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Dissolution mass transfer in porous media phenomena in hydrogeology (modified from Patmonoaji et al. [
In groundwater contamination by NAPL, NAPL could leak from industrial plant and contaminate the groundwater stream [1, 2, 3]. When the NAPL enters the groundwater, it displaces the existing groundwater, and then it was trapped due to capillary trapping. As it is trapped, mass transfer from the trapped NAPL to the groundwater flow occurs, transferring the dissolved NAPL to the groundwater. As a result, groundwater is contaminated, and the period of contamination depends on the mass transfer rate of trapped NAPL to the groundwater. Low mass transfer rate results in a prolong contamination, and vice versa.
In-situ bioremediation is one of the decontamination method of NAPL contamination [4, 5]. This method harnesses the ability of native bacteria in the groundwater to biodegrade the contaminants. To enhance the biodegradation process, O2, CH4, NH3, CO3 and N2 gases are injected at the upstream of the contamination area to supply nutrients and stimulate the bacteria. After the injection, these gases are trapped due to capillary trapping and thus mass transfer occurs. The mass transfer rate is important because it affects the efficiency of the gas transport. Higher mass transfer rate means faster delivery but faster depletion of trapped gases. Therefore, it should be controlled to match the bacteria needs.
In GCS, the captured CO2 from the point source, e.g., coal power plant, steel factory, and cement factory, is injected into deep saline aquifers for sequestration. After the injection, the CO2 is trapped in the groundwater due to capillary trapping [6, 8]. Although the CO2 can be sequestered this way, the reservoir pressure increases due to the injection. However, by the time, the trapped CO2 will dissolve, reducing the reservoir pressure. As a result, the mass transfer rate is critical to approximate the pressure reduction rate for safer CO2 sequestration process.
Given the scale of the phenomena is massive, the observation scale can be categorized into pore-scale, core-scale, and field-scale (Figure 1). Pore-scale is in the order of μm, and core-scale is in the order of cm. Field-scale, however, is in the order of m to km. Each of the scale is used to explain the scales above it. Pore-scale and core-scale are performed in a laboratory, whereas field-scale is performed in the field. Only pore-scale and core-scale will be discussed in this chapter.
Given the opaqueness of porous media system, direct observation of dissolution process is difficult to be performed. To evaluate the mass transfer process in detail, the interfacial area (
Earlier studies mainly relied only on core-scale studies by the measurement of dissolved phase concentration from the effluent and amount of trapped phase through gravimetrical measurement without knowing the trapped phase characteristics inside [1, 2, 3, 4, 5]. However, because the trapped phase characteristics are required to measure the
In this chapter, dissolution mass transfer in porous media is discussed based on experimental observation on the dissolution mass transfer of various trapped gas into flowing water in porous media by using micro-tomography and image processing [7, 9, 10, 11, 12]. By using this methods,
The contents of this chapter are described as follow. In Section 2, the physical and mathematical model are given. In Section 3, the general description about the micro-tomography and image processing methods are described. In Section 4, various parameters affecting this process, i.e., porous media characteristics, capillary trapping characteristics, flow bypassing, TP properties, and FP velocity, are discussed. All of these condition will be investigated based on the point of view of three parameters affecting dissolution mass transfer:
At first, we can start from the basic by describing mass transfer. Mass transfer is a process of a movement of mass from one medium to another medium. The examples can be found easily in the surrounding of our daily lives. For example, the mass transfer of camphor in a bathroom into the surrounding air to improve the smells of the room (Figure 2a). Another example is the mass transfer of oxygen from the air into a lake (Figure 2b). Without the supply of oxygen through mass transfer, the fishes living in the lake cannot breathe the oxygen and stay alive.
(a) Mass transfer of camphor to surrounding air, (b) mass transfer of oxygen from air to water in lake, and (c) illustration of mass transfer process.
