Workshop Pumping Test

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F:\Aquitard workshop\Clay till\Hydraulic conductivity-specific storage of clay tills.doc 1 of 5

Hydraulic conductivity and specific storage of clay tills

Christopher J. Neville

S.S. Papadopulos & Associates, Inc.

Last update: February 20, 2008

1. Hydraulic conductivities of clay till aquitards

Till is a broad descriptor and materials that are logged as tills may have properties that

range from aquifers to aquitards. The hydraulic conductivity of clean gravel can bereduced orders or magnitude with the addition of a relatively small quantity of fine-

grained sediments. In reflection of this, guidance in the literature regarding representative

values of hydraulic conductivity for till suggests a wide range. For example, in the chartbelow reproduced from Freeze and Cherry (1979), the hydraulic conductivity ranges from

about 10-10

to 10-4

cm/s.

Figure 1. General ranges of hydraulic conductivities


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R.E. Gerber and colleagues have conducted superb studies of groundwater flow through

the till aquitards in areas of the Oak Ridges Moraine in southern Ontario. Thecompilation of hydraulic values reproduced below from Gerber and Howard (2000)

illustrates the complexity of the sediments.

Figure 2. Hydraulic conductivities of Oak Ridges Moraine materials

(Gerber and Howard, 2000)


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2. Specific storage values for till

The specific storage is defined as the volume of water that is released from confined

storage for a unit decline in hydraulic head. The specific storage has units of L-1

. The

stored water is released from compaction (consolidation) of the porous medium and

expansion of the water. Jacob (1940; p. 576) derived the following expression for thespecific storage:

( )s wS g n = +

where w is the density of water, g is the acceleration due to gravity, is the

compressibility of the porous medium, is the compressibility of water, and n is theporosity of the porous medium.

Typical values for the properties of water are:

w = 1000 kg/m3; and= 4.410

10m-s

2/kg.

Younger (1993) presented typical order-of-magnitude values of the compressibility of

porous materials. Younger presents a value for clay of 10-6

m-s2/kg (Pa

-1) for clay. Freeze

and Cherry (1979) suggest that the typical range for the compressibility of clay isbetween 10

-8and 10

-6Pa

-1. Therefore, a back-of-the-envelope estimate of the range of

specific storage is 10-4

to 10-2

m-1

.

Some values of specific storage of clay till that have been reported in the literature aretabulated below. The literature values are consistent with the back-of-the-envelope

range of between 10-4 to 10-2 m-1.

Location Ss

(m-1)

Reference Notes

Manitoba till 9.910-3 Grisak and Cherry (1975) Consolidation tests

mean of 34 samples

WRNE, Manitoba 8.210-3 to

1.610-2

Grisak and Cherry (1975) Consolidation tests8 samples

Saskatchewan till 1.110-2 Grisak and Cherry (1975) Consolidation tests

mean of 24 samples

Dalmeny, Saskatchewan 1.110-4 to

1.610-4

Keller et al. (1986) Consolidation tests

3 samples

Warman, Saskatchewan 2.110-4 to3.310-4

Keller et al. (1989) Consolidation tests3 samples

Alberta till 1.010-2 Grisak and Cherry (1975) Consolidation tests

mean of 27 samples

Pine Coulee, Alberta 1.010-3 Smerdon et al. (2005) Model calibration

Grand Forks area, ND 1.010-4 to

1.210-3

Shaver (1998) Consolidation tests

107 samples


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Konikow and Neuzil (2007; Figure 5) have developed a chart indicating the likely range

of specific storage for aquitard materials. Their chart is reproduced on the next page.Konikow and Neuzil suggest that it is a generalized relation for normally consolidated

and overconsolidated clayey confining layers. The chart synthesizes data from several

sources (Domenico and Mifflin, 1965; Skempton, 1970; Cripps and Taylor, 1981; Tellam

and Lloyd, 1981; Burland, 1990; and Neuzil, 1993).

The water compressibility only curve refers to the specific storage if the porous

medium was incompressible. For this case, the specific storage is:

s wS gn =

If the porous medium was incompressible and the porosity was equal to 0.2, the specific

storage would be 8.610-7

m-1

.

Figure 3. Specific storage of clayey materials

(Konikow and Neuzil, 2007)


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3. References

Freeze, R.A., and J.A. Cherry, 1979: Groundwater, Prentice-Hall, Inc., Englewood

Cliffs, New Jersey.

Gerber, R.E., and K. Howard, 2000: Recharge through a regional till aquitard: Three-dimensional flow model water balance approach, Ground Water, 38(3), pp. 410-422.

Jacob, C.E., 1940: On the flow of water in an elastic artesian aquifer, Transactions,American Geophysical Union, 21, pp. 574-586.

Konikow, L.F., and C.E. Neuzil, 2007: A method to estimate groundwater depletion fromconfining layers, Water Resources Research, 43, W07417,

doi: 10.1029/2006WR005597.

Younger, P.L., 1993: Simple generalized methods for estimating aquifer storage

parameters, Quarterly Journal of Engineering Geology, 26, pp. 127-135.


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C:\CJN\Aquitards\Leaky aquifers\Pumping tests in leaky aquifers.doc

Models for interpreting pumping tests in leaky aquifers


S.S. Papadopulos & Associates, Inc.

