Deconvolution of variable-rate reservoir-performance data using B-splines

We use B-splines for representing the derivative of the unknown unit-rate drawdown pressure and numerical inversion of the Laplace transform to formulate a new deconvolution algorithm. When significant errors and inconsistencies are present in the data functions, direct and indirect regularization methods are incorporated. We provide examples of under- and over-regularization, and we discuss procedures for ensuring proper regularization. We validate our method using synthetic examples generated without and with errors (up to 10%). Upon validation, we then demonstrate our deconvolution method using a variety of field cases, including traditional well tests, permanent downhole gauge data, and production data. Our work suggests that the new deconvolution method has broad applicability in variable rate/pressure problems and can be implemented in typical well-test and production-data-analysis applications.