An Object-Based Shale Permeability Model: Non-Darcy Gas Flow,Sorption, and Surface Diffusion Effects

Shale samples consist of two major components: organic matter (OM) and inorganic mineral component (iOM). Each component has its distinct pore network topology and morphology, which necessitates generating a model capable of distinguishing the two media. We constructed an object-based model using the OM and iOM composition of shale samples. In the model, we integrated information such as OM population and size distribution, as well as its associated pore-size distribution. For the iOM part, we used mineralogy and pore-size information derived from X-ray diffraction, scanning electron microscopy, and nitrogen sorption measurements. Our proposed model results in millimeter-scale 2D realizations of shale samples by honoring OM and mineral types, their compositions, shapes, and size distributions. The model can capture heterogeneities smaller than 1 mm. We studied the effects of different gas flow processes and found that Knudsen diffusion and gas slippage dominate the flow, but surface diffusion has little impact on total gas flow.     

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.     

Numerical Simulation of Porosity in Cements

The pores in cementitious materials, their sizes and connectivity have an important influence on the durability of concrete. Several microstructural models have been developed to simulate the three-dimensional pore network in cement pastes. In this article, microstructures with the $\upmu $ μ ic model are compared with experimental results. It is seen that despite having a resolution for the capillary pores very close to reality, the experimentally observed breakthrough diameter from mercury intrusion is much lower than the values obtained by applying an algorithm of mercury intrusion to the simulated microstructure. The effect of some of the most important input parameters on the pore sizes in the simulated microstructure explored. The phenomenon which seems best able to explain this discrepancy is that C–S–H is not in fact a phase with a smooth surface as represented in microstructural models, but a phase which grows as needles into the pore space, leading to very small water-filled capillary pores from quite young ages. The results demonstrate it will be extremely challenging to represent the porosity of real microstructures in microstructural models on the scale of hundreds of microns necessary to study macroscopic transport.     

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.     

Connectivity of Pore Space as a Control on Two-Phase Flow Properties of Tight-Gas Sandstones

We predict capillary-pressure (drainage) curves in tight-gas sandstones which have little matrix or microporosity using a quantitative grain-scale model. The model accounts for the geometric results of some depositional and diagenetic processes important for porosity and permeability reduction in tight-gas sandstones, such as deformation of ductile grains during burial and quartz cementation. The model represents the original sediment as a dense, disordered packing of spheres. We simulated the evolution of this model sediment into a low-porosity sandstone by applying different amounts of ductile grains and quartz precipitation. A substantial fraction of original pore throats in the sediment is closed by the simulated diagenetic alteration. Because the percolation threshold corresponds to closure of half of the pore throats, the pore space in this type of tight-gas sandstone is poorly connected and is often close to being completely disconnected. The drainage curve for different model rocks was computed using invasion percolation in a network taken directly from the grain-scale geometry and topology of the model. Some general trends follow classical expectations and were confirmed by experimental measurements: increasing the amount of cement shifts the drainage curve to larger pressures. This is related to reduction of the connectivity of pore space resulting from closure of throats. Existence of ductile grains in the ductile grain model also reduces the connectivity of pore space but it treats the throats distribution differently causing the drainage curves to be shifted to larger irreducible water saturation when cement is added to the model. The range of porosities in which these connectivity effects are important corresponds to the range of porosities common for tight gas sandstones. Consequently these rocks can exhibit small effective permeability to gas even at large gas saturations. This problem occurs at larger porosities in rocks with significant content of ductile grains because ductile deformation blocks a significant fraction of pore throats even before cementation begins. Predicted drainage curves agree with measurements on two samples with little microporosity, one dominated by rigid grains, the other containing a significant fraction of ductile grains. We conclude that connectivity of the matrix pore space is an important factor for an understanding of flow properties of tight-gas sandstones.     

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.     

Limit Models of Pore Space Structure of Porous Materials for Determination of Limit Pore Size Distributions Based on Mercury Intrusion Data

This paper proposes the application of capillary and chain random models of pore space structure for determination of limit pore diameter distributions of porous materials, based on the mercury intrusion curves. Both distributions determine the range in which the pore diameter distribution of the investigated material occurs and defines the degree of inaccuracy of the method based on the mercury intrusion data caused by the indeterminacy of the sample shape and its pore space architecture. We derived equations describing the quasi-static process of mercury intrusion into the porous layer and porous ball with a random chain pore space structure and analysed the influence of the model parameters on the mercury intrusion curves. It was shown that the distribution of link length in the chain model of the pore space, random location of chain capillaries in the sample and the length distribution of the capillaries do not influence significantly the intrusion process. Therefore, a simple model of the mercury intrusion into the layer is proposed in which chain links of the pore space have random diameters and constant length. This model is used as a basic model of the intrusion process into a sample of any shape and size and with homogeneous and isotropic chain pore space architecture. The thickness of the layer then represents the mean length of chain capillaries in the sample. It was also proved that the capillary and chain models of pore space architecture are limit models of the network model identified in this paper with the pore architecture of the investigated sample. This justifies the application of both models for determination of limit cumulative distributions of pore diameters in porous materials based on the mercury intrusion data.

