Hydrogeology - setting up ODE for tank problem

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In summary, to set up the ODE for this hydrogeologic problem, use the mass balance equation and take the derivative with respect to time to express the change in water storage as a function of time.
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RasmusJes
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Hi there.

I have trouble setting up an ODE for a hydrogeologi problem. I have attached the problem and my answers as pdf as they contain diagrams and graphs to help describe the problem. Hope that someone can/will help me. :-) Thank you.

-Rasmus1. The problem is attached in Eng04-2010.pdf
2. I have trouble setting up the equation in question 5. I would very much appreciate a tip to start out with
3. My answers for the first four questions is attached in Exercises_4.pdf
 

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To start with the ODE for this hydrogeologic problem, you can use the mass balance equation. This equation states that the change in water storage (ΔS) is equal to the inflow minus the outflow, which is expressed mathematically as:ΔS = Q_in - Q_outYou can then express the change in water storage as a function of time by taking the derivative of both sides with respect to time:dS/dt = dQ_in/dt - dQ_out/dtThis equation is the governing ODE for the hydrogeologic problem and can be used to solve for S, Q_in, and Q_out.
 

Question 1: What is hydrogeology?

Hydrogeology is the study of the distribution, movement, and quality of water in the Earth's subsurface. It involves examining the processes and properties of groundwater, aquifers, and their interactions with the surrounding environment.

Question 2: Why is setting up ODE important in hydrogeology?

Setting up ODE (ordinary differential equation) models is important in hydrogeology because it allows us to mathematically describe and simulate the movement of groundwater. This is essential for understanding and predicting the behavior of aquifers and designing effective management and remediation strategies.

Question 3: What is the tank problem in hydrogeology?

The tank problem in hydrogeology refers to a commonly used analogy to describe the movement of groundwater in an aquifer. It involves a tank filled with water that has a small hole at the bottom, representing a recharge area, and an outlet pipe, representing a discharge area. The goal is to determine how much water is in the tank and how long it will take for the tank to reach equilibrium.

Question 4: How do you set up an ODE for the tank problem in hydrogeology?

The ODE for the tank problem in hydrogeology can be set up by using the principles of mass balance and Darcy's Law. This involves writing a differential equation that describes the change in water volume over time, taking into account the inflow and outflow rates, as well as the porosity and hydraulic conductivity of the aquifer.

Question 5: What are the key considerations when setting up an ODE for the tank problem in hydrogeology?

The key considerations when setting up an ODE for the tank problem in hydrogeology include accurately defining the boundary conditions, selecting appropriate values for hydraulic parameters, and properly incorporating any sources or sinks of water. It is also important to verify the model results with field data and adjust the model as needed for better accuracy.

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