Growth models have been widely used to describe behavior in different areas of knowledge; among them the Logistics and Gompertz models, classified as models with a fixed inflection point, have been widely studied and applied. In the present work, a model is proposed that contains these growth models as extreme cases; this model is generalized by including the Caputo-type fractional derivative of order 0<β≤1, resulting in a Fractional Growth Model which could be classified as a growth model with non-fixed inflection point. Moreover, the proposed model is generalized to include multiple sigmoidal behaviors and thereby multiple inflection points. The models developed are applied to describe cumulative confirmed cases of COVID-19 in Mexico, US and Russia, obtaining an excellent adjustment corroborated by a coefficient of determination R2>0.999.

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.

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non- conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behaviour is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.

The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as independent variables and space as a dependent variable, and found the solution as a power series in t1/2 that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in tν/2, where ν∈(0,2) is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil.

Due to the complexity imposed by all the attributes of the fracture network of many naturally fractured reservoirs, it has been observed that fluid flow does not necessarily represent a normal diffusion, i.e., Darcy’s law. Thus, to capture the sub-diffusion process, various tools have been implemented, from fractal geometry to characterize the structure of the porous medium to fractional calculus to include the memory effect in the fluid flow. Considering infinite naturally fractured reservoirs (Type I system of Nelson), a spatial fractional Darcy’s law is proposed, where the spatial derivative is replaced by the Weyl fractional derivative, and the resulting flow model also considers Caputo’s fractional derivative in time. The proposed model maintains its dimensional balance and is solved numerically. The results of analyzing the effect of the spatial fractional Darcy’s law on the pressure drop and its Bourdet derivative are shown, proving that two definitions of fractional derivatives are compatible. Finally, the results of the proposed model are compared with models that consider fractal geometry showing a good agreement. It is shown that modified Darcy’s law, which considers the dependency of the fluid flow path, includes the intrinsic geometry of the porous medium, thus recovering the heterogeneity at the phenomenological level.