Solution of the deconvolution problem in the general statement

Abstract

The deconvolution problem, which arises in the description of well testing processes, is formulated in the form of a Volterra equation of the first kind with discontinuous input data (the kernel is the flow rate and the right-hand side is the pressure change) characterized by large measurement errors. In addition, the solution of this equation has multiscale behavior in its domain. In this situation, the traditional solution algorithms for Volterra equations, as a rule, do not provide satisfactory results. To solve the problem, we use the variational regularization methods and construct a function basis (a system of exponents), which allows us to take into account in the algorithm all a priori constraints known for the desired solution. As a result, we form a family of approximate solutions that satisfies the conditions of smoothness and exactness required for the interpretation of well tests. For the constructed regularizing algorithms, we formulate convergence theorems and describe the details of numerical implementation.