A one-dimensional moving grid solution for the coupled non-linear equations governing multiphase flow in porous media. 1: Model development

A straightforward moving grid finite element method is developed to solve the one-dimensional coupled system of non-linear partial differential equations (PDEs) governing two- and three-phase flow in porous media. The method combines features from a number of self-adaptive grid techniques. These techniques are the equidistribution, the moving grid finite element and the local grid refinement/coarsening methods. Two equidistribution criteria, based on solution gradient and curvature, are employed and nodal distributions are computed iterativcly. Using the developed approach, an intermingle-free nodal distribution is guaranteed. The method involves examination of a single representative gradient to facilitate the application of moving grid algorithms to solve a non-linear coupled set of PDEs and includes a feature to limit mass balance error during nodal redistribution. The finite element part of the developed algorithm is verified against an existing finite difference model. A numerical simulation example involving a single-front two-phase flow problem is presented to illustrate model performance. Additional simulation examples are given in Part 2 of this paper. These examples include single and double moving fronts in two- and three-phase flow systems incorporating source/sink terms. Simulation sensitivity to the moving grid parameters is also explored in Part 2.     

Flow of micropolar fluid between two orthogonally moving porous disks

The unsteady,laminar,incompressible,and two-dimensional flow of a micropolar fluid between two orthogonally moving porous coaxial disks is considered.The extension of von Karman’s similarity transformations is used to reduce the governing partial differential equations(PDEs) to a set of non-linear coupled ordinary differential equations(ODEs) in the dimensionless form.The analytical solutions are obtained by employing the homotopy analysis method(HAM).The effects of various physical parameters such as the expansion ratio and the permeability Reynolds number on the velocity fields are discussed in detail.     

Simulation of multiphase flows in porous media

The ability to numerically simulate single phase and multiphase flow of fluids in porous media is extremely important in developing an understanding of the complex phenomena governing the flow. The flow is complicated by the presence of heterogeneities in the reservoir and by phenomena such as diffusion, dispersion, and viscous fingering. These effects must be modeled by terms in coupled systems of nonlinear partial differential equations which form the basis of the simulator. The simulator must be able to handle both single and multiphase flows and the transition regimes between the two. A discussion of some of the aspects of modeling dispersion and viscous fingering is presented along with directions for future work.The partial differential equation models are convection-dominated and contain important local effects. An operator-splitting technique is used to address these different effects accurately. Convection is treated by time stepping along the characteristics of the associated pure convection problem, and diffusion is modeled via a Galerkin method for single phase flow and a Petrov-Galerkin technique for multiphase regimes. ELLAM (Eulerian-Lagrangian Localized Adjoint Methods) are discussed to effectively treat the advection-dominated processes. Accurate approximations of the fluid velocities needed in the Eulerian-Lagrangian time-stepping procedure are obtained by mixed finite element methods. Adaptive local grid refinement techniques are then indicated to resolve important local phenomena around wells and large heterogeneities or to resolve the moving internal boundary layers which often govern the mass transfer between phases.     

Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis

In the present paper, Lie group symmetry method is used to obtain some exact solutions for a hyperbolic system of partial differential equations (PDEs), which governs an isothermal no-slip drift-flux model for multiphase flow problem. Those symmetries are used for the governing system of equations to obtain infinitesimal transformations, which consequently reduces the governing system of PDEs to a system of ODEs. Further, the solutions of the system of ODEs which in turn produces some exact solutions for the PDEs are presented. Finally, the evolutionary behavior of weak discontinuity is discussed.     

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness,variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.     

An improved SPH model for multiphase flows with large density ratios

This paper presents a new smoothed particle hydrodynamics (SPH) model for simulating multiphase fluid flows with large density ratios. The new SPH model consists of an improved discretization scheme, an enhanced multiphase interface treatment algorithm, and a coupled dynamic boundary treatment technique. The presented SPH discretization scheme is developed from Taylor series analysis with kernel normalization and kernel gradient correction and is then used to discretize the Navier‐Stokes equation to obtain improved SPH equations of motion for multiphase fluid flows. The multiphase interface treatment algorithm involves treating neighboring particles from different phases as virtual particles with specially updated density to maintain pressure consistency and a repulsive interface force between neighboring interface particles into the pressure gradient to keep sharp interface. The coupled dynamic boundary treatment technique includes a soft repulsive force between approaching fluid and solid particles while the information of virtual particles are approximated using the improved SPH discretization scheme. The presented SPH model is applied to 3 typical multiphase flow problems including dam breaking, Rayleigh‐Taylor instability, and air bubble rising in water. It is demonstrated that inherent multiphase flow physics can be well captured while the dynamic evolution of the complex multiphase interfaces is sharp with consistent pressure across the interfaces.     

