Steady-State Radial Flow Modeling through the Production Well in the Confined Aquifer of Monzoungoudo, Benin

This study aims to develop a mathematical analysis for one-dimensional modeling of a radial flow through a production well drilled in a confined aquifer, in the case of steady-state flow conditions. An analytical solution has derived from that expression for estimation of drawdowns according to different flowrates. Through that process, the evaluation of static pressure, the calculation of hydraulic charge due to the waterflow through the well is evaluated, the drawdowns curves are drawn and at last, the obtained curves are analyzed. The curves obtained for the different flow rates have an asymptotic direction, the axis of the hydraulic charges. The variation of the hydraulic charge depends on the radial distance for different flow rates. The P point, is a common point of all curves obtained for different production flowrates in the well. This point is where the well production flowrate is optimum for the optimal hydraulic charge.


Introduction
Groundwater flows from the interconnections of aquifers to the producing well by radial flow (waterflow between aquifers and wells) and obeys to the physical phenomenon based on the relevant physical principles: Darcy's law and mass balance, (Equations (1) and (2) respectively) which are fundamental equations for developing theories in groundwater flow; they are: Darcy's law: and the mass balance This developed theory of flow modeling process is applicable to several areas of flow mechanics, i.e. geothermal reservoirs, hydrodynamics, and mining process.
During the waterflow through the well, the hydraulic charge in the confined aquifer varies from the hydraulic charge h R , due to the well-range R, to the hydraulic charge h W due to the radius of the well a. The present study develops first, the calculation of the drawdown, therefore, the drawdown curve (cone of depression) in the potentiometric surface around the producing well basing on Equation (1) Secondly, it develops a case study of the appearance of groundwater drawdown curves and analyzes through the developed mathematical model, their variation according of the wellhead flowrates during the water production process and their different curves.

Position of the Problem
The objective of the present paper is to determinate the static pressure of the confined aquifer and function of the pressure at the bottom of the drilling, and then simulate the variation of the hydraulic charge of the groundwater in function of the well action radius R, and finally to simulate mathematically the potentiometric surface of the depression cone created by the variation of the radial distance (r) and the variation of the flow rates (Q) at the wellhead, which describes the different levels attended from the static pressure to the hydraulic pressure at the bottom of the well. Figure 1 shows the radial flow in steady-state flow conditions around the well in a confined aquifer [1].  The physical model of the problem is constituted by a confined aquifer, a porous medium which thickness is e; and a producing drilling whose flow rate is Q. The well cross-section is simulated to a circular section which radius is a. In the cylindrical coordinates system ( ) , , O r z ; R h is the hydraulic charge at the limit of the action radius R in the confined aquifer, and W h is the hydraulic charge of the well; the reservoir, the porous medium which thickness is e; and the producing well which flowrate is Q and R h are the hydraulic charge at the limit of the well-range R in the reservoir [1].
The well-range is the locus point for a part of potentiometric surface curve where, the pressure does not influence the potentiometric surface in the reservoir during the groundwater exploitation, comparing to its axis, and the value of the depression cone (pumping-out), i.e. the drawdown is not available;

Materials and Method
The following assumptions for the governing equations are the following: the porous medium is isotropic, homogenous; permeability, porosity (ϕ), hydraulic conductivity (K), and productive thickness (e), are supposed to be constant; the density (ρ), the dynamic viscosity (μ) and the cinematic viscosity (ϑ) of the groundwater are considered as constants.
The water is produced in accordance to Figure 1 with the well diameter is (D), the drilling depth is (H) and its flow rate is (Q).
The boundary conditions for the governing equations are the following.

