A generalized kinetic model of the advection-dispersion process in a sorbing medium

. A new time-fractional derivative with Mittag-Leffler memory kernel, called the generalized Atangana-Baleanu time-fractional derivative is defined along with the associated integral operator. Some properties of the new operators are proved. The new operator is suitable to generate by particularization the known Atangana-Baleanu, Caputo-Fabrizio and Caputo time-fractional derivatives. A generalized mathematical model of the advection-dispersion process with kinetic adsorption is formulated by considering the constitutive equation of the diffusive flux with the new generalized time-fractional derivative. Analytical solutions of the generalized advection-dispersion equation with kinetic adsorption are determined using the Laplace transform method. The solution corresponding to the ordinary model is compared with solutions corresponding to the four models with fractional derivatives.


Introduction
The researchers in the agricultural and soil fields, have been interested in the effectiveness of agricultural chemicals, such as fertilizers, pesticides, etc., that are applied to soil for enhancing the crop grows. These concerns, along with possible contamination of groundwater have provided a major impulse to the study of solute transport in soils.
The movement of dissolved substances is significantly influenced by a large numberof physical, chemical and microbiological processes. Modeling solute transport requires extensive mathematical and physical knowledge, as well as highprecision experimental procedures.
Transport of a solute (a dissolved substance) depends on the magnitude and direction of the solvent flux (for example, the water flux). The determination of solute concentrations is not always straightforward, particularly if the solute is involved in partitioning between different phases or subject to transformations.
The standard transport mechanism is mathematically described by the fundamental advection-dispersion equation (convection-dispersion equation). To note that, the traditional advection-dispersion process is not always adequate to describe the solutes transport in soils.
The motion of a solute that undergoes adsorption by the soil requires a modified mathematical model, especially when several solute species are present that may participate in a few different reactions [1]. Wu et al. [2] have investigated a nonlinear adsorptive transport model through layered soil and developed an analytical solution to a one-dimensional transport problem. The obtained analytical solution is an exact solution for non-dispersive transport and it becomes an approximate one when the dispersion effects are included. The authors found that the shape and non-linearity of the adsorption isotherm could be a controlling tool on the transport characteristics. Kakur and Van Keer [3] proposed a new numerical algorithm to the mathematical model of convection diffusion and adsorption based on the relaxation method and on the method of characteristics. They proved the convergence of the method and applied it to a one-dimensional problem. Their results could be applied to the model of contaminant transport in porous media with different type of adsorption.
Van Kooten [4] developed a method to find solution of the transport equations for a kinetically adsorbing solute in a porous medium with the velocity field and dispersion coefficients depending on the spatial variables. Using the stochastic nature of the kinetic process of first-order, the author has proceeded to de decoupling of the advection-dispersion equation and the adsorption isotherm process. When the solution for a non-adsorbing solute is given, the proposed method provides an exact solution for the kinetically adsorbing solute. Uffink et al. [5] investigated the non-Gaussian spreading of solutes due to advection, dispersion and kinetic sorption (adsorption/desorption). By analyzing the behavior of a single particle and applying a random walk to describe advection/dispersion process plus a Markov chain to describe kinetic sorption, the authors obtained the set of partial differential equations of the model. The authors have shown that two spreading phenomena are active, namely, the Gaussian microdispersive spreading plus the kinetics-induced non-Gaussian spreading. Kurikami et al. [6] developed a modified diffusion-sorptionfixation model, based on the advection-dispersion equation and have applied the model to study the vertical migration of radiocesium in soils. The proposed model introduces kinetics for the reversible binding of radiocesium. They have tested the model's results by comparing them results to depth profiles measured in Fukushima Prefecture, Japan, since 2011. Their results have shown that the proposed model captures the long tails of the radiocesium distribution at large depths, which are caused by different rates for kinetic sorption and desorption.
Lee et al. [7] have investigated how two sorption kinetics of the first-order differ from each other. They compared the sorption parameters of both models estimated from experimental data obtained from the kinetic sorption test, and by simulating the breakthrough curves of a reactive chemical using a solute transport equation coupled with sorption kinetics. Liu et al. [8] studied a fractional diffusion equation with variable coefficients by considering a non-local mathematical model based on Caputo time-fractional derivative, respectively on the Riemann-Liouville space-fractional derivatives. Solutions of the studied problem have been determined with the variational iteration method.
An interesting study of the exothermic reactions model with constant heat source in the porous media with strong memory effects was carried out by Kumar et al. [9]. The mathematical models used Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional operators to induce memory effects of exothermic reactions.
Kumar et al. [10] introduced a new fractional operator based on the Rabotnov fractional-exponential kernels. They provided some properties and applications of these operators.
Povstenko and Kyrylych [11] studied two generalized advection-diffusion equations using mathematical models with space-time-fractional derivatives. Caputo time-fractional derivative and Riesz fractional Laplacian are used in their study. Using the Laplace and Fourier transforms, the authors have determined fundamental solutions to the Cauchy and source problems. Luchko [12] has proved some interesting properties of the fundamental solutions to a multi-dimensional space-timefractional diffusion-wave equation using the Mellin-Barnes representation of the derived fundamental solutions.
In the present paper, we investigate a generalized mathematical model of the solute transport by considering the mass transport of chemicals in porous media with sorbtion as a part of the dispersion mechanism. In such models, the free phase chemical concentration is related to the sorbed chemical concentration in the medium.
On the other hand, in order to implement the memory effects of the diffusion process, a generalized form of the diffusive flux, based on the generalized Atangana-Baleanu time-fractional derivative is proposed.
First, we introduce the new time-fractional derivative whose kernel is the Mittag-Leffler function with one parameter. This new time-fractional derivative generates by customization three other known fractional derivatives, namely, the Atangana-Baleanu fractional derivative, Caputo-Fabrizio fractional derivative and Caputo fractional derivative. This important property of the new time-fractional derivative makes it possible the comparison between a solute transport based on the mathematical model with derivatives of integer order (the ordinary model) and four other models with different memory kernels. Some basic properties of the generalized time-fractional Atangana-Baleanu derivative are proved.
A mathematical model of the generalized solute transport is developed. Analytical solutions to the one-dimensional problem are determined using the Laplace transform. Solutions corresponding to the model based on the generalized Atangana-Baleanu fractional derivatives are particularized to obtain solutions for the advection-diffusion equation with the kernel of diffusive flux of Atangana-Baleanu, Caputo-Fabrizio and Caputo type. The solution corresponding to the ordinary advection-diffusion equation has been also obtained as a particular case. An application for the constant concentration on the boundaries is investigated by graphical illustrations. The obtained results lead to the choice of the best model for obtaining the optimal concentrations in a certain practical problem.

