Trapping of Bubbles in Oil Sands Tailing Ponds

Abstract

Oil sands tailings ponds are significant emitters of greenhouse gases (GHGs) in Canada. To move beyond making surface or atmospheric measurements of GHG release, it is necessary to understand the physical mechanisms by which gas is generated, bubbles form and then are either released or remain trapped in the pond. We present a review of the physical description of tailings ponds, relevant to gas release models. In particular, we target rheological variations within a pond and how these directly affect the distribution of trapped gas bubbles with depth. Within the limits of the available data, we show how gas content may vary significantly across ponds, and develop data-driven one-dimensional models of gas distribution and rheology.

1. Introduction

The oil sands in Northeastern Alberta represent a significant energy resource for Canada, and globally. Extraction has been underway since the 1970s, strongly influenced by oil price fluctuations. Surface mining techniques are far cheaper than in situ extraction, but result in large volumes of tailings, i.e., waste by-products. Oil sand companies operate under the policy of zero discharge in the Alberta Environment Act, meaning that all tailings should be stored on site [1,2,3]. This has resulted in more than 170 km² of tailing ponds (settling basins), storing in excess of $1300\times {10}^6$ m³ tailings as of 2021 [4,5,6]. There are many different aspects to the management of tailings ponds. Here, our review is targeted at the production and release of greenhouse gases (GHGs) from these ponds.

Gas production was first noted in the 1980s. The following are some of the concerns, which also motivate our study. (i) As gas bubbles rise from lower in the ponds, they may entrain materials which then enter into the water cap on the lake. This results in increased turbidity and delays the already slow settling process. Lawrence and co-authors have studied lake turbidity, potential mixing methods and entrainment processes [7,8,9,10]. (ii) Below the water cap, the pond liquids and residual solids form clay-like colloidal suspensions. By virtue of their yield stress, bubbles below a critical size can be trapped within these layers. Equally, gas (CO₂ primarily) may remain dissolved in the liquids. Both features are vulnerable to sudden atmospheric pressure changes and have unpredictable release dynamics. For example, Daneshi and Frigaard [11] have shown that certain bubble cloud configurations with modest gas content may be vulnerable to burst-type release, whereas others show modulated release. These states have relevance for tailings pond management strategies. (iii) GHG impacts on global warming and an increased regulatory interest in emissions control. While the spotlight is mainly on CH₄ emissions and their CO₂ equivalents, recent studies suggest that significant amounts of unmonitored organic gases are released, including concerning intermediate-volatility and semi-volatile organic compounds [12].

Our paper has two objectives. First, we provide a targeted review of the literature concerning the structure of oil sands tailings ponds. The focus is on features that are relevant to a hydrodynamic and rheological description of bubble trapping and release. Secondly, we develop a minimalist modelling framework within which we are able to incorporate depthwise variations in pond variables, as measured for annual regulatory reporting, and turn these into predictions of trapped gas distributions.

2. Composition and Structure of Tailings Ponds

Upon delivery from the bitumen extraction plant, the tailings slurry constitutes sand, clay, residual organic compounds, salts, minerals and trace metals [5,6,13]. The exact properties depend on the company operating the plant, the processes used and the specific oil sand ore. Once deposited in the pond, tailing slurries settle and form a stratified pond [5,6,13]. Larger and denser particles settle faster. Densities of the solid phase constituents are generally higher than water. Thus, the industry has adopted a diameter criterion in order to classify solids. Solids with diameters exceeding 44 μm are termed sand. The rest are termed fines, although the fines consist also of a range of larger silt particles and smaller clay particles. Sand particles of the largest sizes settle to the bottom in days. The remaining material, conceptually fines plus water, can be considered a colloidal suspension with a yield stress. The yield stress comes largely from the clay content, as discussed later, and is able to hold suspended both small particles and bubbles.

Over time, three layers emerge in the ponds (Figure 1). The oldest strata with the highest solid concentration, the Mature Fine Tailing (MFTs), sits atop the sand layer which forms the base of the pond. The MFT has a solids content of at least 25%. Above the MFT are Fluid Fine Tailings (FFTs), and above the FFTs is the water layer [5,13,14] (with <5 % fines). The interface between the water and FFTs is referred to as the mud line, often having a layer of settled fines on its surface. Tailings suspensions that consist only of fines are very stable and may take decades to settle, and they reduce the water content sufficiently to solidify.

One of the goals for sustainable land use is to reclaim the lands and start natural vegetation there. The MFT should be consolidated and as much water as possible should be removed to achieve this. Thus, different strategies have been used to control/accelerate the consolidation. Eckert et al. [15] predicted it would take more than 100 years to obtain a consolidated layer with more than 60% solid. Jeeravipoolvarn et al. [16] observed that for a 10 m pipe filled with FFT, over 25 years, the average solid content increased from only 30.6% to 41.8%. Therefore, to expedite the consolidation process, oil sand operators mix MFT with sand, gypsum and other materials to extract the water. The new mixture, known as consolidated tailing by Suncor or composite tailing by Syncrude (CT), can reach solid contents of up to 80% [17]. In comparison, the oldest MFT in the 30 m depth of the Mildred Settling Basin (MLSB) had only 71.6% solid contents after more than 30 years of natural consolidation [18].

Different floatation and separation treatments have been used upstream by different operators, meaning that the delivered tailings stream compositions are variable between ponds, or even within one pond at different times deposited and locations. In some cases, tailings are transported between ponds and processed (e.g., to make CT).

Table 1. Summary of tailings pond information in 2015, from [5].

