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On a generalized diffusion equation arising in petroleum engineering
Advances in Difference Equations volume 2013, Article number: 349 (2013)
In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.
In order to modernize the public water service of the town of Dijon, France, Henry Darcy made several experiments and wrote an interesting document which soon formed the basis of the theory of fluid conduction. His work was about the download flow of water through filter sands. He established a (diffusion) equation, well known nowadays as ‘Darcy’s law’, which plays the same role as Fourier’s law in heat conduction theory and Ohm’s law in the electricity conduction theory (see [1, 2]). Of course, this law has its limitations. It is restricted to the situations where the flow through the pores can be modeled as Stokes flows. In particular, for large values of the Reynolds number (or at high values of flow rates) Darcy’s law is not valid anymore. It losses its accuracy. Apart from that, Darcy’s law has proved its usefulness in many applications such as the recovery of fuel from underground oil reservoirs. It defines the rock permeability (which controls the directional movement of the flow rate).
Many researchers have shown interest in this law and have worked extensively on extending and generalizing it to more complicated situations and in different contexts. For instance, they observed that this flow is ‘linear’: Darcy himself assumed that the flow is weak and as such the pressure drop is linearly related to the flow discharge rate. This motivated them to extend this law to the ‘nonlinear’ case. They also considered the case when the inertial forces (and/or the deformation of the solid) are not negligible as compared with those arising from viscosity. In particular, they generalized the law from Stokes to ‘non-Stokes’ flows in porous media. In fact, it has been found that more general physical formulations may be obtained by including new parameters and forces in the equation itself. Moreover, we may mention here the transition from a single phase to multiphase and from fixed to growing porous media.
In  Caputo modified the standard diffusion equation
by introducing a fractional derivative (in the sense of Caputo) in Darcy’s law  to take into account the memory effects that may prevail in the fluid. The equation became
where denotes the Caputo fractional derivative defined by
Similar equations have been derived in other situations like the one in  which describes anomalous subdiffusion particles (and p is the probability density function) and in  when studying the Stokes first problem (see also ). A natural question which arises is: what about the commonly used Riemann-Liouville fractional derivative,
or other types of fractional derivatives?
It is well known by now that the Caputo fractional derivative is often preferred over the Riemann-Liouville fractional derivative at least for two main reasons. The Caputo derivative allows us to use the usual initial data (like ), whereas the Riemann-Liouville derivative requires the prior knowledge of when the order of the equation is α (say ). In addition to that, the Caputo derivative of a constant is zero which is not the case for the Riemann-Liouville derivative.
In the present note, we consider a generalized fractional derivative which encompasses both fractional derivatives as special cases and provides a whole range of other types of fractional derivatives in between. This derivative was introduced by Hilfer in [7–9]
and therefore we name it after him. This kind of fractional derivative has already proved its usefulness (see [7–9]), and we are witnessing a growing interest in it. The parameter β when equal to zero gives the Riemann-Liouville derivative and when it takes the value one gives the Caputo derivative. For , we obtain many fractional derivatives interpolating these two types of well-known derivatives.
Using the Laplace-Fourier transform, we find the explicit solution for our equation under appropriate boundary conditions. We prove here that, in our case, all these derivatives when considered in diffusion equation (1) lead to similar solutions. Consequently, there is no major benefit in treating separately these derivatives unless there is a need to unify these treatments.
In the next section, we present some material needed in our arguments. In Section 3 we consider the modified diffusion equation with Neumann boundary conditions and indicate how to find an explicit solution. Section 4 contains the Dirichlet boundary conditions case. Finally, in Section 5 we treat the infinite reservoir case. Our results are illustrated by some graphs.
Definition 1 The Riemann-Liouville fractional integral of order α of f is defined by
when the right-hand side exists.
Definition 2 The Riemann-Liouville fractional derivative of order α of f is defined by
when the right-hand side exists. Note that
Definition 3 The Caputo fractional derivative of order α of f is defined by
(the prime here is for the derivative) when the right-hand side exists. Note that
The relationship between these two types of derivatives is given by the following theorem.
Theorem 1 
Definition 4 The Hilfer fractional derivative of f of order α and type β is defined by
whenever the right-hand side exists.
Note that when ,
which is the Riemann-Liouville fractional derivative (see Definition 2), and when ,
which is the Caputo fractional derivative (see Definition 3).
