A new robust algorithm for solution of pressure/rate deconvolution problem

Abstract

Summary: A new robust algorithm for the pressure/rate deconvolution problem, described by Duhamel's convolution integral, which is a first-kind linear Volterra integral equation, has been developed. A transformation of the convolution integral to a nonlinear one is used to impose explicitly the positivity constraint on the solution. The weighted least-squares method with regularization on the solution by a curvature constraint has been used for computation of the convolution kernel (impulse function or deconvolved pressure) of the system. The algorithm takes into account the errors (or noise) in both the left-hand-side (measured pressures) and flow rate measurements (normally, the time dependent inner boundary condition) of the convolution integral. The solution algorithm also allows one to adjust flow rates and/or the initial reservoir pressure (an initial condition for the solution) during calculations, where both flow rate and the initial pressure may contain some level of uncertainty. For validation of the results of the algorithm, three synthetic examples are presented.