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1 GROUNDWATER HYDROLOGY prof. dr. ir. F. De Smedt
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Page 1: Groundwater hydrology part 2libvolume3.xyz/civil/btech/semester6/groundwater...i Groundwater Hydrology Course notes – October 2009 PART 2: Annex with additional comments prof. dr.

1

GROUNDWATER HYDROLOGY

prof. dr. ir. F. De Smedt

Page 2: Groundwater hydrology part 2libvolume3.xyz/civil/btech/semester6/groundwater...i Groundwater Hydrology Course notes – October 2009 PART 2: Annex with additional comments prof. dr.

i

Groundwater Hydrology

Course notes – October 2009

PART 2: Annex with additional comments prof. dr. ir. F. De Smedt

Department of Hydrology and Hydraulic Engineering Faculty of Applied Sciences

Free University Brussel Pleinlaan 2, 1050 Brussel

Bureau 6G306D - tel. 02/629.35.47 Email: [email protected]

Secretariat T115 tel. 02/629.30.21 fax. 02/629.30.22

Email: [email protected]

Page 3: Groundwater hydrology part 2libvolume3.xyz/civil/btech/semester6/groundwater...i Groundwater Hydrology Course notes – October 2009 PART 2: Annex with additional comments prof. dr.

ii

Letting the days go by, let the water hold me down. Letting the days go by, water flowing underground. Into the blue again, in the silent water. Under the rocks and stones, there is water underground. Same as it ever was...same as it ever was...same as it ever was... Same as it ever was...same as it ever was...same as it ever was... Same as it ever was...same as it ever was... Talking Heads – Once in a lifetime http://www.youtube.com/watch?v=EYbUCvz1LYE&feature=related

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Contents

1 SIMPLE MEASUREMENT TECHNIQUES FOR HYDRAULIC CONDUCTIVITY ............... 1 1.1 USING PARTICLE SIZE DIMENSIONS .................................................................................................. 1 1.2 IN THE LABORATORY: PERMEAMETER ............................................................................................. 2 1.3 IN THE FIELD: SLUG TEST OR PIEZOMETER TECHNIQUE .................................................................... 4

2 STEADY STATE GROUNDWATER FLOW TO PUMPING WELLS ....................................... 6 2.1 INTRODUCTION ................................................................................................................................ 6 2.2 PUMPING IN A CONFINED OR SEMI-CONFINED AQUIFER .................................................................... 6 2.3 PUMPING IN AN UNCONFINED AQUIFER ............................................................................................ 8

3 RADIUS OF INFLUENCE OF STEADY STATE PUMPING WELLS ..................................... 11 3.1 PUMPING WELL IN A PHREATIC AQUIFER ....................................................................................... 11 3.2 PUMPING WELL IN A PHREATIC AQUIFER NEAR A STREAM ............................................................. 11 3.3 PUMPING WELL IN A CONFINED AQUIFER ....................................................................................... 12 3.4 PUMPING WELL IN A SEMI-CONFINED AQUIFER .............................................................................. 13

4 ANALYSIS OF A PUMPING TEST .............................................................................................. 15 4.1 PUMPING TEST IN A CONFINED AQUIFER ........................................................................................ 15 4.2 SIMPLIFICATIONS OF THE THEIS SOLUTION .................................................................................... 17 4.3 PUMPING TEST IN A SEMI-CONFINED AQUIFER ............................................................................... 19 4.4 PUMPING TEST IN A PHREATIC AQUIFER ......................................................................................... 22 4.5 RECOVERY METHOD ...................................................................................................................... 24 4.6 PARTIALLY PENETRATING WELLS .................................................................................................. 26

5 MAXIMUM AND SPECIFIC CAPACITY OF A PUMPING WELL ........................................ 27 5.1 MAXIMUM CAPACITY .................................................................................................................... 27 5.2 SPECIFIC CAPACITY ....................................................................................................................... 29 5.3 INFLUENCE OF WELL LOSSES ......................................................................................................... 31

6 DESIGN OF WELL FIELDS .......................................................................................................... 33 6.1 BASIC CONSIDERATIONS ................................................................................................................ 33 6.2 WELLS ARRANGED ON A SINGLE LINE ............................................................................................ 35 6.3 OTHER ARRANGEMENTS ................................................................................................................ 36

7 GROUNDWATER CHEMISTRY ................................................................................................. 38 7.1 CHEMICAL COMPOSITION OF GROUNDWATER ................................................................................ 38 7.2 TRANSPORT OF SOLUTES IN GROUNDWATER ................................................................................. 40 7.3 ADSORPTION OF CHEMICALS ......................................................................................................... 43 7.4 DECAY OF CHEMICALS .................................................................................................................. 45 7.5 PROTECTION ZONES AROUND PUMPING INSTALLATIONS ................................................................ 47

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1 SIMPLE MEASUREMENT TECHNIQUES FOR HYDRAULIC CONDUCTIVITY

1.1 Using particle size dimensions The hydraulic conductivity is given by

ν

=gkK (1.1)

where K is the hydraulic conductivity [L/T], g the acceleration due to gravity ≈ 9.81 m/s2, k the permeability [L2], and ν the kinematic viscosity [L2/T], equal to about 1.10-6 m2/s at 20°C; the kinematic viscosity is given by µ/ρ where µ is the dynamic viscosity [M/LT], equal to about 1.10-3 Pa·s at 20°C, and ρ is the density of water, equal to about 1000 kg/m3 at 20°C. Values of the density and kinematic and dynamic viscosity of water versus temperature are given in Table 1.1.

Table 1.1. Properties of water in function of the temperature at normal atmospheric pressure.

Temperature Density Dynamic

viscosity Kinematic viscosity

T (°C)

ρ (kg/m3)

µ (10-3 Pa⋅s)

ν (10-6 m2/s)

0 999,9 1,787 1,787 5 1000,0 1,519 1,519

10 999,7 1,307 1,307 15 999,1 1,140 1,141 20 998,2 1,002 1,004 25 997,1 0,894 0,897 30 995,7 0,801 0,804 35 994,1 0,723 0,727 40 992,2 0,656 0,661 45 990,2 0,599 0,605 50 988,1 0,549 0,556 55 985,7 0,506 0,513 60 983,2 0,469 0,477 65 980,6 0,436 0,444 70 977,8 0,406 0,415 75 974,9 0,380 0,390 80 971,8 0,357 0,367 85 968,6 0,336 0,347 90 965,3 0,317 0,328 95 961,9 0,299 0,311

100 958,4 0,284 0,296

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The permeability depends upon the size of the pores, and one would be tempted to relate this to the size of the particles. Unfortunately, the exact relationship is complicated by the particles size distribution, the packing of the particles, and the shape of the particles. Therefore, only crude estimations are possible. The first one is the Kozeny-Carman equation, which is valid for a porous material of equal sized spherical grains

( ) 180

²Dn1

nk 2

3

−≈ (1.2)

where n is the porosity [L3/L3], which represents in this equation the effect of the packing of the grains (dense packing means low porosity and loose packing high porosity), and D is the diameter of the grains [L]. Another relation is due to Hazen and is applicable to sandy soils. Hazen showed by means of experiments that the 10% smallest grains determines the permeability 2

10D0013.0k ×≈ (1.3) where D10 is the diameter [L] determined by a particle size analyses of the sand such that 10% by weight of the grains are smaller; the equation is only valid if 0.1 mm < D10 < 3 mm. These methods, based on particle size analyses, are considered to not very accurate, due to the approximate nature of the equations that relate particle sizes to permeability. Hence, these can only considered as orders of magnitude

1.2 In the laboratory: permeameter A permeameter is a laboratory set-up to measure the hydraulic conductivity on undisturbed soil samples. There are two types of permeameter set-ups as shown in Fig. 1.1. The first type is the constant head permeameter (Fig. 1.1a). This is a set-up as used by Darcy. The potential difference (head) and flow through the sample are constant in time, hence, the name constant head permeameter. The flow is usually from the bottom to the top of the sample in order for air to escape. It is sufficient to measure the dimensions of the sample, the potential difference, and the flow rate. Using Darcy’s law the hydraulic conductivity can be calculated as

hA

QLK∆

= (1.4)

where Q is the flow rate [L3/T], L the length of the sample [L], A the cross-sectional area of the sample [L2], and ∆h the constant potential or head difference [L].

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Fig. 1.1 Determination of the hydraulic conductivity with a permeameter: (a) constant head permeameter, and (b) falling head permeameter.

A problem with this set-up is that it becomes very difficult to use when the hydraulic conductivity is very small, because the flow rate Q cannot be measured accurately. In that case it is more convenient to use a falling head permeameter (Fig. 1.1b), for which the flow rate Q needs not to be measured. There is no inlet reservoir and when the water passes through the sample the potential difference ∆h drops gradually in time, hence the name falling head permeameter. The hydraulic conductivity can be calculated as

dt

)h(daLhKAQ ∆

−=∆

= (1.5)

where a is the cross-sectional area of the inlet tube [L2], t the time [T], and ∆h the potential difference [L] measured at a certain time since the start of the experiment. The equation can be solved to obtain the conductivity as

⎟⎠⎞

⎜⎝⎛∆∆

=h

hn1

AtaLK 0 (1.6)

with ∆h0 the initial potential difference [L] at the start of the experiment. There are three problems associated with such type of measurements: Samples need to be taken and installed in the permeameter undisturbed. This is not

so easy to do. Undisturbed sampling in the fields requires specialised equipment and experimental skill.

Soil samples are usually in order of magnitude of 1 dm3 in size and do not necessarily represent the through nature of the soil material in the field from which these were taken; hence, more samples are needed to improve the accuracy.