For , the Laplace transforms of these derivatives are given by
It is clear that the differences in these Laplace transforms are in the ‘initial’ data , and (this last one is natural initial data for the Hilfer derivative).
The natural space for the Hilfer fractional derivative is
However, here, since our problem is of order one, solutions must be much smoother.
3 The problem
We shall investigate the following linear generalized fractional diffusion problem:
The different parameters μ, ϕ, , k and A are positive constants which account for the viscosity, porosity, total compressibility, permeability and the area, respectively. This model describes the flow of oil in a finite reservoir. A closely related and interesting work is in  but for the normal form of the diffusion.
This problem, for the Caputo fractional derivative, was derived and studied in . Here we consider the more general (Hilfer) fractional derivative which covers both the Riemann-Liouville derivative and the Caputo derivative (in addition to many others for β in the interval ). Following the argument in , we introduce the dimensionless variables: , and , where .
The next two lemmas will, in particular, show that we obtain the same explicit solution using the Laplace transform as in the case of Caputo derivative (see ).
Lemma 1 If , then , , , , .
Proof From Definition 4, we have, for , ,
Let , we see that
Put , then clearly
Put also , it appears that
The proof is complete. □
Lemma 2 Assume that is continuous on for some , then
Proof Let , then
where M is a bound for on . □
In view of our problem (5), the solution must be at least absolutely continuous. Its continuity nearby zero implies by Lemma 2 that . Therefore all three derivatives have equal Laplace transforms (see (2)-(4)) (if in addition ). This fact together with Lemma 1 shows that the argument in  is valid without any changes.
Observe that the change of variables has lead to , which by Theorem 1 implies that both the Riemann-Liouville derivative and the Caputo derivative are equal. Another way to see this fact is from the next two lemmas.
Definition 5 A function (the space of summable functions) is said to have a summable fractional derivative if (the space of absolutely continuous functions).
Lemma 3 [, p.48]
If has a summable fractional derivative with , then we have
Lemma 4 [, p.95]
If , , then , .
Now, Definition 3, Definition 4 and Lemma 3 imply that
This holds if has a summable fractional derivative that is , which is the case when f itself is absolutely continuous on for some . In our case, is absolutely continuous. Lemma 2 ensures that . Therefore,
by Lemma 4 if is continuous. This is the case if and f is continuously differentiable because maps into (it also maps into , see ).
4 Dirichlet boundary conditions
Consider the problem
where , are continuously differentiable and is a forcing term.
Putting , with
we see that
In addition to that, we have
Therefore from (8)-(11), satisfies
Definition 6 The finite Fourier sine transform of , , is defined by
and its inverse is
Theorem 2 The following holds:
where is the finite Fourier cosine transform and
Applying first the Laplace transform to the equation in (12), we find
where and denote the Laplace transforms of P and , respectively.
In view of Lemma 2, equation (13) reduces to
Next, we apply the Fourier sine transform to (14) to get
where the ‘hat’ is for the Fourier sine transform. Therefore,
From the relation
it follows that
Furthermore, it is easy to see that
Using the inverse of the Fourier sine transform, we find
and then, inverting the Laplace transform, we obtain
In view of (15) we have
Now, appealing to the formula
we see that
Taking into account (20)-(22) in (19), we obtain
Finally, relation (18) leads to
and gives the explicit solution of problem (7).
5 Explicit solution in an infinite reservoir
In this section we consider a similar problem but in an infinite reservoir, which is the case in many practical situations.
The Laplace transform with respect to the time variable applied to the equation in (25) gives
As and , by the continuity of nearby (see Lemma 2), we get
Next, we apply the Fourier transform () to this last equation (26) to arrive at
Now, the inverse Laplace transform of (27) yields
and the inverse Fourier transform gives
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The authors are grateful for the financial support and the facilities provided by King Fahd University of Petroleum and Minerals and King Abdulaziz City for Science and Technology (KACST) NSTIP research grant No. 11-OIL1663-04.
The authors declare that they have no competing interests.
SA and NT solved the problems. RAG participated in establishing the explicit solution and carried out the numerical treatment. All authors read and approved the final manuscript.
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Al-Homidan, S., Ghanam, R.A. & Tatar, N. On a generalized diffusion equation arising in petroleum engineering. Adv Differ Equ 2013, 349 (2013). https://doi.org/10.1186/1687-1847-2013-349
- Fractional Derivative
- Diffusion Equation
- Caputo Fractional Derivative
- Caputo Derivative
- Fourier Sine