Evaluation of inclined arc trajectory parameters for design and control of 3D directional well

Abstract

The circular-arc trajectory on inclined the plane is one of the models for the orbit design of directional drilling. The key issue of the design and control for the circular-arc trajectory is how to judge the theoretical calculated values of the target-hitting azimuth and tool face angle, because the target-hitting azimuth and tool face angle toward targets of 3D directional well may exist in four quadrants of 0 ${}^{\circ}$ $\sim$ 360 ${}^{\circ}$ . Based on the spatial geometric model of the circular-arc trajectory on the inclined plane, a new evaluation method of target azimuth change is proposed according to coordinates of the target relative to the horizontal projection of two lines’ intersection. These two lines are tangents at the starting and ending points of the complete build-up curve section respectively. Therefore, the analytic expression of a criterion for target azimuth change in the deflecting section is obtained. For control of circular-arc trajectory, a judgmental method of the tool face angle is also discussed in the paper, and its criterion is worked out using quadrant overlap of the value calculated by two formulas of tool face angle. The practical engineering cases have demonstrated that the new evaluation methods of the target azimuth and the tool face angle are effective. And the trajectory data of back calculation shows high accuracy to hit the target. Compared with traditional evaluation modes, the methods proposed in this paper avoid the ambiguity of 0 ${}^{\circ}$ azimuth and the partial increase of tool face angle. Simultaneously they can be applied directly in practical project as well as algorithm of directional drilling software.

Keywords

Directional drilling well trajectory inclined arc target azimuth tool face angle evaluation model

1. Introduction

In directional drilling, the circular-arc trajectory on the inclined plane has been applied widely to designing and controlling well orbit by its inherent advantages including the simple structure of track bending, the minimum overall curve angle of hitting target, and etc. The theoretical design and practical control of circular-arc trajectory are more complex because the projections of inclined arc trajectory on the vertical section and horizontal plane are the curve of non-constant curvature [1, 2, 3]. And some parameters of the trajectory design and control are multiplicity periodic solutions as the target of well may exist in any position in three-dimensional space. During the past decades, many scholars have done many studies about the design and control method by use of level projection, simplified analysis method, vector analysis, coordinate transformation, lagrange algorithm, coordinate increment identity, etc. [4, 5, 6, 7, 8, 9, 10, 11, 12]. The research on the design and control of well-path in space, revealed the well deflection evolution rules of inclined arc trajectory, and different modes of design and control were proposed in their papers [1, 3, 13]. In view of the four-quadrant evaluation of target-hitting azimuth and tool face angle, many approaches were presented, including target closure azimuth, azimuth comparison and compensation, judging positive or negative of the numerator and the denominator in the functional equation. In addition, some other methods such as the effect rule of setting angle in different quadrants, the positive or negative rotation of target azimuth, had been researched and applied to judging the value of tool face angle and controlling the circular-arc trajectory [12, 13, 14, 15, 16, 17, 18, 19]. Although the design theory and control technology of inclined arc trajectory have been continuously developed, most evaluation models are complex and difficult to understand. Some estimation models are simple, but there may be situations where the trajectory control produces deviation sometimes as a result of the “partial increase” of tool face angle and the ambiguity of the 0 ${}^{\circ}$ azimuth not being taken into consideration [3, 20, 21, 22]. Therefore, it is necessary to further study the evaluation models of the target-hitting azimuth for trajectory design and the tool face angle for trajectory control. Based on the geometric relationship of circular-arc trajectory on the inclined plane in space, the evaluation model according to coordinates of the given target relative to the horizontal projection of the PTI (point of tangents intersection, which is the intersection of two tangents at the starting and ending point of the complete build-up curve section respectively) is studied for design of the target-hitting azimuth. Also the judgment model utilizing quadrant overlap of the numerical results calculated by two different equations (the numerical overlap of double formulas) is studied to determine the tool face angle in this paper.

