specific capacity well

Mace dog-barking draft 9/11/00

1

Estimating Transmissivity Using Specific-Capacity Data1

Robert E. Mace Bureau of Economic Geology

The University of Texas at Austin University Station, Box X Austin, TX 78713-8924

[email protected] (512) 471-6246

Abstract

Specific-capacity data are very useful for estimating transmissivity and should be

used whenever possible in hydrogeologic studies. Including specific-capacity data will

increase the number of transmissivity estimates in aquifers, sometimes dramatically,

allowing more accurate statistical and spatial descriptions of transmissivity and hydraulic

conductivity. There are three main approaches for estimating transmissivity from specific-

capacity data: analytical, empirical, and geostatistical techniques. The most commonly

used analytical approach is an equation derived from the Theis nonequilibrium formula

which requires corrections for well loss, vertical flow due to partial penetration, and

reduced saturated thickness. The empirical approach involves empirically relating

transmissivity to specific capacity measured in the same well. This approach usually

requires at least 25 data pairs and does not require a correction for turbulent well loss or

vertical flow resulting from partial penetration. Geostatistical techniques are useful for

estimating transmissivity from specific capacity, developing interpolated maps of

1 This report is in press at the Bureau of Economic Geology, The University of Texas at Austin, and is expected to be formally published in late fall of 2000. There may be minor editing changes between this copy and the final published copy (8/28/2000)


2

transmissivity, and quantifying the estimation uncertainty. However, geostatistical

techniques are mathematically complicated and require substantial data to define

semivariograms and cross-semivariograms. Techniques to estimate transmissivity from

specific-capacity data have been successfully applied in many aquifers in Texas and

elsewhere to provide valuable information for input to numerical models and for

evaluating water-resources.

1.0 Introduction

Transmissivity is one of the most fundamental parameters of an aquifer.

Transmissivity allows hydrogeologists to estimate water levels in and around pumping

wells, to estimate ground-water flow and contaminant transport times, to characterize

aquifer heterogeneity, and to parameterize numerical ground-water flow models. Because

transmissivity in aquifers can range over 13 orders of magnitude (Freeze and Cherry,

1979, p. 29, assuming a constant formation thickness), it is imperative that the most

accurate values possible are attained and that the distribution of the values are known to

characterize heterogeneity.

There are many techniques available to accurately assess the transmissivity of

aquifers using time-drawdown aquifer tests (e.g. Ferris, 1962; Hantush, 1964; Walton,

1970; Kruseman and de Ridder, 1990). In a time-drawdown test, the change in water level

over time is measured for a constant pumping rate (fig. 1), usually for at least 24 hours,

and is then analyzed using type curves (e.g. Theis, 1935) or other graphical (e.g Cooper

and Jacob, 1946) or numerical techniques to determine transmissivity. Because of the

costs of performing a well-designed aquifer test and the expertise required to perform and


3

analyze the data, most water-supply wells, especially private wells, have not had time-

drawdown tests performed on them. This leaves the hydrogeologist with only a few tests

to characterize the transmissivity of an aquifer resulting in ill-defined averages, especially

in heterogeneous aquifers.

Water-well drillers often conduct a well-performance test after a well is drilled

and completed to determine the specific capacity of a well. During a well-performance, or

specific-capacity, test, a well is pumped at a constant rate and the amount of drawdown is

noted at the end of pumping. Specific capacity is then calculated by dividing the pumping

rate by the amount of drawdown. In the United States, specific capacity is usually

reported in units of gallons per minute per foot of drawdown (gpm/ft). Ideally, the well is

pumped until the change in water level is small (pseudo steady state), and the duration of

the pumping is noted at the time the drawdown is measured in the well. The data

collected for a specific-capacity test is essentially one point on the drawdown curve for a

time-drawdown test (fig. 1).

The quantitative use of specific capacity probably began in the 19th century when

Michal, in a paper published by the Paris Academy in 1863, developed two equations to

describe the relationship between the discharge of an artesian well in Paris and drawdown

using the ratio of production to drawdown (Davis, 1988). Water-well drillers have

historically used specific capacity to quantify the productivity of a well to determine

where to install the pump to attain the desired yield. More recently, specific capacity has

been proposed to help characterize contaminant sites (Heeley and Mabee, 1983; Wynne,

1992).

Specific capacity is, in part, a function of the hydraulic properties of the aquifer.

Therefore, numerous researchers have tried to relate specific capacity to transmissivity.


4

Because specific-capacity data are typically much more abundant and readily available

than time-drawdown data, relating specific capacity to transmissivity can increase the

number of transmissivity estimates in an aquifer by an order of magnitude. For example,

in the Edwards aquifer of central Texas, over 1,000 specific-capacity tests were found in

over 1,000 wells compared to only 71 time-drawdown tests in 21 wells (Mace, 1997). In

the Cretaceous sandstone aquifers of north-central Texas, about 2,000 specific-capacity

tests were found compared to 291 time-drawdown tests (Mace and others, 1994; Dutton

and others, 1996). In the Carrizo-Wilcox aquifer of Texas, over 5,000 specific-capacity

tests were found compared to about 200 time-drawdown tests (Mace and others, 2000).

Incorporating specific-capacity data into hydrogeologic assessments allows a more

rigorous characterization of the hydraulic properties of a regional aquifer and a better

understanding of flow in the aquifer (e.g. Hovorka and others, 1998).

There are several different approaches for estimating transmissivity from specific

capacity and for correcting specific capacity for non-ideal conditions such as partial

penetration, turbulent well losses, or fracture flow. These approaches include analytical,

semi-analytical, empirical, geostatistical, and hybrid approaches. Thomasson and others

(1960) developed one of the first techniques to relate transmissivity to specific capacity

based on the Dupuit-Thiem steady-state equation. Theis (1963) developed a more

definitive approach using the Theis nonequilibrium equation. Empirical relationships

were first used by Eagon and Johe (1972) and improved upon and popularized by Razack

and Huntley, 1991. Geostatistical techniques were first used by Delhomme (1974, 1976;

linear regression with kriging) and Aboufirassi and Marino (1984; cokriging). The

appropriate technique for relating specific capacity to transmissivity depends on well


5

construction, aquifer setting, pumping rates, number of available tests, and, ultimately,

the accuracy of the applied technique.

The purpose of this paper is to categorize and summarize the different techniques

available for relating specific capacity to transmissivity. The paper begins with the

mathematical definition of specific capacity and a discussion of the factors that influence

specific-capacity measurements. The paper then presents analytical, empirical, and

geostatistical approaches for estimating transmissivity from specific capacity. Subsequent

sections discuss correcting specific capacity for partial penetration and turbulent well

loss, fractured rocks, and data-quality issues. The paper concludes with recommendations

on the appropriate approach for estimating transmissivity from specific capacity. In

addition to a review of the literature, this paper also presents new insights into estimating

transmissivity from specific-capacity data and new relationships for several sandstone

aquifers in Texas.

2.0 Specific Capacity

Specific capacity is defined as the well production per unit decline in head.

Mathematically, specific capacity, cS , is defined as

w

c sQS = (1)

where Q is the pumping rate [L3 t-1] and ws is the measured drawdown (change in

hydraulic head) in the well [L]. Specific capacity is generally reported as yield per unit of

drawdown. For example, a well pumped at 6 l s-1 (100 gpm) with 6 m (20 ft) of drawdown

would have specific capacity reported to be 1 l s-1 m-1 (5 gpm/ft). Note that although


6

specific capacity is generally reported in units of volume per time per length, specific

capacity has the same units as transmissivity: length squared per time [L2 t-1].

Specific capacity is a function of the aquifer setting, well setting, and pumping

duration. Attributes of the aquifer setting that can influence specific capacity include

transmissivity, storativity, and aquifer type (i.e. confined, unconfined, semiconfined,

boundaries, fractured, etc.). Greater transmissivity leads to greater specific capacity.

Greater storativity also leads to greater specific capacity because a greater storativity

results in less drawdown. Specific capacity in an unconfined aquifer of the same

transmissivity as a confined aquifer of similar thickness will have a lower specific

capacity at longer pumping times due to delayed yield caused by partially dewatering the

aquifer (e.g. Boulton, 1954; Neuman, 1972), a decrease in the saturated thickness, and a

higher storativity value. Similarly, specific capacity in a semi-confined or leaky-confined

aquifer of the same transmissivity as a confined aquifer will tend to have a lower specific

capacity due to less drawdown caused by additional flow into the well.

The well setting, such as well radius, degree of penetration, and well losses due to

well construction, affect specific capacity. A larger well radius leads to a greater specific

capacity. This is because a larger radius creates a greater surface area that makes it easier

for water to flow into the well. Therefore, not as much drawdown is required to produce

the same amount of water. Degree of penetration refers to how much of the aquifer is

penetrated by the well. In general, a well that only penetrates part of the aquifer (a

partially penetrating well) has a lower specific capacity than if the well penetrated the

entire aquifer. Furthermore, a well that partially penetrates an aquifer may overestimate

the specific capacity for the penetrated portion of the aquifer owing to vertical flow

components caused by the partial penetration. Laminar and turbulent well loss cause


7

specific capacity to be underestimated because the measured drawdown at the well is

greater than the actual water-level decline in the aquifer.

For transient conditions, specific capacity is also a function of pumping duration

(Jacob and Lohman, 1952; McWhorter and Sunada, 1977). As pumping continues in a

confined aquifer, the drawdown in the well continues to increase leading to lower specific

capacity for greater pumping times (fig. 2). Although water levels never theoretically

reach a steady state in a confined aquifer, the change in water level for a given time

period becomes small at large pumping times and is effectively at a ‘pseudo steady state.’

The pumping time required to reach a pseudo steady state depends on the definition of

pseudo steady state and properties of the well and aquifer. The time to reach pseudo-

steady state is less for greater transmissivity values and greater for greater storativity and

well radius values (fig. 3). Water levels in confined, semi-confined, and fractured aquifers

behave differently over time than confined aquifers, and therefore specific capacity varies

differently over time in these aquifer settings.

Specific capacity can be normalized to aquifer thickness using the specific-

capacity index (Davis and DeWeist, 1966),

a

ci b

SS =

, (2)

where ab is the aquifer thickness [L]. Poland (1959) and Thomasson and others (1960)

calculate specific-capacity index using units of gpm and ft, multiply it by 100 ft, and call

the result the ‘yield factor,’ which normalizes specific capacity to a 100 ft thick aquifer.

Specific-capacity index has the same units [L t-1] and is somewhat analogous to the

hydraulic conductivity. Specific-capacity index is not commonly used, although it has


8

been used instead of specific capacity to remove the effect of aquifer thickness variation

on aquifer productivity (e.g. Siddiqui and Parizek, 1971; LaRiccia and Rauch, 1977;

Gelbaum, 1981).

3.0 Analytical Methods

Analytical methods to relate transmissivity to specific capacity involve using

mathematical equations based on the theory of ground-water flow. Analytical methods are

advantageous because they are exact. However, their application can be limited due to (1)

unrealistic assumptions about the aquifer and well hydraulics and (2) limited information

on the aquifer or the well.

3.1 Solutions based on the Dupuit-Thiem equation

Thomasson and others (1960) were among the first to analytically relate

transmissivity to specific capacity. They used the Dupuit-Thiem equation,

w

w rR

TQs ln

2π= , (3)

where T is the transmissivity of the aquifer [L2 t-1], R radius of influence of the pumping

well [L], and wr is the radius of the well [L]. They solved equation 3 for T

ww s

QrRT

= ln

21π

, (4a)

to show that transmissivity is linearly related to specific capacity by a constant cC ,

cc SCT = , (4b)


9

where

=

wc r

RC ln21π

. (4c)

This approach assumes that water levels are in steady state and that storativity, partial

penetration, and well loss do not influence results. Note that equation 4b is the equation

of a line with an intercept at the origin and a slope of cC .

Use of the Dupuit-Thiem equation (eqn. 4a) requires an assumption on the radius

of influence, which is defined as the lateral distance from the center of the pumping well

to the edge of the cone of depression (where drawdown is close to zero). The steady-state

radius of influence is dependent on aquifer properties and aquifer setting and is greater for

greater transmissivity and comparatively greater for confined aquifers than for unconfined

aquifers with similar transmissivities (Driscoll, 1986). Therefore, cC is partially a

function of transmissivity which results in a nonlinear relationship between transmissivity

and specific capacity.

Assuming a radius of influence ranging from about 100 to 1,000 m (300 to 3,000

ft) and typical well radii, Thomasson and others (1960) found that cC should range from

1.01 to 1.53. For tests in the valley-fill sediments in California, they found cC to range

from 0.9 to 1.5 with an average of about 1.2. Others have investigated what cC might be

for other geologic settings. Johnson and others (1966) found cC to be 1.10 for wells in

confined alluvial aquifers assuming a unit well radius and a radius of influence of 1,000

m (300 ft). Adyalkar and Mani (1972) determined cC to be between 0.23 and 0.34 and as

high as 0.44 for large-diameter wells in a fractured aquifer. They expected that cC would


10

be lower for less permeable fractured formations. Adyalkar and others (1981) calculated

cC to be 0.42 for the weathered zone of massive and vesicular basalts of the Deccan Trap

in India. Figure 5 shows the relationship between transmissivity and specific capacity

using the Thomasson and others (1960) relationship for alluvial and fractured hard-rock

aquifers.

Mace and others (1994) assumed that the general equation from the Thomasson

and others (1972) relationship (eqn. 4b) applied to three sandstone aquifers and fit a line

with one end fixed at the origin to the measured pairs of transmissivity and specific

capacity values. They found cC values of 0.59, 0.63, and 0.75 for the three sandstone

aquifers.

For alluvial sediments, cC is approximately 1 for consistent units between

specific capacity and transmissivity. Therefore, a good rule-of-thumb for making back-of-

the-envelope estimates is that transmissivity is approximately equal to specific capacity.

This estimate should be within a factor of 2 of the actual transmissivity. For fractured

hard rocks, the transmissivity should be about half the measured specific capacity.

Because in practice the radius of influence is assumed, the Thomasson and others

(1960) approach is actually semi-analytical: a relationship based on theory with

parameters determined by assumption or observation.

3.2 Solutions based on the Theis nonequilibrium equation

Theis and others (1963) presented an equation to relate specific capacity to

transmissivity,


11

=

SrTt

TS

w

pc

2

25.2ln

4π , (5)

based on the Theis nonequilibrium equation where S is the storativity of the aquifer [-]

and pt is the pumping time [t]. This equation assumes (1) a fully-penetrating well, (2)

homogeneous, isotropic porous media, (3) negligible well loss, (4) and an effective radius

equal to the radius of the production well (Walton, 1970).

