Low Capillary Numbers Research Articles

Numerical investigation of multiphase flow through self-affine rough fractures

Multiphase flow through fractures has great significance in subsurface energy recovery and gas storage applications. Different fracture and flow properties affect flow through a fracture which is difficult to control in laboratory experiments. Here, we perform lattice Boltzmann simulations in an ensemble of synthetically generated fractures. Drainage simulations are performed at different capillary numbers, wettability, and viscosity ratios. We track the invading front and quantify breakthrough saturations and show that roughness and wettability have a strong effect on fluid invasion through a complex fracture. Invading a more viscous fluid results in more stable displacement regardless of the capillary number while at very low capillary numbers, fluid migration is dependent on the inherent structure of the fracture. We develop a fluid displacement phase diagram in a single rough fracture and compare our results from that in the literature. Finally, we extend the phase diagrams across multiple fractures and demonstrate the importance of natural fracture features of roughness and wettability in identifying stable versus unstable displacement regimes during multiphase flow through rough fractures. Our work presents an end-to-end numerical pathway for testing on experimental data and expanding numerical data sets for testing combinations of different physical phenomenon and make valuable predictions on fluid flow through rough fractures.

Lattice Boltzmann modeling of forced imbibition dynamics in dual-wetted porous media

Forced imbibition dynamics are critical for enhancing recovery rates in reservoirs, as efficient fluid displacement directly impacts resource extraction. This study employs the Zou-He velocity boundary condition and a modified convective boundary condition (mCBC) within the multi-component Shan-Chen lattice Boltzmann method (LBM), thus facilitating unobstructed flow of multi-component fluids at the outlet while maintaining constant outlet pressure. The model's validity was established through outflow tests of immiscible droplets in channels. Its accuracy was further confirmed by comparing results from forced imbibition dynamics tests in dual-wetted pores with theoretical predictions. A symmetrical porous medium was constructed using the stacking method, achieving dual wettability by fixing the contact angle in the upper region and varying it in the lower region. We performed 20 simulation sets under unfavorable viscosity ratios and varying capillary numbers, focusing on overall displacement efficiency before and after breakthrough, and employing energy balance equations to evaluate the dominant forces. Results reveal that capillary forces predominantly dictate forced imbibition dynamics in low capillary number scenarios. In strongly wetted regions, the invading fluid fully occupies pore spaces; however, in weakly wetted regions, displacement efficiency significantly declines as wettability approaches neutrality, even nearing zero. As capillary numbers increase, viscous forces become more prominent, controlling dynamics and leading to fingering and trapping of defending fluids. In weakly wetted areas, the increasing influence of viscous forces enhances fluid displacement, resulting in significant improvements in overall displacement efficiency compared to conditions dominated by capillary forces.

Surface-active microrobots can propel through blood faster than inert microrobots

Abstract Microrobots that can move through a network of blood vessels have promising medical applications. Blood contains a high volume fraction of blood cells, so in order for a microrobot to move through the blood, it must propel itself by rearranging the surrounding blood cells. However, swimming form effective for propulsion in blood is unknown. This study shows numerically that a surface-active microrobot, such as a squirmer, is more efficient in moving through blood than an inert microrobot. This is because the surface velocity of the microrobot steers the blood cells laterally, allowing them to propel themselves into the hole they are digging. When the microrobot size is comparable to a red blood cell or when the microrobot operates under a low Capillary number, the puller microrobot swims faster than the pusher microrobot. The trend reverses under considerably smaller microrobot sizes or high Capillary number scenarios. Additionally, the swimming speed is strongly dependent on the hematocrit and magnetic torque used to control the microrobot orientation. A comparative analysis between the squirmer and Janus squirmer models underscores the extensive applicability of the squirmer model. The obtained results provide new insight into the design of microrobots propelled efficiently through blood, paving the way for innovative medical applications.

