A Research of Perforation Plan-decision Based on Grey Cluster Relation

Perforated well completion As the most extensive and major method of the well’s completion, the reasonable selection of parameters for the program has great meaning of improving efficiency and reducing costs[1][2]. By establishing a quantitative regression model to study the relationship between the parameters of the perforation and the production ratio, this algorithm can also analysis how different factors (perforation elasticity, perforation penetration, shot density, perforation diameter, perforation phase angle) act on the production ratio and casing strength coefficient. It provides a reliable theoretical basis for the perforation parameter optimization, and gives different perforation completion optimization schemes [3].

Due to the mutual restriction of different parameters, the current subjective decision-making for perforation program can’t make all the factors to achieve the best at the same time. In order to solve the above problems and reduce the subjective influence of the decision maker, maximize the productivity ratio[4], a Perforation Plan-decision Based on Grey Cluster Relations proposed[5-7].

II.

PERFORATION PLAN-DECISION BASED ON GREY CLUSTER RELATION

Perforation optimization needs to confirm a solution to maximize the production capacity. This solution depends on many factors and the main influencing factors are hole depth, pore size, pore density, phase angle, formation heterogeneity, drilling pollution degree and depth, perforation compaction thickness and degree. All these factors are acting on the decision-making of the solution on the same time.

Perforation Plan-decision based on Grey Cluster has made the model of perforation parameters and the oil well productivity. Gray parameters are clustered in the parameters of the perforation scheme, and the evaluation function is established to design the optimal scheme [8-10].

Building of model

First simulating and calculating the productivity ratio of oil and gas, then making a non-linear regression analysis, According to whether perforation penetration penetrate the drilling zone or not, an equation can be established, it indicates the relationship between perforating parameters and capacity.

The regression equation when the perforation penetration does not penetrate the drilling zone:(1) $\begin{array}{l} PR = 0.000156 Y_{h}^{2} - 0.000452 W_{h} + 0.000000205 W_{h}^{2} + 0.319 K_{z r} - 0.0009512 r_{w} + 0.000296 K_{j}^{2} \\ - 0.25 + 0.00148 K_{s} - 0.00000123 K_{s}^{2} + 0.00958 K_{m} - 0.0000998 K_{m}^{2} - 0.00467 X \lg (K_{z r}) + 0.0178 K_{j} - 0.000296 K_{j}^{2} \\ + 0.00195 X_{w} - 0.00001 X_{x}^{2} - 2 * 0.000828 X_{w} * \lg (k_{z r}) + 0.3488 Y_{c} - 0.1745 Y_{c}^{2} + {0.69}_{c} - 0.35 w_{c}^{2} \\ - 0.00778 Y_{h} \end{array}$

The regression equation that perforation penetration has penetrated the contaminated zone of drilling:(2) $\begin{array}{l} PR = - 0.25 + 0.00191 K_{s} - 0.00000136 K_{s}^{2} + 0.01665 K_{m} - 0.000173 K_{m}^{2} - 0.00856 K_{m}^{2} \lg (K_{zr}) + 0.0201 K_{j} \\ - 0.000335 K_{j}^{2} + 0.00211 X_{w} - 0.0000108 X_{w}^{2} - 0.001774 X_{w} \lg (K_{zr}) + {0.51}_{g} (K_{zr}) + 0.512 Y_{c} - 0.253 Y_{c}^{2} \\ + 0.3315 W_{c} - 0.168 W_{c}^{2} - 0.00963 Y_{h} + 0.000193 Y_{h}^{2} - 0.000841 W_{h} + 0.000000714 W_{h}^{2} + 0.406 K_{zr} - {0.135}_{zr}^{2} \\ - 0.0009736 r_{w} + 0.00000243 r_{w}^{2} \end{array}$

The quantitative relationships between parameters (perforation penetration KS, perforation aperture Kj, perforation phase Xw, perforation compaction degree Yc, perforation compaction thickness Yh, drilling damage thickness Wh, drilling pollution degree Wc, shot density Km, borehole radius rw, formation permeability Kzr) and the oil production ratio PR is the basis for the optimization of perforating parameters.