Experiments show that hydraulic conductivity measurements are not normally distributed. This is due to the skewness of possible K-values, because negative values are impossible, but extreme large values are possible although not very likely,

∆h

Q

L

A

A

Q

B

∆h0

L A

∆ht

a

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as for instance worm holes, cracks, etc. Hence, K-values are rather log-normally distributed and this has to be taken into account in the statistics. For instance, the best estimate is not the arithmetic mean value but the geometric mean value.

1.3 In the field: slug test or piezometer technique The most accurate field method is the pumping test, but this will be discussed when dealing with pumping wells. A simpler, method is the slug test, also called piezometer technique, shown in Fig. 1.2.

Fig. 1.2 Set-up of the slug test or piezometer technique.

In a piezometer or observation well, some volume of water is taken out from the tube with a bailer or pump, and the subsequent rise of the water back to its original position is recorded in time. These readings are plotted in time on semi-log plot as shown in Fig. 1.3, and decrease exponentially in time )ttexp(hh 00 −∆=∆ (1.7) where ∆h is the potential or head difference [L] at time t, t0 a characteristic time interval depending upon the conductivity of the soil, and ∆h0 the initial potential difference [L] at the start of the experiment. The initial potential difference ∆h0 is usually unknown, because it is difficult to take out water and measure the water level at the same time. Hence, the data are extrapolated to time zero to find the initial difference in water level ∆h0. With this value the characteristic time interval t0 for the rise of the water level can be determined from the graph, as shown in Fig. 1.3 000 ttforh37.0e/hh =∆≈∆=∆ (1.8) Finally, the hydraulic conductivity can be calculated as

L

∆h

2ro

2ri

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⎟⎟⎠

⎞⎜⎜⎝

⎛=

o0

2i

rLn1

Lt2r

K (1.9)

where ri is the inside radius [L] of the tube or well, ro is the outside radius [L] of the tube or well screen (including a filter pack if present), and L is the length [L] of the screened section.

Fig. 1.3 Graphical analysis of the slug test or piezometer technique.

The accuracy and validity of this technique is rather good, as long as experiments are carried out after a sufficiently long time since the piezometer was installed, such that the soil has settled around the filter in his original state. Also, the tested volume of soil is large, in order of magnitude about 1 m3 depending upon the size of the filter. A necessary condition is that the dimensions of the piezometer need to be known precisely. Often installation records of piezometers are not available or have been lost. The depth can always be measured anew but the position and length of the filter and presence of a filter pack not.

∆h

∆h0

0.37∆h0

t t0 0

Log

axis

Linear axis

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2 STEADY STATE GROUNDWATER FLOW TO PUMPING WELLS

2.1 Introduction When wells have been pumping for an appreciable time steady state conditions may prevail. It is only under state conditions that continued abstraction of ground water is feasible. If not, groundwater levels will continue to decline, which finally will results in the well falling dry or in a complete exhaustion of the groundwater reservoir. Hence, it is important to understand the relationships under which steady state groundwater flow to pumping wells can occur.

2.2 Pumping in a confined or semi-confined aquifer Consider a well that is pumping continuously with a constant rate Q in a confined or semi-confined aquifer with constant thickness b and homogeneous hydraulic conductivity K. The situation is depicted in Fig. 2.1.

Fig. 2.1 Steady state groundwater flow towards a pumping well in a confined or semi-

confined aquifer. Originally, when the well was not pumping the groundwater head was at a level h0, which can be assumed more or less constant in the vicinity of the well. When the well is pumping a cone of depression is formed that enables groundwater flow towards the well. The flow can be considered completely radial towards the well if the well is screened throughout the entire thickness of the aquifer. If qr is the radial groundwater flux at a distance r from the well, it follows from the mass balance equation that the total radial flow towards the well should be equal to the pumping rate rrbq2Q π−= (2.1)

Q

h

r

h0

2rw

r0

qr b

s

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where the minus sign expresses the fact that qr is negative as it is directed against the positive sense of the radial axis r. Using Darcy’s law to express the groundwater flux this becomes

drdhrT2

drdhKrb2Q π=⎟

⎠⎞

⎜⎝⎛−π−= (2.2)

where T is the transmissivity of the aquifer. From this equation it follows

r

drT2

Qdhπ

= (2.3)

This equation can be integrated to obtain an expression for the groundwater head h

crlnT2

Qh +π

= (2.4)

where c is an integration constant, whose value can be obtained by stating that at a distance r0 from the well the groundwater head is equal to its original natural level h0

⎟⎠⎞

⎜⎝⎛

π−=

rr

lnT2

Qhh 00 (2.5)

This radius r0 is called the radius of influence; it determines the zone in which the pumping well creates a cone of depression and influences the groundwater flow and head. Outside this zone for r > r0 there is no influence and h equals h0. The drawdown s is defined as the difference in groundwater head due to the pumping well

⎟⎠⎞

⎜⎝⎛

π=−=

rr

lnT2

Qhhs 00 (2.6)

This equation states that the drawdown is proportional to the pumping rate Q and inversely proportional to the transmissivity of the aquifer. Hence, large drawdown will occur in aquifers with a low transmissivity and for wells with a high pumping rate. This equation also states that the drawdown increases towards the well according to the logarithm of the distance. The maximum drawdown occurs at the well screen and is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=w

0a r

rln

T2Qs (2.7)

where sa is the drawdown in the aquifer at the well screen and rw is the outer radius of the well screen (including a filter pack if present). The relationship between drawdown and the logarithm of the distance is shown in Fig. 2.2.

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Fig. 2.2 Drawdown versus logarithm of the distance in case of a confined or semi-

confined aquifer.

2.3 Pumping in an unconfined aquifer Consider now a well that is pumping continuously with a constant rate Q in an unconfined aquifer with homogeneous hydraulic conductivity K. The situation is depicted in Fig. 2.3.

Fig. 2.3 Steady state groundwater flow towards a pumping well in a unconfined aquifer. Originally when the well was not pumping the groundwater table was at a level h0, or H0 above the base of the aquifer, which can be assumed more or less horizontal in the vicinity of the well. When the well is pumping a cone of depression is formed that enables groundwater flow towards the well. This also causes a decrease of the water table such that the position H of the water table measured from the base of the aquifer becomes variable. The mass balance equation now becomes

s

sa

0 ln(rw) ln(r0)

ln(r)

Q

h

r

h0

2rw

r0

qr H

s

H0

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rrHq2Q π−= (2.8) Using Darcy’s law to express the groundwater flux this becomes

drdHrKH2

drdhKrH2Q π=⎟

⎠⎞

⎜⎝⎛−π−= (2.9)

From this equation it follows

r

drK2

QHdHπ

= (2.10)

This equation can be integrated to obtain an expression for the water table position H

crlnK2

Q2

H 2

= (2.11)

where c is an integration constant, whose value can be obtained by stating that at a distance r0 from the well the groundwater table is equal to its original natural position H0

⎟⎠⎞

⎜⎝⎛

π−=

rr

lnK

QHH 020

2 (2.12)

The drawdown s is the difference in water table position due to the pumping well sHH 0 −= (2.13) or ( ) ( )00

20

20

20

20

2 H2s1sH2HssH2HsHH −−=+−=−= (2.14) When this result is substituted in equation 12 it follows

( ) ⎟⎠⎞

⎜⎝⎛

π=−

rr

lnKH2QH2s1s 0

00 (2.15)

This shows that the relationship between drawdown and logarithm of the distance is slightly non-linear; this results from the fact that the drawdown also reduces the transmissivity of the aquifer. Usually there are two ways to deal with this. The first way is an approximation: if the drawdown is small with respect to the original thickness of the aquifer s/2H0 can be neglected compared to 1, hence

⎟⎠⎞

⎜⎝⎛

π≈

rr

lnKH2Qs 0

0

(2.16)

In this case the result becomes identical as for a confined aquifer, given by equation 2.7. The second approach is to define a pseudo drawdown s’ given by

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( )0

2

0 H2ssH2s1s's −=−= (2.17)

so that

⎟⎠⎞

⎜⎝⎛

π=

rr

lnKH2Q's 0

0

(2.18)

which is also similar to equation 2.7. The relationship between the pseudo drawdown, the drawdown, and the logarithm of the distance is shown in Fig. 2.4.

Fig. 2.4 Drawdown versus logarithm of the distance for an unconfined aquifer.

s’ = s - s2/2H0

s’a

0 ln(rw) ln(r0)

ln(r)

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3 RADIUS OF INFLUENCE OF STEADY STATE PUMPING WELLS

3.1 Pumping well in a phreatic aquifer

Fig. 3.1 Steady state groundwater flow towards a pumping well in a phreatic aquifer. In normal conditions all groundwater being pumped originates from recharge. Hence, the total water balance for the zone influence by the pumped well is RrQ 2

0π= (3.1) where Q is the pumping rate, r0 the radius of influence, and R the recharge rate. From this relationship r0 can be obtained

R

Qr0 π= (3.2)

3.2 Pumping well in a phreatic aquifer near a stream The pumped well creates a drawdown that affects the river. It is assumed that the river carries sufficient water to be able to supply the aquifer with water (induced recharge) without running dry. The situation can be modelled by introducing a mirror image well at the other side of the river that injects water into the aquifer with the same rate Q as the pumped well. Using the principle of superposition the drawdown around the well can be calculated as

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⎟⎟⎠

⎞⎜⎜⎝

⎛π

−⎟⎟⎠

⎞⎜⎜⎝

⎛π

=1

2

02

0

01

0

0 rrln

KH2Q

rr

lnKH2Q

rr

lnKH2Q's (3.3)

r0 well

Q

r0

H0

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where s’ is the pseudo drawdown, K the aquifer conductivity, H0 the original aquifer thickness, r1 the distance to the pumped well, and r2 the distance to the image well. Close around the well, r2 can be approximated as 2L, where L is the distance of the well to the river. Hence

⎟⎟⎠

⎞⎜⎜⎝

⎛π

≈10 rL2ln

KH2Q's (3.4)

Fig. 3.2 Steady state groundwater flow towards a pumping well in a phreatic aquifer near a stream.

