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An analytical model for optimization of perforation scheme to control water movement profile.


Water serves as one means of maintaining formation pressure as well as displacing crude oil towards production wells, namely, water flooding. In layered and heterogeneous reservoirs water movement front is not always uniform due to different permeability in layers, leading to breakthrough of water in high permeable layers while in some layers oil not being displaced efficiently. This phenomenon during water displacing oil is called 'tongue effect'. High water cut in the produced liquid could reduce total oil recovery, lower well productivity and increase surface oil processing cost. So it is of paramount importance to avoid premature breakthrough of water. Currently common methods used to mitigate this phenomenon include mechanical treatment, chemical treatment and downhole water separation, etc. These methods are also known as water shutoff (WSO) methods (Thomas et al., 1998; Ghedan et al., 2009). These methods can be technically and economically very effective if they are applied appropriately in the near wellbore region. However it was found that with the complexity of pay zones, these methods cannot always fulfill the need of oil field. As a supplementary method, limited entry perforating using perforation scheme optimization has drawn more attention. It is defined as optimization of perforation parameters within the process of perforated completion, based on filtration resistance distribution in the formation to homogenize water movement front profile across multiple layers.

The energy loss is caused by a so-called resistance which includes resistance to fluid flow in the formation, across perforation and along wellbore, see Figure 1. Among these three resistances what we can control is only pressure loss across perforations, generated by perforation tools like bullet gun, water jets or more popular shaped charges. This paper introduces a simplified mathematical model to combine the concept of perforation optimization with productivity index. By means of adjusting these perforation parameters, e.g., shot density, perforation penetration, pressure loss across perforation can be controlled. For high permeable zone, flow density and penetration should be reduced and vice versa, so that flow resistance in this region can be adjusted to homogenize the flow resistance in every single layer, reach a uniform distribution of water movement front and get an optimal productivity.


Factors influencing productivity index ratio

The factors influencing productivity index include shot density, perforation penetration, phasing angle, shot diameter, etc. Shot density is the measurement of the perforations made per unit length of a gun, normally represented as number of shots per foot (SPF) or shots per meter (SPM). In most instances a density of 1 to 4 SPF (3 to 13 SPM) is enough. Perforation guns with a density of more than 4 SPF (13 SPM) are called High-Shot-Density guns. A higher shot density in one of the multiple layers can reduce pressure loss and make the flow across the vicinity of wellbore easier. Perforation penetration can range from essentially zero to several inches, depending on the perforator used and mechanical and physical properties of the materials being penetrated. The holes are distributed in an angular pattern around the interior of the wellbore. This dispersion is called phasing angle.

The influences of the reservoir properties, well geometry and perforation parameters on the productivity of perforated wells have been investigated by many researchers (Harris, 1966; Hong, 1975; McLeod, 1983; Karakas and Tariq, 1991; Bell et al., 1995; El-Bermawy and El-Assal, 2003; Asheim and Oudeman, 1997). Perrin et al. (1999) has summarized these factors influencing productivity of perforated wellbore. In case only the steady state flow is considered, shot density has a great impact on productivity index within a certain range. After reaching a peak value, an increasing shot density cannot increase productivity index more. The productivity index increases with perforation penetration within a range as well. It has been proved that phase angle of 60[degrees] and 90[degrees] can lead to best productivity index, then 120[degrees] and 180[degrees] to a lesser extent, and the worst phase angle is 0[degrees]. Hole diameter does not have much impact on productivity index.

Model development Assumptions

Several assumptions are made before building a model including:

* Flow from the vicinity of the well to the wellbore is assumed to be steady state, meeting the Darcy flow requirements and the effect of gravity is ignored;

* The reservoir has multiple layers with non-uniform permeability and thickness distribution;

* In the horizontal direction the layers are isotropic, homogeneous with uniform thickness.

Total Filtration Resistance

As the reservoir fluids are produced, the pressure across the complete system drops, as discussed in the introduction. Based on permeability values in different reservoir layers, resistance to fluid filtration in different layers will be produced. In the low permeable reservoir the filtration resistance is larger because of small pore throat of the void space (Baoquan et al., 2010). Total filtration resistance is a sum of resistances in formation ([R.sub.r]), across perforations ([S.sub.t]) and in the wellbore ([R.sub.b]).

Filtration Resistance Adjustment Coefficient (FRAC)

Filtration resistance adjustment coefficient (FRAC) is defined as the ratio of total filtration resistance after perforation optimization to the filtration resistance of an open-hole without any skin near wellbore. It reflects the degree of perforation optimization. Mathematically it can be written as:

FRAC = [R.sub.b] + [R.sub.r] + [S.sub.t]/ [R.sub.r] + [R.sub.b] (1)

The filtration resistance of fluid flow in the wellbore can be eliminated at this point as we consider the fluid flow in the formation and the perforation system only. Hence the above equation can be simplified as:

FRAC = [R.sub.r] + [S.sub.t]/[R.sub.r] (2)

Considering the reservoir section, three zones can be divided between water injector and oil producer as: water zone (mainly water), oil-water co-existence zone and oil zone (mainly oil), see Figure 2.