Last update: November 17, 2008

Overview

Hydrogeologists are frequently charged with interpreting the data from pumping tests in

which the effects of flow processes in the confining layers are significant. To interpretthe data from these tests we must use solutions from the literature on leaky aquifers. One

of the many strengths of computer-assisted interpretation packages is that they support

several models of pumping tests in leaky aquifers, of increasing complexity. However,the large number of options may also be a source of confusion. These notes have been

prepared to provide a systematic development of the conceptual models that underlie the

most widely used of the solutions.

Outline

1. Introduction2. General conceptual model for analytical solutions3. Representation of the pumped aquifer4. Hantush and Jacob (1955) analysis5. Hantush (1960) analysis6. Neuman and Witherspoon (1969) analysis7. Evaluation of alternative conceptual models for aquitard storage8. Moench (1985) analysis9. Cooley and Case (1973) analysis10.References


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1. IntroductionA fundamental assumption underlying the Theis solution is that the aquifer is perfectly

confined, so that the release of water from confined storage is the only source of water.

This is a relatively restrictive assumption, and may hold only if the duration of pumping

is relatively brief, and the hydraulic conductivities of the confining layers are relativelysmall. In the longer term, this assumption is generally grossly violated, and in

multiaquifer settings the bulk of the water withdrawn by a pumping well will be derived

by transmission across the confining layers. In the intermediate term, water may bederived from storage in the confining layers as well as by storage across them. Aquifers

with significant transmission of water from confining layers are designated as leaky.

When introducing the interpretation of aquifer tests in leaky aquifers, it is first importantto issue a clarification: it is not the aquifers that are leaky. Rather it is the over and

underlying confining layers that are doing the leaking.

One of the many strengths of computer-assisted aquifer test interpretation packages is

that they support several models of pumping tests in leaky aquifers, of increasingcomplexity. However, the large number of options may also be a source of confusion. In

these notes we proceed systematically from the simplest to the most complex conceptualmodels of the responses to pumping in aquifers with flow from confining units.

Although published nearly 40 years ago, Hantushs 1964 monographHydraulics of Wellsremains the starting point for understanding aquifer test solutions. The coverage of leaky

aquifers is especially good, reflecting the fact that Hantush was a giant in the subject.

Significant developments on the interpretation of aquifer tests in leaky aquifers haveoccurred since 1964. There are two textbooks with particularly good treatments of the

interpretation of aquifer tests in leaky aquifers. V. Batus Aquifer Hydraulics (1998)has a very good presentation of the theory underlying the different methods of analysis,

with illustrative examples, and Kruseman and de Ridders Analysis and Evaluation of

Pumping Test Data (2nd

edition, 1990) remains an indispensable reference.


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2. General conceptual model for analytical solutionsThe results of most aquifer tests are interpreted with analytical solutions. Although it is

simpler to use an analytical rather than a numerical solution for the interpretation of

routine tests, it is important to note that analytical solutions are usually based on highly

idealized conceptual models. A key assumption that underlies the solutions developedfor leaky aquifers concerns the directions of flow in the aquifers and aquitards. In

particular, it is assumed that flow is horizontal in the aquifers and vertical in the

aquitards. This conceptual model is illustrated below.

Figure 1. Conceptual model for pumping in a multiaquifer system


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The assumed flow directions are realistic if there is a relatively large contrast between the

hydraulic conductivities of the aquifers and aquitards, designated Kand K, respectively.Hantush (1964) suggested that any horizontal component of flow in an aquitard would be

negligible forK < K/500. Neuman and Witherspoon (1971) conducted extensive

analyses with a general numerical solution, and suggested that if the hydraulic

conductivities differ by at least 100, the error introduced by simplifying the flowdirections is small. We will use their results to provide a general rule-of-thumb for the

applicability of the simplified analyses. The analytical solutions will be considered to be

applicable if:

'100

KK < (1)


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3. Representation of the pumped aquiferThe methods generally applied to interpret the results from pumping tests in leaky

aquifers start from the Theis solution, and share many of its underlying assumptions. In

particular, the aquifer is assumed to be:

Horizontal; Homogeneous; Isotropic; Infinite in areal extent; Fully saturated; and Pumped by a fully penetrating well.

The conceptual model for the aquifer is shown below:

Figure 2. Conceptual model for the pumped aquifer


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The governing equation for flow in the aquifer is:

1L

h hS T r q

t r r r

= +

(2)

where:

h : head in the aquifer [L]

r : radial distance from the center of the pumping well [L]t : time elapsed since the start of pumping [T]T : transmissivity [L

2T

-1]; and

S : confined storage coefficient [-].

The term qL designates leakage from the aquitard, expressed as a flow rate per unit area

[L3T

-1/L

2].

If we define the drawdown as the difference between the static head in the aquifer and thehead at any time tand distance r:

( ) ( ), ,is r t h h r t = (3)

the governing equation for flow in the aquifer can be written as:

1L

s sS T r q

t r r r

=

(4)

Equation (4) represents the fundamental governing equation for flow in the aquifer. The

essential differences among the leaky aquifer solutions arise from the way that the

leakage term qL is evaluated; that is, how flow in the aquitard is represented.


