Understanding fingering, as a challenge to stable displacement during the immiscible flow, has become a crucial phenomenon for geological carbon sequestration, enhanced oil recovery, and groundwater protection. Typica...Understanding fingering, as a challenge to stable displacement during the immiscible flow, has become a crucial phenomenon for geological carbon sequestration, enhanced oil recovery, and groundwater protection. Typically governed by gravity, viscous and capillary forces, these factors lead invasive fluids to occupy pore space irregularly and incompletely. Previous studies have demonstrated capillary numbers,describing the viscous and capillary forces, to quantificationally induce evolution of invasion patterns.While the evolution mechanisms of invasive patterns have not been deeply elucidated under the constant capillary number and three variable parameters including velocity, viscosity, and interfacial tension.Our research employs two horizontal visualization systems and a two-phase laminar flow simulation to investigate the tendency of invasive pattern transition by various parameters at the pore scale. We showed that increasing invasive viscosity or reducing interfacial tension in a homogeneous pore space significantly enhanced sweep efficiency, under constant capillary number. Additionally, in the fingering crossover pattern, the region near the inlet was prone to capillary fingering with multi-directional invasion, while the viscous fingering with unidirectional invasion was more susceptible occurred in the region near the outlet. Furthermore, increasing invasive viscosity or decreasing invasive velocity and interfacial tension promoted the extension of viscous fingering from the outlet to the inlet, presenting that the subsequent invasive fluid flows toward the outlet. In the case of invasive trunk along a unidirectional path, the invasive flow increased exponentially closer to the outlet, resulting in a significant decrease in the width of the invasive interface. Our work holds promising applications for optimizing invasive patterns in heterogeneous porous media.展开更多
The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generali...The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .展开更多
基金supported by the National Natural Science Foundation of China Joint Fund Project (Grant/Award Number: U20B6003)National Natural Science Foundation of China (Grant/Award Number: 52304054)。
文摘Understanding fingering, as a challenge to stable displacement during the immiscible flow, has become a crucial phenomenon for geological carbon sequestration, enhanced oil recovery, and groundwater protection. Typically governed by gravity, viscous and capillary forces, these factors lead invasive fluids to occupy pore space irregularly and incompletely. Previous studies have demonstrated capillary numbers,describing the viscous and capillary forces, to quantificationally induce evolution of invasion patterns.While the evolution mechanisms of invasive patterns have not been deeply elucidated under the constant capillary number and three variable parameters including velocity, viscosity, and interfacial tension.Our research employs two horizontal visualization systems and a two-phase laminar flow simulation to investigate the tendency of invasive pattern transition by various parameters at the pore scale. We showed that increasing invasive viscosity or reducing interfacial tension in a homogeneous pore space significantly enhanced sweep efficiency, under constant capillary number. Additionally, in the fingering crossover pattern, the region near the inlet was prone to capillary fingering with multi-directional invasion, while the viscous fingering with unidirectional invasion was more susceptible occurred in the region near the outlet. Furthermore, increasing invasive viscosity or decreasing invasive velocity and interfacial tension promoted the extension of viscous fingering from the outlet to the inlet, presenting that the subsequent invasive fluid flows toward the outlet. In the case of invasive trunk along a unidirectional path, the invasive flow increased exponentially closer to the outlet, resulting in a significant decrease in the width of the invasive interface. Our work holds promising applications for optimizing invasive patterns in heterogeneous porous media.
文摘The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .
