Institute for Groundwater Studies (IGS)

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Open Access
Development of a groundwater recharge model
(University of the Free State, 2017-01) Spannenberg, Jescica M.; Atangana, Abdon
English: Existing groundwater recharge estimation methods appear to mainly generate site specific groundwater recharge estimates. These methods fail to yield reliable recharge estimates on a regional scale. This is due to failure in accounting for the concepts of heterogeneity, viscoelasticity, and the memory effect. Accordingly, this study was aimed at developing a new approach to groundwater recharge estimation by means of taking these concepts into account. Literature proves that these concepts have been well accounted for in the field of fractional differentiation. This study’s methodological approach entailed obtaining an exact solution to a selected groundwater recharge equation by applying the Laplace and inverse Laplace transform. Upon doing an uncertainty analysis and statistical analysis of the parameters within the solution, it was found that storativity and drainage resistance both require accurate estimation when estimating recharge from the selected equation. Following this, the Caputo derivative, Caputo-Fabrizio derivative, and the Atangana-Baleanu derivative were applied and an exact solution was obtained for each derivative; and upon doing a numerical simulation for each of these solutions, the results depict the behaviour of a particular real world problem. It was concluded that groundwater recharge within a heterogeneous and viscoelastic geological formation is well described with the concept of fractional differentiation with the generalised Mittag-Leffler law or the Atangana-Baleanu fractional derivative. To add, recharge via elastic geological formations can be model via the Caputo and Caputo-Fabrizio derivatives. Furthermore, hydraulic head is assumed to be influenced by uncertain factors which are not accounted for in the general recharge equations. The Eton approach was thus applied, and reveals the uncertain function has a significant effect on hydraulic head distribution. Ultimately, this study concludes that a groundwater recharge model incorporating heterogeneity, viscoelasticity, the memory effect, and uncertainties, will generate a new and improved understanding to groundwater recharge investigations.
Open Access
A generic assessment of waste disposal at Douala City practice, prinicipals and uncertainties
(University of the Free State, 2013-01) Atangana, Abdon; Botha, Joseph François
English: One reason why groundwater, so often constitutes the main source of drinking water in many cities and towns around the world, is because it is frequently present in sufficient quantities at the point of demand. However, this seemingly advantage may sometimes be its greatest disadvantage, especially in situations where the groundwater occurs at shallow depths and the area overlying the aquifer is populated densely. This problem is particularly relevant in the present technological age with its vast quantities of waste that is often disposed in an uncontrolled manner. Such a situation occurs at Douala the economic capital of Cameroon in central Africa. The city not only host more than 80% of industries in the country, but also has the largest urban population of approximately 3 000 000 with a population density of approximately 350 persons per square kilometre, which continue to increase at a rate of approximately 120 000 migrants per year from the rural areas, while the groundwater level is very shallow and may sometimes rise above the soil surface, especially during floods, which occur not too infrequently. Although the pollution problem is not restricted to groundwater as such, it is aggravated here, because of the ancient belief that wastes are safely disposed of, if buried below the earth’s surface. It took disasters like Love Canal and the Price Landfill to discover the detrimental effects that this practice may have on the population living on or near polluted aquifers. Extreme care therefore should be exercised to prevent the pollution of any aquifer that may pose problems to living organisms or to try and restore a polluted aquifer threatening the natural environment. Groundwater pollution should therefore receive urgent attention when discovered. This thesis describes an attempt to develop a set of guidelines for the restoration of the groundwater resources at Douala, based on the relatively new technique of permeable reactive barriers for groundwater remediation—a technique that is also increasingly applied in the restoration of the Superfund sites in the United States of America. Modern attempts to clean up contaminated aquifers, relies heavily on the use of suitable computational numerical models. Such models have in the past always been based on the classical hydrodynamic dispersion equation. However, an analysis of the equation in this thesis has shown that the equation cannot account for the long‐tail contamination plumes characteristic of fractured rock aquifers. Fortunately, it is not too difficult to develop a more suitable equation. For, as shown in the thesis, all that one has to do is to replace the ordinary derivatives in the classical equation with fractional derivatives. Mechanistic modeling of physical systems is often complicated by the presence of uncertainties, which was in the past usually neglected in the models used in the restoration of aquifers.While these uncertainties have regularly been neglected in the past, it is nowadays imperative that any groundwater model be accompanied by estimates of uncertainties associated with the model. Although a large number of approaches are available for this purpose, they often require exorbitant computing resources. The present investigation was consequently limited to the application of the Latin Hypercube Sampling method applied to an analytical solution of the hydrodynamic dispersion equation. It has been known for years that the hydrodynamic dispersion equation discussed in Chapter 5, is not able to account for the long‐tail plumes often observed in studies of contaminated fractured‐rock aquifers. An approach frequently used to account for this is to replace the ordinary spatial and temporal derivatives in the hydrodynamic equation with fractional derivatives—a procedure confirmed in this thesis.