2. Judgment of the target azimuth angle for trajectory design

2.1 Trajectory model

Illustrated by the case of the trajectory design of the three dimensional drilling towards a given target, commonly used in directional well, the space model of the borehole trajectory on the inclined plane can be built as shown in the Fig. 2, $O$ is the wellhead on the surface of the earth; $E$ is the given target point in trajectory design; OA is the known section of upper well (during pre-drilling design, OA can be a straight well section with fixed inclination and azimuth; in tracking design while drilling, OA can be a curved well section with different inclination and azimuth; branch drilling, OA can be the upper section of the main hole); $A$ is the kick-off point (KOP) or selected branch point; AB is the build-up section or deflection section and its radius of curvature is $R$ in design; BE is steady inclined straight section (holding section). Plane I is horizontal plane, which point $E$ is in. Plane II is vertical plane which is determined by trajectory tangent at the point $A$ . The inclined plane III, which the trajectory ABE is in, is determined by trajectory tangent at the point $A$ and the target $E$ . According to geometric relationship of the circular-arc trajectory parameters, the calculation formula of the target-hitting azimuth is derived [23]:

$\displaystyle\Delta\phi=\pm\arccos\frac{\cos\gamma-\cos\alpha_{A}\cos\alpha_{B% }}{\sin\alpha_{A}\sin\alpha_{B}}+2k\pi\quad(k=0,1)$ (1) $\displaystyle\phi_{E}=\phi_{B}=\phi_{A}+\Delta\phi$ (2)

Figure 1.

Design model of trajectory for 3-D directional well on the inclined plane in space.

Figure 2.

Space model of the tool face angle and well trajectory parameters.

Here, $\Delta\phi$ is the azimuth change of target $E$ relative to point $A$ ; $\gamma$ is the overall angle change of section AB; $\alpha$ ${}_{A}$ is the inclination of trajectory at the point $A$ ; $\alpha$ ${}_{B}$ is the inclination of trajectory at the point $B$ ; $\phi$ ${}_{E}$ is the target-hitting azimuth; $\phi$ ${}_{A}$ is the azimuth of trajectory at the point $A$ ; $\phi$ ${}_{B}$ is the azimuth of trajectory at the point B. Due to BE is the holding section of inclined line, $\phi_{B}=\phi_{E}$ .

2.2 Analysis of target-hitting azimuth change

In practical engineering, the azimuth change $\Delta\phi$ may exist in the four quadrants of 0 ${}^{\circ}$ $\sim$ 360 ${}^{\circ}$ , target E, especially the branch target in three-dimensional space, could be in any direction relative to point A, And two corresponding values could be solved by Eq. (1). Therefore, it is necessary to make reasonable judgment on the value of $\Delta\phi$ .

On the basis of the trajectory space model in Fig. 2, the essence of $\Delta\phi$ is the azimuth change (increment/decrement) on the horizontal plane when the trajectory direction of point A is turned to the direction of target E. Hence, extending the tangent of point $A$ , the point $D$ is its intersection with the plane I. Extending BE, and it cuts AD at the point $C$ (the point $C$ is the intersection of the two tangents at endpoints of AB, and defined as PTI, for short). On the horizontal plane I, the value of $\Delta\alpha$ could be judged and determined by the quadrant location of target E relative to projection $C^{\prime}$ which is the horizontal projection of the PTI. Assume that, $\phi$ ${}_{A}$ and $\phi$ ${}_{B}$ are geographical azimuth and have been modified by magnetic declination and meridian convergence, so the geographical coordinates of point $C$ can be calculated by Eq. (3).

$\displaystyle\left\{{\begin{array}[]{l}Y_{C}=Y_{A}+\frac{1}{K}\tan\frac{\gamma% }{2}\sin\alpha_{A}\sin\phi_{A}\\ \\ X_{C}=X_{A}+\frac{1}{K}\tan\frac{\gamma}{2}\sin\alpha_{A}\cos\phi_{A}\\ \\ Z_{C}=Z_{A}+\frac{1}{K}\tan\frac{\gamma}{2}\cos\alpha_{A}\\ \end{array}}\right.$ (3)

In Eq. (3), $X_{A}$ , $Y_{A}$ and $Z_{A}$ are relative geographical coordinates of the point $A$ , which can be determined by the inclination and azimuth of the section OA; $K$ is the curvature of the section AB, which can be assigned as required.

