EPA/600/R-93/202
February 1994
A REVIEW AND ASSESSMENT OF FACTORS AFFECTING HYDRAULIC
CONDUCTIVITY VALUES DETERMINED FROM SLUG TESTS
by
Gordon K. Binkhorst and Gary A. Robbins
Department of Geology and Geophysics
University of Connecticut
Storrs, CT 06269
Cooperative Agreement
CR-817587-01-1 and CR-817587-02-0
Project Officer
Katrina Varner
Advanced Monitoring Systems Division
Environmental Monitoring Systems Laboratory
Office of Research and Development
Las Vegas, Nevada 89193
ENVIRONMENTAL MONITORING SYSTEMS LABORATORY-LAS VEGAS
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
LAS VEGAS, NEVADA 89193-3478
9311-0102amd93
^g/y Printed on Recycled Paper
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NOTICE
The information in this document has been funded by the United States Environmental
Protection Agency under cooperative agreement CR-817587 to the University of Connecticut.
It has been subjected to the Agency's peer and administrative review, and it has been
approved for publication as an EPA document. Mention of trade names or commercial
products does not constitute endorsement or recommendation for use.
11
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PREFACE
This review is part of a cooperative agreement project between the U.S. Environmental
Protection Agency, Environmental Monitoring Systems Laboratory, Las Vegas and the
University of Connecticut. The overall goal of the project is to improve site assessments for
gasoline contamination at service station facilities. This review provides an evaluation of
current practices for determining hydraulic conductivity using slug tests in typical monitoring
wells. To further evaluate and improve current practices, field and laboratory testing have
also been conducted as part of this research. Results of these tests may at this time be found
in Binkhorst, O.K. 1992. Evaluation of Slug Test Methods to Determine Hydraulic
Conductivity in Unconfined Aquifers, Unpublished M.S. Thesis, Department of Geology and
Geophysics, University of Connecticut.
111
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ABSTRACT
The purpose of this study was to evaluate factors that influence hydraulic conductivity
values determined by slug tests in typical monitoring wells. Procedures for conducting tests
and methods for analyzing test results are summarized and compared. Sensitivity analyses
were conducted to identify and evaluate the factors influencing the calculated hydraulic
conductivity values for available slug test solutions. These factors include mathematical
approximation of the shape of the screen intake, the presence of a less or more permeable
well skin, determination of the radius of influence, formation anisotropies and heterogeneities,
partial penetration and boundary conditions, and filter sand pack drainage from wells with
screens that span the water table (partially submerged well screens). Of these factors, the
most significant are the presence of a less permeable well skin, and filter sand pack drainage.
The presence of a less permeable well skin around the well screen can result in orders of
magnitude under-estimation of the formation conductivity. The effect of a less permeable
well skin may be reduced by designing wells to minimize entrapment of fines in the filter
pack, and by vigorously developing wells to remove fines associated with well installation.
Conducting slug tests in wells with partially submerged screens causes rapid filter sand
drainage and extreme reduction of the initial head drop. The initial head reduction can reduce
the radius of influence of the test, thereby accentuating well skin effects. Drainage of the
filter sand necessitates accounting for the filter sand recharge in the analysis of slug tests.
This can be accomplished by using early time slug test data to estimate the specific yield of
the filter sand. A method is presented to calculate the filter sand's specific yield from
hydraulic head versus time curves for incorporation into existing slug test solutions.
However, more reliable hydraulic conductivity values may be obtained from wells with screen
sections that remain below the water table during the slug test. This avoids the complexities
incurred by filter sand drainage.
IV
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CONTENTS
Preface , . iii
Abstract iv
Figures . . . . . ......... ... ..... ... .V. . . . . . . . vi
List of Symbols and Abbreviations ......; viii
1. Introduction . 1
2. Analysis of Slug TestData . . . .5
Slug Test Interpretation in Unconfined Aquifers . . . 5
Slug Test Interpretation in Confined Aquifers . . M. .......... 14
3. Theoretical Assessments of Factors Affecting Slug Tests 21
Shape Factors for the 1/2 and Full Ellipse, and
Doughnut Models 21
Radius of Influence . 23
Well Skin Effects 34
Anisotropy and Heterogeneities 42
Partial Penetration and Boundary Effects 46
Filter Pack Drainage and Resaturation in Partially
Submerged Well Screens 48
Reduction of Screen Intake Length in Partially
Submerged Screen Wells . 57
4. Summary, Conclusions and Recommendations 60
References 63
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FIGURES
Number
1 Typical Groundwater Monitoring Well Geometry ; 3
2 Slug Test Geometries for Doughnut, 1/2 Ellipse, and Full Ellipse
Models 7
3 Hydraulic Head Representation Following a Slug Out Test 10
4 Typical Slug Test Recovery Curve for a Monitoring Well with a
Fully Submerged Screen 12
5 Hvorslev Basic Time Lag Method 13
6 Comparison of Dimensionless Discharge Curves for Different
Slug Test Models (after Dagan, 1978) 15
7 Confined Aquifer Type Curves (after Papadopulos et al., 1973) 18
8 Variation in Dimensionless Discharge for Different Degrees of
Partial Penetration (after Widdowson et al., 1990) 20
9 Comparison of Analytical Shape Factors for the 1/2 and Full
Ellipses Models 22
10 Comparison of Analytical, Electric Analog and Numeric Shape
Factors for Cylindrical Intakes (after Chapuis, 1989) 24
11 Impact of the Radius of Influence and Screen Length on the
Doughnut Model Shape Factor : 26
12 Well and Aquifer Geometry Symbols for the Bouwer and Rice
(1976) Model 28
13 Dimensionless Parameters A, B and C for the Bouwer and
Rice (1976) Model 29
14 Impact of Screen Length and Screen Radius on the Radius of
Influence (Bouwer and Rice Method) 30
15 Comparison of Shape Factors for the Doughnut Model using Bouwer and Rice
(1976) Radii of Influence, and for the Full Ellipse Model 32
16 Recovery Curves for Varying Skin Hydraulic Conductivities (after Faust
and Mercer, 1984) 36
17 Comparison of Cylindrical Intake Dimensionless Discharge Terms
(after Braester and Thunvik, 1984) 37
18 Apparent Formation Permeability for Different Well Skin Permeabilities and
Thicknesses (data of Braester and Thunvik, 1984) 38
19 Effects of Borehole Skin on Recovery Curves for a Fully Penetrating Well in
a Confined Aquifer (after McElwee et al., 1990) 39
VI
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FIGURES (Continued)
Number Page
20 Plan View of a Well Identifying Parameters used to Derive
Effective Hydraulic Conductivity Equation 41
21 Effects of Well Skin and Radius of Influence on the Effective
Hydraulic Conductivity . . . 43
22 Influence of Anisotropy on the Full Ellipse Shape Factor 44
23 Comparison of the Isotropic Hydraulic Conductivity of Dagan (1978) with the
Anisotropic Conductivity of Braester and Thunvik (1984) 47
24 Effects of Partial Penetration on hXR/RJ Term Using the Method of Bouwer
and Rice (1976) 49
25 Slug Test Recovery Curve from a Typical Monitoring Well with a Partially
Submerged Well Screen . 51
26 Illustration of Parameters Needed to Calculate the Initial Head Drop 53
27 Determination of the Transition Point (H,) 55
28 Illustration of Parameters Used to Estimate Specific Yield of the Filter
Sand Pack 56
29 Changes in the Full Ellipse Model Shape Factor Caused by Screen
Length Reduction ; 58
vu
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LIST OF SYMBOLS AND ABBREVIATIONS
A Cross-sectional area
As Area of a spherical intake
Aw Area of a well casing
b Aquifer thickness
BI Length of bailer
B.H. Outside radius of bailer
D Diameter
D* Distance from lower confining
unit to bottom of screen (after
Bouwer and Rice, 1916)
Ds Screen Diameter
H Hydraulic head
Hj Length of desaturated filter
sand column
H(t) Head drop at any time, t
Hh Head at the borehole
HI Head at the radius of influence
H0 Initial head drop
HO,. Calculated initial head drop
Hj Head at the well screen
dH/dl Hydraulic gradient
H/HQ Normalized head
k Formation permeability
K Hydraulic conductivity
KCIT Effective hydraulic conductivity
Kf Formation hydraulic conductivity
Kh Horizontal hydraulic conductivity
K, Well skin hydraulic conductivity
kg Well skin permeability
Ky Vertical hydraulic conductivity
Kxtyf Hydraulic conductivity in given
Cartesian coordinate
L Length
Ls Screen length
Lw Distance from water table to
bottom of screen (after Bouwer
and Rice, 1976)
m Slope
ma Anisotropy factor (after
Hvorslev, 1951)
n Porosity
P Dimensionless discharge (after
Widdowson et al., 1990)
Q Discharge
Q' Dimensionless discharge (after
Dagan, 1978)
Qf Flow through formation
Qh Flow borehole formation
Qs Flow through well screen
r Radial distance
Rc Well casing radius
Rce Effective well casing radius
Rci Inside radius of well casing
Rco Outside radius of well casing
Rh Borehole radius
RI Radius of influence
RS Well screen radius
S Aquifer storativity
Sf Shape factor
Ss Specific storage
Sy Specific yield
t Time
T Transmissivity
TL Basic Time Lag
V Volume
Vb Volume of bailer
Vc Volume in well casing
Vs Volume of drained sand pack
Vw Volume of water released from
sand pack
vm
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SECTION 1
INTRODUCTION
The hydraulic conductivity is a measure of a formation's ability to transmit water.
The hydraulic conductivity is one of the most important hydrogeologic parameters to be
determined at a contamination site. TypicaL values can vary from 102 cm/sec for gravel to
10"10 cm/sec for silts and glacial tills (Freeze and Cherry, 1979). Estimates of groundwater
flow velocities, well yields, and rates of contaminant migration will be directly proportional to
the value of the hydraulic conductivity and associated errors in determining this value.
Accurate determination of hydraulic conductivity and its spatial distribution are essential for
modeling both adjective and dispersive contaminant migration. An order of magnitude error
in the conductivity would result in an equal magnitude of error in predicting the position of
an adjective front of a contaminant plume. A distribution of varying conductivities at a site
may create preferential pathways and result in large scale dispersion of contaminated
groundwater (Molz et al., 1988). Other important uses of hydraulic conductivity are: design
of remediation wells, evaluating capture zones of remediation or production wells, and
assessing risk associated with subsurface contamination.
The hydraulic conductivity can be determined by several laboratory and field methods.
Laboratory methods involve collecting either an undisturbed core sample, or remolding a
sample to its approximate in situ bulk density. These samples can be tested by either
constant or falling head permeameter tests. Core samples, collected in a vertical orientation,
are difficult to collect and preserve in a truly undisturbed manner. Hydraulic conductivity
values obtained in vertically oriented cores can be strongly affected by anisotropies associated
with stratification. Therefore, laboratory tests may not be reflective of field conditions, where
flow may be predominantly horizontal. Remolding of samples may provide a better overall
estimate, but disturbs the natural properties of the sediment, such as layering, small scale
structures and packing (Taylor et al., 1990). Rough estimates of hydraulic conductivities may
also be obtained from core samples based on correlations with grain size distribution
(Sheperd, 1989).
