a. slug test review and theory...outline 1. what is a slug test? 2. how are slug tests performed? 3....
TRANSCRIPT
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A. Slug Test Review and Theory
Author
Gary A. Robbins
Copyright © 2016 Gary A. Robbins. All rights reserved.
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Outline
1. What is a slug test?
2. How are slug tests performed?
3. Slug test responses
4. Theory and models of overdamped
responses in low K formations
5. Theory and models of underdamped
responses in high K formations
6. Examples
7. References
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What is a Slug Test?
• One well field test for determining hydraulic
conductivity (K)
• Entails:
– “Instantaneously” adding or removing a slug of
water in a well and monitoring water level
recovery (head vs. time)
– Substitute water level vs. time data and well
geometry parameters into a mathematical model
to solve for K
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Why Determine K?
• Recall Darcy’s Law
Q = KAdh/dl
Q/An = v = (K/n)dh/dl
• K Influences:
Magnitude and direction of ground water flow and
contaminant migration
Whether a formation is an aquifer
Magnitude and rate of contaminant attenuation processes
(e.g., DO flux impact on MNA)
Risk
Remediation Method (e.g., pumping or excavation?)
Investigation and remediation costs
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K Range in Nature
K (cm/s) K(ft/day) Material
<10-7 <2.8e-4 Crystalline rocks, clays
10-4 – 10-6 0.28 – 2.8e-3 Clay, fine sand, poor ly sorted (well graded ) material, till
10-1 – 10-3 283 – 2.83 Sand, gravel, well sorted,(poorly graded) material
>10-1 >283 Coarse gravels, cobbles, very clean sand (no fines), karst limestone
Approximately 12 orders of magnitude variation in K
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Alternative Ways of Determining K
Laboratory test Empirical relations
based on grain size analysis
Single or multiple well pump test
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Slug Test Advantages
One well field test
Easy to perform
Minimal time and equipment
No or minimal water for
disposal, if contaminated
Most popular test performed
at contamination sites
Developed for Low K formations
(K< 10-4 cm/s)
High frequency pressure transducers and
pneumatic slug testing > 10-2 cm/s
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How are Slug Tests Performed?
–Slug of water in
–Slug of water out
–Displacement (slug out)
–Pneumatic (slug out or slug in)
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Slug In Bucket
Or Pump
Flow out
Monitor water level fall
with water level sounder and time
with a stopwatch if K is low, or use
pressure transducer raise
water
level
Modified after McCall, Geoprobe System, Inc.
static
Use background water if use
water level sounder, or distilled
water with pressure transducer
Main advantages
1. Deep well input rapid
2. Prevent water level from
lowering into screen section
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Flow in
Slug Out
lower bailer
wait for water level
to recover pull bailer
monitor water level
rise with time
Modified after McCall, Geoprobe System, Inc.
static
Main advantages
1. No water use
2. Well controlled and
instantaneous start
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Flow out Flow in
Displacement (Mandrel) Slug Out
lower solid rod
monitor water level
rise with time
wait for water level to recover
pull solid rod
Modified after McCall, Geoprobe System, Inc.
static
Main advantages
1. No water to
dispose
2. Very reproducible
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Valve
Closed
Valve
Open
Valve
Closed
Water
Level
Rises
Flow in Flow out
Valve
Open
Air In
regulated at
constant P=
1 ft
Air Out
Pressure
Transducer
Water
Level
Falls
Eq
uil
ibri
um
Pneumatic Slug Out
Modified after McCall, Geoprobe System, Inc.
1 ft
Air Pressure Gage
Static
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Valve
Closed
Valve
Open
Valve
Closed
Water
Level
Falls
Flow in Flow out
Valve
Open
Air out
regulated at
constant P= 1 ft
Air In
Pressure
Transducer
Water
Level
Rises
Eq
uil
ibri
um
Pneumatic Slug In
Modified after McCall, Geoprobe System, Inc.
1 ft
Vacuum Pressure Gage
Static
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Original UCONN Modified
Geoprobe
System for Slug-In
Added vacuum/pressure pump
instead of foot pump
Added isolation and
regulator valves
Added vacuum gauge
Added isolation valve
Details in Bartlett et al.