Perforation program base on Grey Cluster Relation

The main factors in the decision-making of the perforation plan are six factors: perforation ratio, perforation phase angle, shot density, perforation penetration, perforation diameter and casing strength decreasing coefficient, which are expressed by attributes x₁, x₂, x₃, x₄, x₅, x₆ respectively. Initial feature object matrix D is made like this:(3) $D = (\begin{matrix} x_{11} & x_{12} & x_{13} & \dots & x_{1 n} \\ x_{21} & x_{22} & x_{23} & \dots & x_{2 n} \\ x_{31} & x_{32} & x_{33} & \dots & x_{3 n} \\ x_{41} & x_{42} & x_{43} & \dots & x_{4 n} \\ x_{51} & x_{52} & x_{53} & \dots & x_{5 n} \\ x_{61} & x_{62} & x_{63} & \dots & x_{6 n} \end{matrix})$

In the formula, x_ij represents j th attribute of the ith scheme; in the j scheme x_1j represents the productivity ratio, x_2j is the phase angle, x_3j is the perforation diameter, x_4j is the hole depth, x_5j is the aperture, and x_6j is the casing strength reduction coefficient. There are n scheme and 6 attributes.

As the different dimensions will have an impact on decision-making, so the formula (4) - (6) are used to D for normalization.

The normalization of attribute data based on the different effects caused by different attributes, the formula (4) shows the method to normalize production ratio, which called upper limit method. Inherent properties such perforation phase angle, shot density, perforation penetration, perforation diameter are concluded by extreme conversion method, shown as formula (5). Casing strength decreasing coefficient, as a cost-type attribute, calculated by the lower limit method, shown as formula (6).

(4)

r_{1 j} = \frac{x_{1 j}}{\max (x_{1 j})}

(5)

r_{i j} = \frac{x_{i j}}{\max (x_{i j}) - \min (x_{i j})}

(6)

r_{6 j} = \frac{x_{6 j}}{\min (x_{6 j})}

In the formula, 2 ≤ i ≤ 5, j ≤ n, the normalized decision matrix can be calculated: R = (r_ij)_{6 × n}.

The Grey Clustering analysis is used to classify the attributes and the similar factors can be classified and simplified.

Initialize processing:(7) $\begin{array}{l} r_{i j} = r_{i j} - r_{i 1} \\ 2 \leq i \leq 5, j \leq n \end{array}$

Calculate the gray absolute correlation degree ε_ik of any two parameter index data Ri and Rk sequence (1 ≤ k, 1 ≤ i ≤ 6, j ≤ n):(8) ${\begin{array}{l} | S_{i} | = | \sum_{j = 2}^{n - 1} r_{i j} + 0.5 * r_{i n} | \\ | S_{k} - S_{i} | = | \sum_{j = 2}^{n - 1} (r_{k j} - r_{i j}) r_{i j} + 0.5 * (r_{k n} - r_{i n}) | \\ ε_{i k} = \frac{1 + | S_{i} | + | S_{k} |}{1 + | S_{i} | + | S_{k} | + | S_{i} - S_{k} |} \end{array}$

Establishing attribute correlative sequence matrix according to the above gray absolute correlation degree:(9) $ε = [\begin{matrix} 1 & ε_{12} & ε_{13} & ε_{14} & ε_{15} & ε_{16} \\ 1 & ε_{23} & ε_{24} & ε_{25} & ε_{26} \\ 1 & ε_{34} & ε_{35} & ε_{36} \\ 1 & ε_{45} & ε_{46} \\ 1 & ε_{56} \\ 1 \end{matrix}]$

The critical value r ∈ (0, 1), in pursuit of accuracy the value of r is higher than 0.5, the higher the r value, the more accurate the classification is, and the accurate value of r is determined by actual data, the Ri and Rk classified as similar attributes; when ε_ij ≥ r.

Several attributes can be merged by the calculation above, and an attribute can be chosen to instead of other similar attributes.

A new feature matrix D’ and new normalization matrix R′ = (r′_ij)_{m × n} is established according to the Grey Clustering analysis, where m is the number of attributes and n is the number of schemes.

Computing information entropy E, and weight ω_i(1 ≤ i ≤ m, j ≤ n):(10) ${\begin{array}{l} E_{i} = - \frac{1}{\ln (n)} \sum_{j = 1}^{n} r^{''}_{i j} \ln (r^{''}_{i j}) \\ r^{''}_{i j} = \frac{r^{'}_{i j}}{\sum_{j = 1}^{m} r^{'}_{i j}} \end{array}$

In particular, when $r_{i j}^{''} = 0$ , let $r_{i j}^{''} \ln (r_{i j}^{''}) = 0$ .

(11) $ω_{i} = \frac{1 - E_{i}}{\sum_{i = 1}^{m} (1 - E_{i})} 1 \leq i \leq m$

And 0 ≤ ω_i ≤ 1, ω₁ + ω₂ + … ω_m = 1, 1 ≤ m ≤ 6.

