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.     

Waves in partially saturated porous media

The propagation of compressional waves in a porous medium is investigated in case the pore liquid contains a small volume fraction of gas. The effect of oscillating gas bubbles is taken into account by introducing a frequency-dependent fluid bulk modulus, which is incorporated in the Biot theory. Using a shock tube technique, new experimental data are obtained for a porous column subjected to a pressure step wave. An oscillatory behaviour is observed, consisting of two distinct frequency bands, which is predicted by the theoretical analysis.     

Impact of Capillary-Driven Liquid Films on Salt Crystallization

Flow-through drying of ionic liquids in porous media can lead to super saturation and hence crystallization of salts. A model for the evolution of solid and liquid concentrations of salt, in porous media, due to evaporation by gas flow is presented. The model takes into account the impact of capillary-driven liquid film flow on the evaporation rates as well as the rate of transport of salt through those films. It is shown that at high capillary wicking numbers and high dimensionless pressure drops, supersaturation of brine takes place in the higher drying rate regions in the porous medium. This leads to solid salt crystallization and accumulation in the higher drying rate region. In the absence of wicking, there is no transport and accumulation of solid salt. Results from experiments of flow-through drying in rock cores are compared with model prediction of salt crystallization and accumulation.     

On the stability of Taylor bubbles inside a confined highly porous medium

This work focuses on the local hydrodynamics of a multiphase gas–liquid flow forced into an innovative medium of high porosity (96%): an open cell solid foam. The gas (nitrogen) and liquid (ethanol) phases are injected at constant flow-rates in a millichannel to form a well-controlled Taylor flow which enters the porous medium. Based on a fluorescence technique, the apparent liquid holdup in the porous medium is quantified, and its evolution in time and along the porous medium extracted from spatiotemporal diagrams. The analysis of the main frequency, when varying the gas–liquid flow-rate ratio, leads to the identification of two hydrodynamic regimes. A model based on a scaling analysis is proposed to quantify the dimensionless numbers describing the transition between both regimes. It points out that the bubble length fixed by the Taylor flow is the control parameter. The model prediction of the critical bubble length at which the transition occurs is in good agreement with the experimental observations.     

Kinetic Models for a Gas Filled Porous Matrix. WAVE PROPAGATION

The propagation of small perturbation in a gas filled porous matrix is investigated. The skeleton is supposed rigid and governed by the energy balance equation, where the heat exchanged between the two phases is taken into account. The Boltzmann equation is written for the gas where the integrals of the collisions between gas and solid particles are evaluated as those for the particles of a mixture. Different choices of the time and space scales lead to models equations which hold for different rarefaction regimes. The wave propagation characteristics are then dealt with in various situations.     

Theoretical and Experimental Investigations of Miscible Displacement in Fractured Porous Media

In this paper a mathematical model for miscible displacement in fractured porous media is developed. The model takes into account mechanisms of mass transfer between fracture and matrix. The model is normalized by using the dimensionless parameters, which characterize the process, and the analytical solutions of the resulting system of equations are provided by utilizing the method of characteristics. For comparison the results of model with experimental results, laboratory displacement tests have been performed in fractured systems under miscible displacement. The porous media used were cylindrical Asmari cores from Iranian reservoirs containing an artificially vertical fracture. Normal heptane and kerosene were two miscible fluid used. There is very good agreement between experiments and model prediction.     

Numerical Simulation of Diagenetic Alteration and Its Effect on Residual Gas in Tight Gas Sandstones

In this study, we numerically cemented a segmented X-ray microtomography image of a sandstone to understand changes to pore space connectivity, capillary control on gas, and water distributions, and ultimately production behavior in tight gas sandstone reservoirs. Level set method-based progressive quasi-static algorithm (a state-of-the-art direct simulation of capillarity-dominated fluid displacement) was used to find the gas/water configurations during drainage and imbibition cycles. Further, we account for gas?Cwater interfacial tension changes using 1D burial history model based on available geologic data. We have found the displacement simulation method robust, and that diagenetic changes impart a significantly larger effect on gas trapping compared with interfacial tension changes.     