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Newer Version of the UCONN
Pneumatic Slug Test System
PVC
Quick Connects for P/Vac Cable Connector
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Pneumatic Test Pros and Cons • Pros
– High K > 10-3 cm/s (test lasts only seconds)
– Alternative to pumping test
– Use it to profile
– No water used
– “Instantaneous” input
– Very reproducible
– Can readily do multiple tests and get statistics
– Equipment can be used for low K and constant head pumping tests
• Cons
– Cost (keep in mind cost for pumping test)
– Requires higher level of training
– Low K formations you have to wait for equilibration
– Screen must be submerged
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Water Level Responses During Recovery
Overdamped Response
Low K < 10-3 cm/s exponential response (H or Wd vs, time)
Plot log Head or normalized Head [(H(Wd) = H(t)/Ho] vs time
= linear
Note in late time log plot becomes non-linear owing to drawdown in the
Formation
H = driving head = drawdown = abs [DTW (t) – DTW(static)]
Ho = initial head
H(Wd) = Wd = normalized head = H(t)/Ho
t = 0 t = t
static
H
Overdamped
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L
rc
rs
dh
Overdamped Model Derivation (Quasi Steady State)
1. In the casing at any time t
Qin =dV/dt =rc2dh/dt
2. Flow out of the formation into the well follows a steady state form of
Darcy’s Law
Qout =dV/dt =-KSh
S = shape factor (depends on intake geometry)
3. Equate 1 and 2 and rearrange:
dh/h = - (K/rc2)Sdt
4. Integrate 3 from ho to h and to = 0 to t
ln (h/ h0 ) = - (K/rc2 )St
5. Rearrange, convert ln to log
log(h) = - [(K/(2.303rc2 ))S]t – log(ho)
y = - m x + b
K = slope* 2.303rc2 /S
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Overdamped Models Use quasi-steady state forms of Darcy’s law that depends on intake geometry (fully vs partially penetrating)
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Hvorslev Equations
impermeable
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Hvorslev Equations
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Bouwer and Rice Equations
How to find Re?
• Guess?
• 50 x hole radius
• 200 x hole radius (NAVY)
• Bouwer and Rice formulas
based on electric analog modeling
• Note Re in log term (so 100 x
difference = 2*K)
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From Bouwer, 1989, Ground Water, v. 27, n. 3, p. 304-309
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Model vs. Reality
• Model predicts log-linear recovery – Assumes water table stays at initial level
– In reality, in late time cone of depression forms and slows recovery
• In analyzing data only use data points that are log-linear
• low K, use h > 15%ho
• high K, use h> 30%ho
• Best to plot all data then delete late time non-linear points
Model Reality
Log h
time
Log h
time
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Critically Damped
Water Level Responses During Recovery
High K formations K > 10-3 cm/s
Rapid return to static but some small oscillation
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How to Tell If Critically Damped
Plot log of H vs. time
If straight line = overdamped
If concave downward = critically damped
Overdamped Critically Damped
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Underdamped
Water Level Responses During Recovery
High K formations K > about 5x10-3 cm/s
Rapid return to static but oscillatory
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What Causes Oscillation in High K Wells?
• Inertia is key factor in causing oscillation
Inertia is the resistance of an object to a change in its state of motion.
Newton’s first law: An object in motion will stay in motion, unless acted upon by an outside force
• Automobile analogy • Car accelerates driver resists motion and is pressed to seat (driver moves slower than car)
• Car constant speed driver and car moving at same speed
• Car suddenly decelerates, driver thrown forward due to going faster than car
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What Causes Oscillation in High K Wells?
• Water oscillates in and out of well
• Causes mounding and depression in water table
outside the well
• Water moving at “high” velocity and has inertia and overshoots static
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Underdamped Response Theory Damped Spring-Mass System
• A spring develops a restoring force proportional to how far it is stretched (and acting in the opposite
direction to the stretch)
Fspring = −k × x (Hook’s Law)
where: k = spring constant
x = location of the spring, at rest x =0.
• In addition, there is a damping (friction) force that resists the motion that is proportional to the velocity (v)
Fdamping = −b v
where: v = velocity
b = friction coeff, (viscous damping coeff.)
• Ftotal = Fdamping + Fspring = − bv − kx
• F = ma = - bv - kx
• mdv/dt = - bv - kx (conservation of momentum (mv))
• md2x/dt2 = - bdx/dt – kx
• Divide by m: d2x/dt2 = - (b/m)dx/dt – (k/m)x
• d2x/dt2 + (b/m)dx/dt + (k/m)x = 0
• Define wo = undamped angular frequency = (k/m).5
Cd = damping ratio =b/2mwo
• d2x/dt2 + 2Cd wo dx/dt + wo 2x = 0
• Cd >1 overdamped, spring returns to equilibrium without oscillating
• Cd =1 critically damped, spring returns quickly to equilibrium without oscillating
• Cd <1 underdamped, spring returns to equilibrium but oscillates
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Underdamped Slug Test Model Solutions
• Solve conservation of momemtum (differential equation) using physics of damped spring
• d2wd /dt2d + Cddwd /dtd + wd = 0 and hd = d2wd /dt2d + wd
Where wd = dimensionless displacement = w/Ho
w = displacement from static
ho = initial displacement from static at t = 0
td = dimensionless time = (g/Le).5 t
hd = dimenionless head = h/ho
• Cd = dimensionless damping parameter (fttting parameter)
• Cd = function of K, pipe friction, well geometry parameters (rc, rs, L, Le),
fully or partially penetrating well screen
• Le = effective length of the water column above screen (fitting parameter)
W Le
rc
rs
L
y
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Underdamped Slug Test Model Solutions
How is K determined?
