Well HydraulicFlow to the well
• Assoc. Prof. Dr Ismail Yusoff (Geology,UM)
• 03-79674141/017-2420372
Lecture Outcomes
• Understand the concept of Well Hydraulic-Radial Flow
• Understand the unsteady and steady radial flow system
• Outlines the analytical solution to the radial flow system
References• 1)Fetter, C.W. (2001): Applied Hydrogeology,
Prentice Hall, New Jersey; 598pp.
• 2)Todd, D.K. and Mays L.W., (2005): Groundwater Hydrology.(3rd Edition). John Wiley & Sons, New York
• 3) Freeze, R.A. and Cherry, J.A. (1979): Groundwater, Prentice Hall, New Jersey,604pp.
• Details Ref.: Krusemen G.P. and deRidder, N.A. 1983.Analysis and evaluation of pumping test data (3rd ed.)
Groundwater Hydrology• It is the study of characteristics,
movement and occurance of water found below the surface
Aquifer
• Water bearing geological formation that can store and yield
usable amount of water
GWater Flow Equation
• Equation is based on Darcy’s Law• Flow equations are partial differential
equations in which head, h, is described in terms of x, y, z, and t.
• Equation is derived from “law of conservation of mass and energy”/ continuity law
• Assume the aquifer is homogeneous and isotropic
• Assume the fluid moves in only one direction
Flow types• Steady flow and Unsteady flow
• Unsteady flow - flow in which head changes with time
• Steady Flow -flow in which head does not changes with time, equilibrium
• Unconfined and confined aquifer flow equation
• Unconfined and confined aquifer differ in hydraulic properties
Ground-water Flow to Wells
• ”Radial Flow”
• Unsteady flow and Steady flow
• Cone of depression - area around a discharging well where the hydraulic head in the aquifer is lowered by pumping
• Produce “Time Drawdown Data”
• compute aquifer properties : T, S, Ss and K
Radial Flow and Cone of Depression
Defination T, S, Ss• Transmissivity
The rate at which water of a specific density and viscosity is transmitted through a unit width of an aquifer or confining bed under a unit hydraulic gradient. (1)
• T = Kb (m2/d) ; b = saturated thickness K = Hyd.conductivity
• Storativity /(Storage Coefficient) (S)
• The volume of water that a permeable unit will adsorb or expel from the storage per unit surface area per unit change in head.
• Confined aquifer S ≤ 0.005
• Unconfined S≈Sy in the range of 0.02-0.30
Radial Flow
Assumptions
Graphical Explanation (Time-Drawdown)
Unsteady/Transient/Nonequilibrium Radial
Flow
Steady Radial Flow
Examples
Assumptions• (1.) Bottom confining layer• (2.) All geologic units are horizontal and of infinite extent.• (3.) Potentiometric surface is horizontal prior to the start of
pumping.• (4.) Potentiometric surface is not changing with time prior to the
start of pumping.• (5.) All changes in potentiometric surface position are due to the
effect of the pumping well.• (6.) Aquifer is homogeneous and isotropic.• (7.) All flow is radial toward the well.• (8.) Ground-water flow is horizontal.• (9.) Darcy’s Law is valid.• (10.) Ground-water has constant density and viscosity.• (11.) Pumping well and observation wells are fully penetrating.• (12.) Well has an infinitesimal diameter and is 100% efficient.
Test Setup• Single Well
• Pumping well and Observation well
• Field note
Time
Depth to water level/Auto measure
Compute Drawdown
• Graph construction
“Time Drawdown Graph”
• Time Drawdown data analysis
Obtaining T, K, S
Field Setup
900 V-Notch Q=2.5H5/2
Confined Aquifer-Aquifer Test
Unconfined Aquifer-Aquifer Test
Curve shape = aquifer type
Unsteady Radial Flow
• Time Drawdown data analysis• Curve Matching technique• Confined aquifer
Theis MethodJacob Approximation method
• Unconfined aquiferNeuman/Boulton/Walton
• Leaky aquifer• Partially penetrating well-geometrical case• Pumping at the boundary-geometrical case• Recovery Test (T)
Unsteady Flow to Wells in Confined Aquifers
Confined Aq.:Theis Method
C.V. Theis 1935
Additional Assumption
• (1.) Aquifer is confined top and bottom.
