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Bulk foam generalities
Liquid foams describe a complex state of matter defined as “gas bubbles which are closely packed within a liquid carrier matrix” (Drenckhan, Saint-Jalmes 2015). In this section we give an overview of bulk foam properties, which are also carried through to porous media.
Interfacial considerations
As with all types of dispersed media, it is fundamental to understand the nature of the two-phase interactions that exist at the boundary between the gas and liquid phases.
Interfacial tension
Within a volume of liquid, attractive intramolecular forces ensure the cohesion of particles with each other. A single molecule is surrounded by an isotropic force of attraction created by its neighbors. In a volume of gas, the situation is analogous albeit for a less dense fluid and hence weaker force. At the interface between the two fluids, the bordering molecules on the liquid experience a higher energy state as on one it sides it no longer shares bonds with water molecules. Because of this state, the molecules will attempt to create the minimum energy surface for the given conditions. For a small volume of gas in a large volume of water, a spherical bubble is created. Each added molecule to the boundary will increase the total Gibbs free energy of the system by a specific amount at a given temperature , pressure and density . The interfacial tension is simply the measure of the added energy per unit area of boundary created.
In other words, interfacial tension results from the cohesive energy of the molecules at the interface, who limit as much as energetically possible interface expansion (Schramm 2005).
Young-Laplace equation
The curvature of the interface separating the two phases is closely linked to the pressure difference between them. For interfacial curves, including foam bubbles, the mean curvature is related to the pressure difference ∆ across interface by the interfacial tension γ via the Young-Laplace equation. This relationship will hold locally for any smooth section of a bubble and give an expression in terms of principal radii of curvature and : ∆ = 1 + 1 (2)
This equation can be simplified for spherical bubbles of radius R in which = = as: ∆ = (3)
Bulk foam structure
Foam structure depends on the proportion of water and gas that composes it. Liquid fraction is defined as the fraction of the total liquid volume over the total foam volume such as = .
For dry foams ( < 0.05) Plateau’s laws become relevant to study of foams, where the lack of aqueous solution forces the bubbles into close contact. Dense packing minimal surface problems arise from the interfacial cost. Plateau made a series of experimental observations for dry foams that dictate the behavior of foams at very low water content. He discovered that dry foam bubbles always meet in threes and that lamella junctions at Plateau borders between bubbles always form 120° angles between each other. Figure 1-1 shows the basic elements of foam structure.
In turn, Plateau borders branch out in different directions and cross in fours at vertices at an angle of 109.49°, also known as the tetrahedral angle (Plateau 1873).
For a monodisperse foam restricted to a two-dimensional plane, the lowest energy stable foam structure is the honeycomb pattern, which minimizes the total interface perimeter. The morphology of the discontinuous gas phase depends on the gas fraction in the two-phase state. For lower liquid saturations (high foam quality), the gas bubbles tend to create angular intersections that produce polyhedral volumes as opposed to the spherical “ball type” foam brought about by high liquid saturation (low foam quality). These two extreme cases serve as boundaries for the intermediate shapes situated between polyhedra and spheres, depending on increasing liquid saturation. Figure 1-2 shows these two extreme foam types with an intermediate state.
Figure 1-2: Varying foam structure with decreasing water saturation: the monodisperse foam bubbles are almost spherical when a high quantity of interstitial liquid is present but progress towards shapes with sharp hexagonal faces at lower saturation, taken from (Höhler et al. 2008).
Foam Stability
Bulk foam lamellae are thermodynamically metastable. The use of ionic surfactants modifies surface interaction forces. Film stability is ensured by the adsorption of surfactants on the gas-liquid interface, creating a situation in which it is energetically favorable to maintain the given film thickness. Ionic surfactants reduce the surface tension between two phases because of their polar nature. They are most often molecules made up of two separate components, a hydrophilic head and a hydrophobic tail. In the context of foam applications, the lower surface tension can facilitate creation of foam bubbles (although a surface tension too low in fact inhibits foam formation) but they serve primarily the purpose of stabilizing foam lamella. The lifetime of a foam film may be influenced by a number of physical parameters, and multiple phenomena exist that account for the thinning or stabilizing of thin stabilized films in bulk foam. Sheng (2013) gives a list of these.