Field methods for determining the hydraulic conductivity include tracer, pump and
slug tests. Tracer tests can provide both hydraulic conductivity and dispersion characteristics
at the scale of the injection and monitoring points. However, tracer tests involve injecting a
foreign, readily detectable compound, such as sodium bromide or a radionuclide into the
aquifer. Both conducting and analyzing these tests can be complex, time-consuming and
expensive. Pumping tests generally provide information on the large scale hydrogeologic
characteristics, hydraulic conductivity and storage properties of an aquifer (Butler, 1990).
Pumping tests can be expensive, of long duration, require one or more observation wells for
data analysis, and involve disposal of large quantities of contaminated water. Slug tests may
provide spatially discrete values of the hydraulic conductivity using a single well point or
multi-level sampler (Widdowson et al., 1990). They are relatively quick, inexpensive, and
involve removal of little or no potentially contaminated water. As such, slug tests in
1
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groundwater monitoring wells are the most widely used method by the environmental
consulting industry for determining the hydraulic conductivity at contamination sites.
A slug test is performed by creating an instantaneous hydraulic head difference
between the well and the adjacent formation. This head difference can be created by the
'instantaneous' addition or removal of either a slug of water or a solid slug (mandrel
displacement). The recovery of this head difference is subsequently monitored by recording
the change in water level in the well with time using either a water level indicator or pressure
transducer. The head difference generally recovers exponentially or log-linearly with time
(Dax, 1987). The slope of the logarithm of head versus time relationship can be used to
determine the hydraulic conductivity. Interpretation of slug test data to derive a hydraulic
conductivity value is a function of the geometry of the well, hydrogeologic parameters, and
well and borehole conditions. Geometric parameters of the well include casing, screen, and
filter pack dimensions. These parameters are incorporated as "shape factors" into
mathematical solutions in order to analyze slug tests. Hydrogeologic parameters which may
affect slug tests include: confined versus unconfined conditions; degree of screen section
penetration into the aquifer; aquifer storativity; presence and proximity of semi- or
impermeable boundaries; anisotropy of aquifer properties; and the radius of influence of the
test Well and borehole conditions include skin effects due to borehole disturbance, the
presence of a filter pack around the screen, well storage, and well friction losses. Conducting
and interpreting slug tests in typical groundwater monitoring wells with screens and filter
packs spanning the water table in unconfined aquifers can be additionally complicated by the
drainage and subsequent resaturation of the filter pack.
A typical monitoring well in which slug tests are commonly conducted by the
environmental consulting industry is shown in Figure 1. A typical well consists of a 5.08 cm
(2 inch) screen installed in a 22,86 cm (9 inch) diameter hole in an unconfined aquifer. The
annular space between the well screen and the borehole is typically filled with a filter or sand
pack of fine- to coarse-grained sand. The well screen is commonly partially submerged, i.e.,
both the screen and the filter pack extend from above the water table to some depth below the
water table. In order to analyze a slug test conducted in this type of well, the water table is
generally considered to be a no flow boundary, i.e., no drawdown occurs in the water table
adjacent to the well. A partially submerged well screen creates several additional problems in
both conducting and analyzing slug tests. These problems include filter pack drainage, a
variable intake length during water level recovery, re-saturation of an unknown portion of the
filter pack, a small head difference after filter pack drainage, and potential propagation of a
cone of depression.
Analysis of slug test data is based on application of either Darcy's Law and the Thiem
Equation for the case of no storage as summarized by Hvorslev (1951) or numerical solutions
to the partial differential equation for non-steady radial flow with aquifer storage as first
proposed by Ferris and Knowles (1954) and later by Cooper et al. (1967). Both approaches
have been adapted to incorporate such effects as borehole skin, partial penetration, anisotropy,
impermeable boundaries, the radius of influence, and the intake geometry (e.g., Bouwer and
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Well Cover
Ground Surface
Native Material
Bentonite Seal
T Water Table
Screen Section
Filter Sand
Undisturbed
Formation
Figure 1. Typical ground water monitoring well with partially submerged well screen. R = screen radius; R
= borehole radius; Rc = well casing radius. n
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Rice, 1976; Braester and Thunvik, 1984; Widdowson et al., 1990). A review of the
derivation and application of the most widely used models for determining hydraulic
conductivity from slug tests in both confined and unconfined aquifers is contained hi
Chapter 2. Chapter 3 is an assessment of the factors affecting slug tests and provides
methods for accounting for these factors. Chapter 4 provides recommendations for
conducting and interpreting slug tests.
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SECTION 2
ANALYSIS OF SLUG TEST DATA
Slug^test analysis techniques can be differentiated into two'main categories based on
whether the aquifer is unconfined or confined. For the unconfined case, the solution is based
on steady state flow to the well due to a hydraulic gradient, and is derived using Darcy's Law
or the Thiem Equation integrated over time. In the case of a confined aquifer, the solutions
incorporate the release of water from storage due to compression/expansion of the aquifer
matrix and expansion/compression of the water. The confined solutions are based primarily
on numerical solutions of the transient saturated flow equation. If the storage coefficient is
low, or the amount of water released from storage is small or instantaneous, then the solutions
for the confined case will converge on the solutions for the unconfined case. The available
solutions for both the unconfined and confined cases are presented below.
SLUG TEST INTERPRETATION IN UNCONFINED AQUIFERS
Derivation of the hydraulic conductivity from a slug test in a well in an unconfined
aquifer using a modified form of Darcy's Law was first proposed by Kirkham (1946),
following the auger hole seepage theory of Hooghoudt (1936). Darcy's Law (1) states that
one dimensional flow through a porous media of cross-sectional area (A) due to a hydraulic
gradient (dH/dl) is proportional to the hydraulic conductivity (K).
Q = -K dH/dl A
(1)
To illustrate the derivation of a slug test solution using Equation 1, consider steady state flow
to a spherical, intake in an unconfined, homogeneous and isotropic aquifer of infinite extent.
Darcy's Law in spherical coordinates for this case may be expressed as:
Q = -
dH
As
dr
(2)
In this form, the cross-sectional area has been replaced by the area of a spherical intake, As,
equal to 47tr2. Rearranging and integrating Equation 2 yields:
0
/(dr/r2) = -4reK JdH
(3)
The solution of Equation 3 for steady state flow is:
Q = -K4itRsH
(4)
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In Equation 4, R. is the radius of a spherical intake (for naturally developed wells with no
filter pack, Rj is the screen radius; for wells with a filter sand pack, Rs should be the borehole
radius, R,,). Imagine that this spherical intake is connected to a vertical standpipe of radius Rc
(where R^ « RS). Consider a slug test being performed in the standpipe. During water level
recovery, the flow rate, Q, into the spherical intake is equal to the rise (or fall) of the water
level with time in the standpipe (i.e., JcR02dH/dt). Substituting for Q, rearranging, and
integrating (5) over time and head yields (6). Equation 6 is the equation for determining the
hydraulic conductivity from slug test data collected from a well with a spherical intake.
H2
JdH =
R47tR3 1,
J dt
(5)
K = -
7UR,2
(6)
4JTR,
Equation 6 is exact for a spherical intake assuming the aforementioned conditions, no cone of
depression around the well, and a fully submerged screen section at all times. The term
in Equation 6 is called the shape factor (Sf) for a spherical intake. As such, (6) can be
re-written as:
K = -
7tRc2 ln(H2/H')
(7)
Equation 7 is the general form of slug test solutions. For the idealized case of a spherical
intake, Sf is a simple term. For other intake geometries, Sf can be quite complex, and no
simple mathematical solution can be derived. Therefore, shape factors for Equation 7 have
been approximated by closed form analytical methods, 2-D and 3-D electrical analog
methods, and numerical methods for a wide variety of well and aquifer geometries. A full
presentation of currently available shape factors and their implications will be presented in
Chapters.!.
The two most common shape factors for use in (7) which are applicable to slug tests in .
unconfined aquifers are 1) the '1/2 ellipse' shape factor for flow to a cylindrical intake open
below an impermeable boundary and 2) the full ellipse shape factor for flow to a cylindrical
intake hi an infinite aquifer. The applicable well and aquifer geometries are depicted in
Figure 2. The earliest derived shape factor for the 1/2 ellipse model of a cylindrical intake^
with length Ls and radius R,., open below an impermeable boundary was first approximated by
Forchiemer (1930) and Dachler (1936) using closed form analytical solutions of the Laplace
6
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(a)
\ 7
Water Table
Impermeable
Boundary
(b)
(C)
Figure 2. Slug test geometries for (a) doughnut, (b) 1/2 ellipse, and (c) full ellipse models.
7
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Equation for an ellipsoid equivalent of a cylindrical injection zone. The solution is based on
flow from a line source for which the equipotential surfaces are semi-ellipsoids:
Sf =
2riLs
(8)
+ /I +
The denominator of Equation 8 can be simplified to ln(2Ls/Rs) for lyR, > 4, resulting in error
of less than 1% (Hvorslev, 1951). It is emphasized by Dachler (1936) that this formula for
the 1/2 ellipse shape factor can provide only approximate results when applied to a cylindrical
intake open below an impermeable boundary. The full ellipse shape factor applies to a
cylindrical intake in an infinite medium (i.e., no boundaries). For this case, the assumption
was made that flow lines are symmetrical with respect to a horizontal plane through the
center of' the intake, and then the formula from (8) was applied to the upper and lower halves
of the intake to obtain the following shape factor:
Sf =
(9)
(LA,)2)
Similar to Equation 8, the denominator of Equation 9 can be simplified to In^s/R,.) for
> 4. If the ratio of screen length to screen radius is unity in (9), the shape factor is
approximately the same as that for a spherical intake.
While the 1/2 ellipse and full ellipse models account for both horizontal and vertical
flow to the well screen, as the screen length gets increasingly larger relative to the well
diameter, the influence of vertical flow diminishes relative to horizontal flow. As stated in
Hvorslev (1951), the accuracy of these shape factors probably decrease with increasing values
of LS/R,. Assuming that for most monitoring well geometries, L/2R,. > 4, and substituting (8)
and (9) into (7), yields the 1/2 ellipse model (10) and the full ellipse model (11), respectively:
(R.2) ln(2Ls^s)
K = -
(10)
K = -
(Rc2)
(11)
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If flow is restricted to radial flow, this may be described by the Thiem Equation. The'model
geometry for this case, the doughnut model, is also presented in Figure 2. The radial flow
assumption is ideal for an aquifer bounded at the top and bottom by an impermeable
boundary, but may similarly apply to a partially penetrating well in an anisotropic aquifer
were the horizontal conductivity is significantly greater than the vertical. The doughnut
model is frequently applied to wells containing a partially submerged screen section. This
implicitly assumes radial flow and no drawdown of the water table adjacent to the well.
- ----., . f
Derivation of the doughnut model parallels the derivation of the models described
above. For this case, the area of a sphere in Equation 2 is replaced by the area of a cylinder,
2juR,.Ls. The resulting equation is then integrated from R,. to Ri, the radius of influence. The
remaining derivation follows the steps above. The resulting doughnut model is:
(Rc2)
K = -
2L.
