Transcript Ground-water flow to wells

Ground-water flow to wells
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Extract water
Remove contaminated water
Lower water table for constructions
Relieve pressures under dams
Injections – recharges
Control slat-water intrusion
Our purpose of well studies
• Compute the decline in the water level, or
drawdown, around a pumping well whose
hydraulic properties are known.
• Determine the hydraulic properties of an
aquifer by performing an aquifer test in
which a well is pumped at a constant rate
and either the stabilized drawdown or the
change in drawdown over time is measured.
Our Wells
• Fully penetrate aquifers
• Radial symmetric
• Aquifers are homogeneous and isotropic
Basic Assumptions
• The aquifer is bounded on the bottom by a
confining layer.
• All geologic formations are horizontal and have
infinite horizontal extent.
• The potentiometric surface of the aquifer is
horizontal prior to the start of the pumping.
• The potentiometric surface of the aquifer is not
recharging with time prior to the start of the
pumping.
• All charges in the position of the potentiometric
surface are due to the effect of the pumping alone.
Basic Assumptions (cont.)
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The aquifer is homogeneous and isotropic.
All flow is radial toward the well.
Ground water flow is horizontal.
Darcy’s law is valid.
Ground water has a constant density and viscosity.
The pumping well and the observation wells are
fully penetrating.
• The pumping well has an infinitesimal diameter
and is 100% efficient.
A completely confined aquifer
• Addition assumptions:
The aquifer is confined top and bottom.
The is no source of recharge to the aquifer.
The aquifer is compressible.
Water is released instantaneously.
Constant pumping rate of the well.
Theis (nonequilibrium) Equation
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h0 – h = (Q/4πT) W(u)
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(u) = well function.
T = transmissivity (L2/T; m2/d or ft2/d)
Theis (nonequilibrium) Equation
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u = (r2S/4Tt)
T = transmissivity (L2/T; m2/d or ft2/d)
S = storativity (dimensionless)
t = time since pumping began (T; d)
r = radial distance from the pumping well
(L; m or ft)
Well Function
• W(u) = u (e-a/a) da
= -0.5772 + ln u + u – u2/22! + u3/33! – u4/44! + …
A Leaky, Confined Aquifer
• Equations
A Leaky, Confined Aquifer – no
water drains from the confining layer
• The aquifer is bounded on the top by an
aquitard.
• The aquitard is overlain by an unconfined
aquifer, know as the source bed.
• The water table in the source bed is initially
horizontal.
• The water table in the source bed does not
fall during pumping of the aquifer.
A Leaky, Confined Aquifer – no
water drains from the confining layer
• Ground water flow in the aquitard is
vertical.
• The aquifer is compressible, and water
drains instantaneously with a decline in
head
• The aquitard is incompressible, so that no
water is released from storage in the
aquitard when the aquifer is pumped.
Wells in Confined Aquifers
• Completely confined aquifer.
• Confined, leaky with no elastic storage in
the leaky layer.
• Confined, leaky with elastic storage in the
leaky layer.
Hantush-Jacob Formula
Confined with no elastic storage
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h0 – h = (Q/4πT) W(u,r/B)
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(u,r/B) = leaky artesian well function
T = transmissivity (L2/T; m2/d or ft2/d)
B = (Tb’/K’)1/2
Drawdown Formula
Confined with elastic storage
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h0 – h = (Q/4πT) H(u,)
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
H(u,) = a modified leaky artesian well function
T = transmissivity (L2/T; m2/d or ft2/d)
 = r/4B (S’/S)1/2; B = (Tb’/K’)1/2
Unconfined aquifer – 3 phases
• Early stage –
pressure drops
specific storage as a major contribution
behaves as an artesian aquifer
flow is horizontal
time-drawdown follows Theis curve
S - the elastic storativity.
Unconfined aquifer – 3 phases
• Second stage –
water table declines
specific yield as a major contribution
flow is both horizontal and vertical
time-drawdown is a function of Kv/Kh r, b
Unconfined aquifer – 3 phases
• Later stage –
rate of drawdown decreases
flow is again horizontal
time-drawdown again follows Theis curve
S - the specific yield.
Neuman’ assumptions
• Aquifer is unconfined.
• Vadose zone has no influence on the
drawdown.
• Water initially pumped comes from the
instantaneous release of water from elastic
storage.
• Eventually water comes from storage due to
gravity drainage of interconnected pores.
Neuman’ assumptions (cont.)
• The drawdown is negligible compared to the
saturated thickness.
• The specific yield is at least 10 times the elastic
storativity.
• The aquifer may be – but does not have to be –
anisotropic with the radial hydraulic conductivity
different than the vertical hydraulic conductivity.
