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© 2014 HydroSOLVE, Inc.

Using Derivative Analysis to Improve Pumping Test Interpretation with the Cooper and Jacob Method

Glenn M. Duffield

HydroSOLVE, Inc.

703.264.9024

www.aqtesolv.com

[email protected]


What Is a Pumping Test?

An aquifer test performed with a controlled pumping rate

– constant-rate test

– step-drawdown test (well performance)

– recovery test

Water-level response (drawdown) measured in control well and one or more observation wells


What Are the Objectives of a Pumping Test?

Estimation of hydraulic properties (aquifers and aquitards)

Detection of boundaries

Evaluation of well performance (well loss)


Analysis of Pumping Test Data

Traditional Methods


Theis (1935) introduced a type-curve matching technique for estimating aquifer properties from a constant-rate pumping test assuming a fully penetrating pumping well in a homogeneous and isotropic nonleaky confined aquifer of infinite extent and constant thickness…

In The Beginning…

there was Theis!


In The Beginning… there was Theis!

100

101

102

10310

-2

10-1

100

101

1/u

w(u

)

100

101

102

10310

-1

100

101

102

Time (min)

Dra

wd

ow

n (

ft)

)(

4uw

s

QT

2

4

r

uTtS

s*, t*, w(u)*, u*


Cooper and Jacob (1946) subsequently discovered that the Theis solution, drawn on semilog axes, plots as a straight line after sufficiently long periods of pumping…

And Then Came…

Cooper and Jacob!


And Then Came… Cooper-Jacob!

100

101

102

1030.

2.

4.

6.

8.

10.

12.

Time (min)

Dra

wd

ow

n (

ft)

2

025.2

r

TtS

s

QT

4

303.2

t0

s

valid when

u ≤ 0.01 (i.e., when

slope is constant)

where u=r²S/4Tt


Pumping Test Data Analysis

How often is the Cooper and Jacob method the first step in your analysis of pumping test data?

Are there techniques you could use to get more reliable results?


A Different Approach...

A more productive approach to pumping test data analysis begins with the application of derivative analysis that helps you to:

– identify common flow regimes

– guide subsequent curve matching

What is derivative analysis?


Derivative Analysis

Technique popularized in the petroleum industry (Bourdet et al. 1983)

Plot of ∂s/∂lnt vs t

Derivatives are calculated from field data

A derivative plot, which combines the display of drawdown and derivative data, is a powerful diagnostic and curve matching tool


Interpretation of Derivative slope of drawdown data on semilog plot

10-1

100

101

102

103

1040.

2.

4.

6.

8.

10.

Dimensionless Time

Dim

en

sio

nle

ss

Dra

wd

ow

nTheis drawdown

solution derivative is

constant at

late time when

Cooper and

Jacob solution

is valid

(infinite-acting

radial flow

regime)

IARF (plateau)


Derivative Smoothing

Derivatives computed directly from field data are often noisy

Four smoothing options are available in AQTESOLV to reduce noise

– nearest neighbor (no smoothing)

– Bourdet method

– Spane method

– smoothing

Begin with

nearest

neighbor method.

Avoid excessive

smoothing!


Effect of Smoothing Nearest Neighbor Option Bourdet Option


Simple Example: Single-Well Test

12-hour constant-rate test (Q = 86.9 gpm)

Recovery monitored for 1 hour

Aquifer–fractured bedrock (Triassic sandstone, siltstone, shale sequence)

Upper boundary–water table

Lower boundary–unknown (total depth of well is 465 ft)

Assume test well is fully penetrating but most of water may be coming from lower 150 ft

rc = rw = 4 inches


Nonleaky Confined Aquifer

(e.g., Theis 1935;

Cooper and

Jacob 1946;

Papadopulos and

Cooper 1967)

radial flow


Cooper and Jacob Match

100

101

102

1030.

40.

80.

120.

160.

200.

