Effect of Mean Network Coordination Number on Dispersivity Characteristics

In this study, we investigate the role of topology on the macroscopic (centimeter scale) dispersion characteristics derived from pore-network models. We consider 3D random porous networks extracted from a regular cubic lattice with coordination number distributed in accordance with real porous structures. We use physically consistent rules including ideal mixing in pore bodies, molecular diffusion, and Taylor dispersion in pore throats to simulate transport at the pore-scale level. Fundamental properties of porous networks are based on statistical distributions of basic parameters. Theoretical calculations demonstrate strong correspondence with data obtained from numerical experiments. For low coordination numbers, we observe normal transport in porous networks. Anomalous effects expressed by tailing in concentration evolution are seen for higher coordination numbers. We find that the mean network coordination number has significant influence on averaged characteristics of porous networks such as geometric and hydraulic dispersivity, while other topological properties are of less significance. We give an explicit formula that describes the decrease of geometric dispersivity with growing mean coordination number. The results demonstrate the importance of network topology for models for flow and transport in porous media.     

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 Method for Generating Pore-Network Models of Porous Media

In this study, we have developed a new method to generate a multi-directional pore network for representing a porous medium. The method is based on a regular cubic lattice network, which has two elements: pore bodies located at the regular lattice points and pore throats connecting the pore bodies. One of the main features of our network is that pore throats can be oriented in 13 different directions, allowing a maximum coordination number of 26 that is possible in a regular lattice in 3D space. The coordination number of pore bodies ranges from 0 to 26, with a pre-specified average value for the whole network. We have applied this method to reconstruct real sandstone and granular sand samples through utilizing information on their coordination number distributions. Good agreement was found between simulation results and observation data on coordination number distribution and other network properties, such as number of pore bodies and pore throats and average coordination number. Our method can be especially useful in studying the effect of structure and coordination number distribution of pore networks on transport and multiphase flow in porous media systems.     

A characterization of the coupled evolution of grain fabric and pore space using complex networks: Pore connectivity and optimized flows in the presence of shear bands

A framework for the multiscale characterization of the coupled evolution of the solid grain fabric and its associated pore space in dense granular media is developed. In this framework, a pseudo-dual graph transformation of the grain contact network produces a graph of pores which can be readily interpreted as a pore space network. Survivability, a new metric succinctly summarizing the connectivity of the solid grain and pore space networks, measures material robustness. The size distribution and the connectivity of pores can be characterized quantitatively through various network properties. Assortativity characterizes the pore space with respect to the parity of the number of particles enclosing the pore. Multiscale clusters of odd parity versus even parity contact cycles alternate spatially along the shear band: these represent, respectively, local jamming and unjamming regions that continually switch positions in time throughout the failure regime. Optimal paths, established using network shortest paths in favor of large pores, provide clues on preferential paths for interstitial matter transport. In systems with higher rolling resistance at contacts, less tortuous shortest paths thread through larger pores in shear bands. Notably the structural patterns uncovered in the pore space suggest that more robust models of interstitial pore flow through deforming granular systems require a proper consideration of the evolution of in situ shear band and fracture patterns – not just globally, but also inside these localized failure zones.     

Multiscale Structures to Describe Porous Media Part II: Transport Properties and Application to Test Materials

A renormalization method for the computation of the transport properties of a porous medium modelled as a multiscale random network is proposed. The method applies to electrical conduction, molecular diffusion, hydraulic transport under low Reynolds number, transport of condensable vapour, in the medium fully or partially saturated by one or two immiscible fluids. For 31 test materials, the method previously exposed by the authors for the reconstitution of the pore structure from the mercury intrusion curve is applied. Then, the intrinsic permeability is computed. The results are in good agreement with the measured permeability.     

