SEES 503 - 8. Groundwaterusers.metu.edu.tr/bertug/SEES503/SEES 503 - 8... · SEES 503 Sustainable...

SEES 503 Sustainable Water Resources 1/70

Instructor

Assist. Prof. Dr. Bertuğ Akıntuğ

Civil Engineering ProgramMiddle East Technical University

Northern Cyprus Campus

SEES 503SEES 503SUSTAINABLE WATER RESOURCESSUSTAINABLE WATER RESOURCES

GROUNDWATERGROUNDWATER


8. GROUNDWATER 8. GROUNDWATER

Overview

IntroductionOccurrence of subsurface waterStorage characteristics of aquifersFundamentals of groundwater flowGroundwater flow equationsUnsteady radial flow (well hydraulics)Generalization of solutions



Introduction

Groundwater Hydrology: deals with theexistence,movement,quantity, andquality of water in the soil formations below the ground surface.

Application areasthe water supply through the wells,water storage in underground reservoirs,solution of groundwater contamination problems,lowering the groundwater table etc….



IntroductionNumber of wells and their depths in different regions of Turkey (DSI, 1995)

Groundwater: • 0.76% of all waters• 30% of the total fresh water• generally free from pollution• useful for domestic and irrigation use

In Turkey• 40% of total fresh water

In USA• 25% of total fresh water

The surface water and groundwater feed each other. Therefore the exchange between ground and surface water source must be understood clearly.



Overview




Occurrence of Subsurface Water

In general, the water below the surface of the soil occurs in two major zones separated by ground table.

Zone of aeration. The pores contain water and air.

The pores completely full of water




Aeration Zone:There are three different parts in aeration zone.

Root zone: Thickness depends on the type of vegetation.

Thickness depends on the size of the soil particles.

There are four types of water in the aeration zone.• gravity water• capillary water• hygroscopic water• water vapor




Field Capacity: The amount of water remaining in the soil after percolation.Wilting Point: The lower limit of water content in the soil at which plants cannot extract water any more.Irrigation should be applied before wilting point is reached.

Equilibrium points in the soil




Saturation Zone:Water in the saturation zone is called groundwater.

The geologic formations which contain water are called aquifers.

They are generally sand and gravel formations and classified as

• Confined Aquifers• Unconfined Aquifers




There are mainly four different formations in saturated zones:Aquifer is the saturated permeable geologic formation, which contains and transmits water in economic amounts, under ordinary hydraulic gradients for water supply and has generally sand and gravel.

Aquifuge is sold granite type formation and it neither contains nor transmits water, therefore totally accepted as totally impervious.

Aquiclude is relatively impermeable saturated material like clay. It generally contains but not transmit it, therefore it is also accepted as impervious.

Aquitard is a formation, which may not transmit water to a well in economic amounts but may feed an adjacent aquifer through the leakage especially if it is thick. Such formations generally contain sandy clay or gravel in them.




Types of Aquifers:



Overview




Storage Characteristics of Aquifer

Porosity: The ratio of pore volume to the total volume.The amount of water stored in an aquifer is a function of its porosity.Pore size vary greatly in different soils.

Clay and shale very small poresLimestone and lava very large pores




The productivity of the aquifer is not directly related to its porosity.A different characteristics is defined to represent aquifers total water storage and also ability to transmit it.Storage coefficient (storativity, S) = volume of water released from storage (or added to it) / unit horizontal area of the aquifer / unit decline (or rise) in the piezometric head.




S = b Ss

Ss : Specific Storativity

b: thickness of the aquifer.

The mechanism of releasing water in confined and unconfined aquifer are completely different.




Water Release in Unconfined AquiferAs water is pumped from a well (water release) the piezometric surface, which is ground water table, drops.However pores are not completely drain.(The specific storativity of unconfined aquifer) < (the porosity)Specific storativity Specific Yiled




Water Release in Unconfined AquiferSpecific Retention = Porosity - Specific YieldRetantionclay = 42 – 3 = 39%Retantionsand(coarse) = 39 – 27 = 12%




Water Release in Confined AquiferThe storage coefficient is a function of the elasticity and compressibility of the aquifer.

Total stress: σT = σs + pσs: skeleton stressp: water pressure

When water is pumped from a well, the water pressure drops (dp).

Since the load above is unchanged and has to be carried, the extra part is carried by the skeleton of the medium and therefore there will be an increase in the skeleton stress with in the same amount.dσs = - dp



Overview




Fundamentals of Groundwater Flow

For easy processing, the actual porous medium is replaced by an imaginary medium which has the same characteristics.