Non‐intrusive reduced‐order modeling for multiphase porous media flows using Smolyak sparse grids

In this article, we describe a non‐intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non‐intrusive reduced‐order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non‐intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced‐porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high‐fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.     

A Potential Formulation of Non-Linear Models of Flow through Anisotropic Porous Media

A theoretical analysis, based on the search for a normal dissipation potential, is performed in order to generalize the empirical non-Darcy one-dimensional flow models to 3-D flows through anisotropic porous media. In an abstract framework, it is proven that a large number of heuristic non-linear equations governing the multidimensional flow through isotropic porous media can be derived starting from a potential strictly related to the mechanical power dissipated by the fluid. Such a formulation allows to define, for the tensor permeability case, a wide class of filtration models according to the Onsager's generalized theory of dissipative mechanical systems. A consistent generalization to anisotropic permeability case of polynomial flow models is proposed. Both primal and dual mixed variational formulations associated to the proposed quadratic and incomplete cubic flow models are introduced and discussed.     

Multiphase non‐Newtonian effects on pulsatile hemodynamics in a coronary artery

Hemodynamic stresses are involved in the development and progression of vascular diseases. This study investigates the influence of mechanical factors on the hemodynamics of the curved coronary artery in an attempt to identify critical factors of non‐Newtonian models. Multiphase non‐Newtonian fluid simulations of pulsatile flow were performed and compared with the standard Newtonian fluid models. Different inlet hematocrit levels were used with the simulations to analyze the relationship that hematocrit levels have with red blood cell (RBC) viscosity, shear stress, velocity, and secondary flow. Our results demonstrated that high hematocrit levels induce secondary flow on the inside curvature of the vessel. In addition, RBC viscosity and wall shear stress (WSS) vary as a function of hematocrit level. Low WSS was found to be associated with areas of high hematocrit. These results describe how RBCs interact with the curvature of artery walls. It is concluded that although all models have a good approximation in blood behavior, the multiphase non‐Newtonian viscosity model is optimal to demonstrate effects of changes in hematocrit. They provide a better stimulation of realistic blood flow analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation of Single-Phase Multicomponent Flow Problems in Gas Reservoirs by Eulerian–Lagrangian Techniques

Over the past two decades most discussions of the simulation of miscible displacement in porous media were related to incompressible flow problems; recently, however, attention has shifted to compressible problems. The first goal of this paper is the derivation of the governing equations (mathematical models) for a hierarchy of miscible isothermal displacements in porous media, starting from a very general single-phase, multicomponent, compressible flow problem; these models are then compared with previously proposed models. Next, we formulate an extension of the modified method of characteristics with adjusted advection to treat the transport and dispersion of the components of the miscible fluid; the fluid displacement must be coupled in a two-stage operator-splitting procedure with a pressure equation to define the Darcy velocity field required for transport and dispersion, with the outer stage incorporating an implicit solution of the nonlinear parabolic pressure equation and an inner stage for transport and diffussion in which the mass fraction equations are solved sequentially by first applying a globally conservative Eulerian–Lagrangian scheme to solve for transport, followed by a standard implicit procedure for including the diffusive effects. The third objective is a careful investigation of the underlying physics in compressible displacements in porous media through several high resolution numerical experiments. We consider real binary gas mixtures, with realistic thermodynamic correlations, in homogeneous and heterogeneous formations.     

