This chapter provides an extensive review of numerical methods tailored for solving fractional-order ordinary differential equations (ODEs), fractional partial differential equations (FPDEs), and systems of such equations. We discuss several numerical schemes, including explicit and implicit finite difference, Galerkin and mixed finite element, spectral element methods, and meshless methods, highlighting their application to both time and space fractional-order PDEs. Starting with the adaptation of classical numerical techniques, such as the Euler method and Runge-Kutta methods, to the fractional calculus framework, demonstrating their effectiveness through the introduction of fractional versions like the Fractional Forward Euler Method and Explicit Fractional Order Runge-Kutta (EFORK) methods, through examples and pseudocodes.

As the primary focus of this book is employing fractional modeling in the study of fluid dynamics and transport phenomena, this chapter presents the basics of fractional modeling of fluid flow. It begins with exploring fractional differential equations and discusses their advantages and challenges. Subsequent sections focus on the derivations of fractional-order formulations for conserving mass and momentum. The chapter also introduces the derivation of the fractional energy conservation equation, including models for heat conduction, heat convection-conduction, and general transport phenomena. Additionally, the discussion extends to fluid flow in porous media, featuring adaptations of Darcy’s Law that incorporate time and space memory effects and address anomalous diffusion processes.

This chapter presents the concepts of fractional calculus used in the field of fluid mechanics required in the rest of the book. The chapter begins with an overview and then introduces preliminary concepts crucial for understanding fractional calculus, including the Gamma and Beta functions, the Mittag-Leffler function, and various fractional operators. These foundational elements are essential for grasping the more complex aspects of fractional calculus in fluid mechanics. Moreover, the chapter examines different fractional derivative models, providing the basic definition of several key types. These include the Riemann-Liouville, Caputo, Grünwald-Letnikov, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. Each model is explored, offering insights into their unique characteristics and applications. Also, a significant portion of the chapter is dedicated to the Laplace transform, which covers its definition, basic principles, and properties, along with a list of common Laplace transforms and techniques for applying the inverse Laplace transform. This comprehensive coverage equips readers with the tools to use the Laplace transform in various contexts of fractional calculus. The chapter ends with exercises designed to reinforce the concepts covered.

This chapter presents the concepts of fractional calculus used in the field of fluid mechanics required in the rest of the book. The chapter begins with an overview and then introduces preliminary concepts crucial for understanding fractional calculus, including the Gamma and Beta functions, the Mittag-Leffler function, and various fractional operators. These foundational elements are essential for grasping the more complex aspects of fractional calculus in fluid mechanics. Moreover, the chapter examines different fractional derivative models, providing the basic definition of several key types. These include the Riemann-Liouville, Caputo, Grünwald-Letnikov, Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives. Each model is explored, offering insights into their unique characteristics and applications. Also, a significant portion of the chapter is dedicated to the Laplace transform, which covers its definition, basic principles, and properties, along with a list of common Laplace transforms and techniques for applying the inverse Laplace transform. This comprehensive coverage equips readers with the tools to use the Laplace transform in various contexts of fractional calculus. The chapter ends with exercises designed to reinforce the concepts covered.

This chapter provides an introductory overview of the key concepts and principles in fluid mechanics. It begins by exploring the properties and classifications of fluids. We then present the fluid motion equations, covering the principles of mass, momentum, energy conservation, and solute transport. As common cases, the inviscid flow, Euler's equations, and Bernoulli's principle are included, illustrating fundamental concepts in fluid dynamics. The chapter also emphasizes the significance of dimensional analysis. This powerful tool simplifies complex fluid dynamics problems and helps identify parallels between disparate systems. Following this, we examine boundary layer theory, essential for understanding fluid behavior in proximity to solid surfaces. Additionally, the chapter introduces the concept of non-Newtonian fluids. Finally, we discuss the fundamentals of flow in porous media. This includes an overview of Darcy's law, various dispersion models, and the dynamics of two-phase and multiphase flows within porous structures.