Pond Name	Established Date	Area (km2)	Tailing Type	Equivalent CO2 Emissions (tons/ha/year)	Total CO2 Equivalent per Year
1A	1966	0.5	Coarse/MFT	44	2196
2/3		2.8	Coarse/Fine, FTT	258	72,279
5	1995	1.9	CT	51	9599
6	2004	2.9	CT	5	1320
7	2006	3.9	CT	5	1989
8A	2005	1.4	CT/MFT	10	1455
8B	2005	7	MFT	73	50,946
STP	2006	7.9	Coarse/MFT	26	20,690
East In-Pit	1999	2.4	CT	25	5885
West In-Pit/BML	1995	6.5	CT/MFT	158	102,564
MLSB	1977	9.3	Coarse/Fine/FTT/MFT	603	561,227
SWSS	1991	9.7	Coarse/MFT	12	11,407
Aurora In-Pit	2010	1.7	CT/Coarse/MFT	501	85,104
ASB	2000	5.6	Coarse/MFT	32	17,668
MRM ETF	2003	3.1	Coarse/Fine/FTT/MFT	6	1934
MRM In-Pit Cell 1A	2008	1.2	Coarse/CT/MFT/FTTT	167	20,018
JPM sand cell 1	2010	3	Coarse/Fine	47	14,073
JPM thickened tails	2010	1.2	Coarse/Fine/TT	24	2861
Horizon	2008	7.7	Coarse/Fine/FTT/MFT	36	27,982
Total		79.7			1,011,195.4

summarizes the information on tailing ponds in the Athabasca river oil sand deposits. Here, Fluid Fine Tailings (FFTs) refers to the fresh tailings produced in the oil sand extraction plants. These tailings do not include any diluents. Froth Treatment Tailings (FTTs) do, however, include naphtha and other diluents. These residual organic diluents are believed to be the primary source of methanogenesis [2]. In some cases, coagulants and flocculants are added to the FFTs in the processing plants to recover water. The resulting tailings are then termed Thickened Tailings (TTs) [19,20]. For more details on the hot floatation process of oil sands, see Schramm [13].

2.1. Emissions

The last two columns of Table 1 show the information relevant to greenhouse gas (GHG) emissions, in terms of CO₂ equivalents. It can be seen that the severity varies significantly from the GHG perspective. Foght et al. [21] first observed signs of microbial activity in MFT samples with no clear sign of methanogenesis in the lakes. In the 1990s, signs of the first methane release from MLSB were observed [22]. Guo [23] showed that this bacterial activity might increase the consolidation rate of the tailings, i.e., being beneficial in this respect. Fru et al. [4] postulated that chemical changes in pore water due to bacterial activity cause accelerated consolidation. The more recent concern is of course that GHG release is believed to be responsible for climate change. In this context, it has been estimated that tailing ponds account for 2% of the total greenhouse gas emissions from the Canadian oil sands sector [5,24], although this may be an underestimate.

Different questions arise. First, it is noted that after a few years, aged tailing ponds start bubbling [17,18,22,23]. Gaseous emissions from tailing ponds are broadly separated into volatile organic compounds (VOCs) and GHG gas production. The latter is due to bacterial activity [5]. Organic materials in the tailings, mainly diluents such as naphtha, are consumed by methanogenesis bacteria to produce CO₂ and CH₄ [2,17,18,22,25]. The delayed bubbling may be due to the initial gas produced being dissolved, until saturation is reached, and then due to small bubbles being held in the FFT/MFT due to the yield stress, i.e., certain thresholds must be crossed before bubbling is observed. We return to this later.

The most common measurement method is to use a flux chamber, deployed at the pond surface; see Small et al. [5]. Standard measurements include GHG gases (CO₂ and CH₄), and VOCs. He et al. [12] have recently analyzed aerial measurements from a range of oil sands facilities, observing significantly larger emission rates than previously reported and over a wider range of compounds (including intermediate and semi-volatility organic compounds, IVOCs and SVOCs). They point out that these are not routinely included in industry reporting.

Burkus [26] has developed a model that calculates a theoretical total pond GHG emissions intensity (CO₂ equivalent/MJ energy produced). This uses various data on the tailings composition and specific diluent and bitumen residuals, combined with assumed factors describing the efficiencies of the unit processes. Although simplistic, this style of model appears essential to understanding the potential for gas emission and the variability between sites. The difference between this and our work is that we take a mechanistic approach to understanding the trapped gas.

2.2. Density

Tailing pond density can be estimated readily once the solid content distribution is known. As per provisional regulations, oil sand operators submit annual reports detailing their tailing management. These publicly available reports provide detailed measurements from each tailing pond. The regulator also produces an annual summary document tracking the status of different facilities. Typically, samples are taken at various positions on the lake, e.g, in a grid structure, and are taken at successive depths at each point. From these, e.g., as in [27], lake cross-sectional plots are generated showing the distributions of solids, fines, clay content, etc. These plots show strong stratification in depth of most of the variables, but there are also variations laterally across the pond. This type of data distribution suggests that a 1D model is adequate.

The actual depth variation can differ significantly but has common qualitative features. Firstly, the mineral content of the solids is dependent on the tailings source, but the denser and larger particles settle out quickly. Thus, both solids % (wt/wt) and the overall density of the pond material increases with depth. The solids are separated into sands and fines (defined as diameters ≤ 44 μm), which are quantified in these reports via the SFR (sand to fines ratio, wt/wt). The SFR generally increases with depth unless new tailings have been deposited. For positions within any specific pond, one could use solids % data from the tailing management reports directly [6,27,28] to generate density profiles. For simplicity and illustration, one might assume a linear variation in the solid content, starting from the mud line to the bottom of the pond. Assuming that most of the solid content is a kaolinite clay with a density of ${\rho }_s=2650$ kg · m⁻³ [29], we may calculate the solid volume fraction (denoted by by ${\varphi }_s\left(\tilde{z}\right)$). Here, $\tilde{z}=(z-z_{mudline})/(z_{bed}-z_{mudline})$ denotes the non-dimensional depth starting from the mud line with a positive direction being downward. The tailings’ density can be calculated as ${\rho }_{tailing}=(1-{\varphi }_s){\rho }_w+{\varphi }_s{\rho }_s$. Figure 2 illustrates this variation in density. Evidently, the calculation can be made more complex and complete when more detailed data are supplied.

