Pore network modeling
The basic physical principle used in pore network modeling (PNM) is conservation (Bern-abe 1995). This means that there are no sinks or sources inside the pore network. Meshes of pores are built with a particular geometry (e.g., rectangular, hexagonal, triangular) and are connected through a node. The size of the pore needs to be explicit (i.e. radius of the pore). The goal of PNM is to determine the response of the whole network subjected to an exter-nal gradient (for example a voltage) given a particular condition of each pore (for example particular radius).
Maineult et al. (2017) and corrected by Maineult (2018) propose a PNM to obtain the SIP response of randomly sized tubes (see figure 1.7). They randomly distribute the radius size to each pore, given a pore size distribution. They obtain the electrical response of the system using the expression: r 2 ¿ ˘ 2D . (1.39).
In order to measure the polarization of the electrodes (or lack thereof ) an SIP measurement is done. To do so, I fill a recipient with very saline water (around 36.8 mS cm¡1), I locate two current injecting electrodes made of stainless steel at opposing sides of the recipient, with the two potential electrodes to be tested in the middle (all located in a straight line, see figure 2.2). All four electrodes barely touch the water (see figure 2.2). This is a way to test the electrode polarization as very saline water should ideally have a very small phase. Knowing that phase can be represented as: ’ ˘ ¡ar ct an µ¾000 ¶, (2.1) ¾.
where in very saline water ¾0 ¨¨ ¾00, therefore the phase of very saline water should be close to zero. If the measured phase is not close to zero, that means that the measured polarization does not come from the very saline water but from the electrodes themselves. It is worth mentioning that it is normal to have a small polarization signal at high frequencies (>100 Hz), coming from impedance effects of the electrodes (see Huisman et al. 2016; Wang and Slater 2019). The idea is to build electrodes with the least amount of internal polarization possible. All the electrodes batches used in this study were tested in this same manner.
Path to finding a good electrode design Type and amount of gelifying agent
A problem with electrodes of the first kind (including Cu-CuSO4 electrodes) is that the cham-ber holding the aqueous solution (CuSO4 in this case) may leak out of the chamber through the porous plug by diffusion. Kremer (2015) did an extensive study on the type of porous caps that can be used to plug the electrodes. Additionally, Kremer (2015) and Kremer et al. (2016) present a test using a gelifying agent to decrease the leakage of the solution (i.e. re-duce the ionic mobility). I based the electrode construction off of their work. However, I decided to do further tests. I tested the concentration of CuSO4, and type (and amount) of gelifying agent.
On table 2.1 I present a synthesis on the tests performed to obtain a suitable pair of non-polarizable Cu-CuSO4 electrodes. Note, that all solutions were made from an amount of penta-dehydrated CuSO4, 100 ml of de-ionized water, and an amount of a gelifying agent. For the first test, I was advised by Feras Abdulsamad to use 0.9 g of agar-agar. As to the amount of penta-dehydrated CuSO4, I was advised to use an amount that would saturate the solution. The amount of CuSO4 that saturates a solution varies with temperature, since these solutions are heated up to 100 –C, I decided to test the amount of CuSO4 based of the temperature range I would heat the solution to. With the first test, I was able to narrow down the amount of penta-dehydrated CuSO4, as most electrodes showed significant signs of precipitation (negative results, marked as « – » in table 2.1). I decided that visible signs of early precipitation in the electrodes meant the lack of chemical equilibrium in the solution. Constant chemical reactions within the solution itself could potentially affect the measured SIP signal.
For the second test, I decided to slightly increase the amount of agar-agar, as the first batch of electrodes did not all gelatinize. I decided also to narrow down the search on the proper amount of penta-dehydrated CuSO4 needed for the electrode construction. The pair of 32 g of penta-dehydrated CuSO4 still presented precipitation. To test these electrodes, I mea-sured the SIP signal of the 23, 26 and 29 g of CuSO4 electrodes in a recipient filled with saline water. For an improved readability of this text, I will not present the SIP measurements of all the electrode tests, just the final and most relevant tests. I singled out the pair of electrodes using 23 g of CuSO4 because they had the best SIP signal. For the third test, I decided to vary the amount and type of gelifying agent. None of these electrodes precipitated.