Pore-scale study of three-dimensional three-phase dynamic behaviors and displacement processes in porous media

Carbon dioxide geological sequestration is a key method to alleviate global warming and enhance oil recovery, where the three-phase displacement processes of oil, water, and carbon dioxide gas in porous media are frequently encountered. In this study, a three-phase three-dimensional lattice Boltzmann method coupled with special wettability and outlet boundary schemes is adopted to simulate the three-phase displacement processes in porous media. The method is validated by the contact angles on a curved surface and droplet flowing through the outlet boundary. With this method, the influences of capillary number, wettability, and local large pores on three-phase flow are investigated. In particular, different dynamic behaviors of fluids are observed at the pore scale, such as bypass-double displacement, stop-wait displacement, burst displacement, snap-off trapping, and corner flow. Further, Euler number and oil saturation are calculated to quantitatively characterize the fluidic morphology and displacement efficiency under different conditions. For all three phases, the Euler number of low capillary number, strong water-wet, and structures with large and medium pores is relatively low, indicating that the morphology of fluids is more connective. For enhancing oil recovery efficiency, high capillary number and strong water-wet structures are beneficial.

Numerical and experimental investigation of the effect of interstitial liquid viscosity on the collapse of wet granular columns

The collapse of wet granular columns with high- and low-viscosity interstitial liquid is investigated through experiments and numerical simulations. By incorporating both capillary and viscous force models of interstitial liquid in the Discrete Element Method (DEM), simulations have been conducted and reproduced consistent collapse processes with the experiment. The influence of water content, contact angle, particle diameter, liquid surface tension, and viscosity is explored over a large parameter space using DEM. For wet granular columns with low-viscosity interstitial liquid of water, the final profile of the collapsed column can be predicted by Bond number (Bo), a ratio of particle gravity to capillary force. For wet granular columns where the inertial effect dominates and both capillary and viscous forces are significant, dimensional analysis is employed to characterize the collapse, indicating that the collapsed profile can be predicted by Bo and Galileo number (Ga). Using water and silicone oils with different viscosities as interstitial liquids, simulations show that the collapsed profile depends on Bo with a low capillary number (Ca ∼ 10−3), Bo and Ga when Ca is intermediate (Ca ∼ 10−1). When Ca is high (Ca > 100), the final profile is solely controlled by Bo and a rolling-collapsed regime is observed.

Pore-scale investigation of forced imbibition in porous rocks through interface curvature and pore topology analysis

Forced imbibition, the invasion of a wetting fluid into porous rocks, plays an important role in the effective exploitation of hydrocarbon resources and the geological sequestration of carbon dioxide. However, the interface dynamics influenced by complex topology commonly leads to non-wetting fluid trapping. Particularly, the underlying mechanisms under viscously unfavorable conditions remain unclear. This study employs a direct numerical simulation method to simulate forced imbibition through the reconstructed digital rocks of sandstone. The interface dynamics and fluid–fluid interactions are investigated through transient simulations, while the pore topology metrics are introduced to analyze the impact on steady-state residual fluid distribution obtained by a pseudo-transient scheme. The results show that the cooperative pore-filling process promoted by corner flow is dominant at low capillary numbers. This leads to unstable inlet pressure, mass flow, and interface curvature, which correspond to complicated interface dynamics and higher residual fluid saturation. During forced imbibition, the interface curvature gradually increases, with the pore-filling mechanisms involving the cooperation of main terminal meniscus movement and arc menisci filling. Complex topology with small diameter pores may result in the destabilization of interface curvature. The residual fluid saturation is negatively correlated with porosity and pore throat size, and positively correlated with tortuosity and aspect ratio. A large mean coordination number characterizing global connectivity promotes imbibition. However, high connectivity characterized by the standardized Euler number corresponding to small pores is associated with a high probability of non-wetting fluid trapping.

Modeling of Pore-Scale Capillary-Dominated Flow and Bubble Detachment in PEM Water Electrolyzer Anodes Using the Volume of Fluid Method

Fluid dynamics models complement expensive experiments with limited measurement accuracy that investigate the mass transport in PEM water electrolysis. Here, a first-principle microscale model for oxygen transport is successfully validated that accounts for (1) uncertain transport processes in catalyst layers, (2) numerically challenging capillary-dominated two-phase flow and (3) bubble detachments in channels. We developed algorithms for the stochastic generation of geometries and for the coupling of flow and transport processes. The flow model is based on the volume of fluid method and reproduces experimentally measured pressure drops and bubble velocities within minichannels with a 30% and 20% accuracy, respectively, provided that the capillary number is above 2.1 × 10−7. At lower capillary numbers, excessive spurious currents occur. Correspondingly, two-phase flow simulations within the porous transport layers are stable at current densities above 0.5 A cm−2 and match operando gas saturation measurements within a 20% margin at relevant locations. The simulated bubble detachments occur at pore throats that agree with porosimetry and microfluidic experiments. The presented model allows explaining and optimizing mass transport processes in channels and porous transport layers. These were found to be negligibly sensitive to transport resistances within the catalyst layer, providing information on boundary conditions for future catalyst layer models.

