t h e values of T and S can be respectively calculated from eqs. (10.11) and (10.12). It should be noted t h a t , once t h e value of T is computed, t h e value of S c a n also b e calculated from eq. (10.10) and t h e coordinates of any point on t h e straight line.

Example 1. Measurements in one well at various times.

A well fully

penetrating a confined aquifer is pumped at a uniform rate of 300 gal/min (0.668 cfs).

Drawdown variations with t i m e are measured in a n observation

well 310 f t from t h e pumped well (Table 10.1).

For this case, eq. (10.10) can

b e rewritten as

s = ( 2 . 3 Q / 4 f l ) log (2.25T/r2S) + ( 2 . 3 Q / 4 f l ) log t Thus, a plot of s vs. log t (Fig. 10.2) gives a straight line having a slope m = (2.3Q/41rT) = 7 f t .

Therefore,

2 Because t h e drawdown in eq. (10.10) becomes z e r o when 2.25 T t / r S equals

unity, we can use t h e intersect of t h e straight line with t h e zero-drawdown axis, t

0'

to determine t h e value of S .

Thus,

298 20 I

1

TIME AFTER PUMPING STARTED, t (min)

Figure 10.2

Plot of s versus log t for d a t a in Table 4.1.

Figure 10.3

Plot of s versus log r for d a t a in Table 4.2.

S =

2 ' 2 5 TtO

2 -- (2.25)(0.017 f t / s e c ) ( 8 4 0 s e c ) -- o.ooo3

r2 Example 2.

(310 f t l 2 Measurements in t h r e e wells at t h e s a m e time.

A well

fully penetrating a confined aquifer is pumped at a uniform rate of 200 gal/min (0.445 cfs).

Drawdown measurements are taken a t t h r e e nearby

observation wells 30 hr (108,000 see) a f t e r pumping has started. The measured d a t a and t h e distance from t h e pumped well t o each of t h e observation wells

are presented in Table 10.2. Since only t h e distance r from t h e pumped well to each observation well varies, eq. (10.10) can b e rewritten as

299

s = (2.3Q/4nT)

l o g (2.25Tt/S) - ( 2 . 3 Q / 2 f l )

log r

Thus a plot of s vs. log r (Fig. 10.3) gives a s t r a i g h t line whose slope has a n absolute value m = 2.3Q/2nT = 9.1 f t .

Therefore,

Measured drawdown at t h r e e observation wells 30 hrs a f t e r pumping began.

Table 10.2

Observation well

Distance from pumped well (ft)

Measured drawdown (ft)

150 300 600

1 2 3

14.4 11.7 9 .o

T h e value of S c a n b e determined from t h e i n t e r c e p t of t h e straight line In o t h e r words, for s t o b e zero, w e must with t h e zero-drawdown axis, r 0' 2 have 2.25Tt/r S = 1. It follows t h a t

S =

2.25 Tt - (2.25)(0.018 2 "0

2 f t /sec)(108,000 see) = o ~ o o o 1 3 (5800 f t l 2

Example 3.

Measurements in two wells at various times.

Consider

a well fully penetrating a confined aquifer and discharging uniformly at 1700 gal/min (3.786 cfs).

Drawdown measurements are t a k e n at t w o observation

wells at various t i m e s (Table 10.3).

Equation (10.10) c a n now b e rewritten

as S = (2.3Q/4nT)

log (2.25T/S) - (2.3Q/4nT)

log ( r 2 / t )

2

Thus, a composite plot of s vs. log r /t (Fig. 10.4) gives a s t r a i g h t line with a n absolute slope value of m = 2.3Q/4nT = 5.5 f t .

Therefore,

300 T = - 2.3Q 2 - ( 2 * 3 ) ( 3 * 7 8 6 c f s ) - 0.126 f t /see = 81,400 g p d / f t 4mn (4)(3.14)(5.5 f t ) 2 Because t h e i n t e r c e p t (r /tI0 falls outside t h e d a t a s h e e t (Fig. 10.41, S c a n b e computed from eq. (10.10) and t h e coordinates of any point on t h e s t r a i g h t line.

It follows from eq. (10.10) t h a t

20

c

15

v

In

f

x

10

2

a c 5 L3

1 102

I 10

103

r 2 / t (ft2/min) Figure 10.4

Table 10.3

Plot of s versus log r 2/t for d a t a in Table 4.3.

Drawdown measurements taken at t w o observation wells at

various times. ~

T i m e since pumping s t a r t e d t (mid 30 60 90 120 150 300 500 1000 2400 5000

O.W. #1 (rl = 60 f t ) s1 (ft)

7.8 9.2 10.7 ll.2 11.0 12.7 14.0 16.4 18.5

r,Z/t (ft2/min) 120.0 60.0 40.0 30 .O 24.0 12.o 7.2 3.6 1.5

O.W. $2 ( r 2 = 200 f t )

s2 (ft)

r2/t 2 ( f t 2/ m i d

3.3 4.1 5.4 5.9 7.2 8.2 10.5 12.0 14.0

666.7 444.4 333.3 266.7 133.3 80.0 40 .O 16.7 8-0

301

s

= antilog

[log 2.25 T t

47rTs

r Selecting t h e coordinate values (s, r 2 /t) = (13.5 f t , 10 f t 2/min) and substituting them in t h e preceding equation along with t h e previously calculated value of T yields S = 0.0001.

10.2 .D

Theis Recovery Method

As noted in Section 9.3.D, t h e drawdown during recovery o r residual drawdown 2 in observation wells at time t' > 25r S/T since pumping stopped in a fully penetrating and steadily discharging well can be expressed as s' =

-%

3Q (lo, 2 25 T t - log 2.25 T t ' 4fl r 2~

L

)

-

2.3Q l o g

(+)

(10.13)

in which s' is t h e residual drawdown and t and t' a r e respectively t h e times a f t e r pumping started and stopped.

Clearly, eq. (10.13) represents a straight

line on a data plot of s' vs. log (t/t') with a slope As'/Alog (t/t').

If m

denotes t h e absolute value of t h e slope, conveniently taken as As/log cycle, t h e variation of T can b e calculated from t h e expression T = 2.3Q/4mn

(10.14)

The value of S cannot be determined from such data. 10.3

UNCONFINED AQUIFERS

The methods of analyses for tests in confined aquifers c a n be applied t o analyze tests in unconfined aquifers if t h e dewatering is less than 25% of t h e initial depth of saturation (i.e., s < 0.25 hi) and t h e quantities s, T, and 2 S in t h e former equations a r e respectively replaced with s - (s l2h.1, Khi, 1

and S Y' 10.3.A

Boulton Type-Curve Method

As indicated in Section 9.3.B, Boulton (1963) proposed t h e use of eq. (9.41) t o describe t h e drawdown of t h e water table near a fully penetrating and

302

steadily discharging well in an unconfined aquifer.

That is,

s ( r , t ) = ( Q / ~ T T )W(UA, uB, r/D)

(10.15)

where W(uA, uB, r/D) is Boulton's unconfined-well function and r/D = r / n m . Y Variations on t h e original theory of delayed yield from storage were l a t e r reported by Boulton (1970) and Boulton and Pontin (1971).

The theory was

used originally by Boulton (1963) and later by Prickett (1965) t o describe a graphical procedure for determining from aquifer test data t h e transmissivity, T, storativity,

S, specific yield, S

unconfined aquifers.

Y'

and so-called delay index, l/a, of

The procedure of Boulton's type-curve method is as follows: 1) Plot t h e type curves of W(uA, r/D) versus l/uA and W(uB, r/D) versus l/uB on logarithmic paper, as in Fig. 9.10. 2) Plot t h e drawdown s at a given observation well versus t h e values

of time t (field data) on logarithmic paper of t h e same scale as t h a t of t h e type curves. Superimpose t h e field data on t h e type-A curves, keeping t h e horizontal and vertical axes of both graphs parallel t o each other and matching 3)

as much as possible of t h e earliest time-drawdown data t o a particular type curve.

The value of r/D corresponding to this type curve is noted.