Study on Expansion Characteristic of Twin Combustion Gas Jets in Five-Stage Cylindrical Stepped-wall Observation Chamber

The five-stage cylindrical stepped-wall observation chamber is designed to investigate the method of controlling the interior ballistic stability in bulk-loaded propellant guns. The expansion and mixing process of twin combustion gas jets in liquid is studied by means of high speed photographic system. The influence of multiple parameters on jet expansion shape is discussed. Based on the experiment, the three-dimensional mathematics model is established to simulate the expansion process of twin gas jets in liquid. The pressure, density, temperature, velocity contours and evolutionary process of vortices are obtained. Results show that vortices behind the corner of the steps are formed due to the inducing effect of steps. The jets can expand along the axial and radial direction simultaneously, weakening the Kelvin-Helmholtz instability. The numerical simulation results of axial expansion displacement are in good agreement with the experimental data.     

A two-phase, two-component model for vertical upward gas-liquid annular flow

In this paper, a new two-fluid two-component computational fluid dynamics (CFD) model is developed to simulate vertical upward two-phase annular flow. The two-phase VOF scheme is utilized to model the roll wave flow, and the gas core is described by a two-component phase consisting of liquid droplets and gas phase. The entrainment and deposition processes are taken into account by source terms of the governing equations. Unlike the previous models, the newly developed model includes the effect of liquid roll waves directly determined from the CFD code, which is able to provide more detailed and, the most important, more self-standing information for both the gas core flow and the film flow as well as their interactions. Predicted results are compared with experimental data, and a good agreement is achieved.     

Water removal from porous media by gas injection: experiments and simulation

The flow of a saturated gas through a porous medium, partially occupied by a liquid phase, causes evaporation due to gas expansion. This process, referred to as flow-through drying, is important in a wide variety of natural and industrial applications, such as natural gas production, convective drying of paper, catalysts, fuel cells and membranes. X-ray imaging experiments were performed to study the flow-through drying of water-saturated porous media during gas injection. The results show that the liquid saturation profile and the rate of drying are dependent on the viscous pressure drop, the state of saturation of the gas and the capillary characteristics of the porous medium. During the injection of a completely saturated gas, drying occurs only due to gas expansion. Capillary-driven flow from regions of high saturation to regions of low saturation lead to more uniform saturation profiles. During the injection of a dry gas, a drying front develops at the inlet and propagates through the porous medium. The experimental results are compared with numerical results from a continuum model. A good agreement is found for the case of sandstone. The comparison is less satisfactory for the experiments with limestone.     

Capillary Model of Gravity Elution of High Viscosity Liquid from a Porous Bed with a Low-Viscosity Liquid

Modeling of the processes of elution from porous systems is essential importance for development of the removing oily contaminations from the soils and intensification of the oil recovery processes. In the paper, capillary model of gravitational elution of high viscosity substances from the porous medium by using low viscosity liquids was derived. This model allowed for the prediction of changes in time of such parameters like: level of bed saturation with oil, relative bed permeability, liquid flow rate, flow resistance, volume of eluted liquid during the process. When modeling, phenomena and the physical properties associated with the analyzed process, such as, for example, the effects of surface tension, fluids viscosity, specific size of the granular bed, initial oil saturation of bed, variable process driving force, and the flow of liquid through the preferential flow paths, were taken into account. This allowed for the more complete imaging of elution process. The model has been verified on the basis of the results of experimental studies. In addition, the discussion on the behavior of the model due to changes in values of various parameters was carried out.     

AN ALGEBRAIC MULTIGRID METHOD FOR COUPLED THERMO-HYDRO-MECHANICAL PROBLEMS

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Computing the Longtime Behaviour of NMR Propagators in Porous Media Using a Pore Network Random Walk Model

We present a pore network model combined with a random walk algorithm allowing the simulation of molecular displacement distributions in porous media as measured by NMR. A particular feature of this technique is the ability to probe the time evolution of these distributions. The objective is to predict the displacement behaviour for time intervals larger than the experimental observation time and explore the asymptotic dispersion regime at long times. Starting from 3D micro-CT images, we computed the variance of displacement distributions of water molecules in a Fontainebleau sand and found very good agreement of the time evolution of the variance with experimental data, without fitting parameter. The model confirms a weak superdispersion in the asymptotic regime. In addition, we conclude that, since pore network models do not take into account small scale features of the porous medium (e.g., surface roughness and grain shape), the origin of the observed superdispersion is mainly due to the topology and geometry of the porous medium.     

