Quantitative determination of the threshold pressure for a discontinuous phase to pass through a constriction using microscale simulation
Introduction
Although advanced enhanced oil recovery methods such as chemical flooding (Dang et al., 2018; Varel et al., 2021), thermal recovery (Ding et al., 2019; Lyu et al., 2018), and gas flooding (Afzali et al., 2018) have been developed to enhance oil recovery, water flooding is still of interest due to the abundant availability of water. However, the recovery of oil after water flooding is low, typically in the range of 30%-40% (Dai and Wang, 2014; Roberts et al., 2003). At the small scale, oil recovery is limited by its trapping in the pore space. Due to pore structure and wettability, the discontinuous phase can present different configurations such as pendular oil rings, funicular oil and isolated oil (Guo et al., 2019). The mechanism of displacement for each configuration might be different. The isolated oil can flow by entrainment of water and be trapped at the pore constriction that also hinders the flow of water. Therefore, understanding the mechanism of displacement of an isolated discontinuous phase is a key factor for the improvement of water flooding efficiency.
Discontinuous phase motion through constrictions is primarily controlled by two opposing forces: viscous and capillary (Chao et al., 2019; Dai et al., 2016; Liang et al., 2021; Morrow, 1979; Tsai and Miksis, 1994; Wang et al., 2017; Zhang et al., 2011; Zhang et al., 2017). The difference in pressure around the fluid-fluid interface at the pore scale varies due to viscous forces. This pressure must overcome the difference in capillary pressure needed to force the discontinuous phase through the constriction. This phenomenon, known as the Jamin effect, was first observed by Jamin (1860). Gardescu (1930) performed an analytical study of the pressure necessary for a gas bubble to be forced into a constriction and developed a Jamin effect expression of resistance based on the pressure at the interface. Lenormand et al. (1983) studied the displacement of trapped blobs in a duct and showed that there exists a critical pressure for which the blob gains access into the duct. Iassonov and Beresnev (2008) studied the entrapment of droplet due to capillary forces in a converging-diverging channel. However, the trapped droplet was assimiliated to a continuous phase since it spans to several adjacent pores. Rücker et al. (2015) used a three-dimensional micromodel to clarify the process of formation of discontinuous non-wetting phase and the ganglion dynamics. Their results have shown that although viscous forces are insufficient for ganglion mobilization, the oil ganglion can be mobilized due to coalescence that occurs after snap-off during imbibition process. Simon et al. (2012) designed a constricted microfluidic device based on the capillary pressure to study the mobilization of a trapped discontinuous phase. However, they could not establish the critical condition for droplet mobilization since the viscous pressure could not be obtained. Moreover, Helland (2016) stated that the local capillary pressure between the discontinuous and continuous phase changes through the pore and throat, and Singh et al. (2017) showed that the local capillary pressure on the side of trapped oil decreases and creates a capillary pressure gradient across the pore resulting in a snap-off which will lead to a trapping of oil droplet. Although many papers have studied the competition of viscous and capillary forces, they have mainly focused on droplet snap-off in the constriction (Datta et al., 2014a, 2014b; Ransohoff et al., 1987; Roof, 1970; Tsai and Miksis, 1997). Recently, Liang et al. (2015) used these pressures to establish a mathematical model that gives the minimum pressure at which the droplet can pass through a constriction, considering also the inclination of the channel. A relationship to define the critical pressure was established through dimensional analysis by Liu et al. (2021). However, they could not determine the threshold pressure at the microscale level quantitatively since the driving pressure of a droplet is dependent on the scale, that is to say, the smaller the scale, the greater the pressure (Lundstrom, 1996).
