gw lecture 01-03

06.01.2011

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CVNG 2011- Engineering Hydrology –Part I: Groundwater Hydrology

Lecture 1Hydrologic Cycle Fluid Motion & Hydraulic Head

Department of Civil and Environmental Engineering

CVNG 2011 – Engineering Hydrology

- Hydrologic Cycle, Fluid Motion & Hydraulic Head -

What is “Hydrology” ?

The scientific study of the propertiesThe scientific study of the properties, distribution, and effects of water on the earth's surface, in the soil and underlying rocks, and in the atmosphere



atmosphere.

(http://www.thefreedictionary.com)

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Hydrologic cycle



after Chow, Maidment & Mays (1988)

Hydrologic cycle



Chow, Maidment & Mays (1988)

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Hydrologic budget

• also called “water budget” or “ water balance”

• Measurement of continuity of the flow of water

• Consider an “open system” where change of storage of water (dS/dt) is equal to the inputs (I) minus the outputs (O)



Hydrologic budget



Marsh & Dozier (1986)

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Hydrologic budget

• Groundwater System Hydrologic Budget

STEQGGI Δ ggggoutin STEQGGI Δ=−−−−+



Marsh & Dozier (1986)

Groundwater System Hydrologic Budget

• Example:Ground-water budget anal sis

Inflow to ground-water system m3/s

analysis

• Assume equilibrium with no change in storage

Groundwater recharge (I+Gin) 750

Outflow from ground-water system m3/s

Evapotranspiration (Eg+Tg) 50

Discharge to streams (Qg) 700



ggggoutin STEQGGI Δ=−−−−+

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Fluid Motion, Potentials & Hydraulic Head

Potentials in ground-water flow:

• Gravitational Potential - Energy due to positionΦGravitational Potential Energy due to position

– Reference elevation (z0) is arbitrary• Often taken as sea level• any convenient point can be selected depending on the particular

problem (e.g. z0=datum: z0=0)

gΦgzg =Φ



• Pressure Potential - Water under pressure possesses energy

– Using gauge pressure instead of absolute pressure

pΦ

ρρpppwithpp

patmp =Φ=−

=Φ 00

Fluid Motion, Potentials & Hydraulic Head• Kinetic Potential kΦ

2

2vk =Φ

– In groundwater, v is usually very small

• Fluid Potential (in groundwater)

2k

Φ

ρpgzpg +=Φ+Φ=Φ



– with

hydraulic head height of fluid above a point

ρΨ=

Φ= gpand

gh ρ

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Fluid Motion, Potentials & Hydraulic Head• Kinetic Potential kΦ

2

2vk =Φ

– In groundwater, v is usually very small

• Fluid Potential

2k

Φ

ρpgzpg +=Φ+Φ=Φ



– with

hydraulic head height of fluid above a point

ρΨ=

Φ= gpand

gh ρ Ψ+=zh

Fluid Motion, Potentials & Hydraulic Head

• Diagram of head components

In practical terms:

h

z

gpρ

=Ψ

datum, z=0



In practical terms:

z = elevation of the point above the selected datum

= height of water column above the point

h = hydraulic head at a point is the elevation of the top of the water column above the point with respect to the datum elevation

Ψ

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Using Piezometers for Hydraulic Head

• Piezometer is used to measure hydraulic head at a point in a geologic formation



point in a geologic formation.The elevation of the water inside the pipe is an “average” hydraulic head for the screened or opened interval.

Groundwater Movement• Water tends to move in directions

of decreasing energy, i.e. in directions of decreasing hydraulic head or decreasinghydraulic head or decreasing hydraulic potential.

• Rate of groundwater movement is proportional to the rate of change in energy, i.e. rate of change in hydraulic head.



• Rate of change of hydraulic head (with respect to distance in the direction of groundwater flow) is referred to as the hydraulic gradient (i).

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Groundwater Flow Directions



Charles Harvey , 1.72 Groundwater Hydrology, Fall 2005. (MIT OpenCourseWare: Massachusetts Institute of Technology),http://ocw.mit.edu/OcwWeb/Civil-and-Environmental-Engineering/1-72Fall-2005/CourseHome/ (Accessed August 10, 2009). License: Creative commons BY-NC-SA


Lecture 2Types of Aquifers Darcy’s Law Permeability Porosity



- Types of Aquifers, Darcy’s Law, Permeability, Porosity -

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Divisions of Subsurface Water




Divisions of Subsurface Water



after Todd & Mays (2005)

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Aquifers & Aquitards

• Aquifers:Geologic unit with a significant capacity to store and transmit water, i.e.

a pumping well will yield > 1litre/min (enough to supply a home with water)water)

Common aquifers medium to high K (hydraulic conductivity)e.g. unconsolidated sands and gravels

sandstoneslimestones (fractured and/or karstic)

• Aquitards:



May have sufficient capacity to store water, but transmits water very slowly! Does not supply sufficient water to wells when pumped!