Time in min

Dis

pla

ce

me

nt

in f

t

Obs. Wells

AR-09

Aquifer Model

Confined

Solution

Cooper-Jacob

Parameters

T = 1435.8 gal/day/ft

S = 3.788E-6

u criterion

easily met when

r is small

constant derivative

at late time (IARF)


Papadopulos and Cooper Match

100

101

102

1030.

40.

80.

120.

160.

200.

Time in min

Dis

pla

ce

me

nt

in f

t

Obs. Wells

AR-09

Aquifer Model

Confined

Solution

Papadopulos-Cooper

Parameters

T = 1505.1 gal/day/ft

S = 1.727E-6

r(w) = 0.333 ft

r(c) = 0.2791 ft

derivative

curve


Key Concepts and Tips

Combine derivative analysis with the Cooper and Jacob method to

– identify IARF period (derivative plateau)

– improve fitting of straight line

Cooper and Jacob can obtain results comparable with more rigorous methods with less effort



Cooper and Jacob applied to single-well tests can yield reliable estimates of T; however, S often will be biased due to partial penetration and/or well losses.


Case Study: Coastal Aquifer Oude Korendijk, The Netherlands

14-hour constant-rate test (Q = 788 m3/day)

Aquifer–7 m of coarse sand with some gravel

Upper boundary–18 m of clay, peat and clayey fine sand; note clayey fine sand directly above aquifer

Lower boundary–fine sand and clay sediments

Test well is fully penetrating

Observation wells at r = 30, 90 and 215 m from pumped well

Source: Kruseman and de Ridder (1994)


Stratigraphy

from Kruseman and de Ridder (1994)


Cooper and Jacob Analysis

Kruseman and de Ridder assumed a nonleaky confined aquifer for the analysis of the constant-rate pumping test.

Let’s consider interpretations of drawdown data with and without derivative analysis…


Cooper and Jacob, r=30 and 90 m

10-4

10-3

10-2

10-1

1000.

0.4

0.8

1.2

1.6

2.

Adjusted Time, t/r2 (min/m2)

Dra

wd

ow

n (

m)

Obs. Wells

H-30

H-90

Aquifer Model

Confined

Solution

Cooper-Jacob

Parameters

T = 437. m2/day

S = 0.00017

match on composite

plot reported by

Kruseman and

de Ridder (1994)


Cooper and Jacob, r=30 m, late

10-1

100

101

102

1030.

0.2

0.4

0.6

0.8

1.

1.2

Time (min)

Dra

wd

ow

n (

m)

Obs. Wells

H-30

Aquifer Model

Confined

Solution

Cooper-Jacob

Parameters

T = 604.4 m2/day

S = 2.33E-5

u=

0.0

1

match to late-

time data


Cooper and Jacob, r=30 m, early

10-1

100

101

102

1030.

0.2

0.4

0.6

0.8

1.

1.2

Time (min)

Dra

wd

ow

n (

m)

Obs. Wells

H-30

Aquifer Model

Confined

Solution

Cooper-Jacob

Parameters

T = 376.3 m2/day

S = 0.0001744

u=

0.1

match to early-

time data


Cooper and Jacob Results

We have three very different estimates of T and S. Which interpretation is most reliable?

Let’s consider the response of a leaky confined aquifer with aquitard storage and its associated derivative plot…


Leaky Confined Aquifer

(e.g., Hantush

and Jacob 1955;

Hantush 1960)

radial flow


Derivative Plot, Leaky Confined Aquifer


Hantush, r=30 and 90 m, leaky aquitard

10-1

100

101

102

1030.

0.2

0.4

0.6

0.8

1.