• (2.) There is no source of recharge to aquifer.
• (3.) The Aquifer is compressible and water is released instantaneously from the aquifer as the head is lowered.
• (4.) The well is pumping at a constant rate.
Unsteady Flow to a Well in a Confined Aquifer
• Continuity
• Drawdown
• Theis equation
• Well function
s u( ) =Q
4pTW u( )
W u( ) =e-h
hu
¥
ò dh
u =r2S
4Tt
s(r,t) = h0 - h r,t( )
Ground surface
Bedrock
Confined
aquifer
Q
h0
Confining Layer
b
r
h(r)
Q
Pumping
well
¶2h
¶r2+
1
r
¶h
¶r=
S
T
¶h
¶t
Unsteady Flow to a Well in a Confined Aquifer
Well Function
U vs W(u) 1/u vs W(u)
d
euW
u
Tt
Sru
4
2
Unsteady Flow to a Well in a Confined Aquifer
Example - Theis Equation
Q = 1500 m3/dayT = 600 m2/dayS = 4 x 10-4
Find: Drawdown 1 km from well after 1 year
u =r2S
4Tt=
(1000 m)2 (4x10-4 )
4(600m2 /d)(365d)= 4.6x10-4
Ground surface
Bedrock
Confined
aquiferQ
Confining Layer
b
r1
h1
Q
Pumping
well
Unsteady Flow to a Well in a Confined Aquifer
Well Function
u = 4.6x10-4
W(u) = 7.12
Example - Theis EquationQ = 1500 m3/dayT = 600 m2/dayS = 4 x 10-4
Find: Drawdown 1 km from well after 1 year
s =Q
4pTW (u) =
1500 m3 /d
4p (600 m2 /d)* 7.12 =1.42 m
u = 4.6x10-4
Ground surface
Bedrock
Confined
aquiferQ
Confining Layer
b
r1
h1
Q
Pumping
well
W(u) = 7.12
Unsteady Flow to a Well in a Confined Aquifer
Pump Test in Confined AquifersTheis Method
Pump Test Analysis – Theis Method
s = Q
4pT
æ
è ç
ö
ø ÷ * W u( )
• Q/4pT and 4T/S are constant
• Relationship between
– s and r2/t is similar to the relationship between
– W(u) and u
– So if we make 2 plots: W(u) vs u, and s vs r2/t
– We can estimate the constants T, and S
Tt
Sru
4
2
r2
t =
4T
S
æ
è ç
ö
ø ÷ * u
constants
s =Q
4pTW (u)
Ground surface
Bedrock
Confined
aquiferQ
Confining Layer
br1
h1
Q
Pumpin
g well
Example - Theis Method
• Pumping test in a sandy aquifer
• Original water level = 20 m above mean sea level (amsl)
• Q = 1000 m3/hr
• Observation well = 1000 m from pumping well
• Find: S and T
Ground surface
Bedrock
Confined
aquifer
h0 = 20 m
Confining Layer
b
r1 = 1000 m
h1
Q
Pumping
well
Bear, J., Hydraulics of Groundwater, Problem 11-4, pp 539-540, McGraw-Hill, 1979.
