Immiscible Two-phase Flow Research Articles

Application of the Lattice-Boltzmann method to the modeling of population blob dynamics in 2D porous domains

In the present paper, the Lattice-Boltzmann method is employed for the simulation of immiscible two-phase flow through a 2D porous domain when the volume fraction of the non-wetting phase is relatively low and thus it flows in the form of disconnected blobs. The flow problem is solved using an immiscible two-phase LB model where interfacial forces are expressed in terms of the chemical potential through the Gibbs–Duhem equation. We study the population dynamics of the non-wetting fluid blobs, namely the temporal evolution of the average blob size, with respect to the applied body force and the wetting phase volume fraction. Our results show that the system reaches a “steady state” where the average values of the studied parameters, such as the superficial velocities of both phases, and the number and size distribution of the blobs remain practically constant in time, although the temporal fluctuations around average values may be significant. We show that the average volume of the blobs decreases (and the population of the blobs increases) as the body force increases, namely as the viscous forces become dominant over capillary forces. The effect of the wetting volume fraction on the number of the blobs is more complex; as the wetting volume fraction decreases at constant body force, the blobs cover larger areas within the pore space producing larger pressure gradients and the dynamic breakup of blobs intensifies resulting in increasing blob numbers. However, below a critical value of the wetting volume fraction, the number of blobs begins to decrease and the non-wetting phase begins to span the entire pore network.

Hydrodynamic boundary conditions: An emergent behavior of fluid–solid interactions

By using the Onsager principle of minimum energy dissipation, the hydrodynamic boundary conditions at the fluid–solid interface are shown to be the natural emergent behavior of microscopic interactions that lead to the interfacial tension and the tangential friction at the fluid–solid interface [T. Qian, C. Qiu, P. Sheng, J. Fluid Mech. 611 (2008) 333]. This is satisfying because the equations of motion, e.g., the Stokes equation, and the hydrodynamic boundary conditions can now be derived from a unified framework. The resulting continuum hydrodynamic formulation yields predictions for immiscible two-phase flows that are in quantitative agreement with molecular dynamic simulations. In particular, the classical problem of the moving contact line is resolved. We also show results on the moving contact line over chemically patterned surfaces which exhibit striking nanoscale characteristics as well as sub-quadratic dependence of the moving contact line dissipation on its average velocity.

Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures

We design and investigate a sequential discontinuous Galerkin method to approximate two-phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures. The nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation. An accurate reconstruction of the total velocity is considered in the Raviart–Thomas(–Nédélec) finite element spaces, together with diffusivity-dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities. The proposed method is assessed on one-dimensional test cases exhibiting rough solutions, degeneracies, and capillary barriers. Stable and accurate solutions are obtained without limiters.

Asymptotic Behavior of Two-Phase Flows in Heterogeneous Porous Media for Capillarity Depending Only on Space. I. Convergence to the Optimal Entropy Solution

We consider an immiscible two-phase flow in a heterogeneous one-dimensional porous medium. We suppose particularly that the capillary pressure field is discontinuous with respect to the space variable. The dependence of the capillary pressure on the oil saturation is supposed to be weak, at least for saturations which are not too close to 0 or 1. We study the asymptotic behavior when the capillary pressure tends to a function which does not depend on the saturation. In this paper, we show that if the capillary forces at the spatial discontinuities are oriented in the same sense as the gravity forces, or if the two phases move in the same sense, then the saturation profile with capillary diffusion converges toward the unique optimal entropy solution to the hyperbolic scalar conservation law with discontinuous flux functions.

Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

This paper is devoted to the homogenization of a coupled system of diffusion-convection equations in a domain with periodic microstructure, modeling the flow and transport of immiscible compressible, such as water-gas, fluids through porous media. The problem is formulated in terms of a nonlinear parabolic equation for the nonwetting phase pressure and a nonlinear degenerate parabolic diffusion-convection equation for the wetting saturation phase with rapidly oscillating porosity function and absolute permeability tensor. We obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using two-scale convergence. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

