A massive concrete gravity dam is built on a permeable, isotropic, deep sandy foundation with a known hydraulic conductivity ($K$) of $5 \times 10^{-5} \text{ m/s}$. The permeable foundation layer is $20 \text{ m}$ deep, bounded below by totally impermeable, solid granite bedrock. When the reservoir is full, the water level is $15 \text{ m}$ above the downstream tailwater level. The length of the critical seepage path ($L$) directly under the massive base of the dam is $40 \text{ m}$. Calculate the volumetric seepage flow rate ($Q$) per linear meter of the dam's length.

Using the same dam from Problem 1, engineers decide that the seepage rate of $3.75 \times 10^{-4} \text{ m}^3/\text{s/m}$ is unacceptably high and poses a severe risk of internal erosion (piping). They decide to construct a completely impermeable concrete cutoff wall extending vertically downward from the "heel" (upstream edge) of the dam. The wall penetrates $10 \text{ m}$ deep into the $20 \text{ m}$ thick sandy foundation. Calculate the new seepage flow rate ($Q_{new}$).

Before constructing a dam, engineers must decide if the foundation rock requires extensive pressure grouting (injecting cement to seal cracks). In rock mechanics, permeability is often measured in "Lugeon units" (Lu). One Lugeon is defined as a water loss of $1 \text{ liter per minute}$ per meter of borehole length under a constant pressure of $1 \text{ MPa}$ ($10 \text{ bars}$).

During a standard packer test on a $5 \text{ m}$ section of a borehole in fractured limestone, the water loss ($q$) is measured at $45 \text{ liters}$ over a $10 \text{ minute}$ period while a constant pressure ($P$) of $0.5 \text{ MPa}$ is maintained. Calculate the Lugeon value of the rock mass and determine if grouting is necessary.

In 1954, French engineers completed the Malpasset Dam, a strikingly thin, elegant concrete arch dam soaring $66 \text{ meters}$ high across a steep river gorge to provide municipal water and irrigation. The dam was designed to transfer the immense, crushing thrust of the full reservoir almost entirely into the solid rock abutments (the canyon walls) rather than relying on its own weight like a gravity dam.

The Gotthard Base Tunnel in Switzerland is the longest and deepest railway tunnel on Earth, plunging an incredible $57 \text{ km}$ straight through the massive heart of the high Alps, with up to $2,300 \text{ meters}$ of solid rock towering directly overhead.