Mathematical Analysis of Groundwater Flow Models [1 ed.] 1032209941, 9781032209944

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Mathematical Analysis of Groundwater Flow Models

Mathematical Analysis of Groundwater Flow Models Edited by

Abdon Atangana

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 selection and editorial matter, Abdon Atangana; individual chapters, the contributors CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Atangana, Abdon, editor. Title: Mathematical analysis of groundwater flow models / edited by Abdon Atangana. Description: First edition. | Boca Raton : CRC Press, [2022] | Includes bibliographical references and index. Identifiers: LCCN 2021045765 (print) | LCCN 2021045766 (ebook) | ISBN 9781032209944 (hbk) | ISBN 9781032209951 (pbk) | ISBN 9781003266266 (ebk) Subjects: LCSH: Groundwater flow--Mathematical models. Classification: LCC GB1197.7 .M388 2022 (print) | LCC GB1197.7 (ebook) | DDC 551.4901/5118--dc23/eng/20211119 LC record available at https://lccn.loc.gov/2021045765 LC ebook record available at https://lccn.loc.gov/2021045766 ISBN: 978-1-032-20994-4 (hbk) ISBN: 978-1-032-20995-1 (pbk) ISBN: 978-1-003-26626-6 (ebk) DOI: 10.1201/9781003266266 Typeset in Times by SPi Technologies India Pvt Ltd (Straive)

Contents Preface................................................................................................................................................ix Editor..................................................................................................................................................xi Contributors.....................................................................................................................................xiii Chapter 1 A  nalysis of the Existing Model for the Vertical Flow of Groundwater in Saturated–Unsaturated Zones.................................................................................... 1 Rendani Vele Makahane and Abdon Atangana Chapter 2 N  ew Model of the Saturated–Unsaturated Groundwater Flow with Power Law and Scale-Invariant Mean Square Displacement...................................... 17 Rendani Vele Makahane and Abdon Atangana Chapter 3 New model of the 1-d Unsaturated–Saturated Groundwater Flow with Crossover from Usual to Confined Flow Mean Square Displacement........................ 37 Rendani Vele Makahane and Abdon Atangana Chapter 4 A New Model of the 1-d Unsaturated–Saturated Groundwater Flow with Crossover from Usual to Sub-Flow Mean Square Displacement................................ 49 Rendani Vele Makahane and Abdon Atangana Chapter 5 New Model of the 1-d Saturated–Unsaturated Groundwater Flow Using the Fractal-Fractional Derivative................................................................................. 65 Rendani Vele Makahane and Abdon Atangana Chapter 6 Application of the Fractional-Stochastic Approach to a Saturated–Unsaturated Zone Model................................................................................................................. 77 Rendani Vele Makahane and Abdon Atangana Chapter 7 Transfer Function of the Sumudu, Laplace Transforms and Their Application to Groundwater.......................................................................................................... 107 Rendani Vele Makahane and Abdon Atangana Chapter 8 Analyzing the New Generalized Equation of Groundwater Flowing within a Leaky Aquifer Using Power Law, Exponential Decay Law and Mittag–Leffler Law................................................................................................... 117 Amanda Ramotsho and Abdon Atangana

v

viContents

Chapter 9 Application of the New Numerical Method with Caputo Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations.......................................... 149 Amanda Ramotsho and Abdon Atangana Chapter 10 Application of the New Numerical Method with Caputo–Fabrizio Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations............ 167 Amanda Ramotsho and Abdon Atangana Chapter 11 Application of the New Numerical Method with Atangana–Baleanu Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations............ 181 Amanda Ramotsho and Abdon Atangana Chapter 12 Analysis of General Groundwater Flow Equation within a Confined Aquifer Using Caputo Fractional Derivative and Caputo–Fabrizio Fractional Derivative..... 199 Mashudu Mathobo and Abdon Atangana Chapter 13 Analysis of General Groundwater Flow Equation with Fractal Derivative............... 223 Mashudu Mathobo and Abdon Atangana Chapter 14 Analysis of General Groundwater Flow Equation with Fractal-Fractional Differential Operators................................................................................................ 243 Mashudu Mathobo and Abdon Atangana Chapter 15 A New Model for Groundwater Contamination Transport in Dual Media............... 261 Mpafane Deyi and Abdon Atangana Chapter 16 Groundwater Contamination Transport Model with Fading Memory Property........ 279 Mpafane Deyi and Abdon Atangana Chapter 17 A New Groundwater Transport in Dual Media with Power Law Process................. 289 Mpafane Deyi and Abdon Atangana Chapter 18 New Groundwater Transport in Dual Media with the Atangana–Baleanu Differential Operators................................................................................................ 303 Mpafane Deyi and Abdon Atangana Chapter 19 Modeling Soil Moisture Flow: New Proposed Models............................................. 319 Tshanduko Mutandanyi and Abdon Atangana

