Groundwater Flow EquationsGroundwater 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

2x

x

2x

x

2

)( xx

qq xx

2

)( xx

qq xx

xx

y

zControl volume

Mass Flux

• Mass flux = Mass in - Mass out:

• Fluid mass in the volume:

• Continuity: Mass flux = change of mass

mass flux in mass flux out2x

x

2x

x

2

)( x

x

qq xx

2

)( x

x

qq xx

xx

y

z

Aquifer Storage

Continuity Equation

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)

Ground surface

Bedrock

Confined aquiferQx

K

xyz

h

Head in confined aquifer

Confining Layer

b

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

Potentiometric

Surfa

ce

Confining Bed

Confined Aquiferb

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

Ground surface

Bedrock

Confined aquiferQx

K

xyz

h

Head in confined aquifer

Confining Layer

b

Radial Coordinates

Example – Horizontal Flow

• Consider steady flow from left to right in a confined aquifer • Find: Head in the aquifer, h(x)

Ground surface

Bedrock

Confined aquifer

Qx

Kxyz

hB

Confining Layer

b

hA

L

steady flow

Head in the aquifer

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

Steady Flow in an Unconfined Aquifer

• 1-D flow

• Steady State

• K = constant

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 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