Effect of Anisotropy and Stratification on Pressure Transient Analysis of Wells with Restricted Flow Entry
Authors: Kazemi Hossein , Seth Mohan S. 
Published: December 2007

Effect of Anisotropy and Stratification on Pressure Transient Analysis of Wells with Pressure Transient Analysis of Wells with Restricted Flow Entry
A numerical model, devised to study here the effect of anisotropy and stratification on pressure transient analysis of wells with restricted flow entry, utilizes a more realistic boundary condition than do the usual ones appearing in the literature. Nonetheless, the results from both types of models can be correlated.

Introduction
The effects of anisotropy and stratification on flow behavior and production performance of oil and gas wells have been studied by numerous authors. Lefkovits et al. studied the performance of totally penetrated, stratified bounded reservoirs with no penetrated, stratified bounded reservoirs with no crossflow. Others have studied a similar problem, but with crossflow; Russell and Prats have summarized the findings of these papers. Muskat studied the steady-state behavior of liquid flow for a partially penetrating well in an anisotropic system and penetrating well in an anisotropic system and concluded that the productivity weakly depended on the directional permeability ratios (k(z)/k(r) greater than 0.1).
Nisle used the classical method of point source solution to the diffusivity equation to study the effect of partial penetration on pressure buildup of oil wells in an isotropic system. He found that the theoretical pressure buildup curves consisted of two straight pressure buildup curves consisted of two straight lines, the one at early buildup times having a slope proportional to the flow capacity of the open interval proportional to the flow capacity of the open interval (l-d)k(r) and the other having a slope proportional to the flow capacity of the entire formation hk(r). Later Brons and Marting supplemented the findings of Nisle and concluded that partial penetration gave rise to a pseudo damage, which reflected itself as a skin factor in pressure buildup calculations. Using the pseudo skin, they calculated the impairment of pseudo skin, they calculated the impairment of productivity of the production wells, which agreed well productivity of the production wells, which agreed well with the results of Muskat. Pizzi et al., using an electrical analog of a stratified system, concluded that a high pseudo damage resulted if the reservoir was partially penetrated. Harris" studied the optimum spacing and arrangement of perforations in cased holes. Hantush derived an expression for pressure distribution in an unbounded reservoir with restricted flow entry. Seth extended Hantush's work to a bounded reservoir. Odeh has studied a similar problem for steady-state conditions. problem for steady-state conditions. The purpose of this work was to investigate:the combined effects of anisotropy, stratification with crossflow, and restricted flow entry on pressure transient analysis of flow and buildup tests,the feasibility of calculating the horizontal permeabilities of intercommunicating stratified systems, andthe pressure interference tests in such systems.

Theory
The partial differential equation governing the isothermal flow of homogeneous fluids of constant compressibility is given by Eq. 1:
..............................(1)
where
The initial and the boundary conditions are:
1. The reservoir potential Phi is initially constant, say Phi i.
JPT
P. 639