Stabilizing phenomenon
In the Marangoni effect, a sudden decrease of surfactant concentration at an interface, due to interface expansion or due to other loss of surfactant effect, causes a net liquid flow towards this same area. A region with higher surface tension (lower surfactant) will “pull” more liquid towards it. This in turn re-equilibrates the surfactant concentration and stabilizes the film. This effect is illustrated in Figure 1-3b.
Figure 1-3: Dynamic lamella processes: a) destabilizing Laplace capillary suction: the liquid pressure in the Plateau borders (PB) is lower as curvature is higher for constant gas pressure PG; b) stabilizing Marangoni effect in which the lower surfactant concentration causes a migration of fluid that re-establishes thickness. Adapted from (Schramm 2005)
Destabilizing phenomena
Leading to thin film coalescence:
Gravity drainage, in which the force of gravity pushes the liquid out of the thin films downwards.
Laplace capillary suction, in which the difference in curvature between the sharp Plateau borders and the flat thin films creates a difference in pressure, in accordance with the Young-Laplace equation. This pressure gradient phase redirects the fluid into the Plateau borders. This effect is illustrated in Figure 1-3a.
Other foam destabilizing phenomena:
Coarsening, where a larger Laplace pressure will exist in smaller bubbles, causing gas to diffuse through the thin lamella into the more voluminous neighbors due to the concentration gradient. The pressure difference is additionally increased. This leads to a runaway effect that culminates in the disappearance of the smaller bubble.
Bulk foams – Rheology
Foam macroscopically behaves both as a solid under low stress and as a liquid under high stress. It has a characteristic yield stress, when it transitions from one behavior to another (Dollet and Raufaste 2014). The elastic behavior of bulk foam is explained by the small increases in gas-liquid interfacial area that arise from small stresses, which corresponds to a higher energetic state, causing the foam structure to relax back to its initial equilibrium state. We can therefore characterize a bulk modulus of the foam in this elastic state. When the applied stress becomes sufficiently high, the foam bubbles spontaneously rearrange to a new topological state (Höhler and Cohen-Addad 2005), which corresponds to a liquid behavior. The foam reorganization is process illustrated by the application of a constant stress on an ideal 2-dimensional dry foam represented Figure 1-4.
Figure 1-4: Dry foam cell reorganization, a constant stress is applied that initially causes a bubble deformation, before jumping to a new topological state, similar to the initial configuration. Taken from (Höhler and Cohen-Addad 2005).
The yield stress corresponding to cell reorganization is dependent on the foam quality, and it is highest for dry foams (Weaire and Hutzler 2001). Above the yield stress, foams can be considered as shear-thinning fluids and are usually well described by the phenomenological Herschel-Buckley law, seen in equation (4).
This relationship shows that for shear stresses lower than the yield stress , the shear rate ̇is zero. Parameters and are the consistency and flow indexes. For foams, the flow index is found to be somewhere between 0.25 and 1 (Höhler and Cohen-Addad 2005).
Foams in porous media
Foams in porous media share many of the characteristics of bulk foams. They are also composed of thin films and Plateau borders, whose stability and structure can be partially understood in terms of the phenomena described above. However, they exhibit unique behavior in porous media over a series of scales. Foam lamella creation and destruction at the pore-scale is governed by several in situ processes that can affect gas phase viscosity as a function of the lamella density. Macroscopic foam flow in porous media shows multiple regimes, in which measured pressure differences are strong functions of the fluid injection rates, structural properties of the media and surfactant properties.
Basic petrophysical concepts
Here we present some concepts used in porous media studies that are essential to understanding foam dynamics.
Young’s equation and wetting angle
For two static fluids in contact with a third solid material such a rock surface, the tangent of the two fluid’s interface at the intersection with the rock surface is described by a contact angle θ. The wettability, quantified by the contact angle, is a measure of balance of affinity of either fluid to the rock, owing to the energetic preference of interface creation of either fluid with the rock. In Figure 1-5 we give a graphical representation of the contact angle.
