Fractional Operators with Constant and Variable Order with Application to Geo-hydrology

By Abdon Atangana

Fractional Operators with Constant and Variable Order with Application to Geo-hydrology provides a physical review of fractional operators, fractional variable order operators, and uncertain derivatives to groundwater flow and environmental remediation. It presents a formal set of mathematical equations for the description of groundwater flow and pollution problems using the concept of non-integer order derivative. Both advantages and disadvantages of models with fractional operators are discussed. Based on the author’s analyses, the book proposes new techniques for groundwater remediation, including guidelines on how chemical companies can be positioned in any city to avoid groundwater pollution.

• Proposes new aquifer derivatives for leaky, confined and unconfined formations
• Presents useful aids for applied scientists and engineers seeking to solve complex problems that cannot be handled using constant fractional order derivatives
• Provides a real physical interpretation of operators relevant to groundwater flow problems
• Models both fractional and variable order derivatives, presented together with uncertainties analysis

• Language: English
• Publisher: Academic Press
• Release date: Sep 19, 2017
• ISBN: 9780128097960

Abdon Atangana

Dr. Abdon Atangana is Academic Head of Department and Professor of Applied Mathematics at the University of the Free State, Bloemfontein, Republic of South Africa. He obtained his honours and master’s degrees from the Department of Applied Mathematics at the UFS with distinction. He obtained his PhD in applied mathematics from the Institute for Groundwater Studies. He was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, and he was awarded The World Academy of Sciences (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries. In 2018 Dr. Atangana was elected as a member of the African Academy of Sciences and in 2021 a member of The World Academy of Sciences. He also ranked number one in the world in mathematics, number 186 in the world in all fields, and number one in Africa in all fields, according to the Stanford University list of top 2% scientists in the world. He was one of the first recipients of the Obada Award in 2018. Dr. Atangana published a paper that was ranked by Clarivate in 2017 as the most cited mathematics paper in the world. Dr. Atangana serves as an editor for 20 international journals, lead guest editor for 10 journals, and is also a reviewer of more than 200 international accredited journals. His research interests include methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling. Dr. Atangana is a pioneer in research on fractional calculus with non-local and non-singular kernels popular in applied mathematics today. He is the author of numerous books, including Integral Transforms and Engineering: Theory, Methods, and Applications, Taylor and Francis/CRC Press; Numerical Methods for Fractal-Fractional Differential Equations and Engineering: Simulations and Modeling, Taylor and Francis/CRC Press; Numerical Methods for Fractional Differentiation, Springer; Fractional Stochastic Differential Equations, Springer; Fractional Order Analysis, Wiley; Applications of Fractional Calculus to Modeling in Dynamics and Chaos, Taylor and Francis/CRC Press; Fractional Operators with Constant and Variable Order with Application to Geo-hydrology, Elsevier/Academic Press; Derivative with a New Parameter: Theory, Methods, and Applications, Elsevier/Academic Press; and New Numerical Scheme with Newton Polynomial: Theory, Methods, and Applications, Elsevier/Academic Press; among others.

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Chapter 1

Aquifers and Their Properties

Abstract

This chapter presents the general definition of the geological formation called aquifers together with their mathematical properties. Some reported examples of these aquifers around the world are presented.

Keywords

Confined aquifers; Unconfined aquifers; Semi-confined aquifers; Example of aquifers in the world; Mathematical properties of aquifers

1.1 Introduction

An aquifer is a geological formation in which groundwater flows through with ease. Aquifers should therefore have both permeability and porosity. Examples of these geological formations which form aquifers include sandstone, conglomerate, fractured limestone, and unconsolidated sand and gravel formations. Another example of an aquifer system is a fractured volcanic rock formation such as columnar basalt. The rubble zone where volcanic flow exists is usually both porous and permeable, thus allowing for good aquifer systems [1,2]. Furthermore, geological formations such as granite and schist have low porosity, and so are usually classified as poor aquifers. However, once they become fractured, they can produce good aquifers [1,2]. Furthermore, for a well to be productive, the well itself should be drilled into the ground to penetrate the aquifer. If groundwater is abstracted from the well at a rate faster than the aquifer is recovered, there is a decline in the water table, sometimes to a point that the well dries. As pumping occurs, the water table normally declines, resulting in a cone of depression at the well. Moreover, groundwater flow generally follows the slope of the water table, thus in this case, groundwater flows in the direction of the well being pumped. Not all aquifers can be seen as groundwater reservoirs [1–3]. These reservoirs are, however, found underground but only in cavernous geological formations where the formations surrounding fractures or cracks have undergone dissolution, forming open channels which allow rapid water movement similar to that of a river. Furthermore, since groundwater migrates at a slow rate through pore spaces of aquifer material, the only living organisms, that could float as it would in an actual river, are bacteria or viruses which are minute enough to migrate through pore spaces. Movement of groundwater through an aquifer occurs as groundwater is forced through a pore space of geological formations. Hence, porosity essentially defines an aquifer. To add, porosity of certain aquifers also allows them to act as good filters generating natural purification [2,3]. Since effort is required for forcing water movement through small pores, there is a loss in groundwater energy as it flows. This eventually results in decreased hydraulic head in the direction of groundwater flow. On the other hand, when pores are large in size, there is increased permeability, less energy loss, and rapid groundwater movement. Subsequently, groundwater migration is rapid for aquifers with large pores, such as in the case of the lower Portneuf River aquifer or in cases where porosity is a result of fractures which are interconnected. It is also significantly rapid in fractured rock aquifers such as the basalts of the eastern Snake River Plain. Despite being good aquifers, they are vulnerable to spreading of contamination which is challenging and often impossible to prevent [3].