Establish an evaluation function Z_k:(12) $Z_{k} = \sum_{i = 1}^{m} ω_{i} r_{i k}^{'}, 1 \leq i \leq m, 1 \leq k \leq n$

When the evaluation function value Z(Rk) is larger, the corresponding scheme is better. The program has the largest value of Z(k) is chosen as the final construction program.

III.

SIMULATION EXPERIMENT

White XX well in Chang-qing Oilfield, the reservoir depth of middle layer is 1 884.5m, the total thickness is 9.5 m, the thickness of the perforated zone is 3.0 m, the porosity is 13.41%, reservoir drainage radius is 200m, well-bore radius is 0.111 m, the pressure of formation is 13.073 MPa, the crude oil saturation pressure is 9.86 MPa, drilling pollution depth is 69.5mm, the drilling pollution degree is 0.6. The casing strength is 47.8MPa, reservoir heterogeneity is 0.7 (vertical permeability / horizontal permeability), the water saturation is 30.21%, rock Poisson’s ratio is 0.5, the inclination is 5º, the oil viscosity is 1.03 MPa.S, the perforation optimization scheme is shown as Table 1.

The initial feature matrix D = (X_ij)_{6 × 24} can be constructed from the data in Table 1 and the results are shown in Table 2.

The feature object matrix R = (r_ij)_{6 × 24} is established by the above equation (4) - (6) and the initial feature matrix D, is shown as table 3. The index data association matrix is established by the above equations (7) and (8): $ε = [\begin{matrix} 1 & 0 .9953 & 0 .9499 & 0 .9937 & 0 .9937 & 0 .9976 \\ 1 & 0 .5841 & 0 .8571 & 0 .8571 & 0 .9176 \\ 1 & 0 .7747 & 0 .7747 & 0 .9602 \\ 1 & 1 & 0.9696 \\ 1 & 0.9696 \\ 1 \end{matrix}]$

ESTABLISHMENT OF FEATURE OBJECT MATRIX R

According to the correlation degree matrix, take the critical value r = 0.8, R2, R4 and R5 can be regarded as same class, then take R2 represent this class. Then the influencing attributes of perforation program are adjusted to: productivity ratio R1, perforation phase angle R2, shot density R3, casing strength decreasing coefficient R6. Establishing new normalization matrix R’=(r_ij)4 ×24, shown as table 4.

DEALS WITH THE FEATURE MATRIX BY GREY CLUSTER RELATION R’

The attribute weight vectors ω =(0.0036,0.2826, 0.2797,0.4340) are calculated according to formulas (10) and (11).

Then the evaluation function Z is established according to (12): Z={2.2090,2.2348,2.4716,2.1051,2.2282,2.0103,2.2211,2.0032,2.1068,2.0288,1.6710,2.0375,1.8364,1.8451,1.6 527,1.7238,1.5569,1.4562,1.2894,1.3900,1.1225,1.2519,1.1105,0.9437}. The optimal scheme is A3 because the Z value of scenario A3 is the largest. It means under the existing formation conditions, the best perforation program is: perforation bullet SYD127-1, phase angle 120º, hole density 32m, wearing depth 328.68mm, aperture 10.68mm.

IV.

CONCLUSION

In this paper, a Perforation plan-decision based on Grey Cluster Relation is putted forward. This method can be widely used to predict the productivity of wells under different perforation conditions, determine the perforating efficiency of perforated bombs, and study how different factors (the perforation elasticity, perforation penetration, shot density, perforation diameter, perforation phase angle) impose influence to productivity ratio, and casing strength decreasing coefficient. According to the pending reservoir, it also let the oil production capacity to achieve the higher perforation operating parameters and process of excellent combination. It also saves a lot of manpower, materials and time cost, and provide the theoretical basis for the design of completion perforation construction.

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A Research of Perforation Plan-decision Based on Grey Cluster Relation

Online veröffentlicht: 07. Mai 2018

Seitenbereich: 86 - 91

DOI: https://doi.org/10.21307/ijanmc-2018-014

Schlüsselwörter
Perforating Operation, Grey Cluster Relation, Perforation Plan-decision

© Xue Jijun 2018 et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

TABLE I.

TABLE II.

TABLE III.

TABLE IV.

A Research of Perforation Plan-decision Based on Grey Cluster Relation

Online veröffentlicht: 07. Mai 2018

Seitenbereich: 86 - 91

DOI: https://doi.org/10.21307/ijanmc-2018-014

SchlüsselwörterPerforating Operation, Grey Cluster Relation, Perforation Plan-decision

© Xue Jijun 2018 et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

TABLE I.

TABLE II.

TABLE III.

TABLE IV.

Schlüsselwörter
Perforating Operation, Grey Cluster Relation, Perforation Plan-decision