This paper proposes the difference-value plotting function (DVPF) for the diagnostic analysis and interpretation of pressure transient test data in low-permeability reservoirs. Specifically, this work uses the approximation of the analytical solution for the performance of a vertical well with a single finite conductivity vertical fracture, where a Taylor Series expansion is used to obtain an asymptotic solution for early-time flow, which includes terms for wellbore storage and fracture conductivity. The well-testing derivative of this result is then obtained and is of a similar form.
By subtracting the derivative form from the pressure form, we remove the "dominant" wellbore storage term from the asymptotic solution. We then need to normalize that difference by the square root of time (or dimensionless time) to obtain the final formulation of the DVPF which leaves a single constant parameter multiplied by time on the right-hand-side. Our contention is that this formulation leaves us with a diagnostic plotting function which provides a unique and contrasting behavior compared to using the pressure drop and/or pressure drop derivative functions alone for diagnostics and interpretations.
As is typical of pressure transient or well testing data at early times, the observed pressures often exhibit random data noise. As such, we have adapted a noise reduction algorithm that was originally used for signal processing to smooth both the pressure and derivative functions.
Lastly, we demonstrate the difference-value plotting function (DVPF) on several cases of synthetic and field-derived data to illustrate the utility of this methodology. Specifically, we have applied this method to cases in which it is difficult to determine unique interpretations using traditional methods (e.g., insufficient duration tests, lengthy WBS distortion, and effects of ultra-low permeability). The proposed DVPF allows us to observe underlying characteristics that are obscured at early times in traditional pressure and derivative analysis, and for the demonstration examples provided in this work, the DVPF does provide a strong auxiliary means of interpretation.