In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Molera et al. (2003), in the following referred to as Mo03. We thus assess the 13 points indicated here

Mo03 performed both chloride and iodide through-diffusion tests on “MX-80” bentonite, but here we focus on the chloride results. However, since the only example in the paper of an outflux evolution and corresponding concentration profile is for iodide, this particular result will also be investigated. The tests were performed at background concentrations of 0.01 M or 0.1 M NaClO₄, and nominal sample densities of 0.4, 0.8, 1.2, 1.6, and 1.8 g/cm³. We refer to a single test by stating “nominal density/background concentration”, e.g. a test performed at nominal density 1.6 and background concentration 0.1 M is referred to as “1.6/0.1”.

Uncertainty of samples

The material used is discussed only briefly, and the only reference given for its properties is (Müller-Von Moos and Kahr, 1983). I don’t find any reason to believe that the “MX-80” batch used in this study actually is the one investigated in this reference, and have to assume the same type of uncertainty regarding the material as we did in the assessment of Muurinen et al (1988). I therefore refer to that blog post for a discussion on uncertainty in montmorillonite content, cation population, and soluble calcium minerals.

Density

The samples in Mo03 are cylindrical with radius 0.5 cm and length 0.5 cm, giving a volume of 0.39 cm³. This is quite small, and corresponds e.g. only to about 4% of the sample size used in Muurinen et al (1988). With such a small volume, the samples are at the limit for being considered as a homogeneous material, especially for the lowest densities: the samples of density 0.4 g/cm³ contain 0.157 g dry substance in total, while a single 1 mm³ accessory grain weighs about 0.002 — 0.003 g.

Furthermore, as the samples are sectioned after termination, the amount substance in each piece may be very small. This could cause additional problems, e.g. enhancing the effect of drying. The reported profile (1.6/0.1, iodide diffusion) has 10 sections in the first 2 mm. As the total mass dry substance in this sample is 0.628 g, these sections have about 0.025 g dry substance each (corresponding to the mass of about ten 1 mm³ grains). For the lowest density, a similar sectioning corresponds to slices of dry mass 0.006 g (the paper does not give any information on how the low density samples were sectioned).

Mo03 only report nominal densities for the samples, but from the above considerations it is clear that a substantial (but unknown) variation may be expected in densities and concentrations.

A common feature of many through-diffusion studies is that the sample density appears to decrease in the first few millimeters near the confining filters. We saw this effect in the profiles of Muurinen et al (1988), and it has been the topic of some studies, including Mo03. Here, we don’t consider any possible cause, but simply note that the samples seem to show this feature quite generally (below we discuss how Mo03 handle this). Since the samples of Mo03 are only of length 5 mm, we may expect that the major part of them are affected by this effect. Of course, this increases the uncertainty of the actual density of the used samples.

Uncertainty of external solutions

Mo03 do not describe how the external solutions were prepared, other than that they used high grade chemicals. We assume here that the preparation did not introduce any significant uncertainty.

Since “MX-80” contains a substantial amount of divalent ions, connecting this material with (initially) pure sodium solutions inevitably initiates cation exchange processes. The extent of this exchange depends on details such as solution concentrations, reservoir volumes, number of solution replacements, time, etc…

Very little information is given on the volume of the external solution reservoirs. It is only hinted that the outlet reservoir may be 25 ml, and for the inlet reservoir the only information is

The volume of the inlet reservoir was sufficient to keep the concentration nearly constant (within a few percent) throughout the experiments.

Consequently, we do not have enough information to assess the exact ion population during the course of the tests. We can, however, simulate this process of “unintentional exchange” to get some appreciation for the amount of divalent ions still left in the sample, as we did in the assessment of Muurinen et al. (1988). Here are the results from calculating the exchange equilibrium between a sample initially containing 30% exchangeable charge in form of calcium (70% sodium), and external NaClO₄ solutions of various concentrations and volumes

In these calculations we assume a sample of density 1.6 g/cm³ (except when indicated), a volume of 0.39 cm³, a cation exchange capacity of 0.75 eq/kg, and a Ca/Na selectivity coefficient of 5.

These simulations make it clear that the tests performed at 0.01 M most probably contain most of the divalent ions initially present in the “MX-80” material: even with an external solution volume of 1000 ml, or with density 0.4 g/cm³, exchange is quite limited. For the tests performed at 0.1 M we expect some exchange of the divalent ions, but we really can’t tell to what extent, as the exact value strongly depends on handling (solution volumes, if solutions were replaced, etc.). That the exact ion population is unknown, and that the divalent/monovalent ratio probably is different for different samples, are obviously major problems of the study (the same problems were identified in Muurinen et al (1988)).

Uncertainty of diffusion parameters

Diffusion model

Mo03 determine diffusion parameters by fitting a model to all available data, i.e the outflux evolution and the concentration profile across the sample at termination. The model is solved by a numerical code (“ANADIFF”) that takes into account transport both in clay samples and filters. The fitted parameters are an apparent diffusivity, \(D_a\), and a so-called “capacity factor”, \(\alpha\). \(\alpha\) is vaguely interpreted as being the combination of a porosity factor \(\epsilon\), and a sorption distribution coefficient \(K_d\), described as “a generic term devoid of mechanism”

\begin{equation} \alpha = \epsilon + \rho\cdot K_d \end{equation}

It is claimed that for anions, \(K_d\) can be treated as negative, giving \(\alpha < \epsilon\). I have criticized this mixing of what actually are incompatible models in an earlier blog post. Strictly, this use of a “generic term devoid of mechanism” means that the evaluated \(\alpha\) should not be interpreted in any particular way. Nevertheless, the way this study is referenced in other publications, \(\alpha\) is interpreted as an effective porosity. It should be noticed, however, that this study is performed with a background electrolyte of NaClO₄. The only chloride (or iodide) present is therefore at trace level, and it cannot be excluded that a mechanism of true sorption influences the results (there are indications that this is the case in other studies).

