Images Of Porous Media Research Articles

Improved algorithm for estimating pore size distribution from pore space images of porous media.

Pore size distribution (PSD), which is defined as the percentage of the pores of each size to the total volume of the pores, is a key parameter to characterize the microstructure of porous media. Based on the pore geometry definition of three-dimensional (3D) maximal ball and two-dimensional (2D) maximal disk in discrete space, an improved algorithm is proposed to obtain PSD from pore space images of porous media. To validate the accuracy of our algorithm, a synthetic porous medium with a known PSD is generated by computer simulation. In addition, other 2D and 3D images of various types of samples, including sandstones, carbonates, and man-made materials, were also used to obtain PSD. The PSDs were compared quantitatively with that obtained by using the existing state of the art algorithm. The results show that our algorithm is more accurate in characterizing the pore structure and more consistent with the direct visual identification. In addition, the efficiency of our algorithm was also evaluated by calculating the PSD of large-scale 3D images, the results of which indicate that all calculations can be completed in a relatively short time.

Calculation of tensorial flow properties on pore level: Exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions.

While it is well known that permeability is a tensorial property, it is usually reported as a scalar property or only diagonal values are reported. However, experimental evaluation of tensorial flow properties is problematic. Pore-scale modeling using three-dimensional (3D) images of porous media with subsequent upscaling to a continuum scale (homogenization) is a valuable alternative. In this study, we explore the influence of different types of boundary conditions on the external walls of the representative modeling domain along the applied pressure gradient on the magnitude and orientation of the computed permeability tensor. To implement periodic flow boundary conditions, we utilized stochastic reconstruction methodology to create statistically similar (to real porous media structures) geometrically periodic 3D structures. Stochastic reconstructions are similar to encapsulation of the porous media into statistically similar geometrically periodic one with the same permeability tensor. Seven main boundary conditions (BC) were implemented: closed walls, periodic flow, slip on the walls, linear pressure, translation, symmetry, and immersion. The different combinations of BCs amounted to a total number of 15 BC variations. All these BCs significantly influenced the resulting tensorial permeabilities, including both magnitude and orientation. Periodic boundary conditions produced the most physical flow patterns, while other classical BCs either suppressed crucial transversal flows or resulted in unphysical currents. Our results are crucial to performing flow properties upscaling and will be relevant to computing not only single-phase but also multiphase flow properties. Moreover, other calculation of physical properties such as some mechanical, transport, or heat conduction properties may benefit from the technique described in this study.

Two-phase flow of CO2-brine in a heterogeneous sandstone: Characterization of the rock and comparison of the lattice-Boltzmann, pore-network, and direct numerical simulation methods

Understanding the physics of two-phase flow of CO2 and brine in porous geological formations is essential to sequestration of carbon dioxide in deep saline reservoirs, as well as the older problem of enhanced oil recovery from hydrocarbon reservoirs by CO2 injection. A pilot CO2 injection in Decatur, Illinois, was undertaken, with the injection zone being the highly saline and heterogeneous Mt. Simon sandstone, in order to better understand the feasibility of full-scale sequestration process. This paper reports the results of an extensive study of the morphology of the sandstone and its heterogeneity, and simulation of single-phase and two-phase flow of CO2 and brine in the formation's three-dimensional images. As we demonstrate by extensive analysis, the formation is much more heterogeneous than the typical sandstone, such as Berea sandstone. In addition to characterizing the morphology of the sandstone and computing its important flow characteristics, an important goal of the study is to compare the accuracy and computational efficiency of three distinct simulation approaches, namely, the lattice-Boltzmann (LB) approach, direct numerical simulation (DNS) of the governing equations of fluid flow that uses the finite-volume method coupled with the OpenFOAM simulator, and pore-network (PN) simulation. After validating the simulators by comparing the computed relative permeabilities that they produce for Berea sandstone, we simulate displacement of brine by CO2 at low and relatively high capillary numbers, and compute the relative permeabilities and other quantities of interest. We demonstrate that all the three methods provide consistent relative permeability-saturation functions that are in close agreement with one another. However, although the LB and DNS both produce similar relative permeabilities, the DNS approach is computationally more efficient because it simulates drainage by only a single set of computations over the entire saturation range, whereas the LB simulation requires separate simulation for each set saturation. Thus, the question of what method to use for simulating such flow processes at the scale of core plugs should mainly be addressed based on the computational time that one can afford and the computational resources that one has access to. Another important question addressed is the effect of the resolution of the computational grids or lattices used, particularly when one uses the LB method with voxelized images of porous media. We show that, unlike many claims in the past, one may need many lattice units per voxel in order to obtain reliable, lattice-independent results.