2.3 Judgment criterion of azimuth change

The right-handed relative coordinate system is established with $C^{\prime}$ for the origin and the direction of the azimuth $\phi$ ${}_{A}$ for positive direction of axis $X$ . And coordinates of the target E relative to point $C^{\prime}$ , $\Delta X_{EC}$ and $\Delta Y_{EC}$ , can be calculated by Eq. (4).

$\displaystyle\left\{{\begin{array}[]{l}\Delta X_{EC}=(X_{E}-X_{C})\cos\phi_{A}% +(Y_{E}-Y_{C})\sin\phi_{A}\\ \Delta Y_{EC}=(Y_{E}-Y_{C})\cos\phi_{A}-(X_{E}-X_{C})\sin\phi_{A}\\ \end{array}}\right.$ (4)

The judgment criterion of azimuth change $\Delta\phi$ is derived as follow

$\displaystyle\left\{{\begin{array}[]{l}\Delta\phi\epsilon(0^{\circ},90^{\circ}% ),\text{when}\;\Delta X_{EC}>0,\Delta Y_{EC}>0\\ \Delta\phi\epsilon(90^{\circ},180^{\circ}),\text{when}\;\Delta X_{EC}<0,\Delta Y% _{EC}>0\\ \Delta\phi\epsilon(180^{\circ},270^{\circ}),\text{when}\;\Delta X_{EC}<0,% \Delta Y_{EC}<0\\ \Delta\phi\epsilon(270^{\circ},360^{\circ}),\text{when}\;\Delta X_{EC}>0,% \Delta Y_{EC}<0\\ \end{array}}\right.$ (5)

3. Judgment of the tool face

In the directional well, the method to control tool face angle for downhole motor drilling is often adopted to achieve circular-arc trajectory on the inclined plane. The tool face angle, also called setting angle, includes gravity tool face angle and magnetic tool face angle.

3.1 Geometric relationship model of tool face angle

Taking gravity tool face angle for example, the space model of the corresponding relationship between tool face angle and geometric parameters of the well trajectory is shown in Fig. 2.

In Fig. 2, CA is the tangent at the starting point $A$ in the section AB; CB is the tangent at the endpoint $B$ in the section AB; $\textit{CC}^{\prime}$ is vertical line; $\alpha$ ${}_{A}$ and $\alpha$ ${}_{B}$ are their inclinations respectively; $\gamma$ is the overall angle of the build-up section AB; $M^{\prime}$ is the point of gravity high side at well trajectory $A$ ; $\omega$ is the gravity tool face angle; $C^{\prime}A^{\prime}$ and $C^{\prime}B^{\prime}$ are the projection of CA and CB on the horizontal plane; $\phi$ ${}_{A}$ and $\phi$ ${}_{B}$ are there azimuth angles respectively, $\Delta\phi=\phi_{B}-\phi_{A}$ ; According to the spatial geometric relationships, the equation for $\omega$ can be derived as follow [2, 21, 23, 24, 25]

$\displaystyle\omega=\arctan\frac{\sin\Delta\phi}{\cos\alpha_{A}\cos\Delta\phi-% \sin\alpha_{A}\cot\alpha_{B}}+k\pi\quad(k=0,1,2)$ (6)

Obviously, the tool face angle $\omega$ will have two calculation values in 0 $\sim$ 360 ${}^{\circ}$ by Eq. (6). In accordance with control principle of the unique path for the circular-arc trajectory on the inclined plane, some judgment criteria must be supplemented and used to pick out the correct value from them.

According to the corresponding geometric relationships between tool face angle and geometric parameters of well trajectory, another equation for $\omega$ is given by

$\displaystyle\omega=\pm\arccos\frac{\cos\gamma\cos\alpha_{A}-\cos\alpha_{B}}{% \sin\gamma\sin\alpha_{A}}+2k\pi\quad(k=0,1)$ (7)

3.2 Judgmental method of tool face angle

The method is to construct a criterion using quadrant overlap of the value calculated by two formulas, namely the numerical overlap of double formulas.