4) Choose a matching point anywhere on t h e overlapping portion of t h e two sheets of paper. Record t h e values of W(uA, r/D), l/uA, s, and t from t h e coordinates of t h e matching point. 5)

The transmissivity is now calculated from

and t h e storativity from 2

S = 4Tt/r (l/uA)

6)

(10.17)

Superimpose t h e field data on t h e type-B curves, keeping the

horizontal and vertical axes of both graphs parallel to each other and matching

303

as much as possible of t h e latest time-drawdown d a t a to a particular type curve. The value of r/D corresponding to this type curve must be t h e s a m e as t h a t obtained earlier from t h e type-A curves. 7) Choose a new matching point on t h e overlapping portion of the two s h e e t s of paper. Record t h e values of W(uB, r/D), l/uB, s, and t from t h e coordinates of this matching point. 8) The transmissivity is calculated from T = ( Q / 4 ~ r s ) W(uB, r/D)

(10.18)

Its value should be approximately equal to t h a t previously calculated with eq. (10.16). S

Y

The specific yield is obtained from

2 = 4Tt/r (l/uB)

(10.19)

9) The r/D value from t h e best-fitting type curve (as noted in steps 3 and 6) is used t o calculate D. Then c1 is calculated from t h e relation a = T / D2 S Y 10.3.B

(10.20)

Neuman Type-Curve Method

Neuman (1973a) showed t h a t when a pumping well and a n observation well fully p e n e t r a t e an anisotropic unconfined aquifer, t h e drawdown in t h e observation well is given by eq. (9.47). That is,

W(u u $) is Neuman's unconfined-well function and Q = 2 % B' (KZ/Kr)(r /hi 1. The procedure of Neuman's type-curve method is very similar to t h a t where

of Boulton's type-curve method, just described. 1) Plot t h e type curves of W(uA, Q) versus l/uA and of W(uB, $) versus l/uB on logarithmic paper, as in Fig. 9.11. 2) Plot t h e drawdown s at a given observation well versus t h e values

304

of t i m e t (field data) on logarithmic paper of t h e s a m e scale as t h a t of t h e type curves. 3)

Superimpose t h e field d a t a on t h e type-B curves, keeping t h e

horizontal and vertical axes of both graphs parallel to each other and matching

as much of t h e latest time-drawdown d a t a as possible to a particular type curve.

The value of $ corresponding to this type curve is noted. Choose a matching point anywhere on t h e overlapping portion of

4)

Record t h e values of W(uB, $1, l/uB, s, and t from t h e coordinates of t h e matching point.

t h e two s h e e t s of paper. 5)

The transmissivity is now calculated from

and t h e specific yield from

S = 4Tt/r 2(l/uB) Y

(10.23)

Superimpose t h e field d a t a on t h e type-A curves keeping t h e horizontal and vertical axes of both graphs parallel t o each other and matching 6)

as much as possible of t h e earliest time-drawdown d a t a to a particular type curve.

The value of

corresponding to this type curve must be t h e s a m e

as t h a t obtained earlier from t h e type-B curves. 7) Choose a new matching point on t h e overlapping portion of t h e two sheets of paper. Record t h e values of W(uA, $1, l/uA, s, and t from t h e coordinates of this matching point. 8) The transmissivity is calculated from

Its value should be approximately equal to t h a t previously calculated from t h e late drswdown data.

S = 4Tt/r 2(l/uA) 9)

The storativity is obtained from (10.25)

Having determined T, t h e horizontal hydraulic conductivity is

calculated from

305 Kr = T / h i

(10.26)

T h e d e g r e e of anisotropy

K D = KZ/Kr is obtained from t h e values of

)I

according t o

% = KZ/Kr

2

= $hi / r

2

(10.27)

T h e v e r t i c a l hydraulic conductivity c a n now be obtained from (10.28)

KZ = KD Kr

The specific storage of t h e aquifer is calculated from Ss = S / h .

(10.29)

1

Consider a n aquifer test described by Bonnet et al. (1970) and Neuman (1975). An unconfined aquifer has a depth of 13.75 m and a n Example.

initial s a t u r a t e d thickness of 8.24 m.

The pumping well is perforated within

t h e depth interval 7-13.75 m and has a d i a m e t e r of 0.32 m.

The well is pumped f o r 48 hr and 50 min at a r a t e oscillating between 51 and 54.6 m 3 /hr 3 and averaging about 53 m /hr. Drawdowns are measured a t two observation wells 10 m and 30 m from t h e pumping well.

T h e observation wells are

assumed to be perforated throughout t h e e n t i r e thickness of t h e aquifer. Owing t o t h e large amount of penetration of t h e pumping well, t h e e f f e c t of p a r t i a l penetration can be neglected at r = 10 m and r = 30 m. Drawdown variations with t i m e in t h e pumping well and observation wells are shown by t h e open circles in Fig. 10.5.

T h e solid lines are traces

of t h e type curves t h a t appeared to give t h e b e s t visual f i t with t h e d a t a . The corresponding matching points are also shown in t h e figure. T h e coordinates of t h e matching point corresponding t o r = 10 m and t h e type-B c u r v e f o r $ = 0.01 are W(uB, $1 = 1, l/ug = 4, s = 0.06 m, and

t = 200 sec = 0.0556 hr.

By using eqs. (10.22) and (10.23) o n e t h e r e f o r e

calculates T = (53)(1)/(4)(3.14)(0.06)

= 70.3 m2/hr

306

Type B r =30meters

TypeA TypeB r=lOmeters r=lOmeters

10-2

0

MEASURED

0

MATCH POINTS

-THEORETICAL

1

1

10'

103

102

104

105

106

t (seconds)

Figure 10.5 S

Y

Example of Neuman type-curve method (Neuman, 1975).

= (4)(70.3)(0.0556)/(10)2(4)

= 3.9 x

By using Boulton's theory, Bonnet et al. (1970) obtained T = 68.0 m 2 /hr and

sY

= 4.5 x

The coordinates of t h e matching point corresponding to r = 10 m and

t h e type-A curve for $ = 0.01 are W(uA, $) = 1, l/uA = 40, s = 0.064 m, and

t = 80 sec = 0.022 hr.

From eqs. (10.24) and (10.25) one obtains

T = (53)(1)/(4)(3.14)(0.064)

= 65.9 m2/hr

2 Bonnet et al. obtained from Boulton's theory T = 69.0 m /hr and S = 1.5 x 3 10 Neuman remarks t h a t since t h e late d a t a give a b e t t e r f i t with t h e

.

type curves than t h e early d a t a , t h e results from t h e late d a t a appear to 2 be more reliable. W e therefore adopt t h e value T = 70.3 m /hr in t h e calculation below

.

Having determined $, T, S and S, one can now obtain all t h e remaining Y' aquifer parameters from eqs. (10.26) through (10.29): Kr = ( 7 0 . 3 ) / ( 8 . 2 4 )

= 8.53 m/hr

307

E;D =

(0.01)(8.24)

2

= 6.79 x

Kz = (6.79 x lC1-~)(8.53) = 5.79 x Ss = (1.45

m/hr

x 10-3)/(8.24) = 1.76 x 10-4 m-1

The d a t a at r = 30 m appear t o fit a type-B curve for $ = 0.18. The coordinates of t h e corresponding matching point a r e W(uB, $1 = 1, l/ug = 4,

s = 0.06 m, and t = 3100 see = 0.86 hr.

From eqs. (10.22) and (10.23) one

obtains 2 T = ( 5 3 ) ( 1 ) / ( 4 ) 3.14)(0.06) = 70.3 m /hr S

Y

= ( 4 ) ( 7 0 . 3 ) ( 0 . 8 6 ) / ( 3 0 ) 2 ( 4 ) = 6.72 x

By using Boulton's theory, Bonnet et al. calculated T = 65.0 m 2 /hr and S -2 Y = 8.0 x 10 , From eqs. (10.26) through (10.28) one gets Kr = ( 7 0 . 3 ) / ( 8 . 2 4 ) = 8.53 m/hr

%=

( 0 . 1 8 ) ( 8 . 2 4 ) 2 / ( 3 0 ) 2 = 1.36 x

Kz = (1.36 x 10-')(8.53)

= 1.16 x 10-1 m/hr

Neuman remarks t h a t $ in t h e two observation wells at r = 10 m and

r = 30 m does not vary as t h e square of these radii (see definition of $1 as a result of such diverse causes as aquifer heterogeneity, partial penetration, or lack of sensitivity of the data match t o t h e values of $. 10.3.C Neuman Semilogarithmic Method Neuman's (1975) semilogarithmic approach t o determine unconfined aquifer parameters is faster and simpler t o use than t h e type-curve method just described.

tB, where t

When 4vTs/Q = W(uA, uB, $1 is plotted versus log tA and log A

= 1/4uA = Tt/r 2 S and tB = 1/4uB = Tt/r 2 S the result is as Y'

308

0.0

-

10-1

Tt

,t

Sr2

I00

10'

I02

lo3

10-4

10-3

10-2

10-1

lo4

7.0 6.0

3 5.0 m

3

-3

4

3

4.0

3.0 2 .o I .o

0

I02

lo3

Tt tg=Syr2

Figure 10.6

Plot of W(uA, uB, $1 versus log t A and log tB (Neuman, 1975). It is evident t h a t t h e late drawdown d a t a tend to fall

shown in Fig. 10.6.

on a straight line t h a t , according to Cooper and Jacob (1946), is given by 4nTs/Q = 2.3 log (2.25tB)

(10.30)

The intermediate d a t a tend t o f a l l on a horizontal line, whereas some of t h e early d a t a tend t o fall near t h e line 4nTslQ = 2.3 log (2.25tA) Let t

BJI

(10.31)

be t h e value of tg corresponding to t h e intersection of any

horizontal line with t h e inclined line described by eq. (10.30).

