An Improved IMPES Method for Two-Phase Flow in Porous Media

In this paper we develop and numerically study an improved IMPES method for solving a partial differential coupled system for two-phase flow in a three-dimensional porous medium. This improved method utilizes an adaptive control strategy on the choice of a time step for saturation and takes a much larger time step for pressure than for the saturation. Through a stability analysis and a comparison with a simultaneous solution method, we show that this improved IMPES method is effective and efficient for the numerical simulation of two-phase flow and it is capable of solving two-phase coning problems.     

并列圆柱绕流的格子Boltzmann数值模拟

采用非均匀不可压格子Boltzmann模型对低雷诺数下并列圆柱绕流进行了数值模拟,给出了数值计算结果,分析了间距g对圆柱尾流及升力、阻力的影响,并在此基础上得到了4种尾迹模式.此外,研究了流场的初始扰动对流动分岔现象的影响,发现在适当的扰动下可以很快得到同步同相的尾流.对Re=160和200下圆柱的升、阻力进行了对比,结果表明升力和阻力受间距g的影响大于雷诺数.     

Analysis of Thermal Flow in a Rotating Porous U-Turn Duct Using Lattice Boltzmann Method

The lattice Boltzmann method is carried out to investigate the heat transfer enhancement in a U-turn duct which is partially filled with a porous media. The porous layer is inserted at the core of the duct and is modeled using the Brinkman–Forchheimer assumptions. In order to validate the results, first a channel flow problem without any porous layer is compared with available data. Second, the porous Couette flow and partially porous channel flow are successfully compared with the studies of other researchers. Then, fluid flow in a clear U-turn duct is studied looking carefully at the velocity, curvature and rotation effects. Finally, the effects of porous layer thickness on the rate of heat transfer and pressure drop are investigated. Parametric studies are conducted to evaluate the effects of various parameters (i.e., Reynolds number, Darcy number, rotation number), highlighting their influences on the thermo-hydrodynamics behavior of the flow. The optimum values of porous layer thickness are presented for specific flow parameters.     

A Study of the Role of Microfractures in Counter-Current Spontaneous Imbibition by Lattice Boltzmann Simulation

Transport in Porous Media - During waterflooding, spontaneous imbibition is a fundamental recovery mechanism in fractured reservoirs. A large number of numerical and experimental studies have been...     

An Orthorhombic Lattice Boltzmann Model for Pore-Scale Simulation of Fluid Flow in Porous Media

One application of the lattice Boltzmann equation (LBE) models is in combination with tomography to simulate pore-scale flow and transport processes in porous media. Most LBE models in the literature are based on cubic lattice, and if the voxels in a tomography image are not cubic or cannot be divided into cubes due to computational limitations, these models will lose most of their advantages. How to deal with such images is, hence, an interest in use of the LBE model to simulate pore-scale processes. In this paper, we present an orthorhombic LBE model based on the single-relaxation time approach with the relaxation parameter varying with lattice directions. The equilibrium distribution functions in the standard LBE model were modified to correct the anisotropy induced by the non-cubic lattice, and the calculations of the fluid density and momentum were also redefined in order to maintain the conservation of mass and momentum during the collision. We tested the model against analytical solution for fluid flow in a tube, and against the standard cubic-based LBE model for fluid flow in a duct with an island inside. The model was then applied to simulate fluid flow in a 3D image in attempts to analyse the errors if the voxels in the image are not cubic but are assumed to be cubic.     

