Why are we interested in radial flow?
Generally interact with groundwater through wells
Pump = cone of depression
If system homogenous and isotropic, and aquifer is sufficiently large, flow = axisymmetric
Flow to production wells is
Radially convergent
Flow to injection wells is
Radially divergent
Qw =
flow rate out of a well
Derivation of the Thiem equation, basic gist:
- Recall Darcy’s Law
- A = 2piHr
- Separate and integrate w.r.t r
- Impose boundary conditions to eliminate C
- Rearrange for (he-h) =
(6. Recall law of logs to invert fraction and remove -ve) - S =
What is the radius of influence?
re, the well has an insignificant effect on Q
At the extent of the radius of influence r=re and h=he
S =
drawdown - the amount hydraulic head has been drawn down as a consequence of water flowing out
= he-h
When is Q = -Qw
When water is moving in the opposite direction to r
Thiem equation
S = (Qw/2piHK)ln(re/r)
or
S = (Qw/2piT)ln(re/r)
Assumptions for Thiem equation
Homogenous
Isotropic
1D radial flow (head get only in radial direction)
Infinite aquifer
S-S conditions (no variation in time)
Darcy’s law applies i.e. head gradient linearly proportional to flow rate
How can we use Thiem’s equation?
- For given well radius rw and production rate Qw we can forecast the drawdown Sw
- For a given well radius rw and drawdown Sw we can forecast the production rate Qw
Principle of superposition
Can describe how a system will respond to multiple simultaneous events by adding together the responses that would be expected if each event occurred by itself
Estimating T from Thiem, basic gist:
- Form Thiem’s equation for the production well and observation well separately
- Separate the ln(x/y) into lnx - lny
- Take one away from the other to eliminate re
- Rearrange so that T =
Estimating re from Thiem, basic gist:
- Form Thiem’s equation for the production well
2. Rearrange to that re =
Flow to a well near a river, assumptions:
Constant stream stage therefore interface with aquifer = constant head/equipotential boundary
BOTH well and stream fully penetrate aquifer (not actually in practice but ~small error)
No sealing layer of fine sediment on streamed (full hydraulic connection)
Pseudo-steady state conditions
Flow to a well near a river/impermeable boundary, basic gist:
“Method of images”
- whatever you have, do an image of the opposite
- Drawdown due to production/injection well (Thiem’s equation)
- Drawdown due to image well
- Drawdown due to both = add
Calculating the radii from the river/impermeable boundaries to the wells
Draw diagram
Use Pythagoras
What is the drawdown at the river? (and how)
S = 0
- river exists where x = 0
- substitute this into Thiem equation with pythag radii
- Qw/2piT ln(1) = 0
Sources of well losses
Near pumping well, flow velocities are artificially enhanced
Turbulent effects associated with flow through the well screen and the standpipe up to the well pump
= inertial/turbulent effects significant
= non-linearity in drawdown flow currents
Step drawdown test
Pump a well at sequentially increasing rate (Qw)
After certain amount of time, drawdown reaches a steady value
Plot these against production rate
Sw = AQw + BQw^2 (Jacob’s equation)
Using Jacob’s equation
Sw = AQw + BQw^2
- Plot Sw/Qw by Qw:
Sw/Qw = A + BQw^2
- A and B can also be found = well quality (smaller B better)
2. When Qw <=1, Sw~=AQw
3. Compare to Thiem’s equation
4. Rearrange to find T or A
Good well, highly developed - B =
<675
Mediocre well - B =
675 < B < 1350
Clogged/deteriorated well - B =
1350 < B < 5400