For example,

Fig. 10.6 shows t h a t t h e value of t

for $ = 0.03 is equal t o 5.2. When B$ t h e result is a unique curve as shown in

log 1/$ is plotted versus log t BJI' Fig. 10.7. The values from which this curve was plotted are given in Table 10.4.

Neuman (1975) reports t h a t a good approximation for t h e relationship

between JI and t

JI = 0.195/t

BJI

within t h e range 4.0 < tB

5 100.0 is

1 .lo53 B$

(10.32)

309

’*

1

10’

Figure 10.7

Plot of 1/J, versus t

B4J

(Neuman, 1975).

which is represented by t h e dashed line in Fig. 10.7. The procedure of Neuman’s semilogarithmic method is as follows:

1)

P%$ the drwdmw s at B gj’jver,9 b ~ e ~ ’ ~ ~ well t i o nversus 103 t -

Fit a straight line t o the l a t e portion of t h e time-drawdown data. The intersection of this line with t h e horizontal axis corresponding t o s = 0 2)

is denoted by tL. The change in s along this line corresponding t o a tenfold increase in t (i.e., to one logarithmic cycle) is denoted by AsL. Then, according t o eq. (10.301, t h e transmissivity of the aquifer is determined from T = 2.3 Q/4nAsL

( 10.33)

and t h e specific yield from S

Y

= 2.25 TtL/r 2

(10.34)

310

Table 10.4

2.50 1.67 2.00 2.50 3.33 4.00 5.00 6.67 1.00 1.25 1.67 2.50 5.00 1.00 1.67 3.33 1.00 2.50 1.00

x x x x x x x x x x x x x x x x x x x

Values of l/$ and t

10::

4.52 4.55 4.59 4.67 4.81 4.94 5.13 5.45 6.11 6.60 7.39 8.93 1.31 2.10 3.10 5.42 1.42 3.22 1.23

loo loo loo loo loo lo1 lo1 lo1

lo2 lo2 lo3 10

3)

BJ,

x x x x x x x x x x x x x x x x x x x

used in plotting Fig. 10.7 (Neuman, 1975).

10:: low1

loo loo loo loo lo1 lo1 lo2

10

Fit a horizontal line to t h e intermediate portion of t h e time-

drawdown data.

The value of t corresponding to t h e intersection of this

horizontal line with t h e straight line passing through t h e late d a t a is denoted by t$. Knowing T and S from s t e p 2, compute t h e dimensionless time t B$ Y with t h e formula

tw = Tt,,,/r 2 S Y

(10.35)

The value of $ c a n now be obtained directly from t h e curve in Fig. 10.7 or, f o r a limited range of t

values, from eq. (10.32). B$ 4) Fit a straight line to a portion of t h e early time-drawdown data.

If t h e slope of this line differs markedly from t h a t of t h e line passing through t h e late d a t a , s t e p 5 must be skipped, and in this case S must b e determined by t h e type-curve method. If t h e t w o lines are nearly parallel to each other,

311

t h e intersection of t h e early line with t h e horizontal axis at s = 0 is denoted by tE. The change in s along this line, corresponding to a tenfold increase in t, is denoted by AsE. The transmissivity is then calculated from T

2.3 Q/4TAsE

=

(10.36)

This value of T should be approximately equal to t h a t previously obtained from t h e late drawdown data.

The storativity is obtained from (10.37)

S = 2.25 T t E / r 2

5) The values of t h e parameters Kr, KD, KZ, and SS can now be calculated from eqs. (10.26) through (10.29). Example.

Consider t h e plot of s vs. log t presented by Neuman (1975)

and shown in Fig. 10.8.

The d a t a are t h e s a m e as in Fig. 10.5 but on a

0.9

0.8 0.7

-

-

Pumping w e l l

0.6 -

v)

& 0.5

0

+

Q)

E

v)

0.4

0.3 0.2

0 .I 0

10'

'L102

1 0 3 t ~~ J I ~ J I 104

105

I06

t (seconds) Figure 10.8

Example of Neuman semilogarithmic method (Neuman, 1975).

312

semilogarithmic scale.

At r = 10 m, two parallel straight lines are f i t t e d

to t h e late and early d a t a and a horizontal line i s f i t t e d to t h e intermediate data. These lines give AsL = 0.137 m, tL = 70 see = 0.0194 hr, t = 2250 4J see = 0.625 hr, AsE = 0.138 m, and tE = 4.25 sec = 0.00118 hr. By using eqs. (10.33) and (10.34), one calculates, respectively: T = (2.3)(53)/(4)(3.14)(0.137)

2

= 70.8 m / h r

S = ( 2 . 2 5 ) ( 7 0 . 8 ) ( 0 . 0 1 9 4 ) / ( 1 0 ) 2 = 3.08 x lo-' Y

2

T h e results obtained by Bonnet et al. (1970) are T = 67.0 m /hr and S

Y

3.5 x lo? From eq. (10.351, t h e dimensionless t i m e t

t

B4J

= (70.8)(0.625)/(10)

2 (3.08 x lo-')

B$

=

is

= 14.37

Since this falls within t h e range of values for which eq. (10.32) applies, $ can b e determined either by eq. (10.32) or from Fig. 10.7.

According to eq.

(10.321,

4J =

0.195/(14.37) l S 1 O 5 = 0.01 The early d a t a are analyzed with eqs. (10.36) and (10.37).

T = (2.3)(53)/(4)(3.14)(0.138) S = (2.25)(70.4)(0.00118)/(10)2

Thus,

2

= 70.4 m / h r = 1.87 x

2 By using t h e average value of transmissivity T = 70.6 m /hr, w e c a n now calculate all t h e remaining aquifer p a r a m e t e r s with t h e aid of eqs. (10.26) through (10.29), Kr = ( 7 0 . 6 ) / ( 8 . 2 4 )

= 8.57 m/hr

'(f> = ( 0 . 0 1 ) ( 8 . 2 4 ) 2 / ( 1 0 ) 2 = 6.79 x

313

Kz = (6.79 x 10-3)(8.57) = 5.82 x -4

Ss = (1.87 x 10-3)/(8.24) = 2.27 x 10

m -1

At r = 30 m, a straight line is fitted t o t h e late d a t a and a horizontal These lines give A s L = 0.137 m, tL = 1300

line t o t h e intermediate data.

sec = 0.361 hr, and t

4J =

5200 sec = 1.444 hr.

The use of eqs. (10.33) and

(10.34) gives, respectively: = 70.8 m 2 /hr

T = (2.3)(53)/(4)(3.14)(0.137)

S

Y

= 6.38 x

= (2.25)(70.8)(0.361)/(30)2

2 From t h e s a m e data, Bonnet et al. obtained T = 63.0 m /hr and S

Y

From eq. (10.35), the dimensionless time t

x

= (70.8)(1.444)/(30)2(6.38

BQ

= 6.5 x

is

= 1.78

This falls outside t h e range of values for which eq. (10.32) is applicable, and

so Q must be determined from Fig. 10.7. According t o this figure, l/$ = 7.8, and therefore Q = 1/7.8 = 0.128. From eqs. (10.26) through (10.28) one calculates Kr = ( 7 0 . 8 ) / ( 8 . 2 4 ) = 8.59 m/hr

2

K,, = (0.128)(8.24) / ( 3 0 1 2 = 9.66 x

Kz = (9.66 x 10-3)(8.59) = 8.29 x

m/hr

A straight line can also be fitted t o t h e late drawdown in the pumping

well, giving A s

L

= 0.136 m.

T = (2.3)(53)/(4)(3.14)(0.136)

By using this value in eq. (10.33) we obtain = 71.43 m2/hr

2 T h e result of Bonnet et al. for t h e same d a t a is T = 69.0 m /hr.

314 10.3.D

Neuman Recovery Method

Neuman's

(1975) recovery

method

allows aquifer transmissivity t o be

determined with recovery test d a t a from t h e pumping well or from t h e observation wells. Let t be t h e t i m e since pumping s t a r t e d and let t' be t h e t i m e since t h e pump was shut off and recovery began. 0

2

.\

I

By plotting t h e

I

I

I

0 r = 10 m e t e r s 0 Pumplng Pumping w e l l

0.2

01

-E 0.3 c

al

m

0.4-

a I

I

Figure 10.9

I

1

i ,

Example of Neuman recovery method (Neuman, 1975).

residual drawdown versus log t/t' one finds t h a t at large values of t' (i.e.,

at small values of t/t') these d a t a tend to fall on a straight line. If AsL is t h e residual drawdown corresponding to a tenfold increase in t/t' along this straight line, then T can be calculated from eq. (10.33). I t should be pointed out t h a t t h e recovery d a t a from a given well will not fall on a straight line

as long as t h e e f f e c t of elastic storage, created by t h e cessation of pumping, has not dissipated at this well. Example.