For the present assessment we anyway assume that \(\alpha\) directly quantifies the anion equilibrium between clay and the external solution (i.e. equivalent to the incorrect way of assuming that \(\alpha\) quantifies a volume accessible to chloride). It should be kept in mind, though, that effects of anion equilibrium and potential true sorption is not resolved by the single parameter \(\alpha\).

In practice, then, the model is

\begin{equation} \frac{\partial c}{\partial t} = D_p\frac{\partial^2 c}{\partial x^2} \tag{1} \end{equation}

where \(c\) is the concentration in the clay of the isotope under consideration, and the diffusion coefficient is written \(D_p\) to acknowledge that it is a pore diffusivity (when referring to models and parameter evaluations in Mo03 we will use the notation “\(D_a\)”). The boundary conditions are

\begin{equation} c(0,t) = \alpha \cdot C_0 \;\;\;\; c(L,t) = 0 \tag{2} \end{equation}

where \(C_0\) is the concentration in the source reservoir,¹ and \(L\) is the sample length.

This model — that we have discussed before — has a relatively simple analytical solution, and the outflux can be written

\begin{equation} j^\mathrm{out}(t) = j^\mathrm{ss}\left (1 + 2 \sum_{n=1}^\infty \left (-1 \right )^n e^{-\frac{\pi^2n^2D_pt}{L^2}} \right) \end{equation}

where \(j^\mathrm{ss}\) is the corresponding steady-state flux. Here, the steady-state flux is related to the other parameters as

\begin{equation} j^\mathrm{ss} = \alpha\cdot D_p \frac{C_0}{L} \tag{3} \end{equation}

“Fast” and “slow” processes

Oddly, Mo03 model the system as if two independent diffusion processes are simultaneously active. They refer to these as the “fast” and the “slow” processes, and hypothesize that they relate to diffusion in interlayer water² and “interparticle water”,³ respectively.

The “fast” process is the “ordinary” process that is assumed to reach steady state during the course of the test, and that is the focus of other through-diffusion studies. The “slow” process, on the other hand, is introduced to account for the frequent observation that measured tracer profiles are usually significantly non-linear near the interface to the source reservoir (discussed briefly above). I guess that the reason for this concentration variation is due to swelling when the sample is unloaded. But even if the reason is not fully clear, it can be directly ruled out that it is the effect of a second, independent, diffusion process — because this is not how diffusion works!

If anions move both in interlayers and “interparticle water”, they reasonably transfer back and forth between these domains, resulting in a single diffusion process (the diffusivity of such a process depends on the diffusivity of the individual domains and their geometrical configuration). To instead treat diffusion in each domain as independent means that these processes are assumed to occur without transfer between the domains, i.e. that the bentonite is supposed to contain isolated “interlayer pipes”, and “interparticle pipes”, that don’t interact. It should be obvious that this is not a reasonable assumption. Incidentally, this is how all multi-porous models assume diffusion to occur (while simultaneously assuming that the domains are in local equilibrium…).

We will thus focus on the “fast” process in this assessment, although we also use the information provided by the parameters for the “slow” process. Mo03 report the fitted values for \(D_a\) and \(\alpha\) in a table (and diagrams), and only show a comparison between model and measured data in a single case: for iodide diffusion at 0.1 M background concentration and density 1.6 g/cm³. To make any kind of assessment of the quality of these estimations we therefore have to focus on this experiment (the article states that these results are “typical high clay density data”).

Outflux

The first thing to note is that the modeled accumulated diffusive substance does not correspond to the analytical solution for the diffusion process. Here is a figure of the experimental data and the reported model (as presented in the article), that also include the solution to eqs. 1 and 2.

In fact, the model presented in Mo03 has an incorrect time dependency in the early stages. Here is a comparison between the presented model and analytical solutions in the transient stage

With the given boundary conditions, the solutions to the diffusion equation inevitably has zero slope at \(t = 0\),⁴ reflecting that it takes a finite amount of time for any substance to reach the outflux boundary. The models presented in Mo03, on the other hand, has a non-zero slope in this limit. I cannot understand the reason for this (is it an underlying problem with the model, or just a graphical error?), but it certainly puts all reported parameter values in doubt.

The preferred way to evaluate diffusion data is, in my opinion, to look at the flux evolution rather than the evolution of the accumulated amount of diffused substance. Converting the reported data to flux, gives the following picture.⁵

From a flux evolution it is easier to establish the steady-state, as it reaches a constant. It furthermore gives a better understanding for how well constrained the model is by the data. As is seen from the figure, the model is not at all very well constrained, as the experimental data almost completely miss the transient stage. (And, again, it is seen that the model in the paper with \(D_a= 9\cdot 10^{-11}\) m/s² does not correspond to the analytical solution.)

