Computational Transport Phenomena Lab
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Numerical approximation of a binary fluidsurfactant phase field model of twophase incompressible flow(arXiv, 20180417)In this paper, we consider the numerical approximation of a binary fluidsurfactant phase field model of twophase incompressible flow. The nonlinearly coupled model consists of two CahnHilliard type equations and incompressible NavierStokes equations. Using the Invariant Energy Quadratization (IEQ) approach, the governing system is transformed into an equivalent form, which allows the nonlinear potentials to be treated efficiently and semiexplicitly. we construct a first and a secondorder time marching schemes, which are extremely efficient and easytoimplement, for the transformed governing system. At each time step, the schemes involve solving a sequence of linear elliptic equations, and computations of phase variables, velocity and pressure are totally decoupled. We further establish a rigorous proof of unconditional energy stability for the semiimplicit schemes. Numerical results in both two and three dimensions are obtained, which demonstrate that the proposed schemes are accurate, efficient and unconditionally energy stable. Using our schemes, we investigate the effect of surfactants on droplet deformation and collision under a shear flow. The increase of surfactant concentration can enhance droplet deformation and inhibit droplet coalescence.

Global mass conservation method for dualcontinuum gas reservoir simulation(Elsevier BV, 20180317)In this paper, we find that the numerical simulation of gas flow in dualcontinuum porous media may generate unphysical or nonrobust results using regular finite difference method. The reason is the unphysical mass loss caused by the gas compressibility and the nondiagonal dominance of the discretized equations caused by the nonlinear well term. The well term contains the product of density and pressure. For oil flow, density is independent of pressure so that the well term is linear. For gas flow, density is related to pressure by the gas law so that the well term is nonlinear. To avoid these two problems, numerical methods are proposed using the mass balance relation and the local linearization of the nonlinear source term to ensure the global mass conservation and the diagonal dominance of discretized equations in the computation. The proposed numerical methods are successfully applied to dualcontinuum gas reservoir simulation. Mass conservation is satisfied while the computation becomes robust. Numerical results show that the location of the production well relative to the largepermeability region is very sensitive to the production efficiency. It decreases apparently when the production well is moved from the largepermeability region to the smallpermeability region, even though the well is very close to the interface of the two regions. The production well is suggested to be placed inside the largepermeability region regardless of the specific position.

Entropy stable modeling of nonisothermal multicomponent diffuseinterface twophase flows with realistic equations of state(arXiv, 20180225)In this paper, we consider mathematical modeling and numerical simulation of nonisothermal compressible multicomponent diffuseinterface twophase flows with realistic equations of state. A general model with general reference velocity is derived rigorously through thermodynamical laws and Onsager's reciprocal principle, and it is capable of characterizing compressibility and partial miscibility between multiple fluids. We prove a novel relation among the pressure, temperature and chemical potentials, which results in a new formulation of the momentum conservation equation indicating that the gradients of chemical potentials and temperature become the primary driving force of the fluid motion except for the external forces. A key challenge in numerical simulation is to develop entropy stable numerical schemes preserving the laws of thermodynamics. Based on the convexconcave splitting of Helmholtz free energy density with respect to molar densities and temperature, we propose an entropy stable numerical method, which solves the total energy balance equation directly, and thus, naturally satisfies the first law of thermodynamics. Unconditional entropy stability (the second law of thermodynamics) of the proposed method is proved by estimating the variations of Helmholtz free energy and kinetic energy with time steps. Numerical results validate the proposed method.

A fully discrete energy stable scheme for a phase filed moving contact line model with variable densities and viscosities(arXiv, 20180126)In this paper, a fully discrete scheme which considers temporal and spatial discretizations is presented for the coupled CahnHilliard equation in conserved form with the dynamic contact line condition and the NavierStokes equation with the generalized Navier boundary condition. Variable densities and viscosities are incorporated in this model. A rigorous proof of energy stability is provided for the fully discrete scheme based on a semiimplicit temporal discretization and a finite difference method on the staggered grids for the spatial discretization. A splitting method based on the pressure stabilization is implemented to solve the NavierStokes equation, while the stabilization approach is also used for the CahnHilliard equation. Numerical results in both 2D and 3D demonstrate the accuracy, efficiency and decaying property of discrete energy of the proposed scheme.

