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