Because equation 5 cannot be solved directly for transmissivity (note that T is in

both the numerator and denominator), it must be solved graphically or iteratively. Meyer

(1963) and Theis and others (1963) presented a chart based on equation 5 for estimating

transmissivity from specific capacity for 1 day of pumping for different values of

storativity and well diameter. Theis (1963) presented a chart, also based on equation 5,

for estimating transmissivity from specific capacity for constant storativity but different

pumping times and well diameters. Narasimhan (1967) presented curves to more easily

estimate transmissivity from specific capacity, storativity, well radius, and time. Walton

(1970) proposed a graphical solution for the Theis and others (1963) equation.

Equation 5 can also be solved iteratively. This can be done by hand or using a

calculator by (1) guessing an initial transmissivity value (the value of the specific capacity

is a good initial guess), (2) calculating the right hand side of equation 5, and (3)

comparing the resulting number to the measured specific capacity. If the calculated and

measured values of specific capacity are close, then the transmissivity value used in step

2 is the correct answer. If the two numbers do not agree, then the transmissivity value

must be adjusted up or down and steps 2 and 3 repeated. Equation 5 can also be easily

solved using digital spreadsheets such as Microsoft Excel (appendix A). Equation 5 can


12

also be solved iteratively using computers. Bradbury and Rothschild (1985) describe one

such approach and provide a FORTRAN program to make the appropriate calculations.

Kasenow and Pare (1995) also present a numerical approach.

The Theis and others (1963) approach is in many ways superior to the Thomasson

and others (1960) approach because the radius of influence is not needed and transient

conditions are considered. Because of this, the Theis and others (1963) approach is

probably the most frequently used technique for estimating transmissivity from specific

capacity. Gabrysch (1968), Walton (1970), Lohman (1972), Macpherson (1983), and

Bradbury and Rothschild (1985), among others, offer examples of using this approach.

In practice, storativity is generally assumed for the aquifer (preferably based on

actual measurements elsewhere in the aquifer) which are generally accurate within an

order of magnitude (0.01 to 0.3 for unconfined aquifers, 0.005 to 0.00005 for confined

aquifers, and 0.005 to 0.01 for semi-confined aquifers).

The Theis and others (1963) equation can be used to gage the sensitivity of

specific capacity to variations in transmissivity, pumping time, well radius, and

storativity. To do the sensitivity analysis, the time, well radius, and storativity variables of

equation 5 were grouped into a single variable, C ′ , defined as

Sr

tCw2=′ (6)

so that equation 5 becomes

( )[ ]TCTSc ′

=25.2ln

4π . (7)


13

Using this relationship, specific capacity was calculated for different transmissivities and

values of C ′ . Representative ranges of values for the variables were used for the

sensitivity analysis: 10-6 to 105 m2 d-1 (10-5 to 106 ft2 d-1) for transmissivity, 0.1 to 0.45 m

(0.33 to 1.5 ft) for well radius, 0.00001 to 0.2 for storativity, and 1 to 48 hr for pumping

time. According to this relationship, specific capacity is most sensitive to transmissivity

where several orders of magnitude of variation in transmissivity leads to several orders of

magnitude variation in specific capacity (fig. 4). For a given transmissivity, specific

capacity varies 0.5 to 1 orders of magnitude depending on the value of C ′ (fig. 4).

Storativity causes the most variation in C ′ because storativity values can vary four orders

of magnitude depending on aquifer setting (unconfined, confined, or semi-confined). In

comparison, pumping time may vary two orders of magnitude and well radius may vary

an order of magnitude. If the aquifer setting, and hence the storativity, is known, pumping

time may cause the greatest amount of variability in C ′ .

Bradbury and Rothschild (1985) derived an equation based on Theis and others

(1963) that also considers partial penetration and well loss,

( )

+

−= p

wLw

sSrTt

ssQT 225.2ln

4 2π, (8)

where Ls is the well loss (discussed in section 7.0), and ps is the partial penetration

factor (discussed in section 6.0).

3.5 Other analytical approaches

A number of other, less frequently used, analytical approaches have been

proposed. Bedinger and Emmett (1963) presented equations and a chart based on a


14

combination of the Thiem and Theis equations and average transmissivity and specific

capacity for a specific area for estimating transmissivity from specific capacity. Hurr

(1966) developed a method assuming a 0.15 m (0.5 ft) well radius using the Theis (1935)

and Boulton (1954) equations, which is not as sensitive to storativity and allows for

delayed yield.

Ogden (1965) developed a method of estimating transmissivity from one

drawdown measurement in a well using the Theis (1935) nonequilibrium equation,

( )uWT

Qsw π4= , (9)

where ( )uW is the well function, estimated by

( ) ...!44!33!22

ln577.0432

+×

−×

+×

−+−−= uuuuuuW , (10)

and u is defined as

TtSr

u w

4

2

= . (11)

Solving equation 11 for T, substituting the result for T in equation 9, and solving for

uW(u), results in

( )Qt

SsruuW ww

44 2π

= . (12)

For each value of uW(u) there is a unique value of u that can be determined from a table

(Ogden, 1965), a curve, or fitting an equation,

( )[ ]638.5

061.1uuWu = , (13)

to the tabular data.


15

Once u is known, a rearranged version of equation 11,

utSr

T w

4

2

= , (14)

can be used to calculate transmissivity.

This approach is simply a more rigorous version of the Theis and others (1963)

relationship (eqn. 5) in that the full estimation of ( )uW (eqn. 10) is used instead of only

the first two terms as was done by Theis and others (1963) in their solution. In most

cases, the first two terms of equation 10 are adequate for estimating ( )uW and therefore

there is little benefit of the Odgen (1965) approach over the Theis and others (1963)

approach.

El-Naqa (1994) presents an equation that relates transmissivity to specific capacity

for a semiconfined aquifer using the Hantush (1956) inflection point method,

cw SBr

KT

= 02

1π

, (15)

where 0K is the zero-order modified Bessel function of the second kind and B is the

laminar well-loss coefficient,

=

wrR

TB ln

21π

. (16)

Adyalkar and Mani (1972) show how specific capacity can be calculated using

Slichter’s (1906) recovery equation,

=

2

1log25.17ss

tA

Ss

wc , (17)


16

where cS is in gpm/ft, wA is the cross-sectional area of the well in ft2, st is the time since

pumping stopped in minutes, 1s is the residual drawdown after st , 2s is the maximum

drawdown. This approach can be used to calculate specific capacity when equilibrium has

not been reached. Chandrashekhar and others (1976) also used Slichter’s equation to

estimate specific capacity.

The Theis (1935) nonequilibrium equation can be rearranged to solve for

theoretical specific capacity (e.g. Gabrysch, 1968),

( )uWT

sQS

wc

π4== , (18)

with all the assumptions of using the Theis (1935) equation.

4.0 Empirical Methods

Empirical methods involve statistically relating transmissivity to specific capacity

using paired values of both parameters measured in the same well. Data that are used to

determine transmissivity in any well can also be used to determine specific capacity,

usually for a specified pumping time. Empirical methods are advantageous because the

uncertainty in the estimate can be estimated and because many nonideal conditions, such

as turbulent well loss, are indirectly considered. However, their application can be limited

due to too few measurements of transmissivity or too much uncertainty in the relationship

compared to actual heterogeneity of the aquifer.


17

4.1 Statistically derived linear relationships

Razack and Huntley (1991) found that the analytical relationships (Thomasson

and others, 1960; Theis and others, 1963) uncorrected for well loss did not always agree

with measured values of transmissivity and that empirical relationships produced less

error than analytical solutions. The relationship is usually defined between log-

transformed values of transmissivity and log-transformed values of specific capacity

because transmissivity and specific capacity are generally log-normally distributed and, in

this case, log-log relationships result in better correlation coefficients than using

untransformed values (e.g. Razack and Huntley, 1991; El-Naqa, 1994).

The empirical approach involves (1) compiling all available aquifer-test

information for an aquifer, (2) determining the transmissivity and specific capacity for

each of the tests, (3) using regression to fit a line to the plotted pairs of log transmissivity

and log specific capacity, and (4) calculating the uncertainty in the linear relationship.

Once the data are compiled, transmissivity can be determined using standard techniques

(e.g. Ferris and others, 1962; Hantush, 1964; Walton, 1970; Kruseman and de Ridder,

1990) and specific capacity can be determined from the aquifer-test data for a specified

time. This time may be defined by the mean test time of the all the specific-capacity tests

in the aquifer. In this manner, errors due to variation in time can be minimized. Short-

term or long term specific-capacity tests will be in more error than those tests with

pumping times closer to the mean.

Once the transmissivity and specific-capacity pairs are compiled, least-squares

regression can be used to fit a line to the log-transformed values. This is done by

defining:


18

ii XbbY 10ˆ += (19)

where, in the present case,

( )ii TY logˆ = and (20a)

( )( )ici SX log= (20b)

and, in general,

x

xy

SSSS

b =1 , (20c)

∑ ∑∑= ==

−=

n

i

n

ii

n

iiiixy YX

nYXSS

1 11

1 , (20d)

2

11

2 1

−= ∑∑

==

n

ii

n

iix X

nXSS , (20e)

XbYb 10 −= , (20f)

∑=

=n

iiY

nY

1

1 , and (20g)

∑=

=n

iiX

nX

1

1 . (20h)

By solving for 0b and 1b using equations 20f and 20c, respectively, log transmissivity can

be directly estimated using equation 19. Equation 19 can be rearranged into

( ) 1010 bc

b ST = (21)

so that an untransformed transmissivity can be directly calculated. Note that this equation

has essentially the same form as the Thomasson and others (1960) relationship (eqn. 4b)

except that specific capacity is raised to a power and, therefore, the relationship between

transmissivity and specific capacity may be nonlinear.

Once the best-fit line is found, how well the line fits the data can be estimated.

The coefficient of determination (also called the goodness of fit), R2, describes how much


19

of the observed variability of a parameter can be explained by the regression model. The

coefficient of determination is found from

YSS

SSR ε−= 12 (22)

where SSY is the total corrected sum of squares or the ‘total variation’ of Y,

( )∑=

−=n

iiY YYSS

1

2 , (23a)

and SSε is the residual sum of squares or the ‘unexplained variation’ from the predicted

value, iY , and the observed value, Yi ,

( )∑=

−=n

iii YYSS

1

2ˆε . (23b)

The uncertainty of the regression can be quantified using prediction intervals that

define an envelope around the line that defines how certain an estimate of transmissivity

is for a given specific-capacity value. The 100(1-α ) percent prediction interval for iY

when iX = pX is

( )

x

pi SS

XXn

tY2

2

11ˆ −++± α (24)

where α is the level of significance and 2

αt is based on (n-1) degrees of freedom where

2αt is defined from a statistical table of critical values of t for the t-distribution (e.g.

Mendenhall, 1987; or any other introductory statistics book).

An example is shown in figure 6a from Mace (1997). In this case, Mace (1997)

plotted 71 pairs of transmissivity and specific-capacity values and calculated a best-fit

line and 95 percent prediction intervals. The best-fit line to these data, in the form of

equation 21, is:

( ) 08.196.0 cST = (25)


20

where T and Sc are in m2 d–1. The coefficient of determination, R2, equals 0.89. The 95-

percent prediction interval spans about 1.4 log cycles (upper limit minus the lower limit)

(fig. 6a), indicating that the range of probable transmissivities for a given measured

specific capacity is more than one and a half orders of magnitude. Though the 95-percent

prediction interval is large, this relationship extends over five orders of magnitude of

specific-capacity and transmissivity values (fig. 6a) and therefore allows rough estimates

of transmissivity from specific-capacity values.

The linear relationship overestimates transmissivity for lower values of specific

capacity in the range of 1 to 10 m2 d-1 (fig. 6a). A slightly better fit (R2 equals 0.91) can

be obtained through the multiple linear regression of a second order polynomial (fig. 6b),

( ) ( )[ ]{ }2ln060.0ln80.104.2exp cc SST −+−= . (26a)

An even better fit (R2 equals 0.92) can be obtained through the multiple linear regression

of a third order polynomial (fig. 6b),

( ) ( )[ ] ( )[ ]{ }32 ln011.0ln25.0ln79.246.3exp ccc SSST +−+−= . (26b)

The coefficients of determination for each of these fits are very close to each other with

slight improvements for more complex lines. The coefficient of determination does not

indicate the appropriateness of a model. Better fits (lower R2) can be obtained with more

complex polynomials, but these polynomials may cause difficulties when predicting

outside the range of the observed data. It is better to use the simplest model that explains

a response (Jensen and others, 1997, p. 205). In this case, the linear fit is adequate except

for values of specific capacity less than 10 m2 d-1. For these low values of specific

capacity, resulting transmissivities may be off by 1 to 3 m2 d-1, which is quite small


21

considering that the mean value of transmissivity in the aquifer is about 500 m2d-1. Meier

and others (1999) found that the nonlinearity between specific capacity and transmissivity

are likely caused by heterogeneity and the analyzed time period.

4.2 Published empirical relationships

Several authors have developed empirical linear relationships between

transmissivity and specific capacity for a range of hydrologic settings (table 1, fig. 7).

Most of the relationships have prediction intervals that span a little more than an order of

magnitude (fig. 7). As part of defining their relationship, Razack and Huntley (1991)

performed a sensitivity analysis on their data set of 215 pairs to determine how many data

pairs would be required to arrive at a reasonable relationship and found that at least 25

pairs would be needed. Relationships based on less than 25 pairs may result in prediction

intervals that span the entire range of transmissivity values. Huntley and others (1992)

found that measured transmissivities for wells in fractured rocks were lower than values

determined using the theoretical relationships and therefore developed an empirical

relationship (table 1c, fig. 7c). Mace (1997) developed an empirical relationship for a

karstic aquifer and showed that data from karstic aquifers in Florida and Ohio overlaid

the relationship, suggesting that the relationship may be applied to other karstic aquifers.

Fabbri (1997) developed an empirical relationship a limestone fractured aquifer in

Italy (table 1g, fig. 7g). Fabbri (1997) believed his equation differed from Huntley and

others (1992) due to partial penetration of his wells.

Figure 8 compares and contrasts most of the different empirical relationships and

shows how they relate to the Thomasson and others (1960) and Theis and others (1963)

approaches. Most of the relationships rest within an order of magnitude of each other


22

with the exception of the Razack and Huntley (1991) relationship for a heterogeneous

alluvial aquifer (line 2 in fig. 8). Slopes are also very similar between the relationships

except for the Razack and Huntley (1991) and Huntley and others (1992) relationships,

which have slopes that are lower and greater, respectively, than the rest. The relationship

by Eagon and Johe (1972) has a lower slope than the rest but has been corrected for

turbulent well loss and is therefore derived differently than the others. The empirical

relationships derived for fractured or carbonate rocks (lines 1, 4, 5, 6, 7 in fig. 8) all rest

within half an order of magnitude of each other. The Razack and Huntley (1991)

relationship for alluvium (line 2 in fig. 8) is considerably different than the analytical

predictions of the Thomasson and others (1960) and Theis and others (1963) approaches

for alluvium (lines 11 and 13, respectively, in fig. 8) and also the sandstone aquifers of

north-central Texas (lines 8, 9, and 10 in fig. 8).