The creeping movement of a soft colloidal particle normal to a planar interface

A methodological blend of analytical and numerical strategies employing collocation techniques is presented to investigate the task of describing the Stokes flow generated by a soft particle (composite sphere) moving perpendicularly to a planar interface of infinite extent, separating two semi-infinite, immiscible viscous fluid domains. The particle consists of a solid core enclosed by a porous membrane allowing fluid passage. The movement of the soft nanoparticle has been examined through a continuum mathematical model. This model incorporates the Stokes and Brinkman equations, accounting for the hydrodynamic fields both outside and within the porous membrane layer, respectively. The motion is investigated under conditions characterized by low Reynolds and capillary numbers, where the interface experiences negligible deformation. The solution combines cylindrical and spherical fundamental solutions via superposition. Initially, the boundary conditions at the fluid–fluid interface are satisfied utilizing Fourier–Bessel transforms, subsequently addressing the conditions at the soft particle's surface through a collocation method. The normalized drag force exerted on the particle is accurately calculated, exhibiting robust convergence across various geometric and physical parameters. These findings are effectively visualized via graphs and tables. We juxtapose our drag force coefficient results with established literature data, particularly focusing on the extreme cases. The findings highlight the substantial impact of the interface on the drag force coefficient. Across the full range of viscosity ratios, the normalized drag force decreases as the relative thickness of the porous layer increases. These results enhance the understanding of practical systems and industrial processes such as sedimentation, flotation, electrophoresis, and agglomeration.

Mechanism of temporal interface evolution and internal circulations during the droplet formation in a planar slit T-microchannel

The present study has numerically explored the mechanism of interface evolution and internal flow circulations during the droplet formation in two-phase flow through a planar T-microchannel. The two-dimensional unsteady form of the conservative level set equation coupled with Navier–Stokes equations has been solved using the finite element method. The range of parameters include the contact angle (θ) from 120° to 180°, and the flow rate ratio (Qr) from 0.1 to 10 for the low capillary number (Cac≤10−2). The present study indicates that surface wettability plays a crucial role in influencing the temporal evolution of the interface. The internal flow circulation in the droplet is controlled by the axial and radial velocities primarily influenced by shear stress. The newly introduced novel “interface-to-neck ratio” parameter has provided another platform to investigate the pinch-off dynamics of droplets. Moreover, the phenomenon of droplet pinch-off is primarily initiated and driven by the Laplace pressure, defined by three distinct approaches: the pressure difference method, the determination of the minimum local radius of curvature on the rear side, and a calculation of the neck width. The predictive correlations have been established to estimate the droplet characteristics as a function of the flow rate ratio and contact angle. The findings reported have significant implications for the design of droplet dispensing systems that depend on surface wettability as a critical regulating parameter.

High-Speed Generation of Microbubbles with Constant Cumulative Production in a Glass Capillary Microfluidic Bubble Generator.

This work reports a simple bubble generator for the high-speed generation of microbubbles with constant cumulative production. To achieve this, a gas-liquid co-flowing microfluidic device with a tiny capillary orifice as small as 5 μm is fabricated to produce monodisperse microbubbles. The diameter of the microbubbles can be adjusted precisely by tuning the input gas pressure and flow rate of the continuous liquid phase. The co-flowing structure ensures the uniformity of the generated microbubbles, and the surfactant in the liquid phase prevents coalescence of the collected microbubbles. The diameter coefficient of variation (CV) of the generated microbubbles can reach a minimum of 1.3%. Additionally, the relationship between microbubble diameter and the gas channel orifice is studied using the low Capillary number (Ca) and Weber number (We) of the liquid phase. Moreover, by maintaining a consistent gas input pressure, the CV of the cumulative microbubble volume can reach 3.6% regardless of the flow rate of the liquid phase. This method not only facilitates the generation of microbubbles with morphologic stability under variable flow conditions, but also ensures that the cumulative microbubble production over a certain period of time remains constant, which is important for the volume-dominated application of chromatographic analysis and the component analysis of natural gas.

