A Model for the Apparent Gas Permeability of Shale Matrix Organic Nanopore Considering Multiple Physical Phenomena

The flow of shale gas in nano scale pores is affected by multiple physical phenomena. At present, the influence of multiple physical phenomena on the transport mechanism of gas in nano-pores is not clear, and a unified mathematical model to describe these multiple physical phenomena is still not available. In this paper, an apparent permeability model was established, after comprehensively considering three gas flow mechanisms in shale matrix organic pores, including viscous slippage Flow, Knudsen diffusion and surface diffusion of adsorbed gas, and real gas effect and confinement effect, and at the same time considering the effects of matrix shrinkage, stress sensitivity, adsorption layer thinning, confinement effect and real gas effect on pore radius. The contribution of three flow mechanisms to apparent permeability under different pore pressure and pore size is analyzed. The effects of adsorption layer thinning, stress sensitivity, matrix shrinkage effect, real gas effect and confinement effect on apparent permeability were also systematically analyzed. The results show that the apparent permeability first decreases and then increases with the decrease of pore pressure. With the decrease of pore pressure, matrix shrinkage, Knudsen diffusion, slippage effect and surface diffusion effect increase gradually. These four effects will not only make up for the permeability loss caused by stress sensitivity and adsorption layer, but also significantly increase the permeability. With the decrease of pore radius, the contribution of slippage flow decreases, and the contributions of Knudsen diffusion and surface diffusion increase gradually. With the decrease of pore radius and the increase of pore pressure, the influence of real gas effect and confinement effect on permeability increases significantly. Considering real gas and confinement effect, the apparent permeability of pores with radius of 5 nm is increased by 13.2%, and the apparent permeability of pores with radius of 1 nm is increased by 61.3%. The apparent permeability model obtained in this paper can provide a theoretical basis for more accurate measurement of permeability of shale matrix and accurate evaluation of productivity of shale gas horizontal wells.

Direct Numerical Simulation of Real-gas Effects within Turbulent Boundary Layers for Fully-developed Channel Flows

Real-gas effects have a significant impact on compressible turbulent flows of dense gases, especially when flow properties are in proximity of the saturation line and/or the thermodynamic critical point. Understanding of these effects is key for the analysis and improvement of performance for many industrial components, including expanders and heat exchangers in organic Rankine cycle systems. This work analyzes the real-gas effect on the turbulent boundary layer of fully developed channel flow of two organic gases, R1233zd(E) and MDM - two candidate working fluids for ORC systems. Compressible direct numerical simulations (DNS) with real-gas equations of state are used in this research. Three cases are set up for each organic vapour, representing thermodynamic states far from, close to and inside the supercritical region, and these cases refer to weak, normal and strong real-gas effect in each fluid. The results within this work show that the real-gas effect can significantly influence the profile of averaged thermodynamic properties, relative to an air baseline case. This effect has a reverse impact on the distribution of averaged temperature and density. As the real-gas effect gets stronger, the averaged centre-to-wall temperature ratio decreases but the density drop increases. In a strong real-gas effect case, the dynamic viscosity at the channel center point can be lower than at channel wall. This phenomenon can not be found in a perfect gas flow. The real-gas effect increases the normal Reynolds stress in the wall-normal direction by 7–20% and in the spanwise direction by 10–21%, which is caused by its impact on the viscosity profile. It also increases the Reynolds shear stress by 5–8%. The real-gas effect increases the turbulence kinetic energy dissipation in the viscous sublayer and buffer sublayer <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo><</mml:mo><mml:mn>30</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> but not in the outer layer. The turbulent viscosity hypthesis is checked in these two fluids, and the result shows that the standard two-function RANS model (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mi>ϵ</mml:mi></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mi>ω</mml:mi></mml:math></inline-formula>) with a constant <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:msub><mml:mi>C</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.09</mml:mn></mml:math></inline-formula> is still suitable in the outer layer <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>></mml:mo><mml:mn>70</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>, with an error in ±10%.

Real Gas Effect on Gas Slippage Phenomenon in Shale

The past decade has seen the rapid development of shale gas across the world, as the record-breaking success and on-going surge of commercial shale gas production in such unconventional reservoirs pose a tremendous potential to meet the global energy supply. However, questions have been raised about the intricate gas transport mechanisms in the shale matrix, of which the gas slippage phenomenon is one of the key mechanisms for enhancing the fluid transport capacity and, therefore, the overall gas production. Given that shale reservoirs are often naturally deposited in the deep underground formations at high pressure and temperature conditions (much deeper than most typical conventional deposits), the real gas effect cannot be ignored as gas properties may vary significantly under such conditions. The purpose of this study is thus to investigate the real gas effect on the gas slippage phenomenon in shale by taking into account the gas compressibility factor (Z) and Knudsen number (Kn). This study begins with a specific determination of Z for natural gas at various pressures and temperatures under the real gas effect, followed by several calculations of the gas molecular mean free path at in-situ conditions. Following this, the real gas effect on gas slippage phenomenon in shale is specifically analyzed by examining the change in Knudsen number. Also discussed are the permeability deviation from Darcy flux (non-Darcy flow) due to the combination of gas slippage and real gas effect and the specific range of pressure and pore size for gas slippage phenomenon in shale reservoirs. The results show that the gas molecular mean free path generally increases with decreasing pressure, especially at relatively low pressures (&lt; 20 MPa). And, increasing temperature will cause the gas molecular mean free path to rise, also at low pressures. Knudsen number of an ideal gas is greater than that of a real gas; while lower than that of a real gas as pressure continues to rise. That is, the real gas effect suppresses the gas slippage phenomenon at low pressures, while enhancing it at high pressures. Also, Darcy’s law starts deviating when Kn &gt; 0.01 and becomes invalid at high Knudsen numbers, and this deviation increases with decreasing pore size. No matter how pore size varies, this deviation increases with decreasing pressure, meaning that the gas slippage effect is significant at low pressures. Finally, slip flow dominates in the various gas transport mechanisms given the typical range of pressure and pore size in shale reservoirs (1 MPa &lt; P &lt; 80 MPa; 3 nm &lt; d &lt; 3000 nm). Gas transport in shale is predominantly controlled by the slippage effect that mostly occurs in micro- or meso-pores (10 to 200 nm). Moreover, considering the real gas effect would improve the accuracy for determining the specific pressure range of the gas slippage phenomenon in shale.

