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Flow in Aquifers

CE374L Groundwater Hydraulics

Daene C. McKinney

www.ce.utexas.edu/prof/mckinney/ce374l/ce374l.html

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Summary

Confined Aquifer Flow

Continuity Equation

Steady Horizontal Flow

Transmissivity

Unconfined Aquifer Flow

Continuity Equation

Steady Horizontal Flow

Steady Horizontal Flow with Infiltration

Drains

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Confined Aquifer Flow

CE374L Groundwater Hydraulics

Daene C. McKinney

www.ce.utexas.edu/prof/mckinney/ce374l/ce374l.html

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Recall: Continuity Equation

Continuity equation

Darcys Law

1-D Continuity

3-D Continuity

qx Kh

x

qxx

Sht

x

Khx

S

ht

x

Kx hx

yKy hy

zKz hz

Sh

t

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Horizontal Aquifer Flow

Most aquifers are thin

compared to horizontal

extent

Flow is horizontal, qx and qy

No vertical flow, qz = 0

Average properties over

aquifer thickness (b)

h(x,y,t)= 1b

h(x,y,z,t)dz0

b

Ground surface

Bedrock

Confined aquiferQx

K

xyz

h

Head in confined aquifer

Confining Layer

b

qx(x,y,t)=1

bqx(x,y,z,t)dz

0

b

Qx bqx

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Aquifer Transmissivity

Transmissivity (T)

Discharge through thickness of

aquifer per unit width per unit

head gradient Product of conductivity and

thickness

T =K b

Hydraulicgradient = 1 m/m

b

1 m

1 m

1 m

Transmissivity, T, volumeof water flowing an area 1m x b under hydraulic

gradient of 1 m/m

Conductivity, K, volume of waterflowing an area 1 m x 1 m underhydraulic gradient of 1 m/m

K(x,y)=1

bK(x,y,z)dz

0

b

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Recall: Continuity Equation

Continuity equation

Darcys Law

Continuity

Qxx

Sht

Qx Txhx

x

Txhx

S

ht

Ground surface

Bedrock

Confined aquiferQx

K

xyz

h

Head in confined aquifer

Confining Layer

b

1

r

r

rhr

ST

ht

Radial Coordinates

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Example Horizontal Flow

Consider steady flow from left to right in a confined aquifer

Find: Head in the aquifer, h(x)

xTx

h

x

S

h

t 0

Td2h

x2 0

h(x) hA hB hA

Lx

Ground surface

Bedrock

Confined aquiferQx

Kxyz

hB

Confining Layer

b

hA

L

steady flow

Head in the aquifer

h(x)

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Example Horizontal Flow

L = 1000 m, hA = 100 m, hB = 80 m, K= 20 m/d, f= 0.35 Find: head, specific discharge, and average velocity

h(x) hA hB hA

Lx 100 0.02x m

q KhB hA

L

(20 m /d) 801001000

0.4 m /day

v q

f

1.14 m /day

Ground surface

Bedrock

Confined aquiferQx

Kxyz

hB

Confining Layer

b

hA

L

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Unconfined Aquifer Flow

CE374L Groundwater Hydraulics

Daene C. McKinney

www.ce.utexas.edu/prof/mckinney/ce374l/ce374l.html

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Flow in an Unconfined Aquifer

Dupuit approximations

Slope of the water table is small

Velocities are horizontal

Ground surface

Bedrock

Unconfined aquifer

Water table

Dx

Qx

K

h

xyz

Qx qxh (Khx

)h

Qx

x

Syh

t

x

Khhx

Sy

ht

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Steady Flow in an Unconfined Aquifer

1-D flow

Steady State

K= constant

d2 h2 dx2

0

x

Khhx

Sy

ht

d

dxKh

dh

dx

0

h2(x) hA2 (hB2

hA2

L)x

h

FlowhA

hB

Water Table

Ground Surface

Bedrock Lx

Q (Kdh

dx)h

K

2

dh2

dx

K

2

hB2 hA

2

L

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Steady Flow in an Unconfined Aquifer

K= 10-1 cm/sec

hA = 6.5 m

hB = 4 m

x= 150 m

Find Q hFlowhA

hB

Water Table

Ground Surface

Bedrock Lx

Q K

2

hB2 hA

2

L

86.4 m /d

2

6.52 42

150

7.56 m

3/d/m

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Steady Flow in an Unconfined Aquifer

With Infiltration

hhMax

FlowhA

hB

Water Table

Ground Surface

Bedrock

N, Infiltration

Lx

h2(x) hA

2

hB2 hA

2

L

x

N

K(L x)x

d2 h2 dx2

2N

K

x

Khhx

N Sy

ht

Steady Flow Constant K

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h hMax FlowhL

hR

Water TableGround Surface

Bedrock

N, Infiltration

Lx

Steady flow in an unconfined aquifer

with infiltration

Q(x) K

2

d(h2 )

dx

xdivide L

2

K

2NL(hR

2 hL2)

dh2

dx

hB2 hA

2

L

N

K(L 2x)

Q(x) KhR2 hL

2

2LN(

L

2 x)

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Example (Embankment)

Given

L = 3000 m

K= 20 m/day

hL = 30 m

hR = 20 m

N = 500 mm/yr

Find: Flow to the streams and shape of water table

h

hMax

FlowhL

hR

Water Table

Ground Surface

Bedrock

N, Infiltration

Lx

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Example (Cont.)

xdivide 30002 20*(500)2*1.37*3

283.5m

Q(0) 20

2(1.37

203

302 202

3000)

3.72m3 /day

Q(L) 3.721.37*3

0.39m3 /day

h hMax

FlowhL

hR

Water Table

Ground Surface

Bedrock

N, Infiltration

Lx

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Example (Drains)

Subsurface drains are often used to prevent the water table elevation from

increasing to an elevation where it would waterlog the roots of crops or bring

unwanted salts and other chemicals in to the root zone.

Steady, 1-D flow , homogeneous,

isotropic aquifer, w/infiltration

hd

hMax

Water Table

Ground Surface

LowerConfiningLayer

N, Infiltration

Lx

d2 h2 dx2

2N

K

x

Khhx

N Sy

ht

h2 Nx

K(L x) hd

2

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Example (Cont.)

Solve for max water table elevation

hd

hMax

Water Table

Ground Surface

Lower ConfiningLayer

N, Infiltration

Lx

h2

Nx

K (L x) hd2

hMax2 h 0.5L 2

NL2

4K

hd2

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Summary

Confined Aquifer Flow

Continuity Equation

Steady Horizontal Flow

Transmissivity

Unconfined Aquifer Flow

Continuity Equation

Steady Horizontal Flow Steady Horizontal Flow with Infiltration

Drains