Robust Procedures for Parameter Estimation by Automated Type-Curve Matching

Abstract

Type curve analysis of pressure transient data influenced by wellbore storage effects can yield non-unique answers due to similarity in the shapes of the curves. In order to alleviate this problem, several researchers have proposed methods for unbiased parameter estimation using nonlinear regression analysis techniques. Typically, these methods are sensitive to initial guesses for the parameter values; thus, with poor initial estimates, it is possible that the schemes may diverge or converge to wrong answers. The probability of convergence to correct parameter values is further decreased if the data contains outliers; i.e., measured data whose behavior is significantly different from the "average" behavior of the data set. In this work, we focus on practical methods of applying the nonlinear regression to the analysis of pressure transient data. Our aim is to provide procedures that enhance the probability of obtaining a unique match, and which detect and properly account for the presence of outliers. A new two-step procedure which utilizes the pressure-pressure derivative ratio in the first step, is demonstrated to increase the chances of obtaining a unique fit. Also, we present a new method of applying "robust" parameter estimation (which accounts for data outliers) that uses commonly available algorithms based on least squares (LS) regression. We demonstrate applicability of the proposed methods by analyzing several sets of field data. Introduction Well test analysis is an important tool in the petroleum industry. Interpretation of transient pressure tests has been one of the most important sources for estimating reservoir properties such as transmissibility and storativity, and for detecting the presence of boundaries and formation heterogeneities. Further, well test analysis provides important information about the well condition, e.g., formation damage, evaluation of hydraulic fracture jobs and restricted-entry wells. The engineer can successfully evaluate the performance of a well or a prospect based on the results of transient pressure tests in conjunction with available information from geology and logs. Interpretation of well test data consists of defining an appropriate well/reservoir model which best represents the system under study, and adjusting the reservoir-model parameters to obtain the closest match between field and model data. Conventional type curve analysis relies on manual alignment of field pressure (and pressure derivative) responses with corresponding (idealized) model pressure responses to obtain system parameters. This approach is inherently subjective; thus, it is possible that a single set of pressure transient data may be matched differently (by different analysts, or by the same analyst at different times), to yield inconsistent estimates of the system parameters. Much of the subjectiveness of the classical type curve matching procedure can be removed by applying computer-aided automated type curve matching. There are many additional advantages of automated type-curve matching over the conventional graphical techniques: it is superior for analyzing tests with a variable flow-rate history, (since rate variations can be directly incorporated into the model pressure response); it can handle complex reservoir models (e.g., horizontal well, bounded systems) for which type curves are inadequate or nonexistent; and it is immune to noisy pressure derivative data when applied to the analysis of pressure data only. An efficient automated method does not require subjective adjustments of parameters from the analyst and decreases the probability of numerical and procedural errors. This work represents the beginning of our effort to delineate reliable, efficient procedures for estimation of reservoir parameters using regression techniques based on publicly available software. The majority of public-domain codes are based on nonlinear least-squares (LS) regression analysis, and this is the focal point of our analysis strategy, (although we often employ it in a way that renders it equivalent to a robust estimator). Among the many arguments in favor of using the LS method is that reliable software (such as the package we use, LMDER from the Argonne National Laboratory's MINPACK library) is readily available. Although nonlinear regression analysis (i.e., automatic type curve matching), is generally more applicable than classical type curve matching, it too suffers from non uniqueness problems. Different initial guesses may yield different results, and for some initial guesses, the schemes may diverge. P. 749^

Nonlinear Regression: The Information Content of Pressure and Pressure-Derivative Data

“…Commonly used procedures [3][4][5][6][7] incorporate derivative data in nonlinear regression using the incorrect objective function based on Eqs. 7 or 8.…”

“…This rigorous theoretical result does not contradict results in the literature that suggest that the number of convergence failures or the number of iterations required for convergence may be reduced, if we regress first on derivative data to obtain initial parameter estimates and then regress on pressure. Carvalho et al 4,5 suggested that it might be best to regress first on the pressure/pressure-derivative ratio (see Onur and Reynolds 18 ) and then regress on pressure. They also suggested that outliers could be removed by applying least absolute value (LAV) regression before applying regression based on least-squares to pressure data.…”

“…However, they did not compute the confidence intervals for the parameters estimated from pressure or pressure-derivative matching. Carvalho 4 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.…”

Robust Well-Test Interpretation by Using Nonlinear Regression With Parameter and Data Transformations

“…These authors used only the significant wavelet coefficients of permeability distribution to reduce the number of parameters, hence stabilizing the parameter-estimation algorithm. In another reparameterization application, Carvalho et al (1992) investigated the effect of using the grouped parameter set {k, k/C, Ce 2s /}, typically used in traditional type curves, instead of the primitive parameters {k, C, s}. Using the grouped parameter set did not provide any advantage over the primitive parameters.…”

Fractured Reservoir Parameters Estimation Considering Multiple Block Size

“…Many works [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] have addressed the inverse problem, involving several petrophysical parameters. The confidence of the results has been dealt with in some of them 6-12, 28, 37 .…”