Experimental study of liquid–liquid dispersion patterns in T-inlet microchannels with different junction angles

Liquid-liquid two-phase flow in T-inlet microchannels with different junction angles (θ = 30°, 60°, 90°, 120° and 150°) was investigated experimentally. Four flow regimes of the dispersed phase were identified, i.e., parallel flow, jetting, dripping and squeezing, and the distribution of flow regimes for the dispersed phase corresponding to variations in the junction angle was plotted. The consequences of varying junction angle and flow conditions in the squeezing regime on the generated droplet size were analyzed. The results indicate that low capillary number and large flow rate ratio are conducive to the formation of large-size droplets. For constant flow conditions, junction angle θ = 90° is detrimental to the formation of squeezing microdroplets. The increase in microchannel junction angle causes the droplet size to decrease until θ > 90°, where the droplet size increases with the junction angle. On the basis of experimental results, the scaling law correlation equations containing the junction angle for predicting the droplet length and droplet volume are proposed, respectively. The predicted values match well with the experimental data. The results of this work contribute to the enhancement of the monodispersity of microdroplets and the precise control over a wide range of the generated droplet size by adjusting the junction angle.

Dynamics of hydrogen storage in subsurface saline aquifers: A computational and experimental pore-scale displacement study

The catastrophic effects of climate change can be mitigated by transitioning to renewable energy sources such as wind, solar, and hydropower while utilizing hydrogen (H2) as an energy carrier. As renewable fuel sources are intermittent and location-specific, large-scale, long-term storage options for H2 must be explored with high necessity. Subsurface hydrogen storage in saline aquifers provides a vital scope to store H2 gas in available pore voids with minimal environmental risk. In this work, a highly heterogeneous porous micromodel was used to study the immiscible two-phase dynamics at high-temperature, high-pressure saline aquifer conditions using computational fluid dynamics (CFD) simulation based on volume of fluid (VOF) method. A set of microfluidic experiments were carried out under atmospheric conditions to validate the numerical model. The VOF model was observed to show good agreement with microfluidic experiments. An unstable displacement pattern with viscous fingering was observed during various flow conditions that captured high pressure (15 and 25 MPa) and temperature conditions (323 and 343 K). It was observed that fingering displacement of brine limits the storage spaces for hydrogen, which consequently follows the high permeable path. As a result, only 0.272 fraction of brine was invaded by the H2 at a temperature of 323 K and pressure of 15 MPa. A comparison between H2-brine and nitrogen (N2)-brine flow dynamics revealed a 46% less storage capacity of porous media for H2. Formations bearing high temperature (343 K) showed an 11.27% increase in H2 storage capacity, while pressure increase had negligible effect. At low capillary number (Ca), more snap-off effects resulted in higher trapping of H2 gas. This study aims to provide meaningful insights into the complexities of flow dynamics and displacement patterns that are crucial for optimizing hydrogen storage efficiency in saline aquifers and for potential ‘white’ hydrogen production.

Understanding droplet formation in T-shaped channels with magnetic field influence: A computational investigation

This study investigates the production of ferrofluid droplets in a T-junction geometry using the level set method and magnetic force manipulation in the three-dimensional. The analysis reveals key insights into droplet formation processes in four stages: entering, blocking, necking, and detachment. The results show that increasing the Capillary number leads to a significant decrease in volume for non-ferrofluid droplets. Application of a magnetic force enhances the balance of forces during droplet formation, directly impacting droplet volume. Moreover, increasing the magnetic Bond number substantially increases droplet volume, with a more pronounced effect at lower Capillary numbers. Modifying magnetic properties influences droplet volume, with doubling the magnetization results in a significant volume increase. Overall, magnetic forces emerge as a crucial control parameter for droplet volume in ferrofluid systems, offering potential applications in droplet-based technologies and microfluidic devices.

