Pore-scale Flow Simulations Research Articles

Significance of shape factor on permeability anisotropy of sand: representative elementary volume study for pore-scale analysis

This paper is a study of determination of representative elementary volume (REV) size suitable for pore-scale flow simulation (PFS) and evaluation of permeability anisotropy for sand based on Kozeny–Carman (KC) equation. We extracted small digitized samples for PFS from the 3D structures of Toyoura sand and glass beads obtained by X-ray computed tomography. Using the digitized samples with various sizes, we performed a series of flow simulations. We characterized the sample size using the ratio of the porous length L to the median grain diameter $$D_{50}$$ . The $$L/D_{50}$$ of REV was determined on the basis of statistical evaluation on coefficients of variations of the simulated values of void ratio, specific surface, hydraulic conductivity and hydraulic tortuosity. We confirmed that the hydraulic properties obtained from the PFS using the REV size determined agreed well with the past experimental studies. We also conducted PFSs in the horizontal direction to evaluate the permeability anisotropy. We clarified that the shape factor in the KC equation has a strong influence on permeability anisotropy even though previous studies treated hydraulic tortuosity as the only factor that affects the permeability anisotropy.

Anomalous transport through free-flow-porous media interface: Pore-scale simulation and predictive modeling

Pore-scale velocity and turbulence structures near streambeds may control solute transport and dispersion in streams. In this study, pore-scale flow and transport simulations are performed to investigate the effects of pore-scale processes on anomalous transport, which is often manifested by remarkably long residence times of solute particles, in coupled free-flow and porous media systems. By solving the 2D Reynolds-averaged Navier–Stokes equations integrated with a renormalization group k − ε turbulence model, we resolve complex pore-scale flows featured by vortices and preferential flows. Then, we incorporate the simulated velocity and turbulence fields into a Lagrangian particle tracking model that considers advection, turbulent diffusion, and molecular diffusion. Simulation results reveal that the interplay between pore-scale vortices and turbulence structures near the free-flow-permeable bed interface controls anomalous transport. High porosity induces strong turbulence penetration and preferential flow paths within permeable beds. The enhanced subsurface turbulence facilitates the escape of solute particles from recirculation zones via turbulent diffusion, causing steep power-law slopes in breakthrough curves (BTCs). In contrast, low porosity introduces heavy tailing in BTCs from particles that are trapped in the near-interface recirculation zones characterized by low velocities and limited turbulence. We upscale and predict particle transport via a Spatial Markov model (SMM) honoring the interplay between Lagrangian velocity distribution and velocity correlation. The SMM reproduces anomalous transport behaviors obtained from the numerical simulations. These results demonstrate that Lagrangian velocity statistics effectively encode anomalous transport mechanisms in the coupled systems.

Control mechanisms of self-affine, rough cleat networks on flow dynamics in coal reservoir

Cleat networks dominate the migration of coalbed methane (CBM), which is assembled by cleats of different type following special configuration and possessing rough, self-affine surfaces. Nowadays, configuration implications, scale-invariant properties, fractal control mechanisms, and their relations to flow dynamics have not yet been fully clarified. Herein we explore these issues numerically using effective modeling of cleat networks and pore-scale simulations of fluid flow through them. Firstly, the control mechanics of fractal dynamics was clarified by fractal topography theory, a new definition of Weierstrass-Mandelbrot (W-M) function was proposed to characterize the self-affine surface geometries, an algorithm was developed to effectively construct cleat networks similar in coal, and Lattice Boltzmann method (LBM) was used to reproduce the fluid flow in numerical cleat networks at the pore scale. Afterwards, the implications of spatial configuration of cleats and fractal control mechanisms of surface geometries on the permeability were systematically analyzed and quantified. Finally, an empirical model was established to predict the permeability of self-affine, rough cleat networks, rather than a rough estimation by a power-law proportionality in previous research. The performance of the proposed model was fully verified by comparative analysis and numerical simulations. Theoretical analysis denotes that our model can generalize several traditional and newly developed models from the literature.

