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 ^{[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

2.1 Fractional calculus in fractal continuum

The FCFC of authors of ^{[33},^{34]}, is built on the basis of Tarazov’s
aproximation to the continuum physics and mechanics ^{[26},^{27]}, and it
basically consist in the transformation of a problem of a intrinsically
discontinuous medium (fractal) onto a problem in a continuous space (Euclidean)
in which this fractal is embedded ^{[30]},
dealing in the process with linear, superficial and volume fractional
infinitesimal coefficients, this coefficients are written in terms of fractal
dimensionalities proper of the medium and are supported by a specific metric
well defined as we can see in ^{[34]} and
its function is to vinculate the Euclidean differential elements with fractals
ones, they rewrite the concept of Hausdorff derivative given in ^{[32]} in terms of an ordinary derivative
multiplied by a power law function of the variable

where the function ^{[33},^{34]}. The DOS describes in this case, how permitted states of
particles are closely packed in the ^{[34]}. Now, Hausdorff’s partial derivative
is defined as:

and definition of fractional Laplacian is:

where:

this Hausdorff Laplacian turns to ordinary Laplacian when

respectively, where ^{[34]} in the 3D case, the DOS, is defined analogous to

where ^{[34]}. A useful and clarifying definition of ^{[30]}. More
definitions of operators of FCFC can be consulted in ^{[33},^{34]}, we have
included just those ones we are going to employ in the next sections.

2.2 Pressure transient equation for fractal continuum flow

In order to get the transient pressure equation for fractal continuum flow, as in the classical case, it is necessary to relate the generalized Darcy equation:

with equation for slightly compressible liquids:

and continuity equation:

then, susbtituing and into the result reads:

where is assume that characteristic tensor property of the fractal continuum flow ^{[34]}. Equation (11) is the well known pressure diffusion equation
for the case of an anisotropic three-dimensional fractal continuum flow as is
referred in ^{[34]},

and ^{[34]}.

2.3 Formula derivation

Using Eqs. (5) and (6) to rewrite (11) we obtain the partial differential equation:

where:

with ^{[35]}, multiplying (13) by

where ^{[35]}. We
can assume that

writing

applying the surface natural boundary condition:

with ^{[35]}. Taking into account that general solution over an
element has the form:

we get:

Taking into consideration expression (12) and arranging terms, (15) turns to:

the three terms inside the second integral of volume of (16), can be expressed in matrix form as follows:

where:

and:

therefore, the finite element equations are:

that has the typical form:

where:

(17) represents a system of first
order ordinary differential equations, also is the MFEE of (11) for general weighting
functions ^{[37},^{38]}. In the case that ^{[36},^{37]}, it would be simple to see that (18) includes information of the
tetrahedral coordinates

where ^{[38]} and explicit value of

according with ^{[33]}. Analog expressions
can be arise for the remaining terms of (18). Term

which, in this work, we solved analitically for the spatial variables of the
particular case of a canonical tetrahedron in the Euclidean reference frame
(vertices

details of calculations that we made can be read in Appendix.

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, ^{[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 >

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