In the aquifer test described by Bonnet et al. (1970) and

Neuman (19751, t h e residual drawdown during recovery was measured in t h e pumping well and at r = 10 m for a period of 6 hr and 20 min. Figure 10.9 shows t h e corresponding plot of residual drawdown versus log (t/t') as presented by Neuman (1975). Two parallel straight lines are fitted to t h e late recovery d a t a from both wells, giving AsL = 0.137 m.

From eq. (10.331,

T = ( 2 . 3 ) ( 5 3 ) / ( 4 ) ( 3 . 1 4 > ( 0 . 1 3 7 ) = 7 0 . 8 m2 / h r By t h e s a m e method, Bonnet et al. obtained T = 72.0 m 2 /hr.

315

Neuman's Relationship Between Boulton's Delay Index and

10.3.E

Aquifer Character istics Neuman (1975) derived an explicit mathematical relationship between Boulton's semiempirical quantity a (the reciprocal of which is called delay index) and the physical characteristics of t h e aquifer. This relationship makes it possible

to reinterpret the results of aquifer tests that were previously obtained with t h e aid of Boulton's (1963) theory without necessarily reexamining the original drawdown data.

The relationship can be obtained in the following manner.

As we have previously seen, Boulton's type curves a r e expressed in terms of t h e dimensionless parameters (r/D) = r(a% /T)1'2, 2

whereas Neuman's type

2

curves a r e expressed in terms of Q = (KZ/Kr)(r /hi 1. Considering the horizontal portions of these type curves, one can plot a curve of Q versus W(uA, uB, 10'

10'

loo

loo

(

+ lo-'

10'' ;)2

Io-2

Io-2

lo-? .-

10-2

10-1

W (uA,us, Figure 10.10

I00

+ or r / D )

10'

IO-~ .-

Plots of $ versus W(uA, uB, $) and r/D versus W(uA, uB, r/D)

corresponding t o horizontal portion of type curves (Neuman, 1975).

316 5.0

I

I

I

(r/D)'

-~3.063-0.567logy JI

Cor r e Ia t ion c o e f f i c i e n t = 0.99

4 .o

a 3.o

2.o

10-3

Figure 10.U

10-2

+

10-1

10 I

100

2 Relationship between (r/D) /$ and $ (Neurnan, 1975).

$1 and another curve of r/D versus W(uA, uB, r/D) as shown in Fig. 10.10. By plotting the ratio between (r/D)

2

and $ for given W(uA, uB, $ or r/D)

values versus $ on semilogarithmic paper one obtains the set of points shown in Fig. 10.11.

Linear regression yields t h e straight line

2 ( r / D ) /$ = 3.063 - 0.567 l o g $

(10.38)

with a correlation coefficient of 0.99. From the definitions of r/D and $ it follows that eq. (10.38) can be rewritten as (10.39)

Equation (10.39) indicates t h a t in a given homogeneous aquifer, in direct proportion t o log r.

c1

decreases

This contradicts Boulton's theory, in which

c1

is assumed t o be a characteristic constant of the aquifer. Streltsova (1972b) used a finite-difference delayed-response process and concluded that

c1

decreases with r and increases

with t. Equation (10.39) supports the conclusion t h a t it contradicts t h e conclusion that

c1

approximation for t h e c1

varies with time.

decreases with r, but

317

Equation (10.39) can be interpreted t o mean that in a given aquifer the effect of delayed gravity drainage decreases linearly with t h e logarithm of t h e radial distance from t h e pumping well.

In other words, the influence

of elastic storage becomes less important as t h e radial distance increases. With eq. (10.38) one can reevaluate the results of aquifer tests obtained with the aid of Boulton's theory in light of Neuman's theory, with no need t o reexamine t h e original drawdown data. Since t h e values of T, S and S Y' a r e determined from early and late drawdown data that respectively fit the early and late Theis curves, results obtained with the aid of Boulton's theory will be practically identical t o results obtained with Neuman's methods.

The

difference between Boulton's theory and Neuman's theory (as far as fully penetrating wells a r e concerned) is that t h e former enables one t o calculate

a, whereas the l a t t e r enables one t o determine the degree of anisotropy of the aquifer, KD, as well as its horizontal and vertical hydraulic conductivities, K r and KZ. By knowing the values of T, S and a as calculated on the

Y' basis of Boulton's theory for a given value of r, one first determines the The value of Q can then be corresponding value of (r/D) = r(aS /T)1/2 Y obtained directly from eq. (10.38) by using the iterative Newton-Raphson

.

method (10.40)

where f(Q)" =

(r/D)2

(3.063 - 0.567 l o g Q) - 1

Q=P

-

3.063 - 0.567 log

Q-

and n is t h e number of iterations. According t o Neuman (19751, a good initial 0 estimate for J, is obtained by setting log Q = 0 in eq. (10.38), so that Qo = (r/DI2/3.063. accuracy of

Usually, not more than three iterations are required for an

lan - an-'1

5 0.01 $n.

318

Once J, has been determined in this manner, all the unknown aquifer parameters can be calculated by s t e p 9 of t h e Neuman type-curve method, described earlier. Example.

Consider t h e aquifer test described earlier and analyzed by

Neuman [19751. At r = 10 m, Bonnet et al. [19701 obtained by using Boulton's 2 S = theory t h e following average values: T = 68.25 m /hr, S = 4.0 x

Y

l/a = 6000 sec = 1.667 hr, and (r/D) = 0.2. By using t h e 1 Newton-Raphson method as described in eq. (10.40) we obtain J,' = 0.013, J, = 2 3 4 0.088, J, = 0.0096, J, = 0.0095, and J, = 0.0095. This result is very similar

1.5 x

t o the value J, = 0.01 previously obtained by Neuman's type-curve method. Thus, from eqs. (10.26) through (10.28) one can now obtain Kr = (68.25)/(8.24)

%=

= 8.28 m/hr

(0.0095)(8.24)2/(10)2 = 6.45 x

Kz = (6.45 x 10-3)(8.28)

= 5.34 x

m/hr

These results a r e practically t h e same as those previously calculated. At r = 30 m, Bonnet et al. obtained from Boulton's theory T = 64.0

2

m /hr, S = 7.25 x l/a = 6250 sec = 1.736 hr, and (r/D) = 0.8. From Y 1 2 3 eq. (10.40) we obtain = 0.209, J, = 0.178, J, = 0.185, J, = 0.184, 4~~ = 0.184, which, again, is very similar to the value J, = 0.18 as previously

qp

determined by Neuman's type-curve method. (10.28) one calculates

Thus, from eqs. (10.26) through

Kr = ( 6 4 . 0 ) / ( 8 . 2 4 ) = 7.77 m/hr K,

= ( 0 . 1 8 4 ) ( 8 . 2 4 ) 2 / ( 3 0 ) 2 = 1.39 x

Kz = (1.39 x 10-2)(7.77) = 1.08 x 10-1 m/hr

10.3.F

Partial Penetration Methods

Neuman (1974) developed equations for analyzing field drawdown data when the pumping well or the observation well partially penetrates t h e saturated

319

thickness of t h e aquifer.

The large number of dimensionless parameters in

these equations, however, makes i t practically impossible to construct a sufficient number of type curves to cover t h e e n t i r e range of values necessary for field application. expressed

in

For a set of t y p e curves to be useful, they should be

t e r m s of

not

more than t w o independent dimensionless

parameters. Neuman (1975) gives a procedure by which t h e number of independent parameters can be reduced to two.

This procedure, however, requires t h a t

a special set of theoretical curves (such as those in Fig. 9.11 or Fig. 10.7) be developed for each observation well in t h e field. Theoretical curves useful for analyzing field drawdown d a t a under partially penetrating conditions can be constructed also from tables presented by Dagan (1967a, 1967b) and Streltsova (1974). Another approach is, of course, to design t h e aquifer test beforehand

so as to minimize t h e e f f e c t of partial penetration on t h e drawdown in t h e observation wells. Neuman (1974) showed t h a t t h e e f f e c t of partial penetration on t h e drawdown in a n unconfined aquifer decreases with radial distance from t h e pumping well and with t h e r a t i o K D = KZ/Kr. At distances r > 2

this e f f e c t disappears completely when t i m e t > 10r S /T and t h e Y drawdown d a t a follow t h e late Theis curve in t e r m s of l/ug. Thus, if t h e hi/Kki2

observation well is located f a r from t h e pumping well, t h e late drawdown d a t a may eventually b e used to determine T and S by a conventional method. Y The early and intermediate d a t a , however, can be used to determine additional aquifer p a r a m e t e r s only when a special set of theoretical curves is developed f o r each observation well. Neuman (1975) emphasizes t h a t t h e Theis curve should not b e used to analyze late drawdown d a t a without having first verified t h a t t h e e f f e c t of partial penetration has actually dissipated at r.

