Pore-Scale Simulation of Shear Thinning Fluid Flow Using Lattice Boltzmann Method

The present work attempts to identify the roles of flow and geometric variables on the scaling factor which is a necessary parameter for modeling the apparent viscosity of non-Newtonian fluid in porous media. While idealizing the porous media microstructure as arrays of circular and square cylinders, the present study uses multi-relaxation time lattice Boltzmann method to conduct pore-scale simulation of shear thinning non-Newtonian fluid flow. Variation in the size and inclusion ratio of the solid cylinders generates wide range of porous media with varying porosity and permeability. The present study also used stochastic reconstruction technique to generate realistic, random porous microstructures. For each case, pore-scale fluid flow simulation enables the calculation of equivalent viscosity based on the computed shear rate within the pores. It is observed that the scaling factor has strong dependence on porosity, permeability, tortuosity and the percolation threshold, while approaching the maximum value at the percolation threshold porosity. The present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.     

Finite-Difference Approximation for Fluid-Flow Simulation and Calculation of Permeability in Porous Media

We introduce a finite-difference method to simulate pore scale steady-state creeping fluid flow in porous media. First, a geometrical approximation is invoked to describe the interstitial space of grid-based images of porous media. Subsequently, a generalized Laplace equation is derived and solved to calculate fluid pressure and velocity distributions in the interstitial space domain. We use a previously validated lattice-Boltzmann method (LBM) as ground truth for modeling comparison purposes. Our method requires on average 17 % of the CPU time used by LBM to calculate permeability in the same pore-scale distributions. After grid refinement, calculations of permeability performed from velocity distributions converge with both methods, and our modeling results differ within 6 % from those yielded by LBM. However, without grid refinement, permeability calculations differ within 20 % from those yielded by LBM for the case of high-porosity rocks and by as much as 100 % in low-porosity and highly tortuous porous media. We confirm that grid refinement is essential to secure reliable results when modeling fluid flow in porous media. Without grid refinement, permeability results obtained with our modeling method are closer to converged results than those yielded by LBM in low-porosity and highly tortuous media. However, the accuracy of the presented model decreases in pores with elongated cross sections.     

Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media

As a typical multiphase fluid flow process, drainage in porous media is of fundamental interest both in nature and in industrial applications. During drainage processes in unsaturated soils and porous media in general, saturated regions, or clusters, in which a liquid phase fully occupies the pore space between solid grains, affect the relative permeability and effective stress of the system. Here, we experimentally study drainage processes in unsaturated granular media as a model porous system. The distribution of saturated clusters is analysed by optical imaging under different drainage conditions, with pore-scale information from Voronoi and Delaunay tessellation used to characterise the topology of saturated cluster distributions. By employing statistical analyses, we describe the observed spatial and temporal evolution of multiphase flow and fluid entrapment in granular media. Results indicate that the distributions of both the crystallised cell size and pore size are positively correlated to the spatial and temporal distribution of saturated cluster sizes. The saturated cluster size is found to follow a lognormal distribution, in which the generalised Bond number (\( Bo^{*} \)) correlates negatively to the scale parameter (μ) and positively to the shape parameter (σ). With further consideration of the total surface energy obtained based on liquid–air interfaces, we were able to include additional grain-scale information in the constitutive modelling of unsaturated soils using both the degree of saturation and generalised Bond number. These findings can be used to connect pore-scale behaviour with overall hydro-mechanical characteristics in granular systems.     

A lattice model of foam flow in porous media: A percolation approach

Because fluid flow in porous media is opaque to most observational techniques simulations of the processes occurring in porous media have become important. Typical reservoir simulations treat the flow as taking place in some averaged (Darcy-scale) medium but simulations can also be carried out at the level of the network of pores and throats of the porous medium. We report the results of a pore-scale investigation of mechanisms for the alteration of mobility by foam lamella blockage in a network of these spaces and channels of porous media. Saturation and relative permeability curves are obtained using well-known power-law expressions of percolation theory and a rescaling of the percolation parameter readily permits a number of lamella-blocking mechanisms to be treated. An explanation of the shift in breakthrough gas saturation and the deformation of the shape of permeabilityvs saturation curves upon introduction of foam is provided for a variety of blocking mechanisms. The qualitatively different features seen in experimental studies of modification of gas mobility by foam can be rationalized using only two parameters which characterize the throat-size at which blockage commences and the degree of blockage.     