4.3 Hybrid approaches

Prudic (1991), in a study of hydraulic conductivity in the Gulf Coast regional

aquifer system, used an analytical approach to estimate transmissivity and found that

transmissivity estimated in this manner tended to be underestimated compared to values

determined from aquifer tests. His approach was to (1) estimate transmissivity with the

Theis and others (1963) relationship (cST ), (2) develop an empirical relationship between

transmissivity estimated from specific capacity using the analytical approach and

transmissivity determined from aquifer tests, and (3) adjust estimated transmissivity

according to the relationship (R2 equals 0.82):

( ) 86.089.3cSTT = . (27)


23

Darling and others (1994) found estimates of transmissivity using the Theis and

others (1963) relationship to result in transmissivity values much higher than those

calculated from aquifer tests. They multiplied each estimated transmissivity by a uniform

factor determined from the mean of the percent difference between measured

transmissivity and transmissivity estimated from specific capacity in the same well using

the analytical relationship.

5.0 Geostatistical Methods

Geostatistics is a statistical approach for working with spatially distributed data

that considers the spatial location of a point and its correlation with nearby points. In

addition to quantifying spatial characteristics with semivariograms and interpolating

using kriging, geostatistics can be used to quantify the estimation variance of interpolated

points and to estimate transmissivity from specific capacity. The estimation variance is

quantified by combining kriging with linear regression, and transmissivity is estimated

from specific capacity using cokriging. Each of these techniques, plus some others, are

discussed in more detail below.

5.1 Kriging with linear regression

Delhomme (1974, 1976; discussed in de Marsily, 1986, p. 318; and Ahmed and de

Marsily, 1987) suggested an approach that uses linear regression (as discussed in section

4.1) and kriging to estimate transmissivity and the uncertainty of the estimate at unknown

points. Similar to the empirical approach described above, log transmissivity and log

specific capacity are linearly related with a regression in those wells with both types of


24

information. The regressed relationship is then used to estimate transmissivity for

locations where only specific capacity is measured or to interpolate between known

points. This technique has been used by Binsariti (1980), Clifton and Neuman (1982),

Ahmed and de Marsily (1987), and de Marsily and Ahmed (1987).

The variance of the prediction error for the regression model, 2jσ , is defined as

( ) ( )[ ]

( ) ( )[ ]

−

−++=

∑=

n

icc

ccj

SS

SSn

1

2

222

lnln

lnln11σσ j = 1,...,m (28)

where ( )1−m is the number of wells that only have specific-capacity values, n is the

number of pairs of transmissivity and specific-capacity values used in the regression, and

cSln is mean of the log-transformed specific-capacity values. The residual variance is

defined as

[ ]∑=

−−−

=n

ici bSbT

n 1

201

2 lnln2

1σ . (29)

Delhomme (1974, 1976, 1978) modified the standard kriging equations (e.g.

Journel and Huijbregts, 1978) to account for the variance of the prediction errors from the

regression model. Ahmed and de Marsily (1987) summarized the derivation of the

modified kriging equations:

Let ( )ixZ be any realization (estimate) of any intrinsic random function

(transmissivity) at a point ix that is uncertain and composed of two terms:

( ) ( ) ( )iii xexvxZ += (30)


25

where ( )ixv is the true value and ( )ixe is the uncertainty or error whose variance is given

by 2iσ . The errors considered by ( )ixe can represent measurement errors and regression

errors and are assumed to be nonsystematic, uncorrelated between themselves, and

uncorrelated with the variable.

The estimation of v at 0x from m measurements is

( ) ( ) ( ) ( )[ ]∑ ∑= =

+==m

i

m

iiiiii xexvxZxv

1 10

* λλ (31)

where * signifies an estimate, iλ is a weight, and the goal is to estimate the true value,

( )0xv , from the uncertain data, ( )ixZ . The kriging equations remain an unbiased

estimator after this modification. The variance of the estimation errors are minimized

from

( ) ( )( ) ( ) ( ) ( )[ ]∑ ∑ −+=− 000* varvar xvxexvxvxv iiii λλ , (32a)

( ) ( )( ) ( ) ( )[ ] ( )[ ]∑∑ +−=− iiii xexvxvxvxv λλ varvarvar 000* , and (32b)

( ) ( )( ) ( ) ( )[ ] [ ]∑∑ +−=− 22000

* varvarvar iiii xvxvxvxv σλλ (32c)

which results in a modified system of kriging equations,

∑=

=+−m

jiiiiji

10

2 γµσλγλ i = 1,...,m, (33a)

where ∑=

=m

jj

1

1λ , (33b)

µ is the Lagrange multiplier, and the semivariogram for the measured values of

transmissivity is


26

( ) ( )( )[ ]25.0 jiij xvxvE −=γ (34)

where the E[x] is the expectation of x.

The variance of the estimation error is

∑=

+=m

iiiE

10

2 µγλσ (35)

for the points where transmissivity is estimated. Although this equation for the variance

of the estimation error is the same as that for ordinary kriging, it includes the regression

errors because it is based on weights that have been adjusted in the kriging system.

Kriging with linear regression can only be used if the correlation coefficient is

high between transmissivity and specific capacity and if the residuals of the regression are

uncorrelated. Also, a sufficient number of measured transmissivity values are required to

define a semivariogram. This may be a problem because while there may be enough data

to define a regression equation between transmissivity and specific capacity, there may

not be enough data to define a semivariogram for transmissivity. De Smedt and others

(1985) suggest that the semivariogram can be determined using the uncertain values (i.e.

transmissivity estimated from specific capacity using the regression model) if the nugget

is entirely due to measurement errors and is subtracted from the semivariogram.

However, Ahmed and de Marsily (1987) believe that the nugget is also a result of random

small-scale variations of transmissivity in many cases and that the de Smedt and others

(1985) approach cannot be used in most cases.


27

5.2 Cokriging

Aboufirassi and Marino (1984) used cokriging (e.g. Matheron, 1971; Journel and

Huijbregts, 1978) to improve the estimation of transmissivity from more abundant

specific-capacity values. Cokriging has a better theoretical foundation than kriging with

regression because it does not make any assumptions on the correlation between

transmissivity and specific capacity (Ahmed and de Marsily, 1987) as is done with linear

regression. In cokriging, the degree of correlation between transmissivity and specific

capacity and the spatial structure of the correlation are considered in the cross

semivariogram. Ahmed and de Marsily (1987) summarized the cokriging approach:

A transmissivity, ( )0* xZ , can be estimated at a point, 0x , from the measured

values of transmissivity ( ( )ixZ for i = 1,...,n) and specific capacity ( ( )kxY for k = 1,...,m)

using,

( ) ( ) ( )∑ ∑= =

+=n

i

m

kkkii xYxZxZ

1 10

* λλ & (36)

where the weights, λ , must satisfy

∑=

=n

ii

1

1λ and (37a)

∑=

=m

kk

1

0λ& (37b)

for unbiased conditions and estimation variance is minimized using

∑ ∑= =

=++n

j

m

kiikkijj

1 101 γµγλγλ &&& i = 1,...,n and (38a)


28

∑∑==

=++m

iklki

n

iiki

102

1

γµγλγλ &&&&& k = 1,...,m (38b)

where γ is the semivariogram for transmissivity,

( ) ( ) ( )( )[ ]25.0 ji xZxZEh −=γ , (39)

γ& is the semivariogram for specific capacity,

( ) ( ) ( )( )[ ]25.0 ji xYxYEh −=γ& , (40)

γ&&is the cross-semivariogram between transmissivity and specific capacity,

( ) ( ) ( )( ) ( ) ( )( )[ ]jiji xYxYxZxZEh −−= 5.0γ&& , (41)

and 1µ and 2µ are Lagrange multipliers. The variance of the estimation error is

∑ ∑= =

++=n

i

m

kkkii

1 1100

2 µγλγλσ &&& . (42)

Ahmed and de Marsily (1987) suggested that cokriging was the more rigorous

approach and required fewer assumptions. However, there needs to be enough data to

define semivariograms for both measured transmissivity and specific capacity and enough

common points of transmissivity and specific-capacity measurements to define a cross-

semivariogram. The cross-semivariogram can be difficult to calculate and fit with a

theoretical model and must also be shown to be positive definite (Ahmed and de Marsily,

1987). Furthermore, Neuman (1984) suggested that cokriging only provides a slight

improvement over kriging with linear regression if the log transform of transmissivity and

specific capacity are normally distributed and highly correlated. de Marsily and Ahmed

(1987) found cokriging and kriging with linear regression to yield similar results.


29

Hughson and others (1996), in a study on an alluvial aquifer, found cokriging to provide

better estimates if there were 50 or more pairs of transmissivity and specific-capacity data

available. For less than 30 pairs, kriging with linear regression provided better results. In

his study of fractured carbonate rocks, Fabbri (1997) found that cokriging gave poor

results.

5.3 Kriging with transformed values

Because kriging with linear regression and cokriging gave poor results due to too

few data points to define the semivariogram for measured transmissivity, Fabbri (1997)

simply used the regression equation to estimate transmissivity, determined a

semivariogram for pooled measured and estimated transmissivity, and kriged using the

pooled data. In this case, kriging considers all the data, including the estimated

transmissivities, as exact values. Mace (1995) and Hovorka and others (1998) used this

same approach, although without showing that cokriging gave poor results. Using this

approach, the uncertainties are underestimated at interpolated points because the kriging

system does not consider the uncertainty of the linear regression.

5.4 Other geostatistical approaches

Ahmed and de Marsily (1987) present a technique of kriging with an external drift

that does not require any common data points between the transmissivity and specific-

capacity measurements. Ahmed and de Marsily (1987) also present a technique of kriging

with a guess field but found no advantage to the approach over other techniques.


30

Bardossy and others (1986) geostatistically analyzed geoelectric estimates of specific

capacity. Wallroth and Rosenbaum (1996) used multi-Gaussian kriging, which involves a

normal score transformation, to interpolate specific capacity in fractured crystalline rock.

6.0 Partial Penetration

Partial penetration is when the well is not exposed to the entire thickness of the

aquifer. When a partially penetrating well is pumped, vertical flow to the well results and

causes drawdowns to be less than expected, thus overestimating specific capacity for the

penetrated section. However, because the well does not completely penetrate the aquifer,

the specific capacity for the entire thickness of the aquifer is underestimated.

There are a few techniques that correct for partial penetration effects. Muskat

(1946) and also Turcan (1963) present one possible solution,

+

=′

a

w

w

w

a

w

cc

bL

Lr

bL

SS

2cos

271

2/1π

, (43)

where 'cS is specific capacity corrected for partial penetration, Lw is the length of the well

screened in the aquifer, ba is the aquifer thickness, and rw is the well radius. This

equation accounts for vertical flow to the well and estimates the specific capacity for

entire aquifer thickness assuming that the unpenetrated interval of the aquifer has the

same characteristics as the penetrated interval of the aquifer. To only correct for the

vertical components of flow,


31

+

=′

a

w

w

w

cc

bL

Lr

SS

2cos

271

2/1π

(44)

can be used. If there are no vertical components of flow and the aquifer is vertically

homogeneous, the following can be used:

cw

ac S

Lb

S

=′ . (45)

Equation 43 and 44 were derived for partial penetration from the top of the

aquifer, although it is commonly used wherever the well is screened in the aquifer

(Walton, 1970). These equations are not valid when the aquifer thickness is small, the

percent of penetration is large, and the well radius is large (Driscoll, 1986).

Specific capacity can be related to transmissivity considering the effects of partial

penetration by rearranging an equation by Sternberg (1973),

+

=

pw

c

sSrTt

TS225.2ln

4

2

π , (46)

where sp is the partial penetration factor defined by Brons and Marting (1961) as

−

−

=a

w

w

a

a

w

a

w

p bL

Grb

bL

bL

s ln1

(47)

where G is a function of the ratio of wL to ab . Brons and Marting (1961) evaluated G for

different values of the ratio of wL to ab . Bradbury and Rothschild (1985) fit a polynomial

(correlation coefficient = 0.992) to the Brons and Marting (1961) data to get


32

32

675.4447.11363.7948.2

−

+

−=

a

w

a

w

a

w

a

w

bL

bL

bL

bL

G . (48)

7.0 Well Loss

The water-level drawdown in a well has two components: (1) drawdown due to

head loss in the formation and (2) drawdown due to head loss from the resistance to flow

into and inside the well. Drawdown due to head loss from the resistance to flow into and

inside the well is termed ‘well loss’ and may be laminar and turbulent. Well losses are

generally due to well screen with insufficient open area, poor distribution of screen

openings, and a poorly designed filter pack in the well annulus (after Driscoll, 1986, p.

245). Well loss causes observed drawdowns in a well to be greater than those in the

aquifer. Drawdown observed in the well, ws , is the sum of the head loss in the aquifer,

as , and the well loss, Ls ,

Law sss += . (49)

Therefore, where well loss occurs, the specific capacity of the aquifer will be

underestimated. Specific capacity corrected for well loss, cS ′ , can be calculated using

aLw

c sQ

ssQS =−

=′ . (50)

As mentioned earlier, well loss can be either laminar or turbulent. Equation 49 can

be expressed in terms of flow equations as:

( )2CQQCBQsw +′′+= (51)


33

where B is a proportionality constant for laminar flow in the formation, C ′′ is a

proportionality constant for laminar well loss, and C is a proportionality constant for

turbulent well loss. It is implicitly assumed that there is no turbulent flow component in

the aquifer although there may be turbulent flow in fractured and karstic systems.

The proportionality constant for laminar flow in the formation is based on flow

equations for either steady-state or transient conditions. For example, for steady radial

flow in a confined aquifer,

TrR

B w

π2

ln

= , (52)

and for transient conditions,

−−=

p

w

TtSr

TB

4ln5772.0

41 2

π. (53)

The proportionality constant for laminar flow in the formation may be defined differently

for other hydrogeologic settings (e.g. unconfined, leaky-confined, and fractured aquifers).

Laminar well loss can be estimated using a combination of the step-drawdown

test, discussed below, and a distance-drawdown plot. The step-drawdown test is used to

estimate laminar well loss, and a distance-drawdown plot is used to estimate total well

loss. The difference between the total well loss and the turbulent well loss is the laminar

well loss. This approach requires at least two observation wells and assumes a

homogeneous aquifer. In most cases, laminar well loss is assumed to be negligible and

C ′′ is zero.


34

Turbulent well loss is more of a concern at higher pumping rates: the higher the

pumping rate the more likely there is turbulent well loss. Furthermore, turbulent well loss

increases as the square of the pumping rate. In many cases, higher specific capacities are

positively correlated with higher pumping rates because wells that can support greater

yields tend to be pumped at higher rates (e.g. the Edwards aquifer in Texas, fig. 9).

Greater turbulent well losses for greater specific capacities should result in specific

capacities that are more in error for large values than for small values.

