Simulation of a Perforated Vertical Wellbore with Near Wall Porous Media Effect

This paper aims to predict the effect of porous media permeability and perforations parameters on the pressure drop and productivity index, for the perforated vertical wellbore with six perforations and for two phase angles. The first 60o phase angle with helical distribution and the second 180o phase angle with normal distribution. In this study, a Computational Fluid Dynamics (CFD) software has been used to simulate a model of 3-D turbulent fluid flow with stander k−∈, steady-state, and single phase. The effect of the permeability of porous media, inlet mass flow rate from porous media, perforations length, and diameter of perforations are studied, for two cases of the phase angles. The results of this study show that, the pressure drop decreases with increasing permeability pf porous media, so the productivity index increasing. Also, increase of inlet mass flow rate from porous media causing an increase in the pressure drop. The perforations length has a few effects on the pressure drop and productivity index, while the diameter of perforations has a greater effect on the pressure drop.


Introduction
The formation damage is a serious problem in oil and gas industry. The formation damage caused by drilling-fluid invasion, production, or injection can leads to positive skin factors and affect fluid flow by reducing the permeability of the formation surrounding a wellbore. Many researchers focused on their study on the effects of perforation parameters and permeability zone surrounding the wellbore. The effect of perforation parameters on the main inflow efficiency are studied using analytical calculations derived by Muskat [1]. He analytical calculations to determine the effect of spacing between perforations and shot density on productivity of perforated vertical wells. He determined that the well productivity can be identified by the shot density and the distribution of perforations. Locke [2] presented a new theoretical method to predict the productivity ratio of a perforated vertical wellbore by constructing a more accurate simulation model, a finer finite element method was used to run simulations. He focused on the effect of perforation length, perforation phase angle, perforation diameter, shot density, crushed zone effect and damaged zone effect on productivity ratio. Deo et al. [3] studied two types of flow targets; linear target (unperforated) and radial target (perforated) in a cylindrical laboratory sample to determine which of these targets best represents downhole conditions, using a 3D finite element to calculate inflow into perforations. The effects of shot density and perforation phase angle were investigated by them. Their results showed that neither linear nor radial targets provided a perfect model for inflow from perforations. Karakas and Tariq [4] presented a semi-analytical solution to predict the productivity ratio of perforated vertical wellbore, using 2D and 3D finite element for flowrate performance of the perforations. Their results showed that the productivity ratio increases with increasing perforation length and the crushed zone around perforations essentially increases the vertical resistance to flow. Ihara et al. [5] presented an experimental study for single phase flow in a channel of rectangular cross-section with inflow through porous walls to predict the pressure drop in a horizontal wellbore.
The frictional pressure drop in the model was treated as that for flow in a pipe.
Dogulu [6] presented a numerical model to estimate the productivity of perforated horizontal, vertical and slanted wells as a function of shot density, perforations length, and phasing angles. He used FDM for a single phase and algebraic grid generation technique to build the grid of perforated wellbore. Tang et al. [7] presented a comprehensive study on a horizontal well with slotted-liners or perforation completion to obtain the productivity ratio, based on semi analytical model that couples a reservoir and wellbore fluid flow equations. they showed that both perforation length and density has a significant effect on productivity. Ansah et al. [8] presented a new 3D model for a vertical wellbore to predict the effect of the perforations length, casing (pipe) diameter, perforations density, perforations phase angles, and the degree of damage inside and around the perforations on the productivity ratio and skin factor, using a 3D finite element model ANSYS 5.7 to obtain a result to demonstrate the improvements of the flow rate predictions. Guerra and Yildiz [9] presented a simple approximate model to predict the inflow performance of perforated vertical wellbore, using a programmable calculator to solve algebraic equations and compared with analytical solution of SPAN software. Yildiz [10] reviewed the methods used to predict the productivity ratio and total skin factor of perforated vertical, horizontal and inclined wellbore, and compared the results with experimental analysis. The results showed that the productivity ratio increases and the total skin factor decrease with increasing perforation length. Hagoort [11] presented an analytical model to predict the productivity ratio of a perforated vertical wellbore, based on the analytical solution for a single phase Darcy flow for a single perforation with considered the effect of perforation damage. Kuljabekov et al. [12] presented a numerical solution for the technology of multistage filters setting in porous media near a vertical wellbore. Using Darcy and conservation laws to describe the fluid flow in a homogeneous and isotropic medium. Elsharafi et al. [13] explained how to evaluate the different perforation parameters of the production vertical oil wells by using well test reservoir description and perforation information. The necessary data have been collected from Hungarian oil wells including reservoir description data from the MOL Company files. This study was concentrated on the effects of damaged skin factor, crushed zone skin factor and perforation skin factor. Also, calculation method for the perforation depth and flow rate for different kinds of the gun are used. The aims of this study is to assess the effect of porous zone permeability surrounding vertical wellbore and inlet mass flow rate from porous media on the pressure drop, productivity index, pressure and velocity distribution surrounding the wellbore. Also, the effect of diameter and length of perforations are studied. Also, the comparison between normal and helical distribution for the perforations will be viewed.

Description of the Cases
In this work, the effect of porous media surrounded the wellbore and the parameters of perforation on the pressure drop and productivity index are studied. The simulation performed for two models as shown in Figure (

Governing Equations
The The Forchheimer's equation is used to describe the flow through the porous media at perforation and expressed mathematically as follows [17]; ( ) | | ( ) where is the pressure gradient, k is the permeability of porous media,  is a fluid viscosity, is the Forchheimer coefficient and is the Darcy/Forchheimer velocity.

CFD Model Solution Method
Simulation of fluid flow using CFD involves the following five steps: creating the computational model (2D or 3D geometry); surface or volume meshing; preprocessing (definition of fluid domains, physical models, and boundary and initial

Boundary Condition
(i) The inlet velocity of the wellbore (u 1 ) is 1.5 m/s and the inlet velocity from porous media is 0.5 kg/s.
The pressure (P 3 ) at exit of the wellbore is equal to zero. As shown in Figure   (2).
(iii) The roughness of the casing wall is equal to 0.02 mm.
(iv) No slip velocity at the walls.

CFD model validation
In order to verify the accuracy of the CFD model. A comparison with the results of

Results and Discussion
The effect of the porous media surrounding the wellbore and perforation parameters was studied. The numerical simulation for a perforated vertical wellbore with 6 conical shape perforations and two phase angle was performed. The first was 60ᵒ

Conclusion
The effect of pressure drop and productivity index on the performance of perforated vertical wellbores was studied. Also, the effect of the permeability of the zone surrounding the wellbore and parameters perforation. It was simulated by using CFD software. From the results obtained in this study, the following conclusions can be presented:

i.
Increase the permeability of the porous media causing a decrease in the pressure drop. Also, productivity index increases with increasing the permeability of the porous media.

ii.
The pressure drop increases with increasing inlet mass flow rate from porous media, while the productivity index decreasing.

iii.
Increase of the perforations length has a few effects on the pressure drop and productivity index.

iv.
Increase of the perforations diameter causing an increase in pressure drop.
Also, the productivity index increases with increasing diameter of perforations.

v.
In all cases, the 60ᵒ phase angle with helical distribution is better and has less losses than 180ᵒ phase angle with normal distribution.