Gas Well Back-Pressure Deliverability (Rawlins and Schellhardt)

This relation says how much gas, the rate $q$, a well will deliver if the flowing pressure at the bottom of the hole is held at a chosen value. The shut-in reservoir pressure $p_r$ is the pressure with the well closed; the flowing bottom-hole pressure $p_{wf}$ is lower while gas is being drawn off. Both are absolute pressures, and it is the difference of their squares that sets the rate, because pressure squared, not pressure, is the natural variable for gas at reservoir pressures.

The relation is empirical: the exponent and the coefficient come from a back-pressure test. The exponent $n$ is 1.0 for Darcy flow and falls toward 0.5 as the flow becomes turbulent near the well. The coefficient $C$ carries the size of the well and the properties of the gas, and its unit depends on $n$, so it cannot have one unit; this page fixes the convention of $q$ in thousands of cubic metres a day and pressures in kilopascals absolute. The state regulators' manual (IOGCC) tells testers to treat an exponent outside 0.5 to 1.0 as 0.5 or 1.0 when drawing the line, and finds the exponent from two points a decade apart on the log-log plot: 1 450 thousand cubic feet a day at a pressure-squared difference of 300 (in thousands of psia squared) and 8 750 at 3 000 give $n = 0.781$. The page reproduces that value: fix $C$ from the first point, enter the second, and it returns the exponent.

With the pressure at the bottom of the hole at zero, the rate is the absolute open flow, the figure regulators ask for. It is a calculated limit, not a rate any well is flowed at. The page's own example is that case: with $C$ of 1e-4, $n$ of 0.8 and a reservoir pressure of 10 000 kPa, the rate is 251.18864 thousand m³/d. Raise the flowing pressure to 2 000 kPa and the same well delivers 243.11793 thousand m³/d, so the two rates are close, because the reservoir pressure squared dominates the difference.

Rawlins and Schellhardt published the equation in the 1930s as Monograph 7 of the United States Bureau of Mines, and the page takes its use from the IOGCC manual, Back-Pressure Testing of Gas Wells (2000), whose multipoint test finds the absolute open flow and the exponent from the same equation. Because $C$ and $n$ are fitted to one well, they describe that well alone, and the relation gives the rate at a moment, not how it declines over the years.