Investigating rock micro-structure of sandstones by pattern recognition on their X-ray images

Analysing rock micro-structure from micro-computed tomographic images of porous media is vital to understand fluid-flow and estimating petrophysical properties like permeability. The two main appro...

Pore-scale simulations of rarefied gas flows in ultra-tight porous media

An in-depth understanding of gas transport in ultra-tight porous media is the key to quantifying flow properties of shale rocks with pore space as small as a few nanometers, where the gas rarefaction effects play a major role. As the conventional fluid mechanics theory fails to describe non-equilibrium rarefied flow, we resort to the gas kinetic theory and directly simulate gas flow inside the porous media utilising the digital images of porous media where the pore space is resolved. The Boltzmann model equation is solved by the discrete velocity method (DVM), which can accurately predict the permeability enhancement caused by rarefaction effects. Our simulations for different porous media show that the commonly-used standard lattice Boltzmann method (LBM) cannot describe rarefaction effects, although the kinetic boundary condition, which helps to capture velocity-slip, can extend the validity of the LBM to the slip flow regime. The heuristic Klinkenberg-type models proposed for all the flow regimes often involve many unknown empirical parameters, which may be calibrated by our simulations. However, these parameters are different for each porous medium and also depend on flow conditions, so these models are not of any practical use. By contrast, our kinetic solver can accurately predict apparent permeability without introducing any empirical parameters, which lays firm foundation for upscaling. As the large flow paths with least flow resistance dominate the overall permeability, the requirement on the velocity-space resolution is significantly reduced for our DVM simulations to predict accurate permeability with affordable computational costs, which offers a promising new way for digital rock analysis.

Porous Media Characterization Using Minkowski Functionals: Theories, Applications and Future Directions

An elementary question in porous media research is in regard to the relationship between structure and function. In most fields, the porosity and permeability of porous media are properties of key interest. There is, however, no universal relationship between porosity and permeability since not only does the fraction of void space matter for permeability but also the connectivity of the void fraction. With the evolution of modern day X-ray microcomputed tomography (micro-CT) and advanced computing, it is now possible to visualize porous media at an unprecedented level of detail. Approaches in analyzing micro-CT data of porous structures vary in the literature from phenomenological characterization to network analysis to geometrical and/or topological measurements. This leads to a question about how to consistently characterize porous media in a way that facilitates theoretical developments. In this effort, the Minkowski functionals (MF) emerge from the field of statistical physics where it is evident that many physical processes depend on the geometry and topology of bodies or multiple bodies in 3D space. Herein we review the theoretical basis of the MF, mathematical theorems and methods necessary for porous media characterization, common measurement errors when using micro-CT data and recent findings relating the MF to macroscale porous media properties. This paper is written to provide the basics necessary for porous media characterization and theoretical developments. With the wealth of information generated from 3D imaging of porous media, it is necessary to develop an understanding of the limitations and opportunities in this exciting area of research.

Permeability from 3D Porous Media Images: a Fast Two-Step Approach

An efficient methodology to calculate absolute permeability of porous media using a two-step algorithm is developed. In the first step, the creeping flow equations over the pore space are translated into a Darcy flow problem with the pore space being represented by appropriately chosen local flow conductivities. In the next step, a combined renormalization group and multi-level iterative Laplace solver approach is used to upscale the local conductivities to obtain the effective permeability for the full domain. The accuracy and computational efficiency of the proposed two-step local conductivity–Laplace scheme (LC-LAP) are tested against a FFT (fast Fourier transform) accelerated solver which uses a semi-implicit method for the pressure-linked equation (SIMPLE-FFT) and against a solver that features the GPGPU implementation of the multiple-relaxation-time lattice Boltzmann method (MRT-LBM). A detailed comparison is made by computing permeabilities from all three methods over model geometries and digitized images obtained from micron-scale-resolution computerized tomography (micro-CT) of sandstone rocks of varying porosities and heterogeneity levels. We observe an agreement between our method and either benchmark methods (SIMPLE-FFT and MRT-LBM) that is similar to the agreement between both benchmarks. On the samples tested, the computational performance advantage of the LC-LAP approach ranges from 10- to 40-fold compered to SIMPLE-FFT and 8- to 25-fold compared to MRT-LBM. The proposed method is suitable for fast computations and for computations over very large volumes (due to much lower memory and compute resource requirements) for determining single-phase permeabilities of medium- to high-permeability rocks.

