Rheology of foam in porous media (Darcy scale)

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Statement of the problem

This research is a part of the project entitled “Using the foams with blocking effect and the desorption/iron delivery foams for the treatment of heterogeneous high-velocity groundwater polluted by heavy chloride compounds,” with an acronym of FAMOUS (in French: “Utilisation des mousses à effets bloquants et des mousses de désorption/vectorisation du Fer pour le traitement de nappes phréatiques hétérogènes à forte vélocité polluées par des composés chlorés lourds”). This project is conducted by SOLVAY, SERPOL, BRGM (French Geological Survey) companies, I2M laboratory in Bordeaux, the University of Franche – Comté (UTINAM laboratory) in the framework of GESIPOL call for projects of ADEME (Environment and Energy Management Agency). The aim of the FAMOUS project is the remediation of DNAPL contaminated soil using two different foam types: 1) to divert the high-speed groundwater flow from polluted zones, 2) to dissolve and treat the contaminants.
The contaminated site is located in southeastern France on the western foothills of the Alps mountain range system shown in Fig. 1.1. The water table of the site is 10 m deep with ±1 m amplitude, and the thickness of the aquifer is about 65 m. The aquifer is heterogeneous with the hydraulic conductivity from 10-4 to 10-2 m/s (corresponding permeability from 10-11 to 10-9 m2), and the average groundwater velocity is 10 m/day. The direction of the aquifer flow is from the south-southeast (SSE) to the north-northwest (NNW).
We also know that the mean porosity of the site is equal to 40%. The contaminated site with an area of 80 by 60 meters has been polluted by DNAPL. The objective of this PhD work within the FAMOUS project was to study foam flow for diverting the flow of groundwater from contaminated soil areas without coming into contact with contaminants. The conceptual illustration of the foam injection process in the contaminated site is presented in Fig. 1.2.

Bulk foam

Foams are close to concentrated emulsions, and the disperse phase in foams is gas. The bulk foam structure is presented in Fig. 1.3, where the presence of gas and liquid phases are shown. The liquid phase of foam consists of surfactants, in which high-molecular-weight substances can be added to increase the stability of the surfactant solutions (Perepelkin & Matveev, 1979). Foam, also called the gas-emulsion, is considered as one of the numerous classes of colloidal systems in the classification of Bikerman (Bikerman, 1973).

Surfactant

Surfactant is the essential component of the foam and plays a crucial role in foaming. Surfactants are organic molecules with different polarity parts that lower the interfacial (or surface) tension between two fluids, or between a liquid and a solid (Guozhong, 2004). One aspect of the surfactant is hydrophilic (often called the hydrophilic head) and polar. The other part is a hydrocarbon chain, which is hydrophobic (often called the hydrophobic tail) and lipophilic, hence non-polar (see Fig. 1.3). Thus in foams, the hydrophilic part is directed towards the surfactant solution and the lipophilic part towards the gas phase.
According to Gibbs’ law (Gibbs, 1879), the interfacial tension decreases sharply with an increase in the surfactant concentration, thus saturating the interface between two phases with surfactants. When the surfactant concentration reaches a particular concentration called critical micelle concentration (CMC), the surface tension remains relatively constant or changes with a lower slope (see Fig. 1.4).

Foam formation

Foams can be generated by two methods: dispersion and nucleation (Bikerman, 1973). In the dispersion process, the initial dispersion medium is liquid (the continuous phase). The gas is the dispersed phase introduced in this medium. Thus, foam is the opposite of fog or aerosol, in which a liquid phase in dispersed in a continuous gas phase. Dispersion foams can be made by injection of gas into a surfactant solute, or also by shaking of the surfactant solution by, for example, agitating a can containing solution and air. This foam generation method is mechanical, and there is no chemical modification of the substances present (see Fig. 1.5a). There are numerous foam generation experiments by dispersion method, for instance, air injection into a rotary cylinder, gas injection through a soap or surfactant solution, simultaneous injection of gas and surfactant solution (Bragg & Nye, 1947; Smith, 1949; Neppiras, 1969).
The second type of foaming method is the nucleation method, where the gas molecules are initially present in the liquid phase as a dissolved phase. When the pressure or temperature changes to a specific value, the dissolved gas molecules are transformed into bubbles, thus generating foam. For example, beer and soft drinks (see Fig. 1.5b). Nucleation methods involve more chemical effects than dispersion. The nucleation of gas from a liquid solution requires particular thermodynamic conditions in terms of pressure or temperature.
Regarding the dispersion method by injection of gas, Bikerman (1973) emphasized the importance of gas injection speed and, therefore, the Reynolds number on the properties of the foams generated. Moreover, the foam generation techniques must be adequately described, since foam quality (the gas fraction) and texture of foam (bubble density) can vary depending on the type of method used. Typically, foam texture is defined as a number of lamellae per unit volume of the gas phase, which is also inversely proportional to the bubble size. In addition, depending on the field of application, it is crucial to choose the best foam formation techniques to have a well-characterized stable foam. The most well-known foam formation method is the simultaneous injection of gas and surfactant solution through a porous medium, where the porous medium can be stacked glass beads (glass frits) or sand-pack (Burley & Shakarin, 1992; Enzendorfer, et al., 1995; Raza & Marsden, 1967; Patton, et al., 1983).