1.2

Time (min)

Dra

wd

ow

n (

m)

Obs. Wells

H-30

H-90

Aquifer Model

Leaky

Solution

Hantush

Parameters

T = 347.5 m2/day

S = 0.0001704

1/B' = 0.001683 m-1

ß'/r = 0.0008895 m-1

1/B" = 0. m-1

ß"/r = 0. m-1


Hantush, r=30 and 90 m, leaky aquitard

10-1

100

101

102

10310

-2

10-1

100

101

Time (min)

Dra

wd

ow

n (

m)

Obs. Wells

H-30

H-90

Aquifer Model

Leaky

Solution

Hantush

Parameters

T = 347.5 m2/day

S = 0.0001704

1/B' = 0.001683 m-1

ß'/r = 0.0008895 m-1

1/B" = 0. m-1

ß"/r = 0. m-1


Summary of Results

Estimates of T (leaky confined):

– 348 m2/day (compressible aquitard)

Estimates of T (nonleaky confined):

– 375 m2/day (Cooper-Jacob, early)

– 437 m2/day (Cooper-Jacob, composite)

– 600 m2/day (Cooper-Jacob, late)



Apply the Cooper and Jacob method in conjunction with derivative analysis to provide reasonable preliminary estimates of T and S for leaky confined aquifers.

Use derivative analysis to choose and refine conceptual model(s) of groundwater flow system.


Case Study: Channel Aquifer Estevan, Saskatchewan

Walton (1970) presented data and results from an eight-day pumping test conducted in a buried sand-and-gravel channel aquifer near Estevan, Saskatchewan, Canada

– Q = 457 to 464 imperial gallons-per-minute

– b = 30 to 90 ft (typical)

– width of channel = 3,000 to 12,000 ft (typical)


Well Locations

three observation wells

– r = 84 ft

– r = 250 ft

– r = 729 ft

from Walton (1970)


Well Logs

from Walton (1970)


Walton’s Analysis

10-1

100

101

102

103

104

10510

-2

10-1

100

101

102

Time (min)

Dra

wd

ow

n (

ft)

Obs. Wells

PW

OW 1

OW 2

OW 3

Aquifer Model

Confined

Solution

Theis

Parameters

T = 3.02E+4 ft2/day

S = 0.00022

Kz/Kr = 1.

b = 50. ft

Walton only used a few

early-time observations

from OW3 in his

analysis…


Composite Plot

10-7

10-5

10-3

10-1

101

103

1050.

4.

8.

12.

16.

20.

Time, t/r2 (min/ft2)

Dra

wd

ow

n (

ft)

all wells show same

slope at intermediate

time suggesting

infinite-acting aquifer

conditions


Derivative Plot

10-7

10-5

10-3

10-1

101

103

1050.

4.

8.

12.

16.

20.


Dra

wd

ow

n (

ft)

derivative plot

confirms constant

slope during infinite-

acting period


Cooper and Jacob Match

10-7

10-5

10-3

10-1

101

103

1050.

4.

8.

12.

16.

20.


Dra

wd

ow

n (

ft)

Obs. Wells

PW

OW 1

OW 2

OW 3

Aquifer Model

Confined

Solution

Cooper-Jacob

Parameters

T = 1.983E+4 ft2/day

S = 0.0002836

aquifer properties

determined from

infinite-acting

period


Image Well Arrays

from Kruseman and de Ridder (1994)

typical channel aquifer

no-flow boundaries


Theis Analysis w/Channel

10-7

10-5

10-3

10-1

101

103

1050.

4.

8.

12.

16.

20.


Dra

wd

ow

n (

ft)

Obs. Wells

PW

OW 1

OW 2

OW 3

Aquifer Model

Confined

Solution

Theis

Parameters

T = 1.983E+4 ft2/day

S = 0.0002836

Kz/Kr = 1.

b = 50. ft

estimated channel

width = 8000 ft



Buried channel aquifer inferred from late-time derivative response

Aquifer properties (T and S) estimated efficiently from the infinite-acting period with composite plot and Cooper and Jacob solution

Channel width identified easily by trial-and-error using Theis solution and image wells


Lessons

Combine derivative analysis with Cooper and Jacob for more reliable estimation of aquifer properties

Look for infinite-acting radial flow regime to match Cooper and Jacob

Use derivative analysis to select aquifer models and identify boundaries


Lessons

When applied carefully, Cooper and Jacob can provide reliable estimates of T and S in confined aquifers with or without leakage

Do not rely on Cooper and Jacob to determine S from single-well tests due to well loss

using derivative analysis to improve pumping test ...ky.aipg.org/pdf/glenn...

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