Radial Flow to a Well

  Groundwater resources in a confined aquifer with a nonsteady-state flow can be evaluated for the consideration of the construction of wells. A confined aquifer is a primary source for well tapping and is defined to be an aquifer bounded by an upper and lower bed of material that allows only small amounts of groundwater to penetrate through. A nonsteadystate flow is a flow in which the velocity changes direction or magnitude at some point in time. Equation 2 is the diffusion equation. The solution gives the value of the hydraulic head at any point in a flow field at any time. In this form, the equation describes a flow through a saturated anistrophic porous medium. Incorportating the two basic assumptions of essentially horizontal flow obeying Darcy's Law which reduces the equation to a two dimensional form and that the flow is in an aquifer that is homogeneous, saturated and isotropic (i.e. ), Equation 2 reduces to:

It is assumed that the flow towards the well, where water is being removed, is radial. Hence, we convert Equation 3 into radial coordinates (Appendix D) to describe the hydraulic head in terms of the drawdown around the well.

where

s      = drawdown() [L]

r      = radial distance from the well [L]

t      = time since pumping began [L]

      = hydraulic diffusivity T/S [LT]

T      = transmissivity [LT]
       = function of Hydraulic Conductivity and the thickness of the saturated aquifer

S      = storativity
       = function of the thickness of the saturated aquifer and the specific storativity

  
Figure 3: Cone of Depression in a Confined Aquifer

Equation 4 provides solutions in terms of the drawdown based on the physics of the movement of the flow towards a well during pumping. The drawdown is computed in terms of the radial distance from the cone of depression which surrounds the pumping well. The cone of depression is the potentiometric or gravity free pressure surface and extends radially outward from the well and is a function of storativity, transmissivity, the pumping rate of the well and time. RADIAL

  
Figure 4: Radial Flow from a Well

Figure 4 shows the radial flow from r = 0 at the well to as the cone of depression radiates out from the well.

For the theoretical analysis of the hydraulic head drawdown in a proposed well, we will make the following assumptions:

• A single pumping well in the aquifer
• Pumping rate is constant with time
• Well diameter is infinitesimally small such that the volume of water removed from the well bore during pumping does not affect the total amount of water in the aquifer.
• Well penetrates the entire thickness of the aquifer and receives water by a horizontal flow
• Hydraulic head or initial water level is horizontal throughout the aquifer prior to pumping [8]

Initial Condition

(drawdown at any radial distance at time = 0)
Boundary Conditions for

(drawdown at an infinite distance for any time)

(discharge condition at the well).

• Theis Equation