Storativity of leaky layers

[MattBrst]

While working on another version of the WATEX crosssection I noticed a subtle detail in the plot: In the hinterland, the storativity S is written without the subscript s. This is caused by a combination of topboundary="semi" and phreatictop=True, so the S value is supposed to be a specific yield instead of a specific storage coefficient.

In my application ('short' changes in hydraulic head which come from an underlying aquifer and are propagating slowly into an aquitard), I would say the storage is not taking place in the top of the clay layer (around the phreatic groundwaterlevel), but rather in the lowest part of the clay layer, which is completely saturated.

So the quick solution for me is to set phreatictop=False for this section. However, more generally spoken I doubt whether there are situations where the combination of topboundary="semi" and phreatictop=True makes sense. If not, we might want to block this combination.

[HendrikMe]

Please mind that in Dutch hydraulic engineering the “indringingslengte” is a common method to schematise the depth where a hydraulic head change occurs. Depth from the top of the aquifer. See paragraph 7.13 in “schematiserings handleiding macrostabiliteit”.

To schematise this “indringingslengte” you will need several layers, and a combination of semi and phreatic top, I assume.

[mbakker7]

Thanks for bringing this up, @HendrikMe. It is important to point out that the leaky layers don't need to be divided into several layers to simulate the correct effect, as vertical flow in the leaky layers is already simulated analytically. See, for example, the theory explained here: https://github.com/mbakker7/ttim/blob/master/papers/2013_bakker_ttim_theory.pdf (you need to download the pdf, as github doesn't show it in most browswers).

For dike stability, it is important to compute the pressure change in the leaky layer. Although ttim simulates this, there is no function to compute the pressure distribution inside a leaky layer at the moment. This would be an important additional function to be added for the computation of dike stability. Maybe you can add an Issue for an Enhancement?

[dbrakenhoff]

Issue #101 opened to keep track of this development.

[dbrakenhoff]

From the email conversation preceding this discussion:

TTim supports 3 types of top-boundaries: confined, semi-confined and leaky. The phreatictop option is only supported for the confined and leaky top boundaries. The difference between the leaky and semi-confined top boundaries, as explained by @mbakker7:

• "semi" is a semi-confined aquifer covered by an aquitard with a constant water level above the aquitard.
• "leaky" is a semi-confined aquifer covered by an aquitard (and no constant water level above). The aquifer is fully saturated and there is (phreatic) storage in the top aquitard.

An issue was made to address the plot labels and alert users that phreatictop=True does not do anything when topboundary="semi" (#98).

[MattBrst]

In the email conversation @mbakker7 referred to the MLU documentation as an inspiration source for the modeling of the topboundary.

Quoting from pages 9 and 10 of the MLU Tutorial:

Storage and leakage
When a leaky (semi-confined) aquifer is pumped, the pumped water comes from different
sources that gradually change with time. In the case of a single aquifer with an overlying
aquitard and an impervious base, five different sources can be identified. In the order of their
effects becoming noticeable, these sources are:

• “a” – Wellbore storage (due to the falling water level in the well casing)

• “b” – Aquifer storage (due to the compressibility of the aquifer material and that of water)

• “c” – Aquitard storage (due to the compressibility of the aquitard material and that of
  water)

• “d” – Leakage through the aquitard (from overlying surface water or from an adjacent
  aquifer)

• “e” – Water released by a falling water table in the aquitard.

Many analytical solutions account for only a few of these sources. The simplest of them is the
steady-state solution where all water comes from leakage. Such solutions require that the
unaccounted for sources are negligible, which limits their applicability to a restricted time
period.

In analytical solutions, aquitard storage is usually assumed negligible and consequently its
effects are often neglected or attributed to the storativity of adjacent aquifer layers. Aquitard
storage is used in two of the Section 5 examples (Dalem, Schroth).

MLU uses a generalized analytical solution, which as a rule accounts for the sources “a”
through “d”. The water released by a falling water table can also be simulated by MLU, but
such conditions require an additional layer at the top of the system. This ‘water-table layer’ or
‘phreatic layer’ is characterized by a low transmissivity and a high storativity (the specific
yield). Further details about ‘water-table layers’ are discussed in the Section 5 examples
(Moench, Two-aquifers, Vennebulten).

I am curious how the terms "c" to "e" map to the topboundary usage in TIM. Thanks in advance for your help!

I would say:

• topboundary = "confined", then none of the terms $c$, $d$ and $e$ are taken into account
• topboundary = "semi", then term $c$ is taken into account
• topboundary = "leaky", then terms $c$ and $d$ are taken into account (so: leakage from a leakagelayer is facilitated)

This would mean that in order to account for water from a falling water table in the aquitard, the MLU approach needs to be followed by adding an additional layer.

[dbrakenhoff]

I think the last two should be the other way around:

• topboundary = "confined", then none of the terms $c$ , $d$ and $e$ are taken into account
• topboundary = "semi", then terms $c$ and $d$ are taken into account:
  • With "semi" there is a constant water level above the top aquitard that can act as a source of water, representing $d$
  • When setting Sll aquitard storage can be taken into account ($c$)
• topboundary = "leaky", then term $c$ can be taken into account when Sll is set, but there is no source of water in or above the aquitard.

This would mean that in order to account for water from a falling water table in the aquitard, the MLU approach needs to be followed by adding an additional layer

Agreed!

[MattBrst]

I hope @mbakker7 can help us out by looking at these lines, where it is happening: https://github.com/mbakker7/ttim/blob/4af45958a5f3e64f031f1a131dc9f7f5adbd758d/ttim/aquifer.py#L152-L155

[MattBrst]

My calculation results seem to confirm your definition of 'semi' and 'leaky': When I model a block response, in the 'leaky' case there is not much damping in the far field, whereas in the 'semi' case the heads are (at larger times) damping out with increasing distance.