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4. Hantush and Jacob (1955) analysisHantush and Jacob (1955) were the first researchers to develop an effective method for

interpreting pumping tests affected by the leakage from the confining aquitards. In their

analysis, Hantush and Jacob assumed that the contribution from storage in the aquitards

was negligible.

Hantush and Jacobs conceptual model for an incompressible aquitard is shown below.

For an incompressible aquitard, the vertical head profile is linear at any distance from thepumping well rand at any time t.

Figure 3. Hantush and Jacob (1955) conceptual model for the aquitard


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Referring to the previous sketch, the leakage flux is given by:

( ),'

'

i

L

h h r t q K

b

=

(5)

where hi is the head at the top of the aquitard (constant through time), h(r,t) is the head in

the aquifer, corresponding to the head at the base of the aquitard, and K and b are thehydraulic conductivity and thickness of the aquitard, respectively. A negative sign is

placed in front of the leakage flux; according to the sign convention on thez-coordinate, a

flux in the negativez-direction represents a source to the aquifer.

Writing (4) in terms of the drawdown in the aquifer:

''

' 'L

s Kq K s

b b= = (6)

The quotient K/b is referred to as the leakance, with units of T-1

. If we substitute thisexpression forqL in the governing equation for the aquifer (3), we obtain:

1 '

'

s s KS T r s

t r r r b

=

(7)

The initial and boundary conditions for the aquifer are:

( ),0 0s r = (8a)

0lim 2r

srT Q

r

=

(8b)

( ), 0s t = (8c)

The sign convention adopted for the inner boundary condition (8b) assigns a positive

value forQ for groundwater withdrawals, that is, for pumping that induces declines in thehead in the aquifer (positive drawdowns).


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The analytical solution for (6) subject to (7a-c) is:

2

2

1

4 4u

Q rs EXP y dy

T y B y

=

(9)

with u andB2 given by:

2

4

r Su

Tt= (10a)

2 '

'

bB T

K= (10b)

We recognize the term u immediately it is the same as the argument of the Theis wellfunction W(u). The termB is inversely related to the leakance.

The solution is generally written as:

,4

Q rs W u

T B

=

(11)

with the integral designated as the Hantush leaky well function W(u,r/B).

In the limit, for a perfectly confined aquifer, r/B 0, we have:

{ }1lim ( , ) ( ,0) ( )B

u

rW u W u EXP y dy W uB y

= = = (12)

As expected, forK 0 we recover the Theis well function.

The Hantush leaky well function is plotted in Figure 4. Because the solution is a functionof two dimensionless parameters, u and r/B, the Hantush leaky well function is presented

a family of curves for different values ofr/B. In reality, the parameterr/B is continuous,

and a computer-based analysis package that implements (8) may report any value ofr/B,

not just the values shown on type curves reproduced in textbooks.


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Figure 4. Hantush-Jacob (1955) leaky well type curves

For relatively early times, we see that a portion of the Hantush-Jacob leaky aquifer curves

follow the Theis solution (the Theis curve is approximated closely by the curve forr/B = 0.001). During this period of the response the pumped water is derived fromstorage within the pumped aquifer and leakage across the aquitard is insignificant. This

means that we can still use the Theis analysis to obtain a preliminary estimate of the

transmissivity, provided we restrict our fit to the drawdown data for which nostabilization is observed.

At late time, the Hantush-Jacob leaky aquifer curves become flat. This indicates that the

pumped water is derived primarily from leakage across the aquitard.


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It is interesting to note that for a given r/B, the deviation from the Theis curve is greater

at larger radial distances from the well. This statement may seem counter-intuitive. Thecloser to the well, the greater is the head difference between the pumped and unpumped

aquifers. Consequently, the closer to the well, the greater is the leakage. However, the

solution shows that the greater the leakage, the less deviation from the Theis solution.

Why? The answer lies in the fact that the head difference between the pumped andunpumped aquifers gives the leakage per unit area. Near the well, the leakage per unit

area is large, but the area over which the leakage occurs is small. Therefore, the totalamount of leakage in the vicinity of the well is a small portion of the well discharge, and

therefore drawdown is close to the Theis solution. Far from the well, the situation is

reversed, and the total amount of leakage accounts for a large amount of the welldischarge, so the Theis solution is no longer appropriate.

In principle, Tand Sof the pumped aquifer and K/b of the aquitard can be computed by

matching drawdown data at a single observation well to a type curve. In practice, thetype curves in Figure 4 have very similar shapes, and it is often difficult to match the data

to a unique type curve. For reliable determination of aquifer and aquitard parameters, itis desirable to have two observation wells, one close to the pumped well, and the other farfrom it. A composite plot (s versus t/r2) is made using both sets of data [they will not

form a single curve.] As explained above, drawdown data from the close to the pumping

well should show little effect of leakage, so that much of the data (except for late time)should fit the Theis curve, thus yielding Tand Sof the pumped aquifer. Next, without

moving the data plot relative to the type-curve plot, one can choose the appropriate type

curve that matches the data from the distant observation well. This should yield r/B,

from which one can calculate K/b. K can also be calculated ifb is known.

Checks on the results of a Hantush-Jacob analysis

Whether the Hantush-Jacob analysis is accomplished with pencil and paper, or with a

computer-aided interpretation package, it is not complete without some reality-checks.

