If you look at a 2D representation of curved space-time around a black hole, it stretches out into an infinite tube, asymptotically approaching the Schwarzschild radius.

image from sciencenews.org image from sciencenews.org
The usual way to explain why light bends in the vicinity of a large object is to say that it actually goes straight along a geodesic if you consider space to be bent into a higher dimension (2D space bent into 3D in the picture, though its evidently mathematically equivalent to say the bending is into the dimension of time).

image from astronomynotes.com image from sciencenews.org
When you get closer to a black hole than the point at which light has a circular orbit (traveling a straight line according to this model), its orbit will decay. Any object with mass will actually fall in faster if you increase its orbital speed, as if centrifugal force is reversed.

The problem is that there doesn't actually seem to be a position in the diagram which has this property unless you make the end of the tube balloon outwards. I'd like to know what concepts have to be added to the model in order to make it accurately describe this case.

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    $\begingroup$ These rubber-sheet images are just crude visualizations to give beginners a little intuition. They’re not really models. If you want a visual aid that is actually correct and useful, learn about the “Schwarzschild effective potential”. $\endgroup$ – G. Smith Dec 20 '18 at 6:44
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    $\begingroup$ Your physical intuition is good! Yes, something is wrong with the rubber sheets: they are simply not valid in GR. Compare the classical and GR Schw. terms in the effective potential of a massive test particle (negligible but non-zero mass) on a momentarily circular orbit. The latter has a GR-specific attracting term related to orbital momentum as $-L^2/r^3$, or $-v^2/r$; the minus means that the "fast" particle will "steer" more on the sheet downwards, ever more parallel to radial gridlines (better think polar than cartesian grid). But graphing the potential is more revealing. $\endgroup$ – kkm Dec 20 '18 at 12:50
  • $\begingroup$ @kkm Yes, something is wrong with the rubber sheets Something? Those are absolute mystification, a real shame. An "elastic" sheet instead of spacetime. A ball whose weight (coming from where?) should warp this pseudo-spacetime. I always wonder how could a minimally competent person have conceived such an idea, and how may it go on spreading. $\endgroup$ – Elio Fabri Dec 20 '18 at 15:02
  • $\begingroup$ OP asks I'd like to know what concepts have to be added to the model in order to make it accurately describe this case. My answer is: the only thing you can do is to throw anything away. $\endgroup$ – Elio Fabri Dec 20 '18 at 15:02
  • $\begingroup$ @ElioFabri, “you are right, too!” One way to understand what is wrong with a model is to attempt at fixing it (and fail at it, feeling how it resists fixing). Another is throw it away and go back to clear whiteboard--and often a blank confused mind. I've seen people arriving at a good answer and understanding in all twisted ways. It just seemed to me that the OP is already well underway in the first approach, and with a good intuitions, so why not? :) $\endgroup$ – kkm Dec 20 '18 at 22:05

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