For shots like the ones below (top image is from Dragon Wars
and bottom is from Armageddon
) the most ideal way to achieve that shot is make a matte painting of the crater and composite it into the shot, blend the edges of the crater into your original footage, and then composite some smoke rising in a few areas.
If you can't create a matte painting you should get a chroma green colored poster board, cut a hole in the middle of it in the shape of your crater, go outside and lay it over some soft damp dirt (damp because it will be easier to mold and it will stick together better so it won't run everywhere and make a mess, don't make it too damp or it may get into your poster board and discolor the chroma green). Through the hole in the poster board, dig a hole in the dirt with a small shovel and place the dirt you dig out around the edges of the poster board hole around the area that the dirt would blast up if a big object slammed into the ground. After you dig out the dirt, you can mold the damp dirt a bit to look more dramatic and realistic as if its a bigger scale than what it really is.
Since nothing in this stage of the shot is moving, it would be best to set up a still image camera on a tripod and take a hi-rez picture of the crater with the green base around, well lit, and at the right angle required to match your original footage. Now just take this image and key out the green and composite it onto your original plate footage and touch up the edges to blend with your original plate and adjust the color grading to match it, then just add small smoke stacks rising where needed/wanted.
In The Hills Have Eyes
(2006) theres a scene where a character discovers a car graveyard where the mutants drive all the cars of their victims and store them, which is a huge crater from one of the bombs tested in the desert. Its an awesome vfx shot which has a shot breakdown in the special features section of the DVD, unfortunately I couldn't find a frame of the shot though on the net.