Leakage and Rotordynamic Characteristics for Three Types of Annular Gas Seals Operating in Supercritical CO2 Turbomachinery

This paper presents a comprehensive assessment and comparison on the leakage and rotordynamic performance of three types of annular gas seals for application in a 14 MW supercritical CO2 turbine. These three seals represent the main seal types used in high-speed rotating machines at the balance piston location in efforts to limit internal leakage flow and achieve rotordynamic stability, including a labyrinth seal (LABY), a fully-partitioned pocket damper seal (FPDS), and a hole-pattern seal (HPS). These three seals were designed to have the same sealing clearance and similar axial lengths. To enhance the seal net damping capability at high inlet preswirl condition, a straight swirl brake also was designed and employed at seal entrance for each type seal to reduce the seal inlet pre-swirl velocity. Numerical results of leakage flow rates, rotordynamic force coefficients, cavity dynamic pressure and swirl velocity developments were analyzed and compared for three seal designs at high positive inlet preswirl (in the direction of shaft rotation), using a proposed transient CFD-based perturbation method based on the multiple-frequency elliptical-orbit rotor whirling model and the mesh deformation technique. To take into account of real gas effect with high accuracy, a table look-up procedure based on the NIST database was implemented, using an in-house code, for the fluid properties of CO2 in both supercritical and subcritical conditions.

Numerical Simulation of a GH2/LOx Single Injector Combustor and the Effect of the Turbulent Schmidt Number

A large-eddy simulation (LES) of a gaseous hydrogen/liquid oxygen (GH2/LOX) single-injector rocket combustor is performed in this study. The Redlich–Kwong–Peng–Robinson (RK–PR) equation of state is used to simulate the real-gas effect under high-pressure conditions, and the steady laminar flamelet model (SLFM) is implemented to simulate fast chemistry, such as a H2/O2 reaction. From the numerical simulation, the characteristics of time-averaged flow and flame fields are obtained, and their relationship with the real-gas effect is investigated. It is possible to investigate unsteady flame features and the mixing mechanism of propellants in detail by examining multiple snapshots of the field contour. Another purpose of the study is to investigate the differences in flow and flame structures according to the variation in the turbulent Schmidt number. By comparing the simulation result with the natural OH* emission image and temperature profiles from experimental data, the appropriate range of the turbulent Schmidt number for the simulation is obtained. Furthermore, this paper suggests the usefulness and validity of the current research by quantitatively comparing (i.e., temperature profiles) numerical results with those of existing literature.

A fractal permeability model for the dual-porosity media of tight gas reservoirs

Hydraulic fracturing is a crucial method for the exploitation of tight gas reservoirs. The matrix permeability is a key factor influencing the fracturing result. This paper assumes that the matrix permeability is provided by a series of capillary bundles and tree-like networks, fully considering the stress sensitivity to establish a single-capillary (fracture) flow equation in terms of factors such as the water saturation, threshold pressure gradient (TPG), fracture width dynamic changes and real gas effect. The established permeability model after fracturing is generalized by Darcy’s law with the fractal theory. The apparent permeability model shows that (1) the gas flow in capillaries and fractures is single-phase flow considering the connate water saturation, stress sensitivity, real gas effect, TPG, and fracture width dynamic changes. The fracture permeability is much higher than the capillary permeability. When the production pressure gradient is lower than the TPG, the flow rate is 0. As the formation pressure decreases, the dual-porosity medium permeability increases. (2) As the water saturation increases, the permeability decreases, and with increasing stress sensitivity and real gas effect, the permeability decreases. (3) The parameters of the tree-like fractal structure greatly affect the permeability. The larger the number and series of bifurcations are, the higher the permeability is. The fracture length ratio is K∝γ, and the fracture width ratio is α∝K. The negative correlation becomes increasingly profound with increasing number and series of bifurcations. This fractal model fully considers TPG, stress sensitivity, and real gas effects, making the dual-porous medium reservoir permeability calculation model more complete, which can provide a more accurate calculation method for the permeability of the reservoir stimulation area after fracturing.