The short transient stage is a consequence of using thin samples (0.5 cm). Compared e.g. to Muurinen et al (1988), who used three times as long samples, the breakthrough time is here expected to be \(3^2 = 9\) times shorter. As Muurinen et al. (1988) evaluated breakthrough times in the range 1 — 9 days, we here expect very short times. Here are the breakthrough times for all chloride diffusion tests, evaluated from the reported diffusion coefficients (“fast” process) using the formula \(t_\mathrm{bt} = L^2/(6D_a)\).

Test	\(D_a\)	\(t_\mathrm{bt}\)
	(m2/s)	(days)
0.4/0.01	\(8\cdot 10^{-10}\)	0.06
0.4/0.1	\(9\cdot 10^{-10}\)	0.05
0.4/0.1	\(8\cdot 10^{-10}\)	0.06
0.8/0.01	\(3.5\cdot 10^{-10}\)	0.14
0.8/0.1	\(3.5\cdot 10^{-10}\)	0.14
0.8/0.1	\(3.7\cdot 10^{-10}\)	0.13
1.2/0.01	\(1.4\cdot 10^{-10}\)	0.34
1.2/0.1	\(2.3\cdot 10^{-10}\)	0.21
1.2/0.1	\(2.0\cdot 10^{-10}\)	0.24
1.6/0.1	\(1.0\cdot 10^{-10}\)	0.48
1.8/0.01	\(2\cdot 10^{-11}\)	2.41
1.8/0.1	\(5\cdot 10^{-11}\)	0.96
1.8/0.1	\(5.5\cdot 10^{-11}\)	0.88

The breakthrough time is much shorter than a day in almost all tests! To sample the transient stage properly requires a sampling frequency higher than \(1/t_{bt}\). As seen from the provided example of a outflux evolution, this is not the case: The second measurement is done after about 1 day, while the breakthrough time is about 0.5 days (moreover, the first measurement appears as an outlier). We have no information on sampling frequency in the other tests, but note that to properly sample e.g. the tests at 0.8 g/cm³ requires measurements at least every third hour or so. For 0.4 g/cm³, the required sample frequency is once an hour! This design choice puts more doubt on the quality of the evaluated parameters.

Concentration profile

The measured concentration profile across the 1.6/0.1 iodide sample, and corresponding model results are presented in Mo03 in a figure very similar to this

Here the two models correspond to the “slow” and “fast” process discussed above (a division, remember, that don’t make sense). Zooming in on the “linear” part of the profile, we can compare the “fast” process with analytical solutions (eqs. 1 and 2)

The analytical solutions correspond directly to the outflux curves presented above. We note that the analytical solution with \(D_p = 9\cdot 10^{-11}\) m/s² corresponds almost exactly to the model presented by Mo03. As this model basically has the same steady state flux and diffusion coefficient, we expect this similarity. It is, however, still a bit surprising, since the corresponding outflux curve of the model in Mo03 was seen to not correspond to the analytical solution. This continues to cast doubt on the model used for evaluating the parameters.

We furthermore note that the evolution of the activity of the source reservoir is not reported. Once in the text is mentioned that the “carrier concentration” is \(10^{-6}\) M, but since we don’t know how much of this concentration corresponds to the radioactive isotope, we can not directly compare with reported concentration profile across the sample (whose concentration unit is counts per minute per cm³). By extrapolating the above model curve with \(\alpha = 0.15\), we can however deduce that the corresponding source activity for this particular sample is \(C_0 = 1.26\cdot 10^5/0.15\) cpu/cm³ \(= 8.40\cdot 10^5\) cpu/cm³. But it is unsatisfying that we cannot check this independently. Also, we can of course not assume that this value of \(C_0\) is the same in any other of the tests (in particular those involving chloride). We thus lack vital information (\(C_0\)) to be able to make a full assessment of the model fitting.

It should furthermore be noticed that the experimental concentration profile does not constrain the models very well. Indeed, the adopted model (diffusivity \(9\cdot 10^{-11}\) m/s²) misses the two rightmost concentration points (which corresponds to half the sample!). A model that fits this part of the profile has a considerable higher diffusivity, and a correspondingly lower \(\alpha\) (note that the product \(D_p\cdot \alpha\) is constrained by the steady-state flux, eq. 3).

More peculiarities of the modeling is found if looking at the “slow” process (remember that this is not a real diffusion process!). Zooming in on the interface part of the profile and comparing with analytical solutions gives this picture

Here we note that an analytical solution coincides with the model presented in Mo03 with parameters \(D_a = 6\cdot 10^{-14}\) m²/s and \(\alpha = 1.12\) only if it is propagated for about 15 days! Given that no outflux measurements seem to have been performed after about 4 days (see above), I don’t now what to make of this. Was the test actually conducted for 15 days? If so, why is not more of the outflux measured/reported? (And why were the samples then designed to give a breakthrough time of only a few hours?)

Without knowledge of for how long the tests were conducted, the reported diffusion parameters becomes rather arbitrary, especially for the low density samples. For e.g. the samples of density 0.4 g/cm³, even the “slow” process has a diffusivity high enough to reach steady-state within a few days. Simulating the processes with the reported parameters gives the following profiles if evaluated after 1 and 4 days, respectively

The line denoted “total” is what should resemble the measured (unreported) data. It should be clear from these plots that the division of the profile into two separate parts is quite arbitrary. It follows that the evaluated diffusion parameters for the process of which we are interested (“fast”) has little value.

Summary and verdict

We have seen that the reported model fitting leaves a lot of unanswered questions: some of the model curves don’t correspond to the analytical solutions, information on evolution times and source concentrations is missing, and the modeled profiles are divided quite arbitrary into two separate contributions (which are not two independent diffusion process).