When we compare this equation with the steady state well flow equation, it follows that in this case the radius of influence equals 2L, hence L2r0 = (3.5)

3.3 Pumping well in a confined aquifer The confined aquifer is completely closed and no input or output is possible in the vicinity of the well. However, the aquifer must receive water from somewhere further away where it no longer confined; this is called the recharge area. We will assume that this area is at a distance L from the pumped well and we will approximate this recharge by a well that injects water into the aquifer with the same rate Q as the pumped well. Using the principle of superposition it follows

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⎟⎟⎠

⎞⎜⎜⎝

⎛π

−⎟⎟⎠

⎞⎜⎜⎝

⎛π

=1

2

2

0

1

0

rrln

T2Q

rr

lnT2

Qrr

lnT2

Qs (3.6)

where s is the drawdown, T the aquifer transmissivity, r1 the distance to the pumped well, and r2 the distance to the recharge well. Close around the well, r2 can be approximated as L

L L

well image well

Q

L

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⎟⎟⎠

⎞⎜⎜⎝

⎛π

≈1rLln

T2Qs (3.7)

When we compare this equation with the steady state well flow equation is follows that the radius of influence equals L, hence Lr0 = (3.8)

Fig. 3.3 Steady state groundwater flow towards a pumping well in confined aquifer.

3.4 Pumping well in a semi-confined aquifer The semi-confined aquifer receives its water from an overlaying aquitard characterised by its leakage coefficient C. The drawdown in the semi-confined aquifer creates a potential difference over the aquitard, which is equal to the drawdown in the semi-confined aquifer if the groundwater potentials at the other side of the aquitard remain unchanged. In this case it can be shown that the drawdown is given by following formula

⎟⎠⎞

⎜⎝⎛

π=

BrK

T2Qs 0 (3.9)

where K0 is the modified Bessel function of the second kind and order zero, and B is the leakage factor given as CTB = (3.10) For small values of the argument K0 can be approximated as follows γ−≈ )x2ln()x(K 0 (3.11) where γ is the Euler constant (0.5772...). Hence, the drawdown equation can be approximated as

Q

L

Recharge area

well

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⎟⎠⎞

⎜⎝⎛

π≈⎟⎟

⎞⎜⎜⎝

⎛π

=⎥⎦

⎤⎢⎣

⎡γ−⎟

⎠⎞

⎜⎝⎛

π≈

γ−

rB123.1ln

T2Q

rBe2ln

T2Q

rB2ln

T2Qs (3.12)

When we compare this equation with the standard steady state well flow equation it follows that the radius of influence is given by B123.1r0 = (3.13)

Fig. 3.4 Steady state groundwater flow towards a pumping well in a semi-confined

aquifer.

Q

s

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4 ANALYSIS OF A PUMPING TEST

4.1 Pumping test in a confined aquifer Theis gave a solution for a pumping test in a confined aquifer under following assumptions: • The aquifer is confined, horizontal, homogeneous, isotropic, and has a constant

thickness. • The pumping well is screened over the entire thickness of the aquifer and pumping

with a known constant pumping rate starting from time zero onwards. • Drawdown is measured in observation wells at different directions and distances (r1,

r2, ...) from the pumping well. Because of previous assumptions, groundwater flow towards the well will be completely horizontal and, hence, observation wells do not have to be screened over the entire thickness of the aquifer.

The situation is depicted in Fig. 4.1.

Fig. 4.1 Pumping test in a confined aquifer.

The solution of Theis is as follows

( )uWT4

Qsπ

= (4.1)

where u is given by

Tt4

²Sru = (4.2)

with S is the aquifer storage coefficient and other symbols are as previously explained. W is the Theis or well-function given by

∫∞ λ−

λλ

=u

de)u(W (4.3)

Q

t = 0

t

r

s

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Values of the Theis function are given in many groundwater handbooks, either evaluated approximately by numerical integration or by using a series expansion

...!44

u!33

³u!22

²uu)u(n1)u(W4

+⋅

−⋅

+⋅

−+−γ−= (4.4)

with γ = 0.5772… the Euler constant. Notice that when we take the logarithm of Eq. 4.1 this yields

( ) ( )WlogT4

Qlogslog +⎟⎠⎞

⎜⎝⎛π

= (4.5)

and the log of 1/u gives

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

T4Slog

u1log

rtlog 2 (4.6)

where log stands for the logarithm base 10. Equations 4.5 and 4.65 show that when observed drawdown is plotted versus t/r2 on a double log graph, this should be similar to a graph of W versus 1/u with only a certain vertical and horizontal translation depending upon the values of Q/4πT and S/4T which are constant. Hence, the needed shifts will reveal the values of Q/4πT and S/4T and because Q is known T and S can be calculated.

Fig. 4.2 Analysis of a pumping test according to the method of Theis. In practice the method is as illustrated in Fig. 4.2. Observations s are plotted versus t/r2 on double logarithmic graph paper; this is the so called data graph. Next, W versus 1/u is plotted on similar double log paper, forming the so called type curve or Theis curve.

log(t/r2)

log(s)

log(1/u)

log(W)

Theis curve Drawdown at observation well 1Drawdown at observation well 2…

Match pointWmp

(1/u)mp

smp

(t/r2)mp

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The two sheets are superimposed and shifted vertically and horizontally until the two graphs match. In order to determine the vertical and horizontal shifts between the curves a common point is selected on both graphs; this is the matching point and can be arbitrarily chosen as it does not need to lye on the curves. The four coordinates of the matching point, ump, (t/r2)mp, Wmp, and (1/u)mp, are noted and T and S are calculated as

mp

mp

s4QW

= (4.7)

( )( )mp

mp2

u1rtT4

S = (4.8)

Notice that s is plotted versus t/r2 because this will enable that the drawdown observed in different observation wells at different distances from the pumping well will coincide. If there is only one observation well s can be plotted versus t.

4.2 Simplifications of the Theis solution For u < 0.05 the first two terms of the series expansions, Eq. 4.4, are sufficient to approximate the Theis function; hence

)u561.0ln(u

eln)u(n1)u(W ≈⎟⎟⎠

⎞⎜⎜⎝

⎛=−γ−≈

γ−

(4.9)

It follows that the Theis solution, Eq. 4.1, can be approximated as

⎟⎠⎞

⎜⎝⎛

π=⎟

⎠⎞

⎜⎝⎛ ⋅

π≈ 22 Sr

Tt25.2og1T4Q3.2

SrTt4561.0n1

T4Qs (4.10)

This is the approximation of Jacob, which can also be used to analyse a pumping test. In this case observations of drawdown s are plotted versus t/r2 on a semi-log graph as shown in Fig. 4.3. For values of t/r2 sufficiently large, a straight line can be drawn through the data points as indicated by Eq. 4.10. The slope of the line can be quantified by reading the value of ∆s on the s-axis corresponding to one log-interval on the t/r2-axis (one log-interval is the distance between any value and 10 times that value). This enables to calculate T as

s4

Q3.2T∆π

= (4.11)

The value of S can be obtained from the intercept (t/r2)0 of the fitted straight line with the t/r2-axis ( )02rtT25.2S = (4.12) Afterward, one has to verify whether condition u < 0.05 holds for all observation points on the straight line; in view of Eq. 4.2 this implies that

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TS5rt 2 > (4.13)

Fig. 4.3 Analysis of a pumping test with the Jacob method. The Jacob method is simple to apply, but has the disadvantage that only part of the observations are used to determine T and S. Another simplification is based on the fact that when pumping continues for a long time a steady state condition may be reached. Mathematically the Theis solution never converges to a steady state solution, because of the assumption that the aquifer is fully confined without any input from outside. However, the Jacob approximation shows that a pseudo steady state solution is reached, when Eq. 4.10 is written as

⎟⎠⎞

⎜⎝⎛

π=⎟

⎠⎞

⎜⎝⎛

π=⎟⎟

⎞⎜⎜⎝

π=

rr

og1T4Q3.2

rRn1

T2Q

SrTt25.2n1

T2Qs 0

2 (4.14)

with R the radius of influence given by

S

Tt25.2r0 = (4.15)

Equation 4.15 predicts that the radius of influence keeps on increasing with time as a consequence of the assumption that the aquifer is fully confined. In practice, the radius of influence can become constant provided the pumping time is long enough and the pumping rate can be compensated by recharge into the aquifer. Whether the radius of influence becomes constant or not is not the point, because Eq. 4.14 predicts that when

s

log(t/r2)

∆s = 2,3Q/4πKD

1 log-interval

(t/r2)0 0

Jacob straight line Drawdown at observation well 1Drawdown at observation well 2…

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the condition given by Eq. 4.13 is satisfied, the drawdown becomes linearly related to the log of the distance from the pumping well. Hence, the drawdown observed at a certain time in different observation wells can be plotted versus the log(r) as shown in Fig. 4.4.

Fig. 4.4 Analysis of a pumping test by the pseudo steady state method.