Rorabaugh (1953; as cited by Bennett and Patten, 1960) suggested that for a 30

cm (12 inch) well, turbulent well loss becomes important for pumping rates greater than

21 liters per second (340 gpm). Errors in specific capacity due to turbulent well loss are

minimal (<10 percent) for dimensionless well loss, C*Q/B, less than 0.1 (fig. 10).

There are several approaches for estimating turbulent well loss including step-

drawdown tests, time-drawdown tests, pipe-flow theory, and empirical relationships.

Most specific-capacity tests do not include enough information to directly quantify

turbulent well loss. Therefore, indirect methods such as pipe-flow theory and empirical

approaches must be used.

7.1 Estimating turbulent well-loss from step-drawdown tests

Step-drawdown tests were developed to determine the performance of wells

having turbulent flow (Jacob, 1947). Step-drawdown tests are essentially multiple

specific-capacity tests run in the same well at different pumping rates. These tests are

generally performed in one day where the well is pumped at successively higher rates

over 1 to 3 hour intervals. Although it is recommended that water levels be allowed to


35

recover between each pumping interval (Driscoll, 1986, p. 556), continuous pumping is

more common.

Several authors have developed different approaches for interpreting step-

drawdown tests (e.g. Csallany and Walton, 1963), but Bierschenk (1964) developed a

simple graphical technique that is now commonly used. Dividing equation 51 by Q,

assuming that C ′′ equals 0, rearranging the terms that include B and C , and noting that

Qsw / is the inverse specific capacity results in

BCQQs

Sw

c

+==1 . (54)

Note that this equation has the same form of a line,

bxmy a += , (55)

where y and x are variables, am is the slope, and b is the y intercept. If the inverse of

specific capacity is plotted against pumping rate, the slope of the line equals the well-loss

constant and the intercept equals B (fig. 11a). A slope of zero (i.e., a horizontal line)

indicates no turbulent well loss.

In a step-drawdown test with two steps or pumping episodes, the well is pumped

at a rate 1−iQ and the steady-state drawdown, 1−iws , is recorded. The pumping rate is then

increased to iQ and the new drawdown, iws , is recorded. The well-loss constant can then

be estimated from (Jacob, 1947)

ii

i

iw

i

iw

QQQs

Qs

C∆+∆

∆−

∆=

−

−

−

1

1

1

. (56)


36

In general, more than two steps are preferable to account for measurement errors.

In practice, a step-drawdown test is conducted with several pumping rates. Each step is

plotted and a line is fit to the points (fig. 11b). The resulting equation is a function of the

units used to derive the line. For this particular example (taken from the Edwards aquifer

in south-central Texas), the amount of turbulent well loss can be estimated using the

estimated well-loss constant of 1.3×10-6 s2 m-5 (2.3×10-5 ft gpm-2). For a pumping rate of

30 l s-1 (500 gpm), the amount of drawdown in the well due to turbulent well loss is about

2 m (6 ft). For a pumping rate of 63 l s-1 (1000 gpm), the amount of drawdown in the well

due to turbulent well loss is about 7 m (23 ft).

The well-loss constant may not be a constant for new or especially old wells

(Walton, 1970). At greater pumping rates, the well-loss constant may be greater or lower.

For example, the well screen or aquifer may become plugged at greater pumping rates

thus resulting in a higher well-loss constant than at lower pumping rates, or material may

be removed from the screen resulting in a lower well-loss constant.

Rorabaugh (1953) suggested that the turbulent component may not be exactly

proportional to the square of the discharge rate and presented a more general relationship,

PL CQs = , (57)

where the turbulent component is directly proportional to the Pth power of the pumping

rate. The exponent P and the proportionality constants B and C can be found from step-

drawdown plots by fitting

BCQQs Pw += −1 (58)


37

to the observed measurements. If the data points are in a straight line, then the exponent,

P, equals 2 and Jacob’s relationship (eqn. 54) can be used. Nonlinearity suggests that

equation 58 may be a better approximation to behavior in the well. Figure 11c shows how

the step-drawdown plot is affected by changes in n for constant B and C. Note that the

lines in figure 11c appear linear even though P does not equal 2. Therefore, it is difficult

to arrive at a unique combination of P and C for a given step-drawdown test. Because of

this, most practitioners assume that P equals 2.

Other methods for estimating well-loss constants are found in Lennox (1966) and

Campbell and Lehr (1973). Rorabaugh (1953) discusses more exact methods for

determining well-loss constants for a broad range of pumping rates. Sheahan (1971)

presents a set of type curves that simplifies the Rorabaugh (1953) method. Eagon and

Johe (1972) investigated the use of step-drawdown tests in carbonate rock aquifers.

The step-drawdown test is used to quantify the drawdown due to all turbulent

losses including those induced by the well and in the formation. In most cases, there is no

turbulent flow in the formation. However, in some extreme cases, such as very high flow

rates through permeable sand and fractured and karstic rocks, flow in the formation itself

may have a turbulent component. In this case, equation 49 should be written as

wtatwlalw sssss −−−− +++= (59)

where als − is drawdown due to laminar flow in the aquifer, wls − is drawdown due to

laminar flow into and through the well, ats − is drawdown due to turbulent flow in the

aquifer, and wts − is due to turbulent flow into and through the well. If there is turbulent

flow in the aquifer, then ‘well-loss constant’ is a misnomer because it then also includes

turbulent losses in the aquifer. Turbulent aquifer losses are not differentiated during a


38

step-drawdown test. This may not be a problem because the techniques discussed later in

this document assume laminar flow in the aquifer. In most cases in clastic formations,

wls − and ats − are negligible.

Transmissivity and storativity can be determined from one of the pumping periods

(Driscoll, 1986, p. 557). Bisroy and Summers (1980) present a technique to estimate

transmissivity and storativity using all the drawdown and recovery data in a step-

drawdown test. Harrill (1971) developed an approach to estimate transmissivity from

water-level recovery of a step-drawdown test.

7.2 Estimating well-loss from time-drawdown tests

Gabrysch (1968) presented a method of estimating well efficiency from time-

drawdown tests. Well efficiency is defined as the percentage of the total head loss that is

laminar. His technique involves (1) calculating transmissivity from the time-drawdown

data, (2) calculating a measured specific capacity, meascS , at the end of the time-drawdown

test, (3) calculating a theoretical specific capacity, theorcS , from the calculated

transmissivity (using eqn. 18), and (4) calculating well efficiency, pL , from the measured

and theoretical specific capacity,

%100•= theorc

measc

p SS

L . (60)

Using equation 54 and equation 60, the well-loss constant can be determined

using

( )

QLLB

Cp

p−=

100. (61)


39

Turbulent well loss can then be determined using

2CQsL = . (62)

Well efficiency is also determined using

%100•+

=CQB

BLp . (63)

7.3 Graphically estimating well loss

Rao and others (1991) prepared a dimensionless curve from the curves of

Rorabaugh (1953) and Todd (1959, p. 110) to graphically estimate turbulent well loss

given a pumping rate (fig. 12). The total drawdown due to turbulent well loss plotted

against pumping rate. This plot agrees with Rorabaugh’s (1953; as cited by Bennett and

Patten, 1960) suggestion that turbulent well loss is not important for pumping rates less

than 20 l s-1 (340 gpm). According to this plot, turbulent well loss accounts for less than 5

percent of total drawdown for pumping rates less than about 36 l s-1 (600 gpm). Rao and

others (1991) acknowledge that this method provides a first approximation to well loss in

a well. I fit the following equation to the line shown in figure 12 to make the relationship

easier to use:

44.23

12.1Qss

w

L = , (64)

This equation is only valid if Q is in cubic feet per second (cfs).

7.4 Estimating turbulent well-loss using empirical relationships

Turbulent well-loss can also be estimated with empirical relationships between

specific capacity and well loss constant. Zeizel and others (1962) were among the first to


40

define such a relationship. Eagon and Johe (1972) made a cross plot of well-loss constant

and specific capacity for a carbonate aquifer. Mace (1997) found a relationship similar to

Eagon and Johe’s (1972) in the Edwards aquifer (fig. 13) but for higher specific-capacity

values. The best-fit line for the pooled data is

29.160.0 −= cSC (65)

where C is in d2m-5 and cS is in m2 d-1. The 2R for the relationship is 0.82 and has a 95

percent prediction interval that spans 1.4 log cycles.

Using the relationship involves calculating the well-loss constant from specific

capacity using equation 65 and then calculating well loss using equation 62. Mace (1997)

noted that this approach, because of the large prediction interval, can lead to greatly

under- and over-estimated well losses. Mace (1997) found that equation 65 predicted well

losses that exceeded total measured drawdown in about 20 percent of the specific-

capacity tests.

7.5 Estimating well-loss using pipe-flow theory

Well loss due to the flow of water through the well bore to the pump, also called

friction loss, can be approximated using equations that describe laminar and turbulent

flow of fluids through pipes. Different equations describe friction loss for laminar and

turbulent conditions. The Reynolds number, Re, the ratio of inertial to viscous forces, is

used to determined whether flow is laminar or turbulent. The Reynolds number is defined

as

d

evdR

µρ= (66)


41

where ρ is the fluid density [M L-3], v is the specific discharge [L t-1], d is the diameter of

the pipe [L], and dµ is the dynamic fluid viscosity [M L-1 t-1]. If the Reynolds number is

less than 2100, flow is laminar. If the Reynolds number is greater than 4000, flow is

turbulent. Critical flow (transition between laminar and turbulent flow) occurs between

Reynolds numbers of 2100 and 4000.

Friction loss, fh , is found from

dg

fLvh f 2

2

= (67)

where f is friction factor [-], L is the pipe length the fluid flows through [L], and g is

gravitational acceleration [L t-1]. For laminar flow, the friction factor can be determined

from

eR

f 64= (68)

For turbulent flow, friction factor must determined from friction tables (e.g. White, 1994,

p. 316).

The Hazen-Williams equation can also be used to approximate friction loss for

turbulent flow

165.185.1

85.1022.3dC

Lvh

hw

ff = (69)

where hwC is the Hazen-Williams roughness coefficient. The Hazen-Williams equation is

unit dependent where v must be in gallons per minute and d must be in inches. L and hf

are unit independent. The roughness coefficient is 140 for new pipe and 100 for old pipe.

The Hazel-Williams equation is applicable for water at about 60° F with 20 percent error


42

for very high and very low temperatures. For wells, Lf can be approximated by the

distance from the well screen to the pump intake. However, flow to a pump in a well is

more complicated because of the contribution of flow from the formation into the well

occurs along the well bore such that v changes over the completed length of the well.

Mace (1997) used this approach to estimate well losses from specific-capacity

tests in the Edwards aquifer. Pipe-flow theory underestimated well loss (fig. 14) because

it did not account for the turbulent flow of water from the formation into the well. In this

study, pipe-flow theory resulted in well losses that exceeded total measured drawdown in

about 10 percent of specific-capacity tests.

7.6 Incorporating well-loss corrections

Well-loss corrections can be incorporated into approaches for estimating

transmissivity from specific capacity in different ways. One approach is to correct

measured specific capacity and using the corrected specific capacity in the analytical

equations. Correcting the measured specific capacity involves calculating well loss using

one of the above methods and then calculating corrected specific capacity, cS ′ , using

LwLw

Lcc ss

Qss

sSS

−=

−=′ . (70)

Another approach is to make the correction within the analytical relationship

between transmissivity and specific capacity. Razack and Huntley (1991), using an

equation that describes total drawdown in a well due to laminar and turbulent flow

(Jacob, 1947), developed an equation,


43

−=

ww rR

CQQs

T ln2

1

π, (71)

that is similar to the Thomasson and others (1960) solution (eqn. 4a) and considers

turbulent well losses. Bradbury and Rothschild (1985) derived an equation based on Theis

and others (1963) that considers well loss and partial penetration (eqn. 8).

The empirical relationship between transmissivity and specific capacity developed

by Eagon and Johe (1972, equation fitted in this document, table 1a) uses specific

capacity corrected for well loss. Note that the other empirical relationships (table 1b-m)

use uncorrected specific capacity to estimate transmissivity. These relationships assume

that the uncertainty due to well loss is included in the relationship. Another approach may

be to assume a unique value for the well-loss coefficient, C, for all the other tests based

on step-drawdown tests. However, variability of well construction may invalidate this

approach.

8.0 Fractured and karstic rocks

Estimating transmissivity from specific capacity in fractured and karstic rocks

may be problematic because the analytical equations in section 3 assume porous-media

flow. Therefore, the Theis equation may not accurately represent transmissivity of

fractured rocks (Huntley and others, 1992; Knopman and Hollyday, 1993). White (1988)

states that “In general, pump test data are of marginal value in evaluating water resources

in karstic aquifers except for the diffuse flow part of the system.”


44

Although there may be limitations in the standard porous-media approach,

investigators have related variations in specific capacity to well construction and

variability in fractured rocks. Walton and Neill (1963) noted the correlation between

specific capacity and geology for dolomite aquifer in Illinois. Zeizel (1963) looked at

variation in yields in Silurian dolomite aquifer, northeastern Illinois. Siddiqui and Parizek

(1971) and Yin and Brook (1992) have shown that fault zones, fracture size and density,

dip of rock, and folding capacity affect specific capacity in fractured rock. Cederstrom

(1972) evaluated well yields in consolidated rocks and found greater yields in structurally

deformed areas. Chandrashekhar and others (1976) investigated the variation of specific

capacity in basalt and laterites and found that specific capacities were controlled by depth

of the well, depth to water, topographic setting, and well radius. LaRiccia and Rauch

(1977) related specific capacity to distance from photolineaments. Viswanathiah and

Sastri (1978) investigated variations of specific capacity in the hard rocks of India.

Daniel (1987, 1989a,b) used statistical analysis of well yields in the fractured,

crystalline rocks of the Piedmont and Blue Ridge provinces of North Carolina and found

that wells in draws or valleys had average yields three times higher than those of wells on

hills and ridges and that well yield was directly proportional to well diameter. Knopman

(1990) and Knopman and Hollyday (1993) investigated the variation in specific capacity

in fractured rocks and described the variability of specific capacity values in terms of

formation heterogeneities and variability in well construction and field measurement

factors. They found that casing diameter, primary water use, and duration of pumping

accounted for 24 percent of the variation observed in specific-capacity data, lithology

alone accounted for about 12 percent. They developed a classification based on lithology

and differences in casing diameter, water use, duration of discharge, topographic setting,


45

well depth, and casing depth that could explain about half of the variation in specific

capacity values. Wallroth and Rosenbaum (1996) looked at variations in specific capacity

in the fractured crystalline rock of Sweden.

Based on the Gringarten and Witherspoon (1972) solution for drawdown in a well

drilled through a vertical fracture, Huntley and others (1992) developed a normal

nonlinear regression model to relate specific capacity to transmissivity for a fractured

aquifer,

( ) εθ += ,ec TgS , (72)

where eT is the effective transmissivity, e is a normally distributed error term and

( )θ,eTg is a nonlinear function defined by

( )15.0

41

212,

−

−−

=

tTEi

tTerf

tT

Tgee

ee θθθ

πθ (73)

where

( ) 5.0yxe TTT = , (74)

2

/

f

yx

Sx

TT=θ , (75)

erf is the error function, Ei is the exponential integral and fx is the horizontal distance

the fracture extends from the well [L].