vii

Contents

Chapter 20 Deterministic and Stochastic Analysis of Groundwater in Unconfined Aquifer Model........................................................................................................... 353 Dineo Ramakatsa and Abdon Atangana Chapter 21 A New Method for Modeling Groundwater Flow Problems: Fractional-Stochastic Modeling................................................................................ 385 Mohau Mahantane and Abdon Atangana Chapter 22 Modelling a Conversion of a Confined to an Unconfined Aquifer Flow with Classical and Fractional Derivatives......................................................................... 413 Awodwa Magingi and Abdon Atangana Chapter 23 New Model to Capture the Conversion of Flow from Confined to Unconfined Aquifers................................................................................................. 437 Makosha Ishmaeline Charlotte Morakaladi and Abdon Atangana Chapter 24 Modeling the Diffusion of Chemical Contamination in Soil with Non-Conventional Differential Operators................................................................. 459 Palesa Myeko and Abdon Atangana Chapter 25 Modelling Groundwater Flow in a Confined Aquifer with Dual Layers..................489 Disebo Venoliah Chaka and Abdon Atangana Chapter 26 The Dual Porosity Model.......................................................................................... 515 Siphokazi Simnikiwe Manundu and Abdon Atangana Chapter 27 One-Dimensional Modelling of Reactive Pollutant Transport in Groundwater: The Case of Two Species.......................................................................................... 555 Hans Tah Mbah and Abdon Atangana Chapter 28 Stochastic Modeling in Confined and Leaky Aquifers.............................................. 587 Sarti Amakali and Abdon Atangana Index............................................................................................................................................... 619

Preface Groundwater is a natural resource present below the Earth’s surface; stored within rocks and soil pore spaces; and it makes up the largest portion of the existing body of freshwater on Earth and is highly useful to sustain life for both humans and other ecosystems. Groundwater is recharged from the surface; and is naturally discharged on to the surface through springs, seeps and/or rivers forming important sources of water known as oases. Because of its availability and accessibility, it is often withdrawn via boreholes for various uses including agricultural, industrial, mining, municipal and domestic. This limited resource has been for decades subjected to over-use, over-abstraction or overdraft leading to significant problems encountered by human users in different parts of the world. In addition, some groundwater sources are affected significantly by pollution, which reduces the availability of clean and healthy water. As a result several related environmental issues have been observed on a large scale around the world. The worst case scenario is that subsurface water pollution is hardly noticeable, and more difficult to purify, than pollution occurring in surface water. Hence the protection, regulation and monitoring of these sources of fresh water has recently become a focal point of human beings to ensure their sustainability. However, the realization of this process incorporates several steps needing to be performed. The first step includes data collection, and the second consists of analyzing collected data to identify which law the recorded data may follow. The last step is based on the conversion from observation to mathematical model, derivation of solutions, and finally the comparison of obtained solutions with experimental data. In the event, an agreement between collected data and the solution of mathematical models is obtained, and a good prediction can be performed. The book is devoted to discussions underpinning modeling groundwater problems such as flow in different geological formations, and artificial and natural recharge as well as the flow of contamination plumes. Existing mathematical models are analyzed and modified using new concepts to include the models in mathematical equations related to complexities of geological formations. Classical differential and integral operators are considered in some cases to model local behaviors observed in groundwater flow, recharge and pollution problems. Additionally, different types of nonlocal operators, including fractal derivative and integral, fractional derivatives and integral based on power law kernel, fractional derivative and integral based on exponential decay, and the generalized Mittag–Leffler functions are used to include into mathematical equations heterogeneous properties of subsurface formation. Thus several analytical and numerical techniques are accordingly utilized to derive exact and approximated solutions, in which numerical solutions are depicted using different software such as maple, Mathematica and Matlab. MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

