The Combined Effects Of Storage, Skin, And Partial Penetration On Well Test Analysis
Authors: Bilhartz H.L. , Ramey Henry J. 
Published: December 2007

Abstract
Although there have been many studies on unsteady behavior of wells with a partially-penetrating wellbore, there has been no study of the combined effects of wellbore damage and storage on the behavior of partially-penetrating wells. The purpose of this study is to fill this existing gap in knowledge. Specifically, this study concerned a bounded, anisotropic, cylindrical reservoir with a partially-penetrating, infinite-conductivity partially-penetrating, infinite-conductivity cylindrical inner boundary. This inner boundary also contained wellbore storage and a flux-dependent, infinitesimal (zero storage capacity) skin effect.
An analysis of pressure versus time indicated the possible existence of two semi-log straight line periods assuming a very small wellbore storage effect. Either straight line might be interpreted to yield horizontal permeability and the skin effect. Vertical permeability might be evaluated from correlations relating either the ending time of the first straight line, the intersection time of the two straight lines, or the beginning time to the second straight line to the effective vertical permeability. All three methods will fail if permeability. All three methods will fail if wellbore storage is significant.

Introduction
In many oil and gas reservoirs, producing wells are completed as partially-penetrating wells; that is, only a portion of the zone is perforated. This may be done for many reasons, but the most common one is to prevent or delay the intrusion of unwanted fluids (gas or water) into the wellbore. Partial-penetration will cause performance which, Partial-penetration will cause performance which, if not properly evaluated, can be mistaken for formation damage and can lead to errors in the interpretation of well-test data.
The purpose of this study is to reach general conclusions concerning the effects of wellbore damage and storage on the behavior of partially-penetrating wells, and to determine how these partially-penetrating wells, and to determine how these effects combine to influence the interpretation of short-time well tests. The combined effects of partial-penetration, wellbore storage, and partial-penetration, wellbore storage, and wellbore damage have never been studied.
The partially-penetrating well problem was first studied by Muskat for steady-state conditions. He calculated pressure distributions and productive capacities for an anisotropic system, productive capacities for an anisotropic system, and concluded that the productivity depended slightly on the directional permeability ratio (kz/kr greater than 0.1).
In 1958, Nisle used the instantaneous point source solution to the diffusivity equation to solve the constant flux, isotropic, partial-penetration problem. He constructed synthetic partial-penetration problem. He constructed synthetic pressure buildup curves for various penetration pressure buildup curves for various penetration ratios, and found that theoretical buildup curves consisted of two semi-log straight line portions: an early time straight line having a slope inversely proportional to the flow capacity of the open proportional to the flow capacity of the open interval khw, and a later semi-log straight line which had a slope inversely proportional to the flow capacity of the entire formation kh. Nisle showed that it was theoretically possible to calculate the penetration ratio from the ratio of the slope of the late part to that of the early part of the buildup curve. From the calculated penetration ratio and the thickness of the known penetration ratio and the thickness of the known producing interval, the effective formation thickness producing interval, the effective formation thickness might be obtained. Later Brons and Marting computed pseudoskin effects caused by either a partially-penetrating or limited-entry line source partially-penetrating or limited-entry line source well. Their results compared closely with the steady-state solutions of Muskat.
Odeh used a finite cosine transform to arrive at a solution for the steady-state flow problem where the open interval was located anywhere within the producing formation. Hantush solved the transient, anisotropic, partial-penetration problem by the successive use of LaPlace and Fourier transforms for the infinite reservoir case. He assumed the wellbore radius vanishing and the flux to be uniform for each point along the vertical section open to flow.