Mathematical Analysis of Groundwater Flow Models

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Editor
Contributors
Chapter 1: Analysis of the Existing Model for the Vertical Flow of Groundwater in Saturated–Unsaturated Zones
	1.1 Introduction
	1.2 Background Review
	1.3 Governing Saturated Groundwater Flow Equation
		1.3.1 Analytical Solution Using the Integral Transform
		1.3.2 Analytical Solution Using the Method of Separation of Variables
	1.4 Numerical Solution
		1.4.1 Numerical Solution Using the Forward Euler Method (FTCS)
		1.4.2 Numerical Solution Using the Backward Euler Method (BTCS)
		1.4.3 Numerical Solution Using the Crank–Nicolson Method
	1.5 Numerical Stability Analysis
		1.5.1 Stability Analysis of a Forward Euler Method (FTCS)
		1.5.2 Stability Analysis of a Backward Euler Method (BTCS)
		1.5.3 Stability Analysis of the Crank–Nicolson Method
	1.6 Governing Unsaturated Groundwater Flow Equation
		1.6.1 Numerical Solution for the Unsaturated Groundwater Flow Model
	1.7 Numerical Simulations
	1.8 Conclusion
	References
Chapter 2: New Model of the Saturated–Unsaturated Groundwater Flow with Power Law and Scale-Invariant Mean Square Displacement
	2.1 Introduction
	2.2 Numerical Solution for the Saturated–Unsaturated Zone Using the Caputo Fractional Derivative
		2.2.1 Numerical Solution of the Caputo Fractional Derivative
		2.2.2 Numerical Solution of the 1-d Saturated–Unsaturated Groundwater Flow Equation Using the Caputo Fractional Derivative
			2.2.2.1 Numerical Solution of the 1-d Saturated Groundwater Flow Equation Using the Caputo Fractional Derivative
	2.3 Numerical Solution of the New Saturated–Unsaturated Groundwater Flow Model Using the New Numerical Scheme
		2.3.1 Numerical Solution of the Saturated Zone Model Using the New Numerical Scheme
		2.3.2 Numerical Solution of the Unsaturated Zone Using the New Numerical Scheme
	2.4 Conclusion
	References
Chapter 3: New Model of the 1-d Unsaturated–Saturated Groundwater Flow with Crossover from Usual to Confined Flow Mean Square Displacement
	3.1 Introduction
	3.2 The Caputo–Fabrizio Fractional-Order Derivative
	3.3 Governing Equation
	3.4 Numerical Solutions for the Saturated–Unsaturated Zone Using the Caputo–Fabrizio Fractional Derivative
		3.4.1 Numerical Solution for the Saturated Zone Using the Caputo–Fabrizio Fractional Derivative
		3.4.2 Stability Analysis Using Von Neumann
		3.4.3 Numerical Solution for the Unsaturated Zone Using Caputo–Fabrizio Fractional Derivative
	3.5 Conclusion
	References
Chapter 4: A New Model of the 1-d Unsaturated–Saturated Groundwater Flow with Crossover from Usual to Sub-Flow Mean Square Displacement
	4.1 Introduction
	4.2 A-B Derivative with Fractional Order
	4.3 Numerical Solution of the Saturated–Unsaturated Groundwater Flow Equation Using the A-B Fractional Derivative
		4.3.1 Numerical Solution of the Saturated Zone Using the A-B Fractional Derivative
		4.3.2 Numerical Solution of the Unsaturated Zone Using the A-B Fractional Derivative
		4.3.3 Numerical Solution of the Saturated–Unsaturated Groundwater Flow Equation Using the Ghanbari–Atangana Numerical Scheme
	4.4 Conclusion
	References
Chapter 5: New Model of the 1-d Saturated–Unsaturated Groundwater Flow Using the Fractal-Fractional Derivative
	5.1 Introduction
	5.2 Numerical Solution of the New Saturated–Unsaturated Groundwater Flow Model Using the Fractal Derivative
		5.2.1 Numerical Solution for the 1-d Saturated Zone Using the Fractal-Fractional Derivative
		5.2.2 Numerical Solution of the 1-d Unsaturated Zone Using the Fractal-Fractional Derivative
	5.3 Numerical Simulations, Results and Discussion
	5.4 Conclusion
	References