Moreover, the ion population (divalent vs. monovalent cations) of the samples are not known, but there are strong reasons to believe that the 0.01 M tests contain a significant amount of divalent ions, while the 0.1 M samples are partly converted to a more pure sodium state.

Also, the small size of the samples contributes to more uncertainty, both in terms of density, but also for the flux evolution because the breakthrough times becomes very short.

Based on all of these uncertainties, I mean that the results of Mo03 does not contribute to quantitative process understanding and my decision is to not to use the study for e.g. validating models of anion exclusion.

A confirmation of the uncertainty in this study is given by considering the density dependence on the chloride equilibrium concentrations for constant background concentration, evaluated from the reported diffusion parameters (\(\alpha\) for the “fast” process).

If these results should be taken at face value, we have to accept a very intricate density dependence: for 0.1 M background, the equilibrium concentration is mainly constant between densities 0.3 g/cm³ and 0.7 g/cm³, and increases between densities 1.0 g/cm³ and 1.45 g/cm³ (or, at least, does not decrease). For 0.01 M background, the equilibrium concentration instead falls quite dramatically between between densities 0.3 g/cm³ and 0.7 g/cm³, and thereafter displays only a minor density dependence.

To accept such dependencies, I require a considerably more rigorous experimental procedure and evaluation. In this case, I rather view the above plot as a confirmation of large uncertainties in parameter evaluation and sample properties.

• Chloride content: UNKNOWN
• Extracting anion equilibrium concentrations from through-diffusion tests
• Assessment of chloride equilibrium concentrations: Muurinen et al. (1988)

Footnotes

[1] Strictly, \(c(0,t)\) relates to the concentration in the endpoint of the inlet filter. But we ignore filter resistance in this assessment, which is valid for the 1.6/0.1 sample. Moreover, the filter diffusivities are not reported in Mo03.

[2] Mo03 refer to interlayer pores as “intralayer” pores, which may cause some confusion.

[3] Apparently, the authors assume an underlying stack view of the material.

[4] It may be objected that the analytical solution do not include the filter resistance. But note that filter resistance only will increase the delay. Moreover, the transport capacity of the sample in this test is so low that filters have no significant influence.

[5] The model by Mo03 looks noisy because I have read off values of accumulated concentration from the published graph. The “noise” occurs because the flux is evaluated from the concentration data by the difference formula:

\begin{equation} \bar{j}(\bar{t}_i) =\frac{1}{A} \frac{a(t_{i+1})-a(t_i)}{t_{i+1}-t_{i}} \end{equation}

where \(t_i\) and \(t_{i+1}\) are the time coordinates for two consequitive data points, \(a(t)\) is the accumulated amount diffused substance at time \(t\), \(A\) is the cross sectional area of the sample, \(\bar{t}_i = (t_{i+1} + t_i)/2\) is the average time of the considered time interval, and \(\bar{j}\) denotes the average flux during this time interval.

In the ongoing assessment of chloride equilibrium concentrations in bentonite, we here take a closer look at the study by Muurinen et al. (1988), in the following referred to as Mu88.¹ In the quite messy plot containing all reported chloride equilibrium concentrations, we thus investigate the twelve points indicated here

Mu88 performed both chloride and uranium through-diffusion tests on “MX-80” bentonite, as well as sorption tests. Here we focus solely on the chloride diffusion. We also disregard one diffusion test series that does not vary external concentration (it was conducted with an unspecified “artificial groundwater” and varied sample density).

Left are two test series performed with nominal sample densities 1.2 g/cm³ and 1.8 g/cm³, respectively. For each of these densities, chloride through-diffusion tests were performed with external NaCl concentrations of 0.01 M, 0.1 M, and 1.0 M, respectively. The samples were cylindrical with a diameter of 3.0 cm, and a length of 1.5 cm, giving a volume of 10.6 cm³. To refer to a specific test or sample, we use the nomenclature “nominal density/external concentration”, e.g. the test performed at nominal density 1.2 g/cm³ and external solution 0.1 M is referred to as “1.2/0.1”.

Uncertainty of bentonite samples

“MX-80” is not the name of some specific standardized material, but simply a product name.² It is quite peculiar that that “MX-80” nevertheless is a de facto standard in the research field for clay buffers in radwaste repositories. But, being a de facto standard, several batches of bentonite with this name have been investigated and reported throughout the years. We consequently have some appreciation for its constitution, and the associated variation.

In Mu88, the material used is only mentioned by name, and it is only mentioned once (in the abstract!). We therefore can’t tell which of the studies that is more appropriate to refer to. Instead, let’s take a look at how “MX-80” has been reported generally.

Report	Batch year	Mmt content	CEC	Na-content
		(%)	(eq/kg)	(%)
TR-06-30 (“WySt”)	1980	82.5	0.76	83
NTR 83-12	< 1983	75.5	0.76	86
TR-06-30 (“WyL1”)	1995	79.5	0.77	–
TR-06-30 (“WyL2”)	1999	79.8	0.75	71
TR-06-30 (“WyR1”)	2001	82.7	0.75	75
TR-06-30 (“WyR2”)	2001	80.0	0.71	71
NTB 01-08	< 2002	–	0.79*	85
WR 2004-023	< 2004	80 — 85	0.84*	65

Montmorillonite content

Reported montmorillonite content varies in the range 75 — 85%. For the present context, this primarily gives an uncertainty in adopted effective montmorillonite dry density, which, in turn, is important for making relevant comparison between bentonite materials with different montmorillonite content. For the “MX-80” used in Mu88 we here assume a montmorillonite content of 80%. In the table below is listed the corresponding effective montmorillonite densities when varying the montmorillonite content in the range \(x =\) 0.75 — 0.85, for the two nominal dry densities.

