## Steady Radial Flow to a Well

When soil is pumped, water is removed from the aquifer surrounding the well, and water table or piezometric surface, depending on the type of aquifer, is lowered.

### Confined Aquifer

To derive the radial flow equation (which relates the well discharge to drawdown) for a well completely penetrating a confined aquifer will prove helpful. The flow is assumed two-dimensional to a well centered on a circular island and penetrating a homogeneous and isotropic aquifer. Because the flow is everywhere horizontal, the Dupuit assumptions apply without error. Using place polar coordinates with the well as the origin, we obtain the well discharge Q at any distance r as-

**Q= Av = -2πrbK (dh/dr)**

For steady radial flow to the well, Rearranging and integrating for thew boundary conditions at the well, **h=h _{w,} **and

**r=r**and at the edge of the island,

_{w}**h=h**and

_{0,}**r=r**yield,

_{0},**Q= 2πKb{(h _{0}-h_{w})/ln(r_{0}/r_{w})}**

With negative sign neglected.

In the general case of a well penetrating an extensive confined aquifer, there is no external limit for r, from the above derivation at any given value of r,

**Q= 2πKb{(h _{0}-h_{w})/ln(r/r_{w})}**

Which shows that h increases indefinitely with increasing r. Yet, the maximum h is the initial uniform head h_{0}. Thus, from a theoretical aspect, steady radial flow in an extensive aquifer does not exist because the cone of depression must expand indefinitely with time. However, from a practical standpoint, h approaches h_{0} with distance from the well, and the drawdown varies with the logarithm of the distance from the well.

Theoretically, h_{w} at the pumped well can serve as one measurement point; however, well loses caused by flow through the well screen and inside the well introduce errors so that h_{w} should be avoided. The transmissivity is given by-

**T= Kb = {Q / 2π(h _{2}-h_{1})} ln(r_{2}/r_{1})**

### Unconfined Aquifer

An equation for steady radial flow to a well in an unconfined aquifer also can be derived with the help of the Dupuit assumptions. The well completely penetrates the aquifer to the horizontal base and a concentric boundary of constant head surrounds the well. The well discharge is

**Q= -2πrhK (dh/dr)**

Which, when integrated between the limits **h=h _{w}** at

**r=r**and

_{w}**h=h**at

_{0}**r=r**, yields

_{0}**Q= πK{(h _{0}^{2}-h_{w}^{2})/ln(r0/r_{w})}**

Converting to heads and radii at two observation wells

**Q= πK{(h _{2}^{2}-h_{1}^{2})/ln(r_{2}/r_{1})}**

And rearranging to solve for the hydraulic conductivity

**K= {Q / π(h _{2}^{2}-h_{1}^{2})} ln(r_{2}/r_{1})**

This equation fails to accurately describe the drawdown curve near the well because the large vertical flow components contradict the **Dupuit assumptions**; however, estimates of K for given heads are good. In practice, drawdown should be small in relation to the saturated thickness of the unconfined aquifer.

The transmissivity can be approximated by-

**T = K (h _{1}+ h_{2}) / 2**

### Unconfined Aquifer with Uniform Recharge

A well penetrating an unconfined aquifer that is recharged uniformly at rate W from rainfall, excess irrigation water, or other surface-water source. The flow Q toward the well increases as the well is approached, reaching a maximum of Qw at the well. The increase in flow dQ through a cylinder of thickness dr and radius r comes from the recharged water entering the cylinder from above: hence,

**dQ = -2πr (dr) W**

Integration we Obtain,

**Q= – πr ^{2}W + C**

But at the well **r=0** and **Q=Q _{w}**, so that,

**Q= – πr ^{2}W + Q_{w}**

Substituting this flow in the equation for the flow to the well gives-

**-2 πKh(dh/dr) = – πr ^{2}W + Q_{w}**

Integrating, and nothing that h=h_{0} at r=r_{0}, yield the equation for the drawdown curve;

**h _{0}^{2}-h^{2}= W(r^{2}-r_{0}^{2}) / 2K + Q_{w}ln(r_{0}/r) / πK**

It follows that when r=r_{0}, Q=0, so that from equation-

**Q _{w}= πr_{0}^{2}W**

Thus, the total flow of the well equals the recharge within the circle defined by the radius of influence; conversely, the radius of influence is a function of the well pumpage and the recharge rate only.

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