Assuming that

$\displaystyle F_{1}=\frac{\sin\Delta\phi}{\cos\alpha_{A}\cos\Delta\phi-\sin% \alpha_{A}\cot\alpha_{B}}\quad F_{1}\epsilon[-\infty,\infty];$ $\displaystyle F_{2}=\frac{\cos\gamma\cos\alpha_{A}-\cos\alpha_{B}}{\sin\gamma% \sin\alpha_{A}}\quad F_{2}\epsilon[-1,1].$

The judgmental criterion is:

$\displaystyle\left\{{\begin{array}[]{l}\text{If}\;F_{1}\epsilon[0,\infty]\;% \text{and}\;F_{2}\epsilon[0,1],\text{then arctan}F_{1}\cap\text{arccos}F_{2},% \omega\epsilon[0^{\circ},90^{\circ}]\\ \text{If}\;F_{1}\epsilon[0,\infty]\;\text{and}\;F_{2}\epsilon[-1,0],\text{then% arctan}F_{1}\cap\text{arccos}F_{2},\omega\epsilon[180^{\circ},270^{\circ}]\\ \text{If}\;F_{1}\epsilon[-\infty,0]\;\text{and}\;F_{2}\epsilon[0,1],\text{then% arctan}F_{1}\cap\text{arccos}F_{2},\omega\epsilon[270^{\circ},360^{\circ}]\\ \text{If}\;F_{1}\epsilon[-\infty,0]\;\text{and}\;F_{2}\epsilon[-1,0],\text{% then arctan}F_{1}\cap\text{arccos}F_{2},\omega\epsilon[80^{\circ},180^{\circ}]% \\ \end{array}}\right.$ (8)

4. Engineering case verification

A project is about drlling eight multi-branch wells at the depth of 925–1100 m in the completed well ZK5. The inclination and azimuth (which has been modified by magnetic declination and meridian convergence) in the upper section of well ZK5 are shown in Table 1.

Table 1
Trajectory deviation data in the upper section of well ZK5

Well depth (m)	Inclination ( ${}^{\circ}$ )	Azimuth ( ${}^{\circ}$ )	Well depth (m)	Inclination ( ${}^{\circ}$ )	Azimuth ( ${}^{\circ}$ )
0	0		900	4.37	248.33
100	1.26	212.31	925	4.32	251.21
200	2.61	225.26	950	4.26	249.25
300	3.80	237.82	975	4.73	246.32
400	4.55	242.50	1000	5.20	248.15
500	5.23	245.12	1025	4.98	252.46
600	4.76	243.73	1050	5.16	251.65
700	3.92	247.88	1075	5.31	249.22
800	3.88	244.91	1100	5.50	248.96

For simplicity, a relative geographic coordinate system is established with the wellhead of ZK5 for the origin, the geographic north for positive direction of axis X and the geographic vertical line pointing to the center of the earth for positive direction of axis Z. The target coordinates and KOP depth of eight multi-branch wells are shown in Table 5. These targets are radial all-around and distribute in four quadrants of the relative geographic coordinate system. Respectively, the design of target-hitting for eight multi-branch wells involves the azimuth increment/decrement in different quadrants. And it is representative for testing the judgment criterion of the target azimuth and the tool face angle.

4.1 Design and judgment of target-hitting azimuth

According to the data in Table 1, the relative geographic coordinates of KOP can be calculated by using the minimum curvature method which is widely applied in engineering. On the basis of the given target and KOP depth in Tables 2 and 5, the target-hitting inclination $\alpha$ ${}_{B}$ and the overall angle $\gamma$ can be also calculated by using the equations of trajectory design for 3D directional drilling. Taking the calculated $\alpha$ ${}_{B}$ and $\gamma$ into Eq. (1), two numerical values of the azimuth change $\Delta\phi$ can be obtained by calculation as shown in Table 2. Based on Eqs (3)–(5), the correct values judging on $\Delta\phi$ are the data in brackets.