1. Confirm that the conceptual model for the Hantush-Jacob analysis is appropriate foryour situation. In particular, the aquifer must be extensive and overlain by arelatively incompressible aquitard. The water level at the top of the aquitard should

remain constant during the pumping test. A drawdown response that is diagnostic of

leaky aquifer response can be mimicked by other conditions for example, aconfined aquifer that intersects a constant-head boundary.

2. Confirm that your fitted parameters are consistent with the basic assumption:'

100

KK <


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5. Hantush (1960) analysisHantush (1960) derived a general solution for pumping from a leaky aquifer. In this

more general formulation the assumption of an incompressible aquitard is relaxed. The

aquitard can supply water both by transmission from an overlying aquifer and through

changes in storage. The full Hantush (1960) solution is relatively general, and canaccommodate an aquitard with a finite thickness with two alternate boundary conditions

at the top of the aquitard, zero-drawdown or zero-flow. However, the final solution isrelatively complicated and Hantush did not evaluate specific results with it. Instead,

Hantush considered the asymptotic cases of early time and late time. The late-time case

is very similar to the Hantush-Jacob (1955) solution late time in effect means, afterthe effects of storage in the aquitard have dissipated. We will focus here on the early

time results.

During the early period of pumping the pressure pulse moving upwards into the aquitardhas not yet had time to reach the upper boundary of the aquitard. Therefore, the aquitard

can be idealized as being infinitely thick. The conceptual model for the aquitard is shownbelow.

Figure 5. Conceptual model for an infinitely thick compressible aquitard


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The leakage flux from the aquitard to the aquifer at any time tand radial distance ris

given by:

'

'L

sq K

z

=

(13)

The drawdown in the aquitard s is derived from a consideration of transient flow in the

aquitard. The governing equation for flow in the aquitard is:

2'

2

' ''

s

s sS K

t z

=

(14)

The initial and boundary conditions for the aquitard are:

( )' ,0 0s r = (15a)

( ) ( )' ,0, ,s r t s r t = (15b)

( )' , , 0s r t = (15c)

The analytical solution for the coupled set of equations (7) subject to (8a-c) and (14)

subject to (15a-c) is:

{ }

( )

1/ 2

1/ 2

1

4u

Q us EXP y ERFC dy

T yy y u

=

(16)

with u and given by:

2

4

r Su

Tt= , as before

1/ 2' '

1

4

sK SrTS

=

The term should not be confused with the previousB. We saw previously that for an

impermeable aquitardB. For the Hantush (1960) the corresponding case would be 0.


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The solution is generally written as:

( ),4

Qs H u

T

= (17)

where the integral is designated the Hantush well functionH(u,). The Hantush (1960)leaky well function is plotted in the next figure. As with the Hantush-Jacob (1955)

solution, the early-time solution for a compressible aquitard solution is a function of two

dimensionless parameters (u and ) and the well function is presented a family of curves

for different values of.

Figure 6. Hantush (1960) early-time type curves

For the case of an impermeable aquitard, 0, the solution reduces to:

{ } { }

{ }

01lim ( , ) 0

1( )

u

u

H u EXP y ERFC dyy

EXP y dy W uy

=

= =

(18)

That is, we again recover the Theis well function, as expected.


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Apart from the limiting case small , the type curves bear little resemblance to theHantush-Jacob type curves for a leaky aquitard with no storage. This indicates that if the

aquitard is compressible, the effects of leakage will be exhibited throughout thedrawdown history of a well in the aquifer, and not just at later times. The implication is

that when storage in the aquitard is significant there may not be any portion of the dataover which we can use the Theis analysis to obtain a preliminary estimate of thetransmissivity.

Checks on the results of a Hantush (1960) analysis

1. Again we must confirm that the fitted parameters are consistent with the basicassumption:

'100

KK <

2. We must also confirm that the portion of the data that have been analyzed is restrictedto early time, when the idealization of the aquitard as an infinitely thick layer is valid.

Hantush provided the following criterion for early time:

( )2

' '

'0.1

sS bt

K<

Warning regarding the Hantush (1960) analysis

The Hantush (1960) type curve analysis is restricted to early times. In a multiaquifer

system the effects of storage in the aquitard will eventually dissipate and there will be

steady transmission of water across the aquitard. When this occurs drawdowns willstabilize in the pumped aquifer. The Hantush (1960) type curves do not consider this

possibility. If the drawdowns do stabilize the Hantush-Jacob (1955) solution is

appropriate.

AQTESOLV implements two solutions for the Hantush (1960) problem. The first

solution it calls Hantush (1960) and is actually the asymptotic solution for early time.This solution should not be used for predicting long-term drawdowns. The second

solution it calls Neuman and Witherspoon (1969). This solution represents the sameconceptual model as Hantush (1960), but covers the entire range of time.


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6. Neuman and Witherspoon (1969) analysisNeuman and Witherspoon (1969a) continued the development of the Hantush (1960)

analysis, generalizing it for the case of a pumped and unpumped aquifer separated by a

compressible aquitard. The conceptual model is illustrated below:

Figure 7. Conceptual model for the Neuman and Witherspoon analysis

As in the Hantush (1960) solution, the leakage fluxes from the aquitard to the aquifer at

any time tand radial distance ris given by:

''L

sq K

z

=

(19)

Leakage terms must be evaluated at the top and bottom of the aquitard. At its base, water

flows from the aquitard to the pumped aquifer. At its top, water flows from the

unpumped aquifer to the aquitard in response to pumping.

