Determination of Permeability Distribution From Well-Test Pressure Data
Authors: Feitosa G.S. , Chu Lifu , Thompson L.G. , Reynolds A.C. 
Published: December 2007

Summary
This work considers the analysis of pressure data, both drawdown and buildup, obtained at a well producing a reservoir in which the absolute permeability varies with position. A new inverse-solution algorithm is presented that can be applied to estimate the reservoir permeability distribution as a function of distance from the well.

Introduction
The emergence of reservoir characterization has stimulated efforts to obtain improved information on reservoir heterogeneities. This work considers single-phase flow to a well in a reservoir where permeability varies with distance from the well. We consider methods for estimating the permeability distribution from well-test pressure data. The methods considered were obtained by modifying and extending elegant seminal works of Oliver and Yeh and Agarwal.
Oliver used a perturbation theory technique to obtain the wellbore pressure drawdown solution at a single well in an infinite-acting reservoir where absolute permeability varies with position. His solution assumes 2D flow in an (r,) coordinate system and that permeability is a function of r and i.e., k = k (r,). As presented, his solution assumes that permeability varies slightly about a reference, base, or "average" value, kref. In Ref. 3, Oliver used the Backus-Gilbert method to approximate the permeability distribution under the assumption that a reference permeability value can be determined from a semilog plot of pressure vs. time. He applied the method to a three-zone, composite, infinite-acting reservoir where the permeability in the inner and outer zones is k = 2,000 md and in the middle zone is k = 1,500 md. In Refs. 1 and 3, Oliver considers only the analysis of pressure-drawdown data. In this work, we remove Oliver's restrictions and consider the analysis of both drawdown and buildup data obtained at a well with an arbitrary variation in absolute permeability in the radial direction. Most importantly, we derive an inverse-solution algorithm to estimate this permeability distribution directly from well-test pressure data. Unlike Oliver's application of the Backus-Gilbert procedure, our inverse-solution algorithm does not assume that we can compute a reference or base permeability value from a semilog plot of pressure vs. time. In fact, we show that the base permeability value controls only the shifting of the time scale used to evaluate the kernel weighting function in Oliver's solution. Our inverse-solution algorithm, which is recursive but stable, can be applied for large variations in permeabilities and in cases where pressure data exhibit no semilog straight lines.
Rosa and Horne examined the same problem as Oliver. While they noted that the pressure response for a multirate test was more sensitive to reservoir heterogeneities, like Oliver, they concluded that the inverse problem (i.e., the determination of permeability distributions) does not have a unique solution. Ref. 5 indicated that, for a multicomposite reservoir, the permeability distribution could be determined (by nonlinear regression analysis) only if the inner and outer radii of each zone were known. Kamal et al. also used a multicomposite model consisting of a few zones to analyze data from a damaged well. They matched pressure-buildup data with the model using nonlinear regression analysis to determine estimates of permeability in each zone.