(12)
The use of this model requires the determination or estimation of a radius of influence. The
shape factor for this model is:
Sf =
27lLs
ln(R/Rs)
(13)
The radius of influence (Rj) appears equivalent to screen length (Ls) of the full ellipse model,
or twice the screen length of the 1/2 ellipse model, as was pointed out by Bouwer and Rice
(1976). The shape factors for the elliptical solutions are approximations, whereas the
doughnut shape factor is exact for the model geometry. In the derivation of this model, it
was necessary to reduce the constant head radius from infinity to R; in order to keep head
finite. Theoretically, the radius of influence is infinite in an aquifer of infinite extent.
Practically, a slug test involves small quantities of water aiid lasts for a short period of time.
Consequently, the radius of influence must be relatively close to the well. The radius of
influence of a slug test has been defined and evaluated by a number of authors, e.g.,
Hvorslev, 1951; Bouwer and Rice, 1976; Dagan," 1978; Sageev, 1986; Dax, 1987; Karasaki et
al., 1988; Chirlin, 1989. A full discussion of the radius of influence and its implications is
included in Chapter 3.2.
Application of the above models to determine the hydraulic conductivity from slug test
data may be conducted in three ways, by linear regression, using two representative head and
corresponding time measurements, and by the Hvorslev (1951) Basic Time Lag method. The
recovering head in the well following either a slug out (or in) test is equivalent to the
changing (recharging or discharging) water level in the well. The equilibrium water level is
taken as a fixed datum point. A depiction of the recovering head following a slug out test is
included as Figure 3. The head versus time data may then be plotted on semi-log paper, or
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m>
m*
m
\
7
\/
Rc
Water Table
H(t)
.'..vy
R
h
Figure 3. Hydraulic head representation following a slug out test. L. = screen length, R. = borehole radius,
Rt = well screen radius, R0 = well casing radius, Hoc = calculated initial drawdown, and H(t) = head at any time
T»
10
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similarly, the natural log (In) of head versus time may be plotted on normal paper. Figure 4
is an example of such a plot for a well with a fully submerged well screen. The plot exhibits
log-linear recovery in early time, then tails off. The tailing off of the recovery curve in late
time is most likely due to the recovery of a cone of depression in the adjacent formation, i.e.,
an ever diminishing hydraulic gradient between the well and the formation. It may also be .
accentuated in wells with partially submerged screens by the resaturation of the filter sand
pack. The slope of the log-linear portion of the recovery curve is directly proportional to the
product of the shape factor and the hydraulic conductivity, and inversely proportional to the
square of the radius of the casing in which the water level is recovering. This may be seen
by rearranging Equation 7 in log-linear form, In y = mx + In b yielding Equation 14:
In Hj, =
KSf
+ hi
(14)
The slope of (14) is equivalent to:
slope = m = -
KSf
(15)
and the radius of the casing hi which the water level is rising, the hydraulic conductivity may
be calculated. Similarly, the hydraulic conductivity can be calculated using (14) and two
pairs of head and time measurements. Alternatively, the 'Basic Time Lag' method can be
used as developed by Hvorslev (1951). This method is a simple graphical manipulation of
(7), and can facilitate incorporation of differing shape factors. By rearranging (7) and
substituting the term Aw for the area of the casing:
KSf
Setting tx = 0, tj = t, H2 = H, and Ht = H0 (the initial head drop):
(16)
t = -
(17)
A plot of H/H0 (normalized head) versus time on semi-log paper, as shown in Figure 5, is
generated for this method. The initial head drop (Ho) should be established as the Y axis
intercept of the linear portion of the semi-log plot of head versus time data (Figure 4).
Calculation of the initial head drop using the volume of the removed slug or basing the initial
11
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&
1
H-l
-2.5 -r
0 200 400 600 800 1000 1200
Time (sec)
Figure 4. Typical slug out test recovery curve for a monitoring well with a fully submerged well screen. Well
TJ-1; UCONN Beach Hall; March 27,1991. After Binkhorst, 1992.
12
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s
£
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.37
TL = 125 sec
0
r~
50
100 150 200 250 300
Time (sec)
Figure 5. Hvorslev Basic Time Lag method. H = head at given time. H = initial head drop, and T = the Basic
Time Lag. ° L
13
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head drop on the first collected readings may incorporate errors stemming from filter pack
drainage, water level disturbance, and lack of accuracy in early time measurements. At H/H,,
= 0.37, at which time 63% of the initial head drop has dissipated, ln(H/H0) = -1. The time (t)
at which this occurs is termed the Basic Time Lag (TL) and (17) becomes:
(18)
From Figure 5, TL may be graphically determined. Aside from facilitating the use of different
shape factors, no other advantage is gained from this method.
In addition to the methods summarized by Hvorslev (1951), Dagan (1978) presented an
analytical solution for steady state flow to partially penetrating wells in unconfmed aquifers.
Dagan's solution considers the effects of vertical and horizontal flow, but assumes no storage
and a constant head condition at the water table. The solution is only applicable for cases
where the screen intake length is greater than 50 times the well radius. Following Muskat
(1937), Dagan obtained the steady state flow solution by assuming flow is generated by a
continuous distribution of sources of unknown strength along the well axis. He used the
theory of images to account for a lower impermeable boundary. The intake length is divided
into a finite number of intervals and the total flow to the well is calculated as the sum of the
flow to each interval. Solving a series of equations, Dagan generated values of dimensionless
discharge (Q}) versus log(L/R^) for different degrees of partial penetration of the screen. He
compared his method for packers with slug test and Bouwer and Rice (1976) solutions as is '
shown in Figure 6 (after Dagan, 1978). The dimensionless discharge Q' is:
Q' =
2jcKLsH
(19)
Using the determined dimensionless discharge based on the well and aquifer geometry, and
the slope of the natural log of head versus time curve, the hydraulic conductivity may be
calculated using Dagan's method from:
R.2
K = -
(20)
2Q'LS Vt.
SLUG TEST INTERPRETATION IN CONFINED AQUIFERS
A head difference from a slug test in a confined aquifer results in flow to the well not
only due to the induced hydraulic gradient, but also from a release of water from storage.
The release of water from storage due to a head difference during a slug out test is due to
14
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I
wa
a
W3
09
0.4
0.3
0.2
0.1
0.0
1.0
b/Ls = 1
b/Ls = 1.5
Packer
Slug
Bouwer and Rice
(b/Ls = oo)
b/Ls =
1.5
2.0
2.5
3.0
3.5
Log (L/Rs)
4.0
Figure 6. Comparison of dlmensionless discharge (Q1) curves for different slug test models, b = aquifer
thickness, R. = well screen radius, and L. = well screen length. After Dagan, 1978.
15
-------
compression of the aquifer matrix and expansion of the water. Flow to a well in a confined
aquifer is governed by:
d dH d dH d dH dH
(K, ) + (Ky ) + (K, ) = Ss
dx dx dy dy dz dz
(21)
dt
where Ss is equal to the specific storage and Kx>y>z is equal to the hydraulic conductivity in the
x, y, and z directions of the given Cartesian coordinate system. For the case of. a
homogeneous and isotropic media (K^ = Ky = K,, = K), Equation 21 becomes:
d2H d2H d2H Ss dH
dx2
dy2 dz2 K dt
(22)
For the case of a fully penetrating well in a confined aquifer of thickness b, ,
Equation 22 becomes 23 in Cartesian coordinates or Equation 24 in radial coordinates,
where the storativity, S = Ssb and transmissivity, T = Kb:
d2H d2!! S dH
dx2
dy2 T dt
(23)
1 dH
r dr
S dH
dr2 T dt
(24)
As shown in Lohman (1972), the equations for transient, saturated flow in confined aquifers,
shown here as Equations 21-24, are based on a combination of Darcy's Law, the clarification
of hydraulic potential by Hubbert (1940), the concepts of aquifer elasticity by Meinzer (1923),
and effective stress by Terzaghi (1925). The classical development of these equations was
first completed by Jacob (1940), and can be found in complete form in Jacob (1950).
Ferris and Knowles (1954) first applied these equations to determine the hydraulic
conductivity from slug tests in confined aquifers by approximating the well with an
instantaneous line source. However, Bredehoeft et al. (1966) demonstrated by electrical
analog that the solution of Ferris and Knowles was only satisfactory when time was large
enough for the ratio of head to initial head (H/H,,) to be less than 0.0025. The error
associated with Ferris and Knowles method is due to the approximation of the well as a line
source.
16
-------
Cooper et al. (1967) solved Equation 24 numerically with the Laplace transform with
respect to time for a finite diameter, fully penetrating well in a confined aquifer. The Cooper
et al. method comprises, perhaps, the most commonly used solution for interpretation of slug
tests in confined aquifers. Using the inverse transform, as provided by Carslaw and Jaeger
(1959) for the analogous heat flow problem, Cooper et al. showed:
where:
a =
B =
u =
Vu =
n =
Y, =
a -Bu2 du
Jexp()
2 a uVu
o
R/S/R,2
Tt/R/
Variable of integration
(uJ0(u) -2aJ,(u))2 + (uY0(u) -2aY,(u))2
Bessel function of the first kind, n* order
Bessel function of the second kind, n* order
(25)
Values of H/H0 versus Tt/Rc2 for five different values of a were obtained by numerical
simulation of Equation 25 and were plotted as a family of semi-logarithmic curves. In order
to obtain a transmissivity and storage coefficient, slug test data are plotted as H/H0 versus
time on semi-log paper at the same scale as the Cooper et al. type curves. The slug test data
curve is then superimposed on the best fit type curve and a value of t is determined at T/RC2
= 1.0. Each type curve is assigned a value of a from 10'1 to 10'5 in Cooper et al. (1967); the°
range of a was extended for cases of low storativity from W5 to 10'10 by Papadopulos et al.
(1973). A combination of the Cooper et al. curves and the Papadopulos et al. curves is
included as Figure 7 (after Papadopulos et al., 1973). The transmissivity and storativity may
then be calculated from Equations 26 and 27 respectively:
T =
(26)
S =
a
(27)
Cooper et al. concluded that the determination of storage coefficients by this method was of
questionable reliability due to the similarity of the shapes of the type curves. A shift to an
adjacent type curve results in an order of magnitude change in a, and consequently in the
value of S. According to the authors, the value of T is much less sensitive to this problem.
According to Papadopulos et al. (1973), an analysis in the range of a <10'5 indicates that if the
17
-------
0.1 -
Tt/Rc2
Figure 7. Confined aquifer type curves. After Papadopulosetal., 1973.
18
-------
value of a for the chosen type curve is within two orders of magnitude of the actual value,
the error in the determined value of T would be less than 30%.
Slug tests in multi-level wells with small screened sections in anisptropic confined
aquifers was considered by Widdowson et al. (1990) using the finite element flow model
(EFLOW) to simulate multi-level slug tests. EFLOW may be utilized to solve two
dimensional flow in an axisymrnetric domain using the Galerkin method. EFLOW was
modified to accommodate transient hydraulic head and fluid flow conditions in the screened
section. Using a modified form of EFLOW, slug tests were simulated for different well
geometries for different screen positions relative to the nearest impermeable boundary. Slug
tests were simulated for different degrees of anisotropy, storage coefficients, and radii of
influence. Storage was determined to have a negligible effect on the dimensionless discharge
after 1 second for storativity coefficients in the range of 0 to 10"4. The storage effects were
neglected by Widdowson et al. as the steady state region was judged to correspond to the
log-linear portion of slug test recovery data. Given the negligible influence of storage, the
method of Widdowson et al. would appear applicable to unconfined conditions as well.
