How to solidify photogrammetry model by filling it from inside? I struggle with making a photogrammetry model a solid. 
I tried a Solidify modifier, which works quite well until the thickness value is big enough to intersect trough the outer shell and destroy the fine structure. And unfotunately the safe value (for not intersecting the outer shell) is not enough for a desired thickness of my solid to be 3D printed. 

Then I just made a solid by intersecting the model with a cube, triangulating the new solid and trying to scrape out the material using sculpt. OK, still I cannot control the distance between the model shell and my digging. This results in occasionaly intersecting the outer shell. And I want to optimize its volume to keep the printing cost lower. 

So, is there any way to fill the mesh (viewed from bottom on the attached picture) like "pouring some clay" inside to stick to the faces in a given distance? Any parameters to Sculpt tools that would not affect the outer original shell?

Please, help me to solve this issue as it seems to be out of my Blender experience. 

 

EDIT: After 10 Replies answer I need to add a visualization of the final result of the recommended transformations. The lighter shaded faces are those of a "shrinked" extruded inner shell.
Try going into edit mode after selecting the model. Press E and then Z to extend only in the z direction. Now, if you want the bottom to be completely flat, press S, and then Z to only scale in the z direction, and then type 0 on your keyboard.