Hydrogeology

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Goals of this unit

Handouts for out class discussion

Handout - application of integration: arc length.

Handout: 1D groundwater flow. Note: there are some sample calculations you can do here.

Handout - Darcy's Law and Well Production. Note: there are some sample calculations you can do here.

Reference Figure: Unconfined aquifer during pumping. How would we describe the water flow?

unconfined

Handout: Determining Hydraulic Conductivity via a Pump Test

 

Reference Figure: Confined aquifer during pumping. How would we describe the water flow?

ca1

ca2

Well Intake (surface) Area

wisa

Deriving the flow equation for the confined aquifer case

Here is a sketch of the confined aquifer case:

ca2

FIGURE A (above)

Assumptions we will make:

ca1

FIGURE B (above)

HERE WE GO

Darcy's Law says:

Q=KAhAhBL. Using Figure A, we can rewrite this as Q=KAΔhΔr, or replacing the Δ with differentials (as we set up for integration below) we have: Q=KAdhdr

The area A of the well-screen is the cylinder area: A=2πrB. Substituting this for A in we can how get: Q=K2πrBdhdr. This is a nice little differential equation.

Now I advertised an integral, and integrals add things up. What are we adding up?.

Looking at Figure A and our last equation, the flow rate Q is made up of contributions across the radii and corresponding head, h changes. So we want to add all of that together -- i.e. integrate.

But in what form will we integrate the last equation? Let's rearrange to separate the differentials:

Qdrr=2πBKdh.

We can now integrate each side - but what are the limits? On the left, r1 to r2, and on the right the corresponding h1 to h2.

r1r2Qrdr=h1h22πKBdh

This gives us, after integrating and subbing in the limits of integration:

$ Q \cdot \ln \frac{r_1}{r_2} = 2 \pi K B (h_2 - h_1)$

Finally, solving for Q gives us:

Q=2πBK(h2h1)ln(r1r2)

Remember this is for the confined aquifer case.

For the unconfined case, there is a modification because now the drawdown curve physically occurs within the aquifer (unlike the hypothetical potentiometric surface of the confined aquifer), so the height of the intake screen on the well is not constantly equal to B. In other words, the height of the aquifer is variable, so B gets replaced with h.

This will then integrate out to: Q=πK(h22h12)ln(r1r2)

 

 

 

 


The above material is motivated by this from the book Geology for Engineers & Environmental Scientists by Alan E. Kehew (3rd edition).