We use cookies to help provide and enhance our service and tailor content and ads. An alternative method is used here to derive the exact solution of the new groundwater flow equation. Note that [6-1] and [6-2] represent exactly the same thing. Finally, the solution Φ0(r,t) is approximated by the truncated series. Finally, the formulation given in (6.6) brings about a different type of generalization of the Caputo derivative. Homogeneous, isotropic, confined aquifer; the well is fully penetrating (open to the entire thickness (b) of aquifer); the well has zero radius (it is approximated as a vertical line) – therefore no water can be stored in the well; the head loss over the well screen is negligible; horizontal (not sloping), flat, impermeable (non-leaky) top and bottom boundaries of aquifer; no other wells or long-term changes; in regional water levels all changes in potentiometric surface are the result of the pumping well alone. Therefore, calculation of the left terms of equation (2) gives groundwater recharge. If vertical components of flow are negligible or small, we can use the Dupuit assumptions to simplify the solution of the equations. In this instance, since the new groundwater flow equation is linear, it follows that the total drawdown at any time t > tk will be given by, Note that, in the above equation, the summation can be transformed into an integral if Δt → 0. It is no longer of essence to know the initial point a, and subsequently label, the initial point a. The formulation in (6.6) appears to be similar to the one-dimensional fractional Laplacian; however, exception is given to the fact that integration occurs solely from one side. The available data will provide only an incomplete picture of the actual subsurface system. In this chapter, a polar coordinate system with radial geometry, describing the radial groundwater flow towards a well, is considered. In order to include explicitly the variability of the medium through which the flow takes place, the standard version of the partial derivative with respect to time is replaced here with variable-order (VO) fractional to obtain. Accurate simulation of the conceptual system is often taken to mean accurate simulation of the real system. 2 can be solved analytically in various geometries, provided that certain hypotheses are satisfied. The behavior of such an aquifer, often referred to as a leaky or semi-confined aquifer, needs thus not be the same as that of a confined aquifer. The assumption made in (6.9) is comparable to that of (6.9) for the kernel G(x,y,t) of L. Similarly, the condition (6.8) is comparable to G(x,y,t)=G(y,x,t) and was a requirement in [206] as a result of the equation being of divergence form. Therefore, using the beta-Laplace transform on both sides of Equation (5.2), we obtain, Again, applying the Laplace transform in respect to r, we obtain, Applying the boundary condition together with the initial condition, we obtain the following, Taking double inverse Laplace transform on both sides of above equation yields, Applying again the initial condition, we obtain the following exact solution of the new groundwater flowing within a confined aquifer, The above derivation can be found in [114]. ∆M= VρwSs∆ψ from Eq. by multiplying the flow rate by the aquifer thickness, D (9.10) The seepage into the aquifer at x = O is then found by substituting x = O into Equation 9.10. 2.3.1 Derivative of Theis Groundwater Flow Equation To derive the groundwater flow equation, we make use of the principle of continuity equation of the flow, that is the difference between the rate inflow and the rate outflow from annular cylinder which is the equation … For this matter if m=1, we obtained the following iteration formula: This infers that before water is pumped out of a borehole, the water level in the groundwater system is the same. the numerical representations of the exact solution of function of time and space for different values of beta. We are following work by Theis in 1935, in which he suggested an analytical solution to the normal groundwater flow equation. For instance, refer to [206] where a kernel G(t,s,x) does not satisfy the condition in (6.8) and (6.9) is utilized. These statements characteristically engage the bearing of flow, geometry of the aquifer, and the heterogeneity or anisotropy of sediments or bedrock within the aquifer. With sophisticated model-generated graphics, there is a tendency to forget the unavoidable uncertainties in representing the real system with a simpler conceptual model. However, very few geological formations are completely impermeable to fluids. We are following work by Theis in 1935, in which he suggested an analytical solution to the normal, Limitations of Groundwater Models With Local Derivative, Actual groundwater flow systems are much more complex than the conceptual models can typically represent. Comprehensive statement of groundwater flow. This method is often used to solve some class of parabolic partial differential equations. In a well-constructed mathematical model, most of the uncertainty in the results stems from discrepancies between the real system and the conceptual system. This is considered to be zero level. the discharge rate of fluid will often be different through different formation materials (or even through the same material, in different directions) even if the same pressure gradient exists in both cases. The derivative with integer order cannot portray such real-world problem. Now from the definition of storage coefficient (S), S is the volume of water released per unit surface area per unit change in head normal to the surface. Leakage of the water could thus occur, should a confined aquifer be over- or under-lain by another aquifer. The accuracy of predictive simulations is thus difficult to assess, so it's wise to assume a fair amount of uncertainty when using models to make predictions. Let us consider now the following function [114]. Therefore, the change in volume is given as: Here t is the time since the beginning of pumping. The Rigorous Theory, Water Resources Research, 10.1029/WR009i004p01022, 9 , 4, (1022-1028), (2010). The simplest oversimplification of, Physica A: Statistical Mechanics and its Applications. For a partial differential equation with two parameters, the method assumes that the solution is in the form of [113]: The above is then replaced in the main equation and two different equations are obtained with inclusion of an Eigen-value. Rearranging the above equation, we obtain two separated equation linked with a parameters called Eigen-values and they are provided as: The spatial ordinary different equation can be solved using the well-known Laplace–Carson transform, defined as: Properties of this operator can be found in [114].

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