A Dynamic Pore Network Model for Oil Displacement by Wettability-Altering Surfactant Solution

A dynamic pore network model, capable of predicting the displacement of oil from a porous medium by a wettability-altering and interfacial tension reducing surfactant solution, is presented. The key ingredients of the model are (1) a dynamic network model for the displacement of oil by aqueous phase taking account of capillary and viscous effects, (2) a simulation of the transport of surfactant through the network by advection and diffusion taking account of adsorption on the solid surface, and (3) the coupling of these two by linking the contact angle and interfacial tension appearing in the dynamic network simulation to the local concentration of surfactant computed in the transport simulation. The coupling is two-way: The flow field used to advect the surfactant concentration is that associated with the displacement of oil by the injected aqueous phase, and the surfactant concentration influences the flow field through its effect on the capillarity parameters. We present results obtained using the model to validate that it reproduces the displacement patterns observed by other authors in two-dimensional networks as capillary number and mobility ratio are varied, and to illustrate the effects of surfactant on displacement patterns. A mechanism is demonstrated whereby in an initially mixed-wet medium, surfactant-induced wettability alteration can lead to stabilization of displacement fronts.     

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.     

Reliability of Algorithms Interpreting Topological and Geometric Properties of Porous Media for Pore Network Modelling

Pore network models (PNMs) offer a computationally efficient way to analyse transport in porous media. Their effectiveness depends on how well they represent the topology and geometry of real pore systems, for example as imaged by X-ray CT. The performance of two popular algorithms, maximum ball and watershed, is evaluated for three porous systems: an idealised medium with known pore throat properties and two rocks with different morphogenesis—carbonate and sandstone. It is demonstrated that while the extracted PNM simulates simple flow (permeability) with acceptable accuracy, their topological and geometric properties are significantly different. This suggests that such PNM may not serve more complex studies, such as reactive/convective transport of contaminants or bacteria, and further research is necessary to improve the interpretation of real pore spaces with networks. Linear topology–geometry relations are derived and presented to stimulate development of more realistic PNM.

     

Effect of degree correlation on the thermal transport in complex networks

Nonlinear Dynamics - In this work, we present a complex network model to study thermal transport in the nanotube and nanowire networks, with nodes forming a random network. It is shown that the...     

Effect of Network Structure on Characterization and Flow Modeling Using X-ray Micro-Tomography Images of Granular and Fibrous Porous Media

Image-based network modeling has become a powerful tool for modeling transport in real materials that have been imaged using X-ray computed micro-tomography (XCT) or other three-dimensional imaging techniques. Network generation is an essential part of image-based network modeling, but little quantitative work has been done to understand the influence of different network structures on modeling. We use XCT images of three different porous materials (disordered packings of spheres, sand, and cylinders) to create a series of four networks for each material. Despite originating from the same data, the networks can be made to vary over two orders of magnitude in pore density, which in turn affects network properties such as pore-size distribution and pore connectivity. Despite the orders-of-magnitude difference in pore density, single-phase permeability predictions remain remarkably consistent for a given material, even for the simplest throat conductance formulas. Detailed explanations for this beneficial attribute are given in the article; in general, it is a consequence of using physically representative network models. The capillary pressure curve generated from quasi-static drainage is more sensitive to network structure than permeability. However, using the capillary pressure curve to extract pore-size distributions gives reasonably consistent results even though the networks vary significantly. These results provide encouraging evidence that robust network modeling algorithms are not overly sensitive to the specific structure of the underlying physically representative network, which is important given the variety image-based network-generation strategies that have been developed in recent years.     

Network model evaluation of permeability and spatial correlation in a real random sphere packing

In principle, network models can replicate exactly the microstructure of porous media. In practice, however, network models have been constructed using various assumptions concerning pore structure. This paper presents a network model of a real, disordered porous medium that invokes no assumptions regarding pore structure. The calculated permeability of the model agrees well with measured permeabilities, providing a new and more rigorous confirmation of the validity of the network approach. Several assumptions commonly used in constructing network models are found to be invalid for a random packing of equal spheres. In addition, the model permits quantification of the effect of pore-scale correlation (departure from randomness) upon permeability. The effect is comparable to reported discrepancies between measured permeabilities and predictions of other network models. The implications of this finding are twofold. First, a key assumption of several theories of transport in porous media, namely that pore dimensions are randomly distributed upon a network, may be invalid for real porous systems. Second, efforts both to model and to measure pore-scale correlations could yield more accurate predictions of permeability.     

