# Groundwater and Well Hydraulics: Steady Unidirectional Flow

## Groundwater and Well Hydraulics

Darcy’s law and the fundamental equations governing groundwater movement can now be applied to particular situations. Solutions of groundwater flow to wells rank highest in importance. From pumping tests of wells, storage coefficients and transmissivities of aquifers can be determined.

## Steady Unidirectional Flow

Steady flow implies that no change occurs with time. Flow conditions differ for confined and unconfined aquifers and hence need to be considered separately, beginning with flow in one direction.

### Confined Aquifer

Let the groundwater flow with a velocity v in the x-direction of a confined aquifer of uniform thickness. Then for one-dimensional steady flow-

**∂ ^{2}h/ **

**∂x**

^{2}= 0Which has for its solution

**h= C _{1}x + C_{2}**

where h is the head above a given datum and C1 and C2 are constants of integration. Assuming h=0 when x=0 and from Darcy’s Law-

**∂h/ ****∂x= (-w/K)**

Then we have-

**h= -vx/K**

This states that the head decreases linearly with flow in the x-direction-

### Unconfined Aquifer

For the similar flow situation in an unconfined aquifer, the direct analytic solution of the Laplace equation is not possible. The difficulty arises from the fact that the water table in the two-dimensional case represents a flow line. The shape of the water table determines the flow distribution, but at the same time, the flow distribution governs the water table shape. To obtain a solution, Dupuit assumed the velocity of the flow to be proportional to the tangent of the hydraulic gradient instead of the sine as defined in Darcy’s law and the flow to be horizontal and uniform everywhere in a vertical section.

These assumptions made, although permitting a solution to be obtained, limit the application of the results. for unidirectional flow the discharge per unit width q at any vertical section can be given as

**q= -Kh ^{2}/2 + C**

and if h= h_{0} where x=0 then the Dupuit equation

**q= (K/2x) (h _{0}^{2}-h^{2})**

results which indicates that the water table is parabolic in form.

### Base Flow to a stream

Estimates of the base flow to streams or average groundwater recharge can be computed by applying the above analysis of one-directional flow in an unconfined aquifer. For example, picture the idealized boundaries with a continuous recharge rate W occurring uniformly over the aquifer. With the Dupuit assumptions, the flow per unit thickness is

**q= -Kh(dh/dx)**

and by continuity

**q= Wx**

Combining these equations and integrating leads to the result

**h ^{2} =h_{a}^{2} + W(a^{2}-x^{2})/K**

where h, h_{a}, a and x are defined in fig and K is the hydraulic conductivity. From symmetry and continuity

**Q _{b}= 2aW**

Where Q_{b} is the base flow entering each stream per unit length of the stream channel. If **h** is known at any point Q_{b} or W can be computed, provided K is known.

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