Oscillations of a spherical particle in the presence of a flat interface separating two fluid phases

An analytical–numerical approach based on collocation techniques is introduced to examine the challenge of characterizing Stokes flow induced by a spherical particle in perpendicular motion to a plane interface that divides two semi-infinite, immiscible viscous fluid phases. The movement is examined under the conditions of low Reynolds and capillary numbers, where the interface undergoes negligible deformation. We contemplate both the rectilinear oscillations along and the rotational oscillation around an axis that is perpendicular to the interface. A comprehensive solution is formed by combining basic solutions in both cylindrical and spherical coordinate systems through superposition. Firstly, the boundary conditions at the fluid–fluid interface are fulfilled through Fourier–Bessel transforms, followed by addressing the conditions at the particle’s surface using a boundary collocation scheme. The drag force and torque coefficients, both in-phase and out-of-phase, acting on the particle are determined with satisfactory convergence across different geometric and physical parameter values. These results are illustrated through graphs and tables. We compare our results of drag force and torque coefficients with existing data in the literature, specifically for the limiting cases. The outcomes of this study highlight the substantial impact of the interface on both drag force and torque coefficients. This inquiry is driven by the necessity to enhance our comprehension of the fluid tapping mode in atomic force microscope devices, particularly when this mode involves an interface substrate with another fluid.

A modified forcing approach in the Rothman–Keller method for simulations of flow phenomena at low capillary numbers

AbstractThe lattice‐Boltzmann method (LBM) is becoming increasingly popular for simulating multi‐phase flows on the microscale because of its advantages in terms of computational efficiency. Many applications of the method are restricted to relatively simple geometries. When a more complex geometry is considered—circular and inclined microchannels—some important physical phenomena may not be accurately captured, especially at low capillary numbers. A Y‐Y micro‐fluidic channel, widely used for a range of applications, is an example of a more complex geometry. This work aims to capture the various flow phenomena, with an emphasis on parallel flow and leakage, using the Rothman–Keller (RK) model of the LBM. To this purpose, we modify the forcing term to implement the surface tension for use at low capillary numbers. We compare the simulation results of the RK model with and without the force modification with experiments, Volume of Fluid and the phase field method and observe that the modified forcing term is an improvement over the current RK model at low capillary numbers, and it also captures parallel flow and leakage more accurately than the other simulation techniques.

4D microvelocimetry reveals multiphase flow field perturbations in porous media

Many environmental and industrial processes depend on how fluids displace each other in porous materials. However, the flow dynamics that govern this process are still poorly understood, hampered by the lack of methods to measure flows in optically opaque, microscopic geometries. We introduce a 4D microvelocimetry method based on high-resolution X-ray computed tomography with fast imaging rates (up to 4 Hz). We use this to measure flow fields during unsteady-state drainage, injecting a viscous fluid into rock and filter samples. This provides experimental insight into the nonequilibrium energy dynamics of this process. We show that fluid displacements convert surface energy into kinetic energy. The latter corresponds to velocity perturbations in the pore-scale flow field behind the invading fluid front, reaching local velocities more than 40 times faster than the constant pump rate. The characteristic length scale of these perturbations exceeds the characteristic pore size by more than an order of magnitude. These flow field observations suggest that nonlocal dynamic effects may be long-ranged even at low capillary numbers, impacting the local viscous-capillary force balance and the representative elementary volume. Furthermore, the velocity perturbations can enhance unsaturated dispersive mixing and colloid transport and yet, are not accounted for in current models. Overall, this work shows that 4D X-ray velocimetry opens the way to solve long-standing fundamental questions regarding flow and transport in porous materials, underlying models of, e.g., groundwater pollution remediation and subsurface storage of CO2 and hydrogen.

An Experimental Investigation on the Energy Signature Associated With Multiphase Flow in Porous Media Displacement Regimes

AbstractThis study investigates the energy signature associated with multiphase flow in porous media displacement regimes. We proposed that net efficiency, as the conversion of external work applied to surface energy generated, provides new insights into the transition of flow regimes. The combined effects of wettability and flow rates on immiscible fluid‐fluid displacement is experimentally investigated using high‐resolution imaging in microfluidic flow cells, which allows for tracking the evolution of interfacial area under applied external work. Our study focuses on the morphology of invasion patterns and the efficiency of conversion of external work to surface energy. The results show that the morphology of invasion patterns under unfavorable displacement condition is sharper than that under favorable displacement condition with larger ratios of length and width for fingers. The efficiency of conversion of external work to surface energy decreases with the increase of contact angles and reduces greatly with the increase of Capillary numbers. With favorable displacement conditions, the efficiency of conversion is consistently higher than that for unfavorable displacement. The trends of external work and surface energy serve as a signature of the transition between different flow regimes, which exhibits alteration especially at low Capillary numbers. The findings highlight the importance of wettability and flow rates in determining the efficiency of fluid‐fluid displacement in porous media. Understanding the energy signature associated with multiphase flow can have implications for various geoscience applications, such as oil/gas recovery, groundwater remediation, and underground energy storage.

