Esercitazione su test con pozzi (Falde confinate e non)

STEADY STATE UNSTEADY STATE

Thiem I Thiem II Theis Jacob I Jacob II Jacob III

H30 H90 H30 H90

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T=KB (m /d) 370 370 348 369 492 534 371 486

-4 -4 -5 -4 -4 -4

S (-) 1.9·10 2.6·10 9.9·10 1.6·10 5.3·10 1.4·10 2

If we compare the results we can notice that the value of transmissivity lies in a range of 350 ÷ 530 m /d

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while the storativity range is 1÷5 ·10 which are both physically consistent with the literature values for this

type of soil.

Let’s start with a general comparison between the steady-state solutions and the unsteady-state solutions.

The main difference between the two conditions is the number of aquifer parameters that can be

determined: in particular, we prefer all the unsteady-state methods since there is the possibility to determine

both the transmissivity and the storativity. Moreover, we know that the steady state condition is impossible

to be reached in a confined aquifer so the unsteady state analysis in more representative of the real flow

condition in the Oude Korendijk confined aquifer.

We want to mention again, that we considered the transmissivity as a characteristic of the aquifer, and not

its hydraulic conductivity, since the T value thus determined is valid to determine K at a field scale rather

than its absolute correct value, moreover it can be easily obtained dividing T for the aquifer thickness that is

assumed to be constant.

Going back to the comparison of flow condition: another difference is the fact that the steady-state solution

(that in reality is not fully reached so we should talk of pseudo-steady-state) accounts on fewer measurement

data. For example, in this case, Thiem method relies only on data measured after 10 minutes from the start

of pumping.

For what concerns the numeric value: we notice a close agreement between the Thiem (I and II) and the

Theis value of transmissivity even if the assumption, on the flow condition, is different.

Anyway, we need to mention that since the Theis method is graphic, the accuracy of these results lies in part

on the precision and sensitivity of the user since even little displacement leads to quite different results.

In this case, it was quite simple to overlap the two curves since there was a great number of drawdown

measurements in both H30 and H90 well.

Now, if we compare the different procedures of the unsteady state method of Jacob, we can notice that

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while Jacob I and III give us a transmissivity of around 500 m /d, on the other hand Jacob II gives a lower

2

value of around 370 m /d (almost identical to the steady-state condition). This can be explained by the fact

that we were forced to use only two reliable sets of data instead of three so the line was interpolated only

on two points.

For what concerns the storativity, we see that Jacob II is the storativity is overdimensioned. This was the

reason why we decided to not rely on this method.

In the end, we can establish that the aquifer properties in the geological site of Oude Korendijk have

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averagely the following values: the transmissivity T=KB= 413 m /d and the storativity S=1.7·10 .

2. The Dalem case study: a confined leaky aquifer

In this second paragraph, we will briefly introduce the geological and geometric characteristics of the

leaky aquifer of Dalem.

As we can see in Figure 10.a, the site is characterized by a three-layered geological structure: the clayey

aquitard between the ground surface and 10 m below (in green) and two coarse sandy confined aquifers,

the lower one in contact with an impermeable aquiclude (in orange).

Even if not shown in Figure 10.a, we are guessing that there will be some system of water replenishment

which allows the aquitard to constantly transmit water to the lower aquifers.

Therefore, we can schematize the system as shown in Figure 10.b

The pumping well is fully penetrating the lowest aquifer, but the test has been conducted sealing the

lower screen and allowing water to filter only from the upper aquifer between 11 and 19 m.

Finally, there are four measurement wells at distances of 30, 60, 90, and 120 m from the pumped well

Figure 10.a: geological section of the leaky aquifer of Dalem.

Figure 10.b scheme of a typical leaky confined aquifer. In Red the

impermeable aquiclude. In Orange the confined aquifer. In Green the

leaky aquitard. In Blue the recharge system.

2.1. Initial assumptions

- The first assumption is to consider a leaky confined aquifer: this means that the pumped water is derived

from leakage as well as storage.

- The aquifer and the aquitard have an infinite lateral extension: it means that compared to their horizontal

dimension they are considerably thin.

- - The aquifer and the aquitard have a constant horizontal thickness of respectively 11 and 8 m: it means

that local thickness variations are neglected.

- The aquifer and the aquitard are homogeneous and isotropic.

- The flow toward the aquifer is vertical while in the aquifer is horizontal.

Besides, we consider that:

- The initial piezometric surface H is horizontal.

- The well is fully penetrating the aquifer. 3

- The well is pumping at a constant rate of 31.7 m /hr (for 8 hours).

- The head losses over the well screen are neglected.

