In this chapter, we present a discussion about the practical application of the fractal properties of the medium in the mathematical model through the use of fractional partial derivatives. We present one of the known models for the flow in saturated media and its generalization in fractional order derivatives. In the middle section, we present one of the main arguments that motivate the use of fractional derivatives in the porous media models, this is the Professor Nigmatullin’s work. The final part describes the process for obtaining the coupled system of three equations for the monophase flow with triple porosity and triple permeability, briefly mentioning the method used for the first solutions of the system.
- fractional calculus
- fractional derivatives
- anomalous diffusion
- porous media
- fractal dimension
The objects of nature rarely have a classical geometric form; in the particular case of oil reservoirs, the ground where the wells are found has been considered with Euclidean geometry; this is not sufficient in many cases to give good approximations in the mathematical models. Since its forms are closer to the fractal geometry, the knowledge of this can be useful to develop models that allow us to better manage the wells. This work presents an approach in fractional derivatives for the triple porosity and triple permeability monophasic saturated model, based on the one proposed by Camacho et al. [1, 2] and generalized partially by Fuentes et al. . The main contribution is to consider the link between fractional equations and fractal geometry through the revision of Alexander-Orbach’s conjecture , taken to the particular case.
2. Background of the approach of models of diffusion on fractal media
Fractional calculus was originated as a way to generalize classic calculus; however, it is more difficult to find a direct physical interpretation than in the classical version. When we consider an oil well as a fractal, it is important to choose which of its properties can be useful for elaborating a mathematical model [20, 21, 26, 27].
Alexander and Orbach  calculated the “spectral dimension (fracton)”; this parameter is associated to volume and fractal connectivity by being considered as an elastic fractal net of particles connected by harmonic strings. Thus, we consider the particle movement over this fractal and we find a relation of root mean square of an
O’Shaughnessy and Procaccia  used the concepts of Alexander and Orbach to formulate their fractal diffusion equation:
of which one finds a power law
Metzler et al.  started with the characterization of an anomalous diffusion process 1. Here, they consider as the
valid in the asyntotic range , and
Thus, it is possible to obtain the solution of the fractional derivative diffusion equation:
3. Brief history of fractional calculus
In mathematics, one way to obtain new concept is to generalize by extending one definition or context for values not previously considered. For example, it is possible to generalize the power concept of
In his correspondence with Johan Bernoulli, Leibniz mentioned to him general order derivatives. In 1697, he established that differential calculus can be used to achieve these generalizations and used the notation to denote order derivative .
The (fractional) non-integer order derivative motivated Euler to introduce the Gamma function. Euler knew that he needed to generalize (or, as he said, interpolate), the product for non-integer
and used it to partially solve the Leibniz paradox. He also gave the basic fractional derivative (with modern notation ):
which is valid for non-integer
for an integer
This integral is, except for the factor, a fractional integral of 1/2 order, Abel wrote the left part as , thus he worked with both sides of the equation as
The first integral equation in history had been solved. Two facts may be observed: the regard for the sum of the orders, and that unlike in classical calculus, the derivative of a constant is not zero [8, 22].
from which he got
that can be obtained using the extension
for an arbitrary number
of which, after a change of variable and a suitable rewriting is obtained
Liouville also tackled the tautochrone problem and proposed differential equations of arbitrary order.
In 1832, he wrote about a generalization of Leibnitz’s rule about the
Liouville expanded the coefficients in Eq. (18) as
And inserted those equations in Eq. (18) to get
These formulas would be retaken by Grünwald in 1867.
be valid for . The factor is determined by the condition, and he found that it was . Riemann then derived Eq. (25) expression for negative
They worked using the contour integral method, with the contribution of Laurent (1884), they achieved the definition:
For an integration to an arbitrary order, when
Using integration properties, more definitions will be given.