Dry density	EMDD (\(x\)=0.75)	EMDD (\(x\)=0.80)	EMDD (\(x\)=0.85)
(g/cm3)	(g/cm3)	(g/cm3)	(g/cm3)
1.2	1.01	1.05	1.09
1.8	1.61	1.66	1.70

The uncertainty in montmorillonite content thus translates to an uncertainty in effective montmorillonite dry density on the order of 0.1 g/cm³.

Cation population

While reported values of the cation exchange capacity of “MX-80” are relatively constant, of around 0.75 eq/kg,⁴ the reported fraction of sodium ions is seen to vary, in the range 70 — 85 %. The remaining population is mainly di-valent rare-earth metal ions (calcium and magnesium). This does not only mean that different studies on “MX-80” may give results for quite different types of systems, as the mono- to di-valent ion ratio may vary, but also that samples within the study may represent quite different systems. We examine this uncertainty below, when discussing the external solutions.

Soluble calcium minerals

The uncertainty of how much divalent cations are available is in fact larger than just discussed. “MX-80” is reported to contain a certain amount of soluble calcium minerals, in particular gypsum. These provide additional sources for divalent ions, which certainly will be involved in the chemical equilibration as the samples are water saturated. Reported values of gypsum content in “MX-80” are on the order of 1%. With a molar mass of 0.172 kg/mol, this contributes to the calcium content by \(2\cdot 0.01/0.172\) eq/kg \(\approx 0.12\) eq/kg, or about 16% of the cation exchange capacity.

Sample density

The samples in Mu88 that we focus on have nominal dry density of 1.2 and 1.8 g/cm³. The paper also reports measured porosities on each individual sample, listed in the below table together with corresponding values of dry density⁵

Test	\(\phi\)	\(\rho_d\)
	(-)	(g/cm3)
1.2/0.01	0.54	1.27
1.2/0.1	0.52	1.32
1.2/1.0	0.49	1.40
1.8/0.01	0.37	1.73
1.8/0.1	0.31	1.89
1.8/1.0	0.34	1.81

We note a substantial variation in measured density for samples with the same nominal density: for the 1.2 g/cm³ samples, the standard deviation is 0.06 g/cm³, and for the 1.8 g/cm³ samples it is 0.07 g/cm³. Moreover, while the mean value for the 1.8 g/cm³ samples is close to the nominal value (1.81 g/cm³), that for the 1.2 g/cm³ samples is substantially higher (1.33 g/cm³).

It is impossible to know from the information provided in Mu88 if this uncertainty is intrinsic to the procedure of preparing the samples, or if it is more related to the procedure of measuring the density at test termination.⁶

Uncertainty of external solutions

Mu88 do not describe how the external solutions were prepared. We assume here, however, that preparing pure NaCl solutions gives no significant uncertainty.

Further, the paper contains no information on how the samples were water saturated, nor on the external solution volumes. Since samples with an appreciable amount of di-valent cations are contacted with pure sodium solutions, it is unavoidable that an ion exchange process is initiated. As we don’t know any detail of the preparation process, this introduces an uncertainty of the exact aqueous chemistry during the course of a test.

To illustrate this problem, here are the results from calculating the exchange equilibrium between a sample initially containing 30% exchangeable charge in form of calcium (70% sodium), and external NaCl solutions of various concentrations and volumes

In these calculations we assume a sample of density 1.8 g/cm³ with the same volume as in Mu88 (10.6 cm³), a cation exchange capacity of 0.75 eq/kg, and a Ca/Na selectivity coefficient of 5.

In a main series, we varied the external volume between 50 and 1000 ml (solid lines). While the solution volume naturally has a significant influence on the process, it is seen that the initial calcium content essentially remain for the lowest concentration (0.01 M). In contrast, for a 1.0 M solution, a significant amount of calcium is exchanged for all the solution volumes.

The figure also shows a case for sample density 1.2 g/cm³ (dashed line), and a scenario where equilibrium has been obtained twice, with a replacement of the first solution (to a once again pure NaCl solution) (dot-dashed line).

The main lesson from these simulations is that the actual amount of di-valent ions present during a diffusion test depends on many details: the way samples were saturated, volume of external solutions, if and how often solutions were replaced, time, etc. It is therefore impossible to state the exact ion population in any of the tests in Mu88. But, guided by the simulations, it seems very probable that the tests performed at 0.01 M contain a substantial amount of di-valent ions, while those performed at 1.0 M probably resemble more pure sodium systems.

The only information on external solutions in Mu88 is that the “solution on the low concentration side was changed regularly” during the course of a test. This implies that the amount of di-valent cations may not even be constant during the tests.

Uncertainty of diffusion parameters

The diffusion parameters explicitly listed in Mu88 are \(D_e\) and “\(D_a\)”, while it is implicitly understood that they have been obtained by fitting the effective porosity model to outflux data and the measured clay concentration profile in the final state. “\(D_a\)” is thus really the pore diffusivity \(D_p\),⁷ and relates to \(D_e\) as \(D_e = \epsilon_\mathrm{eff} D_p\), where \(\epsilon_\mathrm{eff}\) is the so-called “effective porosity”. In a previous blog post, we discussed in detail how anion equilibrium concentrations can be extracted from through-diffusion tests, and the results derived there is used extensively in this section.

