Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows

We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.     

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.     

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.     

Estimation of Shale Intrinsic Permeability with Process-Based Pore Network Modeling Approach

A multi-scale pore network model is developed for shale with the process-based method (PBM). The pore network comprises three types of sub-networks: the \(\upmu \)m-scale sub-network, the nm-scale pore sub-network in organic matter (OM) particles and the nm-scale pore sub-network in clay aggregates. Process-based simulations mimic shale-forming geological processes and generate a \(\upmu \)m-scale sub-network which connects interparticle pores, OM particles and clay aggregates. The nm-scale pore sub-networks in OM and clay are extracted from monodisperse sphere packing. Nm-scale throats in OM and clay are simplified to be cylindrical and cuboid-shaped, respectively. The nm-scale pore sub-networks are inserted into selected OM particles and clay aggregates in the \(\upmu \)m-scale sub-network to form an integrated multi-scale pore network. No-slip permeability is evaluated on multi-scale pore networks. Permeability calculations verify that shales permeability keeps decreasing when nm-scale pores and throats replace \(\upmu \)m-scale pores. Soft shales may have higher porosity but similar range of permeability with hard shales. Small compaction leads to higher permeability when nm-scale pores dominate a pore network. Nm-scale pore networks with higher interconnectivity contribute to higher permeability. Under constant shale porosity, the shale matrix with cuboid-shaped nm-scale throats has lower no-slip permeability than that with cylindrical throats. Different from previous reconstruction processes, the new reconstruction process first considers the porous OM and clay distribution with PBM. The influence of geological processes on the multi-scale pore networks is also first analyzed for shale. Moreover, this study considers the effect of OM porosities and different pore morphologies in OM and clay on shale permeability.     

Laboratory Investigation of LNAPL Migration in Double-Porosity Soil Under Fractured Condition Using Digital Image Analysis

Gas production from shale gas reservoirs plays a significant role in satisfying increasing energy demands. Compared with conventional sandstone and carbonate reservoirs, shale gas reservoirs are characterized by extremely low porosity, ultra-low permeability and high clay content. Slip flow, diffusion, adsorption and desorption are the primary gas transport processes in shale matrix, while Darcy flow is restricted to fractures. Understanding methane diffusion and adsorption, and gas flow and equilibrium in the low-permeability matrix of shale is crucial for shale formation evaluation and for predicting gas production. Modeling of diffusion in low-permeability shale rocks requires use of the Dusty gas model (DGM) rather than Fick’s law. The DGM is incorporated in the TOUGH2 module EOS7C-ECBM, a modified version of EOS7C that simulates multicomponent gas mixture transport in porous media. Also included in EOS7C-ECBM is the extended Langmuir model for adsorption and desorption of gases. In this study, a column shale model was constructed to simulate methane diffusion and adsorption through shale rocks. The process of binary \(\hbox {CH}_{4}{-}\hbox {N}_{2}\) diffusion and adsorption was analyzed. A sensitivity study was performed to investigate the effects of pressure, temperature and permeability on diffusion and adsorption in shale rocks. The results show that methane gas diffusion and adsorption in shale is a slow process of dynamic equilibrium, which can be illustrated by the slope of a curve in \(\hbox {CH}_{4}\) mass variation. The amount of adsorption increases with the pressure increase at the low pressure, and the mass change by gas diffusion will decrease due to the decrease in the compressibility factor of the gas. With the elevated temperature, the gas molecules move faster and then the greater gas diffusion rates make the process duration shorter. The gas diffusion rate decreases with the permeability decrease, and there is a limit of gas diffusion if the permeability is less than \(1.0\,\times \,10^{-15}\, \hbox { m}^{2}\). The results can provide insights for a better understanding of methane diffusion and adsorption in the shale rocks so as to optimize gas production performance of shale gas reservoirs.     

A Network Model for Diffusion in Media with Partially Resolvable Pore Space Characteristics

A microstructure-informed meso-scale model for diffusion of foreign species in porous media is proposed. The model is intended for media where the pore geometry data acquired experimentally represent a fraction of total porosity. A cellular complex, with a cell representing the average pore neighbourhood, is used to generate 3D graphs of sites at cell centres and bonds between neighbouring cells. The novel interpretation of pore systems as graphs allows for clear separation between topology (here connectedness) and physics (here diffusion) in the mathematical formulation of transport. Further, it allows for easy introduction of dynamics into the system, i.e., local changes in topology due to other physical mechanisms, such as micro-cracking or blockage of pores. A mapping between microstructure features and graph elements is used for model construction. The mapping is based on data for clays, where the experimentally resolved pore system comprises isolated elongated pores of preferred orientation with a large volume fraction of unresolved pores. Both the resolved and the “hidden” systems are accounted for. The graph geometry is described by a principal length, the cell size in the preferred orientation, and a secondary length, the cell size out of preferred orientation. This is considered as a representation of mineralogical heterogeneity of clays. Analysis on graphs, a specialisation of the discrete exterior calculus, is used to obtain connectivity and diffusivity properties of formed networks. Since the experimental data are not sufficient to determine the principal length, upper and lower limits are determined from the limited information. Effects of the principal cell size between limits and of the secondary cell size are studied. The results are within the range of experimentally measured macroscopic (bulk) diffusivity for the material studied, including anisotropic diffusion coefficients. The variation of calculated diffusivity coefficients with principal and secondary lengths provides an explanation for the variability in experimentally measured coefficients across different clays.     

