EPA
Introduction to Fractured
Rock Hydrogeology
Presented by Joseph Alfano
July 10, 1997
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How fractures form
¦ Fractures are formed by tectonic stresses within the
earth's crust
• Isostatic adjustment
—Up and down movements of the crust to come to
equilibrium with gravity
—Uplift from removal of overburden by erosion or
unloading of glacial ice sheets
—Downward movement by loading of sediments or
the advancement of a glacial ice sheet
— Isostatic adjustments create predominately
tensional fractures (example — Joints with vertical
orientation)
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How fractures form (cont'd.)
¦ Stress relief fractures are also caused by removal of
overburden
¦ The rock under pressure at depth expands as it nears
the earth's surface (overburden pressure is removed)
¦ Results predominantly in horizontal tensional fractures
¦ Stress relief fractures may occur at pre-existing planes
of weakness
¦ Bedding planes in sedimentary rock or foliation in
metamorphic rock
¦ Stress relief fractures decrease in number and increase
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How fractures form (cont'd.)
(After H. Cloos, 1992; From Balk, 1937; From Dennis, 1992)
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How fractures form (cont'd.)
¦ Lateral tectonic forces cause faults
H Faults are fractures with a displacement and can extend
for miles
¦ Faults occur at any angle and are compressional and
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How fractures form (cont'd.)
¦ Whether a fault or fracture is more or less permeable
than the surrounding rock depends on the material that
fills the fracture, if any
• Clays low permeability
• Broken rock higher permeability
¦ Shear zones are narrow zones in a rock body where
shear stress is accommodated by plastic deformation in
the mineral grains
¦ Shear zones can be zones of increased permeability
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How fractures form (cont'd.)
¦ The orientation of a fracture or any planar feature is
described by strike and dip
¦ Strike is the intersection of a horizontal plane and the
plane of a fracture
¦ Strike is a horizontal line that is measured as an angle
from geographic north or south, or as an azimuth with
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¦
How fractures form (cont'd.)
N
Representation of a plane by stereographic
projection. From Spencer (1977)
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How fractures form (cont'd.)
¦ Dip is the angle the fracture plane makes with the
horizontal in a direction perpendicular to strike
Dip direction
angle
¦ Any planar feature is oriented by its strike and dip
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Size of fractures
¦ Micro fractures (fissures)
• Narrow aperature — less than 100 microns (micron =
10"6 meters)
• Often limited to a single layer
• Limited length and width
• If interconnected can be significant
¦ Macro fractures
• Wide aperature — greater than 100 microns
• Develop across various layers
• Can be of considerable length (miles)
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Size of fractures (cont'd.)
50 100
Fracture Openings (microns)
Statistical frequency curve of opening width . Reprinted with
permission of Elsevier Scientific Publishing Company. From
van Golf Racht (1982)
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Size of fractures (cont'd.)
Table 1. Statistical parameters of the distri-
bution of fracture apertures at exposures of
Pikes Peak granite
(range 0* 12-5* 41 mm)
Parameter log, (mm) (mm)
Arith. mean
Geom. mean
Median
S.D.
Skewness
Kurtosis
-0-700
0-000
-0-100
0-043
-0-060
3-051
0-932
1-000
0-905
1-045
0 I
02
OS
1 0
20
SO
IOO
200 -
300
4ao-
500
•OO
100
800-
900
95 0
980
990
99 3
99 8
99 9.
1 1 1 1—I I I I I
1 1—i—r
/
"/
:/
/
j i i i i i i i i
OS 10
Aperture (2b), mm
J I I L
SO
Log-probability plot of cumulative frequency of 256 apertures of
exposed fractures, Pike's Peak granite (replotted from BIANCHI
[24]). From Snow (1970)
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Size of fractures (cont'd.)
(Zq;)3- q
12* i>-1m
(2ai )3-g
(k/ Dh—0.032)
(k/Dh>0.032)
20:
0.1 mm %
fl 9 mm
u-'mm
0.4 mm T
//////""
¦777777777
0.7 mm
'//////,
1.0 mm
7777777777,
k / Dh
s 0.032
0.25
* 0.032
0.25
* 0.032
0.25
s 0.032
0.25
* 0.032
0.25
k0 [ rn/s 1 | =k (soil)
0.6-10
-6
03 - W
os-1a5
02- 1a5
0.4 -10
,-4
0.2 • 10
0.2-10'
0.1 -10
-j
0.6 • 10
0.3-10
silt
sand
gravel
From Wittke (1990)
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Porosity
¦ Primary porosity
• Voids formed at the same time as the rock formation
• Spaces between the sand grains
voids
p= : X100
V total
¦ Secondary porosity
• Voids formed after rock formation
—Faults, joints and solution cavities
P = P + P
1 total 1 primary ' secondary
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Characterization of fractures
¦ Orientation — strike and dip — measured with a
Brunton Compass
¦ Size of fractures — aperature and length
• Is the fracture filled?
—Minerals
—Clay
—Gravel
• To what degree is the fracture weathered?
¦ Fracture density
• Linear fracture density (average fracture frequency)
• Number of fractures per unit length of a straight line
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Characterization of fractures (cont'd.)
*. _ Total number of fractures _ 20 fractures _ 4 fractures
~ Total length of sampling line 5 meters ~ meter
• The inverse of fracture frequency is fracture spacing
(d)
. 0.25 meters
d =
fracture
Roughness — is the degree fracture walls are not smooth
(Rr)
• Roughness is defined as the ratio of mean irregularity
(I) to the hydraulic aperature (Hr). (Sen, 1995)
Rr < 0.032 is smooth (
R.l
Rr > 0.032 is rough
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Lineament study
¦ Fracture traces and lineaments are the surface
expression of the intersection of fractures and the
ground surface
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Lineament study (cont'd.)
From Lattman and Nickelsen (1958)
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Lineament study (cont'd.)
40*50:— N
I MILE
Bedrock fault
directions (opprox)
-1 Photogeologic fracture
traces ¦
Bedrock |oint directions •
Photogeologic fracture traces and bedrock
joints. From Lattman and Nickelson (1958)
10
A Bedrock joints
76 observotions
10
9 Photogeologic frocture traces
U4 observations
Histograms of photogeologic fracture traces and
bedrock joints. From Lattman and Nickelsen
(1958)
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Lineament study (cont'd.)
Trends of Linear Stream Segments and
Major Erosional Features
w
From Alfano (1993)
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Follow-up lineament study with field study
¦ Check nature of observed lineament
• Are they manmade or natural?
¦ Collect outcrop data on fracture orientation and
characteristics
¦ Use Brunton Compass for strike and dip measurements
¦ Include orientation of any bedrock fabrics — Bedding
planes in sediments rock and foliation in metamorphic
rock
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Scanline analysis
¦ A series of scanlines are established on the rock face.
The orientation of the rock face is measured. The
traverses (scanlines) are established is different
direction to compensate for directional bias.
Perpendicular traverses are good
¦ Measure the angle between the fracture trace and the
scanline. Measure the distance along the scanline to
the fracture trace and any cross cutting relationship
between fracture traces (which fracture cut off other
fractures). If the strike and dip of the fracture trace can
be measured do so
¦ The fracture traces are plotted on a stereonet and the
fracture sets established by grouping fracture traces
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Scanline analysis (cont'd.)
The number of fractures in a fracture set are corrected
to account for the directional bias imparted by the angle
the fracture trace is to the scanline. The corrected
fracture frequency (Nc) is determined by
fVLu*. ® = Acute angle between
"sin# scan,'ne anc*
fracture trace
A scanline analysis is useful when a statistical analysis
of the fracture data is performed. (Priest and Hudson,
1976 and Priest, 1993)
¦ The 2 dimensional scanline fracture groups can be
correlated with 3 dimensional orientation (strike and dip)
fracture group data
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Problem with surface measurements
¦ Problem with field survey
• Fracture characteristic, such as aperature,
weathering measured at the surface will be different
then fractures at depth under the surface because
fractures open up (aperature increases) as fracture
near the surface (over burden pressure is decreased)
and weathering is more intense near the surface
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Directional bias
¦ The directional bias experienced in scanline analyses is
also encountered when boreholes are drilled
¦ The vertical orientation of most boreholes will
selectively locate low angle fractures and miss high
angle fractures (joints)
From Aquilera (1980)
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Directional bias (cont'd)
Outcrop versus core fractures
Field and core data
0-10
11-20
81-90
Horizontal
Vertical
BEB
Perpend Fol
sss
Against Fol
uz
With Foliation
21-30 31-40 41-50 51-60 61-70 71-80
Fracture dip angles (degrees) FromAifano (1993)
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Core analysis
¦ Bedrock coring is an expensive operation and oriented
coring is as much as three times more expensive than
ordinary coring
¦ Without oriented core the orientation of the fractures are
difficult to determine
¦ Coring can create fractures that are hard to distinguish
from original fractures. Although weathering along the
fracture is an indication of original fractures and which
fractures are conducting water
¦ Intensely weathered zone obscure individual fractures
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Core analysis (cont'd.)
