A $2m$ thick fill $(=15\frac{kN}{m^3})$ of large lateral extend is placed on the surface of soil deposit having the subsurface profile as illustrated in Figure A$.$ By using Figure B$,$ determine the following.

$($a$)$ Time factor $T_v$ and the corresponding consolidation $U$ at the center of the clay layer after $100$ days.

$($b$)$ Ultimate settlement of the clay layer by placing the thick fill by using ${}_w=10\frac{kN}{m^3}.$

$($c$)$ Settlement of the clay layer $1$ year after the placement of the fill.