1.2 Classification of Aquifers

Hydrogeology is known as the field investigating the flow of water through aquifers. Terms related to aquifers include aquitard, which is a formation associated with low permeability along an aquifer; and aquiclude or aquifuge, which are associated with an impermeable formation existing above or below an aquifer values would still differ. This can also be associated with hydraulic transmissivity and hydraulic resistance. Furthermore, in the event of calculating flow to drains and wells penetrating aquifer systems, the anisotropy should be accounted for in case there may be fault associated with the design of the drainage system. As indicated, there also exist aquitards and aquicludes. Aquitards are merely formations restricting groundwater flow from one aquifer to the next, thus they generally are defined by clay or non-porous rock having low hydraulic conductivity. These formations become aquicludes once the formation is completely impermeable.

1.3 Example of Some Aquifers in the World

The following are a number of aquifers from around the world:

•  The Great Artesian Basin , and has a fundamental role of water supply for Queensland and isolated locations of South Australia.

•  The Guarani Aquifer , a thickness ranging between 50 m and 800 m, and its maximum depth is more or less 1800 m.

•  The Ogallala Aquifer is another large of aquifer system of the world, whose location is within the United States. It lies beneath the surface of eight different states. Although one of the largest aquifers, some parts of the aquifer are undergoing rapid depletion due to expanding municipal and agricultural activity. The aquifer is comprised mainly of fossil water originating from the previous glaciation. The average annual groundwater recharge for the more arid areas of the aquifer receives approximately 10% of the annual abstractions.

•  The Edwards Aquifer ), and is therefore one of the world's most productive artesian aquifers, providing water to about two million people. To add, this aquifer is essential to many different unique and endangered species.

•  The Arkell Spring Grounds is found in southwestern Ontario, Canada, is a highly productive bedrock aquifer, yielding several freshwater springs along the Eramosa River situated in the northeast of the village of Arkell. In 1903, the City of Guelph started using these springs for municipal use, and then during 1963 and 1967 four additional boreholes were formed. In addition, this source of water primarily goes to the City of Guelph.

•  The Laurentian River is an old river located in southern Ontario, Canada. The river system formed prior to the recent ice ages, and glacial debris filled this ancient river valley. Furthermore, the water still migrates down this river valley, and then eventually underground. To add, the source of this water is assumed to be located near Georgian Bay which is about 200 kilometers (120 mi) away.

•  The Oak Ridges Moraine (730 sq mi). Moraine is a highly significant landform which obtained its name from the rolling hills and river valleys which extend about 160 km (99 mi) from the Niagara Escarpment toward the east, to Rice Lake. This landform was created more or less 12,000 years ago through glaciers that were advancing and retreating. Additionally, Moraine is presently of utmost importance because it is essential to large-scale urban development.

•  The Biscayne Aquifer ). To add, it is a shallow aquifer existing within a layer of significantly permeable limestone.

•  The Snake River Aquifer is a large groundwater system found beneath the Snake River Plain which is located in the southern part of Idaho, United States. This aquifer system's water is sourced from rain and melting snow which migrate onto the Plain from the Snake River, Big Lost River, Bruneau River, as well as additional watercourses found in southern Idaho. Furthermore, the aquifer extends from east to west at a distance of about 400 miles (640 km). It is an essential water source for agricultural irrigation activity occurring in the Plain. Moreover, it is an aquifer subdivided into two aquifers due to the Salmon Falls Creek, namely the Eastern Snake River Plain Aquifer and Western Snake River Plain Aquifer.

•  The Floridan aquifer forms part of a major artesian aquifer which extends into Florida. The aquifer lies beneath all of Florida and also large portions of coastal Georgia and regions within coastal Alabama and South Carolina. In essence, it exists beneath the coastal areas of south east United States. In addition, the Floridan aquifer is comprised of carbonate lithology and is also one of the world's most productive aquifers.

•  The Mahomet Aquifer ) of groundwater per day for public water use, as well as industrial and irrigational use.