According to Neuman, one

way of doing t h a t is to install two piezometers at t h e s a m e radial distance

r, one at a shallow depth beneath t h e water table and t h e other at a substantially greater depth. Plotting s from both piezometers on a single s h e e t of logarithmic paper will give two curves t h a t tend to merge at large values of t. When t h e distance between these two curves becomes very small, one has an indication t h a t from a practical standpoint no vertical flow is taking place and t h e e f f e c t of partial penetration is thus nil.

320 Neuman (1974) also showed t h a t t h e influence of partial penetration on early and late drawdown d a t a can be minimized by perforating t h e observation well throughout t h e e n t i r e saturated depth of t h e aquifer. In such

a case, t h e drawdown at distances r > hi/KD 1/2 will follow t h e late Theis curve at times t > r 2 S /T, and t h e drawdown at distances r < 0.03 hi/KD 1/2 Y 2 will follow t h e early Theis curve at t i m e t < r S/T. Thus if a fully penetrating observation well is located f a r from t h e pumping well, i t s late drawdown d a t a c a n b e used to determine T and S by conventional methods. If, on Y t h e other hand, t h e observation well is close enough to t h e pumping well, i t s early drawdown d a t a may enable one to determine T and S.

Here, again,

t h e intermediate d a t a from both wells c a n b e used to determine additional aquifer p a r a m e t e r s only if theoretical curves are developed t h a t fit t h e practical situation at hand. 10.3.G

Concluding Remarks

Neuman (1979) recently presented a perspective on fundamental aspects of t h e hydraulics of unconfined wells.

He concluded t h a t in analyzing time-

drawdown d a t a from wells t h a t fully p e n e t r a t e an unconfined aquifer, t h e models of Boulton (1954b, 19631, Neuman (1972, 1973a, 1974,19751, and Streltsova (1972a, 1972b, 1973) will yield practically identical values of T, S, and S (or Y Kr, Ss, and S if hi is known). The model of Neuman will also yield a value Y for KZ t h a t can be expected to b e more a c c u r a t e than t h e value obtained from Streltsova's model.

Boulton's model, on t h e other hand, will yield a

value not f o r K Z but only for t h e lumped parameter a.

All t h r e e models

are applicable to t h e s a m e class of unconfined-flow problems. In addition, in analyzing time-drawdown d a t a from partially penetrating wells, t h e models of Boulton (1954b, 1963) and Streltsova (1972a, 1972b, 1973) are no longer applicable, and one must use t h e method of Neuman. 10.4

LEAKY AQUIFERS

Several procedures have been developed, steady-state as well as transient-state methods, to determine t h e hydraulic properties of leaky aquifers. 10.4.A

Jacob Type-Curve Method

The Jacob type-curve method is based on t h e solution of eq. (8.1111, namely

321

(10.41) where B = /

T m )

( 10.42)

This method, as well as t h e Hantush semilogarithmic approach discussed in Section 10.4.B, requires t h a t drawdown d a t a b e collected from t h r e e or more observation wells a f t e r flow toward t h e pumping well has a t t a i n e d essential stability within t h e region of observation. The procedure of solution is as follows: Plot a type curve consisting of values of KO(r/B) versus r/B on

1)

logarithmic paper (see Ta.ble 8.1). 2)

Plot a d a t a curve consisting of observed values of s versus r on

logarithmic paper of t h e s a m e scale as t h a t of t h e type curve. 3) Superimpose t h e d a t a curve on t h e type curve (or vice versa) and move horizontally and/or vertically, keeping t h e coordinate axes of t h e two curves parallel, to a position t h a t gives t h e best f i t of t h e data curve to t h e type curve. 4)

Select a n arbitrary matching point anywhere on t h e overlapping

s h e e t s and record t h e values of KO(r/B), r/B, s, and r from t h e coordinates of t h e matching point. 5) Substitute t h e coordinates of t h e r n a t c h b g point into eqs. (10.41) and (10.42) to solve f o r T and K'/b'. 10.4.B

Hantush Semilogarithmic Method

This method is based c n t h e solution of eq. (10.41) when r/E < G.05. In this

case, t h e drawdown c a n be approximated by

s = (2.3Q/27~T) log (1.123B/r)

(10.43)

The procedure of solution consists in plotting s (from t h r e e or more observation wells) versus log r on semilogarithmic paper. straight line where r/B < 0.05.

The d a t a points will form a

A line f i t t e d through t h e points t h a t appear

to define a straight-line variation will have a slope As/Alog r with a n absolute

magnitude m equal to 2.3Q/21rT and a n r-intercept, ro, on t h e zero-drawdown axis equal to 1.123B.

Thus, T and K'/b' c a n respectively be calculated from

T = 2.3Q/2mn and K ' / b ' = T(1.123/r0) 10.4.C

2

Walton Type-Curve Method

The procedure of solution of this method (Walton, 1960) is essentially t h a t of t h e Theis (1935) type-curve approach. The method involves t h e superposition and matching of field d a t a to a family of type curves given by t h e HantushJacob formula for leaky artesian aquifers when both storage in t h e aquitard and drawdown in t h e unpumped aquifer are neglected. As reported by Neuman and Witherspoon (1969b), t h e assumption t h a t storage in t h e aquitard is negligible can lead to significant errors when

= (r/4byK'S's /KSs of t h e

pumped aquifer is greater than 0.01. The second assumption of zero drawdown in t h e umpumped aquifer can also lead to significant errors at large values of time.

These errors cannot be neglected unless t h e T of t h e unpumped

aquifer is significantly greater than t h a t of t h e pumped aquifer.

Since t h e

Hantush-Jacob formula relies on both of these assumptions, one must be cautious in using it to analyze field data. Recall t h a t t h e Hantush-Jacob formula is s = (Q/4nT) W(u, r/B)

(10.44)

where u = r 2 Sl4Tt

( 10.45)

B =

(10.46)

Observe t h a t if T, S, and Q are constants, then eqs. (10.44) and (10.45) can be rewritten as

log s

-

log W(u, r/B) = log

323

& = constant

t - log 1 S log = log = constant 2 U 4T r Clearly, a logarithmic plot of W(u, r/B) vs. l/u is similar to a logarithmic plot, on t h e s a m e scale, of s vs. log t/r

2

.

The procedure of t h e method is as follows:

1)

Plot a family of type curves of W(u, r/B) vs. I/u on logarithmic

paper in which r/B is t i e running parameter of t h e family of curves, as in Fig. 9.13. Plot a d a t a curve consisting of observed values of s vs. t/r 2 (or against t for t h e case of one observation well) on logarithmic paper of t h e 2)

s a m e scale as t h a t of t h e family of type curves. 3) Superimpose t h e d a t a curve on t h e family of type curves (or vice versa), keeping t h e s axis parallel t o t h e W(u, r/B) axis and t h e t/rz axis parallel to t h e l/u axis, and adjust until a matching position is obtained. The d a t a curve will follow one of t h e family of type curves. 4) Select an arbitrary matching point anywhere on t h e overlapping s h e e t s and record t h e values of r/B, W(u, r/B), l/u, s, and t/r 2 from t h e coordinates of t h e matching point. 5) Substitute t h e coordinates of t h e matching point into eqs. (10.441,

(10.45), and (10.46) to solve for T, S, and K1/bl. As noted by Hantush (19641, a unique fitting position is difficult to obtain with this method unless enough d a t a points fall within t h e period during which leakage e f f e c t s are insignificant.

Because leakage e f f e c t s may

be insignificant during t h e early period of pumping, t h e Theis type curve

serves as a guide in obtaining t h e best-fitting position.

A major disadvantage

of t h e method is t h a t a graphical solution becomes practically indeterminate if t h e field d a t a (plotted on logarithmic paper) exhibit a flat curvature.

As

observed by Hantush (19641, for a f l a t curvature, such as t h a t of t h e type curve f o r l/u > 100, several apparently reasonably good matching positions can be obtained; thus, resort must be made t o other methods of analyses (see, e.g., Marifio and Yeh, 1973b).

324

Example. Consider a leaky artesian aquifer system in which t h e aquifer is overlain by an aquitard and underlain by an impermeable bed. A well fully

penetrating t h e aquifer is pumped at a uniform rate of 500 gal/min (1.14 cfs). Drawdown variations with time are measured in a n observation well t h a t fully penetrates t h e pumped aquifer and is located 700 f t from t h e pumped well.