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Editor Abdon Atangana works at the Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa as a full Professor. His research interests are, but not limited to, fractional calculus and applications, numerical and analytical methods, and modeling. He is the author of more than 250 research papers and four books in top tier journals of applied mathematics and groundwater modeling. He was elected Highly Cited Mathematician in 2019 and Highly Cited Mathematician with Crossfield Impact in 2020. He is a recipient of the World Academia of Science Award for Mathematics 2020. He serves as editor in top tier journals in various fields of study.

xi

Contributors Abdon Atangana Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa and Department of Medical Research, China Medical University Hospital China Medical University Taichung, Taiwan

Mohau Mahantane Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Amanda Ramotsho Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Mpafane Deyi Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Awodwa Magingi Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Palesa Myeko Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Dineo Ramakatsa Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Rendani Vele Makahane Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Disebo Venoliah Chaka Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Sarti Amakali Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Hans Tah Mbah Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Siphokazi Simnikiwe Manundu Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Mashudu Mathobo Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Tshanduko Mutandanyi Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

Makosha Ishmaeline Charlotte Morakaladi Institute of Groundwater Studies University of the Free State Bloemfontein, South Africa

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1

Analysis of the Existing Model for the Vertical Flow of Groundwater in Saturated–Unsaturated Zones Rendani Vele Makahane and Abdon Atangana University of the Free State, Bloemfontein, South Africa

CONTENTS 1.1 Introduction................................................................................................................................1 1.2 Background Review................................................................................................................... 2 1.3 Governing Saturated Groundwater Flow Equation.................................................................... 3 1.3.1 Analytical Solution Using the Integral Transform......................................................... 4 1.3.2 Analytical Solution Using the Method of Separation of Variables................................ 6 1.4 Numerical Solution.................................................................................................................... 8 1.4.1 Numerical Solution Using the Forward Euler Method (FTCS)..................................... 8 1.4.2 Numerical Solution Using the Backward Euler Method (BTCS).................................8 1.4.3 Numerical Solution Using the Crank–Nicolson Method............................................... 8 1.5 Numerical Stability Analysis..................................................................................................... 8 1.5.1 Stability Analysis of a Forward Euler Method (FTCS)................................................. 9 1.5.2 Stability Analysis of a Backward Euler Method (BTCS)............................................ 10 1.5.3 Stability Analysis of the Crank–Nicolson Method...................................................... 11 1.6 Governing Unsaturated Groundwater Flow Equation............................................................. 12 1.6.1 Numerical Solution for the Unsaturated Groundwater Flow Model........................... 13 1.7 Numerical Simulations............................................................................................................ 13 1.8 Conclusion...............................................................................................................................14 References......................................................................................................................................... 15

1.1 INTRODUCTION Since studying the interaction between geology and the movement of groundwater can be quite complex, a model expressing the nature of the system must be introduced. A model is viewed as an approximation and not an exact solution of the physical process; nonetheless, even as an approximation, it can be a useful investigation tool (Atangana & Botha, 2012, 2013). A groundwater model describing the movement of water in a porous media is defined in mathematical terms by combining the law of mass conservation and Darcy’s law. The advantage of this equation is that it can be used for the whole flow region and can handle both unconfined and confined saturated aquifers (Freeze, 1971). The resulting equation is linear for a saturated flow and nonlinear for the unsaturated flow (Nishigaki & Kono, 1980; Cheng & Gulliksson, 2003). This chapter focuses on the mathematical modeling of problems related to groundwater flow in the saturated–unsaturated zone. The main objective is to accomplish both analytical and numerical solutions for the classical saturated and

DOI: 10.1201/9781003266266-1

1

2

Mathematical Analysis of Groundwater Flow Models

unsaturated groundwater flow equation, where applicable. Different solutions will be compared to see which one can best describe saturated–unsaturated groundwater flow problems.

1.2 BACKGROUND REVIEW The rate at which water flows through a porous medium is associated with the properties of a porous medium, the properties of the water and change in hydraulic head. This relationship is well described by Darcy’s law (Konikow, 1996). By considering that the air pressure is always constant, i.e., zero, the movement of water through the saturated–unsaturated porous media can be described mathematically (List & Radu, 2015; Zimmerman & Bodvarsson, 1989). The saturated–unsaturated equation is obtained by merging the mass conservation equation with Darcy’s law (Danesfaraz & Kaya, 2009). The equation includes the hydraulic properties of the soil, which are a function of the suction head of the soil and therefore are nonlinear (Allepalli & Govindaraju, 2009). The 1-d saturated–unsaturated groundwater flow equation is given in (1.1):