Evaluation of the Darcy's law performance for two-fluid flow hydrodynamics in a particle debris bed using a lattice-Boltzmann model

In the present paper, multiphase flow dynamics in a porous medium are analyzed by employing the lattice-Boltzmann modeling approach. A two-dimensional formulation of a lattice-Boltzmann model, using a D2Q9 scheme, is used. Results of the FlowLab code simulation for single phase flow in porous media and for two-phase flow in a channel are compared with analytical solutions. Excellent agreement is obtained. Additionally, fluid-fluid interaction and fluid-solid interaction (wettability) are modeled and examined. Calculations are performed to simulate two-fluid dynamics in porous media, in a wide range of physical parameters of porous media and flowing fluids. It is shown that the model is capable of determining the minimum body force needed for the nonwetting fluid to percolate through the porous medium. Dependence of the force on the pore size, and geometry, as well as on the saturation of the nonwetting fluid is predicted by the model. In these simulations, the results obtained for the relative permeability coefficients indicate the validity of the reciprocity for the two coupling terms in the modified Darcy's law equations. Implication of the simulation results on two-fluid flow hydrodynamics in a decay-heated particle debris bed is discussed. Received on 1 December 1999     

Comparison of GMRES and ORTHOMIN for the black oil model on unstructured grids

This paper addresses an application of ORTHOMIN and GMRES to petroleum reservoir simulation using the black oil model on unstructured grids. Comparisons between these two algorithms are presented in terms of storage and total flops per restart step. Numerical results indicate that GMRES is faster than ORTHOMIN for all tested petroleum reservoir problems, particularly for large scale problems. The control volume function approximation method is utilized in the discretization of the governing equations of the black oil model. This method can accurately approximate both the pressure and velocity in the simulation of multiphase flow in porous media, effectively reduce grid orientation effects, and be easily applied to arbitrarily shaped control volumes. It is particularly suitable for hybrid grid reservoir simulation. Copyright © 2006 John Wiley & Sons, Ltd.     

LBM Simulation of Viscous Fingering Phenomenon in Immiscible Displacement of Two Fluids in Porous Media

This article presents the lattice Boltzmann simulation of viscous fingering phenomenon in immiscible displacement of two fluids in porous media. Such phenomenon generally takes place when a less viscous fluid is used to displace a more viscous fluid, and it can be found in many industrial fields. Dimensionless quantities, such as capillary number, Bond number and viscosity ratio between displaced fluid and displacing fluid are introduced to illustrate the effects of capillary force, viscous force, and gravity on the fluid behaviour. The surface wettability, which has an impact on the finger pattern, is also considered in the simulation. The numerical procedure is validated against the experiment about viscous fingering in a Hele-Shaw cell. The displacement efficiency is investigated using the parameter, areal sweep efficiency. The present simulation shows an additional evidence to demonstrate that the lattice Boltzmann method is a useful method for simulating some multiphase flow problems in porous media.     

Multiphase flow in heterogeneous porous media: A classical finite element method versus an implicit pressure–explicit saturation‐based mixed finite element–finite volume approach

Various discretization methods exist for the numerical simulation of multiphase flow in porous media. In this paper, two methods are introduced and analyzed—a full‐upwind Galerkin method which belongs to the classical finite element methods, and a mixed‐hybrid finite element method based on an implicit pressure–explicit saturation (IMPES) approach. Both methods are derived from the governing equations of two‐phase flow. Their discretization concepts are compared in detail. Their efficiency is discussed using several examples. Copyright © 1999 John Wiley & Sons, Ltd.     

一种基于黎曼解处理大密度比多相流SPH的改进算法

多相流界面存在密度、黏性等物理场间断,直接采用传统光滑粒子水动力学(smoothedparticle hydrodynamics,SPH)方法进行数值模拟,界面附近的压力和速度存在震荡.一套基于黎曼解能够处理大密度比的多相流SPH计算模型被提出,该模型利用黎曼解在处理接触间断问题方面的优势,将黎曼解引入到SPH多相流计算模型中,为了能够准确求解多相流体物理黏性、减小黎曼耗散,对黎曼形式的SPH动量方程进行了改进,又将Adami固壁边界与黎曼单侧问题相结合来施加多相流SPH固壁边界,同时模型中考虑了表面张力对小尺度异相界面的影响,该模型没有添加任何人工黏性、人工耗散和非物理人工处理技术,能够反应多相流真实物理黏性和物理演变状态.采用该模型首先对三种不同粒子间距离散下方形液滴震荡问题进行了数值模拟,验证了该模型在处理异相界面的正确性和模型本身的收敛性;后又通过对Rayleigh--Taylor不稳定、单气泡上浮、双气泡上浮问题进行了模拟计算,结果与文献对比吻合度高,异相界面捕捉清晰,结果表明,本文改进的多相流SPH模型能够稳定、有效的模拟大密度比和黏性比的多相流问题.      