Drawdown Formula
unconfined with elastic storage
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h0 – h = (Q/4πT) W(uA,uB,)
h0 = initial hydraulic head (L; m or ft)
h = hydraulic head (L; m or ft)
h0 – h = drawdown (L; m or ft)
Q = constant pumping rate (L3/T; m3/d or ft3/d)
W(uA,uB,) = the well function for water-table
aquifer
• T = transmissivity (L2/T; m2/d or ft2/d)
• uA =r2S/(4Tt); uB =r2Sy/(4Tt); =r2Kv/(r2Kh)
Drawdown
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Steady-radial flow in a confined
Aquifer
• The aquifer is confined top and bottom.
• Well is pumped at a constant rate.
• Equilibrium has reached.
Steady-radial flow in a
unconfined Aquifer
• The aquifer is unconfined and underlain by
a horizontal aquiclude.
• Well is pumped at a constant rate.
• Equilibrium has reached.
Drawdown
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Aquifer test
• Steady-state conditions.
Cone of depression stabilizes.
• Nonequilibrium flow conditions.
Cone of depression changes.
Needs a pumping well and at least one
observational well.
Transient flow in a confined
Aquifer – Theis Method
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/4(h0-h) log (2.25Tt/(r2S)).
• Only valid when u, (r2S/4Tt) < 0.05
--- after some time of pumping
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/ 4(h0-h)
S = 2.25Tt0/r2
• (h0-h) – drawdown per log cycle of time
• t0 - is the time, where the straight line
intersects the zero-drawdown axis.
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/ 2(h0-h)
S = 2.25Tt/r02
• (h0-h) – drawdown per log cycle of
distance
• r0 - is the distance, where the straight line
intersects the zero-drawdown axis.
Transient flow in a leaky,
confined aquifer with no storage.
• Walton Graphic method
• Hantush inflection-point method.
Transient flow in a leaky,
confined aquifer with no storage
• T = Q/ 4(h0-h)W(u,r/B)
S = 4Tut/r2
r/B = r /(Tb’/K’)1/2
K’ = [Tb’(r/B)2]/r2
Transient flow in a leaky,
confined aquifer with storage
• T = Q/ 4(h0-h)H(u,)
S = 4Tut/r2
2 = r2S’ / (16 B2S)
B = (Tb’/K’)1/2
K’S’ = [16 2Tb’S]/r2
Transient flow in an unconfined
aquifer
• T = Q/ 4(h0-h)W(uA,uB,)
S = 4TuAt/r2 (for early drawdown)
Sy = 4TuBt/r2 (for later drawdown)
 = r2Kv/b2Kh
Aquifer tests
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Aquifer tests
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/4(h0-h) log (2.25Tt/(r2S)).
• Only valid when u, (r2S/4Tt) < 0.05
--- after some time of pumping
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/ 4(h0-h)
S = 2.25Tt0/r2
• (h0-h) – drawdown per log cycle of time
• t0 - is the time, where the straight line
intersects the zero-drawdown axis.
Transient flow in a confined
aquifer – Cooper-Jacob Method
• T = 2.3Q/ 2(h0-h)
S = 2.25Tt/r02
• (h0-h) – drawdown per log cycle of
distance
• r0 - is the distance, where the straight line
intersects the zero-drawdown axis.
Aquifer test
• Steady-state conditions.
Cone of depression stabilizes.
• Nonequilibrium flow conditions.
Cone of depression changes.
Needs a pumping well and at least one
observational well.
Steady-radial flow in a confined
Aquifer
• The aquifer is confined top and bottom.
• Well is pumped at a constant rate.
• Equilibrium has reached.
Steady-radial flow in a
unconfined Aquifer
• The aquifer is unconfined and underlain by
a horizontal aquiclude.
• Well is pumped at a constant rate.
• Equilibrium has reached.
Our purpose of well studies
• Compute the decline in the water level, or
drawdown, around a pumping well whose
hydraulic properties are known.
• Determine the hydraulic properties of an
aquifer by performing an aquifer test in
which a well is pumped at a constant rate
and either the stabilized drawdown or the
change in drawdown over time is measured.
Drawdown
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Slug test
• Goal – to determine the hydraulic
conductivity of the formation in the
immediate vicinity of a monitoring well.
• Means – A known volume of water is
quickly drawn from or added to the
monitoring, the rate which the water level
rises or falls is measured and analyzed.
Slug test
• Overdamped – water level recovers to the
initial static level in a smooth manner that is
approximately exponential.
• Underdamped – water level oscillates about
the static water level with the magnitude of
oscillation decreasing with time until the
oscillations cease.
Cooper-Bredehoeft-Papadopulos
Method (confined aquifer)
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H/H0 = F(,)
H – head at time t.
H0 – head at time t = 0.
 = T t/rc2
 = rs2S/rc2
Cooper-Bredehoeft-Papadopulos
Method (confined aquifer)
• H/H0 = 1,  = 1 at match point.
