Time-Dependent Diffusion and Surface-Enhanced Relaxation in Stochastic Replicas of Porous Rock

Understanding the connection between pore structure and NMR behavior of fluid-saturated porous rock is essential in interpreting the results of NMR measurements in the field or laboratory and in establishing correlations between NMR parameters and petrophysical properties. In this paper we use random-walk simulation to study NMR relaxation and time-dependent diffusion in 3D stochastic replicas of real porous media. The microstructures are generated using low-order statistical information (porosity, void–void autocorrelation function) obtained from 2D images of thepore space. Pore size distributions obtained directly by a 3D pore space partitioning method and indirectly by inversion of NMR relaxation data are compared for the first time. For surface relaxation conditions typical of reservoir rock, diffusional coupling between pores of different size is observed to cause considerable deviations between the two distributions. Nevertheless, the pore space correlation length and the size of surface asperity are mirrored in the NMR relaxation data for the media studied. This observation is used to explain the performance of NMR-based permeability correlations. Additionally, the early time behavior of the time-dependent diffusion coefficient is shown to reflect the average pore surface-to-volume ratio. For sufficiently high values of the self-diffusion coefficient, the tortuosity of the pore space is also recovered from the long-time behavior of the time-dependent diffusion coefficient, even in the presence of surface relaxation. Finally, the simulations expose key limitations of the stochastic reconstruction method, and allow suggestions for future development to be made.     

Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media

A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archie's law. This behavior results from the relatively high pore space percolation threshold (0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.     

An Algorithm for 3D Pore Space Reconstruction from a 2D Image Using Sequential Simulation and Gradual Deformation with the Probability Perturbation Sampler

The construction of a faithful 3D pore space model of a porous medium that could reproduce the macroscopic behavior of that medium is of great interest in various fields including medicine, material science, hydrology and petroleum engineering. A computationally efficient algorithm is developed that uses the probability perturbation method and sequential multiple-point statistics simulations to generate 3D stochastic and equiprobable representations of random porous media when only a 2D thin section image is available. By employing the probability perturbation method as a gradual deformation technique, the pore patterns of a single 2D image are deformed to generate a series of 2D stochastically simulated images. The 3D pore structure is then generated by simply stacking the 2D-simulated images. The quality of the 3D reconstruction is critically dependent on the rate of deformation and a simple general procedure for choosing this parameter is presented. Various criteria such as porosity, two-point auto-correlation function, multiple-point connectivity function, local percolation probability, absolute permeability obtained by lattice-Boltzmann method (LBM), formation factor and two-phase relative permeability calculations are used to validate the results. The method is tested on two random porous solids; Berea Sandstone and synthetic Silica, for which directly measured 3D micro-CT images are available. The stochastically reconstructed 3D pore space preserves the low- and high-order spatial statistics, the macroscopic flow properties and the microstructure of the 3D micro-CT images.     

Digitally Reconstructed Porous Media: Transport and Sorption Properties

The basic aim of this work is to present a combination of techniques for the reconstruction of the porous structure and the study of transport properties in porous media. The disordered structure of porous systems like random sphere packing, Vycor glass and North Sea chalk, is represented by three-dimensional binary images. The random sphere pack is generated by a standard ballistic deposition procedure, while the chalk and the Vycor matrices by a stochastic reconstruction technique. The transport properties (Knudsen diffusivity, molecular diffusivity and permeability) of the resulting 3-dimensional binary domains are investigated through computer simulations. Furthermore, physically sound spatial distributions of two phases filling the pore space are determined by the use of a simulated annealing algorithm. The wetting and the non-wetting phases are initially randomly distributed in the pore space and trial-and-error swaps are performed in order to attain the global minimum of the total interfacial energy. The effective diffusivities of the resulting domains are then computed and a parametric study with respect to the pore volume fraction occupied by each phase is performed. Reasonable agreement with available data is obtained in the single- and multi-phase transport cases.     

Effect of 2D Image Resolution on 3D Stochastic Reconstruction and Developing Petrophysical Trend

Multi-resolution digital rock physics (DRP) makes it possible to up-scale petrophysical properties from micron size to core sample size using two-dimensional (2D) thin section images. Resolution of 3D images and sample size are challenging problems in DRP where high-resolution images are acquired from small samples using inefficient and expensive micro-CT facilities. Three-dimensional stochastic reconstruction is an alternative approach to overcome these challenges. In this paper, we use multi-resolution images and investigate effect of 2D image resolution on 3D stochastic reconstruction and development of petrophysical trends for our two sandstone and carbonate original representative volume elements (RVEs). The proposed method includes three steps. In the first step, the spatial resolution of our original RVEs is decreased synthetically. In the second step, stochastic RVEs are realized for each resolution using two perpendicular images, correlation functions, and phase recovery algorithm. In the reconstruction method, a full set of two-point correlation functions (TPCFs) is extracted from two perpendicular 2D images. Then TPCF vectors are decomposed and averaged to realize 3D stochastic RVEs. In the third step, petrophysical properties like relative and absolute permeability as well as porosity and formation factor are computed. The output is used to develop trends for petrophysical properties in different resolutions. Experimental results illustrate that the proposed method can be used to predict petrophysical properties and reconstruct 3D RVEs for resolutions unavailable in the acquired 2D or 3D data.     