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.     

Pore Connectivity Between Organic and Inorganic Matter in Shale: Network Modeling of Mercury Capillary Pressure

Each of the two major mineral components found in shale samples—organic matter (OM) and inorganic matter (iOM)—has a distinct pore system revealed by scanning electron microscope images, low-pressure nitrogen adsorption, and high-pressure mercury injection tests. Although a vast amount of research has been conducted to detect and measure pore sizes in OM and iOM separately, the connectivity of the pores in these two components remains unclear. In permeability models, pore connectivity between OM and iOM components plays an important role in studying and predicting fluid flow. We studied pore connectivity between OM and iOM by developing pore-network models to mimic the composite nature of distributed OM patches in shale. Input parameters to generate network models were porosity, pore- and throat-size distribution, and total organic content. Mercury injection and capillary-pressure curves were then simulated through generated network models using percolation theory. To study the effects of pore connectivity between OM and iOM, we changed the size and locale of OM patches in the generated network models. Simulation results showed that the locale of OM affects mercury saturation (location and numbers of invaded pores) at given applied pressures. To study the effect of pore-size overlap between OM and iOM pores, we simulated mercury injection for two groups of constructed pore networks: non-overlapping and overlapping. In non-overlapping cases, first all iOM pores were invaded with mercury; then, only OM pores at very high pressure were invaded. In overlapping cases, OM and iOM pores can be invaded simultaneously because some of the pores have similar sizes in both components. The simulated capillary-pressure curves show distinct behavior in the non-overlapping and overlapping cases. Non-overlapping capillary-pressure curves show a sudden increase when OM pores are invaded, whereas overlapping capillary-pressure curves are smoother. Results of this work increase understanding of the connectivity of pores from measured capillary-pressure curves for further implementation in permeability-predictive models.     

Application of generalised effective-medium theory to transport in porous media

The use of effective-medium treatments to estimate bulk properties pertaining to transport (of, for example, fluids, heat, particles or electricity) through random composite media (such as reservoir rocks), is widespread. This is because they are relatively simple, often reasonably accurate (on occasion, remarkably so) and in many cases yield closed-form expressions for the properties concerned. However, the single-bond effective-medium treatment (EMT) of random resistor networks that has been used to determine transport coefficients for various transport problems in pore networks is limited to some special isotropic networks with nearest-neighbour connections. We demonstrate here that transport through two different fracture system models, with stress-induced anisotropy, can be treated using an EMT originally applied to anisotropic resistor networks. The main purpose of the present contribution, however, is to present a new, more general effective medium formalism applicable to networks of arbitrary topology. This new generalised EMT is used to obtain a new criterion for percolation of an arbitrary conducting network under random dilution. A specific application to unsaturated flow through a pore network with nearest- and next-nearest-neighbour connections is also given.     

Investigation of Various Pressure Transient Techniques on Permeability Measurement of Unconventional Gas Reservoirs

Chloride ingress into concrete is a major cause for material degradation, such as cracking due to corrosion-induced steel reinforcement expansion. Corresponding transport processes encompass diffusion, convection, and migration, and their mathematical quantification as a function of the concrete composition remains an unrevealed enigma. Approaching the problem step by step, we here concentrate on the diffusivity of cement paste, and how it follows from the microstructural features of the material and from the chloride diffusivity in the capillary pore spaces. For this purpose, we employ advanced self-consistent homogenization theory as recently used for permeability upscaling, based on the resolution of the pore space as pore channels being oriented in all space directions, resulting in a quite compact analytical relation between porosity, pore diffusivity, and the overall diffusivity of the cement paste. This relation is supported by experiments and reconfirms the pivotal role that layered water most probably plays for the reduction of the pore diffusivity, with respect to the diffusivity found under the chemical condition of a bulk solution.     

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.