Huntley and others (1992) successfully used equation 72 to explain the variation

in observed specific-capacity values for a fractured rock aquifer. However, they point out

that it is too difficult to use the equation to estimate transmissivity from specific capacity.


46

Eagon and Johe (1972) empirically related transmissivity and specific capacity in

a karstic aquifer in northwestern Ohio (table 1a). Huntley and others (1992) developed an

empirical relationship between specific capacity and transmissivity for fractured-rock

aquifers (table 1c), although its applicability to other aquifers is unclear. El-Naqa (1994)

developed an empirical relationship between transmissivity and specific capacity for a

fractured carbonate rock aquifer in central Jordan (table 1d). Mace (1995, 1996, 1997)

empirically related transmissivity to specific capacity for a fractured and karstic aquifer in

Central Texas (table 1e). Mace (1997) also developed an empirical equation for the

fractured and karstic Floridan aquifer (table 1f). Fabbri (1997) developed a relationship

for a fractured, limestone aquifer in Italy (table 1g). These empirical relationships are

discussed in more detail in section 4.2.

9.0 Data Quality Issues

Specific-capacity data may not be collected under the most ideal of conditions.

Nonscientists, usually drillers, make most of the measurements of specific-capacity data.

Different techniques of varying quality and reliability are used to produce water and

measure water levels and production rate. In the best circumstances, a well is pumped for

a finite (and recorded) amount of time and the drawdown is measured directly. In

mediocre circumstances, pumping rate is estimated by an experienced driller or the well

is bailed rather than pumped. In worst circumstances, estimates are grossly incorrect or

estimated purely based on experience with no testing at the well. Less-than-honest drillers

may over-report the productivity of a well to impress a client with the high-quality of

their work or to compensate for a well that produces more poorly than promised.


47

Specific-capacity tests are generally performed at the time of well construction

and therefore the well may not be fully developed thus underestimating specific capacity.

Gabrysch (1968) noted that some well drillers in the Houston area improperly calculate

specific capacity with the ‘active static’ water level, which is the water level measured 10

minutes after the pump has been turned off thus overestimating specific capacity.

Some ways to filter data include only choosing those wells that were reportedly

pumped or jetted. Bailing does not result in a uniform removal of water and water levels

are generally measured after the bailer has been removed which allows time for water

levels to recover. However, bail tests may provide useful data in low permeability

formations. Another way to filter data is to only choose those tests that also include

production time. This assumes that a driller that also notes the production time in addition

to production rate and drawdown has a greater appreciation of the test and is more

attentive to data quality. Because pumping times for well-performance tests are usually

less than pumping times for pumping tests, transmissivity estimated from specific-

capacity data are dominated by the transmissivity near the well (Meier and others, 1999).

Given that the standard error in estimating hydraulic conductivity from aquifer

tests is often 100 percent or higher (Winter, 1981), estimates from specific capacity are

probably not too much worse. It has been my experience, as well as others (e.g.

Suvagondha and Singharajwarapan, 1987), that transmissivity values estimated from

specific capacity generally agree with transmissivity values determined from pumping

tests.


48

10.0 Other Issues

There are a number of other issues that may impact specific-capacity

measurements including decreasing saturated thickness during testing and considering

tests with incomplete data, no measurable drawdown, or production from multiple zones.

10.1 Correcting for decreased saturated thickness

In unconfined aquifers, a decline in water level decreases the saturated thickness

of the aquifer and thus decreases the transmissivity. This decrease in transmissivity leads

to greater drawdowns and thus underestimates actual specific capacity. In many cases, the

amount of drawdown relative to the initial saturated thickness of the aquifer is small. In

cases where this is not true, the drawdown must be corrected for the decrease in saturated

thickness. Walton (1970, p. 224) presents an equation derived by Jacob (1944) to correct

drawdown measurements for decreases in saturated thickness,

wt

wtwta b

sss

2

2

−= , (76)

where as is the drawdown that would occur in an equivalent nonleaky artesian aquifer,

wts is the observed drawdown in the water-table aquifer, and wtb is the initial saturated

thickness of the aquifer. If the measured drawdown is less than 20 percent of the initial

saturated thickness, the actual drawdown will be in error by less than 10 percent.

10.2 Including tests with no reported pumping time or well radius

The Theis and others (1963) equation (eqn. 5) requires information on the

production time and well radius. Many tests may not report the pumping time and/or the


49

well radius. Oftentimes, these tests can be excluded from the analysis. However,

sometimes it is important to include these tests if there are not a large number of tests that

report production time and well radius. Recall from section 3.2 that transmissivity is the

most important variable that influences specific-capacity values (see fig. 4). Therefore,

pumping time and well radius can be assumed, which results in increased uncertainty in

the estimated transmissivity. These values can be assigned according to experience or,

preferably, based on the mean or geometric mean (depending on the distribution) of the

reported values. The uncertainty of assumed values of pumping time and well radius can

be investigated by using the upper and lower standard deviation on the resulting

distributions.

10.3 Including tests with no measurable drawdown

It is not unusual to find some specific-capacity tests where the well is pumped but

there is no measurable drawdown. These are cases where the aquifer is not stressed

enough to develop a measurable amount of drawdown. In most cases, specific-capacity

tests with no drawdown are discarded. However, in some cases, discarding these tests can

create a bias toward lower transmissivity values. Hovorka and others (1995, 1998) and

Mace (1995, 1996, in review) noted that about 20 percent of specific capacity tests in the

Edwards aquifer did not produce a measurable amount of drawdown. In most cases,

especially in clastic aquifers, the production rate is too low. However, production rates in

the Edwards aquifer can be as great as 8,000 gpm and still not produce drawdown (fig.

15).

Hovorka and others (1995, 1998) and Mace (1995, 1996, in review) included

specific-capacity tests with no measurable drawdown by assuming that drillers could


50

measure a 0.3 m (1 ft) decline in water level in the well. They then calculated specific

capacity based on this 0.3 m (1 ft) assumed drawdown and noted that the resulting

specific capacity (and any estimate of transmissivity from that specific capacity) was a

minimum value. The tests with no drawdown had a mean transmissivity that was 10 times

greater than the tests with measurable drawdown. Including the tests with no drawdown

increased the geometric mean by 50 percent.

Halihan and others (1997) noted that tests with no drawdown should be expected

in karstic aquifers where there are fractures and conduits that can result in extremely high

transmissivity. The minimum amount of pumping, mQ , required to observe the minimum

amount of measurable drawdown, cs , can be defined from a modified version of the

Theis and others (1963) equation (eqn. 5),

=

SrTt

TsQ

w

p

cm

2

25.2ln

4π. (77)

Assuming a pumping time of 8 hr, a well radius of 4 in, and a storativity of 10-4, Figure 16

shows the threshold of pumping rate required to observe measurable drawdown.

10.4 Multiple production zones

Some wells will be screened in multiple production zones in an aquifer or group

of aquifers to achieve the desired yield. For example, a well might be screened from 30 to

40 m, 50 to 60 m, and 65 to 80 m. Therefore, the production of the well, and thus the

value of specific capacity, is from a combination of producing zones,

( )∑=

=n

incc SS

1

, (78)


51

where n is the number of production zones. Back calculating the specific capacity of each

zone is not possible unless specific capacity is measured at different well depths as the

well was drilled or after the well was drilled by isolating each well section.

Walton (1970) describes an approach to qualitatively determine if deeper units are

less or more permeable than upper units. This is done by first calculating the specific-

capacity index (eqn. 2) for each well, segregating the wells into categories based on

formations penetrated, and comparing the distributions of specific-capacity index for the

different categories. If lower specific-capacity indices are found for wells that intersect

more of the formation, then the lower units are less productive. If the specific-capacity

index increases, then the lower units are more productive. If specific-capacity index

remains the same, then the formations have similar productivity. A similar comparison

can be done with the geometric means of the specific-capacity index.

Bennett and Patten (1960) used geophysical techniques to investigate the specific

capacity of multiple contributing zones. Hovorka and others (1995, 1998) used equation

30 and the mathematical definition of the harmonic mean to determine specific capacity

and ultimately transmissivity for wells that were tested after each 15 m (50 ft) penetration

and for wells tested with packers. Westly (1993) suggests an approach to measure specific

capacity with increasing depth while drilling a well.

One issue with multiple production zones is that the well may have been

strategically completed in the most productive part of the formation. Therefore, the

transmissivity of the entire formation thickness may be overestimated if the test is

corrected for partial penetration.


52

11.0 Recommended approach

There are three general approaches to estimating transmissivity from specific

capacity: analytical, empirical, and geostatistical. The approach that is used depends

primarily on how many specific capacity and pumping tests are available and which

approach results in more accurate predictions. The analytical approach only requires a

single specific-capacity test to estimate transmissivity. The empirical approach only

requires a single specific-capacity test to estimate transmissivity for a predefined

relationship but requires at least 25 pairs of specific-capacity and transmissivity tests to

define a new relationship. Geostatistics requires enough specific-capacity and measured

transmissivity data to define a linear relationship between transmissivity and specific

capacity (at least 25 pairs [Razack and Huntley, 1991]), semivariograms for specific

capacity and measured transmissivity, and a cross-semivariogram between specific

capacity and measured transmissivity, depending on which geostatistical approach is

used. The number of transmissivity and specific-capacity measurements required to

define semivariograms depends on how spatially correlated transmissivity and specific

capacity are and the spacing and density of points in the aquifer. With limited data, the

cross-semivariogram can be difficult to calculate, fit with a theoretical model, and shown

to be positive definite (Ahmed and de Marsily, 1987).

If there are less than 25 points, then two choices are available: the analytical

approach or an existing empirical equation. An existing empirical equation should only

be used if you are reasonably confidant that the relationship will apply to your

hydrogeologic setting. Currently, only carbonate aquifers have had a number of empirical

relationships defined that are very similar to each other. The relationship used should be


53

based on data that encompass the value or values of specific capacity. The empirical

relationship of Mace (1997) (table 1e) agrees well with data from three different aquifers

for a wide range of specific-capacity values and is similar to other relationships defined

for carbonate aquifers.

Because there have not been many published empirical relationships for clastic or

hard-rock aquifers, it is not advisable to use published relationships on other aquifers,

especially considering the differences between the relationships defined by Huntley and

others (1992) (table 1c) and those defined here (table 1h, i, j, l, and m) for clastic aquifers.

For these hydrogeologic settings, the analytical approach should be used.

The Theis and others (1963) approach (eqn. 5) is the preferred analytical

approach. In general, when using the analytical solution, one should correct for partial

penetration effects and turbulent well loss. When correcting for partial penetration, it is

recommended to only correct for the vertical flow components of partial penetration (eqn.

44). This is because the full partial penetration correction (eqn. 43), which results in a

specific capacity estimate for the entire aquifer thickness, assumes that the unpenetrated

thickness of the aquifer has the same production properties as the penetrated thickness of

the aquifer.

Because turbulent well loss is difficult to estimate, it is important to first

determine if a well-loss correction is necessary. One way of doing this is to inspect the

production rates for the specific-capacity tests: if production is less than 600 gpm, then

turbulent well losses can be ignored in most cases. For wells with production rates greater

than 600 gpm, it might be useful to determine well losses from step-drawdown or

pumping tests and determine if well loss at higher pumping rates greatly affects specific

capacity. If there is a consistent well-loss constant among the tested wells, then that well-


54

loss constant can be used to estimate well losses in the other wells with pumping greater

than 600 gpm. If this is not possible, then it is recommended that Figure 12 or equation

64 be used to estimate well loss. While it is acknowledged that this graph is an

approximation, it will not result in well losses that exceed measured drawdown in most

cases, unlike the empirical approach in section 7.4 and the pipe-flow approach in section

7.5. Note that the empirical relationships do not require well-loss or partial-penetration

corrections (although the empirical relationship only includes the vertical flow

component of the partial penetration correction). For unconfined aquifers, it might be

necessary to correct for decreased saturated thickness (eqn. 76).

If pumping tests are available, then it is appropriate to apply the selected approach

on specific-capacity values from the pumping tests to compare transmissivity determined

from pumping-test analysis to transmissivity estimated from specific capacity. At the very

least, transmissivities estimated from specific-capacity tests should approximate

transmissivity values determined from pumping tests. If there seems to be a consistent

error between transmissivity measured from pumping tests and transmissivity estimated

from specific capacity tests, then a correction factor can be applied following the

technique of Prudic (1991) or Darling and others (1994) (see section 4.3).

When a greater number of tests (>25) are available, then the ultimate purpose of

the data becomes important along with the number of data points. If the purpose is to

simply estimate transmissivity for statistical analysis or provide estimates of

transmissivity at measured specific-capacity points, then the empirical approach is the

simplest.

If the purpose of the data is to develop an interpolated map of transmissivity, then

the geostatistical approach should be considered. The choice of geostatistical approach,


55

kriging with linear regression or cokriging, depends on the number of transmissivity and

specific capacity pairs. If there are greater than 50 transmissivity and specific capacity

pairs, then cokriging may offer better results. If there are less than 50 pairs, then kriging

with linear regression might be better. Ideally, both approaches should be compared.

Before deciding to use the geostatistical approach, the first step is to determine if

there are enough transmissivity and specific capacity values for the analysis. Generally,

the limitation is the number of transmissivity values because these values are rarer than

specific-capacity values. For kriging with linear regression, there needs to be (1) enough

transmissivity and specific-capacity pairs to define a linear relationship between

transmissivity and specific capacity (similar to the empirical approach) with a high

correlation coefficient and uncorrelated residuals and (2) enough specific-capacity values

and measured values of transmissivity to define a semivariogram for each.

For cokriging, there needs to be enough specific-capacity values and measured

values of transmissivity to define a semivariogram for each and enough transmissivity

and specific-capacity data pairs to define a cross-semivariogram. If there is not enough

specific capacity and measured transmissivity data to do the above, then the geostatistical

approach cannot be used.

Transmissivity values can still be interpolated using kriging as described in

section 5.3, but the uncertainties of the estimates will be underestimated because the

uncertainty of using the linear regression or correlating transmissivity to specific capacity

are not included. Before using the geostatistical approach, it is important to either correct

for partial penetration or only include those tests that penetrate most of the aquifer. This

is because the semivariograms should represent the spatial correlation of hydraulic

properties rather than artifacts of well construction and completion.


56

Ultimately, the approach that is used should minimize the error of estimating

transmissivity from specific capacity. In the following example, the analytical and

empirical approaches are compared for a sandstone aquifer (Carrizo-Wilcox) in east-

central Texas to determine which approach provides the best estimates of transmissivity

from specific capacity.