Foam quality and structure

As mentioned above, in foams, gas bubbles are separated by liquid films called lamellae. The main concept of these lamella or foam films is a sandwich-like structure with three layers in which the inner layer has the same viscosity as a liquid in the system. The two exterior layers, which are in contact with the gas phase, are more viscous and have a non-Newtonian behavior. These outer layers’ structures are more like a solid material; they do not flow under minimal stress. For example, consider foam based on soap or high molecular weight surfactant solution and allow it to dry for sufficient time. After a while, we can observe that the liquid between two outer layers of lamella evaporates. A double sheet of solid soap or solid plastic may exist for an uncertain period. It was concluded that there is no hydrostatic equilibrium in the foam films since the pressure varied at each point of a lamella (Bikerman, 1973).
Foams are classified as wet or dry, depending on the gas and liquid fractions (see Fig. 1.6a), which is also called the foam quality. Hence, in a static state, the foam quality can be expressed as, where, and are the volumetric contents of liquid and gas in foams, respectively.
The gas fraction of wet foams vary from 64% to 95%, and when the foam quality >95% foams become dry. Foams transform into a more bubbly liquid if <64% (see Fig. 1.6a). The transition from wet foam to bubbly liquid is called the jamming transition.
The shape of bubbles changes from spherical to polyhedral depending on the type of foams. Wet foams produce spherical shape bubbles, while dry foams favored generating polyhedral form. For the polyhedral foams, if three bubbles meet symmetrically, the connection between lamellae called the Plateaus border (Plateau, 1873) creates a three-fold border with a 120° angle (Fig. 1.6b). If four Plateaus borders symmetrically intersect in the space, it forms tetrahedral angles with almost 109° (see Fig. 1.6b).
The change of foam structure also depends on the arrangement of foam bubbles and the size differences between them. Fig. 1.6a shows the most common types of foam structure, depending on the gas fraction. For example, the difference in the structure of dry foams can be polydisperse-disordered (a), monodisperse-disordered (b), and monodisperse-ordered (c). Thus, wet foams (d-f), wet foams at jamming transition (g-i), and bubbly liquid (j-l) can be classified according to the bubble size distribution and the arrangement of foam bubbles (Drenckhan & Hutzler, 2015).

Foam density

Essentially, foam is a two-phase system, and to calculate the foam density, one must consider the densities of gas and liquid. Therefore it can be expressed as follows: = + (1.2)
In which is the volume of foam, and are the mass of liquid and gas, respectively. The mass of foam is usually negligible in very high-quality foams (Zhang, et al., 2009).