The governing equation for transient flow in the aquitard is:

2'

2

' ''s

s sS K

t z

=

(20)


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The initial and boundary conditions for the aquitard are:

( )' ,0 0s r = (21a)

( ) ( )1' ,0, ,s r t s r t = (21b)

( ) ( )2' , ', ,s r b t s r t = (21c)

The boundary conditions for the aquitard differ from the Hantush (1960) model presentedpreviously, in that there are linkages with two aquifers. As written in (21a), the pumped

aquifer (aquifer #1) is located at the boundary of the aquitard designated z = 0. As

written in (21b), the unpumped aquifer (aquifer #2) is located at the boundary of theaquitard designatedz = b.

The governing equation for the pumped aquifer is:

1 11 1 1

1L

s sS T r q

t r r r

=

(22)

subject to:

( )1 ,0 0s r = (23a)

1

0lim 2r

srT Q

r

=

(23b)

( )1 , 0s t = (23c)

The governing equation for the unpumped aquifer is:

2 22 2 2

1L

s sS T r q

t r r r

=

(24)

subject to:

( )2 ,0 0s r = (25a)

2

0lim 2 0r

srT

r

=

(25b)

( )2 , 0s t = (25c)


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Neuman and Witherspoons final solution is a set of relatively complicated expressions

involving integrals with Bessel functions. Their solution is expressed in terms ofdimensionless time defined as:

11 2

1

D

T tt

S r

= (26)

and three dimensionless groupings in the case of a single aquitard:

1/ 2

111

'

'

T bB

K

=

(27a)

1/ 2

221

'

'

T bB

K

=

(27b)

1/ 2

11

1 1

' '

4

sK Sr

T S

=

(27c)

Neuman and Witherspoon were able to evaluate their solution over a full range of times

and conditions. However, their solution involves too many parameters to support

analyses with type curves. Their solution has been implemented in the popular aquifertest interpretation package AQTESOLV for Windows (Duffield, 2000). AQTESOLV

supports the application of the solution with automated parameter estimation, using a

robust nonlinear fitting routine.

If the unpumped aquifer is sufficiently transmissive that there are no drawdowns, theNeuman and Witherspoon (1969a) solution reduces to a generalization of the

Hantush (1960) solution over the full range of time. The solution can be expressed interms of two dimensionless parameters:

1/ 2'

'

r Kr

B Tb

=

(28a)

1/ 2' '

4

sK Sr

TS

= (28b)


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A plot of dimensionless drawdown sDversus dimensionless time tDfor = 0.01 and

various values ofr/B is shown in Figure 8. For a small value of such as 0.01, aquitardstorage is insignificant and the Hantush (1960) solution is virtually identical to the

Hantush-Jacob (1955) solution (compare Figure 8 with Figure 4). The drawdown

initially follows the Theis curve, and then deviates to a steady state value that depends on

the value of the parameterr/B.

Figure 8. Neuman and Witherspoon type curves for = 0.01To illustrate the effect of aquitard storage, Figure9 shows a plot of dimensionless

drawdown versus dimensionless time for various values ofwhile keeping r/B = 0.3.Note that larger values ofcause the drawdown curve to deviate significantly below theTheis curve. However, at late time, all the solutions still converge to the steady statevalue determined by r/B.

Figure 9. Neuman and Witherspoon type curves for r/B = 0.3


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Figures 8 and 9 illustrate that the parameters and r/B control the drawdown in thepumped aquifer in different ways. The parametercontrols the early behavior. Ifissmall (insignificant aquitard storage), the early time drawdown follows the Theis

solution. Ifis large (significant aquitard storage), the early time drawdown isconsiderably less than that predicted by the Theis solution. In contrast, the parameterr/B

controls the late time, steady state drawdown. When the drawdown curve reaches theflat, steady-state portion, a linear gradient is established across the aquitard, and the

pumped water is derived primarily from the unpumped aquifer.

Neuman and Witherspoon (1969b) present a thorough evaluation of the Neuman and

Witherspoon solution. Batu (1999) also presents an excellent summary of the results of

the solution.


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7. Evaluation of alternative conceptual models for aquitard leakageLet us review the conceptual models for the conventional leaky aquifer solutions.

Figure 10. Conventional conceptual model for leaky aquifer solutions

The solutions that we will compare are listed below, along with the key assumptions for

each solution.

Theis (1935) solution

Perfect confinement: K = 0

Hantush and Jacob (1955) solution

K> 0, Ss = 0No drawdown in the unpumped aquifer

Hantush (1960) solutionK> 0, Ss > 0No drawdown in the unpumped aquifer

Neuman and Witherspoon (1969) solution

K> 0, Ss > 0Drawdown in the unpumped aquifer


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The differences between the solutions implemented in AQTESOLV for the

conventional conceptual model are illustrated in terms of dimensionless parameters.