Chapter 6: Application of the Fractional-Stochastic Approach to a Saturated–Unsaturated Zone Model
	6.1 Introduction
	6.2 Application of the Stochastic Approach
		6.2.1 The Mean and Variance of the Hydraulic Conductivity
		6.2.2 The Mean and Variance of the Specific Storage
		6.2.3 The Stochastic 1-D Saturated–Unsaturated Groundwater Flow Equation
	6.3 Application of the Fractional-Stochastic Approach
		6.3.1 Stochastic Differential Equation Using the Caputo Fractional Derivative
			6.3.1.1 Explicit Forward Euler Method
			6.3.1.2 Implicit Backward Euler Method
			6.3.1.3 Implicit Crank–Nicolson Method
			6.3.1.4 New Model of the Unsaturated Zone in the Caputo Sense
		6.3.2 Stochastic Differential Equation Using the Caputo–Fabrizio Fractional Derivative
		6.3.3 Stochastic Differential Equation Using the Atangana-Baleanu Fractional Derivative
	6.4 Conclusion
	References
Chapter 7: Transfer Function of the Sumudu, Laplace Transforms and Their Application to Groundwater
	7.1 Introduction
	7.2 Application of the Laplace Transform to the Saturated Groundwater Equation
	7.3 Application of the Sumudu Transform to the Saturated Groundwater Equation
	7.4 Bode Plots of the Laplace and Sumudu Transform
	7.5 Conclusion
	References
Chapter 8: Analyzing the New Generalized Equation of Groundwater Flowing within a Leaky Aquifer Using Power Law, Exponential Decay Law and Mittag–Leffler Law
	8.1 Introduction
	8.2 Power Law Operators
		8.2.1 Riemann–Liouville Fractional Derivative
		8.2.2 Caputo Fractional Derivative
			8.2.2.1 Applying the Crank–Nicolson Scheme into the Classical New Groundwater Equation of Flow within a Leaky Aquifer
				8.2.2.1.1 Stability Analysis
			8.2.2.2 Applying the New Numerical Approximation Compiled by Atangana and Toufik
	8.3 Exponential Decay Law
		8.3.1 Caputo–Fabrizio Fractional Derivative
			8.3.1.1 Numerical Approximation Using the Adam–Bashforth Method
				8.3.1.1.1 Stability Analysis Using the Von Neumann Method
	8.4 Mittag–Leffler
		8.4.1 Mittag–Leffler Special Function and Its General Form
			8.4.1.1 Applying the Atangana–Baleanu (A–B)Fractional Derivative
				8.4.1.1.1 Stability Analysis
	8.5 Simulations
		8.5.1 Caputo Numerical Figures and Interpretation
	8.6 Conclusion
	References
Chapter 9: Application of the New Numerical Method with Caputo Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations
	9.1 Introduction
	9.2 Definitions in Terms of Differentiation
	9.3 New Numerical Method with Caputo Fractal-Fractional Derivative by Atangana and Araz
		9.3.1 Application of the New Caputo Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equation: Scenario 1
		9.3.2 Application of the New Caputo Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equation: Scenario 2
	9.4 Simulation
	References
Chapter 10: Application of the New Numerical Method with Caputo–Fabrizio Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations
	10.1 Introduction
	10.2 Definitions: Fractal-Fractional Derivative in Caputo–Fabrizio Sense
	10.3 The New Numerical Scheme for Ordinary Differential Equations and Partial Differential Equations with Caputo–Fabrizio Fractional Derivative by Atangana and Araz
	10.4 Discretizing Using the Caputo–Fabrizio Derivative and Applying the Numerical Scheme Given Above on the Self-Similar Leaky Aquifer Equation Scenario 1
	10.5 Implementation of Caputo–Fabrizio Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equation Scenario 2
	10.6 Simulations and Interpretation
	10.7 Conclusion
	References
Chapter 11: Application of the New Numerical Method with Atangana–Baleanu Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations
	11.1 Introduction