Foam stability

Foam stability is a state of being stable of foam texture under static or dynamic conditions. Commonly, foam stability in a static state is estimated by measuring the time necessary for the foam to collapse down to half of the volume initially generated (half-life time). In addition, two other methods can also be used to examine the foam stability: 1) the lifetime of a single bubble; 2) the change of foam volume under dynamic conditions, such as gas flow, shaking or shearing (dynamic) (Schramm, 2006). It is essential to consider both static and dynamic stability aspects to understand the properties of foam better.
The value of the surfactant concentration is also significant for the behavior of foam since the behavior of foam at a high concentration is very different compared to that at a low concentration (Schramm, 2006). For instance, Mulligan and Eftekhari (2003) investigated the foam stability and performed quality tests 60 for foams made with ten different commercial surfactants. They found a better ability to foam Triton X-100 and JBR425 surfactants when compared to other surfactants. Rothmel et al. (1998) assumed that foam stability is independent of essential properties like hydrophile-lipophile balance and critical micelle concentration. However, some studies showed a significant dependence on foam stability on surfactant concentration and foam quality (Zhang, et al., 2009). Moreover, the highest foam stability was obtained between 90% and 99% of foam qualities.
Since the foam is an unstable disperse system, one of the most common causes of foam instability is the effect of gravity drainage. The drainage of continuous liquid phase through lamellae that leads to the close coming of the bubble surfaces, thus causing coalescence and collapse of the foam. Moreover, foam cannot be considered as a homogeneous system when liquid accumulates at the bottom of the sample (Princen, 1990), which mostly occurs for wet foams. Another reason for foam instability can be caused by the coalescence effect and Ostwald ripening (Voorhees, 1985), where a gas diffuses from small bubbles to large bubbles. The leading root of this effect is the presence of high pressure in small bubbles rather than large ones. Capillary pressure plays an essential role in the coalescence of bubbles, that is to say, that two or more bubbles merge during contact to form a single bubble. As illustrated in Fig. 1.7, the liquid surface between two bubbles is concave to the gas phase, as the pressure at points b is higher than at point a, the higher pressure from two sides pushes the floating bubbles to each other (Bikerman, 1970; Bikerman, 1973).
Fig. 1.7 Capillary attraction of two bubbles (Bikerman, 1973)

Bubble size and shape

Foam is a complex fluid, and one of the essential foam parameters is the bubble size that has substantial interconnections with bubble shape. The bubble size can vary according to the gas diffusion (Ostwald ripening) and bubble coalescence effects, which also depend on the foam stability; therefore, the type of surfactant and gas. Moreover, the gas injection rate, the interfacial tension, and the gas quality influence foam bubble size and shape. As already mentioned above, wet foams produce spherical shape bubbles, while dry foams generate polyhedral forms. The bubble shape can also be deformed due to gravitational forces, which change the ideal form of the stationary bubble. Moreover, the resistance of the liquid distorts this shape. For modeling purposes, if the bubble size is relatively small compared to the sample volume, foam can be considered as a continuous phase (Herzhaft, 1999). However, bubble size was ignored as a parameter during some experimental studies (Bikerman, 1968; Achorn Jr. & Schwab, 1948).

Foam rheology and yield stress

Rheology of the foam is also very intricate and requires extensive study to characterize. Moreover, to understand foams thoroughly and, in particular, their flow near a solid surface or in a confined domain, one needs to consider numerous parameters, for instance, foam quality, wall slip velocity, bubble size, viscosity variation, and gas compressibility.
In most of the studies reported in the literature, the bulk foam was highlighted as a yield-stress fluid, and the majority of the results are commonly well fitted by the Herschel-Bulkley law (Cohen-Addad, et al., 2013; Denkov, et al., 2009; Hohler & Cohen-Addad, 2005; Katgert, et al., 2013; Kraynik, 1988; Dollet & Raufaste, 2014; Khan, et al., 1988; Herzhaft, 1999; Denkov, et al., 2005). Mainly, measuring yield stress is not an easy task, especially in capillary geometry. For instance, Khan et al. (1988) measured foam rheology through a rheometer with parallel-plate geometry. They covered the solid plates with sandpaper to avoid wall slip velocity. Elsewhere, the yield stress was measured with the “stress relaxation” method in coaxial geometry (Khan, et al., 1988). They found increasing in yield stress with the foam quality. Some authors experimentally showed the presence of a yield stress in foam by observing behaviors of foam bubbles near the walls of a transparent pipe (Kraynik, 1988). Moreover, when the shear stress is smaller than yield stress and if wall slip velocity exists, the foam flows entirely by ‘plug flow,’ and many authors experimentally confirmed this behavior (Kraynik, 1988; Beyer, et al., 1972; Thondavadi & Lemlich, 1985).
Denkov et al. (2005) studied foam rheology and wall slip velocity for a foam with 90% gas fraction using a rheometer. The foam used was generated using a syringe with a needle with id (inner diameter) of 2.5
mm. They found a shear-thinning behavior where the shear stress was determined to be a power-law function with the power-law index equal to 0.25 and 0.42 for tangentially mobile and immobile bubble surfaces, respectively. Denkov et al. (2009) studied the effects of surfactant type and bubble surfaces on bulk foam rheology with ≥80%. They classified the rheological behavior of foam into two different classes depending on the values of the power-law index , by taking into account the viscous friction between bubbles and also between bubbles and solid walls, qualitatively. Their results with ≈0.5 referred to a system of the first type corresponding to friction dominance in foam films. The second type <0.5 (mostly between 0.2 and 0.25) was defined as systems with essential energy dissipation on the bubble surfaces. These studies show solid, plastic, or viscoelastic behavior of foam below a yield stress and non-Newtonian liquid regime above the yield stress. The transition from solid-like to liquid-like mechanical behavior is called yielding.
The apparent viscosity of foam is several times larger than the viscosity of the continuous phase, i.e. surfactant solution, even at low shear rates. This can be explained from a description of foam at the molecular and bubble scale (Hohler & Cohen-Addad, 2005).