Widdowson et al. state that this method is only suitable for naturally developed wells where
the screened section is fully submerged. Similar to the interpretation method of Dagan
(1978), a family of curves of dimensionless discharge (P equal to Q' in Equation 19) versus
log(LJRs) was generated for a range of aquifer thickness to screen length ratios as shown in
Figure 8. The slug test data analysis technique is identical to Dagan (1978), again using
Equation 20. As stated by Widdowson et al., their technique may be viewed as an extension
of Dagan's solution for small ratios of screen length to radius (L^). A favorable
comparison between the two methods was achieved for the zero storage, isotropic case for
Ls/Rs > 50. The absolute difference between the two methods ranged from 0 to 4%, with the
maximum difference occurring at L,/R^ = 50.
19
-------
to
o
tt>
rH
P
I
o
0.70
0.60
0.50
0.40
0.30
Q
0.20
0.8
T 1 1 I I I 1 1 I I I I | 1 I I I | I I I 'I | I I I ' I "'""
-e- b/Ls = 1.25
b/Ls = 1.5
b/Ls = 2
b/Ls = 4
b/Ls = 16
!_...« i i i I i i » 1 I t I I I » I I I t I 1 » 1 1.
1.0
1.2 1.4
Log (L/Rs)
1.6
1.8
2.0
Figure 8. Variation In dlmenslonless discharge (P) for different degrees of partial penetration, b = aquifer
thickness, R, = well screen radius, and L. = well screen length. After Widdowson et al., 1990.
20
-------
SECTIONS
THEORETICAL ASSESSMENTS OF FACTORS AFFECTING SLUG TESTS
The analysis of slug tests may be adversely affected by borehole skin effects,
anisotropy, determination of a radius of influence, partial penetration, approximation of intake
shape factors, drainage of the filter pack in partially submerged screen wells, and poor curve
matching. The potential effects of these factors on the calculated hydraulic conductivity value
and methods for accounting for these factors is presented in this section. A new method is
also presented to account for the drainage and re-saturation of the filter pack in wells with
partially submerged well screens.
SHAPE FACTORS FOR THE 1/2 AND FULL ELLIPSE, AND DOUGHNUT MODELS
Shape factors constitute an unknown, or at least approximated, component of the
solution to obtain hydraulic conductivity from slug tests in unconfined aquifers (Kirkham,
1946). In addition to the closed form analytical shape factors discussed in Chapter 2, shape
factors have been approximated by electric analog and numerical methods. The analytical
shape factors for the 1/2 ellipse (Equation 8) and the full ellipse (Equation 9), are compared
as a function of the ratio of screen length to diameter (LS/2RS) in Figure 9. The l^s/2Rs ratios
for typical 5.08 cm (2 inch) diameter monitoring wells with 0.5, 1.0, 3.0 and 5.0 meter
screens are 9.84, 19.69, 59.06, and 98.43, respectively. The difference between the two shape
factors is small, increasing to a percentage difference of approximately 18% at the largest
screen length to diameter ratio. Since the hydraulic conductivity is inversely proportional to
the shape factor (see Equation 18), the difference between a calculated conductivity using
either of these models would be relatively small.
Subsequent to the work of Kirkham (1946), additional shape factors for a cylindrical
intake were, obtained by 3-D electrical analog. Using a 3-D electrical analog in the
laboratory, Frevert and Kirkham (1948) obtained values for the shape factor for the case when
the soil is flush with the bottom of the casing. Luthin and Kirkham (1949) obtained values of
the shape factor for the case of a 2.54 cm (1 inch) diameter cylindrical cavity open 1.27 to
10.16 cm below the end of the casing. Smiles and Young (1965) re-investigated these shape
factors and found them to be consistently 12% too high. Smiles and Young attributed this to
a constant erroneous measurement of the specific conductivity of the electrolyte. Smiles and
Young (1965) also investigated the case of a hemi-spherical cavity below a casing, and
provide a summary of previously determined shape factors.
The shape factor for a cylindrical intake has also been investigated by numerical
methods. The earliest numerical shape factors were derived from finite difference methods
(Wilkinson, 1968; Al-Dhahir and Morgenstern, 1969; Raymond and Azzouz, 1969).
Al-Dhahir and Morgenstern (1969), for example, replaced the Laplace Equation in radial
coordinates by a finite difference form and solved the resulting set of simultaneous linear
21
-------
S2
&
.1
u
cs
I
5.5 -]
5
4.5 H
4
3.5 -
3
2.5 -
2
1.5 -
1
0.5 -
0
Full Ellipse (equation 9)
1/2 Ellipse (equation 8)
i i i i i i i
i i i i i i i r
10 20 30 40 50 60 70 80 90 100
Screen Length/Diameter (L/ZR^
Figure 9. Comparison of analytical shape factors for the 1/2 and full ellipse models for 2.54 cm (1 inch)
radius well screen and screen lengths ranging from 0.01 to 5 meters.
22
-------
equations by an iterative method. Potential problems with these methods include adjustments
required to solve convergence and the discontinuity problems in the corners of the physical
domains where the method was applied (Chapuis, 1989). Shape factors have also been
.derived by finite element methods (Randolph and Booker, 1982: Tavenas et al., 1986a,
1986b). According to Chapuis (1989), these methods are known to have limited accuracy
because conventional finite elements with nodes at then- apexes do not insure continuity of
both pore pressures and flow rates.
A comparison of the shape factors for cylindrical injection zones in an aquifer of
infinite extent as proposed by the above authors is presented as a function of L/2R,. in
Figure 10 (after Chapuis, 1989). The majority of the shape factors fall into a narrow-range
with increasing divergence with increasing screen length to diameter ratios. The maximum
percent difference among the shape factors shown in Figure 10 is approximately 40% (or a
factor of less than 2) for L/2R, equal to 16, excluding the results of Randolph and Booker
(1982). This percent difference remains approximately the same if the curves are linearly
extrapolated to a typical monitoring well L/2R,. ratio of approximately 59 (3 m screen length
and 5.08 cm screen diameter). As such, it appears that the choice of a particular shape
factors is not a major source of error in calculating hydraulic conductivity values. Recall that
the hydraulic conductivity is inversely proportional to the shape factor.
Chapuis (1989) attributes the main differences in shape factors for cylindrical intake
zones to the limitations of numerical methods, whether the shape factors incorporate either a
pervious or impervious bottom, and the finite domains used in numerical methods. For
cylindrical cavities with a pervious bottom, Chapuis (1989) recommends the use of Equation
9 for L^RS > 1. If the bottom of the cylindrical intake is impervious, Chapuis recommends
subtracting 5.5R,. from the shape factor calculated using Equation 9, based on electric analog
methods from Taylor et al. (1948). This method is approximately equivalent to subtracting
the circumference of the bottom of the intake (2riRJ,
RADIUS OF INFLUENCE
Determination of the radius of influence of a slug test is needed to solve for hydraulic
conductivity using the doughnut model and to estimate the volume of aquifer affected by a
slug test. The radialjlistance influenced by the test is also related to the degree of well skin
effects (Chapter 3.3) and to the influence of boundary conditions (Chapter 3.5).
Determination of the radius of influence depends on how it is defined. The radius of
influence has been defined as the distance to the source of supply (Hvorslev, 1951), the radial
distance over which the initial head drop is dissipated (Bouwer and Rice, 1976), as
unbounded storage at some radial distance (Chhiin, 1989), and as the distance at which some
fraction of the initial head drop occurs (Barker and Black, 1983). Sageev (1986) suggests
that the practical radius of influence is the detectable radius of investigation and should be set
according to the accuracy of the pressure or head recording devices used to detect it. From a
23
-------
P
g
O
"
43
00
0 2 4 6 8 10 12
Screen Length/Diameter (L^IRJ)
14 16
Figure 10. Comparison of analytical, electric analog, and numerical shape factors (normalized to the screen
diameter) for cylindrical Intakes. A = Randolph and Booker, 1982; B = Wilkinson, 1968; C = Tavenas et al.,
1968a; D a Brand and Premchltt, 1980; E = Kallstenlus and Wallgren, 1956; F = Raymond and Azzouz, 1969; G
s AI-Dhahlr and Morgenstern, 1969; H = Smiles and Young, 1956. After Chapuis, 1989.
24
-------
mass balance perspective, a slug of water removed from a well will be replaced by an equal
volume of water from the aquifer around the well. This volume in turn will be replaced by a
subsequent volume of water coming from a radial distance further from the well. Assuming
no storage and an infinite aquifer, mass balance implies that this replacement process must
continue to infinity. While the radius of influence may be theoretically infinite, the actual
radius of influence of a slug test is most likely finite due to storage or boundary effects.
The volume of aquifer which contributes to the recharge of a well is dependent on the
volume of water removed from the well, the hydraulic conductivity of the aquifer, the screen
length contributing to flow, aquifer storage, and boundary conditions. Larger head drops
created in a well imply that larger volumes of the aquifer are affected. For the same head
drop, a well with a longer screen would have a smaller radius of influence than one with a
shorter screen. This is a consequence of water entering the well over a thicker portion of the
aquifer. In addition, the radius of influence would decrease as the storage coefficient of the
aquifer increases. Larger amounts of water are released from storage as the storage
coefficient increases. If low permeable boundaries restrict vertical flow near the well, the
radius of influence would increase relative to a homogeneous, isotropic aquifer of infinite
extent The same effect would occur if the aquifer is highly anisotropic, such that the
horizontal conductivity is greater than the vertical conductivity. The Theis (1935) Equation
for flow to a pumping well predicts a steep cone of depression diminishing rapidly away from
a well in a low conductivity formation. Similarly, a slug test in a formation of low hydraulic
conductivity would also affect a relatively small volume of aquifer.
The radius of influence appears in the shape factor for the doughnut model (see
Equation 13). The shape factor decreases exponentially as the radius of influence increases.
This can be seen in Figure 11 for a 5.08 cm diameter well with a 0.2, 0.5, 1.5 and 3.0 meter
screen section using an arbitrary range of 0 to 30 meters for radii of influence. Reference, to
Equation 18 indicates that a reduction in the shape factor would increase the resulting
hydraulic conductivity value, i.e., the larger the radius of influence the larger the hydraulic
conductivity. The shape factor is very sensitive at radii of influence less than approximately
2 meters and appears to approach the screen length in the limit.
Bouwer and Rice (1976) investigated the radius of influence using a 2-D electrical
resistance network analog for different well and aquifer geometries. The radial extent of the
analog was 60,000 times the largest screen radius used in the analyses. Current flow to a
'well' for varied well and aquifer geometries was converted to volumetric flow (Q). Values
of ln(Ri/Rs) were calculated from a modified Thiem Equation for steady state flow at given
values of constant head (H):
H
= 2jcKLs.