The slope of the fitted line can be determined by reading the value of ∆s on the s-axis corresponding to one log-interval on the log(r)-axis, from which the transmissivity can be calculated

s2

Q3.2T∆π

= (4.16)

The intercept of the straight line with the log(r)-axis gives the value of the radius of influence. If steady state has not been reached one can estimate the storage coefficient from this as 2

0rTt25.2S = (4.17)

4.3 Pumping test in a semi-confined aquifer Hantush has proposed two methods to analyse pumping tests in a semi-confined aquifer. The first approach is applicable for early times, when due to the pumping well water is leaking from the aquitard which is in contact with the semi-confined aquifer. The situation is depicted in Fig. 4.5. Hantush showed that in case the aquitard has an infinite thickness the drawdown in the semi-confined aquifer is given by

( )βπ

= ,uHT4

Qs (4.18)

where β is

log(r)

∆s

log(r0)0

Straight line fit Drawdown at observation well 1Drawdown at observation well 2…

s

1 log-interval

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S'S

B4r

=β (4.19)

with S’ the storage coefficient of the aquitard, and B the leakage factor as defined in Eq. 3.10. H(u,β) is the Hantush well function given by

( )∫

∞ λ−

λ⎟⎟⎠

⎞⎜⎜⎝

−λλβ

λ=β

ud

uuerfce),u(H (4.20)

where erfc is the co-error function. Values of H versus u and β can be found in the specialised literature.

Fig. 4.5 Pumping test in a semi-confined aquifer, with water leaking from the aquitard.

In practice the method works as follows. Observations of the drawdown s are plotted versus t/r2 on double logarithmic graph paper similar as for the Theis method. However, in this case the data of different observation wells will not coincide on the same curve. Next, values of H versus 1/u are plotted on similar double log paper for different values of β; this forms a series of type curves (β = 0 corresponds to the Theis curve). The two sheets are superimposed and each data curve is fitted to one of the H-curves by shifting vertically and horizontally until the graphs match. A matching point is selected and T and S are calculated as given by Eqs. 4.7 and 4.8. The method is not used very much in practice because the assumption of an infinite thick aquitard is not very realistic. The second approach of Hantush applies to larger pumping times, when it can be assumed that the aquitard no longer releases water from storage, but transmits water coming from another aquifer, as shown in Fig. 4.6. In addition it is assumed that this other aquifer has the same initial groundwater head as the pumped semi-confined aquifer and that this head remains fixed during the pumping test. Usually, these conditions apply to the case of a thin aquitard and an above lying phreatic aquifer, which has sufficient water storage and input, such that the groundwater heads remains unaffected by the loss of water by leakage through the aquitard.

Q

t = 0

t

r

s

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The solution of Hantush is

( )βπ

= ,uWT4

Qs (4.21)

where now β is given by Br=β (4.22) The so called leaky well function is given by

( )∫∞ − λλβ−λ−λ=βu

21 d4exp),u(W (4.23)

Values of W versus u and β can be found in the specialised literature. For large values of time W(u,β) becomes equals to the steady state solution K0(β) as discussed in paragraph 3.4.

Fig. 4.5 Pumping test in a semi-confined aquifer, with water leaking from another

aquifer through the aquitard. The procedure to analyse such a pumping test is similar as explained for the previous case, but there are some interesting features of practical value that need to be mentioned. The analysis of a pumping test in a semi-confined aquifer with the method of Hantush is illustrated in Fig. 4.6. For β = 0, W(u,β) equals the Theis function W(u); for other values of β, the W(u,β) curves are lying below the Theis curve, because the drawdown in a semi-confined aquifer is smaller than in a confined aquifer (for same T, S, and Q values) because of the leakage effect. Furthermore, all W(u,β) curves for β ≠ 0 become constant for 1/u sufficiently large; this means that the drawdown in a semi-confined aquifer becomes constant after sufficient time, again because of the leakage. This steady state drawdown is given by Eq. 3.9, and is a typical indication that the pumped aquifer is semi-confined. Once a match point is chosen and T and S determined, one can also note for each set of observations in a particular observation

Q

t = 0t

r

s

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well the distance r to the pumping well and the β-value of the corresponding type curve, and calculate the leakage coefficient C of the aquitard using Eqs. 3.10 and 4.22, i.e.

2

2

rTC β

= (4.24)

Fig. 4.6 Analysis of a pumping test in a semi-confined aquifer with the method of

Hantush.

4.4 Pumping test in a phreatic aquifer The situation is depicted in Fig. 4.7. A first approximation is to assume that a pumping test in a phreatic aquifer can analysed with the Theis technique provided the aquifer storage S is replaced by the specific yield Sy. In addition, to compensate for the fact that the thickness of the aquifer is not constant, but decreases around the pumped well because of the drawdown, is to use the pseudo drawdown s’ as defined in Eq. 2.17; hence

( )yuWT4

Q'sπ

= (4.24)

with uy given by

Tt4

²rSu y

y = (4.25)

where the transmissivity T = KhH0 (horizontal hydraulic conductivity times saturated thickness of the aquifer). However, it turns out that the situation is more complicated. This has to do with the fact that the water released from storage is occurring at the water table and needs some time to be distributed over the aquifer before it arrives at the pumped well. This causes a delay in response of the aquifer to the pumping. The delay

log(t/r2)

log(s)

log(1/u)

log(W)

Hantush leaky well function W(u,β)Drawdown at observation well 1 Drawdown at observation well 2 …

Match pointWmp

(1/u)mp

smp

(t/r2)mp

β=0

β3

β1

β2

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is manifested by the early response of the aquifer behaving as if it were confined, i.e. the drawdown at early times is given by Eq. 4.1. However, gradually the water released by decline of the water table becomes effective and the drawdown is governed by Eq. 4.24. Hence, the drawdown varies between two Theis functions as shown in Fig. 4.8.

Fig. 4.7 Pumping test in a phreatic (unconfined) aquifer.

Fig. 4.8 Analysis of a pumping test in a phreatic aquifer.

This behaviour makes the analysis of a pumping test in a phreatic aquifer very difficult. One possibility is to fit the data points by enveloping the data by two Theis curves, one

log(t/r2)

log(s)

log(1/u) or log(1/uy)

log(W)

Theis function W(u) or W(uy) Neuman well function W(u,uy,β)Drawdown at observation well 1Drawdown at observation well 2…

β1

β2

Q

t = 0

t

r

s

H0

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on the left and one on the right, as shown in Fig. 4.8. The Theis curves should be similar except for a horizontal shift, because the T-value remains the same for both Eqs. 4.1 and 4.21, and only the storage coefficients are different. After fitting, T, S and Sy can be determined by the matching point method. The same approach could also be done using the Jacob approximation, but it is not sure that 1/u values will be large enough to apply this technique. Also, one should keep in mind that for the long time behaviour for the water released from water table decline to become fully active, long pumping times are needed, often several weeks or months! Another method is the technique of Neuman, who succeeded in solving this problem mathematically. The solution can be written as

( )βπ

= ,u,uWT4

Q's y (4.26)

where β is given by

TH

Kr

0

v2

=β (4.27)

with Kv the vertical hydraulic conductivity of the aquifer. The mathematical expression of W(u,uy,β) is very complicated as it involves an integral with infinite boundaries of an infinite series with terms that depend upon the solution of a transcendental equation. Values can be found only in specialised literature or in some commercial software packages. Using these type cures, one can fit each series of observations for a particular observation well to one of the type curves with a particular β-value; in addition one has to stretch the type curves to match the early and late time Theis type behaviour. After fitting, values for T, S and Sy can be determined with the matching point method, and for each observation well one can also note the distance to the pumping well and the corresponding β-value to determine the vertical hydraulic conductivity of the aquifer

20

v rTH

= (4.28)

4.5 Recovery method The recovery method is a simple technique to obtain the transmissivity of any aquifer. The technique consists of pumping for a small time interval ∆t before any special aquifer properties have an effect such that any aquifer behaves as if it were confined. The drawdown is recorded after the pumping has stopped and can be recorded in the pumping well itself. The drawdown in the so called recorvery phase can be optained by the principle of superposition, assuming that the pumping does not stop but an imaginary well injects the same amount of water back into the aquifer after time ∆t

( ) ttttT4

SrWT4

QTt4

SrWT4

Qs22

∆>⎟⎟⎠

⎞⎜⎜⎝

⎛∆−π

−⎟⎟⎠

⎞⎜⎜⎝

⎛π

= (4.29)

Using the Jacob approximation, this becomes

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( )

⎟⎠⎞

⎜⎝⎛

∆−π=⎥

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ∆−

−⎟⎠⎞

⎜⎝⎛

π≈

tttlog

T4Q3.2

SrttT25.2log

SrTt25.2log

T4Q3.2s 22 (4.30)

Hence, the drawdown recorded during the recovery can be plotted versus log(t/t- ∆t) yielding a straight line relationsip as shown in Fig. 4.9. Theoretically the line should pass trough the origin as for large time: s = 0 and t/(t- ∆t) ≈ 1. The slope of the line can be determined to obtain T, which can done for instance by noting ∆s for t/(t- ∆t) = 2; the value of T can bedetermined as

( )s4

Q2log3.2T∆π

= (4.31)

In a phreatic aquifer it often happens that during the recovery the stoage coefficient becomes different because of air inclosures. In such case Eq. 4.30 should be modified as

( )

( )⎟⎟⎠⎞

⎜⎜⎝

⎛∆−π

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ∆−

−⎟⎠⎞

⎜⎝⎛

π≈

tt'SStlog

T4Q3.2

r'SttT25.2log

SrTt25.2log

T4Q3.2s 22 (4.30)

where S’ is the storage coefficient during recovery. In this case, the straight line fit should not pass through the origin, but has an intercept of S/S’ on the x-axis, and the value of ∆s should be read of at 2S/S’ as shown in Fig. 4.9.