Governing Equations
The governing equations are [1] [2]: The generalized law of Darcy: in the case of an incompressible fluid, The continuity equation: The diffusivity equation: Considering the porosity ϕ independent of time, the porous medium non-deformable, taking only the hydrodynamic part and the temperature constant, ρ depending only on p, we got: is the storage-specific coefficient which shows the ability of the porous medium to release fluid when increasing the pressure. By introducing Equation (6) in Equation (4), we obtain: Since the fluid is incompressible, thus, 0 we got the mass balance equation for an incompressible fluid in a non-deformable medium: the diffusivity equation for the confined groundwater aquifer.
In steady flow condition, and thus: This Equation (9) allows to estimate the (hydraulic charge) potentiometric surface created hydraulic charge expression ( ) h r in function of the hydrogeological parameters. In cylindrical coordinates, the diffusivity equation can be written as follows: As the flow is axisymmetric flow, and there is no horizontal flow so Thus the previous differential equation is reduced to: Using the boundary conditions: The general solution is then: The flow velocity into the well is calculated by using Darcy's equation and taking into account the previous boundary conditions: The flow rate can be determined by evaluating the flow at any radial distance r and integrating over the flow surface. Taking into account r a = , we get: Open Journal of Fluid Dynamics According to Laurent et al. [3], in a steady flow, the hydrodynamic flow pres-

Evaluation of the Static Pressure in the Aquifer
The static pressure of the water at any point of the aquifer is defined with the equation of Bernoulli [5] [6]. To formulate its expression, the simplifying assumptions are the following [7]: -The reservoir is at a constant temperature of 20˚C, and so negligible, At the wellhead, the pressure can be expressed as [ The calculation of the static pressure assumed the use of the parameters of the well and aquifer. Just before the pumping operation ( 0 t = ), the hydraulic charge is uniformly distributed throughout the aquifer, the initial condition is The two boundary conditions associated with the problem are the following: -At the aquifer: r R = , the charge is R h -At the well: r a = , the charge is W h

Hydrodynamic Parameters of the Reservoir
From the test pumping carried out on the aquifer of Monzoungoudo by DGEau, we obtained some values of hydrodynamic parameters whose are the following: -Temperature of the well is constant at 20˚C,

Geometric and Hydrodynamic of the Well of Monzoungoudo
Geometric and hydrodynamics data of Monzoungoudo well, are provided and collected in Table 1

Results and Analysis
The following Figure 3(a) and Figure 3(b) reveal the evolution of the hydraulic charge as a function of radial distance when water flowrates vary; 0.002 Q  0.005 in Figure 3(a) and 0.006  Q  0.009 in Figure 3    In the first interval, no matter the flow rates, all curves get a common part where all the curves are identic and mixed; and in the second one, the curves are different, in accordance to water flowrates.
Observing curves, we note that, the drawdown (s) tends to zero (by adopting as asymptotic direction, the horizontal direction i.e. the static pressure piezometric surface level) while the radial distance tends to well-range. Figure 3 proves that the drawdown occurs significantly in the close proximity of the well, for variable different flowrates. It belongs to the vertical line delimiting the influence zone of the drawdown in two intervals. Thus, the following remarks are: -In the case of 0.063 m 33.74 m r ≤ < , the vertical asymptotic direction, is the hydraulic charges axis. In this zone, the drawdown is appreciable until the radial distance 33.74 m r = .
-At the P point (33.74 m; 286.65m), the drawdown is identic no matter the value of flowrate; when 33.74 m r = , all hydraulic charge curves intersect regardless of the value of the flowrate; thus; at this point, the common optimum value of drawdown opt s is available. -When 33.74 m r > , for higher production flow rate, the closer representative hydraulic charge curve is in the undisturbed aquifer charge (initial piezometric surface). The water production does not influence the initial piezometric surface. When the production rate increases, then the corresponding drawdowns are low compared to those of previous flow rates. In the same way, the different drawdowns observed for the different hydraulic charge curves (according to the different flow rates) decrease when the radial distance (r) increases and are cancelled while the radius of action (R) of the well is reached.

Conclusion
This paper attended to the mathematical steady-state radial modelling simulation of flow around the production well in the confined aquifer of Monzoun-