A total of 217 wells had time-drawdown tests to determine transmissivity and

specific capacity. To define an empirical relationship between transmissivity and specific

capacity, log-transformed values of each parameter were plotted against each other and a

line was fit through the data using least squares (fig. 17). The best-fit line through the

data was

84.020.0 cST = (79)

where the units of T and Sc are in m2d-1 and the correlation coefficient, R2, is 0.91. The

relationship has a 90 percent prediction interval that spans a little less than about an order

of magnitude.

To compare transmissivity estimated using the empirical relationship to values

estimated using the analytical relationship (eqn. 5), the mean absolute error and the mean

error between the estimates and the measured values of log-transformed transmissivity

were determined. The appropriate information needed to use the analytical equation

(discharge rate, drawdown, pumping time, and well radius) were available for 57 of the

217 tests. Mean absolute error, ε , is defined by:

( ) ( )[ ]∑=

−=n

iem TT

n 1

loglog1ε (80)


57

where n is the number of values, mT is the transmissivity determined from the pumping

test, and eT is the value of transmissivity estimated from specific capacity. Mean error,

ε , is defined by:

( ) ( )[ ]∑=

−=n

iem TT

n 1

loglog1ε (81)

The mean absolute error and mean error for transmissivity estimated using the

analytical approach are 0.17 and -0.002, respectively. A mean absolute error of 0.17

means that, on average, the estimated value of transmissivity is within a factor of 1.5 of

the measured value (determined by taking the inverse log of 0.17). Because the mean

error is close to zero, estimates of transmissivity made with the analytical approach are

collectively unbiased and do not have a systematic error toward underestimating or

overestimating transmissivity.

The mean absolute error and mean error for transmissivity estimated using the

empirical relationship are 0.33 and 0.17, respectively. A mean absolute error of 0.33

means that, on average, the estimated value of transmissivity is within a factor of 2.1 of

the measured value. The positive mean error indicates a bias toward overpredicting

transmissivity.

Based on the mean absolute errors, the analytical approach provides more accurate

estimates of transmissivity than the empirical approach. Because many specific-capacity

tests do not include information on pumping time and well radius to use the analytical

equation, mean values based on wells that include this information were used. Using this

approach increases the number of wells available for analysis to 107 and increases the

mean absolute error and mean error slightly to 0.173 and -0.02, respectively. Therefore,


58

even with assumed values, the analytical approach is superior. However, both methods

can result in errors as much as a factor of 5 (fig. 18).

12.0 Conclusions

Specific-capacity data are very useful for estimating transmissivity and should be

used in water-resources investigations. Including transmissivity estimated from specific-

capacity data can dramatically increase the number of transmissivity estimates in an

aquifer. Studies in Texas have shown that estimating transmissivity from specific

capacity increases the number of transmissivity data points from 71 to 1,083 in the

Edwards aquifer, from 291 to 1,973 in the sandstone aquifers of North-Central Texas, and

from 200 to over 9,500 in the Carrizo-Wilcox aquifer. These data allow the variability of

transmissivity values to be better defined and allow better correlation to geology and

better interpolation of transmissivity values.

There are three main approaches for estimating transmissivity from specific

capacity data: analytical techniques, empirical techniques, and geostatistical techniques.

The most commonly used analytical approach is an equation derived from the Theis

nonequilibrium formula which requires specific capacity, well radius, production time,

and an estimate of storativity for estimating transmissivity. When using the analytical

approach, it is appropriate to correct for well loss, vertical flow due to partial penetration,

and, if important in unconfined aquifers, decreased saturated thickness. Well loss does

not become important in most wells until the production rate exceeds 600 gpm. Because

estimating well loss is difficult, a simple graphical approach is recommended.


59

The empirical approach involves empirically relating transmissivity to specific

capacity measured in the same well. This approach usually requires at least 25 data pairs

before a useful relationship can be defined. This approach is advantageous because

specific-capacity data do not have to be corrected for turbulent well loss or vertical

components of partial penetration. Existing relationships can probably be used for

carbonate aquifers but should not be used for other aquifer settings because they have not

been shown to be applicable in other aquifers.

Geostatistical techniques are useful for estimating transmissivity, developing

interpolated maps of transmissivity, and quantifying the uncertainty of the estimates. Two

geostatistical techniques are commonly used: kriging with linear regression or cokriging.

The choice of technique depends on the number of transmissivity and specific-capacity

data pairs, and, ultimately, how each performs in estimating transmissivity. If there are

greater than 50 transmissivity and specific capacity pairs, cokriging may offer better

results than kriging with linear regression. Whether or not geostatistical techniques can be

used depends on whether there are enough specific capacity and measured transmissivity

values to define a linear relationship between transmissivity and specific capacity,

semivariograms for specific capacity and measured transmissivity, and a cross-

semivariogram between specific capacity and transmissivity.

13.0 Acknowledgments

I thank Jinhuo Liang for the initial Excel program to calculate transmissivity from

specific capacity, Liying Xu for her dedicated assistance in the literature review, and

Mike Harren and Ted Way at the Texas Water Develoment Board for data on the Carrizo-


60

Wilcox aquifer. I also thank Dr. Alan R. Dutton and Dr. Bridget Scanlon of the Bureau of

Economic Geology and Dr. Shao-Chih (Ted) Way of the Texas Water Development

Board for their thorough reviews of the manuscript. This publication includes work

performed for the Edwards Aquifer Authority, Texas Water Development Board, the

Lower Colorado River Authority, and the Texas National Research Laboratory

Commission: I am grateful for their support.

14.0 Symbols

Units are included with the brackets [] where L is length, M is mass, and t is time. A dash, [-], means the parameter is unitless, and a tildi, [~], means units of the parameter can be variable.

α level of significance [-]

wA cross-sectional area of the well [L2] b y-intercept of a line [~]

0b regression model parameter [~]

1b regression coefficient [~] ba aquifer thickness [L]

wtb initial saturated thickness of the water-table aquifer [L] B laminar well-loss coefficient [t L-2] C well-loss constant, proportionality constant for turbulent flow [t2 L-5,

assuming n = 2] C ′′ proportionality constant for laminar well loss [t L-2] C ′ sensitivity parameter [t L2]

cC constant for the Thomasson and others (1960) relationship [-] hwC Hazen-Williams roughness coefficient [-]

d pipe diameter [L] ( )ixe uncertainty of ( )ixZ [~] [ ]xE expectation of x [~]

f friction factor [-] ijγ semivariogram [~]

γ semivariogram for transmissivity [L4 t-2] γ& semivariogram for specific capacity [L4 t-2] γ&& cross-semivariogram between transmissivity and specific capacity [L4 t-2]


61

g gravitational acceleration constant [L t-2] ( )θ,eTg nonlinear function describing fracture flow [L2 t-1]

G function for partial-penetration correction [-] fh friction loss [L]

i index number [-] j index number [-]

0K zero-order modified Bessel function of the second kind λ weighting parameter [-]

pL well efficiency [-] Lf length of pipe through which the fluid flows [L] Lw length of the well screened to the aquifer [L] µ Lagrange multiplier [-]

1µ Lagrange multiplier [-]

2µ Lagrange multiplier [-]

dµ dynamic fluid viscosity [M L-1 t-1] m number of specific-capacity measurements [-]

am slope of a line [~] n number of data points [-] Q pumping rate [L3 t-1] P turbulent flow exponent [-] ρ fluid density [M L-3] rw well radius [L] R radius of influence [L] Re Reynolds number, ratio of inertial to viscous forces [-]

2R coefficient of determination [-] 2σ variance [~] 2Eσ variance of the estimation error [~] 2iσ variance of the uncertainty [~]

1s residual drawdown after st minutes [L]

2s maximum drawdown [L]

as drawdown due to head loss in the aquifer [L]

als − drawdown due to laminar flow in the aquifer [L]

wls − drawdown due to laminar flow into and through the well [L]

Ls drawdown due to well loss [L]

ps partial penetration factor [-]

ats − drawdown due to turbulent flow in the aquifer [L]

wts − drawdown due to turbulent flow into and through the well [L]

ws measured drawdown in the well [L]

wts observed drawdown in the water-table aquifer [L]


62

S storativity [-] cS specific capacity [L2 t-1]

cS ′ specific capacity corrected for partial penetration [L2 t-1]

cS ′ specific capacity corrected for well loss [L2 t-1] measuredcS measured value of specific capacity [L2 t-1]

ltheoreticacS theoretical value of specific capacity [L2 t-1]

iS specific-capacity index [L t-1]

εSS sum of squares of the errors [~]

xSS sum of squares for iX [~]

xySS sum of squares for iX and iY [~]

ySS sum of squares for iY [~] θ factor for calculating ( )θ,eTg [L-1]

2/αt values of the t-distribution [-]

pt production time [t]

st time since pumping stopped [t] T transmissivity [L2 t-1]

eT effective transmissivity [L2 t-1]

iT transmissivity for measurement i [L2 t-1]

cST transmissivity determined from specific capacity [L2 t-1] u dimensionless time used with the well function [-] v specific discharge [L t-1]

( )ixv true value of ( )ixZ [~]

( )0xv∗

estimate of a true value at point 0x [~] ( )uW well function, dimensionless drawdown [-]

x variable [~] ix point i [-]

iX predictor variable for linear regression for index i [~] X mean value of iX [~]

pX a specific value of iX [~] y variable [~]

iY observed value for linear regression for index i [~]

iY predicted value of iY [~]

Y mean of iY [~] ( )ixZ realization (estimate) of any intrinsic random function at a point ix [~]


63

15.0 References

Aboufirassi, Mohamed, and Marino, M. A., 1984, Cokriging of aquifer transmissivities from field measurements of transmissivity and specific capacity: Journal of the International Association for Mathematical Geology, v. 16, no. 1, p. 19-35.

Adyalkar, P. G., Dias, J. P., and Srihari Rao, S., 1981, Empirical methods for evaluating hydraulic properties of basaltic water table aquifers with specific capacity values: Indian Journal of Earth Sciences, v. 8, no. 1, p. 69-75.

Adyalkar, P. G., and Mani, V. V. S., 1972, An attempt at estimating the transmissibilities of trappean aquifers from specific capacity values: Journal of Hydrology, v. 17 no. 3, p. 237-241.

Ahmed, Shakeel, and De Marsily, Ghislain, 1987, Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity: Water Resources Research, v. 23, no. 9, p. 1717-1737.

Bardossy, A., Bogardi, I., and Kelly, W. E., 1986, Geostatistical analysis of geoelectric estimates for specific capacity: Journal of Hydrology, v. 84, no. 1-2, p. 81-95.

Bear, Jacob, 1953, Hydraulics of groundwater: New York, New York, McGraw-Hill International Book Company, 567 p.

Bedinger, M. S., and Emmett, L. F., 1963, Mapping transmissibility of alluvium in the lower Arkansas River valley, Arkansas: in Short papers in geology and hydrology: U.S. Geological Survey Professional Paper 475-C, p. C188-C190.

Bennett, G. D., and Patten, E. P., Jr., 1960, Borehole geophysical methods for analyzing specific capacity of multiaquifer wells: U. S. Geological Survey Water-Supply Paper, W 1536-A, p. 1-25.

Bierschenk, W. H., 1964, Determining well efficiency by multiple step-drawdown tests: International Association of Scientific Hydrology, Publication 64.

Binsariti, A. A., 1980, Statistical analysis and stochastic modeling of the Cortaroaquifer in southern Arizona: Ph.D. Thesis, University of Arizona, 243 p.

Bisroy, Y. K., and Summers, W. K., 1980, Determination of aquifer parameters from step tests and intermittent pumping data: Ground Water, v. 18, n. 2, p. 137-146.

Boulton, N. S., 1954, Unsteady radial flow to a pumped well allowing for delayed yield from storage: International Assication of Sci. Hydrology Publication 37, p. 472-477.

Bradbury, K. R., and Rothschild, E. R., 1985, A computerized technique for estimating the hydraulic conductivity of aquifers from specific capacity data: Ground Water, v. 23, no. 2, p. 240-246.

Brons, F., and Marting, V. E., 1961, The effect of restricted fluid entry on well productivity: Journal of Petroleum Technology, v. 13, no. 2, p. 172-174.

Brown, R. H., 1963, Estimating the transmissivity of an artesian aquifer from the specific capacity of a well: U.S. Geological Survey Water Supply Paper 1536-I p. 336-338.

Campbell, M. D., and Lehr, J. H., 1973, Water well technology: McGraw-Hill, New York, 681 p.


64

Cederstrom, D. J., 1972, Evaluation of yields of wells in consolidated rocks, Virginia to Maine: U. S. Geological Survey Water-Supply Paper W 2021, 38 p.

Chandrashekhar, H., Jayapal, H. N., and Ramaprasad, B. K., 1976, Specific capacity of wells in basalt and laterite of the Karanja Basin; a comparative study: Prasad, Balmiki, and Manjrekar, B. S. (eds.), Proceedings of a Symposium on Deccan Trap and bauxite, Special Publication Series - Geological Survey of India, v. 14, p. 125-131.

Clifton, P. M., and Neuman, S. P., 1982, Effects of kriging and inverse modeling on conditional simulation of the Avra Valley aquifer in southern Arizona: Water Resources Research, v. 18, no. 4, p. 1215-1234.

Cooper, H. H., Jr., and Jacob, C., E., 1946, A generalized graphical method for evaluating formation constants and summarizing well field history: Transactions of the American Geophysical Union, v. 27, p. 526-534.

Csallany, S., and Walton, W. C., 1963, Yields of shallow dolomite wells in northern Illinois: Report of Investigations 46, Illinois State Water Survey, Champaign, Illinois, p. 14.

Daniel, C. C., III, 1987, Statistical analysis relating well yield to construction practices and siting of wells in the Piedmont and Blue Ridge provinces of North Carolina: U. S. Geological Survey Water-Resources Investigations, WRI 86-4132, 54 p.

Daniel, C. C., III, 1989a, Statistical analysis relating well yield to construction practices and siting of wells in the Piedmont and Blue Ridge provinces of North Carolina: U. S. Geological Survey Water-Supply Paper, W 2341-A, 27 p.

Daniel, C. C., III, 1989b, Correlation of well yield to well depth and diameter in fractured crystalline rocks, North Carolina: in Proceedings of a conference on Ground water in the Piedmont of the Eastern United States: Daniel, C. C., III, White, R. K., and Stone, P. A., eds., 638-653.

Darling, B. K., Hibbs, B. J., and Dutton, A. R., 1994, Ground-water hydrology and hydrogeochemistry of Eagle Flat and surrounding area: Bureau of Economic Geology, The University of Texas at Austin, Final Report prepared for the Texas Low-Level Radioactive Waste Disposal Authority, 137 p.

Daugherty, R. L., and Ingersoll, A. C., 1954, Fluid mechanics: McGraw Hill, New York.

Davis, S. N., and DeWeist, R. J. M., 1966, Hydrogeology: New York, New York, John Wiley and Sons, Inc., 463 p.

Davis, S. N., 1988, Artesian wells of France, Darcy's law and specific capacity: Eos, v. 69, no. 44, p. 1203.

