1. Introduction

Origins of Fractional Order Calculus (FOC) back in time to the end of XVII century in the famous question of L’Hospital to Leibnitz; “What if n be 1/2?” (question obviously inspired in the very known notation invented by Leibnitz for derivatives), Leibnitz’s response to L’Hospital was; “It will lead to a paradox, from this apparent paradox, one day useful consequences will be drawn” ^[1]. This was the point in time line in which seed of FOC had been planted. Due to the Liouville’s works together with those of Riemann, the current definitions of the differential and integral fractional operators of Riemann-Liouville were published in the 1800’s, in the same period the definition of the fractional integral of Gründwald-Letnikov also emerges. In the twentieth century, the definitions of the fractional operators of Weyl, Riesz and Caputo arise. The operators mentioned above are among many more definitions of fractional operators, the ones that are currently used or common the most ^[2,^3]. In the last fifty years, many works based on this fractional calculus operators have been published, to name; Kilbas et al. ^[3], Miller and Ross ^[4], Oldham and Spanier ^[5] and Samko et al. ^[6] in the rigorous mathematical context and some others like Strichartz ^[7] and Kigami ^[8] have been started to solve partial differential equations on mathematical fractal sets. Recently, important studies related to the application of FOC have been reported, for example; Gómez et al. ^[9] in the modeling of electrical circuits; Coronel et al. ^[13] stuying fractional behavior of BFT and CK oscillators; Atangana and Gómez in the study of the fundamental differences between power law, exponential decay, Mittag-Leffler law and their possible applications to real problems ^[10]; Atangana ^[11] in the application of the semigroup principle to the analysis of fractional derivatives of evolutions equations; Morales et al. ^[12] in the discussion of generalized Cauchy problems in diffusion wave processes. Authors like Herrmann ^[14] and West et al. ^[15] had focussed the FOC to some engineering applications. On the other hand, many researchers have reported findings based on Mandelbrot’s ideas for fractal characterization of natural systems ^[17], ^18]; for example, from biological systems ^[16], computer simulation ^[19], geological sciences ^[20-^22], folded and crumpled of thin matter ^[23-^25] to fluid flow ^[27-^34], but from the point of view of physics, there was not a proposal on fractality and fractional calculus in the continuum until continuum-type equations for fractal media were proposed by Tarasov ^[26`, that essentially links the fractal dimension of a fractal set with the order of the derivative (or integral). The works in the same line are ^[26-³³,³⁵,^37]. In ^[33,^34] the explicit proposal of the FCFC is done.

In the present work, we used the results published in about the fractional calculus operators in the fractal continuum in order to discretize the pressure diffusion equation. Section 2 is devoted to resume important definitions of FCFC together with the pressure transient equation for fractal continuum flow, also derivation of master finite element equation is included in this section. Section 3 includes the discussion of our results and potential uses. We wrote our conclusions in Sec. 4 and finally, details of calculations are shown in Appendix.

2. Basic Theory and Formula Derivation

3. Discussion

Actually, problems dealing with transport phenomena are very important in science and engineering, particularly, in the study of porous media there is a great research activity both theoretical and experimental ^[27,³³,³⁴,⁴⁰-⁴³,⁴⁵-^54]. On the other hand, since researchers began to apply fractional calculus in order to solve diverse engineering problems, many authors have made important contributions as we have referred before because of that, importance of modelling this type of systems lies in the successful forecast of the behavior that have quantities like flows, speeds, amounts of matter, pressure drops, etc. In real systems, the difficulty is that big because the medium in question is characterized by very complex geometric shapes, turning the modelling in a strong mathematical challenge, for that reason, the FCFC has special significance ^[34]. In that sense, we can notice that differential equation (22) contains the geometry information associated with the fractal medium under study through the corresponding fractal dimensions, ζi, D, di, cut off lower limits li and transformation function c3, ^[33,^34]. In the present case, we have employed the FCFC in the discretization process of the three-dimensional pressure diffusion equation for the anisotropic continuum fractal flow published in ^[34], it can be written in computer codes in any programming language and be of great interest in the field of computer simulation.

The discretization process of the parabolic equation (11) was written in (18) for general form functions >Ni, rewritten for arbitrary linear tetrahedron in (19) and solved analytically for spatial variables over the canonical tetrahedron in (22). We have shown explicitly the process to be followed with other types of finite elements. We also mention that the integral formulas that have been obtained analytically for the spatial case are general in the sense that they were solved for non-particular fractional parameter values, such parameters will depend on the geometry of the system to be simulated.

The fractional transient-pressure equation for flow in a porous medium has been solved analytically in ^[34], its solution corresponds to the specific case of radial contribution in a cylindrical symmetry domain with isotropic porosity. This type of results are helpful, for example, in the oil industry (well production analysis) or in the characterization of aquifers. From the point of view of software tools, it is useful to have numerical procedures for the solution of this typeof equations moreover, in the computational field, one can aspire to solve more complex cases like anisotropic one. In the present work, we have focused on the application of FEM for the most generic resolution of such pressure equation.

The results, by themselves, are already of significance for the computational implementation and allow the more accurate calculation of the integrals that appear in the matrix elements of the formulation, reducing computational complexity and also clarifies the panorama of the applicability of such method in this case of relative novelty.

4. Conclusions

We employ the FCFC defined by means of fractional operators (1), (2) and (6) of ^[33,^34] that relate a discontinuous system with a continuous one through the transformation function defined by (7) in order to get the MFEE for the transient-pressure equation in a three-dimensional continuum fractal flow. Explicit form of coefficient c3 for the geometry of a linear tetrahedron is given in (20). We have solved analitically the integral formulas for the spatial variables of (17) for the case of a canonical tetrahedron anchored in vertices -lx,-ly,-lz, 1-lx,-ly,-lz, -lx,1-ly,-lz and -lx,-ly,1-lz using a very similar process to the one carried out in the literature of mathematical methods to obtain the Dirichlet’s integral formula . We also mention that the results we obtained in this work can be linked to real field data that allow the development of adequate computer simulations. As a continuity of the present work, in a future publication, we will report a robust implementation that allows to see graphically the contrast that has the inclusion of the intrinsic geometry of the medium in the modeling of real application pressure diffusion problems, in contrast with the usual Euclideans aproximations implemented in commercial simulations softwares that not include fractional and fractal features.