Ground-water flow to wells

• 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.)

• 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

• 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

• 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 FormulaConfined with no elastic storage

• 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 FormulaConfined with elastic storage

• 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 Formulaunconfined with elastic storage

• 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)

• H/H0 = F(,)

• H – head at time t.

• H0 – head at time t = 0.

= T t/rc2

= rs2S/rc

2

Cooper-Bredehoeft-Papadopulos Method (confined aquifer)

• H/H0 = 1, = 1 at match point.

= T t1/rc2

= rs2S/rc

2

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)

= ln[H(t1)/H(t2)]/ (t2 – t1)

= 2/(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/rs

2)(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.

x = -y/tan(2Kbiy/Q)

Q - pumping rateK - conductivityb – initial thicknessi – initial h gradient

x0 = -Q/tan(2Kbi)

ymax = Q/(2Kbi)

Confined

Capture Zone Analysis (unconfined aquifer)

• x = -y / tan[K[h12-h2

2)y/QL]

• x0 = -QL/[K(h12-h2

2)]

• ymax = QL/[K (h12-h2

2)]

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.

z = [2Gq’x/K]1/2

Dupit-Ghyben-Herzberg model

z = [G2q’2/K2 + 2Gq’x/K]1/2

x0 = - Gq’/2K

Hx = H0 exp[-x(S/t0T)1/2]

tT = x(t0S/4T)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(h-Lw)/R)/(Le/R)]-1 (Lw < h)

• ln (Re/R) = [1.1/ln(Lw/R) + C/(Le/R)]-1

(Lw = h)

(1/t) ln(H0/Ht) = [1/(t2-t1)]ln(H1/H2)

t1 t2

H1

H2

This reflects K of The undisturbed aquifer

Transmissivity from specific capacity data

• Specific capacity = yield/drawdown.

• T = Q/(h0-h) 2.3/4log (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]