An efficient numerical algorithm for solving viscosity contrast Cahn–Hilliard–Navier–Stokes system in porous media

Two-phase flow with viscosity contrast at the pore scale is modeled by a time-dependent Cahn–Hilliard–Navier–Stokes model and belongs to the class of diffuse interface method. The model allows for moving contact line and varying wettability. The numerical scheme utilizes an efficient pressure-correction projection algorithm, in conjunction with interior penalty discontinuous Galerkin schemes for space discretization developed within the framework of a distributed parallel pore-scale flow simulation system. The effect of viscosity contrast on the phase distribution is studied in relation with capillary forces and wettability. The algorithm is numerically robust and lends itself naturally to large-scale 3D numerical simulations.

Pore-scale simulation of shale oil flow based on pore network model

The pore-grain characteristics of shale oil reservoir rocks are complex: nano-size pore throats, multi-scale of pore size and topology, and rich in organic matter. Generally, shale oil flow in nanoporous media is significantly affected by microscopic fluid slippage and adsorption on the pore surfaces, which give rise to variable slip length, viscosity ratio of adsorbed phase to bulk phase, and thickness of adsorption layer in organic & inorganic pore throats. In this study, a new pore network model is developed based on a modified shale oil flow equation to consider those combinational effects on shale oil permeability under different organic matter contents, and then is applied to provide an order analysis of those effects on the permeability of a representative shale model. Different flow patterns characterizing fluid-solid interaction are also discussed. Analysis results show that shale oil permeability is strongly influenced by the slip length. The effect of adsorption on permeability is negligible for cases considered. Shale oil flow is mainly controlled by inorganic pores when volumetric TOC is low. The influence of TOC on permeability depends on slippage and adsorption conditions in organic and inorganic matters, referred in this paper to as flow patterns. As the organic matter content increases, the connectivity of organic pores and throats improves, which further affects network permeability.

Pore-scale simulation of non-Newtonian power-law fluid flow and forced convection in partially porous media: Thermal lattice Boltzmann method

In this paper, the two-dimensional forced convection heat transfer of non-Newtonian power-law fluid flow between two parallel plates filled with partially porous media is studied numerically using the thermal lattice Boltzmann method (TLBM). Shear-thinning (n=0.8), Newtonian (n=1.0), and shear-thickening (n=1.2) fluid are used to investigate the non-Newtonian behavior of power-law fluids. The porous media is prepared by the arrangement of circular obstacles. The effect of Reynolds numbers between 100<Re<300 and different power indices n (0.8, 1.0, 1.2) on the fluid flow and heat transfer characteristics in the porous media are investigated. The obtained results indicated that by increment the Reynolds number, the length of vortexes and rate of heat transfer increased. Upon decrement of the power index n, the vortexes grow and reaching the maximum size at n=0.8 and Re=300. In addition, increasing the power index leads to a decrease in the rate of heat transfer.

Coupled Processes in Charged Porous Media: From Theory to Applications

Charged porous media are pervasive, and modeling such systems is mathematically and computationally challenging due to the highly coupled hydrodynamic and electrochemical interactions caused by the presence of charged solid surfaces, ions in the fluid, and chemical reactions between the ions in the fluid and the solid surface. In addition to the microscopic physics, applied external potentials, such as hydrodynamic, electrical, and chemical potential gradients, control the macroscopic dynamics of the system. This paper aims to give fresh overview of modeling pore-scale and Darcy-scale coupled processes for different applications. At the microscale, fundamental microscopic concepts and corresponding mass and momentum balance equations for charged porous media are presented. Given the highly coupled nonlinear physiochemical processes in charged porous media as well as the huge discrepancy in length scales of these physiochemical phenomena versus the application, numerical simulation of these processes at the Darcy scale is even more challenging than the direct pore-scale simulation of multiphase flow in porous media. Thus, upscaling the microscopic processes up to the Darcy scale is essential and highly required for large-scale applications. Hence, we provide and discuss Darcy-scale porous medium theories obtained using the hybrid mixture theory and homogenization along with their corresponding assumptions. Then, application of these theoretical developments in clays, batteries, enhanced oil recovery, and biological systems is discussed.