In the imaginary medium, variables and the characteristics are averaged and assumed to represent the whole porous medium for groundwater flow.

This is called continuum approach.

A sample of porous medium




The flows in a actual porous medium occurs between particles as seen in figure (a), having and actual velocity distribution as shown in figure (b).

The actual velocity will be a function of all three cartesian coordinates plus time, which is difficult to deal with.

Therefore, a simple velocity definition is given for groundwater flow analysis. It is named as discharge velocity, specific discharge or superficial velocity.




From here on in the following parts the term velocity will mean discharge velocity as it is defined above.

area tiveRepresentamedium porous of area tiverepresenta a through Discharge

dischargespecific velocity lsuperficiavelocity discharge

=




Total head, h is constant for a liquid at rest or in uniform horizontal flow. This is also valid for a saturated continuous porous medium. Velocity head (v2/2g) is neglected, since the velocity is very small.

constant)(hpz =+γ

Heads in unconfined and confined aquifers




Darcy Law

dldhAK

LhhAKQ −=

−−= 12

Darcy’s experimentWhereQ: flow rate (m3/s)A: cross sectional area (m2)K: Hydraulic conductivity or permeability (m/s)h: Hydraulic headL: Lengthdh/dl: hydraulic gradient




Permeability or hydraulic conductivity is the measure of resistance of the medium to the flow and it is a function of the characteristics of both fluid and porous medium.

Permeability or hydraulic conductivity change with time and location.




Heterogeneity and AnisotropyIsotropic – anisotropic: If the hydraulic conductivity is the same in all directions offlow at a certain point, the medium is said to be isotropic, otherwise anisotropic.

In an anisotropic medium directions at which the hydraulic conductivity gets its maximum and minimum value are called principal directions of anisotropy.

Homogeneous – Heterogeneous: If in a medium hydraulic conductivity does not change from one point to the other, the medium is said to be homogeneous, otherwise heterogeneous




Possible combinations of heterogeneity and isotropy (Sevuk, 1986)




When hydraulic conductivity is considered to be different in the three cartesiandirections, then the velocity components in these directions will be given with corresponding conductivity value in Darcy’s Law.

dydhKV yy −=

dzdhKV zz −=

dxdhKV xx −=



Overview




Groundwater Flow Equations

Confined AquifersGroundwater flow equations = Continuity Equation + Darcy’s Law

Control Volume

Thickness of the aquifer

qv = Leakage into the aquifer / unit horizontal area.




Confined AquifersThe net flux through the control volume is equal to the rate of change of mass, which is given as:

whereρ : density of the fluidS: storage coefficient of the aquifer

: change of head with timedx, dy: elementary horizontal distance

th ∂∂ /

dydxthSFLUXNET∂∂

= ρ




Confined AquifersFor two-dimensional unsteady flow case in confined aquifers, the equation will be as

when velocity components are expressed as given by Darcy’s Law, and the aquifer thickness times the permeability (b x K) is defined as transmissivity, T of the aquifer the following equation will be obtained

( ) ( )thSqbV

ybV

x vyx ∂∂

=+∂∂

−∂∂

−

thSq

yhT

yxhT

x vyx ∂∂

=+

∂∂

∂∂

−

∂∂

∂∂

−




Confined AquifersThis equation is called general differential equation of groundwater

When the aquifer is homogeneous, transmissivity does not change with location

thSq

yhT

yxhT

x vyx ∂∂

=+

∂∂

∂∂

−

∂∂

∂∂

−

thSq

yhT

xhT vyx ∂

∂=+

∂∂

−∂∂

− 2

2

2

2




Confined AquifersIn addition to homogenity, if aquifer is also isotropic then Tx will be equal to Ty which will be a constant transmissivity value T

If there is no leakage to the aquifer, the term qv will also drop

Finally if the flow is steady, the change in head with respect to time will be zero and Laplace equation for two dimensional groundwater flow will be obtained.

th

TS

Tq

yh

xh v

∂∂

=+∂∂

+∂∂

2

2

2

2

th

TS

yh

xh

∂∂

=∂∂

+∂∂

2

2

2

2

02

2

2

2=

∂∂

+∂∂

yh

xh




Unconfined AquifersIn unconfined aquifers there is also a vertical velocity component.To avoid this difficulty, Dupuit approximation is made which assume the slope of the water table as negligible. This way the flow is assumed to be horizontal as seen in the figure.