Direct geometrical simulation of pore space evolution through hydrate dissociation in methane hydrate reservoirs

The identification of methane hydrate behavior in porous media is one of the most challenging yet rewarding pore-level visualization and simulation tasks. The hydrate morphology influences the physical characteristics of the host sediments as during the hydrate formation and dissociation processes the pore space and flow pathways constantly evolve. Here, a direct three-phase pore morphological simulation approach is proposed, verified and utilized to simulate hydrate deformities and predict fluid occupancies and absolute and effective permeabilities of hydrate-bearing geological formations. The proposed technique simulates capillary-dominant displacements by applying a set of geomaterial rules directly to the pixels of pore-level porous media images. The case studies are sandy microstructures generated based on the particle size distributions of the Mallik gas hydrate deposit. The fluid occupancy profiles, absolute permeability, and hydraulic tortuosity curves are comparable with the experimental datasets. The sensitivity analysis shows that although the gas relative permeability is less sensitive to the hydrate specifications, the porosity, grain size distribution, and hydrate content and occupancy remarkably influence the rock absolute permeability.

Reconstruction of three-dimensional porous media using generative adversarial neural networks.

To evaluate the variability of multiphase flow properties of porous media at the pore scale, it is necessary to acquire a number of representative samples of the void-solid structure. While modern x-ray computer tomography has made it possible to extract three-dimensional images of the pore space, assessment of the variability in the inherent material properties is often experimentally not feasible. We present a method to reconstruct the solid-void structure of porous media by applying a generative neural network that allows an implicit description of the probability distribution represented by three-dimensional image data sets. We show, by using an adversarial learning approach for neural networks, that this method of unsupervised learning is able to generate representative samples of porous media that honor their statistics. We successfully compare measures of pore morphology, such as the Euler characteristic, two-point statistics, and directional single-phase permeability of synthetic realizations with the calculated properties of a bead pack, Berea sandstone, and Ketton limestone. Results show that generative adversarial networks can be used to reconstruct high-resolution three-dimensional images of porous media at different scales that are representative of the morphology of the images used to train the neural network. The fully convolutional nature of the trained neural network allows the generation of large samples while maintaining computational efficiency. Compared to classical stochastic methods of image reconstruction, the implicit representation of the learned data distribution can be stored and reused to generate multiple realizations of the pore structure very rapidly.

References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks

We generate a novel reference dataset to quantify the impact of numerical solvers, boundary conditions, and simulation platforms. We consider a variety of microstructures ranging from idealized pipes to digital rocks. Pore throats of the digital rocks considered are large enough to be well resolved with state-of-the-art micro-computerized tomography technology. Permeability is computed using multiple numerical engines, 12 in total, including, Lattice-Boltzmann, computational fluid dynamics, voxel based, fast semi-analytical, and known empirical models. Thus, we provide a measure of uncertainty associated with flow computations of digital media. Moreover, the reference and standards dataset generated is the first of its kind and can be used to test and improve new fluid flow algorithms. We find that there is an overall good agreement between solvers for idealized cross-section shape pipes. As expected, the disagreement increases with increase in complexity of the pore space. Numerical solutions for pipes with sinusoidal variation of cross section show larger variability compared to pipes of constant cross-section shapes. We notice relatively larger variability in computed permeability of digital rocks with coefficient of variation (of up to 25%) in computed values between various solvers. Still, these differences are small given other subsurface uncertainties. The observed differences between solvers can be attributed to several causes including, differences in boundary conditions, numerical convergence criteria, and parameterization of fundamental physics equations. Solvers that perform additional meshing of irregular pore shapes require an additional step in practical workflows which involves skill and can introduce further uncertainty. Computation times for digital rocks vary from minutes to several days depending on the algorithm and available computational resources. We find that more stringent convergence criteria can improve solver accuracy but at the expense of longer computation time.

Generalized network modeling: Network extraction as a coarse-scale discretization of the void space of porous media.

A generalized network extraction workflow is developed for parameterizing three-dimensional (3D) images of porous media. The aim of this workflow is to reduce the uncertainties in conventional network modeling predictions introduced due to the oversimplification of complex pore geometries encountered in natural porous media. The generalized network serves as a coarse discretization of the surface generated from a medial-axis transformation of the 3D image. This discretization divides the void space into individual pores and then subdivides each pore into sub-elements called half-throat connections. Each half-throat connection is further segmented into corners by analyzing the medial axis curves of its axial plane. The parameters approximating each corner-corner angle, volume, and conductivity-are extracted at different discretization levels, corresponding to different wetting layer thickness and local capillary pressures during multiphase flow simulations. Conductivities are calculated using direct single-phase flow simulation so that the network can reproduce the single-phase flow permeability of the underlying image exactly. We first validate the algorithm by using it to discretize synthetic angular pore geometries and show that the network model reproduces the corner angles accurately. We then extract network models from micro-CT images of porous rocks and show that the network extraction preserves macroscopic properties, the permeability and formation factor, and the statistics of the micro-CT images.