     

Multi-scale Analysis of Gas Transport Mechanisms in Kerogen

Quantification of natural gas transport in organic-rich shale is important in predicting natural gas production. However, laboratory measurements are challenging due to tight nature of the rock and include large uncertainties. The emphasis of this work is to understand mass transport mechanisms inside the organic nanoporous material known as kerogen under subsurface conditions and describe its permeability. This requires a multi-scale theoretical approach that includes flow measurements in model nanocapillaries and within their network. Molecular dynamics simulation results of steady-state supercritical methane flow in single-wall carbon nanotube are presented in this article. A transition from convection to molecular diffusion is observed. The simulation results show that the adsorbed methane molecules are mobile and contribute a significant portion to the total mass flux in nanocapillaries with diameter \({<}\)10 nm. They experience cluster diffusion that is dependent on the applied pressure drop across the capillary. A modified Hagen–Poiseuille equation is proposed considering the convective–diffusive nature of the overall transport in nanocapillary. The molecular-level study of steady-state transport is extended to a simple network of interconnected nanocapillaries representing kerogen. The modified Hagen–Poiseuille equation leads to a representative elementary volume of the model kerogen. The estimated permeability of the volume is sensitive to compressed and adsorbed fluids density ratio and to surface properties of the nanocapillary walls, indicating that fluid–wall interactions driven by molecular forces could be significant during the large-scale transport within shale. A modified Kozeny–Carman correlation is proposed, relating kerogen porosity and tortuosity to the permeability.     

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.     

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.     

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.     

页岩储层体积压裂的地应力变化研究

页岩储层属于致密超低渗透储层,需改造形成复杂缝网才有经济产能.体积压裂是页岩储层增产改造的主要措施,而地应力场特别是水平主应力差值是体积压裂的关键控制因素. 理论研究表明:(1)当初始两向水平主应力差较小时,容易形成缝网,反之不易产生缝网;(2)人工裂缝的形成能够改变地层初始应力场. 因此应在前人研究的基础上优化设计压裂方式,以克服和翻转初始水平主应力差值,产生体积缝网.基于此,建立了页岩气藏水平井体积压裂数值模型,模型中采用多孔介质流固耦合单元模拟页岩基质的行为,采用带有孔压的"cohesive"单元描述水力裂缝的性质,模型对"Texas Two-Step" 压裂方法进行了数值模拟,模拟结果得到了压裂过程中地层应力场的分布及其变化,模拟结果和解析公式计算结果吻合良好.模拟结果表明:(1)裂缝的产生减弱了地层应力场的各向异性;(2 对于低水平应力差页岩储层,采用"Texas Two-Step"压裂方法可以产生缝网. 对于采用"Texas Two-Step"压裂方法无法产生缝网的高应力差页岩储层,提出了三次应力"共振" 和四次应力"共振" 压裂方法并进行了数值模拟,模拟结果得到了压裂过程中页岩储层应力场的分布及其变化,得到了缝网形成的区域,模拟结果表明:(1)对于高应力差页岩储层,采用"Texas Two-Step" 压裂方法无法产生缝网;(2)对于高应力差页岩储层,三次应力"共振" 和四次应力"共振"压裂方法是有效的体积压裂缝网形成的方法.      

Effects of serial and parallel pore nonuniformities: Results from two models of the porous structure

The effects of parallel-type and serial-type pore nonuniformities on the effective diffusivity and the permeability of a porous material were evaluated, constant porosity and constant specific surface area being assumed. Two structural models were considered. In the first model, the porous structure was described as a bundle of cylindrical capillaries penetrating the whole thickness of the material and in the other it was described instead as a collection of randomly distributed obstacles hindering transport. Both models predicted that parallel-type pore nonuniformities produce an increase in permeability compared with uniform structures having the same porosity and specific surface area. Both models also predicted that the increase in permeability due to parallel-type pore nonuniformities would be larger than the increase in effective diffusivity. Regarding serial-type pore nonuniformities, both models predicted a decrease in permeability and that this decrease would be greater than the decrease in effective diffusivity. The predicted changes in effective diffusivity due to nonuniformities of the sample differed for the two structural models.     

Effective Diffusivity of Porous Materials with Microcracks: Self-Similar Mean-Field Homogenization and Pixel Finite Element Simulations

We investigate the influence of distributed microcracks on the overall diffusion properties of a porous material using the self-similar cascade continuum micromechanics model within the framework of mean-field homogenization and computational homogenization of diffusion simulations using a high-resolution pixel finite element method. In addition to isotropic, also anisotropic crack distributions are considered. The comparison of the results from the cascade continuum micromechanics model and the numerical simulations provides a deeper insight into the qualitative transport characteristics such as the influence of the crack density on the complexity and connectivity of crack networks. The analysis shows that the effective diffusivity for a disordered microcrack distribution is independent of the absolute length scale of the cracks. It is observed that the overall effective diffusivity of a microcracked material with the microcracks oriented in the direction of transport is not necessarily higher than that of a material with a random orientation of microcracks, independent of the microcrack density.