Model Validation, Refinement, and Parameter Estimation
As stated by Civan (1994), confidence in the model cannot be established without validating it by experimental data. However, the microscopic phenomena is too complex to study each detail individually. Thus, a practical method is to test the system for various conditions to generate its input-output response data. Then, determine the model parameters such that model predictions match the actual measurements within an acceptable tolerance. However, some parameters may be directly measurable. A general block diagram for parameter identification and model development and verification is given in this article (Civan, 1994).
The experimental system is a reservoir core sample subjected to fluid flow. The input variables are injection flow rate or pressure differential and its particles concentration, temperature, pressure, pH, etc. The output
variables are the measured pressure differential, pH, and species concentration of the effluent. Frequently, data filtering and smoothing are required to remove noise from the data as indicated in this article. However, some important information may be lost in the process. Millan-Arcia and Civan (1990) reported that frequent breakage of particle bridges at the pore throat may cause temporary permeability improvements, which are real and not just a noise. Baghdikian et al. (1989) reported that accumulation and flushing of particle floccules can cause an oscillatory behavior during permeability damage.
An integration of the model equations over the length of core yields the equations of a macroscopic model called the "parity equations." However, for a complicated model of rock-fluid-particle interactions in geological porous formations it is impractical to carry out such an integration analytically. Hence, an appropriate numerical method, such as described in this article, is facilitated to generate the model response (pressure differential across the core or sectional pressure differentials, and the effluent conditions) for a range of input conditions (i.e., the conditions of the influent, confining stress, temperature, pressure, pH, etc.).
Parameter Estimation with Linearized Models
Luckert (1994) points out that estimating parameters using linearized model equations obtained by transformation is subject to uncertainties and errors because of the errors introduced by numerical transformation of the experimental data. Especially, numerical differentiation is prone to larger errors than numerical integration. Luckert (1994) explains this problem on the determination of the parameters K and q of the following filtration model:
Thus, a straightline plot of Eq. 17-41 using a least-squares fit provides the values of log K and q as the intercept and slope of this line, respectively. Luckert (1994) points out, however, this approach leads to highly uncertain results because numerical differentiation of the experimental data involves some errors, second differentiation involves more errors than the first derivative, and numerical calculation of logarithms of the second numerical derivatives introduce further errors.
Therefore, Luckert (1994) recommends linearization only for preliminary parameter estimation, when the linearization requires numerical processing of experimental data for differentiation. Luckert (1994) states that "From a statistical point of view, experimental values should not be transformed in order that the error distribution remains unchanged." Using the parameter values determined this way, Civan (1998) has shown that the model predictions compared well with the measured filtrate volumes and cake thicknesses.
History Matching for Parameter Identification
The model equations contain various parameters dealing with the rate equations. They are determined by a procedure similar to history matching commonly used in reservoir simulation. In this method, an objective function is defined as
where Ym is the measured values, Yc is the calculated values, W is the weighting matrix, W = V"1, V is the variance-covariance matrix of the measurement error, and n is the number of data points. Szucs and Civan (1996) facilitated an alternative formulation of the objective function, called the p-form, which lessens the effect of the outliers in the measured data points. Then, a suitable method is used for minimizing the objective function to obtain the best estimates of the unknown model parameters (Ohen and Civan, 1990, 1993).
Note that the number of measurements should be equal or greater than the number of unknown parameters. When there are less measurements than the unknown parameters, additional data can be generated by interpolation between the existing data points. However, for meaningful estimates of the model parameters the range of the data points should cover a sufficiently long test period to reflect the effect of the governing formation damage mechanisms. The above described method has difficulties. First, it may require a lot of effort to converge on the best estimates of the parameter values. Second, there is no guarantee concerning the uniqueness of the parameter values determined with nonlinear models.
However, some parameters can be eliminated for less important mechanisms for a given formation and fluid system. The remaining parameters are determined by a history matching procedure. In this method, the best estimates of the unknown parameters are determined in such a way that the model predictions match the measurements obtained by laboratory testing of cores within a reasonable accuracy (Civan, 1994).
Frequently used optimization methods are:
1 trial-and-error [tedious and time consuming];
2 Levenberg-Marquardt (Marquardt, 1963) method [requires derivative evaluation, computationally intensive];
3 simulated annealing [algebraic and practical (Szucs and Civan, 1996; Ucan et al., 1997)].
Detailed examples of the history matching method by Ohen and Civan (1990, 1993) have been presented in this article.
Once the model is developed and verified with experimental data, studies of the sensitivity of the model predictions with respect to the assumptions, considerations, and parameters of the model can be conducted. Consequently, the factors having negligible effects can be determined and the model can be simplified accordingly. Models simplified this way are preferred for routine, specific applications. The solid line represents the best fit of the experimental data using the best estimates of the model parameters obtained by history matching. When the parameter values were
perturbed in a random manner, a significantly different trend, represented by the dashed line, was obtained.
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