¦ If original fractures are distinguishable an average
fracture frequency is easy to determine (fractures per
meter)
¦ Weathering and mineralization along fractures is
directly observable
¦ Good lithological description
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Core analysis (cont'd.)
¦ Problems of core analysis:
• Coring is expensive and the analysis of the core is
time consuming. Borehole geophysical methods are
less expensive and less time consuming to analyze
• Vertical cores are bias toward intersecting low angle
fractures
• High angle fractures are underrepresented
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635
630
625
620
6)5
610
605
600
595
590
585
580
is (cont'd.)
From Alfano (1993)
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Core analysis (cont'd.)
Rock quality designation — RQD used to express the
degree to which the rock is competent
n__Sum of the length of all pieces over 4 inches (100mm)
KUU~ Total length recovered * 100
¦ Problem:
• If a five foot core section has a fracture every foot
there are four fractures but the RQD is 100%
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Geophysical method
¦ Conventional well logs
• Neutron logs detect porosity (fracture) by detecting
the hydrogen in water
• Density logs detects porosity by measuring scattered
gamma radiation
¦ The above logs have a resolution of about 0.1 meters.
The shallow penetration of these logs may only
measure the gouging and wash out area adjacent to the
borehole
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Geophysical method (cont'd.)
¦ Caliper logs measure variation in the diameter of the
borehole
¦ Best response for areas of washout where fracture
zones (more than one fracture) or large fractures are
located
¦ Small fractures will not show up
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Geophysical method (cont'd.)
¦ Fluid conductivity logs
• Detects a change in the electrical conductivity of the
water
• Need to replace the water in the well bore with low
conductivity (deionized) water first so water entering
the wellbore from the fractures will have higher
conductance
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Geophysical method (cont'd.)
¦ Acoustic televiewer
• Uses reflected sonic waves to detect fracture
openings or soft weathered zones
• Can be used in clear or murky water
• Fracture orientation is easy to determine because it is
equipped with internal compass
Determination of dip angle and azimuth of fracture from the ATV log. D = diameter of borehole. From Cohen (1995)
dip angle=arctan(H/D)
dip azimuth
N E S W N
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Geophysical method (cont'd.)
ACOUSTC TB-EVIEWET? LOGS
Q.
o
SW-4
SW-3 SW-2 SW-1 0-0 SE-1 SE-2 SE-3
SE-4
R
j
r
S
¦100 J .
Acoustic televiewer logs of all nine wells. From Cohen (1995)
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Geophysical method (cont'd.)
¦ Problems of televiewer
• Gouged out closed fracture appear open
• Individual fractures cannot be determined in intensely
weathered or washed out areas
• Fill material in fractures cannot be directly seen
• Can only be used in water filled portion of well
• Not all fractures conduct water
• Must use other methods to determine conductive
fractures
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Geophysical method (cont'd.)
¦ Downhole television camera
• Inexpensive method to get video picture of the
borehole
• Fractures and fill material directly observable
• Borehole does not have to have water in it
¦ Problems of downhole television
• Cannot see in murky water
• Many downhole cameras do not have orientation
devises (wire line)
• Fractures hard to see in dark rocks
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Geophysical method (cont'd.)
¦ All borehole detection methods are limited by ability to
only measure conditions near the borehole where
drilling damage is present
¦ Measurements near the borehole may not be
representative of condition in the rest of the rock
¦ Geophysical methods for detection between boreholes
or a borehole and the surface do exist
¦ Seismic, radar, electromagnetic tomography
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Surface methods
¦ Electromagnetic methods
• Uses electric and magnetic fields to detect anomalies
in conductance
• In the sounding mode (different depth but same
location) horizontal fractures are detectable
• In the profiling mode (same depth but different
location) vertical and dipping fractures are detected
• Can penetrate to 100 meters if bedrock is exposed
• Clay-rich overburden greatly reduces depth of
penetration
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Surface methods (cont'd.)
¦ Ground-penetrating radar
• Also uses electromagnetic energy to detect changes
in electrical conductance
• Can penetrate to 100 meters if bedrock is exposed
• Clay-rich overburden greatly reduces depth of
penetration
¦ If overburden obscures detail in the bedrock
electromagnetic method can still define the bedrock
surface
¦ Depression in the bedrock surface may indicate high
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Surface methods (cont'd.)
Natural Resistivity
Gamma (16 In)
SE-2
Caliper ATV
Fluid Identified
Electrical Conductive
Conductivity Fractures
Reference from Cohen (1995)
SW-1
ATV HPF FEC
SE-1
I ATV HPF FEC
SE-2
I ATV HPF FEC I
e
*
g
I §
5 3"
ATV s Acoustic HPF ~ Mea* Pl'5C
Televiewer Log ~ Flowmeter
5
-m.
%
I
FEC
FLuid Bectrtaal
Conduct Ivily
Reference from Cohen (1995)
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Borehole flow logs
¦ Impeller flowmeter
• Water in the borehole flows past the impeller
• The flow rate (liters/minute) is calculated from the
RPMs and the diameter of the borehole
• Natural flow rates are usually too low to turn the
impeller so a pump and pressure transducer are
placed near the top of the water surface to induce
higher upward flow and to measure the drawdown
• As the impeller is lower in the borehole the flow rate
through the impeller will decrease as conductive
fracture zones are passed
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Borehole flow logs (cont'd.)
Q out-<
flow meter
NOT TO SCALE
0
I
fracture zone
~to Instrumentation
pressure
transducer
pump
downhole impeller
flow meter
packer
MEASUREMENTS: CALCULATED INFLOWS:
Q [l/minj
5 10 15
H h
H
pump rate:
12.5 L/min
z
(m]
-20-
-40-
-60-
-80-
pump
• data
all flow is upward
Q [l/min]
5 10 15
H 1 1
(12.5 - 5 = 7.5)
Schematic of impeller flowmeter test Example flow profile and calculation of wellbore
configuration. From Cohen (1995) inflows.
From Cohen (1993)
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Borehole flow logs (cont'd.)
The transmissivity (T) for the entire borehole interval is
calculated from (Cohen, 1995)
Q = Flow rate
23Q
T= r As ) As = Change in drawdown over one log
Uogcydej cycle on drawdown(s) versus log
time (t) plot (ft)
¦ The transmissivity for each conductive fracture internal
(T|) is calculated by
to Q| = Flow rate for each conductive
Q interval
¦ Relative conductivity for each interval is established by
values of ^
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Borehole flow logs (cont'd.)
¦ Problems of impeller flowmeters:
• Assume porous media equivalence — radial flow will
may not be valid with some fracture geometries
• Transmissivity values should be considered order of
magnitude estimations. Cohen (1995)
• Relative conductivity for each interval is established
by values of _Ql
q
• Near the bottom of the borehole flow can decrease
below the stall rate of the impeller giving the
appearance of no flow
• If the rock formation being tested has a low yield the
water level will drop below the pump or conducting
fractures
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Borehole flow logs (cont'd.)
¦ Thermal — pulse flowmeter
• Used in observations wells while another well is
pumped
• A pulse of heat is generated by a heat grid
• Sensors above and below the heat grid detect the
time it takes the heat to flow to one or the other
sensor giving the flow rate and direction of flow (up
and down)
• Can measure lower flow rates than the impeller type
so natural flow may be measured
• Interval transmissivity is calculated with the same
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Borehole flow logs (cont'd.)
-2
y-
Q [l/min]
¦ 1 0
H
+ 1
2
lm]
-20
-40
-60
-80
downward flow
upward and downward flow
J converging and exiting borehole
upward flow
} water entering borehole
$ upward flow
water entering borehole
Hypothetical flow profile in a well during pumping in an adjacent
From Cohen (1993)
well
Problems of thermal — pulse flowmeter:
• Same as impeller method but sensitivity of thermal
pulse flowmeter detects turbulence, eddies and
changes in the flow field that develop over time
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Displaying the fractures
¦ Histogram graphical display of frequency distribution
c
®
U
w
0)
a.
120
150
180° Strike
From Dennis (1972)
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Displaying the fractures (cont'd.)
¦ Rose diagram
• A specific type of histogram with circular or semi-
circular shape for displaying a frequency distribution
in relation to compass bearings
Strikes of Outcrop Fractures
From Alfano (1993)
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Displaying the fractures (cont'd.)
¦ Rose diagrams represent 2 dimensional features
¦ Stereonets allow the representation of features in 3
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Stereonet
¦ There are equal area (Schmidt) and equal angle (Wulff)
stereonets
The equal area stereonet is used for fracture analysis
Polar and equatorial prvjtcnon
Equatorial equal -angle net
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Stereonet (cont'd.)