Rather than fitting the model to the full set of data (i.e. outflux evolution and final state concentration profile), diffusion parameters in Mu88 have been extracted in various limits.

Evaluation of \(D_e\) in Mu88

The effective diffusivity was obtained by estimating the steady-state flux, dividing by external concentration difference of the tracer, and multiplying by sample length \begin{equation} D_e = \frac{j^\mathrm{ss}\cdot L}{c^\mathrm{source}}\tag{1} \end{equation}

Here it is assumed that the target reservoir tracer concentration can be neglected (we assume this throughout). Eq. 1 is basically eq. 1 in Mu88 (and eq. 8 in the earlier blog post), from which we can evaluate the values of the steady-state flux that was used for the reported values of \(D_e\) (\(A \approx 7.1\) cm² denotes sample cross sectional area)

Test	\(D_e\)	\(A\cdot j^\mathrm{ss}/c^\mathrm{source}\)
	(\(\mathrm{m^2/s}\))	(ml/day)
1.2/0.01	\(7.7\cdot 10^{-12}\)	0.031
1.2/0.1	\(2.9\cdot 10^{-11}\)	0.118
1.2/1.0	\(1.2\cdot 10^{-10}\)	0.489
1.8/0.01	\(3.3\cdot 10^{-13}\)	0.001
1.8/0.1	\(4.8\cdot 10^{-13}\)	0.002
1.8/1.0	\(4.0\cdot 10^{-12}\)	0.016

The figure below compares the evaluated values of the steady-state flux with the flux evaluated from the measured target concentration evolution,⁸ for samples with nominal dry density 1.8 g/cm³ (no concentration data was reported for the 1.2 g/cm³ samples)

These plots clearly show that the transition to steady-state is only resolved properly for the test with highest background concentration (1.0 M). It follows that the uncertainty of the evaluated steady-state — and, consequently, of the evaluated \(D_e\) values — increases dramatically with decreasing background concentration for these samples.

Evaluation of \(D_p\) in Mu88

Pore diffusivities were obtained in two different ways. One method was to relate the steady-state flux to the clay concentration profile at the end of the test, giving \begin{equation} D_{p,c} = \frac{j^\mathrm{ss}\cdot L}{\phi\cdot\bar{c}(0)} \tag{2} \end{equation}

where \(\bar{c}(0)\) denotes the chloride clay concentration at the interface to the source reservoir. The quantity in eq. 2 is called “\(D_{ac}\)”⁷ in Mu88, and this equation is essentially the same as eq. 2 in Mu88⁹ (and eq. 10 in the previous blog post). Using the steady-state fluxes, we can back-calculate the values of \(\bar{c}(0)\) used for this evaluation of \(D_{p,c}\)

Test	\(D_{p,c}\)	\(A\cdot j^\mathrm{ss}/c^\mathrm{source}\)	\(\phi\)	\(\bar{c}(0)/c^\mathrm{source}\)
	(\(\mathrm{m^2/s}\))	(ml/day)	(-)	(-)
1.2/0.01	\(7.0\cdot 10^{-11}\)	0.031	0.54	0.204
1.2/0.1	\(2.8\cdot 10^{-10}\)	0.118	0.52	0.199
1.2/1.0	\(5.1\cdot 10^{-10}\)	0.489	0.49	0.480
1.8/0.01	\(2.0\cdot 10^{-11}\)	0.001	0.37	0.045
1.8/0.1	\(3.1\cdot 10^{-11}\)	0.002	0.31	0.050
1.8/1.0	\(5.2\cdot 10^{-11}\)	0.016	0.34	0.226

Note that, although we did some calculations to obtain them, the values for \(\bar{c}(0)/c^\mathrm{source}\) in this table are closer to the actual measured raw data (concentrations). We made the calculation above to “de-derive” these values from the reported diffusion coefficients (combining eqs. 1 and 2 shows that \(\bar{c}(0)\) is obtained from the reported parameters as \(\bar{c}(0)/c^\mathrm{source} = D_e/(\phi D_{p,c})\)).

Here are compared the measured concentration profiles for the samples of nominal density 1.8 g/cm³ and the corresponding slopes used to evaluate \(D_{p,c}\) (profiles for the 1.2 g/cm³ samples are not provided in Mu88)

For background concentrations 1.0 M and 0.1 M, the evaluated slope corresponds quite well to the raw data. For the 0.01 M sample, however, the match not very satisfactory. I suspect that a detection limit may have been reached for the analysis of the profile of this sample. Needless to say, the evaluated value of \(\bar{c}(0)\) is very uncertain for the 0.01 M sample.

It may also be noted that all measured concentration profiles deviates from linearity near the interface to the source reservoir. This is a general behavior in through-diffusion tests, which I am quite convinced of is related to sample swelling during dismantling, but there are also other suggested explanations. Here we neglect this effect and relate diffusion quantities to the linear parts of profiles, but this issue should certainly be treated in a separate discussion. Update (220407): non-linear profiles are discussed here.

\(D_p\) was also evaluated in a different way in Mu88, by measuring what we here will call the breakthrough time, \(t_\mathrm{bt}\) (Mu88 call it “time-lag”). This quantity is fairly abstract, and relates to the asymptotic behavior of the analytical expression for the outflux that apply for constant boundary concentrations (we here assume them to be \(c^\mathrm{source}\) and 0, respectively). This expression is displayed in eq. 7 in the previous blog post.