Effects of grain size and shape distribution on pore-scale numerical simulation of two-phase flow in a heterogeneous porous medium

Series of numerical simulations were done to quantify the effects of pore structure approximations on micro-scale displacement mechanisms. Coupled Cahn–Hilliard phase field and Navier–Stokes equations were solved using finite element method to simulate two-phase flow in a heterogeneous model -having a real pore network- as the base case, compared to two other approximated models; in one of those, the complex grain geometries were simplified to circular shapes; while in the other one, only the sizes of the real-shaped grains were gently (less than ±10%) modified. In both approximated models, the medium porosity and absolute permeability were kept unchanged, compared to the base case. The simplification of the grain shapes changed the morphologies of the formed fingers and the trapped oil volumes. The general trends of the oil recovery factor and macro-scale capillary pressure variations were similar during displacement at different medium wettabilities for the base and the simplified models. However, the simplified model showed higher displacement efficiency at water-wet conditions and lower capillary pressures at oil-wet conditions; due to less complexity of its pore network geometry. Different micro-scale events were captured similarly in both models, including reverse displacement, interface coalescence, water bursting and stick-slip motion. But the oil trapping mechanisms were totally different. Slight modification of the grain sizes resulted in different displacement profiles, especially at neutral and oil-wetting conditions and low capillary numbers (log Ca < −3.69), due to modification of capillarity in the medium, hence variation of the preferred paths of the displacing phase finger(s).

Pore-scale lattice Boltzmann simulation of two-component shale gas flow

Shale gas is usually a multi-component gas mixture dominated by CH4. Moreover, CO2 sequestration in shale reservoirs and CO2-enhanced shale-gas recovery also result in multi-component gas flow in shale reservoirs. Therefore, compared with single-component gas flow, multi-component gas flow is more often encountered in practice. Furthermore, shale rock contains a lot of nano-pores in which the micro-scale effect makes the multi-component gas flow become more complex. In this paper, the lattice Boltzmann method is employed to simulate the two-component shale gas flow in a two-dimensional micropore under different conditions. The pore-scale transport mechanism of the two-component shale gas is investigated and the gas separation phenomenon for the shale gas flow is discussed in detail. It is found that the molar fraction of each species in shale gas does not distribute uniformly along the micropore and the gas separation phenomenon exists in the two-component pressure-driven shale gas flow. The molar fraction distribution of each species along the micropore is affected by the Knudsen number, pressure ratio, shale gas composition and molar fraction of each species for the pressure-driven gas flows. In particular, we find that the molar fraction distribution and pressure distribution for two-component shale gas along the micropore are related to the pressure ratio and are unrelated to the pressure gradient. With increasing the molar fraction of CH4 in shale gas, both the gas-mixture velocity at the outlet and the slip velocity along the micropore become larger. Both the Knudsen number and pressure ratio affect the molar fraction distribution of CO2 for the CH4CO2 mixture in the micropore during CO2 injection, while the influence of them works oppositely. Furthermore, the concentration diffusion without the external force is affected by the micropore width and the concentration difference between the gas mixtures.

Anomalous Dispersion in Pore-Scale Simulations of Two-Phase Flow

We investigate anomalous dispersion in steady-state two-phase flow though a random, artificial porous domain. A natural distribution of trapped wetting-phase fluid was obtained via two-phase lattice Boltzmann drainage simulations. To avoid spurious velocities, accurate inter-pore velocity fields were derived via additional one-phase lattice Boltzmann simulations incorporating slip boundary conditions imposed at various interfaces. The nature of the active dispersion at various timescales was subsequently studied via random walk particle tracking. For our system, results show persistent anomalous dispersion that depends strongly on the assumed molecular diffusivity and the initial positions of tracer particles. Imposing slip versus no-slip boundary conditions on fluid–fluid interfaces made no observable difference to results, indicating that observed anomalous dispersion resulted primarily from the complex flow network induced by the trapped fluid phase.