P N
Stereographic projection of an inclined plane, (a) Projection to the
horizontal equatorial plane, (b) Corresponding stereogram. (After Phillips,
1971. From Ragan, 1973)
¦ Planes (fracture, bedding planes, foliation) are
represented as curved lines
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Stereonet (cont'd.)
¦ When there are a lot of fracture it is convenient to plot
the "poles" to the fractures
¦ Poles are a line perpendicular to the fracture plane.
When the pole is projected on the stereonet it is
represented by a point
¦ The pole is 90° from the curved line representing the
fracture
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Stereonet (cont'd.)
From Ragan (1973)
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Stereonet (cont'd.)
Use a
Kalsbeek counting
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Stereonet (cont'd.)
MW 22 Core
Poles to the Fractures n
From Alfano (1993)
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Stereonet
(cont'
d.)
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Stereonet
(cont'd.)
M W 25 Co r-e
Poles to the Fractures
From Alfano (1993)
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Stereonet (cont'd.)
N
Surface Measurements
0 MW 25 Poles to the Fractures
° MW 22 Poles to the Fractures
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Fractured rock aquifer systems
Fractured rock aquifer systems can have three or more
layers with different hydraulic characteristics (e.g.
hydraulic conductivity, storage, porosity)
(After Lattman and Parizek, 1964. From Freeze and
Cherry, 1979)
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Fractured rock aquifer systems (cont'd.)
¦ Shales, metamorphic and igneous rocks are
impermeable for all practical purposes unless they are
fractured
¦ Sandstone and limestones can have primary porosity
¦ Fracture density and size typically increase as you
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Fractured rock aquifer systems (cont'd.)
¦ Bedrock transitions into the partially weathered rock
(PWR) which is typically highly fractured and has
increased rock matrix porosity from weathering
¦ PWR zone is usually the thinnest layer but can be the
most permeable
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Fractured rock aquifer systems (cont'd.)
¦ The PWR grades into the weathered residual soil
(residuum)
¦ The weathered residuum from igneous and
metamorphic rock is called saprolite when it retains the
original rock fabric (layering)
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Fractured rock aquifer systems (cont'd.)
¦ The transition from bedrock to PWR or PWR to
saprolite is more abrupt if associated with a horizontal
(low angle) fracture zone
¦ Boulders of competent bedrock are found in the
weathered residuum — floaters
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Fractured rock aquifer systems (cont'd.)
¦ Monitoring wells are needed in all layers of a fractured
rock aquifer system to determine the hydraulic
interaction of the layers
¦ Upward or downward hydraulic gradients can exist
between them
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Fractured rock aquifer systems (cont'd.)
M 650 _
615 _
610 _
635 _
630 _
625 _
620 _
615 _
61 O _
605 _
600 _
5Q5 _
500 _
585 _
A
MW8
MW15
MW6
MW1 6
A '
MW2Q MW1
__ 650
_ 615
_ 610
_ 635
_ 630
_ 625
_ 620
_ 615
_ 6 1 O
_ 605
_ 600
_ 595
_ 5QO
_ 585
= 25 M
LEGEND: CROSS SECTION A
JRESIDUUM
JPWR
BEDROCK
Static Water Level
SAPROLITE
iiSFILL
From Alfano (1993)
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Fractured rock aquifer systems (cont'd.)
From Alfano (1993)
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Where to locate monitoring wells
¦ In lineaments identified from aerial photographs
¦ Bedrock surface maps and overburden thickness maps
are useful for locating depressions in the bedrock
surface that can indicate a high angle fracture zone
(joints)
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Where to locate monitoring wells (cont'd.)
\%\v\ \ \
* \ \ \ v \ ^ W\\T Cr
\t&> 72
From Alfano (1993)
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Where
to locate monitoring wells
(cont'd.)
From Alfano (1993)
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Where to locate monitoring wells (cont'd.)
¦ Extrapolation from nearby outcrops. Good for high
angle and low angle fractures
¦ Horizontal (low angle) fracture are much easier to hit
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Potentiometric surface maps
¦ Only monitoring wells screened entirely in the fractured
bedrock are used to produce a potentiometric surface
map and only if it is proved that they are interconnected
¦ Pump a bedrock well and observe the response in the
other bedrock monitoring wells
¦ Separate fracture systems are treated as separate flow
systems
¦ Open borehole bedrock wells run the risk of connecting
formerly unconnected fracture systems allowing
contamination or DNALPs into uncontaminated portion
-------
Potentiometric surface maps (cont'd
From Alfano (1993)
-------
Potentiometric surface maps (cont'd.)
From Alfano (1993)
-------
Potentiometric surface maps (cont'd.)
¦ Strictly speaking, a potentiometric surface map
represents horizontal flow only so wells used to produce
it should be screened at approximately the same
elevation
¦ Monitoring wells located along a dipping fracture zone
are not measuring horizontal flow but flow along a
fracture zone. This information is useful but not a
potentiometric surface map by definition
¦ Know what the water level measurements in fractured
rock aquifer are showing you
¦ Distinguish between true horizontal flow and flow
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Anisotropy in fractured rock aquifers
B Weathered overburden aquifers are typically considered
homogeneous isotropic so that groundwater flow (q) is
in the same direction as hydraulic gradient (J)
-------
Anisotropy in fractured rock aquifers
(cont'd.)
¦ Fractured bedrock aquifers are often anisotropic so flow
is usually not in the same direction as hydraulic gradient
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Anisotropy in fractured rock aquifers
(cont'd.)
¦ When a single well is pumped in a homogenous
isotropic aquifer the cone of depression forms a circle
¦ When a single well is pumped in an anisotropic aquifer
the cone of depression forms an ellipse
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Directional transmissivity
A vector has magnitude and direction
y = North
A
vx = 5 cos 60°
vw = 5 sin 60° :
x = East
¦2.5 m/sec
4.3 m/sec
v direction N 30° E
vl magnitude 5.0 m/sec.
vx =1 vl cos 0
vy M vl sin 0
"Vx"
"I vl cos 60t
2.5
m/sec
V -
Lvyj
Jvlsin 60°.
4.3
m/sec
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Directional transmissivity (cont'd.)
0 = angle counterclockwise from x-axis
V
0 = arc tanrp if 0 lies in quadrants 1 and 4
vx
0 = arc tan^- + 180 if 0 lies in quadrants 2 and 3
1
-------
Directional transmissivity (cont'd.)
¦ Magnitude of vector is
I vl = ^Vx2 + vy2
Ivl = yVx2 + Vy2 + ^z2
¦ Unit vector (e)
cos 0
sin 0_
_ y JVIagnitude and direction
e~lvl~ Magnitude
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Directional transmissivity (cont'd.)
¦ Anisotropic aquifer — groundwater flow qjs not in the
same direction as the hydraulic gradient J except in the
direction of the Principal Hydraulic Conductivity
Directions
¦ Td(0) is the ratio between the magnitude of the
groundwater flow to the component of hydraulic
conductivity J in the direction of q
y = North
i
Td (9)
^~x = East
-------
Directional transmissivity (cont'd.)
A unit vector e in the direction of q is
e =
cos 0"
sin 0
Groundwater flow q can be written
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Directional transmissivity (cont'd.)
¦ The component of the hydraulic gradient J in the
direction of the gradient flow q~is the dot product of J
with unit vector e
J.
a
cos#
A
w
sin#
0 = angle between x-axis and q
-------
Directional transmissivity (cont'd.)
W) = =A=
J • e
T.(0) =
i i
J, cos#+ J, sin <9
-------
Directional transmissivity (cont'd.)
Another way to write the directional transmissivity is as
a tensor
TTxx Txylsymmetrical tensor
q = | | J T _ T
|_Tyx TyyJ 1 XV ~ 1 V*
Q< _ T Txx Txyl I Jxl
Q/ LTyx TyyJ [_Jyj
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Directional transmissivity (cont'd.)
¦ Values for Txx, Tyy, Txy are obtained from aquifer
testings
|"y c] Txx = 7 m/sec
T = I 5 gl Tyy = 3 m/sec
Txy = Tyx = 5 m/sec
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Directional transmissivity (cont'd.)
You know hydraulic gradient (J) from potentiometric
surface map can find groundwater flow
y = North
x = East
J = N 60 E or 9 = 30°
magnitude is gradient = .5
cos 30°
Jx = .5 cos 30°
Jx = .43
Jy = J sin 30°
Jy = .5 sin 30°
Jy = .25
J =
"j;
7m3
A
.15
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Directional transmissivity (cont'd.)
¦ Groundwater flow discharge per unit width q induced by
the hydraulic gradient J is
Qc _ I Txx Txyl I Jx I
Q/ |_Tyx TyyJ [_ Jyj
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Directional transmissivity (cont'd.)