       SS Sa    C      K z     1 t z   z 

(1.1)

where: ψ = pressure head n = porosity SS = specific storage of the soil Sa = saturation of the aqueous phase C(ψ) = capillary capacity of soil = dθ/dψ Kz = hydraulic conductivity z = vertical co-ordinate. Since the movement of water in the unsaturated zone must be distinguished from the movement of water in the saturated zone, we shall provide a brief literature on how flow in these two zones will differ, with the most complex being the unsaturated zone. Overcoming this complexity will require knowledge of the nonlinear relationship that exists between the soil hydraulic function (Cattaneo et al., 2016). Soil hydraulic functions refer to the hydraulic conductivity function, K(ψ), and the soil water content function, θ(ψ), that are required to explain the movement of water. Numerous functions have been suggested to define the soil hydraulic properties empirically. Popular models are the equations of Brook and Corey (Allepalli & Gavindraraju, 1996):



  K    K s  b   



K    K s



2  3

       0   s   0   b   

where: ψ = pressure head ψb = air entry suction pressure head

 0

(1.2)

 0



(1.3)



 0

(1.4)

3

Analysis of the Existing Model for the Vertical Flow

(Unsaturated zone)

( )

Pressure head ( )

( )=

( )=

0

+(

=0 0

Volumetric V l t i moisture it content t t( )

2+3

−

≤ 0,

∴

( )

≤0

> 0,

( )=

=

0)

>0

=1

(Saturated zone)

FIGURE 1.1  Relationship between the pressure head and volumetric water content for saturated–unsaturated flow (Modified after Nishigaki & Kono, 1980).



θ(ψ) = soil water content θs = saturated water content θ0 = residual water content λ = pore size distribution index Ks = saturated hydraulic conductivity.

The functional relationship between ψ and θ is called the water retention curve and is shown in Figure 1.1. Incorporated in this figure are the soil hydraulic functions (K(ψ) and θ(ψ)) showing how they differ between unsaturated and saturated zones. This figure is modified after Nishigaki & Kono, 1980. Now that the flow in the unsaturated and saturated zones has been distinguished, classical equations suitable for each zone are derived using powerful methods that result in explicit forms of a solution. In the case of the saturated zone, analytical and numerical solutions will be derived using integral transform, the methods of separation of variables and numerical schemes, respectively. The conditions under which the numerical method used converges will be derived and presented in detail. Because of the complexity of the unsaturated model, we relied only on the numerical method to derive an approximate solution. Detailed results of the analytical and numerical solutions are documented below for the classical equations where applicable.

1.3 GOVERNING SATURATED GROUNDWATER FLOW EQUATION When considering flow in the saturated zone only, it is assumed that the volumetric water content is equal to the porosity, the hydraulic conductivity is constant, and the capillary capacity becomes zero. Therefore, Equation (1.1) is written such that (Maslouhi et al., 2009):

 SS 

       Ks   1 t z   z 

(1.5)

4

Mathematical Analysis of Groundwater Flow Models

Which is further simplified to:  K s  2  t Ss z 2



(1.6)

where: ψ = is the pressure head Ks = saturated hydraulic conductivity Ss = specific yield z = vertical flow t = time.

1.3.1 Analytical Solution Using the Integral Transform The above saturated flow equation (1.6) can be solved analytically using an integral transform, for example, the Fourier transform or Laplace; for this case we will make use of both the Laplace and Fourier transform in attempt to provide an exact solution. Therefore, applying the Laplace transform on both sides of Equation (1.6) we obtain the following:  K  2        s 2    t   Ss z 



(1.7)

Applying the Laplace transform of a derivative into Equation (1.7) we obtain the following with s being considered as the Laplace variable:  K  2  s  z, s    z, 0    s 2   z, s   Ss z 



(1.8)

To eliminate the space component in order to obtain an algebraic, we can use the Fourier transform on the above equation:





 s  z, s    z, 0  

2 K s     z, s     Ss  z 2 

(1.9)

Using the properties of the Fourier transform, the right-hand side of the above equation can be simplified further into:



s  , s    , 0  

2 Ks  i    , s  Ss

(1.10)

The above equation can be factorized to obtain the equation below:



2  Ks  s  S  i    , s     , 0  s  

(1.11)

5

Analysis of the Existing Model for the Vertical Flow

We further simplify the above equation into:

  , s  

   , 0  2  Ks  s  S  i   s  

(1.12)

