Three-dimensional transient flow in heterogeneous reservoirs: Integral transform solution of a generalized point-source problem

Abstract

Appraisal well testing plays a key role in optimizing a complex field’s exploitation strategy — and increasing complexity often calls for more elaborate reservoir models for the proper interpretation of each test’s results. The classical analytical methods usually require restrictive simplifying assumptions, which limit their usefulness when dealing with reservoir heterogeneity. We seek to solve this issue by applying the hybrid analytical–numerical Generalized Integral Transform Technique (GITT) to the pressure diffusivity equation — and specifically, to the point-source problem extended to a reservoir with arbitrary permeability variation irregularly distributed throughout the 3D domain. The technique is first demonstrated to produce an exact alternative form of the well-known solution to the classical point-source in a homogeneous reservoir. This equivalence is then used to derive a computationally efficient working expression for the generalized point-source in a heterogeneous reservoir. Finally, this building block is applied to construct a uniform flow solution for a limited entry vertical well through spatial superposition — thus demonstrating the usage of the GITT for other wellbore geometries. Synthetic examples are used to show that the obtained expressions are in good agreement with results from a commercial reservoir simulator. In all cases presented, the eigenfunction expansions for both the drawdown pressure and its logarithmic derivative converge to at least four and two significant digits, respectively, within the adopted practical ranges of the series’ truncation orders. It is clear that the pressure expansions have better convergence characteristics, which are also influenced by the time value and by the distance from the position under consideration to the point-source. The resulting novel solution to the point-source provided in the present work is the most general and least restrictive expression presented so far to this single problem in a heterogeneous domain, and it is suitable for pressure transient analysis. • Generalized Integral Transform Technique (GITT) for pressure transient analysis. • Generalization of the point-source solution for a heterogeneous reservoir. • Building block for any wellbore geometry — illustrated with the limited entry well.