Superposition in Time

Key Takeaways

• The Horner time ratio is the most widely used implementation of superposition in time for buildup test analysis, but its accuracy depends on the production history being representable as an equivalent single-rate production period — the Horner plot plots the buildup pressure versus log[(tp + delta-t)/delta-t], where tp is the equivalent producing time (total cumulative production divided by the last producing rate before shut-in) and delta-t is the shut-in time; the straight-line slope of the Horner plot during the middle-time radial flow regime allows calculation of permeability thickness (kh) and skin; the extrapolation of this line to infinite shut-in time (Horner time ratio = 1) gives an estimate of the average static reservoir pressure; the Horner approximation is exact only when the well has been producing at a single constant rate for the entire production period tp before shut-in, and becomes increasingly inaccurate when the actual production rate has varied significantly before the test; when the production rate immediately before shut-in is very different from the average rate over the entire producing history, the Horner's tp calculation (which uses average rate implicitly) introduces error that can affect the pressure extrapolation by 100-500 psi in a long-producing well, leading to significant overestimation or underestimation of initial reservoir pressure; in these situations, the full multirate superposition function should be used instead of the Horner approximation.
• Multirate superposition generalizes the Horner approach to handle any arbitrary rate history before a buildup or drawdown test, at the cost of additional computational complexity — the multirate superposition time function for a buildup test is: SUM(i=1 to n) of [(qi - qi-1)/qn] x log(delta-t + tn - ti-1), where qi is the rate during period i, tn is the total production time, ti-1 is the start time of period i, and delta-t is the shut-in time; this function replaces the horizontal axis of the Horner plot with a more complex time variable that correctly accounts for every rate change in the production history; when this multirate time function is used as the x-axis, the buildup pressure data plots as a straight line with slope proportional to kh and skin during the radial flow period, just as in the Horner plot, but without the approximation error from ignoring rate history variability; modern well testing software computes the multirate superposition function automatically from the production history database, making the computational burden trivial compared to the manual calculations required before computing technology was available; the availability of accurate multirate analysis has reduced one of the most common sources of systematic error in well test interpretation, which was incorrectly applying the Horner approximation to wells with highly variable production histories.
• Superposition in space (the image well method) is a complementary application of the superposition principle that accounts for boundary effects — while superposition in time accounts for the production history of a single well, superposition in space places fictitious "image wells" at geometric reflections of the real well across reservoir boundaries to satisfy the no-flow or constant-pressure boundary conditions without explicitly solving the boundary value problem; the combination of time superposition (for rate history) and spatial superposition (for boundary effects) allows the pressure response of any well in a bounded reservoir with arbitrary production history to be computed analytically as the sum of contributions from the real well's rate history and the image well's contributions reflected at each boundary; this combined superposition is the theoretical foundation of all analytical pressure transient analysis methods used in commercial well testing software, and its validity is guaranteed by the linearity of the governing diffusivity equation — a requirement that breaks down when fluid properties change significantly with pressure (as in gas wells or reservoirs near bubble point) and requires pseudopressure transformation to restore the linearity needed for superposition to work correctly.
• Deconvolution is the inverse of superposition — it recovers the unit impulse response of the reservoir from the measured pressure and rate history, effectively removing the rate history complexity from the buildup or drawdown data — traditional well test analysis using Horner or multirate superposition requires assuming a specific reservoir model and then adjusting model parameters to match the measured pressure data; deconvolution approaches the problem differently by mathematically inverting the superposition integral (pressure equals rate convolved with the unit impulse response) to extract the unit response function (how the reservoir would respond to a unit step in rate) directly from the measured data without assuming a model; once the unit response is extracted, it can be differentiated to produce the derivative response that identifies flow regimes (radial flow, boundary effects, dual porosity) and interpreted with conventional type curve matching or analytical methods; deconvolution is powerful because it extends the effective duration of any rate period's data beyond its actual duration (using information from all rate changes in the history), revealing boundary effects and flow regime transitions that might not be visible in the raw data of any individual rate period; its limitation is sensitivity to rate measurement errors and systematic biases in the pressure data, which the mathematical inversion amplifies rather than suppresses.
• Gas well pressure transient analysis requires pseudopressure transformation before superposition can be rigorously applied because gas pressure-dependent properties violate the linearity assumption of the diffusivity equation — gas viscosity and compressibility both change significantly with pressure (gas compressibility, in particular, varies inversely with pressure over the range of typical reservoir pressures), which means the diffusivity equation for gas is nonlinear; superposition, which requires linear equations, produces systematic errors when applied directly to gas pressure data without transformation; the pseudopressure (also called the real gas potential or m(p)) is defined as the integral of (2p / mu x Z) dp from a reference pressure to p, where Z is the gas compressibility factor and mu is the gas viscosity at pressure p; replacing the actual pressure with pseudopressure in all time superposition calculations restores the linearity of the governing equation and allows rigorous application of superposition, Horner analysis, and type curve matching using the same techniques as for oil wells but operating in the pseudopressure domain rather than the actual pressure domain; the practical difference in interpreted reservoir parameters between a correct pseudopressure analysis and an incorrect direct-pressure analysis for a high-pressure gas well can be 10-30% in kh and skin, which is significant for completions decision-making.