Table 2
Evaluation of the target-hitting azimuth change $\Delta\phi$ and tool face angle $\omega$

Well	Design and judgment on the azimuth change $\Delta\phi$					Design and judgment on tool face angle $\omega$
name	$X_{C}$ (m)	$Y_{C}$ (m)	$\Delta X_{EC}$ (m)	$\Delta Y_{EC}$ (m)	$\Delta\phi$ ( ${}^{\circ}$ )	Eq. (5) $\omega$ ( ${}^{\circ}$ )	Eq. (6) $\omega$ ( ${}^{\circ}$ )
ZK5-1	$-$ 32.200	$-$ 63.834	51.010	$-$ 77.868	56.772 ( $-$ 56.772)	79.518 (259.518)	100.482 (259.518)
ZK5-2	$-$ 33.011	$-$ 65.938	$-$ 3.298	6.350	(117.442) $-$ 117.442	(175.361) 355.361	(175.361) 184.639
ZK5-3	$-$ 31.703	$-$ 62.690	73.477	91.065	(51.101) $-$ 51.101	(71.815 ) 251.815	(71.815) 288.185
ZK5-4	$-$ 31.500	$-$ 61.148	$-$ 30.646	151.261	(101.453) $-$ 101.453	(113.274) 293.274	(113.274) 246.726
ZK5-5	$-$ 30.126	$-$ 58.175	$-$ 116.658	122.544	(133.591) $-$ 133.591	(138.452) 318.452	(138.452) 221.548
ZK5-6	$-$ 28.842	$-$ 55.165	$-$ 158.430	53.777	(161.251) $-$ 161.251	(164.213) 344.213	(164.213) 195.787
ZK5-7	$-$ 30.342	$-$ 57.986	$-$ 142.820	$-$ 60.732	156.963 ( $-$ 156.963)	17.954 (197.954)	162.046 (197.954)
ZK5-8	$-$ 31.171	$-$ 61.178	$-$ 58.643	$-$ 109.703	118.127 ( $-$ 118.127)	44.283 (224.283)	135.717 (224.283 )

4.2 Design and judgment of tool face angle

Bringing the target-hitting azimuth change $\Delta\phi$ (which has been identified and listed in the brackets of Table 2), inclination $\alpha$ ${}_{A}$ and $\alpha$ ${}_{B}$ , overall angle $\gamma$ into Eqs (6) and (7), two numerical values of the tool face angles $\omega$ can be obtained by calculation as shown in Table 2. The correct values about $\omega$ judged by the criterion (8) about the numerical overlap of double formulas are the data in brackets.

4.3 Comparing the azimuth change and the tool face angle with some references’ results

Using the methods studied [1, 11, 12, 18, 19, 21, 25], the targeting azimuth change $\Delta\varphi$ and the tool face angle $\omega$ of the Engineering case are shown in Table 3. Here, their principles are not illustrated due to the limitation of the paper’s texts. The different data from the paper’s results are shown by italic in Table 3. Because the problems of the “partial increase” of the tool face angle and the inclination extreme of the circular-arc trajectory on the inclined plane, six tool face angles of the reference [19] can’t be confirmed. Although the results of the reference [11, 12, 18] are completely consistent with the data of the paper, the paper’s methods are simpler.

Table 3
Comparing the azimuth change and the tool face angle with some references’ results

Well	Azimuth change $\Delta\varphi$ ( ${}^{\circ}$ )		Tool face angle $\omega$ ( ${}^{\circ}$ )
name	Ref.[11]	Ref. [18]	Ref. [1]	Ref. [12]	Ref. [19]	Ref. [21]	Ref. [25]
ZK5-1	$-$ 56.772	$-$ 56.772	100.482	259.518	uncertain	79.518	259.518
ZK5-2	117.442	117.442	4.639	175.361	175.361	175.361	59.740
ZK5-3	51.101	51.101	71.815	71.815	71.815	71.815	71.815
ZK5-4	101.453	101.453	66.726	113.274	uncertain	293.274	113.274
ZK5-5	133.590	133.590	41.548	138.452	uncertain	318.452	138.452
ZK5-6	161.251	161.251	15.787	164.213	uncertain	344.213	164.213
ZK5-7	$-$ 156.963	$-$ 156.963	162.046	197.954	uncertain	7.954	197.954
ZK5-8	$-$ 118.127	$-$ 118.127	135.717	224.283	uncertain	44.283	224.283
Compare results	consistent	consistent	partial	consistent	partial	partial	partial
			consistent		consistent	consistent	consistent

Table 4

Trajectory data of backward calculating of ZK5-3

Well depth	Design data of		Trajectory data of			Trajectory coordinates of
(m)	target-hitting		back calculation			back calculation