•  = T t1/rc2
•  = rs2S/rc2
Underdamped Response Slug Test
Underdamped Response Slug
Test
• Van der Kamp Method – confined aquifer
and well fully penetrating.
• H(t) = H0 e-t cos t
H(t) - hydraulic head (L) at time t (T)
H0 - the instantaneous change in head (L)
 - damping constant (T-1)
 - an angular frequency (T-1)
Underdamped Response Slug
Test (cont.)
• T = c + a ln T
c = -a ln[0.79 rs2S(g/L)1/2]
a = [rc2(g/L)1/2] / (8d)
d = /(g/L)1/2
L = g / (2 + 2)
 = 2/(t2-t1)
 = ln[H(t1)/H(t2)]/ (t2 – t1)
Underdamped Response Slug
Test (cont.)
• T1 = c + a ln c
• T2 = c + a ln T1
Till, L computed from
L = g / (2 + 2)
With 20% of the value as computed by
L = Lc + (rc2/rs2)(b/2)
Aquifer tests
• T = Q/ 4(h0-h)G(u)
• G(u) =
W(u)
- completely confined.
W(u,r/B) – leaky, confined, no storage.
H(u,) – leaky, confined, with storage.
W(uA,uB,) - unconfined.
Slug test
• Overdamped
– water level recovers to the initial static level in a
smooth manner that is approximately exponential.
• Underdamped
– water level oscillates about the static water level
with the magnitude of oscillation decreasing with
time until the oscillations cease.
Confined
x = -y/tan(2Kbiy/Q)
Q - pumping rate
K - conductivity
b – initial thickness
i – initial h gradient
x0 = -Q/tan(2Kbi)
ymax =  Q/(2Kbi)
Capture Zone Analysis
(unconfined aquifer)
• x = -y / tan[K[h12-h22)y/QL]
• x0 = -QL/[K(h12-h22)]
• ymax =  QL/[K (h12-h22)]
Static fresh and slat water
Ghyben-Herzberg principle
Dupuit assumptions
• Hydraulic gradient is equal to the slope of
the water table.
• For small water-table gradients, the
streamlines are horizontal and equipotential
lines are vertical.
Dupit-Ghyben-Herzberg model
z = [2Gq’x/K]1/2
x0 = - Gq’/2K
z = [G2q’2/K2 + 2Gq’x/K]1/2
Hx = H0 exp[-x(S/t0T)1/2]
tT = x(t0S/4T)1/2
t0 tide period
Hvorslev Method (partially
penetrated well)
• Log (h/h0) ~ time t (Le/R > 8)
• h – head at time t; h0 – head at time t = 0.
• K = (r2 ln (Le/R))/(2Let37)
K – hydraulic conductivity (L/T; ft/d, m/d);
r – radius of the well casing (L; ft, m);
R – radius of well screen (L; ft, m);
Le – length of the well screen (L; ft, m);
t37 – time it takes for water level to rise or fall to
37% of the initial change, (T; d, s).
Hvorslev Method (partially
penetrated well)
• Log (h/h0) ~ time t (Le/R > 8)
• h – head at time t; h0 – head at time t = 0.
• K = (r2 ln (Le/R))/(2Let37)
K – hydraulic conductivity (L/T; ft/d, m/d);
r – radius of the well casing (L; ft, m);
R – radius of well screen (L; ft, m);
Le – length of the well screen (L; ft, m);
t37 – time it takes for water level to rise or fall to
37% of the initial change, (T; d, s).
Bouwer and Rice Method
• K = rc2 ln (Re/R)/2Le 1/t ln(H0/Ht)
K – hydraulic conductivity (L/T; ft/d, m/d);
r – radius of the well casing (L; ft, m);
R – radius of gravel envelope (L; ft, m);
Re – effective radial distance over which head is
dissipated (L; ft, m);
Le – length of the well screen (L; ft, m);
t – time since H = H0
H – head at time t; H0 – head at time t = 0.
Bouwer and Rice Method
• ln (Re/R) = [1.1/ln(Lw/R) + (A+B ln(hLw)/R)/(Le/R)]-1 (Lw < h)
• ln (Re/R) = [1.1/ln(Lw/R) + C/(Le/R)]-1
(Lw = h)
H1
H2
(1/t) ln(H0/Ht)
= [1/(t2-t1)]ln(H1/H2)
t1 t2
This reflects K of
The undisturbed
aquifer
Transmissivity from specific
capacity data
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Specific capacity = yield/drawdown.
T = Q/(h0-h) 2.3/4log (2.25Tt/(r2S)).
T = 15.3 [Q/(h0-h)]0.67 [m,d]
T = 33.6 [Q/(h0-h)]0.67 [ft,d]
T = 0.76 [Q/(h0-h)]1.08 [m,d]