Steady-state Gas Flow Research Articles

Multiscale modeling of gas-induced fracturing in anisotropic clayey rocks

In the context of repositories for nuclear waste, understanding the behavior of gas migration through clayey rocks with inherent anisotropy is crucial for assessing the safety of geological disposal facilities. The primary mechanism for gas breakthrough is the opening of micro-fractures due to high gas pressure. This occurs at gas pressures lower than the combined strength of the rock and its minimum principal stress under external loading conditions. To investigate the mechanism of microscale mode-I ruptures, it is essential to incorporate a multiscale approach that includes subcritical microcracks in the modeling framework. In this contribution, we derive the model from microstructures that contain periodically distributed microcracks within a porous material. The damage evolution law is coupled with the macroscopic poroelastic system by employing the asymptotic homogenization method and considering the inherent hydro-mechanical (HM) anisotropy at the microscale. The resulting permeability change induced by fracture opening is implicitly integrated into the gas flow equation. Verification examples are presented to validate the developed model step by step. An analysis of local macroscopic response is undertaken to underscore the influence of factors such as strain rate, initial damage, and applied stress, on the gas migration process. Numerical examples of direct tension tests are used to demonstrate the model's efficacy in describing localized failure characteristics. Finally, the simulation results for preferential gas flow reveal the robustness of the two-scale model in explicitly depicting gas-induced fracturing in anisotropic clayey rocks. The model successfully captures the common behaviors observed in laboratory experiments, such as a sudden drop in gas injection pressure, rapid build-up of downstream gas pressure, and steady-state gas flow following gas breakthrough.

Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

Ramping up the hydrogen sector: An energy system modeling framework

With the transition towards a decarbonized society, energy system integration is becoming ever more essential. In this transition, the energy vector hydrogen is expected to play a key role as it can be produced from (renewable) power and natural gas with carbon capture and storage and utilized in a plethora of applications and processes across sectors. Despite global hydrogen demand reaching 94 million tonnes in 2021, still less than 0.7% of it is supplied by low-emission hydrogen. Moreover, to date hydrogen production is mostly located in close proximity to where it is used. In order to link future production and demand sites, it is planned to re-purpose existing natural gas and expand dedicated hydrogen pipelines. During the early stages of ramping up the hydrogen sector (2020s and early 2030s), however, blending natural gas with hydrogen for joint pipeline transmission has been suggested. Against this background, this paper studies hydrogen blending from a modeling perspective, both in terms of the implications of considering (or omitting) technical modeling details and in terms of the potential impact on the ramp-up of the hydrogen sector. To this end, we present a highly modular and flexible integrated sector-coupled energy system optimization model of the power, natural gas, and hydrogen sectors with a novel gas flow formulation for modeling blending in the context of steady-state gas flows. A stylized case study illustrates that hydrogen blending has the potential to initiate and to facilitate the ramp-up of the hydrogen sector under certain assumptions, while omitting the technical realities of gas flows – particularly in the context of blending – can result in suboptimal expansion planning not only in the hydrogen, but also in the power sector, as well as in an operationally infeasible system.

Visualization and Analysis of Temporal and Steady-State Gas Concentration in Process Chamber Using 70-dB SNR 1000 fps Absorption Imaging System

We are developing a 70-dB SNR 1,000 frame-per-second absorption imaging system, and it enables us to visualize gas concentration distribution in a vacuum chamber for semiconductor manufacturing. This article presents the measurement and analysis of temporal and steady-state NO2 gas concentration distribution using this system. In the steady-state gas flow condition which supplied from a shower plate, the spectrometry of NO2 gas shows a good linear relationship between the absorbance and the partial pressure with the system. The system enables the visualization of steady-state of NO2 gas concentration distribution which supplied from one hole of shower plate, and the results are compared with simulation to discuss the reasonability of the measurement. The system also enables to visualize the temporal NO2 gas concentration distribution supplied from a single nozzle, and the advantages of the system’s features of 70-dB and 1,000 fps are discussed.