A Simplified Model and Similarity Solutions for Interfacial Evolution of Two-Phase Flow in Porous Media

For two-phase flows of immiscible displacement processes in porous media, we proposed a simplified model to capture the interfacial fronts, which is given by explicit expressions and satisfies the continuity conditions of pressure and normal velocity across the interface. A new similarity solution for the interfacial evolution in the rectangular coordinate system was derived by postulating a first-order approximation of the velocity distribution in the region that the two-phase fluids co-exist. The interfacial evolution equation can be explicitly expressed as a linear function, where the slope of the interfacial equation is simply related to the mobility ratio of two-phase fluids in porous media. The application of the proposed solutions to predictions of interfacial evolutions in carbon dioxide injected into saline aquifers was illustrated under different mobility ratios and operational parameters. For the purpose of comparison, the numerical solutions obtained by level set method and the similarity solutions based on the Dupuit assumptions were presented. The results show that the proposed solution can give a better approximation of interfacial evolution than the currently available similarity solutions, especially in the situation that the mobility ratio is large. The proposed approximate solutions can provide physical insight into the interfacial phenomenon and be readily used for rapidly screening carbon dioxide storage capacity in subsurface formations and monitoring the migration of carbon dioxide plume.

     

Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method

Numerical models that solve transport of pollutants at the macroscopic scale in unsaturated porous media need the effective diffusion dependence on saturation as an input. We conducted numerical computations at the pore scale in order to obtain the effective diffusion curve as a function of saturation for an academic sphere packing porous medium and for a real porous medium where pore structure knowledge was obtained through X-ray tomography. The computations were performed using a combination of lattice Boltzmann models based on two relaxation time (TRT) scheme. The first stage of the calculations consisted in recovering the water spatial distribution into the pore structure for several fixed saturations using a phase separation TRT lattice Boltzmann model. Then, we performed diffusion computation of a non-reactive solute in the connected water structure using a diffusion TRT lattice Boltzmann model. Finally, the effective diffusion for each selected saturation value was estimated through inversion of a macroscopic classical analytical solution.     

Study of Slip Flow in Unconventional Shale Rocks Using Lattice Boltzmann Method: Effects of Boundary Conditions and TMAC

Steady, laminar, fully developed flows of a Newtonian fluid driven by a constant pressure gradient in (1) a curvilinear constant cross section triangle bounded by two straight no-slip segments and a circular meniscus and (2) a wedge bounded by two rays and an adjacent film bulging near the corner are studied analytically by the theory of holomorphic functions and numerically by finite elements. The analytical solution of the first problem is obtained by reducing the Poisson equation for the longitudinal flow velocity to the Laplace equation, conformal mapping of the corresponding transformed physical domain onto an auxiliary half-plane and solving there the Signorini mixed boundary value problem (BVP). The numerical solution is obtained by meshing the circular sector and solving a system of linear equations ensuing from the Poisson equation. Comparisons are made with known solutions for flows in a rectangular conduit, circular annulus and Philip’s circular duct with a no-shear sector. Problem (2) is treated by the Saint-Venant semi-inverse method: the free surface (quasi-meniscus) is reconstructed by a one-parametric family, which specifies a holomorphic function of the first derivative of the physical coordinate with respect to an auxiliary variable. The latter maps the flow domain onto a quarter of a unit disc where a mixed BVP for a characteristic function is solved by the Zhukovsky–Chaplygin method. Velocity distributions in a cross section perpendicular to the flow direction are obtained. It is shown that the change of the type of the boundary condition from no slip to perfect slip (along the meniscus) causes a dramatic increase of the total flow rate (conductance). For example, the classical Saint-Venant formulae for a sector, with all three boundaries being no-slip segments, predict up to four times smaller rate as compared to a free surface meniscus. Mathematically equivalent problems of unconfined flows in aquifers recharged by a constant-intensity infiltration are also addressed.     