	11.2 Mittag-Leffler Law Type
	11.3 Numerical Scheme: Using Atangana–Baleanu Fractal-Fractional Derivative
	11.4 Implementation of Atangana–Baleanu Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equation Scenario 1
	11.5 Implementation of Atangana–Baleanu Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equation Scenario 2
	11.6 Simulations and Interpretation
	11.7 Conclusion
	References
Chapter 12: Analysis of General Groundwater Flow Equation within a Confined Aquifer Using Caputo Fractional Derivative and Caputo–Fabrizio Fractional Derivative
	12.1 Introduction
	12.2 Analysis of General Groundwater Flow with Caputo Fractional Derivative
	12.3 Analysis of General Groundwater Flow Equation with Caputo–Fabrizio Fractional Derivative
		12.3.1 Properties and Applications of Caputo–Fabrizio Fractional Derivative
		12.3.2 Analysis of General Groundwater Flow with Caputo–Fabrizio Fractional Derivative
	12.4 Numerical Simulations and Discussion
	12.5 Conclusion
	References
Chapter 13: Analysis of General Groundwater Flow Equation with Fractal Derivative
	13.1 Introduction
	13.2 Properties of Fractals
	13.3 Analysis of General Groundwater Flow With Fractal Derivative
	13.4 Numerical Simulations and Discussion
	13.5 Conclusion
	References
Chapter 14: Analysis of General Groundwater Flow Equation with Fractal-Fractional Differential Operators
	14.1 Introduction
	14.2 Application of Fractal-Fractional Derivative
		14.2.1 Analysis with Atangana–Baleanu Fractal-Fractional Derivative
		14.2.2 Analysis with Caputo Fractal-Fractional Derivatives
	14.3 Numerical Simulation and Discussion
	14.4 Conclusion
	References
Chapter 15: A New Model for Groundwater Contamination Transport in Dual Media
	15.1 Introduction
	15.2 Groundwater Contamination
	15.3 Contamination Transport in Dual Media
	15.4 Derivation of Equations and Numerical Analysis
	15.5 Relationship Between Hydraulic Conductivity and Intrinsic Permeability
	15.6 Hydrodynamic Dispersion
	15.7 Retardation Factor
	15.8 Groundwater Transport in Fracture
	15.9 Solving for an Aperture
	15.10 Uniqueness of the Proposed Equations
	15.11 Numerical Analysis of System of Equations
		15.11.1 Solving 1-d Diffusion with Advection for Steady Flow
	15.12 Stability Analysis Using von Neumann’s Method
	15.13 Conclusion
	References
Chapter 16: Groundwater Contamination Transport Model with Fading Memory Property
	16.1 Introduction
	16.2 Introducing a Caputo–Fabrizio Operator into Matrix–Fracture Equations
	16.3 Caputo and Fabrizio Derivative
	16.4 Laplace Transform
	16.5 Applying the Laplace Transform Technique to the Caputo–Fabrizio Integral
	16.6 Numerical Approximation
	16.7 Numerical Approximation of Caputo–Fabrizio Derivative
	16.8 Numerical Approximation of Caputo–Fabrizio Integral
	16.9 Model with Caputo–Fabrizio
	16.10 Conclusion
	References
Chapter 17: A New Groundwater Transport in Dual Media with Power Law Process
	17.1 Introduction
	17.2 Introducing the Caputo Operator into the Matrix–Fracture Equations
	17.3 Riemann–Liouville Power Law
	17.4 Mittag-Leffler Law
	17.5 Caputo Derivative
	17.6 Caputo Derivative Integral and Applying the Laplace Transform
	17.7 Numerical Approximation of the Caputo Derivatives
	17.8 Numerical Approximation of Integrals
	17.9 Lagrange Approximation
	17.10 Model with Power Law Process
	17.11 Conclusion
	References
Chapter 18: New Groundwater Transport in Dual Media with the Atangana–Baleanu Differential Operators
	18.1 Introduction
	18.2 Introducing Atangana–Baleanu Operators into the Matrix–Fracture Equations
	18.3 Atangana–Baleanu Derivative and Integral
	18.4 Laplace Transform
		18.4.1 Applying the Laplace Transform Technique to the Atangana–Baleanu Integral
