yhd-12.3105 soil and groundwater hydrology steady-state flow teemu kokkonenemail:...

Yhd-12.3105 Soil and Groundwater Hydrology

Steady-state flow

Teemu Kokkonen Email: [email protected]. 09-470 23838Room: 272 (Tietotie 1 E)

Water EngineeringDepartment of Civil and Environmental EngineeringAalto University School of Engineering


Steady-state flow

Aquifer types

• Aquifer– Latin: aqua (water) + ferre (bear, carry)– An underground bed or layer of permeable rock, sediment,

or soil that yields water

• Confined aquifer– Between two impermeable layers– Groundwater is under pressure and will rise in a borehole

above the confining layer

• Unconfined aquifer (phreatic aquifer)– Groundwater table forms the upper boundary


Steady-state flow

Aquifer Types


Steady-state flow

Some Terms

• Saturated zone (vadose zone)− Pore space fully saturated with water

• Groundwater level (water table)– Is defined as the surface where the water pressure is equal to the

atmospheric pressure– In groundwater studies the atmospheric pressure is typically used

as the reference point and assigned with the value of zero• Capillary fringe

– Saturated (or almost saturated) layer just above the grounwater level

• Unsaturated zone– Both water and air are present in the pore space


Steady-state flow

Hydraulisia johtavuuksia


Steady-state flow

Equation for Groundwater Flow Steady state 1D

Inflow per unit time Qi = 2 l s-1

Outflow per unit time Qo = 2 l/s

Inflow per unit time and unit area (influx) qi = 2 l s-1 / 0.4 m2 = 0.5 cm s-1

Outflow per unit time and unit area (outflux) qo = 0.5 cm s-1

• When the water level in the container does not change it is in steady-state

– The influx and outflux must then be equal to each other


Steady-state flow


• Darcy’s law– Conservation of momentum

• Continuity equation– Conservation of mass

In steady state conditions the amount of stored water does not change

dx

dHKq

The outgoing flux must equal the incoming flux


Steady-state flow


• Conservation of mass

• Inserting Darcy’s law to describe the flux q yields:

• Under the assumption of homogeneity:

0dx

dq

00

dx

dHK

dx

d

dx

dHK

dx

d

dx

dq

002

2

2

2

dx

Hd

dx

HdK

dx

dH

dx

dK

dx

dHK

dx

d

Laplace equation in 1D


Steady-state flow


• The groundwater equation just derived in one dimension is easy to generalise to three dimensions

0dz

dq

dy

dq

dx

dq zyx

Analogous analysis to the previous slide yields for the homogeneous 3D case:

02

2

2

2

2

2

dz

Hd

dy

Hd

dx

Hd


Steady-state flow

• Typically thicknes of aquifers is relatively small compared to their areal extent, which justifies the assumption of essentially horizontal flow


0zq


Steady-state flow

Exchange of Water: Sink or Source

• An aquifer can receive (source) or loose (sink) water in interaction with the world beyond its domain– Source: recharge from precipitation, injection wells – Sink: pumping wells

• In the groundwater equation added or removed water is described using a sink / source term


Steady-state flow

Equation for Groundwater Flow Steady state 2D, Sink / Source

yxRy

dy

xbqdx

dx

ybqd yx

qx(x) = 9 ? qx(x+Dx) = 6 ?

qy(y) = 2 ? qy(y+Dy) = 5 ?

Dx = 3 ? Dy = 2 ? b = 2 ?

R = -1 ?

Explain in your own words what water balance components the circled terms in the above equation represent. Use then the values given below to compute their values assuming that the derivatives are constant within the rectangular control volume. Give also units to the quantities listed below.


Steady-state flow


R

dy

bqd

dx

bqd yx

yxRy

dy

xbqdx

dx

ybqd yx

Rdy

dHbK

dy

d

dx

dHbK

dx

dyx

yxyxRyx

dy

bqdyx

dx

bqd yx :||


Steady-state flow


Rdy

dHbK

dy

d

dx

dHbK

dx

dyx

How does the equation change if the aquifer is homogeneous?R

dy

dH

dy

dbK

dx

dH

dx

dbK yx

How does the equation change if the aquifer is isotropic?

Rdy

dH

dy

dKb

dx

dH

dx

dKb

Defining transmissivity T to be the product of hydraulic conductivity K and the thickness of the water conducting layer b yields:

T

R

dy

Hd

dx

Hd

2

2

2

2


Steady-state flow


T

R

dy

Hd

dx

Hd

2

2

2

2

Does R vary in space? When?

Does T vary in space? When?

Rdy

dHbK

dy

d

dx

dHbK

dx

dyx

Rdy

dHT

dy

d

dx

dHT

dx

dyx


Steady-state flow

Boundary Conditions

• Governing equation for groundwater flow– Describes how the water flux depends on the gradient of the

hydraulic head (Darcy’s law)– Requires the mass to be conserved

• To represent a particular aquifer boundary conditions need to be defined– Boundary conditions describe how the studied aquifer

interacts with the regions surrounding the aquifer


Steady-state flow

Boundary Conditions

• Two main categories – Constant head (fixed head,

prescribed head)• Dirichlet condition• Water bodies (lakes, rivers)

– Constant flux • Neumann condition• Impermeable boundary is a

common special case (clay, rock, artificial liners...)


Steady-state flow

Numerical SolutionSteady-state 1D

x

HH ii

1

02

2

dx

Hd

211

iii

HHH

Homogeneous aquifer

Let us derive a numerical approximation for the steady-state 1D groundwater flow equation.

Step 1. How would you approximate the spatial derivative between nodes i and i+1?

Step 2. How would you approximate the spatial derivative between nodes i-1 and i?

Step 3. How would you approximate the 2nd spatial derivative around node i?

x

HH ii

1

x

H i H i+1H i-1

Dx Dx

0)(

22

11

x

HHH iii


Steady-state flow

Numerical SolutionSteady-state 1D

Heterogeneous aquifer

Let us again derive a numerical approximation for the steady-state 1D groundwater flow equation.

How to compute and ?

0

dx

dHK

dx

d

H i H i+1H i-1

Dx Dx

½iK ½iK

0

1½

1½

xxHH

KxHH

K iii

iii

½iK ½iK

Geometric average

iii KKK 1½

iii KKK 1½


Steady-state flow

Numerical Solution2D Steady-state Flow, Sink/source Term

Ry

HT

yx

HT

x

ji

jijiji

jijiii

jijiji

jijiji

Ry

y

HHT

y

HHT

x

x

HHT

x

HHT

,

1,,½,

,1,½,

,1,½,

,,1½,

H i,j H i+1,jH i-1,j

H i,j-1

H i,j+1

i

j

i

j


Steady-state flow

Boundary ConditionsNumerical Solution

Hydraulic head to be computed from the groundwater flow equation.

Lake H = 10 m

Clay (almost impermeable)

Hydraulic head set to a fixed value representing the level of the lake.

Hydraulic head value is ”mirrorred” across the no-flow boundary.

I

II

HII = HIHydraulic gradient across this line becomes zero => no flow

yhd-12.3105 soil and groundwater hydrology steady-state flow teemu kokkonenemail:...

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