Many factors such as the wettability condition of the rock surface (Alyafei and Blunt, 2016; Avendaño et al., 2019; Blunt, 1997; Dai and Lin, 2020; Li et al., 2020; Wang et al., 2009), the pore structure (Liu et al., 2019; Long et al., 2019), and the fluid properties are involved in the determination of threshold pressure since they are responsible for droplet entrapment in the pore space. Johannesen and Graue (2007) investigated the effects of wettability on mobilization of remaining oil and found that trapped oil could be mobilized most easily at moderately water-wet to neutrally-wet conditions. The effect of pore constriction size ratio has been extensively studied by Wardlaw (1982) who concluded that the entrapment of residual oil in a simplified pore having a constriction is a function of the shape of the meniscus as it enters and crosses a restriction in the pore space. The interface shape is dependent on wettability and pore geometry (Tsai and Miksis, 1994). Nguyen et al. (2006) revealed that a significant difference in size between the radius of a pore-body and the radius of a constriction favors capillary entrapment. Similarly, Pentland et al. (2012) demonstrated that more trapping occurs more in narrower pore throats than in wider pore throats. The condition of breakthrough of a discontinuous phase through a constriction is also dependent on the droplet size. Long et al. (2019) investigated the critical condition at which a long droplet could be displaced through a constriction. These studies have deepened the understanding of the conditions of discontinuous phase flow. However, in porous media, the droplets and pore constrictions sizes are in the range of nanometers to micrometers, which are difficult to implement experimentally. Moreover, the viscous pressure drop and capillary pressure are hard to measure (Asadi et al., 2014; Raeini et al., 2014). With these considerations, numerical approaches have been developed to study the fundamental mechanisms of residual oil movement at the microscale and have shown significant advantages over experimental studies.
Recently, the study of porous media has expanded considerably due to the rapid development of computational capacity (Blunt et al., 1992; Blunt et al., 2013; Shams et al., 2018; Zhu et al., 2016). A series of pore-scale numerical approaches have been developed for two-phase flow. The lattice/particle based methods such as the lattice Boltzmann method (LBM), topological methods such as pore-network models (PNM), and direct numerical simulation (DNS) method are the most widely used methods. Among these methods, the PNM has high computational efficiency because of the idealization of pore space, but it also restricts the predictive capability and accuracy of PNM. Zhao et al. (2019) found that only LBM could model leading films and corner flow in strong imbibition, however, computing demand remains a challenge. Furthermore, they showed that no single method performed well in all situations. In this study, a DNS method has been used due to its consistency. Also, the displacement mechanisms and flow characteristics are covered in the DNS method and it can be applied to simulate complex multiphase flow like imbibition processes in complex heterogeneous pore space. However, the physics of the interface movement is not included in Navier−Stokes (N−S) equations and it should be coupled with interface capturing models for multiphase flow, such as phase field method (PFM), volume of fluid (VOF), and level set method (LS). Although, the level set model can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects, it often significantly violates mass conservation. The VOF method makes it very easy to follow shapes that change topology, but it is less elegant than the level set method in terms of computational accuracy. The phase-field method treats the interface as a physically diffuse thin layer and can guarantee mass conservation in the simulation. Considering numerical instability arising at the interface region when the interfacial tension becomes a dominant factor if using the VOF method or level set method for pore-scale simulation, the phase-field method is applied to track the two-phase interface during our simulation. Raeini et al. (2013) used a finite volume method to model two-phase flow in a star-shaped pore-throat system. In their work, they combined the viscous-capillary pressure competition and the Darcy-like equation to predict the trapping and displacement of a discontinuous phase. However, a discontinuous phase movement is a piston-like displacement and when one include the layer flow, such as in their work, the flow will no longer be piston-like. Among the existing grid-based methods, the phase field has become a popular tool for numerical simulation of complex interfaces (Ganapathy et al., 2013; Sun and Beckermann, 2007; Takada and Tomiyama, 2006; Zhu et al., 2019b). Zhu et al. (2016) applied this approach to investigate the dynamic contact angle on an oil slug displacement with water injection. Another direct simulation method is density functional hydrodynamics (DFH) which is based on density functional theory and continuum mechanics (Dinariev, 1996). Dinariev and Evseev (2018) studied different multiphase flow phenomena applying the density functional method in three dimensions. Their results have shown that the DFH can be efficiently applied in solving multiphase flow problems. Although, it has the advantage of increasing the computational accuracy without increasing computation time, the computational cost is still larger than that of two-dimensional models.
