Pressure Derivative Calculation Using Dynamic Window Length
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Solution Overview
Problem
Existing methods for calculating pressure derivatives from noisy field data are prone to noise amplification and distortion, leading to inaccurate flow regime identification in formations due to the use of forward, backward, or central difference methods, which are unsuitable for noise-contaminated data with uneven spacing in the log-time domain.
Innovation Solution
A method that determines an optimal window length for pressure derivative calculations using piecewise linear regression, where the window length is adjusted based on the derivative of the pressure derivative with respect to the smoothing interval, and a Padé approximant is used to fit the integral curve, allowing for accurate suppression of noise while maintaining data trend accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If forward, backward or central difference methods are used for pressure derivative calculation, then the calculation is simple and works well for noise-free data, but the noise is greatly magnified when applied to noisy field data
Solution Approach 1:
The patent implements a dynamic window length adaptation mechanism where the smoothing window size is automatically adjusted based on local data characteristics. The window length varies with the local derivative magnitude and noise level, allowing the algorithm to maintain simplicity while adapting to different data conditions - using smaller windows in low-noise regions and larger windows in high-noise regions, thus resolving the contradiction between computational simplicity and noise robustness
Solution Approach 2:
The patent changes the key parameter of window length from a fixed value to a dynamically determined value based on local data statistics. By computing the optimal window length at each point based on local noise estimates and derivative magnitudes, the method transforms the static difference method into an adaptive algorithm that maintains simplicity in computation while improving reliability through parameter optimization
2Measurement precision
If a small differentiation interval L is used in the Bourdet algorithm, then the derivative calculation is less distorted by overall trend, but the derivative is dominated by noise because fluctuations become comparable or overwhelm the data trend
Solution Approach 1:
The patent makes the differentiation interval L dynamic by computing an optimal window length at each data point based on local noise characteristics and derivative magnitude. This dynamic adaptation allows the algorithm to select the appropriate smoothing level locally - using smaller L where noise is low and larger L where noise is high - thus simultaneously achieving local precision and overall reliability
Solution Approach 2:
The patent optimizes the parameter L by deriving an expression for optimal window length that balances noise suppression and local feature preservation. The optimal L is determined as a function of local noise estimates and data characteristics, transforming the fixed-parameter Bourdet method into an adaptive algorithm that resolves the contradiction between local accuracy and noise robustness
3Reliability
If a large differentiation interval L is used in the Bourdet algorithm, then the derivative curve is less affected by noise, but the derivative curve is distorted by the overall trend of the data instead of reflecting the local value
Solution Approach 1:
The patent implements dynamic window length adjustment that adapts to local data conditions. In regions with high noise levels, the algorithm automatically increases the window length L to improve noise suppression, while in regions with low noise, it decreases L to preserve local features and avoid trend distortion. This dynamic behavior resolves the contradiction between noise robustness and local accuracy
Solution Approach 2:
The patent derives and applies an optimal window length formula that balances the competing requirements of noise suppression and local feature preservation. By changing L from a fixed parameter to an optimally determined value based on local data statistics, the method simultaneously achieves both reliability in noisy regions and precision in clean regions
4Ease of operation
If pressure data are spaced uniformly in time but the independent variable is ln(t) or log(t), then the data are easy to record, but the data are very sparse at the beginning of a test and dense at a later stage, magnifying noise effects
Solution Approach 1:
The patent implements a dynamic window length mechanism that automatically adapts to the non-uniform data distribution in the log-time domain. The algorithm computes optimal window lengths that account for the varying data density - using larger effective windows in sparse early-time regions and smaller windows in dense later-time regions - thus maintaining calculation reliability despite the convenience of uniform time sampling
Solution Approach 2:
The patent optimizes the differentiation parameters by deriving window length expressions that explicitly account for the non-uniform spacing in the log-time domain. By changing from fixed-parameter differentiation to optimally adapted parameters that reflect the actual data distribution, the method resolves the contradiction between recording convenience and calculation accuracy
Data Source
AI summary
A method of investigating an earth formation. A tool having a pressure sensor is used in a borehole to collect formation fluid pressure data over time. A pressure derivative curve is generated from the formation fluid pressure data by conducting a piecewise linear regression of the data having optimal window length values L determined by calculating a derivative with respect to L of a pressure derivative value (DD), and selecting values of L where DD has a transition that departs from oscillatory behavior to gradual change. The pressure derivative is calculated with piecewise linear regression with the optimal window length values 2L. Different L values are generated for different groups of data points obtained over time. The pressure derivative is then used for flow regime determination.