Fig. 4.9 Analysis of a recovery pumping test.

s

log(t/t-∆t)∆s

S/S’1

0

2 3 4

straight line fits In case of no storage effectsIn case of starage effects

2S/S’

∆s

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4.6 Partially penetrating wells In all classical pumping test analyses it is assumed that the pumping well has a screen over the total thickness of the aquifer. In practice, this is often not the case. As a result, the groundwater flow to the well is not horizontal what complicates the analysis. For instance, drawdown in an observation well can also depend upon depth. Analytical solutions for such problems also exist, as for instance Hantush or Neuman solutions for partially penetrating wells, but these are very complicated, hence, difficult to evaluate and to apply, and mostly only available in commercial software packages. However, vertical flow effects are only important close to the pumping well, and as a rule of thumb one accepts that for distances larger than 1.5 times the thickness of the aquifer these effects can be ignored, as shown in Fig. 4.10, and classical solutions remain valid. In case of the recovery technique, when the drawdown in measured the pumping well itself, one assumes that the thickness of the aquifer should be substituted by the length L of the well screen

Fig. 4.10 A partially penetrating well.

Q

1.5D

D L

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5 MAXIMUM AND SPECIFIC CAPACITY OF A PUMPING WELL

5.1 Maximum capacity In order to dimension groundwater abstraction installations it is very important to predict the maximum yield that a well can deliver and its economical feasibility. This can be predicted starting from the well flow equation. In case of a (semi-)confined aquifer this is

⎟⎠⎞

⎜⎝⎛

π=

rr

lnT2

Qs 0 (5.1)

The maximum drawdown in the aquifer sa occurs at the well screen and is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=w

0a r

rln

T2Qs (5.2)

From this equation we can calculate the pumping rate as

)rr(ln

Ts2Q

w0

aπ= (5.3)

A similar equation is obtained in case of a pumping well in a phreatic aquifer

)rr(ln

)H2ss(KH2)rr(ln'sKH2

Qw0

02aa0

w0

a0 −π=

π= (5.4)

The question is now how the pumping rate can be maximised. The only material property involved is the transmissivity of the (semi-)confined aquifer, or conductivity times original aquifer thickness in case of a phreatic aquifer. Obviously, the pumping rate is proportional to T or KH0. Hence, pumping wells should be installed in aquifers with large conductivity and large thickness, but this conclusion is no surprise. The pumping rate also depends upon the radius of influence and the radius of the well. However, both parameters appear in the log term and, hence, their influence is limited. Furthermore, the radius of influence depends upon environmental conditions and the radius of the well on drilling costs and equipment. The pumping rate is positively influenced by a large well radius and a small radius of influence. But a well with a large screen diameter is very costly to install, and does not yield a proportional increase in pumping rate, because of the logarithmic dependence. Hence, there is little benefit in making the well diameter as large as possible, and in practice the well diameter will be chosen in function of the drilling costs and available drilling equipment. A similar reasoning applies to the radius of influence. Evidently, it is beneficial to install wells in aquifers with a small radius of influence. These are for instance phreatic

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aquifers with a large recharge rate or when the well is installed close to a river. The same applies to other type of aquifers, i.e. semi-confined aquifer bounded by aquitards with large leakage coefficients and confined aquifers with close nearby recharge areas. But again because of the logarithmic dependence the influence is rather limited. Hence, what remains is to maximise the pumping rate by making the drawdown in the aquifer as large as possible, but this depends upon the type of aquifer. For a (semi-)confined aquifer the situation is depicted in Fig. 5.1. The maximum possible drawdown in a (semi-)confined aquifer is determined by the original groundwater potential h0 and the top of the aquifer zt.

Fig. 5.1 Maximum pumping in a (semi-)confined aquifer.

The groundwater level cannot become lower than the top of the aquifer, otherwise the aquifer would be no longer (semi-)confined, and the transmissivity would decline because the aquifer becomes unconfined. Hence, the capacity of a well or maximum pumping rate is given by

)rr(ln

)zh(T2Q

w0

t0max

−π= (5.5)

For a phreatic aquifer the situation is depicted in Fig. 5.2. The situation is not very clear because any decline in the water table decreases the overall capacity of the aquifer to conduct the water to the well, because the saturated thickness of the aquifer becomes smaller. Theoretically, the maximum drawdown could be equal to the entire aquifer thickness, so sa = H0 and the capacity or maximum pumping rate would be given by

)rr(ln

KHQ

w0

20

max

π= (5.6)

Obviously this solution is not very convincing as the entire well runs dry. The problem arises from the assumption that the flow towards the well is horizontal, which is clearly not the case anymore if the drawdown becomes too much. Hence, we conclude that for a well in a phreatic aquifer the obtained result is only theoretical and other considerations should be taken into account to find the real feasible maximum pumping

Q

sa

h0

zt

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rate. For this we need also to consider the gains versus the costs of the groundwater abstraction.

Fig. 5.2 Maximum pumping in a phreatic aquifer (?)

Obviously this solution is not very convincing as the entire well runs dry. The problem arises from the assumption that the flow towards the well is horizontal, which is clearly not the case anymore if the drawdown becomes too much. Hence, we conclude that for a well in a phreatic aquifer the obtained result is only theoretical and other considerations should be taken into account to find the real feasible maximum pumping rate. For this we need also to consider the gains versus the costs of the groundwater abstraction.

5.2 Specific capacity Groundwater abstraction by pumping wells has a clear drawback, i.e. to induce groundwater flow towards the well we need to create drawdown, which, however, decreases the water level in the well and forces us to spend more effort and energy to pump the water to the surface. Hence, to compare the gain with the cost, we should consider the pumping rate per drawdown. For a (semi-)confined aquifer this becomes

)rr(ln

T2sQ

w0a

π= (5.7)

and for a phreatic aquifer

)rr(ln

)H2s1(KH2sQ

w0

0a0

a

−π= (5.8)

This is called the specific capacity or specific pumping rate of a well. The specific capacity predominantly depends upon the transmissivity of the (semi-)confined aquifer, or conductivity times original aquifer thickness in case of a phreatic aquifer. Other parameters are less important. Notice from the equations that the specific capacity is proportional to the transmissivity (T or KH0) of the aquifer. Hence, the transmissivity tells us already a lot about the groundwater abstraction possibilities in a certain aquifer. In fact, because the diameter of a well has a rather limited range in practice, say

Q

H0

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between 20 and 100 cm, while the radius of influence is in the range of 400 to 2000 m or about 2000 times larger, one could calculate an approximate proportionality coefficient between the specific capacity of a well and the transmissivity of an aquifer

T8.0T)2000ln(

2sQ

a

×≈π

≈ (5.9)

This is Logan’s rule for a quick estimation of the specific capacity. It can also be used as a crude estimation of the transmissivity from the specific capacity of a pumping well. Next, we will focus on ways to maximise the specific capacity of a well. For a (semi-)confined aquifer there is no maximisation possible. The transmissvity is the only determining factor and this is fixed. But for a phreatic aquifer, the specific capacity also depends upon the drawdown. From the formula is follows that the specific capacity decreases with increasing drawdown. Hence, the maximum specific capacity is obtained for sa = 0, or

)rr(ln

KH2sQ

w0

0

maxa

π=⎟⎟

⎞⎜⎜⎝

⎛ (5.10)

However, this result is rather absurd, because there will be no discharge when the drawdown is zero. Hence, for a phreatic aquifer there seems to be a dilemma as maximising the pumping rate requires large drawdown, but maximising the specific pumping rate requires the opposite. The only possible solution is to find a reasonable trade off between these conflicting requirements. Therefore, we investigate the relative increase in pumping rate and the relative decrease in specific capacity as a function of the relative drawdown sa/H0

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

a

0

a

max H2s

1Hs2

QQ (5.11)

0

a

maxa

a

H2s

1)sQ(

sQ−= (5.12)

Both relationships are shown in Fig. 5.3. From the figure one can notice that a good compromise would be obtained by having a 50% relative drawdown, or sa equal to half of the aquifer thickness H0. This would yield a relative total capacity and specific capacity of both 75%. Other investigators have other ideas about this, and suggestion vary between 33% and 66% relative drawdown whether one is more focussed on yield or efficiency. In case of 50% relative drawdown we obtain following optimum expressions for the capacity of a well in a phreatic aquifer

)rr(ln4

KH3Q

w0

20

opt

π= (5.13)

)rr(ln2

KH3sQ

w0

0

opta

π=⎟⎟

⎞⎜⎜⎝

⎛ (5.14)

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Fig. 5.3 Relative capacity of a well in a phreatic aquifer versus relative drawdown. The optimum situation is shown in Fig. 5.5.

Fig. 5.4 Optimum pumping in a phreatic aquifer.

5.3 Influence of well losses In case of well loss the total water level drawdown st inside the well is larger than the drawdown sa in the aquifer, or

Q

H0

sa = H0/2

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Relative drawdown

Effic

ienc

y (%

)

Pumping rate

Specific capacity

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2wat CQBQsss +=+= (5.15)

where sw is the well loss, which is proportional to the pumping rate squared because it is caused by friction and turbulent flow through the well screen; coefficient B is given by the well flow equation and coefficient C is the well loss coefficient, depending upon the number and size of the openings in the well screen. Ideally one would like sw to be negligible small but this is not always the case in practice, because wells might be wrongly designed, badly installed, or degenerated after some time by corrosion or clogging. The well efficiency coefficient E is defined as the ratio between sa and st

t

a

ss

E = (5.16)

All considerations discussed in previous pages about maximum and specific capacity of a well should be reconsidered taking into account the efficiency of the well. Hence, in all formulae sa should be replaced by Est and all criteria should be based on st instead of sa. Hence, in case of a (semi-)confined aquifer, this becomes

)rr(ln

)zh(T2EQ

w0

t0max

−π= (5.17)

)rr(ln

T2EsQ

w0t

π= (5.18)

and for a phreatic aquifer we obtain similar expressions (st = H0/2 or sa = EH0/2 should be substituted in equation 5.4)

)rr(ln

KH)4E1(EQ

w0

20

opt

π−= (5.19)

)rr(ln

KH2)4E1(E

sQ

w0

0

optt

π−=⎟⎟

⎞⎜⎜⎝

⎛ (5.20)

It is clear that the efficiency is an important factor with respect to well capacity, and for an inefficient well the maximum and specific capacity will be reduced accordingly. In practice E varies from 60% to 80%, the latter value can be assumed for design of wells if one is optimistic and the former value if one is rather pessimistic about the final performance of the well. The efficiency of existing wells can be determined by a well performance test, as will be explained later on.