And then:

  , s  

   , 0  2  Ks  Ss  K s     S  s  s

(1.13)

The above formula is the multiplication of two Fourier transforms of two functions; thus one can use the convolution theorem to obtain the inverse transform as follow:





 1   , 0     z, 0 



(1.14)



      1 1   z, s     2   2  Ss   s         Ks   



  z, s  

  z, s  

2

(1.15)

Ss s Ks

(1.16)

2   Ss   S 2 s s 2   s   Ks   Ks     

1   S S 2 s s exp   s s  z     Ks Ks  

(1.17)

By integrating the above equation, we obtain:

  z, s  

z

  S 1    , 0  exp   s s  z     d  S  K s  2 s s 0 Ks



(1.18)

To obtain the exact solution of the saturated zone, we apply the inverse Laplace transform on both sides of Equation (1.18):

  z, t    1  z, s  

(1.19)

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Mathematical Analysis of Groundwater Flow Models

1.3.2 Analytical Solution Using the Method of Separation of Variables We shall now solve the above equation (1.6) using the analytical method of separation of variables. This method is used to solve a wide variety of linear and homogeneous differential equations, such as the saturated groundwater flow equation. The boundary conditions are linear and homogenous, such that:





z 0



zL

0

(1.20)



The dependence of ψ on z and t can be written as a product of function z and function t, such that:

  z, t   F  z  G  t 



(1.21)



Substitution ψ back into Equation (1.6) using the product rule, we obtain:



 F  z  G t  t



K

s



2 F  z  G t  z

Ss

2



(1.22)

Now we can separate the variables as follows: F z



d 2F  z  Ss dG  t   G t  K s dt dz 2

(1.23)

Variables can be further separated to obtain the following: d 2F  z  Ss dG  t  2 K s dt  dz G t  F z



(1.24)

The RHS and the LHS of Equation (1.24) can be a function of z and t, respectively, if they both equate to a constant value α, such that:



Ss dG  t   G t  K s dt

(1.25)

d 2F  z    F z dz 2

(1.26)

and:



To find a solution that satisfies boundary conditions and is not identically zero, let’s assume that for the constant α  0. Then there exists the real number A, B, C such that: G  t   Ae



S  s t Ks

(1.30)



and F  z   B sin







 z  C cos



z



(1.31)

From the above information C = 0 and that for some positive integer n:

 n



 L

(1.32)

Therefore, a general solution can be given as:  2 2 Ss  n K n z s   z, t   Dn sin exp   2 L L  n 1   





 t    

(1.33)

where Dn can be evaluated using the Fourier series, given the following initial condition:





t 0

 f z

(1.34)



So, we obtain: v z 



D sin n

n 1

Multiplying both sides with sin

n z L

(1.35)

nπ z and integrating over [0, L] results in: L L

2 n z Dn  f  z  sin dz L L





0

(1.36)

Hence the complete solution for Equation (1.33) is given by:  2 2 Ss  2 L   n K   n z n z s   exp     z, t   f  z  sin dz sin 2 L L L L    n 1  0    

 



 t    

(1.37)

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Mathematical Analysis of Groundwater Flow Models

1.4 NUMERICAL SOLUTION The analytical solution of our model is limited to less complexity such as the assumption of homogeneity, isotropy, simple initial condition, and simple geological formation. However, natural systems can have a more complex geological formation, a complex initial condition and they can be heterogeneous and anisotropic (Zhang, 2016). Such complexities require numerical solutions (Igboekwe & Amos-Uhegbu, 2011; Konikow, 1996). Depending on how ψt is approximated, we have three basic finite difference schemes: implicit Crank–Nicolson scheme, explicit, and implicit. Using these schemes, we attempt to find numerical solutions for the 1-dimension saturated groundwater flow equation (1.6).