To decide which pair of electrodes were optimal, I measured their SIP response (see figure 2.3), at different times after the making (the day after making the electrodes, a month later, and two months after construction). It is worth mentioning that in figure 2.3c and f, the SIP signal of the agar-agar electrodes is not presented, because the electrodes had visibly- Table 2.1: Chronological tests to obtain a suitable pair of non-polarizable Cu-CuSO4 electrodes. All these tests were done in 100 ml of de-ionized water. Note that by + and – in this table, I mean positive or negative results. By positive, I mean that the electrodes did not show any visible sign of precipita-tion, by negative I mean that they did show visible signs of precipitation.
Brass electrodes testing
Building Cu-CuSO4 electrodes is a tedious task that needs to be repeated every other month or so. This is why some researchers such as Huisman et al. (2016) have proposed to use brass electrodes (a brass rod) in contact with fluid from the same salinity as the one saturating the rock sample (see figure 2.7b and c).
The brass rod is retracted in a tube (figure 2.7b), so the streamlines of the SIP measurements will not pass through the metal, as the metal is connected to the sample through the liquid of the pore solution but is not inserted in the sample. This is ideally supposed to work, as what would create polarization is the contact between a metal and the rock sample. If the current lines are not in contact with the metal, no polarization should occur.
I purchased a few brass rods with a 5 mm diameter, with standards BS2874/CZ121M (1986)BS EN 12164/CW614N (from the specification sheet of the manufacturer). I retracted the elec-trodes by 1 cm from the end of the tube (see figure 2.7b) and filled this space with the same salted water as in the recipient (figure 2.7). I measured the SIP signal with these electrodes (see figure 2.8, this dataset corresponds to the date 25-06-19). I also attempted sanding the rods (in case the brass rods had some coating, see figure 2.7a), re-filled the recipient with new very saline water and re-measured the SIP signal (see figure 2.8, this dataset corresponds to the date 26-06-19). The SIP signals I measured are presented in figure 2.8. The errorbars of these measurements reached 1500 mrad, and the lowest measured phases were around 400 mrad; significantly higher than the SIP signals I had measured in the clay samples. Some-thing was obviously very wrong with my set-up. Many researchers (e.g., Huisman et al. 2016; Izumoto et al. 2020) have been able to use the brass electrodes without problems. I am not attempting to discourage the reader on the use of these electrodes, I am merely presenting a way that does not work so the reader does not follow this path. I am not sure why this at-tempt did not work. Although, brass is an alloy of metals, I wonder if a specific type of alloy is needed in order to get good measurements. I did not further investigate these types of electrodes for SIP, for time purposes. In the future, I would like to contact these researchers for more details on the use of their electrodes.
Another electrode-related improvement I attempted was the SIP signal electrode correction. Huisman et al. (2016) proposed a way to correct SIP measurements in a rock sample. They measure the SIP signal in the traditional configuration with 4 electrodes, then they measure the reciprocal. They use both measurements to determine the electrode impedance and substract it from the measured SIP signal. Later Wang and Slater (2019) proposed another correction in which the SIP signal is measured in 4 different configurations, then the metal of the electrode (copper wire for Cu-CuSO4 electrodes) is inserted deeper gradually. At each metal insertion step, the 4 SIP configurations are measured. All of these measurements are stored and then plotted all together in a phase vs electrode impedance plot for each fre-quency. Allowing to infer the phase for zero electrode impedance. In the results shown by Wang and Slater (2019) they were able to remove the high frequency noise of the measure-ments. I attempted to do this, however, after all the calculations I obtained a noisier signal than the original one I measured (see figure 2.9). I therefore did not remove the electrode error from the SIP measurements in this study. Again, this does not mean that the protocol proposed by Wang and Slater (2019) does not work. Here, I have some suggestions at ele-ments that have to be taken into account for this test. Elements I had a hard time with in the measuring part of this test.