¦ Matrix multiplication
l~7 5l [.431 _ T7(.43)46(.25l
L5 3j L.25J ~ L5(.43)+3(.25j
2x© Qx 1
can do _ T 3.01 +1.251
~ L2.15 + .75J
_ r 4.31
L2.9J
-------
Directional transmissivity (cont'd.)
q = TJ
q =
qxl
[qyj =
f| - 7{4.2t) +
|cj = 72679
jcj = 5.2rtfsec
"4.5
2.9
rtfse
rrfse
(2.3)
Direction:
9 = arctan^
qx
2 9
6 = arctarH-
4.3
0 = 4CP
J = 0.5 @ N 60° E
q = 5.2 m/sec @ N 50° E
x = East
-------
Graphical method to estimate difference
between groundwater flow (q) and
hydraulic gradient (J)
From Freeze and Cherry (1979)
Kx = maximum hydraulic conductivity axis
Kz = minimum hydraulic conductivity axis
Determination of direction of flow in an anisotropic region with KJKZ = 5. Freeze and Cherry (1979)
z
Equipotential line
Direction of flow
Direction of hydraulic gradient
-------
Does groundwater and contaminants flow
under the stream?
0.2 S
0.1 S
0
jj^
|g§|
V/n
'T—- . \ \CJ_L>^ \ / ^T- 1-1
, 11,' ,rnT* ^ ' • 1 1 -i -j 11
"1 i mi*'—l Ht" i i i" \ ii ~T~-m
\y w /
\>A /
r*-r~^ M
0 0.1 S 0.2 S 0.3 S 0.4 S 0.5 S 0.6 S 0.7 S 0.8 S 0.9 S S
Regional and local flow. Freeze and Cherry (1979)
-------
Does groundwater and contaminants flow
under the stream (cont'd.)?
Most likely to occur in area of regional faulting where
significant fracture zones exist at depth
-«io.ooo ft
~5.000
SEA lEVEl
-5.000
10.000
From Berg, 1962 & Dennis, 1972
-------
Mathematical Models for Determining the
Hydraulic Properties of Fractures and
Fractured Rock Bodies
¦ Two approaches:
• Discrete modeling
-------
Discrete model
¦ The equation for one-dimensional flow through a single
fracture is:
q = Kf dh
dx
q = volume of flow per unit of time per unit length of the
cross-sectional area of the fracture (b x w) with
w= 1, (L/T)
Kf = hydraulic conductivity of the fracture, (L/T)
dh/hx = hydraulic head gradient, (no units)
-------
Cubic law
Q b2pg dh
bw 12 ^ dx
Q b3pg dh
w 12 n dx
¦ The hydraulic conductivity of the fracture (Kf) with units
of length/time is given by
Kf = b2pg
12 JLl
kf = b2
12
;1_b3pg dh
-------
Cubic law (cont'd.)
Q = volume of flow per unit of time, (L3/T)
q1 = volume of flow per unit of time per unit width of
the fracture (L2/T)
K = hydraulic conductivity of the fracture, (L/T)
k = permeability of fracture (L2)
dh/dx = hydraulic head gradient, (no units)
w = unit width of the fracture, (L)
b = thickness of the fracture aperature, (L)
p = density of water, (M/L3)
H = dynamic viscosity of water, (M/TL)
g = acceleration due to gravity (L/T2)
-------
Discrete fractures
¦ For a number (n) of parallel fractures with equal
aperture (b) the discharge per unit width is
q,=Knb^h
dx
12 ju
-------
Discrete fractures (cont'd.)
If the aperture varies the hydraulic conductivity can be
written for the average aperture (b1)
b-r^gib3
i=*
K- (b1)3 PQ
12 "
-------
Rock mass permeability
¦ The formulation of a fractured rock mass permeability
was produced by Romm and Pozinenko (1963), Snow
(1965, 1969), and Bianchi and Snow (1969). The
permeability of a rock mass with continuous fractures is
described by a second rank tensor. Snow (1969)
defined this tensor in relation to a sampling line
designated by (D) by
b = fracture apertures, (L)
Pj = direction cosines of normal to the fracture planes
D| = direction cosines of the sampling line
my = PiPj, matrix formed by the direction cosines of the
normal to the fractured planes
8:: = Kronecker's delta which is 1 when i = j and 0 when
-------
Rock mass permeability (cont'd.)
¦ The permeability of a rock mass along several sampling
lines is determined by summing the permeability
tensors of each sampling line.
¦ Once the permeability tensor is determined, the
principal permeability axes are parallel to the
eigenvectors and the values of the principal
permeability are equal to the eigenvalues
-------
Problems with the discrete model
¦ These mathematical formulations of fractured rock
mass permeability and flow are limited by the
assumption of infinite (in relation to the test area)
fractures that are continuous in their own plane. In
reality, these conditions are only found in small test
areas
¦ The discrete formulae are also limited by the
dependence of these equations on the knowledge of
fracture aperatures (b). Real fractures do not have
constant aperatures and are in contact in some areas
and open in others. The aperature values would have
to reflect the effective aperature, that is the aperature
-------
Problems with the discrete model (cont'd.)
9 Fracture apertures are extremely difficult to define in the
field. Even measurements of fracture apertures from
down-hole televiewers may not represent aperatures
away from the borehole
¦ Not all fractures are interconnected
¦ Not all fractures conduct water
¦ Fracture aperature is effected by changes in stress
¦ Numerical computer models with statistical packages
are being developed for these types of analysis
-------
Porous media equivalent models
¦ SINGLE POROSITY models created for porous
granular aquifers
¦ Theis solutions for confined transient radial flow and
modifications for unconfined, leaky confined, and
anisotropic aquifers
¦ DOUBLE POROSITY models for two overlapping
continuum. Barenblatt and others (1969)
¦ Hydraulic conductivity and storage values for the
fracture systems
-------
Porous media equivalent models (cont'd.)
¦ Porous media equivalent models are valid within a
representative elemental volume
¦ The volume of aquifer for which a single value for a
parameter (hydraulic conductivity, porosity) is measured
with an increase in volume
Macroscopic
Heterogeneous
Homogeneous
Volume
Microscopic and macroscopic domains and the representative elementary volume
V3 (after Hubbert, 1956; Bear, 1972)
From Freeze and Cherry (1979)
(/)
O
o
-------
Fractured rock aquifers as porous media
¦ Long and others (1982) determined that fractured
crystalline rock aquifers behave more like porous media
when:
• The fracture density increases
• Aperatures are constant rather than distributed
• Orientations are variable rather than constant
• Large volumes of rock are tested
¦ Long and Witherspoon (1985) investigated the influence
of fracture length using two-dimensional computer
models
• Fracture length is more important than fracture
frequency in the ability of the fracture system to
behave like a porous medium
• Increase in the fracture lengths increase the
-------
Fractured rock aquifers as porous media
(cont'd.)
B The porous behavior of a fracture set is determined by
a polar coordinate plot of the square root of the
directional hydraulic conductivity (K(0))1/2
¦ If (K(0))1/2 plots as an approximate ellipse then the
fracture system is behaving as an equivalent porous
medium (Long and Witherspoon, 1985)
-------
Porous media equivalent models (cont'd.)
/Meosuicdvalueiof /.. , .
1-K|jX,Xt
Plot of the measured values of 1/(Kg)1/2, and the corresponding "best fit" ellipse
From Long and Witherspoon (1985)
Directional hydraulic conductivity ellipsoid. The semiaxes of the ellipsoid are the
square roots of the principal hydraulic conductivities Kf, K2P and K3P
From Hsieh and others (1985)
-------
Porous media equivalent models (cont'd.)
An equivalent porous medium is described by a
symmetrical hydraulic conductivity (K) or transmissivity
(T) tensor _ _
q = K J
qx
Kxx
Kxy
K
qy
Kyx
Kyy
K
qz
Kzx
Kyz
K
XZ
yz
zz
In a symmetrical tensor Kyx = K
'xy
XZ =
XZ
Kzy , K,2
There are six unknowns in a 3D symmetrical tensor
-------
Single porosity models
¦ Developed for primary porosity aquifer
or
¦ Secondary porosity (fracture) only when matrix
permeability is negligible compared to fracture
-------
Single porosity models (cont'd.)
Theis assumptions
1. Aquifer is confined by impermeable layers
2. Flow is laminar not turbulent
3. Aquifer is horizontal and only horizontal flow occurs
4. Aquifer is of infinite extent (large in relation to tested
area)
5. Pumping well is fully penetrating and fully screened
-------
Single porosity models (cont'd.)
Theis assumptions (cont'd.)
6. Pumping well diameter is small (no significant wellbore
storage)
7. Aquifer is pumped at a constant rate (Q is constant)
8. Aquifer is of constant thickness
9. Aquifer is Homogeneous and Isotropic
Homogeneity = Transmissivity or transmissivity tensor is
the same everywhere in the aquifer
Isotropy = Transmissivity value is the same in all three
-------
Single porosity models (cont'd.)