Pump Test Analysis – Theis Method
Theis Method
Time
Water level,
h(1000)
Drawdown,
s(1000)
min m m
0 20.00 0.00
3 19.92 0.08
4 19.85 0.15
5 19.78 0.22
6 19.70 0.30
7 19.64 0.36
8 19.57 0.43
10 19.45 0.55
…
60 18.00 2.00
70 17.87 2.13
…
100 17.50 2.50
…
1000 15.25 4.75
…
4000 13.80 6.20
Pump Test Analysis – Theis Method
Theis Method
Time r2/t s u W(u)
(min) (m2/min) (m)
0 0.00 1.0E-04 8.63
3 333333 0.08 2.0E-04 7.94
4 250000 0.15 3.0E-04 7.53
5 200000 0.22 4.0E-04 7.25
6 166667 0.30 5.0E-04 7.02
7 142857 0.36 6.0E-04 6.84
8 125000 0.43 7.0E-04 6.69
10 100000 0.55 8.0E-04 6.55
…
3000 333 5.85 8.0E-01 0.31
4000 250 6.20 9.0E-01 0.26
s vs r2/t
W(u) vs u
Pump Test Analysis – Theis Method
r2/t
s
u
W(u)
r2/t s W(u)u
0.01
0.1
1
10
10 100 1000 10000 100000 1000000
s
r2/t
0.01
0.1
1
10
0.0001 0.0010 0.0100 0.1000 1.0000 10.0000
W(u
)
u
Match PointW(u) = 1, u = 0.10s = 1, r2/t = 20000
Theis MethodPump Test Analysis – Theis Method
Theis Method
• Match Point
• W(u) = 1, u = 0.10
• s = 1, r2/t = 20000
T =Q
4p
Wmp
smp
æ
è ç ç
ö
ø ÷ ÷ =
1000 m3 /hr
4p
1
1 m
æ
è ç
ö
ø ÷ = 79.58 m2 /hr (=1910 m2 /d)
S = 4Tump
r2
tmp
æ
è
ç ç ç
ö
ø
÷ ÷ ÷
= 4(79.58 m2 /hr)0.1
20000 m2 /min* 60 min/hr
æ
è ç
ö
ø ÷ = 2.65x10-5
Pump Test Analysis – Theis Method
Pump Test in Confined AquifersJacob Method
Jacob’s Approximation
Two approximation;
a) For one pumped well and one observation well
• Time- Drawdown Method
b) For one pumped well and more than three observation wells
• Distance-Drawdown method
Jacob Approximation
• Drawdown, s
• Well Function, W(u)
• Series approximation of W(u)
• Approximation of s
s u( ) =Q
4pTW u( )
W u( ) =e-h
hu
¥
ò dh » -0.5772 - ln(u)+ u -u2
2!+
u =r2S
4Tt
W u( ) » -0.5772 - ln(u) for small u < 0.01
s(r,t) »Q
4pT-0.5772 - ln
r2S
4Tt
æ
è ç
ö
ø ÷
é
ë ê ê
ù
û ú ú
Pump Test Analysis – Jacob Method
Jacob Approximation
s =2.3Q
4pTlog(
2.25Tt
r2S)
0 =2.3Q
4pTlog(
2.25Tt0
r2S)
t0
1=2.25Tt0
r2S
S =2.25Tt0
r2
Pump Test Analysis – Jacob Method
Jacob Approximation
t0
S =2.25Tt0
r2
t1 t2
s1
s2
Ds
logt2
t1
æ
è ç
ö
ø ÷ = log
10* t1t1
æ
è ç
ö
ø ÷ =1
1 LOG CYCLE
1 LOG CYCLE
Pump Test Analysis – Jacob Method
Jacob Approximation
S =2.25Tt0
r2=
2.25(76.26 m2/hr)(8 min*1 hr /60 min)
(1000 m)2
= 2.29x10-5
t0
t1 t2
s1
s2
Ds
t0 = 8 min
s2 = 5 ms1 = 2.6 mDs = 2.4 m
Pump Test Analysis – Jacob Method
Jacob Distance-Drawdown Method
• In case more than three observation wells
• Modification to Jacob-Time drawdown method
• Plot semilog graph drawdown(y normal axis) Vs. distance (x log axis)
• Extend the straight line to ZERO drawdown= ro
T and S calculation
• T = 2.3Q
• 2Π∆(h0-h) where
• T - ft2/d or m2/d
• Q - ft3/d or m3/d
• ∆ (h0 - h) - Drawdown / log cycle (ft)
• S = 2.25Tt where
• ro2
• S = storativity
• t= time of pumping
• ro = distance where straight line intersects the zero drawdown axis (days)
Unsteady Flow to Wells in Unconfined Aquifers
Unsteady Flow to a Well in an Unconfined Aquifer
• Water is produced by
– Dewatering of unconfined aquifer
– Compressibility factors as in a confined aquifer
– Lateral movement from other formations
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Prepumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
Unsteady Flow to Wells in Unconfined Aquifers
Analyzing Drawdown in An Unconfined Aquifer
• Early– Release of water is from
compaction of aquifer and expansion of water – like confined aquifer.
