Application of the Laplace Transform to Discrete Time-Rate Data for the Analysis and Forecasting of Well Performance Behavior in Unconventional Reservoirs

Abstract

The numerical Laplace transformation of discrete data has been discussed extensively in the petroleum literature in applications related to well-test analysis. This approach has been shown to be a useful tool for the deconvolution of variable-rate pressure responses, although the success of this method heavily relies on the algorithms used to transform the discretely sampled data into the Laplace domain and then to invert these results (numerically) back into the time domain (Onur and Reynolds, 1998). Onur and Reynolds [1998] note that the major limitation for the numerical Laplace transformation of discrete data is the need for an "extrapolation" of the data function in the real domain (both for "early" (near zero) times and "late" times (beyond existing data)). The most important distinction of the present work is that it focuses on rate functions, which are inherently decreasing and positive functions. The fact that the function of interest is decreasing and positive is not an issue in terms of the mathematics of this scenario, but there are several challenges — e.g., the applicability of various extrapolation schemes. Simply stated, the primary objective of this research is to examine the application of existing algorithms for discrete-data Laplace Transforms, and to develop an appropriate workflow utilizing the numerical Laplace transform of discrete data for the analysis and forecasting of well performance behavior. The concept of this research is both simple and straightforward — for a given discrete data set of time and rate values, we use the Laplace transformation to generate the following: ● The Laplace transform smoothed rate function ● The Laplace transform smoothed cumulative function ● The Laplace transform smoothed rate derivative function ● A Laplace transform smoothed D-parameter function ● A Laplace transform smoothed b-parameter function The traditional approach to this problem is to use numerical integration (most typically, the Trapezoidal Rule) and numerical differentiation (most typically, the Bourdet (weighted difference) Algorithm). We believe that the Laplace transform has the potential to generate smoother and more mathematically rigorous integration and differentiation. We know that this approach has been used on occasion for such analyses, but neither systematically nor exhaustively as we propose in this work.