¦ Conditions of horizontal laminar flow in a homogeneous
equal thickness aquifer is problematic in fractured rock
aquifers
¦ At best, porous media equivalence solution gives an
order of magnitude estimate of aquifer properties
¦ Use with other methods of investigation (geochemical,
geological, and qualitative data)
-------
Fracture system as porous media
equivalent
¦ Confined radial flow — Theis solution log drawdown(s)
versus log time (t)
1
B
1
•
A
fracture
1
B
•
A
•
unfractured
rock
A: pumped interval B: observation interval
From National Research Council (1996)
-------
Fracture system as porous media
equivalent (cont'd.)
h0- h (m) 0.1
0.01
10
0.1
-Field data
-Type curve
I I IK I I
N
P
latch
oint
H 7
,o-'
sf-
o-
!
i ~
y
/1
t \
0 = 40
r = 55 m
10"3 m3/s
/io*
103
104
t(s)
At match point
W(u) = 1.0
u = 1.0
t = 250 s
h0 "h = 0.14 m
10
W (u)
0.1
10 102
1/u
0.01
10-
10"
T =
0W(u) (4.0 x 10"3)(1.0)
47r(h0-h) (4 0)(3.14)(0.14)
0.0023 m2/s (15,700 U.S.gal/doy/f 1)
_ 4uT t (4.0)0.0) (0.0023) (2 50) _ _4
5 1 :— = =7.5 x 10
rZ (55.0)
Determination of T and S from hQ — h versus t data using the log-log curve-
matching procedure and the W(u) versus 1/u-type curve
From Freeze and Cherry (1979)
-------
Fracture system as porous media
equivalent (cont'd.)
¦ Cooper-Jacob straight-line method (1946) drawdown(s)
versus log time (t)
0 75
-0 50
¦C
I
o
x:
0.25
10
T =
10*
2 3Q
4>rAh
Q-4 0 *1
r = 55 m
0"3 m3/s (c
(18011)
10 //s ,63
U S. gal/mi
n)
/
/ L
t
=0 32m-
| (1 06 ft)
/
*
/
= 440 s
30
2 0
10
•c
I
o
103 I04
Ms)
(2 3)(4 0 »10~s)
(4)(3.14)(0 32)
10=
= 00023 m2/s
10s
1
(t>>
„ 2.25T t0 (2.251(0 00231(440) c
S = s—* - ¦, = 7 5 * (0
155)
(a)
(a) Determination of T and S from h0 — h versus t data using semilog method;
(b) semilog plot in the vicinity of an impermeable boundary. From Freeze and
Cherry (1979)
-------
Fracture system as porous media
equivalent (cont'd.)
¦ A good fall-back method for fractured rock
• Several porous media equivalent solutions exhibit this
behavior including anisotropic and double porosity
models
¦ Change in the slope of the line indicates possible
hetergeneities or boundaries
• Increase in slope
—Impermeable boundary — closing of fracture
system
• Decrease in slope
—Recharge boundary — increase in permeability of
-------
Fracture system as porous media
equivalent (cont'd.)
¦ Problem:
¦ No all straight lines on the drawdown versus log time
plot in fracture systems is actually laminar radial flow
¦ The following method is used to determine if the values
calculated with the straight line method are valid. Sen
(1995)
-------
Fracture system as porous media
equivalent (cont'd.)
Using the T and S value calculated from the straight
line method plot values of dimensionless time (uf) and
dimensionless drawdown (wf) on semi-log graph paper
from the following equations
uf= r2_S
4tf T
and
wf = 4rrT s?
Q
¦ Plot the dimensionless time on the log x-axis and
dimensionless drawdown on the y-axis. Graph paper
must have same scale as original straight-line plot
-------
Fracture system as porous media
equivalent (cont'd.)
¦ The field drawdown (sf) and time (tf) values for the
above equation are obtained from the original field data
with the drawdown corresponding to several arbitrarily
picked times
¦ The semi-log plot of dimensionless drawdown (wf)
versus log dimensionless time (uf) must have
approximately the same slope as the original drawdown
versus log time field data plot for the calculated T and S
values to be valid
-------
Fracture system as porous media
equivalent (cont'd.)
Semi-confined (leaky) aquifers
¦ Hantush (1960)
• Includes water released from storage from confining
layers
• Not a good assumption for impermeable rock matrix
like crystalline rock
Hantush and Jacob (1955) and Hantush (1956)
• Water just passes ^
through confining layers ^
(K values only)
• Better assumption for
fractured impermeable
_ lesi tranaminive fracture
rO /*** 1^ A: pumped interval
' ^ ¦ » B: observation interval straddling pumped fracture
C: observation interval straddling fractured rock
From National Research
Council(1996)
-------
Fracture system as porous media
equivalent (cont'd.)
1/u
Theoretical curves of W (u, r/BO versus 1/u for a leaky aquifer (after Walton,
1960)
From Freeze and Cherry(1979)
-------
Fracture system as porous media
equivalent (cont'd.)
¦ Multiple Aquifer System
• Neuman and
Witherspoon (1969)
• Hantush (1967)
highly transmissive bedding plane fracture
less transmissive vertical fracture
A: pumped interval
B: observation interval straddling pumped fracture
C: observation interval straddling unpumped fracture
From National Research Council (1996)
-------
Fracture system as porous media
equivalent (cont'd.)
¦ Complex Fracture Systems
• Beyond simple analytical models
• Numerical computer modeling
fracture zone
— fracture
A: pumped interval
B: observation interval
From National Research Council (1996)
-------
Fracture-induced anisotropy
¦ All porous media equivalent models discussed so far
are isotropic models
¦ Fractures often induce preferred directions of flow —
anisotrophy
¦ K and T values obtained from isotropic models are the
effective K or effective T value of an anisotropic aquifer
* K effective =\/Kx Ky 2 dimensions
•T effective = ^Tx Ty Tz 3 dimensions
-------
Fracture-induced anisotropy (cont'd.)
¦ Isotropic values are the geometric mean of the
directional values
¦ The maximum transmissivity value will be larger than
-------
Fracture induced anisotrophy (cont'd.)
¦ Anisotropic aquifers
• An elliptical cone of depression is created when an
anisotropic aquifer is pumped
• The long axis of the ellipse is in the direction of the
maximum transmissivity
• The short ellipse axis is the minimum transmissivity
(90° from the maximum T) direction
-------
Fracture induced anisotrophy (cont'd.)
¦ Directional transmissivity Td (0)
• Two dimensional Td (0) is written in the form
(Popadopulous, 1965)
D
Td (0) = Tyy cos2 0 + Txx sin2 0 - 2 Txy cos 0 sin 0
• D is the determinant of the transmissivity tensor
-------
Fracture induced anisotrophy (cont'd.)
Dof
T
T
yx
T
T
xy
yy
= T T
1 xx 1 yy
T T
xy 1 yx
Symmetrical tensor so Txy = Tyx
D = T..T...-(T
-------
Fracture induced anisotrophy (cont'd.)
¦ When Td (0) is calculated for different angles (0) a
polar coordinate plot of V Td (0) versus direction (0)
yields an ellipse
¦ The long axis of the ellipse is equal to the square root
of the maximum transmissivity (/ Tmax ) and the short
axis is equal to the square root of the minimum
transmissivity (V Tmin )
-------
Anisotropic aquifer methods
¦ Popadopulous Method (1965)
• All Theis assumptions except anisotropic
• Must have one pumping well and at least three
observation wells. More are better
• Three approaches
—Type curve matching to Theis type curve
—Straight-line, semi-log based on Cooper-Jacob
method
—Modified graphical approach using polar plot of
7Td (0) versus direction (0). Easiest method
¦ Problems with the Popadopulous Method
• Aquifer must be confined (no leakage)
-------
Anisotropic aquifer methods (cont'd.)
¦ Hantush (1966) and Hantush-Thomas (1966)
• Same assumptions as Theis method except
anisotropic and can be a leaky aquifer
• Need one pumping well and at least three
observation wells
• Uses values of (T/S) from abundant isotropic
methods. S = storage
¦ Problems with the Hantush-Thomas method
• Method assumes that Td (0) versus direction (0) is an
ellipse and calculates values of Td (0) accordingly
• Cannot check if aquifer is behaving as a
homogeneous anisotropic aquifer unlike the
Popadopulous method
-------
Anisotropic aquifer methods (cont'd.)
¦ Problem with all porous media equivalent
homogeneous anisotropic methods when applied to
fractured rock aquifers
• Hetergencities can make aquifer behave as an
anisotropic aquifer making the calculated directions
and values incorrect (National Research Council,
1996)
• If observation wells are screened in areas of higher
(more or larger fractures) or lower (less or smaller
fractures) transmissivity the discharge (Q) is not
constant throughout the test area
• Anisotropic calculation is invalid
-------
Anisotropic aquifer methods (cont'd.)
