Problem $7(15$ points$)$ From guest lecture. An earth pressure balance TBM with $6\mathrm{.}5$ m diameter will excavate through clay soil with $w=22\%,PL=21\%$ and LL

$=47\%.$ The clay soil is below the groundwater table and therefore is fully saturated. Assume the specific gravity of the clay solids $=2\mathrm{.}7.$

a$.$ Calculate the increment of gravimetric water content required to increase the water content to halfway between the PL and LL$.$

b$.$ Calculate the volume of water that must be injected per meter of advance of the TBM$.$ Remember the TBM has a diameter of $6\mathrm{.}5$ m $.$