The Spurious Deflection on Log-Log Superposition-Time Derivative Plots of Diagnostic Fracture-Injection Tests

Abstract

Summary Log-log superposition-time derivative plots are used to identify flow regimes in well tests with variable rate. The use of superposition time adjusts for the effect of the prior rate history, and (under some conditions) shows what the transient would have looked like if the test had been performed at a constant rate. In this report, I show that if these plots are used to interpret shut-in transients from diagnostic fracture-injection tests (DFITs), the superposition-time derivative has an upward deflection that does not represent actual reservoir or transient behavior. I review mathematical properties of the superposition-time derivative. I derive equations for the pressure transient in a simplified model DFIT in which closure does not occur. I show that the onset of late-time impulse flow is controlled by injection volume and formation, wellbore, and fracture properties, not the duration of injection (as implied by the definition of superposition time). Log-log superposition-time derivative plots of DFITs exhibit a slope of 3/2 at intermediate time. However, pressure change never scales with a 3/2 power of time. One form of the G-function superficially resembles a superposition-time function constructed by summing constant-rate solutions with 3/2 power scaling. However, this is not a mathematically or physically valid interpretation. The 3/2 power arises from a spatial integration of the Carter leakoff solution. There is not a mathematical, physical, or practical justification for plotting DFIT pressure-time data in a way that creates a 3/2 slope. I conclude by providing a field example and practical recommendations for DFIT interpretation.