Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs

Chang, Ailian; Sun, HongGuang; Zhang, Yong; Chen, Kewei; Min, Fanlu

doi:10.1016/j.physa.2018.11.040

Cited by 47 publications

(17 citation statements)

References 33 publications

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“…This section will present several synthetic results of solving the fractional triple permeability and triple porosity model, Equations (21)- (23), subject to the initial and boundary conditions (25)- (27). The results show the value of the pressure at the boundary of the well in real space p Dw and its respective Bourdet derivative, p Dw [8]; pressure results will be shown with a solid line, while its derivative will be with a dashed line; both results will be drawn with the same color.…”

Section: Resultsmentioning

confidence: 99%

“…Equation (27) represents the external border condition when considering an infinite reservoir. Next, the fractional triple permeability and triple porosity model will be analytically solved, Equations (21)-(23), with boundary conditions (25)-(27).…”

Section: Fractional Triple Permeability and Triple Porosity Modelmentioning

confidence: 99%

“…Chang et al [27], starting from the fact that integer derivatives cannot reliably quantify fluid flow with time memory and/or spatial path dependency, proposed and evaluated a Darcy's law model with a spatial fractional derivative in the sense of Riemann-Liouville; showed how their model can felicitously depict flow properties; explored subdiffusive flows and saw how these flows could represent porous media with small pore sizes or dead pores.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

Alcántara-López

Fuentes

Brambila-Paz

2020

MCA

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The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered in a full-penetrating wellbore. Furthermore, the model relaxes classic hypotheses considering that matrix permeability has a significant impact on the pressure deficit from the wellbore, reaching the triple permeability and triple porosity model wich allows the wellbore to be fed by all the porous media and not exclusively by the fractured system; where it is considered a pseudostable interporous flow. In addition, it is considered the anomalous flow phenomenon from the pressure of each independent porous medium and as a whole, through the temporal fractional derivative of Caputo type; the resulting phenomenon is studied for orders in the fractional derivatives in (0, 2), known as superdiffusive and subdiffusive phenomena. Synthetic results highlight the effect of anomalous flows throughout the entire transient behavior considering a significant permeability in the matrix and it is contrasted with the effect of an almost negligible matrix permeability. The model is solved analytically in the Laplace space, incorporating the Tartaglia–Cardano equations.

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Section: Resultsmentioning

confidence: 99%

Section: Fractional Triple Permeability and Triple Porosity Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

Alcántara-López

Fuentes

Brambila-Paz

2020

MCA

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“…Because of this advantage, the fractional calculus has taken on the importance and popularity of modeling realistic cases, especially those with memory effects [20,21,22,23]. Furthermore, the application of the fractional calculus is used in many fields of social sciences, engineering, and biology [24,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Ullah

Băleanu

2020

Preprint

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Everyone is talking about coronavirus from last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed and as many as 180 countries are so far affected with 928,287 (14 September, 2020) deaths within couple of months. Ironically, 29,185,779 are still active cases. Having seen such drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. Major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 01 April to 15 March, 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam-Hyers and its generalized version. Best parameters have been obtained via nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than classical version of the SIR model [4]. Average absolute relative error under the Caputo operator is about 48% smaller than the one obtained in the classical case (ν = 1). Time series and 3D contour plots offer social distancing to be the most effective measure to control the epidemic.

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“…On the other hand, the construction of diffusion equation is obtained with the Atangana-Baleanu fractional integral [32]. Existence and uniqueness of Caputo time fractional Navier-Stokes equation is proven [33], whereas flow characteristics of gas and oil are discussed for a heterogeneous media [34].…”

Section: Introductionmentioning

confidence: 99%

Optimal Control Problem of a Non-integer Order Waterborne Pathogen Model in Case of Environmental Stressors

Yıldız

2019

Front. Phys.

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In this work, we extend a mathematical model, which has been proposed for susceptible and infected compartments together with pathogen population, by including recovered subgroup. It is known that environmental pollution, such as contaminated drinking water and lack of an ordinary toilet, affects individuals and such negative impacts can be defined as "stressors." In order to include the influence of such stressors, susceptible subpopulation has been divided into two groups as the one affected by stress or not. Thus, spread of the disease is expressed in terms of a five-dimensional system. Moreover, we extend this model with the use of a time fractional derivative due to non-local effects of water pollution and we prove that the solution is non-negative and bounded from above. Then, we perform stability analysis for the disease-free equilibrium point. Afterward, the next step is to apply optimal control theory to optimize the decay rate of pathogens and the stress related parameters so that the number of infected individuals and the pathogen population can be minimized. Finally, we present some numerical results to find out the most appropriate control policy and the effect of the fractional order.

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Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs

Cited by 47 publications

References 33 publications

Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

Caputo SIR Model for COVID-19 under Optimized Fractional Order

Optimal Control Problem of a Non-integer Order Waterborne Pathogen Model in Case of Environmental Stressors

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