The filtration flow resistance in the formation from the injector to the producer is therefore the sum of resistance in the three zones mentioned above. Therefore, it can be mathematically expressed as:

R = [[mu].sub.w]([D.sub.e]/ [D.sub.o])/2[pi][K.sub.w]h + [alpha] [[mu].sub.ow] ln([D.sub.0]/ [D.sub.f])/ 2[pi][K.sub.ow]h + [[mu].sub.0]ln([D.sub.f] / [D.sub.w])/ 2[pi][K.sub.o]h (3)

Where, [[mu].sub.w], [[mu].sub.0w], p.0 is the water, oil-water, and oil viscosity, respectively; [K.sub.w], [K.sub.0w], [K.sub.0] is the water, oil- water, and oil permeability in the respective zones; h is the height of the zone; [D.sub.w] is the wellbore radius; [D.sub.f] is the radial distance of the oil zone from the wellbore; D0 is the radial distance of the oil-water zone from the wellbore; De is the radial distance of the water zone from the wellbore; [alpha] is expressed as (Ge, 2001):

[alpha] = [[mu].sub.w]/[[mu].sub.o](1.7 + 8[z.sub.[phi]] + 25[z.sub.[phi].sup.2]) (4)

Where [z.sub.[PHI]] represents the movable oil saturation on the front of oil- water contact and is given by following equation:

[z.sub.[phi]] = 0.1[square root of [[mu].sub.r]/1.5(1 - [S.sub.or] - [S.sub.wi])-[z.sub.[phi]] (5)

[z.sub.[PHI]] can be obtained by solving this implicit equation. Residual oil saturation S0r and irreducible water saturation [S.sub.wi] can be obtained from the saturation distribution curves for water displacing oil. [[mu].sub.r] is defined as relative viscosity, viz. [[mu].sub.w]/[[mu].sub.o]. Total skin factor [S.sub.t] is defined as:

Where [S.sub.d] represents skin due to damage; [s.sub.c+[theta]] represents skin due to partial completion and slant; sp represents perforation skin effect; [summation][s.sub.pseudo] represents pseudo-skin group defined as a sum of all phase and rate-dependent effects.

Filtration Resistance Intensity Adjustment Coefficient (FRIAC)

The purpose of perforation optimization is to homogenize permeability in different layers with different filtration resistances. It is achieved by reaching a uniform distribution of filtration resistance and avoiding the premature water breakthrough by means of optimizing shot density and perforation penetration. Permeability values in the layers are expected to approach an average value which is called a benchmark value. Therefore, the filtration resistance intensity adjustment coefficient is defined as the ratio of the benchmark value of filtration resistance to the filtration resistance in a single layer (see Equation 7 and Figure 3). In case it is bigger than 1, the layer is usually high permeable and the resistance in that layer should be increased after perforation; In case smaller than 1, after perforation the resistance in that layer should be decreased, normally in low permeable layer.

FRIAC = R(benchmark)/R(i) (i = o, 1, 2,...,z) (7)

Where R(benchmark) is benchmark value to be achieved; R(i) filtration resistance of 1st, 2nd, 3rd......zth formation layer.


Relation between FRAC and FRIAC

Filtration Resistance Intensity Adjustment Coefficient (FRIAC) reflects the filtration resistance difference between layers while Filtration Resistance Adjustment Coefficient FRAC reflects the resistance difference before and after perforation in one single layer. Apparently we should equalize them to reach our goal, and then we have


Productivity index ratio

Darcy equation for steady-state flow can be expressed as Equation 9. A rearrangement of the equation can yield an expression for the productivity index (PI) as Equation 10. In a cased and perforated well, the productivity index is affected by the formation parameters as well as perforation parameters. Productivity index ratio (PIR) is referred to as the ratio of actual productivity index to the ideal open-hole productivity index, see Equation 11 and Equation 12.

P - [] = q[mu]/2[pi]kh (1n r/[r.sub.w] + s) (9)



Where, [] is bottom hole flowing pressure; [r.sub.w] is wellbore radius. By combining Equation 8 and Equation 11 an expression of PIR in terms of [B.sub.ta] will be obtained.


Productivity index ratio and perforation parameters

A large quantity of data from the oil well perforation model has been obtained by researchers, which is regressed and calculated with the help of non-linear regression method. Without consideration of influence of contamination and compaction, the following expression has been derived:

PIR = a + b * [P.sub.A] + c * [m.sub.k] + d * [j.sub.k] + e * [P.sub.HA] + f * [D.sub.w] + g * [K.sub.v] / [K.sub.h] - h * [K.sub.v] /([K.sub.r] [D.sub.w]) - i * [P.sub.HA.sup.2] (13)

Where, a, b,..., i are coefficient related to perforation; Pa is perforation penetration, mm; [m.sub.k] is perforation density, mm; [j.sub.k] is perforation diameter; [K.sub.v]/[K.sub.h] is ratio of vertical and horizontal permeability; [P.sub.HA] is phase angle, [degrees]; [D.sub.w] is wellbore diameter.

