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Chapter 5: Numerical Methods for Solving Fractional PDEs in the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis chapter provides an extensive review of numerical methods tailored for solving fractionalorder 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 fractionalorder PDEs. Starting with the adaptation of classical numerical techniques, such as the Euler method and RungeKutta 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 RungeKutta (EFORK) methods, through examples and pseudocodes.

Chapter 4: Analytical Solutions of Fractional PDEs, in the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis chapter introduces the analytical solutions of fractional partial differential equations (PDEs), focusing on their significance in modeling transport phenomena that exhibit anomalous diffusion or nonlocal dynamics. The chapter begins by exploring powerseries methods for solving fractional differential equations (FDEs), illustrating the technique through examples such as gas flow in porous media and boundarylayer flow. It then transitions to the Adomian Decomposition Method (ADM), a semianalytical approach that simplifies the solution of nonlinear, fractionalorder differential equations. Through detailed examples, including the timefractional convectionconduction equation, the timefractional diffusionreaction equation, time and timespace fractional advectiondiffusion equation, the chapter showcases the versatility and efficiency of ADM in handling complex fractional PDEs. Finally, the diffusion equation with time CaputoFabrizio fractional derivative has been solved using Laplace transform method.

Chapter 3: Fundamentals of Fractional Modeling of Fluid Flow, of the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaAs 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 fractionalorder formulations for conserving mass and momentum. The chapter also introduces the derivation of the fractional energy conservation equation, including models for heat conduction, heat convectionconduction, 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.

Chapter 12: Fractional Models in Renewable Energy Systems, in the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis chapter explores the application of fractional calculus in modeling and controlling renewable energy systems, including wind energy, solar energy, and biochemical reactions. For wind turbines, it has been used in the modeling of turbulent flows and improves control systems through fractionalorder PID controllers. In solar energy, it refines the thermal dynamics models of solar heating systems, while in biochemical processes, it offers a detailed analysis of reaction kinetics in anaerobic digestion.

Chapter 2: Fundamentals of Fractional CalculusThis 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 MittagLeffler 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 RiemannLiouville, Caputo, GrünwaldLetnikov, CaputoFabrizio, and AtanganaBaleanu 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.

Traditional Modeling of Fluid Flow and Transport PhenomenaThis 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 nonNewtonian 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 twophase and multiphase flows within porous structures.

Chapter 2: Fundamentals of Fractional Calculus, of the book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis 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 MittagLeffler 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 RiemannLiouville, Caputo, GrünwaldLetnikov, CaputoFabrizio, and AtanganaBaleanu 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.

Traditional Modeling of Fluid Flow and Transport Phenom enaThis 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 equa tions, 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 nonNewtonian 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 twophase and multiphase flows within porous structures.

Chapter 1: Traditional Modeling of Fluid Flow and Transport Phenomena, in the Book: Fractional Modelling of Fluid Flow and Transport PhenomenaThis 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 nonNewtonian 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 twophase and multiphase flows within porous structures.

10  Other nanoparticles transport interactionsThis chapter presents some the most special aspects of nanoparticles interactions, e.g., nanoparticle cotransport and/or interaction with nonaqueous phase liquids (NAPLs) and nanoparticles–polymers transport in porous media and nanoparticles associated with heat transfer. The concept of stability of nanoparticles suspensions is discussed in Section 10.2, while Section 10.3 presents the nanoparticles interaction with NAPL transport. After that Section 10.4 introduces the polymer transport under magnetic field in porous media with analytical solutions. The nanoparticles interactions with heat transfer are discussed in Section 10.5. Finally, the nanofluids in boundary layer flow are discussed in Section 10.6 and similarity solutions are introduced in both analytical and numerical modes.

8  Magnetic nanoparticles transport in porous mediaThis chapter presents the mathematical modeling of magnetic nanoparticle transport with singlephase 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 twophase flow, followed by the development of a corresponding numerical solution. Finally, we introduce analytical solutions to the singlephase 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 singlephase case, the twophase case is provided in Section 8.5 with numerical solutions. Finally, analytical solutions are presented in Section 8.6.

