Abstract
Nonlinear regression methods (least squares, least absolute value, etc.) have gained acceptance as practical technology for analyzing well-test pressure data. Even for relatively simple problems, however, commonly used algorithms sometimes converge to nonfeasible parameter estimates (e.g., negative permeabilities) resulting in a failure of the method. To avoid this problem, the optimization method must be constrained. Several methods for constrained optimization exist, but nonlinear least squares with penally functions used to impose constraints is usually the algorithm of choice for pressure transient analysis problems. With some experience, penalty functions can be used effectively, but their use requires the user to determine the weighting given the penalty function. Moreover, if initial estimates are far from the solution, non-physical estimates of the parameters may still be obtained.
The primary objective of this work is to present a new method for imaging the objective function across all boundaries imposed to satisfy physical constraints on the parameters. The algorithm is extremely simple and reliable. The method uses an equivalent unconstrained objective function to impose the physical constraints required in the original problem. Thus, it can be used with standard unconstrained least squares software without reprogramming and provides a viable alternative to penalty functions for imposing constraints when estimating well and reservoir parameters from pressure transient data.
In this work, we also present two methods of implementing the penalty function approach for imposing parameter constraints in a general unconstrained least squares algorithm. Based on our experience, the new imaging method always converges to a feasible solution in less time than the penalty function methods.
