scholarly journals The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017 ◽  
Vol 5 ◽  
Author(s):  
Angel A. Tateishi ◽  
Haroldo V. Ribeiro ◽  
Ervin K. Lenzi
Keyword(s):  
Anomalous Diffusion ◽  
Time Derivative ◽  
Fractional Time Derivative

Production-Decline Models Using Anomalous Diffusion Stemming From a Complex Fracture Network

SPE Journal ◽  
2019 ◽  
Vol 24 (06) ◽  
pp. 2609-2634 ◽  
Author(s):  
Shuai Liu ◽  
Peter P. Valkó
Keyword(s):  
Anomalous Diffusion ◽  
Time Derivative ◽  
Horizontal Wells ◽  
Fracture Network ◽  
Production Data ◽  
Fracture Networks ◽  
Type Curves ◽  
Complex Fracture ◽  
Complex Fracture Network ◽  
Fractional Time Derivative

Summary In this work, we develop a well–based fractional–production–decline model, which resorts to anomalous diffusion to characterize the heterogeneity of complex fracture networks and can be conveniently used to interpret the rate and cumulative data in the early and middle times of production from multifractured horizontal wells in unconventional reservoirs. We first develop a model (the fracture–based model) to be used in inverse problems by simplifying the fractional–time–derivative model using the traditional multiple–planar–fracture framework to essentially contain three parameters (α,Ar, and Ix). After performing tests with elaborate synthetic cases, the trace of anomalous diffusion is captured but the drawbacks of this model are also detected and analyzed. Then, by focusing on improving the fracture–based model, we consider an infinite–conductivity horizontal well draining a complex fracture network through a fractional–flux–related skin. The improved model (the well–based model) is bolstered with the corresponding type curves regarding production rate and cumulative production, which can be applied to interpret the single–phase production in the early and middle times and to mainly characterize the fracture–network–related heterogeneity reflected in the production data during this early–to–middle period. The well–based model is applied to two synthetic cases with explicitly modeled complex fracture networks and to the cases of multifracture horizontal wells in the Permian Basin. The results of all case studies display good matches between our model and the production data, which indicates the model's capability to accurately describe the transient regime of the flow in the extremely heterogeneous fracture networks on the basis of average values of the formation properties.

Dynamic Viscoelastic Rod Stability Modeling by Fractional Differential Operator

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
D. Ingman ◽  
J. Suzdalnitsky
Keyword(s):  
Damping Ratio ◽  
Time Derivative ◽  
Additional Term ◽  
Bifurcation Line ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
The Stability ◽  
Fractional Time Derivative ◽  
Stability Modeling

The need to take into account the oscillation of a system is a special feature in the linear problem of the stability of a cantilevered rod under a follower force. Involvement of viscoelastic materials leads to damping of the oscillation hence to overestimation of critical loads. This new problem is solved here by means of an additional term introduced into the constitutive equation and proportional to the fractional time derivative with complex order—besides the inertial one. The effects contributed by the damping ratio, the real part of the order and the corrective role of its imaginary part on the shape of the bifurcation line, on its maximum and on the disposition of the inflection and maximal deflection points on the centerline of the deformed rod during the secondary loss of stability, are discussed.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

scholarly journals New Iterative Method for Fractional Gas Dynamics and Coupled Burger’s Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mohamed S. Al-luhaibi
Keyword(s):  
Iterative Method ◽  
Analytical Solutions ◽  
Gas Dynamics ◽  
Time Derivative ◽  
Explicit Solutions ◽  
Fractional Partial Differential Equations ◽  
Approximate Analytical Solutions ◽  
Fractional Time Derivative ◽  
Burger's Equations ◽  
New Iterative Method

This paper presents the approximate analytical solutions to solve the nonlinear gas dynamics and coupled Burger’s equations with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm. Numerical results show that the new iterative method is easy to implement and accurate when applied to time-fractional partial differential equations.

Application of Fractional Derivatives for Simulating Diffusion Into Porous Matrix in Mathematical Modeling of the Contaminant Transport in a Confined Fractured Porous Aquifer

2006 ◽  
Author(s):  
Sergei Fomin ◽  
Vladimir Chugunov ◽  
Toshiyuki Hashida
Keyword(s):  
Solute Transport ◽  
Contaminant Transport ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Porous Matrix ◽  
Confined Aquifer ◽  
Closed Form Solutions ◽  
Advection Dispersion ◽  
Porous Aquifer ◽  
Fractional Time Derivative

Solute transport in the fractured porous confined aquifer is modeled by the advection-dispersion equation with fractional time derivative of order γ, which may vary from 0 to 1. Accounting for diffusion in the surrounding rock mass leads to the introduction of an additional fractional time derivative of order 1/2 in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties are modeled and analyzed.

scholarly journals Fractional Time Derivative Seismic Wave Equation Modeling for Natural Gas Hydrate

Energies ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 5901
Author(s):  
Yanfei Wang ◽  
Yaxin Ning ◽  
Yibo Wang
Keyword(s):  
Wave Equation ◽  
Natural Gas ◽  
Seismic Wave ◽  
Gas Hydrate ◽  
Wave Equations ◽  
Time Derivative ◽  
Seismic Attenuation ◽  
Equation Modeling ◽  
Natural Gas Hydrate ◽  
Fractional Time Derivative

Simulation of the seismic wave propagation in natural gas hydrate (NGH) is of great importance. To finely portray the propagation of seismic wave in NGH, attenuation properties of the earth’s medium which causes reduced amplitude and dispersion need to be considered. The traditional viscoacoustic wave equations described by integer-order derivatives can only nearly describe the seismic attenuation. Differently, the fractional time derivative seismic wave-equation, which was rigorously derived from the Kjartansson’s constant-Q model, could be used to accurately describe the attenuation behavior in realistic media. We propose a new fractional finite-difference method, which is more accurate and faster with the short memory length. Numerical experiments are performed to show the feasibility of the proposed simulation scheme for NGH, which will be useful for next stage of seismic imaging of NGH.

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