Saturating with saline solution

When discussing semi-permeability, we noted that a bentonite sample that is saturated with a saline solution probably contains more salt in the initial stages of the process than what is dictated by the final state Donnan equilibrium. This salt must consequently diffuse out of the sample before equilibrium is reached.

The reason for such a possible “overshoot” of the clay concentration is that an infiltrating solution is not subject to a Donnan effect (between sample and external solution) when it fills out the air-filled voids of an unsaturated sample. Also, even if the region near the interface to the external solution becomes saturated — so that a Donnan effect is active — a sample may still take up more salt than prescribed by the final state, due to hyperfiltration: with a net inflow of water and an active Donnan effect, salt will accumulate at the inlet interface (unless the interface is flushed). This increased concentration, in turn, alters the Donnan equilibrium at the interface, with the effect that more salt diffuses into the clay.

These effects are relevant for our ongoing assessment of studies of chloride equilibrium concentrations. If bentonite samples are saturated with saline solutions, without taking precautions against these effects, evaluated equilibrium concentrations may be overestimated. Note that, even if saturating a sample may be relatively fast, it may take a long time for salt to reach full equilibrium, depending on details of the experimental set-up. In particular, if the set-up is such that the external solution does not flow past the inlet, equilibration may take a very long time, being limited by diffusion in filters and tubing.

Interface excess salt

Another way for evaluated salt concentrations to overestimate the true equilibrium value — which is independent of whether or not the sample has been saturated with a saline solution — is due to excess salt at the sample interfaces.

Suppose that you determine the equilibrium salt concentration in a bentonite sample in the following way. First you prepare the sample in a test cell and contact it with an external salt solution via filters. When the system (bentonite + solution) has reached equilibrium (taking all the precautions against overestimation discussed above), the concentration profile may be conceptualized like this

The aim is to determine \(\bar{c}_\mathrm{clay}\), the clay concentration of the species of interest (e.g. chloride), and to relate it to the corresponding concentration in the external solution (\(c_ \mathrm{ext}\)).

After ensuring the value of \(c_\mathrm{ext}\) (e.g. by sampling or controlling the external solution), you unload the test cell and isolate the bentonite sample. In doing so, we must keep in mind that the sample will begin to swell as soon as the force on it is released, if only water is available. In the present example it is difficult not to imagine that some water is available, e.g. in the filters.¹

It is thus plausible that the actual concentration profile look something like this directly after the sample has been isolated

We will refer to the elevated concentration at the interfaces as the interface excess. The exact shape of the resulting concentration profile depends reasonably on the detailed procedure for isolating the sample.² If the ion content of the sample is measured as a whole, and/or if the sample is stored for an appreciable amount of time before further analysis (so that the profile evens out due to diffusion), it is clear that the evaluated ion content will be larger than the actual clay concentration.

To quantify how much the clay concentration may be overestimated due to the interface excess, we introduce an effective penetration depth, \(\delta\)

\(\delta\) corresponds to a depth of the external concentration that gives the same interface excess as the actual distribution. Using this parameter, it is easy to see that the clay concentration evaluated as the average over the entire sample is

\begin{equation} \bar{c}_\mathrm{eval} = \bar{c}_\mathrm{clay}+\frac{2\cdot\delta} {L} \cdot \left (c_\mathrm{ext} – \bar{c}_\mathrm{clay} \right ) \end{equation}

By dividing by the actual value \(\bar{c}_\mathrm{clay}\), we get an expression for the relative overestimation

\begin{equation} \frac{\bar{c}_\mathrm{eval}}{\bar{c}_\mathrm{clay}} = 1 + \frac{2\cdot\delta} {L} \cdot \left (\frac{c_\mathrm{ext}}{\bar{c}_\mathrm{clay}} – 1 \right ) \tag{1} \end{equation}

This expression is quite interesting. We see that the relative overestimation, reasonably, depends linearly on \(\delta\) and on the inverse of sample length. But the expression also contains the ratio \(r \equiv c_\mathrm{ext}/\bar{c}_\mathrm{clay}\), indicating that the effect may be more severe for systems where the clay concentration is small in comparison to the external concentration (high density, low \(c_\mathrm{ext}\)).

