Tank Spillover — Gooseneck Cascade Flow
Steady-state flow through the gooseneck (inverted-U) spillover pipes that cascade produced water between oilfield disposal tanks — two parallel trains, T1→T3 and T2→T4. Each pipe is the smaller of a weir limit at the crest and a full Darcy–Weisbach + Colebrook–Whiteenergy balance; the floating oil pad adds (lighter) head that still helps push water over the crest. A bisection solver finds the source level that passes exactly the inlet flow, flags overflow when even a full tank can’t, and reports settling and storage-to-alarm times alongside a live schematic.
Cascade schematic
Train 1 — T1 → T3
927 bbl (water + oil pad)
Capacity 11,345 m³/d · weir (crest-limited)
Effective (oil-corrected) head over the crest
Total differential driving the flow
Re = 233,874
Water retention at this inlet
Source + downstream tank, to high-level alarm
…to high-level shutdown
Train 2 — T2 → T4
994 bbl (water + oil pad)
Capacity 10,246 m³/d · weir (crest-limited)
Effective (oil-corrected) head over the crest
Total differential driving the flow
Re = 233,865
Water retention at this inlet
Source + downstream tank, to high-level alarm
…to high-level shutdown
Equivalent length
Up-leg + down-leg + connecting run
ΣK = 3.00, as ΣK·D/f
Colebrook–White at the duty point
Straight + fittings
How the numbers are made & assumptions
Flow model — min(weir, Darcy)
Two limits are computed and the smaller governs. Weir (crest-limited): when the effective head above the crest is below the pipe diameter the pipe runs partially full at the crest and behaves like a weir, Q = Cd·Aseg(h)·√(2gh) with Cd = 0.62. Darcy (energy-limited): a full Darcy–Weisbach energy balance from source to receiving tank using the driving head, pipe friction (Colebrook–White) and the minor (fitting) losses. Low downstream level ⇒ weir regime; high downstream level ⇒ back-pressure regime; head above crest past the diameter ⇒ full pipe.
Oil-pad head correction
Oil floats and is skimmed separately, so only water goes through the pipe — but the oil pad still presses down. Because it’s lighter (SGoil/SGwater< 1) it adds head at a reduced rate: effective head above the crest = (water − crest) + (SGoil/SGwater)·oil pad. That lets flow start even when the water surface sits below the crest.
Steady-state solver & overflow
A bisection search finds the source-tank water level whose cascade flow equals the inlet flow (mass balance). If even a full tank can’t pass the inlet, the train overflows and the reported cascade flow drops to the actual full-tank capacity.
Settling & storage times
Settling = water volume in the source tank ÷ inlet flow (retention for oil/water separation, water only). Storage to HLA / HLSD = the spare volume from the current levels up to the alarm / shutdown setpoint, across the source anddownstream tanks of the train, ÷ inlet flow.
The math runs internally in US units (ft, ft³/s); flow is shown in m³/d to match the field convention. Pipe IDs come from standard NPS/schedule tables.
Read before you rely on these numbers
- These are steady-state estimates for planning — verify against a detailed hydraulic model before procurement.
- The weir coefficient (Cd = 0.62) is a sharp-crested approximation; a real gooseneck crest may differ.
- Fitting K-values are nominal; foul/scaled pipe and partial-vacuum siphon effects aren’t modelled.
- Oil/water separation quality drives the real settling requirement — the settling time here is retention only.