The uniform thicknesses of t h e aquifer and overlying aquitard have

been respectively estimated to be 60 f t and 25 ft.

Because time-drawdown

d a t a are available only for one observation well, t h e d a t a curve consists of

a plot of s vs. t on logarithmic paper (Fig. 10.12).

The superposition of t h e

d a t a curve on t h e family of type curves closely follows t h e trace of t h e r/B = 0.2 type curve.

The coordinates of t h e match point (W(u, r/B) = 1.0,

l/u = 10, s = 0.77 f t , and t = 54 min = 3240 see) and t h e r/B value of 0.2

are substituted into eqs. (10.44) through (10.46) to calculate t h e values of t h e aquifer parameters. That is,

TIME AFTER PUMPING STARTED (minutes)

Figure 1012

Drawdown variations with t i m e in observation well.

325 = T - 1.15 x 10-1

6-

60

= 1.92 x

f t / s e c = 1240 g p d / f t

2

B = r / 0 . 2 = 700/0.2 = 3 . 5 x 103 f t

K ' = (9.4 x 10-')(25)

10.4.D

= 2.35 x

f t / s e c = 0.15 g p d / f t 2

Hantush Type-Curve Methods

These type-curve methods are based on t h e drawdown equations presented in Section 9.3.C.1,

where storage in t h e aquitard is taken into account but

drawdown in t h e unpumped aquifer in neglected. Recall t h a t , for large values of time, t > both 2b'S'/K' and 3061rw 2 /(T/S) 2 [l - 10(rw/B) 1 , with rw/B < 0.1, Hantush's (1960a) asymptotic solution is s = (Q/47rT) W(u

*

,

r/B)

(10.47)

where (10.48) ( 10.49)

The procedure for determining t h e values of t h e aquifer parameters is essentially t h e same as with other type-curve methods discussed earlier. In this case, t h e type curves are constructed by plotting on logarithmic paper t h e function W(u

*

, r/B)

vs. l/u

*

with r/B as t h e running parameter of t h e

type curves (see Table 9.6). The d a t a plot is t h a t of s vs. t, also on logarithmic paper. In t h e matching position, t h e d a t a curve will follow one of t h e family of t h e type curves. The match-point coordinates of W(u

*

, r/B)

326

and s are substituted into eq. (10.47) to solve for T.

Knowing T and t h e r/B value of t h e matched curve, one can calculate t h e value of K'. From t h e

*

and t, one * can solve for 3 s + S' by using eq. (10.481, i.e., 3 s + S' = 12Ttu /r 2 If t h e calculated value of T and t h e match-point coordinates of l/u

value of S' or S

S

.

can be determined from field or laboratory measurements,

then one can solve for S or Ss. Also, recall t h a t for small values of time, t < b'S'/lOK',

Hantush's

(1960a) asymptotic solution is

s = ( Q / ~ I T TW(U, ) B)

(10.50)

where 2 u = r S/4Tt

(10.51)

In this case, t h e type curves consist of a logarithmic plot of W(u, B) vs. l/u with B as t h e running p a r a m e t e r of t h e type curves (see Table 9.7). The d a t a curve consists of a logarithmic plot of s vs. t.

From t h e match-point

coordinates of W(u, B) and s, one can determine T by using eq. (10.50). From t h e calculated value of T and t h e match-point coordinates of l/u and t, one can determine S by using eq. (10.51).

Knowing t h e values of T, S, and B

(from t h e matched curves), one can calculate t h e product K'S' by using eq. 2 2 (10.521, i.e., K'S' = 16 B TSb'/r If t h e value of S' or Ss' can be determined

.

from field or laboratory measurements, one c a n solve for K'. 10.4.E

Neuman-Witherspoon Ratio Method f o r Aquitard Evaluation

As we have seen, conventional methods of analyzing leaky aquifers usually rely on drawdown d a t a from t h e pumped aquifer alone.

Neuman and

Witherspoon (1972) devised a method t h a t requires observation wells to b e placed not only in t h e aquifer being pumped but also in t h e aquitards above and/or below (Fig. 10.13).

The procedure is called t h e r a t i o method since

t h e ratio of t h e drawdown in t h e aquitard t o t h a t measured in t h e aquifer

327

r

Aquitard K '

-

I

ill

A

I

I

I Aquifer K

I

I

*

b

Aquitard K'

Figure 1013 S c h e m a t i c of leaky a q u i f e r system. 10'

10-1

S' -

S

10-2

10-2

lo-'

Nfllf

10'

Figure 1014

I

I

1

I00

lo'

102

3

Variation of s'/s with l/u' (Neuman and Witherspoon, 1972).

328

at t h e s a m e time and t h e s a m e radial distance from t h e pumping well, s'/s, is used to evaluate t h e hydraulic properties of t h e aquitard. The method is applicable to arbitrary multiple aquifer systems, provided t h a t t h e sum of values with respect t o t h e overlying and t h e 6..9 = (r/4bi)(Kj'Ss .'/K.S 1 s. underlying aquitards bk of drder 1 or less. It relies on a family of curves of s'/s versus l/u', each curve corresponding to a different value of l/u (Fig 10.14).

The curves in Fig. 1014 have been prepared from values given in

Table 10.5 (Witherspoon et al., 1967). The procedure of t h e Neuman-Witherspoon ratio method is as follows: 1)

Calculate t h e value of s'/s at a given radial distance from t h e

pumping well r and at a given instant of t i m e t. 2)

Determine t h e magnitude of l/u for t h e particular values of r and

t at which s'/s has been measured. When l/u < 400, t h e curves in Fig. 10.14 are sensitive t o minor changes in t h e magnitude of this parameter, and therefore a good e s t i m a t e of l/u is desirable. When l/u > 400, these curves are so close to each other t h a t they can be assumed to be practically independent of l/u. Then even a crude e s t i m a t e of l/u will be sufficient f o r t h e ratio method to yield satisfactory results. 3) Read off a value of l/u' corresponding to t h e computed r a t i o of s'/s

.

4) The hydraulic diffusivity a' = K1/Ss' of t h e aquitard is determined from

The critical quantity determining t h e value a' at a given elevation z is not t h e actual magnitude of t h e drawdown in t h e aquitard but t h e lag time t between t h e start of t h e test and t h e t i m e when t h e aquitard observation well begins to respond.

Thus, in using t h e ratio method, one need not worry

about having extremely sensitive measurements of drawdown in t h e aquitard observation wells. A conventional piezometer with a standing water column will usually give information accurate enough f o r most field situations. To determine t h e hydraulic conductivity, K', and specific storage, Ssl, of an aquitard from its hydraulic diffusivity, a', one of these p a r a m e t e r s

Table 10.5

Values of s'/s for different values of l/u' and l/u (Witherspoon et al., 1967). 8.0 x 10-1

2.8 x 10'

4.0 x 10'

5.00

lo3

9.57 x 10-1

2.50

lo3

9.40 x 10-1 9.56 x 10-l 9.58 x 10-1

1.67

lo3

9.27 x 10-1

1.25

lo3

9.16 x 10-1 9.38 x 10-1

9.68 x 10-1 9.71 x 10-1

9.46 x 10-1

8.0 x 10'

1.6 x lo1

4.0 x lo1

4.0 x 10'

4.0 x lo3

9.74 x 10-1 9.76 x 10-1 9.78 x 10-1 9.80 x 10-1 9.81 x 10-1

4.0 x lo4 9.82 x 10-1

4.0 x lo7

4.0 x l o l o

9.83 x 10-1 9.83 x 10-1

9.66 x 10-1 9.68 x 10-1 9.72 x 10-1

9.73 x 10-1 9.74 x 10-1

9.76 x 10-1

9.76 x 10-1

9.49 x 10-1 9.54 x 10-1 9.58 x 10-1 9.61 x 10-1 9.66 x 10-1

9.68 x 10-1 9.69 x 10-1

9.70 x 10-1

9.71 x 10-1

9.42 x 10-1 9.47 x 10-1

9.63 x 10-1

9.63 x 10-1

9.52 x 10-1 9.55 x 10-1 9.60 x 10-1

5.00 x lo2

8.70 x 10-1

9.03 x 10-1 9.09 x 10-1

2.50 x 102

8.20 x 10-1

8.65 x 10-1

1.67 x lo2

7.84 x 10-1

8.36 x 10-1 8.46 x 10-1 8.60 x 10-1 8.71 x 10-1

9.18 x 10-1 9.24 x 10-1 9.30 x 10-1 9.37 x 10-1 9.41 x la-'