9  Nanoferrofluids transport in porous mediaNanoferrofluids type is one of the prospective applications of nanoparticles that work with the magnetic field. This chapter will discuss nanoferrofluids transport in porous media. This chapter presents the properties of ferrofluids without repeating the main equations that presented in Chapter 8. The ferrofluids transport in singlephase flow has been introduced and possible analytical solution for some cases. After that, the modeling of nonisothermal ferrofluids transport in porous media has been established, and an appropriate numerical algorithm has been developed. Finally, the model and numerical method of ferrofluids transport in twophase flow are provided.

11  Machine learning techniques for nanoparticles transportMachine 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 opensource 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.

12  Applications of nanoparticles in porous mediaSome applications of using nanoparticle transport in porous media are presented in this chapter. The applications include using nanoparticles in enhanced oil recovery in addition to the very wide area of application of nanoparticles in the field of heat transfer. Moreover, the combination of nanoparticles and surfactants is also presented. Another very recent application is the harvesting of atmosphere water with aid of using nanoparticles. Also, we discussed the carbon dioxide capture by nanoporous materials and its sequestration in the geological underground. Another important application presented of nanofluid in porous media is the metal hydride hydrogen storage.

7  Nanoparticles transport in anisotropic porous mediaAnisotropy 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.

6  Nanoparticles transport in fractured porous mediaThis chapter discusses the modeling of flow fractured porous media. It covers the most common fundamental approaches and presents their physical, mathematical and numerical aspects. Several approaches are introduced including the dualcontinuum, boundary conditions, shape factor, and discrete fracture model (DFM). After that, we focus on the DFM for nanoparticle transport in singlephase flow and twophase flow. Numerical multiscale timesplitting scheme has been developed to solve the DFM model. Finally, the hybrid embedded fracture model has been discussed.

5: Iterative schemes and convergence analysisIterative 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 porewalls, and entrapped nanoparticles concentration in porethroats. In the case of the twophase flow, the saturation equation is also considered. Iterative methods are often employed to solve such kinds of timedependent complicated systems. The nonlinear iterative numerical scheme such as iterative Implicit Pressure Concentration, and the iterative Implicit Pressure Explicit Saturation–Implicit Concentration (IMPESIMC) has been introduced to solve the model under consideration. The iterative IMPIMC scheme is devoted to solving the problem of nanoparticles transport with singlephase flow in porous media, while the iterative IMPESIMC treats the twophase 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.

4  Temporal numerical discretization schemesThis chapter is concerned with the numerical methods frequently used for temporal discretization in the problems of nanoparticles transport in porous media. The forward and backward Euler difference schemes are presented in the following section. Then, the Courant–Friedrichs–Lewy (CFL) stability condition was introduced. After that, we discussed the possibility of using a multiscale timesplitting scheme. Also, we defined the relaxation factor and how it can be used with the CFL condition. Then, we presented the Implicit Pressure Implicit Concentration scheme as well as the Implicit Pressure Explicit Saturation Implicit Concentration scheme. Finally, a stability analysis for the Implicit Pressure Explicit Saturation scheme has been provided.

3  Spatial numerical discretization methods for nanoparticles transport in porous mediaNowadays, 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 cellcentered 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 (shiftingmatrix) 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.

2  Dimensional analysis and analytical solutionsAn analytical solution helps in the validation of numerical methods/solutions as well as the comprehension of mechanisms and physical effects. Analytical solutions of the problem of flow in porous media, magnetic flow in porous media, nanoparticles transport in porous media, and magnetic nanoparticles transport in porous media are addressed in this chapter. Also, dimensional analysis is vital in handling problems regardless of their actual dimensions, which is helpful in largescale problems such as hydrocarbon reservoirs. Therefore, this chapter also introduces a simplified onedimensional model of nanoparticle transport in porous media and its generalized nondimensional form that will be solved analytically and numerically to obtain physical insight.