Liquid seeping into porous medium

 A mathematical model is presented to describe the hydrodynamics behavior of liquid seeping into porous medium. The model takes into account the inertia and the evaporation effects. Analytical solution is obtained for the hydrodynamics behavior of the seeping process when the inertia effects are excluded and numerical solutions are obtained when these effects are included. It is found that both evaporation and inertia have significant effects on the seeping process. These effects are investigated qualitatively and quantitatively. Received on 10 June 1999     

Lamb's integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil,drainage is common.In this paper,the saturated porous soil with a two-phase saturated medium is simulated,and Lamb's integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot's equation and Betti's theorem(the reciprocal theorem).According to the basic solution to Biot's equation,Green's function Gij and three terms of Green's function G4i,Gi4,and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented.The displacement field with drainage,the magnitude of drainage,and the pore pressure of the center explosion source are obtained in computation.The results of the classical Sharpe's solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared.Good agreement is observed.     

Low Pressure Gas Percolation Characteristic in Ultra-low Permeability Porous Media

Low pressure gas percolation characteristic in ultra-low permeability porous media is investigated in this article through core flow experiments. The results show that the wall-slip layer covers more than 10% of the average porous channel radius on account of minimum pore size when the permeability is below 0.1 × 10^?3μ m ² order, and seepage behavior is contrasted to that in mid-high permeability pore media. When the gas pressure is not high enough, the flow regime turns into transitional flow instead of slip flow, and nonlinear relationship between the measured gas permeability and the reciprocal of average pressure exists. The gas measuring permeability experiment would be influenced by the non-linear relationship. If Klinkenberg-corrected method is applied to speculate the equivalent liquid permeability, the extrapolated value will become less or minus. Simultaneously, actual gas flow velocity at the outlet is beyond the calculated value with Klinkenberg formula. A new gas seepage model based on the general slip boundary condition is derived from the homogenization technique in this article. At last the flow model is examined to be suitable for representing the gas flow behavior in ultra-low permeability media and estimating the absolute permeability from single-point, steady-states measurements.     

Equilibrium of a crack in a porous medium with injection of the filtering liquid

In connection with the exploitation of petroleum deposits, the article discusses the equilibrium of a porous medium with a crack under conditions of plane deformation, with the steady-state filtration of a liquid injected into the porous medium through a crack. It is assumed that the crack, which has initial zero dimensions, can become wider and longer with a rise in the pressure. The displacement of the sides of the crack is determined on the basis of the theory of elasticity, taking account of the deformation properties of a saturated porous medium. The stress and the displacement are expressed in terms of two analytical Muskhelishvili functions and the complex filtration potential. A change in the volume of the porous medium leads to a discontinuity of the displacements at the feed contour, and to distortion in the filtration region. For a circular stratum, the dimensions of the crack and the mass flow rate of the liquid are determined in the first approximation. The region of values of the pressure in which there exists a stable equilibrium state of the open crack and a steady-state flow of the liquid is found.     

Pore Network Simulation of Evaporation of a Binary Liquid from a Capillary Porous Medium

A pore-network model of evaporation of a binary liquid mixture into a ternary gas phase is developed. The model is applied to study the influence of surface tension gradients induced by composition variations of the liquid on the phase distribution within a capillary porous medium. Numerical simulations based on the proposed model show that the surface tension gradients lead to the accumulation of liquid near the open edge of the network. This surface tension gradient effect is only significant for weakly disordered porous media.     

一维流体饱和粘弹性多孔介质层的动力响应

本文研究了不可压流体饱和粘弹性多孔介质层的一维动力响应问题。基于粘弹性理论和多孔介质理论,在流相和固相微观不可压、固相骨架服从粘弹性积分型本构关系和小变形的假定下,建立了不可压流体饱和粘弹性多孔介质层一维动力响应的数学模型,利用Laplace变换,求得了原初边值问题在变换空间中的解析解,并利用Laplace逆变换的Crump数值反演方法,得到原动力响应问题的数值解。数值研究了饱和标准线性粘弹性多孔介质层的动力响应,分析了固相位移、渗流速度、孔隙压力及固相有效应力等的响应特征。结果表明,与不可压流体饱和弹性多孔介质相同,不可压流体饱和粘弹性多孔介质中亦只存在一个纵波,并且固相骨架的粘性对动力行为有显著的影响。