     

Simulation of the dynamic (frequency-dependent) permeability and electrical conductivity in porous materials based on the concept of an ensemble of pores

The class of models of porous media based on the concept of an ensemble of pores with a certain distribution of the main geometrical parameters (e.g., pore size) is studied. The case of the saturation of the pore space with a single-phase multicomponent fluid mixture is studied with and without taking into account the transfer of electric charges. Transfer laws are derived from the condition of decreasing free energy. The hydrodynamic connectivity of pores (and electrical conductivity) is described by two kernels: one kernel describes the connectivity of pores in space, and the other describes the connectivity of pores in the elementary macrovolume. The frequency dependences of the dynamic permeability determined in laboratory experiments and the electrical conductivity of the porous medium were determined using the concept of an ensemble of pores. The relationship between the models considered and relaxation filtration models is established.     

A Mathematical Model for Determining Oil Migration Characteristics in Low-Permeability Porous Media Based on Fractal Theory

Shale reservoirs are characterized by very low permeability in the scale of nano-Darcy. This is due to the nanometer scale of pores and throats in shale reservoirs, which causes a difference in flow behavior from conventional reservoirs. Slip flow is considered to be one of the main flow regimes affecting the flow behavior in shale gas reservoirs and has been widely studied in the literature. However, the important mechanism of gas desorption or adsorption that happens in shale reservoirs has not been investigated thoroughly in the literature. This paper aims to study slip flow together with gas desorption in shale gas reservoirs using pore network modeling. To do so, the compressible Stokes equation with proper boundary conditions was applied to model gas flow in a pore network that properly represents the pore size distribution of typical shale reservoirs. A pore network model was created using the digitized image of a thin section of a Berea sandstone and scaled down to represent the pore size range of shale reservoirs. Based on the size of pores in the network and the pore pressure applied, the Knudsen number which controls the flow regimes was within the slip flow regime range. Compressible Stokes equation with proper boundary conditions at the pore’s walls was applied to model the gas flow. The desorption mechanism was also included through a boundary condition by deriving a velocity term using Langmuir-type isotherm. It was observed that when the slip flow was activated together with desorption in the model, their contributions were not summative. That, is the slippage effect limited the desorption mechanism through a reduction of pressure drop. Eagle Ford and Barnett shale samples were investigated in this study when the measured adsorption isotherm data from the literature were used. Barnett sample showed larger contribution of gas desorption toward gas recovery as compared to Eagle Ford sample. This paper has produced a pore network model to further understand the gas desorption and the slip flow effects in recovery of shale gas reservoirs.

     

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.     

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.     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

Modeling of isothermal processes in porous materials on the basis of the concept of an ensemble of pores

A class of models of porous media based on the concept of an ensemble of pores with a certain distribution of the main geometric parameters (for example, the pore size) is considered. The cases of pores saturated with single-and two-phase multicomponent liquid mixtures are investigated. The properties of equilibrium states of the mixture are derived from the minimum free energy condition and the transfer laws from the decreasing free energy condition. The hydrodynamic connectivity of the pores is described by two types of kernels: one describes the spatial connectivity and the other the connectivity in an elementary macrovolume. Analytically and numerically, the one-dimensional problems of establishment of a steady-state regime, propagation of a passive admixture, and two-phase flow (an analog of the Buckley-Leverett problem) are investigated. A relationship between the models in question and relaxational filtration models is demonstrated. A simple model of capillary hysteresis related with the non-monotonicity of the pore area to volume ratio function is proposed.     