Formulation of the problem 2.1. The ordinary mathematical model
Consider the transport of a chemical species (the solute) in a spatial porous media. The solute transport may be highly affected by interactions between the solute and the solid matrix. At the macroscopic level, the balance equations for the solute species subject to arbitrary reactions are given as [3,4], where, ( , ) C X t is the concentration of the solute in the free phase, ( , ) S X t is the concentration in the adsorbed phase, ( , ) s J X t is the solute flux density vector, () RX is the rate of zero-order production at the point X , 1 k is the forward reaction rate, 2 k is the backward reaction rate, ( , ) v X t is the fluid velocity vector, () DX is the dispersion tensor, ( , ) adv J X t and ( , ) dif J X t are the advective, respectively dispersive components of the solute flux vector, and  is the gradient operator.
In the present article, we consider the one-dimensional problem defined for x e being the unit vector in the x-direction. In this case, the above equations become Introducing the non-dimensional variables into Eqs. (4)-(6) and neglecting the star notation, we obtain the problem written in dimensionless form, In the following, we aim to develop a mathematical model based on the generalized form of the Atangana-Baleanu time-fractional derivative. This model is suitable for particularizations to describe the memory effects with Caputo kernel, Caputo-Fabrizio kernel, and Atangana-Baleanu kernel. In the next section, we present the basic definitions and properties of the generalized Atangana-Baleanu time-fractional derivatives.
For the one-parameter Mittag-Leffler function, the following integral representations are useful [13]: Remark 1 (The Laplace transform [14] of function

Definition 1 (Caputo kernel). The function
The Laplace transform of Caputo kernel is given by Using (23), the kernel (22) can be defined for 1 therefore, where, ()   is Dirac's distribution.
and "  " denotes the convolution product.

Remark 2.
The following properties are obvious: is called Caputo-Fabrizio kernel. The Laplace transform of the Caputo-Fabrizio kernel is given by Using Eq. (29) we extend definition (28) for 1 therefore, (31) Remark 3. The following particular cases are true: therefore, Remark 4. Based on the properties (36) and (37), we obtain ( ) Remark 5. Even if the exponential function is a particular case of the one-parametric Mittag-Leffler function, the Caputo-fabrizio fractional derivative is not a particular case of the Atangana-Baleanu derivative defined above.
In this paper, we will define a generalized Atangana-Baleanu derivative which contains as particular cases Caputo fractional derivative, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives.