– Water table doesn’t drop significantly
• Middle– Release of water is from gravity
drainage
– Decrease in slope of time-drawdown curve relative to Theiscurve
• Late– Release of water is due to drainage
of formation over large area
– Water table decline slows and flow is essentially horizontal
Unsteady Flow to Wells in Unconfined Aquifers
Early
Late
Unconfined Aquifer (Neuman Solution)
s =Q
4pTW (ua ,h)
ua =r2S
4Tt
s =Q
4pTW (uy ,h)
uy =r2Sy
4Tt
h =r
b
æ
è ç
ö
ø ÷
2 Kz
Kr
Early (a)
Late (y)
Unsteady Flow to Wells in Unconfined Aquifers
Procedure - Unconfined Aquifer (Neuman Solution)
• Get Neuman Well Function Curves
• Plot pump test data (drawdown s vs time t)
• Match early-time data with “a-type” curve. Note the value of
• Select the match point (a) on the two graphs. Note the values of s, t, 1/ua, and W(ua, )
• Solve for T and S
• Match late-time points with “y-type” curve with the same as the a-type curve
• Select the match point (y) on the two graphs. Note s, t, 1/uy, and W(uy, )
• Solve for T and Sy
T =Q
4psW (ua ,h)
S =4Ttua
r2
T =Q
4psW (uy ,h)
Sy =4Ttuy
r2
Unsteady Flow to Wells in Unconfined Aquifers
Procedure - Unconfined Aquifer (Neuman Solution)
• From the T value and the initial (pre-pumping) saturated thickness of the aquifer b, calculate Kr
• Calculate Kz
Kr =T
b
Kz =hKrb2
r2
Unsteady Flow to Wells in Unconfined Aquifers
Example – Unconfined Aquifer Pump Test
• Q = 144.4 ft3/min
• Initial aquifer thickness = 25 ft
• Observation well 73 ft away
• Find: T, S, Sy, Kr, Kz Ground surface
Bedrock
Unconfined aquiferQ
h0=25 ft
Prepumping
Water level
r1=73 ft
h1
hw
Observation
wells
Water Table
Q= 144.4 ft3/min
Pumping
well
Unsteady Flow to Wells in Unconfined Aquifers
Pump Test data
Unsteady Flow to Wells in Unconfined Aquifers
Early-Time Data
h = 0.06
t = 0.17 min; s = 0.57 ft
1/ua =1.0; W =1.0
Unsteady Flow to Wells in Unconfined Aquifers
Early-Time Analysis
T =Q
4psW (ua ,h)
=144.4 ft 3 /min
4p (0.57 ft)(1.0)
= 20.16 ft2 /min
(29,900 ft2 /day)
S =4Ttua
r2
=4(20.16 ft2 /min)(1.0)(0.17 min)
(73 ft)2
= 0.00257
h = 0.06
t = 0.17 min; s = 0.57 ft
1/ua =1.0; W (ua ,h) =1.0
Unsteady Flow to Wells in Unconfined Aquifers
Late-Time Data
h = 0.06
t =13 min; s = 0.57 ft
1/uy = 0.1; W =1.0
Unsteady Flow to Wells in Unconfined Aquifers
Late-Time Analysis
h = 0.06
t =13 min; s = 0.57 ft
1/uy = 0.1; W =1.0
T =Q
4psW (uy ,h)
= 20.16 ft2 /min
(29,900 ft2 /day)
Sy =4Ttuy
r2
=4(20.16 ft2 /min)(0.1)(13 min)
(73 ft)2
= 0.02
Kr =T
b
=20.16 ft2 /min
25 ft
= 0.806 ft /min
Kz =(0.06)(0.806 ft /min)(25 ft)2
(73 ft)2
= 5.67x10-3 ft /min
Unsteady Flow to Wells in Unconfined Aquifers
Unsteady Flow to Wells in Leaky/Semi Confined Aquifers
Radial Flow in a Leaky AquiferWalton’s method
K b
ground surface
bedrock
aquitard
confined aquifer
initial head
Well
s(r)
r
Q
R
h0
Cone of Depression leakage
h(r)
unconfined aquifer
B
ruW
T
Qs ,
4p
bK
T
r
B
r
/
dzz
e
B
ruW
u
zB
rz
2
2
4,
When there is leakage from other layers, the drawdown from a pumping test will be less than the fully confined case.
