Pressure-Squared Plot

Key Takeaways

• Gas well deliverability testing uses the pressure-squared plot to construct the backpressure curve (also called the deliverability curve or IPR for gas), which shows the gas production rate that a well can deliver at any given flowing wellbore pressure — the deliverability test involves flowing the well at multiple stabilized rates (or using the modified isochronal test to approximate stabilization in low-permeability wells where true stabilization would require weeks of flow) and plotting (p_r² - p_wf²) on the vertical axis against gas rate q on the horizontal axis on a log-log scale; the slope n of the backpressure curve (which ranges from 0.5 for fully turbulent flow to 1.0 for fully laminar Darcy flow) characterizes the turbulence intensity in the near-wellbore region; the deliverability curve allows the engineer to predict the stabilized gas rate the well can produce against any pipeline pressure (backpressure), which is essential for designing compression requirements, sizing surface facilities, and forecasting field deliverability in gas reservoir management.
• The Darcy and non-Darcy (turbulent) flow components of gas pressure drop are both revealed by the pressure-squared analysis through the two-term backpressure equation: (p_r² - p_wf²) = a*q + b*q², where the first term a*q represents Darcy laminar flow pressure drop (proportional to rate) and b*q² represents the non-Darcy turbulent flow pressure drop (proportional to rate squared); plotting (p_r² - p_wf²)/q versus q on a Cartesian graph produces a straight line with intercept a and slope b, from which the Darcy skin and the turbulence factor D (which determines how rapidly the non-Darcy pressure drop grows with rate) can be extracted; the non-Darcy term dominates in high-rate gas wells with small perforation areas, tight perforation tunnels, or high-velocity flow near the wellbore, and failing to account for it leads to overoptimistic deliverability predictions; in fractured wells, turbulent flow in the fracture itself (particularly near the fracture-wellbore interface) adds a rate-squared term to the pressure drop that is captured in the b*q² component of the pressure-squared deliverability analysis.
• Horner buildup analysis for gas wells uses the pressure-squared transformation to convert the shut-in buildup pressure data into a form compatible with the semilog straight-line method — in a conventional oil well buildup, the Horner plot of shut-in pressure versus log[(tp+delta_t)/delta_t] produces a straight line whose slope gives permeability directly; for a low-pressure gas well, the equivalent analysis uses (p_ws²) on the vertical axis rather than p_ws, and the slope of the resulting semilog straight line against the Horner time ratio gives the formation permeability-thickness product through the modified Darcy flow equations for gas; the pressure-squared Horner analysis is the standard method for gas well test interpretation at reservoir pressures below 2,000 psia and is embedded in most commercial well test interpretation software as a standard analysis module; the interpretation sequence for a gas well buildup is the same as for an oil well buildup (identify radial flow on the pressure derivative, read slope from the semilog straight line, calculate permeability and skin) but uses the pressure-squared transform throughout rather than pressure itself.
• Reservoir pressure decline in gas wells is monitored and analyzed using the pressure-squared framework to convert individual shut-in wellhead pressure readings (taken routinely without formal test equipment) into estimates of average reservoir pressure — by applying the appropriate wellbore fluid gradient correction to convert wellhead shut-in pressure to bottomhole shut-in pressure, and plotting the resulting p_ws² values against cumulative production on a p²/z plot (also called a p/z plot for volumetric depletion analysis), the engineer can estimate the original gas in place (OGIP) from the linear extrapolation to zero pressure and track whether the reservoir is producing under volumetric depletion (a straight line declining linearly with cumulative production) or is receiving aquifer influx or is pressure-connected to a larger reservoir system (deviations from the linear decline); the p²/z plot or p/z plot is one of the most powerful and simple diagnostic tools in gas reservoir engineering, and pressure-squared plots are a natural extension of this framework for wells where absolute pressure terms are too nonlinear for direct analysis.
• The transition from pressure-squared to pseudopressure analysis at higher reservoir pressures is a practical engineering judgment that balances accuracy against complexity — pseudopressure (symbolized as psi or m(p)) is always more accurate than the pressure-squared approximation, but computing pseudopressure requires integrating the function 2p/[mu(p)*z(p)] numerically from a reference pressure to the pressure of interest, which requires knowing the pressure-dependent gas properties (viscosity and z-factor) as a function of pressure throughout the reservoir pressure range; this computation is straightforward with modern software but requires accurate gas composition data to generate the pressure-dependent PVT properties; the pressure-squared approximation requires no such PVT integration and can be applied with only the reservoir pressure and the wellbore flowing pressure; in low-pressure gas basins like the Appalachian basin (where reservoir pressures are often below 1,500 psia), the pressure-squared approximation is excellent and pseudopressure analysis adds no meaningful accuracy; in high-pressure deep gas plays like the Gulf Coast Tuscaloosa Marine Shale or the Haynesville (reservoir pressures of 7,000-12,000 psia), pseudopressure is required and the pressure-squared approximation should not be used without checking the mu*z constancy assumption first.