Archie’s Law in Microsystems

Since 1942 Archie??s law is used every day to estimate, from electrical measurements, the quantity of oil present in oil fields. In this article, we perform the first experimental analysis of electric conductivity in well controlled models of porous media. We used microfluidic networks (called micromodels in the oil industry jargon), incorporating thousands of pores, with controlled wettability. Different electrode and pore geometries are considered. In all cases the evolution of the conductivity with the conductive fluid fraction (??saturation??) clearly reveals the presence of percolation thresholds, signaling that as the fraction of the conductive fluid decreases below some critical value, there are no more pathways involving only channels entirely filled with the conductive fluid that connect the electrodes. This behavior is observed in all cases, for all the network/electrode geometries and wetting properties we investigated, and is consequently likely to reflect a genuine behavior for microfluidic ??2D?? networks. The existing models??based on percolation theory or on mean field approach??reproduce correctly the structure of this behavior, but generally at a semi-quantitative level. The most successful case is obtained with the effective medium theory (EMT) model, with drainage and perpendicular electrodes. This outcome suggests that, despite the complexity of these systems, very simple models can describe correctly the physics of the system. Nonetheless, more precise modeling requires case-by-case studies. Our results are consistent with the current body of knowledge accumulated for decades on three-dimensional samples. The key point is that in 3D systems, owing to topological reasons, the threshold is extremely low in terms of water saturations. Archie??s law completely neglects the threshold effect. Nonetheless the percolation threshold should not be overlooked, and modeling should take this aspect systematically into account, as it is already done by several investigators.     

Improving the Estimations of Petrophysical Transport Behavior of Carbonate Rocks Using a Dual Pore Network Approach Combined with Computed Microtomography

Due to the intricate structure of carbonate rocks, relationships between porosity or saturation and petrophysical transport properties classically used for reservoir estimation and recovery strategies are either very complex or nonexistent. Thus, further understanding of the influence of the rock structure on the petrophysical transport properties becomes relevant. We therefore present a Dual Pore Network approach (D-PNM) applied to???-CT images of bimodal porous media. The major advantage of this method lies in the fact that it takes into account the real architecture of the connected macropore network as well as the microporosity unresolved by???-CT imaging. Whereas governing equations are solved in each individual macropore, transport behavior of microporosity is simulated by average quantities. Thus, D-PNM is particularly suited for the investigation of carbonate rocks, characterized by broad pore size distributions. We describe the principles of the image acquisition and network extraction procedure and the governing equations of D-PNM. The model is tested on three carbonate samples, two outcrop, and one reservoir carbonate. Calculated petrophysical transport properties are compared to experimental data and we show that D-PNM correctly reproduces conventional as well as unconventional electrical transport behavior. A major restriction of D-PNM is the requirement of a connected macropore network, that is, especially in the case of carbonates, not always available. Solutions to that are presented.     

Effect of Network Topology on Relative Permeability

We consider the role of topology on drainage relative permeabilities derived from network models. We describe the topological properties of rock networks derived from a suite of tomographic images of Fontainbleau sandstone (Lindquist et al., 2000, J. Geophys. Res. 105B, 21508). All rock networks display a broad distribution of coordination number and the presence of long-range topological bonds. We show the importance of accurately reproducing sample topology when deriving relative permeability curves from the model networks. Comparisons between the relative permeability curves for the rock networks and those computed on a regular cubic lattice with identical geometric characteristics (pore and throat size distributions) show poor agreement. Relative permeabilities computed on regular lattices and on diluted lattices with a similar average coordination number to the rock networks also display poor agreement. We find that relative permeability curves computed on stochastic networks which honour the full coordination number distribution of the rock networks produce reasonable agreement with the rock networks. We show that random and regular lattices with the same coordination number distribution produce similar relative permeabilities and that the introduction of longer-range topological bonds has only a small effect. We show that relative permeabilities for networks exhibiting pore–throat size correlations and sizes up to the core-scale still exhibit a significant dependence on network topology. The results show the importance of incorporating realistic 3D topologies in network models for predicting multiphase flow properties.     