Delhomme, J. P., 1974, La cartographie d’une grandeur physique á partir de données de différentes qualités: Proc. Meet. Int. Assoc. Hydrogeol., Mem. 10, 1, p. 185-194.

Delhomme, J. P., 1976, Application de la théorie des variables régionalisées dans les sciences de l’eau: Ph.d. thesis, Ecole des Mines de Paris, Fontainebleau, France.

Delhomme, J. P., 1978, Kriging in hydrosciences: Advances in Water Resources, v. 1, no. 5, p. 251-266.

de Marsily, G., 1986, Quantitative hydrogeology: San Diego, Academic Press, Inc., 440 p.


65

de Marsily, Ghislain, and Ahmed, Shakeel, 1987, Application of kriging techniques in groundwater hydrology: Ramesam, V., ed., Trends in groundwater research, Journal of the Geological Society of India, v. 29, no. 1, p. 57-82.

de Smedt, F., Belhadi, M., and Nurul, I. K., 1985, Study of spatial variability of the hydraulic conductvity with different measurement techniques: in Proceedings of the International Symposium on the Stochastic Approach to Subsurface Flow: de Marsily, G., ed., International Association for Hydraulic Research, Montvillargenne, France, June 3-6.

Driscoll, F. G., 1986, Groundwater and wells: second edition, U.S. Filter/Johnson Screens, St, Paul, Minnesota, 1089 p.

Dutton, A. R., Mace, R. E., Nance, H. S., and Blüm, Martina, 1996, Geologic hydrogeologic framework of regional aquifers in the Twin Mountains, Paluxy, and Woodbine Formations near the SSC site, North-Central Texas: The University of Texas at Austin, Bureau of Economic Geology, topical report prepared for Texas National Research Laboratory Commission under contract no. IAC(92-93)-0301.

Eagon, H. B., Jr., and Johe, D. E., 1972, Practical solutions for pumping tests in carbonate-rock aquifers: Ground Water, v. 10, no. 4, p. 6-13.

El-Naqa, A., 1994, Estimation of transmissivity from specific capacity data in fractured carbonate rock aquifer, central Jordan: Environmental Geology, v. 23, no. 1, p. 73-80.

Fabbri, Paolo, 1997, Transmissivity in the geothermal Euganean Basin; a geostatistical analysis: Ground Water, v. 35, no. 5, p. 881-887.

Ferris, J. G., Knowles, R. H., Browne, R. H., and Stallman, R. W., 1962, Theory of aquifer tests: U.S. Geological Survey Water Supply Paper 1536-E.

Freeze, R. A., and Cherry, J. A., 1979, Groundwater: Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 604 p.

Gabrysch, R. K., 1968, The relationship between specific capacity and aquifer transmissibility in the Houston area, Texas: Ground Water, v. 6 no. 4, p. 9-14.

Gelbaum, Carol, 1981, Using the specific capacity index to analyze ground-water productivity in the Gulf Trough area of Georgia: Fifty-eighth annual meeting of the Georgia Academy of Science, Georgia Journal of Science, v. 39, no. 2, p. 69-70.

Gringarten, A. C., and Witherspoon, P. A., 1972, A method of analyzing pump test data from fractured rock aquifers: Proceedings of a Symposium on Percolation Through Fissured Rock, International Association of Rock Mechanics, Stuttgart, Germany, p. T3-B1-T3-B8.

Halihan, Todd, Mace, R. E., and Sharp, J. M., Jr., 1997, Using laboratory-scale permeability to interpret well-test data in a fractured karst aquifer: (abs.), Geological Society of America, Abstracts with Programs, v. 29, no. 7, p. 227.

Hantush, M. S., 1956, Analysis of data from pumping tests in leaky aquifers: Tranactions of the American Geophysical Union, v. 37, p. 702-714.

Hantush, M. S., 1964, Hydraulics of wells: Adv. Hydrosci., v. 1, p. 281-432. Harrill, J. R., 1971, Determining transmissivity from water-level recovery of a step-

drawdown test: U.S. Geological Survey Professional Paper 700-C, p. C212-213. Heeley, R. W., Mabee, S. B., 1983, A cost-effective technique for reconnaissance

evaluation of aquifers: Nielsen, D. M., ed., Proceedings of the Third national symposium on aquifer restoration and ground-water monitoring, Proceedings of


66

the National Symposium on Aquifer Restoration and Ground-Water Monitoring, v. 3, p. 213-219.

Hovorka, S. D., Mace, R. E., and Collins, E. W., 1995, Regional distribution of permeability in the Edwards Aquifer: Bureau of Economic Geology, The University of Texas at Austin, Final contract report to the Edwards Underground Water District under contract no. 93-17-FO, 126 p.

Hovorka, S. D., Mace, R. E., and Collins, E. W., 1998, Permeability structure of the Edwards aquifer, south Texas- Implications for aquifer management: Bureau of Economic Geology, The University of Texas at Austin, Report of Investigations No. 250, 55 p.

Hughson, L. , Huntley, D., and Razack, M., 1996, Cokriging limited transmissivity data using widely sampled specific capacity from pump tests in an alluvial aquifer: Ground Water, v. 1, no. 1, p. 12-18.

Huntley, David, Nommensen, Roger, and Steffey, Duane, 1992, The use of specific capacity to assess transmissivity in fractured-rock aquifers: Ground Water, v. 30 no. 3, p. 396-402.

Hurr, R. T., 1966, A new approach for estimating transmissivity from specific capacity: Water Resources Research, v. 2, no. 4, p. 657-664.

Hurr, R. T., 1967, Reply to discussion of 'A new approach for estimating transmissibility from specific capacity' (1966) by T. N. Narasimhan: Water Resources Research, v. 3 no. 4, p. 1095.

Jacob, C. E., 1944, Notes on determining permeability by pumping tests under water-table conditions: U.S. Geological Survey, mimeographed report.

Jacob, C. E., 1947, Drawdown test to determine effective radius of artesian well: Transactions of the American Society of Civil Engineers, v. 112, p. 1047-1070.

Jacob, C. E., 1950, Flow of groundwater, Engineering hydraulics: ed. H. Rouse, New York, John Wiley, p. 321-386.

Jacob, C. E., and Lohman, S. W., 1952, Nonsteady flow to a well of constant drawdown in an extensive aquifer: American Geophysical Union Transactions, v. 33, no. 4, 559-569.

Jensen, J. L., Lake, L. W., Corbett, P. W. M., and Goggin, D. J., 1997, Statistics for petroleum engineers and geoscientists: Upper Saddle River, New Jersey, Pentice Hall PTR, 390 p.

Johnson, A. I., Moston, R. P., and Versaw, S. F., 1966, Laboratory study of aquifer properties and well design for an artificial recharge site: U.S. Geological Sureby Water Supply Paper, no. 1615-H, p. H23-H25.

Journel, A. G., and Huijbregts, J. C., 1978, Mining geostatistics: Academic, Orlando, Florida.

Kasenow, M., and Pare, P., 1995, Using specific capacity to estimate transmissivity- field and computer methods: Highlands Ranch, Colorado, Water Resources Publication, variously paginated.

Knopman, D. S., 1990, Factors related to the water-yielding potential of rocks in the Piedmont and Valley and Ridge provinces of Pennsylvania: U. S. Geological Survey Water-Resources Investigations, WRI 90-4174, 52 p.

Knopman, D. S., and Hollyday, E. F., 1993, Variation in specific capacity in fractured rocks, Pennsylvania, Ground Water, v. 31 no. 1, p. 135-145.


67

Kruseman, G. P., and de Ridder, N. A., 1990, Analysis and evaluation of pumping test data, second edition: International Institute for Land Reclamation and Improvement, Publication 47, Wageningen, The Nederlands, 377 p.

LaRiccia, M. P., Rauch, H. W., 1977, Water well productivity related to photo-lineaments in carbonates of Frederick Valley, Maryland: Dilamarter, R. R., and Csallany, S. C., eds., Hydrologic problems in karst regions, p. 228-234.

Lennox, D. H., 1966, Analysis and application of step-drawdown test: ASCE, v. 92. Lohman, S. W., 1972, Ground-water hydraulics: U.S. Geological Survey Pofessional

Paper 708, 70 p. Mace, R. E., 1995, Geostatistical description of hydraulic properties in karst aquifers,

a case study in the Edwards Aquifer: in Groundwater Management, Proceedings of the International Symposium sponsored by the Water Resources Engineering Division, American Society of Civil Engineers, ed., Charbeneau, R. J., American Society of Civil Engineers, p. 193-198.

Mace, R. E., 1996, Distribution of specific capacity and transmissivity in the Edwards aquifer, in Woodruff, Jr., C. M., and Sherrod, C. L, Urban Karst, Geologic Excursions in Travis and Williamson Counties: Austin Geological Society Guidebook 16, p. 11-15.

Mace, R. E., 1997, Determination of transmissivity from specific capacity tests in a karst aquifer, Ground Water, v. 35 no. 5, p. 738-742.

Mace, R. E., in review, Regional distribution and spatial correlation of transmissivity in a karstic aquifer, Edwards Group, South-Central Texas: submitted to Ground Water.

Mace, R. E., Nance, H. S., and Dutton, A. R., 1994, Geologic and hydrogeologic framework of regional aquifers in the Twin Mountains, Paluxy, and Woodbine Formations near the SSC site, North-Central Texas: Draft topical report submitted to the TNRLC under contract no. IAC(92-93)-0301 and no. IAC 94-0108, 48 p.

Mace, R. E., Smyth, R., Xu, Liying, and Liang, Jinhuo, 2000, Hydraulic conductivity and storativity of the Carrizo-Wilcox aquifer in Texas: Bureau of Economic Geology, The University of Texas at Austin, submitted to the Texas Water Development Board.

Macpherson, G. L., 1983, Regional trends in transmissivity and hydraulic conductivity, Lower Cretaceous sands, north-central Texas: Ground Water, v. 21, no. 5, p. 577-583.

Matheron, G., 1971, The theory of regionalized variables and its applications: Cahiers du Centre de Morphologie Mathématique, v. 5, Ecole des Mines de Paris, Fontainebleau, France.

McWhorter, D. B., and Sunada, D. K., 1977, Ground-water hydrology and hydraulics: Fort Collins, Colorado, Water Resources Publications, 290 p.

Meier, P. M., Carrera, Jesus, and Sanchez-Vila, Xavier, 1999, A numerical study on the relationship between transmissivity and specific capacity in heterogeneous aquifers: Ground Water, v. 37, no. 4, p. 611-617.

Mendenhall, W., 1987, Introduction to probability and statistics: Duxbury Press, Boston, Massachesetts, 783 p.

Meyer, R. R., 1963, A chart relating well diameter, specific capacity, and the cofficient of transmissibility and storage: in Bentall, R., (ed.), Methods of


68

determining permeability, transmissibility and drawdown: U.S. Geological Survey Water-Supply Paper 1536-I, p. 338-340.

Muskat, M., 1946, The flow of homogeneous fluids through porous media: Ann Arbor, Michigan, J. W. Edwards.

Narasimhan, T. N., 1967, Note on "A new approach for estimating transmissibility from specific capacity' by R. Theodore Hurr: Water Resources Research, v. 3 no. 4, p. 1093-1095.

Neuman, S. P. , 1972, Theory of flow in unconfined aquifers considering delayed response of the watertable: Water Resources Research, v. 8, p. 1031-1045.

Neuman, S. P., 1984, Role of geostatistics in subsurface hydrology: in Geostatistics for natural resource characterization, part 2G, Verly, G., ed., p. 787-816.

Ogden, L., 1965, Estimating transmissibility with one drawdown: Ground Water, v. 3, no. 3, p. 51-55.

Poland, J. F., 1959, Hydrology of the Long Beach-Santa Ana area, California, with special reference to the watertightness of the Newport-Inglewood structural zone: U. S. Geological Survey Water-Supply Paper, W 1471, 257 p.

Prudic, D. E., 1991, Estimates of hydraulic conductivity from aquifer-test analyses and specific-capacity data, Gulf Coast regional aquifer systems, south-central United States: Water-Resources Investigations - U. S. Geological Survey, WRI 90-4121, 38 p.

Rao, G. N., Beck, J. N., Murray, H. E., and Nyman, D. J., 1991, Estimating transmissivity and hydraulic conductivity of Chicot Aquifer from specific capacity data: Water Resources Bulletin, v. 27, no. 1, p. 47-58.

Razack, M., and Huntley, David, 1991, Assessing transmissivity from specific capacity in a large and heterogeneous alluvial aquifer: Ground Water, v. 29, no. 6, p. 856-861.

Rorabaugh, M. I., 1953, Graphical and theoretical analysis of step-drawdown test of artesian well: Proceedings of the American Society of Civil Engineers, v. 79, separte no. 362, 23 p.

Rose, H. G., and Smith, H. F., 1957, Particles and permeability; a method of determining permeability and specific capacity from effective grain size: Water Well Journal, v. 11, no. 3, p. 10.

Sheahan, N. T., 1971, Type-curve solution of step-drawdown test: Ground Water, v. 9, no. 1, p. 25-29.

Siddiqui, S. H., and Parizek, R. R., 1971, Hydrogeologic factors influencing well yields in folded carbonate rocks in central Pennsylvania: Water Resources Research, v. 7, no. 5, p. 1295-1312.

Slichter, S. C., 1906, The underflow in Arkansas valley in western Kansas: U.S. Geological Survey Water Supply Paper no. 153, 61 p.

Sternberg, Y. M., 1973, Efficiency of partially penetrating wells: Ground Water, v. 11, no. 3, p. 5-7.

Suvagondha, F., and Singharajwarapan, S., 1987, Estimating the hydraulic conductivity from specific capacity data of the eastern part of Chiang Mai Basin: Awadalla, S., and Noor, I. M., eds., Groundwater and the Environment; proceedings of the International Groundwater Conference, p. C39-C46.


69

Theis, C. V., 1935, The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage: Transactions of the American Geophysical Union, v. 16, p. 519-524.

Theis, C. V., 1963, Estimating the transmissivity of a water-table aquifer from the specific capacity of a well: U.S. Geological Survey Water Supply Paper 1536-I, p. 332-336.

Theis, C. V., Brown, R. H., and Myers, R. R., 1963, Estimating the transmissibility of aquifers from the specific capacity of wells. Methods of determining permeability, transmissivity, and drawdown: U.S. Geological Survey Water Supply Papers, 1536-I.

Thomasson, H. J., Olmstead, F. H., LeRoux, E. R., 1960, Geology, water resources, and usable ground water storage capacity of part of Solano County, CA: U.S. Geological Survey Water Supply Paper 1464, 693 p.

Todd, D. K., 1959, Ground water hydrology: New York, Wiley, 336 p. Turcan, A. N., Jr., 1963, Estimating the specific capacity of a well: U.S. Geological

Survey Propfessional Paper 450-E. Viswanathiah, M. N., and Sastri, J. C. V., 1978, Specific capacity of wells in some

hard rocks of Karnataka: Journal of the Geological Society of India, v. 19 no. 9, p. 426-430.