B=Leakage factor
Unsteady Flow to Wells in Leaky Aquifers
Leaky Well Function dzz
e
B
ruW
u
zB
rz
2
2
4,
r/B = 0.01
r/B = 3
cleveland1.cive.uh.edu/software/spreadsheets/ssgwhydro/MODEL6.XLS
Unsteady Flow to Wells in Leaky Aquifers
r/B=0 ≈Theis curve
Leaky Aquifer Example• Given:
– Well pumping in a confined aquifer
– Confining layer b’ = 14 ft. thick
– Observation well r = 96 ft. form well
– Well Q = 25 gal/min
• Find:
– T, S, and K’
From: Fetter, Example, pg. 179
t (min) s (ft)5 0.76
28 3.341 3.5960 4.0875 4.39
244 5.47493 5.96669 6.11958 6.27
1129 6.41185 6.42
K b
ground surface
bedrock
aquitard
confined aquifer
initial head
Well
s(r)
r
Q
R
h0
Cone of Depression leakage
h(r)
unconfined aquifer
Unsteady Flow to Wells in Leaky Aquifers
= 0.15
= 0.20
= 0.30
= 0.40
r/B
Match PointW(u, r/B) = 1, 1/u = 10s = 1.6 ft, t = 26 min, r/B = 0.15
Unsteady Flow to Wells in Leaky Aquifers
Leaky Aquifer Example
• Match Point
• Wmp = 1, (1/u)mp = 10
• smp = 1.6 ft, tmp = 26 min, r/Bmp = 0.15
• Q = 25 gal/min * 1/7.48 ft3/gal*1440 min/d = 4800 ft3/d
• t = 26 min*1/1440 d/min = 0.01806 d
T =Q
4psmp
Wmp =4800 ft 3 /d
4p (1.6 ft)*1= 238.7 ft2/d
S =4Tump
r2
tæ
è ç
ö
ø ÷ mp
=4(238.7 ft2 /d)(0.1)(0.01806)
(96 ft)2=1.87 x10-4
¢ K =T ¢ b (r /B)2
r2=
(238.7 ft2 /d)(14 ft)(0.15)2
(96 ft)2= 0.0081 ft /d
Unsteady Flow to Wells in Leaky Aquifers
Steady Flow to Wells in Confined Aquifers
Steady Flow to a Well in a Confined Aquifer
2rw
Ground surface
Bedrock
Confined
aquifer
Q
h0
Pre-pumping
head
Confining Layer
b
r1
r2
h2
h1
hw
Observation
wells
Drawdown curve
Q
Pumping
well
Q = Aq
= (2prb)Kdh
dr
rdh
dr=
Q
2pT
h2 = h1 +Q
2pTln(
r2r1
)
Theim Equation
In terms of head (we can write it in terms of drawdown also)
h = h1 at r = r1
h = h2 at r = r2
Example - Theim Equation
• Q = 400 m3/hr
• b = 40 m.
