Testing for sediment mass conservation

[elbeejay]

From discussions in #291:

The model is supposed to be mass conservative of sediment, at least generally and in it's overall formulation. More specifically, the model is actually volume conservative, since there is no porosity term. The model is explicitly not mass conversative of water, though, and explicitly not conservative of the difference between sand and mud (but rather just total sediment volume).

Volume leaving the domain is okay, so long as we know about it. I forget what happens when stepmax is reached, but I thought that material is just dropped in place, not discarded. So in either case, mass (volume) should be conserved (not creating new mass or losing track of mass).

We could check that by differencing topography after 10 time steps from the initial topography, and comparing that to the volume input times elapsed time. They should be the same. The exception is likely to be the inlet cells, where a boundary condition on elevation is applied, violating conservation there.

We could also do this by loading up the checkpoint run, just to see how close things are after a long period of time.

You don't need to do these for this PR/issue, just talking out load.

Originally posted by @amoodie in #291 (comment)

[elbeejay]

I'm also a bit fuzzy on my memory of the exact process, but I thought the parcels just "vanish" when they hit the stepmax criteria (https://github.com/DeltaRCM/pyDeltaRCM/blob/develop/pyDeltaRCM/sed_tools.py#L753) and therefore take any sediment they are holding out of the model; thus mass is not explicitly conserved.

With the original MATLAB model I do recall doing some testing to see if we were losing too much mass, and my recollection is that the losses were on the order of single-digit percentages of the input sediment volume. As noted in the documentation where suggestions regarding model stability and sediment routing are made (https://deltarcm.org/pyDeltaRCM/info/morphodynamics.html#model-stability), I think "mass conservation" was sensitive to things like base level change as things like those "scour holes" meant sediment parcels were sucking up a lot of mass and then disappearing.

[elbeejay]

Ran the tests mocked up here just to get a relative sense of the scale of the "missing mass". The volume discrepancy appears to be roughly 1%, which tracks with my recollection that it was a small amount of sediment being "lost" (although "lost" could also be viewed as "tranported off-shore" or "compacted", two real processes we do not model).