Similar with other transport phenomena process, i.e., heat transfer and momentum transfer, mass transfer occurs because of non-equilibrium condition [13, 14, 15]. In case of mass transfer, equilibrium can be achieved once the system possesses equal chemical potential (
with subscripts 1 and 2 represent the phases. This equation can also be described with partition coefficient or Henry’s coefficient (Figure 2c) as follow:
with
The concept of mass transfer nowadays is mainly based on the works of Thomas Graham (1805–1869) and Adolf Fick (1829–1901). Thomas Graham was the pioneer in the study of mass transfer through his works in the diffusion of H2 gas in a tube [16]. By using analogies with Fourier’s work [17] in heat conduction, Adolf Fick formulated the based equation for mass transfer that is the Fick’s law of diffusion mass transfer [18] defined as follow:
with
with
In case of dissolution mass transfer, an additional process present. Dissolution here refers to the decrease in volume due to mass transfer. As a result, the Eq. (4) can be modified to volume perspective as follow:
with
There are several mechanisms models explaining mass transfer from an interface into a bulk solution [14]. Due to the analogy between mass transfer and heat transfer, similar models with heat transfer are also used. However, because most of heat transfer application is in fluid–solid system, whereas mass transfer applications occur in fluid–fluid system, not all model can be used for both.
The simplest physical model is the stagnant film theory (Figure 3a) [19] that assumes the existence of a stagnant film near the interface of mass transfer process. This stagnant film is also referred as an unstirred layer because no fluid motion exists in this film. However, this film is hypothetical because fluid motions still occur beyond the no-slip condition at interface. The effects of velocity and other parameters are then represented by the thickness of this stagnant film. Higher mass transfer rate results in thinner δ, and vice versa.
(a) Illustration of stagnant film model, (b) penetration theory model at low and high velocity, and (c) surface renewal theory model.
Second mechanisms model is penetration theory (Figure 3b) [20]. The key assumption is that a thick moving film is continuously generated in the bulk fluid, which the mass transfer occurs by diffusion across this film. The effect of velocity to the mass transfer is represented by the concentration distribution. As the solute diffuse into the bulk liquid, it will be flushed away by the flow. Higher velocity results in faster removal of the solute. As a result, higher velocity results in steeper concentration distribution, leading to higher mass transfer rate, and vice versa.
Third mechanisms model is surface renewal theory (Figure 3c) [21], which is the further development of penetration theory. The bulk liquid is depicted into two regions, a region near the interface that is renewed quickly and a region of well-mixed bulk fluid. The renewed region is renewed by the flow and similar with penetration theory. However, it is constantly exchanged with new elements from a second bulk region, making this theory closer to the realistic physical situation.
Among these models, however, no theory is completely satisfactory. Nevertheless, these models provide valuable insight to the mechanisms in mass transfer. For more detail, readers are advised to check the articles in the references [19, 20, 21].
For more empirical modeling of the mass transfer process, an empirical equation based on dimensionless number is usually used. The experimental data of mass transfer is represented by power-law relation of dimensionless number as follow:
with
with
In case of porous media, the physical mechanism of mass transfer is actually similar. In an open space, when dissolution occurs, the solute can be flushed away easily. In porous media, on the other hand, these processes are governed by the solute transport in porous media. In addition, in the application of hydrogeology, the TP clusters are distributed throughout the porous media (Figure 4), resulting in continuous mass transfer process along the porous media. As a result, a special attention to the solute transport throughout the porous media is required before assessing the mass transfer process with Eq. (5).
Illustration of dissolution mass transfer in porous media and the predicted
The fluid flow in hydrogeology application is relatively slow and can be classified as Stokes flow or creeping flow, which the ratio between fluid velocity and viscosities is relatively low. This condition is represented by
with
with
For the solute transport in porous media, it is affected by the tortuosity and velocity profile in the pore networks. Nevertheless, under Darcy regime, it can be predicted by using advection-dispersion equation [26] as follow:
with,
To evaluate the dissolution mass transfer in porous media, combination of modified version of Eqs. (5) and (11) are used [1, 2, 3, 4, 5, 9]. To calculate the mass transfer coefficient in porous media, Eq. (5) is slightly modified as follow:
with
with
with
By discretizing
The average of concentration in between
This
with
For the modeling of mass transfer,
As mentioned in Section 1, this chapter is based on the experimental observation by using micro-tomography and image processing [7, 9, 10, 11, 12]. For the micro-tomography, an X-ray microtomography scanner (micro-CT) was used. This micro-CT is able to generate 3D images in the form of 992 cross section slice image with the dimension of 992 × 992 voxels with the voxel size of 16.427 μm. As a result, observation area with the size of 16.34 × 16.34 × 16.34 mm3 was generated.
To generate the porous media, cylindrical container with an inner diameter of 10 mm was used as the casing of porous media. Inside this container, various particles will be packed inside to generate various porous media. All of the packing was performed under vibration to generate closed random packing classification [28]. As a result, the quality of the packing can be controlled. Additionally, sintered glass discs were placed at inlet and outlet to generate uniform flow.