A Capillary Water-Retention Framework for the Effective Stress Parameter Considering Hydraulic Hysteresis

Transport in Porous Media - The establishment of the representative elementary volume (REV) in studies of porous media is crucial for linking microscopic structure and pore-scale fluid...     

Filtration in a Porous Granular Medium: 1. Simulation of Pore-Scale Particle Deposition and Clogging

This paper presents a numerical model for simulating the pore-scale transport and infiltration of dilute suspensions of particles in a granular porous medium under the action of hydrodynamic and gravitational forces. The formulation solves the Stokes’ flow equations for an incompressible fluid using a fixed grid, multigrid finite difference method and an embedded boundary technique for modeling particle–fluid coupling. The analyses simulate a constant flux of the fluid suspension through a cylindrical model pore. Randomly generated particles are collected within the model pore, initially through contact and attachment at the grain surface (pore wall) and later through mounding close to the pore inlet. Simple correlations have been derived from extensive numerical simulations in order to estimate the volume of filtered particles that accumulate in the pore and the differential pressure needed to maintain a constant flux through the pore. The results show that particle collection efficiency is correlated with the Stokes’ settling velocity and indirectly through the attachment probability with the particle–grain surface roughness. The differential pressure is correlated directly with the maximum mound height and indirectly with particle size and settling velocity that affect mound packing density. Simple modification factors are introduced to account for pore length and dip angle. These parameters are used to characterize pore-scale infiltration processes within larger scale network models of particle transport in granular porous media in a companion paper. Articlenote: Currently at GZA GeoEnvironmental Inc., 1 Edgewater Drive, Norwood, MA 02062, U.S.A.     

Measurement of 3D pore-scale flow in index-matched porous media

We present the experimental analysis of fluid flow at the pore-scale of a transparent porous medium with matched refractive indices of the solid and liquid phases. The planar laser-induced fluorescence (PLIF) technique described is the first to simultaneously visualize the 3D pore-scale flow of two immiscible liquid phases in porous media. Through the application of a highly precise index matching method and the employment of up-to-date CCD imaging hardware, the system features a high spatial resolution and sampling rate. The method was used to study the dispersion of a tracer dye in single-phase flow and the displacement of oil by water in an imbibition process.     

Foreword of the Founding Editor of the Journal Transport in Porous Media,Jacob Bear

The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.

     

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

     

Hydrodynamic Model for the Evolution of an Invaded Zone in Borehole Drilling

The evolution of an invaded zone during borehole drilling in water- and oil-saturated sand formations is studied by mathematical modeling of hydrodynamic processes in porous media filled with a two-component fluid. The use of hydrodynamic modeling to interpret high-frequency electromagnetic logs makes it possible to construct consistent geoelectric and hydrodynamic models for formations with different fluid saturations. The results obtained are in good agreement with geological and geophysical data.     

Stationary discontinuity and flutter instability of wave propagation in elasto-plastic saturated porous media

In the present work a model based on the Biot theory for simulating coupled hydrodynamic behavior in saturated porous media is utilized with integration of the inertial coupling effect between the solid-fluid phases of the media into the model. The non-associated Drucker-Prager criterion to describe nonlinear constitutive behavior of pressure dependent elasto-plasticity for the media is particularly considered. With no consideration of compressibility of solid grains and the pore fluid, the discontinuity and instability of the wave propagation in saturated porous media are analyzed for the plane strain problems in detail. The critical conditions of stationary discontinuity and flutter instability in the wave propagation are given. The results and conclusions obtained by the present work will provide some bases or clues for overcoming the difficulties in numerical modeling of wave propagation in the media subjected to dynamic loading. The project supported by the National Natural Science Foundation of China (19832010)     

Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage

The aim of the paper is to model at a large scale, the formation damage in porous media by erosion and deposition of solid particles. We start from the equations governing the pore-scale processes of erosion, deposition, convection and diffusion. The macroscopic equivalent behaviour is investigated by using a homogenization method. Four characteristic models with different dominating phenomena at the pore scale are determined. The main results are twofold: first dispersion-deposition and dispersion-erosion phenomena are shown at the macroscopic scale for peculiar values of the dimensionless numbers; furthermore, and contrarily to phenomenological models, erosion and deposition generally occur in regions of intense and slow flow, respectively.     

