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</html>";s:4:"text";s:5059:"We shall use this method to derive the solution of the new groundwater flow equation. In this chapter, a polar coordinate system with radial geometry, describing the radial groundwater flow towards a well, is considered. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000072, URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000023, URL: https://www.sciencedirect.com/science/article/pii/B9780081006443000052, URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000126, URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000047, URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000060, URL: https://www.sciencedirect.com/science/article/pii/B9780128096703000114, Applications of Fractional Operators to Groundwater Models, Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology, Applications of local derivative with new parameter, We devote this section to the derivation of the exact solution of, Modeling Groundwater Pollution With Variable Order Derivatives, Values of the variational iteration method as well as its applications for a variety of categories of differentials equations are documented in several research papers in the literature. However, very few geological formations are completely impermeable to fluids. Therefore, in order to well replicate the movement of water by means of porous media in different layers in the aquifer, we reformulate the well-known Darcy law in order to produce a new groundwater flow equation within a confined aquifer [107,108,111]. capitulates the following Lagrange multipliers, giving up to the following Lagrange multipliers: λ=−1 for the case where m=1 and λ=t−τ for m=2. In this matter two components of the decomposition series were obtained, whereby Φ(r,t) was evaluated to have the following expansion: Groundwater flow models are necessarily simplified mathematical representations of complex natural systems. Finally, the formulation given in (6.6) brings about a different type of generalization of the Caputo derivative. According to Hantush and Jacob [172,173], the drawdown due to pumping a leaky aquifer can be described by the following equation: where u is the drawdown or change in the level of water, U is the specific storativity of the aquifer, and T is the transmissivity: with C and C′ the hydraulic conductivities of the main and confining layers, respectively; B and d the thicknesses of the main and confining layers, respectively; and Θ the discharge rate of the pumping. Solution of the equations for unconfined groundwater flow is complicated by the fact that the aquifer thickness changes as groundwater is withdrawn; i.e., removal of water from the aquifer lowers the water table. The correction functional can be approximately expressed for this matter as follows: where λ is a general Lagrange multiplier, which can be recognized optimally by means of variation assumption, and (Φ1(r,t))ˆ2r, (Φ1(r,t))ˆr, and h(r,τ)ˆ are considered as constrained variations. This partial differential equation describing the movement of water through the geological formation during the pumping is subjected to the following initial and boundary conditions: However, when we consider the diffusion process in the porous medium, if the medium structure or external field changes with time, in this situation, the ordinary integer order and constant-order fractional diffusion equation model cannot be used to properly characterize such a phenomenon [180,184]. 1) Darcy’s law, continuity, and the groundwater flow equation q x q y q y+dy q x+dx ( − )+ ( − ) =0 x x+dx x x y y+dy y y In 1935, Theis [183] was the first to extend a modus operandi for unsteady-state stream that brought in the time factor and the storativity. These statements characteristically engage the bearing of flow, geometry of the aquifer, and the heterogeneity or anisotropy of sediments or bedrock within the aquifer. Then Equation (5.25) becomes, A particularly important solution arises when t0 is considered at the origin zero and at the point the discharge rate is independent of time. This is the case of the groundwater flow in the deformable aquifer, the medium through which the flow changes with time and space [165,166]. Groundwater representations portray the subsurface movement and transportation processes by means of mathematical equations supported by positive simplifying suppositions. It follows that the right side of (12.54) and (12.54) is known. The Caputo derivative as presented earlier is defined as, For C1 functions, there may be used integration by parts for showing the equivalent formula, Upon defining q(t)=q(a) for t<a then as in [205] this gives birth to the equivalent formulation, This one-sided nonlocal derivative is referred to as the Marchaud derivative [216], which was recently looked at in [205]. 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