This work focuses on different methods to generate confidence regions for nonlinear parameter identification problems. Three methods for confidence region estimation are considered: a linear approximation method, an F test method, and a log likelihood method. Each of these methods are applied to three case studies. One case study is a problem with synthetic data, and the other two case studies identify hydraulic parameters in groundwater flow problems based on experimental well test results. The confidence regions for each case study are analyzed and compared. Although the F test and log likelihood methods result in similar regions, there are differences between these regions and the regions generated by the linear approximation method for nonlinear problems. The differing results, capabilities, and drawbacks of all three methods are discussed.