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Porous media

Porous media are generally a solid material that contains pores (i.e., voids) in which ordinarily the solid matrix and the pore spaces are continuous. However, there are also certain concepts of porous media where naturally closed pores exist. Hence, in a subsurface environment, porous media commonly can be occupied by fluids such as water, oil, and gases. A solid matrix in this medium can be classified as follows:
Consolidated porous media, where the solid matrix is composed of cemented grains (e.g., sandstones), and the pores in such a media may be connected (permeable) or unconnected (impermeable);
Unconsolidated porous media, in which solid particles are not bound to each other, therefore the fluid can pass through the pores between the particles (e.g., sand).

Fluid flow in porous media

The study of fluid behavior in porous media is a subject of interest for many industries. For example, filtration technologies commonly used to purify water; reactors filled with porous catalyst support used for chemical reactions; membranes for gas separation; in the oil and gas industry, and soil remediation processes. The main governing law of fluid flow in porous media is Darcy’s law, which was formulated by French civil engineer Henry Darcy in 1856 (Darcy, 1856). Hence, Darcy velocity (m/s) for single-phase fluid flow through a porous medium is presented as follows: = = − • ∇( + ) (1.3) where (m2) is the absolute permeability as a 3×3 tensor, (m2) is the cross-sectional area of the porous sample, (Pa.s) is the fluid viscosity, (kg/m3) is the fluid mass density, (m/s2) is the gravitational constant, is the vertical axis oriented up, and ∇ (Pa/m) is pressure gradient linearly dependent on volumetric flow flux (m3/s).
In the case of multiphase flow, the porous medium is saturated (occupied) with several immiscible fluids.
The volume fraction of each phase is expressed by a term called saturation, which is defined as follows: in which is the volume of a fluid within the porous medium, and (or pore volume, PV) is the fixed volume of the porous space.
Consequently, the flow model of multiphase transport in porous media consists of mass and momentum balance Eqs. (1.5) and (1.6): ( Ø ) +∇∙( )=0 (1.5) where Ø (-) is the porosity of porous media. Depending on the number of phases occupied the porous medium, the general form of Darcy’s law (Muskat & Meres, 1936; Muskat, et al., 1937) is: ( ) (1.6) = − • ∇( + ) where, ( ) are the relative permeabilities (dimensionless).
In the two-phase theory ( =1, 2), the relative permeabilities are the unique functions of saturation that have a non-linear form (see Fig. 1.8). In order to have a closure relationship of 4 equations (1.5) and (1.6) for two-phase flow system, we need to define equations of state of each phase:
and capillary law.
Capillary pressure is the pressure between two immiscible fluids where the pressure is proportional to the curvature of the interface between two fluids, and when the interface is a plate, =0.

Foam in porous media

As it was discussed in the previous sections, foam is a complex two-phase fluid. However, it should be noted that the presence of foam in porous media differs from bulk foam because it does not exist as a continuous liquid/film structure that includes gas bubbles (Hirasaki, 1989). Classically, foam flow through porous media is considered as a bubble train, and the bubbles are considered either of the same sizes as the pores or exceeding one pore size. Therefore, it is essential to study the characteristics of foam and porous media, and also the interaction between them. Foam behavior in porous media depends on several factors such as pore structure (shape, pore body-to-throat ratio, pore size distribution) and wettability of porous media, which is mostly (naturally) hydrophilic in aquifers, and that is the case studied here. As shown in Fig. 1.9, the structure of foam in porous media has been considered either as a continuous-gas or as a discontinuous-gas foam. In continuous-gas foams, lamellae block certain gas flow paths by reducing gas mobility, but some gases can still circulate freely. In discontinuous-gas foams, lamellae block all ways of gas flow. These foams can flow as a train of bubbles or can be trapped, as shown in Fig. 1.9.
Fig. 1.9 Schematic of foam structures in porous media (Adebanjo & Udofia, 2015)