	18.5 Numerical Approximation
		18.5.1 Numerical Approximation of the Atangana–Baleanu Derivative
		18.5.2 Numerical Approximation of the Atangana–Baleanu Integral
	18.6 Model with Atangana–Baleanu
	18.7 Conclusion
	References
Chapter 19: Modeling Soil Moisture Flow: New Proposed Models
	19.1 Introduction
	19.2 The Unsaturated Flow Model
	19.3 Methods and Materials
		19.3.1 Development of a Linear Unsaturated Hydraulic Conductivity Model
			19.3.1.1 The Linear Unsaturated Flow Model
		19.3.2 The Exact Solution to Richards Equation
		19.3.3 Numerical Analysis
			19.3.3.1 Numerical Analysis of Richards Equation Combined with Pre-Existing Nonlinear Models
				19.3.3.1.1 Crank–Nicolson Scheme
				19.3.3.1.2 Laplace Adams–Bashforth Scheme
			19.3.3.2 Numerical Analysis of the Proposed Linear Model
				19.3.3.2.1 Crank–Nicolson Finite-Difference Approximation Scheme
				19.3.3.2.2 Laplace Adams–Bashforth Scheme
		19.3.4 Numerical Stability Analysis
			19.3.4.1 Crank–Nicolson Finite-Difference Approximation Scheme
				19.3.4.1.1 The Laplace Adams–Bashforth Scheme
	19.4 Numerical Simulations
		19.4.1 Results and Discussion
	19.5 Conclusion
	References
Chapter 20: Deterministic and Stochastic Analysis of Groundwater in Unconfined Aquifer Model
	20.1 Introduction
	20.2 Deterministic Approach
	20.3 Stochastic Approach
	20.4 Numerical Approximation
	20.5 Analysis of the Deterministic Model
		20.5.1 Von Neumann Stability Analysis
	20.6 Analysis of the Stochastic Model
		20.6.1 Log-Normal Distribution
		20.6.2 Notation
		20.6.3 Probability Density Function
		20.6.4 Cumulative Distributive Function
		20.6.5 The Stochastic Model
		20.6.6 Von Neumann Stability Analysis
	20.7 NEW Numerical Scheme: Lagrange Polynomial Interpolation and the Trapezoidal Rule
	20.8 Numerical Simulations
	20.9 Results and Discussions
	20.10 Conclusion
	References
Chapter 21: A New Method for Modeling Groundwater Flow Problems: Fractional–Stochastic Modeling
	21.1 Introduction
	21.2 Fractional–Stochastic Modeling
	21.3 Numerical Solutions
		21.3.1 Numerical Solution of the New Model with Caputo Fractional Derivative
		21.3.2 Numerical Solution of the New Model with Caputo–Fabrizio Fractional Derivative
		21.3.3 Numerical Solution of the New Model with Atangana–Baleanu Fractional Derivative Caputo Sense
		21.3.4 Numerical Stability Analysis of the New Model Using the von Neumann Method
			21.3.4.1 Stability Analysis of the New Numerical Scheme for Solution of PDEs Derived in Terms of the Caputo–Fabrizio Fractional Derivative
			21.3.4.2 Stability Analysis of the New Numerical Scheme for Solution of PDEs Derived in Terms of the Atangana–Baleanu Fractional Derivative in the Caputo Sense
		21.3.5 Numerical Simulations
		21.3.6 Results and Discussions
	21.4 Conclusion
	References
Chapter 22: Modelling a Conversion of a Confined to an Unconfined Aquifer Flow with Classical and Fractional Derivatives
	22.1 Introduction
	22.2 Model Outline
	22.3 Numerical Solutions
		22.3.1 Adams–Bashforth Method (AB)
		22.3.2 Atangana–Gnitchogna Numerical Method (New Two-Step Laplace Adam-Bashforth Method)
		22.3.3 Numerical Solution for the Unconfined Aquifer Zone
	22.4 Application of the Non-Classic Atangana–Batogna Numerical Scheme
	22.5 Fractional Differentiation
		22.5.1 Application of the Atangana–Baleanu Derivative
		22.5.2 Stability Analysis
	22.6 Numerical Simulations
	22.7 Conclusion
	References
Chapter 23: New Model to Capture the Conversion of Flow from Confined to Unconfined Aquifers
	23.1 Introduction
	23.2 An Existing Model: The Moench and Prickett Model (MP Model)
	23.3 A New Mathematical Model to Capture the Conversion with Delay