Three dimensional anisotropy
• In fractured rock systems, the third principal
transmissivity axis may not be vertical (perpendicular
to horizontal flow)
A: pumped interval
B: observation interval
From National Research Council (1996)
-------
Anisotropic aquifer methods (cont'd.)
¦ Hsieh-Neuman Cross-hole method (1985)
• Can calculate the three-dimensional transmissivity
ellipsoid if fractured rock system behaves as
homogeneous anisotropic aquifer
• Move packers to obtain at least six measurements
from at least three boreholes that do not lie in a plane
• Simplest approach to the method is for packer
intervals that are small compared to the distance
between the boreholes
-------
Anisotropic aquifer methods (cont'd.)
Ahpo= erfc
r T]
4b
AhPD = dimensionless head
tD = dimensionless time
erfc = complimentary error function
Log-log plot of 2hpd versus td. From Hsieh and Neuman (1985)
-------
Anisotropic aquifer methods (cont'd.)
¦ Develop at least six (six unknowns) simultaneous
equation to determine the three dimensional
conductivity tensor
¦ The degree to which an ellipsoid is formed is the
degree to which the rock is an anisotropic porous
medium equivalent
-------
Double porosity models
¦ Used when there is primary (rock matrix) and secondary (fracture)
porosity (fractured limestone or sandstone)
¦ Barenblatt and others (1960) developed the double porosity
concept
¦ A homogeneous isotropic porous rock matrix continuum overlaps a
homogeneous isotropic fracture system continuum
¦ The fractures have high transmissivity and low storage. The rock
blocks have low transmissivity and high storage
¦ Water is pumped from the fractures lowing the pressure in the
fractures. The rock matrix then releases water from storage into
the fractures
-------
Double porosity models (cont'd.)
¦ Warren and Root model (1963) — for pumping well only
• Fracture system is idealized as orthogonal system
with cubic rock blocks (homogeneous isotropic)
actual reservoir model reservoir
After Warren and Root. From Aquilera (1980)
-------
Double porosity models (cont'd.)
¦ The log-log type curve resembles the unconfined
aquifer type curve. The geology of the aquifer must be
known to distinguish which behavior is being observed
¦ The semi-log plot of dimensionless pressure (PD)
versus log dimensionless time (tD) clearly shows the
three parts of the double porosity model
Prciiur* build-up
AP(»)
i
y
/
/
s
/
/
s
s
/
/
/
/
/
/
/
A P
/
/T / '
¦ /
\ /
1 X
1 /
1 /
V
In t
—~
-------
Double porosity models (cont'd.)
¦ First straight line segment (early time)
• Radial flow through the fracture system. Storage from fracture
system only. Can analyze with Cooper-Jacob straight line
method
¦ Flat segment (intermediate time)
• Water released from storage from the rock matrix
• The length of this segment depends on the difference in the
storage in the rock matrix and the fracture system (w)
¦ Second straight line segment (late time)
• Radial flow through fracture system
• Same slope as first straight line segment (same transmissivity)
-------
Double porosity models (cont'd.)
Double porosity models have variables to represent the
interaction of the fracture system and the rock matrix
Permeability contrast ratio
A = oc d*?
Kf
Coefficient of block surface
km = permeability of rock matrix
kf = permeability of fractures
rw = radius of pumping well
_4n(2n+1)
n = number of fracture planes
L = length of block (cubes)
-------
Double porosity models (cont'd.)
¦ Specific storage ratio (w)
_ S Sf = specific storage in fractures
S + St, Sm = specific storage in rock matrix
As Sf -> 0 then w -> 0 purely rock matrix porosity
As Sm -> 0 then w-> 1 purely fracture porosity (blocks
impermeable)
t,
w = - from semi leg plot
-------
Double porosity models (cont'd.)
Kazemi model - For observation well
• Fracture system idealized as horizontal fractures with
slab of homogeneous isotropic rock
MODEL RESERVOIR
ACTUAL RESERVOIR
FRACTURE
MATRIX
MODEL RESERVOIR
MATRIX
A FINITE RESERVOIR WITH CENTRALLY LOCATED WELL
Idealization of naturally fractured porous medium. II, Warren-Root model; III, Kazemi model (after Kazemi). From Aquilera (1980)
-------
Double porosity models (cont'd.)
¦ Kazemi model (cont'd.)
• If there is not a large contrast between hydraulic
properties (K and S) of the fractures and rock matrix,
or
• If observation well is far from the pumping well:
—The first straight line segment will be missing
—The flat intermediate segment will be short or
missing
—The third straight line segment is analyzed with
the Cooper-Jacob straight line method
¦ More details on the solution to double porosity models
are in "Applied Hydrogeology for Scientists and
Engineers" by Zekai Sen (1995)
-------
Vertical Fracture Model - Not a double
porosity model
Vertical fracture in a homogeneous isotropic porous
aquifer
• Gringarten and others (1974)
• Gringarten and Witherspoon (1972)
• The vertical fracture is of finite size with the pumping
well in the middle of the fracture
• The pumping well and any observation wells
screened in the fracture will have a 1/2 unit slope on
a log drawdown versus log time plot
• At a late time in the pumping well or for an
observation well far from the pumping well (within the
equivalent porous medium part of the system), the
aquifer will behave like horizontal radial flow (Cooper-
Jacob) straight line method when plotted on
drawdown versus log time graph
-------
Horizontal Fracture Model - Not a double
porosity model
B Horizontal fracture in a homogeneous isotropic porous
aquifer
• Gringarten and Ramey (1974)
• The pumping well is in the middle of a horizontal
fracture surrounded by an equivalent porous medium
aquifer
• The initial response can be the 1/2 unit slope on the
log drawdown versus log time plot but altered by a
dimensionless head factor (HD)
= thickness of equivalent porous medium
rf v km rr = half the length of the fracture
kr = fracture permeability
km = equivalent porous medium permeability
-------
Summary of characteristic well responses
if)
10
o
c
Z
£
o
~o
101
£
03
Q
10°
101 10
Time (f)
Log drawdown versus log time plot
• Wellbore storage — unit slope (45°)
Linear flow
• High conductivity fracture
• Low conductivity fracture
1/2 slope
1/4 slope
-------
Summary of characteristic well responses
(contld.)
i/)
o
"D
£
03
Decrease in
permeability oj:
discharge ^
boundary
\ Increase in
ermeability or
recharge boundary
104
103
Time (t)
Drawdown versus log time plot (semi-log)
Radial flow — transmissivity inversely proportional to
the slope of the straight line
T =
2.3 Q
4k (As / log cycle)
-------
Packer Test
¦ Packer test
• Zanger (1953)
• U.S. Dept of Interior (1977)
¦ Have been used for years in geotechnical engineering
to determine the hydraulic conductivity of discrete
fracture zones
¦ Problems
• Increase in pressure during injection test can open
fractures, causing an increase in conductivity during
testing
-------
Packer Tests (cont'd.)
¦ Must be careful to measure pressure changes above
and below packer interval that may be caused by "short
circuiting" or bad seal on the borehole wall
Invalidation of test results: a) leakage at the packer; b) flow around the packer
From Wittke (1990)
-------
Problems with porous media equivalent
aquifer test methods — Cohen (1995)
¦ Glosses over detail to give you the composite behavior of the
aquifer
• There can be features (fractures) within the system with a order
of magnitude higher hydraulic conductivities than calculated with
porous media equivalent models
¦ For pumping well
• Well bore storage and skin effects (damaged or enhanced
conditions caused by the drilling or development of well) can
mask important early time data
¦ For observation well
• Results may not be from the area of the system you are
interested in
-------
REFERENCES
Alfano, J., 1993, M.S. Thesis, Hydrogeological Evaluation
of a Fractured Crystalline Rock Aquifer, Hendersonville,
North Carolina: Geology Department, Georgia State
University.
Armstrong, R.L., and Dick, 1974, A model for the
development of thin overthrust sheets of crystalline
rock: Geology, January, p. 35-40.
Arnold, M.D., Gonzalez, H.J., and Crawford, P.B., 1962,
Estimation of reservoir anisotropy From production data:
Journal of Petroleum Technology, August, p. 909-912.
Atobrah, K., 1985. On the relationships of flow in crystalline
rock aquifers, a solution to the fracture problem:
Hydrogeology of Rocks of Low Permeability in Memoires of
the International Association of Hydrogeologist, v. XVII,
p. 527-537.
Barker, J.A., 1988, A generalized radial flow model for
hydraulic tests in fractured rock: Water Resources
Research, v. 24, n. 10, p. 1796-1804.
Berkowitz, B., Bear, J., and Braester, C. , 1988, Continuum
models for contaminant transport in fractured porous
formations: Water Resources Research, v. 24, n. 8,
p. 1225-1236.
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Wilson, C.W., and Witherspoon, P.A., 1970, An investigation of
laminar flow in fractured rocks: Geotechnical Report No.