Figure 11. Comparison of leaky aquifer solutions

For this conceptual model, in the long-term the contributions from storage in the aquitard

dissipate and the there is steady transmission across the aquitard. Since the water level in

the unpumped aquifer is assumed to remain constant through time, drawdowns in the

pumped aquifer stabilize. We see that the Hantush solution matches the Neuman andWitherspoon solution at early time, but for later time it does not stabilize. Once the

Hantush departs from the Neuman and Witherspoon solution, its results are incorrect asthe solution is beyond its range of applicability.


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8. Moench (1985) analysisMoench (1985) considered a conceptual model that is similar to the Neuman and

Witherspoon (1969a) analysis, but incorporated wellbore storage in the pumping well.

The Moench (1985) analysis appears to be more general in that it accommodatesaquitards both above and below the pumped aquifer. However, it should be noted that it

is straightforward to generalize the Hantush and Neuman-Witherspoon analyses to also

consider the second aquitard. Furthermore, although the two aquitards can be assigneddifferent properties, in practice it is not possible to distinguish the leakage from the two,

and the system responds as if there was one equivalent aquitard.

In the Moench (1985) analysis, the boundary conditions at the top of the overlying

aquitard and bottom of the underlying aquitard can be either no-drawdown (constant-

head) or no-flow. In this regard, the Neuman-Witherspoon analysis is more general sinceit can represent the spectrum of conditions that could occur at the top. The corresponding

end-member cases considered in the Moench (1985) analysis are represented as follows:

No-drawdown: Kof unpumped aquifer relatively high; and No-flow: Kof unpumped aquifer set to 0.0.

The key innovation in the Moench (1985) analysis is the incorporation of wellbore

storage. Let us recall the inner boundary condition for the pumped aquifer assumed inthe Neuman-Witherspoon solution:

1

0lim 2r

srT Q

r

=

(29)

This is identical to the Theis solution.

Relaxing the assumption that the well has an infinitesimal diameter allows us to consider

wellbore storage in the analysis. The inner boundary condition is written as a statement

of mass conservation at the wellbore:

212 w cs H

r T r Qr r

+ =

(30)

whereH(t) is the head in the well.

Moench (1985) writes the boundary condition at the well in a somewhat more general

form, allowing for the consideration of a thin skin (a zone with properties that are altered

with respect to the formation) around the pumping well.


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Moench (1985) used the Laplace transform to derive the solution and obtained final

results by numerical inversion of the Laplace transform solutions. This is a simplerapproach that is generally more accurate and certainly more efficient. This approach e

solution has been implemented in AQTESOLV for Windows (Duffield, 2000). As with

the Neuman-Witherspoon analysis, the Moench 91985) solutions involve too many

parameters to support analyses with type curves. AQTESOLV supports the applicationof the solution with automated parameter estimation, using a robust fitting routine.


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9. Cooley and Case (1973) analysisCooley and Case (1973) considered a setting that is similar to that analyzed by Neuman

and Witherspoon (1969a). However, in their conceptual model the pumped aquifer is

overlain by an unconfined aquitard. The conceptual model is illustrated below:

Figure 12. Conceptual model for the Cooley and Case analysis

The leakage fluxes from the aquitard to the aquifer at any time tand radial distance ris

given by:

''L

sq K

z

=

(31)

The drawdown in the aquitard s must be derived from a consideration of transient flow

in the aquitard. The governing equation for flow in the aquitard is:

2'

2

' ''s

s sS K

t z

=

(32)

The initial conditions for the aquitard are:

( )' ,0 0s r = (33a)


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The boundary conditions at the interface between the aquitard and the aquifer are:

( ) ( )1' ,0, ,s r t s r t = (33b)

So far, the formulation is identical to the models of Hantush, Neuman and Witherspoon,

and Moench.

The upper boundary of the aquitard is assigned a boundary condition that accounts for the

decline of the water table. Cooley and Case (1973) represent this drainage process using

the Boulton (1954) integral boundary condition:

( ) ( ) ( ){ }1 2 2 1 2 20

' '' , , , ,

t

y

s sK r b b t S r b b EXP t d

z

+ = +

(33c)

where K and Sy are the vertical hydraulic conductivity and specific yield of the aquitard,

and 2 is the delayed-yield parameter. The thickness of the aquifer is b1 and the initialsaturated thickness of the aquitard is b2, so b1+b2 represents the top surface of theaquitard. The delayed yield parameter is defined as:

2

y

K

S L

=

(34)

whereL provides an approximate measure of the height of the capillary fringe.

The Cooley and Case (1973) present an exact form of their solution, but it is cast as acomplex integral involving Bessel functions. The solution has recently been

implemented in AQTESOLV for Windows (Duffield, 2000), with results obtained usingnumerical inversion of the Laplace transform solutions.

The results of some experiments with the Cooley and Case (1973) are plotted in

Figures 13 and 14. The results are presented in Figure 11 with log-log axes to illustratethe differences with respect to the results shown in the previous figure. The Cooley and

Case solution includes a parameter to represent the capillary fringe in the aquitard. The

differences between the solutions for different values ofL/b are very small, which

suggests that for this example the capillary fringe in the aquitard is not significant.


