I am not quite sure how the data looks like but here is a wild guess

select 

       BLOCK_ID

     , DENSITY

     , AVG_THICK

     , BLOCK_SIDE

 , (case when BLOCK_SIDE ='top' then 23 else 0 end)

  - (case when BLOCK_SIDE ='top' then 1 else -1 end) * 

    (select sum(AVG_THICK)

     from dbo.THICKNESS_TABLE y

      where y.[TIME]  <= x.[TIME]

      and y.BLOCK_ID = x.BLOCK_ID)

  as running_total

from dbo.THICKNESS_TABLE x

where BLOCK_ID like [conditional parameter]

group by BLOCK_ID, DENSITY, AVG_THICK, BLOCK_SIDE 

I assumed BLOCK_SIDE  can only be 'top' or 'bottom', right?

hth