Multiplying the outflux by the sample cross sectional area \(A\) and integrating, gives the accumulated amount of diffused tracers. In the limit of long times, this quantity is, not surprisingly, linear in \(t\) \begin{equation} A\cdot j^\mathrm{ss} \cdot \left(t – \frac{L^2}{6\cdot D_p} \right ) \end{equation}

\(t_\mathrm{bt}\) is defined as the time for which this asymptotic expression is zero. Determining \(t_\mathrm{bt}\) from the measured outflux evolution consequently allows for an estimation of \(D_p\) as \begin{equation} D_{p,t} = \frac{L^2}{6t_\mathrm{bt}} \tag{3} \end{equation}

This quantity is called “\(D_{at}\)” in Mu88⁷ (eq. 3 is eq. 3 in Mu88). With another back calculation we can extract the values of \(t_\mathrm{bt}\) determined from the raw data

Test	\(D_{p,t}\)	\(t_\mathrm{bt}\)
	(\(\mathrm{m^2/s}\))	(days)
1.2/0.01	\(1.4\cdot 10^{-10}\)	3.1
1.2/0.1	\(2.0\cdot 10^{-10}\)	2.2
1.2/1.0	\(3.2\cdot 10^{-10}\)	1.4
1.8/0.01	\(5.0\cdot 10^{-11}\)	8.7
1.8/0.1	\(5.4\cdot 10^{-11}\)	8.0
1.8/1.0	\(7.7\cdot 10^{-11}\)	5.6

These evaluated breakthrough times are indicated in the flux plots above for samples of nominal dry density 1.8 g/cm³. For the 0.1 M and 0.01 M samples it is obvious that this value is very uncertain — without a certain steady-state flux it is impossible to achieve a certain breakthrough time. The breakthrough time for the 1.8/1.0 test, on the other hand, simply appears to be incorrectly evaluated: in terms of outflux vs. time, the breakthrough time should be the time where the flux has reached 62% of the steady-state value.¹⁰

As no raw data is reported for the 1.2 g/cm³ tests, the quality of the evaluated breakthrough times cannot be checked for them. It may be noted, however, that the evaluated breakthrough times are significantly shorter in this case as compared with the 1.8 g/cm³ tests. Consequently, while the sampling frequency is high enough to properly resolve the transient stage of the outflux evolution for the 1.8g/cm³ tests, it must be substantially higher in order to resolve this stage in the 1.2g/cm³ tests (I guess a rule of thumb is that sampling frequency must be at least higher than \(1/t_{bt}\)).

With the pore diffusivities evaluated from \(t_\mathrm{bt}\) we get a second estimation of \(\bar{c}(0)/c^\mathrm{source}\), using eq. 2. These values are listed in the table below and compared with the direct evaluation from the steady-state concentration profiles.

Test	\(\bar{c}(0)/c^\mathrm{source}\)	\(\bar{c}(0)/c^\mathrm{source}\)
	(breakthrough)	(profile)
1.2/0.01	0.102	0.204
1.2/0.1	0.279	0.199
1.2/1.0	0.765	0.480
1.8/0.01	0.018	0.045
1.8/0.1	0.029	0.050
1.8/1.0	0.153	0.226

In a well conducted study these estimates should be similar; \(D_{p,c}\) and \(D_{p,t}\) are, after all, estimations of the same quantity: the pore diffusivity \(D_p\).⁷ But here we note a discrepancy of approximately a factor 2 between several values of \(\bar{c}(0)\).

It is difficult to judge generally which of the estimations are more accurate, but we have seen that for the 1.8/0.1 and 1.8/0.01 tests, the flux data is not very well resolved, giving a corresponding uncertainty on the equilibrium concentration estimated from the breakthrough time. On the other hand, also the concentration profile is poorly resolved in the case of 0.01 M at 1.8 g/cm³.

However, in cases where the value of \(\bar{c}(0)/c^\mathrm{source}\) is substantial (as for the 1.8/1.0 test and, reasonably, for all tests at 1.2 g/cm³), we expect the estimation directly from the concentration profile to be accurate and robust (as for the 1.8 g/cm³ test at high NaCl concentration). For the 1.2 g/cm³ samples we cannot say much more than this, since Mu88 don’t provide the concentration raw data. For the 1.8/1.0 test, however, we can continue the analysis by fitting the model to all available data.

Re-evaluation by fitting to the full data set

Note that all evaluations in Mu88 are based on making an initial estimation of the steady-state flux, giving \(D_e\) (eq. 1). This value of \(D_e\) (or \(j^{ss}\)) is thereafter fixed in the subsequent estimation of \(D_{p,c}\) (eq. 2). Likewise, an estimation of the steady-state flux is required for estimating the breakthrough time. Here is an animation showing the variation of the model when transitioning from the value of the pore diffusivity estimated from breakthrough time (\(7.7\cdot 10^{-11}\) m²/s), to the value estimated from concentration profile (\(5.2\cdot 10^{-11}\) m²/s) for the 1.8/1.0 test, keeping the steady-state flux fixed at the initial estimation

Note that the axes for the flux is on top (time) and to the right (accumulation rate). This animation confirms that the diffusivity evaluated from breakthrough time in Mu88 gives a way too fast process: the slope of the steady-state concentration profile is too small, and the outflux evolution has a too short transient stage. On the other hand, using the diffusivity estimated from the concentration profiles still doesn’t give a flux that fit very well. The problem is that this fitting is performed with a fixed value of the steady-state flux. By instead keeping the slope fixed at the experimental values, while varying diffusivity (and thus steady state flux), we get the following variation