Open Access
The investigation of groundwater transport in a geological formation: West Park Cemetery, Johannesburg, South Africa
(University of the Free State, 2023) Vava, Bomkazi; Atangana, Abdon
Groundwater is a source of fresh water for many people who live in communities with little surface water around the world. The major sectors of groundwater use are municipal; rural; agricultural irrigation; agricultural-livestock watering; industry and mining. The amount and quality of groundwater have nevertheless decreased because of anthropogenic activities, global climate change, and poor groundwater management. The use of fertilizers and pesticides, urban development, the dumping of household waste on the land, municipal waste discharge and laboratory waste disposal, the aquifer close to contaminated streams, mining operations, and the discharge of effluents with high concentrations of industrial chemicals and sludge on the land pose a concern on groundwater. Due to the migration of solutions from leachate to the soil, dumpsites are also regarded as important sources of groundwater pollution. Pollution from naturally existing toxins, such as arsenic or fluoride, should not be overlooked. Historically, contaminated drinking water has been known to be capable of transmitting dangerous chemicals and deadly diseases. In this thesis, we have considered contamination from cemeteries because they also contribute to contaminating groundwater, which spreads infectious diseases and dangerous substances. Many lives are lost because of the spread of some infectious diseases, resulting in a large number of burials that could pollute groundwater. The Johannesburg West Park Cemetery was selected as our case study location. We gathered a variety of data regarding the cemetery expect for water samples as the boreholes were dry. The collected data were subjected to numerous analyses, and a detailed presentation of the geological structure under the cemetery was made. A high rate of burials was seen between 2020 and 2022, primarily because of the spread of COVID-19. We presented evidence to support our claim that the crossover seen during the decay process cannot be replicated by the decay model with the classical derivative. To simulate the transition from fast decay to slow decay, a fractional model was utilized using a Mittag-Leffler function as the solution. The acquired results assisted us in choosing an appropriate mathematical formulation of the advection-dispersion equation, which was also numerically examined. Under certain hypotheses, we ran some numerical simulations utilizing the mass released within the geological formation as the beginning concentration. The cemetery's boreholes should be checked monthly to improve the environment, and additional boreholes should be dug to figure out the type of aquifer beneath the cemetery and the direction of flow since these details aren't currently known. This will help to ensure that the cemetery does not contaminate the environment, or the water supplies nearby.
Open Access
Mathematical modelling of pressure build-in due to geological carbon dioxide storage in deep Saline Aquifers, using non-local operators: the context of groundwater protection in the climate change mitigation era
(University of the Free State, 2023) Mbah, Hans Tah; Atangana, Abdon
Man-caused Green House Gas (GHG) emissions have perturbed the earth’s energy balance, and the need to achieve deep emission reduction is a pressing challenge, faced by humankind. Carbon Capture and Storage (CCS) in deep geological saline aquifers is a viable option for Green House Gas (GHG) mitigation. Industrial-scale scenarios may induce large-scale reservoir pressurization and displacement of native fluids. Especially in closed systems, the pressure buildup can quickly elevate beyond the reservoir fracture threshold and potentially fracture and or reactivate existing faults in the cap rock. This can create conduits for focused leakage and mobilization of heavy metals and harmful trace elements, into capture zones of freshwater wells. Risk assessment in CCS requires that careful safety and environmental impact evaluations be considered. Through sustainable pressure management, the reservoir's hydraulic integrity is maintained. This is theoretically achievable with the help of a numerical simulation, using robust mathematical models that provide consistent and effective ways to understand and predict pressure buildup in such complex systems. This thesis focuses on modelling two aspects: (𝑖) the pressure buildup and (𝑖𝑖) CO₂ saturation evolution in the two-phase flow zone, by extending the classical pressure diffusivity and the Buckley-Leverette fractional flow models to the framework of non-local differential and integral operators. The extended models describe the pressure behavior during CO₂ injection through a vertical well, open in a saline aquifer. To include in the mathematical models the effect long-range, fading memory and weak crossover from stretched exponential to power-law, several differential operators were considered. For each extended model, different numerical schemes were adapted to derive numerical solutions. The presented numerical results provide an overview of subsurface transient pressure response and are useful tools for accurate risk assessment and sustainable reservoir management and operational safety during geo-sequestration in basins with shallow freshwater capture zones.
