A geotechnical engineer needs to stabilize a layer of medium sand to reduce its permeability. Sieve analysis of the sand yields a $D_{15}$ (the diameter at which $15\%$ of the soil mass is finer) of $0.8\text{ mm}$. The proposed grout is a standard Portland cement suspension with a $D_{85}$ (the diameter at which $85\%$ of the cement particles are finer) of $0.05\text{ mm}$. Determine the Groutability Ratio ($N_R$) and assess whether this grout will successfully permeate the soil.

To solve the filtration issue identified in Problem 1, the engineer specifies a microfine (or ultrafine) cement grout for the same medium sand ($D_{15(\text{soil})} = 0.8\text{ mm}$). The microfine cement has been extensively milled, resulting in a much smaller particle size distribution where $D_{85(\text{grout})} = 0.015\text{ mm}$. Calculate the new Groutability Ratio and re-evaluate the feasibility.

A contractor is tasked with solidifying a deposit of fine silty sand under an existing bridge pier. The soil has a very small grain size, with $D_{15(\text{soil})} = 0.12\text{ mm}$. To ensure successful permeation grouting, the contractor must order a specific grade of cement or chemical grout. Calculate the maximum allowable $D_{85}$ for the grout to achieve a minimum Groutability Ratio of $25$.

An earth-fill dam is experiencing dangerous levels of seepage through a highly permeable gravel layer located beneath its foundation. Excavating the gravel is impossible without draining the reservoir. The engineer must create an impermeable barrier (cut-off wall) in situ.

A multi-story masonry building has experienced $50\text{ mm}$ of differential settlement at one corner due to a localized pocket of loose, collapsible silty soil beneath the footing. The building is actively cracking, and the foundation must be stabilized and lifted back to level.