An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: II. Finite Difference Treatment
Authors: Wattenbarger Robert A. , Ramey H.J. 
Published: December 2007

Abstract
An investigation of the effect of wellbore storage and skin effect on transient flow was conducted using a finite-difference solution to the basic partial differential equation. The concept of skin partial differential equation. The concept of skin effect was generalized to include a composite annular region adjacent to the wellbore (a composite reservoir). The numerical solutions were compared with analytical solutions for cases with the usual steady-state skin effect. It was found that the solutions for a finite-capacity skin effect compared closely with analytical solutions at short times (wellbore storage controlled) and at long times after the usual straight line was reached. For intermediate times, presence of a unite-capacity skin effect caused significant departures from the infinitesimal skin solutions. Two straight lines occurred on the drawdown plot for cases of large radius of damage. The first had a slope characteristic of the flow, capacity of the damaged region; the second straight line had a slope characteristic of the flow capacity of the undamaged region. Results are presented both in tabular form and as log-log plots of dimensionless pressures vs dimensionless times. The log-log pressures vs dimensionless times. The log-log plot may be used in a type-curve matching plot may be used in a type-curve matching procedure to analyze short-time (before normal procedure to analyze short-time (before normal straight line) well-test data.

Introduction
Skin effect was defined by van Everdingen and Hurst as being an impediment to flow that is caused by an infinitesimally thin damaged region around the wellbore. The additional pressure drop through this skin is proportional to the wellbore flow rate and behaves as though flow through the skin were steady-state. Wellbore storage is caused by having a moving liquid level in a wellbore, or by simply having a volume of compressible fluid stored in the wellbore. When surface flow rates change abruptly, wellbore storage causes a time lag in formation flow rates and a corresponding damped pressure response. A recent study was made to determine the combined effects of infinitesimally thin skin and wellbore storage. Analytical methods were used along with numerical integration of a Laplace transformation inversion integral. Tabular and graphical results were presented for various cases. It was recognized during the study that this representation of skin was oversimplified; that skin effect should be thought of as a result of formation damage or improvement to a finite region adjacent to the wellbore. It was suggested that a skin effect could arise physical in a number of ways. One simple example physical in a number of ways. One simple example would be to assume that an annular volume adjacent to the wellbore is reduced uniformly to a lower permeability than the original value. This would be similar to the composite reservoir problem. Perhaps a better example would be to problem. Perhaps a better example would be to assume that the permeability increases continuously from a low value at the wellbore to a constant value in the undamaged reservoir. In either case, the damaged region would have a finite storage capacity and would lead to transient behavior within the skin region. A negative skin effect could arise from an increase in permeability within an annular region adjacent to the wellbore. This might physically result from acidizing. But it is believed that cases of more practical importance are those in which negative skin effects are caused by hydraulic fracturing. A high-permeability fracture communicating with the wellbore gives the appearance of a negative skin effect. For the purposes of this study, it was decided to represent a skin effect, either positive or negative as an annular region adjacent to the wellbore with either decreased or increased permeability. permeability. SPEJ
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