Prediction of Effective Properties of Porous Carbon Electrodes from a Parametric 3D Random Morphological Model

Pore structures have a major impact on the transport and electrical properties of electrochemical devices, such as batteries and electric double-layer capacitors (EDLCs). In this work we are concerned with the prediction of the electrical conductivity, ion diffusivity and volumetric capacitance of EDLC electrodes, manufactured from hierarchically porous carbons. To investigate the dependence of the effective properties on the pore structures, we use a structurally resolved parametric model of a random medium. Our approach starts from 3D FIB-SEM imaging, combined with automatic segmentation. Then, a random set model is fitted to the segmented structures and the effective transport properties are predicted using full field simulations by iterations of FFT on 3D pore space images and calculations based on the geometric properties of the structure model. A parameter study of the model is used to investigate the sensitivity of the effective conductivity and diffusivity to changes in the model parameters. Finally, we investigate the volumetric capacitance of the EDLC electrodes with a geometric model, make a comparison with experimental measurements and do a parameter study to suggest improved microstructures.     

基于规则网络的碳酸盐岩多尺度网络模型构建方法研究

碳酸盐岩油藏非均质性强,孔隙大小变化可达好几个数量级,描述碳酸盐岩油藏多尺度孔隙特征具有重要意义.本文首先基于三维规则网络模型建立了不同物理尺寸的溶洞网络、大孔隙网络和微孔隙网络;然后提出一种耦合算法,以溶洞网络为基础,通过添加适当比例的大孔隙和微孔隙,构建出碳酸盐岩多尺度网络模型;最后对比分析了各网络模型的几何性质、拓扑性质和绝对渗透率.结果表明,碳酸盐岩多尺度网络模型能够同时描述不同尺度孔隙的几何和拓扑特征;且相比各单一尺度的孔隙网络模型,多尺度网络模型有着较高的绝对渗透率,这是由于各尺度孔隙之间的相互连通极大地提高了网络的整体连通性和流动能力,为碳酸盐岩油藏微观渗流模拟提供了重要的研究平台.     

Parameter Determination Using 3D FIB-SEM Images for Development of Effective Model of Shale Gas Flow in Nanoscale Pore Clusters

A large amount of nano-pores exists in pore clusters in shale gas reservoirs. In addition to the multiple transport regimes that occur on the nanoscale, the pore space is another major factor that significantly affects the shale gas recoverability. An investigation of the pore-scale shale gas flow is therefore important, and the results can be used to develop an effective cluster-scale pore network model for the convenient examination of the process efficiency. Focused ion beam scanning electron microscope imaging, which enables the acquisition of nanometre-resolution images that facilitate nano-pore identification, was used in conjunction with a high-precision pore network extraction algorithm to generate the equivalent pore network for the simulation of Darcy and shale gas flows through the pores. The characteristic parameters of the pores and the gas transport features were determined and analysed to obtain a deeper understanding of shale gas flow through nanoscale pore clusters, such as the importance of the throat flux–radius distribution and the variation of the tortuosity with pressure. The best parameter scheme for the proposed effective model of shale gas flow was selected out of three derived schemes based on the pore-scale prediction results. The model is applicable to pore-scale to cluster-scale shale gas flows and can be used to avoid the multiple-solution problems in the study of gas flows. It affords a foundation for further study to develop models for shale gas flows on larger scales.