(28)
25
-------
fi
I
£4
I
iiiiiiiiiiiii i i ni i i
Q 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Radius of Influence (meters)
Figure 11. Impact of the radius of influence and screen length (L,) on the doughnut model shape factor for a
2.54 cm radius well screen.
26
-------
The test geometry is illustrated in Figure 12. Different values of the radius of influence, as
hXRj/RJ, were determined for different sets of aquifer thickness, screen radius, and position
of the bottom of the screen relative to both the water table (LW-LS) and the lower confining
unit (b-Lw = D'). The position of the top of the screen immediately at the water table was
not evaluated directly, as a short between the water table as a source and the well as a sink
would have occurred. Current flow to the well was found to vary linearly with screen length.
This allowed extrapolation for the case where the top of the screen was at the water table.
Based on the results of the electric analog work, empirical formula were derived for k^Rj/RJ
for the case of fully and partially penetrating well screens, presented here as Equations 29 and
30, respectively:
1.1
A +
Rs
(29)
1.1
R,
(30)
The terms A, B, and C are empirical constants. These constants are a function of (L./RJ and
can be selected graphically from Figure 13 (after Bouwer and Rice, 1976). Bouwer and Rice
(1976) report increasing error as the screen length decreased relative to the steady state head.
They report 10% error for Ls > 0.4H, and 25% error if Ls « H. For one case, the calculated
radius of influence was compared to the equipotential lines radiating from the electric analog
of the well. The calculated radius of influence corresponded to equipotential lines
representing 85% of the initial head drop in the horizontal and 80% in the vertical directions.
They concluded that this radius is indicative of the portion of the aquifer sampled for
hydraulic conductivity from slug tests. The method of Bouwer and Rice for determination of
Ri is only 2 dimensional and only for steady state conditions. Additional error may result
from having to read empirical constants from a graph, the discrete nature of their network,
and imperfect calibration of the individual resistance elements in their model (Dagan, 1978).
The Bouwer and Rice method for calculating the radius of influence is also independent of
the hydraulic conductivity and specific yield of the aquifer, as well as the initial head drop
created by the slug test.
The Bouwer and Rice method was applied here to typical monitoring well geometries in
order to estimate radii of influence. The radii of influence for two fully penetrating wells
with screen radii of 2.54 cm (typical casing radius) and 11.43 cm (typical hole radius) was
calculated using Equation 30 for variable screen lengths ranging from 1 to 10 meters. The
results are presented in Figure 14. The radius of influence was found to range from 0.27 to
2.65 meters, and from 0.37 to 3.35 meters for the 2.54 and 11.43 cm radii well screens,
respectively. The radius of influence was found to increase linearly with screen length with a
correlation coefficient of 0.999 for both well screen radii. The direct correlation between
27
-------
1
Lower Confining Unit
Water Table
1
D'
Figure 12. Well and aquifer geometry symbols for the Bouwer and Rice (1976) model.
28
-------
A
AND
C
14
12
10
4
2
0
< ' < I
B
3
2
1
5 10
50 100
500 1000
5000
Screen Length//Radius (L,/Rs)
Figure 13. Dimensionless parameters A, B, and C for the Bouwer and Rice (1976) model.
29
-------
1
1
3.5
3 -
2.5 -
8 2 -
1
13
C/3
=1
1.5-
0.5 -
Screen Radius = 11.43 cm
Screen Radius = 2.54 cm
34567
Screen Length (meters)
10
Figure 14. Impact of screen length and screen radius on the radius of influence, calculated using the Bouwer
and Rica (1976) method.
30
-------
screen length and radii of influence is a consequence of having wells that are fully penetrating
and having radial flow. The radius of influence was found to be approximately one third the
screen length.
The screen length variable (Ls) in the full ellipse model and the radius of influence
variable (Rj) in the doughnut model are present in the terms InCL^) and ln(Ri/Rs),
respectively, as presented in Equations 11 and 12. Differences between hydraulic
conductivity values calculated using the two models are only a function of the difference
between these two terms. The presence of the natural log function in each of the terms will
further reduce the difference between the two methods. The doughnut model shape factor
was calculated using Equation 13, incorporating the radius of influence calculated using
Equation 30 for a fully penetrating 2.54 cm radius well with a screen length ranging from 1
to 10 meters. The full ellipse model shape factor was calculated for the same well geometry
using Equation 9. Figure 15 is a. comparison of the two shape factors. Shape factors derived
from the doughnut model were consistently higher and increased with screen length at a faster
rate than the full ellipse model. However, the percent difference between the two shape
factors increased linearly from only 3.95% for a 0.61 meter screen to 32% for a 9.14 meter
screen. Hence, hydraulic conductivity values determined using either model will not be
significantly different, especially at smaller screen length to diameter ratios.
Widdowson et al. (1990) using the modified form of EFLOW found that setting the
radius of influence between 18.29 and 182.88 meters (60 to 600 feet) resulted in
2% to 5% difference in the resulting dimensionless discharge (P) from numerically simulated
slug tests. Widdowson et al. held the value of Rj constant for generation of dimensionless
discharge values for given well geometries. They found that the dimensionless discharge
term is equivalent to the inverse of the .ln(R/Rs) term of the doughnut model. Therefore,
calculation of hydraulic conductivity values using the Widdowson et al. method will yield
approximately the same value as the Bouwer and Rice method for isotropic conditions far
from an impermeable boundary.
Determination of a representative radius .of influence may also be approached from a
mass balance performed on the volume of the slug of water removed from a well. Assuming
that no water is released from storage, the volume of water removed from the well (V0) must
;be replaced by an equal volume of water (Vt) extending from the well screen to a radial
.distance Rt. This volume must be similarly replaced by volume V2 extending to a radial '
distance R^ The equations for the volumes of each replacing cylinder follow as Equations
31-34, and are shown in generic form for any volume, Vn, in Equation 35. In the following
.equations, n equals the formation porosity. Note that any incremental volume must be equal
to the original volume of water added or removed from the well (Equation 36).
31
-------
52
S
2
3
CO
14
12 -
10 -
8 -
6-
4-
2-
0
Doughnut model with
Bouwer and Rice Ri
Full Ellipse (equation 9)
0 20 40 60 80 100 120 140 160 180 200
Screen Length/Diameter (L^R^
Figure 15. Comparison of shape factors for the doughnut model using Bouwer and Rice (1976) radii of
influence, and for the full ellipse model for a 2.54 cm radius well screen.
32
-------
V,
V2
V
- V =
n . jrR02H0
= 7iR/Lsn - V, - V0 =
7iRc2H0
= 7iR32Lsn - V2 - V, - V0 = JiR32Lsn - 7iR22Lsn
sn - TtR/L.n + TiR/H,, - Tt
= itR32Lsn -
(31)
(32)
(33)
(34)
. (35)
(36)
While the replacing volumes must remain equal, the incremental increase in radial distance
will decrease with each additional replaced volume. The radius of influence of a slug test
may then be regarded as the distance at which the percent difference between the increasing
radial distances becomes negligible. Dividing through by TT, and rearranging Equation 36
allows for setting the percent difference between two adjacent radial distances.
(37)
R,2 -
Selection of a limit for the percent difference between the increasing radial
distances is arbitrary. If the percent increase in radial distance is only 1%, then
equivalent to 0.99R,,, and R,, becomes the radius of influence.
R,2 - (.99RJ2 = Rc2H,/Lsn
50.25(RC2)H0
is
x
Lsn
(38)
(39)
If the difference between the two radii is selected as 0.1% or 10%, the resultant equations for
calculation of the radius of influence would be Equations 40 and 41 respectively.
5.26(RC2)H0
Lsn
(40)
500.25(RC2)H0
Lsn
(41)
33
-------
Examination of the above equations indicates that the radius of influence is proportional to
the volume of water removed from the casing, and inversely proportional to both the screen
length and the porosity. Using the above equations, with an initial drawdown equal to the
screen length and a porosity of 30%, yields a radius of influence equal to 40.84RC, 12.94R,.
and 4.15R,. for a 0.1, 1.0 and 10% difference in increasing radii, respectively. For a typical
5.08 cm diameter well casing, these values correspond to radii of influence of 1.04, 0.33, and
0.11 meters, respectively, from the well screen. This range is significantly less than those
calculated by the Bouwer and Rice method for a fully penetrating well. These calculations
further support that a slug test only influences a small volume of aquifer and emphasizes the
need to carefully construct and develop wells for slug testing to minimize adverse factors
such as well skin effects, clogged screens, and formation disturbance.
WELL SKIN EFFECTS
The process of well drilling, emplacement of a filter pack around the screen, penetration
of drilling mud into the formation, and the well development process may produce a disturbed
zone around a well commonly referred to as a well skin. Well skins may be more permeable
than the formation, e.g., a high permeability filter pack around the screen section; or less
permeable than the formation, e.g., borehole damage due to smearing of clays or other fines
along the borehole. More permeable well skins would increase the apparent hydraulic
conductivity as determined from a slug test, and a less permeable well skin would have the
opposite effect. The impact of well skin effects on slug tests may be considerable, especially
if the disturbed zone is large relative to the volume of aquifer tested by the slug test.
Kipp (1985) describes the skin effect as an empirical concept based on postulating an
additional pressure drop through a thin annulus around the well bore that is proportional to
the flow rate into the well. Van Everdingen (1953) discussed the effects that well bore skin
have on the pressure response of a well for the case of constant discharge. Jaeger (1956)
considered the solution to a heat flow slug test that included well bore storage and skin. The
skin effect was also described by Carslaw and Jaeger (1960). The analytical solution for a
slug test in a well surrounded by an infinitesimally thin skin was originally published by
Ramey and Agarwal (1972), and updated by reducing the number of dimensionless parameters
by Ramey et al. (1975). Faust and Mercer (1984) presented a ignite difference model of a
slug test response in a well having a finite skin thickness. Moench and Hsieh (1985) contend
that the results of Ramey and Agarwal (1972) and Faust and Mercer (1984) are actually
identical. A finite element solution for quasi-steady state flow to a double packer system was
presented by Braester and Thunvik (1984) which included an evaluation of the effects of well
skin and anisotropy. The effects of well skin, as well as anisotropy, boundaries and partial
penetration, was also considered by McElwee et al. (1990).
Faust and Mercer (1984) numerically simulated four slug test runs with a well skin
hydraulic conductivity of 0.01, 0.1, 1.0, and 10 times the hydraulic conductivity of the
formation. For their simulations, the well radius was 0.385 m, the casing radius was 0.0205
m, the initial head was 5.0 m, and the storativity was 2.25 x 10"8. The thickness of the well
34
-------
skin was equivalent to the radius of the borehole. Their results for a formation having a
hydraulic conductivity of 10~9 are presented in Figure 16. The Figure indicates that a more
permeable well skin has little effect on the position or shape of the recovery curve. Hence,
their results suggest that a more permeable well skin would have little effect on the calculated
hydraulic conductivity. However, less permeable well skins significantly influenced the
recovery curves. Using the type curve method of Cooper (1967), this would lead to
underestimating the formation's hydraulic conductivity. The similar shape of the curves
would not indicate the presence of a well skin. Moench and Hsieh (1985) indicate that the
hydraulic head response in a slug test interval may respond to the skin properties at early time
and to aquifer properties at late time. Moench and Hsieh suggest that Faust and Mercer
would have found the same results, if they considered a typical aquifer with storativity values
in the range of 10'3 to 10'6.