Mechanisms of foam generation in porous media

Besides the methods mentioned in section 1.2.2, foam can also be generated in porous media, owing to the creation and destruction processes of the lamellae at the pore-scale level. Foam formation in porous media can involve mainly three mechanisms, such as snap-off, lamella division, and leave-behind (Rossen, 1999) (see Fig. 1.10).
a) The snap-off mechanism occurs when the gas enters into a pore while liquid begins to gather in a throat, thereby blocking the throat. This can also be explained by the capillary pressure that increases during gas injection and falls by creating lamellae in pore throats. This mechanism usually generates bubbles of the same size as the pores of the porous medium, and it reduces gas mobility by a factor of one hundred through creating discontinuous-gas foam. Moreover, the snap-off process dominates foam blockage ability in porous media (Rossen, 2003; Kovscek, et al., 2007).
b) The division of lamella (bubble) mostly occurs when bubbles are larger than penetrating pores. Consequently, lamellae form by subdividing the existing foam bubbles.
c) The leave-behind mechanism takes place when gas approaches pore throats from both facing pore bodies. Lamellae created by this process makes the gas phase continues, and an enormous number of lamellae can be created in this way.
Rossen (1999) assumed the existence of the fourth mechanism of lamella creation that can be the production of gas in the middle of the liquid by chemical reaction (changing of temperature and pressure) leave behind, adapted from (Almajid & Kovscek, 2020)
In practice, foam can be generated in situ by co-injection of surfactant solution and gas. However, when porous media are very highly permeable ( >1000 darcy), as is the case in soil remediation, foam cannot be generated in situ due to the low capillary pressure. Therefore, in this case, ex-situ generation (i.e. pre-generation) of foam applied in order to make a strong foam.

Foam flow in porous media

Foam flow through porous media is inevitably accompanied by foam formation and destruction, and the foam texture (i.e., bubble density) is a consequence of these two processes. Thus, relying on aqueous and gas flow rates, foam flow in porous media has two regimes: weak (coarse) and strong (fine) foams (see Fig. 1.11) (Gauglitz, et al., 2002). Weak foam is a coarsely textured foam with large bubbles, and the reduction of gas mobility is low or moderate, ranging from 10 to 100 times. Strong foam is a fine-textured foam with small bubbles that has a significant gas mobility reduction, which can range up to 10,000 times. Gauglitz et al. (2002) also pointed out on an unstable intermediate regime between strong and weak foam regimes.
The real application of this foam is doubtful due to its instability. In order to generate a stable foam at a low-pressure gradient, one must look for a surfactant that gives stable lamellae, which is very crucial for soil remediation applications.
As mentioned above, foam is considered as a yield stress fluid in some studies in the literature (Herzhaft, 1999; Denkov, et al., 2009; Hohler & Cohen-Addad, 2005). The yield stress that is related to the bubble size plays a significant role in the mobility of foam through porous media (Falls, et al., 1989). The main issue to keep in mind is a notable effect of the yield stress on foam flow, where sufficiently large pressure gradients are needed before the flow can take place (Friedmann, et al., 1994). This issue has subsequently been controverted by other researchers (Patzek, 1996).
In order to study the foam in porous media, different simplifications can be applied depending on the conditions of foam flow (e.g., dry or wet). Commonly, in a two-phase flow system, the capillary pressure ( ) goes to infinity when water saturation is near zero. However, in the case of foam flow in porous media, the system always has a nearly zero liquid saturation at high gas fractional flow (i.e., dry foam). According to Khatib et al. (1988), the capillary pressure at first increases and then approaches a specific value at a relatively high fractional gas flow which called “limiting capillary pressure ( ∗)” (see Fig. 1.12). When the fractional flow of gas increases further, the capillary pressure was found to remain at the limiting value. Nevertheless, they observed the process of coalescence and displacement of the coarse-textured foam near the limiting value. It means that collapsing of foam is not gradual as the rises, but happens abruptly at a single ∗. If this hypothesis justifies, many simplifications are possible for strong foams flowing under dry conditions (Rossen, 1999).
For wet foams, Rossen and Wang (1997) proposed a model in which the foam bubbles were considered to be of fixed size regardless of liquid and gas flow rates. This assumption is based on the idea that bubbles smaller than the pore size disappear due to the diffusion process between the bubbles. Eventually, it must be noted that most of the simplifications are reasonable if assumptions are justified (Rossen, 1999).