Prediction of fluid topology and relative permeability in imbibition in sandstone rock by direct numerical simulation

Pore-to-Darcy scale upscaling of multiphase flow is one of the major unresolved problems in many fields of porous media research. While this problem involves very fundamental aspects, there are many practical and application-driven challenges as well, such as the accurate prediction of Darcy-scale multiphase effective properties, e.g., relative permeability by pore-scale flow simulation on the basis of the imaged pore geometry, e.g., via X-ray computed micro-tomography. Validation of pore-scale modeling methods against experimental data by comparison of measured against simulated relative permeability curves has proven to be insufficient. Comparison of the fluid topology, in particular, the non-wetting phase topology, is a much more reliable criteria as relative permeability shows a very strong correlation with connectivity. While percolation-based quasi-static modeling approaches operating in the capillary limit have proven moderately successful in drainage, they largely fail to predict fluid connectivity in imbibition. We show that a fully coupled visco-capillary simulation, using a free-energy based lattice Boltzmann approach, correctly predicts fluid connectivity in imbibition, which was not predicted correctly by a quasi-static approach in a previous study on the same dataset. The respective relative permeability data shows a close match with results from Darcy-scale core flooding experiments while there is a major mismatch from quasi-static approach. The results once more highlight the close connection between relative permeability and fluid topology and suggest that topological measures are much more meaningful validation criteria for pore-scale simulation. The results also show that the direct simulation approach described in this paper, using appropriate initial and boundary conditions, correctly predicts Darcy-scale effective properties.

Pore-scale simulation of flow and turbulence characteristics in three-dimensional randomly packed beds

Fluid flow and turbulence characteristics were numerically simulated at the pore scale in actual packed beds under different operating conditions. Real geometrical structures of randomly packed beds were modeled using the discrete element software LIGGGHTS (LAMMPS Improved for General Granular and Granular Heat Transfer Simulations). The numerical model was validated against experimental data. Good agreements were achieved between the predicted results and measurements. The influences of different parameters, such as the gas velocity and particle, size were studied and discussed in detail. The results of a macroscopic analysis of the mean parameters show that inside the packed beds, both the mean porosity and flow parameters had fluctuate variations. As the pellet radius increased, the fluctuations in the mean velocity and pressure increased. The results of a local detail analysis demonstrate that complicated turbulent eddy structures exist in the pores of the packed beds. Eddies with large sizes were primarily distributed at the entrance of the main channel and near the spherical wall, and eddies with high intensities were mainly distributed near the above spherical wall and on the right side of the main channel.

Fluid Interfaces during Viscous-Dominated Primary Drainage in 2D Micromodels Using Pore-Scale SPH Simulations

We perform pore-scale resolved direct numerical simulations of immiscible two-phase flow in porous media to study the evolution of fluid interfaces. Using a Smoothed-Particle Hydrodynamics approach, we simulate saturation-controlled primary drainage in heterogeneous, partially wettable 2D porous microstructures. While imaging the evolution of fluid interfaces near capillary equilibrium becomes more feasible as fast X-ray tomography techniques mature, imaging methods with suitable temporal resolution for viscous-dominated flow have only recently emerged. In this work, we study viscous fingering and stable displacement processes. During viscous fingering, pore-scale flow fields are reminiscent of Bretherton annular flow, that is, the less viscous phase percolates through the core of a pore-throat forming a hydrodynamic wetting film. Even in simple microstructures wetting films have major impact on the evolution of fluid interfacial area and are observed to give rise to nonnegligible interfacial viscous coupling. Although macroscopically appearing flat, saturation fronts during stable displacement extend over the length of the capillary dispersion zone. While far from the dispersion zone fluid permeation obeys Darcy’s law, the interplay of viscous and capillary forces is found to render fluid flow within complex. Here we show that the characteristic length scale of the capillary dispersion zone increases with the heterogeneity of the microstructure.