Common aquifers low K (hydraulic conductivity)e.g. clays, silts, silty clays

Types of Aquifers

• Confined Aquifers:Positioned between two aquitards and the elevation of its piezometric surface

must be above the elevation of the top of this unit.




Water is under pressure !Water flows to the surface under artesian pressure.

• Unconfined Aquifers:“Water Table Aquifer”, i.e. water table defines the upper boundary; water is in

contact with athmospheric pressure

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Aquifer Contamination




• Although unconfined aquifers are used for water supply, they are often contaminated by wastes and chemicals at the surface.

• Confined aquifers are less likely to be contaminated and thereby provide supplies of good quality.

Darcy’s Law

• Henri Darcy (1803-1858)Published 1856 results from

studies of water flow through gρstudies of water flow through sand filters (used to pre-treat or “purify” water prior to use). He showed that the volumetric flow rate of water through the sand filter was proportional to the cross-sectional area of the cylinder

gρ

gρ



and the hydraulic gradient.This is considered to mark the

beginning of “groundwater” as a science.


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Darcy’s Law

• Flow through column is Q in [L3/T] m3/s

• Flow per unit area is specific discharge Q/A i [L/T] /

lhKqΔΔ

−= )1Re( <for

q=Q/A in [L/T] m/s

Note: called Darcian velocity or Darcian flux, but not actual velocity of the fluid!

• Q is in direction of decreasing head. (from high to low)

• Proportional to

gρ

gρ

12 hhhi −−=

Δ−=−



• K is the proportionality constant and incorporates the characteristics of both the fluid and the porous medium.

Known as Hydraulic Conductivity [L/T] m/safter Todd & Mays (2005)

12 llli

−Δ

Hydraulic Conductivity

• Hydraulic Conductivity (K) is an empirical parameter and depends on properties of both soil 1qp p pthe geologic material and the fluid.

Here: only water is regarded

• K represents the ease with which a particular fluid will pass through

soil 2

soil 3

K1> K2

K2

K3< K2

hΔ



a particular porous medium.

• Intrinsic Permeability (k):

lhΔΔ

μρgkK =

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Effect of Fluid and Porous Medium on Hydraulic Gradient

alcohol ?water ?molasses ?

dldhKq −=

molasses ?

d ?



sand ?silt ?gravel ?

Typical Values of Hydraulic Conductivity




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How to evaluate ‘Hydraulic Conductivity’ K?

1. Visual correlation (sediment type and literature values)2. Grain size methods (10+ “methods” or equations)3. Permeameter tests (Laboratory)4. Single-Well Tests or “Slug Tests” (Field5. Pumping Tests (Field):

- Pump production well and monitor changes in water level in piezometer(s).An integrated value of K over a large volume of the



- An integrated value of K over a large volume of the aquifer.

Porosity

• Typical symbols used for porosity: after Todd & Mays (2005)

αθφ ,,, nαθφ ,,, n

t

stV

VVV

VV −

==α

a) Well sorted (high)

φ ,,,

10 <≤α



• only connected pores contribute to groundwater flow

Effective Porosity

a) Well sorted (high)

b) Poorly sorted (low)

c) Well sorted with porous pebbles (high)

d) Deposition of mineral matter (low)

e) Rock rendered porous by dissolution.

f) Rock rendered porous by fracturing.

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Typical Values of ‘Porosity’



Average Linear Porewater Velocity• Also named “groundwater velocity” or “seepage velocity”

important in contaminant transport !

• Recall: q, the specific discharge (Darcy flux or Darcy “velocity”) has dimensions of a velocity [L/T] but is not a true velocity!i.e. obtained from volume/(area x time) not distance/time !

• However, the “groundwater velocity” (va) can be calculated from q.



• Because only a portion of the cross-sectional area is accessible to the water, the “groundwater velocity” is greater than the specific discharge: (va > q)

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Average Linear Porewater Velocity

1. Consider flow Q through a cross-sectional pipe of cross-sectional area A:

A Qqv ==

2. If the cross-sectional area is reduced by ½ but Q remains constant, then

3. Thus, by analogy, va for a porous medium can be calculated using the Darcy’s Law:

5.0qvBUT

AQq a ==

Q

Aqva ==



lhKqv

effeffa Δ

Δ−==αα

Tutorial Week 1:Example Calculations for Travel Time in a Porous Medium

As a result of a spill, contaminants entered the groundwater at a location 500m up-gradient from a water supply well.



p g pp y

The average hydraulic gradient is 0.01 and the hydraulic conductivity of the medium-fine grained sand is 10-3 cm/s.

How long will it take for contaminants to reach the well (in average) ?