Soil contamination and remediation

Environmental pollution, particularly soil contamination, is a burning problem nowadays. Since the soil is an essential part of the entire ecosystem, it bears the greatest burden of environmental pollution. The presence of contaminants in soils at high concentrations show high potential health and ecological risk. There are many ways of soil pollution, which are directly linked to human activities in the past. In order to eliminate the failures and errors of the past, significant research and technological advances have been performed in the area of soil and groundwater remediation over the past three decades. Nevertheless, the characterization of subsurface contamination is very complex, and this also makes soil remediation very challenging (Fig. 1.13).

Table of contents :

Part 1 – Substantial summary in French
1. Introduction générale
2. Considerations théoriques
3. Etude expérimentale
3.1. Fluides et matériels
3.2. Dispositifs expérimentales
3.3. Procédures expérimentales
4. Etude théorique : technique de changement d’échelle
4.1. Description de l’écoulement de la mousse à l’échelle du pore
4.2. Caractéristiques des cellules unitaires
4.3. Changement d’échelle
5. Résultats et discussion
5.1. Etude expérimentale
5.1.1. Génération de mousse en milieu poreux
5.1.2. Rhéologie de la mousse
5.1.3. Écoulement de mousse en milieu poreux (échelle de Darcy)
5.1.4. Rhéologie de la mousse dans les tubes capillaires : effet du diamètre du tube, du matériau et de la textur
5.2. Etude théorique : changement d’échelle de l’écoulement de mousse en milieu poreux
5.2.1. Influence de la qualité de la mousse
5.2.2. Effet de la géométrie des milieux poreux
6. Conclusion générale
Part 2 – Dissertation in English
1.1. Statement of the problem
1.2. Bulk foam
1.2.1. Surfactant
1.2.2. Foam formation
1.2.3. Foam quality and structure
1.2.4. Foam density
1.2.5. Foam stability
1.2.6. Bubble size and shape
1.2.7. Foam rheology and yield stress
1.3. Porous media
1.3.1. Fluid flow in porous media
1.4. Foam in porous media
1.4.1. Mechanisms of foam generation in porous media
1.4.2. Foam flow in porous media
1.5. Soil contamination and remediation
1.6. Foam for soil remediation
1.6.1. Benefits of foam application for contaminated-soil remediation
1.6.2. Review of foam applications on contaminated-soil remediation
1.7. Modeling foam flow in porous media
1.8. Upscaling techniques and upscaling of foam flow in porous media
1.9. Scope of the thesis and organization
2.1. Introduction
2.2. Theoretical considerations
2.3. Foam characterization
2.3.1. Selection of a surfactant and the surfactant concentration
2.3.2. Gas selection
2.4. Experimental details
2.4.1. Materials
2.4.2. Experimental setup
2.4.3. Experimental procedure
2.4.4. Strategy
2.5. Results and Discussion
2.5.1. Foam generation in highly permeable porous media
2.5.2. Effect of foam quality on foam flow behavior
2.5.3. Effect of foam bubble size on foam rheology
2.5.4. Effect of grain size (permeability) on foam rheology
2.6. Conclusions
3.1. Introduction
3.2. Theoretical considerations
3.3. Experimental approach
3.3.1. Fluids and materials
3.3.2. Experimental setups
3.3.3. Experimental procedures
3.4. Theoretical approach: upscaling technique
3.4.1. Description of foam flow at the pore scale
3.4.2. Characteristics of the microstructure
3.4.3. Upscaling
3.5. Results and discussion
3.5.1. Rheology of bulk foam
3.5.2. Rheology of foam in porous media (Darcy scale)
3.5.3. Upscaling of foam flow
3.6. Conclusion
4.1. Introduction
4.2. Theoretical considerations
4.3. Experimental study
4.3.1. Fluids and materials
4.3.2. Experimental setup
4.3.3. Experimental procedure
4.4. Results and discussion
4.4.1. Foam quality in capillary tubes: effect of tube material and inside diameter on foam flow
4.4.2. Foam rheology in capillary tubes: effect of tube diameter, material and foam texture
4.4.3. Consistency of the foam behavior in capillary tubes with the rheology of bulk foam
4.5. Conclusion
4.6. Perspectives
5.1. Conclusions
5.2. Perspectives
5.2.1. Surfactant and gas for blocking foam
5.2.2. Foam generator and main columns
5.2.3. Bubble size
5.2.4. Compressibility
5.2.5. Upscaling
5.2.6. Wall slip velocity and surface roughness in capillary tubes
Appendix A
References

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