Open Access
Modelling reactive pollutant transport in ground water: the case of two species
(University of the Free State, 2019-06) Hans Tah, Mbah; Atangana, Abdon
The locations of a significant number of industrial facilities, landfills and almost all mineral ore bodies are characterised by high in situ stresses and fractures and fissures act as flow paths for fluids underground. Regional scale fracture systems that transport pollution from spatially isolated source locations can cause mixing of chemical pollutants from different source origins due to fracture- fracture flux across two or more intersecting fractures, hence reactive transport. Alerts of groundwater contamination in response to multicomponent pollution transport have been investigated using a mathematical model of the hydrodynamic response of incompressible fluids such as groundwater flow. Fundamental to the model is the conservation of mass associated with the applied source strength and the concentration velocity field redistributions after source has released pollution, assuming the formation is homogenous. Solute distribution depends on the formation porosity and generally, fluids travel faster in fractures than in sedimentary formations. Different variations of the deterministic advection dispersion equation have been employed to predict coupled transport/reactive processes by substitution of a reaction term (retardation factor, etc..) which accounts for the changing concentration of the solid face components through time due to chemical reaction. However, fractured aquifers are inherently heterogeneous due to interconnecting fractures. The results in a non-homogenous mathematical formulation which difficult to solve analytically. As a result, most research endeavours have tended to depend on numerical solutions, increasingly made possible through advanced computational power. Even though it is questionable to what extend numerical models of groundwater reactive transport can be useful in making accurate quantitative predictions, it is still possible for a reactive model to predict the outcome of a particular chemical representation in an aquifer. Nevertheless, the linear, non-homogenous advection dispersion equation can still be solved analytical using the Green’s function method. In this thesis, we show how the advection-dispersion transport equation can be extended to account for geochemical reaction processes in a heterogeneous media. For the hypothetical case study, the system was made of a homogenous and a non-homogenous sub-component. The study’s methodological approach involved coupling of the homogenous transport phase with the non- homogenous system. The solution of the homogenous equation is obtained using Laplace transform and the exact solution of the new non-homogenous equation is obtained analytically using Laplace transform and the Green’s Function method. Both sub-models were solved numerically using the Crank-Nicolson discretization scheme and their stability conditions also established. For the proposed fracture flow system, the linear non-homogenous model was able to approximate the contribution of reactive transport processes in the system. Chemical reactions can attenuate the spread of a contaminant plume due to processes such as sorption and precipitation. The model presented in this thesis was able to predict fate of each species within the system. Mass transfer during and after the reaction resulted in the depletion of one source with respect to another, which the model showed positive results in capturing. The thesis concludes with a chapter on chemical equilibria which is the basis of kinetic modelling and the understanding of the progress of chemical reactions.
Open Access
New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model
(Vinča Institute of Nuclear Sciences, Belgrade, 2016) Atangana, Abdon
In this paper a new fractional derivative with non-local and no-singular kernel is proposed. Some useful properties of the new derivative are presented and applied to solve the fractional heat transfer model.