A new method to reconstruct structured mesh model from micro-computed tomography images of porous media and its application

Pore-scale modeling of porous media has been emphasized as a platform to study the mechanical properties, fluid storage and transport properties in it. A universal and easy-to-realize approach of structured mesh model is presented in this paper to reconstruct the three-dimensional (3D) pore scale models from the micro computed tomography (micro-CT) images. By substituting the structured hexahedron elements for the voxels in the 3D images, codes are developed to reconstruct the micro structure of porous media. The structured model is tested by the micro-CT images of four rock samples. The advantages of topology and mesh quality of this model is validated by comparisons between the PNM and unstructured mesh models. Then numerical simulations of single- and two- phase flow are conducted in Fluent software. The models are validated between the experimental data and simulation results of PNM on the absolute permeability and relative permeability curves under different conditions of sample wettability.

What is the role of tortuosity in the Kozeny-Carman equation?

Hydraulic tortuosity is an important parameter in characterizing fluid-flow heterogeneity in porous media. The most basic definition of tortuosity is the ratio of the average flow path length to the sample length. Although this definition seems straightforward, the lack of understanding and the lack of proper ways to measure tortuosity make it one of the most abused parameters in rock physics. Hydraulic tortuosity is often treated merely as a fitting factor, or worse, it is neglected by being combined with a geometric factor in the Kozeny-Carman (KC) equation. Often, the tortuosity is obtained from laboratory measurements of porosity, permeability, and specific surface area by inverting the KC equation. This approach has a major pitfall because it treats tortuosity as a fitting factor, and the inverted tortuosity is often unphysically high. In contrast, we obtained the tortuosity from 3D segmented binary images of porous media using streamlines extracted from a local flux, the output from the lattice Boltzmann method (LBM) flow simulation. After obtaining streamlines from each sample, we calculated the distribution of tortuosities and flux-weighted average tortuosity. With the tortuosity measurement from streamlines, every parameter in the KC equation can be measured accurately from 3D segmented binary images. We found, however, that the KC equation is still missing some important geometric information needed to predict permeability. With known parameters and without a fitting factor, the KC equation predicts permeability higher by one to two orders of magnitude than that predicted by the LBM. We searched for a missing parameter by exploring various concepts such as connected pore space and pore throat distribution. We found that the connected pore space does not contribute to the difference between the KC permeability and LBM permeability, whereas, as we learn with sinusoidal pipe examples, the pore throat distribution captures what is missing from the KC equation.

Three-dimensional imaging of porous media using confocal laser scanning microscopy.

In the last decade, imaging techniques capable of reconstructing three-dimensional (3-D) pore-scale model have played a pivotal role in the study of fluid flow through complex porous media. In this study, we present advances in the application of confocal laser scanning microscopy (CLSM) to image, reconstruct and characterize complex porous geological materials with hydrocarbon reservoir and CO2 storage potential. CLSM has a unique capability of producing 3-D thin optical sections of a material, with a wide field of view and submicron resolution in the lateral and axial planes. However, CLSM is limited in the depth (z-dimension) that can be imaged in porous materials. In this study, we introduce a 'grind and slice' technique to overcome this limitation. We discuss the practical and technical aspects of the confocal imaging technique with application to complex rock samples including Mt. Gambier and Ketton carbonates. We then describe the complete workflow of image processing to filtering and segmenting the raw 3-D confocal volumetric data into pores and grains. Finally, we use the resulting 3-D pore-scale binarized confocal data obtained to quantitatively determine petrophysical pore-scale properties such as total porosity, macro- and microporosity and single-phase permeability using lattice Boltzmann (LB) simulations, validated by experiments.

Universal Stochastic Multiscale Image Fusion: An Example Application for Shale Rock.