An interface excess is more than a theoretical concept, and is frequently observed e.g. in anion through-diffusion studies. We have previously encountered them when assessing the diffusion studies of Muurinen et al. (1988) and Molera et al. (2003).³ Van Loon et al. (2007) clearly demonstrate the phenomenon, as they evaluate the distribution of stable chloride (the background electrolyte) in the samples after performing the diffusion tests.⁴ Here is an example of the chloride distribution in a sample of density 1.6 g/cm³ and background concentration of 0.1 M⁵

The line labeled \(\bar{c}_\mathrm{clay}\) is evaluated from the average of only the interior sections (0.0066 M), while the line labeled \(\bar{c}_\mathrm{eval}\) is the average of all sections (0.0104 M). Using the full sample to evaluate the chloride clay concentration thus overestimates the value by a factor 1.6. From eq. 1, we see that this corresponds to \(\delta = 0.2\) mm. For a sample of length 5 mm with the same penetration depth, the corresponding overestimation is a factor of 2.1.

Here is plotted the relative overestimation (eq. 1) as a function of \(\delta\) for several systems of varying length and \(r\) (\(= c^\mathrm{ext}/\bar{c}_\mathrm{clay}\))

We see that systems with large \(r\) and/or small \(L\) become hypersensitive to this effect. Thus, even if it may be expected that \(\delta\) decreases with increasing \(r\)⁶, we may still expect an increased overestimation for such systems.

To avoid this potential overestimation of the clay concentration, I guess the best practice is to quickly remove the first couple of millimeters on both sides of a sample after it has been unloaded. In many through-diffusion tests, this is done as part of the study, as the concentration profile across the sample often is measured. In studies where samples are merely equilibrated with an external solution, however, removing the interface regions may not be considered.

Summary

We have here discussed some plausible reasons for why an evaluated equilibrium salt concentration in a clay sample may be overestimated:

• If samples are saturated directly with a saline solution. Better practice is to first saturate the sample with pure water (or a dilute solution) and then to equilibrate with respect to salt in a second stage.
• If the external solution is not circulated. Diffusion may then occur over very long distances (depending on test design). The reasonable practice is to always circulate external solutions.
• If interface excess is not handled. This is an issue even if saturation is done with pure water. The most convenient way to deal with this is to section off the first millimeters on both sides of the samples as quickly as possible after they are unloaded.

Footnotes

[1] One way to minimize this possible effect could be to empty the filter before unloading the test cell. This may, however, be difficult unless the filter itself is flushable. Also, you may run into the problem of beginning to dry the sample.

[2] The only study I’m aware of that has systematically investigated these types of concentration profiles is Glaus et al. (2011). They claim, if I understand correctly, that the interface excess is not caused by swelling during dismantling. Rather, they mean that the profile is the result of an intrinsic density decrease that occurs in interface regions. Still, they don’t discuss how swelling are supposed to be inhibited, neither during dismantling, nor in order for the density inhomogeneity to remain. Under any circumstance, the conclusions in this blog post are not dependent on the cause for the presence of a salt interface excess.

[3] In through-diffusion tests, the problem of the interface excess is usually not that the equilibrium clay concentration is systematically overestimated, since the detailed concentration profile often is sampled in the final state. Instead, the problem becomes how to separate the linear and non-linear parts of the profile.

[4] Van Loon et al. (2007) will be assessed regarding evaluated chloride equilibrium concentrations in a future blog post. However, the study was considered in the post on the failure of Archie’s law in bentonite. Update (220721): Van Loon et al. (2007) is assessed in detail here.

[5] Van Loon et al. (2007) reports evaluated values of “effective porosity”, \(\epsilon_\mathrm{eff}\). I have calculated the clay concentration from these as \(\bar{c}_\mathrm{clay} = c_\mathrm{ext}\cdot \epsilon_\mathrm{eff}/\phi\), where \(\phi\) is the physical porosity. Note that \(\bar{c}_\mathrm{clay}\) is a model independent parameter, while \(\epsilon_\mathrm{eff}\) certainly is not.

[6] Because \(r\) and \(\delta\) may co-vary with density.