Error Analysis: Propagation, Impact, Estimation
Although the phenomenological descriptions of processes are generally accomplished in terms of differential equations, called mathematical models, solutions can be obtained analytically for simplified cases and numerically for complicated cases. Accuracy of model predictions is dependent on various factors, including
1 the adequacy of the model,
2 the accuracy of the input data, and
3 the accuracy of the solution technique.
Various sources of uncertainties affect the reliability of the predictions of models, as described in this article by Bu and Damsleth (1996). Experimental measurements taken under controlled test conditions to determine the input-output (or cause-and-affect or the parity relationship) response of systems (such as core plugs undergoing a flow test) also involve uncertainties. In general, solutions of models, called model predictions, and the response of the test systems under prescribed conditions can be represented numerically or analytically by functional relationships, mathematically expressed as:
in which / is a system response and xl,x2,x3,... denote the various input variables and parameters. Uncertainties involved in actual calculations (predictions) or measurements (experimental testing) lead to estimated or approximate results, the accuracy of which depend on the errors involved. Therefore, the actual values are the sum of the estimates and the errors. Thus, if f, x1,x2,..., xn indicate the estimated values of the function and its variables, and Δf,ΔAxl,Δx2,...,Δxn represent the errors or uncertainties associated with these quantities, the following equations, expressing the actual quantities as a sum of the estimated values and the errors associated with them, can be written:
The estimation of the propagation and impact of errors is usually based on a Taylor series expansion (Chapra and Canale, 1998):
Neglecting the higher order terms for relatively small errors, Eq. 17-11 can be written in a compact form as:
Then, applying Eqs. 17-7-10 into Eq. 17-12, the error or the uncertainty in the function value can be estimated by (Chapra and Canale, 1998):
The uncertainty associated with summation and/or subtraction of numbers, defined by Eqs. 17-7 through 9, is the square root of the squares of the uncertainties in these numbers (Reilly, 1992). Thus, if
The relative uncertainty in a multiplication or division of numbers is the square root of the sum of the squares of the relative uncertainties in these numbers (Reilly, 1992). Thus, if
the error in the function value as a result of using an erroneous measured value of x = 0.5±0.1 can be estimated by applying Eq. 17-13 as:
Thus, substituting x = 0.5 and Δx = 0.1 into Eqs. 17-20 and 21 yields f = 0.6 and Δf = 0.1. Therefore, the calculated value is expressed according to Eq. 17-10 as:
Thus, Eqs. 17-23 through 25 lead to the following relative error expression:
Applying Eq. 17-14 for Eq. 17-23 results in
Thus, they express relative error in the calculated K value as a function of the measurements involving errors as (apply Eq. 17-14):
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