Modeling the Effects of Hyporheic Flow on Stream
Temperature
Zachary Salem Enrique Thomann, Jorge Ramirez, Julia Jones Im going
to present to you today a model we have been working on this
summer.With the help of Enrique Thomann, Jorge Ramirez, and Julia
Jones we developed a very simplified model for the effects
hyporheic flow has on stream temperatures. Modeling the Effects of
Hyporheic Flow on Stream Temperatures
Potential and Irrotational Flow Velocity Streamlines Stream
Temperature Model Conclusions I will give a brief description of
what exactly hyporheic flow is an how it will affect temperatures
and also how it varies from stream to stream.I will talk a bit
about the modeling that we did using fluid dynamics and potential
flow for the hyporheic flow itself.Ill then show how those
equations can be related to the ad equation to explain temperature.
Hyporheic Flow Hyporheic flow occurs when water leaves the stream
channel and enters the soil.From there the water is warmed or
cooled before returning to the stream channel where it affects
water temperature in the stream.The flow can either go down and
enter the stream bed or it can move laterally and into the side of
the channel due to meandering or other aspects.Hyporheic flow is
not to be confused with groundwater flow as groundwater comes from
water that has not previously been in the stream channel. Hyporheic
Flow Photo Zack Salem 2007
Hyporheic flow can be controlled by such as stream substrate, the
shape of the stream, the velocity of the stream, and many other
factors.In streams such as this one, where it is entirely exposed
bedrock, there is little to no hyporheic flow.Even though pools
build up behind steps there is no place for the water to go other
than over the step. Photo Zack Salem 2007 Hyporheic Flow Photo by
Mike Gooseff
This stream is a much different case.As you can see, the stream is
composed of loose rocks and sediment.Streams like this will have
more hyporheic flow than the previous one. Even though they both
have pools then steps down, the different streambed material allows
flow to go through it rather than just over or around. Photo by
Mike Gooseff Hyporheic Flow Gooseff et. al. 2005
This is an image of hyporheic flow made by performing tracer tests
at the HJ Andrews.As you can see by the lines, hyporheic flow seems
to center itself around the steps in each stream reach. Gooseff et.
al. 2005 System This is what our end product looks like. From here
I will explain how we gathered all of this information. Potential
Flow http://dehesa.freeshell.org/FDLIB/mdc.html
To develop this model we wished to explain the stream velocity
through and around an obstacle in the stream channel.We used fluid
mechanics and the theory of potential flow to do this.Potential
flow is defined by that equation which is called the complex
potential.Phi is what is known at the velocity potential, which can
be used to compute the velocity of the particle while Psi is called
the stream function which can be used to determine streamlines, the
paths the water will take in the system. Potential Flow Hyporheic
Zone Stream Channel
This is how we are considering our stream.We have a level stream
with perfect semicircular bump in it.The formula for potential flow
around a porous circular object is these.This is the formula for
the stream channel and this is for the hyporheic flow.Z is a
complex variable and when you convert these equations to polar
coordinates you end up with these. U_c is the velocity of the
stream up some infinite distance and is a known value.U_h is not a
known value and is the velocity in the hyporheic flow. Now from the
prior equation, we had a real part phi which was the velocity
potential and we had the imaginary part psi which was the stream
function.Solving these equations two equations for phi and psi
allow us to determine the velocity in both regions. Stream Channel
Velocity
Since we know phi and psi we can determine two components u and v
of a velocity vector.This is for the stream channel and we know
that this equation is the x-component and this is the y
component.So what this tells us is that at for example this point,
you simply plug in the coordinates to each of these equations and
you get the x component which is the velocity going downstream this
way and the y component which is the velocity this way. Hyporheic
Flow Velocity
Now for the hyporheic flow the equation is a little simpler and you
get these equations.So in this entire region the velocity of the
water does not change. It does not have a y component either, the
water simply moves in the x direction with some velocity.This
velocity is determined by using this condition.This is based on the
interface between the hyporheic flow and the stream channel. All
these numbers here are known or can be tested so you can relate the
velocity of the hyporheic flow to the velocity of the channel. This
is the porosity of the channel and this is the porosity of the
hyporheic flow.Kappa is the difference between the permeability of
the channel and that of the hyporheic zone.So now we are able to
determine the velocity of the stream any hyporheic flow at any
point. Streamlines Stream Channel
Streamlines are found using the psi function from before.We found
that this is the equation for the streamlines.Which look like this.
Streamlines Hyporheic Zone
In the hyporheic flow its a little simpler because there is no y
component to the velocity so the water just flows in the positive x
direction. Potential Flow This is what everything looks like
together.We dont have any velocity in the soil down here but we
have a velocity vector of Uh in the hyporheic zone, and Uc in the
stream channel.Some interesting points are evident from the
equations, we find that at these two points there is no flow.They
are what we call points of stagnation.Also at this point up here
the river is moving fastest.Right on the top of the semicircle the
stream is moving at 2Uc, 2 times as fast as it is upstream. Heat
Equation Now we wish to take what we have found and apply that to a
heat equation and stream temperature.This equations is the
advection-dispersion equation and the basis for the following
work.D in this equations is a diffusivity matrix which is dependent
on the region you are describing.Additionally U is dependent on the
region you are describing.Now we can develop a set of heat
equations for each region, Heat Equations This is the heat
equations for the water in the channel.This is the equation for the
water in the hyporheic, and this is for the temperature in the
soil. Heat Equations Now what this equation tells us is that if
some heat is added right here its movement is dependent on the
diffusivity of the material and the velocity.So by this part, the
faster the water is moving the faster the heat is moved down stream
or up in the stream.Since in the hyporheic flow we do not have any
vertical movement of water we dont have this term.In the soil we do
not have any movement of water so we have neither term, only the
diffusivity within the soil. In order to solve these equations we
need boundary conditions between each region Boundary Conditions
Continuous heat transfer
Conservation of mass of water Temp of water entering stream channel
Temp of soil at a depth Now we have these equations which ill
briefly describe.We have a set of these top two equations for each
interface, between the channel and the soil, the channel and the
hyporheic zone, and the hyporheic zone and the soil. We have
continuous heat transfer, which means that heat is going to move,
theres no switch to turn it off.We also have conservation of mass
of water, so any water that enters one of these regions is going to
leave.We also have a constant temperature of water entering the
stream reach in addition to a constant temperature of the soil at
some depth and any depth below that. Initial Conditions Temperature
Function at time t=0
We also need some initial conditions at time zero.Here we need to
know the temperature in each region and because none of the regions
are going to be a constant temperature it needs to be a function,
which we call f.These equations are the formulas for this model.
From them we can make some conclusions Conclusions Future Work
Complex Model Time Limitations Data Multiple Obstacles
Atmospheric Conditions This is a very complex model and because of
time limitations I was not able to use real data to solve this but
I think future expansion should include data to solve.Additionally
it can be expanded to behave more like a real stream ecosystem by
adding more obstacles in the stream.It would also be possible down
the line to add air temperature and other factors.