In recent years, many authors have simulated multiphase flow in complex three-dimensional geometries to study a variety of displacement phenomena (Aljasmi and Sahimi, 2021; Blunt et al., 2013; Piri and Blunt, 2005; Raeini et al., 2012). Shams et al. (2021) performed three-dimensional (3-D) computational fluid dynamics simulations directly on images obtained from an X-ray micro-CT two-phase flow experiment in order to predict snap-off and trapping events. Anderson (2018) performed a capillary-trapping experiment to identify crucial pore-throat and pore-body dimensions using 3-D image data taken with X-ray micro-computed tomography after implementing a morphological opening transform with a structural element of increasing size in the pore space. Herring et al. (2016) performed an experiment on a 3-D model to study the wettability alteration and trapping of a non-wetting phase. They found that the capillary pressure decreases with the increase of contact angle resulting in increase of buoyancy effects. However, in our work, similarly to Lenormand et al. (1983), we neglect the effect of gravity, therefore, the buoyancy effect is not considered. Armstrong et al. (2016) applied the density functional method to study different multiphase flow phenomena. They used a 3-D model to study the snap-off of a non-wetting phase passing through a constriction. They have shown that when the non-wetting phase is injected in a capillary filled with a wetting phase, while the non-wetting occupies the center of the pore, a film of the wetting phase is formed on contact with the solid boundary. The presence of this film in the constriction results in the snap-off. While this work is valuable to predict the behavior in specific circumstances, it does not directly lead to a general quantification of processes of interest, such as, in our case, the mobilization of trapped oil. Krummel et al. (2013) described a method for fully visualizing the flow of two immiscible liquids at the pore scale using a model of a three-dimensional porous medium. They demonstrated that as the capillary number Ca increases, the size of the discontinuous phase, and even the total amount of residual oil, significantly reduces. This trend includes an opposition between the viscous force and the capillary force necessary to push oil through the pores of the channel. However, they did not consider the effect of wettability. Hoang et al. (2017) used a three-dimensional numerical model to examine the behavior of a droplet in a planar constriction microchannel. However, their study was limited in the characterization of three regimes of the droplet dynamics, namely, trap, squeeze and breakup, depending on capillary number (Ca) and contraction ratio (C). Zinchenko and Davis (2006) performed a 3-D simulation of a droplet passing through constriction made of spheres. They suggested that there exists a critical capillary number below which the droplet gets trapped. In this paper, while we use two-dimensional simulation on simple geometries, we are able to derive criteria for ganglion mobilization applicable to a wide range of pore geometry and wettability.
Understanding two-phase flow hydrodynamics in microchannel is not only relevant for enhanced oil recovery but many fields including surface cleaning, medical purposes, and chemical and inkjet engineering applications (Seemann et al., 2012). The dynamics of the motion of the drop in constricted capillary channel has been extensively investigated in multiphase flow community. For instance, in biological applications, 3D bioprinting employs the notion of microfluidics for cell contact and deposition where cell is considered as a discontinuous phase while blood is seen as a continuous phase. Based on this, Nath et al. (2019) investigated the detection of cancer. Chai et al. (2014) experimentally studied the effect of geometry on pressure drop and on two-phase flow patterns. One straight microchannel and two microchannels with alternated expansions and constrictions were explored. Depending on the flow rates of both phases, they observed slug flow, annular, or single-phase liquid flow patterns. Using a boundary element technique, Khayat et al. (1997) investigated the effect of constriction entry geometry and rheology on droplet deformation. Later, they also investigated the effects of shear and elongation on droplet deformation numerically and experimentally in a hyperbolic convergent-divergent microchannel (Khayat et al., 2000). Using a finite element approach, Chung et al. (2009) examined the effect of viscoelasticity on drop dynamics in a planar contraction-expansion microchannel. Christafakis and Tsangaris (2008) studied the effect of capillary number (Ca), Reynolds number (Re), Weber number (We) and viscosity ratio on the droplet dynamics in a two-dimensional contraction microchannel. Similarly, Harvie et al. (2006, 2008) studied droplet dynamics in an axisymmetric contraction microchannel investigating the influence of the Re, Ca, and viscosity ratios on droplet deformation. They have shown that when surface tension strength is small and the Reynolds number is low, the droplet assumes the form of a “string of sausages”, due to the influence of large amplitude instabilities on the droplet surface. Mulligan et al. (2011) experimentally investigated droplet deformation and breakup in a planar hyperbolic contraction microchannel. Most of the works focused on droplet deformation, coalescence, break-up and pressure drop. Wu et al. (2015) studied the critical pressure for driving a red blood cell through a contracting microfluidic channel. They established a quantitative connection between the minimum pressure needed to drive a red blood cell through a contracting microfluidic channel and the rigidity of the cell membrane. However, the cell membrane is elastic and differs considerably from a fluid interface.