This chapter presents the mathematical modeling of magnetic nanoparticle transport with single-phase flow in porous media under the effect of an external magnetic field. We developed a mathematical model for the transport of magnetic nanoparticles in a two-phase flow, followed by the development of a corresponding numerical solution. Finally, we introduce analytical solutions to the single-phase case. The rest of this chapter is organized as follows: Section 8.3 presents the mathematical modeling of the transport of magnetic nanoparticles; while Section 8.4 focuses on the single-phase case, the two-phase case is provided in Section 8.5 with numerical solutions. Finally, analytical solutions are presented in Section 8.6.

Machine learning is a branch of artificial intelligence concerned with creating and developing algorithms that enable computers to learn behaviors or patterns from empirical data. The aim of this chapter is the implementation of machine learning algorithms in predicting nanoparticle transport in the oil reservoir. We used Jupyter Notebook for the implementation, which utilizes Python programming language. Jupyter Notebook is an open-source web tool that allows you to write live code while creating statistics and machine learning models. This chapter contains selected machine learning techniques that can be used for nanoparticle transport in porous media. It starts with the fundamentals of a number of machine learning methods, followed by basic metrics that are frequently used. After that, we discuss datasets and their analysis. Finally, we explain how to implement machine learning techniques in the Jupyter Notebook environment using Python programming language.

Anisotropy of porous media is an essential feature in subsurface formations. In this chapter, nanoparticle transport in anisotropic porous media will be discussed. The next section presents the nature of the anisotropic porous media. After that, we introduce the mathematical modeling of the flow in anisotropic porous media. Moreover, the model of nanoparticle transport in anisotropic porous media has been developed. Then, the numerical techniques that are appropriate for anisotropic porous media have been discussed, particularly the multipoint flux approximation, followed by a numerical example.

Iterative approaches are frequently used to solve intricate coupled and highly nonlinear systems. As stated in the previous chapters the mathematical model that governed the nanoparticles transport in porous media consists of equations of pressure, nanoparticles concentration, deposited nanoparticles concentration on the pore-walls, and entrapped nanoparticles concentration in pore-throats. In the case of the two-phase flow, the saturation equation is also considered. Iterative methods are often employed to solve such kinds of time-dependent complicated systems. The nonlinear iterative numerical scheme such as iterative Implicit Pressure Concentration, and the iterative Implicit Pressure Explicit Saturation–Implicit Concentration (IMPES-IMC) has been introduced to solve the model under consideration. The iterative IMP-IMC scheme is devoted to solving the problem of nanoparticles transport with single-phase flow in porous media, while the iterative IMPES-IMC treats the two-phase flow case. This chapter focuses on the iterative methods and investigates their theoretical and numerical convergence. Moreover, a theoretical foundation for the convergence of the iterative approach has been proved using the mathematical induction method.

Nowadays, computational methods are becoming increasingly more of a third science research means, parallel with experimental and theoretical methods. Especially in oil engineering and the research of groundwater flow and transport phenomena, numerical simulation is turning the most essential method, due to the fast development of computers. Generally, when adopting numerical simulation to research problems, the first step is establishing a mathematical model according to some physical laws of the problems. The second procedure is discretizing the mathematical model, followed by the third step, which is to code and run it on the computer to get the results. Finally, we can understand the original problem through simulation results. In this chapter, we introduce numerical methods that will be used for spatial discretization in nanoparticle transport in porous media. This chapter starts with mesh generation using MATLAB, including uniform and nonuniform 1D/2D/3D grids. After that, we introduce the cell-centered finite difference method (CCFD), including the discretization of the pressure equation, Darcy's law, and how to treat the boundary conditions. Then, the vectored implementation (shifting-matrix) of the CCFD method was presented. Therefore, the harmonic mean of permeability and transmissibility matrices has been listed. Moreover, the finite element method (FEM) is discussed by highlighting its discretization and weak formulation. The theoretical foundation of the FEM requires presenting Raviart–Thomas space. Also, the mixed FEM has been discussed with some numerical examples as it is essential in solving partial differential equations (PDEs) that govern the flow and transport in porous media.