1.4.1 Numerical Solution Using the Forward Euler Method (FTCS) Appling the explicit forward Euler method to Equation (1.6), we obtain the following numerical solution:



 i n 1  i n K s  t Ss

 n  2 n   n  i i 1  i 1 . 2    z 

(1.38)

1.4.2 Numerical Solution Using the Backward Euler Method (BTCS) We obtain another numerical solution by applying the implicit backward Euler method on Equation (1.6):



 i n 1  i n K s  t Ss

 n 1  2 n 1   n 1  i i 1  i 1  2    z  

(1.39)

1.4.3 Numerical Solution Using the Crank–Nicolson Method Now, we apply the implicit Crank–Nicolson discretization on Equation (1.6) such that:



 i n 1   i n K s  t Ss

 n 1  2 n 1   n 1  n  2 n   n  i i 1 i i 1  i 1   i 1 2 2   2  z   2z 

(1.40)

1.5 NUMERICAL STABILITY ANALYSIS In physical problems such as groundwater flow, stability analysis of an equation is fundamental. A procedure called von Neumann stability analysis based on the Fourier series is used to analyze the stability of the finite difference schemes. The stability of finite difference schemes is linked to numerical errors. A scheme is said to be von Neumann stable if its amplification factor is less or equal to 1. Accuracy, however, requires that the amplification factor be as close to 1 as possible. In this section, the numerical solutions provided in Equations (1.38)–(1.40) for the 1-d saturated groundwater flow Equation (1.6) are subjected under von Neumann stability analysis. A detailed procedure for each solution is given below.

9

Analysis of the Existing Model for the Vertical Flow

1.5.1 Stability Analysis of a Forward Euler Method (FTCS) Let us recall the forward Euler method (Equation 1.38), which can also be written as:



 i n 1   i n 1  2     i 1n   i 1n





(1.41)

K s  t  Ss   z 2    If we consider an initial harmonic perturbation:

where  

 i 0  eiki z

which with time evolves as:

 i n   neikm z

To analyze the stability of this scheme, let us find the amplification factor σ by inserting the above assumption into Equation (1.41) such that:





ik z  z ik z  z  n 1eikm z   neikm z 1  2     ne m     ne m  



(1.42)

We can simplify Equation (1.42) by pulling out the common factor, such that:







 n 1eikm z   neikm z 1  2   eikm z  e ikm z   

(1.43)

(1.44)



  eikm z  e ikm z   n 1eikm z   neikm z 1  2   1 2    

Considering the definition for a hyperbolic cosine, we can rewrite the above equation (1.44) as:

 n 1eikm z   neikm z 1  2  cos  km z   1 



(1.45)

Using the double angle identity for cos, Equation (1.45) is written such that: (1.46)



   k z     n 1eikm z   neikm z 1  2  1  2 sin 2  m   1    2    

(1.47)



k z    n 1eikm z   neikm z 1  4 sin 2 m  2  

10

Mathematical Analysis of Groundwater Flow Models

Dividing both sides by σ neikiΔz we get:

  1  4 sin 2



km z 2

(1.48)

The above equation gives us the amplification factor σ for the forward Euler method, and the stability condition is written as:

 1



(1.49)



However, if km∆z = π ⇒ σ = 1 − 4α the stability of the above scheme (1.41) for all k will only 1 hold if   . Therefore, the forward Euler method (1.41) is conditionally stable or von Neumann 2 unstable as applied to the 1-d saturated groundwater flow equation (1.6).

1.5.2 Stability Analysis of a Backward Euler Method (BTCS) For the above backward Euler method (1.39), we can obtain the following from solving a system of linear equations:  i n   i 1n 1  1  2  i n 1   i 1n 1



(1.50)



K s  t  . Ss   z 2    If we consider an initial harmonic perturbation from the previous section, we can find the amplification factor σ for this scheme by inserting it into Equation (1.50), such that: where  



 neiki z   n 1eiki1z  1  2   n 1eiki z   n 1eiki1z



(1.51)

We can simplify Equation (5.13) by pulling out the common factor, such that:







 neiki z   n 1eiki z   1  2    eik z  e ik z   

(1.52)

Further simplification of the above equation results in the following:



  eik z  e ik z  neiki z   n 1eiki z   1  2   2  2  

   

(1.53)

Considering the definition for a hyperbolic cosine, we can rewrite the above equation (1.53) as:

 n 1eiki z   neiki z  1  2  2 cos  k z  



(1.54)

Using the double angle identity for cos, Equation (1.55) is written such that:



  k z    n 1eiki z   neiki z 1  2  2  4 sin 2    2  

(1.55)

11

Analysis of the Existing Model for the Vertical Flow



k z    n 1eiki z   neiki z 1  4 sin 2 2  

(1.56)

Dividing both sides by σ neikiΔz : k z      1  4 sin 2 2  



1

(1.57)

The above equation gives us the amplification factor σ for the backward Euler method, and the stability condition is written as:

 1



(1.58)



This equation is unconditionally stable, since the value of σ will always be less than or equal to 1.