A Hierarchical Sampling for Capturing Permeability Trend in Rock Physics

Among all properties of reservoir rocks, permeability is an important parameter which is typically measured from core samples in the laboratory. Due to limitations of core drilling all over a reservoir, simulation of rock porous media is demanded to explore more scenarios not seen in the available data. One of the most accurate methods is cross correlation based simulation (CCSIM) which recently has broadly applied in geoscience and porous media. The purpose of this study is producing realizations with the same permeability trend to a real sample. Berea sandstone sample is selected for this aim. Permeability results, extracted from smaller sub-samples of the original sample, showed that classic Kozeny–Carman permeability trend is not suitable for this sample. One reason can be due to lack of including geometrical and fractal properties of pore-space distribution in this equation. Thus, a general trend based on fractal dimensions of pore-space and tortuosity of the Berea sample is applied in this paper. Results show that direct 3D stochastic modeling of porous media preserves porous structure and fractal behavior of rock. On the other hand, using only 2D images for constructing the 3D pore structures does not reproduce the measured experimental permeability. For this aim, a hierarchical sampling is implemented in two and three steps using both 2D and 3D stochastic modeling. Results showed that two-step sampling is not suitable enough, while the utilized three-step sampling occurs to be show excellent performance by which different models of porous media with the same permeability trend as the Berea sandstone sample can be generated.     

Transport Properties of Stochastically Reconstructed Porous Media with Improved Pore Connectivity

Results from stochastic reconstruction of porous solids and from a direct comparison of calculated and experimental effective transport properties are presented. Eight porous solids of different microstructures were selected to evaluate the performance of two reconstruction methods based on simulated annealing. The common method was constrained by the two-point probability function and the lineal-path function for the void phase, whilst the constraints of our new method were further supplemented by the lineal-path function for the solid phase and by two adjustable parameters. The new method was capable of reproducing the void and solid phases as large clusters spanning the entire replicas. Non-percolating clusters formed minor volume fractions of both phases. Although the common method reproduced the microstructures quite well, their pore space connectivity was significantly poorer. Therefore, effective permeability, effective ordinary diffusivity, and effective Knudsen diffusivity calculated for the replicas obtained using the new method were always much greater than the same quantities related to the common reconstruction method. For most of the porous solids, values of the effective properties calculated on the basis of the new reconstruction method better matched their experimental counterparts than the corresponding values derived from the microstructures reproduced using the common reconstruction method.     

Stochastic Pore Network Generation from 3D Rock Images

Pore networks can be extracted from 3D rock images to accurately predict multi-phase flow properties of rocks by network flow simulation. However, the predicted flow properties may be sensitive to the extracted pore network if it is small, even though its underlying characteristics are representative. Therefore, it is a challenge to investigate the effects on flow properties of microscopic rock features individually and collectively based on small samples. In this article, a new approach is introduced to generate from an initial network a stochastic network of arbitrary size that has the same flow properties as the parent network. Firstly, we characterise the realistic parent network in terms of distributions of the geometrical pore properties and correlations between these properties, as well as the connectivity function describing the detailed network topology. Secondly, to create a stochastic network of arbitrary size, we generate the required number of nodes and bonds with the correlated properties of the original network. The nodes are randomly located in the given network domain and connected by bonds according to the strongest correlation between node and bond properties, while honouring the connectivity function. Thirdly, using a state-of-the-art two-phase flow network model, we demonstrate for two samples that the rock flow properties (capillary pressure, absolute and relative permeability) are preserved in the stochastic networks, in particular, if the latter are larger than the original, or the method reveals that the size of the original sample is not representative. We also show the information that is necessary to reproduce the realistic networks correctly, in particular the connectivity function. This approach forms the basis for the stochastic generation of networks from multiple rock images at different resolutions by combining the relevant statistics from the corresponding networks, which will be presented in a future publication.     