A heterogeneous modeling method for porous media flows

This paper presents a new heterogeneous multiscale modeling method for porous media flows. Physics at the global level is governed by one set of PDEs, while features in the solution that are beyond the resolution capacity of the global model are accounted for by the next refined set of governing equations. In this method, the global or coarse model is given by the Darcy equation, while the local or refined model is given by the Darcy–Stokes equation. Concurrent domain decomposition where global and local models are applied to adjacent subdomains, as well as overlapping domain decomposition where global and local models coexist on overlapping domains, is considered. An interface operator is developed for the case where global and local models commute along the common interface. For the overlapping decomposition, a residual‐based coupling technique is developed that consistently facilitates bottom‐up embedding of scale effects from the local Darcy–Stokes model into the global Darcy model. Numerical results are presented for nonoverlapping and overlapping domain decompositions for various benchmark problems. Computed results show that the hierarchically coupled models accurately account for the heterogeneity of the medium and efficiently incorporate local features into the global response. Copyright © 2014 John Wiley & Sons, Ltd.     

Comment on the paper “Transient MHD free convective flow past an infinite vertical plate embedded in a porous medium with viscous dissipation,Siva Reddy Sheri,R. Srinivasa Raju,Meccanica, DOI 10.1007/s11012-015-0285-y”

A numerical investigation of transient magnetohydrodynamic free convection flow past an infinite vertical plate embedded in a porous medium with viscous dissipation is presented in the above paper. The governing differential equations are transformed into a set of non-linear coupled partial differential equations and are solved numerically using the finite element method. Numerical results for the velocity, temperature and concentration profiles within the boundary layer are presented and discussed.     

Unsteady flow of a fourth-grade fluid due to an oscillating plate

The governing non-linear high-order, sixth-order in space and third-order in time, differential equation is constructed for the unsteady flow of an incompressible conducting fourth-grade fluid in a semi-infinite domain. The unsteady flow is induced by a periodically oscillating two-dimensional infinite porous plate with suction/blowing, located in a uniform magnetic field. It is shown that by augmenting additional boundary conditions at infinity based on asymptotic structures and transforming the semi-infinite physical space to a bounded computational domain by means of a coordinate transformation, it is possible to obtain numerical solutions of the non-linear magnetohydrodynamic equation. In particular, due to the unsymmetry of the boundary conditions, in numerical simulations non-central difference schemes are constructed and employed to approximate the emerging higher-order spatial derivatives. Effects of material parameters, uniform suction or blowing past the porous plate, exerted magnetic field and oscillation frequency of the plate on the time-dependent flow, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviour of the fourth-grade non-Newtonian fluid is also compared with those of the Newtonian fluid.     

Multicomponent, Multiphase Thermodynamics of Swelling Porous Media with Electroquasistatics: II. Constitutive Theory

In Part I macroscopic field equations of mass, linear and angular momentum, energy, and the quasistatic form of Maxwell's equations for a multiphase, multicomponent medium were derived. Here we exploit the entropy inequality to obtain restrictions on constitutive relations at the macroscale for a 2-phase, multiple-constituent, polarizable mixture of fluids and solids. Specific emphasis is placed on charged porous media in the presence of electrolytes. The governing equations for the stress tensors of each phase, flow of the fluid through a deforming medium, and diffusion of constituents through such a medium are derived. The results have applications in swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, chromotography, drug delivery, and other swelling systems.     

ON FLOW IN POROUS MEDIA

渗流是流体在多孔介质中的流动,渗流现象广泛地存在于自然界、工程材料、动物、植物中。多孔介质种类繁多,包括岩石(含各类矿藏)、土壤、生物材料和人工多孔介质材料等。渗流理论已经成为人类开发地下水、地热、石油、天然气、煤炭与煤层气等诸多地下资源的重要理论基础。本文从渗流的基本概念、渗流的分类、渗流的影响因素、渗流的特征以及渗流的研究意义等方面进行了阐述。