Penner and Foght [18] measured the properties of samples taken from MLSB. Stasik et al. [30] also report the distribution of ${\varphi }_s$ for two unnamed tailing ponds. Jeeravipoolvarn et al. [16] filled a 10 m pipe with fine tailings sampled from a 25 m depth of MLSB in 1982 and let it settle for 25 years. They show a slight increase and then a decrease in solid content in the middle layers of the column. Although these measurements are performed on settling tailings from different depths, all show a somewhat linear increase in solid content in the bottom half of the tailings. For our concerns, density is important primarily in determining buoyancy forces on solid particles and bubbles. When bubbles are able to move and migrate, the timescale of motion is significantly shorter than any consolidation timescale. Hence, the density can be regarded as stationary in time for our purposes.

2.3. Rheology and Modelling of FFT/MFT Yield Stress

Numerous studies have reported on the rheology of fluid fine tailings. Wang et al. [31] summarize some of these studies. Suspension rheology is also an active research field in general. The difficulty with FFTs and MFTs is that the composition varies considerably, and in all cases, there is a wide distribution of particle sizes. At the micro-mechanical level, attractive colloidal forces generally dominate Brownian forces and are also responsible for the yield stress of the cement slurry. Typically, for particle sizes in the ⪆10 μm range, buoyancy forces begin to dominate colloidal forces, so that the particle size range for sands (>44 μmm) likely represents a range where the colloidal forces on particles may be neglected. There are various micro-mechanical models for the yield stress [32,33], based on colloidal forces. For polydispersed systems, such as FFTs/MFTs with significant volume fractions of micron size particles (e.g., 20–50% [31]), the dominant contribution to the yield stress comes from these micron size particles. In mineralogy terms, these are termed clays and, sensibly, the clay fractions are monitored in tailings management reports.

Different companies report this in different ways. For example, Syncrude report the clay content as the fraction below 2 μm, whereas Suncor report this as a clay to water ratio (CWR). The content of fines is largest nearest the surface, i.e., SFR increases with depth, and the solids content also increases with depth. Thus, the water fraction is smaller deeper in the pond. The CWR may then either increase with depth due to the reduction in water content, or may be larger near the surface. The relevance of the CWR is that it correlates with the yield strength (stress) of the pond materials. Wells and Kaminsky [34] use the methylene blue index as an indicator of the CWR, providing calibration expressions to compare with % clay. For CWR $⪅0.7$, the material is very much slurry-like and the yield stress is determined using laboratory rheometry, leading to yield stresses typically in the 1–100 Pa range. For CWR $>1$, the material becomes much more soil-like and yield strengths are measured using geotechnical tests such as the vane test, with strengths covering the range 1–100 kPa. In terms of laboratory rheometry testing, Derakhshandeh [35] reported on the properties of FFTs and proposed that kaolinite suspensions are a suitable choice as model fluid for FFT behaviour. Piette et al. [29] reported the rheological properties of two MFT samples, also finding yield stresses in the typical range for slurries.

To be consistent with the different physical behaviours of the different particles sizes, we propose to consider a form of bi-disperse suspension rheology model as a simplification. For the clay fraction, we consider this to be part of a basal viscoplastic liquid. Particles at this scale are dominated by colloidal forces. The yield stress may be determined via a model correlation, such as that of Wells and Kaminsky [34], or from laboratory testing such as that of Piette et al. [29], Derakhshandeh [35], or eventually from a micro-mechanical model such as Kapur et al. [32], Scales et al. [33]. Generally for such colloidal models, since the yield stress contributions depend on the square of the separation distance, there is some lower percolation threshold on the volume fraction (${\varphi }_{perc}$), below which there is no yield stress. Hence, we simply assume a model in which the base yield stress ${\tau }_{Y,0}$ depends on ${\varphi }_{perc}$ and ${\varphi }_{clay}$. Here, both ${\varphi }_{perc}$ and ${\varphi }_{clay}$ should be considered as volumetric solid fractions for a suspension consisting only of clay particles and water. This same basal fluid therefore also has a density determined by only the clay particles and water.

The solids in the diameter range >2 μm are considered to combine with the basal viscoplastic liquid to form a (viscoplastic) suspension. As the particle size increases, the number density decreases, and with it, the contribution to the underlying liquid properties. On the other hand, these larger particles do contribute to a volumetric effect on the yield stress (and any viscous stress), as well as to the suspension density. As the ponds are largely static, we are more concerned with yield stress than viscous effects, although we may assume that the basal viscoplastic liquid has Herschel–Bulkley behaviour [29,35]. We denote the solids fraction (of silts and sands) by $\varphi$.

The use of a relative suspension viscosity, as a multiplier to the basal effective viscosity, dates back to the Einstein–Sutherland expressions, with perhaps the most commonly used due to Krieger and Dougherty [36]. For yield stress suspensions, a similar methodology is widely found [37,38]. Consistent with the idea that the larger particles contribute non-colloidally, we adopt the homogenization approach developed by Chateau et al. [39], who provide a method to describe the overall rheology of a suspension of particles in a non-linear fluid, such as the Herschel–Bulkley. They propose to calculate homogenized yield stress and consistency of the fluid using

$$\begin{array}{ccc}\hfill {\tau }_Y/{\tau }_{Y,0}& =& \sqrt{(1-\varphi )g\left(\varphi \right)},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \kappa /{\kappa }_0& =& \sqrt{\left(g{\left(\varphi \right)}^{(n+1)}\right)/(1-\varphi )(n-1)},\hfill \end{array}$$

(2)

with ${\tau }_{Y,0}$ and ${\kappa }_0$ the yield stress and consistency of the basal viscoplastic liquid. The dimensionless modulus, $g\left(\varphi \right)$, is calculated using the Krieger–Dougherty law, with ${\varphi }_{max}$ measured through experiments [39,40]. As noted, the baseline ${\tau }_{Y,0}$, which varies with the CWR, may not always be monotonic in a particular pond. For any pond, if $\varphi$ and the CWR are known, we can calculate a basal yield stress using data from Wells and Kaminsky [34], and then use Equation (1) to find ${\tau }_Y$. Using data from Suncor reported on South West In-Pit (SWIP) at Mildred Lake site Sun [28], we can estimate the ${\tau }_Y$ variation in depth.

