First-Order and Monadic Second-Order Model-Checking on Ordered Structures

2012 ◽  
Author(s):  
Viktor Engelmann ◽  
Stephan Kreutzer ◽  
Sebastian Siebertz
Keyword(s):  
Model Checking ◽  
Second Order ◽  
Order Model ◽  
Ordered Structures ◽  
First Order ◽  
Second Order Model

Second-Order Correlation of Klinkenberg-Corrected Permeability and its Experimental Verification on Heterogeneously-Stressed Gas Shale

2020 ◽  
Author(s):  
Yufei Chen ◽  
Changbao Jiang ◽  
Juliana Y. Leung ◽  
Andrew K. Wojtanowicz ◽  
Dongming Zhang ◽  
...  
Keyword(s):  
Experimental Data ◽  
Second Order ◽  
Gas Shale ◽  
Order Model ◽  
First Order ◽  
Gas Slippage ◽  
Steady State Method ◽  
Slippage Effect ◽  
Stress Heterogeneity ◽  
Second Order Model

Abstract Shale is an extremely tight and fine-grained sedimentary rock with nanometer-scale pore sizes. The nanopore structure within a shale system contributes not only to the low to ultra-low permeability coefficients (10−18 to 10−22 m2), but also to the significant gas slippage effect. The Klinkenberg equation, a first-order correlation, offers a satisfying solution to describe this particular phenomenon for decades. However, in recent years, several scholars and engineers have found that the linear relation from the Klinkenberg equation is invalid for most gas shale reservoirs, and a need for a second-order model is, therefore, proceeding apace. In this regard, the purpose of this study was to develop a second-order approach with experimental verifications. The study involved a derivation of a second-order correlation of the Klinkenberg-corrected permeability, followed by experimental verifications on a cubic shale sample sourced from the Sichuan Basin in southwestern China. We utilized a newly developed multi-functional true triaxial geophysical (TTG) apparatus to carry out permeability measurements with the steady-state method in the presence of heterogeneous stresses. Also discussed were the effects of two gas slippage factors, Klinkenberg-corrected permeability, and heterogeneous stress. Finally, based on the second-order slip theory, we analyzed the deviation of permeability from Darcy flux. The results showed that the apparent permeability increased more rapidly as the pore pressure declined when the pore pressures are relatively low, which is a strong evidence of the gas slippage effect. The second-order model could reasonably match the experimental data, resulting in a lower Klinkenberg-corrected permeability compared with that from the linear Klinkenberg equation. That is, the second-order approach improves the intrinsic permeability estimation of gas shales with the result being closer to the liquid permeability compared with the Klinkenberg approach. Analysis of the experimental data reported that both the first-order slippage factor A and the second-order slippage factor B increased with increasing stress heterogeneity, and that A was likely to be more sensitive to stress heterogeneity compared with B. Interestingly, both A and B first slightly increased and then significantly as the permeability declined. It is recommended that when the shale permeability is below 10−18 m2, the second-order approach should be taken into account. Darcy’s law starts to deviate when Kn > 0.01 and is invalid at high Knudsen numbers. The second-order approach seems to alleviate the problem of overestimation compared with the Klinkenberg approach and is more accurate in permeability evolution.

Effectiveness of parallel second order model over second and first order models

2011 ◽  
Vol 32 (1-3) ◽  
pp. 107-114 ◽  
Author(s):  
Ahmad Jabari Kohpaei ◽  
Arumugam Sathasivan ◽  
Hanieh Aboutalebi
Keyword(s):  
Second Order ◽  
Order Model ◽  
First Order ◽  
Second Order Model

scholarly journals Analytical Deriving of Second Order Model of Payne from First Order Lighthil-Whitham-Richards Model

2013 ◽  
Vol 13 (4) ◽  
pp. 54-62
Author(s):  
Vasilka Stoilova ◽  
Emil Nikolov ◽  
Nina Nikolova
Keyword(s):  
Second Order ◽  
Order Model ◽  
First Order ◽  
Richards Model ◽  
Payne Model ◽  
Second Order Model

Abstract Analytical deriving has been realized to obtain the second order Payne model from the first order Lighthil-Witham-Richards model.