Direct Numerical Simulation of Flow on Pore-Scale Images Using the Phase-Field Method

SummaryAdvances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase-flow simulator that solves mass- and momentum-balance equations valid at the pore scale (i.e., at scales where the Darcy velocity homogenization starts to break down). The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling work flow.A Helmholtz free-energy-driven, thermodynamically based diffuse-interface/phase-field method is used for the effective simulation of numerous advecting interfaces, while honoring the interfacial tension (IFT). The advective Cahn-Hilliard (CH) (mass-balance, energy dissipation) and Navier-Stokes (NS) (momentum-balance, incompressibility) equations are coupled to each other within the phase-field framework. Wettability on rock/fluid interfaces is accounted for by means of an energy-penalty-based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass-balance equation is semi-implicit in time using a convex/concave splitting of the energy term. The momentum-balance equation is split from the incompressibility constraint by a projection method and linearized with a Picard splitting. Mass- and momentum-balance equations are coupled to each other by means of operator splitting, and are solved sequentially.We discuss the mathematical model and its DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical-validation tests show optimal convergence rates for the DG discretization, indicating the correctness of the numerical scheme and its implementation. Physical-validation tests demonstrate the consistency of the phase distribution and velocity fields simulated within our framework. Finally, two-phase-flow simulations on two real pore-scale images demonstrate the usefulness of the pore-scale simulator. The direct pore-scale numerical-simulation methodology rigorously considers the flow physics by directly acting on pore-scale images of rocks without remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.

Improved skeleton extraction method considering surface feature of natural micro fractures in unconventional shale/tight reservoirs

Massive micro fractures (MFs) developed in the ultra-tight formations (such as shale/tight reservoirs), which provide preferential channels for the fluids flow. Accurate characterization of such pore-fracture systems and suitable pore network models are the fundamentals of pore structure characterization and micro scale flow simulation. Conventional medial axis (MA) skeleton extraction method cannot preserve the fracture surface feature and connectivity information, which is not suitable for accurate pore scale simulation for these porous media with MFs.In this paper, a new skeleton model was proposed to distinguish MFs from pore space via extraction of surface points set of MFs. In the procedure of points set extraction, we improved the classic “MA based” shrink method to “medial surface (MS) based” method for the MFs characterization through introducing a new set of skeleton points (i.e., surface points and edge points of the micro fractures). The former describes their apertures and the latter is used for collecting connectivity information and determining the extension ranges of the MFs. Comparison of connectivity index, fracture length, Euclidean distance showed enhanced effectiveness and accuracy of the proposed method. The proposed method was applied in four ideal models and one field shale core sample. Results show that the proposed skeleton model can show more comprehensible forms of the real connected junction instead of the conventional ideal model. The extracted skeleton can also satisfy demands of the traditional skeleton extraction model and preserve the topology of the original pore-fracture space. This work proposed a more accurate method for pore-scale modeling in cores with natural MFs, and potentially applicable for pore scale flow simulations for tight/shale reservoirs.

Petrographic characterization to build an accurate rock model using micro-CT: Case study on low-permeable to tight turbidite sandstone from Eocene Shahejie Formation

Pore scale flow simulations heavily depend on petrographic characterizing and modeling of reservoir rocks. Mineral phase segmentation and pore network modeling are crucial stages in micro-CT based rock modeling. The success of the pore network model (PNM) to predict petrophysical properties relies on image segmentation, image resolution and most importantly nature of rock (homogenous, complex or microporous). The pore network modeling has experienced extensive research and development during last decade, however the application of these models to a variety of naturally heterogenous reservoir rock is still a challenge. In this paper, four samples from a low permeable to tight sandstone reservoir were used to characterize their petrographic and petrophysical properties using high-resolution micro-CT imaging. The phase segmentation analysis from micro-CT images shows that 5–6% microporous regions are present in kaolinite rich sandstone (E3 and E4), while 1.7–1.8% are present in illite rich sandstone (E1 and E2). The pore system percolates without micropores in E1 and E2 while it does not percolate without micropores in E3 and E4. In E1 and E2, total MICP porosity is equal to the volume percent of macrospores determined from micro-CT images, which indicate that the macropores are well connected and microspores do not play any role in non-wetting fluid (mercury) displacement process. Whereas in E3 and E4 sandstones, the volume percent of micropores is far less (almost 50%) than the total MICP porosity which means that almost half of the pore space was not detected by the micro-CT scan. PNM behaved well in E1 and E2 where better agreement exists in PNM and MICP measurements. While E3 and E4 exhibit multiscale pore space which cannot be addressed with single scale PNM method, a multiscale approach is needed to characterize such complex rocks. This study provides helpful insights towards the application of existing micro-CT based petrographic characterization methodology to naturally complex petroleum reservoir rocks.

