At a planned construction site, a $2-$m thick stratum of normally consolidated clay underlies a surface layer of compacted granular soil $3$ m deep. The unit weight for the compacted granular soil is $20\mathrm{.}2k\frac{N}{m^3}.$ The clay material has a unit weight of $17\mathrm{.}6k\frac{N}{m^3}.$ The groundwater table is very deep. Lab testing of the clay indicates an in$-$place void ratio of $1\mathrm{.}35$ and a compression index of $0\mathrm{.}42.$ The building planned for the site will create an applied stress of $24\mathrm{.}5k\frac{N}{m^2}$ at the center of the clay layer.

a$)$ Assume the foundations for the building will be situated near the surface of the upper compacted granular soil layer. Determine the foundation settlement due to primary compression occurring in the clay layer because of the stress increase.

b$)$ Calculate the settlement to be expected if the groundwater table were at the soil surface and compare to a$).$