Modeling Large-deformation-induced Microflow in Soft Biological Tissues

The homogenization approach to multiscale modeling of soft biological tissues is presented. The homogenized model describes the relationship between the macroscopic hereditary creep behavior and the microflow in a fluid-saturated dual-porous medium at the microscopic level. The micromodel is based on Biot’s system for quasistatic deformation processes, modified for the updated Lagrangian formulation to account for coupling the fluid diffusion through a porous solid undergoing large deformation. Its microstructure is constituted by fluid-filled inclusions embedded in the porous matrix. The tangential stiffness coefficients and the retardation stress for the macromodel are derived for a time-stepping algorithm. Numerical examples are discussed, showing the strong potential of the model for simulations of deformation-driven physiological processes at the microscopic scale.     

Modeling Geometric State for Fluids in Porous Media: Evolution of the Euler Characteristic

Multiphase flow in porous media is strongly influenced by the pore-scale arrangement of fluids. Reservoir-scale constitutive relationships capture these effects in a phenomenological way, relying only on fluid saturation to characterize the macroscopic behavior. Working toward a more rigorous framework, we make use of the fact that the momentary state of such a system is uniquely characterized by the geometry of the pore-scale fluid distribution. We consider how fluids evolve as they undergo topological changes induced by pore-scale displacement events. Changes to the topology of an object are fundamentally discrete events. We describe how discontinuities arise, characterize the possible topological transformations and analyze the associated source terms based on geometric evolution equations. Geometric evolution is shown to be hierarchical in nature, with a topological source term that constrains how a structure can evolve with time. The challenge associated with predicting topological changes is addressed by constructing a universal geometric state function that predicts the possible states based on a non-dimensional relationship with two degrees of freedom. The approach is validated using fluid configurations from both capillary and viscous regimes in ten different porous media with porosity between 0.10 and 0.38. We show that the non-dimensional relationship is independent of both the material type and flow regime. We demonstrate that the state function can be used to predict history-dependent behavior associated with the evolution of the Euler characteristic during two-fluid flow.

     

Macroscopic Behavior of Swelling Porous Media Derived from Micromechanical Analysis

A new macroscopic model for swelling porous media is derived based on a rigorous upscaling of the microstructure. Considering that at the microscale the medium is composed of a charged solid phase (e.g. clay platelets, bio-macromolecules, colloidal or polymeric particles) saturated by a binary monovalent aqueous electrolyte solution composed of cations + and anions – of an entirely dissociated salt, the homogenization procedure is applied to scale up the pore-scale model. The microscopic system of governing equations consists of the local electro-hydrodynamics governing the movement of the electrolyte solution (Poisson–Boltzmann coupled with a modified Stokes problem including an additional body force of Coulombic interaction) together with modified convection–diffusion equations governing cations and anions transport. This system is coupled with the elasticity problem which describes the deformation of the solid phase. Novel forms of Terzaghi's effective principle and Darcy's law are derived including the effects of swelling pressure and osmotically induced flows, respectively. Micromechanical representations are provided for the macroscopic physico-chemical quantities.     

考虑热流固耦合作用的多孔介质孔隙尺度两相流动模拟

为了研究深层油气资源在岩石多孔介质内的运移过程, 使用一种基于Darcy-Brinkman-Biot的流固耦合数值方法, 结合传热模型, 完成了Duhamel-Neumann热弹性应力的计算, 实现了在孔隙模拟多孔介质内的考虑热流固耦合作用的两相流动过程. 模型通过求解Navier-Stokes方程完成对孔隙空间内多相流体的计算, 通过求解Darcy方程完成流体在岩石固体颗粒内的计算, 二者通过以动能方式耦合的形式, 计算出岩石固体颗粒质点的位移, 从而实现了流固耦合计算. 在此基础上, 加入传热模型考虑温度场对两相渗流过程的影响. 温度场通过以产生热弹性应力的形式作用于岩石固体颗粒, 总体上实现热流固耦合过程. 基于数值模型, 模拟油水两相流体在二维多孔介质模型内受热流固耦合作用的流动过程. 研究结果表明: 热应力与流固耦合作用产生的应力方向相反, 使得总应力比单独考虑流固耦合作用下的应力小; 温度的增加使得模型孔隙度增加, 但当注入温差达到150 K后, 孔隙度不再有明显增加; 温度的增加使得水相的相对渗流能力增加, 等渗点左移.      

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.