Definition 7 (The generalized Atangana-Baleanu kernel). The function
is called the generalized Atangana-Baleanu kernel. The Laplace transform of the kernel (40) is given by The following properties of the kernel (40) are easily deduced: therefore, (43) Remark 5. Using Eqs. (43) and (44), we obtain the following properties of the generalized Atangana-Baleanu time-fractional derivative: Associated with the generalized Atangana-Baleanu derivative, we define the following fractional integral operator: where the kernel 0 ( , ) t  is defined as Using the property (52), the fractional integral operator can be defined for 0  = .
Remark 6. The fractional integral operator has the following properties: Regarding the generalized Atangana-Baleanu derivative and associated fractional integral operator, we prove the proposition Proposition1. The following relationships are fulfilled: Proof. To demonstrate relations (54) we use the Laplace transform. We have The generalized fractional integral operator (50) contains the following particular cases: i.e. the well known Riemann-Liouville fractional integral operator [24].

The generalized mathematical model with fractional diffusion flux
In the following we will consider a generalized mathematical model based on the new definition of the Atangana-Baleanu fractional derivative, namely, we will consider the non-dimensional diffusive flux having the following definition: where, , () In the following we will deal with the dimensionless problem described by equations  To find analytical or semi-analytical solutions of the problem, the Laplace transform is employed. Applying the Laplace transform [14] to Eqs. (8) -(10)1 and (60), using the initial conditions (61), we obtain the transformed problem ( ) where, Along with the differential equation (65), the boundary conditions are Using the transformation we obtain for function ( , ) xp  the following differential equation

Transport of contaminant without spatial source
In this section we will determine the analytical solution of Eq. (69) in the assumption that no contaminant is added to system or extracted from the system, so that ( ) 0 Fx= . The boundary conditions are The solution of this problem is given by To find the solution ( , ) C x t we need the inverse Laplace transform of the function given by (73).
First, we will present the inverse Laplace transforms of some auxiliary functions that will be used further, namely: (0,1), 0.

 =
(85) As in the previous case, we analyze the following situations: whose the inverse Laplace transform is For the inverse Laplace transforms of function (92) we have: [28] .
Now, the inverse Laplace of the function (97) is given by whose inverse Laplace transform is   (105) whose inverse Laplace transform is given by  The above boundary conditions show that in the spatial position 0 x = the concentrations are kept at zero value, while in the spatial position 1 x = solute concentrations are kept at the constant value (1, ) 1, 0 To make a comparison between the values of the free phase concentration In Fig. 1 are presented profiles of the solute concentration ( , ) C x t corresponding to the models with generalized Atangana-Baleanu derivative, Atangana-Baleanu and the derivative of integer order. It is observed in Fig. 1 that the lowest concentration is obtained for the ordinary diffusive process. Generally, the model based on the Atangana-Baleanu derivative leads to a smaller concentration than the model with generalized Atangana-Baleanu memory kernel. However, there are values of the fractional parameter for which the model with generalized Atangana-Baleanu derivative leads to smaller values than the Atangana-Baleanu model. These changing behaviors of the solute concentration are generated by the generalized diffusive flux because different memory kernels lead to different dumping of the concentration gradient, therefore to a different diffusion process.

Conclusions
A generalized mathematical model of the solute transport by considering the mass transport of chemicals in porous media with sorbtion as a part of the dispersion mechanism has been investigated.
The memory effects of the diffusion process have been considered in the mathematical model introducing a new form of the diffusive flux based on the generalized Atangana-Baleanu time-fractional derivative.
The new time-fractional derivative/integral operators with Mittag-Leffler function as kernel have been presented together with their properties. The new timefractional derivative generates by customization three other known fractional derivatives, namely, the Atangana-Baleanu fractional derivative, Caputo-Fabrizio fractional derivative and Caputo fractional derivative. This important property of the new time-fractional derivative makes it possible the comparison between a solute transport based on the mathematical model with derivatives of integer order (the ordinary model) and four other models with different memory kernels.
A mathematical model of the generalized solute transport based on the new Atangana-Baleanu derivative has been formulated and studied. Analytical solutions to the one-dimensional problem are determined using the Laplace transform.
Solutions corresponding to the generalized model are particularized to obtain solutions for the advection-diffusion equation with the kernel of diffusive flux of Atangana-Baleanu, Caputo-Fabrizio and Caputo type. The solution corresponding to the ordinary advection-diffusion equation has been also obtained as a particular case. An application for the constant concentration on the boundaries is investigated by graphical illustrations.
The obtained results lead to the choice of the best model for obtaining the optimal concentrations in a certain practical problem