Physicspreserving averaging scheme based on GrunwaldLetnikov formula for gas flow in fractured media(Elsevier BV, 20180102)The heterogeneous natures of rock fabrics, due to the existence of multiscale fractures and geological formations, led to the deviations from unity in the fluxequations fractionalexponent magnitudes. In this paper, the resulting nonNewtonian nonDarcy fractionalderivatives flux equations are solved using physicspreserving averaging schemes that incorporates both, original and shifted, GrunwaldLetnikov (GL) approximation formulas preserving the physics, by reducing the shifting effects, while maintaining the stability of the system, by keeping one shifted expansion. The proposed way of using the GL expansions also generate symmetrical coefficient matrices that significantly reduces the discretization complexities appearing with all shifted cases from literature, and help considerably in 2D and 3D systems. Systems equations derivations and discretization details are discussed. Then, the physicspreserving averaging scheme is explained and illustrated. Finally, results are presented and reviewed. Edgebased original GL expansions are unstable as also illustrated in literatures. Shifted GL expansions are stable but add a lot of additional weights to both discretization sides affecting the physical accuracy. In comparison, the physicspreserving averaging scheme balances the physical accuracy and stability requirements leading to a more physically conservative scheme that is more stable than the original GL approximation but might be slightly less stable than the shifted GL approximations. It is a locally conservative SingleContinuum averaging scheme that applies a finitevolume viewpoint.

Thermodynamically consistent modeling and simulation of multicomponent twophase flow with partial miscibility(Elsevier BV, 20171209)A general diffuse interface model with a realistic equation of state (e.g. PengRobinson equation of state) is proposed to describe the multicomponent twophase fluid flow based on the principles of the NVTbased framework which is an attractive alternative recently over the NPTbased framework to model the realistic fluids. The proposed model uses the Helmholtz free energy rather than Gibbs free energy in the NPTbased framework. Different from the classical routines, we combine the first law of thermodynamics and related thermodynamical relations to derive the entropy balance equation, and then we derive a transport equation of the Helmholtz free energy density. Furthermore, by using the second law of thermodynamics, we derive a set of unified equations for both interfaces and bulk phases that can describe the partial miscibility of multiple fluids. A relation between the pressure gradient and chemical potential gradients is established, and this relation leads to a new formulation of the momentum balance equation, which demonstrates that chemical potential gradients become the primary driving force of fluid motion. Moreover, we prove that the proposed model satisfies the total (free) energy dissipation with time. For numerical simulation of the proposed model, the key difficulties result from the strong nonlinearity of Helmholtz free energy density and tight coupling relations between molar densities and velocity. To resolve these problems, we propose a novel convexconcave splitting of Helmholtz free energy density and deal well with the coupling relations between molar densities and velocity through very careful physical observations with a mathematical rigor. We prove that the proposed numerical scheme can preserve the discrete (free) energy dissipation. Numerical tests are carried out to verify the effectiveness of the proposed method.

Linearly decoupled energystable numerical methods for multicomponent twophase compressible flow(arXiv, 20171206)In this paper, for the first time we propose two linear, decoupled, energystable numerical schemes for multicomponent twophase compressible flow with a realistic equation of state (e.g. PengRobinson equation of state). The methods are constructed based on the scalar auxiliary variable (SAV) approaches for Helmholtz free energy and the intermediate velocities that are designed to decouple the tight relationship between velocity and molar densities. The intermediate velocities are also involved in the discrete momentum equation to ensure a consistency relationship with the mass balance equations. Moreover, we propose a componentwise SAV approach for a multicomponent fluid, which requires solving a sequence of linear, separate mass balance equations. We prove that the methods have the unconditional energydissipation feature. Numerical results are presented to verify the effectiveness of the proposed methods.

Thermodynamically consistent simulation of nonisothermal diffuseinterface twophase flow with PengRobinson equation of state(arXiv, 20171206)In this paper, we consider a diffuseinterface gasliquid twophase flow model with inhomogeneous temperatures, in which we employ the PengRobinson equation of state and the temperaturedependent influence parameter instead of the van der Waals equation of state and the constant influence parameter used in the existing models. As a result, our model can characterize accurately the physical behaviors of numerous realistic gasliquid fluids, especially hydrocarbons. Furthermore, we prove a relation associating the pressure gradient with the gradients of temperature and chemical potential, and thereby derive a new formulation of the momentum balance equation, which shows that gradients of the chemical potential and temperature become the primary driving force of the fluid motion. It is rigorously proved that the new formulations of the model obey the first and second laws of thermodynamics. To design efficient numerical methods, we prove that Helmholtz free energy density is a concave function with respect to the temperature under certain physical conditions. Based on the proposed modeling formulations and the convexconcave splitting of Helmholtz free energy density, we propose a novel thermodynamically stable numerical scheme. We rigorously prove that the proposed method satisfies the first and second laws of thermodynamics. Finally, numerical tests are carried out to verify the effectiveness of the proposed simulation method.