The general equation for anisotropic, heterogeneous and unconfined aquifer

thSq

yhhK

yxhhK

x yvyx ∂∂

=+

∂∂

∂∂

−

∂∂

∂∂ Sy: specific yield

h: depth of the aquifer



Overview




Unsteady Radial Flow (Well Hydraulics)

For the solution of the problems, groundwater equation is used considering the following assumptions.1. Aquifer is homogeneous, isotropic and infinite in areal extent.2. The thickness of the aquifer is constant.3. Pumping is continuous with a constant rate.4. Well diameter is infinitely small.5. Initially piezometric surface is horizontal.

The solution for the following two case will be presented here.Fully penetrating well in a confined aquifer.Fully penetrating well in a leaky confined aquifer.




Fully Penetrating Well in a Confined AquiferWhen the well penetrates, the flow towards the well will be in horizontal direction only.

Theis and Cooper - Jacob Methods

An infinite, homogenous, and isotropic aquifer having constant thickness.

The drawdown, s, can be found by the solution of the following equation:

ts

TS

rs

rrs

∂∂

=∂∂

+∂∂ 1

2

2

Fully penetrating well in a confined aquifer.




Fully Penetrating Well in a Confined AquiferTheis and Cooper - Jacob Methods (continued)

Equation is subject to the following initial

and boundary conditions:

1. s(r,0)=0 Drawdown is zero at any point at time zero.

2. s(∞,t)=0 Drawdown is zero at any time when the point is infinitely far away.

3. Point sink condition

ts

TS

rs

rrs

∂∂

=∂∂

+∂∂ 1

2

2

TQ

rsr

r π2lim

0−=

∂∂

→





For the solution, Boltzman variable is defined

Then the drawdown, s:

where

Q: discharge (m3/s), s: drawdown (m)r : radial distance (m), S: storage coefficient,T: transmissivity (m2/s) t: time from the start of pumping (s)

TtSru

4

2

=

)(4

uWTQsπ

=

dxxeuW

u

x

∫∞ −

=)( This solution type is called Theis method.





W(u) well function: W(u) – u for a confined aquifer.





Well function can be expressed as an infinite series as

When u is small, u<0.01, the terms after the first two, become very small and therefore well function can be approximated by the first two terms.

Then drawdown equation will be

−−=

TtSr

TQs

4ln5772.0

4

2

π

K±×

+×

−+−−=!33!22

ln5772.0)(32 uuuuuW





After rearranging and converting the ten-base logarithm the drawdown equation will be

This solution is first given by Cooper and Jacob (1946) and is valid for small u values. As it is seen from the definition of u, it will be small if r is small or t is large.

=

SrTt

TQs 2

25.2log4

3.2π

TtSru

2

2

=




Fully Penetrating Well in a Confined AquiferTheis and Cooper - Jacob Methods (continued)Both methods has some practical applications such as

Determination of aquifer characteristics, S and T, by performing pumping tests with observations of discharge and corresponding drawdown and time.

For an aquifer with known characteristics, S and T, determination of drawdown for a certain discharge at a certain location and time.

Determination of maximum discharge for a maximum permissible drawdown at a certain location and time within an aquifer, whose characteristics are known.




Fully Penetrating Well in a Confined AquiferDetermination of Aquifer CharacteristicsAquifer characteristics, S and T, can be determined graphically by

1. Theis Method, or2. Cooper – Jacob Method

Theis Method

)(4

uWTQsπ

=TtSru

4

2

=

)(log4

loglog uWTQs +=π

uST

tr 42

=

uST

tr log4loglog

2

+=




Fully Penetrating Well in a Confined AquiferDetermination of Aquifer CharacteristicsTheis Method

Using the similarity of these two equations and conducting a pumping test with constant rate for a long duration S and T can be determined with the following procedure

1. A plot of W(u) – u (type curve) is prepared on log-log paper graph a.

Theis graphical method




Determination of Aquifer CharacteristicsTheis Method

2. From the pumping test data, a plot of r2/t – s is prepared on a transparent log-log paper (graph b).

The length of cycle in both graphs should be the same.





Determination of Aquifer CharacteristicsTheis Method

3. The plot of data is superimposed on type curve keeping the coordinate axes parallel to each other and adjusting as many points as possible on the curve.

4. An arbitrary point is selected and corresponding four coordinate values areobtained form the four axes:W(u)*, u*, values from graph aand (r2/t)* and s* values fromgraph b.