Stick-Slip Motion of Moving Contact Line on Chemically Patterned Surfaces

Based on our continuum hydrodynamic model for immiscible two-phase flows at solid surfaces, the stick-slip motion has been predicted for moving contact line at chemically patterned surfaces [Wang et al., J. Fluid Mech., 605 (2008), pp. 59-78]. In this paper we show that the continuum predictions can be quantitatively verified by molecular dynamics (MD) simulations. Our MD simulations are carried out for two immiscible Lennard-Jones fluids confined by two planar solid walls in Poiseuille flow geometry. In particular, one solid surface is chemically patterned with alternating stripes. For comparison, the continuum model is numerically solved using material parameters directly measured in MD simulations. From oscillatory fluid-fluid interface to intermittent stick-slip motion of moving contact line, we have quantitative agreement between the continuum and MD results. This agreement is attributed to the accurate description down to molecular scale by the generalized Navier boundary condition in our continuum model. Numerical results are also presented for the relaxational dynamics of fluid-fluid interface, in agreement with a theoretical analysis based on the Onsager principle of minimum energy dissipation. PACS: 68.08.-p, 83.50.Rp, 83.10.Mj, 83.10.Ff

On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

We consider a simplified model for two-phase flows in one-dimensional heterogeneous porous media made of two different rocks.We focus on the effects induced by the discontinuity of thecapillarity field at interface. We first consider a model withcapillarity forces within the rocks, stating an existence/uniquenessresult. Then we look for the asymptotic problem for vanishingcapillarity within the rocks, remaining only on the interface. Weshow that either the solution to the asymptotic problem is theoptimal entropy solution to a scalar conservation law withdiscontinuous flux, or it admits a non-classical shock at theinterface modeling oil-trapping.

The Velocity and Shape of Convected Elongated Liquid Drops in Vertical Narrow Gaps

Abstract The motion and shape of a liquid drop through another continuous liquid phase (conveying phase) in a vertical Hele-Shaw cell with two different apertures were investigated experimentally. Two different liquid/liquid systems were tested. In all cases, the continuous phase was more viscous and wetted the bounding walls. In the capillarity-dominated region, the irregular shape of the discontinuous phase changed with time and distance, with much lower velocity than that of the conveying phase. In contrast to gas/liquid systems, the velocity of these stabilized, elongated drops was 2.5 to almost 5 times higher than that of conveying liquid. Despite the similarities between flow in vertical and horizontal Hele-Shaw cells, the velocity of droplets in a vertical fracture is different from that of a horizontal fracture. A new correlation is derived from dimensionless analysis and the experimental data to predict the elongated drop velocity as a function of the dimensionless parameters governing the system. Introduction Two-phase flow in micro-fractures is fundamental to many different fields of advanced science and technology, such as chemical process engineering, bioengineering, medical and genetic engineering, as well as petroleum engineering. For instance, understanding the flow of two-phase fluids in near-parallel gaps through fractured rocks has a significant effect on design of different recovery methods for naturally fractured reservoir. The flow pattern of two-phase immiscible flow in a fracture depends on the flow rates of the phases, the geometry, aperture, roughness of the fracture, the flow properties of the phases and interfacial tension between the phases. The flow patterns in a fracture are different from that in macro-sized rectangular ducts or pipes because of the small aperture, which can enhance capillary effects. The flow structure in the fracture affects the flow and transport through the surrounding porous matrix blocks. The slug flow pattern in a fracture, which occurs over a wide range of parameters, is frequently encountered in oil-wet fractured reservoirs during the immiscible displacement of viscous oil. It also occurs in natural gas reservoirs during displacement of water during gas production.

Modeling and Numerical Simulations of Immiscible Compressible Two-Phase Flow in Porous Media by the Concept of Global Pressure

A new formulation is presented for the modeling of immiscible compressible two-phase flow in porous media taking into account gravity, capillary effects, and heterogeneity. The formulation is intended for the numerical simulation of multidimensional flows and is fully equivalent to the original equations, contrary to the one introduced in Chavent and Jaffre (Mathematical Models and Finite Elements for Reservoir Simulation, 1986). The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. A finite volume method is used to solve the global pressure equation and the saturation equation for the water and gas phase in the context of gas migration through engineered and geological barriers for a deep repository for radioactive waste. Numerical results for the one-dimensional problem are presented. The accuracy of the fully equivalent fractional flow model is demonstrated through comparison with the simplified model already developed in Chavent and Jaffre (Mathematical Models and Finite Elements for Reservoir Simulation, 1986).