Figure 1-5: Contact angle θ at the intersection of two fluids (white and grey) in contact with a flat solid surface (black).
Young’s equation is an application of the force balance created at the intersection of the three components for the specific case of a flat solid surface. For the example of a liquid drop on a flat surface, surrounded by gas, the interfacial tensions at the three interfaces projected onto the direction parallel to the surface, we obtain: / cos = / − / (5)
In which / is the surface tension between the liquid and gas phase, / is the surface tension between the solid and gas phase and / is the interfacial tension between the solid and liquid phase. For an angle of = 0 the solid is completely wetted by the liquid, but for < 90° in practice is considered to represent wetting conditions. Inversely, > 90° represents non-wetting conditions (Schramm 2005).
Darcy’s law
Flow in porous media can be understood via Darcy’s law, a constitutive equation that enables us to explain the viscous flow of fluids through a permeable medium. If we exclude any external forces acting upon the fluid apart from a pressure difference, for single phase of viscosity , transport in a medium with a given permeability , Darcy velocity , volumic flow rate and pressure difference ∆ over the horizontal transporting medium of length and sectional area , then Darcy’s law relates these parameters as: = / = ∆ (6)
Darcy’s law is only applicable in laminar flow situations, and typical reservoir injections have a Reynold’s number small enough to allow its use. Despite being established in a simplified set of experimental conditions (incompressible flow, saturated flow, steady-state-flow, isotropic media, laminar flow), Darcy’s law can be applied to a variety of situations outside of its initial scope (Freeze and Cherry 1979). Certain simplifications are made to model foam using Darcy’s law. Usually, foams are represented as a high-viscosity gas phase and not considered as a third phase, distinct from foamer solution and gas. For multiphase flow, Darcy’s law expresses the flow rate , component for each phase .
In which , , , and , represent the relative permeability, viscosity and Darcy velocity of each phase, while ∆ , represents the pressure drop over each phase.
Capillary pressure
In the context of a two-phase interface, the difference between wetting phase pressure and non- wetting phase pressuresuch as aqueous solution and gas given by =− is called the capillary pressure or . Capillary pressure is established similarly as is ∆ in the Young-Laplace equation, by balancing forces across the interface (equation (2)). In porous media, this pressure difference is defined by incorporating the wetting angle of the three phase interaction as (Bear 1988):
Through this section we will look at the effect of injecting a foaming solution and gas or a preformed foam into a porous medium. We proceed initially from the Darcy-scale corefloods, to understanding in detail how the foam comes into existence and is transported in the porous network on a microscopic level. We will follow this decomposition to explain each phenomenon.
Gas mobility reduction
Foam affects flow in porous media in multiple ways. To overcome the problems inherent to gas or water injections in oil reservoirs such as channeling and gravity segregation, foam is used to create a thicker fluid that will propagate in a more uniform manner through the rock. In this manner, foams can increase sweep efficiency. This enables a significant amount of fluid diversion into low permeability areas in heterogeneous or fractured media (Li et al. 2013). The effect of foam is usually quantified by a mobility reduction factor defined simply as (Schramm 1994): Where ∆ is the measured steady state foam pressure drop and ∆ is the pressure drop for identical injection conditions without foaming agent. In Figure 1-6 we show some typical mobility reduction factors in terms of gas types at 50°C and 30 bars in Berea cores.
Figure 1-6: Mobility reduction factors in terms of gas type at 50°C and 30 bars. Taken from Aarra et al. (2014). PV signifies the Pore Volume inside the core.