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10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

t/r2 (min/ft2)

10-2

10-1

100

101

102

Drawdown(t)

Cooley-Case, L/b' = 0.06

Cooley-Case, L/b' = 0.0Cooley-Case, L/b' = 1.0

Theis, T and S

Hantush

Theis, T and Sy

Figure 13. Results of the Cooley and Case analysis


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10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

t/r2 (min/ft2)

0

5

10

15

20

Drawdown(

ft)

Cooley-Case, L/b' = 0.06

Cooley-Case, L/b' = 0.0Cooley-Case, L/b' = 1.0

Theis, T and S

Hantush

Theis, T and Sy

Figure 14. Results of the Cooley and Case analysis (semilog plot)


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The two preceding figures are a bit busy, but the key points are straightforward.

1. At very early times, the drawdown in the aquifer is approximated by the Theissolution, with the transmissivity and storativity (confined storage coefficient) of the

pumped aquifer. We see that for this example the effects of leakage occur so early

that we never see this confined portion of the response.

2. During early to middle times, the drawdown in the aquifer predicted with the Cooleyand Case (1973) solution is the same as the Hantush (1960) solution for a leakyaquitard with storage.

3. The Hantush (1960) solution predicts that drawdowns in the aquifer will stabilize; theunpumped aquifer above the aquitard acts as an inexhaustible source of water. For

later times this assumption is clearly inappropriate, and leads to a significant

underprediction of drawdowns in the pumped aquifer.

4.

During late time, the drawdown in the aquifer is approximated closely by the Theissolution again, but this time using the transmissivity of the pumped aquifer and thespecific yield of the aquitard. This may be the most important lesson from the

experiments with the Cooley and Case (1973) solution. If our objective is to estimate

the long-term drawdowns in the pumped aquifer, we can gauge the relative

magnitudes of the later-time drawdowns using the Theis solution with arepresentative estimate of the transmissivity and considering a range of specific

yields.


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10.ReferencesBatu, V., 1998: Aquifer Hydraulics, John Wiley & Sons, Inc.

Boulton, N.S., Unsteady radial flow to a pumped well allowing for delayed yield from

storage, Association Internationale dHydrologie Scientifique, Assemble Gnerale deRome, Tome II, 472-477, 1954.

Cooley, R.L., and C.M. Case, 1973: Effect of a water table aquitard on drawdown in anunderlying pumped aquifer, Water Resources Research, 9(2), pp. 434-447.

Duffield, G.M., 2000: AQTESOLV for Windows Users Guide, HydroSOLVE, Inc.,Reston, VA.

Hantush. M.S., 1960: Modification of the theory of leaky aquifers,Journal of

Geophysical Research, 65(11), pp. 3713-3725.

Hantush, M.S., and C.E. Jacob, 1955: Non-steady radial flow in an infinite leaky aquifer,Transactions of the American Geophysical Union, 36(1), pp. 95-100.

Kruseman, G.P., and N/A. de Ridder, 1990: Analysis and Evaluation of Pumping Test

Data, 2nd

Edition, Publication 47, International Institute for Land Reclamation andImprovement, Wageningen, The Netherlands.

Moench, A.F., 1985: Transient flow to a large-diameter well in an aquifer with storativesemiconfining layers, Water Resources Research, 21(8), pp. 1121-1131.

Neuman, S.P., and P.A. Witherspoon, 1969a: Theory of flow in a confined two aquifer

system, Water Resources Research, 5(4), pp. 803-816.

Neuman, S.P., and P.A. Witherspoon, 1969b: Applicability of current theories of flow in

leaky aquifers, Water Resources Research, 5(4), pp. 816-829.


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Page 1 of 10

F:\Aquitards\NW-ratio-method\Neuman-Witherspoon ratio method.doc

Interpretation of aquitard properties from pumping-induced aquifer

and aquifer drawdowns:

Neuman and Witherspoon ratio method


S.S. Papadopulos & Associates, Inc.Last update: June 28, 2010

1. Introduction

Although the solutions of Hantush (1960) and Neuman and Witherspoon (1969)

implicitly contain within them the solutions for the drawdown in the aquitard, they weredeveloped specifically to interpret the drawdowns in the pumped aquifer. Neuman and

Witherspoon (1972) conducted additional analyses with their solution to develop a

practical method for identifying the properties of the aquitard from pumping data. Theirmethod is based on a simplified version of their solution, which assumes an aquitard that

is infinitely thick. The method is referred to as theNeuman-Witherspoon ratio method.

In these notes we review the foundations of the Neuman-Witherspoon ratio method andillustrate its application with an example calculation.


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2. Neuman and Witherspoon (1972) ratio method conceptual model

The conceptual model for the Neuman-Witherspoon ratio method is shown schematically

in Figure 1. The conceptual model is a simplification with respect to the analysis of

Neuman and Witherspoon (1969). In particular, it is assumed that the pumped single

aquifer is overlain by a relatively thick aquitard. We recognize that this is the sameconceptual model that underlies the Hantush (1960) early-time (thick aquitard) solution.

Figure 1. Conceptual model for the Neuman-Witherspoon ratio method


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3. Results from the Neuman and Witherspoon (1969) solution

For pumping from an aquifer overlain by a compressible, infinitely thick aquitard, the

solution of Neuman and Witherspoon (1969) is expressed in terms of three dimensionless

parameters , tD, and tD, defined as:

1/ 2' '1

4

sK S

rTS

=

(1)

2D

Ttt

Sr= (2)

'

' 2

'D

s

K tt

S z= (3)

Neuman and Witherspoon investigated the variation in the ratio of the drawdowns in the

aquitard and the pumped aquifer at the same distance from the pumping well,s(r,z,t)/s(r,t), as a function of the elapsed time (t) and the distance above the top of theaquifer (z). They summarized their results in the form of the remarkable plot reproduced

in Figure 2.