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6 DESIGN OF WELL FIELDS

6.1 Basic considerations Suppose we need to pump a total amount of ground water QT. We also know the properties of the aquifer, which we will assume to be confined, as transmissivity T, radius of influence r0, and maximum allowable drawdown smax. With the well flow equation given by

⎟⎠⎞

⎜⎝⎛

π=

rr

lnT2

Qs 0 (6.1)

we can estimate the maximum pumping rate Qmax that one well can yield

)rr(ln

Ts2EQ

w0

maxmax

π= (6.2)

where rw is the radius of the well and E the efficiency of the well. In case Qmax is smaller than QT one well will not be sufficient to obtain the desired amount of groundwater. The only thing that still can be changed in equation 6.2 is the radius of the well. However, increasing the radius of a well will invoke a large increase in the installation cost. Hence, the only alternative is that more wells are needed. One would be tempted to estimate the number n of wells needed as

max

T

QQn = (6.3)

However, this would only work in case there is no interference between the wells, i.e. if the wells are positioned sufficiently far from each other, theoretically at a distance of more than 2 times the radius of influence from each other. Because r0 will usually be large (the only exception would probably be wells in a phreatic aquifer positioned close to a river) this will not be a very practical solution. Hence, in practice wells will be positioned closer to each other, also in view of size of the terrain, connection tubing, power supply, maintenance, etc. A series of wells that operate together is called a well field. In such case the wells will interfere with each other and the drawdown is given by the superposition principle

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=n

1i i

0i

rr

lnT2

Qs (6.4)

where Qi is the pumping rate of the i-th well, ri the distance to the i-th well, and n is the number of wells. It is clear that none of the wells can pump at maximum capacity, because if any well would be pumping at maximum capacity the drawdown in that well will exceed the maximum drawdown due to the additional drawdown created by the other wells. Hence, more wells will be needed than predicted by equation 6.3.

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Fig. 6.1 Example of a well field. How can we now calculate the necessary number of wells, their pumping rate and their location? Mostly people will use trail and error and assume a number of wells and pumping rates and verify with equation 6.4 what the drawdown will be in a number of points. If the drawdown is found to be everywhere lower than smax that arrangement would work. However, there is no guarantee that this arrangement is the most optimal. So, we need to examine this situation somewhat closer, and assume some general working hypotheses. It is first of all evident to assume that all wells are similar; hence, they will have the same radius and efficiency, and all will pump at the same rate. This enables us to calculate the pumping rate from the number of wells and total required pumping rate

n

QQ T= (6.5)

This will simplify equation 6.4 as follows

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⎟⎟⎠

⎞⎜⎜⎝

⎛π

= ∑∑==

n1n21

0T

n21

n0T

n

1i i

0Tn

1i i

0

)rrr(r

lnT2

Qrrr

rln

Tn2Q

rr

lnTn2

Qrr

lnT2

QsKK

(6.6)

If we define an equivalent distance re as the geometric mean of all the distances n1

n21e )rrr(r K= (6.7) the equation to calculate the total drawdown simplifies considerably to

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=e

0T

rr

lnT2

Qs (6.8)

This equation resembles equation 6.1, except that the radius of the well is replaced by re. Hence, it is as if we would have to design one equivalent well with a large radius re, but this radius is actually obtained by having more wells! The next question is where the maximum drawdown would occur in case of a well field. The answer is straight forward, i.e. in the most middle well, because (1) the drawdown is already largest inside the wells, and (2) the middle well will have the highest interference from all the other wells. Obviously, the maximum drawdown has to be

well 1well 2

well 3 well 5

well 4r1r2

r3r5

r4

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lower than the maximum allowed drawdown. Hence, taking into account the drawdown in the well and the efficiency E, we find

⎟⎟⎠

⎞⎜⎜⎝

⎛π

=e

0Tmax r

rln

TE2Q

s (6.9)

which allows us to determine the equivalent distance re as

⎟⎟⎠

⎞⎜⎜⎝

⎛ π−=

T

max0e Q

TEs2exprr (6.10)

Notice, that all terms in the right hand side are supposedly known, and that we do not need to know the number of wells to calculate re. Now, the problem has been reduced to a geometric problem: arrange n wells, so that for the most central well the correct re value is obtained. Unfortunately, there is no exact solution for this, as the problem depends upon the available space to arrange the wells. Solutions should be found by trail and error. If we have some freedom in positing the wells things become somewhat simpler. The question is now how from point of view of the central well can the other wells be arranged so that there will be the least of interference. The solution is trivial as the answer is to put the other wells as far away as possible. But we are also constrained by the fact that the arrangement should be practically feasible and limited in space. Hence, the form and extend of the terrain plays a role, and also to cost of connecting tubing and power lines.

6.2 Wells arranged on a single line A fair and very practical arrangement would be to put the wells on a single line at equal distances.

Fig. 6.2 Wells arranged one a line at equal distances.

Let this distance be d, so that the equivalent distance from the middle well becomes [ ] n122

we )d2(drr K= (6.11) where it is assumed that there are more than 5 wells. More precisely, the equation is

dd d d

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n12

1nwe !

21ndrr

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

= − (6.12)

in case n is an odd number, and

n1

1nwe !

2n!

22ndrr ⎥

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ −

= − (6.13)

in case n is an even number. If the number of wells is given, one can solve the equation or d

⎟⎟⎠

⎞⎜⎜⎝

⎛ π−

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

=−−

−−

T

max1n

1

e

w2

0

1n1

e

w2

e QTEs2

exprr

!2

1nrrr

!2

1nrd (6.14)

in case n is odd, and

⎟⎟⎠

⎞⎜⎜⎝

⎛ π−

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

=⎥⎥⎦

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −

=−−

−−

T

max1n

1

e

w2

0

1n1

e

w2

e QTEs2

exprr

!2n!

22nr

rr

!2n!

22nrd (6.15)

in case n is even. Alternatively, one can determine the number of needed wells if the total length of the well field L = nd is given, but in this case the equations cannot be solved explicitly; trail and error should be used to obtain the solution.

6.3 Other arrangements Other arrangements can be considered in case of spatial constraints. Some elementary arrangements are shown in Fig. 6.3. For 3 wells placed on an equilateral triangle with side d, the equivalent radius is the same as for 3 wells placed on a line at equal distance d, i.e.

312

we )dr(r = (6.16) or

ew

e

rrr

d = (6.17)

In case of 4 wells placed on the corners of a square with side d, the equations become

413

we )2dr(r = (6.18) or

31ew

83e

)rr(2r

d = (6.19)

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Fig. 6.3 Wells arranged on the corners of an equilateral triangle or of a square.

If the terrain has a fixed shape and area, it is not sure whether the required amount of water can be obtained even with any number of wells. To estimate the maximum amount of groundwater that can be abstracted, the whole terrain can be replaced in the limit as one giant pumping well, in which case the maximum equivalent radius can be estimated from Ar 2

w =π (6.20) where A is the area of the terrain. Hence, it follows π== Arr we (6.21) This value can be used to estimate the maximum abstraction rate

)rr(ln

Ts2EQ

e0

maxmax

π= (6.22)

If this is sufficient, the amount of wells needed, can be determined by trail and error. However, notice that the optimal position of the wells is on the boundary of the terrain, because this will yield the least of interference. Note: in case of a phreatic aquifer, all above solution methods and equations are similar, but E has to be replaced by E(1-E/4) and T by KH0.

d d

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7 GROUNDWATER CHEMISTRY

7.1 Chemical composition of groundwater The majority of the constituents dissolved in ground water are ions, which result from the dissolution of the various soil, ground, and rock minerals. Due to the abundance of certain mineral types, six anions are predominant in natural waters. These are called the major ions; there are 3 major cations Ca2

+, Mg2+, Na+, and 3 major anions HCO3-, SO4

2- and Cl-, with concentrations usually higher than 5 mg/l. Minor ions have concentrations in the range of 0.01 to 10 g/m3; examples are K+, Fe2+, Fe+3, F-, B-, NO3

- and CO32-. Virtually all

other ions can be present, but under natural conditions these trace ions have concentrations less than 0.1 g/m3. These are called trace elements. Besides ions, non-ionic solutes, especially Si, can be present with concentrations in the range of 5 g/m3, as well as organic matter derived from plants residues, usually measured as DOC (dissolved organic carbon) with a concentration in the range of 0.1 to 10 g/m3, and dissolved gases, as O2, N2, and CO2, which originate from the atmosphere. The total amount of solutes present in groundwater is expressed as TDS (Total Dissolved Solids) as mass of solutes per volume of water. The salinity is the total amount of ions present in the water, expressed as mass of solutes per volume of water. The electrical conductivity, which is measured Siemens (S), is an indirect measurement of the salinity or overall ionic composition, and is normally in the range of zero to 1 mS. The relationship between electrical conductivity and ionic composition can be approximated by the following expression )mS(ECa)mg(S 3 ⋅= (7.1) where S is the salinity or total ion content [M/L3], EC the electrical conductivity [S], and the proportionality factor a is between 550 and 750 depending upon the ionic composition, or on average about 650. Hardness is a measure of the amount of calcium and magnesium ions present in the water, expressed as equivalent mass of calcium carbonate per volume of water, which can be calculated as )Mg(1.4)Ca(5.2)mg(H 223 ++ += (7.2) where the concentrations of Ca2+ and Mg2+ are expressed in g/m3. Water quality can also be determined by some indirect measurements as pH, a measure of the acidity, which is usually in the range of 7 to 8; hence, slightly basic because of dissolution of carbonate minerals. Dissolved oxygen, DO, is a measure of the oxidation-reduction status of the solution, which yields insight in biochemical reactions. It can be measured directly by a special probe; values of dissolved oxygen are usually less than 10 g/m3.