Gas mobility reduction – Microscopic explanation
For foam transport in water-wet media, the water and gas phases rearrange upon entering the medium. The wetting phase occupies predominantly the smaller pores and channels and can propagate through this network unaffected by the existence of a gas phase, which is situated in larger pores as isolated bubbles. Liquid can also circulate through the continuous network of Plateau Borders. Therefore if we describe the foam by its individual constituents, both the viscosity and relative permeability of the aqueous phase are unchanged by the existence of the foam (Bernard and Jacobs 1965). However it is established that the creation and propagation of a foam phase in porous media can affect the viscosity and permeability of the gas phase (Bernard and Holm 1964). The mobility reduction is therefore explained by two foam effects in parallel. We can see this by looking at the definition of the gas phase mobility:
We can make the distinction between continuous and discontinuous gas foams. Falls et al. (1988) summarize foam classification in porous media and provide explanatory diagrams. We display these diagrams in Figure 1-7. They describe two types of discontinuous-gas foams, where bubble trains transport gas across the medium, in opposition to continuous-gas foams, in which at least one percolating passage exists across the medium uninhibited by liquid lamellae. They note that in terms of mobility, the continuous gas foams only affect the relative permeability of the gas by blocking certain channels, and leave the bubbles trapped and immobile. On the contrary, discontinuous gas foams affect both relative permeability of the gas and viscosity of the gas, by forcing lamellae to propagate through the medium.
Table of contents :
Introduction
Motivation
Definition of the Problem
Thesis structure
State of the Art
Bulk foam generalities
Interfacial considerations
Bulk foam structure
Foam Stability
Bulk foams – Rheology
Foams in porous media
Basic petrophysical concepts
Foam effect on flow
Modelling foam in porous media
Population balance models
Synthesis and proposed research themes
Foam trapping and flow heterogeneity
Foam bubble size distributions and their effect on flow
The appropriate scale for describing foam phenomena
Foam Flow in a Micromodel: Data Acquisition and Transformation
Preamble
Article: New insights of foam flow dynamics in a high-complexity 2D micromodel
Introduction
Materials and Methods
Micromodel
Fluids
Microfluidic setup
Data acquisition and analysis tools
Experimental procedure
Image acquisition strategy
Image processing and exploitation
Results
Bubble creation/destruction mechanisms
Bubble size distributions
Comparison of velocity maps
Discussion
Bubble velocity and size relationship
Table of contents
Preferential path flow for larger bubbles
Local structural relationships
Conclusion and perspectives
Micromodel Experiments: Parameter Exploration
Preamble
Injection parameters
Injection rate
Gas fraction
Reversed
Injection method
Inlet foam distributions
Correlation of injection parameters
Foam distribution correlations
Foam distributions – macroscopic injection parameter correlations
Observables
Pressure drop
Viscosity
Longitudinal section flow distribution
Dataset flow intermittency
Passage activation
Trapped fraction
Bubble flow deviation from pressure gradient
Outlet foam distributions
Inlet/Outlet evolution ratios
Total bubble perimeter
Bubble specific surface area (SSA)
Correlation of observables
Correlation of trapped foam variables
Pressure gradient and bubble size distribution parameters
Viscosity and bubble deviation angle
Injection – Observable relationships
Viscosity relationships
Trapped fraction sources
Conclusion
Predicting Local Flow from Structural Parameters: A Machine Learning Approach
Preamble
Modelling framework and goals
Classification types
Machine learning algorithms
K-nearest neighbors
Logistic regression
Decision tree
Table of contents
Random Forest
Structural feature space
Throat structural features:
Pore structural features
Algorithm selection
Metric used
Preliminary algorithm comparison
Random Forest feature importance
Pore activity classification
Throat activity classification
Generalization
Conclusion
Describing High-Velocity Flow Areas Using a Network-Spanning Graph Model
Preamble
Article: Accessing preferential foam flow paths in 2D micromodel using a graph-based 2-parameter model
Introduction
Materials and methods
Foam data acquisition procedure and injection conditions
Micromodel structure and decomposition
Path-proposing algorithm
Experimental path match
Results
1-parameter model – description and path properties
Experimental match: 1-parameter model
2-parameter model – description and path properties
Experimental match: 2-parameter model
Discussion
Path-based flow characterization viability
Difference between experiments
Conclusion
Model Generalization
Application to a reversed orientation experiment
Generalization to a different 2D porous medium
Experiment 7, Géraud et al. (2016)
Experiment 43, Géraud et al. (2016)
Discussion on optimal exponents
Returning to local structural features from graph-based characteristics
Conclusion
General Conclusion
