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Yhd 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

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Yhd 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

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Yhd Soil and Groundwater Hydrology Steady-state flow Aquifer Types

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Yhd 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

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Yhd Soil and Groundwater Hydrology Steady-state flow Hydraulisia johtavuuksia

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Yhd Soil and Groundwater Hydrology Steady-state flow Equation for Groundwater Flow Steady state 1D Inflow per unit time Q i = 2 l s -1 Outflow per unit time Q o = 2 l/s Inflow per unit time and unit area (influx) q i = 2 l s -1 / 0.4 m 2 = 0.5 cm s -1 Outflow per unit time and unit area (outflux) q o = 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

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Yhd 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 The outgoing flux must equal the incoming flux

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Yhd 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: Laplace equation in 1D

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

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Yhd 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

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Yhd 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

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Yhd Soil and Groundwater Hydrology Steady-state flow Equation for Groundwater Flow Steady state 2D, Sink / Source q x (x) = 9 ? q x (x+ x) = 6 ? q y (y) = 2 ? q y (y+ y) = 5 ? x = 3 ? y = 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.

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Yhd Soil and Groundwater Hydrology Steady-state flow Equation for Groundwater Flow Steady state 2D, Sink / Source

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Yhd Soil and Groundwater Hydrology Steady-state flow 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:

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Yhd Soil and Groundwater Hydrology Steady-state flow Equation for Groundwater Flow Steady state 2D, Sink / Source Does R vary in space? When? Does T vary in space? When?

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Yhd 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

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Yhd 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...)

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Yhd Soil and Groundwater Hydrology Steady-state flow Numerical Solution Steady-state 1D 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? xx xx

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

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Yhd Soil and Groundwater Hydrology Steady-state flow Numerical Solution 2D Steady-state Flow, Sink/source Term i j

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Yhd Soil and Groundwater Hydrology Steady-state flow Boundary Conditions Numerical 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 H II = H I Hydraulic gradient across this line becomes zero => no flow

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