2.2. Steady-state flow condition

The main steady-state assumption for a leaky aquifer is that water is assumed to leak without affecting the

existing water pressures: this means that the water table of the upper unconfined aquifer remains constant.

The aquitard then acts as a transmitting medium without suffering any compressibility due to the changes in

water pressures.

This is known as leakage without releasing water from storage from within the aquitard.

Relying on this assumption, the general solution for steady-state flow is given by:

() = ℎ − ℎ() = (/ )

0 0

2 Equation 9

Where s(r) is the equilibrium drawdown at distance r from the well

B is the leakage factor

k

K (r/B ) is the modified Bessel function of the second kind and zero-order.

0 k

In particular, the leakage factor is defined as: ′

√

= = √

′ Equation 10

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Where BK is the transmissivity T of the aquifer while b’ [L] and K’ [LT ] are respectively the thickness and the

hydraulic conductivity of the aquitard. The ratio between b’ and K’ is also called the hydraulic resistance of

the aquitard c [T].

In this particular case of Dalem, as Figure 11 shows, the steady-state condition has not been fully reached

since the line does not tend to the horizontal.

Figure 11: Analysis of the steady state condition.

So, as we did in Paragraph 1, we established that after 0.125 days there will be a pseudo-steady-state

condition since this is the time at which the curves start to run parallel.

In the following methods, we use the latest time drawdown as a measure of the equilibrium drawdown s .

m

With this value of s we determine the characteristic of the aquifer (T transmissivity ) and the aquitard (c

m

hydraulic resistance) under the steady-state assumption.

2.2.1) De Glee’s method

If we rearrange Equation 9 and the definition of B (Equation 10) in a log form, it is possible to determine a

k 10

graphic relationship between the quantities s, r, K (r/B ) and r/B :

0 k k

)

log( = log ( ) + log ( ( ))

0

4

1 ′

log() = log + log ( )

( )

2 ′

{

Where s is the stabilized drawdown: in this case, we choose the latest time value

m H30 H60 H90 H120

s (m) 0.2280 0.1640 0.1430 0.1290

m

If we plot both the data curve (s - r) and the type curve (K (r/B ) - r/B ) on a log paper we observe that the

m 0 k k 10

log ( )

two curves are similar in form, except for a certain displacement determined by the quantities and

4

′

log ( ).

′

So, if we place the type curve over the field data curve and we search the best fit, then we can record the

four coordinates of an arbitrary match point and use them to determine the properties of the aquifer T and

of the aquitard c.

The calculation is really simplified if we consider the match point K (r/B ) = 1- r/B = 1.

0 k k

The matching of the two curves is shown in Figure 12:

Figure 12: De Glee’s method, matching of the data curve and the Bessel function.

So if we re-arrange the system of above, the aquifer properties would be determined by the expression:

∗

( )

0

= ∗

2

∗

=

∗

( )

{

Where the * values are determined through the graphic procedure explained above.

The results obtained with the De Glee’s method are:

2

T (m /d) 1730

c (d) 832

However we should notice that with only four piezometers is very difficult to find the best fit manually.

Even with little position changes the results can be very different.

Another consideration is that, as we already demonstrated, the steady state condition has not been reached.

This means that the four values of drawdown that we used to create the data curve, are maybe a little bit

underestimating the real stabilised drawdown value.

2.2.2) Hantush – Jacob’s method

In some cases, when r/B values are less than 0.05, the solution does not require a matching procedure.

k

In fact, when r/B values are less than 0.05, Equation 9 reduces to:

k

() = ln (1.12 ∗ )

2

Thus T and c values can be determined considering the drawdown-distance curve.

If we plot the stabilized drawdown (as we did before, we choose the latest measured values s ) versus the

m

corresponding radius, in a log scale, we obtain a straight line:

10

Figure 13: Hantush and Jacob method. Plot of the stabilized drawdown versus the corresponding radius

= ∆/∆ln ()

We can consider its (which value is 0.072) by mean of the regression equation (in a

natural logarithm form) and then, obtain T from the equation above as follows:

= 2 ∗

The regression straight line can be extended until it reaches the horizontal axis to determine the value of r 0

(radius corresponding at s=0) and so determine B as:

k

0

=

1.12

In this case, the result does not show a close agreement for what concern the c value if we compare it with

the results obtained with the previous method:

2

T (m /d) 1682

c (d) 207

The reason of this lack of correspondence can be found in the following explanation.

We need to remember the first assumption that this method requires: r/B <0.05.

k

Then, substituting the 4 different radius, we can check at which piezometer the limit is respected.

As we can see from the following