On the Integration of Hydrogen Into Integrated Energy Systems: Modeling, Optimal Operation, and Reliability Assessment

The proliferation of renewable energy (RE) brings tremendous challenges to integrated energy systems (IESs). Converting RE into hydrogen, one of the cleanest energy carriers, provides an appealing alternative for decarbonized IESs. Among the various hydrogen applications, blending hydrogen into natural gas systems is already applicable. However, the disparate hydrogen physical properties trigger concerns about hydrogen integration. This paper investigates the integration of hydrogen into the IESs, focusing on blending hydrogen into natural gas systems. The literature on hydrogen modeling, control, operation, planning, and markets are first reviewed. Based on the convex combination methods, a power-to-hydrogen-heat-methane (P2HHM) model with unit commitment is proposed. The steady-state and dynamic gas flow models with the explicit consideration of hydrogen effects are then proposed. Two applications are given based on the proposed hydrogen modeling. A Wasserstein metric-based distributionally robust optimal operation model is proposed first based on the developed P2HHM and gas flow models. Then a sequential Monte Carlo simulation-based reliability assessment model is formulated to analyze the effects of hydrogen physical properties and hydrogen fractions on the optimal operation and reliability of the IESs. Numerical simulations are conducted to analyze the integration of hydrogen and verify the effectiveness of the proposed model.

Steady-State Gas Flow from Tight Shale Matrix Subject to Water Blocking

Summary This work studies 1D steady-state flow of gas from compressible shale matrix subject to water blocking toward a neighboring fracture. Water blocking is a capillary end effect causing wetting phase (e.g., water) to accumulate near the transition from matrix to fracture. Hydraulic fracturing is essential for economical shale gas production. Water is frequently used as fracturing fluid, but its accumulation in the matrix can reduce gas mobility and production rate. Gas transport is considered at a defined pressure drop. The model accounts for apparent permeability (slip), compressibility of gas and shale, permeability reduction, saturation tortuosity (reduced relative permeability upon compaction), and multiphase flow parameters like relative permeability and capillary pressure, which depend on wettability. The behavior of gas flow rate and distributions of gas saturation, pressure, and permeability subject to different conditions and the stated mechanisms is explored. Water blockage reduces gas relative permeability over a large zone and reduces the gas flow rate. Despite gas flowing, strong capillary forces sustain mobile water over the entire system. Reducing drawdown gave lower driving force and higher resistance (by water blockage) for gas flow. The results show that 75% reduction of drawdown made the gas flow rate a couple orders of magnitude lower compared to if there was no blockage. The impact was most severe in more water-wet systems. The blockage caused most of the pressure drop to occur near the outlet. High pressure in the rest of the system reduced effects from gas decompression, matrix compression, and slip-enhanced permeability, whereas rapid gradients in all these effects occurred near the outlet. Gas decompression resulted in an approximately 10 times higher Darcy velocity and pressure gradient near the outlet compared to inlet, which contributed to removing blockage, but the added resistance reduced the gas production rate. Similarly, higher gas Corey exponent associated gas flow with higher pressure drop. The result was less blockage but lower gas production. Slip increased permeability, especially toward the outlet, and contributed to increase in gas production by 16%. Significant matrix compression was associated with permeability reduction and increased Corey exponent in some examples. These effects reduced production and shifted more of the pressure drop toward the outlet. Upstream pressure was more uniform, and less compression and permeability reduction were seen overall compared to a system without water blockage.

Integrated electricity and gas system modelling with hydrogen injections and gas composition tracking

Despite the direct physical coupling between them, electricity and gas networks were traditionally modelled and operated independently. However, the heavy reliance on gas-fired generators to balance intermittent generation from renewable energy sources (RES), and the promising role of green hydrogen in decarbonising the natural gas system, have prompted a paradigm shift towards integrated electricity and gas system (IEGS) modelling. While many previous studies investigated the role of hydrogen in future low-carbon energy systems, a detailed assessment of hydrogen system integration into the electricity and gas transmission networks is still not addressed. Therefore, this paper presents a novel IEGS model with green hydrogen injections and gas composition tracking. The electricity system is modelled as a unit commitment model, formulated as a mixed-integer linear programming problem, and the gas system is modelled as a steady-state optimal gas flow (OGF). The developed model is demonstrated on two sets of case studies. The first case study validates the proposed OGF methodology on a small meshed gas test network, whereas the second case study demonstrates the applicability of the overall IEGS model with green hydrogen injection on the large-scale, real-world electricity and gas transmission networks of the state of Victoria (Australia). Results show that the proposed methodology can accurately capture the variations in gas flow direction while maintaining the hydrogen molar fraction within limits under hydrogen injections from multiple locations. Moreover, the amount of injected hydrogen not only depends on the level of RES curtailment, but also on local gas network constraints and local demand.