Microfluidic analysis of 3T3 cellular transport in a photonic crystal fiber: part I.

This microfluidic-optical fiber sensor is an experimental system designed to transport and monitor 3D cell cultures, facilitating medical research and technology. This system includes a photonic crystal fiber with a hollow core diameter of 22µm, which functions as a bridge between two microfluidic devices. The purpose of this system was to transport 3T3 cells (of diameters from 15µm to 23µm) between the two devices. At low Reynold's and capillary numbers, spectroscopic analysis confirmed the presence of cellular aggregation at the interface of the fiber and microfluidic device. The transcapillary conductance, T C, is a separate analysis that models the behavior of a cellular aggregate through the hollow channel of a photonic crystal fiber. For the experimental system, conventional fluid mechanics theory is limited and requires special treatment of conditions at the microscale, such that transcapillary conductance treatment was employed. The transcapillary conductance, T C, was empirically derived to model cellular transport at the microfluidic scale and is useful for comparing transport events. For example, for a pressure differential of Δ p=1.5⋅103 c m H 2 O, the transcapillary conductance values were determined to be 10-12<T C<10-9, which were then compared to other literature values, such as the transport of circulating tumor cells (CTCs) at 33<Δ p<80c m H 2 O, with corresponding transcapillary conductance values at 10-7<T C<10-5. These transcapillary conductance values for both the literature and the experimental system are consistent, indicating that an increase in pressure differential does not promote microfluidic transport.

CO2-brine mass transfer patterns and interface dynamics under geological storage conditions

CO2 geological storage is a promising carbon negative technology with important application prospects. Dissolution trapping is a reliable method throughout the life cycle of a storage project, because it plays a significant role in minimizing risks. A comprehensive understanding of its mechanism is challenging due to the complex mass transfer process controlled by numerous physical and chemical factors, which needs to be clarified at different scales but still significantly limited in current research. Investigating CO2-brine interfacial mass transfer in real rock cores during imbibition provides valuable insights into the behavior of CO2 in geological formations. This study conducted a series of pore-scale CO2-brine interfacial mass transfer experiments during imbibition in three rock cores using an X-ray micro-computed tomography machine. The dissolution patterns and interface dynamics during the dissolving processes were discussed systematically. The results indicate that the overall slice-averaged and per-cluster mass transfer coefficient between CO2 and brine is in the order of 10−8 m/s, but its specific value depends on the dissolution patterns and bubble sizes. Various dissolution patterns were found in different rock cores. A core-length dissolution finger was formed in the Berea core, causing higher per-cluster mass transfer coefficients around the finger, while lower coefficients elsewhere in the dissolution experiments. The interfacial mass transfer process, coupled with gravity instability, induced remobilization of isolated CO2 bubbles at a much lower capillary number than the critical capillary number. The remobilization of CO2 bubbles can even occur at single-pore scale. CO2 remobilization also created a phenomenon similar to intermittent pathway flow. Interface dynamics of CO2 dissolving processes were revealed for the first time, giving rise to non-uniform dissolutions in CO2 singlets, ganglia, and clusters, respectively.

Experimental visualization of mass transfer from a slug bubble during co-current flow in a conventional channel

Process intensification during gas–liquid flows is of paramount importance to optimize the size of the chemical reactors. This paper results from an investigation of the mass transfer visualization in a millimetric tube by using the colorimetric technique. A slug bubble train has been established in a 3.8 mm diameter tube for different gas superficial velocities and a constant liquid superficial velocity resulting in the low capillary number range between 0.0004<Ca<0.0006. Color-inducing dye Resazurin solution interacts with the Oxygen bubble to induce the pink color in the solution which has been captured by a high-speed camera. The liquid slug entrapped between the bubbles has three distinct regions namely, the central stem which has a stagnation point ahead of the bubble, the wall lubrication layer, and an entrapped Taylor vortex between the central stem and wall lubrication layer. Flow visualization results have been correlated with the penetration theory-based mass transfer models.