Finally, we consider the stability of the larger particles in the basal viscoplastic liquid (clay suspension). It is well known that yield stress fluids may trap solid particles. A solid sphere of diameter $d_p$ in an infinite medium will not settle provided that ${\tau }_{Y,0}>0.046\Delta \rho g{d}_p$ [41]. The numerical factors in such limits depend on the particle aspect ratio, shape and orientation [42]. However, for yield stresses in the 1–10 Pa range, millimetre size solids with significant density differences $\Delta \rho$ are likely to remain suspended. Thus, although buoyancy dominates over colloidal forces, it is not enough for the particles to settle in yield stress suspensions. Lastly, note that residual bitumen, with a density similar to water, is unlikely to separate in these suspensions.

As noted above, there are a number of different models for estimating the tailing pond’s basal yield stress, ${\tau }_{Y,0}$, according to the role played by different particle sizes and the available data. The homogenization approach of Chateau et al. [39] can then be used to find ${\tau }_Y$, based on the solid volume fraction of non-colloidal particles. To illustrate, we consider the following methods.

• Apply the Wells and Kaminsky [34] correlation between yield stress and CWR, using the CWR values calculated from [27] for Suncor SWIP. From the reported clay content of the solids, one can find $CWR={\displaystyle \frac{Clay\%\times Solids\%}{1-Solids\%}}$
• Use input fluids tailings composition of SWIP and then the Wells and Kaminsky [34] correlation to find the basal yield stress.
• Use solid content and SFR reported in [27] to find the volume fraction of solids below 44 μm, but now use a higher cut-off diameter for colloidal particles, e.g., assume that particles smaller than 20 μm are dominated by colloidal forces. Assume a uniform particle size distribution and find the volume fraction of particles below 20 μm. Then, apply Wells and Kaminsky [34] to the CWR calculated based on the mass fraction of the particles smaller than 20 μm.
• Find the volume fraction of particles below 20 μm similar to method 3. Apply the Yodel model [38] to the volume fraction of particles smaller than 20 μm. The Yodel model should be calibrated based on the data from Derakhshandeh [35] for the yield stress of fluid tailings with different solid volume fractions.

These results are shown in Figure 3. All methods generally capture the expected yield stress trends in a tailing pond. The methods based on Wells and Kaminsky [34] are sensitive to the amount of pore water present, which reduces in the depths of a pond with high solid volume fractions. Moreover, for the methods using Yodel and Equation (1), as ${\varphi }_S$ gets closer to the ${\varphi }_{max}$, the effects will be more pronounced. The results are not particularly conclusive, as they depend on the accuracy of any in situ measurements made on the tailings at the different depths of the pond, e.g., the non-monotone results in Figure 3 come from the underlying data taken from [27]. With more data, any of these methods could be suitable to estimate the yield stress distribution of the tailing pond. The main point is to illustrate that there are likely significant variations through the depth of the ponds.

3. Modelling Gas Content

Two-phase flow modelling can be arranged in several ways based on the chosen variables. One common framework is known as the two-fluid model; see Ishii and Hibiki [43]. In this model, continuity and momentum equations for each phase are solved independently. Phase interactions are accounted for by mass and momentum transfer terms in the equations. We base our model on a simplification of this general framework. The model is one-dimensional (1D), with z measuring depth downwards from the mud line at a fixed position in the pond. The liquid carrier phase is assumed to have negligible velocity. The bubble phase total mass is negligible compared to the liquid mass. The gas phase is therefore decoupled from the liquid phase in the momentum balance. The resulting model is:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \left(\alpha {\rho }_g\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\alpha {\rho }_g{v}_g\right)}{\partial z}}& =& {\Gamma }_g,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill (1-\alpha ){\displaystyle \frac{\partial P_l}{\partial z}}+(1-\alpha ){\rho }_l{g}_z& =& 0.\hfill \end{array}$$

(4)

In these equations, subscripts g and l are used for the gas phase and liquid phase, respectively, $\rho$ is the density, and $v_g$ is the gas velocity. Here $\alpha$ is the volume fraction of the gas phase and ${\Gamma }_g$ is the gas mass generation rate. The only variables of interest are the gas volume fraction $\alpha$, from the gas mass conservation equation, and the hydrostatic pressure, $P_l$, from the liquid momentum equation. A more sophisticated description of the carrier phase could be used, to include the normal stress term ${\tau }_{L,zz}$ evolving depthwise as the pond liquids develop strength and become semi-solid.

The bubble phase momentum, as with the bubble mass, is negligible. However, the momentum balance should be replaced by a closure law that determines bubble phase velocity $v_g$, e.g., a Stokes drag law most simply. In the scenarios of interest, bubbles of different sizes exist in the pond. However, the current formulation of the two-fluid model does not include any size information about the bubbles. Bubble size is important since in a yield stress fluid, bubbles below a critical diameter $d_c$ will be stationary, while others will rise. We discuss bubble motion further in Section 3.2. To solve the equations above, one must know the closure expression for $v_g$, the fluid properties of the gas and carrier phase, as well as ${\Gamma }_g$. The pond is then modelled by considering a density profile for the carrier phase, stratified in layers of MFT, FFT, and water. Gas density can be approximated through the ideal gas law, provided the temperature and hydrodynamic pressure are known.

