Model Assisted Analysis and Interpretation of Laboratory and Field Tests
J. Willard Gibbs stated that "The purpose of a theory is to find that viewpoint from which experimental observations appear to fit the pattern" (Duda, 1990). Toward this end, we carry out theoretical analysis and modeling, and experiments under controlled conditions in an effort to predict the behavior of the interactions of the reservoir rock and fluid systems. However, model predictions usually involve uncertainties because of the approximations in systems description and uncertainties in the parameters and measurements. Luckert (1994) draws attention to the fact that "the models often contain only differential values while the experimental values are integrals." Thus, for direct comparison with experimental data, the models must be transformed into integral forms either by analytical or numerical solution methods.
As explained by Frenklach and Miller (1985), the predictive equations of natural phenomena, frequently called mathematical models, are usually
derived in the form of differential and/or integral equations, and often solutions can only be obtained by numerical methods. Frenklach and Miller (1985) stress that the dynamic model building process has to deal with several important issues:
1 adequacy,
2 statistical reliability of the proposed model, and
3 determination of its parameters.
Frenklach and Miller (1985) describe the usual approach taken to determine the model parameters as an iterative adjustment of the parameter values until the numerical solution of the model, called the model response or prediction, fits the experimental data. They add that frequently the adjustment of the parameter values is guided by the sensitivity analysis based on the partial derivatives of the predictions of the model with respect to its parameters. Frenklach and Miller (1985) draw attention to several problems associated with this approach:
1 in the statistical sense, the sensitivity is physically meaningful only if the model is adequate,
2 sensitivity varies during a dynamic process and, therefore, point estimates of sensitivities in the parameter space are not adequate, and
3 correlating sensitivities independently of each other complicates the interpretation of the sensitivities.
Frenklach and Miller (1985) circumvent these problems by incorporating the parameter estimation, adequacy test, and sensitivity analysis tasks into mathematical modeling. Their parameter estimation approach is based on developing an analytically or numerically determined functional relationship between the response and parameters of the model. For this purpose the individual responses of the dynamic model for different prescribed values of the parameters are obtained by means of the numerical solution of the model. Then a functional relationship between the model responses and prescribed parameter values is developed by applying a statistical analysis. They recommend the application of the experimental design techniques to improve the efficiency of this method.
The objectives of model assisted analysis and interpretation include the identification of the governing formation damage mechanisms and their relative contributions and importance; estimation and correlation of model parameters; model calibration via history matching; model verification and improvement; and sensitivity and simulation studies (Civan, 1996). Direct measurement of all the model parameters is usually not feasible when the model involves many parameters. Therefore, many researchers (Civan et al., 1989; Ohen and Civan, 1990, 1993; Millan-Arcia and Civan, 1992; Chang and Civan, 1991, 1992, 1997; and Liu and Civan, 1995, 1996; Civan, 1994, 1996; Wilhite et al., 1991; Vittal et al., 1988) have resorted to indirect methods of inferring the values of such parameters by history matching of some experimental data.
Although others (Gruesbeck and Collins, 1982; Amaefele et al., 1988; Sharma and Yortsos, 1987; Khilar and Fogler, 1987; Civan, 1998) offer some analytical expressions and/or direct measurement methods, these apply only to extremely simplified models having only a few model parameters. For complicated models, history matching appears the best choice in lack of a better method. However, some parameters may be measured and the remainder can be estimated by an optimal history matching method to minimize an objective function expressing the weighted sum of the squares of the deviations between the directly measured and model predicted formation damage indicators such as pressure loss, permeability impairment, and effluent conditions (Civan, 1996). For this purpose, simulated annealing is appealing as a practical optimization method (Szucs and Civan, 1996) because it does not require any derivative evaluations and it leads to global minimum without being trapped in one of the local minimal. However, the achievability of the uniqueness of the estimated parameter values depends on and increases by the amount of the measured data. Ucan et al. (1997) have demonstrated that uniqueness can be achieved if both the external and internal core fluid data are used simultaneously. Typical internal data include the sectional pressure difference and fluid saturations along the core plug. Typical external data includes the pressures at the core inlet and outlet, and the effluent solution properties.
Measurement Error
Measurements are uncertain numbers that are random and independent variables (Reilly, 1992). As stated by Cook (1980):
Error is the uncertainty in a measured quantity. An opposite expression is accuracy which is the reliability of the measurement. Precision, on the other hand, is the repeatability or reproducibility of a measurement. Hence, the measurements can be precise but not accurate; meaning that there is a systematic error.
There are three main sources of errors that affect the accuracy of measurements (Civan, 1989). The first is the human errors resulting from improper handling of instruments and incorrect readings of indicators such as the manometer, clock, pressure gauge, etc. The second source of error is the systematic errors in the instruments themselves. Errors involving various elements of instruments can accumulate and lead to pronounced errors in the value of the measurements. The third source of errors is the statistical errors, which are not predictable. Fluctuations in ambient pressure and temperature, and in electric power supply are examples of statistical errors. Statistical and human errors can be referred conveniently to as random errors. Human errors can be minimized by using the instruments carefully and maintaining them in good condition, but errors in instruments are systematic and often are undetected. Statistical errors are unavoidable. Thus, repeated measurements of variables should be taken to obtain statistically good results (although this will not change the systematic errors).
Random Error
In the following, first the relations for the error estimate referring only to the random error are discussed. The values of variables, xt, measured at z = l,2,...,«, repeated tests differ somewhat from each other. Thus, an arithmetic mean value of the measured values should be used, defined by
However, the mean value alone does not indicate the extent of reproducibility of the measurements. Therefore, an estimate of the confidence limits should also be given. In this respect, one of the following forms of random error estimates can be used, assuming the random errors are normally distributed:
Hence, the measured value is reported in terms of its mean value and the random error or confidence limits as x ± E, where E can be represented
by one of the random error expressions. See Mickley et al. (1957), Spiegel (1961), and Schenck, Jr. (1961) for details. Composite random error in a calculated value obtained by a series of calculational steps can be estimated analytically based on the accuracy of various quantities involving these calculation steps. For this purpose, the upper bounds of the overall random error can be determined by applying the rules given in the next section.
Systematic Error
As stated by Ku (1969) "systematic error is a fixed deviation that is inherent in each and every measurement." Hence, the measurements can be corrected for the systematic error if the magnitude and direction of the systematic error are known. Complex devices make it difficult to predict their accuracy. Leaks, variation of temperature, and pressure also influence the accuracy. Large volume flow-type tests suffer from sudden variation of species composition due to their larger residence times. All of these are sources of errors. The mechanical design parameters and dimensions of experimental systems immensely effect the accuracy of measurements. Careful analysis of their design and innovative improvements to increase their accuracy are vital for measurement with better accuracy.
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