•  The Ogallala Aquifer ) of the Great Plains of the United States. In essence, the aquifer lies beneath parts of eight different states: South Dakota, Wyoming, Nebraska, Kansas, Colorado, Oklahoma, Texas, and New Mexico.

•  The Permian Basin is a sedimentary basin primarily located in the western parts of Texas and south eastern parts of New Mexico, in the United States. The aquifer extends from south of Midland and Odessa toward the west, and then into the southeast portion of the neighboring New Mexico state.

•  The San Diego Formation is a geological formation comprised of coastal transitional marine and non-marine pebble, cobble conglomerate deposits, marine sandstone material, and the Pliocene marine fossils, from a bay that was previously deposited during the Middle to Late Pliocene of the Cenozoic era (2 to 3 million years ago). This geological formation covers an area from the southern end of Mount Soledad located in San Diego County to Rosarito Beach in northern Baja California. This thus includes Mexico and Tijuana as well as the southwest end of San Diego County, from San Ysidro to Pacific Beach.

•  The San Joaquin River is the biggest river in central California, United States. It extends up to a length of 366 miles (589 km), whereby the source is in the high regions of Sierra Nevada. The river migrates through agricultural areas of the northern San Joaquin Valley to where it discharges in Suisun Bay, San Francisco Bay, and the Pacific Ocean. The San Joaquin River is essential to irrigation and also acts as a wildlife corridor. To add, the river is one of the significantly dammed and diverted rivers of California.

•  The Sankoty aquifer is located in Illinois, United States. It supplies groundwater to northwest and central communities of Illinois. Furthermore, it is an unconsolidated deposit found in a bedrock valley which was previously associated with the ancestral Mississippi River.

•  The Amazon basin (2,670,000 sq mi), or about 40% of the continent of South America. Moreover, the Amazon basin exists within a few different countries of South America, and which are Bolivia, Brazil, Colombia, Ecuador, Guyana, Peru, Suriname, and Ven.

•  The Hamza River is the unofficial name given to an aquifer which appears to flow slowly through Brazil. The river is about 6000 km (3700 mi) in length. This water body was established in 2011, and its unofficial name is associated with named Valiya Mannathal Hamza, a scientist of Brazil's National Observator, who conducted four decades of research in the area.

•  The Canning Basin is located on land. The basin is known for having abundant oil and gas, and thus has undergone extensive research. Moreover, in June 2003, 250 wells were drilled and there was 78,000 km of seismic shot.

•  The Jandakot Mound, or Jandakot Groundwater Mound .

•  The Yarragadee Aquifer is an important freshwater aquifer found in the south west regions of Western Australia. It exists primarily below the Swan Coastal Plain which is found west of the Darling Scarp. Furthermore, the Yarragadee Aquifer has a north–south range from near Geraldton to the southern coast. However, the formation south of Perth is divided into two, whereby the southern portion is called the South West Yarragadee Aquifer. Moreover, the aquifer is significantly deep, located hundreds of meters below surface, and has a thickness extending up to more or less two kilometers.

•  The Bas Saharan Basin is an artesian aquifer covering a large part of the Algerian and Tunisian Sahara. It extends up to Morocco and Libya, thus encompassing the entire Grand Erg Oriental.

•  The Lotikipi Basin Aquifer . To add, it has the capacity to provide water to the population as it has sufficient fresh water which can last up to 70 years, provided there is adequate management of the aquifer.

•  The Nubian Sandstone Aquifer System (NSAS) groundwater.

•  Alnarpsströmmen is an artesian aquifer located in the Swedish province of Skåne. It is a subterranean aquifer which has been used for wells since the 18th century. Moreover, since 1901, it has been a freshwater resource for Malmö; and today it is used by numerous additional towns. Since it is artesian, it often results in the formation of fountains. Furthermore, more or less 400 liters per second of groundwater migrates from Alnarpsströmmen to Öresund through quaternary moraine. In addition, the aquifer has a width of about 20 km, and the flow of groundwater is only 10 meters per year.

•  Schwyll Aquifer was initially called the Great Spring of Glamorgan. However, recently Welsh Water used the Schwyll Spring close to Ewenny as the primary water source for the Bridgend region. The aquifer now acts as a backup resource, and therefore has several related source protection zones which the Environmental Agency governs. The spring's outflow exceeds all other freshwater springs found in Wales; and it is larger than that of Wookey Hole or Cheddar Gorge risings. Finally, the aquifer has an underground waterway located in carboniferous limestone formations of South Wales.

•  The Upper Rhine aquifer (110,000 cu mi) of fresh water. The aquifer provides water for about 3 million people in both France and Germany, and it accounts for 75% of potable water and 50% of industrial water use. To add, since the 1970s, the aquifer underwent pollution due to addition of nitrates, pesticides, chloride, and