1. Modeling the Effects of Hyporheic Flow on Stream Temperature
Zachary Salem
Enrique Thomann, Jorge Ramirez, Julia Jones
I’m 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.

2. 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. I’ll then show how those equations can be related to the ad equation to explain temperature.

3. 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.

4. 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

5. 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

6. 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

7. System
This is what our end product looks like. From here I will explain how we gathered all of this information.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. Streamlines Hyporheic Zone
In the hyporheic flow it’s a little simpler because there is no y component to the velocity so the water just flows in the positive x direction.

14. Potential Flow
This is what everything looks like together. We don’t 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.

15. 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,

16. 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.

17. 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 don’t 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

18. Boundary Conditions Continuous heat transfer
Conservation of mass of water
Temp of water entering stream channel
Temp of soil at a depth α
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.

19. 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

20. Conclusions

21. 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.