Braester and Thunvik (1984), using a finite element model, evaluated the effect of well
skin on steady state flow to packer tests using input parameters reproduced from a series of
tests on a granite. The input parameters include a porosity of 0.3%, total water and rock
compressibility of 10~10 Pa'1, borehole radius of 0.028 m, a packer spacing of 2 m, and an
injection overpressure of 4 x 10s Pa. Their numerical results compared well with the full
ellipse model (Equation 11), Dagan's (1978) analytical solution to the double packer flow,
and with Moye (1967) as shown in Figure 17. The maximum relative difference between the
numerical results and Dagan's formula was 4% at the smallest ratio of well screen length to
radius ratio examined (log (L,/RS) = 1.85). The maximum relative difference between the
results of the full ellipse and Moye's formula with those of Dagan's was 8.8 and 14.6%,
respectively, also at log (L/RJ = 1.85. To evaluate the effect of borehole skin, apparent
permeability values were numerically computed for skin radii ranging from 0.11 to 0.33 m
from a 0.028 m radius borehole. The skin permeability values used were 0.001, 0.01, and 0.1
times the intrinsic permeability of the formation (10"12 m2). The numerically, generated steady
state flows were then used in Dagan's formula to evaluate the composite apparent
permeability. Using the data tabulated in Braester and Thunvik (1984), a graph of the
apparent permeability versus the well skin radius was generated as Figure 18. As shown on
the Figure, the apparent permeability closely mimics the well skin permeability. The case of
a more permeable well skin was not investigated.
McElwee et al. (1990) investigated the effects of well skin in both a fully penetrating
and partially penetrating well in an isotropic medium. They used an analytical solution
obtained in Fourier-Laplace space and inverted for a solution in space-time domain using the
Stehfest algorithm for a numerical approximation of the inversion. McElwee et al. indicate
that for a fully penetrating well with no skin, their solution is identical to Cooper et al.
(1967). For the isotropic case with well skin, their solution reduces to the Laplace-space
solution of Moench and Hsieh (1985). Simulations were conducted for the case of a fully
penetrating well, with a skin radius 10 times the well radius and a skin hydraulic conductivity
two orders of magnitude greater and less than the formation. As shown in Figure 19, then-
results also indicate that a less permeable well skin has a far greater impact than the more
permeable well skin. The more permeable well skin, however, did affect the curves. The case
35
-------
A
1.0
0.8 -
0.6 -
0.4 -
0.0
0.01
0.1
1.0
Kbt/R
10
Figure 16. Recovery curves for varying skin hydraulic conductivities. Numbers adjacent to curves represent
akin conductivities In m/s. K = formation hydraulic conductivity, b = aquifer thickness, t = time, Rc = well
casing radius, and H/H = normalized head. After Faust and Mercer, 1984.
36
-------
I
CO
CO
CO
CO
0.4
0.3
0.2
0.1
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Log L/Rs
Figure 17. Comparison of cylindrical intake dimensionless discharge terms. (1) Braester and Thunvik's
(1984) numerical solutions; (2) Dagan, 1978; (3) Hvorslev, 1951; and (4) Move. 1967. L = well screen lenath-
R. = well screen radius. After Braester and Thunvik, 1984. ' , M '
37
-------
1E-12
8P
B
.9
I
8
&
1E-13 -
1E-14 -
IE-IS
ks = k formation = 1e-12
ks = 1e-13
ks = 1e-15
O.l
i
0.15
I
0.2
0.25
l
0.3
0.35
Well Skin Radius (meters)
Figure 18. Apparent formation permeability for different well skin permeabilities (k.) and thicknesses, using
data from Brassier and Thunvlk, 1984.
38
-------
s
i.oo
0.80
0.60
0.40 -
0.20 -
0.00
\
\
\
\
\
\
Ks/Kf
Ks/Kf
Ka/Kf
\
\
\
10
-3
~1 1 I I Illl) 1 1 I I llll| 1 1 I I llll|
10-2 10-1 i 10 102 103
Dimensionless Time (Kfbt/Rc2)
Figure 19. Effects of borehole skin on recovery curves for a fully penetrating well In a confined aquifer. K
and K, are the skin and formation conductivities, respectively. After McElwee et al., 1990. *
39
-------
of the partially penetrating well appears to have still included a fully penetrating well skin.
The effect of the more permeable skin in this case was accentuated due to a fully penetrating
preferential pathway to the well screen.
The influence of a well skin on the hydraulic conductivity value determined from a slug
test may also be treated using radial averaging. Recall the Thiem Equation for steady state
radial flow:
2bK(Hs-Hi)
Q = " (42)
ln(Ri/Rs)
If a well skin is present, the determined hydraulic conductivity will be a composite of both
the well skin and the formation, which may be considered as an effective hydraulic
conductivity (Ken). The total head drop in the system will be the same, with a portion of the
head drop taken up in the well skin and a portion in the formation. The flow rate (Q.)
coming into the screen will be equivalent to the flow coming through the borehole or well
skin (Q,), which in turn will be equivalent to the flow coming through the formation (Qf).
The conservation of flow regime is depicted in Figure 20 for reference. Three steady state
flow equations may therefore be written representing the composite flow through the entire
system from the radius of influence to the screen (43), flow through the well skin to the
screen (44), and flow through the formation to the well skin (45). Note that the radius of the
well skin is given as the borehole radius, R,,. In the following equations, the subscripts s, h,
f, i relate to the screen, borehole or well skin, formation, and radius of influence, respectively.
(43)
(44)
(45)
ln(Ri/Rh)
As stated above, the total head drop across the system is equivalent to the head drop
across the well skin plus the head drop across the formation:
= (HfH,,) + (H
Re-writing Equations 43, 44, and 45 into Equation 46 yields:
(46)
40
-------
ff*v * f * ... \t : .*..*«f f * *» f \.*«..*\**A
r-1-^_ % t v > * \:\: *s\:\. s"*«" .*i
^^ ^^ « * . % -. V*. "" ". " \* \' ". " '. " ". * *«V ". " ". * ". ' *. !
^ t ^ \."-\V"-* V*«. .*.*».".*.*"-..*'
X. v. f .?" .> Vf .*\f\t.\t\f \*\f**f if\f.\f
Well
Screen
Radius of Influence
Figure 20. Plan view of a well identifying parameters used to derive effective hydraulic conductivity
equation. Q, R, K, and H represent the flow, radius, hydraulic conductivity and hydraulic head at the labeled
boundaries, respectively.
41
-------
27cbKf
(47)
ln(R/R.)
Solving Equation 47 for K^ yields:
ln(R/Rh)
Kj Kf
Equation 48 may also be written as a ratio of the effective to formation hydraulic
conductivity:
(48)
(49)
+ ln(R/Rh)
The effect of the radius of influence on the ratio of effective to formation conductivity was
calculated using Equation 49, for a 2.54 cm radius well screen with either a 5.08 and 10.16
cm radius well skin. A K/Kf ratio equal to 10 and 0.1 was used. Results are shown in
Figure 21. The Figure indicates that the conductivity will be overestimated in the case of a
more permeable well skin, and underestimated for the case of a less permeable well skin.
The maximum degree of over- or underestimation approaches the ratio of the skin to
formation conductivity as the radius of influence diminishes. It should be noted that the
maximum possible difference for the situation presented in Figure 21 is a factor of 10. As
the radius of influence increases, the effective conductivity approaches the formation
conductivity in an exponential manner. This occurs to a greater degree and more rapidly for
the case of the well skin with a higher conductivity than the formation. The effect of a less
permeable well skin is more pronounced and cannot be readily overcome by increasing the
radius of influence (i.e., by increasing the initial head drop). A more permeable well skin
will affect the results of a slug test, but with a radius of influence of less than one half of a
meter, this effect may be reduced to less than a factor of 2.
ANISOTROPY AND HETEROGENEITIES
Most natural geologic formations have a strong horizontal to vertical anisotropy ratio
with respect to hydraulic conductivity (Taylor et al., 1990). Anisotropy ratios of 10:1 or
more are common (Freeze and Cherry, 1979). Formation heterogeneities due to natural
layering may also be present. Measurements of hydraulic conductivity determined from slug.
tests may reflect these anisotropies, either under- or overestimating the hydraulic conductivity
42
-------
I
10
0.1
Ks/Kf = 10; Rh = 5.08 cm
Ks/Kf = 10: Rh = 10.16 cm
Keff = Kf
Ks/Kf = 0.1; Rh = 5.08 cm
Ks/Kf = 0.1; Rh = 10.16 cm
0.1
1 10
Ri (meters)
100
Figure 21. Effects of well skin and radius of influence on the effective hydraulic conductivity for a 2 54 cm
radius well screen. The radius of the well skin is given as the hole radius (F^).
43
-------
12
11
10 -
9
-£
I 8
&
§ 7
o
£ 6
tsJ
00 J "
I4'
!"3 o
2 -
1
0
0
Kh/Kv = 0.01
Kh/Kv = 0.1
X"
Kh/Kv = 1
^
Kh/Kv = 10
^
Kh/Kv = 100
\ - 1 - 1 - 1 - 1 - 1 - 1 r
40 50 60 70 80 90 100 110 120
10 20 30
Screen Length/Diameter (LJ2Rs)
Figure 22. Influence of anlsotropy on the full ellipse shape factor for a 2.54 cm radius well screen. « and K^
correspond to the horizontal and vertical hydraulic conductivities, respectively.
44
-------
in the direction of fluid flow. Slug tests in formations with layered heterogeneities will
therefore reflect a weighted average of the hydraulic conductivities of the layers intercepted
by the screen section of the well. Several authors have evaluated and/or incorporated the
effects of anisotropy on slug tests (Maasland, 1955; Widdowson et al., 1990; McElwee et al.,
1990; Hvorslev, 1951; and Braester and Thunvik, 1984). Unfortunately, anisotropy ratios are
scale dependent and difficult to measure. Small scale anisotropy ratios may be determined
with oriented, undisturbed core samples, while large scale anisotropy ratios may be estimated
with slug tests conducted in multi-level piezometers or from boring logs.
The effects of anisotropy in soil was first considered by Samsioe (1931) and Dachler
(1936), and has been summarized in Hvorslev (1951). The influence of different horizontal
(K^ and vertical (IQ hydraulic conductivities in two dimensions may be accounted for by
dividing the horizontal dimensions of the half and full ellipse shape factors by an anisotropy
correction factor n:
m.-JK./K,
For the half and full ellipse models with L/2R,. > 4, the horizontal hydraulic
conductivity, Kh may be expressed as Equations 51 and 52, respectively.
(Rc2) ln(2maLs^c)
(50)
(51)
ln(H,/H2)
2L.
