Welcome! My name is Hubert Savenije and I am a hydrologist.
In this Module we'll discuss the runoff generating mechanisms
Maybe the most predictable part of a hydrograph is the recession curve
which represents the slow groundwater depletion during the base flow of a river
The recession flow obeys the water balance of the renewable (deep blue) groundwater
In the dry season the deep blue groundwater has no recharge and in the river there is
no surface runoff.
So the catchment discharge consists of groundwater and the active storage is the groundwater
storage
It is the water balance of the dark blue box
This results in a very simple water balance for dry weather
On top of that , we may assume that the groundwater discharge is directly proportional to the
storage
The proportionality coefficient k represents the average residence time of the water in
the groundwater reservoir
Mathematically , it can be easily shown that the solution of these two equations is an
exponential function
(please see the example and do it yourself)
One can easily recognise the residence time k from the graph,
because it is the time where the tangent to the curve hits the time axis
Here you see this illustrated.
The exponential function has special properties:
the time derivative of Q is –Q/k
and the integral of Q (being the remaining groundwater storage) is k*Q
Check it yourself!
Now let's discuss the flood generating mechanisms
When we want to determine the runoff as a result of a large precipitation event,
we only should consider that part of the rain which contributed to runoff:
the effective precipitation Pa.
We thus have to subtract that part of the precipitation that replenishes the soil moisture
or that fills stagnant pools, later to be evaporated.
The evaporation itself during a large precipitation events we may neglect.
So how do we subtract this increase of temporary storage from the precipitation?
There are different methods.
Some subtract a certain percentage
Some subtract a fixed threshold
Some distinguish between a short term buffer (pool formation) and a longer term threshold
(soil moisture storage)
And others assume a sort of maximum infiltration capacity.
We generally take the fixed threshold for its simplicity.
We saw that groundwater depletion can be simulated as a linear reservoir.
If groundwater is dominant in a catchment, we could simulate both the fill and the depletion
by a linear reservoir
We call that the storage principle.
The solution is an exponential function with a fixed time scale k
For every time increment, we can apply this principle, whereby the runoff tends exponentially
to the effective precipitation Pa,
starting from the discharge in the previous time step
We can do this analytically,
but also numerically
The numerical approach can be easily done in a spreadsheet (please look at the example)
The numerical solution is that the discharge Q2 at the next time step
is a function of the effective precipitation and the discharge in the previous time step.
The coefficients depend on both the time scale and the time step.
It is easy to see that the sum of the two coefficients is one.
Why is that?
Here is what the result look like for different values of k
If a catchment is not groundwater dominated, but rather dominated by fast runoff
(for instance a paved area, or relatively impervious area, or a hillslope with underlying
impervious hard rock)
then the runtime principle can be very useful.
This approach is also called the Rational method,
whereby the runoff is proportional to the surface area contributing to the runoff.
This area increases over time as an ever larger area of the catchment contributes to the runoff,
depending on the time needed for the water to reach the outfall.
Imagine you are in an amphitheater and ping-pong balls come falling from the sky at a continuous
rate
The balls that fall closest to the stage discharge first.
Only when the balls from the highest seats in the amphitheater have reached the stage,
is the discharge equal to the flux coming from the sky.
This is how it would look.
The top graph shows the precipitation rate
The middle one the contributing area
and the lower one the discharge.
At the end, the discharge equals the precipitation rate
Of course this is only correct under a range of limiting assumptions,
But in relatively small catchments in urban or impervious areas, it may work very well.
Mathematically it can be described by a simple threshold function, described bij the MIN
operator
And the interesting part is that the runoff can be described purely as a function of only
effective precipitation and the so-called time of concentration:
the time it takes for the most remote part of the catchment to contribute to the discharge
If you have subsequent precipitation events of fixed time steps, then we can also transfer
this solution into a numerical scheme
which can be readily incorporated in a spreadsheet
Here you see outputs of the spreadsheet for the same effective precipitation event, but
with different times of concentration
Here are some questions for you to reflect on
The storage principle and the runtime principle are primarily educational tools,
although engineers may use them for particular circumstances, as long as they realise the
limitations.
But real catchments are more complex
Scientific hydrologists still don’t exactly know how water moves through the terrestrial
system.
I plan to make a follow-up course in hydrological modeling to help you to further explore runoff
generating processes
“There still is a world to discover in Hydrology”