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Lecture 3Continuity & Flow Nets



- Continuity & Flow Nets -

Equations of Groundwater Flow

1. Darcy’s Law in three dimensions:hKq ∂

−=

hKq

yhKq

xKq

zz

yy

xx

∂∂

−=

∂∂

−=

∂

Here we have 3 equations and 4 unknowns.We need another equation!



2. Conservation of Mass (Equation of Continuity):

For steady-state Flow:rate of fluid mass flow OUT = rate of fluid mass flow IN

zq zz ∂

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Equations of Groundwater Flow

assuming that the density is constant, this leads to: 0=⎥

⎦

⎤⎢⎣

⎡∂

∂+

∂

∂+

∂∂

−yq

yq

xq yyx

Substitution of Darcy’s Law for e.g.xhKq xx ∂∂

−=

Steady-state Flow through an Heterogeneous & Anisotropic Porous Medium (2D):

0=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

yhK

yxhK

x yx



Steady-state Flow through a Homogeneous & Isotropic Porous Medium (2D):

02

2

2

2

=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

yh

xh

(Laplace’s Equation)

Flow Nets

H t fi d ki d f l ti t thi ti

0=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

yhK

yxhK

x yx

• Have to find some kind of solution to this equation.

• Solution will be based on boundary conditions and in the transient case, on initial conditions.

• Relatively straightforward graphical technique can be usedconstruction of a flow net



• Flow net:– is a set of equipotential lines (constant head) and the associated

flow lines (lines along which groundwater moves)– needs a set of boundary conditions

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Boundary Conditions• no-flow

– Flow is parallel to the boundary.– Equipotentials are perpendicular to the

boundaryboundary.

• constant head– h=constant– Flow is perpendicular to the boundary.– Equipotentials are parallel to the boundary.



• water-table boundary– h=z (not a constant)– Neither flow nor equipotentials are

necessarily perpendicular to the boundary.Charles Harvey , 1.72 Groundwater Hydrology, Fall 2005. (MIT OpenCourseWare: Massachusetts Institute of Technology),http://ocw.mit.edu/OcwWeb/Civil-and-Environmental-Engineering/1-72Fall-2005/CourseHome/ (Accessed August 10, 2009). License: Creative commons BY-NC-SA

Rules for Flow Nets (Isotropic, Homogeneous System)

1. Flow is perpendicular to equipotentials everywhere.

2. Flow lines never intersect.

3. The areas between flow lines and equipotentials are “curvilinear squares”. i.e. If you draw a circle inside the



curvilinear square, it is tangential to all four sides at some point.

preserves dQ along any stream tube


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Calculations for Flow Nets (Isotropic, Homogeneous System)

dsdhi =

after Todd & Mays (2005)hdh

dhKqdmds

dmdsdhKq

=≅

=

for squares

n: number of head drops



y ( )

nKmhmqQ

ndh

==

= n: number of head drops

m: number of channels

Examples



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Flow Nets in Anisotropic, Homogeneous Systems

• In anisotropic media, flow lines and equipotential lines are not necessarily orthogonal to each other.

I d t l l t fl f thi it ti th b d i f fl ti t• In order to calculate flows for this situation, the boundaries of a flow section must be transformed so that an isotropic medium is obtained.

• For the typical case of , all horizontal dimensions are reduced by the ratio

• This creates a transformed section with an isotropic medium having an equivilant hydraulic conductivity

zxKKzx KK >



• After the flow net has been identified, it can be converted back to the true aniostropic section by multiplying all horizontal directions with K’

zxKKK ='

Examples

10/ =zx KKboth:




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Flow Nets in Heterogeneous Systems

• In a layered heterogeneous system, the same rules apply as in the homogeneous case, but with the following restrictions:– Curvilinear squares can only be

drawn in ONE layer. In a two-layer system, you will only have curvilinear squares in one of the layers.

• At boundaries between layers, flow lines are refracted (in a similar way to the way light is refracted between two different

Θ1

Θ2



gmedia). The relationship between the angles in two layers is given by the “tangent law”:

2

1

2

1

tantan

θθ

=KK after Todd & Mays (2005)

Example for Heterogeneous Systems




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Example

Draw a quantitatively accurate flow net for the case below. Show flow direction arrows on the flow lines. Calculate the ol me discharge per meter thickness of the section ifvolume discharge per meter thickness of the section if

K=1x10-4 m/s.



Tutorial Week 1:Flow Nets

1. Draw a quantitatively accurate flow net for the case below. Show flow direction arrows on the flow lines. Calculate the volume discharge per meter thickness of the section if K=1x10-3 cm/s.



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2. Draw a quantitatively accurate flow net for the case below:




3. Three layer system.K2=K3=1x10-3 m/sK1=1x10-4 /s1

Determine the flow lines (i.e. the angles) in layers K1 and K3.



gw lecture 01-03

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