A new approach to quantifying parameter uncertainty has been implemented in the numerical welltest simulator GTFM. The method incorporates inverse techniques, advanced statistical analysis, and probabilistic techniques. The approach is holistic in that it accounts for uncertainty introduced at all stages of the testing and analysis procedure. The methodology expands upon accepted methods by giving greater emphasis to model assumption diagnostics and parameter uncertainty.
Model assumption diagnostics are used to distinguish between competing flow models. Candidate flow models are evaluated based on the distribution of residuals between the measured data and the model response. A flow model is accepted if its residuals are normally distributed and discarded if the residuals deviate from the normal distribution.
Typically, parameter uncertainty from pressure-transient tests is analyzed using a discrete sensitivity analysis in which a single parameter is varied while keeping all other parameters constant. More recently, inverse techniques have focused on the estimation of confidence intervals which put bounds on the values of fitting parameters. This approach is limited by the assumption that there are no correlations among fitting parameters. and that non-fitting parameters are known perfectly.
For our new probabilistic-based approach, joint-confidence regions are calculated to quantify the fitting-parameter uncertainty resulting from parameter correlation. For non-fitting parameters. distributions are assigned, sampled using a Latin Hypercube sampling routine, and an inverse procedure performed for each set of sampled parameters. This process results in a distribution of joint-confidence regions for the fitting parameters (Fig. 1), which in turn is an expression of uncertainty in the non-fitting parameters. The new approach also makes it possible to graphically display the correlations between fitting and non-fitting parameters for sampled populations.
Forward simulations using parameter combinations from Figure 1 result in virtually no change in the goodness-of-fit of the simulations to the data (Fig. 2). This approach to quantifying parameter uncertainty differs from other approaches in that the simulated fit shows no degradation within the parameters uncertainty range because correlations between parameters is not neglected.
Uncertainty in fitting-parameter values can result from uncertainty in the conceptual flow model, data noise, correlations among fitting parameters, and correlations among fitting parameters and imperfectly known nonfitting parameters.