To consider the effect of bubble size, we will use population balance modelling. For this, we replace the conservation of mass with the conservation of bubble numbers. To develop the conservation of bubble numbers, consider $f(z,r,t)$, which is the population density distribution of bubbles with size r at depth z at time t. Bubbles can change their location and size through various dynamics, and thus, $f(z,r,t)$ evolves. Here, for simplicity, let us first consider that bubbles do not move in spatial coordinates and only change size due to growth rate, ${\Gamma }_r(z,r,t)={\displaystyle \frac{dr}{dt}}$. For bubbles within the radius interval $r\in [r_1,r_2]$, the change in bubble number density due to the change in size can be written as ${\Gamma }_r(z,r_1,t)f(z,r_1,t)-{\Gamma }_r(z,r_2,t)f(z,r_2,t)$. These two terms represent particle flux in and particle flux out of $[r_1,r_2]$, respectively. The bubble number balance for the interval $[r_1,r_2]$ is:

$${\displaystyle \frac{d}{dt}}{\int }_{r_1}^{r_2}f(z,r,t)dr={\Gamma }_r(z,r_1,t)f(z,r_1,t)-{\Gamma }_r(z,r_2,t)f(z,r_2,t).$$

(5)

This can be rewritten as

$${\int }_{r_1}^{r_2}[{\displaystyle \frac{df(z,r,t)}{dt}}+{\displaystyle \frac{d}{dr}}\left({\Gamma }_r(z,r,t)f(z,r,t)\right)]dr=0.$$

Assuming smoothness of functions and arbitrarily chosen interval, the integral can be removed and the population balance equation be written as

$${\displaystyle \frac{df(z,r,t)}{dt}}+{\displaystyle \frac{d}{dr}}\left[{\Gamma }_r(z,r,t)f(z,r,t)\right]=0.$$

(6)

While Equation (6) includes bubble growth, it neglects bubble advection and a source term due to bubble nucleation. Considering these effects, the population balance equation can be written as:

$${\displaystyle \frac{\partial f(z,r,t)}{\partial t}}+v_g(z,r,t){\displaystyle \frac{\partial }{\partial z}}f(z,r,t)+{\displaystyle \frac{\partial }{\partial r}}\left({\Gamma }_r(z,r,t)f(z,r,t)\right)={\Gamma }_n(z,r,t)$$

(7)

with $v_g(z,r,t)$ as the bubble velocity and ${\Gamma }_n(z,r,t)$ as the bubble nucleation rate. For details of the derivations above, see Ramkrishna [44], Yeoh et al. [45], Carrica et al. [46] and Guido-Lavalle et al. [47].

To solve Equation (7), we discretize the bubble size domain into N discrete intervals of $[r_i,r_i+\Delta r]$ with $i\in \{1,2,...,N\}$. We define the number of bubbles in the interval $[r_i,r_i+\Delta r]$ as

$$F_i(z,t)={\int }_{r_i}^{r_i+\Delta r}f(r,z,t)dr.$$

(8)

Taking the integral of Equation (7) for $r_i<r<r_i+\Delta r$, we have

$${\displaystyle \frac{\partial F_i(z,t)}{\partial t}}+v_g(z,r,t){\displaystyle \frac{\partial }{\partial z}}{F}_i(z,t)+{\Gamma }_r(z,r,t)f(z,r,t){|}_{r_i}^{r_i+\Delta r}={\overline{\Gamma }}_n(z,t),$$

(9)

where ${\overline{\Gamma }}_n(z,t)={\int }_{r_i}^{r_i+\Delta r}{\Gamma }_n(r,z,t)dr$.

To model the bubble growth term, ${\Gamma }_r(z,r,t)$, let us consider a single bubble at time t with radius $r-\delta r$. As time changes from $t\to t+\Delta t$, $\Delta m=S_0{A}_b(r-\delta r)\Delta t$ is added to the bubble mass. Here, $A_b\left(r\right)$ is the surface area of a bubble with radius r and $S_0$ is the mass transfer rate due to gas generation. The new bubble volume becomes $V_b\left(r\right)=V_b(r-\delta r)+{\displaystyle \frac{\Delta m}{\rho }}$. Geometrically, $\Delta V_b\left(r\right)=A_b\left(r\right)\delta r$. With some rearrangement and taking $\Delta t\to 0$, we have ${\displaystyle \frac{\partial V_b}{\partial t}}={\displaystyle \frac{S_0{A}_b}{\rho }}$ and ${\displaystyle \frac{\partial V_b}{\partial t}}=A_b{\displaystyle \frac{\partial r}{\partial t}}$, leading to:

$${\displaystyle \frac{\partial r}{\partial t}}={\displaystyle \frac{S_0}{\rho }}.$$

(10)

The term $S_0$ can be modelled using the local gas generation rate, ${\Gamma }_g(z,t)$. We expect $S_0$ to depend on the amount of liquid available around the bubble. Hence, we propose $S_0=(1-\alpha ){\Gamma }_g$. To simplify, we assume that bubble velocity within a size group is independent of r. From Equation (8) we can deduce ${\displaystyle \frac{\partial F_i(z,t)}{\partial r_i}}=f(r,z,t){|}_{r_i}^{r_i+\Delta r}$ and substitute along with (10) into Equation (9) to obtain:

$${\displaystyle \frac{\partial F_i(z,t)}{\partial t}}+v_g(z,r,t){\displaystyle \frac{\partial }{\partial z}}{\int }_{r_i}^{r_i+\Delta r}f(z,r,t)dr+{\displaystyle \frac{S_0}{\rho }}{\displaystyle \frac{\partial F_i(z,t)}{\partial r_i}}={\overline{\Gamma }}_n(z,t).$$

(11)

Equation (11) requires two boundary conditions, one for z and another for $r_i$, and an initial condition to be fully defined. We impose zero bubble flux at the lower spatial boundary $z=0$, and zero bubble number for $r<=0$. These translate to

$$u{\displaystyle \frac{\partial F_i}{\partial z}}=0z=0$$

(12)

$$f(z,r<=0,t)=0\to F_{i=0}=0$$

(13)

For the initial condition, we choose zero bubbles in the domain.

$$F_i(z,t=0)=0fori\in \{1,2,3,...,N\}$$

(14)

The only undefined parameter in Equation (11) is the bubble number generation rate, ${\overline{\Gamma }}_n$. Currently, we do not have any insight into the characteristics of ${\overline{\Gamma }}_n$. To proceed with the model, we will assume that ${\overline{\Gamma }}_n$ has the same distribution as ${\Gamma }_r$. However, the value of ${\overline{\Gamma }}_n$ will be used as a calibration constant to match our simulated release rates to the field data.

