Direct numerical simulation of trapped-phase recirculation at low capillary number
Graphical Abstract
Introduction
Transport phenomena at pore-scale have gained great prominence in recent years since they are of significant importance in a variety of areas including enhanced oil recovery (Zhao et al., 2010), soil remediation (Aminnaji et al., 2019), and carbon/hydrogen storage (Ebigbo et al., 2013; Riazi et al., 2011). Various microscale forces including capillary, viscous, inertial, and gravitational forces play an important role in fluid dynamics especially at the interface (Dejam et al., 2014; Ferrari and Lunati, 2014, 2013; Mashayekhizadeh et al., 2011; Méheust et al., 2002; Raeini et al., 2014). The roles of rock and fluid properties that can significantly affect the displacement process are also important (Avraam and Payatakes, 1995).
Specifically, pore-scale investigation is a strong tool that can shed light on the subsurface process at the microscale. To investigate the physics of fluid flow at pore-scale, 2D glass micromodels can be used in conjunction with fluorescence microscopy and micro-particle image velocimetry (micro PIV) to directly observe and measure the pore-scale processes (Jiménez‐Martínez et al., 2017; Roman et al., 2016, Roman et al., 2019). Roman et al. (2016) investigated the velocity profiles in two-phase flow through the heterogeneous sandstone-replica micromodel using micro PIV. The results depicted recirculation motion inside the immobile pockets of the wetting phase due to momentum transmitted by the flowing oil. Kazemifar et al. (2016) reported shear-induced circulation in trapped water ganglia during post-front passage displacement of water by supercritical CO2 in a homogenous micromodel. In concordance with the Kazemifar et al. work, Li et al (2017) observed the shear-induced circulation during post-front passage flow in a heterogeneous micromodel for a supercritical CO2/water system. They determined CO2 flow directions with the aim of circulation within the trapped water phase, as it is very difficult to directly measure CO2 flow. Using micro PIV, Heshmati and Piri (2018) observed the recirculation process in a pore-doublet model in the viscous dominant regime. They reported that the shear stress applied to the trapped non-wetting phase was intensified as the viscous forces increased. The effect of increasing injection velocity on the velocity of the non-wetting phase was less noticeable, indicating the presence of a slip boundary between the fluids. Zarikos et al. (2018) analyzed the interaction of capillarity and momentum transfer between the two immiscible phases for different non-wetting trapped globules with various sizes. In contrast with stagnant areas where the viscous drag applied on the fluid-fluid interface is negligible, in nonzero velocity areas the momentum transfers cause circulations within the trapped phase. The circulation velocity is maximum near the fluid-fluid interface and an increase in capillary number of invading phase increases the circulation velocity as well as the shift in circulation centroid toward high-velocity region. Similar to the circulation phenomenon process for trapped phases in porous media, there are other two-phase systems in which this process is noticeable (Bhaga and Weber, 1981; Maxworthy et al., 1996). The effect of capillary numbers, slug size, and viscosity ratios on internal recirculation within two phases in capillary channels have been investigated in several experimental (Kashid et al., 2007; Liu et al., 2017; Ma et al., 2014) and numerical studies (Kashid et al., 2005; Sarrazin et al., 2006).
Contrary to the recent studies where the velocity fields were observed in only one phase (usually the water phase), two-phase flow can be explored using an appropriate two-phase flow simulator. Pore-network modeling as a low-cost computational tool has advanced during recent years in both quasi-static and dynamic modes (Blunt, 2001; Oren et al., 1998). Direct numerical simulation methods including smoothed particle hydrodynamic (SPH) as a meshless method (Hirschler et al., 2016), the lattice Boltzmann method (LBM) (Kang et al., 2002), and computational fluid dynamic methods (CFD) (Raeini et al., 2012) can perform simulation at high resolution of the real structure of pore spaces. Fakhari et al. (2018) using the lattice Boltzmann model for the simulation of multiphase flows of water/CO2 systems in a micromodel, observed the recirculating flows in the still-flowing CO2 phase as the impacts of inertial forces. Direct numerical simulation of pore-scale processes by CFD methods have gained great prominence due to their ability to simulate fluid flow for wide density and viscosity ratio ranges (Meakin and Tartakovsky, 2009). In a recent study, using direct numerical simulation, the results of drainage processes in a 2D cavity showed better mixing processes and larger interfacial mass transfer in the presence of recirculation inside the cavity (Maes and Soulaine, 2018). Shams et al. (2018) analyzed the viscous coupling effects as a function of the viscosity ratio, wettability, and varying fluid configurations in circular capillary tubes. Raeini et al. (2014) studied the snap-off and layer flow to analyze the impact of geometry and flow rate on the hydraulic conductivity of disconnected ganglia using volume-of-fluid (VOF) based finite-volume method. To consider interfacial tension, various approaches can be applied to VOF; the continuous surface stress (CSS) method (Gueyffier et al., 1999), the continuous surface force (CSF) method (Brackbill et al., 1992), and the sharp surface force (SSF) (Francois et al., 2006; Raeini et al., 2012). In these approaches, it is very difficult to predict flow at low capillary numbers where the capillary forces are dominant, which can result in non-physical velocity currents around the interface (Raeini et al., 2014). To eliminate the non-physical behavior from the numerical solution for the cases of capillary forces dominancy, the filtered surface force (FSF) is introduced (Raeini et al., 2012). Specifically, the capillary pressure, dynamic pressure gradients, and viscous and inertial forces are separately calculated to prevent introducing instabilities in the numerical equations. It is shown that the spurious current is minimum around the interface when the FSF is used (Pavuluri et al., 2018).
