Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs

Alcántara-López, Fernando; Fuentes, Carlos; Camacho-Velázquez, R.; Brambila-Paz, F.; Chávez, Carlos

doi:10.3390/en15134837

Cited by 3 publications

(1 citation statement)

References 23 publications

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“…If α = 1 in Equation ( 2), then Equation ( 1) is obtained. Another formulation relating the flux to the pressure gradient on the sample of length x 1 in the flow direction using fractional calculus was suggested by [28], which has dimensionless variables,…”

Section: Introductionmentioning

confidence: 99%

A Mechanical Picture of Fractal Darcy’s Law

Damián Adame,

Gutiérrez-Torres,

Figueroa-Espinoza

et al. 2023

Fractal Fract

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The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

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Section: Introductionmentioning

confidence: 99%

A Mechanical Picture of Fractal Darcy’s Law

Damián Adame,

Gutiérrez-Torres,

Figueroa-Espinoza

et al. 2023

Fractal Fract

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A Similarity-Based Solution for Nonlinear Gas Fractional Diffusivity Equation with Application to Rate Transient Analysis of Unconventional Heterogeneous Reservoirs

2022

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Summary Production characteristics of fractured wells in unconventional heterogeneous reservoirs have been shown to be effectively captured via anomalous diffusion model in which a partial differential equation (PDE) with fractional derivatives is solved. This paper presents a novel semianalytical solution of the nonlinear fractional diffusivity equation (FDE) applied to compressible fluid (gas) flow toward hydraulic fractures placed in heterogeneous and complex geological porous media. Self-similar theory and scaling transformation are used to solve the nonlinear PDE of fractional derivative written for real gas flow using density as the primary variable. The governing nonlinear partial gas FDE is transformed to ordinary nonlinear fractional differential equation after introducing similarity variables, which is later solved via shooting method coupled with Runge-Kutta integration. Pressure-dependent gas properties are captured straightforwardly in the solution without resorting to any further linearization via pseudopressure or pseudotime functions. The proposed similarity-based semianalytical solution is benchmarked against a Laplace transform-based analytical solution for linear, liquid FDE, and validated against a finely gridded numerical solution for the nonlinear, gas FDE. The proposed solution enables the diagnostic interpretation and characterization of production responses of unconventional gas wells exhibiting power-law behavior on the premise of anomalous diffusion during early transient period, which permits the estimation of important reservoir and fracture properties as shown in the case studies. Field and numerical examples are presented to showcase the capabilities of the proposed approach in the inverse, rate transient analysis.

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Torricelli’s Law in Fractal Space–Time Continuum

Samayoa,

Alvarez-Romero,

Jiménez-Bernal

et al. 2024

Mathematics

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A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

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Spatial Fractional Darcy’s Law on the Diffusion Equation with a Fractional Time Derivative in Single-Porosity Naturally Fractured Reservoirs

Cited by 3 publications

References 23 publications

A Mechanical Picture of Fractal Darcy’s Law

A Mechanical Picture of Fractal Darcy’s Law

A Similarity-Based Solution for Nonlinear Gas Fractional Diffusivity Equation with Application to Rate Transient Analysis of Unconventional Heterogeneous Reservoirs

Torricelli’s Law in Fractal Space–Time Continuum

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