Analytical solutions for one-dimensional diabatic flows with wall friction

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Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles

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Conflicting Aspects in the Flat Plate Boundary Layer Conventional Solution

The solution of the steady-state isothermal gas flow with zero-pressure gradient over a flat plate is classical and was presented by Blasius in 1908. Despite being reduced to a third-order Ordinary Differential Equation, nobody has solved it using just three boundary conditions. In consequence, the “solutions” (exact or numerical) are not unique, do not satisfy Prandtl’s boundary layer concept, and give rise to imprecise criteria of the boundary layer thickness definition (0.99.U∞, where U∞ is the free stream velocity) and to idealizations such as the displacement thickness, δ∗, and the momentum thickness, δI. Even though η, the similarity parameter, is defined as a function of x, it is surprising that it is seen as a constant, η∞, at the limit of the boundary layer, being valid for the entire plate. It is shown that uncountable “solutions” satisfying the classical equation and its three natural boundary layer conditions can be built. The reduction technique of the Boundary Value Problem, BVP, to an Initial Value Problem, IVP, and the Perturbation Analysis Technique, used as an attempt to incorporate a fourth boundary condition to the differential equation, are discussed. A more general and direct method to deduce the classical Blasius’s flat plate equation is considered, as one of the steps to rectify its solution, and shed light on the origin of the conflicting issues involved.

Some Features of Plasma Bunches Throwing to Hypervelocities in a Magnetoplasma Accelerator

In this article, we consider some features of plasma bunches throwing in a rail electromagnetic accelerator [magnetoplasma accelerator (MPA)] associated with the limitation of experimentally achieved muzzle velocity compared to the theoretically expected one. The problem is considered from two points of view: 1) based on a qualitative study of the equations of the 1-D flow of an ideal conductive gas in an electromagnetic field and 2) from the point of view of accelerating a current bridge with lumped parameters. It is shown that the results of the qualitative studies of the steady-state gas flow equations are not applicable to the case of plasma bunches throwing in pulsed MPAs since the theory assumes a continuous effect, in particular of an electromagnetic field, on conductive gas over the entire length of the device. Using a model of the moving current bridge, implying that the current flows only in a narrow current layer (plasma piston), which allows us to accurately reflect the process of its throwing in an MPA until the plasma piston remains compact. Stratification of the plasma piston and secondary breakdowns in the space between the accelerator breech and the piston lead to the cessation of the acceleration process and the transition of the process described by the model similar to the steady-state flow model.

Steady State Gas Flow in Pipeline Networks: Existence and Uniqueness of Solution

In this paper we discuss the uniqueness and existence of solution to a real gas flow network by employing graph theory. A directed graph is an efficient way to represent a gas network. We consider steady state real gas flow network that includes pipelines, compressors, and the connectors. The pipelines and compressors are represented as edges of the graph and the interconnecting points are represented as nodes of the graph representing the network. We show that a unique solution of such a system exists. We use monotonicity property of a mapping to proof uniqueness, and the contraction mapping theorem is used to prove existence.

A Mathematical Model of Gas and Water Flow in a Swelling Geomaterial—Part 1. Verification with Analytical Solution

Gas generation and migration are important processes that must be considered in a safety case for a deep geological repository (DGR) for the long-term containment of radioactive waste. Expansive soils, such as bentonite-based materials, are widely considered as sealing materials. Understanding their long-term performance as barriers to mitigate gas migration is vital in the design and long-term safety assessment of a DGR. Development and the application of numerical models are key to understanding the processes involved in gas migration. This study builds upon the authors’ previous work for developing a hydro-mechanical mathematical model for migration of gas through a low-permeable geomaterial based on the theoretical framework of poromechanics through the contribution of model verification. The study first derives analytical solutions for a 1D steady-state gas flow and 1D transient gas flow problem. Using the finite element method, the model is used to simulate 1D flow through a confined cylindrical sample of near-saturated low-permeable soil under a constant volume boundary stress condition. Verification of the numerical model is performed by comparing the pore-gas pressure evolution and stress evolution to that of the results of the analytical solution. The results of the numerical model closely matched those of the analytical solutions. Future studies will attempt to improve upon the model complexity and investigate processes and material characteristics that can enhance gas migration in a nearly saturated swelling geomaterial.

