How can I Use percent_dV(z) to calculate dV(t) in a geometry(z)?

[@PinkGeology]

So, I have a solution script that predicts how much the volume of a fluid will change as a function of z ... this is a magma expanding as gas in solution is depressurized, if that problem sounds strange. :)

Now, the REST of my problem is that this magma is filling a tall, thin ellipsoid stretching from z = 6000 m (depth in the Earth) to z = 800 m at some rate Q such that the volume of magma in this ellipsoid (which expands like a balloon to be the size of the magma) at any given time = Q * t (where t = time the magma has been running intp the system.

I'd like to write a script that describes the expansion of that final volume (Q * t) over the depth using the first solution, which is ...

(dV% = density_1/density_2)
density_2 = density_gas/mass_frac_of_gas + density_water/mass_frac_water + density_1/mass_frac_magma

... I guess what I want is a graph showing total change in volume over time at some given Q, taking into account expansion as a matter of depth and the geometry of this ellipsoid spread out over that depth.

I am just exhausted this week and even though I know there is a straightforward solution, I don't seem to have the available brain cells to put it together. Ideas?

 

[gtoguy97]

The best way to approach this problem would be to use numerical integration. You need to calculate the change in volume at each depth, and then add them up to get the total change in volume over the entire depth range (from 6000 m to 800 m).You could use a numerical integration method such as the Trapezoidal rule or Simpson's rule to calculate the area under the curve of the volume change at each depth. This should give you the total volume change over time at some given Q.Once you have the total volume change over time at some given Q, you can plot it on a graph to show the expansion of the final volume (Q * t) over the depth.