18.6.1. GATE 2006 Example

Let us consider that there is a soil matrix, that means there are soils that are there which is having porous space, and in that soil component we have the flow. The water is coming, it is Q1, Q2, and Q3 is going out. And at the bottom, there is percolation or seepaging.

In this problem, we are dealing with a soil matrix that is porous. Porous soil allows water to flow through it. The flow is defined by various inputs (inflows) and outputs (outflows). Here, Q1 and Q2 represent the inflows of water into the soil from two different points, while Q3 represents water flowing out of the soil. Additionally, there is also a percolation process occurring at the bottom, which further influences the total flow of water in the system. Understanding this flow interaction is crucial as it helps determine how much water is retained in the soil and how much seeps out.

Think of a sponge placed in water. When you pour water into the sponge (like Q1 and Q2), it soaks up the liquid. However, if you hold the sponge above a bowl, water drips out (like Q3), depending on how full the sponge is. The more water you pour, the more drips out until it reaches a point where it can’t hold any more.

[The soil matrix is filled with water by the two one-dimensional inlets and one outlet with the downwards percolation. Find out the amount of percolation from the given data. Q1 = Q2 = 0.1 lit/sec, Q3 = 0.05 lit/sec and q = f(s) = KS + 0.1 where S is storage and K is hydraulic conductivity.]

To find out how much water is percolating, we apply the conservation of mass principle. This principle states that the total mass (or volume, since we're assuming constant density) entering a system must equal the total mass leaving the system plus any change in storage within that system. With Q1 and Q2 as inflows, and Q3 as the outflow, we can calculate how much water remains in the soil matrix by looking at the differences between these quantities.

Imagine filling a bathtub with two faucets (Q1 and Q2) while there’s a drain (Q3) at the bottom. If the water flowing in from the faucets equals the water flowing out from the drain and any extra water is just filling the bathtub, you would have a steady level of water. But if the drain is slightly larger than the faucets, the water level eventually decreases.

Flow classification: One dimensional, Unsteady, Laminar, Fixed control volume, Incompressible flow.

This classification gives us an idea of the nature of the flow we are dealing with. One-dimensional means the flow can be described along a single line. Unsteady indicates that the flow parameters (like velocity or pressure) change with time. Laminar flow suggests the flow is smooth and layered rather than chaotic. Fixed control volume indicates we are observing the flow within a set boundary. Lastly, incompressible means that the fluid's density remains constant throughout the flow.

Consider a calm stream of water flowing rapidly down a slide. The water moves in layers (laminar), follows a one-way path straight down (one-dimensional), and if you let it flow for a while, the speed of the water does not change much over time (steady).

Applying the control volume approach, equation for the unsteady flow with two inlet and one outlet.

Now, we set up our control volume, which includes the inputs and outputs of water. The mass conservation equation reflects that as the water enters from Q1 and Q2 and leaves through Q3, these two inflow variables and the outflow must relate to changes in storage (or water level) in the soil. We can formulate the equation to account for these variables, thus allowing us to solve for unknowns like water storage or seepage rate.

Think again about the bathtub: if you add water at the same rate you drain it, the bathtub stays at the same level. If you increase the inflow, more water gets stored until you hit overflow! This balance is what we track with our equations.

So, that way, if you look, very complex problems like this, when you have a soil matrix and porous structure and you have the flow of Q1, Q2 inflows, and outflow is there, the seepage is a function of how of water storage within the soil matrix.

In conclusion, through the setup and application of conservation of mass, we can analyze complex systems. By carefully considering the inputs and outputs and applying the steady flow principles, one can determine how much storage occurs within the porous soil. This allows us to make predictions or calculations about water percolation and storage over time based on the defined equations.

If you were to monitor a sponge soaked in water, you'd notice that the initial amount of water would change based on how quickly you squeezed it out (analogous to outflow), how much you added by wetting it (inflow), and how quickly those balance. Over time, you would see a pattern in how full the sponge becomes or retains water.

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