Once Equation (11) is solved, we can calculate the volume concentration of bubbles in any given size range:

$${\alpha }_i(z,t)=\alpha (z,r_i<r<r_i+\Delta r,t)={\int }_{r_i}^{r_i+\Delta r}{F}_i{V}_{b}dr$$

(15)

In particular, we can divide the bubble population into two groups: stationary and moving bubbles. Thus, we define stationary bubble volume fraction ${\alpha }_s$ and moving bubble volume fraction ${\alpha }_m$ as

$${\alpha }_s={\int }_0^{{\displaystyle \frac{d_c}{2}}}{F}_i{V}_{b}dr,$$

(16)

$${\alpha }_m={\int }_{{\displaystyle \frac{d_c}{2}}}^{\infty }{F}_i{V}_{b}dr.$$

(17)

The stationary bubble fraction ${\alpha }_s(z,t)$ can give valuable insights into what happens under the surface of the pond. For example, we can calculate the total volume of stored gas in the pond, $V_{b,stored}={\int }_0^H{\alpha }_{s}dz$.

3.1. Gas Generation

In this paper, we focus mainly on methane production in tailing ponds. Methane production is highly dependent on the composition of the tailing. To estimate a typical gas production rate, we can use the measured emission rates from lakes and find the average gas production rate, assuming both a uniform production and that the average emissions have reached a steady state. For example, for a model case of the Mildred Lake settling basin (MLSB), 26.4 tonnes of CH₄ is released per hectare of lake surface in one year [5]. As of 2022, Syncrude has reported a total of 121.1 Mm³ tailing volume in the MLSB. At the time of CH₄ measurement, there would have been at least 170 Mm³ tailing in MLSB with a 9.3 km² area. Thus, the average gas production rate, ${\Gamma }_g$, is 144 g per cubic meter of tailing over one year [5,27].

As a first step in our model, we will assume a uniform ${\Gamma }_g$ in both MFT and FFT layers. There are of course many variations that can be incorporated into a varying production rate. First, in the early years of a tailing pond, most ponds do not exhibit any significant emissions. This may be partly due to the time taken for dissolved gas to reach saturation levels and for bubbles to grow large enough to be released. Secondly, gas production is generally not uniform throughout a pond as different regions have different biochemical compositions, meaning both bacterial and diluent concentrations. It should be noted that degradation of other residual hydrocarbons also occurs, emitting CO₂ directly. Lastly, as well as stratification due to settling, there is stratification due to filling of tailings and general tailings management, meaning that different layers have different ages, and hence, different times for degradation.

3.2. Bubble Motion

A key part of the model proposed is the gas rise velocity $v_g$. For bubbles in a Newtonian fluid, Wallis [48] gives a review of various terminal velocity correlations. For small isolated bubbles, Stokes approximation gives

$$v_{\infty }={\displaystyle \frac{1}{18}}{\displaystyle \frac{d_e^{2}g({\rho }_f-{\rho }_g)}{{\mu }_f}}.$$

Here $d_e$ is the equivalent spherical bubble diameter of the same size. For the other extreme case of large bubbles, viscosity and surface tension effects can be neglected, giving $v_{\infty }=0.707\sqrt{g{d}_e}$. For a more modern review of the topic, see Kulkarni and Joshi [49]. It should be noted that bubble density and size both vary with surrounding hydrostatic pressure, and thus, for a bubble rising in a pond, $v_{\infty }=v_{\infty }\left(z\right)$. Chhabra [50] provides a comprehensive review of the bubble dynamics in non-Newtonian fluids; however, it lacks details about bubble terminal velocity in yield stress fluids and trapping criteria.

Dubash and Frigaard [51] and Sikorski et al. [52] were some of the early researchers to specifically study the motion of single bubbles in a viscoplastic fluid and report the terminal velocities. Lopez et al. [53] recently reported bubble terminal velocities for various effective diameters and Bingham numbers (Figure 4). Later, Pourzahedi et al. [54] and Zhao et al. [9] verified these results for a more limited number of cases. These results show that once a bubble starts rising, it will have a significant terminal velocity of the order of $v_t=0.1$ m/s. With such a rapid rise, a bubble can travel through the layers of the tailing pond within a few minutes. This suggests that the motion of the bubble has a faster time scale compared to the nucleation and growth of the bubble. For moving bubbles, on balancing drag and buoyancy forces, we have:

$$v_g=v_{\infty }=\sqrt{{\displaystyle \frac{4}{3}}{\displaystyle \frac{g{d}_e}{C_D}}}$$

(18)

where $C_D$ is the drag coefficient of the rising bubble. The above-mentioned results have been collated by Pourzahedi et al. [54], leading to relationships of form $C_D=aR{e}^{-b}$ for various yield stress materials, where $b\approx 1$.

These Stokesian regime closure expressions are probably adequate for most purposes. However, as the timescale for changes in the ponds is driven by the settling behaviour (i.e., months, years, decades ⋯), it is worth pointing out that the precise rise velocity is possibly irrelevant. Small bubbles are trapped in yield stress fluids, as observed in ice cores taken from ponds, which can show both bubbles and residual bitumen frozen in position. If the bubbles are large enough to rise, the effective viscosities of the yielded pond fluids at small strain rates are simply not large enough to prevent migration over a time span of years. Thus, the most important criterion here is whether a bubble is trapped or not. This transition is defined by a critical yield number,

$$Y_c={\displaystyle \frac{{\tau }_y}{({\rho }_l-{\rho }_g)g{d}_c}},$$

(19)

which defines a critical diameter $d_c$ above which bubbles rise. Here, ${\tau }_Y$ is the yield stress of the surrounding fluid, g is the gravitational acceleration, and ${\rho }_l$ and ${\rho }_g$ are the density of the surrounding fluid and the gas, respectively.