	$\alpha$ ${}_{B}$ ( ${}^{\circ}$ )	$\Delta\phi$ ( ${}^{\circ}$ )	Inclination ( ${}^{\circ}$ )	Azimuth	Azimuth ( ${}^{\circ}$ )	$X$ (m)	$Y$ (m)	$Z$ (m)
				change ( ${}^{\circ}$ )

1050.000			5.160		251.650	$-$ 30.910	$-$ 60.300	1047.458
1055.000			5.554	9.865	261.515	$-$ 31.017	$-$ 60.753	1052.436
1060.000			6.088	18.220	269.870	$-$ 31.053	$-$ 61.257	1057.410
1065.000			6.728	25.113	276.763	$-$ 31.019	$-$ 61.813	1062.379
1070.000			7.448	30.749	282.399	$-$ 30.915	$-$ 62.421	1067.341
1075.000			8.226	35.361	287.011	$-$ 30.741	$-$ 63.079	1072.294
1080.000			9.048	39.161	290.811	$-$ 30.496	$-$ 63.789	1077.237
1085.000			9.902	42.322	293.972	$-$ 30.182	$-$ 64.549	1082.169
1090.000			10.781	44.979	296.629	$-$ 29.798	$-$ 65.360	1087.087
1095.000			11.680	47.236	298.886	$-$ 29.344	$-$ 66.221	1091.991
1100.000			12.593	49.172	300.822	$-$ 28.820	$-$ 67.132	1096.879
1105.817	13.671	51.101	13.671	51.101	302.751	$-$ 28.123	$-$ 68.255	1102.544
1572.905			13.671		302.751	31.600	$-$ 161.100	1556.400

Table 5

Comparison of given target coordinates with back calculating coordinates

Well name	KOP depth (m)	Coordinates of given target			Target coordinates of back calculation
		$X_{E}$ (m)	$Y_{E}$ (m)	$Z_{E}$ (m)	$X_{E}$ (m)	$Y_{E}$ (m)	$Z_{E}$ (m)
ZK5-1	1075	$-$ 123.10	$-$ 83.90	1792.80	$-$ 123.10	$-$ 83.90	1792.80
ZK5-2	1100	$-$ 25.90	$-$ 65.10	1893.80	$-$ 25.90	$-$ 65.10	1893.80
ZK5-3	1050	31.60	$-$ 161.10	1556.40	31.60	$-$ 161.10	1556.40
ZK5-4	1000	120.30	$-$ 89.00	1438.40	120.30	$-$ 89.00	1438.40
ZK5-5	950	125.80	7.50	1327.50	125.80	7.50	1327.50
ZK5-6	925	73.10	77.50	1415.30	73.10	77.50	1415.30
ZK5-7	975	$-$ 28.60	97.20	1570.60	$-$ 28.60	97.20	1570.60
ZK5-8	1025	$-$ 118.10	27.80	1681.20	$-$ 118.10	27.80	1681.20

4.4 Back calculation of well trajectory using judgmental value

The correctness of the judgmental value about $\Delta\phi$ and $\omega$ can be confirmed by the method of back calculation on the well trajectory. The inclination and azimuth angles at any well depth point in build-up section can be calculated from Eqs (9) and (10) [21, 23, 25]. If the judgmental value about $\Delta\phi$ and $\omega$ are selected correctly, the inclination and azimuth change calculated from Eqs (9) and (10) at well depth point $B$ are in accord with the $\alpha$ ${}_{B}$ and $\Delta\phi$ which have been worked out anteriority. Also, using the inclination and azimuth derived from Eqs (9) and (10), the coordinates calculated by the minimum curvature method at the true vertical depth of the target are in accord with the coordinates of the given target. Otherwise there exists apparent difference.

