Nonlinear Regression: The Information Content of Pressure and Pressure-Derivative Data
- Mustafa Onur (Istanbul Technical U.) | Albert C. Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 2002
- Document Type
- Journal Paper
- 243 - 249
- 2002. Society of Petroleum Engineers
- 5.6.4 Drillstem/Well Testing, 4.6 Natural Gas, 5.6.3 Pressure Transient Testing, 5.5.8 History Matching, 5.8.6 Naturally Fractured Reservoir
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It is shown that commonly used procedures for estimating parameters by regressing on pressure-derivative data are based on incorrect covariance matrices and thus violate the underlying statistical basis for nonlinear least-squares parameter estimation. Although the resulting estimates of model parameters may be reasonable, calculated confidence intervals are meaningless. Here, we show how to compute the correct derivative data covariance matrix that should be used for estimating parameters by nonlinear least squares. It is also shown that the information content of derivative data cannot be greater than the information content of pressure data, in the sense that regression on derivative data with the proper covariance matrix does not yield smaller confidence intervals than those obtained by regressing on pressure data only. In fact, we prove that matching a data set, including all interior derivative data plus two appropriate pressure points, yields exactly the same estimates and confidence intervals that would be obtained by regressing on pressure data only.
As introduced by Bourdet et al.,1,2 the pressure derivative has proved to be an invaluable diagnostic tool for model and flow regime identification in pressure transient analysis. It is also customary to match pressure and pressure-derivative data simultaneously when estimating formation parameters by type-curve matching. Perhaps, because of these reasons, engineers often regress on pressure-derivative data, or all pressure and pressure-derivative data simultaneously, when estimating parameters using a nonlinear least-squares procedure; see Refs. 3 through 7. To the best of the authors' knowledge, Barua et al.3 were the first to regress on pressure derivatives using nonlinear least squares. They used ordinary (unweighted) least squares to match derivative data for estimating reservoir parameters for a double porosity model. In their work, observed (conditioning) derivative data were computed from the Bourdet et al.2 formula, while true observed derivative data were computed directly from the analytical pressure derivative solution in Laplace space. For this problem, they found that the condition number of the Hessian matrix based on matching pressure-derivative data was on the order of five times smaller than the condition number of the Hessian matrix based on matching pressure data. Based on this result, they concluded that the convergence properties of a minimization algorithm might be better if pressure-derivative data are matched. However, they did not compute the confidence intervals for the parameters estimated from pressure or pressure-derivative matching. Carvalho4 and Carvalho et al.5 investigated the use of pressure-derivative data in nonlinear regression. They considered examples from both single- and double-porosity models. By using a nonlinear least-squares objective function based on equal weighting for both pressure and pressure- derivative data, they concluded that introduction of pressure-derivative data may actually decrease the chance of the scheme converging for single-porosity models. On the other hand, for double-porosity models, they indicated that regression on pressure-derivative data is more likely to converge to a global minimum than regression on pressure data. (Note that this last result is consistent with the conclusion of Barua et al.3 mentioned previously.) Based on this result, they suggest that one should first match pressure-derivative data (or data for the pressure-derivative ratio) to obtain preliminary estimates of model parameters and then match pressure data to improve these estimates and construct confidence intervals. Their results also indicated that confidence intervals obtained for pressure-derivative matching tend to be significantly larger than those obtained from matching pressure data. In his book, Horne6 made similar comments for pressure-derivative matching by nonlinear regression. Specifically, Horne states that the best approach to nonlinear regression for doubleporosity systems is to match first using derivatives and then match a second time using pressures. He also states that the confidence intervals obtained from the pressure-derivative match may be twice as wide as those obtained from the pressure match.
To condition a 3D channel to well-test pressure data, Rahon et al.7 considered an objective function that includes the squares of both pressure and pressure-derivative mismatches. They use equally weighted all-pressure mismatch terms equally, and also weighted all-pressure derivative mismatch terms equally, but formulate the objective function so that pressure mismatch weights and derivative mismatch weights could be different.
As shown here, the aforementioned implementations of regression on derivative data use an incorrect covariance matrix for pressure-derivative data and, hence, calculated estimates and confidence intervals have no statistical basis. As shown in this work, this problem arises because pressure-derivative data are not actually measured, but are constructed from measured pressure data using finite difference approximations for the derivative of pressure with respect to the natural logarithm of time. Because of this, pressure-derivative data are linearly related to pressure data. It follows that the covariance matrix for pressure-derivative errors must be directly related to the covariance matrix for pressure errors and the matrix that defines the linear transformation between pressure- derivative data and pressure data. In a related context, Watson et al.8 recognized that covariance had to be modified for history matching cumulative production data. In this work, we show how to construct the derivative data covariance matrix for use in nonlinear least-squares analysis.
Assuming that measurement errors for pressure data can be modeled as independent normally distributed random variables with mean zero and a given covariance matrix, we show that the covariance (or "weighting") matrix for derivative data is not diagonal. It is shown that when the correct derivative data covariance matrix is used, nonlinear regression is equivalent to a weighted least-squares method for parameter estimation when matching pressure-derivative data. In addition, we prove that matching of a data set, including all interior derivative data plus two appropriate pressure points, yields exactly the same estimates and confidence intervals as matching only pressure data.
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