By combining Equation 12 and Equation 13 we can get the relation between perforation parameters and FRIAC. By tuning the shot density, perforation penetration and phase angle the resistance in every single layer can be controlled.

Development of Perforation Optimization Software

Software for Optimization of Perforation Scheme has been developed by Northeast Petroleum University in China based on principles stated above. This software can be used to obtain optimal perforation schemes to meet the requirement of maximizing productivity and lowering casing strength decrease. The software has been programmed with Microsoft Visual Basic 6.0 in the context of 2000/XP and has accounted for the requirement of accuracy, usability and reliability. The principle of this software towards an optimal perforation scheme can be summarized as three steps shown in Figure 4.

Firstly calculate the flow resistance in every layer based on the formation and fluid parameters;

According to resistance difference in different layers, a benchmark value is determined so that every layer will be adjusted to approach this value, thus the resistance intensity adjustment coefficient in every layer can be calculated.

As resistance intensity adjustment coefficient is a function of perforation parameters, e.g., shot density, perforation penetration, shot diameter, etc., it is possible these parameters could be calculated reversely based on the resistance intensity adjustment coefficient. Finally the perforation scenarios will be generated.


Field application

The software has been tested in one pilot area in Daqing oil field and applied for two wells, viz. Well1 and Well2. The basic data of the test block in the pilot area is shown in Table 1. The basic data of every small layer in the reservoir penetrated by Well 1 and Well 2 as well as the optimization results are shown in Table 2 and Table 3. The water percentage amount in every single layer before and after optimization is shown in Figure 5. We can see the water injection profile has been improved.



A simplified mathematical model has been built to facilitate production or reservoir engineers to optimize perforation schemes, e.g., shot density, perforation penetration, etc. Software based on the mathematical model has been applied for two wells in one pilot area in Daqing oil field. After optimization of perforation scheme, the water movement profile is adjusted and recovery is enhanced. The proposed method was designed to offer some optimized combination of shot density, perforation penetration, and phase angle, and provide some reference on the design of perforation scheme. In the reality, some other factors must be also taken into consideration, e.g., cost effectiveness, casing and cement damage, etc.


The authors would like to thank No. 6 Production Company in Daqing Oilfield for permission to publish this paper.


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Jingyuan Zhao (1), Mingxing Bai (1), (2), Yuxue Sun (1), Fulei Zhao (1)

(1) Department of Petroleum Engineering, Northeast Petroleum University, China, (2) Institute of Petroleum Engineering, Clausthal University of Technology, Germany Corresponding author: Jingyuan Zhao. Email:
Table 1: The basic data of one pay zone in the pilot area

Pressure Radius Density Oil volume factor
(MPa) (m) (g x [cm.sup.3]) ([m.sup.3]/[Nm.sup.3])

10.34 150 0.864 1.118

Pressure Oil viscosity Water viscosity
(MPa) (mPa x s) (mPa x s)

10.34 21.6 1

Table 2: Petrophysical properties of layers and optimization
results for Well1

Nr. Layer Perforated Effective Effective
 ID. interval (m) thickness Perm.
 From To (m) ([mu][m.sup.2])

1 4-7 (5) 990.1 987.9 2.2 0.750 ~
 4-7 (5) 1.018
2 987.9 982.7 1.7 0.038 ~

3 4-7 (3) 981.9 981.4 0.5 0.148 ~
 4-7 (2) 0.188
4 979.3 978.8 0.5 0.153 ~
 4-7 (1) 0.436
5 978.1 977.6 0.5 0.258 ~

Nr. Perforation Perforation
 penetration density
 (cm) (shots / m)

1 11 9

2 14 15

3 23 16

4 21 13

5 20 12

Table 3: Petrophysical properties of layers and optimization results
for Well2

Nr. Layer Perforated Effective Effective
 ID. interval (m) thickness Perm.
 From To (m) ([mu][m.sup.2])

1 4-7 (3) 1004.4 998.8 2.2 0.750 ~
 4-7 (3) 1.018
2 998.8 995.2 1.5 0.038 ~
 4-7 (2) 0.548
3 997.3 995.0 1.2 0.148 ~
 4-7 (1) 0.188
4 993.6 993.1 0.5 0.153 ~

Nr. Perforation Perforation
 penetration density
 (cm) (shots / m)

1 12 9

2 13 13

3 16 15

4 21 16
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Author:Zhao, Jingyuan; Bai, Mingxing; Sun, Yuxue; Zhao, Fulei
Publication:International Journal of Petroleum Science and Technology
Date:Sep 1, 2012
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