Well Test Analysis Using Poles and Residues in Laplace Space
Investigator: Ciro Guimaraes
In complex analysis, a pole is defined as an isolated point of the complex plane where a single-valued function becomes unbounded, even though it remains analytic in the surrounding region. The residue is a complex number proportional to the contour integral around a path enclosing that pole, which determines the function's overall behavior near the singularity. In this research, we show that fundamental solutions to the hydraulic diffusion problem can be formulated using single-valued complex expansions in the Laplace space, efficiently represented by poles and their corresponding residues. This formulation enables us to:
1. Re-examine the convolution theorem of the Laplace transform to deconvolve subsurface signals without making any assumptions about late-time reservoir behavior.
Figure 1: Pressure-rate deconvolution of multi-rate well test using the pole-residue approach.
2. Interpret geometric characteristics of the domain from the optimized set of poles and obtain quantitative insights from the associated residues.
Figure 2: Pole-residue representation of parameterized solutions in Laplace domain.
We expect the pole-residue formulation to serve as a foundational framework for addressing more complex problems, including multi-well deconvolution, pressure-temperature deconvolution, and even problems governed by nonlinear physics.