Groundwater Flow Equations

Groundwater Hydraulics

Daene C. McKinney

Summary

• General Groundwater Flow– Control Volume Analysis– General Continuity Equation

• Confined Aquifer Flow– Continuity Equation– Integrate over vertical dimension– Transmissivity– Continuity– Examples

• Unconfined Aquifer Flow– Darcy Law– Continuity Equation– Examples

Control Volume• Control volume of dimensions Dx, Dy, Dz

• Completely saturated with a fluid of density r

x

yz

Mass flux in Mass flux out

2

xx

D

2

xx

D

2

)( x

x

qq x

x

D

rr

2

)( x

x

qq x

x

D

rr

xD

x

yD

zDControl

volume

Mass Flux

• Mass flux = Mass in - Mass out:

mass flux in mass flux out2

xx

D

2

xx

D

2

)( x

x

qq x

x

D

rr

2

)( x

x

qq x

x

D

rr

xD

x

yD

zD

rqx -¶ rqx( )¶x

Dx

2

é

ëê

ù

ûúDyDz - rqx +

¶ rqx( )¶x

Dx

2

é

ëê

ù

ûúDyDz = -

¶ rqx( )¶x

DV

Mass fluxMass in Mass out

Mass Flux

• Mass flux =

• Continuity: Mass flux = change of mass

• Fluid mass in the volume:

• Continuity

mass flux in mass flux out2

xx

D

2

xx

D

2

)( x

x

qq x

x

D

rr

2

)( x

x

qq x

x

D

rr

xD

x

yD

zD

-¶ rqx( )¶x

DV

m = frDV

Mass flux change of mass

Aquifer Storage

Water compressibility Aquifer compressibility

Chain rule

Now, put it back into the continuity equation

Continuity Equation

 

¶

¶xKx

¶h

¶x

æ

è ç

ö

ø ÷ +

¶

¶yKy

¶h

¶y

æ

è ç

ö

ø ÷ +

¶

¶zKz

¶h

¶z

æ

è ç

ö

ø ÷ = S

¶h

¶t

Confined Aquifer Flow

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)= 1

b h(x,y,z,t)dz0

bò

Ground surface

Bedrock

Confined aquiferQx

K

x

yz

h

Head in confined aquifer

Confining Layer

b

 

qx(x,y,t)= 1

b qx(x,y,z,t)dz0

bò

 

Qx = bqx

Aquifer Transmissivity

• Transmissivity (T) – Discharge through thickness of

aquifer per unit width per unit head gradient

– Product of conductivity and thickness

Hydraulic

gradient = 1 m/m

b

1 m

1 m

1 m

Transmissivity, T, volume

of water flowing an area 1

m x b under hydraulic

gradient of 1 m/m

Conductivity, K, volume of water

flowing an area 1 m x 1 m under

hydraulic gradient of 1 m/m

Continuity Equation

• Continuity equation

• Darcy’s Law

• Continuity

 

-¶Qx

¶x= S

¶h

¶t

 

Qx = -Tx¶h

¶x

¶

¶xTx

¶h

¶x

æ

èç

ö

ø÷ = S

¶h

¶t

Ground surface

Bedrock

Confined aquiferQx

K

x

yz

h

Head in confined aquifer

Confining Layer

b

1

r

¶

¶rr

¶h

¶r

æ

èç

ö

ø÷ =

S

T

¶h

¶t

Radial Coordinates

Example – Horizontal Flow

• Consider steady flow from left to right in a confined aquifer

• Find: Head in the aquifer, h(x)

¶

¶xT

¶h

¶x

æ

èç

ö

ø÷ = S

¶h

¶t= 0

 

Td2h

¶x2= 0

Ground surface

Bedrock

Confined aquifer

Qx

K

xyz

hB

Confining Layer

b

hA

L

steady flow

 

h(x)

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

Ground surface

Bedrock

Confined aquifer

Qx

K=2-m/dxyz

hB=80m

Confining Layer

b

hA=100m

L=1000m

Unconfined Aquifer Flow

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

x

yz

 

Qx = qxh = (-K¶h

¶x)h

 

-¶Qx

¶x= Sy

¶h

¶t

 

¶

¶xKh

¶h

¶x

æ

è ç

ö

ø ÷ = Sy

¶h

¶t

Steady Flow in an Unconfined Aquifer

• 1-D flow

• Steady State,

• K = constant

• Find h(x)

 

¶

¶xKh

¶h

¶x

æ

è ç

ö

ø ÷ = Sy

¶h

¶t

h

FlowhA

hB

Water Table

Ground Surface

Bedrock Lx

Steady Flow in an Unconfined Aquifer

• K = 10-1 cm/sec

• L = 150 m

• hA = 6.5 m

• hB = 4 m

• x = 150 m

• Find h(x), Q

h

FlowhA=6.5m

hB=4m

Water Table

Ground Surface

Bedrock L=150mx

K=0.1cm/s

Summary

• General Groundwater Flow– Control Volume Analysis– General Continuity Equation

• Confined Aquifer Flow– Continuity Equation– Integrate over vertical dimension– Transmissivity– Continuity– Examples

• Unconfined Aquifer Flow– Darcy Law– Continuity Equation– Examples

Groundwater Flow EquationsExamples

Example – Horizontal Flow

• Consider steady flow from left to right in a confined aquifer

• Find: Head in the aquifer, h(x)

¶

¶xT

¶h

¶x

æ

èç

ö

ø÷ = S

¶h

¶t= 0

 

Td2h

¶x2= 0

 

h(x) = hA + hB - hA

Lx

Ground surface

Bedrock

Confined aquifer

Qx

K

xyz

hB

Confining Layer

b

hA

L

steady flow

Head in the aquifer

 

h(x)

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)80 -100

1000

= 0.4 m /day

 

v =q

f

=1.14 m /day

Ground surface

Bedrock

Confined aquifer

Qx

K=2-m/dxyz

hB=80m

Confining Layer

b

hA=100m

L=1000m

Steady Flow in an Unconfined Aquifer

• 1-D flow

• Steady State,

• K = constant

 

¶

¶xKh

¶h

¶x

æ

è ç

ö

ø ÷ = Sy

¶h

¶t

 

d

dxKh

dh

dx

æ

è ç

ö

ø ÷ = 0

 

h2(x) = hA2 + (

hB2 - hA

2

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

æ

è ç

ö

ø ÷

Steady Flow in an Unconfined Aquifer

• K = 10-1 cm/sec

• L = 150 m

• hA = 6.5 m

• hB = 4 m

• x = 150 m

• Find Q

h

FlowhA=6.5m

hB=4m

Water Table

Ground Surface

Bedrock L=150mx

 

Q = -K

2

hB2 - hA

2

L

æ

è ç

ö

ø ÷ = -

86.4 m /d

2

6.52 - 42

150

æ

è

ç ç

ö

ø

÷ ÷

= 7.56 m3 /d /m

K=0.1cm/s