Well Testing Applications of Numerical Laplace Transformation of Sampled-Data

In recent years, the numerical Laplace transformation of sampled-data has proven to be useful for well test analysis applications. However, the success of this approach is highly dependent on the algorithms used to transform sampled-data into Laplace space and to perform the numerical inversion. In this work, we investigate several functional approximations (piecewise linear, quadratic, and log-linear) for sampled-data to achieve the "forward" Laplace transformation and present new methods to deal with the "tail" effects associated with transforming sampled-data. New algorithms that provide accurate transformation of sampled-data into Laplace space are provided. The algorithms presented can be applied to generate accurate pressure-derivatives in the time domain. Three different algorithms investigated for the numerical inversion of sampled-data. Applications of the algorithms to convolution, deconvolution, and parameter estimation in Laplace space are also presented. By using the algorithms presented here, it is shown that performing curve-fitting in the Laplace domain without numerical inversion is computationally more efficient than performing it in the time domain. Both synthetic and field examples are considered to illustrate the applicability of the proposed algorithms. Introduction Due to its efficiency, the Stehfest algorithm for the numerical inversion of the Laplace transform is now a well established tool in pressure transient analysis research and applications. Roumboutsos and Stewart showed that convolution and deconvolution in Laplace domain with the aid of the numerical Laplace transformation of measured pressure and/or rate data is more efficient and stable than techniques based on the discretized form of convolution integral in the time domain. Use of the numerical Laplace transformation of tabulated (pressure and/or rate) data has become increasingly popular in recent years for other well testing analysis purposes in a variety of applications; see for example, Refs. 3-10. Guillot and Horne were the first to use piecewise constant and cubic spline interpolations to represent measured flow rate data in Laplace space for the purpose of analyzing pressure tests under variable (downhole or surface) flow rate history by nonlinear regression. Roumboutsos and Stewart were the first to introduce the idea of using the numerical Laplace transformation of measured data for convolution and deconvolution purposes. They presented an algorithm based on piecewise linear interpolation of sampled-data, which can be used to transform measured pressure or rate data into Laplace space. Mendes et al. presented a Laplace domain deconvolution algorithm based on cubic spline interpolation of sampled-data. By considering deconvolution of DST data, they showed that Laplace domain deconvolution is fast and more stable than deconvolution methods based on the discretized forms of the convolution integral in the time domain. However, they noted that noise in pressure and flow rate measurements can also cause instability in Laplace space deconvolution methods, but they did not present any specific results on this issue. Both Corre and Thompson et al. showed that the convolution methods based on a representation of the linear interpolation of the tabulated unit-rate response solution and numerical inversion to the time domain are far more computationally efficient for generating variable rate solutions for complex well/reservoir systems (e.g., partially penetrating wells and horizontal wells) than convolution methods based on the direct use of analytical solutions in Laplace space. Using the numerical Laplace transformation of measured pressure data, Bourgeois and Horne introduced the so-called Laplace pressure and its derivative, and presented Laplace type curves based on these functions for model recognition and parameter estimation purposes. They also deconvolved data using these Laplace pressure functions in the Laplace domain without inversion to the time domain. Wilkinson investigated the applicability of performing nonlinear regression based on the Laplace pressure as suggested in Ref. 7 for parameter estimation purposes.