A new technique for solving pressure–rate deconvolution problem in pressure transient testing

The transient pressure–rate deconvolution problem is formulated in the form of a linear convolution Volterra equation of the first kind. In addition to its ill-posedness, the problem is characterized by multiscale behavior of the solution and discontinuous input data with possibly large measurement errors (noises). These do not allow us to apply standard algorithms for solving the Volterra equation. Therefore, in most cases, deconvolution algorithms may implicitly or explicitly include regularization and take into account a priori information. In general, the solution has to satisfy certain conditions, such as positivity, monotonicity, and/or convexity. However, as is well known, the solution of the deconvolution problem satisfies an infinite system of inequalities. In this paper, we construct two effective regularization algorithms (methods) to obtain smooth approximate solutions satisfying all a priori constraints for the deconvolution problem. The convergence properties of the methods are proven. Finally, the methods are applied to a few sets of pressure–rate data with large measurement errors, and the deconvolution results of the data are discussed.