The concept and application of the recirculation phenomenon are of great importance especially for processes contending with mass transport (Kazemifar et al., 2016; Li et al., 2017; Maes and Soulaine, 2018). Recently some research has been carried out to investigate this phenomenon in both experimental (Roman et al., 2016; Zarikos et al., 2018) and numerical approaches (Fakhari et al., 2018; Maes and Soulaine, 2018). However, they have not discussed the role of displacement mechanism and fluid configurations on the recirculation process. Also, in the case of experimental investigations (Kazemifar et al., 2016; Roman et al., 2016, Roman et al., 2019; Zarikos et al., 2018), the velocity profile is only presented inside the trapped phase. Maes et al. (2018) investigated the role of recirculation phenomena on the mass transfer process for various viscosity ratios; however, they did not consider the effects of injection velocity and surface tension as other dynamic parameters involved in the capillary number. Zarikos et al. (2018) investigated how the center of circulation and average velocity of the trapped phase moved for various capillary numbers, but they did not discuss the behavior of capillary number for various viscosity ratios. This work sheds light on the recirculation process for both imbibition and drainage mechanisms for various fluid configurations at low capillary number. Also, using drag force analysis on fluid-fluid and fluid-solid interfaces, we comprehensively discuss the behavior of recirculation phenomena. In addition, we discuss the analysis of dynamic parameters including injection velocity, viscosity ratio, and surface tension individually on the behavior of recirculation.
We used the FSF formulation of the VOF method implemented in poreFoam (separately available solver for OpenFOAM) to simulate two-phase flow inside a 2D pore-doublet model at low capillary number. In this study, we addressed the question of how recirculation behavior changes under various dynamic and static conditions. As the multiphase-flow simulation is carried out in the real structure of pore spaces, this study is new to those that use simplified models such as pore network modeling. Also, using the FSF formulation of the VOF method minimizes the spurious currents introduced in numerical equations. On the other hand, the simulation run time is relatively long due to small time steps that should be used, especially at low capillary numbers. The main limitation is that even with this advanced method, the simulation of fluid flow for the invading phase capillary numbers lower than 10−7 cannot be performed. Also, because of high computational cost, investigation of more complex porous media was not feasible. The increase in mesh size and shortness of time steps might have led to a long simulation time.
The results of this work can be applied for cases where recirculation plays a pivotal role along with mass transfer, such as CO2 storage. Analyzing drag force on the fluid-fluid interface is also important in processes where the balance of forces on the interface determines the mobilization or trapping of the droplet during enhanced oil recovery methods (Dejam and Hassanzadeh, 2018; Olayiwola and Dejam, 2019; Yang et al., 2020, 2019). Fig. 1 is a schematic of the problem statement. This paper is organized as follows: Section 2 presents the numerical method, Section 3 gives the model validation and verification, Section 4 discusses the results, and finally, Section 5 summarizes the conclusions.
Section snippets
Mathematical formulation
This section described the mathematical model we used to solve two-phase flow. The Navier-Stokes equation describes the dynamics of two immiscible incompressible flow. Since part of the solutions are interface dynamics, a moving boundary problem is required (Batchelor and Batchelor, 2000). For an isothermal condition, the momentum-balance equation in terms of the viscous, inertial, capillary, and gravity forces is written as (Raeini et al., 2012):where U denotes the
Mesh independency
The mesh density of the network is governed in a way that no dependency between the discretization and variable can be observed. The OpenFOAM snappyHexMesh utility was used for domain meshing. This tool allows good refinement of grids in the various parts of a pore assembly. The mesh quality was confirmed by analysis of grid refinement, where numbers of mesh elements were increased gradually from 8800 to 79,000, corresponding to the number of cells in the narrowest throat from 9 to 11, 13, and
Results
This section considers the simulation results of two individual displacement mechanisms i.e. imbibition and drainage through the pore-doublet model. In both cases three patches of defending phase were trapped in the right and left sides of the pore-doublet model as well as the bottom dead-end. This fluid configuration within the pore assembly not only facilitated the analysis of recirculation behavior in various arrangements to the flow direction but also provided the same conditions for both
Summary and conclusion
The numerical simulation of the immiscible displacement process through a pore-doublet model was performed in such a way to set trapped-phase patches in specified locations of the pore assembly. To capture the pore-scale processes at low capillary number we used a filtered surface formulation of direct pore-scale modeling for drainage and imbibition mechanisms.
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The results showed, in the case of ranges capillary number of invading phase analyzed in this study (in order of 1 × 10−2 to 1 × 10−7),
CRediT authorship contribution statement
Amir Hossein Mohammadi Alamooti: Conceptualization, Formal analysis, Methodology, Writing - original draft, Data curation, Software, Visualization, Investigation, Validation, Resources. Qumars Azizi: Conceptualization, Methodology, Formal analysis, Writing - original draft, Data curation, Software, Validation, Visualization, Investigation, Resources. Hossein Davarzani: Supervision, Funding acquisition, Project administration, Conceptualization, Resources.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We gratefully acknowledge the financial support provided to the PIVOTS project by the “Région Centre – Val de Loire” and the European Regional Development Fund.
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