yhd-12.3105 soil and groundwater hydrology steady-state flow teemu kokkonenemail:...
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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
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Aquifer Types
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Hydraulisia johtavuuksia
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 1D
• 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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 1D
• 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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 3D
• 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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
• Typically thicknes of aquifers is relatively small compared to their areal extent, which justifies the assumption of essentially horizontal flow
Equation for Groundwater Flow Steady state 2D
0zq
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
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.
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 2D, Sink / Source
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 :||
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 2D, Sink / Source
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
Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Equation for Groundwater Flow Steady state 2D, Sink / Source
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
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
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...)
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
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
Yhd-12.3105 Soil and Groundwater Hydrology
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