Nitrogen Displacement

Method

The well fluid is displaced down to the pumping depth; the nitrogen that fills the column, and the tubing left on the reel, is totalled as gas, liquid and mass.

1. Pressure at depth balances the hydrostatic column of the well fluid: P = ρ·g·h. The wellhead pressure is that minus the weight of the nitrogen column, from the mean nitrogen density. The column is taken at the mean of the two pressures and of the two temperatures.
2. Volume factor of nitrogen: B = 2.8416·P / (Z·(273 + T)) in sm³/m³, with P in kPa and T in °C. The constant stands for standard conditions of about 15 °C and 101.3 kPa.
3. Volume filled with gas: the production tubing bore minus the coiled tubing steel, times the depth. That is the annulus and the coiled tubing bore, each counted once.
4. Nitrogen for the well is that volume times the factor at mean conditions. Nitrogen for the tubing still on the reel is its bore volume times the factor at wellhead conditions.
5. Liquid nitrogen: 1 m³ of liquid gives 696.5 sm³ of gas and weighs 808 kg.

Assumptions

• The fluid is displaced completely down to the pumping depth, in a vertical well.
• The compressibility factor is constant along the column; the temperature is the mean of the two ends.
• The mean nitrogen density is an input, not derived from the pressure, so the wellhead pressure follows from it.
• No fluid enters the well and no nitrogen is lost; nothing is allowed for the pumping losses or a reserve.

Reproduces a reference worksheet, with one difference: the worksheet counts the coiled tubing bore twice in the volume filled with gas, which here is counted once, so the nitrogen for the well is lower than the worksheet’s. Check the gas properties against a nitrogen table before you rely on a result.

Compressibility factor of nitrogen

Z corrects the ideal gas law for nitrogen at high pressure: above 1 the gas takes more room than an ideal gas, so a vessel or well holds less nitrogen in standard volumes than with Z = 1. The diagram and the table give Z for nitrogen by absolute pressure and temperature; the nitrogen tools interpolate in the table.

Compressibility factor of nitrogen

Absolute pressure	-20 °C -4 °F	0 °C 32 °F	20 °C 68 °F	40 °C 104 °F	60 °C 140 °F	80 °C 176 °F	100 °C 212 °F	120 °C 248 °F	140 °C 284 °F	160 °C 320 °F
1 bar · 15 psi	0.999	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
25 bar · 363 psi	0.984	0.990	0.995	0.999	1.002	1.004	1.006	1.007	1.008	1.009
50 bar · 725 psi	0.971	0.985	0.994	1.001	1.006	1.010	1.013	1.016	1.017	1.019
100 bar · 1,450 psi	0.963	0.985	1.001	1.013	1.021	1.028	1.033	1.037	1.039	1.041
150 bar · 2,176 psi	0.976	1.002	1.021	1.034	1.045	1.052	1.058	1.062	1.065	1.067
200 bar · 2,901 psi	1.010	1.034	1.052	1.065	1.075	1.082	1.088	1.092	1.094	1.096
250 bar · 3,626 psi	1.058	1.078	1.092	1.104	1.112	1.118	1.122	1.125	1.127	1.128
300 bar · 4,351 psi	1.117	1.130	1.140	1.148	1.153	1.157	1.160	1.161	1.161	1.161
350 bar · 5,076 psi	1.182	1.188	1.193	1.196	1.199	1.200	1.200	1.199	1.198	1.196
400 bar · 5,802 psi	1.251	1.250	1.249	1.248	1.247	1.245	1.242	1.240	1.236	1.233
450 bar · 6,527 psi	1.322	1.314	1.307	1.302	1.297	1.291	1.286	1.281	1.276	1.271
500 bar · 7,252 psi	1.394	1.380	1.368	1.357	1.348	1.340	1.332	1.324	1.317	1.310

Computed from the density of nitrogen in the NIST Chemistry WebBook (equation of state of Span et al., 2000) as Z = P·M/(ρ·R·T). Pressures are absolute, gauge pressure plus 1.013 bar. Between the points the tools interpolate linearly; outside 1–500 bar and −20 to 160 °C they use the edge value.

The tools are aids for engineers. Check the inputs and the results: responsibility for any decision stays with the user.