CFD analysis of gas flow characteristics in a rotating packed bed with randomly arranged spherical packing

In recent years, some unit operations such as absorption and rectification in rotating packed beds (RPBs) have drawn great attention, and single-phase hydrodynamics characteristics became the basis to understand the multiphase flow and mass transfer process. In this work, the 3D steady-state gas flow in the RPB with randomly arranged spherical packing was investigated by computational fluid dynamics (CFD). The gas flow field and the effects of operational parameters were analyzed. Results showed that the errors between the simulated and the experimental data of the total pressure drop were within ±20%. The pressure drop of packing zone and tangential velocity increased with the increase of rotational speed, while the rising of the gas flow rate could result in the increase in the radial velocity, the pressure drop of the inner or outer cavity. The turbulent kinetic energy was predominantly affected by the tangential slip velocity, and the higher values of turbulent kinetic energy were illustrated near the outer edge of the rotor. Furthermore, a semi-empirical correlation based on the numerical simulation was proposed to predict the dry pressure drop of the packing zone combining the wall effect.

Simplified operation models of integrated power and gas systems for vulnerability analysis

This paper selects four frequently used operation models for integrated power and gas systems (IPGSs), including a power network connectivity and gas network connectivity (PNC&GNC) combined model, a power network flow and gas network flow (PNF&GNF) combined model, a direct current power flow and gas network flow (DCPF&GNF) combined model and a direct current power flow and steady-state gas flow (DCPF&SGF) combined model. By applying them to three types of vulnerability-related problems, including calculating system-level vulnerability under random failures, identifying the worst-case attack scenarios and mitigating the worst-case vulnerability, the above four operation models are compared in terms of the differences of the vulnerability results and the computational time. Taking several IPGSs as examples, the experiments show that the DCPF&GNF combined model approximates the DCPF&SGF combined model the best for vulnerability analysis, with the computational time saved by one to five orders of magnitude.

Predicting the rarefied gas flow through circular nano/micro short tubes: A semi-empirical model

A new explicit semi-empirical model was proposed for predicting the steady state gas flow rate through short tubes under rarefied condition. The experimental data produced by Fujimoto and Usami and the DSMC data provided by Varoutis et al. were utilized to obtain the model coefficients. The model was further validated by the published experimental data. The proposed model was capable to make predictions while meeting all the criteria of the rarefied gas flow in the following ranges of the variables: 0<Pr < 0.7, 0<δ < 4000 and 0<ω < 13. The model was capable to predict the reduced flow rate of rarefied systems, not only in the free molecular region and the hydrodynamic region, but also in the transition region, hence covering all the Knudsen number domain within the utilized data. It was also found that by increasing the pressure ratio and decreasing the length to diameter ratio, the transition region shifted to the higher values of the rarefaction parameter. The proposed model is useful in the vacuum gas community for engineering purposes by offering a practical closed form expression for the DSMC data to predict the rarefied gas flow rate.

Viscous Gas Flow between Vertical Walls

A new solution for the Navier–Stokes equations has been found for a plane steady-state shear flow of a viscous gas in the gravity field between two vertical walls. For the temperature dependence of the viscosity factor, Sutherland’s formula is accepted; for the heat conductivity factor, two formulas with the same accuracy are used: a known formula for low temperatures (170–1000 K) and a first-proposed formula for high temperatures (800–1500 K). In each of these temperature ranges, a general solution for the Navier–Stokes equations has been obtained. The solution is expressed in terms of a function which satisfies an ordinary second order differential equation having different forms for low and high temperatures depending on the chosen formula for heat conductivity. For low temperatures, the solution of this equation is obtained numerically; for high temperatures, owing to using the new formula for heat conductivity, analytically. Examples of exact solutions are presented.