To generate capillary trapping and similar condition with hydrogeology conditions in Section 1, the porous media will be sequentially saturated with degassed deionized (DDI) water and the gas. First, the gas was injected into the porous medium to saturate the porous media and remove all of other gases. Second, DDI water was injected upward to saturate the porous media, generating a condition similar with groundwater. Third, the experimental gas was injected again to remove most of the DDI water, resulting in similar condition with gas injection into groundwater. Finally, the DDI water was injected upward at the desired experimental flow rate. With this method, the generated trapped gas is uniform along the sample. Once the DDI water had breakthrough, and no gas flow was observed in the outlet tube, creating a quasi-steady state, the experiment began, and the dissolution process was monitored with micro-CT. The schematic about the experimental setup is given in Figure 5.
Schematics of experimental setup with micro-tomography (modified from Patmonoaji et al. [
For the data processing, image processing techniques were carried out to quantify the data from the micro-CT image. The main method is to differentiate the gas from the surroundings by performing image binarization (Figure 6). Although the contrast between water and particles is indistinguishable, the contrast of the gas among other phases is easily distinguishable. Therefore, a direct thresholding method was enough to separate gas from other phases. With this binarized images, then quantitative data, i.e., amount of trapped gas, and surface area of the trapped gas, can be simply derived with image processing to assess the dissolution mass transfer. Furthermore, by identifying separated trapped gas clusters, each of the trapped gas clusters can be characterized based on its volume and surface area. In addition, by performing similar method to porous media saturated with gas, porosity of the porous media can be obtained. Moreover, by applying watershed (WS) segmentation method [11], the throats and pores of the porous media can be identified and measured.
Binarization image processing methods: (a) cross sectional image of the porous media, (b) histogram of the gray value, (c) binarized image with black as the gas, and (d) 3D binarized image of gas (modified from Patmonoaji et al. [
A porous medium can be described as a solid containing void networks inside it. Structure-wise, the void network can be seen as a series of pores and throats (Figure 7a) [28, 29]. Pores are connected by a throat, which is a constriction in the network. Because capillary trapping mainly occupies pores, pore characteristics affect the morphology of the capillary trapping that eventually controls the amount
(a) Illustration of pore and throat in pore network and (b) effect of porous media characteristics to TP morphology.
These pore and throat in porous media are mainly governed by the shape, size, and distribution of the granular particles forming the porous media. Shown in Table 1 are the properties of nine granular particles used to generate porous media [10, 11]. Those properties
Type | Size range (μm) | ||||
---|---|---|---|---|---|
PB | 125–300 | 184 | 1.5 | 0.9 | 0.43 |
125–710 | 212 | 1.8 | 0.8 | 0.41 | |
180–212 | 196 | 1.1 | 1.0 | 0.44 | |
355–425 | 390 | 1.1 | 1.0 | 0.45 | |
600–710 | 655 | 1.1 | 1.0 | 0.44 | |
GB | 180–212 | 196 | 1.1 | 1.0 | 0.33 |
355–425 | 390 | 1.1 | 1.0 | 0.34 | |
600–710 | 655 | 1.1 | 1.0 | 0.35 | |
TS | 180–212 | 196 | 1.1 | 1.0 | 0.32 |
Properties of nine porous media generated from different shape, size, and distribution of granular particles.
Figure 8a shows the pore size distribution (PSD) of the porous media generated from those particles. The PSD is shown as probability density function (PDF) of pore equivalent diameter (
(a) PSD and (b)
Proportionality of the PSD is also found by normalizing PSD with the
The pore shape can also be evaluated through its surface area (
Capillary, buoyancy, and viscous forces play important roles in the formation of capillary trapping (Figure 10a). These forces can be represented with capillary number (
(a) Illustration of forces exerting on TP in porous media and morphology of (b) trapped NWP and (c) WP in porous media.
Wettability of the porous media also affects the TP clusters morphology (Figure 10b and c). In case the TP is non-wetting phase (NWP), the TP is located in the center of the pore. On the other hand, in case the TP is wetting phase (WP), the TP tends to surround the solid. In this work, we mainly focus on the former condition.
