Applications of Rate Normalization for Well Performance Analysis (IV)
"Prediction is very difficult, especially if it's about the future." — Attributed to Niels Bohr
💠Introduction
The previous article in this series, Part III, covered the basics of model identification and the critical role that the pressure derivative function plays in this step. This article expands the discussion of other flow regimes of interest, provides additional tools for flow regime identification, and outlines the various schemes available to compute the pressure derivative. The following topics are covered:
✅ Power Law Flow Regimes and related Pressure Derivatives
✅Chow Pressure Group
✅Explicit Derivative Computation Schemes
💠More on Pressure Derivatives
⏩ Power Law Formulation
Part III presented the characteristic features of the pressure derivative of the radial flow and PSS regimes. However, several other flow regimes are of practical interest. It turns out that the solutions of the diffusivity equation for the linear, bilinear, and pseudo-steady regimes can be described by a power law of time (Bourdet D., 2002),
The latter implies that Pd describes a straight line in Cartesian coordinates with td^1/n as the dependent variable. or equivalently, a straight line of slope 1/n in log-log coordinates. The constants α and n are flow-regime specific. The latter has a value of 1, 2, or 4 corresponding to pseudo-steady state, linear, and bilinear regimes, respectively. These flow regimes are associated, though not exclusively, with boundary-dominated conditions (PSS), high-conductivity fractures (linear), and low-conductivity fractures (bilinear). Applying the pressure derivative definition to the generic pressure response,
This shows that a log-log representation of the Pd' corresponds to a straight line, having a slope 1/n. Note that is independent of the skin factor, S, included in the constant corresponds to a straight line, as it disappears upon differentiation.
As an illustration, Figure 1 presents the log-log response of a fractured well at the center of a square for different xe/xf ratios. The pressure behavior follows a power-law behavior, being described by a linear flow regime (n = 2) in the early time: both the pressure response and the pressure derivative follow a ½ slope straight line and are parallel to each other. The linear flow regime transitions into a pseudo-radial flow regime. Subsequently, the pressure response reaches the PSS regime (another power-law regime with n = 1) that conforms to a unit-slope line.
A word of caution is pertinent: while it is rigorously correct that a fractured well will yield a ½-slope straight line on the log-log pressure derivative plot during the linear flow regime, the converse is not true. The ½-slope line on the derivative is a necessary but not a sufficient condition to indicate the presence of a fracture. Other well and reservoir configurations produce this feature, among them:
This non-uniqueness highlights the necessity of integrating geological, petrophysical, and production data alongside the PTA results for accurate reservoir modeling.
⏩ Chow Pressure Group (CPG)
Another derivative group using the pressure derivative definition is given by (Onur & Reynolds, 1988)
The ratio is also known as the Chow pressure group (CPG). Substituting the definition for and for the power-law regimes,
The asymptotic approximation is,
Hence, the CPG converges to 2, 1 and ½ for bilinear, linear, and PSS regimes, respectively. Thus, the CPG incorporates a specific metric that can be used to supplement the pressure derivative function for model recognition purposes
💠 Pressure Derivative Computation
The pressure derivative function can be computed using explicit numerical differentiation, least-square fitting, integral methods, and Laplace numerical inversion methods, among others. This article covers the first two methods
⏩ Bourdet
Bourdet's (1989) method is the most commonly used tool for computing the pressure derivative function. It is illustrated in Figure 2. The pressure derivative is computed using points separated by a specific interval, L, in log space; this interval is known as the differentiation interval, typically 0.1 to 0.2 log-cycles.
The algorithm is given by the following expression:
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Note that X = ln(t)
⏩ Least-Squares Fit
Clark and Van Golf-Racht (1985) proposed a linear regression approach as an alternative to slope-weighting. By calculating the slope from a least-squares fit over a moving window (Figure 3), this method mitigates the effect of outliers more effectively than simple finite-difference methods
💠Application
Figure 4 shows a black oil reservoir well exhibiting a declining oil production rate alongside a variable flowing pressure profile
The log-log diagnostic plots for this well using different differentiation windows (L = 0.1, 0.3) are shown for both Bourdet’s and the Least-Squares methods.
✅Bourdet’s Algorithm
➤ Differentiation window L = 0.1
➤ Differentiation window L = 0.3
✅Least-Squares Fit
➤ Differentiation window L = 0.1
➤ Differentiation window L = 0.3
For all plots: ♦ = Chow group. ◾= Pressure derivative. ● = Normalized Pressure drop. ✢=Cartesian derivative.
✅Discussion
The derivative plots provide multiple indicators confirming the presence of boundary-dominated flow. These signatures include:
✔️A unit-slope trend on the pressure derivative curve.
✔️ A plateau on the Cartesian derivative.
✔️The Chow pressure group converging to a value of 0.5.
The transition into the PSS regime is well-defined, as the pressure derivative reaches the unit slope approximately one logarithmic cycle before it merges with the rate-normalized pressure (Δp/q) curve.
The selection of a different window width controls how smooth the resulting derivative is. As L is increased, the derivative is smoother; however, some artifacts are introduced in the late time
💠Takeaways
✅ The methods presented in this article are effective and simple to apply, but share a common shortcoming: the differentiation operation amplifies not only the signal but also the associated noise, which is compounded by multiplying by t.
✅A large L may reduce the noise, but at the expense of an increased distortion of the signal; it is a balancing act. In any case, and this is true for any pressure derivative algorithm, the goal is to extract the reservoir signal while spurious noise is minimized.
The next article in this series will expand on other numerical schemes that don’t require an explicit computation of the derivative or slope.
💠References