First generalizing for a
Grünwald generalizes Eq. (32) for an arbitrary
where the binomial coefficient is
also showing that
and with those previous results, it is possible to establish this important property for and
In the twentieth and twenty-first centuries, more definitions will rise, but they will be given in terms within the Riemann-Liouville fractional integral and will be part of the Modern Fractional Calculus Theory, in all their fundamental definitions .
4. Fractional calculus
We will now present the assorted definitions and notations of fractional derivatives that will be used throughout this work. It is worth pointing out that this is necessary because such notation is currently standardized [18, 19].
4.1. Riemann-Liouville fractional derivative
The Riemann-Liouville derivative is the basis to define most fractional derivatives; it generalizes the Cauchy’s formula for derivatives of high order. For an
Following Riemann’s notion of defining fractional derivatives as the integer order derivative of an fractional integral, we have the left and right derivative proposal as follows:
with , i.e., .
It is also possible to prove that the semigroup propriety about the order of integral operators (i.e., for ) is achieved:
For the derivatives, we have
For , the following relationships are valid:
where and in analogy for the right derivative.
All these properties can be used in the phenomena modeling and its solution; such models have shown to improve usual approaches. However, when using equations with Riemann-Liouville type fractional derivatives, the initial conditions cannot be interpreted physically; a clear example is that the derivative Riemann-Liouville of a constant is not zero, contrary to the impression that the derivatives gives a notion about the change that the function experiences when advancing in the time or to modify its position. This was the motivation for another definition that is better coupled with physical interpretations; this is the derivative of Caputo type.
4.2. Caputo fractional derivative
Michele Caputo  published a book in which he introduced a new derivative, which had been independently discovered by Gerasimov (1948). This derivative is quite important, because it allows for understanding initial conditions, and is used to model fractional time. In some texts, it is known as the Gerasimov-Caputo derivative.
where and , i.e., for and for . And if
The connection between Caputo and Riemann derivatives is given by the relations
However, for , we have
On the other hand, if and
The Caputo derivatives behave like inverted operators for the left Riemann-Liouville fractional integrals and , if and
On the other hand, if and , then for good conditions for
In particular if, , then
In his early articles and several after that, Caputo used a Laplace transformed of the Caputo fractional derivative, which is given by
When , then
These derivatives can be defined over the whole real axis resulting in the expressions:
5. Fractal geometry and fractional calculus
The phenomenon of anomalous diffusion is mathematically modeled by a fractional partial differential equation. The parameters of this equation are uniquely determined by the fractal dimension of the underlying object.
There are some results that show the relationship between fractals and fractional operators ; two of the most important that motivated the particular study of the equations to determine the pressure deficit in oil wells are highlighted below.
5.1. Cantor’s Bars and fractional integral
In 1992, Nigmatullin  presents one of the most distinguished contributions to the search of the concrete relationship between the fractal dimension of a porous medium and the order of the fractional derivative to model a phenomena through such a medium; in this, he achieves the evolution of a physical system of a Cantor set type.
In his research, Nigmatullin proposes a relationship between the fractal dimension of a Cantor type set and the order of a fractional integral of the Riemann-Liouville type. The systems he considers are named phenomena with “memory.” The use of fractional derivatives given by assuming a transference function
Through the result of distribution values, he establishes the relation:
Thus, assuming a porous medium with a
The initial results were strongly questioned by different authors, including Roman Rutman (see Refs. [15, 16]), who asserts that the relation is so artificial. However, recent works suggest that Nigmatullin’s statements are not far from reality, but it is necessary to reduce the set of functions and that of fractals for which the necessary convergence is fulfilled.
6. Fractional calculus for modeling oil pressure
In this section, the Equation Continuity which follows from the law of conservation of mass is established. Darcy’s law is used to relate fluid motion to pressure and gravitational gradients. The combination of the Continuity Equation and Darcy’s Law leads to a heat-conducting differential equation in mathematical physics describing the transfer of the fluid. We obtain a system formed by three partial differential equations, one for each fluid. This multiphase system must be solved considering the relevant boundary and initial conditions .