-------
Witherspoon, P.A., and Neuman, S.P., 1967, Evaluating a
slightly permeable caprock in aquifer gas storage; I.
caprock of infinite thickness, Society of Petroleum
Engineers Journal, July, p. 949-955.
Witherspoon, P.A., and Tsang, Y.W., 1981, New approaches to
problems of fluid flow in fractured rock masses:
Proceedings of the 22nd US Symposium on Rock Mechanics,
Massachusetts Institute of Technology, p. 3-22.
Wittke, W. , 1990, Rock Mechanics- Theory and applications with
case histories, Springer-Verlag, 1066 p.
Zanger, C.N., 1953, Theory and problems of water percolation:
-------
-------
TaDle A1.1 Definitions, Dimensions, and SI Units for Basic
Mechanical Properties
Properly
Symbol
Definition
SI unit
SI symbol
Dimension of unit
Derived Basic
Mass
M
kilogram
kg
Length
I
meter
m
m
Time
i
second
s
s
Area
A
A = r-
m:
Volume
1
r = /-'
m3
Velocity
,
i = /,'/
m/s
Acceleration
u
a = l,t-
m/s:
Force
r
F = Ma
new ton
N
N
kg-m/s2
Weight
>.
ii' = \/c
newton
N
N
kg-m/s:
Pressure
p
p = Ft A
pascal
Pa
N/m;
kg/m-s:
Work
if
IV = F!
joule
J
N-m
kg*m:/s:
Energy
U ork done
joule
J
N • m
kg-m:;'s:
Mass density
p
p = M r
kg/m3
Weight density
V
N'/m3
kg/m:-s:
Stress
c :
Internal
pascal
Pa
N;'m:
kt'm-s:
response to
external p
Strain
e
( -
Dimensionless
Young's modulu-.
E
Hooke's law
N/m:
kg/nvs-
Table A1.2 Definitions. Dimensions, and SI Units for Fluid
Properties and Groundwater Terms
Dimensions of unit
Property
Symbol
Definition
SI unit
SI symbol
Derived
Basic
Volume
I'
v = n
liter
< = m1 >¦ 10"3)
(
(
m!
Discharge
Q
q = i>:<
t s
m; s
Fluid pressure
P
p = r'A
pascal
P*
N'm:
kc'nvs:
Head
h
m
Mass density
P
P = A 1j I ''
kg'rn3
Dynamic
M
Newton's law
centipoise
cP
cP, N-s,m:
k g'm ¦ s
viscosity
(= N-s'm: • I0"3)
Kinematic
V
= M P
centistoke
cSt
cSt
m: \
viscosity
(= m:/s -¦ 10"°)
Compressibility-
a. P
a = I IE
m:/N
m-s:/kg
Hydraulic
K
Dares's law-
cm/s
m s
conductivity
Permeability
k
k = K'vlps
cm:
m:
Porosity
n
Dimensionless
Specific storage
s,
S, = pg(a - nP)
l/m
Storativity
s
S = S,b'
Dimensionless
Transmissivity
T
T = Kb'
m :/s
'b, thickness of confined aquifer
From Freeze and Cherry (1979)
b = thickness of the fracture zone or interval being tested
s = drawdown = hmlijl - hfinil
r = distance from pumping well to observation well
rw = radius of pumping well
K' = hydraulic conductivity of the confining layer
b' = thickness of the confining layer
Tu, T16, T20 = Td (0) = directional transmissivity in the direction
-------
APPENDIX A
THE BOULTON (1954, 1963) UNCONFINED AQUIFER METHOD
s = 4%Tmu"u"i)
(22)
t 2 q
ua = <23)
A 4 TC
Z*Sy
u„ = -
B 4T£
(24)
_r
B
(25)
ttS.,
Using the Early Time A-curves
MW-14
Q = 15150.50 cm3/min
r = 10668.00 cm
b = 3048.00 cm
Match Point Coordinates
W (uA, r/B) = 10.00
l/uA =1.00
s = 240.00 cm
t = 3 0.00 min
-------
Use Equation (22) to solve for transmissivity (T)
Use Equation (23) to solve for storativity (S).
MW-16
Q = 15150.50 cm3/min
r = 13563.60 cm
b = 3048.00 cm
Match Point Coordinates
W(uA, r/3) = 10.00
l/uA =1.00
s = 560.00 cm
t = 14.00 min
r/B = .30
Use Equation (22) to solve for transmissivity (T)
Use Equation (23) to solve for storativity (S).
MW- 20
Q = 15150.50 cm3/min
r = 10668.00 cm
b = 3048.00 cm
Match Point Coordinates
W(uA, r/B) = 10.00
l/uA = 1.00
s = 30.50 cm
t = 10.50 min
r/B .10
Use Equation (22) to solve for transmissivity (T)
-------
APPENDIX E
THREE POINT SOLUTION TO DETERMINE THE ORIENTATION OF THE
FRACTURE ZONE
_ Elevation D - Elevation A
Elevation C - Elevation A
tan 0 = Elev3Ci°n B ~ Elevation A (21)
325
8 = amount of the dip angle
Calculate the length of line AD from Equation (26).
/
as B
The strike of the fracture zone is determined
graphically by the orientation of a line connecting the
intermediate fracture zone well and the point with the same
-------
elevation as the intermediate well fracture zone is
determined by the length of AD.
Calculate the dip angle amount by Equation (27).
Using MW-14. MW-15, and MW-16
A = MW-16 fracture zone elevation = 2045.00 ft
B = MW-14 fracture zone elevation = 2059.00 ft
C = MW-15 fracture zone elevation = 2089.00 ft
Fracture zone orientation = S47W 9.4SE
Using MW-14, MW-16. and MW-22
A = MW-22 fracture zone elevation = 2009.00 ft
B = MW-16 fracture zone elevation = 2045.00 ft
C = MW-14 fracture zone elevation = 2059.00 ft
Fracture zone orientation = S45W 8.1SE
Using MW-14. MW-16. and MW-20
A = MW-20 fracture zone elevation = 2019.00 ft
B = MW-16 fracture zone elevation = 2045.00 ft
C = MW-14 fracture zone elevation = 2059.00 ft
-------
APPENDIX C
COOPER AND JACOB (194 6) STRAIGHT LINE METHOD FOR
A RADIAL FLOW AQUIFER
AS = 2 ' 3 g (28) S = 2 ' 25Tt° (29)
47tT r2
As = change in drawdown per log cycle
t0 = time at which the extended straight line
intersects the time axis (s = 0)
MW-14
Q = 15150.50 cm3/nun
r = 10668.00 cm
b = 152.40 cm
Use Equation (28) to solve for transmissivity (T)
Use Equation (29) to solve for storativity (S).
First Straight Line Segment
As = 35.05 cm
tn = 26.00 min
T = 1.32 cm'/sec
K = 8.65 X 10'3 cm/sec
-------
Second Straight Line Segment
As = 3.66 cm
t0 = negligibly small
T =
K =
S =
12.63 cm:/sec
8.29 X lCT-
negligibly small
MW-16
Q = 15150.50 cm3/niin
r = 13563.60 cm
b = 152.4 cm
Use Equation (28) to solve for transmissivity (T).
Use Equation (29) to solve for storativity (S).
First Straight Line Segment
As = 8 7.17 cm
t0 = 13.00 min
T = 5.30 X 10"' cirr/sec
K = 3.48 X 10"3 cm/sec
S = 5.06 X 10"6
Second Straight Line Segment
As = 2.44 cm
t0 = negligibly small
^ = 75.76 cm:/sec
K = 4.97 X 10"1
-------
MW-20
Q = 15150.50 cm3/nun
r = 10668.00 cm
b = 152.4 cm
Use Equation (28) to solve for transmissivity (T).
Use Equation (29) to solve for storativity (S).
First Straight Line Segment
As = 5.94 cm T
t0 = 11.00 min K
S
= 7.78 crrr/sec
= 5.10 X 10'2 cm/sec
= 1.01 X 10a
Second Straight Line Segment
As = 3.37 cm T=3.37 cm2 /sec
t0 = 100.00 min K = 2.21 X 10":
S = 4.00 X 10
-------
APPENDIX D
HANTUSH (19 56, 19 60) METHOD FOR A SEMI - CONFINED AQUIFER
5 = -p- H(u, P) (30)
4n T
U= — (31) p2 = (£!£f) '32)
4rt H 15 b2 KSs'
MW-14 - Radial Flow Period 50-300 min
Q = 15150.50 cm3/min Match Point Coordinates
r = 10668.00 cm fi = 0.10
H (u, (3) =1.00
1/u = 1.00
s = 25.00 cm
t = 25.30
Use
Use
Use
Equation
Equation
Equation
(30) to solve for transmissivity (T)
(31) to solve for storativity (S).