Fully Penetrating Well in a Confined AquiferDetermination of Aquifer CharacteristicsTheis Method

∗

∗

=suWQT

π4)(

( )∗∗

=tr

TuS/

42




Fully Penetrating Well in a Confined AquiferDetermination of Aquifer CharacteristicsCooper - Jacob Method

Plot s – t on a semi-log paper where s will be on linear axis.Slop of the line gives transmissivity.

sQT∆

=π43.2





Plot s – t on a semi-log paper where s will be on linear axis.The line is extended to find t0.

2025.2

rTtS =





For Cooper – Jacob method, u mast be small.u should be checked after T and S are determined.If u is not less than 0.01, the new straight line should be tried for larger time portion of the data pointsThe procedure is then repeated to determine satisfactory S and T.



Overview




Generalization of Solutions

The following assumptions were made for the analytical solutions:1. There is a single well2. The well is pumping with a constant rate continuously3. Aquifer is infinite in areal extent.

To be able to solve practical cases one more assumption has been made.4. Aquifer system and governing groundwater flow equations are linear and superposition method is applied.




Multiple Well CaseThe drawdown at any point in an aquifer is equal to the summation of all drawdownsoccurring due to each of the wells independently.

Drawdown in a confined aquifer with two pumping wells




Multiple Well Case

)(4

)(4 2

21

121 uW

TQuW

TQsss

ππ+=+=

TtSru

TtSru

4and

4

22

2

21

1 ==

∑=

=n

ii

i uWTQs

1)(

4π

In general




Multiple Well CaseQ1=Q2=Q3

Drawdown in a three well system




Variable PumpingAssuming that pumping started with discharge Q0 at time t0 and increased with amounts ∆Q1, ∆Q2, …, ∆Qn, at time t1, t2, …, tn.The drawdown at a distance r from the well will be:

∑=

∆+=n

iii uWQ

TuW

TQs

10

0 )(4

1)(4 ππ

t* = the time at which drawdown is required

)(4and

4 *

2

*

2

0i

ii ttT

SruTtSru

−==




Variable Pumping

Superposition of drawdowns for stepwise pumping




Variable Pumping (Recovery of a well)

Drawdown for recovery after pumping is stopped

( ))()(4 21 uWuWTQs −=π

)(4and

4 *

2

2*

2

1dttT

SruTtSru

−==td = time after pumping is stopped.




Finite AquiferOne of the assumptions for the solution of the groundwater equation was the infinity of aquifer in areal extent.But this may not be true in reality.There may be a recharge (wet) boundary (i.e. river, lake, reservoir) nearby the well orThere may be a barrier (impervious) boundary (i.e. rocky area) nearby the well. For the solution of such cases image well concept is used together with superposition assumptions.In this concept, the aquifer with a boundary is replaced by an imaginary aquifer with infinite areal extend andan imaginary well located at a point symmetrical to the real well with respect to the boundary.




Finite AquiferIn case of impervious boundary, the image well is assumed to be pumping with the same rate as real well and having the same drawdown curve.Then actual drawdown will be determined by summation of the two drawdowns as shown in the figure.

Effect of impervious boundary

( ))()(4 ir uWuWTQs +=π

TtSru

TtSru i

ir

r 4and

4

22

==




Finite AquiferIn case of recharge boundary, the image well is assumed to be recharging, instead of discharging, with the same rate as the real well so that its effect will decrease the drawdown as shown in the figure.The two drawdowns will cancel their effect at the boundary location.

Effect of recharge boundary

( ))()(4 ir uWuWTQs −=π

TtSru

TtSru i

ir

r 4and

4

22

==




Finite AquiferThe image well approach more than one boundary around the wellA well with two impervious boundaries on both sides. i.e. a well in an alluvial valley with parallel impervious sides

Image wells in a confined valley aquifer




Finite AquiferIn parallel boundary case, if on one side there is a river while the other boundary is impervious, then the type of the images will change.

Image wells system for parallel boundaries




Finite AquiferAnother case may be semi-infinite strips aquifer, where two parallel boundaries end at right angle at a third boundary.

Image wells system for semi infinite aquifer




Finite AquiferAnother case my be a rectangular aquifer.

Image wells system for rectangular aquifer




Finite AquiferWhen the boundaries converge to each other wedge shaped aquifers are produced.

Image wells system for wedge shape aquifers with 90º angle.




Finite AquiferWhen the boundaries converge to each other wedge shaped aquifers are produced.

Image wells system for wedge shape aquifers with 45º angle.

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