In the particular case of naturally fractured reservoirs (see Refs. [1, 2]), usually it is possible to discern three porosity types: matrix, fracture, and vugs; with this conception, it is accepted that the three porosities have associated a solid phase, and with this both Continuity Equation and Darcy’s law can be expressed for each fluid in each geometrical media. If we only consider oil (monophasic) in a isotropic and saturated media, we can obtain a three equations system; for this, we begin with standard continuity equation and standard Darcy’s law, respectively (see Ref. ):
General fluid transfer equation results combining the formulas in Eq. (73):
This differential equation contains two dependent variables, namely the humidity content and fluid pressure, but they are related. For this reason, the saturation
6.1. Triadic media
The porous media is considered to be formed by three porous media: the matrix, fractured media, and vuggy media. The total volume of the porous media (
each of the porous media contains solids and voids so that
The porous medium as everything contains solids and voids, with the following relations:
The volume fraction occupied by the matrix is defined as (
The porosity of the porous media (
From the above equations, we deduce the relation between the porosities:
When the empty space contains fluid partially, the total volumetric content of the fluid (
which is reduced to Eq. (85) when the porous medium is fully saturated with fluid. It is satisfied: . The relation between the total volumetric flow of the fluid per unit area in the porous medium (
The continuity equations for the matrix, the fractured medium, and the vuggy media considering Eq. (86) acquire the form
Darcy’s law for the matrix, the fractured medium, and the vuggy media, takes the form
where Φ represents the potential of Kirchoff which is generically defined as
If there is no fluid gain or loss in the porous medium, then ϒ = 0 and in consequence:
The system of differential equations is defined as follows:
The contributions of fluid in each porous medium are modeled with the following relations:
6.2. Monophasic flow saturated in triadic media
In the case of the monophasic flow saturated in triadic means, the continuity equations in each porous medium can be written as follows:
Darcy’s law for each porous media takes the form
The substitution of Darcy’s law in the continuity equation leads to the following equations:
When the fluid is considered at constant density and viscosity and the means of constant permeability, with
6.3. Triple porosity and triple permeability model
The porosity of each medium has been defined as the volume of the space occupied by the medium. However, the porosity can be defined as the volume of empty space in each medium with respect to the volume of the total space occupied by the porous medium as a whole. These new porosities will be denoted with subscripts in lowercase letters and clearly have
In an analogous way, the corresponding Darcy´s flow can be defined in each medium:
Eq. (113) implies that the permeability of the Darcy’s law in each medium is defined as
in which this system constitutes a triple porosity and triple permeability model. In polar coordinates, the system reduces to
6.4. Dimensionless variables
Now we will give a process of dimensionlessness to better manage the variables. This is a technique commonly used to make the parameters or variables in an equation having no units, bring to a range the possible values of a variable or constant in order that its value is known, and in this way, more manipulable.
Substituting derivatives by Caputo fractional derivativeswith , and by Riemann-Liouville complementary derivative, i. e., with infinite limit of integration, (also called Weyl derivative) , with , ; ; rational numbers.
The choice of the derivatives, Caputo and Riemann-Liouville (Weyl), obeys the nature of the problem and the ease with which they can be manipulated.
The monophase flow model with triple porosity and triple permeability is expressed as follows: For the matrix
for fracture media
We reduce this system by applying semigroup properties with respect to the order of the Weyl derivative, assuming: and . Let , putting
The above approach can be solved by numerical methods as finite differences along with a predictor-corrector, such as Daftardar-Gejji works, for example in  and compared with previous ones, such as that presented by Camacho et al. [18, 28], the approximations are significantly improved. However, there is still work to be completed; the optimal solution method has not been found and the best way to determine the appropriate order, so far numerical methods, has been used to estimate the order that best approximates measurements.
The application of the fractional calculation can be very useful for the modeling of anomalous diffusion phenomena in which the fractal structure better reflects the real conditions of the medium, as it is the case of the reservoirs in which because of its very nature it is difficult to find a structure Euclidian.