$\displaystyle\alpha_{i}=\arccos\left({\cos\alpha_{A}\cos\left({(L_{i}-L_{% \textit{KOP}})\frac{180K}{\pi}}\right)-\sin\alpha_{A}\sin\left({(L_{i}-L_{% \textit{KOP}})\frac{180K}{\pi}}\right)\cos\omega}\right)$ (9) $\displaystyle\Delta\phi_{i}=\arcsin\frac{\sin\left({(L_{i}-L_{\textit{KOP}})% \frac{180K}{\pi}}\right)\sin\omega}{\sin\alpha_{i}}$ (10)

In Eqs (9) and (10), $\alpha$ ${}_{i}$ is the inclination at the well depth $L_{i}$ ; $L_{\textit{KOP}}$ is the well depth at the starting point of the build-up section; $L_{i}$ is the well depth at any point in the build-up section; $\Delta\phi_{i}=\phi_{i}-\phi_{A}$ , $\phi$ ${}_{i}$ is the azimuth at the well depth $L_{i}$ .

Take the branch well of ZK5-3, for instance, the data of back calculation on the well trajectory is shown in Table 4. At the well depth of 1105.817 m (endpoint of build-up section), the inclination of back calculation from Eq. (9) is 13.671 ${}^{\circ}$ , the target-hitting azimuth change of back calculation from Eq. (10) is 51.101 ${}^{\circ}$ , the target-hitting azimuth is 302.751 ${}^{\circ}$ . These numerical values of back calculation are exactly consistent with the $\alpha$ ${}_{B}$ and $\Delta\phi$ which are the ordinal design values anteriority. Similarly, using the minimum curvature method, the target coordinates of back calculation for eight multi-branch wells shown in Table 5 are also completely consistent with the coordinates of the given target.

5. Conclusions

(1)

For the well trajectory design and control of 3D directional drilling, there would be two target-hitting azimuth angles and two tool face angles solved by theoretical equation because they may exist in the four quadrants of 0 ${}^{\circ}$ $\sim$ 360 ${}^{\circ}$ . So it is necessary to judge the results of theoretical calculation accurately base on the reasonable criterion. Otherwise, it will lead to great error.

(2)

Based on the coordinates of the target relative to the PTI, a new evaluation method on the theoretical value of the target-hitting azimuth angle is proposed in this paper. And the judgmental criterion in the four quadrants of 0 ${}^{\circ}$ $\sim$ 360 ${}^{\circ}$ for the change of the target-hitting azimuth is worked out. The research result can be used as the theoretical basis to determine the target-hitting azimuth change for the design of the well trajectory on the inclined plane in space.

(3)

According to the periodic principle of trigonometric function in the four quadrants of 0 ${}^{\circ}$ $\sim$ 360 ${}^{\circ}$ , the judgmental criterion about the numerical overlap of double formulas for the tool face angle is worked out and used to select the reasonable value. In the design of the 3D directional well, it provides a new way to identify the tool face angle for the control of the circular-arc trajectory on the inclined plane.

(4)

The validity of the judgmental criteria for the target-hitting azimuth and the tool face angle has been confirmed in the application of engineering case. The trajectory data of back calculation for eight multi-branch wells is in strictly consistent with the design value of the target-hitting azimuth and the coordinates of the given target.

(5)

The judgmental criteria of the target-hitting azimuth change and the tool face angle for the design and control of the circular-arc trajectory on the inclined plane can be widely applied in the engineering of the geological exploration and oil gas well for 3D directional drilling. And they can also be used as algorithm models for the development of the directional drilling software.

Footnotes

Acknowledgments

The research is supported by the special fund from Land and Resources Public Service Industry of China (No. 201311059) and China Geological Survey Project Fund (No. 2014050011).

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Evaluation of inclined arc trajectory parameters for design and control of 3D directional well

Abstract

Keywords

1. Introduction

2. Judgment of the target azimuth angle for trajectory design

2.1 Trajectory model

3.1 Geometric relationship model of tool face angle

Table 1 Trajectory deviation data in the upper section of well ZK5

Table 2 Evaluation of the target-hitting azimuth change Δ ⁢ ϕ and tool face angle ω

4.3 Comparing the azimuth change and the tool face angle with some references’ results

Table 3 Comparing the azimuth change and the tool face angle with some references’ results

Footnotes

Acknowledgments

References

Table 1
Trajectory deviation data in the upper section of well ZK5

Table 2
Evaluation of the target-hitting azimuth change $\Delta\phi$ and tool face angle $\omega$

Table 3
Comparing the azimuth change and the tool face angle with some references’ results