If a spherical object of certain radius and mass (let's say 10 ft, 3,000kg) is placed in a granular medium such as sand. How can you calculate the max depth to which the object would sink?
As stated, the question is not answerable. There are two missing pieces of information.
1) What is the bulk density of the material, and
2) Is the material static or is it being perturbed (by vibration, for instance)?
If the material is static, and it is at maximum density (that is, it has been firmly packed) the weight will not necessarily sink at all.
At the other extreme, if the material is not compacted and is being constantly vibrated, it will behave pretty much like a liquid, and you can calculate the sink depth by applying Archimedes's principle.
The intermediate stages are extremely difficult to quantify. In a liquid, which can be modeled as a granular medium with molecules as the particles, but zero friction between them, the molecules are easily displaced by the weight, and the weight sinks into the liquid. For real particles, the friction is not zero, although by vibrating the assembly you can get a dynamic condition which approximates it. For low vibration levels there is little effect, but at some point increasing vibration will begin to allow displacement of particles, and the medium will begin to act as a high-viscosity liquid. The greater the vibration levels, the less the viscosity. This will be complicated by the effects of the weight, which will tend to hold the medium directly below it in place due to the effects of pressure on particle irregularities and increased static friction.