This animation shows that the model can be fitted well to all data (at least for the 1.8/1.0 test). The problem with the evaluation in Mu88 is that it assumes the steady-state to be fully reached at the later stages of the test. As the above fitting procedure shows, this is only barely true. The experiments could thus have been designed better by conducting them longer, in order to better sample the steady-state phase (and the steady-state flux should have been fitted to the entire data set). Nevertheless, for this sample, the steady-state flux obtained by allowing for this parameter to vary is only slightly different from that used in Mu88 (17.5 rather than 16.3 \(\mathrm{\mu}\)l/day, corresponding to a change of \(D_p\) from \(5.2\cdot10^{-11}\) to \(5.6\cdot10^{-11}\) m²/s). Moreover, this consideration should not be a problem for the 1.2 g/cm³ tests, if they were conducted for as long time as the 1.8 g/cm³ tests, because steady-state is reached much faster (in those tests, sampling frequency may instead be a problem, as discussed above).

As we were able to fit the full model to all data, we conclude that the value of \(\bar{c}(0)/c^\mathrm{source}\) obtained from \(D_{p,c}\) is probably the more robust estimation¹¹, and that there appears to be a problem with how the breakthrough times have been determined. For the 1.8 g/cm³ samples we have demonstrated that this is the case, for the 1.2 g/cm³ we can only make an educated guess that this is the case.

Summary and verdict

We have seen that the results on chloride diffusion in Mu88 suffer from uncertainty from several sources:

• The “MX-80” material is not that well defined
• Densities vary substantially for samples at the same nominal density
• Without knowledge of e.g water saturation procedures and solution volumes, it is impossible to estimate the proper ion population during the course of a test
• It is, however, highly likely that tests performed at low NaCl concentrations contain substantial amounts of di-valent ions, while those at high NaCl concentration are closer to being pure sodium systems.
• The reported diffusivities give a corresponding uncertainty in the chloride equilibrium concentrations of about a factor of 2. While some tests essentially have a too high noise level to give certain estimations, the problem for the others seems to stem from the estimation of breakthrough times.

Here is an attempt to encapsulate the above information in an updated plot for the chloride equilibrium data in Mu88

The colored squares represent “confidence areas” based on the variation within each nominal density (horizontally), and on the variation of \(\bar{c}(0)/c^\mathrm{source}\) from the two reported values on pore diffusivity⁷ (vertically). The limits of these rectangles are simply the 95% confidence interval, based on these variations, and assuming a normal distribution.

Data points put within parentheses are estimations judged to be improper (based on either re-evaluation of the raw data, or informed guesses).

From the present analysis my decision is to not use the data from Mu88 to e.g. validate models for anion exclusion. Although there seems to be nothing fundamentally wrong with how these test were conducted, they suffer from so many uncertainties of various sources that I judge the data to not contribute to quantitative process understanding.

• Chloride content: UNKNOWN
• Extracting anion equilibrium concentrations from through-diffusion tests
  

Footnotes

[1] This work is referred to as “Muurinen et al. (1989)” by several authors.

[2] MX-80 is not only a brand name, but also a band name.

[3] This report is “Bentonite Mineralogy” by L. Carlson (Posiva WR 2004-02), but it appears to not be included in the INIS database. It can, however, be found with some elementary web searching.

[4] It’s interesting to note that the cation exchange capacity of “MX-80” remains more or less constant, while the montmorillonite content has some variation. This implies that the montmorillonite layer charge varies (and is negatively correlated with montmorillonite content). Could it be that the manufacturer has a specified cation exchange capacity as requirement for this product?

[5] To convert porosity to dry density, I used \(\rho_d = \rho_s\cdot(1-\phi)\), with solid grain density \(\rho_s = 2.75\) g/cm³.

[6] A speculation is that the uncertainty stems from the measurement procedure, as this was done on smaller sections of the full samples. It is not specified in Mu88 what the reported porosity represent, but it is reasonable to assume that it is the average of all sections of a sample.

[7] At the risk of losing some clarity, I refuse to use the term “apparent diffusivity” for something which actually is a real pore diffusivity.

[8] These values were not tabulated, but I have read them off from the graphs in Mu88.

[9] Mu88 use the concentration based on the total volume in their expression, while \(\bar{c}\) is defined in terms of water volume (water mass, strictly). Eq.2 therefore contains the physical porosity. In their concentration profile plots, however, Mu88 use \(\bar{c}\) as variable (called \(c_{pw}\) — the “concentration in the pore water”)

[10] Plugging the breakthrough time \(L^2/6D_p\) into the expression for the flux gives

\begin{equation} j^\mathrm{out}(t_\mathrm{bt}) = j^{ss}\cdot\left ( 1 + 2 \sum (-1)^n e^\frac{-\pi^2 n^2}{6} \right ) \approx 0.616725\cdot j^{ss} \end{equation}

I find it amusing that this value is close to the reciprocal golden ratio (0.618033…). Finding the breakthrough time from a flux vs. time plot thus corresponds (approximately) to splitting the y-axis according to the golden ratio.

[11] Note that the actual evaluated values of $D_{p,c}$ in Mu88 still may be uncertain, because they also depend on the values of the steady-state flux, which we have seen were not optimally evaluated.