Spatial data captured with sensors of different resolution would provide a maximum degree of information if the data were to be merged into a single image representing all scales. We develop a general solution for merging multiscale categorical spatial data into a single dataset using stochastic reconstructions with rescaled correlation functions. The versatility of the method is demonstrated by merging three images of shale rock representing macro, micro and nanoscale spatial information on mineral, organic matter and porosity distribution. Merging multiscale images of shale rock is pivotal to quantify more reliably petrophysical properties needed for production optimization and environmental impacts minimization. Images obtained by X-ray microtomography and scanning electron microscopy were fused into a single image with predefined resolution. The methodology is sufficiently generic for implementation of other stochastic reconstruction techniques, any number of scales, any number of material phases, and any number of images for a given scale. The methodology can be further used to assess effective properties of fused porous media images or to compress voluminous spatial datasets for efficient data storage. Practical applications are not limited to petroleum engineering or more broadly geosciences, but will also find their way in material sciences, climatology, and remote sensing.

Prediction of empirical properties using direct pore-scale simulation of straining through 3D microtomography images of porous media

Summary Understanding the mechanisms of filtration through porous media is relevant in many engineering applications ranging from waste water treatment and aquifer contamination in environmental engineering to estimating the permeability reduction in near wellbore region during drilling or water re-injection in petroleum engineering. In this paper we present a pore-scale approach that models straining through the pore structures extracted from X-ray tomographic images of rock and grain pack samples from the first principles, enabling the examination of current macroscopic models. While continuum models are widely used for fast prediction of the retention profiles and permeability of the host porous medium, they require a number of phenomenological parameters which are derived from matching experimental results. One of these parameters is the rate of entrapment, which is the sink term in the advection–diffusion equation. Here we find the constitutive relationship for the rate of entrapment as a product of the filtration coefficient, velocity, and concentration and validate it by comparing with core flood experiments. Results show that the pore-scale simulation gives close approximations of filtration coefficient when pore bridging and straining are the main particle capture mechanisms.

Predictions of dynamic changes in reaction rates as a consequence of incomplete mixing using pore scale reactive transport modeling on images of porous media

We present a pore scale model capable of simulating fluid/fluid reactive transport on images of porous media from first principles. We use a streamline-based particle tracking method for simulating flow and transport, while for reaction to occur, both reactants must be within a diffusive distance of each other during a time-step. We assign a probability of reaction (Pr), as a function of the reaction rate constant (kr) and the diffusion length. Firstly, we validate our model for reaction against analytical solutions for the bimolecular reaction (A+B→C) in a free fluid. Then, we simulate transport and reaction in a beadpack to validate the model through predicting the fluid/fluid reaction experimental results provided by Gramling et al. (2002). Our model accurately predicts the experimental data, as it takes into account the degree of incomplete mixing present at the sub-pore (image voxel) level, in contrast to advection–dispersion–reaction equation (ADRE) model that over-predicts pore scale mixing. Finally, we show how our model can predict dynamic changes in the reaction rate accurately accounting for the local geometry, topology and flow field at the pore scale. We demonstrate the substantial difference between the predicted early-time reaction rate in comparison to the ADRE model.

Quantitative comparison of segmentation algorithms for FIB-SEM images of porous media.

Focused ion beam tomography has proven to be capable of imaging porous structures on a nano-scale. However, due to shine-through artefacts, common segmentation algorithms often lead to severe dislocation of individual structures in z-direction. Recently, a number of approaches have been developed, which take into account the specific nature of focused ion beam-scanning electron microscope images for porous media. In the present study, we analyse three of these approaches by comparing their performance based on simulated focused ion beam-scanning electron microscope images. Performance is measured by determining the amount of misclassified voxels as well as the fidelity of structural characteristics. Based on this analysis we conclude that each algorithm has certain strengths and weaknesses and we determine the scenarios for which each approach might be the best choice.

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.

Stable-phase method for hierarchical annealing in the reconstruction of porous media images

In this paper, we introduce a stable-phase approach for hierarchical annealing which addresses the very large computational costs associated with simulated annealing for the reconstruction of large-scale binary porous media images. Our presented method, which uses the two-point correlation function as the morphological descriptor, involves the reconstruction of three-phase and two-phase structures. We consider reconstructing the three-phase structures based on standard annealing and the two-phase structures based on standard and hierarchical annealings. From the result of the two-dimensional (2D) reconstruction, we find that the 2D generation does not fully capture the morphological information of the original image, even though the two-point correlation function of the reconstruction is in excellent agreement with that of the reference image. For the reconstructed three-dimensional (3D) microstructure, we calculate its permeability and compare it to that of the reference 3D microstructure. The result indicates that the reconstructed structure has a lower degree of connectedness than that of the actual sandstone. We also compare the computation time of our presented method to that of the standard annealing, which shows that our presented method of orders of magnitude improves the convergence rate. That is because only a small part of the pixels in the overall hierarchy need to be considered for sampling by the annealer.