Semi-Analytic Time-Domain Solution to Reservoir Models

Abstract

Abstract Analytic time-domain solutions are not possible even to some simple reservoir models, such as the finite radial model, which has a Laplace domain closed form solution. Several numerical inversion techniques of Laplace transform are available and find ready application for reservoir simulations. Stehfest is the most applied, however it leads to a purely numerical timedomain solution and all computational effort must be repeated for each required point of the response curve. This work develops a simple and efficient numerical inversion technique that leads to an explicit time-domain solution. The idea is to select a transfer function Gn(s) having an explicit inverse and evaluate its parameters by a curve fitting technique (Weighted Least Squares) with the impulse response of the reservoir model in the Laplace domain. The explicit function gn(t) obtained provides an accurate approximation for the impulse response. As we are usually interested in the step response, the function gn(t) still needs to be integrated in time and we find another analytical function for the step response. Any reservoir model which can be described in the Laplace domain by a combination of Bessel functions that is impossible to by analytically inverted is suitable to this method. The first test consisted of matching the results of this semi-analytic method with a model which has an analytic time-domain solution. The infinite reservoir model was approximated by a semi-analytic solution and the time-domain response was compared with the integral exponential solution. Next, the finite reservoir model was approximated by the same semi-analytic function of the infinite reservoir but with different coefficients. The time domain response was compared with the Stehfest inversion algorithm and results showed good agreement. A method of numerical inversion of Laplace domain solutions is described in which the problem is converted to the solution of linear systems. It leads to a series of explicit equations for the time domain that reasonably matches with the traditional Stehfest inversion algorithm. The semi-analytic reservoir model can be applied to reservoirs in which the quality of the information does not warrant the use of a more sophisticated model.