Pourzahedi et al. [54] reported the critical yield number is in the $0.05<Y_c<0.25$ range for bubbles rising in a Carbopol gel of similar yield stress to tailings. In addition, Pourzahedi et al. [55] have shown that for a spherical bubble, $Y_c\approx 0.1$, independent of the interfacial tension. They also computed the yield limits for ellipsoidal bubbles of varying aspect ratio. For a spherical bubble, $Y_c=0.066$, and for a prolate bubble with 5:1 aspect ratio, $Y_c=0.165$. See Figures 20–22 in Pourzahedi et al. [55] for other aspect ratios and the effects of interfacial tension. The relevance of aspect ratio practically is that although the initial bubbles formed will be spherical, dominated by interfacial tension, as bubbles grow, the surrounding media may elastically deform prior to yielding and flow onset. Daneshi and Frigaard [56] study flow onset for bubbles in Carbopol gels with yield stresses in the range 2.9–30.4 Pa. For the larger yield stresses, larger bubble size is required and the bubbles elongate progressively before yielding. The empirically determined critical yield numbers are in the range $Y_c=0.045-0.23$.

Pond fluids are more clay-like than the Carbopol gels considered above. Using Laponite clays, with relatively short resting times, Daneshi and Frigaard [11] found similar bubble shapes to those in Daneshi and Frigaard [56] at lower Carbopol concentrations. However, both high concentration Carbopols and aged Laponite have significantly more irregular (and larger) bubbles at flow onset. In Daneshi and Frigaard [11], this transition away from symmetric bubble shapes is shown to correlate well with the increased ratio of yield stress to interfacial stress, i.e., as the yield stress dominates, interfacial stresses are only relevant at small radii of curvature, locally along the bubble surfaces. As bubbles grow in the elastic creep regime, the shapes tend to respond to the structure of the surrounding material.

We believe that this irregularity is also likely to occur in tailings ponds, as the CWR (and yield stress) grows, although for yield stresses $⪅10$ Pa, we may see more symmetric bubbles. Certainly, as the solid content increases lower in the pond, and in particular the SFR, we will see a transition from slurry-like to soil-like, where bubbles will adapt to the pore space available. A further complication in fitting real pond behaviour will also come from dissolved gas and modelling bubble nucleation/growth. Generally, it is assumed that dissolved gas is transported via diffusion towards nucleated bubbles, coming out of solution as saturation levels are reached. However, these processes are not modelled here.

For simplicity, we ignore interfacial tension and also consider a single $Y_c$ independent of bubble aspect ratio. Saturation levels for CH₄ are significantly less than those for CO₂, so that we also ignore transport processes into the bubbles from the liquid. One could implement a qualitatively correct model by having $Y_c\to \infty$, dependent on a volume fraction that reflects the saturation limit of dissolved gas. However, this refinement is left for later. Thus, we use the following simple model

$$v_g=\left\{\begin{array}{cc}0\hfill & Y<Y_c\hfill \\ v_{\infty }\hfill & Y\ge Y_c\hfill \end{array}\right.$$

(20)

as velocity input in our model, once the rheology of the fluid is known. Here, $Y={\displaystyle \frac{{\tau }_Y}{({\rho }_l-{\rho }_g)g{d}_e}}$ is the yield number and $Y_c$ is the critical yield number at which bubbles start moving. The density and yield stress will vary with depth.

4. Results and Discussion

As discussed above, our knowledge of pond properties is limited. Specifically, the distribution of ${\Gamma }_g$ and ${\Gamma }_n$ in the pond is unknown. However, to demonstrate the model’s capability, we present the results for test scenarios.

In the following part, unless otherwise stated, the pond depth is chosen to be $H=40$ m, which is similar to the depth profile of the Mildred Lake settling basin. First, we consider a uniform distribution in depth for ${\Gamma }_g$ and ${\Gamma }_n$ and explore what range of ${\Gamma }_g$ and ${\Gamma }_n$ give acceptable results and are stable. Figure 5 shows the mean ${\alpha }_s$ for different values of ${\Gamma }_g$ and ${\Gamma }_n$. The liquid density and yield stress were set to be uniform in depth with ${\rho }_l=1500$ kg · m⁻³ and ${\tau }_y=15$ Pa. In addition, gas generation and nucleation source terms are also uniform with depth. It is clear that for ${\displaystyle \frac{{\Gamma }_g}{{\Gamma }_n}}>{10}^{-14}$, $\overline{{\alpha }_s}$ is log-linear. As ${\Gamma }_g/{\Gamma }_n$ becomes smaller, $\overline{{\alpha }_s}$ approaches a saturation value.

A step function can also be used to define where gas generation occurs and where it does not. It has been postulated that only older parts of a tailing pond will be capable of producing gas through the biodegradation of hydrocarbons. Equations (21) and (22) define ${\Gamma }_g\left(z\right)$ and ${\Gamma }_n\left(z\right)$ based on this hypothesis. Here, $z_c$ is a critical depth below which we have gas generation and nucleation and above which there is no gas generation. ${\Gamma }_{g,0}$ and ${\Gamma }_{n,0}$ are base gas generation and nucleation rates.

$${\Gamma }_g\left(z\right)=\left\{\begin{array}{cc}{\Gamma }_{g,0}\hfill & z\le z_c\hfill \\ 0\hfill & z>z_c\hfill \end{array}\right.$$

(21)

$${\Gamma }_n\left(z\right)=\left\{\begin{array}{cc}{\Gamma }_{n,0}\hfill & z\le z_c\hfill \\ 0\hfill & z>z_c\hfill \end{array}\right.$$

(22)

Figure 6 shows the distribution of ${\alpha }_s$ in depth for the uniform distribution of ${\Gamma }_g$ and ${\Gamma }_n$ with $z_c=H$, ${\Gamma }_{g,0}\left(z\right)={10}^{-9}{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3·\mathrm{s}}}$ and ${\Gamma }_{n,0}={10}^{+3}{\displaystyle \frac{1}{{\mathrm{m}}^3·\mathrm{s}}}$. There is a slight gradient in ${\alpha }_s$, increasing with depth. In this scenario, we have a critical yield number based on a single bubble and the increase in ${\alpha }_s$ results from pressurization of the bubbles.