Comments on sample holder construction
One of the goals for this thesis was to create clay heterogeneities. There was a need for a sample holder where clay heterogeneities could be located without major disruption to the sample (such as inserting air bubbles). That is, a sample holder that could be opened longi-tudinally. The best solution I found for this problem was a « sushi bazooka »; a plastic device created to locate sushi ingredients inside, close it giving the sushi ingredients the cylindri-cal shape, and then pushing it out. I transformed these devices into the sample holders by slightly modifying them. The dimensions of the sample holders are presented in table 2.7 and the complementary figure 2.11 showing the dimensions of the sushi bazooka.
I used a pair of stainless steel cylinders as injecting electrodes. There was a need to create a structure to hold the injecting electrodes in a consistent manner. The first attempt of exter-nal structure involved two parallel plastic rectangles, held together by four plastic threaded rods (one for each corner of the rectangle) and the sample holder (i.e. « sushi bazooka ») lo-cated in the middle. A problem arose because it was not possible to locate the sample holder in the same exact location repeatedly. For this, I created an external structure with acrylic sheets cut by a laser cutter (see figure 2.12a). This allowed the creation of acrylic sheets of the same dimensions with more than millimeter precision. I also cut (with the laser cutter) four holes for the four threaded rods, and a smaller hole located exactly in the middle (to lo-cate a small screw as a guide, see figure 2.12b). The screw guide allows for the sample holder to be located in the same exact position repeatedly. I think that the use of new available technologies for the general public improved the cre-ation of the sample holder (as the laser cutter). I think the SIP-laboratory community should further explore these new available technologies, such as laser cutters and 3D printers, and incorporate them to their laboratory equipment.
Table of contents :
List of Figures
List of Tables
1 Theoretical background
1.1 Background on clay structures
1.2 Background on geo-electrical methods
1.2.1 Maxwell’s laws for induced polarization
1.2.2 Active electrical methods in geophysics
1.3 Background on SIP
1.3.1 The electrical double layer
1.3.3 SIP models
188.8.131.52 Phenomenological models
184.108.40.206 Physical models
1.4 Upscaling techniques
1.4.1 Differential effective medium
1.4.2 Pore networkmodeling
2 Materials and methods
2.1 Non-polarizable electrodes
2.1.1 Electrode testing
2.1.2 Path to finding a good electrode design
2.1.3 Brass electrodes testing
2.1.4 Electrode correction
2.2 Tests for water content
2.3 Sample holder
2.3.1 Comments on sample holder construction
2.3.2 Geometrical factor of sample holder
2.4 Compression and decompression tests
2.5 Water chemistry
2.6 Conclusion of this chapter
3 SIP on individual types of clays
3.1 JGR article in the context of this thesis
3.2 JGR article
3.3 Main results of JGR article
3.4 Supplementary information of JGR article
3.4.1 SIP measurements on additional samples
3.4.2 Differentiation of clay minerals
3.4.3 Repeatability test
3.4.4 Relationship between the imaginary conductivity at a frequency of 1.46 Hz and surface area per unit pore volume
3.5 Comment on pore water equilibrium
4 SIP on heterogeneous clays
4.1 Introduction tomanuscript
4.3 Main results of manuscript
4.4 Complements tomanuscript
4.4.1 Mesh types on complex conductance networks
5.1 Varying pH on individual clay samples
5.2 Clay heterogeneity mixtures
5.3 Clay compaction
5.4.1 Numerical models in SIP
5.4.2 Numerical models of clays
5.5 Various recommendations
Calculation of errorbars for the complex conductivity
Article Jougnot et al. (2019)