Kinetics and Equilibrium Study for Acetazolamide in Soils

2013 ◽  
Vol 781-784 ◽  
pp. 2296-2300
Author(s):  
Jian Qiu Chen ◽  
Zhi Liang Wang ◽  
Guo Ping Li ◽  
Rui Xin Guo ◽  
Yu Zhu Hu
Keyword(s):  
Clay Content ◽  
Second Order ◽  
Sorption Behavior ◽  
Order Model ◽  
Sorption Data ◽  
First Order ◽  
Kd Values ◽  
Pseudo Second Order ◽  
Pseudo First Order ◽  
Second Order Model

Sorption of Acetazolamide was found to be impacted by clay content and soil solution pH, with higher Kd values for soils with higher clay content and higher Kd values as the pH increased. Sorption data on soils were well fitted with Freundlich model (R2> 0.98). The kinetics of adsorption of Acetazolamide has been discussed using four kinetic models, i.e., the pseudo-first-order model, the pseudo-second-order model, the Elovich equation, the intraparticle diffusion model. The pseudo-first-order model and the pseudo-second-order model were both used to describe the kinetics data, and the pseudo-second-order model could describe the sorption behavior of Acetazolamide well.

scholarly journals Overshoot of D-dot sensor in measurement of electric field

2021 ◽  
Vol 2083 (2) ◽  
pp. 022033
Author(s):  
Jinghai Guo ◽  
Liang Ma ◽  
Yinhui Cheng ◽  
Xutong Wang ◽  
Yifei Liu ◽  
...  
Keyword(s):  
Electric Field ◽  
Equivalent Circuit ◽  
Field Measurement ◽  
Equivalent Circuit Model ◽  
Second Order ◽  
Order Model ◽  
Scanning Method ◽  
First Order ◽  
The Given ◽  
Second Order Model

Abstract This paper analyzes the cause of the overshot of the signal measured by D-dot sensor, and points out that the second-order RLC equivalent circuit model can explain the overshot Taking the asymptotic conical antenna as an example, the inductance L of the second-order model is obtained by using CST simulation and Pspice parameter scanning method. The first-order model and the second-order model are used to calculate the electric field waveform. Under the given conditions, the first-order model is used to obtain the electric field waveform, the waveform overshot is serious, the maximum overshoot reaches 27%, while the second-order model is used to improve the ringing phenomenon, And the maximum overshoot is reduced to 5%.The results show that the second-order model is more accurate than the first-order model and can improve the precision of electric field measurement using D-dot sensor.

Isothermal Microtube Heat Transfer With Second-Order Slip Flow and Temperature Jump Boundary Conditions

2006 ◽  
Author(s):  
Nian Xiao ◽  
John Elsnab ◽  
Susan Thomas ◽  
Tim Ameel
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Nusselt Number ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Slip Flow ◽  
Second Order ◽  
Order Model ◽  
First Order ◽  
Second Order Model

Two analytical models are presented in which the continuum momentum and energy equations, coupled with second-order slip flow and temperature jump boundary conditions, are solved. An isothermal boundary condition is applied to a microchannel with a circular cross section. The flow is assumed to be hydrodynamically fully developed and thermal field is either fully developed or thermally developing from the tube entrance. A traditional first-order slip boundary condition is found to over predict the slip velocity compared to the second-order model. Heat transfer increases at the upper limit of the slip regime for the second-order model. The maximum second-order correction to the first-order Nusselt number is on the order of 18% for air. The second-order effect is also more significant in the entrance region of the tube. The Nusselt number decreases relative to the no-slip value when slip and temperature jump effects are of the same order or when temperature jump effects dominate. When temperature jump effects are small, the Nusselt number increases relative to the no-slip value. Comparisons to a previously reported model for an isoflux boundary condition indicate that the Nusselt number for the isoflux boundary condition exceeds that for the isothermal case at all axial locations.

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