.(52)
The effects of anisotropy on the calculated hydraulic conductivity was investigated by
comparing the shape factor for the full ellipse model for isotropic conditions (Equations 9)
with the shape factor for the full ellipse model for anisotropic conditions (Equations 52). The
comparison of the two full ellipse shape factors is shown in Figure 22 for anisotropy ratios
(Ki/JQ equal to 0.01, 0.1, 1.0, 10, and 100. The well used for the calculation was naturally
developed with a 2.54 cm radius screen ranging from 0.15 to 4.57 meters in length. An
anisotropy ratio greater than 1 resulted in reducing the shape factor. An anisotropy ratio less
than 1 resulted in increasing the shape factor. Recall that an increase in the shape factor
results in a proportional decrease in the hydraulic conductivity and vice-versa. For cases
where the horizontal conductivity is greater than the vertical, anisotropy will increase the
calculated horizontal conductivity relative to the isotropic case. As such, ignoring this
anisotropy would have resulted in underestimating the hydraulic conductivity in the horizontal
direction. The opposite effect would be the case for anisotropy ratios less than 1. It should
be re-emphasized that in most cases, the aquifer is assumed to be isotropic in analyzing slug
tests, resulting in underestimating the horizontal hydraulic conductivity.
45
-------
Braester and Thunvik (1984) evaluated the effects of different anisotropy ratios on flow
to a partially penetrating well. Steady state flow to a well was calculated using numerical
methods for different anisotropic conditions using well and aquifer parameters described in
Chapter 3.3. They then compared the hydraulic conductivity obtained using the isotropic
solution of Dagan (1978) to the value of the actual horizontal hydraulic conductivity. This
comparison is shown graphically in Figure 23. For the case when KVK^ was equal to 0.1 and
0.01, they found that Dagan's solution overestimated the horizontal conductivity by a factor
of 1.8 and 3.9, respectively. For the case when Kj/K^ was equal to 10 and 100, Dagan's
solution underestimated the actual horizontal conductivity by a factor of 0.94 and 0.7.
In addition to anisotropy of hydraulic conductivity, layered heterogeneities are also
common due to natural deposition. If a well intercepts more than one layer, the determined
horizontal conductivity will be an average of the intercepted layers, weighted to the thickness
and conductivity of each of the layers. Assuming horizontal layering, radial flow and an
equal head drop radially from the well in each of the layers, the effective hydraulic
conductivity (K^) may be expressed as a simple weighted sum of i layers of thicknesses dj
and conductivities Ki5 with a total thickness dj as shown below (Freeze and Cherry, 1979):
(53)
PARTIAL PENETRATION AND BOUNDARY EFFECTS
Partially penetrating wells occur when a well screen does not penetrate the full thickness
of an aquifer. In this case, flow may no longer be strictly radial and the vertical component
of flow may be significant. If vertical flow is important, the radial flow models (e.g., the
doughnut model) will overestimate the hydraulic conductivity. If, however, the vertical
component is not significant, as may be the case in anisotropic conditions or where the screen
length is long, flow to the partially penetrating well may be considered radial and fully
penetrating slug test solutions may be used. The influence of partial penetration on slug test
results will be a function of the proximity of the screen to the lower impermeable boundary
and the degree of anisotropy.
The effects of partial penetration and boundary conditions have been incorporated into
the shape factors of the 1/2 ellipse, full ellipse, and doughnut models, as well as into the
curve matching solutions. The effect of a lower or upper impermeable boundary on the shape
factor Sf for a cylindrical cavity (simulated screen section) was evaluated by a 3-D electric
analog by Luthin and Kirkham (1949) and Youngs (1968). Luthin and Kirkham found that
the shape factor increased as the cavity approached the water table, and decreased when the
cavity approached the lower confining unit. Youngs (1968), using the same 3-D electric
analog employed by Smiles and Youngs (1965), evaluated the influence of a lower
impermeable substratum for a wider range of well geometries. Youngs found that the shape
46
-------
0
0.01
Kh/Kv
Figure 23. Comparison of the isotropic hydraulic conductivity of Dagan (1978) with the anisotropic
conductivity of Braester and Thunvik (1984); developed using data from Braester and Thunvik.
47
-------
factors, taking a impermeable boundary into consideration, do not vary by more than 20%,
provided the distance between the cavity and the substratum is more than the diameter of the
piezometer. As with Luthin and Kirkham, the shape factor decreased as the impermeable
substratum approached the cavity.
Dagan (1978) found that if the distance from the water table to the lower impermeable
boundary was greater than 1.2 times the distance from the water table to the bottom of the
screened section, the effect of the lower boundary was negligible. Widdowson et al. (1990)
found that the position of the screened section relative to the impermeable boundary had little
effect. The effect was even less pronounced when the horizontal conductivity was greater
than the vertical.
Bouwer and Rice (1976) investigated the effects of partial penetration and boundaries on
the doughnut model for different well and aquifer geometries. A description of Bouwer and
Rice's method is presented in Chapter 2.1. In addition, the method for calculating the
InCRj/R,.) term is presented as Equations 29 and 30 for partially and fully penetrating well
screens, respectively. An assessment of the Bouwer and Rice method was conducted by first
calculating the term ln(Rj/Rs) and the radius of influence for a fully penetrating 11.43 cm
radius, 1.52 meter long screen using Equation 30. The IntRj/R,) term equaled 1.86
(dimensionless) with a corresponding radius of influence of 0.74 meters. The distance to the
lower boundary was then increased from 0 to 20 meters for the same well geometry with the
top of the screen at the upper impermeable boundary (i.e., Lw = Ls, referring to Figure 12).
The InORj/Rj) term and the radius of influence was calculated using Equation 29. As shown
on Figure 24, increasing the distance to the lower boundary had the effect of marginally
reducing the InORj/R,.) term from 1.86 to 1.51 in an exponential manner. The radius of
influence also decreased from 0.74 to 0.51 meters. Noting that the hydraulic conductivity of
the doughnut model is directly proportional to the IntRj/RJ term, compensating for partial
penetration in the method of Bouwer and Rice would therefore appear to have a negligible
effect on the calculated hydraulic conductivity.
FILTER PACK DRAINAGE AND RESATURATION IN PARTIALLY SUBMERGED
WELL SCREENS
Removal of a slug of water from a typical monitoring well with a partially submerged
screen and filter pack will cause drainage of the filter pack into the well. The degree to
which drainage of the filter sand may dominate water level recovery is a function of the
contrast between the hydraulic conductivities of the filter sand and the formation. If the filter
sand is less permeable than the formation, the determined hydraulic conductivity will be
predominantly that of the filter sand (see Chapter 3.3). A well with a filter pack with exactly
the same hydraulic properties as the formation may be considered as a naturally developed
well. Filter sands are typically coarser than the formation and have a higher hydraulic
conductivity, although clogging of the filter pack with fines or biofilm development may
reduce the conductivity of the filter sand. If the filter sand is more conductive than the
formation, drainage of the filter sand will dominate the recovery rate to the well until the
48
-------
1.9
1.85
1.8
1.75
1.7
1,65
1.6
1.55
1.5
1.45
13-
0
Fully Penetrating 1.524 m screen
I I I 1 I
8 10 12 14 16 18 20
Distance to Boundary (m)
Figure 24. Effects of partial penetration on the In(R/R.) term as a function of the distance to a lower
impermeable boundary (D1) for a 11.43 cm radius, 1.524 meter long, well screen. Using method of Bouwer
and Rice, 1976.
49
-------
filter sand reaches its specific retention. This leaves a desaturated filter sand column of
length H!. A typical recovery curve from a well with a partially submerged screen is shown
as Figure 25. As exemplified in the Figure, slug test data from partially submerged wells
with highly conductive filter sands typically exhibit a very steep curve in early time
representing the filter pack drainage, followed by a log-linear curve in mid-time representing
recovery of the casing and the filter sand by the formation, followed by a tailing off of head
with time. It is the mid-time, log-linear portion of the recovery curve that should be used for
determining the hydraulic conductivity (often very early time data is mistakenly used). The
transition between early and mid-time portions of the recovery curve (i.e., the point at which
the filter sand effectively stops draining and formation recharge becomes dominant) is a
function of the contrast in conductivities between the filter sand and the formation. If the
contrast is great, the recovery curve will exhibit a sharp transition. If the contrast in
conductivities is not great, then the transition zone may be temporally broad.
Drainage of the filter sand has the effect of significantly reducing the head difference
left for recharge by the formation. Ideally the initial head drop caused by removal of a slug
of water may be calculated by dividing the volume of a slug removed from the well by the
area of the well casing. If the filter pack does not drain, then the calculated initial head drop
(Ho,.) should compare well with the first water level readings collected after removal of the
slug. In the case of rapid filter pack drainage, the initial water level readings are typically
only a fraction of the calculated initial head drop. The reduction in the initial head drop due
to filter pack drainage is a function of the volume and specific yield of the draining sand pack
relative to the size of the well casing. The head remaining in the well after filter drainage
may be reduced, to less than a few tens of centimeters, which implies limited hydraulic stress
induced in the formation. A small hydraulic stress in the formation is pertinent to the radius
of influence of the test, and consequently. The volume of aquifer tested. A very small radius
of influence may also increase the effects of a well skin as was discussed in Chapter 3.3.
The recharge rate to a well is typically measured as the changing water level in a well
casing of a given radius. This method is appropriate for fully submerged screened wells with
or without a filter pack as long as the filter pack remains saturated at all times. For the case
of partially submerged well screens with filter pack drainage, recharge to the well is not only
filling the casing, but also an unknown portion of the drained filter pack. The recharge rate
of the well, and the resulting hydraulic conductivity, will be underestimated if the desaturated
volume of the filter pack is not considered. The additional volume of the filter pack may be
accounted for by replacing the casing radius in model solutions by an effective casing radius
(Re,). The effective casing radius includes an additional radial increment of account for the
resaturation of the filter pack. Slug test solutions are sensitive to error in the casing radius,
since the hydraulic conductivity is proportional to the square of this radius.
Bouwer and Rice (1976) and Bouwer (1989) have addressed the drainage and
subsequent resaturation of the filter pack. Bouwer suggests increasing the radius of the
50
-------
I
s
-2.5
0
200
400 600
Time (Sec)
800
1000
Figure 25. Slug test recovery curve from a typical monitoring well with a partially submerged well screen.
Well MMW-8; UCONN Motorpool, August 6,1991. After Binkhorst, 1992.
51
-------
casing (RJ by a fraction equal to the radius of hole (Rh) minus the radius of the casing,
multiplied by the porosity (n) of the sand pack:
(54)
This method provides an estimate of the effective casing radius. The hole dimensions and the
porosity of the filter sand must be accurately known. It should be realized that the volume of
water that drains from a filter sand is not a function of its porosity, but of its specific yield.
Specific yield values for fine to coarse sands typically used for filters may range from 1 to
30% (Freeze and Cherry, 1979), while the porosity may range from 15 to 48% (de Marsily,
1986). Using the porosity for the determination of the effective casing radius may
overestimate the volume of sand to be recharged and the hydraulic conductivity. If the
borehole is larger than the auger flights, due to formation caving or drill stem wobble, the
effective casing radius would be underestimated and consequently, the hydraulic conductivity
would be overestimated. The range of error involved in accounting for resaturation of the
filter sand was assessed for a typical monitoring well geometry. The well geometry
consisting of a 5.08 cm diameter well in a 22.86 cm diameter hole with a coarse filter sand
having a 45% porosity. The range of error using this geometry was calculated as
approximately one order of magnitude. For the same geometry, a range of error was
calculated using the filter sand porosity (45%) versus a hypothetical specific yield of 10%.