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Because the chemical composition of natural groundwater is dominated by the major ions, the hydro-chemical balance can be checked by means of the electro-neutrality condition, i.e. the sum of the positive charges should balance the sum of the negative charges, or [ ] [ ] [ ] [ ] [ ] [ ]−−−+++ ++≈++ ClSOHCONaMgCa 2

4322 (7.3)

Where the concentrations of the major ions are expressed in equivalence per litre. Small differences in the chemical balance are possible due to minor ions (or pollutants) that are present in the water, but for natural groundwater the error should generally be less small. Hence, we can define a chemical balance error

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ]−−−+++

−−−+++

+++++

++−++=

ClSOHCONaMgCa

ClSOHCONaMgCaError 2

4322

243

22

(7.4)

and verify whether this error is smaller than 0.05 in order to check the accuracy of hydro-chemical analyses, unless the salinity is less than 5 mg/l, in which case a higher error is acceptable.

Fig. 7.1 Example of a Stiff diagram. Stiff diagrams are a visual method to compare the relative proportions of the major ions. The major ion concentrations in milligrams per litre (mg/l) are converted to milli-equivalents per litre (meq/l) (the concentration of an ion in meq/l is derived by dividing its concentration in milligrams per litre by the gram formula weight of the ion and then multiplying by the charge of the ion). Cations are plotted on the left side of the diagram and anions on the right. Hence, in a Stiff diagram sodium plus potassium (Na+ + K+), calcium (Ca2+), and magnesium (Mg2+) are plotted on the left, and chloride (Cl-), bicarbonate plus carbonate (HCO3

- + CO32-), and sulphate (SO4

2-) on the right. This produces a geometric shape, which defines the geochemical fingerprint of the sample.

0

1

2

3

-15 -10 -5 0 5 10 15

Concentration (meq/l)

HCO3- + CO3

2-

Cl-

Mg2+ SO42-

Ca2+

Na+ + K+

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The chemical composition of different samples of groundwater can be easily compared by the form and size of these shapes. The Piper diagram is a tri-linear plot of the major ions. The chemical composition of water can be approximated in terms of three sets of cations (Ca, Mg, Na plus K) and three sets of anions (bicarbonate plus carbonate, SO4 and Cl) expressed as a percentage of total milli-equivalents. The proportions are plotted as points in separate triangles of cation and anion constituents. These points are then projected into a central diamond-shaped field to identify general compositions in terms of water-chemistry types. Piper diagrams can be used to classify groundwater according to their most dominant cation and anion, as for instance calcium-bicarbonate type water. The diamond top diagram also depicts the hydro-chemical evolution of groundwater, starting from the left, which is calcium-bicarbonate type water, hence usually young (recently infiltrated) groundwater, to the right, being sodium chloride groundwater, i.e. usually old (nearly sea-) water.

Fig. 7.2 Example of a Piper diagram.

7.2 Transport of solutes in groundwater The primary mechanism by which particles in groundwater are transported is advection, which is the movement of a dissolved chemical along with the ground water flow, as determined by Darcy’s law. Hence, knowledge about the direction and magnitude of

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groundwater flow yields a first insight into the transport of solutes. Advection implies that the particles move with the same flow velocity as the groundwater. However, dissolved particles are also subjected to dispersion, a seemingly random spreading similar as diffusion. The reason for this spreading are the actual pore water flow velocities which vary considerably due to the irregular shape of the pore space, as illustrated in Fig. 7.3. Dispersion causes particles to deviate from the groundwater flow as calculated by Darcy’s law, because the flow paths determined in such way do not take into account the geometry and tortuosity of the pore space. Hence, some particles will move faster and some slower due to differences in size of the pores, while there are also deviations in flow direction because particles have to move around the grains as shown in Fig. 7.3. The resulting spreading of the particles is rather random and as such very similar to diffusive spreading, but generally has a much larger impact on transport of dissolved chemicals compared to diffusion.

Fig. 7.3. Dispersion of solutes in a porous medium.

The consequences of dispersion are shown in the hypothetical example depicted in Fig. 7.4. Assume that at time t = 0 we inject a tracer in a groundwater flow as shown in the figure. After some time, the tracer will have moved with the groundwater over a certain distance, but also will have dispersed such that a tracer will form a cloud.

Fig. 7.4. Advection and dispersion of a tracer injected in groundwater.

x = vt

Groundwater flow

Tracer 2 × 1.96σL

2 × 1.96σT

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The centre of the cloud has moved along with groundwater over a distance x = vt, where v is the groundwater flow velocity. However, because of dispersion, some tracer particles will deviate from this position. The highest concentration of tracer particles will be in the centre of the cloud and the concentration will decrease away from the centre. Hence, dispersion can be considered as a random process similar as diffusion. From statistics we know that for a normal distribution 95% is contained in a zone ±1.96σ, where σ is the standard deviation. From diffusion theory, we know that the standard deviation for diffusive movement is given by Dt2=σ (7.5) where D is the diffusion coefficient [L/T2]. Notice that the spreading increases with the square root of time. The same principles can be applied to dispersion, but there are some complications. First of all, dispersion is more intensive in the longitudinal direction of groundwater flow than in the transverse direction. This means that the tracer cloud is more elongated in the longitudinal direction, as shown in Fig. 7.4. Therefore, two standard deviations need to be considered, one in the longitudinal direction σL and one in the transversal direction σT, and 95% of the tracer cloud is contained in an ellipsoid with dimensions as indicated in Fig. 7.4. In view of Eq. 7.5, these standard deviations relate to dispersion coefficients as follows tD2 LL =σ (7.6a) tD2 TT =σ (7.6b)

where DL and DT are longitudinal and transverse dispersion coefficients that quantify the dispersive properties of the porous medium. Experiments have shown that these dispersion coefficients are not constant, but depend upon the magnitude of the groundwater flow velocity and the diffusive properties of the particles, i.e. vDD L0L ε+= (7.7a) vDD T0T ε+= (7.7b)

where D0 is the effective diffusion coefficient, which is smaller than the diffusion coefficient in open water because of the effect of the size and tortuosity of the pore space, and εL and εL are longitudinal and transverse dispersivities [L], which relate to scale length of the heterogeneity of the pore space. Unfortunately, more complications arise from the fact that ground layers have heterogeneous properties that are scale depending. On a small scale, say laboratory samples, heterogeneity of a porous medium relates to the size of the pores and solid particles, but in the field heterogeneity of a soil relates to the succession of sand, loam and clay lenses or layers, are for groundwater system heterogeneity depends on the occurrence of pervious, semi-pervious and impervious zones or layers. Hence, dispersivity increases with the scale of the considered problem. Research in this field is still ongoing, but for the time being we use following rule of thumb: the longitudinal

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dispersivity εL is about 5 to 10% of the advective flow distance, i.e. x = vt, while the ratio of transversal to longitudinal dispersivity εT/εL is about 1/3.

7.3 Adsorption of chemicals In addition to transport by advection and dispersion, other factors may affect the flow and concentration of dissolved chemicals. An important process is adsorption of chemicals on the solid particles of a porous medium (Fig. 7.5).

Fig. 7.5 Adsorption and de-sorption of dissolved particles.

Adsorption reactions in groundwater are normally very fast relative to the ground water movement and reversal, such that at any instant there is equilibrium between the mass of the chemical dissolved in the groundwater and the mass of the chemical adsorbed on the solid material. A distribution coefficient Kd can be defined as the ratio of the mass adsorbed and the mass in solution dad MMK = (7.8) whereKd is the distribution coefficient [-], Ma the adsorbed mass [M], and Md the dissolved mass [M]. This distribution coefficient depends upon the type of chemical and the adsorptive capacity of the solid material. Adsorption leads to retardation of the transport of the chemical. An apparent flow velocity can be calculated as the average velocity of all particles, also the ones that are adsorbed adtap M0vMMv ⋅+= (7.9) where vap is the apparent velocity [L/T], and Mt the total mass [M] equal to Ma + Md, or

Rv

K1v

MMvM

vdad

dap =

+=

+= (7.10)

The term R =1 + Kd is called the retardation factor and is a measure of the retardation effect of the adsorption process on the transport of the chemical; it is as if the real groundwater flow velocity is reduced by a value equal to the retardation factor. The retardation factor can be in the range of 1 to 100 or more. When chemicals are strongly adsorbed they become essentially immobile. Calculations of advective and dispersive

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transport can easily be combined with adsorption, it is sufficient to substitute the groundwater velocity by the apparent velocity. Chemicals that can be strongly adsorbed are ions as heavy metals and micro-organic chemicals. Ions are adsorbed by clay particles, metallic oxides and humic substances, by means of ionic exchange. Non-polar micro-organic pollutants are absorbed by natural organic material present in the solid matrix. The absorption process is a simple partitioning reaction, where hydrophobic organic pollutants tend to accumulate in natural organic material. Normally, a soil can contain about 2% organic material and on average about 10% of clay particles, both percentages being expressed on a dry soil weight basis. Distribution coefficients can be determined by laboratory experiments, where solutions containing different concentrations of a certain chemical are mixed with soil samples. After equilibration, the concentration C in the solution and the amount of adsorbed mass per mass of soil S are measured and the distribution coefficient is calculated as nknCSKd ρ=ρ= (7.11) where S is the absorbed concentration, i.e. mass of chemical absorbed per mass of soil M/M], ρ the dry density of the soil material [M/L3], and k the partitioning coefficient [L3/M], equal to S/C. In the literature, many graphs of S versus C have been presented, or values of partitioning coefficients have been tabulated. Also such results can easily be obtained from laboratory experiments.

