Applications of Rate Normalization for Well Performance Analysis (Part V)
"Entities must not be multiplied beyond necessity" Williams of Ockham (c.a. 1300)
💠Introduction
Part IV covered two common methods used for calculating the pressure derivative that rely on explicit computation, such as Finite differences and Linear regression. The foundations of these rely on explicitly computing the derivative. They are simple and convenient; however, they require a judicious use of the differentiation window, L, parameter. On the other hand, there are alternatives to compute the pressure derivative function without directly calculating the derivative, and thus, do not require the use of a differentiation window. This article deals with such methods to perform the pressure derivative computation, namely,
✅ Pressure Integral
✅Numerical Laplace transform inversion
💠More on Pressure Derivatives
The most salient feature of the techniques presented in this article is that they don’t rely on direct computation of the derivative. Rather, as it will be seen, the calculation is done indirectly.
⏩ Pressure Integral
Blasingame et al. (1989) and Onur (1989, 1993) suggested integrating the pressure data function to alleviate the noise problem introduced by the differentiation operation. To accomplish this, the following function is defined,
PDᵢ is known as the pressure integral function and represents the cumulative average value of PD over the interval 0 < τ < tD . Likewise, the pressure integral difference function is defined as
Then, the pressure integral derivative can be computed by
The latter expression gives a convenient means of computing the derivative without explicit differentiation
⏩ Numerical Laplace Transform Inversion
The viability of the Laplace transform for computing pressure derivatives rests on specific mathematical properties that allow time-domain data to be manipulated algebraically in the Laplace domain. The Laplace transform of the time derivative is given by the following relationship
Since the pressure change at time zero is usually zero, it follows that
This property allows the logarithmic time derivative, used in diagnostic plots, to be computed by
As it is customary, the Stehfest’s algorithm (1970) is used to obtain the inverse of sΔ(s)
▶️ Piecewise Linear Approximation of f(t)
Roumboutsos and Stewart (1988) presented a piecewise linear approximation method that allows taking tabulated (real-world) data and moving it into the analytical Laplace domain, thus allowing numerically computing the Laplace transform of a data set described by a sectionally continuous function f(t) given by
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In each interval Tᵢ-₁ < T < Tᵢ, the function can be represented by a linear approximation
Where f*ᵢ-₁ represents the chord between two individual time steps,
By definition, the Laplace transform of the function f(t) is given by
Then f(s) is obtained by substituting the linear approximation of f(t) into the previous expression (Stewart 2011),
Note that the data exists in the interval 0 < T < Tn, but the transform is defined for 0 < T < ∞. Thus, a linear extrapolation is assumed in the region Tn < T < ∞. However, other forms of approximations give a better treatment of f(t) in the intervals 0 < T < T₁ and Tn < T < ∞ [See Onur (1998) for further details]. The Roumboutsos and Stewart method, simple and easy to apply, is effective for most situations.
💠Application
With reference to the example of Part IV, Figure 1 shows the Pressure Integral method. Note thst Pressure-Integral function is smooth; likewise, the pressure integral difference is also a smooth function. The Bourdet's derivative computed for L = 0.3 is shown for comparison; observe the inherent variability for t < 2000, while the pressure integral difference provides a clear trend. For t < 2000 hrs, model identification is challenging due to the noise in the derivative. A cleaner signal can be obtained by increasing L, but at the expense of potentially introducing an artifact. On the other hand, the pressure integral does not require "guessing" a proper L. Note that the pressure derivative of the original data appears vertically shifted with respect to the pressure integral derivative function; the integral function is an average over time, effectively introducing a time-lag
For the same well, Figure 2 presents the original normalized data, the pressure derivative using the numerical Laplace inversion technique, and Bourdet’s derivative. The Laplace inverted derivative is smooth across the whole time domain, being of superior quality for t < 2000 hrs with respect to Bourdet’s Derivative. Also, there is no end-effect in the Laplace inverted derivative. All this despite the inherent noise of the field data.
💠Takeaways
✅ Elimination of Explicit Differentiation: The pressure integral method allows for the computation of the derivative function without the need for explicit differentiation, bypassing the noise amplification inherent in finite-difference methods.
✅ Natural Signal Smoothing: Because integration is an averaging process, it naturally smooths out the signal and minimizes the impact of high-frequency noise without requiring a "judicious" choice of a smoothing window. Laplace numerical inversion has inherent smoothing properties; the process acts like a filter, suppressing high-frequency noise without requiring the arbitrary choice of a differentiation smoothing window
✅ Continuous Analytical Curves: Unlike finite difference methods that produce scattered points only at specific sample times, Laplace-based methods generate smooth, continuous derivative curves that reduce early and late-time distortions.
✅ Superior Quality in Noisy Data: The Laplace inverted derivative maintains superior quality and a clear trend even in the presence of significant field noise, where Bourdet’s derivative might show instability.
✅ No "End-Effect" Artifacts: Laplace-based techniques avoid the sudden "end-effect" increases often seen in Bourdet's derivative at late times, providing a more reliable interpretation of boundary effects.
💠References
Great post! I didn't know the use of Laplace transforms for derivative calculations. I think it could be helpful for RTA. Thanks for sharing!
Good job Adolfo. Well received!