A new stochastic method of reconstructing porous media

We present a new stochastic method of reconstructing porous medium from limited morphological information obtained from two-dimensional micro- images of real porous medium. The method is similar to simulated annealing method in the capability of reconstructing both isotropic and anisotropic structures of multi-phase but differs from the latter in that voxels for exchange are not selected completely randomly as their neighborhood will also be checked and this new method is much simpler to implement and program. We applied it to reconstruct real sandstone utilizing morphological information contained in porosity, two-point probability function and linear-path function. Good agreement of those references verifies our developed method’s powerful capability. The existing isolated regions of both pore phase and matrix phase do quite minor harm to their good connectivity. The lattice Boltzmann method (LBM) is used to compute the permeability of the reconstructed system and the results show its good isotropy and conductivity. However, due to the disadvantage of this method that the connectivity of the reconstructed system’s pore space will decrease when porosity becomes small, we suggest the porosity of the system to be reconstructed be no less than 0.2 to ensure its connectivity and conductivity.     

Accurate Reconstruction of Porous Materials via Stochastic Fusion of Limited Bimodal Microstructural Data

Porous materials such as sandstones have important applications in petroleum engineering and geosciences. An accurate knowledge of the porous microstructure of such materials is crucial for the understanding of their physical properties and performance. Here, we present a procedure for accurate reconstruction of porous materials by stochastically fusing limited bimodal microstructural data including limited-angle X-ray tomographic radiographs and 2D optical micrographs. The key microstructural information contained in the micrographs is statistically extracted and represented using certain lower-order spatial correlation functions associated with the pore phase, and a probabilistic interpretation of the attenuated intensity in the tomographic radiographs is developed. A stochastic procedure based on simulated annealing that generalizes the widely used Yeong–Torquato framework is devised to efficiently incorporate and fuse the complementary bimodal imaging data for accurate microstructure reconstruction. The information content of the complementary microstructural data is systematically investigated using a 2D model system. Our procedure is subsequently applied to accurately reconstruct a variety of 3D sandstone microstructures with a wide range of porosities from limited X-ray tomographic radiographs and 2D optical micrographs. The accuracy of the reconstructions is quantitatively ascertained by directly comparing the original and reconstructed microstructures and their corresponding clustering statistics.     

平稳随机过程模拟的一个快速Hartley变换方法

由平稳随机过程谱表示定理导出了平稳随机过程蒙特卡罗模拟的一个快速Hartley变换方法。按照该方法，样本过程可直接由一正弦和余弦函数和的级数公式计算产生。可以证明，当级数项数N足够大时，模拟的样本过程可准确地反映平稳随机过程规定的性质；当样本过程足够多时，其总体均值和总体自相关函数均趋于相应目标函数。方法有如下一些特点：模拟的样本过程由相互正交的两个过程迭加而成，样本过程具备各态历经性质，样本过程随着级数项数N趋于无穷而渐近呈正态分布，可直接利用快速Hartley变换来大大提高模拟的计算效率。     

Pore space network characterization with sub-voxel definition

Morphological measurements in 3D for pore space characterization (connectivity pore-body/throat classification, shape factors, virtual fluid intrusion) are based on computed intensive digital-thinning operations for skeletonization and medial axis extraction from 3D digital images. We present an alternative method that is measurably faster and allows sub-voxel definition of the pore space network. The method allows extracting—based on morphological considerations only—the centered and shortest stream-lines—i.e., the paths—to follow in order to go through the pore space from one given point to another and to exit. In addition the method penalizes long and narrow pore-throats in favor of short stubby/ones—i.e., it has a built-in exemplification capacity. It exploits well-established mathematical methods successfully applied in medical endoscopy.     

Microtomography and Pore-Scale Modeling of Two-Phase Fluid Distribution

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces three-dimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the pore-scale mechanisms of CO₂ injection into an aquifer, as well as more general multi-phase flows.     

Connectivity in Vuggy Carbonates, New Experimental Methods and Applications

The length and spatial distribution of the touching-vugs channels affect the degree of permeability variations and is the main contributor to heterogeneity in vuggy carbonates. Hence, this article focuses on vug connectivity characterization and its impact on fluid flow. A whole core sample was scanned by X-ray computed tomography (CT). Image segmentation was used to obtain a binarized three-dimensional (3D) model of the vuggy pore space. Analysis of the binarized 3D model is used to calculate the correlation function and correlation length for the vuggy pore space. Connectivity analysis of the binarized 3D model shows that 79% of the vugs connected network spanning along the sample. The remaining 21% vug porosity exists in a large number of isolated vugs. The correlation length for the connected vug network is found to be larger than for vugs in general. NMR T ₂ measurements at increasing capillary pressure is tested on the vuggy material and used to investigate the amount of connected- and isolated vugs. The results verify the large fraction of connected vugs. Application of NMR T ₂ measurements in combination with capillary pressure experiments can also reveal matrix properties that play an important role in recovery processes. The transition between non-Fickian and Fickian regimes for tracer/solute transport is studied by laboratory experiments performed at various sample lengths, from cm to m scale. For the largest sample measured in our experiments show that effluent concentration curve conform to the CDE solution, suggesting that the Fickian regime has been established.