The reason this plot is so remarkable is that the results demonstrate that the ratio s/s is

essentially independent of the value of, for values ofranging between 0.0 and 1.0.

The results for= 10.0 are significantly different, so the results of their analysis can besummarized as follows:

For all practical values of tD, the ratio s/s is independent of, as long as the

value ofis less than about 1.0.

This result has very important practical implications. It suggests that it may be possible toestimate the hydraulic diffusivity of the aquitard (K/Ss) knowing only the ratio of the

aquitard and aquifer drawdowns, and the elapsed dimensionless time with respect to the

pumped aquifer, tD (defined here in Equation 2). In other words, if the assumptionsunderlying Neuman and Witherspoons analysis are satisfied, and independent estimates

of the aquifer properties T and S are available, the properties of the aquitard can be

estimated knowing only the ratio of the drawdowns in the aquifer and the aquitard

observed at the same time and same radial distance from the pumping well.


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Page 4 of 10


Figure 2. Ratio of aquitard and aquifer drawdowns


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Page 5 of 10


4. Neuman and Witherspoon (1972) ratio method analysis

To facilitate the analysis, the results shown in Figure 2 have been extended to consider a

more complete set of curves for values of tD. More complete results are presented in

Figure 3.

Since it is assumed that the aquitard is infinitely thick, the analysis is not applicable after

the effects of pumping reach the top of the aquitard. The limit of applicability is

expressed in Hantush (1960) as:

( )2

' '

'0.1

sS b

tK

< (4)

The steps in the application of the Neuman-Witherspoon ratio method are summarized

below.

1. Determine the values of the transmissivity and storage coefficient, Tand S, for the

pumped aquifer, using an appropriate method.

2. At a selected radial distance from the pumping well (r), determine the ratios(r,z,t)/s(r,t) at a given early value of time. Repeat this calculation for other values ofr,z, and tif possible.

3. Determine the values oftD for the particular values ofrand t.

4. With the known values ofs/s and tD, determine the corresponding value oftD from

the Neuman and Witherspoon plot. This plot is reproduced in Figure 3.

5. Calculate the diffusivity of the aquitard from the estimated value of tD, using a re-

arranged version of Equation (3):

2'

'

'D

s

K zt

S t= (5)


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Figure 3. Plot for the Neuman-Witherspoon ratio method


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5. Example calculations

The results from a pumping test are shown in Figure 4. The drawdowns are presented for

the pumping well, for a piezometer in the pumped aquifer 345 m from the pumping well,

and for a piezometer in the overlying aquitard at the same distance, screened 2 m above

the base of the aquitard.

Figure 4. Results of pumping test

The transmissivity and storage coefficient of the aquifer are estimated from a

Cooper-Jacob straight-line analysis of the drawdowns for piezometer a:

( )( )

3

4 2

12.303

4

0.002 m /s 12.303 7.3 10 m /s

4 0.5m

QT

s

=

= =

( )

( )

0

2

4 2

6

2

2.25

60 s2.25 7.3 10 m /s 2 min

min1.7 10

345 m

TtSr

=

= =


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After 2,500 seconds of pumping, the drawdowns in the aquifer (piezometer a) and the

aquitard (piezometer b) are:

s(r= 345 m, t=2,500 s) = 1.65 m; and s(r= 345 m,z = 2 m, t=2,500 s) = 0.05 m

Therefore:

0.05 m0.030

1.65 m

s

s

= =

At t= 2,500 s, the dimensionless time for the aquifer is:

( )( )( )( )

2

4 2

267.3 10 m /s 2500 s 560

1.7 10 345 m

D

Ttt

Sr

=

= =

Referring to the ratio method chart, fors/s = 0.30 and tD = 560, we estimate tD = 0.11.This estimation is shown in Figure 5.

Recalling the definition of the dimensionless time for the aquitard:

( )( )( )( )

2

4 2

26

7.3 10 m /s 2500 s560

1.7 10 345 m

sD

S zt

K t

=

= =

we can solve for the hydraulic diffusivity of the aquitard:

( )( )

( )

2

2

6 22 m

0.11 2.1 10 m /s2500 s

D

s

K zt

S t

=

= =

Assuming Ss =10-3

m-1

yields K = 2.110-9

m/s.


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Page 9 of 10


0.11Dt = 0.11Dt =

Figure 5. Estimation of dimensionless time for the aquitard


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6. References

Hantush. M.S., 1960: Modification of the theory of leaky aquifers,Journal of

Geophysical Research, vol. 65, no. 11, pp. 3713-3725.

Neuman, S.P., and P.A. Witherspoon, 1969: Theory of flow in a confined two aquifersystem, Water Resources Research, vol. 5, no. 4, pp. 803-816.

Neuman, S.P., and P.A. Witherspoon, 1972: Field determination of the hydraulicproperties of leaky multiple aquifer systems, Water Resources Research, vol.8, no. 5,

pp. 1284-1298.

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