-------
MW-16 - Radial Flow Period 30-300 min
Q = 15150.50 cm3/min Match Point Coordinates
r = 13563 . 60 cm j8 = 0 . 30
H (u, /?) = 1.00
1/u = 1.00
s = 86.00 cm
t = 11.20
Use Equation (30) to solve for transmissivity (T).
Use Equation (31) to solve for storativity (S).
Use Equation (32) to solve for K'Ss'
MW-20 - Radial flow Period 40-600 min
Q = 15150.50 cmJ/min Match Point Coordinates
r = 10668.00 cm 0 = 0.20
H(u ,0) = 1.00
1/u = 1.00
s = 5.00 cm
t = 10.80
Use Equation
Use Equation
Use Equation
(30) to solve for transmissivity (T)
(31) to solve for storativity (S).
-------
APPENDIX E
HANTUSH AND JACOB (1960) METHOD FOR A SEMI-CONFINED AQUIFER
S = —~l L(u, v) (33)
4jxT
(—)
u = (34) — = 4 T— = S — (35)
4 Tt b'r't
MW-14 - Radial Flow Period 50-300 min
Q = 15150.50 cmVmin Match Point Coordinates
r=10668.00cm v=0.30
b = 152.40 cm L(u,v) = 1.00
1/u = 1.00
s = 29.00 cm
t = 34.00
MW-14 - Extended Radial Flow Period 50-800 min
Q = 15150.50 cm3/min Match Point Coordinates
r=10668.00cm v=0.20
b = 152.40 cm L(u,v) = 1.00
-------
s = 23.50 cm
t = 2S.uO
Use Equation
Use Equation
Use Equation
(33) to solve for transmissivity (T).
(34) to solve for storativity (S).
(35) to solve for K'
MW-16 - Radial Flow Period 30-300 min
Q = 15150.50 cm3/min Match Point Coordinates
r= 13563.60 cm v=0.20
b = 152.40 cm L(u,v) = 1.00
1/u =1.00
s = 63.00 cm
t = 16.00
MW-16 - Extended Radial Flow Period 30-650 min
Q = 15150.50 cm3/min Match Point Coordinates
r=13563.60cm v=0.20
b = 152.40 cm L(u,v) = 1.00
1/u = 1.00
s = 64.00 cm
t = 16.20
-------
Use Equation (34) to solve
Use Equation (35) to solve
MW-20 - Radial Flow Period
Q = 15150.50 cm3/min
r = 10668.00 cm
b = 152.40 cm
for storativity (S).
for K'
40-600 min
Match Point Coordinates
v = 0.10
L(u,v) = 1.00
1/u =1.00
s = 3.48 cm
t = 12.30
Use
Use
Use
Equation
Equation
Equation
(33)
(34)
(35)
to solve for transmissivity (T)
to solve for storativity (S).
-------
APPENDIX F
CALCULATION FOR MAPPING THE CONE OF DEPRESSION
HANTUSH (1966 a&b) AND
HANTUSH AND THOMAS (1966)
S = —L(u,v) (36)
4 nr.
u = (37)
4 t Tn
MW-14
From the Hantush Anisotropic Method
Tc = 41.57 cm:/min
T|4/S = 836810.50 cirr/min
v = 0.30
Q = 15150.50 cm
Solve Equation (36) for L(u,v).
A value for "u" is found from Leaky Aquifer Type Curve or a
table of L(u,v), v, u values.
For any given time (t) and any given drawdown (s) Equation
-------
t = 200.00 min
s = 5.00 cm r = 28343.20 cm = 929.90 ft
s = 10.00 cm r = 21647.50 cm = 710.20 ft
s=100.00cm r= 258.70 cm =<8.50 ft
MW-16
From the Hantush Anisotropic Method
Tc = 19 .14 cm2/nu.n
T16/S = 2874550.70 cm2/min
v = 0.20
Q = 15150.50 cm
Solve Equation (36) for L(u,v).
A value for "u" is found from Leaky Aquifer Type Curve or a
table of L(u,v), v, u values.
For any given time (t) and any given drawdown (s) Equation
(37) is solved for the distance (r).
t = 200.00 min
s = 5.00 cm
s = 10.00 cm
s = 100.00 cm
r = 64337.80
r = 55303.90
r = 14780.60
cm = 2110.82 ft
cm = 1814.40 ft
cm = 484.90 ft
-------
From the Hantush Anisotropic Method
Tc = 3 8 8.92 cmVmin
T20/S = 2789368.20 cm2/min
v = 0.05
Q = 15150.50 cm
Solve Equation (36) for L(u,v).
A value for "u" is found from Leaky Aquifer Type Curve or a
table of L(u,v), v, u values.
For any given time (t) and any given drawdown (s) Equation
(37) is solved for the distance (r).
t = 200.00 min
S = 5.00 cm r = 14938.19 cm =490.10 ft
s = 10.00 cm r=6680.56cm=219.18ft
s = 100.00 cm r = 0.0 cm = 0.0 ft
The calculated distances (r) are marked along the rays
from the pumping well to the observation wells. The same
drawdown value (s) along each ray are joined by an arc of an
ellipse with the equipotential line forming an ellipse in a
homogeneous anisotropic aquifer. The result is a map of the
-------
APPENDIX G
HANTUSH (1966 a & b) AND HANTUSH-THOMAS (1966) METHOD FOR A
HOMOGENEOUS ANISOTROPIC AQUIFER
The transmissivity determined from any radial flow
solution is taken as the effective transmissivity (Tc) .
Ta = yfT^Ty (38)
T, = maximum transmissivity
Ty = minimum transmissivity
s = —2—RT(u') (39)
4 nT0
u' = (40)
4tTn
Tn is the transmissivity in the direction (0) of a ray (n)
originating at the pumping well.
From Maasland (1957) and Hantush (1966)
Tn = T(B) = ^ (41)
cos2(0 + a) + —^ sin2(0 + a)
-------
The illustration above shows the relationship of the
rays (defined by a line from the pumping well to each
observation well) to the x-axis (maximum transmissivity
direction). 6 is the angle from the x-axis to the first
ray. Since is the angle from the first ray to another
arbitrary ray a, = 0. The angle from ray 1 to ray 2 is
designated as a2 and the angle from ray 1 to ray 3 is a3 and
so on. So the (8 + 0^) term refers to the angle between the
nth ray and the x-axis.
-------
From Equations (41) and (43;
cos2 (0 + a.) + m sin2 (0 + an)
—i = a = 2 2_ (44)
Tb cos20 + m sin26
so that al = 1
Solving Equation (44) for m yields
r_ Tx _ T An COS28 - COS2 (8 + c„)
T T ¦ - -
V Ty sin2 (0 + an) - an sin20
For three rays (observation wells), solving Equation
[44) for 6 yields
„ (a3 - 1) sin2a, - (a, - 1) sin2a3
tan (20) = -2 2—— J —2 —i — J -1- (46)
(a3 - l) sin 2o2 - (a2 - 1) sin 2a3
From the Hantush (1956, 1960) semi - confined homogeneous
isotropic method for the short radial flow period.
tc14/s
Tel6/S
Te20/S
= 1124567.40 cm:/min
= 4106501.00 cm2/niin
-------
From the geometry of the wells
a2 =39°
a, = 0°
a3 = -30°
Calculated with Equation (44)
a, = 1.0000
a, = 3.6516
a3 = 1.5588
Calculated with Equation (46)
0 = 8.6135°
Calculated with Equation (45)
m = Tx/Ty = 6.715
Tx and Ty are calculated with Equation (42).
T,4, T16, and T20 are calculated with Equation (41) .
S is calculated with the original values of Tel4/S, Tel6/S,
-------
From the Hantush and Jacob (1955) semi - confined
homogeneous isotropic method for the short radial flow
period.
Tcl4/S = 836810.50 cm2/min
Tc16/S = 2874550.70 cur/nun
Tc2o/s = 2313134.6 cm2/min
From the geometry of the wells
= 39°
a, = 0°
a3 = -30°
Calculated with Equation (44)
a, = 1.0000
a: = 3.4351
a3 = 1.2427
Calculated with Equation (46)
e = ii.8°
Calculated with Equation (45)
-------
From the Hantush and Jacob (1955) semi-confined
homogeneous isotropic method for the extended radial flow
period.
Tc14/S = 981088.10 cm2/min
Tc16/S = 2839062.40 cm2/min
TC2o/S = 2313134.6 cm2/min
From the geometry of the wells
a2 =3 9°
a, = 0°
a3 = -3 0°
Calculated with Equation (44)
a, = 1.0000
a: = 2.894
a3 = 1.227
Calculated with Equation (46)
6 = 11.2°
Calculated with Equation (45)
-------
Tx and Ty are calculated with Equation (42) .
T14, T16i and T20 are calculated with Equation (41) .
S is calculated with the original values of TeU/S, Te]6/S,
Te20/S and the calculated values of TU/ T16, and T20.
For a system to be considered truly homogeneous and
anisotropic the values of Te and S from all observation
-------
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