8.73 x 10-1 8.85 x 10-1 8.94 x 10-1 9.01 x 10-1

9.12 x 10-1

9.17 x 10-1

9.64 x 10-1 9.66 x 10-1 9.66 x 10-1 9.43 x 10-1 9.46 x 10-1 9.20 x 10-1

9.47 x 10-1

9.23 x 10-l 9.25 x 10-1

8.80 x 10-1 8.93 x 10-1 8.99 x 10-1 9.02 x 10-1 9.06 x 10-1

9.08 x 10-1

1.25 x 10'

7.54 x 10-1 8.13 x 10-1

5.00 x 101

6.37 x 10-1 7.16 x 10-1 7.32 x 10-1 7.55 x 10-1

7.72 x 10-1 7.87 x 10-1 8.08 x 10-1 8.18 x 10-1 8.23 x 10-1 8.31 x 10-1 8.34 x 10-1

2.50 x lo1

5.23 x 10-1 6.18 x 10-1

6.87 x 10-1 7.07 x 10-1

1.67 x lo1

4.48 x 10-1

8.24 x 10-1 8.40 x 10-1 8.52 x 10-1 8.62 x 10-1

6.37 x 10-1

6.66 x 10-1

8.77 x 10-1 8.83 x 10-1 8.87 x 10-1

7.34 x 10-1 7.47 x 10-1

5.50 x 10-1 5.71 x 10-1 6.03 x 10-1 6.27 x 10-1 6.49 x 10-1 6.80 x 10-1

8.92 x 10-1 8.94 x 10-1

7.54 x 10-1 7.63 x 10-1 7.67 x 10-1

6.94 x 10-1 7.02 x 10-1 7.13 x 10-1 7.18 x 10-1

1.25 x 101

3.92 x 10-1 4.98 x 10-1

5.20 x 10-1 5.54 x 10-1 5.79 x 10-1 6.03 x 10-1 6.36 x 10-1 6.51 x 10-1 6.60 x 10-1 6.72 x 10-1 6.77 x 10-1

5.00 x 10'

2.14 x 10-1 3.13 x 10-1

3.36 x 10-1 3.72 x 10-1 3.99 x 10-1 4.26 x 10-1 4.64 x 10-1

2.50 x 10'

1.02 x 10-1

1.67 x 10'

5.55 x

1.09 x 10-l 1.23 x 10-l

1.25 x 100

3.23 x lo-'

7.09 x lo-'

1.76 x 10-1

4.83 x 10-1

1.95 x 10-1 2.25 x 10-1 2.50 x 10-1 2.75 x 10-1 3.11 x 10-1 3.28 x 10-1 8.18 x lo-'

4.93 x 10-1 5.07 x 10-1

5.13 x 10-1

3.38 x 10-1 3.51

3.57

1.48 x 10-1

1.68 x 10-1 1.89 x 10-1 2.20 x 10-1 2.36 x 10-1 2.44 x 10-l

1.01 x 10-1

1.18 x 10-1 1.35 x 10-1

1.61 x lo-'

10-l

2.56 x 10-l

1.74 x 10-1

1.81 x 10-1 1.91 x 10-1

10-l

2.61 x 10-l 1.95 x 10-1

5.00 x 10-1

2.29 x lo-'

8.19 x

1.03 x lo-'

1.46 x lo-'

1.87 x lo-'

2.33 x

3.11 x

3.52 x

3.76 x lo-'

4.08 x lo-'

4.21 x lo-'

2.50 x 10-1

6.38 x

4.04 x

5.60 x

9.33 x

1.33 x

1.84 x

2.79 x la-'

3.32 x

3.63 x

4.05 x

4.23 x

1.67 x 10-1 4.10 1.25 x 10-1

5.46

I O - ~ 2.60

4.06

7.80

1.17

1.72

10-4

2.87

3.55

1.89

3.93

5.73 x

1.12

1.78

10-5 3.12

10-5 4.04

10-4

3.95

1 0 ' ~ 4.50

10-4 4.74

10-4

10-5 4.55

10-5 5.27

10-5 5.57

10-5

must first b e evaluated by means other than t h e ratio method.

Hydraulic

conductivity may vary by several orders of magnitude from o n e aquitard to another and even from one elevation to another in t h e s a m e aquitard. A more stable range of values is usually encountered when dealing with specific storage.

Measurements of Ssl can be made in t h e field by using borehole

extensometers.

Values of Si c a n be determined also from a consolidation

test on core samples in t h e laboratory. In t h e absence of field or laboratory measurements, Ss' can be estimated by correlating published results on similar sediments.

Once t h e value of S

S

'

is determined, K' is calculated from K' =

cl'SsI.

Having determined t h e hydraulic properties of t h e aquitard, t h e only remaining unknowns to be determined are t h e aquifer transmissivity and t h e storativity.

As shown by Neuman and Witherspoon (19721, t h e T and S of a

leaky aquifer can be evaluated by using conventional methods of analysis based on t h e Theis solution (e.g., t h e Cooper-Jacob semilogarithmic method). The errors introduced by these methods will be small if t h e d a t a are collected

close to t h e pumping well but can be significant if t h e observation well is too f a r away. As indicated by Neuman and Witherspoon (19721, early drawdown d a t a

are generally a f f e c t e d by leakage less than are d a t a taken at a later time. They recommend t h a t in performing t h e analysis most of t h e weight should be given to t h e earliest d a t a available, if, of course, t h e r e is confidence in their reliability.

Once T and S are determined, one c a n calculate l/u at any

given radial distance from t h e pumping well by 2 l / u = 4Tt/r S

(10.54)

Equation (10.54) can then be used with t h e ratio method as discussed earlier. Example.

The following analysis of field data given by Neuman and Witherspoon (1972) illustrates use of t h e ratio method in aquitard evaluation. The aquifer system underlies t h e city of Oxnard, California, and consists of t h e Oxnard aquifer at a depth of 105 f t , with overlying and underlying deposits. The Oxnard aquifer is composed of fine-to-coarse-grained sand and gravel and has a thickness of 93 f t . I t is overlain by a silty and sandy clay aquitard

331

0

10-2

I

I

I

10-1

10'

100

I

I

lo3

lo2

I

10'

1

5x10'

PUMPING T I M E (minutes)

Figure 1015

Response of t h e observation wells in t h e Oxnard aquifer

(Neuman and Witherspoon, 1972). 45 f t thick, which is itself overlain by a semiperched aquifer composed of fine-to-medium-grained

sand with interbedded silty clay lenses.

The Oxnard

aquifer is underlain by a 30-ft-thick

aquitard composed of silty clay with

some interbedded sandy clay lenses.

This lower aquitard is itself underlain

by t h e Mugu aquifer.

The latter is composed of fine-to-coarse-grained

sand

and gravel with some interbedded silty clay. A 31-day aquifer test was conducted with a pumping well in t h e Oxnard

aquifer and observation wells in t h e Oxnard aquifer, upper and lower aquitards, The response of t h e observation

semiperched aquifer, and Mugu aquifer.

wells in t h e Oxnard aquifer to pumping at 1000 gal/min is shown in Fig. 10.15. Neuman and Witherspoon calculated values of T and S from these d a t a by t h e Cooper-Jacob semilogarithmic method (see Table 10.6). Table 10.6

They selected

Values of T and S in t h e Oxnard aquifer calculated by t h e

Cooper-Jacob semilogarithmic method (Neuman and Witherspoon, 1972).

Well

r (ft)

1 22H2 22B5 22K2 23E2

100 502 722 748 1060

T (gpd/ft) 130,600 139,000 142,600 136,700 157,000

S

1.12 3.22 3.08 2.48 2.53

x

-4

x x x x 10

332

5 t h e results from observation well 1 of T = 1.30 x 10 gal/day/ft and S = 1.12

as being most representative of the Oxnard aquifer, at least in t h e area of the aquifer test. The hydraulic diffusivity of the Oxnard aquifer is therefore

x

5 9 ct = T/S = 1.30 x 10 /1.12 x l o v 4 = 1.17 x 10 g p d / f t The hydraulic conductivity and specific storage of t h e aquifer a r e respectively 2 5 K = T/b = 1.30 x 10 /93 = 1405 g p d / f t

and Ss = S/b = 1.12 x 1C1-~/93= 1.20 x

ft-'

Note that the values of T and S can be calculated also by using t h e type-curve method associated with eq. (10.50).

Particular attention should

be given t o the time-drawdown data before the effects of pumping have reached t h e bottom of t h e lower aquitard and observation wells in the upper aquitard. Having estimated t h e hydraulic properties of t h e pumped aquifer, we now consider the results from other parts of this three-aquifer subsystem. Figure 10.16 shows t h e response of the observation wells in the lower aquitard and in the Oxnard and Mugu aquifers.