• Fit head vs time data to model to solve for Cd
• Methods to solve for Cd
Curve Matching, match to type curves
Least squares (minimize sum of the square differences), e.g., AQTESOLV, other software
Develop own spreadsheets use curve matching or least squares
Solve numerical model
• Solve for K given Cd
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Curve Matching
Butler and Garnett
Use these equations to generate type curves to match your data and to determine CD, td*, and t*
td* = pick time
t* = time corresponds to td*
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Raw data Normalize to baseline P Normalize to initial Headrop (H/Ho)
Steps- first normalize data to base line P, then normalize to initial head drop (Ho), then match
Curve Matching
Butler and Garnett
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The dimensionless time axis is expanded or contracted until a reasonable match is obtained
between a CD type curve and the test data . Determine CD, td*, and t*
Curve Matching
Butler and Garnett
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Curve Matching
Butler and Garnett
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Oscillatory Slug Test Model
Differences
• Hvorslev (partial penetrating) vs. Bouwer and Rice ( fully penetrating)
• Confined or unconfined
• Apply to all responses or just oscillatory (low values of Cd )
• Corrects for pipe friction or not ( added term within Cd)
• Corrects for depth of pressure transducer
• Corrects for well skins
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Issues In Conducting and Analyzing Underdamped Slug Tests
• Test lasts only a few seconds
– Need p-transducer that responses rapidly (< 1 sec)
• Hard to make ho instantaneous
– Can use time of peak or trough for start time
• P-transducer should be located near water surface
– Most models based on measuring the difference in water level from static
– pressure measurement are affected by the acceleration of the water column and is a function of the
depth of placement of the pressure transducer
– Best to keep p-trans depth close to static if not it leads to under estimation of K by a factor of 2 or
more.
– Butler suggests put p-trans less than 1.5 ft below static avoids the need for correction. Test by
observing maximum normalized H (h/ho), if less than 0.9 move p-trans closer to static.
• Have to handle a lot of data to isolate test
– E.g, 10 Hz, takes 2 minutes for set up = 1200 rows of data for a spreadsheet
• Are model assumptions met?
• Friction appears only important for small diameter wells
– Butler (2002) found wells r > 0.012 m (1.2 cm) or diameter less than 1 inch, no correction is needed
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Example Test
Time (sec)
H (
feet
)
Three tests
27590 lines of data on spreadsheet
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Test Example P-trans within 1.5 feet of water surface
Rc = 1”
K (cm/s) K(ft/day)
test 1 0.1046 297
test2 0.1039 295
test 3 0.1041 295
average 0.1042 295.37
Std Dev 0.000361 1.022046
%RSD 0.35 0.35
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Select References • Bouwer, H., and R.C. Rice. 1976. A slug test for determining hydraulic conductivity of unconfined aquifers with completely or
partially penetrating wells. Water Resour. Res. 12, no. 3: 423-428.
• Butler, J., 1998, The design performance and analysis of slug tests. Lewis Pub.
• Butler, J., and Garnett, 2000, Simple Procedures for Analysis of Slug Tests in Formations of High Hydraulic Conductivity using
Spreadsheet and Scientific Graphics Software, KGS open file report 2000-40,
http://www.kgs.ku.edu/Hydro/Publications/OFR00_40/index.html
• Butler, JJ, Jr., EJ Garnett, and JM Healey. 2003. “Analysis of slug tests in formations of high hydraulic
• conductivity.” Ground Water 41(5):620-630.
• Butler, J.J., Jr. and X. Zhan, 2004. Hydraulic tests in highly permeable aquifers, Water Resources Research, vol. 40, W12402.
• Chen, C., and C. Wu, 2006, Analysis of Depth-Dependent Pressure Head of Slug Tests in Highly Permeable Aquifers, Ground
Water, 44 (3) :472-477
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