Many studies, through analytical analysis and experimental design have established critical conditions of droplet breakthrough in the constriction but could not quantitatively determine the threshold pressure. Therefore, this paper puts forward more accurate quantification of trapping and the onset of mobilization, which is vital for models of oil production and carbon storage. Water-oil two phase flow in microscale porous media is directly described based on the Navier-Stokes equation, and the two-phase interface position of the water continuous phase and oil discontinuous phase during a displacement is captured by the phase field method. The effects of physical parameters and flow conditions are then analyzed to improve the ability of the oil discontinuous phase passing through the constriction. The competition between the viscous pressure and the capillary pressure is investigated. Through regression analysis, a mathematical model is presented to quantitatively determine the threshold pressure.
Section snippets
Governing equations
The flow of an incompressible viscous fluid can be described by the continuity equation and the momentum equation. The continuity equation satisfies the mass conservation for fluid flow and is given by Bird et al. (1960):where u is the flow velocity of the fluid, m/s.
In the analysis of fluid motion, the Navier-Stokes is the starting equation which describes the momentum conservation in fluid flow. For an incompressible fluid, it can be expressed as:
Results and discussion
To study different characteristics of oil discontinuous flow, a capillary channel with a constriction is adopted, as shown in Fig. 4 . The oil as a discontinuous phase is placed in a capillary channel filled with water. Water is injected from left to right to generate enough pressure to displace the oil. The capillary channel has diameter and length , while the constriction has radius =0.48a and length of =0.8a and it is situated at position 4a. An oil droplet is initially placed
Implications and limitations
Quantitative studies at the level of a pore with simplified shapes can provide qualitative explanations for experiments performed with porous media. For instance, our results are relevant to understand the reasons behind the permeability jail observed in tight sandstones. The permeability jail refers to a range of water saturation values, where flow conductance is significantly reduced resulting in a low relative permeability (Mo et al., 2019). The presence of discontinuous phases and low
Conclusions
In this paper, two-dimensional numerical simulations were performed at the microscale based on the Navier Stokes equations and the phase field method to study the conditions required for a discontinuous phase (oil) to pass through a constriction and its dynamic behavior. The injection of water into a capillary channel under an imposed pressure has been studied, and the effect of wettability, radius of the constriction and radius of the droplet have been investigated, together with their coupled
CRediT authorship contribution statement
Gloire Imani: Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. Lei Zhang: Conceptualization, Writing – review & editing, Supervision. Martin J. Blunt: Resources, Supervision, Writing – review & editing. Chao Xu: Investigation, Visualization. Yaohao Guo: Investigation. Hai Sun: Investigation, Visualization. Jun Yao: Resources, Supervision.
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgements
We would like to express appreciation to the following financial support: The National Natural Science Foundation of China (No. 12172334, 52122402, 52034010, 52174051, 51936001), Shandong Provincial Natural Science Foundation (No. ZR2021ME029), Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R69).
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