A study of the upper limit of solid scatters density for gray Lattice Boltzmann Method

The upper limit of the solid scatters density ns （x）, a key parameter for the simulation of flows in porous media with a gray Lattice Boltzmann Method, is studied by an analytical way for the infiltration Poiseuille flow between two infinite parallel plates. Analyses of three different gray Lattice Boltzmann schemes, separately proposed by Gao and Sharma et al., Dardis and McCloskey, and Thorne and Sukop, indicate that the effective domain of Gao and Sharma＇s scheme is restricted to ns 〈 1/2√3≈0.289, Dardis and McCloskey＇s scheme is restricted to ns 〈 （√57-1）/28≈0.234, and that there is no extra restriction on ns（x） with Thorne and Sukop＇s scheme. These results are obtained for the dimensionless relaxation time τ= 1. The above analytical results are verified by our numerical simulations. The use of a gray LBM is further illustrated by simulating the flow at the interface of a porous medium. Simulation results yield velocity profiles which agree very well with Brinkman＇s prediction.     

Computation of Saturation Dependence of Effective Diffusion Coefficient in Unsaturated Argillite Micro-fracture by Lattice Boltzmann Method

Getting access to the effective diffusion coefficient is a key point to provide realistic predictions of migration of radionuclides from radioactive waste repository in deep argillaceous geological formations. In the present work, the effective diffusion coefficient was computed inside an argillite micro-fracture as a function of its saturation level. The micrometric fracture geometry was extracted from the X-ray \(\mu \)-tomography image (\(0.7\,\upmu \mathrm{m}\) voxel resolution) of an Opalinus clay sample. It was collected in the host rock excavated damaged zone surrounding a borehole in the Mont Terri laboratory. The computations were performed using two two-relaxation-time lattice Boltzmann models. The first one, a phase separation model, was used to extract the connected liquid phase inside the fracture for given saturations. The second, a diffusion model, was used to compute non-reactive tracer diffusion in the connected liquid phase of the fracture and to calculate the effective diffusion coefficient for the associated saturations. The dependence of the effective diffusion coefficient on saturation was found to be quasi-linear and to qualitatively match the Maxwell expression for saturations lower than 0.8.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Accounting for Polydispersion in the Eulerian Large Eddy Simulation of the Two-Phase Flow in an Aeronautical-type Burner

A major issue for the simulation of two-phase flows in engines concerns the modeling of the liquid disperse phase, either in the Lagrangian or the Eulerian approach. In the perspective of massively parallel computing, the Eulerian approach seems to be more suitable, as it uses the same algorithms as the gaseous phase solver. However taking into account the whole physics of a turbulent spray, especially in terms of polydispersity, requires an additional modeling effort. The Mesoscopic Eulerian Formalism (MEF) [13] accounts for the effect of turbulence on the disperse phase, and was extended to the Large Eddy Simulation framework [41], but is limited to monodisperse flows. In [38], the influence of polydispersity on resolved and unresolved turbulent motions of the disperse phase was highlighted, and a first model was proposed, based on size-conditioned statistics. Starting from this idea, a coupling between the MEF and the Multifluid Approach (MA) [30] is proposed. The MA decomposes the Eulerian phase into several fluid classes called sections, and corresponding to size intervals. Each section uses then size-conditioned closures. The original idea of this work is to use the MEF closures independently in each section, taking into account the mean droplet size of this section. This new approach, called Multifluid Mesoscopic Eulerian Formalism (MMEF), is then able to capture polydispersion with associated size-conditioned turbulent dynamics. First, the importance of polydispersity and the ability of MMEF to capture it are highlighted with a 0D evaporation case and a 2D vortex case, showing its impact on dynamics in both size and physical spaces. Then, the MMEF is applied to the MERCATO configuration of ONERA [18]. Results are compared to monodisperse Eulerian, Lagrangian and experimental results.     