The trapped phase size distribution (TPSD) in the porous media described in Table 1 is given in Figure 11 as PDF of TP equivalent diameter (
(a) TPSD and (b)
The TP cluster morphology can also be characterized through its volume (
The average
For the
(a)
For the linearity between
with
As described in this section, the
In general, flow bypassing can be divided into two categories. First is bypassing due to heterogeneity of porous media [32, 33], and second is bypassing induced by dissolution fingering in homogeneous porous media [34, 35, 36]. Both of them direct water flow to preferential flow path, resulting in non-uniform dissolution mass transfer. As a result, it will affect the
Size range (μm) | 125–150 | 180–212 | 355–425 | 600–710 | 125–300 | 250–425 | 125–710 |
---|---|---|---|---|---|---|---|
low | low | high | high | low | high | high | |
4.806 | 6.308 | 12.935 | 21.68 | 5.745 | 12.001 | 7.582 |
Characteristics of particle, porous medium, flow, and mass transfer inside each porous medium sample.
In general, the mechanism of dissolution fingering development can be explained as follow. Initially, capillary trapping is relatively homogeneous, and thus the permeability tends to be homogeneous as well. However, a slight difference in permeability remains present. This slight difference generates preferential water flow, directing the water to flow to region with higher permeability. At this higher permeability region, water flow slightly faster, and thus the TP clusters dissolve slightly faster as well. As a consequence, it gradually increases the permeability of this slightly permeable region, enhancing the permeability differences gradually. This results in a positive feedback, and thus the region of this preferential path grows with the dissolution time, resulting in the development of dissolution finger.
In short, the possibility of dissolution fingering development can be evaluated by measuring the approximated width of dissolution finger (
with
The function of
The dissolution process inside PB 600–710 μm and PB 180–212 μm are given in Figure 15a as 3D images. In PB 600–710 μm, dissolution process occurs uniformly, whereas in PB 180–212 μm, dissolution occurs non-uniformly. The
(a) Dissolution progress of (top) particle 600-710 μm and (bottom) particle 180-212 μm and (b)
As explained in this section, in the presence of bypassing induced by dissolution fingering,
Another parameter that affect the dissolution mass transfer rate is
Dissolution in porous media is not only mass transfer process, but also solute transport. When mass transfer occurs from TP, the dissolve solute need to be transported away from the surrounding TP by the FP. Therefore, high value in
To compare the effect
The axial average TP saturation throughout the dissolution process is given in Figure 16. The dissolution of N2, and CO2 gases behave differently. In case of N2 gas, as the time proceeds,
By observing the dissolution process from the 3D images of segmented TP clusters as shown in Figure 17, the different dissolution behavior can be observed in more detail. In case of N2, TP clusters dissolve completely from multi-pores size. However, in case of CO2, at first, the multi-pores TP clusters dissolve to single-pore TP clusters. Afterwards, second stage occurs, dissolving the remaining clusters.
TP cluster of N2 and CO2 observed at different dissolution time. Different colors indicate different clusters (modified from Patmonoaji et al. [
For the
Average
Compared with other gases, CO2 exhibit about fifty times larger. When TP cluster dissolves, the solute should be transported away by fresh incoming FP. If the solute is not transported, the solute concentration surrounding the TP cluster accumulates, slowing down the mass transfer. During the dissolution of CO2, this rapid dissolution is believed to generate area with high CO2 solute concentration due to much faster mass transfer rate than advection rate. In addition, as the trapped gas dissolves rapidly, it quickly retracts to the center of pores, increasing the difficulty of the solute to be removed by FP. As the dissolution occurs rapidly, the incoming FP cannot compete with the rapid dissolution of TP. As a result, two dissolution stages occur. First stage occurs before the accumulation of solute, whereas the second stage occurs after the accumulated solute is advected away from the surrounding of TP.
As discussed in this section, although higher
To investigate the effect of FP velocity, the experiments were also performed under different FP velocity (
Comparison of predicted-
The Eq. (40) is for N2 gas, whereas Eq. (41) is for CO2 gas. The difference is due to the effect of high
In this section, the effect of FP velocity was shown. With the increase in FP velocity,
Throughout this work, the physical mechanism and mathematical model of dissolution mass transfer of trapped phase in porous media was described. Various condition affecting it were also discussed from experiment data obtained from micro-tomography and a series of image processing techniques. The effects of these conditions were represented in three parameters:
In case of porous media characteristics, the characteristics of porous media generated from various particle shape, size, and distribution were investigated. Smaller
Bypassing induced by dissolution fingering was also described. When the predicted
In addition, the effect of
Eventually, the effect of FP velocity based on
This work was supported by JSPS KAKENHI with grant numbers 17H00790 and 20 J14975.
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