Figure 10.17 shows the response at

two different elevations in t h e upper aquitard and t h e overlying semiperched aquifer. To evaluate t h e lower aquitard, Neuman and Witherspoon (1972) used data from t h e Oxnard aquifer observation well (r = 100 f t ) and from the lower aquitard observation well (r = 81 f t and z = 6 f t ) and determined t h e ratio s'/s at two early values of time, t = 80 min and t = 200 min. A t t = 80 min, one can read on Fig. 10.16 that s' = 0.078 f t and s = 6.6 ft. the ratio s'/s is 0.078/6.6 = 118 x which can be rewritten as l / u = 3.71 x l o e 4 T t / r 2 S

Thus

To obtain l/u we use eq. (10.541,

333

s

1

k

0

I

UPPER A Q U I T A R D

2n 10-1-

-"""y LOWER LOOFT-

oo

AOUITARO

MUGUAOUIFER

Figure 1016 Response of t h e observation well in t h e lower aquitard to t h a t in t h e Oxnard and Mugu aquifers (Neuman and Witherspoon, 1972). 1

-__-------

I

I

1

I

0 0

62FT+O

d Ip72FT-.O 31 I

8

1 1 FT j Z F T

0

0 0 OXNARD A Q U I F E R

n i,

I

1 10' Figure 10.17

mn

0

0

I

A

00

LOWER A Q U I T A R D

0

MUGU AQUIFER

I

I

I

102

103

104

PUMPING TIME (minutes)

Response of t h e observation wells in t h e upper aquitard and

t h e semiperched aquifer (Neuman and Witherspoon, 1972).

105

334

where T i s i n g a l / d a y / f t , t i s i n min, and r i s i n f t .

1= U

PIUS

(3.71 X 10-4)(1.30 x 10-5)(80) - 5.28 (81)2(1.12 x

From Fig. 10.14, these values of s'/s and l/u correspond to l/u' = 3.44 x 10-1. To obtain t h e hydraulic diffusivity of t h e aquitard, we use eq. (10.531, which c a n be rewritten as 3

2

a' = 2.69 x 10 ( z / t ) ( l / u ' ) where a' is i n g a l / d a y / f t , z is i n f t , and t i s i n min.

a' = (2.69 x 103 )[(612/801 (3.44 x lo-')

nus

= 4.17 g p d / f t

2 = 5.99 x 10-1 an /see 2 Similarly, one finds t h a t , at t = 200 min, a' = 3.39 x 10 gpd/ft.

Since t h e

ratio method gives more reliable results when t is small, Neuman and Witherspoon adopted 4.17 gpd/ft as t h e representative value for t h e top 6 f t of t h e lower aquitard. Neuman and Witherspoon calculated t h e values of Ss' from laboratory consolidation tests by using t h e formula

where a

V

is t h e coefficient of compressibility, equal to -Ae/Ap, e is t h e

void ratio, p denotes pressure, and yw is t h e specific weight of water. These

values were then used to calculate K' from t h e relation K' = a'sS'. The results of similar calculations for both aquitards are summarized in Table 10.7. Note t h a t t h e diffusivity of t h e Oxnard aquifer ( a = 1.17 x 9 10 gpd/ft) is more than 1 million times t h e values obtained for t h e aquitards. Also, t h e hydraulic conductivity of t h e aquifer exceeds t h a t of t h e aquitards by more than 4 orders of magnitude. The specific storage of t h e aquifer, however, is less than Ssl in t h e aquitards above and below by 2 orders of

Table 10.7

Hydraulic properties of aquitard layers (Neuman and Witherspoon, 1972).

Layer

Section Tested

Upper

bottom

aqui t a r d

22 f e e t

Upper

bottom

aqui t a r d

11 f e e t

Lower

top

aqui t a r d

6 feet

Hydraulic D i f f u s i v i t y K'/S;

Specific Storage S;

Hydraulic Conductivity K '

gpdlft

2 cm /sec

1.02 x 10

1.47 x 10-1

2.4 x

7.88 x

2.45 x

1.11 x

2.44 x 10

3.51 x 10-1

2.4 x

7.88 x

5.85 x

2.66 x

5.99 x 10-1

1.0 x

3.28 x

4.17 x

1.89 x 10-0

4.17 x

lo2

ft-1

cm-l

gpdlft

cm/sec

co co cn

336

magnitude.

This means t h a t , for t h e s a m e change in head, a unit volume

of aquitard material c a n contribute about 100 times as much water from storage as can t h e s a m e volume of t h e aquifer. 10.5

WELL LOSSES

The drawdown in a pumping well, so, is made up of t h e head loss sw associated with laminar flow in t h e aquifer and t h e head loss se associated with turbulent flcw of water through t h e screens and into t h e pump intake. The loss sw is called t h e aquifer or formation loss, while se is called t h e well loss. The drawdown in a pumping well can be expressed as (10.55)

s o = CaQ + CwQn

where C a = sW/Q is t h e aquifer constant relating t h e discharge of t h e well Q t o sw, Cw is t h e well-loss constant relating Qn to se, and n is t h e exponent due to turbulence relating discharge to se.

Jacob (1947) proposed t h a t n =

2, while Rorabaugh (1953) suggested a n average value of about 2.5.

(1966) reported n values as high as 3.5.

Lennox

Values of n can be less than 2 at

relatively low rates of discharge. The values of Ca, Cw, and n for a given well can be determined best by a step-drawdown test (Section 11.3.B), where so is measured while t h e well is operated during successive periods at a constant fraction of full capacity.

The well is pumped at a given flow rate Q until so changes only relatively

little; Q is then increased, and so is measured over t h e s a m e period used for t h e first flow rate. This procedure is repeated until so is known for at least four different Q values. The step-drawdown test gives information regarding t h e relation between Q and so of a given well, which is important in selecting t h e optimum pump and depth of pumping (Section 11.3.C.4).

Also,

t h e Ca value yielded by t h e test can be used to e s t i m a t e t h e transmissivity of t h e aquifer, using t h e appropriate equation relating sw to Q. Jacob (1947) presented equations for evaluating Ca and Cw from step-drawdown data, assuming n = 2.

Rorabaugh (1953) devised a graphical

method for evaluating Ca, Cw, and n from step-drawdown results. Rorabaugh's method is based on eq. (10.551, which c a n be rearranged as

337

log

(%- ca)

= log

cw+

(n

-

The procedure consists of plotting so/Q for assumed values of Ca.

(10.56)

1) log

-

Ca versus Q on logarithmic paper The value of Ca t h a t gives a straight line on

this plot will b e t h e required one.

The slope of t h e straight line is equal

to n - 1, from which t h e value of n is obtained. The intercept of t h e line on t h e axis Q = 1 gives t h e value of Cw. Alternatively, Cw can be calculated by substituting Cay n, and a certain combination of t h e measured so and Q into eq. (10.55). In addition to Rorabaugh’s (1953) graphical procedure, Sheahan (1971)

developed a method for direct analysis of step-drawdown d a t a using type curves.

For additional details and field application of t h e step-drawdown

test, see Lennox (1966). Example. The graphical trial-and-error

procedure of Rorabaugh is

illustrated in Fig. 10.18, using t h e following hypothetical step-drawdown test:

Figure 10.18

Example of Rorabaugh’s graphical method.

338

Q (ft3/sec) so ( f t )

1 .o 7.02

0.5 2.95

2 .o 18.94

4.0 60.12

Based on these data, so/Q - Ca is plotted versus Q on logarithmic paper, first taking Ca = 0 and then larger values for Ca until a straight line is obtained.

A straight line is obtained when Ca = 5.0.

is 1.16, giving n = 2.16.

The slope of this line

The intercept of this line on t h e axis corresponding

t o Q = 1 cfs gives a value of Cw = 2.0.

Using these values of Ca, Cw, and

n, t h e aquifer losses and well losses for t h e Q values in this example are calculated as follows: Q (ft3/sec) sw ( f t ) s (ft) e

10.6

0.5 2.5 0.45

1 .o 5 .O 2.02

2 .o 10 .o 8.94

4.0 20 .o 40.12

SPECIFIC CAPACITY

The productivity of a well is often expressed in terms of specific capacity, Cs, which is defined as

Clearly, t h e specific capacity is not a constant, but decreases with time and discharge.

A decline in Cs in a certain well-aquifer system may indicate,

for example, deterioration of t h e well screen or declining S or T values as

a result of declining water tables or piezometric surfaces. The specific capacities of wells in a certain aquifer system are sometimes used t o estimate t h e distribution of T in t h e aqiufer a f t e r t h e relation between T and Cs is determined for a few wells in t h e same aquifer system.

This may be a valid

procedure if t h e wells a r e of similar construction and depth and situated in the same aquifer system.

Summers (19721, however, reported substantially

different values of Cs in wells located close together in crystalline rocks.