Using the porosity, the resultant hydraulic conductivity would have been overestimated by
approximately one third an order of magnitude or 330%.
As part of this study, a method has been developed to determine an effective specific
yield of the filter pack surrounding the well, and to calculate an effective casing radius. This
method assumes that hysteresis effects between wetting and drying is negligible and that the
volume of water released from the filter sand is small at late time relative to flow from the
formation. The method uses the early recovery data from typical monitoring wells to
calculate the volume of water released from the filter sand, as well as the volume of filter
sand from which it was released. Parameters are illustrated in Figure 26.
An initial drawdown, B^., in a well without filter pack drainage can be calculated by
equating the volume of water removed from the well (Vb) to an equal volume in the well
casing (V,.), Equation 55. The volume of the well casing is equivalent to the Hoc multiplied
by the area of the well casing (Aw), using the inside radius of the casing (Rci) as in Equation
56. The volume of the bailer (Vb), may be calculated from Equation 57 using the outside
radius (Bor) and length (B,) of the bailer. Equating 56 and 57 and solving in terms H,,,. yields
Equation 59 to calculate the initial drawdown.
52
-------
Rc
Water Table
H
oc
vb
BI
BOR
Figure 26. Illustration of parameters needed to calculate the initial head drop.
53
-------
Vb = 7tBOT2 * B,
(55)
(56)
(57)
(58)
Aw
The bailer must be fully submerged, and the aquifer re-equilibrated prior to bailer removal or
the calculated initial drawdown will be overestimated. The specific yield of a sand is equal
to the volume of water that may be drained per volume of sand.
The effective specific yield of the sand pack can be determined by using the initial drawdown
(H^.), as calculated from the bailer dimensions, and the head at which the
sand effectively stops draining. This head may be determined graphically from the natural
log of head versus time plot as the head (H,) at which log-linear recovery of the well
commences., This method is illustrated in Figure 27. The volume of water released from the
sand (Vw) may be calculated from the difference between Hoc and H1? multiplied by the inside
area of the well casing:
Vw =
2 * (Hoc - H,)
(59)
As shown in Figure 28, the volume of sand from which the water drained must be situated
above H,. In cases of highly conductive formations, a fraction of water may also come from
the formation during filter sand drainage, resulting in a broad transition zone and potential
over-estimation of the volume of water that actually drained from the sand pack. The volume
of sand (Vs) in the filter pack above the head at which linear recovery commenced may be
calculated from the radius of the hole (Rh), the outside radius of the screen (Rco), and H,:
Vs = 7r(Rh2 - Rco2)
2 *
(60)
An effective specific yield of the sand (Sy) may then be calculated using Equation 61. This
method assumes that the specific yield of the filter pack is uniform. Error in the hole
dimensions will be transferred to error in the specific yield. If the hole diameter and, thus,
the actual volume of sand is underestimated, the specific yield of the sand as calculated from
Equation 61 will be overestimated. This error is self-correcting when the effective casing
radius is calculated (see below).
100%
(61)
54
-------
a
Hoc = 4.17 ft =127.1 cm
0.5
0
= 0.75 ft =22.86 cm
0
200
400 600
Time (sec)
800
1000
Figure 27. Determination of the transition point (H,) oh the recovery curve of a typical monitoring well with a
partially submerged screen exhibiting filter sand pack drainage in early time.
55
-------
WATER
TABLE
Flgur* 28. Illustration of parameters used to estimate the specific yield of the filter sand pack from early time
recovery data.
56
-------
Take for an example, a 5.21 cm I.D./5.97 cm O.D. well screen surrounded by a filter
pack installed in a 22.86 cm hole. A slug test in this well using a 0.91 meter, 4.13 cm O.D.
bailer resulted in rapid sand pack drainage early in the test. Linear recovery was observed to
commence at 0.12 meters of head in the well. The total volume of the bailer (Vb) was
calculated using Equation 57 as 1.22 x 10"3 m3. The area of the inside of the well casing (Aw)
was calculated (Equation 56) to be 6.49 x 10"4 m3. Using these values and Equation 58, the
calculated initial drawdown in the well (ELJ was estimated to be 0.575 meters. The volume
of water released from the sand using Equation 59 is 9.63 x 10"4 m3. The volume of sand
where the water was released is 4.75 x 10"3 m3 using Equation 60. From Equation 61, the
specific yield is calculated to be 20.3%.
Once the effective specific yield of the sand pack has been determined, an effective
casing radius (Rce) can be calculated using Equation 54, by substituting the specific yield for
the porosity term, and by using the inside and outside radii of the well casing:
Rce =
Sy (Rh2 - Rc
2)
(62)
For the above example, the effective casing radius would be calculated as 5.7 cm using
Equation 62. Calculation of the effective casing radius is independent of potential error in the
specific yield resulting from poor control on the actual hole dimensions. This error will be
canceled out in Equation 62, since both the specific yield and the hole radius terms include
the inaccurate hole dimensions.
REDUCTION OF SCREEN INTAKE LENGTH IN PARTIALLY SUBMERGED
SCREEN WELLS
The screen intake length of a fully submerged screen well remains constant during a
slug test. However, removal of a slug of water from a well with a partially submerged screen
causes denaturation of the screen and filter sand. The formation can contribute to the
recovery of the well only through the saturated portion of the well screen. Hence,
denaturation of the filter sand effectively reduces the screen length. The effective screen
length would be at a minimum at the beginning of the test and increase as the screen and
filter sand are resaturated.
The importance of this effect would be a function of the desaturated screen length
relative to the overall screen length. In typical 5.08 cm diameter monitoring wells in 22.86
cm diameter holes, the desaturated screen length after filter sand drainage can typically be
less than 15 cm, while the overall screen length may range from 1 to 5 meters. To assess the
importance of screen denaturation, Figure 29 was generated. The Figure shows how shape
factor for the full ellipse model would decrease with increasing screen denaturation for wells
having initial screen lengths equal to" 1, 2, and 5 meters. Recall that the hydraulic
conductivity is inversely proportional to the shape factor. Therefore, if screen denaturation is
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I
1
Uc
I
co
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not accounted for, the hydraulic conductivity will be underestimated. Since the effective
screen length will vary during water level recovery, the log-linearity of the recovery curves
can be affected. This further complicates determining a meaningful hydraulic conductivity
value for wells with partially submerged screens.
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SECTION 4
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
Slug tests provide a useful means of estimating a formation's hydraulic conductivity in
the near vicinity of the screened interval of the well. Slug tests have the advantage of being
relatively rapid, inexpensive, and involve removal of little or no potentially contaminated
water. However, it has been shown that the evaluated hydraulic conductivity may be affected
by many factors, including the approximation of shape factors, evaluation of the radius of
influence, well skin effects, formation anisotropy and heterogeneities, and development of a
cone of depression. Also, in typical monitoring wells with partially submerged screen
sections calculation of hydraulic conductivity may be affected by drainage of the filter sand,
resaturation of an unknown percentage of the filter pack, a dramatic reduction in the initial
head drop due to filter sand pack drainage, and a non-constant screen intake length over the
course of the test. The effect of these factors on the evaluated hydraulic conductivity is a
function of the well geometry and construction, formation properties and the mathematical
models used for interpretation of the slug test data.
Sensitivity analyses yielded the following conclusions:
1) Available full ellipse shape factors vary by less than a factor of 2. The full ellipse
shape factor summarized by Hvorslev (1951) provides a workable average for a typical
monitoring well geometry.
2) The doughnut shape factor is exact for the model geometry, but contains an unknown
radius of influence. The doughnut shape factor is sensitive to changes in the radius of
influence, if the radius of influence is small, e.g., less than approximately 2 meters. A
mass balance approach to the volume of water removed indicates that the radius of
influence for a typical monitoring well may be less than this value.
3) The Bouwer and Rice method is the only available method to calculate the radius of
influence for the doughnut model. However, the method does not account for formation
properties and the magnitude of the initial head exchange. Using the Bouwer and Rice
method, the radius of influence was found to be equal to approximately one third of the
screen length for the typical monitoring well geometry.
4) Slug tests are very sensitive to a less permeable well skin, resulting in significant
underestimation of the formation's hydraulic conductivity. Given this sensitivity, wells
must be carefully constructed and drilled to minimize formation disturbance and
entrapment of fines. In addition, well should be well developed.
5) Overestimation of the formation conductivity due to a more permeable well skin may be
overcome by increasing the radius of influence with a greater initial head change.
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6) Slug tests provide a weighted average conductivity of formations that intersect the well
screen provided that the screen section remains fully submerged. This effect may be
minimized by constructing wells screened in discrete formations.
7) Slug tests do not appear sensitive to formation anisotropy, if the horizontal conductivity
is greater than the vertical.
8) For wells with filter sand packs, the radius of the borehole should be used as the intake
radius in slug test solutions.
With specific reference to a well with a partially submerged screen section:
9) The problems associated with this well geometry are best overcome by only using fully
submerged well screens for slug testing.
10) Water level recovery in a typical monitoring well in early time is dominated by rapid
draining of the filter sand evidenced by a steep portion of the recovery curve in early
time. Hence, this portion of the recovery curve should not be used in calculating
hydraulic conductivity. Calculation of hydraulic conductivity using the early portion of
the recovery curve may overestimate the formation conductivity by over an order of
magnitude.
11) Tailing off of recovery data in late time is a result of the development of a cone of
depression. Using the late time data may underestimate the formation conductivity.
12) The middle log-linear portion of recovery data from typical monitoring wells represents
recharge of the well and the filter sand by the formation. Interpretation of this portion
of the recovery curve must include resaturation of the sand pack's specific yield, or the
calculated hydraulic conductivity will be underestimated. Use of the filter sand porosity
as opposed to the specific yield may result in significant overestimation of the formation
conductivity.
13) Slug-in or mandrel-in tests should not be conducted in wells with partially submerged
screen sections due to saturation of the unsaturated filter sand and formation.
14) A method was presented to calculate the sand pack's specific yield from early recovery
data, and to incorporate the specific yield into slug tests solutions with an effective
casing radius. This method, assumes a uniform specific yield along the length of the
desaturated filter sand column. The method may be used when there is a sharp
transition in the recovery curve between filter sand drainage and formation recharge.
15) To use Hvorslev's (1951) Time Lag Method for slug tests in typical monitoring wells
with a rapid initial recharge, the initial head drop should be regressed as the Y-intercept
of the middle log-linear recovery data.
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16) The great reduction in initial head drop associated with filter pack drainage may
increase well skin effects by reducing the radius of influence. This head reduction may
be reduced using filter sands with lower specific yields and by reducing the annular
space between the well screen and the hole. This effect may also be reduced by
increasing the volume of the slug out.
17) A reduction in screen intake length due to filter pack drainage was determined not to be
significant unless the desaturated screen length becomes large relative to the overall
screen length.
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67 -&U.S. GOVERNMENT PRINTING OFFICE: 1994 - 590-001/80337
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