Scaling law for slip flow of gases in nanoporous media from nanofluidics, rocks, and pore-scale simulations

In unconventional reservoirs, as the effective pore size becomes close to the mean free path of gas molecules, gas transport in porous media begins to deviate from Darcy’s law. The objective of this study is to explore the similarities of gas flows in nanochannels and core samples as well as those simulated by direct simulation BGK (DSBGK), a particle-based method that solves the Bhatnagar-Gross-Krook (BGK) equation.Due to difficulties in fabrication and experimentation, previous study on gas flow experiments in nanochannels is very limited. In this work, steady-state gas flow was measured in reactive-ion etched nanochannels with a controlled channel size on a sillicon wafer. A core-based permeability measurement apparatus was used to perform steady-state gas flow measurements on carbonate and shale samples. Klinkenberg permeability was obtained under varying pore pressures but constant temperature and effective stress. Methane was used in nanofluidic and rock experiments, making them directly comparable. Results from both experiments were then compared to gas flow simulations by DSBGK method carried out on several independently constructed geometry models. DSBGK uses hundreds of millions of simulated molecules to approximate gas flow inside the pore space. The intermolecular collisions are handled by directly integrating the BGK equation along each molecules trajectory, rather than through a sampling scheme like that in the direct simulation Monte-Carlo (DSMC) method. Consequently, the stochastic noise is significantly reduced, and simulation of nano-scale gas flows in complex geometries becomes computationally affordable.The slippage factors obtained from these independent studies varied across three orders of magnitude, yet they all appear to collapse on a single scaling relation where the slippage factor in the slip flow regime is inversely proportional to the square root of intrinsic permeability over porosity. Our correlation could also fit the data in the literature, which were often obtained using nitrogen, after correcting for temperature and gas properties. This study contributes to rock characterization, well testing analysis as well as the understanding of rarefied gas transport in porous media.

Decentralized Optimization of Multi-Area Electricity-Natural Gas Flows Based on Cone Reformulation

A large-scale integrated energy system can represent several subsystems representing areas that are tied by electricity and natural gas networks. Accordingly, we propose a decentralized optimal energy flow (DOEF) calculation as compared with a centralized solution method. The proposed approach demonstrates the merits of the decentralized operation and control of a multi-area integrated electricity-natural gas system (IEGS), in terms of large-scale modeling requirements, faster computations, and data management for local sensitivity analyses. Using the proposed decentralized structure, the communication burden is relatively light as individual area operators in a multi-area IEGS will make optimal dispatch decisions independently and the corresponding information is shared with adjacent subsystems. The reformulation of the second-order cone (SOC) is proposed using advanced sequential cone programming (SCP) to handle the nonlinear steady-state natural gas flow, which provides a feasible solution with a high degree of computational efficiency. Furthermore, an iterative alternating direction method of multipliers (I-ADMM) is adopted to manage the nonconvexity of integer variables, which guarantees a satisfactory convergence performance. Case studies on three multi-area IEGS validate the effectiveness of the proposed model in a multi-area IEGS.

Steady state and transient simulation for electricity‐gas integrated energy systems by using convex optimisation

The increasing interdependence between electric power and natural-gas systems requires integrated simulation tools applicable to the coupled system. Therefore, this study proposes steady-state and transient simulation tools for electricity-gas integrated energy systems (IESs) by using convex optimisation. Steady-state IES simulation is designed for long-term planning studies while transient IES simulation, which includes line-pack storage, is designed for real-time operational research. The proposed convex steady-state gas flow model introduces binary variables to represent gas flow directions, and employs the second-order cone (SOC) to relax non-convex pipeline gas flow equations. As a result, steady-state gas flow analysis is formulated as a mixed integer SOC programming problem. The proposed transient gas flow model employs the implicit finite difference method to transform partial differential equations into algebraic difference equations. Moreover, the short-term gas flow directions are specified and SOC relaxations are used. Simulation results on an integrated IEEE 24-node system and Belgian 20-node gas system verify the effectiveness of the proposed SOC models. The impacts of wind power forecast errors on multi-period gas network operations are also investigated.