• Two observation wells,
1. r1 = 25 m; h1 = 85.3 m
2. r2 = 75 m; h2 = 89.6 m
• Find: Transmissivity (T)
T =Q
2p h2 - h1( )ln
r2r1
æ
è ç
ö
ø ÷ =
400 m3/hr
2p 89.6 m - 85.3m( )ln
75 m
25 m
æ
è ç
ö
ø ÷ =16.3 m2 /hr
h2 = h1 +Q
2pTln(
r2r1
)2rw
Ground surface
Bedrock
Confine
d
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Pumping
well
Steady Flow to a Well in a Confined Aquifer
Steady Radial Flow in a Confined Aquifer
• Head
• Drawdown
h = h0 +Q
2pTln
r
R
æ
èç
ö
ø÷
s =Q
2pTln
R
r
æ
èç
ö
ø÷
Steady Flow to a Well in a Confined Aquifer
Theim Equation In terms of drawdown (we can write it in terms of head also)
s = h0 - h
= h0 - h0 +Q
2pTln
r
R
æ
èç
ö
ø÷
é
ëê
ù
ûú
Example - Theim Equation
• 1-m diameter well
• Q = 113 m3/hr
• b = 30 m
• h0= 40 m
• Two observation wells,
1. r1 = 15 m; h1 = 38.2 m
2. r2 = 50 m; h2 = 39.5 m
• Find: Head and drawdown in the well
2rw
Ground surface
Bedrock
Confined
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Pumping
wellDrawdown
Adapted from Todd and Mays, Groundwater Hydrology
T =Q
2p s1 - s2( )ln
r2r1
æ
è ç
ö
ø ÷ =
113m3/hr
2p 1.8 m - 0.5 m( )ln
50 m
15 m
æ
è ç
ö
ø ÷ =16.66 m2 /hr
s r( ) =Q
2pTln
R
r
æ
è ç
ö
ø ÷
Steady Flow to a Well in a Confined Aquifer
Example - Theim Equation
2rw
Ground surface
Bedrock
Confine
d
aquifer
Q
h0
Confining Layer
b
r1
r2
h2 h1
hw
Q
Drawdown
@ well
Adapted from Todd and Mays, Groundwater Hydrology
hw = h2 +Q
2pTln(
rwr2
) = 39.5 m +113m3 /hr
2p *16.66 m2 /hrln(
0.5 m
50 m) = 34.5 m
sw = h0 - hw = 40 m- 34.5 m = 5.5 m
h2 = h1 +Q
2pTln(
r2r1
)
Steady Flow to a Well in a Confined Aquifer
Drawdown at the well
Steady Flow to Wells in Unconfined Aquifers
Steady Flow to a Well in an Unconfined Aquifer
Q = Aq = (2prh)Kdh
dr
= prKdh2
dr
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Pre-pumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
rd h2( )
dr=
Q
pK
h02 - h2 =
Q
pKln
R
r
æ
è ç
ö
ø ÷
h2(r) = h02 +
Q
pKln
r
R
æ
è ç
ö
ø ÷
Unconfined aquifer
h = h0 at r = R
h 1 and h2 = aquifer saturated thickness; can be b1 and b2
Steady Flow to a Well in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Prepumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
2 observation wells: h1 m @ r1 m h2 m @ r2 m
K =Q
p h22 - h1
2( )ln
r2r1
æ
è ç
ö
ø ÷
h2(r) = h02 +
Q
pKln
r
R
æ
è ç
ö
ø ÷
h22 = h1
2 +Q
pKln
r2r1
æ
è ç
ö
ø ÷
• Given: – Q = 300 m3/hr
– Unconfined aquifer
– 2 observation wells,
• r1 = 50 m, h = 40 m
• r2 = 100 m, h = 43 m
• Find: K
K =Q
p h22 - h1
2( )ln
r2r1
æ
è ç
ö
ø ÷ =
300 m3 /hr / 3600 s /hr
p (43m)2 - (40 m)2[ ]ln
100 m
50 m
æ
è ç
ö
ø ÷ = 7.3x10-5 m /sec
Example – Two Observation Wells in an Unconfined Aquifer
2rw
Ground surface
Bedrock
Unconfined
aquifer
Q
h0
Prepumping
Water level
r1
r2
h2 h1
hw
Observation
wells
Water Table
Q
Pumping
well
Steady Flow to a Well in an Unconfined Aquifer
More Calculation Examples
• Fetter, C.W. (2001)
• Pg 166-169
WL Recovery/ Residual
• Water level measurements taken immediately after test.
• Water levels not influenced by erratic pumping rates.
• Recovery analysis compared to constant rate test.
• Check valve on pump essential to eliminating slugging effects (after the pump is switched off)
• Example: Todd and May (2005) pg. 170-172