Fig. 7.6 Linear and Langmuir absorption isotherms. Sometimes, relations between S and C are not linear, such that no unique value for the partitioning coefficient exists. This occurs for instance when solute concentrations are high and the adsorption capacity of the solid material is limited. Usually, such adsorption relationship can be described by a Langmuir equation (Fig. 7.6)

maxSkC1

kCS+

= (7.12)

where Smax is the maximum soil adsorption capacity [M/M]. In such case, distribution and retardation coefficients will vary with the concentration. However, an

C

S

Smax

0

Langmuir

linear

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approximation is possible by linearization of the isotherm; an effective equilibrium distribution coefficient can be defined as

)CC(n)SS(K

12

12d −

−ρ= (7.13)

where C1 and C2 are respectively the minimum and maximum chemical concentrations, and S2 and S1 the corresponding absorbed concentrations. For many practical applications the linearization technique is sufficiently accurate for predictive purposes. The distribution coefficient of micro-organic chemicals can be related to the partitioning coefficient of the soil organic matter fraction, or soil organic carbon fraction, as ococomom kfkfk == (7.14) where fom is the weight ratio of soil organic matter [M/M], kom the partitioning coefficient for soil organic matter [L3/M], foc the weight ratio of soil organic carbon [M/M], ≈ 0.62fom, and koc the partitioning coefficient for soil organic carbon [L3/M], ≈1.72kom. The partitioning coefficient for soil organic matter (or organic carbon) can be related to the partitioning coefficient of an octanol-water mixture, which can easily be determined in a laboratory. An example of such an empirical relation is ( ) ( ) 78.0klog904.0klog owom −⋅≈ (7.15) where kow is the partitioning coefficient in a water-octanol mixture [-], and kom is in units [l/kg]. Also empirical relationships exist that relate the distribution coefficient for natural organic matter to the solubility of the micro-organic chemical, as for instance ( ) ( ) 001.0Sblog729.0klog om +⋅−≈ (7.16) where Sb is the solubility of the chemical in water [mol/l], and the units of kom are [l/kg].

7.4 Decay of chemicals A dissolved chemical in groundwater can be decomposed by various mechanisms as hydrolyses, biochemical decomposition, radioactive decay, etc. The exact description of these processes is usually very difficult and we tend to simply this by using first order reactions, where the degradation rate is considered to be proportional to the concentration

CdtdC

λ−= (7.17)

with λ a first order reaction rate coefficient [T-1]. This results in an exponential decrease of the concentration )texp(CC 0 λ−= (7.18)

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where C0 is the original concentration [M/L3]. Such a reaction is also characterized by a half-life reaction time, t1/2, which is the time period in which half of the mass is being degraded and removed from the system

λ

=)2ln(t 21 (7.19)

The exact evaluation of this parameter is usually very speculative; values should be considered as possible orders of magnitude. Values can be found in the literature or can be measured in the laboratory. Calculations of concentration variations resulting from decay in combination with advective and dispersive transport, and adsorption reactions, are possible by considering C0 as the concentration in case there is no degradation present. Hence, this concentration should simply be multiplied by the exponential term, which decreases in function of the transport time. Of particular interest is the natural biochemical decomposition of organic material. Micro-organisms that are present in the soil are decomposers, i.e. they obtain their energy through the decomposition of organic substances. In addition they will also use carbon from organic sources for the production of their own biomass (usually 20 to 60% is for biomass production and the rest is respired for energy production). However, such biochemical reactions also require electron acceptors, such as O2, NO3

-, SO42-, etc., and

enzymes (proteins produced by living cells), which catalyze the bioconversion reactions. Furthermore, micro-organisms need other nutrients for their biomass production. A general biomass formula is C106H180O45N16P1, such that for every 106 C-atoms 180 H-atoms are required (usually from hydrocarbons), 45 oxygen atoms, 16 nitrogen atoms and 1 phosphorus atom, in order to build biomass. In addition electron acceptors are also needed for energy production. The most effective electron acceptor is free oxygen from the atmosphere, but free oxygen is limited in groundwater due to restricted exchange with the atmosphere. If free oxygen is lacking, other substances can be used as nitrates, sulphates, and iron hydroxides. These compounds are reduced and result in NH4 or H2S gasses. Hence, a deficiency in free oxygen or the presence of gasses as methane is a strong indication of severe pollution by organic material. From previous discussion, it follows that many factors can influence the reaction rate of a biodegradation process. In order to take into consideration the fact that decomposition reaction rates can be limited, by lack of some necessary substances, a Michaelis-Menten reaction equation can be used

CkCr

dtdC

m

max

+−

= (7.20)

where rmax is the maximum possible reaction rate [M/L3T], and km the half-reaction constant [M/L3]. This equation has the property that the reaction rate is limited for large values of the concentration, due to environmental constraints. Notice that for C equal to km the reaction rate becomes half of the maximum reaction rate. Also notice the resemblance with a Langmuir isotherm. In case concentrations are low, C << km, the reaction equation becomes essentially equal to a first order equation.

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7.5 Protection zones around pumping installations Land-use planning is a major form of prevention in which the producers of hazardous wastes are kept away from areas overlying groundwater resources so that, in the event of an accidental spill, little damage will occur. In order to obtain and maintain high quality water, the environment near groundwater pumping installations should be protected against any kind of contamination. This can be done by delineation of protection zones surrounding the points of abstraction. In principle, the whole area from which groundwater can reach the pumping wells should be included in such protection zones. However, in general this would mean that a total groundwater system, with all its interconnected aquifers should be considered for protective measures. In practice however, a distinction should be made between types of pumping wells and groundwater layers, as phreatic or confined, and the possible groundwater contaminants, as degradable and persistent compounds. A criterion that can be used for determining the size of the protection zone is the travel time that a conservative pollutant needs to flow from any injection point in the aquifer to the pumped well. Other techniques are based on fixed distances, because travel times are often difficult to determine with sufficient accuracy. Many times, combinations of both techniques are used. Usually three protection zones are defined: 1) The exploitation area: this is the region in the immediate vicinity of the pumped wells.

It can be defined as the area were in case of an accidental pollution the wells certainly will be contaminated, without any possibility of taking protective measures or even closing down the well. Consequently, in such a protection zone all activities should be banned except groundwater abstraction and purification;

2) The bacterial protection zone: this is the zone in which any pollution by micro organisms can reach the wells, without the bacteria being degraded or stopped during their movement through the aquifer; in this zone protective measures should be taken in order to avoid contamination, as for instance: • a ban on all types of waste disposal or sewage treatment systems; • a ban on storage facilities for dangerous products; • banning all excavation or drilling activities; • restriction on the use of fertilizers and pesticides.

3) The chemical protection zone: the area from which any type of pollution can reach the pumping well; in principle this includes the total zone influenced by the well; protective measures should be taken for this zone; major ones are: • limitations or regulations concerning waste dumps and sewage treatment systems; • safety regulations for storage facilities of chemical products; • limitations on fertilizer and pesticide use; • duty to report all activities that can pose a threat to groundwater quality.

Zone 1 is usually demarcated as the area characterized by groundwater travel times of less than 24h, or distances of 10 m up to 100 m around the wells. For zone 2 most countries accept travel times of 50 to 60 days, sometimes with maximum distances in the range of 100 to 300 m. The demarcation of zone 3 is very different from country to country. Fixed or minimum distances in the order of kilometres are used, and/or travel times of 15 to 25 years. Exact calculations of flow paths and travel times are usually very difficult. These require powerful computer models and extensive field investigations. Approximate equations can

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be obtained for simple situations and homogeneous groundwater systems. Table 1 lists some equations for estimating the extend of protection zones around pumping installations, based on following assumptions: • all groundwater flows are considered radial around individual wells, or uniform around

batteries of wells; • for travel times in zones 1 and 2 no recharge is taken into consideration, all water

pumped comes from the groundwater layer, having a constant thickness and effective porosity;

• the size of zone 3 is obtained by considering an area such that the total recharge rate becomes equal to the pumping, i.e. the total influence region.

Table 7.1 Approximate equations for protection zones around pumping installation

Protection zone

One individual well or a cluster of wells

Battery of wells

1 or 2 nDQtRπ

= π

−⎟⎠⎞

⎜⎝⎛π

=LL

nDQtR

2

3 qQRπ

= π

−⎟⎠⎞

⎜⎝⎛π

=LL

qQR

2

R: maximum distance from one individual well or from the centre of a cluster of wells, or from the axis of a battery of wells [L]; Q: total pumping rate [L3/T]; t: travel time [T]; n: porosity [L3/L3]; D: thickness of the aquifer [L]; L: length of the battery; q: recharge rate [L/T].

RR


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