Finite-difference method Stokes solver (FDMSS) for 3D pore geometries: Software development, validation and case studies

Permeability is one of the fundamental properties of porous media and is required for large-scale Darcian fluid flow and mass transport models. Whilst permeability can be measured directly at a range of scales, there are increasing opportunities to evaluate permeability from pore-scale fluid flow simulations. We introduce the free software Finite-Difference Method Stokes Solver (FDMSS) that solves Stokes equation using a finite-difference method (FDM) directly on voxelized 3D pore geometries (i.e. without meshing). Based on explicit convergence studies, validation on sphere packings with analytically known permeabilities, and comparison against lattice-Boltzmann and other published FDM studies, we conclude that FDMSS provides a computationally efficient and accurate basis for single-phase pore-scale flow simulations. By implementing an efficient parallelization and code optimization scheme, permeability inferences can now be made from 3D images of up to 109 voxels using modern desktop computers. Case studies demonstrate the broad applicability of the FDMSS software for both natural and artificial porous media.

3D reconstruction and pore-scale modeling of coated catalytic filters for automotive exhaust gas aftertreatment

This paper introduces a newly developed methodology for the pore-scale simulation of flow, diffusion and reaction in the coated catalytic filter. 3D morphology of the porous filter wall including the actual distribution of catalytic material is reconstructed from X-ray tomography (XRT) images and further validated with the mercury intrusion porosimetry (MIP). The reconstructed medium is then transformed into simulation mesh for OpenFOAM. Flow through free pores in the substrate as well as through the coated zones is simulated by porousSimpleFoam solver, while an in-house developed solver is used for component diffusion and reactions. Three cordierite filter samples with different distribution of alumina-based coating ranging from in-wall to on-wall are examined. Velocity, pressure and component concentration profiles are calculated enabling the prediction of permeability and component conversion depending on the actual microstructure of the wall. The simulation results suggest that the gas predominantly flows through remaining free pores in the filter wall and cracks in the coated layer. The mass transport into the coated domains inside the filter wall is enabled mainly by diffusion. Large domains of compact catalytic coating covering complete channel wall result in a significant increase of pressure drop as the local permeability of the coating is two orders of magnitude smaller than that of bare filter wall.

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

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, 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. 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. Present investigation quantifies and proposes meaningful correlations between the scaling factor and the macroscopic properties of the porous media.

Research on Stress Sensitivity of Fractured Carbonate Reservoirs Based on CT Technology

Fracture aperture change under stress has long been considered as one of primary causes of stress sensitivity of fractured gas reservoirs. However, little is known about the evolution of the morphology of fracture apertures on flow property in loading and unloading cycles. This paper reports a stress sensitivity experiment on carbonate core plugs in which Computed Tomography (CT) technology is applied to visualize and quantitatively evaluate morphological changes to the fracture aperture with respect to confining pressure. Fracture models were obtained at selected confining pressures on which pore-scale flow simulations were performed to estimate the equivalent absolute permeability. The results showed that with the increase of confining pressure from 0 to 0.6 MPa, the fracture aperture and equivalent permeability decreased at a greater gradient than their counterparts after 0.6 MPa. This meant that the rock sample is more stress-sensitive at low effective stress than at high effective stress. On the loading path, an exponential fitting was found to fit well between the effective confining pressure and the calculated permeability. On the unloading path, the relationship is found partially reversible, which can evidently be attributed to plastic deformation of the fracture as observed in CT images.