Relative permeabilities and coupling effects in steady-state gas-liquid flow in porous media: A lattice Boltzmann study

In this paper, the viscous coupling effects for immiscible two-phase (gas-liquid) flow in porous media were studied using the Shan–Chen-type single-component multiphase lattice Boltzmann model. Using the model, the two-phase flows in porous media with density ratio as high as 56 could be simulated and the contact angle of the gas-liquid interface at a solid wall is adjustable. To investigate viscous coupling effects, the co- and countercurrent steady-state two-phase flow patterns and relative permeabilities as a function of wetting saturation were obtained for different capillary numbers, wettabilities, and viscosity ratios. The cocurrent relative permeabilities seem usually larger than the countercurrent ones. The opposing drag-force effect and different pore-level saturation distributions in co- and countercurrent flows may contribute to this difference. It is found that for both co- and countercurrent flows, for strongly wet cases and viscosity ratio M>1, knw increase with the driving force and the viscosity ratio. However, for neutrally wet cases, the variations of knw and kw are more complex. It is also observed that different initial pore-level saturation distributions may affect final steady-state distribution, and hence the relative permeabilities. Using the cocurrent and countercurrent steady flow experiments to determine the generalized relative permeabilities seems not correct.

A combined lattice BGK/level set method for immiscible two-phase flows

We present a lattice Boltzmann method (LBM) for simulating immiscible multi-phase flows which is based on a coupling of LBM with the level set method. The computation of immiscible flows using the LBM with BGK collision operator is done separately in each of the fluid domains and coupled at the interface by an appropriate boundary condition. In this way we preserve sharp interfaces between different fluid phases. We apply a new interface condition that represents the fluid mechanical jump conditions at the interface in the kinetic LBM framework. The level set method is applied to compute the evolution of the interface between fluids. Numerical results demonstrate the applicability of the method even in the presence of large viscosity and density ratios.

Flux-dependent percolation transition in immiscible two-phase flows in porous media

Using numerical simulations, we study immiscible two-phase flow in a pore network reconstructed from Berea sandstone under flow conditions that are statistically invariant under translation. Under such conditions, the flow is a state function which is not dependent on initial conditions. We find a second-order phase transition resembling the phase inversion transition found in emulsions. The flow regimes under consideration are those of low surface tension-hence high capillary numbers Ca-where viscous forces dominate. Nevertheless, capillary forces are imminent, we observe a critical stage in saturation where the transition takes place. We determine polydispersity critical exponent tau=2.27+/-0.08 and find that the critical saturation depends on how fast the fluids flow.

Response to “Comments on Abrupt-Interface Solution for Carbon Dioxide Injection into Porous Media by Dentz and Tartakovsky (2008)” by Lu et al.

In their Comment, the authors raise an important issue of the validity of the analytical (Dentz and Tartakovsky 2008) and semi-analytical (Nordbotten et al. 2005) solutions describing migration of supercritical CO2 that is being injected into a deep geological formation. Indeed, any mathematical model (including the two-phase immiscible flow model employed in the Comment) is an abstraction, whose fidelity to a physical phenomenon it purports to describe can be established only through observation. Unlike standardmulti-phase flowmodels, which rely on a number of implicit and often unstated physical conjectures whose validity is hard to ascertain a priori (e.g., Adler and Brenner1988; Gray andHassanizadeh 1998, among others), the solutions of Dentz and Tartakovsky (2008) and Nordbotten et al. (2005) are based on the following premise:

A multi-flowpath model for the interpretation of immiscible displacement experiments in heterogeneous soil columns

This work focuses on the phenomenon of the immiscible two-phase flow of water and oil in saturated heterogeneous soil columns. The goal is to develop a fast and reliable method for quantifying soil heterogeneities for incorporation into the relevant capillary pressure and relative permeability functions. Such data are commonly used as input data in simulators of contaminant transport in the subsurface. Rate-controlled drainage experiments are performed on undisturbed soil columns and the transient response of the axial distribution of water saturation is determined from electrical measurements. The transient responses of the axial distribution of water saturation and total pressure drop are fitted with the multi-flowpath model (MFPM) where the pore space is regarded as a system of parallel paths of different permeability. The MFPM enables us to quantify soil heterogeneity at two scales: the micro-scale parameters describe on average the effects of pore network heterogeneities on the two-phase flow pattern; the macro-scale parameters indicate the variability of permeability at the scale of interconnected pore networks. The capillary pressure curve is consistent with that measured with mercury intrusion porosimetry over the low pressure range. The oil relative permeability increases sharply at a very low oil saturation (< 10 − 3 ) and tends to a high end value. The water relative permeability decreases abruptly at a low oil saturation (~ 0.1), whereas the irreducible wetting phase saturation is quite high. The foregoing characteristics of the two-phase flow properties are associated with critical (preferential) flowpaths that comprise a very small percentage of the total pore volume, control the overall hydraulic conductivity, and are consistent with the very broad range of pore-length scales usually probed in soil porous matrix.

