Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard hydraulic conductivity

Li, Yabing; Zhou, Zhifang; Zhang, Ning

doi:10.1016/j.watres.2023.120957

Water Research

2024

DOI: 10.1016/j.watres.2023.120957

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Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard hydraulic conductivity

Yabing Li,

Zhifang Zhou,

Ning Zhang

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2024

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Cited by 2 publications

References 80 publications

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A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

Tu,

Wu,

Zhang

et al. 2024

Num Anal Meth Geomechanics

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The Forchheimer and Izbash equations have been long employed to investigate the behavior of non‐Darcian flow toward a well in various aquifer systems, but both two equations inevitably introduce problems such as more or less empirical nature, and dimensional unbalance. Therefore, this work makes the attempt to introduce the fractional Darcian model for characterizing the non‐Darcian behavior flow toward a fully penetrating well in a confined aquifer instead of the Forchheimer and Izbash equations. In this study, a fractional Darcian model‐based analytical solution in the time domain is obtained by means of Laplace transform and linearization approximation. The proposed analytical solution of this study can be readily reduced to the classical Theis solution for Darcian flow. Meanwhile, the late‐times and steady‐state analytical solutions for non‐Darcian flow described using the fractional Darcian model are also developed. Moreover, a comparison with previous analytical solutions shows that the newly derived analytical solution in this study is sufficiently accurate at later times. The influences of different parameters on transient drawdown are investigated. The results indicate that the fractional derivative order and hydraulic conductivity have a large influence on drawdown compared with other parameters. The introduction of fractional Darcian model in this study could also provide potential application for further investigations of non‐Darian flow behavior toward fully or partially penetrating wells in different aquifer systems, which can be very beneficial for hydrology and other related fields.

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A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

Tu,

Wu,

Zhang

et al. 2024

Num Anal Meth Geomechanics

View full text Add to dashboard Cite

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Fractional Calculus Applied to the Generalized Model and Control of an Electrohydraulic System

Robles,

Torres,

Balvantín-García

et al. 2024

Fractal Fract

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In this paper, fractional calculus is used to develop a generalized fractional dynamic model of an electrohydraulic system composed of a servo valve and a hydraulic cylinder, where a fractional position controller PIγDμ is proposed for minimizing the performance index according to the integral of the time-weighted absolute error (ITAE). First, the general mathematical equations of the cylinder and servo valve are used to obtain the transfer functions in fractional order by applying Caputo’s definition and a Laplace transform. Then, through a block diagram of the closed-loop system without a controller, the fractional model is validated by comparing its performance concerning the integer-order electrohydraulic system model reported in the literature. Subsequently, a fractional PID controller is designed to control the cylinder position. This controller is included in the closed-loop system to determine the fractional exponents of the transfer functions of the servo valve, cylinder, and control, as well as to tune the controller gains, by using the ITAE objective function, with a comparison of the following: (1) the electrohydraulic system model in integer order and the controller in fractional order; (2) the electrohydraulic system model in fractional order and the controller in integer order; and (3) both the system model and the controller in fractional order. For each of the above alternatives, numerical simulations were carried out using MATLAB®/Simulink® R2023b and adding white noise as a perturbation. The results show that strategy (3), where electrohydraulic system and controller model are given in fractional order, develops the best performance because it generates the minimum value of ITAE.

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Fractional derivative method for anomalous aquitard flow in a leaky aquifer system with depth-decaying aquitard hydraulic conductivity

Cited by 2 publications

References 80 publications

A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

Fractional Calculus Applied to the Generalized Model and Control of an Electrohydraulic System

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