Direct numerical simulation of noninvasive channel healing in electrical field(SAGE Publications, 20171125)Noninvasive channel healing is a new idea to repair the broken pipe wall, using external electric fields to drive iron particles to the destination. The repair can be done in the normal operation of the pipe flow without any shutdown of the pipeline so that this method can be a potentially efficient and safe technology of pipe healing. However, the real application needs full knowledge of healing details. Numerical simulation is an effective method. Thus, in this research, we first established a numerical model for noninvasive channel healing technology to represent fluid–particle interaction. The iron particles can be attached to a cracking area by external electrostatic forces or can also be detached by mechanical forces from the fluid. When enough particles are permanently attached on the cracking area, the pipe wall can be healed. The numerical criterion of the permanent attachment is discussed. A fully threedimensional finite difference framework of direct numerical simulation is established and applied to different cases to simulate the full process of channel healing. The impact of Reynolds number and particle concentration on the healing process is discussed. This numerical investigation provides valuable reference and tools for further simulation of real pipe healing in engineering.

Discretefracturemodel of multi–scale timesplitting two–phase flow including nanoparticles transport in fractured porous media(Elsevier BV, 20171123)In this article, we consider a twophase immiscible incompressible flow including nanoparticles transport in fractured heterogeneous porous media. The system of the governing equations consists of water saturation, Darcy’s law, nanoparticles concentration in water, deposited nanoparticles concentration on the porewall, and entrapped nanoparticles concentration in the porethroat, as well as, porosity and permeability variation due to the nanoparticles deposition/entrapment on/in the pores. The discretefracture model (DFM) is used to describe the flow and transport in fractured porous media. Moreover, multiscale timesplitting strategy has been employed to manage different timestep sizes for different physics, such as saturation, concentration, etc. Numerical examples are provided to demonstrate the efficiency of the proposed multiscale time splitting approach.

Molecular Dynamics Simulation Study of Carbon Dioxide, Methane, and Their Mixture in the Presence of Brine(American Chemical Society (ACS), 20171003)We perform molecular dynamics simulation study of CO2, methane, and their mixture in the presence of brine over a broad range of temperature (311–473 K), pressure (up to about 100 MPa), and NaCl concentration (up to about 14 wt %). The general decrease in the interfacial tension (IFT) values of the CH4–brine system with pressure and temperature is similar to that obtained for the corresponding CH4–water system. The IFT of methane and brine is a linearly increasing function of salt concentration, and the resulting slopes are dependent on the pressure. A similar behavior as methane is observed for such systems containing CO2 and CO2–CH4 mixture. The IFT of CO2 and brine increases linearly with increasing salt content; however, the resulting slopes are independent of pressure. The simulations show that the presence of CO2 decreases the IFT values of the CH4–water and CH4–brine systems, but the degree of reduction depends on the amount of CO2 in each sample, which is consistent with experimental evidence. These IFT values show a linear correlation with the amount of CO2, and the resulting slopes are dependent on the temperature and pressure. Furthermore, our results for the mole fractions of the different species in the CO2–CH4–water system at 323 K and 9 MPa are in agreement with those of experiments. The mole fractions of methane and CO2 in the waterrich phase decrease with increasing salt concentration, whereas that of H2O in the methane or CO2rich phases remains almost unaffected in all of the studied cases. Our results could be useful because of the importance of carbon dioxide sequestration and shale gas production.

A stable algorithm for calculating phase equilibria with capillarity at specified moles, volume and temperature using a dynamic model(Elsevier BV, 20170930)Capillary pressure can significantly affect the phase properties and flow of liquidgas fluids in porous media, and thus, the phase equilibrium calculation incorporating capillary pressure is crucial to simulate such problems accurately. Recently, the phase equilibrium calculation at specified moles, volume and temperature (NVTflash) becomes an attractive issue. In this paper, capillarity is incorporated into the phase equilibrium calculation at specified moles, volume and temperature. A dynamical model for such problem is developed for the first time by using the laws of thermodynamics and Onsager's reciprocal principle. This model consists of the evolutionary equations for moles and volume, and it can characterize the evolutionary process from a nonequilibrium state to an equilibrium state in the presence of capillarity effect at specified moles, volume and temperature. The phase equilibrium equations are naturally derived. To simulate the proposed dynamical model efficiently, we adopt the convexconcave splitting of the total Helmholtz energy, and propose a thermodynamically stable numerical algorithm, which is proved to preserve the second law of thermodynamics at the discrete level. Using the thermodynamical relations, we derive a phase stability condition with capillarity effect at specified moles, volume and temperature. Moreover, we propose a stable numerical algorithm for the phase stability testing, which can provide the feasible initial conditions. The performance of the proposed methods in predicting phase properties under capillarity effect is demonstrated on various cases of pure substance and mixture systems.

Acceleration of Gas Flow Simulations in DualContinuum Porous Media Based on the MassConservation POD Method(MDPI AG, 20170912)Reducedorder modeling approaches for gas flow in dualporosity dualpermeability porous media are studied based on the proper orthogonal decomposition (POD) method combined with Galerkin projection. The typical modeling approach for nonporousmedium liquid flow problems is not appropriate for this compressible gas flow in a dualcontinuum porous media. The reason is that nonzero mass transfer for the dualcontinuum system can be generated artificially via the typical POD projection, violating the massconservation nature and causing the failure of the POD modeling. A new POD modeling approach is proposed considering the mass conservation of the whole matrix fracture system. Computation can be accelerated as much as 720 times with high precision (reconstruction errors as slow as 7.69 × 10−4%~3.87% for the matrix and 8.27 × 10−4%~2.84% for the fracture).

Energy Stability Analysis of Some Fully Discrete Numerical Schemes for Incompressible Navier–Stokes Equations on Staggered Grids(Springer Nature, 20170901)In this paper we consider the energy stability estimates for some fully discrete schemes which both consider time and spatial discretizations for the incompressible Navier–Stokes equations. We focus on three kinds of fully discrete schemes, i.e., the linear implicit scheme for time discretization with the finite difference method (FDM) on staggered grids for spatial discretization, pressurecorrection schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations, and pressurestabilization schemes for time discretization with the FDM on staggered grids for the solutions of the decoupled velocity and pressure equations. The energy stability estimates are obtained for the above each fully discrete scheme. The upwind scheme is used in the discretization of the convection term which plays an important role in the design of unconditionally stable discrete schemes. Numerical results are given to verify the theoretical analysis.

Modeling and Analysis of Magnetic Nanoparticles Injection in WaterOil TwoPhase Flow in Porous Media under Magnetic Field Effect(Hindawi Limited, 20170828)In this paper, the magnetic nanoparticles are injected into a wateroil, twophase system under the influence of an external permanent magnetic field. We lay down the mathematical model and provide a set of numerical exercises of hypothetical cases to show how an external magnetic field can influence the transport of nanoparticles in the proposed twophase system in porous media. We treat the waternanoparticles suspension as a miscible mixture, whereas it is immiscible with the oil phase. The magnetization properties, the density, and the viscosity of the ferrofluids are obtained based on mixture theory relationships. In the mathematical model, the phase pressure contains additional term to account for the extra pressures due to fluid magnetization effect and the magnetostrictive effect. As a proof of concept, the proposed model is applied on a countercurrent imbibition flow system in which both the displacing and the displaced fluids move in opposite directions. Physical variables, including waternanoparticles suspension saturation, nanoparticles concentration, and pore wall/throat concentrations of deposited nanoparticles, are investigated under the influence of the magnetic field. Two different locations of the magnet are studied numerically, and variations in permeability and porosity are considered.

Complexation Behavior of Polyelectrolytes and Polyampholytes(American Chemical Society (ACS), 20170725)We perform grand canonical Monte Carlo simulations to study the pH titrations of isolated polyampholytes and polyelectrolytepolyampholyte complexes in dilute solutions. Our simulations indicate that the electrostatic interactions promote the coexistence of opposite charges along the polyampholyte chain during titration. The repulsion between excess charges typically dominates the electrostatic interaction and leads to polymer stretching. Salt ions can screen the repulsion between excess charges as well as the fluctuationinduced attraction between opposite charges, and therefore make the variation between titration curves of polyampholytes and the ideal (no electrostatic interactions) curves less significant. We observe that this screening of charge repulsion decreases the chain size. The presence of pearlnecklace configuration of polyampholytes is diminished by the addition of salt. Similar simulations for the polyelectrolytepolyampholyte system show that the resulting complexes are generally stable in the low pH region. In comparison to ideal case, electrostatic interactions strongly influence the acidbase properties of polyampholyte chains in the adsorbed state by reducing the presence of the coexistence domain of both positive and negative charges in the titration curves. We attribute the complex formation between polyelectrolyte and polyampholyte chains in the high pH region to, e.g., the high salt content. The pH variation leads to abrupt transition between adsorbed and desorbed states. Independent of charge sequence, a polyampholyte chain in a complex is usually located at one of the ends of the polyelectrolyte chain.