Figure 7 shows the results for a non-uniform gas generation with $z_c=0.2H$ with other parameters similar to Figure 6. We can see that stationary bubbles are only in the gas generation region. This follows since once bubbles are mobilized, they are assumed to rise quickly. We again see the increase in trapped bubbles with depth within the bubble generation layer.

Figure 8 depicts a case for a real pond, in which much thicker tailings result in a near exponential distribution for the yield stress, increasing with depth. Figure 9 shows results based on SWIP data. Density and yield stress for this case are estimated based on the solid fraction distribution data available for the Mildred Lake settling basin. What is interesting in these models is that the static gas content increases with the yield stress, even up to significant volume fractions. Realistically, at large volume fractions, the bubbles are no longer trapped and are released independently. The stability of bubble clouds has been studied recently by Daneshi and Frigaard [11], who show that at larger gas volume fractions, a bubbly liquid can become vulnerable to a type of “burst” release, in which the bubble release results in a drop in hydrostatic pressure and further acceleration/release.

While we cannot validate the results of our model against field or experimental data, the meticulous analysis of different test cases shows the potential of the model in studying gas-generating ponds. In general, rather simple assumptions are made regarding the distributions of solids in ponds and consequent variations in density. Here, we have built the degree of complexity through the different examples. In particular, use of a more complex description of the solids’ distribution allows for a more realistic model of FFT/MFT rheology. Since oil sands companies do regularly sample ponds at different depths, one could use these samples to validate any of the modelling approaches we have used.

Although liquid sampling is possible in the field, we know of no reliable method for assessing the gas distribution at depth in a pond. Visualization is difficult and sampling that neither disturbs the liquids nor preserves the sample pressure is a challenge. Our results simply show that there is a complexity in the pond that should not be ignored. Experiments in a large-scale pressurized bubble column could be used to probe the effects of significant gas content stratification, which our results suggest is a possible scenario.

By making some assumptions, we can also apply our 1D model to a pond with varying properties in the $[x,y]$ plane. We split the pond into an $M\times N$ grid and consider each grid point as a column of tailings. We assume that there is no mass or momentum interaction between the columns and apply our 1D model to each column with its own distinct properties. The result will provide an insight into the surface distribution of gas releases, a parameter which we might have field measurements for, i.e., different locations are observed to bubble at different rates.

Using the same parameters we had for Figure 8, we apply the 1D model to a model pond of a 10 × 10 grid. Here, gas generation is chosen to be uniform. The top 20 m of the pond is considered a water cap without any gas generation. The left graph in Figure 10 provides the depth of a tailing pond in different locations with $H_{max}=140$ m. The graph on the right in Figure 10 shows how various parts of the pond store different amounts of trapped gas. Although the specific data for this example are fictitious, it is not hard to imagine how such a model could be coupled to a pond sampling strategy, i.e., with gas depth distributions generated at each sampling location. Using this model, oil sand operators can predict when a pond will reach a critical gas volume fraction, allowing them to plan for burst release. Furthermore, if selective gas release is an option, the model allows operators to target areas with higher gas concentrations, e.g., to purge.

5. Conclusions

Canadian oil sand extraction has a significant environmental impact, which is partly felt through the long-term storage and treatment of tailings. These tailings may seep toxic substances into nearby aquifers over time. Long-term storage of tailings to increase the solid content consolidation also allows bacteria to digest residual organic compounds and release CO₂ and CH₄. The rate of methane release from ponds is at a concerning level, requiring a clear plan for tackling the problem in the near future.

A significant challenge comes from the tailings’ viscoplastic behavior that traps small bubbles and releases them once they become large enough. There is no clear understanding of where and how these bubbles are generated or released. Partly, the proper consideration of rheological effects on bubble trapping/release is in its infancy. Partly, there is very little measurement of gas content in ponds, which is admittedly much harder than measuring solids content.

In this paper, we have proposed a framework to model the pond’s gas generation and release distribution. We use a basic drift flux model for this, assuming that bubbles rise with terminal velocity and that both carrier phase and mixture velocity are negligible. The gas generation rate is estimated using the available emission data and assumed to be uniform throughout the tailing layer. Our model has focused on realistically representing fluid rheology, which can have a controlling effect on the gas content. The model is an advance on that of Burkus [26], in terms of being able to predict gas distributions as opposed to total gas potential, but really these are complementary approaches. The form of model developed could also be coupled to that of Zhao et al. [57] who study gas ebullition variations in response to atmospheric pressure variations. In their model, there is a prescribed “critical pressure”, which evidently translates to a critical bubble size. Our modelling approach allows us to better define what the critical pressure actually means and tie this back to both atmospheric pressure variations and depthwise rheological variation. Although atmospheric pressure variations do likely affect gas content near the surface, deeper in the ponds will be much less sensitive, and here, it is important to understand the rheological variations.

A more detailed approach might also be needed in the future, in order to explore different control strategies. Different ponds have quite different emissions rates [5] due to differences in both the mined material and its subsequent treatment. Different operators have slightly different data collection methods and levels of transparency/accessibility, all within regulatory constraints. Further, although we have targeted CH₄ here, due to the higher global warming potential, many more ponds produce CO₂ significantly. From the modelling perspective, the main difference comes in gas solubility, which should also be incorporated in advanced models. Lastly, the style of model developed is able to include biochemical closure models that directly predict bubble nucleation rates and mass production. These developments are left for the future.