	23.4 Derivation of an Exact and Numerical Solution of the New Model
	23.5 Applying the Laplace Transform to our Equation
	23.6 Linear Differential Equations
	23.7 New Numerical Scheme Using the Adams–Bashforth Method
	23.8 Von Neumann Stability Analysis
	23.9 Numerical Simulations
	23.10 Results and Discussion
	23.11 Conclusion
	References
Chapter 24: Modeling the Diffusion of Chemical Contamination in Soil with Non-Conventional Differential Operators
	24.1 Introduction
	24.2 Numerical Solutions for the Classical Case
		24.2.1 Forward Euler Numerical Scheme
		24.2.2 Backward Euler Numerical Scheme
		24.2.3 Crank–Nicolson Numerical Scheme
		24.2.4 Discretize the Convective-Diffusive Equation Based on Time
		24.2.5 Numerical Analysis with the Two-Step Laplace Adam–Bashforth Method
	24.3 Fractal Formulation
		24.3.1 Fractal Formulation of the Convective-Diffusive Equation
			24.3.1.1 Numerical Analysis with the Forward Euler Method
			24.3.1.2 Numerical Analysis with Backward Euler
			24.3.1.3 Numerical Analysis with a Crank–Nicolson Numerical Scheme
	24.4 Caputo–Fabrizio Fractional Differential Operator
		24.4.1 New Numerical Scheme That Combines the Trapezoidal Rule and the Lagrange Polynomial
	24.5 Numerical Simulations
	24.6 Conclusion
	References
Chapter 25: Modelling Groundwater Flow in a Confined Aquifer with Dual Layers
	25.1 Introduction
	25.2 Fractal Calculus
	25.3 Connecting Fractional and Fractal Derivations
	25.4 Numerical Solutions
	25.5 Stability Analysis
	25.6 Numerical Simulations
	25.7 Conclusion
	References
Chapter 26: The Dual Porosity Model
	26.1 Introduction
		26.1.1 Different Types of Aquifers
		26.1.2 Dual Media System
		26.1.3 Existing Mathematical Models of the Dual Media System
	26.2 Piecewise Modelling
		26.2.1 Numerical Solution Using the Newton Polynomial Scheme
	26.3 Stochastic Model
		26.3.1 Modified Model with the Stochastic Approach
	26.4 Application of Caputo–Fabrizio and Caputo Fractional Derivatives to the Piecewise Model
		26.4.1 Application of Caputo–Fabrizio and Caputo Derivative
	26.5 Numerical Simulations
	26.6 Results and Discussion
	26.7 Conclusion
	References
Chapter 27: One-Dimensional Modelling of Reactive Pollutant Transport in Groundwater: The Case of Two Species
	27.1 Introduction
	27.2 Conceptual Model and Mathematical Formulation
		27.2.1 Case Study: Solution Derived Using the Laplace Transform Method
		27.2.2 Solutions Obtained Using Green’s Function Method
		27.2.3 Solution of the Homogeneous System
		27.2.4 Solution of the Heterogeneous Part Using Green’s Function
	27.3 Numerical Analysis
		27.3.1 Crank–Nicolson Scheme
	27.4 Central Difference Reaction Constant
	27.5 Discretization Scheme for the Second Equation
	27.6 Stability Analysis
	27.7 Discussion
	27.8 Conclusion
	References
Chapter 28: Stochastic Modeling in Confined and Leaky Aquifers
	28.1 Introduction
	28.2 Groundwater Flow in Confined Aquifers
	28.3 A Groundwater Flow Equation for a Leaky Aquifer
	28.4 Analysis of Stochastic Models of Groundwater Flow: Confined and Leaky Aquifers
	28.5 Analysis of Stochastic Model of Groundwater Flow: Confined Aquifers
	28.6 Analysis of a Stochastic Model of Groundwater Flow: Leaky Aquifers
	28.7 Application of the Newton Method on Stochastic Groundwater Flow Models for Confined and Leaky Aquifers
		28.7.1 Application of the Newton Method to a Stochastic Theis’s Confined Aquifer
		28.7.2 Application of the Newton Method to a Stochastic Hantush’s Leaky Aquifer
		28.7.3 Stability of the Stochastic Confined Aquifer Equation
	28.8 Stability of the Stochastic Leaky Aquifer Equation
	28.9 Simulation
	28.10 Conclusion
	References
Index