1. Yhd-12.3105 Soil and Groundwater Hydrology
Steady-state flow
Teemu Kokkonen
Tel
Room: 272 (Tietotie 1 E)
Water Engineering
Department of Civil and Environmental Engineering
Aalto University School of Engineering

2. Aquifer types Aquifer Confined 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

3. Aquifer Types

4. 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

5. Hydraulisia johtavuuksia

6. Equation for Groundwater Flow Steady state 1D
Inflow per unit time Qi = 2 l s-1
Inflow per unit time and unit area (influx) qi = 2 l s-1 / 0.4 m2 = 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
Outflow per unit time Qo = 2 l/s
Outflow per unit time and unit area (outflux) qo = 0.5 cm s-1

7. 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
The outgoing flux must equal the incoming flux

8. 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:
Laplace equation in 1D

9. Equation for Groundwater Flow Steady state 3D
The groundwater equation just derived in one dimension is easy to generalise to three dimensions
Analogous analysis to the previous slide yields for the homogeneous 3D case:

10. Equation for Groundwater Flow Steady state 2D
Typically thicknes of aquifers is relatively small compared to their areal extent, which justifies the assumption of essentially horizontal flow

11. 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

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

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

14. Equation for Groundwater Flow Steady state 2D, Sink / Source
How does the equation change if the aquifer is homogeneous?
How does the equation change if the aquifer is isotropic?
Defining transmissivity T to be the product of hydraulic conductivity K and the thickness of the water conducting layer b yields:

15. Equation for Groundwater Flow Steady state 2D, Sink / Source
Does R vary in space? When?
Does T vary in space? When?

16. 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

17. 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...)

18. Numerical Solution Steady-state 1DDx
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?

19. Numerical Solution Steady-state 1D
Geometric average
Heterogeneous aquifer Dx
Let us again derive a numerical approximation for the steady-state 1D groundwater flow equation.
How to compute and ?

20. Numerical Solution 2D Steady-state Flow, Sink/source Termj

21. Boundary Conditions Numerical Solution
Hydraulic head to be computed from the groundwater flow equation.
Hydraulic head set to a fixed value representing the level of the lake.
Hydraulic head value is ”mirrorred” across the no-flow boundary.
Lake H = 10 m I II
Clay (almost impermeable)
HII = HI
Hydraulic gradient across this line becomes zero => no flow