Open Access
A new method for modeling groundwater flow problems: fractional-stochastic modeling
(University of the Free State, 2019-06) Mahantane, Mohau L.; Atangana, Abdon
To date, groundwater flow problems are still increasingly becoming a great environmental concern worldwide. This is among some of the reasons that many researchers from various fields of science have focused much of their attention in formulating new mathematical equations and models that could be used to capture and understand the behavior of groundwater flow with respect to space and time. The main aim of this study was to develop a new concept for modeling groundwater flow problems. The approach involved coupling of differential operators with stochastic approach. Literature proves that each of these two concepts has shown a great success in modeling complex real-world problems. But we argued that differential equations with constant coefficient are not fit to capture complexities with statistical setting. Therefore, to solve such a problem in this study, we considered a classical one-dimensional advection-dispersion equation for describing transport in porous medium and then applied stochastic approach to convert groundwater velocity (v), retardation (R) and the dispersion (D) constant coefficients into probability distribution. The next step was to employ the concept of fractional differentiation where we substituted the time derivative with the time fractional differential operator. Thereafter, we applied the Caputo, Caputo-Fabrizio and the Atangana-Baleanu fractional operators and derived conditions under which the exact solution for each derivative can be obtained. We then suggested the numerical solutions using the newly established numerical scheme of the Adams-Bashforth in the case of the aforementioned three (3) different types of differential operators. By combining the two concepts, we developed a new method to capture groundwater flow problems that could not be possible to capture using differential operators or stochastic approach alone. This new approach is believed to be a future technique for modeling complex groundwater flow problems. After solving the new model numerically, the condition for stability was also tested using the Von Neumann stability analysis method. Lastly, we presented numerical simulations using a software package called MATLAB.
Open Access
Quantification and modelling of heterogeneities in aquifers
(University of the Free State, 2017-01) Ahokpossi, Dehouegnon Pacome; Atangana, Abdon; Vermeulen, Danie
English: The future of modelling of heterogeneity in aquifers is definitively in the designing of new in situ testing (hydraulic and mass transport) procedures with new corresponding mathematical models. New trends in mathematical differentiation offer opportunity to explore more flexible and practical mathematical model solutions. This applies to both analytical and numerical modelling. Only a sound understanding of rock structures can clearly pose the problem which will then be used to define hydraulic equations to be solved by mathematical models, and numerical software. The most recent concept of differentiation based on the non-local and non-singular kernel called the generalized Mittag-Leffler function, was employed to reshape the model of fractured aquifer fractal flow. The solution was successfully applied to experimental data collected from four different constant discharge tests. Additionally, a new analytical solution to the fractal flow in a dual media was proposed, where the media could be elastic; heterogeneous; and visco-elastic. The existing dual media fractal flow model was modified by replacing the local derivative with the nonlocal operator (operator with Mittag-Leffler kernel, and Mittag-Leffler-Power law kerne)l. The more accurate numerical scheme known as Upwind was used to numerically solve each model. Heterogeneity in a typical South African crystalline rock aquifer was assessed. From this, a methodical level for quantifying and modelling heterogeneity in an aquifer was deduced. It was demonstrated how spatial heterogeneity in aquifers can be modelled based on the most commonly available tools and data in mining environment. The capability of selected numerical geohydrological softwares were assessed using spatial variability of hydraulic parameters (hydraulic conductivity and recharge). Geostatistical tools were specifically applied. Focus was also given to hydro-geochemical characterization by using bivariate scatter plots, Piper and Expanded Durov diagrams, and PHREEQC hydro-geochemical model as complimentary tools to analyse the groundwater chemistry data to describe different hydro-geochemical process which prevail in the monitored groundwater system. Three manuscripts have been submitted out this thesis, in top tier journals of the Natural and Applied Sciences.
Open Access
Stochastic groundwater flow models in confined and leaky aquifers
(University of the Free State, 2019-06) Amakali, Sarti Rautia; Atangana, Abdon
This dissertation proposes an application of stochastic modeling of groundwater flow in confined and leaky aquifers. We are estimating that aquifer parameters such as transmissivity, storativity and leakage factor vary, not constant, in space at different period especially in heterogeneous environment. Heterogeneous environment are known to be complex because of their uncertainty. Uncertainty referred in modeling includes errors in dataset, which might be bias or variance (under fitting/over fitting), low or not enough data, or unbalanced data, which all affect the model produced if not captured with appropriate model technique. The groundwater flow equation for confined and leaky aquifers derived by the latest version Atangana and Ramotsho, as well asAtangana and Mathobo, which all include scaling matrix of the soil, are considered and further modified to a new scheme of stochastic models for confined and leaky aquifer. We tried to achieve the capture statistical setting of aquifer parameters using the concept of stochastic modeling technique. The aquifer parameters are replaced by distribution for instance, Gaussian or normal distribution.Due to the complexity of the modified models, it is almost impossible to obtain the exact solution by using analytical solution, thus we opt to numerical analysis, in particular the Newton method used to derive the numerical solutions of the modified models. Detailed analysis of stability and convergences, we used method are presented for both models. Numerical simulations are depicted for different distributions.

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