页岩有机质纳米孔隙气体吸附与流动规律研究

页岩储层孔隙结构复杂, 气体赋存方式多样. 有机质孔隙形状对受限空间气体吸附和流动规律的影响尚不明确, 导致难以准确认识页岩气藏气体渗流机理. 为解决该问题, 本文首先采用巨正则蒙特卡洛方法模拟气体在不同形状有机质孔隙(圆形孔隙、狭长孔隙、三角形孔隙、方形孔隙)内吸附过程, 发现不同形状孔隙内吸附规律符合朗格缪尔单层吸附规律, 分析了绝对吸附量、过剩吸附浓量、气体吸附参数随孔隙尺寸、压力的变化, 研究了孔隙形状对气体吸附的影响. 在明确不同形状有机质孔隙内气体热力学吸附规律基础上, 建立不同形状有机质孔隙内吸附气表面扩散数学模型和考虑滑脱效应的自由气流动数学模型, 结合分子吸附模拟结果研究了不同孔隙形状、孔隙尺寸有机质孔隙内吸附气流动与自由气流动对气体渗透率的贡献. 结果表明, 狭长孔隙内最大吸附浓度和朗格缪尔压力最高, 吸附气表面扩散能力最弱. 孔隙半径5 nm以上时, 吸附气表面扩散对气体渗透率影响可忽略. 本文研究揭示了页岩气藏实际生产过程中有机质孔隙形状对页岩气吸附和流动能力的影响机制.      

页岩储层纳微米孔隙CO2/CH4吸附及驱替特性研究进展

利用CO₂开采页岩气不仅能够提高页岩气采收率, 还能够节省水资源并且对CO₂进行地质封存, 有助于实现页岩气开采过程的碳中和. 富有机质页岩储层纳微米孔隙中气体运移机制不同于常规储层, CO₂在储层中具有超临界特性, 致使开采机理复杂, 无法得到CO₂开采页岩气微观机理的准确认识, 所以研究CH₄, CO₂及其二元混合物在页岩储层纳微米孔隙中的吸附及驱替特性对准确评估和高效开采页岩气至关重要. 本文从实验、理论以及模拟方面对页岩储层纳微米孔隙中CH₄的吸附特性、CO₂/CH₄二元混合物竞争吸附特性以及驱替特性进行了综合分析, 对气体在纳微米孔隙中吸附及驱替特性的基础研究及关键问题进行讨论分析并提出了展望. 研究表明CH₄在页岩储层中表现为物理吸附, 有机质特征(丰度、成熟度、类型)、孔隙结构、无机矿物组成、温度和压力、含水率对页岩的CH₄吸附能力均有一定程度的影响. 在相同条件下, CO₂比CH₄更易被页岩储层吸附, 在页岩储层中注入CO₂可以促进CH₄的解吸, 并有利于CO₂的地质埋存. 开采方案的部署可采用井网形式的注采方式, 可以通过调整注入井的位置、数量以及CO₂注入速率对开采方案进行优化.      

A Pore-Scale Investigation of Low-Salinity Waterflooding in Porous Media: Uniformly Wetted Systems

The potential of low-salinity (LS) water injection as an oil recovery technique has been the source of much recent debate within the petroleum industry. Evidence from both laboratory and field-level studies has indicated significant benefits compared to conventional high-salinity (HS) waterflooding, but many conflicting results have also been reported and, to date, the underlying mechanisms remain poorly understood. In this paper, we aim to address this uncertainty by developing a novel, steady-state pore network model in which LS brine displaces oil from a HS-bearing network. The model allows systematic investigation of the crude oil/brine/rock parameter space, with the goal of identifying features that may be critical to the production of incremental oil following LS brine injection. By coupling the displacement model to a salinity-tracking tracer algorithm, and assuming that a reduction of water salinity within the pore network leads to localised wettability alteration, substantial perturbations to standard pore filling sequences are predicted. The results clearly point to two principal effects of dynamic contact angle modification at the pore scale: a “pore sequence” effect, characterised by an alteration to the distribution of displaced pore sizes, and a “sweep efficiency” effect, demonstrated by a change in the overall fraction of pores invaded. Our study indicates that any LS effect will depend on the relative (scenario-dependent) influence of each mechanism, where factors such as the initial wettability state of the system and the pore size distribution of the underlying network are found to play crucial roles. In addition, we highlight the important role played by end-point capillary pressure in determining LS efficacy.     

STUDY OF THE PORE STRUCTURE OF SHALE BASED ON GAS ADSORPTION METHOD AND MERCURY INTRUSION METHOD1)

四川盆地威远、渝西地区下志留统龙马溪组页岩广泛发育,为定量研究该区页岩孔隙结构特征,以威远、渝西地区页岩样品为例,运用二氧化碳吸附、低温氮吸附和高压压汞等实验技术,研究了目标储层的孔隙结构及孔径分布特征。结果表明,微孔发育主要集中于0.45 ~ 0.65 nm之间,介孔发育主要集中于2 ~5 nm之间,宏孔发育相对较差且分布不均匀;比表面积与微孔成正相关性,主要孔隙类型是细颈广体的墨水瓶孔等无定形孔隙,并含有一定量四周开放的平行板孔。