White, W. B., 1988, Geomorphology and hydrology of Karst terrains: New York, New York, Oxford University Press, 464 p.

Wallroth, Thomas, and Rosenbaum, M. S., 1996, Estimating the spatial variability of specific capacity from a Swedish regional database: Middleton, M. F., ed., First Nordic symposium on Petrophysics, Marine and Petroleum Geology, v. 13, no. 4, p. 457-461.

Walton, W. C., 1962, Selected analytical methods for well and aquifer evaluation: Illinois State Water Survey Bulletin 49, 81 p.

Walton, W. C., 1970, Groundwater resource evaluation: New York, McGraw-Hill. Walton, W. C., and Neill, J. C., 1963, Statistical analysis of specific-capacity data for

a dolomite aquifer: Journal of Geophysical Research, v. 68, no. 8, p. 2251-2262. Westly, R. L., 1993, Using specific capacity testing to evaluate aquifer producing

zones during borehole advancement: The Professional Geologist, v. 30, no. 9, p. 5-6.

White, F. M., 1994, Fluid Mechanics, Third Edition: McGraw-Hill, Inc., New York, 736 p.

Winter, T. C., 1981, Uncertainties in estimating the water balance of lakes: Water Resources Bulletin, v. 17, no. 1, p. 82-115.

Wynne, D. B., 1992, Specific capacity and slug testing; an overview and empirical comparison of their uses in preliminarily estimating hydraulic conductivity: Stanley, Anita (ed.), Proceedings of the Sixth national outdoor conference on Aquifer restoration, ground water monitoring, geophysical methods; a conference and exposition, Ground Water Management, v. 11, p. 217-230, 1992.

Yin, Z.-Y., and Brook, G. A., 1992, The topographic appraoch to locating high-yield wells in crystalline rocks- Does it work? Ground Water, v. 30, no. 1, p. 96-102.

Zeizel, A. J., 1963, Hydrogeologic factors influencing well yields in the Silurian dolomite aquifer, northeastern Illinois: Geological Society of America Special Paper, 267 p.


70

Zeizel, A. J., Walton, W. C., Prickett, T. A., and Sasman, R. T., 1962, Ground-water resources of DuPage County, Illinois: Illinois State Water Survey and Geological Survey, Coop. Ground-Water Report No.2.


71

Appendix A: Estimating Transmissivity from Specific Capacity Data using the Theis and

others (1963) Equation Inside a Spreadsheet.

Theis and others (1963) developed an equation for estimating transmissivity, T ,

from specific capacity, cS ,

=

SrTt

TS

w

pc

2

25.2ln

4π (A.1)

where pt is the time of pumping, wr is the well radius, and S is the storativity. However,

transmissivity cannot be directly solved and instead must be solved either using tables or

iteratively. The iterative solution can be done manually, using a computer program

written to solve the equation, or within a spreadsheet. This appendix describes a

technique for solving the above equation using a spreadsheet.

Solving equation A.1 in a spreadsheet takes advantage of a spreadsheet’s ability to

iterate over circular calculations. The first step is to solve equation A.1 for the

transmissivity term in the numerator:

=

SrTtS

Tw

pc2

25.2ln

4π. (A.2)

The idea here is to solve for T in the left-hand side of equation A.2. This can be done

manually (hopefully with a calculator) by substituting the appropriate values for cS , pt ,

wr , and S and an initial guess for T (the value for cS is generally a good start) on the

right-hand side of equation A.2. The resulting T from this calculation is then substituted

back into the right-hand side of the equation. This operation is repeated until the


72

difference between the T substituted into the right-hand side of the equation and the

calculated T is small.

The iterations described above can be performed automatically in a spreadsheet

with an iteration option. First, in an open spreadsheet, activate the iteration function and

set the number of iterations and maximum change (convergence criteria). This is done in

Microsoft Excel 97 for a PC under Tools-Options-Calculation. Check the iteration box,

set the number of iterations to 1,000, and the maximum change to 0.001 for units of

meters and days (this maximum change may be different for different units). Second, four

columns with the input data for cS , pt , wr , and S must be assigned (figure A.1). Third,

two columns for the calculations must be made. In the first of these columns (column E,

figure A.1), the specific capacity should be entered again as a value (not a cell reference

to the input column). This value is the initial guess for the solution of equation A.2. Then,

equation A.2 should be coded into the second of the calculation columns (column F,

figure A.1), with cell references to the input data and to the initial guess in the previous

column (column E, figure A.1). For Microsoft Excel, the equation to calculate

transmissivity for the above example for line 4 should be

=(A4/(4*PI()))*(ln((2.25*E4*B4)/(C4*C4*D4))) (A.3)

Now, overwrite the initial guess in column E with a cell reference to the result in column

4:

=F4 (A.4) The two calculation cells should flicker with calculation until the values in both cells will

be the same with the solution. Remember to use consistent units.

A B C D E F 1


73

2 Sc tp rw S To T 3 4 100 2 0.2 0.00001 170.07 170.07 5 6

Figure A.1. Example spreadsheet to calculate the Theis and others (1963) equation.


74


75

Figures:

Time of pumping0

pump turned on at rate Q

swSc = Q

sw

Specific Capacity:

- measurement point

Figure 1. Data from a pumping test on a well completed in a confined aquifer. Before time 0, the water level in the well is at equilibrium. After the pump is turned on, water levels decline as the aquifer responds. The difference between the pumping level and the equilibrium level is the drawdown. With increasing time, the change in water level decreases. Specific capacity is determined from one point during the test.


76

Time0 Figure 2. Plot showing the decrease in specific capacity with increasing time of

pumping (after Jacob and Lohman, 1952).


77

10410210010-210-410-610-810-1010-1210-3

10-2

10-1

100

101

102

103

104

105

106

107

Stabilization time, ts (days)

10 410310 210110 010 -1.01

.1

1

10

1/u

S=

ts = r 2S4uT

with: u = 10 r = 0.15 m

-510

-410

-310

-210

-110

Theis curve

-4

Figure 3. Time for water levels to stabilize after the initiation of pumping as a

function of transmissivity and storativity. Stabilization times, st , are shown for u = 10-4 (based on a late portion of the Theis curve) and wr = 0.15 m for different storativity values. The relationship describing the lines is shown in the upper right hand corner and was derived from the equations of the Theis (1935) non-equilibrium method for analyzing aquifer tests.


78

6420-2-4-6-8-6

-4

-2

0

2

4

6

Log transmissivity, T (m2 d-1)

′C = trw

2S

′C = 10-1

100

101

102

103

104

105

106

Figure 4. Sensitivity of specific capacity to variation in transmissivity and a

parameter, C ′ , that incorporates the effects of pumping time, well radius, and storativity.


79

6420-2-4-6-8-8

-6

-4

-2

0

2

4

6

Log specific capacity, Sc (m2 d-1)

Range for alluvial aquifers

Range for fractured hard-rock aquifers

Figure 5. Relationship between transmissivity and specific capacity for alluvial and

fractued hard-rock aquifers using the Thomasson and others (1960) approach.


80

Figure 6. Example of an empirical relationship between transmissivity and specific

capacity for a karstic aquifer in Texas (Mace, 1997) showing (a) the best-fit line (solid) and the 95-percent prediction intervals (dashed) for the linear fit and (b) the best-fit lines for second and third order polynomials.


81

432100

1

2

3

4

T = 3.24 Sc( )0.81

R2 = 0.80n = 48

Relationship from Mace (1997)using data from Eagon and Johe (1970)carbonate aquifer in northwestern Ohio

Razack and Huntley (1991)heterogeneous alluvial aquifer

T = 15.3 Sc( )0.67

R2 = 0.63n = 215

Huntley and others (1992)fractured batholith aquifer

n = 129

T = 0.12 Sc( )1.18

R2 = 0.89

El-Naqa (1994)fractured carbonate aquifer

n = 237R2 = 0.95

T = 1.81 Sc( )0.917

3210-1-3

-2

-1

0

1

2

3

Log specific capacity (m2 d-1) Log specific capacity (m2 d-1)


43210

1

2

3

4

(a) (b)

(c) (d)

5432100

1

2

3

4

5


82

6543210-1-1

0

1

2

3

4

5

6

543211

2

3

4

5

Mace (1997)carbonate aquifer, Edwards

n = 71R2 = 0.89

T = 0.76 Sc( )1.08

Mace (1997)carbonate aquifer, Floridan

n = 14R2 = 0.80

T = 1.23 Sc( )1.05

Fabbri (1997)fractured carbonate aquifer

n = 45R2 = 0.95

T = 0.785 Sc( )1.07

this papersandstone aquifer, Trinity

n = 147R2 = 0.55

32100

1

2

3

T = 2.75 Sc( )0.82

43210

1

2

3

4

5


Log specific capacity (m2 d-1)Log specific capacity (m2 d-1)

(e) (f)

(g) (h)


83

2.52.01.51.00.50.0

0.5

1.0

1.5

2.0

2.5

3.0

2.52.01.51.01.0

1.5

2.0

2.5

this papersandstone aquifer, Paluxy

n = 28R2 = 0.57T = 3.16 Sc( )0.79

this papersandstone aquifer, Woodbine

n = 33R2 = 0.45T = 1.51 Sc( )0.91

Log specific capacity (m2 d-1)

3210-1-2

-1

0

1

2

3


this paperlimestone aquifer, Edwards-Trinity

n = 46R2 = 0.82T = 0.78 Sc( )0.98

3210-1

0

1

2

3

4



this papersandstone aquifer, Edwards-Trinity

n = 21R2 = 0.75T = 1.07 Sc( )1.01

(i) (j)

(k) (l)


84

Figure 7. Empirical relationships between transmissivity and specific capacity by (a)

Eagon and Johe (1972) for a carbonate aquifer (using specific capacity corrected for turbulent well loss), (b) Razack and Huntley (1991) for a heterogeneous alluvial aquifer, (c) Huntley and others (1992) for a fractured hard rock aquifer, (d) El-Naqa (1994) for a fractured carbonate aquifer, (e) Mace (1997) for a karstic aquifer in Texas, (f) Mace (1997, this paper) for a karstic aquifer in Florida, (g) Fabbri (1997) for a fractured carbonate aquifer, (h, i, j) this paper for sandstone aquifers, (k) this paper for a limestone aquifer, (l) this paper for a sandstone aquifer, and (m) Mace and others (2000) for a sandstone aquifer.


85

100000100001000100101.1

1

10

100

1000

10000

100000

1000000

Specific capacity (m2 d-1)

1

2

3

12

4

6

5

7

11

13

8

9

10

Figure 8. Comparison among the different empirical relationships and between the

empirical relationships and the analytical appraoches including (1) Eagon and Johe (1972) for a carbonate aquifer, (2) Razack and Huntley (1991) for a heterogeneous alluvial aquifer, (3) Huntley and others (1992) for a fractured hard rock aquifer, (4) El-Naqa (1994) for a fractured carbonate aquifer, (5) Mace (1997) for a karstic aquifer in Texas, (6) Mace (1997, this paper) for a karstic aquifer in Florida, (7) Fabbri (1997) for a fractured carbonate aquifer, (8) this paper for a sandstone aquifer in north-central Texas (Trinity aquifer), (9) this paper for a sandstone aquifer in north-central Texas (Paluxy aquifer), (10) this paper for a sandstone aquifer in north-central Texas (Woodbine aquifer), (11) the Thomasson and others (1960) approach for alluvium, (12) the Thomasson and others (1960) approach for fractured hard rocks, and (13) the Theis and others (1963) approach for an assumed value of C ′ of 105 d m-


86

2. For the empirical relationships (lines 1-7), the length of the line corresponds to the applicable range of the relationship and the prediction intervals are about an order of magnitude. Note that line 7 is coincident with line 5.


87

10-1 10 610 510 410 310 210 110 0

0.001

0.01

0.1

1

10

Specific capacity (m2 d-1)

0.0001

0.00001

10000

100000

1000

100

10

1

Figure 9: Relationship between measured specific capacity and pumping rate in the Edwards aquifer. In general, areas of the aquifer with higher specific capacity are tested with larger pumping rates.


88

Figure 10: Effect of well loss on specific capacity for dimensionless well loss where 1−=Π CQB and

cSR is specific capacity corrected for well loss divided by the measured specific capacity.


89

Q

B

C

110010009008007006005000.0017

Q (gpm)

swQ

= 1.28 ∞10−6( )Q + 1.21 ∞10−3

swQ

1Sc

=swQ

= CQ + B

0.03 0.04 0.05 0.06 0.07Q (m3 s-1)

0.0018

0.0019

0.0020

0.0021

0.0022

0.0023

0.0024

0.0025

0.0026

0.0027

0.0028

for sw in m and Q in m3 d-1

120010008006004002000Q (gpm)

n = 2.2

2.1

2.0

1.91.8

0.060 0.050.040.030.020.01 0.07Q (m3 s-1)

0.001

0.008

0.007

0.006

0.005

0.004

0.003

0.002

Figure 11: (a) Determining B and C from step-drawdown tests, (b) an example of estimating B and C from a step-drawdown test, and (c) sensitivity to n for a constant B and C.


90

Figure 12: Plot to estimate well loss from discharge rate (after Rao and others, 1991).


91

Edwards aquiferKarst aquifer, Ohio(Eagon and Johe, 1972)

Specific capacity, Sc (m2d-1)

Best-fit line

10-4

10-5

10-6

10-7

10-8

10-9

102 103 104 105 106 107

QAb3554c

Figure 13: Relationship between specific capacity and well-loss constant showing the best-fit line (solid) and the 95-percent prediction intervals (dashed). Data points are from the Edwards aquifer (Mace, 1997) and a karst aquifer in northwestern Ohio (Eagon and Johe, 1972).


92

Figure 14: Comparison between measured well loss (determined from step-drawdown tests) and well loss estimated from (1) an empirical relationship between specific capacity and well-loss constant and (2) pipe-flow theory.


93

0

20

40

60

80

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Production rate (gpm)

10000

Figure 15: Production rates for specific-capacity tests in the Edwards aquifer where there was no measurable drawdown. Multiply gpm by 0.00068 to get m3 s-1.


94

10 710 610 510 410 310 210 110 010 -110 -2.0001

.001

.01

.1

1

10

100

1000

10000

100000

Transmissivity (ft2/d)

Figure 16: The minimum production rate required to produce at least a foot of drawdown for a given transmissivity assuming a production time of 8 hrs, a well radius of 4 in, and a storativity of 10-4.


95

Figure 17: Empirical relationship between transmissivity and specific capacity for the Carrizo-Wilcox aquifer of east-central Texas.


96

Figure 18: Comparison on the errors between using the empirical approach and the analytical appraoch to estimating transmissivity from specific capacity in the Carrizo-Wilcox aquifer of east-central Texas.

specific capacity well

Documents

estimating transmissivity

transmissivity of aquifers

specific capacity

introduction transmissivity

specificcapacity data1

geostatistical techniques

substantial data

data pairs