Dualmixed finite elements for the threefield Stokes model as a finite volume method on staggered grids(Elsevier BV, 20170609)In this paper, a new threefield weak formulation for Stokes problems is developed, and from this, a dualmixed finite element method is proposed on a rectangular mesh. In the proposed mixed methods, the components of stress tensor are approximated by piecewise constant functions or Q1 functions, while the velocity and pressure are discretized by the lowestorder RaviartThomas element and the piecewise constant functions, respectively. Using quadrature rules, we demonstrate that this scheme can be reduced into a finite volume method on staggered grid, which is extensively used in computational fluid mechanics and engineering.

Reduced Fracture Finite Element Model Analysis of an Efficient TwoScale Hybrid Embedded Fracture Model(Elsevier BV, 20170609)A Hybrid Embedded Fracture (HEF) model was developed to reduce various computational costs while maintaining physical accuracy (Amir and Sun, 2016). HEF splits the computations into fine scale and coarse scale. Fine scale solves analytically for the matrixfracture flux exchange parameter. Coarse scale solves for the properties of the entire system. In literature, fractures were assumed to be either vertical or horizontal for simplification (Warren and Root, 1963). Matrixfracture flux exchange parameter was given few equations built on that assumption (Kazemi, 1968; Lemonnier and Bourbiaux, 2010). However, such simplified cases do not apply directly for actual random fracture shapes, directions, orientations …etc. This paper shows that the HEF fine scale analytic solution (Amir and Sun, 2016) generates the flux exchange parameter found in literature for vertical and horizontal fracture cases. For other fracture cases, the flux exchange parameter changes according to the angle, slop, direction, … etc. This conclusion rises from the analysis of both: the Discrete Fracture Network (DFN) and the HEF schemes. The behavior of both schemes is analyzed with exactly similar fracture conditions and the results are shown and discussed. Then, a generalization is illustrated for any slightly compressible singlephase fluid within fractured porous media and its results are discussed.

Molecular Simulation Study of Montmorillonite in Contact with Variably Wet Supercritical Carbon Dioxide(American Chemical Society (ACS), 20170307)We perform grand canonical Monte Carlo simulations to study the detailed molecular mechanism of intercalation behavior of CO2 in Na, Ca, and Mg montmorillonite exposed to variably hydrated supercritical CO2 at 323.15 K and 90 bar, The simulations indicate that the intercalation of CO2 strongly depends on the relative humidity (RH). The intercalation of CO2 in the dehydrated interlayer is inhibited, followed by the swelling of the interlayer region due to uptake of water and CO2 as the RH increases. In all of the hydrated clay samples, the amount of the intercalated CO2 generally decreases as a function of increasing RH, which is attributed mainly to the weakening of the interaction between CO2 and clay. At low RH values, Ca and Mg montmorillonite are relatively more efficient in capturing CO2. The amount of CO2 trapped in all clay samples shows similar values above RH of similar to 60%. Molecular dynamics simulations show that the diffusion coefficient of each species generally increases with increasing RH due to the associated expansion of the interlayer distance of the clay. For all the hydrated samples, the diffusion coefficients of CO2 and water in the interlayers are mostly comparable due to the fact that CO2 molecules are well solvated. The diffusion of CO2 in each hydration state is mostly independent of the type of cation in accordance with the fact that CO2 molecules hardly migrate into the first hydration shell of the interlayer cations.

PODGalerkin Model for Incompressible SinglePhase Flow in Porous Media(Walter de Gruyter GmbH, 20170125)Fast prediction modeling via proper orthogonal decomposition method combined with Galerkin projection is applied to incompressible singlephase fluid flow in porous media. Cases for different configurations of porous media, boundary conditions and problem scales are designed to examine the fidelity and robustness of the model. High precision (relative deviation 1.0 x 10(4)% similar to 2.3 x 10(1)%) and large acceleration (speedup 880 similar to 98454 times) of POD model are found in these cases. Moreover, the computational time of POD model is quite insensitive to the complexity of problems. These results indicate POD model is especially suitable for largescale complex problems in engineering.