Hydraulic fracturing, or “fracking,” is widely used to stimulate production from oil or gas deposits. A deep well is lined with steel pipe. A large volume of water is pumped into the well at a pressure sufficient to fracture the rock formation at the bottom. A slurry containing “proppants” (sand or ceramic particles) is then pumped into the formation, the particles acting to keep the cracks open when the applied pressure is released.

Suppose that the pipe diameter is 15 cm and the well depth is 2.3 km. The flow rate during the water-only phase is to be 0.21 m³/s, which is predicted to give P = 44 MPa (6500 psi) at the well bottom, based on what is known about the geology. It is planned to use 60 °C water, for which μ = 0.467 × 10⁻³ Pa · s and ρ = 0.98³ × 103 kg m^–3 at atmospheric pressure. If these properties are constant throughout the pipe, what pressure P must be maintained at the top?