1.5.3 Stability Analysis of the Crank–Nicolson Method So far, we have considered the forward Euler scheme and the backward Euler scheme for the 1-dimension saturated groundwater flow equation. The forward Euler scheme is an explicit method, therefore easy to implement; however, the results obtained for the forward Euler scheme indicates t 1 that it is only stable under the condition that  . The backward Euler scheme, on the other x 2 2 hand, is an implicit method and is unconditionally stable; however, it requires more arithmetic operations to find values at a specific time step (Grigoryan, 2012). It is also essential to note that the two schemes use different sets of points in the computation of ψin + 1. The accuracy and stability of the two schemes can be improved by developing a single implicit scheme which will be a combination of the two schemes with different weights. With this method, a broader set of points can be used to compute the same values (Narasimhan, 2011; Grigoryan, 2012). For us to investigate the possible limits of this single scheme, let’s consider:





 



 i n 1  i n     i 1n 1  2 i n 1   i 1n 1   i 1n  2 i n   i 1n   

(1.59)

We substitute the initial harmonic perturbation from the previous section into Equation (1.59), which leads to:





 n 1eiki z   neiki z     n 1e m    2 n 1eikm z   n 1e m   ik z  z ik z  z   ne m    2 neikm z   ne m    



ik

z  z

z  z 

ik



 (1.60)



We can simplify the above equation to get:









 neiki z   1    neikm z eikm z  2  eikm z   neikm z eikm z  2  eikm z   

(1.61)

Solving for σ, we obtain the following growth factor:



1  2 1  cos k z  1  2 1  cos k z 

(1.62)

12

Mathematical Analysis of Groundwater Flow Models

Then the stability condition for this scheme is given by: 1  2 1  cos k z  1 1  2 1  cos k z 



(1.63)

This scheme was developed in 1947 by John Crank and Phyllis Nicolson, and it is shown as the average of the backward and forward Euler schemes. The denominator of Equation (1.63) will always be greater than the numerator, since α and 1 − cos k∆z are positive. This also means that under every condition, the value of σ will always is less than 1; therefore, the Crank–Nicolson method for the 1-d saturated groundwater flow equation (1.6) is unconditionally stable. It is clear that the Crank–Nicolson scheme is the most stable and accurate of the three schemes.

1.6 GOVERNING UNSATURATED GROUNDWATER FLOW EQUATION In heterogeneous soils, the water content is uneven across layer boundaries because of exceptional unsaturated capillary head relations in different soil layers (Assouline, 2013). Relatively, the capillary head (ψ) is continuous, and can be represented by an equation with ψ as the dependent variable and the moisture content in terms of ψ, θ = θ(ψ) (Farthing & Ogden, 2017). Equation (1.1) can be arranged so as to illustrate the complexity of the unsaturated flow using the equation below:          1   K z    t z   z 



(1.64)

The above Equation (1.64) consists of the soil hydraulic functions (K(ψ) and θ(ψ)). We shall use the famous soil hydraulic property Equations (2 and 5) of Brooks and Corey (1966) to describe these functions. We can now write the above equation as:         0   s   0   b   2  3           b     Ks   1   t z      



(1.65)

Simplifying the equation results in Equation (1.66) below:



 2  3 2  3

   0            s   0   b    K s  b   Ks  b  t t    z     z     

 s  0  b

 s  0 

 b





1   b     Ks   t    z   

 

  1



2  3

    Ks  b  z   

    Ks  b  z  

2  3

2  3

    Ks  b  z  

 2     Ks  b  z 2 z   

2  3

(1.66)

2  3

 2     Ks  b  2 z z   

(1.67) 2  3

(1.68)

13

Analysis of the Existing Model for the Vertical Flow

The final unsaturated groundwater flow equation is given by:   s   0  b

      Ks  b   1 t  i z   

2  3

    Ks  b  z  

2  3

 2     Ks  b  2 z z   

2  3

(1.69)

The above equation is nonlinear and cannot be handled analytically; thus we rely only on numerical methods to provide the numerical solution. This will be done in the next section.

1.6.1 Numerical Solution for the Unsaturated Groundwater Flow Model This section aims to provide a numerical solution to the nonlinear partial differential equation representing the dynamical system underlying the movement of sub-surface water in an unsaturated zone. To achieve this, we substitute the intervals [0, T] to 0 = t0