Investigation on Heat and Mass Transfer in a Evaporator of a Capillary-Pumped Loop with the Lattice Boltzmann Method: Pore Scale Simulation

A two-dimensional transient numerical model based on the lattice Boltzmann method (LBM) for the global evaporator of a capillary-pumped loop (CPL) is proposed to describe heat and mass transfer with evaporation in the porous wick, heat conduction in the cover plate, and heat transfer in the vapor groove. To indicate the stochastic phase distribution characteristics of most porous wick, the quartet structure generation set (QSGS) is introduced for generating more realistic microstructures of porous media. By using the present lattice Boltzmann algorithm along with the porous structure, the heat and mass transfer of an evaporator on pore scale can be predicted without resorting to any empirical parameters determined case by case. The energy equations for entire evaporator are solved as a conjugate problem, which are solved by means of a spatially varying relaxation time in the lattice Boltzmann model and the liquid flow is driven via the interfacial mass flux. A convective boundary condition considering the latent heat during the evaporation on the interface is introduced into the lattice Boltzmann model based on the nonequilibrium extrapolation rule. Especially, the bounce-back rule and the equilibrium rule of the LBM are, respectively, introduced to deal with the momentum boundary conditions inside the porous wick and on the evaporation interface in order to ensure the stability and the efficiency of the LBM model. Numerical results corresponding to different working conditions and different working fluids are presented, which provide guidance for the evaporator design of a CPL system.     

A Least-Squares Mixed Scheme for the Simulation of Two-Phase Flow in Porous Media on Unstructured Grids

A least-squares mixed formulation is developed for simulation of two-phase flow in porous media. Such problems arise in petroleum applications and ground-water flow. An adaptive strategy based on the element residual as an error indicator is developed in conjunction with unstructured remeshing and tested for the two-phase flow of oil and water. An element-by-element conjugate-gradient scheme (EBE-CG) is compared to a band solution algorithm.     

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 Jaffré (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 Jaffré (Mathematical Models and Finite Elements for Reservoir Simulation, 1986).     

On the Role of the Interface Mechanical Interaction in a Gravity-Driven Shear Flow of an Ice-Till Mixture

The ice-till mixtures at the base of glaciers and ice sheets play a very important role in the movement of the glaciers and ice sheets. This mixture is modelled as an isothermal flow which is overlain by a layer of pure ice. In this model, ice is treated as usual as a very viscous fluid with a constant true density, while till, which is assumed to consist of sediment and bound (that is, moving with the sediment) interstitial water and/or ice, is also assumed in a first approximation to behave such as a fluid. For an isothermal flow below the melting point the water component can be neglected. Therefore, only the mass and momentum balances for till and ice are needed. To complete the model, no-slip and stress-free boundary conditions are assumed at the base and free-surface, respectively. The transition from the till-ice mixture layer to the overlying pure ice layer is idealized in the model as a moving interface representing in the simplest case the till material boundary, at which jump balance relations for till and ice apply. The mechanical interactions are considered in the mixture basel layer, as well as at the interface via the surface production. The interface mechanical interaction is supposed to be only a function of the volume fraction jump across the interface. In the context of the thin-layer approximation, numerical solutions of the lowest-order form of the model show a till distribution which is reminiscent to the ice-till layer in geophysical environment.     

Numerical Simulation of Thermal and Concentration Natural Convection in a Porous Cavity in the Presence of an Opposing Flow

Two-dimensional steady-state thermal concentration convection in a rectangular porous cavity is simulated numerically. The temperature and concentration gradients are horizontal and the buoyancy forces act either in the same or in opposite directions. The flow through the porous medium is described by the Darcy-Brinkman or Forchheimer equations. The SIMPLER numerical algorithm based on the finite volume approach is used for solving the problem in the velocity-pressure variables.Numerous series of calculations were carried out over the range Ra _t =3·10⁶ and 3·10⁷, 10^-6 < Da < 1, 1 < N < 20, Le=10 and 100, where Ra, Da, Le, and N are the Rayleigh, Darcy, and Lewis numbers and the buoyancy ratio, respectively. It is shown that the main effect of the presence of the porous medium is to reduce the heat and mass transfer and attenuate the flow field with decrease in permeability. For a certain combination of the Ra, Le, and N numbers the flow has a multicellular structure. The mean Nusselt and Sherwood numbers are presented as functions of the governing parameters.