Precursor simulations in spreading using a multi-mesh adaptive finite element method

Using the phase-field model for immiscible two-phase flows, we have numerically investigated the wetting dynamics. The long-range van der Waals forces towards the solid, which drive the spreading of the wetting phase into the nonwetting phase, have been explicitly taken into account in the governing equations. Our continuum model uses the generalized Navier boundary condition (GNBC) to account for the fluid slipping at the solid surface. The accurate description of the molecular-scale contact-line hydrodynamics makes the numerical simulations cost too much to abide. In this work, we propose an efficient multi-mesh adaptive finite element method which approximates different components of the solution (velocity, pressure and phase variable) on different h-adaptive meshes because of their strongly different local behaviors. That allows us to study the early stage of spreading, wherein the precursor is initiated and developed if the van der Waals forces are strong enough. We find that there is indeed a transition in the spreading behavior across a critical value of the Hamaker constant. In particular, this critical value is noted to be the one that separates the partial wetting from complete wetting.

A finite-volume ELLAM for non-linear flux convection–diffusion problems

In this paper, a modified finite-volume Eulerian–Lagrangian localized adjoint method (FVELLAM) extended for convection–diffusion problems with a non-linear flux function is introduced. Tracking schemes are discussed using viscous Burgers’ equation. It is shown that in order to have smooth results, only the new time level values should be used in tracking process. Then, the proposed method is employed to study immiscible incompressible two-phase flows in porous media. Various one- and two-dimensional test cases involving internal sources and sinks are solved and accuracy of solution and performance of the method are investigated by comparing the results obtained using FVELLAM with those of fine grid solutions. Finally, it is concluded that although proposed FVELLAM produces satisfactory results even on coarse grids and allows fairly large time-steps, its major advantage is in solving convection-dominated problems in heterogeneous media.

Modelling temperature effects on multiphase flow through porous media

We consider strong temperature effects on two-phase immiscible flow through a porous medium. A dependency on temperature is introduced via a temperature-dependent viscosity for one of the phases. The purpose is to develop a simple prototype model for simulating the injection of steam into a porous medium, which is saturated with highly viscous oil. To solve the problem, a Galerkin finite element model is formulated, with special residual-based terms added to stabilise the problem when advective transport processes dominate and to capture the discontinuities that develop at oil-water interfaces. It is also discussed how the mathematical representation of the numerical model can be transformed into computer code automatically, and two test cases are presented that demonstrate that this simplified model can capture typically observed phenomena robustly.

Towards a thermodynamics of immiscible two-phase steady-state flow in porous media

We propose that steady-state two-phase flow in porous media may be described through a formalism closely resembling equilibrium thermodynamics. This leads to a Monte Carlo method that will be highly efficient in studying two-phase flow under steady-state conditions numerically.

A Multiscale Method for Modeling Transport in Porous Media on Unstructured Corner-Point Grids

Multiscale solution methods are currently under active investigation for the simulation of subsurface flow in heterogeneous formations. These procedures capture the effects of fine-scale permeability variations through the calculation of specialized coarse-scale basis functions. Most of the multiscale techniques presented to date address subgrid capturing in pressure solution (elliptic/parabolic equations). In this paper we propose a multiscale method for solving transport equations on a coarse grid. In this method the global flow is computed on a coarse grid scale, but information from a fine scale velocity field is used to improve accuracy. The method is applied to incompressible and immiscible two-phase flow on a synthetic geological model with corner-point grid geometry. Although corner-point grids are given on a logically Cartesian format, the resulting grids are unstructured in physical space. The numerical results demonstrate that the multiscale method gives nearly the same flow characteristics as simulations where the transport equation is solved on the scale of an underlying fine grid.

Moving contact line on chemically patterned surfaces

We simulate the moving contact line in two-dimensional chemically patterned channels using a diffuse-interface model with the generalized Navier boundary condition. The motion of the fluid–fluid interface in confined immiscible two-phase flows is modulated by the chemical pattern on the top and bottom surfaces, leading to a stick–slip behaviour of the contact line. The extra dissipation induced by this oscillatory contact-line motion is significant and increases rapidly with the wettability contrast of the pattern. A critical value of the wettability contrast is identified above which the effect of diffusion becomes important, leading to the interesting behaviour of fluid–fluid interface breaking, with the transport of the non-wetting fluid being assisted and mediated by rapid diffusion through the wetting fluid. Near the critical value, the time-averaged extra dissipation scales as U, the displacement velocity. By decreasing the period of the pattern, we show the solid surface to be characterized by an effective contact angle whose value depends on the material characteristics and composition of the patterned surfaces.